Learning Module 1: Basics of Multiple Regression

20 Learning Module 1 Basics of Multiple Regression and Underlying Assumptions PRACTICE PROBLEMS The following information relates to questions 1-5 You are a junior analyst at an asset management firm. Your supervisor asks you to analyze the return drivers for one of the firm’s portfolios. She asks you to construct a regression model of the portfolio’s monthly excess returns (RET) against three factors: the market excess return (MRKT), a value factor (HML), and the monthly percentage change in a volatility index (VIX). You collect the data and run the regression, and the resulting model is YRET = –0.999 + 1.817XMRKT + 0.489XHML + 0.037XVIX. You then create some diagnostic charts to help determine the model fit. Practice Problems 21 RET vs VIX 30 Portfolio excess returns 20 10 0 –10 –20 –30 –40 –60 –40 –20 0 20 40 60 80 100 120 140 160 % Change in volatility factor RET vs MRKT 30 Portfolio excess returns 20 10 0 –10 –20 –30 –40 –15 –10 –5 0 5 10 15 20 Market excess returns MRKT vs HML MRKT excess returns 10 5 0 –5 –10 –15 –20 –15 –10 –5 0 5 10 15 10.00 20.00 20 HML values Residuals vs Predicted 15.00 RET residuals 10.00 5.00 0.00 –5.00 –10.00 –15.00 –40.00 –30.00 –20.00 –10.00 0.00 RET predicted values 1. Determine the type of regression model you should use. 30.00 22 Learning Module 1 Basics of Multiple Regression and Underlying Assumptions A. Logistic regression B. Simple linear regression C. Multiple linear regression 2. Determine which one of the following statements about the coefficient of the volatility factor (VIX) is true. A. A 1.0% increase in XVIX would result in a –0.962% decrease in YRET. B. A 0.037% increase in XVIX would result in a 1.0% increase in YRET. C. A 1.0% increase in XVIX, holding all the other independent variables constant, would result in a 0.037% increase in YRET. 3. Identify the regression assumption that may be violated based on Chart 1, RET vs. VIX. A. Independence of errors B. Independence of independent variables C. Linearity between dependent variable and explanatory variables 4. Identify which chart, among Charts 2, 3, and 4, is most likely to be used to assess homoskedasticity. A. Chart 2 B. Chart 3 C. Chart 4 5. Identify which chart, among Charts 2, 3, and 4, is most likely to be used to assess independence of independent variables. A. Chart 2 B. Chart 3 C. Chart 4 Solutions 23 SOLUTIONS 1. C is correct. You should use a multiple linear regression model since the dependent variable is continuous (not discrete) and there is more than one explanatory variable. If the dependent variable were discrete, then the model should be estimated as a logistic regression. 2. C is correct. The coefficient of the volatility factor (XVIX) is 0.037. It should be interpreted to mean that holding all the other independent variables constant, a 1% increase (decrease) would result in a 0.037% increase (decrease) in the monthly portfolio excess return (YRET). 3. C is correct. Chart 1 is a scatterplot of RET versus VIX. Linearity between the dependent variable and the independent variables is an assumption underlying multiple linear regression. As shown in the following Revised Chart 1, the relationship appears to be more curved (i.e., quadratic) than linear. RET vs VIX 30 Portfolio excess returns 20 10 0 –10 –20 –30 –40 –60 –40 –20 0 20 40 60 80 100 120 140 160 % Change in volatility factor 4. C is correct. To assess homoskedasticity, we must evaluate whether the variance of the regression residuals is constant for all observations. Chart 4 is a scatterplot of the regression residuals versus the predicted values, so it is very useful for visually assessing the consistency of the variance of the residuals across the observations. Any clusters of high and/or low values of the residuals may indicate a violation of the homoskedasticity assumption. 5. B is correct. Chart 3 is a scatterplot comparing the values of two of the independent variables, MRKT and HML. This chart would most likely be used to assess the independence of these explanatory variables. 46 Learning Module 2 Evaluating Regression Model Fit and Interpreting Model Results PRACTICE PROBLEMS The following information relates to questions 1-5 You are a junior analyst at an asset management firm. Your supervisor asks you to analyze the return drivers for one of the firm’s portfolios. She asks you to construct a regression model of the portfolio’s monthly excess returns (RET) against three factors: the market excess return (MRKT), a value factor (HML), and the monthly percentage change in a volatility index (VIX). You collect the data and run the regression. After completing the first regression (Model 1), you review the ANOVA results with your supervisor. Then, she asks you to create two more models by adding two more explanatory variables: a size factor (SMB) and a momentum factor (MOM). Your three models are as follows: Model 1: RETi = b0 + bMRKTMRKTi + bHMLHMLi + bVIXVIXi + εi. Model 2: RETi = b0 + bMRKTMRKTi + bHMLHMLi + bVIXVIXi + bSMBSMBi + εi. Model 3: RETi = b0 + bMRKTMRKTi + bHMLHMLi + bVIXVIXi + bSMBSMBi + bMOMMOMi + εi. The regression statistics and ANOVA results for the three models are shown in Exhibit 1, Exhibit 2, and Exhibit 3. Exhibit 1: ANOVA Table for Model 1 RETi = b0 + bMRKTMRKTi + bHMLHMLi + bVIXVIXi + εi Regression Statistics Coefficient Std. Error t-Stat. P-Value –0.999 0.414 –2.411 0.018 Multiple R 0.907 Intercept R-Squared 0.823 MRKT 1.817 0.124 14.683 0.000 Adjusted R-Sq. 0.817 HML 0.489 0.118 4.133 0.000 Standard Error 3.438 VIX 0.037 0.018 2.122 0.037 Observations 96.000 ANOVA Df SS MS F Significance F Regression 3 5058.430 1686.143 142.628 0.000 Residual 92 1087.618 11.822 Practice Problems 47 ANOVA Total Df SS MS 95 6146.048 F Significance F Exhibit 2: ANOVA Table for Model 2 RETi = b0 + bMRKTMRKTi + bHMLHMLi + bVIXVIXi + bSMBSMBi + εi Regression Statistics Coefficient Std. Error t-Stat. P-Value –0.820 0.383 –2.139 0.035 Multiple R 0.923 Intercept R-Squared 0.852 MRKT 1.649 0.121 13.683 0.000 Adjusted R-Sq. 0.846 HML 0.434 0.109 3.970 0.000 Standard Error 3.161 VIX 0.025 0.016 1.516 0.133 Observations 96.000 SMB 0.563 0.133 4.223 0.000 ANOVA Df SS MS F Significance F 4 5236.635 1309.159 131.000 0.000 9.994 Regression Residual 91 909.413 Total 95 6146.048 Exhibit 3: ANOVA Table for Model 3 RETi = b0 + bMRKTMRKTi + bHMLHMLi + bVIXVIXi + bSMBSMBi + bMOMMOMi + εi Regression Statistics Coefficient Std. Error Multiple R 0.923 Intercept R-Squared 0.852 Adjusted R-Sq. 0.844 Standard Error Observations t-Stat. P-Value –0.823 0.385 −2.136 0.035 MRKT 1.719 0.280 6.130 0.000 HML 0.412 0.138 2.989 0.004 3.177 VIX 0.026 0.017 1.532 0.129 96.000 SMB 0.553 0.139 3.987 0.000 –0.067 0.242 –0.276 0.783 MOM ANOVA Df SS MS F Significance F 103.751 0.000 Regression 5 5237.402 1047.480 Residual 90 908.647 10.096 Total 95 6146.048 Your supervisor asks for your assessment of the model that provides the best fit as well as the model that is best for predicting values of the monthly portfolio return. So, you calculate Akaike’s information criterion (AIC) and Schwarz’s Bayesian information criterion (BIC) for all three models, as shown in Exhibit 13. 48 Learning Module 2 Evaluating Regression Model Fit and Interpreting Model Results Exhibit 4: Goodness-of-Fit Measures Model 1 Model 2 Model 3 AIC BIC 241.03 251.29 225.85 238.67 227.77 243.16 1. Determine which one of the following reasons for the change in adjusted R2 from Model 2 to Model 3 is most likely to be correct. A. Adjusted R2 decreases since adding MOM does not improve the overall explanatory power of Model 3. B. Adjusted R2 increases since adding SMB improves the overall explanatory power of Model 2. C. Adjusted R2 decreases since adding MOM improves the overall explanatory power of Model 3. 2. Identify the model that provides the best fit. A. Model 1 B. Model 2 C. Model 3 3. Identify the model that should be used for prediction purposes. A. Model 1 B. Model 2 C. Model 3 4. Calculate the predicted RET for Model 3 given the assumed factor values: MRKT = 3, HML = –2, VIX = –5, SMB = 1, MOM = 3. A. 3.732 B. 3.992 C. 4.555 5. Calculate the joint F-statistic and determine whether SMB and MOM together contribute to explaining RET in Model 3 at a 1% significance level (use a critical value of 4.849). A. 2.216, so SMB and MOM together do not contribute to explaining RET B. 8.863, so SMB and MOM together do contribute to explaining RET C. 9.454, so SMB and MOM together do contribute to explaining RET Solutions SOLUTIONS 1. A is correct. Adjusted R2 in Model 3 decreases to 0.844 from 0.846 in Model 2. Model 3 includes all independent variables from Model 2, while adding MOM. Adding variables to a regression model always either increases R2 or causes it to stay the same. But adjusted R2 only increases if the new variable meets a threshold of significance, |t-statistic| > 1. MOM does not meet this threshold, indicating it does not improve the overall explanatory power of Model 3. 2. B is correct. BIC is the preferred measure for determining which model provides the best fit, and a lower BIC is better. Since Model 2 has the lowest BIC value, it provides the best fit among the three models. 3. B is correct. AIC is the preferred measure for determining the model that is best used for prediction purposes. As with BIC, a lower AIC is better. Model 2 also has the lowest AIC value among the three models; thus, it should be used for prediction purposes. 4. A is correct. The regression equation for Model 3 is RET = –0.823 + 1.719MRKT + 0.412HML + 0.026VIX + 0.553SMB – 0.067MOM. Using the assumed values for the independent variables, we have RET = –0.823 + (1.719)(3) + (0.412)(–2) + (0.026)(–5) + (0.553)(1) – (0.067)(3) = 3.732. 5. B is correct. To determine whether SMB and MOM together contribute to the explanation of RET, at least one of the coefficients must be non-zero. So, H0: bSMB = bMOM = 0 and Ha: bSMB ≠ 0 and/or bMOM ≠ 0. We use the F-statistic, where ​(SSE of restricted model − SSE of unrestricted model)​/ q ​F = ________________________________________ ​          ​​, SSE of unrestricted model / ​(n − k − 1)​ with q = 2 and n – k – 1 = 90 degrees of freedom. The test is one-tailed, right side, with α = 1%, so the critical F-value is 4.849. Model 1 does not include SMB and MOM, so it is the restricted model. Model 3 includes all of the variables of Model 1 as well as SMB and MOM, so it is the unrestricted model. Using data in Exhibit 1 and Exhibit 3, the joint F-statistic is calculated as ​(1087.618 − 908.647)​/ 2 89.485       ​ = ​_ ​F = _________________ ​ 10.096 ​ = 8.863​. 908.647 / 90 Since 8.863 > 4.849, we reject H0. Thus, SMB and MOM together do contribute to the explanation of RET in Model 3 at a 1% significance level. 49 74 Learning Module 3 Model Misspecification PRACTICE PROBLEMS The following information relates to questions 1-4 You are a junior analyst at an asset management firm. Your supervisor asks you to analyze the return drivers for one of the firm’s portfolios. She asks you to construct three regression models of the portfolio’s monthly excess returns (RET), starting with the following factors: the market excess return (MRKT), a value factor (HML), and the monthly percentage change in a volatility index (VIX). Next you add a size factor (SMB), and finally you add a momentum factor (MOM). Your three models are as follows: Model 1: RETi = b0 + bMRKTMRKTi + bHMLHMLi + bVIXVIXi + εi. Model 2: RETi = b0 + bMRKTMRKTi + bHMLHMLi + bVIXVIXi + bSMBSMBi + εi. Model 3: RETi = b0 + bMRKTMRKTi + bHMLHMLi + bVIXVIXi + bSMBSMBi + bMOMMOMi + εi. Your supervisor is concerned about conditional heteroskedasticity in Model 3 and asks you to perform the Breusch–Pagan (BP) test. At a 5% confidence level, the BP critical value is 11.07. You run the regression for the BP test; the results are shown in Exhibit 15. Exhibit 1: Testing for Conditional Heteroskedasticity Regression Statistics Multiple R 0.25517 R-Squared 0.06511 Adjusted R-Squared 0.01317 Standard Error 18.22568 Observations 96 Now the chief investment officer (CIO) joins the meeting and asks you to analyze two regression models (A and B) for the portfolio he manages. He gives you the test results for each of the models, shown in Exhibit 16. Exhibit 2: Breusch–Godfrey and Durbin-Watson Test Results Model A Model B Test Type Test Statistic Critical Value Independent Variable Is Lagged Value of Dependent Variable Breusch–Godfrey 12.124 3.927 Yes Durbin–Watson 5.088 4.387 No The CIO also asks you to test a factor model for multicollinearity among its four explanatory variables. You calculate the variance inflation factor (VIF) for each of Practice Problems 75 the four factors; the results are shown in Exhibit 17. Exhibit 3: Multicollinearity Test Results R2 VIF X1 0.748 3.968 X3 Variable X2 0.451 1.820 0.942 17.257 X4 0.926 13.434 1. Calculate the BP test statistic using the data in Exhibit 15 and determine whether there is evidence of heteroskedasticity. A. 1.264, so there is no evidence of heteroskedasticity B. 6.251, so there is no evidence of heteroskedasticity C. 81.792, so there is evidence of heteroskedasticity 2. Identify the type of error and its impacts on regression Model A indicated by the data in Exhibit 16. A. Serial correlation, invalid coefficient estimates, and deflated standard errors. B. Heteroskedasticity, valid coefficient estimates, and deflated standard errors. C. Serial correlation, valid coefficient estimates, and inflated standard errors. 3. Determine using Exhibit 17 which one of the following statements is most likely to be correct. Multicollinearity issues exist for variables: A. X1 and X2. B. X2 and X3. C. X3 and X4. 4. Identify the correct answer related to the following statement. Possible solutions for addressing the multicollinearity issues identified in Exhibit 17 include: 1. excluding one or more of the regression variables. 2. using a different proxy for one of the variables. 3. increasing the sample size. A. Only Solution 1 is correct. B. Only Solution 2 is correct. C. Solutions 1, 2, and 3 are each correct. 76 Learning Module 3 Model Misspecification SOLUTIONS 1. B is correct. The BP test statistic is calculated as nR2, where n is the number of observations and R2 is from the regression for the BP test. So, the BP test statistic = 96 × 0.06511 = 6.251. This is less than the critical value of 11.07, so we cannot reject the null hypothesis of no heteroskedasticity. Thus, there is no evidence of heteroskedasticity. 2. A is correct. The Breusch–Godfrey (BG) test is for serial correlation, and for Model A, the BG test statistic exceeds the critical value. In the presence of serial correlation, if the independent variable is a lagged value of the dependent variable, then regression coefficient estimates are invalid and coefficients’ standard errors are deflated, so t-statistics are inflated. 3. C is correct. A VIF above 10 indicates serious multicollinearity issues requiring correction, while a VIF above 5 warrants further investigation of the given variable. Since X3 and X4 each have VIFs above 10, serious multicollinearity exists for these two variables. VIFs for X1 and X2 are both well below 5, so multicollinearity does not appear to be an issue with these variables. 4. C is correct. Possible solutions for addressing multicollinearity issues include all of the solutions mentioned: excluding one or more of the regression variables, using a different proxy for one of the variables, and increasing the sample size. 102 Learning Module 4 Extensions of Multiple Regression PRACTICE PROBLEMS The following information relates to questions 1-5 The CIO asks you to analyze one of the firm’s portfolios to identify influential outliers that might be skewing regression results of its return drivers. For each observation, you calculate leverage, the studentized residual, and Cook’s D. There are 96 observations and two independent variables (k = 2), and the critical t-statistic is 2.63 at a 1% significance level. Partial results of your calculations are shown in Exhibit 1. Exhibit 1: Regression Data for Detecting Influential Observations hii Studentized Residual Cook’s D Observation 1 0.043 2.784 0.161 Observation 2 0.022 –0.103 0.000 Observation 3 0.036 –0.731 0.010 Observation 4 0.059 –0.122 0.000 Observation 5 0.011 –0.660 0.002 Observation 6 0.101 –2.906 0.347 … … … Observation 45 0.042 2.117 0.094 Observation 46 0.013 0.172 0.000 Observation 47 0.015 –0.672 0.003 Observation 48 0.012 –0.734 0.003 Observation 49 0.064 0.475 0.008 Observation 50 0.141 –2.788 0.594 Observation 51 0.011 1.679 0.016 Observation 52 0.023 –1.218 0.017 … … … Observation 91 0.035 –1.260 0.029 Observation 92 0.025 3.001 0.106 Observation 93 0.017 1.483 0.019 Observation 94 0.097 –0.172 0.001 Observation 95 0.017 0.046 0.000 Observation 96 0.011 1.819 0.019 … … Finally, you are tasked with investigating whether there is any monthly seasonality in the excess portfolio returns. You construct a regression model using dummy variables for the months; your regression statistics and ANOVA results are shown in Exhibit 2. Practice Problems 103 Exhibit 2: Analysis of Monthly Seasonality of Excess Portfolio Returns Regression Statistics Multiple R 0.321 R-Squared 0.103 Adjusted R-Squared –0.014 Standard Error 8.100 Observations 96.000 ANOVA df SS MS F Signif. F Regression 11 634.679 57.698 0.879 0.563 Residual 84 5511.369 65.612 Total 95 6146.048 Coeff. Std. Error t-Stat. P-Value 1.263 2.864 0.441 0.660 Jan 1.311 4.050 0.324 0.747 Feb –3.756 4.050 –0.927 0.356 Mar 3.495 4.050 0.863 0.391 Apr 0.174 4.050 0.043 0.966 May 0.714 4.050 0.176 0.861 Jun 0.944 4.050 0.233 0.816 Jul –0.571 4.050 –0.141 0.888 Aug –0.445 4.050 –0.110 0.913 Sep –1.744 4.050 –0.431 0.668 Oct 4.261 4.050 1.052 0.296 Nov –5.311 4.050 –1.311 0.193 Intercept 1. Determine and justify the potentially influential observation(s) in Exhibit 1 using the leverage measure. A. Observation 50, because it has the highest leverage, at 0.141 B. Observations 6 and 50, because their leverage exceeds 0.100 C. Observations 6, 50, and 94, because their leverage exceeds 3​ (​ _ ​k +n 1 ​) 2. Determine and justify the potentially influential observations in Exhibit 1 using the studentized residuals measure. A. Observations 1 and 92, because the values of their studentized residuals exceed 2.63 B. Observations 1, 6, 50, and 92, because the absolute values of their studentized residuals exceed 2.63 104 Learning Module 4 Extensions of Multiple Regression C. All the observations shown except observation 95, because the absolute value of its studentized residual is less than 0.094 3. Determine and justify the potentially influential observations in Exhibit 1 using the criteria for Cook’s D involving k and n. A. Observations 6 and 50, because their Cook’s D values exceed 0.144 B. Observations 6 and 50, because their Cook’s D values exceed 0.289 C. Observations 1, 6, 50, and 92, because their Cook’s D values exceed 0.100 4. Identify both the base month and the coefficient that represents its returns in Exhibit 2. A. December is the base month, and the intercept coefficient represents its returns. B. November is the base month, and the intercept coefficient represents its returns. C. December is the base month, and the average of the coefficients for the other 11 months represents its returns. 5. Determine using Exhibit 2 which one of the following statements is most likely to be correct. Monthly seasonality in the firm’s portfolio is________. A. highly likely B. highly unlikely C. not able to be determined from the given data The following information relates to questions 6-12 Your second-round interview for the Junior Quantitative Analyst position went well, and the next day, you receive an email from the investment firm congratulating you for making it this far. You are one of four remaining candidates from more than 100 who applied for the position. Because the position involves quantitative analysis, you are given an assignment to complete within 72 hours. You are provided a dataset and tasked with creating two logistic regression models to predict whether an ETF will be a “winning” fund, that is, whether the ETF’s monthly return will be one standard deviation or more above the mean monthly return across all ETFs in the dataset, or whether the ETF will be an “average” fund. The variables in the dataset are as follows: Variable Description 1 net_assets Net assets of ETF in USD 2 small_fund Dummy Variable = 1 if a small-size fund, and 0 otherwise, with large size fund as the base Practice Problems 105 Variable Description 3 medium_fund Dummy Variable =1 if a medium-size fund, and 0 otherwise, with large-size fund as the base 4 portfolio_stocks Percentage of portfolio invested in stocks (1–100) 5 portfolio_bonds Percentage of portfolio invested in bonds (1–100) 6 price_earnings Ratio of price per share to earnings per share 7 price_book Ratio of price per share to book value per share 8 price_sales Ratio of price per share to sales per share 9 price_cashflow Ratio of price per share to cash flow per share 10 label (dependent variable) 1 = winning fund, an ETF whose monthly return is one standard deviation or more above the mean monthly return across all ETFs in the dataset, and 0 = average fund, otherwise For the first logistic regression, you are asked to use all the independent variables, except for the fund size dummy variables (small_fund and medium_fund). For the second logistic regression, you are asked to use all the independent variables except the fund size continuous variable (net assets). You use a standard software package (in Python or R) to develop the logistic regression models. Your results are as follows: Logistic Regression 1 Logistic Regression Results Model: Logit Pseudo-R2: 0.057 Dependent Variable: label No. Observations: 1594 Log-Likelihood: –451.66 Df Model: 7 LL-Null: –478.86 Df Residuals: 1586 LLR P-Value: No. Iterations: Intercept 1.97E−9 10 Coefficient Std.Err. z-Statistic P-Value > |z| [0.025 0.975] –2.0350 0.5221 –3.8979 0.0001 –3.0583 –1.0118 net_assets –0.7667 1.3571 –0.5649 0.5721 –3.4265 1.8932 portfolio_stocks –0.0089 0.0051 –1.7550 0.0793 –0.0188 0.0010 portfolio_bonds –0.1113 0.0729 –1.5263 0.1269 –0.2543 0.0316 0.0292 0.0200 1.4647 0.1430 –0.0099 0.0683 –0.0390 0.1029 –0.3791 0.7046 –0.2407 0.1627 0.3432 0.0777 4.4160 0.0000 0.1909 0.4956 –0.0502 0.0363 –1.3805 0.1674 –0.1214 0.0211 price_earnings price_book price_sales price_cashflow 106 Learning Module 4 Extensions of Multiple Regression Logistic Regression 2 Logistic Regression Results Model: Logit Pseudo-R2: 0.059 Dependent Variable: label No. Observations: 1594 Log-Likelihood: -450.4 Df Model: 8 LL-Null: -478.86 Df Residuals: 1585 LLR P-Value: No. Iterations: Intercept 1.87E−09 10 Coefficient Std.Err. z-Statistic P-Value > |z| [0.025 0.975] –1.9589 0.5254 –3.7283 0.0002 –2.9886 –0.9291 small_fund –0.4794 0.3719 –1.2888 0.1975 –1.2083 0.2496 medium_fund –0.3509 0.2348 –1.4948 0.1350 –0.8111 0.1092 portfolio_stocks –0.0092 0.0051 –1.8099 0.0703 –0.0191 0.0008 portfolio_bonds –0.1121 0.0727 –1.5433 0.1228 –0.2546 0.0303 0.0389 0.0211 1.8467 0.0648 –0.0024 0.0802 –0.0803 0.1068 –0.7519 0.4521 –0.2897 0.1291 price_earnings price_book price_sales price_cashflow 0.3453 0.0796 4.3362 0.0000 0.1892 0.5014 –0.0510 0.0376 –1.3573 0.1747 –0.1247 0.0227 6. Identify which one of the following choices is most likely to be correct: “Logistic regression is the appropriate regression method for your assignment because the __________.” A. independent variables include dummy variables for small- and medium-size funds B. dependent variable is binary rather than continuous C. dependent variable is not continuous, and the independent variables include dummy variables for small- and medium-size funds 7. Identify which one of the following statements best describes the interpretation of an independent variable’s slope coefficient in a logistic regression model: “The slope coefficient is the change in the ___.” A. log odds that the event happens per unit change in the independent variable, while all other independent variables increase by one unit B. odds that the event happens per unit change in the independent variable, holding all other independent variables constant C. log odds that the event happens per unit change in the independent variable, holding all other independent variables constant Practice Problems 8. Determine which one of the following statements is true. “The intercept in these logistic regressions is interpreted as the ___.” A. probability of the ETF being a winning fund if all independent variables are one B. log odds of the ETF being a winning fund if all independent variables are zero C. log odds of the ETF being an average fund if all independent variables are zero 9. Determine for Logistic Regression 1 which of the following is closest to the change in the probability that an ETF will be a winning fund if net_assets increase by one unit, and all else stays constant. A. 31.7% B. 76.7% C. 100.0% 10. Determine for Logistic Regression 2 which of the following is closest to the change in probability that an ETF will be a winning fund if the size of the ETF changes from small to medium, and all else stays constant. A. 3.1% B. 12.9% C. 41.3% 11. Identify which one of the following statements about the logistic regression model fit is most likely to be correct. “Based on the log-likelihood criteria, ____.” A. Model 2 has a better fit because it has a higher log-likelihood value B. Model 2 has a better fit because it has a lower log-likelihood value C. Model 1 has a better fit because it has a higher log-likelihood value 12. Determine, using the significant variable(s) in Logistic Regression 2 and the information provided, which of the following is closest to the probability of the Alpha ETF being a winning fund and whether it would be classified as a winning fund. Alpha ETF variable values: small_fund = 0, medium_fund = 0, portfolio_stocks = 99.3%, portfolio_bonds = 0.7%, price_earnings = 25.0, price_book =1.1, price_ sales = 4.0, and price_cashflow = 5.7. Use significance level of 5% and probability threshold for being a winner of 65%. A. 27.4%, and the Alpha ETF is not classified as a winning fund B. 36.0%, and the Alpha ETF is not classified as a winning fund C. 82.2%, and the Alpha ETF is classified as a winning fund 107 108 Learning Module 4 Extensions of Multiple Regression SOLUTIONS 1. C is correct. The rule of thumb for the leverage measure is that if it exceeds ​3 ​k +n 1 ​)​​, where k is the number of independent variables, then it is a potentially (_ + 1 ​ ​ = 0.09375​. Three influential observation. Since n = 96 and k = 2, then 3 ​ ​(_ ​2 96 ) observations exceed this value: 6, 50, and 94. So, they are potentially influential observations. 2. B is correct. For the studentized residuals measure, the critical t-value is 2.63. So, any observation with a studentized residual whose absolute value exceeds 2.63 is a potentially influential observation. The studentized residuals for observations 1, 6, 50, and 92 have absolute values exceeding 2.63; therefore, they are potentially influential observations. 3. B is correct. The required criteria for using Cook’s D to identify influential observations is _ ​​Di​​ > ​√ _ ​n 2 k which implies the ith observation is _ highly likely to be an influential data point. _ 2 2 2 ​ = 0.289​. Only two observations, 6 and ​nk ​ = ​ _ ​96 Since n = 96 and k = 2, then ​ _ 50, have a Cook’s D that exceeds 0.289, so they are highly likely to be influential observations. √ √ 4. A is correct. December is the base month, and the intercept coefficient represents its returns. We use 11 dummy variables to represent the returns for each month from January through November. December results are measured when each of these dummy variables equals zero, leaving the intercept coefficient to represent December returns. 5. B is correct. Monthly seasonality in the firm’s portfolio is highly unlikely. The variance explained by the model (R-squared) is only 10.3%, and after adjusting for the number of independent variables (adjusted R-squared), it becomes negative. Also, the insignificant F-statistic indicates a 56.3% chance that all variable coefficients are zero. Finally, t-statistics and associated p-values indicate that all the variable coefficients are insignificant (i.e., not significantly different from zero). Consequently, monthly seasonality is highly unlikely to exist in this portfolio. 6. B is correct. Logistic regression is the appropriate regression method because the dependent variable is binary rather than continuous. 7. C is correct. An independent variable’s slope coefficient in a logistic regression model is the change in the log odds that the event happens per unit change in the independent variable, holding all other independent variables constant. 8. B is correct. The intercept in these logistic regressions is interpreted as the log odds of the ETF being a winning fund if all independent variables are zero. 9. A is correct. In Logistic Regression 1, the slope coefficient (i.e., log odds) for net_ assets is –0.7667. Therefore, the odds are e−0.7667 = 0.46454, and the probability (P) is 0.46454/(1.46454) = 0.31719, or 31.72%. So, a one-unit increase in net_assets results in a 31.72% increase in the probability that the EFT is a winning fund, all other variables held constant. Solutions 109 10. A is correct. The net impact on the probability that an ETF will be a winning fund as it grows from a small-size into a medium-size fund is shown in three steps: 1. The probability of a medium-size fund is: odds are e−0.3509 = 0.70405, and probability (P) = 0.70405 ⁄ (1.70405) = 0.41316, or 41.32%. 2. The probability of a small-size fund is: odds are e−0.4794 = 0.61915, and probability (P) = 0.61915 ⁄ (1.61915) = 0.38239, or 38.24%. 3. The net impact on the probability that an ETF will be a winning fund by growing from a small fund into a medium fund is: 41.32% – 38.24% = 3.08%. 11. A is correct. Model 2 has a better fit because it has a higher (less negative) log-likelihood value, –450.40 versus –451.66, compared to Model 1. 12. B is correct. Besides the significant intercept, the only significant (at 5% level) variable in Logistic Regression 2 is price_sales. Using these factors, the probability of this ETF being a winning fund is calculated to be 35.95%, as follows: 1 ​       Probability of being a winning fund = 0.3595 = ___________________________ )( ) ( ) ( 1 + exp​[− ​ − 1.9589 ​+ ​ 0.3453 ​ 4.0 ​]​ Because this probability is well below the 65% threshold for being a winner, the Alpha EFT would not be classified as a winning fund. 170 Learning Module 5 Time-Series Analysis PRACTICE PROBLEMS The following information relates to questions 1-7 Angela Martinez, an energy sector analyst at an investment bank, is concerned about the future level of oil prices and how it might affect portfolio values. She is considering whether to recommend a hedge for the bank portfolio’s exposure to changes in oil prices. Martinez examines West Texas Intermediate (WTI) monthly crude oil price data, expressed in US dollars per barrel, for the 181-month period from August 2000 through August 2015. The end-of-month WTI oil price was $51.16 in July 2015 and $42.86 in August 2015 (Month 181). After reviewing the time-series data, Martinez determines that the mean and variance of the time series of oil prices are not constant over time. She then runs the following four regressions using the WTI time-series data. ■ ■ ■ ■ Linear trend model: Oil pricet = b0 + b1t + et. Log-linear trend model: ln Oil pricet = b0 + b1t + et. AR(1) model: Oil pricet = b0 + b1Oil pricet−1 + et. AR(2) model: Oil pricet= b0 + b1Oil pricet−1 + b2Oil pricet−2 + et. Exhibit 1 presents selected data from all four regressions, and Exhibit 2 presents selected autocorrelation data from the AR(1) models. Exhibit 1: Crude Oil Price per Barrel, August 2000–August 2015 Regression Statistics (t-statistics for coefficients are reported in parentheses) Linear Log-Linear AR(1) AR(2) R2 0.5703 0.6255 0.9583 0.9656 Standard error 18.6327 0.3034 5.7977 5.2799 Observations 181 181 180 179 Durbin–Watson 0.10 0.08 1.16 2.08 2.0787 2.0530 RMSE Coefficients: Intercept t (Trend) Oil pricet−1 Oil pricet−2 28.3278 3.3929 1.5948 2.0017 (10.1846) (74.9091) (1.4610) (1.9957) 0.4086 0.0075 (15.4148) (17.2898) 0.9767 1.3946 (63.9535) (20.2999) −0.4249 (−6.2064) Practice Problems 171 In Exhibit 1, at the 5% significance level, the lower critical value for the Durbin– Watson test statistic is 1.75 for both the linear and log-linear regressions. Exhibit 2: Autocorrelations of the Residual from AR(1) Model Lag Autocorrelation t-Statistic 1 0.4157 5.5768 2 0.2388 3.2045 3 0.0336 0.4512 4 −0.0426 −0.5712 Note: At the 5% significance level, the critical value for a t-statistic is 1.97. After reviewing the data and regression results, Martinez draws the following conclusions. Conclusion 1 The time series for WTI oil prices is covariance stationary. Conclusion 2 Out-of-sample forecasting using the AR(1) model appears to be more accurate than that of the AR(2) model. 1. Based on Exhibit 1, the predicted WTI oil price for October 2015 using the linear trend model is closest to: A. $29.15. B. $74.77. C. $103.10. 2. Based on Exhibit 1, the predicted WTI oil price for September 2015 using the log-linear trend model is closest to: A. $29.75. B. $29.98. C. $116.50. 3. Based on the regression output in Exhibit 1, there is evidence of positive serial correlation in the errors in: A. the linear trend model but not the log-linear trend model. B. both the linear trend model and the log-linear trend model. C. neither the linear trend model nor the log-linear trend model. 4. Martinez’s Conclusion 1 is: A. correct. B. incorrect because the mean and variance of WTI oil prices are not constant over time. 172 Learning Module 5 Time-Series Analysis C. incorrect because the Durbin–Watson statistic of the AR(2) model is greater than 1.75. 5. Based on Exhibit 1, the forecasted oil price in September 2015 based on the AR(2) model is closest to: A. $38.03. B. $40.04. C. $61.77. 6. Based on the data for the AR(1) model in Exhibits 1 and 2, Martinez can conclude that the: A. residuals are not serially correlated. B. autocorrelations do not differ significantly from zero. C. standard error for each of the autocorrelations is 0.0745. 7. Based on the mean-reverting level implied by the AR(1) model regression output in Exhibit 1, the forecasted oil price for September 2015 is most likely to be: A. less than $42.86. B. equal to $42.86. C. greater than $42.86. 8. You have been assigned to analyze automobile manufacturers, and as a first step in your analysis, you decide to model monthly sales of lightweight vehicles to determine sales growth in that part of the industry. Exhibit 3 gives lightweight vehicle monthly sales (annualized) from January 1992 to December 2000. Exhibit 3: Lightweight Vehicle Sales Millions of Units (Annualized) 20 19 18 17 16 15 14 13 12 11 1992 1993 1994 1995 1996 1997 1998 1999 2000 Year Monthly sales in the lightweight vehicle sector, Salest, have been increasing over time, but you suspect that the growth rate of monthly sales is relatively constant. Write the simplest time-series model for Salest that is consistent with your perception. Practice Problems 173 The following information relates to questions 9-10 The civilian unemployment rate (UER) is an important component of many economic models. Exhibit 1 gives regression statistics from estimating a linear trend model of the unemployment rate: UERt = b0 + b1t + εt. Exhibit 1: Estimating a Linear Trend in the Civilian Unemployment Rate: Monthly Observations, January 2013–August 2019 Regression Statistics R2 0.9316 Standard error 0.3227 Observations 80 Durbin–Watson 0.1878 Coefficient Standard Error t-Statistic Intercept 7.2237 0.0728 99.1704 Trend −0.0510 0.0016 −32.6136 9. Using the regression output in the previous table, what is the model’s prediction of the unemployment rate for July 2013? 10. How should we interpret the Durbin–Watson (DW) statistic for this regression? What does the value of the DW statistic say about the validity of a t-test on the coefficient estimates? 11. Exhibit 2 compares the predicted civilian unemployment rate (PRED) with the actual civilian unemployment rate (UER) from January 2013 to August 2019. The predicted results come from estimating the linear time trend model UERt = b0 + b1t + εt. What can we conclude about the appropriateness of this model? Exhibit 1: Predicted and Actual Civilian Unemployment Rates 9 8 7 6 5 4 3 2 1 0 Jan/13 Jan/14 Jan/15 Jan/16 Jan/17 Jan/18 Jan/19 174 Learning Module 5 Time-Series Analysis 12. Exhibit 9 shows a plot of first differences in the log of monthly lightweight vehicle sales over the same period as in Problem 11. Has differencing the data made the resulting series, Δln(Salest) = ln(Salest) − ln(Salest−1), covariance stationary? Exhibit 1: Change in Natural Log of Lightweight Vehicle Sales Δ ln (Millions of Units) (Annualized) 0.15 0.10 0.05 0.00 –0.05 –0.10 –0.15 –0.20 1992 1993 1994 1995 1996 1997 1998 1999 2000 Year The following information relates to questions 13-14 Exhibit 4 shows a plot of the first differences in the civilian unemployment rate (UER) between January 2013 and August 2019, ΔUERt = UERt − UERt−1. Exhibit 1: Change in Civilian Unemployment Rate 0.3 0.2 0.1 0 –0.1 –0.2 –0.3 –0.4 –0.5 –0.6 Jan/13 Jan/14 Jan/15 Jan/16 Jan/17 Jan/18 Jan/19 Practice Problems 175 13. Has differencing the data made the new series, ΔUERt, covariance stationary? Explain your answer. 14. Given the graph of the change in the unemployment rate shown in the figure, describe the steps we should take to determine the appropriate autoregressive time-series model specification for the series ΔUERt. 15. Exhibit 5 gives the regression output of an AR(1) model on first differences in the unemployment rate. Describe how to interpret the DW statistic for this regression. Exhibit 1: Estimating an AR(1) Model of Changes in the Civilian Unemployment Rate: Monthly Observations, February 2013–August 2019 Regression Statistics R2 0.0546 Standard error 0.1309 Observations 79 Durbin–Watson 2.0756 Coefficient Standard Error t-Statistic Intercept −0.0668 0.0158 −4.2278 ΔUERt–1 −0.2320 0.1100 −2.191 The following information relates to questions 16-17 Using monthly data from January 1992 to December 2000, we estimate the following equation for lightweight vehicle sales: Δln(Salest) = 2.7108 + 0.3987Δln(Salest−1) + εt. Exhibit 10 gives sample autocorrelations of the errors from this model. Exhibit 1: Different Order Autocorrelations of Differences in the Logs of Vehicle Sales Lag Autocorrelation Standard Error t-Statistic 1 0.9358 0.0962 9.7247 2 0.8565 0.0962 8.9005 3 0.8083 0.0962 8.4001 4 0.7723 0.0962 8.0257 5 0.7476 0.0962 7.7696 6 0.7326 0.0962 7.6137 7 0.6941 0.0962 7.2138 8 0.6353 0.0962 6.6025 9 0.5867 0.0962 6.0968 176 Learning Module 5 Time-Series Analysis Lag Autocorrelation Standard Error t-Statistic 10 0.5378 0.0962 5.5892 11 0.4745 0.0962 4.9315 12 0.4217 0.0962 4.3827 16. Use the information in the table to assess the appropriateness of the specification given by the equation. 17. If the residuals from the AR(1) model above violate a regression assumption, how would you modify the AR(1) specification? 18. Assume that changes in the civilian unemployment rate are covariance stationary and that an AR(1) model is a good description for the time series of changes in the unemployment rate. Specifically, we have ΔUERt = −0.0668 − 0.2320ΔUERt−1 (using the coefficient estimates given in the previous problem). Given this equation, what is the mean-reverting level to which changes in the unemployment rate converge? The following information relates to questions 19-21 Suppose the following model describes changes in the civilian unemployment rate: ΔUERt = −0.0668 − 0.2320ΔUERt−1. The current change (first difference) in the unemployment rate is 0.0300. Assume that the mean-reverting level for changes in the unemployment rate is −0.0542. 19. What is the best prediction of the next change? 20. What is the prediction of the change following the next change? 21. Explain your answer to Part B in terms of equilibrium. 22. Exhibit 1 gives the actual sales, log of sales, and changes in the log of sales of Cisco Systems for the period 1Q 2019 to 4Q 2019. Exhibit 1 Actual Sales ($ Millions) Log of Sales 1Q 2019 13,072 9.4782 0.0176 2Q 2019 12,446 9.4292 −0.0491 3Q 2019 12,958 9.4695 0.403 4Q 2019 13,428 9.5051 0.0356 Date Changes in Log of Sales Δln(Salest) 1Q 2020 2Q 2020 Forecast the first- and second-quarter sales of Cisco Systems for 2020 using the regression Δln(Salest) = 0.0068 + 0.2633Δln(Salest−1). Practice Problems 177 The following information relates to questions 23-24 Exhibit 1 gives the actual change in the log of sales of Cisco Systems from 1Q 2019 to 4Q 2019, along with the forecasts from the regression model Δln(Salest) = 0.0068 + 0.2633Δln(Salest−1) estimated using data from 1Q 2001 to 4Q 2018. (Note that the observations after the fourth quarter of 2018 are out of sample.) Exhibit 1 Actual Value of Changes in the Log of Sales Δln(Salest) Forecast Value of Changes in the Log of Sales Δln(Salest) 1Q 2019 0.0176 0.0147 2Q 2019 −0.0491 0.0107 3Q 2019 0.4030 0.0096 4Q 2019 0.0356 0.0093 Date 23. Calculate the RMSE for the out-of-sample forecast errors. 24. Compare the forecasting performance of the model given with that of another model having an out-of-sample RMSE of 2%. The following information relates to questions 25-26 25. The AR(1) model for the civilian unemployment rate, ΔUERt = −0.0405 − 0.4674ΔUERt−1, was developed with five years of data. What would be the drawback to using the AR(1) model to predict changes in the civilian unemployment rate 12 months or more ahead, as compared with 1 month ahead? 26. For purposes of estimating a predictive equation, what would be the drawback to using 30 years of civilian unemployment data rather than only 5 years? The following information relates to questions 27-35 Max Busse is an analyst in the research department of a large hedge fund. He was recently asked to develop a model to predict the future exchange rate between two currencies. Busse gathers monthly exchange rate data from the most recent 10-year period and runs a regression based on the following AR(1) model specification: Regression 1: xt = b0 + b1xt−1 + εt, where xt is the exchange rate at time t. Based on his analysis of the time series and the regression results, Busse reaches the following conclusions: 178 Learning Module 5 Time-Series Analysis Conclusion 1 The variance of xt increases over time. Conclusion 2 The mean-reverting level is undefined. Conclusion 3 b0 does not appear to be significantly different from 0. Busse decides to do additional analysis by first-differencing the data and running a new regression. Regression 2: yt = b0 + b1yt−1 + εt, where yt = xt − xt−1. Exhibit 1 shows the regression results. Exhibit 1: First-Differenced Exchange Rate AR(1) Model: Month-End Observations, Last 10 Years Regression Statistics R2 0.0017 Standard error 7.3336 Observations 118 Durbin–Watson 1.9937 Coefficient Standard Error t-Statistic Intercept −0.8803 0.6792 −1.2960 xt−1 − xt−2 0.0412 0.0915 0.4504 Autocorrelation Standard Error t-Statistic 1 0.0028 0.0921 0.0300 2 0.0205 0.0921 0.2223 3 0.0707 0.0921 0.7684 4 0.0485 0.0921 0.5271 Autocorrelations of the Residual Lag Note: The critical t-statistic at the 5% significance level is 1.98. Busse decides that he will need to test the data for nonstationarity using a Dickey–Fuller test. To do so, he knows he must model a transformed version of Regression 1. Busse’s next assignment is to develop a model to predict future quarterly sales for PoweredUP, Inc., a major electronics retailer. He begins by running the following regression: Regression 3: ln Salest − ln Salest−1 = b0 + b1(ln Salest−1 − ln Salest−2) + εt. Exhibit 2 presents the results of this regression. Exhibit 2: Log Differenced Sales AR(1) Model: PoweredUP, Inc., Last 10 Years of Quarterly Sales Regression Statistics R2 0.2011 Practice Problems 179 Regression Statistics Standard error 0.0651 Observations 38 Durbin–Watson 1.9677 Coefficient Standard Error t-Statistic Intercept 0.0408 0.0112 3.6406 ln Salest−1 − ln Salest−2 −0.4311 0.1432 −3.0099 Autocorrelation Standard Error t-Statistic 1 0.0146 0.1622 0.0903 2 −0.1317 0.1622 −0.8119 3 −0.1123 0.1622 −0.6922 4 0.6994 0.1622 4.3111 Autocorrelations of the Residual Lag Note: The critical t-statistic at the 5% significance level is 2.02. Because the regression output from Exhibit 2 raises some concerns, Busse runs a different regression. These regression results, along with quarterly sales data for the past five quarters, are presented in Exhibits 3 and 4, respectively. Exhibit 3: Log Differenced Sales AR(1) Model with Seasonal Lag: PoweredUP, Inc., Last 10 Years of Quarterly Sales Regression Statistics R2 0.6788 Standard error 0.0424 Observations 35 Durbin–Watson 1.8799 Coefficient Standard Error t-Statistic Intercept 0.0092 0.0087 1.0582 ln Salest−1 − ln Salest−2 −0.1279 0.1137 −1.1252 0.7239 0.1093 6.6209 Autocorrelation Standard Error t-Statistic 1 0.0574 0.1690 0.3396 2 0.0440 0.1690 0.2604 3 0.1923 0.1690 1.1379 4 −0.1054 0.1690 −0.6237 ln Salest−4 − ln Salest−5 Autocorrelations of the Residual Lag Note: The critical t-statistic at the 5% significance level is 2.03. 180 Learning Module 5 Time-Series Analysis Exhibit 4: Most Recent Quarterly Sales Data (in billions) Dec 2015 (Salest−1) $3.868 June 2015 (Salest−3) $3.692 Dec 2014 (Salest−5) $3.418 Sep 2015 (Salest−2) $3.780 Mar 2015 (Salest−4) $3.836 After completing his work on PoweredUP, Busse is asked to analyze the relationship of oil prices and the stock prices of three transportation companies. His firm wants to know whether the stock prices can be predicted by the price of oil. Exhibit 5 shows selected information from the results of his analysis. Exhibit 5: Analysis Summary of Stock Prices for Three Transportation Stocks and the Price of Oil Unit Root? Linear or Exponential Trend? Serial Correlation of Residuals in Trend Model? ARCH(1)? Company 1 Yes Exponential Yes Yes Not cointegrated with oil price Company 2 Yes Linear Yes No Cointegrated with oil price Company 3 No Exponential Yes No Not cointegrated with oil price Oil Price Yes Comments To assess the relationship between oil prices and stock prices, Busse runs three regressions using the time series of each company’s stock prices as the dependent variable and the time series of oil prices as the independent variable. 27. Which of Busse’s conclusions regarding the exchange rate time series is consistent with both the properties of a covariance-stationary time series and the properties of a random walk? A. Conclusion 1 B. Conclusion 2 C. Conclusion 3 28. Based on the regression output in Exhibit 1, the first-differenced series used to run Regression 2 is consistent with: A. a random walk. B. covariance stationarity. C. a random walk with drift. 29. Based on the regression results in Exhibit 1, the original time series of exchange rates: A. has a unit root. B. exhibits stationarity. C. can be modeled using linear regression. Practice Problems 30. In order to perform the nonstationarity test, Busse should transform the Regression 1 equation by: A. adding the second lag to the equation. B. changing the regression’s independent variable. C. subtracting the independent variable from both sides of the equation. 31. Based on the regression output in Exhibit 2, what should lead Busse to conclude that the Regression 3 equation is not correctly specified? A. The Durbin–Watson statistic B. The t-statistic for the slope coefficient C. The t-statistics for the autocorrelations of the residual 32. Based on the regression output in Exhibit 3 and sales data in Exhibit 4, the forecasted value of quarterly sales for March 2016 for PoweredUP is closest to: A. $4.193 billion. B. $4.205 billion. C. $4.231 billion. 33. Based on Exhibit 5, Busse should conclude that the variance of the error terms for Company 1: A. is constant. B. can be predicted. C. is homoskedastic. 34. Based on Exhibit 5, for which company would the regression of stock prices on oil prices be expected to yield valid coefficients that could be used to estimate the long-term relationship between stock price and oil price? A. Company 1 B. Company 2 C. Company 3 35. Based on Exhibit 5, which single time-series model would most likely be appropriate for Busse to use in predicting the future stock price of Company 3? A. Log-linear trend model B. First-differenced AR(2) model C. First-differenced log AR(1) model 181 182 Learning Module 5 Time-Series Analysis The following information relates to questions 36-37 Exhibit 1 shows monthly observations on the natural log of lightweight vehicle sales, ln(Salest), for January 1992 to December 2000. Exhibit 1: Lightweight Vehicle Sales ln (Millions of Units) (Annualized) 3.00 2.90 2.80 2.70 2.60 2.50 2.40 2.30 2.20 1992 1993 1994 1995 1996 1997 1998 1999 2000 Year 36. Using the figure, comment on whether the specification ln(Salest) = b0 + b1[ln(Salest−1)] + εt is appropriate. 37. State an appropriate transformation of the time series. 38. Suppose we want to predict the annualized return of the five-year T-bill using the annualized return of the three-month T-bill with monthly observations from January 1993 to December 2002. Our analysis produces the data shown in Exhibit 1. Exhibit 1: Regression with Three-Month T-Bill as the Independent Variable and the Five-Year T-Bill as the Dependent Variable: Monthly Observations, January 1993–December 2002 Regression Statistics R2 0.5829 Standard error 0.6598 Observations 120 Durbin–Watson 0.1130 Coefficient Standard Error t-Statistic Intercept 3.0530 0.2060 14.8181 Three-month 0.5722 0.0446 12.8408 Can we rely on the regression model in Exhibit 1 to produce meaningful predictions? Specify what problem might be a concern with this regression. Practice Problems 183 39. Exhibit 2 shows the quarterly sales of Avon Products from 1Q 1992 to 2Q 2002. Describe the salient features of the data shown. Exhibit 2: Quarterly Sales at Avon Millions of Dollars 1900 1700 1500 1300 1100 900 700 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 Year The following information relates to questions 40-41 Exhibit 1 shows the autocorrelations of the residuals from an AR(1) model fit to the changes in the gross profit margin (GPM) of the Home Depot, Inc. Exhibit 1: Autocorrelations of the Residuals from Estimating the Regression ΔGPMt = 0.0006 − 0.3330ΔGPMt–1 + εt, 1Q 1992–4Q 2001 (40 Observations) Lag Autocorrelation 1 −0.1106 2 −0.5981 3 −0.1525 4 0.8496 5 −0.1099 Exhibit 2 shows the output from a regression on changes in the GPM for Home Depot, where we have changed the specification of the AR regression. Exhibit 2: Change in Gross Profit Margin for Home Depot, 1Q 1992–4Q 2001 Regression Statistics R2 0.9155 Standard error 0.0057 184 Learning Module 5 Time-Series Analysis Regression Statistics Observations 40 Durbin–Watson 2.6464 Coefficient Standard Error t-Statistic Intercept −0.0001 0.0009 −0.0610 ΔGPMt−1 −0.0608 0.0687 −0.8850 0.8720 0.0678 12.8683 ΔGPMt−4 40. Identify the change that was made to the regression model. 41. Discuss the rationale for changing the regression specification. The following information relates to questions 42-43 Suppose we decide to use an autoregressive model with a seasonal lag because of the seasonal autocorrelation in the previous problem. We are modeling quarterly data, so we estimate Equation 15: (ln Salest − ln Salest−1) = b0 + b1(ln Salest−1 − ln Salest−2) + b2(ln Salest−4 − ln Salest−5) + εt. Exhibit 1 shows the regression statistics from this equation. Exhibit 1: Log Differenced Sales: AR(1) Model with Seasonal Lag Johnson & Johnson Quarterly Observations, January 1985–December 2001 Regression Statistics R2 0.4220 Standard error 0.0318 Observations 68 Durbin–Watson 1.8784 Coefficient Standard Error Intercept 0.0121 0.0053 2.3055 Lag 1 −0.0839 0.0958 −0.8757 Lag 4 0.6292 0.0958 6.5693 Autocorrelation Standard Error t-Statistic 1 0.0572 0.1213 0.4720 2 −0.0700 0.1213 −0.5771 3 0.0065 0.1213 −0.0532 4 −0.0368 0.1213 −0.3033 t-Statistic Autocorrelations of the Residual Lag Practice Problems 185 42. Using the information in Exhibit 1, determine whether the model is correctly specified. 43. If sales grew by 1% last quarter and by 2% four quarters ago, use the model to predict the sales growth for this quarter. 44. Describe how to test for autoregressive conditional heteroskedasticity (ARCH) in the residuals from the AR(1) regression on first differences in the civilian unemployment rate, ΔUERt = b0 + b1ΔUERt−1 + εt. The following information relates to questions 45-47 Exhibit 1 shows the quarterly sales of Cisco Systems from 3Q 2001 to 2Q 2019. Exhibit 1: Quarterly Sales at Cisco Millions of Dollars 7000 6000 5000 4000 3000 2000 1000 0 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 Year Exhibit 2 gives the regression statistics from estimating the model Δln(Salest) = b0 + b1Δln(Salest−1) + εt. Exhibit 2: Change in the Natural Log of Sales for Cisco Quarterly Observations, 3Q 1991–4Q 2000 Regression Statistics R2 0.2899 Standard error 0.0408 Observations 38 Durbin–Watson 1.5707 Coefficient Standard Error t-Statistic Intercept 0.0661 0.0175 3.7840 Δln(Salest–1) 0.4698 0.1225 3.8339 186 Learning Module 5 Time-Series Analysis 45. Describe the salient features of the quarterly sales series. 46. Describe the procedures we should use to determine whether the AR(1) specification is correct. 47. Assuming the model is correctly specified, what is the long-run change in the log of sales toward which the series will tend to converge? Solutions SOLUTIONS 1. C is correct. The predicted value for period t from a linear trend is calculated as ˆ y​ ​t​ = b ​ˆ​0​+ b ​ˆ​1​​(​t)​ ​​​. October 2015 is the second month out of sample, or t = 183. So, the predicted value for October 2015 is calculated as ​​yˆ​t​= 28.3278 + 0.4086(183) = $103.10. Therefore, the predicted WTI oil price for October 2015 based on the linear trend model is $103.10. 2. C is correct. The predicted value for period t from a log-linear trend is calculated as ​lnˆ y​ ​t​ = b ​ˆ​0​+ b ​ˆ​1​​(​t)​ ​​​. September 2015 is the first month out of sample, or t = 182. So, the predicted value for September 2015 is calculated as follows: ln ​yˆ​t​= 3.3929 + 0.0075(182). ln ​yˆ​t​= 4.7579. ​yˆ​t​​= e4.7579 = $116.50. Therefore, the predicted WTI oil price for September 2015, based on the log-linear trend model, is $116.50. 3. B is correct. The Durbin–Watson statistic for the linear trend model is 0.10 and for the log-linear trend model is 0.08. Both of these values are below the critical value of 1.75. Therefore, we can reject the hypothesis of no positive serial correlation in the regression errors in both the linear trend model and the log-linear trend model. 4. B is correct. There are three requirements for a time series to be covariance stationary. First, the expected value of the time series must be constant and finite in all periods. Second, the variance of the time series must be constant and finite in all periods. Third, the covariance of the time series with itself for a fixed number of periods in the past or future must be constant and finite in all periods. Martinez concludes that the mean and variance of the time series of WTI oil prices are not constant over time. Therefore, the time series is not covariance stationary. 5. B is correct. The last two observations in the WTI time series are July and August 2015, when the WTI oil price was $51.16 and $42.86, respectively. Therefore, September 2015 represents a one-period-ahead forecast. The one-period-ahead forecast from an AR(2) model is calculated as ​​xˆ​t+1​ = ​bˆ​0​+ ​bˆ​1​​x​t​+ ​bˆ​2​​x​t−1​. So, the one-period-ahead (September 2015) forecast is calculated as ​​xˆ​t+1​= 2.0017 + 1.3946($42.86) − 0.4249($51.16) = $40.04. Therefore, the September 2015 forecast based on the AR(2) model is $40.04. 6. C is correct. The standard error of the autocorrelations is calculated as _ ​ 1_ ​​, where ​√T ​ T represents the number of observations used in the regression. Therefore, the 1 ​= 0.0745. Martinez can _ standard error for each of the autocorrelations is ​_ ​√180 ​ conclude that the residuals are serially correlated and are significantly different from zero because two of the four autocorrelations in Exhibit 2 have a t-statistic 187 188 Learning Module 5 Time-Series Analysis in absolute value that is greater than the critical value of 1.97. Choices A and B are incorrect because two of the four autocorrelations have a t-statistic in absolute value that is greater than the critical value of the t-statistic of 1.97. 7. C is correct. The mean-reverting level from the AR(1) model is calculated as ​b0​ ​ 1.5948 ​1 − b​ ​ ​​ = _ ​1 − 0.9767 ​ = $68.45. ​​xˆ​t​ = _ 1 Therefore, the mean-reverting WTI oil price from the AR(1) model is $68.45. The forecasted oil price in September 2015 will likely be greater than $42.86 because the model predicts that the price will rise in the next period from the August 2015 price of $42.86. 8. A log-linear model captures growth at a constant rate. The log-linear model ln(Salest) = b0 + b1t + εt would be the simplest model consistent with a constant growth rate for monthly sales. Note that we would need to confirm that the regression assumptions are satisfied before accepting the model as valid. 9. The estimated forecasting equation is UERt = 5.5098 − 0.0294(t). The data begin in January 2013, and July 2013 is Period 7. Thus the linear trend model predicts the unemployment rate to be UER7 = 7.2237 − 0.0510(7) = 6.8667, or approximately 6.9%. 10. The DW statistic is designed to detect positive serial correlation of the errors of a regression equation. Under the null hypothesis of no positive serial correlation, the DW statistic is 2.0. Positive serial correlation will lead to a DW statistic that is less than 2.0. From the table in Problem 1, we see that the DW statistic is 0.1878. To see whether this result is significantly less than 2.0, refer to the Durbin– Watson table in Appendix E at the end of this volume, in the column marked k = 1 (one independent variable) and the row corresponding to 60 observations. We see that dl = 1.61. Because our DW statistic is clearly less than dl, we reject the null hypothesis of no serial correlation at the 0.05 significance level. The presence of serial correlation in the error term violates one of the regression assumptions. The standard errors of the estimated coefficients will be biased downward, so we cannot conduct hypothesis testing on the coefficients. 11. The difference between UER and its forecast value, PRED, is the forecast error. In an appropriately specified regression model, the forecast errors are randomly distributed around the regression line and have a constant variance. We can see that the errors from this model specification are persistent. The errors tend first to be above the regression line, and then, starting in 2014, they tend to be below the regression line until 2017, when they again are persistently above the regression line. This persistence suggests that the errors are positively serially correlated. Therefore, we conclude that the model is not appropriate for making estimates. 12. The plot of the series Δln(Salest) appears to fluctuate around a constant mean; its volatility seems constant throughout the period. Differencing the data appears to have made the time series covariance stationary. 13. The plot of the series ΔUERt seems to fluctuate around a constant mean; its volatility appears to be constant throughout the period. Our initial judgment is that the differenced series is covariance stationary. 14. The change in the unemployment rate seems covariance stationary, so we should first estimate an AR(1) model and test to see whether the residuals from this model have significant serial correlation. If the residuals do not display signifi- Solutions 189 cant serial correlation, we should use the AR(1) model. If the residuals do display significant serial correlation, we should try an AR(2) model and test for serial correlation of the residuals of the AR(2) model. We should continue this procedure until the errors from the final AR(p) model are serially uncorrelated. 15. The DW statistic cannot be appropriately used for a regression that has a lagged value of the dependent variable as one of the explanatory variables. To test for serial correlation, we need to examine the autocorrelations. 16. In a correctly specified regression, the residuals must be serially uncorrelated. We _ √ have 108 observations, so the standard error of the autocorrelation is 1 ​ / ​ T , ​ ​ or _ in this case ​1 / √ ​ 108 ​= 0.0962. The t-statistic for each lag is significant at the 0.01 level. We would have to modify the model specification before continuing with the analysis. 17. Because the residuals from the AR(1) specification display significant serial correlation, we should estimate an AR(2) model and test for serial correlation of the residuals of the AR(2) model. If the residuals from the AR(2) model are serially uncorrelated, we should then test for seasonality and ARCH behavior. If any serial correlation remains in the residuals, we should estimate an AR(3) process and test the residuals from that specification for serial correlation. We should continue this procedure until the errors from the final AR(p) model are serially uncorrelated. When serial correlation is eliminated, we should test for seasonality and ARCH behavior. 18. When a covariance-stationary series is at its mean-reverting level, the series will tend not to change until it receives a shock (εt). So, if the series ΔUERt is at the mean-reverting level, ΔUERt = ΔUERt−1. This implies that ΔUERt = −0.0668 − 0.2320ΔUERt, so that (1 + 0.2320)ΔUERt = −0.0668 and ΔUERt = −0.0668/(1 + 0.2320) = −0.0542. The mean-reverting level is −0.0542. In an AR(1) model, the general expression for the mean-reverting level is b0/(1 − b1). 19. The predicted change in the unemployment rate for next period is −7.38%, found by substituting 0.0300 into the forecasting model: −0.0668 − 0.2320(0.03) = −0.0738. 20. If we substitute our one-period-ahead forecast of −0.0738 into the model (using the chain rule of forecasting), we get a two-period-ahead forecast of −0.0497, or −4.97%. 21. The answer to Part B is quite close to the mean-reverting level of −0.0542. A stationary time series may need many periods to return to its equilibrium, mean-reverting level. 22. The forecast of sales is $13,647 million for the first quarter of 2020 and $13,800 million for the second quarter of 2002, as the following table shows. Sales ($ Millions) Log of Sales Actual Value of Changes in the Log of Sales Δln(Salest) 1Q 2019 13,072 9.4782 0.0176 2Q 2019 12,446 9.4292 −0.0491 3Q 2019 12,958 9.4695 0.4030 4Q 2019 13,428 9.5051 0.0356 1Q 2020 13,647 9.5213 Date Forecast Value of Changes in the Log of Sales Δln(Salest) 0.0162 190 Date Learning Module 5 Time-Series Analysis Sales ($ Millions) Log of Sales 13,800 9.5324 2Q 2020 Actual Value of Changes in the Log of Sales Δln(Salest) Forecast Value of Changes in the Log of Sales Δln(Salest) 0.0111 We find the forecasted change in the log of sales for the first quarter of 2020 by inputting the value for the change in the log of sales from the previous quarter into the equation Δln(Salest) = 0.0068 + 0.2633Δln(Salest−1). Specifically, Δln(Salest) = 0.0068 + 0.2633(0.0356) = 0.0162, which means that we forecast the log of sales in the first quarter of 2020 to be 9.5051 + 0.0162 = 9.5213. Next, we forecast the change in the log of sales for the second quarter of 2020 as Δln(Salest) = 0.0068 + 0.2633(0.0162) = 0.0111. Note that we have to use our first-quarter 2020 estimated value of the change in the log of sales as our input for Δln(Salest−1) because we are forecasting past the period for which we have actual data. With a forecasted change of 0.0111, we forecast the log of sales in the second quarter of 2020 to be 9.5213 + 0.0111 = 9.5324. We have forecasted the log of sales in the first and second quarters of 2020 to be 9.5213 and 9.5324, respectively. Finally, we take the antilog of our estimates of the log of sales in the first and second quarters of 2020 to get our estimates of the level of sales: e9.5213 = 13,647 and e9.5324 = 13,800, respectively, for sales of $13,647 million and $13,800 million. 23. The RMSE of the out-of-sample forecast errors is approximately 3.6%. Out-of-sample error refers to the difference between the realized value and the forecasted value of Δln(Salest) for dates beyond the estimation period. In this case, the out-of-sample period is 1Q 2019 to 4Q 2019. These are the four quarters for which we have data that we did not use to obtain the estimated model Δln(Salest) = 0.0068 + 0.2633Δln(Salest−1). The steps to calculate RMSE are as follows: i. Take the difference between the actual and the forecast values. This is the error. ii. Square the error. iii. Sum the squared errors. iv. Divide by the number of forecasts. v. Take the square root of the average. We show the calculations for RMSE in the following table. Actual Values of Changes in the Log of Sales Δln(Salest) Forecast Values of Changes in the Log of Sales Δln(Salest) Error (Column 1 − Column 2) Squared Error (Column 3 Squared) 0.0176 0.0147 0.0029 0.0000 −0.0491 0.0107 −0.0598 0.0036 0.0403 0.0096 0.0307 0.0009 0.0356 0.0093 0.0263 0.0007 Sum 0.0052 Mean 0.0013 RMSE 0.036 Solutions 24. The lower the RMSE, the more accurate the forecasts of a model in forecasting. Therefore, the model with the RMSE of 2% has greater accuracy in forecasting than the model in Part A, which has an RMSE of 3.6%. 25. Predictions too far ahead can be nonsensical. For example, the AR(1) model we have been examining, ΔUERt = −0.0405 − 0.4674ΔUERt−1, taken at face value, predicts declining civilian unemployment into the indefinite future. Because the civilian unemployment rate will probably not go below 3% frictional unemployment and cannot go below 0% unemployment, this model’s long-range forecasts are implausible. The model is designed for short-term forecasting, as are many time-series models. 26. Using more years of data for estimation may lead to nonstationarity even in the series of first differences in the civilian unemployment rate. As we go further back in time, we increase the risk that the underlying civilian unemployment rate series has more than one regime (or true model). If the series has more than one regime, fitting one model to the entire period would not be correct. Note that when we have good reason to believe that a time series is stationary, a longer series of data is generally desirable. 27. C is correct. A random walk can be described by the equation xt = b0 + b1xt−1 + εt, where b0 = 0 and b1 = 1. So b0 = 0 is a characteristic of a random walk time series. A covariance-stationary series must satisfy the following three requirements: 1. The expected value of the time series must be constant and finite in all periods. 2. The variance of the time series must be constant and finite in all periods. 3. The covariance of the time series with itself for a fixed number of periods in the past or future must be constant and finite in all periods. b0 = 0 does not violate any of these three requirements and is thus consistent with the properties of a covariance-stationary time series. 28. B is correct. The critical t-statistic at a 5% confidence level is 1.98. As a result, neither the intercept nor the coefficient on the first lag of the first-differenced exchange rate in Regression 2 differs significantly from zero. Also, the residual autocorrelations do not differ significantly from zero. As a result, Regression 2 can be reduced to yt = εt, with a mean-reverting level of b0/(1 − b1) = 0/1 = 0. Therefore, the variance of yt in each period is var(εt) = σ2. The fact that the residuals are not autocorrelated is consistent with the covariance of the times series with itself being constant and finite at different lags. Because the variance and the mean of ytare constant and finite in each period, we can also conclude that ytis covariance stationary. 29. A is correct. If the exchange rate series is a random walk, then the first-differenced series will yield b0 = 0 and b1 = 0 and the error terms will not be serially correlated. The data in Exhibit 1 show that this is the case: Neither the intercept nor the coefficient on the first lag of the first-differenced exchange rate in Regression 2 differs significantly from zero because the t-statistics of both coefficients are less than the critical t-statistic of 1.98. Also, the residual autocorrelations do not differ significantly from zero because the t-statistics of all autocorrelations are less than the critical t-statistic of 1.98. Therefore, because all random walks have unit roots, the exchange rate time series used to run Regression 1 has a unit root. 191 192 Learning Module 5 Time-Series Analysis 30. C is correct. To conduct the Dickey–Fuller test, one must subtract the independent variable, xt−1,from both sides of the original AR(1) model. This results in a change of the dependent variable (from xt to xt − xt−1) and a change in the regression’s slope coefficient (from b1 to b1 − 1) but not a change in the independent variable. 31. C is correct. The regression output in Exhibit 2 suggests there is serial correlation in the residual errors. The fourth autocorrelation of the residual has a value of 0.6994 and a t-statistic of 4.3111, which is greater than the t-statistic critical value of 2.02. Therefore, the null hypothesis that the fourth autocorrelation is equal to zero can be rejected. This indicates strong and significant seasonal autocorrelation, which means the Regression 3 equation is misspecified. 32. C is correct. The quarterly sales for March 2016 are calculated as follows: ln Salest − ln Salest−1 = b0 + b1(ln Salest−1 − ln Salest−2) + b2(ln Salest−4 − ln Salest−5). ln Salest − ln 3.868 = 0.0092 − 0.1279(ln 3.868 − ln 3.780) + 0.7239(ln 3.836 − ln 3.418). ln Salest − 1.35274 = 0.0092 − 0.1279(1.35274 − 1.32972) + 0.7239(1.34443 − 1.22906). ln Salest = 1.35274 + 0.0092 − 0.1279(0.02301) + 0.7239(0.11538). ln Salest = 1.44251. Salest = e1.44251 = 4.231. 33. B is correct. Exhibit 5 shows that the time series of the stock prices of Company 1 exhibits heteroskedasticity, as evidenced by the fact that the time series is ARCH(1). If a time series is ARCH(1), then the variance of the error in one period depends on the variance of the error in previous periods. Therefore, the variance of the errors in period t + 1 can be predicted in period t using the formula ​​σˆ​2t+1​ = ​aˆ​0​+ ​aˆ​1​​εˆ​2t​​. 34. B is correct. When two time series have a unit root but are cointegrated, the error term in the linear regression of one time series on the other will be covariance stationary. Exhibit 5 shows that the series of stock prices of Company 2 and the oil prices both contain a unit root and the two time series are cointegrated. As a result, the regression coefficients and standard errors are consistent and can be used for hypothesis tests. Although the cointegrated regression estimates the long-term relation between the two series, it may not be the best model of the short-term relationship. 35. C is correct. As a result of the exponential trend in the time series of stock prices for Company 3, Busse would want to take the natural log of the series and then first-difference it. Because the time series also has serial correlation in the residuals from the trend model, Busse should use a more complex model, such as an autoregressive (AR) model. 36. The graph of ln(Salest) appears to trend upward over time. A series that trends upward or downward over time often has a unit root and is thus not covariance stationary. Therefore, using an AR(1) regression on the undifferenced series is probably not correct. In practice, we need to examine regression statistics to confirm such visual impressions. Solutions 37. The most common way to transform a time series with a unit root into a covariance-stationary time series is to difference the data—that is, to create a new series: Δln(Salest) = ln(Salest) − ln(Salest−1). 38. To determine whether we can use linear regression to model more than one time series, we should first determine whether any of the time series has a unit root. If none of the time series has a unit root, then we can safely use linear regression to test the relations between the two time series. Note that if one of the two variables has a unit root, then our analysis would not provide valid results; if both of the variables have unit roots, then we would need to evaluate whether the variables are cointegrated. 39. The quarterly sales of Avon show an upward trend and a clear seasonal pattern, as indicated by the repeated regular cycle. 40. A second explanatory variable, the change in the gross profit margin lagged four quarters, ΔGPMt−4, was added. 41. The model was augmented to account for seasonality in the time series (with quarterly data, significant autocorrelation at the fourth lag indicates seasonality). The standard error of the autocorrelation coefficient equals 1 divided by the _ ​ 40 ​, or 0.1581. The autocorrelation square root of the number of observations: ​1 / √ at the fourth lag (0.8496) is significant: t = 0.8496/0.1581 = 5.37. This indicates seasonality, and accordingly we added ΔGPMt−4. Note that in the augmented regression, the coefficient on ΔGPMt−4 is highly significant. (Although the autocorrelation at second lag is also significant, the fourth lag is more important because of the rationale of seasonality. Once the fourth lag is introduced as an independent variable, we might expect that the second lag in the residuals would not be significant.) 42. In order to determine whether this model is correctly specified, we need to test for serial correlation among the residuals. We want to test whether we can reject the null hypothesis that the value of each autocorrelation is 0 against the alternative hypothesis that each is not equal to 0. At the 0.05 significance level, with 68 observations and three parameters, this model has 65 degrees of freedom. The critical value of the t-statistic needed to reject the null hypothesis is thus about 2.0. The absolute value of the t-statistic for each autocorrelation is below 0.60 (less than 2.0), so we cannot reject the null hypothesis that each autocorrelation is not significantly different from 0. We have determined that the model is correctly specified. 43. If sales grew by 1% last quarter and by 2% four quarters ago, then the model predicts that sales growth this quarter will be 0.0121 − 0.0839[ln(1.01)] + 0.6292[ln(1.02)] = e0.02372 − 1 = 2.40%. 44. We should estimate the regression ΔUERt = b0 + b1ΔUERt−1 + εt and save the residuals from the regression. Then we should create a new variable, ˆ ​ε ​2t​​, by squaring 2 2 the residuals. Finally, we should estimate ˆ ε​ ​t​​ = ​a​0​+ ​a​1​ˆ ε​ ​t−1​+ ​u​t​and test to see whether a1 is statistically different from 0. 45. The series has a steady upward trend of growth, suggesting an exponential growth rate. This finding suggests transforming the series by taking the natural log and differencing the data. 46. First, we should determine whether the residuals from the AR(1) specification are serially uncorrelated. If the residuals are serially correlated, then we should try an AR(2) specification and then test the residuals from the AR(2) model for serial correlation. We should continue in this fashion until the residuals are se- 193 194 Learning Module 5 Time-Series Analysis rially uncorrelated and then look for seasonality in the residuals. If seasonality is present, we should add a seasonal lag. If no seasonality is present, we should test for ARCH. If ARCH is not present, we can conclude that the model is correctly specified. 47. If the model Δln(Salest) = b0 + b1[Δln(Salest−1)] + εt is correctly specified, then the series Δln(Salest) is covariance stationary. So, this series tends to its mean-reverting level, which is b0/(1 − b1), or 0.0661/(1 − 0.4698) = 0.1247. Solutions 195 268 Learning Module 6 Machine Learning PRACTICE PROBLEMS The following information relates to questions 1-10 Alef Associates manages a long-only fund specializing in global smallcap equities. Since its founding a decade ago, Alef maintains a portfolio of 100 stocks (out of an eligible universe of about 10,000 stocks). Some of these holdings are the result of screening the universe for attractive stocks based on several ratios that use readily available market and accounting data; others are the result of investment ideas generated by Alef ’s professional staff of five securities analysts and two portfolio managers. Although Alef ’s investment performance has been good, its Chief Investment Officer, Paul Moresanu, is contemplating a change in the investment process aimed at achieving even better returns. After attending multiple workshops and being approached by data vendors, Moresanu feels that data science should play a role in the way Alef selects its investments. He has also noticed that much of Alef ’s past outperformance is due to stocks that became takeover targets. After some research and reflection, Moresanu writes the following email to the Alef ’s CEO. Exhibit 1: Subject: Investment Process Reorganization I have been thinking about modernizing the way we select stock investments. Given that our past success has put Alef Associates in an excellent financial position, now seems to be a good time to invest in our future. What I propose is that we continue managing a portfolio of 100 global small-cap stocks but restructure our process to benefit from machine learning (ML). Importantly, the new process will still allow a role for human insight, for example, in providing domain knowledge. In addition, I think we should make a special effort to identify companies that are likely to be acquired. Specifically, I suggest following the four steps which would be repeated every quarter. Step 1 We apply ML techniques to a model including fundamental and technical variables (features) to predict next quarter’s return for each of the 100 stocks currently in our portfolio. Then, the 20 stocks with the lowest estimated return are identified for replacement. Step 2 We utilize ML techniques to divide our investable universe of about 10,000 stocks into 20 different groups, based on a wide variety of the most relevant financial and non-financial characteristics. The idea is to prevent unintended portfolio concentration by selecting stocks from each of these distinct groups. Step 3 For each of the 20 different groups, we use labeled data to train a model that will predict the five stocks (in any given group) that are most likely to become acquisition targets in the next one year. Practice Problems 269 Step 4 Our five experienced securities analysts are each assigned four of the groups, and then each analyst selects their one best stock pick from each of their assigned groups. These 20 “high-conviction” stocks will be added to our portfolio (in replacement of the 20 relatively underperforming stocks to be sold in Step 1). A couple of additional comments related to the above: Comment 1 The ML algorithms will require large amounts of data. We would first need to explore using free or inexpensive historical datasets and then evaluate their usefulness for the ML-based stock selection processes before deciding on using data that requires subscription. Comment 2 As time passes, we expect to find additional ways to apply ML techniques to refine Alef ’s investment processes. What do you think? Paul Moresanu 1. The machine learning techniques appropriate for executing Step 1 are most likely to be based on: A. regression B. classification C. clustering 2. Assuming regularization is utilized in the machine learning technique used for executing Step 1, which of the following ML models would be least appropriate: A. Regression tree with pruning. B. LASSO with lambda (λ) equal to 0. C. LASSO with lambda (λ) between 0.5 and 1. 3. Which of the following machine learning techniques is most appropriate for executing Step 2: A. K-Means Clustering B. Principal Components Analysis (PCA) C. Classification and Regression Trees (CART) 4. The hyperparameter in the ML model to be used for accomplishing Step 2 is? A. 100, the number of small-cap stocks in Alef ’s portfolio. B. 10,000, the eligible universe of small-cap stocks in which Alef can potentially invest. C. 20, the number of different groups (i.e. clusters) into which the eligible universe of small-cap stocks will be divided. 5. The target variable for the labelled training data to be used in Step 3 is most likely which one of the following? A. A continuous target variable. 270 Learning Module 6 Machine Learning B. A categorical target variable. C. An ordinal target variable. 6. Comparing two ML models that could be used to accomplish Step 3, which statement(s) best describe(s) the advantages of using Classification and Regression Trees (CART) instead of K-Nearest Neighbor (KNN)? Statement 1 For CART there is no requirement to specify an initial hyperparameter (like K). Statement 2 For CART there is no requirement to specify a similarity (or distance) measure. Statement 3 For CART the output provides a visual explanation for the prediction. A. Statement 1 only. B. Statement 3 only. C. Statements 1, 3 and 3. 7. Assuming a Classification and Regression Tree (CART) model is used to accomplish Step 3, which of the following is most likely to result in model overfitting? A. Using the k-fold cross validation method B. Including an overfitting penalty (i.e., regularization term). C. Using a fitting curve to select a model with low bias error and high variance error. 8. Assuming a Classification and Regression Tree (CART) model is initially used to accomplish Step 3, as a further step which of the following techniques is most likely to result in more accurate predictions? A. Discarding CART and using the predictions of a Support Vector Machine (SVM) model instead. B. Discarding CART and using the predictions of a K-Nearest Neighbor (KNN) model instead. C. Combining the predictions of the CART model with the predictions of other models – such as logistic regression, SVM, and KNN – via ensemble learning. 9. Regarding Comment #2, Moresanu has been thinking about the applications of neural networks (NNs) and deep learning (DL) to investment management. Which statement(s) best describe(s) the tasks for which NNs and DL are well-suited? Statement 1 NNs and DL are well-suited for image and speech recognition, and natural language processing. Statement 2 NNs and DL are well-suited for developing single variable ordinary least squares regression models. Practice Problems Statement 3 271 NNs and DL are well-suited for modelling non-linearities and complex interactions among many features. A. Statement 2 only. B. Statements 1 and 3. C. Statements 1, 2 and 3. 10. Regarding neural networks (NNs) that Alef might potentially implement, which of the following statements is least accurate? A. NNs must have at least 10 hidden layers to be considered deep learning nets. B. The activation function in a node operates like a light dimmer switch since it decreases or increases the strength of the total net input. C. The summation operator receives input values, multiplies each by a weight, sums up the weighted values into the total net input, and passes it to the activation function. 272 Learning Module 6 Machine Learning SOLUTIONS 1. A is correct. The target variable (quarterly return) is continuous, hence this calls for a supervised machine learning based regression model. B is incorrect, since classification uses categorical or ordinal target variables, while in Step 1 the target variable (quarterly return) is continuous. C is incorrect, since clustering involves unsupervised machine learning so does not have a target variable. 2. B is correct. It is least appropriate because with LASSO, when λ = 0 the penalty (i.e., regularization) term reduces to zero, so there is no regularization and the regression is equivalent to an ordinary least squares (OLS) regression. A is incorrect. With Classification and Regression Trees (CART), one way that regularization can be implemented is via pruning which will reduce the size of the regression tree—sections that provide little explanatory power are pruned (i.e., removed). C is incorrect. With LASSO, when λ is between 0.5 and 1 the relatively large penalty (i.e., regularization) term requires that a feature makes a sufficient contribution to model fit to offset the penalty from including it in the model. 3. A is correct. K-Means clustering is an unsupervised machine learning algorithm which repeatedly partitions observations into a fixed number, k, of non-overlapping clusters (i.e., groups). B is incorrect. Principal Components Analysis is a long-established statistical method for dimension reduction, not clustering. PCA aims to summarize or reduce highly correlated features of data into a few main, uncorrelated composite variables. C is incorrect. CART is a supervised machine learning technique that is most commonly applied to binary classification or regression. 4. C is correct. Here, 20 is a hyperparameter (in the K-Means algorithm), which is a parameter whose value must be set by the researcher before learning begins. A is incorrect, because it is not a hyperparameter. It is just the size (number of stocks) of Alef ’s portfolio. B is incorrect, because it is not a hyperparameter. It is just the size (number of stocks) of Alef ’s eligible universe. 5. B is correct. To predict which stocks are likely to become acquisition targets, the ML model would need to be trained on categorical labelled data having the following two categories: “0” for “not acquisition target”, and “1” for “acquisition target”. A is incorrect, because the target variable is categorical, not continuous. C is incorrect, because the target variable is categorical, not ordinal (i.e., 1st, 2nd, 3rd, etc.). 6. C is correct. The advantages of using CART over KNN to classify companies into two categories (“not acquisition target” and “acquisition target”), include all of the following: For CART there are no requirements to specify an initial hyperparameter (like K) or a similarity (or distance) measure as with KNN, and CART provides a visual explanation for the prediction (i.e., the feature variables and their cut-off values at each node). A is incorrect, because CART provides all of the advantages indicated in Statements 1, 2 and 3. Solutions B is incorrect, because CART provides all of the advantages indicated in Statements 1, 2 and 3. 7. C is correct. A fitting curve shows the trade-off between bias error and variance error for various potential models. A model with low bias error and high variance error is, by definition, overfitted. A is incorrect, because there are two common methods to reduce overfitting, one of which is proper data sampling and cross-validation. K-fold cross validation is such a method for estimating out-of-sample error directly by determining the errorin validation samples. B is incorrect, because there are two common methods to reduce overfitting, one of which is preventing the algorithm from getting too complex during selection and training, which requires estimating an overfitting penalty. 8. C is correct. Ensemble learning is the technique of combining the predictions from a collection of models, and it typically produces more accurate and more stable predictions than the best single model. A is incorrect, because a single model will have a certain error rate and will make noisy predictions. By taking the average result of many predictions from many models (i.e., ensemble learning) one can expect to achieve a reduction in noise as the average result converges towards a more accurate prediction. B is incorrect, because a single model will have a certain error rate and will make noisy predictions. By taking the average result of many predictions from many models (i.e., ensemble learning) one can expect to achieve a reduction in noise as the average result converges towards a more accurate prediction. 9. B is correct. NNs and DL are well-suited for addressing highly complex machine learning tasks, such as image classification, face recognition, speech recognition and natural language processing. These complicated tasks are characterized by non-linearities and complex interactions between large numbers of feature inputs. A is incorrect, because NNs and DL are well-suited for addressing highly complex machine learning tasks, not simple single variable OLS regression models. C is incorrect, because NNs and DL are well-suited for addressing highly complex machine learning tasks, not simple single variable OLS regression models. 10. A is correct. It is the least accurate answer because neural networks with many hidden layers—at least 3, but often more than 20 hidden layers—are known as deep learning nets. B is incorrect, because the node’s activation function operates like a light dimmer switch which decreases or increases the strength of the (total net) input. C is incorrect, because the node’s summation operator multiplies each (input) value by a weight and sums up the weighted values to form the total net input. The total net input is then passed to the activation function. 273 Practice Problems 345 PRACTICE PROBLEMS The following information relates to questions 1-15 Aaliyah Schultz is a fixed-income portfolio manager at Aries Investments. Schultz supervises Ameris Steele, a junior analyst. A few years ago, Schultz developed a proprietary machine learning (ML) model that aims to predict downgrades of publicly-traded firms by bond rating agencies. The model currently relies only on structured financial data collected from different sources. Schultz thinks the model’s predictive power may be improved by incorporating sentiment data derived from textual analysis of news articles and Twitter content relating to the subject companies. Schultz and Steele meet to discuss plans for incorporating the sentiment data into the model. They discuss the differences in the steps between building ML models that use traditional structured data and building ML models that use textual big data. Steele tells Schultz: Statement 1 The second step in building text-based ML models is text preparation and wrangling, whereas the second step in building ML models using structured data is data collection. Statement 2 The fourth step in building both types of models encompasses data/text exploration. Steele expresses concern about using Twitter content in the model, noting that research suggests that as much as 10%–15% of social media content is from fake accounts. Schultz tells Steele that she understands her concern but thinks the potential for model improvement outweighs the concern. Steele begins building a model that combines the structured financial data and the sentiment data. She starts with cleansing and wrangling the raw structured financial data. Exhibit 1 presents a small sample of the raw dataset before cleansing: Each row represents data for a particular firm. Exhibit 1: Sample of Raw Structured Data Before Cleansing ID Ticker IPO Date Industry (NAICS) EBIT Interest Expense Total Debt 1 ABC 4/6/17 44 9.4 0.6 10.1 2 BCD November 15, 2004 52 5.5 0.4 6.2 3 HIJ 26-Jun-74 54 8.9 1.2 15.8 4 KLM 14-Mar-15 72 5.7 1.5 0.0 After cleansing the data, Steele then preprocesses the dataset. She creates two new variables: an “Age” variable based on the firm’s IPO date and an “Interest Coverage Ratio” variable equal to EBIT divided by interest expense. She also deletes the “IPO Date” variable from the dataset. After applying these transformations, Steele scales the financial data using normalization. She notes that over the full sample dataset, the “Interest Expense” variable ranges from a minimum of 0.2 346 Learning Module 7 Big Data Projects and a maximum of 12.2, with a mean of 1.1 and a standard deviation of 0.4. Steele and Schultz then discuss how to preprocess the raw text data. Steele tells Schultz that the process can be completed in the following three steps: Step 1 Cleanse the raw text data. Step 2 Split the cleansed data into a collection of words for them to be normalized. Step 3 Normalize the collection of words from Step 2 and create a distinct set of tokens from the normalized words. With respect to Step 1, Steele tells Schultz: “I believe I should remove all html tags, punctuations, numbers, and extra white spaces from the data before normalizing them.” After properly cleansing the raw text data, Steele completes Steps 2 and 3. She then performs exploratory data analysis. To assist in feature selection, she wants to create a visualization that shows the most informative words in the dataset based on their term frequency (TF) values. After creating and analyzing the visualization, Steele is concerned that some tokens are likely to be noise features for ML model training; therefore, she wants to remove them. Steele and Schultz discuss the importance of feature selection and feature engineering in ML model training. Steele tells Schultz: “Appropriate feature selection is a key factor in minimizing model overfitting, whereas feature engineering tends to prevent model underfitting.” Once satisfied with the final set of features, Steele selects and runs a model on the training set that classifies the text as having positive sentiment (Class “1” or negative sentiment (Class “0”). She then evaluates its performance using error analysis. The resulting confusion matrix is presented in Exhibit 2. Exhibit 2: Confusion Matrix Actual Training Results Predicted Results Class “1” Class “0” Class “1” TP = 182 FN = 31 Class “0” FP = 52 TN = 96 1. Which of Steele’s statements relating to the steps in building structured data-based and text-based ML models is correct? A. Only Statement 1 is correct. B. Only Statement 2 is correct. C. Statement 1 and Statement 2 are correct. 2. Steele’s concern about using Twitter data in the model best relates to: A. volume. B. velocity. C. veracity. Practice Problems 3. What type of error appears to be present in the IPO Date column of Exhibit 1? A. invalidity error. B. inconsistency error. C. non-uniformity error. 4. What type of error is most likely present in the last row of data (ID #4) in Exhibit 1? A. Inconsistency error B. Incompleteness error C. Non-uniformity error 5. During the preprocessing of the data in Exhibit 1, what type of data transformation did Steele perform during the data preprocessing step? A. Extraction B. Conversion C. Aggregation 6. Based on Exhibit 1, for the firm with ID #3, Steele should compute the scaled value for the “Interest Expense” variable as: A. 0.008. B. 0.083. C. 0.250. 7. Is Steele’s statement regarding Step 1 of the preprocessing of raw text data correct? A. Yes. B. No, because her suggested treatment of punctuation is incorrect. C. No, because her suggested treatment of extra white spaces is incorrect. 8. Steele’s Step 2 can be best described as: A. tokenization. B. lemmatization. C. standardization. 9. The output created in Steele’s Step 3 can be best described as a: A. bag-of-words. B. set of n-grams. C. document term matrix. 10. Given her objective, the visualization that Steele should create in the exploratory 347 348 Learning Module 7 Big Data Projects data analysis step is a: A. scatter plot. B. word cloud. C. document term matrix. 11. To address her concern in her exploratory data analysis, Steele should focus on those tokens that have: A. low chi-square statistics. B. low mutual information (ML) values. C. very low and very high term frequency (TF) values. 12. Is Steele’s statement regarding the relationship between feature selection/feature engineering and model fit correct? A. Yes. B. No, because she is incorrect with respect to feature selection. C. No, because she is incorrect with respect to feature engineering. 13. Based on Exhibit 2, the model’s precision metric is closest to: A. 78%. B. 81%. C. 85%. 14. Based on Exhibit 2, the model’s F1 score is closest to: A. 77%. B. 81%. C. 85%. 15. Based on Exhibit 2, the model’s accuracy metric is closest to: A. 77%. B. 81%. C. 85%. The following information relates to questions 16-24 Bernadette Rivera is a portfolio manager at Voxkor, a private equity company that provides financing to early-stage start-up businesses. Rivera is working with a data analyst, Tim Achler, on a text-based machine-learning (ML) model to enhance Voxkor’s predictive ability to identify successful start-ups. Voxkor currently uses ML models based only on traditional, structured financial Practice Problems 349 data but would like to develop a new ML model that analyzes textual big data gathered from the internet. The model will classify text information into positive or negative sentiment classes for each respective start-up. Rivera wants to confirm her understanding of any differences in the ML model building steps between data analysis projects that use traditional structured data and projects that involve unstructured, text-based data. Rivera makes the following statements: Statement 1 Some of the methods used in the exploration step are different for structured and unstructured data, but for both types of data, the step involves feature selection and feature engineering. Statement 2 A major difference when developing a text-based ML model is the curation step, which involves cleansing, preprocessing, and converting the data into a structured format usable for model training. Achler uses a web spidering program to obtain the data for the text-based model. The program extracts raw content from social media webpages, which contains English language sentences and special characters. After curating the text, Achler removes unnecessary elements from the raw text using regular expression software and completes additional text cleansing and preprocessing tasks. Next, Achler and Rivera discuss remaining text wrangling tasks—specifically, which tokens to include in the document term matrix (DTM). Achler divides unique tokens into three groups; a sample of each group is shown in Exhibit 1. Exhibit 1: Summary of Sample Tokens Token Group 1 Token Group 2 Token Group 3 "not_increas_market" "not_increased_market" "not," "increased," "market" "currencysign" "currencysign" "EUR" "sale_decreas" "sale_decreased" "Sales," "decreased" The dataset is now ready for the text exploration step. At this point in the process, Rivera wants to better comprehend the collection of unique words. Achler recommends an exploratory data analysis technique that visualizes words by varying their font size proportionately to the number of occurrences of each word in the corpus. As an additional part of the text exploration step, Achler conducts a term frequency analysis to identify outliers. Achler summarizes the analysis in Exhibit 2. Exhibit 2: Words with Highest and Lowest Frequency Value Group 1 Group 2 Word Frequency Word Frequency the 0.04935 naval 1.0123e-05 and 0.04661 stereotype 1.5185e-05 to 0.04179 till 1.5185e-05 that 0.03577 ribbon 2.0247e-05 350 Learning Module 7 Big Data Projects Group 1 Group 2 Word Frequency Word Frequency in 0.03368 deposit 2.5308e-05 Note: “e-05” represents 10–5. Achler has the data ready for the model training process. Rivera asks Achler to include start-up failure rates as a feature. Achler notices that the number of start-ups that fail (majority class) is significantly larger than the number of the start-ups that are successful (minority class). Achler is concerned that because of class imbalance, the model will not be able to discriminate between start-ups that fail and start-ups that are successful. Achler splits the DTM into training, cross-validation, and test datasets. Achler uses a supervised learning approach to train the logistic regression model in predicting sentiment. Applying the receiver operating characteristics (ROC) technique and area under the curve (AUC) metrics, Achler evaluates model performance on both the training and the cross-validation datasets. The trained model performance for three different logistic regressions’ threshold p-values is presented in Exhibit 3. Exhibit 3: AUC for Different Threshold p-values Threshold p-Value Training Set Cross-Validation Set p = 0.57 56.7% 57.3% p = 0.79 91.3% 89.7% p = 0.84 98.4% 87.1% Rivera suggests adjusting the model’s hyperparameters to improve performance. Achler runs a grid search that compares the difference between the prediction error on both the training and the cross-validation datasets for various combinations of hyperparameter values. For the current values of hyperparameters, Achler observes that the prediction error on the training dataset is small, whereas the prediction error on the cross-validation dataset is significantly larger. 16. Which of Rivera’s statements about differences in ML model building steps is correct? A. Only Statement 1 B. Only Statement 2 C. Both Statement 1 and Statement 2 17. Based on the source of the data, as part of the data cleansing and wrangling process, Achler most likely needs to remove: A. html tags and perform scaling. B. numbers and perform lemmatization. C. white spaces and perform winsorization. 18. Based on Exhibit 1, which token group has most likely undergone the text prepa- Practice Problems ration and wrangling process? A. Token Group 1 B. Token Group 2 C. Token Group 3 19. The visual text representation technique that Achler recommends to Rivera is a: A. word cloud. B. bag of words. C. collection frequency. 20. Based on Exhibit 2, Achler should exclude from further analysis words in: A. only Group 1. B. only Group 2. C. both Group 1 and Group 2. 21. Achler’s model training concern related to the model’s ability to discriminate could be addressed by randomly: A. oversampling the failed start-up data. B. oversampling the successful start-up data. C. undersampling the successful start-up data. 22. Based on Exhibit 3, which threshold p-value indicates the best fitting model? A. 0.57 B. 0.79 C. 0.84 23. Based on Exhibit 3, if Achler wants to improve model performance at the threshold p-value of 0.84, he should: A. tune the model to lower the AUC. B. adjust model parameters to decrease ROC convexity. C. apply LASSO regularization to the logistic regression. 24. Based on Achler’s grid search analysis, the current model can be characterized as: A. underfitted. B. having low variance. C. exhibiting slight regularization. 351 352 Learning Module 7 Big Data Projects The following information relates to questions 25-31 Iesha Azarov is a senior analyst at Ganymede Moon Partners (Ganymede), where he works with junior analyst Pàola Bector. Azarov would like to incorporate machine learning (ML) models into the company’s analytical process. Azarov asks Bector to develop ML models for two unstructured stock sentiment datasets, Dataset ABC and Dataset XYZ. Both datasets have been cleaned and preprocessed in preparation for text exploration and model training. Following an exploratory data analysis that revealed Dataset ABC’s most frequent tokens, Bector conducts a collection frequency analysis. Bector then computes TF–IDF (term frequency–inverse document frequency) for several words in the collection and tells Azarov the following: Statement 1 IDF is equal to the inverse of the document frequency measure. Statement 2 TF at the collection level is multiplied by IDF to calculate TF–IDF. Statement 3 TF–IDF values vary by the number of documents in the dataset, and therefore, model performance can vary when applied to a dataset with just a few documents. Bector notes that Dataset ABC is characterized by the absence of ground truth. Bector turns his attention to Dataset XYZ, containing 84,000 tokens and 10,000 sentences. Bector chooses an appropriate feature selection method to identify and remove unnecessary tokens from the dataset and then focuses on model training. For performance evaluation purposes, Dataset XYZ is split into a training set, cross-validation (CV) set, and test set. Each of the sentences has already been labeled as either a positive sentiment (Class “1”) or a negative sentiment (Class “0”) sentence. There is an unequal class distribution between the positive sentiment and negative sentiment sentences in Dataset XYZ. Simple random sampling is applied within levels of the sentiment class labels to balance the class distributions within the splits. Bector’s view is that the false positive and false negative evaluation metrics should be given equal weight. Select performance data from the cross-validation set confusion matrices is presented in Exhibit 1: Exhibit 1: Performance Metrics for Dataset XYZ Confusion Matrix CV Data (threshold p-value) Performance Metrics Precision Recall F1 Score Accuracy A 0.50 0.95 0.87 0.91 0.91 B 0.35 0.93 0.90 0.91 0.92 C 0.65 0.86 0.97 0.92 0.91 Azarov and Bector evaluate the Dataset XYZ performance metrics for Confusion Matrices A, B, and C in Exhibit 1. Azarov says, “For Ganymede’s purposes, we should be most concerned with the cost of Type I errors.” Azarov requests that Bector apply the ML model to the test dataset for Dataset XYZ, assuming a threshold p-value of 0.65. Exhibit 2 contains a sample of results from the test dataset corpus. Practice Problems 353 Exhibit 2: 10 Sample Results of Test Data for Dataset XYZ Sentence # Actual Sentiment Target p-Value 1 1 0.75 2 0 0.45 3 1 0.64 4 1 0.81 5 0 0.43 6 1 0.78 7 0 0.59 8 1 0.60 9 0 0.67 10 0 0.54 Bector makes the following remarks regarding model training: Remark 1: Method selection is governed by such factors as the type of data and the size of data. Remark 2: In the performance evaluation stage, model fitting errors, such as bias error and variance error, are used to measure goodness of fit. 25. Based on the text exploration method used for Dataset ABC, tokens that potentially carry important information useful for differentiating the sentiment embedded in the text are most likely to have values that are: A. low. B. intermediate. C. high. 26. Which of Bector’s statements regarding TF, IDF, and TF–IDF is correct? A. Statement 1 B. Statement 2 C. Statement 3 27. What percentage of Dataset ABC should be allocated to a training subset? A. 0% B. 20% C. 60% 28. Based only on Dataset XYZ’s composition and Bector’s view regarding false positive and false negative evaluation metrics, which performance measure is most appropriate? A. Recall B. F1 score 354 Learning Module 7 Big Data Projects C. Precision 29. Based on Exhibit 1, which confusion matrix demonstrates the most favorable value of the performance metric that best addresses Azarov’s concern? A. Confusion Matrix A B. Confusion Matrix B C. Confusion Matrix C 30. Based on Exhibit 2, the accuracy metric for Dataset XYZ’s test set sample is closest to: A. 0.67. B. 0.70. C. 0.75. 31. Which of Bector’s remarks related to model training is correct? A. Only Remark 1 B. Only Remark 2 C. Both Remark 1 and Remark 2 Solutions 355 SOLUTIONS 1. B is correct. The five steps in building structured data-based ML models are: 1) conceptualization of the modeling task, 2) data collection, 3) data preparation and wrangling, 4) data exploration, and 5) model training. The five steps in building text-based ML models are: 1) text problem formulation, 2) data (text) curation, 3) text preparation and wrangling, 4) text exploration, and 5) model training. Statement 1 is incorrect: Text preparation and wrangling is the third step in building text ML models and occurs after the second data (text) curation step. Statement 2 is correct: The fourth step in building both types of models encompasses data/text exploration. 2. C is correct. Veracity relates to the credibility and reliability of different data sources. Steele is concerned about the credibility and reliability of Twitter content, noting that research suggests that as much as 10%–15% of social media content is from fake accounts. 3. C is correct. A non-uniformity error occurs when the data are not presented in an identical format. The data in the “IPO Date” column represent the IPO date of each firm. While all rows are populated with valid dates in the IPO Date column, the dates are presented in different formats (e.g., mm/dd/yyyy, dd/mm/yyyy). 4. A is correct. There appears to be an inconsistency error in the last row (ID #4). An inconsistency error occurs when a data point conflicts with corresponding data points or reality. In the last row, the interest expense data item has a value of 1.5, and the total debt item has a value of 0.0. This appears to be an error: Firms that have interest expense are likely to have debt in their capital structure, so either the interest expense is incorrect or the total debt value is incorrect. Steele should investigate this issue by using alternative data sources to confirm the correct values for these variables. 5. A is correct. During the data preprocessing step, Steele created a new “Age” variable based on the firm’s IPO date and then deleted the “IPO Date” variable from the dataset. She also created a new “Interest Coverage Ratio” variable equal to EBIT divided by interest expense. Extraction refers to a data transformation where a new variable is extracted from a current variable for ease of analyzing and using for training an ML model, such as creating an age variable from a date variable or a ratio variable. Steele also performed a selection transformation by deleting the IPO Date variable, which refers to deleting the data columns that are not needed for the project. 6. B is correct. Steele uses normalization to scale the financial data. Normalization is the process of rescaling numeric variables in the range of [0, 1]. To normalize variable X, the minimum value (Xmin) is subtracted from each observation (Xi), and then this value is divided by the difference between the maximum and minimum values of X (Xmax – Xmin): ​Xi​​− ​Xmin ​ ​ ​ ​ ​− X ​​Xi​ (normalized)​ = _ ​X ​ ​ ​​​. max min The firm with ID #3 has an interest expense of 1.2. So, its normalized value is calculated as: 1.2 − 0.2 ​12.2 − 0.2 ​ = 0.083​. ​​Xi​ (normalized)​ = _ 7. B is correct. Although most punctuations are not necessary for text analysis and 356 Learning Module 7 Big Data Projects should be removed, some punctuations (e.g., percentage signs, currency symbols, and question marks) may be useful for ML model training. Such punctuations should be substituted with annotations (e.g., /percentSign/, /dollarSign/, and / questionMark/) to preserve their grammatical meaning in the text. Such annotations preserve the semantic meaning of important characters in the text for further text processing and analysis stages. 8. A is correct. Tokenization is the process of splitting a given text into separate tokens. This step takes place after cleansing the raw text data (removing html tags, numbers, extra white spaces, etc.). The tokens are then normalized to create the bag-of-words (BOW). 9. A is correct. After the cleansed text is normalized, a bag-of-words is created. A bag-of-words (BOW) is a collection of a distinct set of tokens from all the texts in a sample dataset. 10. B is correct. Steele wants to create a visualization for Schultz that shows the most informative words in the dataset based on their term frequency (TF, the ratio of the number of times a given token occurs in the dataset to the total number of tokens in the dataset) values. A word cloud is a common visualization when working with text data as it can be made to visualize the most informative words and their TF values. The most commonly occurring words in the dataset can be shown by varying font size, and color is used to add more dimensions, such as frequency and length of words. 11. C is correct. Frequency measures can be used for vocabulary pruning to remove noise features by filtering the tokens with very high and low TF values across all the texts. Noise features are both the most frequent and most sparse (or rare) tokens in the dataset. On one end, noise features can be stop words that are typically present frequently in all the texts across the dataset. On the other end, noise features can be sparse terms that are present in only a few text files. Text classification involves dividing text documents into assigned classes. The frequent tokens strain the ML model to choose a decision boundary among the texts as the terms are present across all the texts (an example of underfitting). The rare tokens mislead the ML model into classifying texts containing the rare terms into a specific class (an example of overfitting). Thus, identifying and removing noise features are critical steps for text classification applications. 12. A is correct. A dataset with a small number of features may not carry all the characteristics that explain relationships between the target variable and the features. Conversely, a large number of features can complicate the model and potentially distort patterns in the data due to low degrees of freedom, causing overfitting. Therefore, appropriate feature selection is a key factor in minimizing such model overfitting. Feature engineering tends to prevent underfitting in the training of the model. New features, when engineered properly, can elevate the underlying data points that better explain the interactions of features. Thus, feature engineering can be critical to overcome underfitting. 13. A is correct. Precision, the ratio of correctly predicted positive classes (true positives) to all predicted positive classes, is calculated as: Precision (P) = TP/(TP + FP) = 182/(182 + 52) = 0.7778 (78%). 14. B is correct. The model’s F1 score, which is the harmonic mean of precision and recall, is calculated as: F1 score = (2 × P × R)/(P + R). Solutions F1 score = (2 × 0.7778 × 0.8545)/(0.7778 + 0.8545) = 0.8143 (81%). 15. A is correct. The model’s accuracy, which is the percentage of correctly predicted classes out of total predictions, is calculated as: Accuracy = (TP + TN)/(TP + FP + TN + FN). Accuracy = (182 + 96)/(182 + 52 + 96 + 31) = 0.7701 (77%). 16. A is correct. Statement 1 is correct because some of the methods used in the fourth step of ML model building (data/text exploration) are different for structured and unstructured data, and for both structured and unstructured data, the exploration step encompasses feature selection and feature engineering. Statement 2 is incorrect because Rivera described the text preparation and wrangling step, not the text curation step. The data (text) curation step involves gathering relevant external text data via web services or programs that extract raw content from a source. B and C are incorrect because Statement 2 is incorrect. Rivera described the text preparation and wrangling step, not the text curation step. The data (text) curation step involves gathering relevant external text data via web services or programs that extract raw content from a source. 17. B is correct. Achler uses a web spidering program that extracts unstructured raw content from social media webpages. Raw text data are a sequence of characters and contain other non-useful elements including html tags, punctuation, and white spaces (including tabs, line breaks, and new lines). Removing numbers is one of the basic operations in the text cleansing/preparation process for unstructured data. When numbers (or digits) are present in the text, they should be removed or substituted with the annotation “/number/.” Lemmatization, which takes places during the text wrangling/preprocessing process for unstructured data, is the process of converting inflected forms of a word into its morphological root (known as lemma). Lemmatization reduces the repetition of words occurring in various forms while maintaining the semantic structure of the text data, thereby aiding in training less complex ML models. A is incorrect because although html tag removal is part of text cleansing/preparation for unstructured data, scaling is a data wrangling/preprocessing process applied to structured data. Scaling adjusts the range of a feature by shifting and changing the scale of data; it is performed on numeric variables, not on text data. C is incorrect because although raw text contains white spaces (including tabs, line breaks, and new lines) that need to be removed as part of the data cleansing/ preparation process for unstructured data, winsorization is a data wrangling/preprocessing task performed on values of data points, not on text data. Winsorization is used for structured numerical data and replaces extreme values and outliers with the maximum (for large-value outliers) and minimum (for small-value outliers) values of data points that are not outliers. 18. A is correct. Data preparation and wrangling involve cleansing and organizing raw data into a consolidated format. Token Group 1 includes n-grams (“not_increas_market,” “sale_decreas”) and the words that have been converted from their inflected forms into their base word (“increas,” “decreas”), and the currency symbol has been replaced with a “currencysign” token. N-gram tokens are helpful for keeping negations intact in the text, which is vital for sentiment prediction. The process of converting inflected forms of a word into its base word is called stemming and helps decrease data sparseness, thereby aiding in training less 357 358 Learning Module 7 Big Data Projects complex ML models. B is incorrect because Token Group 2 includes inflected forms of words (“increased,” “decreased”) before conversion into their base words (known as stems). Stemming (along with lemmatization) decreases data sparseness by aggregating many sparsely occurring words in relatively less sparse stems or lemmas, thereby aiding in training less complex ML models. C is incorrect because Token Group 3 includes inflected forms of words (“increased,” “decreased”) before conversion into their base words (known as stems). In addition, the “EUR” currency symbol has not been replaced with the “currencysign” token and the word “Sales” has not been lowercased. 19. A is correct. Achler recommends creating a word cloud, which is a common text visualization technique at the data exploration phase in ML model building. The most commonly occurring words in the dataset can be visualized by varying font size, and color is used to add more dimensions, such as frequency and length of words. B is incorrect because Achler recommends creating a word cloud and not a bag of words (BOW). A BOW is a collection of a distinct set of tokens from all the texts in a sample dataset. A BOW representation is a basic procedure used primarily to analyze text during Step 3 (text wrangling/preprocessing), although it may also be used in Step 4 during the feature engineering process. In contrast to a word cloud, which visually varies font size and color, BOW is simply a set of words (typically displayed in table). C is incorrect because Achler recommends creating a word cloud and not a collection frequency. Collection frequency (or term frequency) is the ratio of the number of times a given token occurs in all the texts in the dataset to the total number of tokens in the dataset. Collection frequency can be calculated and examined to identify outlier words, but it is not a visual text representation tool. 20. C is correct. Achler should remove words that are in both Group 1 and Group 2. Term frequency values range between 0 and 1. Group 1 consists of the highest frequency values (e.g., “the” = 0.04935), and Group 2 consists of the lowest frequency values (e.g., “naval” = 1.0123e–05). Frequency analysis on the processed text data helps in filtering unnecessary tokens (or features) by quantifying how important tokens are in a sentence and in the corpus as a whole. The most frequent tokens (Group 1) strain the machine-learning model to choose a decision boundary among the texts as the terms are present across all the texts, which leads to model underfitting. The least frequent tokens (Group 2) mislead the machine-learning model into classifying texts containing the rare terms into a specific class, which leads to model overfitting. Identifying and removing noise features is critical for text classification applications. A is incorrect because words in both Group 1 and Group 2 should be removed. The words with high term frequency value are mostly stop words, present in most sentences. Stop words do not carry a semantic meaning for the purpose of text analyses and ML training, so they do not contribute to differentiating sentiment. B is incorrect because words in both Group 1 and Group 2 should be removed. Terms with low term frequency value are mostly rare terms, ones appearing only once or twice in the data. They do not contribute to differentiating sentiment. 21. B is correct. Achler is concerned about class imbalance, which can be resolved by balancing the training data. The majority class (the failed start-up data) can be randomly undersampled, or the minority class (the successful start-up data) can be randomly oversampled. 22. B is correct. The higher the AUC, the better the model performance. For the Solutions threshold p-value of 0.79, the AUC is 91.3% on the training dataset and 89.7% on the cross-validation dataset, and the ROC curves are similar for model performance on both datasets. These findings suggest that the model performs similarly on both training and CV data and thus indicate a good fitting model. A is incorrect because for the threshold p-value of 0.57, the AUC is 56.7% on the training dataset and 57.3% on the cross-validation dataset. The AUC close to 50% signifies random guessing on both the training dataset and the cross-validation dataset. The implication is that for the threshold p-value of 0.57, the model is randomly guessing and is not performing well. C is incorrect because for the threshold p-value of 0.84, there is a substantial difference between the AUC on the training dataset (98.4%) and the AUC on the cross-validation dataset (87.1%). This suggests that the model performs comparatively poorly (with a higher rate of error or misclassification) on the cross-validation dataset when compared with training data. Thus, the implication is that the model is overfitted. 23. C is correct. At the threshold p-value of 0.84, the AUC is 98.4% for the training dataset and 87.1% for the cross-validation dataset, which suggests that the model is currently overfitted. Least absolute shrinkage and selection operator (LASSO) regularization can be applied to the logistic regression to prevent overfitting of logistic regression models. A is incorrect because the higher the AUC, the better the model performance. B is incorrect because the more convex the ROC curve and the higher the AUC, the better the model performance. Adjusting model parameters with the aim of achieving lower ROC convexity would result in worse model performance on the cross-validation dataset. 24. C is correct. Slight regularization occurs when the prediction error on the training dataset is small, while the prediction error on the cross-validation data set is significantly larger. This difference in error is variance. High variance error, which typically is due to too many features and model complexity, results in model overfitting. A is incorrect. The current model has high variance which results in model overfitting, not underfitting. B is incorrect. The difference between the prediction error on the training dataset and the prediction error on the cross-validation dataset is high, which means that the current model has high variance, not low. 25. B is correct. When analyzing term frequency at the corpus level, also known as collection frequency, tokens with intermediate term frequency (TF) values potentially carry important information useful for differentiating the sentiment embedded in the text. Tokens with the highest TF values are mostly stop words that do not contribute to differentiating the sentiment embedded in the text, and tokens with the lowest TF values are mostly proper nouns or sparse terms that are also not important to the meaning of the text. A is incorrect because tokens with the lowest TF values are mostly proper nouns or sparse terms (noisy terms) that are not important to the meaning of the text. C is incorrect because tokens with the highest TF values are mostly stop words (noisy terms) that do not contribute to differentiating the sentiment embedded in the text. 26. C is correct. Statement 3 is correct. TF–IDF values vary by the number of documents in the dataset, and therefore, the model performance can vary when 359 360 Learning Module 7 Big Data Projects applied to a dataset with just a few documents. Statement 1 is incorrect because IDF is calculated as the log of the inverse, or reciprocal, of the document frequency measure. Statement 2 is incorrect because TF at the sentence (not collection) level is multiplied by IDF to calculate TF–IDF. A is incorrect because Statement 1 is incorrect. IDF is calculated as the log of the inverse, or reciprocal, of the document frequency (DF) measure. B is incorrect because Statement 2 is incorrect. TF at the sentence (not collection) level is multiplied by IDF to calculate TF–IDF. 27. A is correct; 0% of the master dataset of Dataset ABC should be allocated to a training subset. Dataset ABC is characterized by the absence of ground truth (i.e., no known outcome or target variable) and is therefore an unsupervised ML model. For unsupervised learning models, no splitting of the master dataset is needed, because of the absence of labeled training data. Supervised ML datasets (with labeled training data) contain ground truth, the known outcome (target variable) of each observation in the dataset. B is incorrect because 20% is the commonly recommended split for the cross-validation set and test set in supervised training ML datasets. Dataset ABC is an unsupervised ML dataset, for which no splitting (0%) of the master dataset is needed, because of the absence of labeled training data. In supervised ML models (which contain labeled training data), the master dataset is split into three subsets (a training set, cross-validation set, and test set), which are used for model training and testing purposes. C is incorrect because 60% is the commonly recommended split for the training set in supervised training ML datasets. Dataset ABC is an unsupervised ML dataset, for which no splitting (0%) of the master dataset is needed, because of the absence of labeled training data. In supervised ML models (which contain labeled training data), the master dataset is split into three subsets (a training set, cross-validation set, and test set), which are used for model training and testing purposes. 28. B is correct. F1 score is the most appropriate performance measure for Dataset XYZ. Bector gives equal weight to false positives and false negatives. Accuracy and F1 score are overall performance measures that give equal weight to false positives and false negatives. Accuracy is considered an appropriate performance measure for balanced datasets, where the number of “1” and “0” classes are equal. F1 score is considered more appropriate than accuracy when there is unequal class distribution in the dataset and it is necessary to measure the equilibrium of precision and recall. Since Dataset XYZ contains an unequal class distribution between positive and negative sentiment sentences, F1 score is the most appropriate performance measure. Precision is the ratio of correctly predicted positive classes to all predicted positive classes and is useful in situations where the cost of false positives or Type I errors is high. Recall is the ratio of correctly predicted positive classes to all actual positive classes and is useful in situations where the cost of false negatives or Type II errors is high. A is incorrect because Bector gives equal weight to false positives and false negatives. Accuracy and F1 score are overall performance measures that give equal weight to false positives and false negatives. Recall is the ratio of correctly predicted positive classes to all actual positive classes and is useful in situations where the cost of false negatives or Type II errors is high. C is incorrect because Bector gives equal weight to false positive and false negatives. Accuracy and F1 score are overall performance measures that give equal weight to false positives and false negatives. Precision is the ratio of correctly predicted positive classes to all predicted positive classes and is useful in situa- Solutions 361 tions where the cost of false positives or Type-I error is high. 29. A is correct. Precision is the ratio of correctly predicted positive classes to all predicted positive classes and is useful in situations where the cost of false positives or Type I errors is high. Confusion Matrix A has the highest precision and therefore demonstrates the most favorable value of the performance metric that best addresses Azarov’s concern about the cost of Type I errors. Confusion Matrix A has a precision score of 0.95, which is higher than the precision scores of Confusion Matrix B (0.93) and Confusion Matrix C (0.86). B is incorrect because precision, not accuracy, is the performance measure that best addresses Azarov’s concern about the cost of Type I errors. Confusion Matrix B demonstrates the most favorable value for the accuracy score (0.92), which is higher than the accuracy scores of Confusion Matrix A (0.91) and Confusion Matrix C (0.91). Accuracy is a performance measure that gives equal weight to false positives and false negatives and is considered an appropriate performance measure when the class distribution in the dataset is equal (a balanced dataset). However, Azarov is most concerned with the cost of false positives, or Type I errors, and not with finding the equilibrium between precision and recall. Furthermore, Dataset XYZ has an unequal (unbalanced) class distribution between positive sentiment and negative sentiment sentences. C is incorrect because precision, not recall or F1 score, is the performance measure that best addresses Azarov’s concern about the cost of Type I errors. Confusion Matrix C demonstrates the most favorable value for the recall score (0.97), which is higher than the recall scores of Confusion Matrix A (0.87) and Confusion Matrix B (0.90). Recall is the ratio of correctly predicted positive classes to all actual positive classes and is useful in situations where the cost of false negatives, or Type II errors, is high. However, Azarov is most concerned with the cost of Type I errors, not Type II errors. F1 score is more appropriate (than accuracy) when there is unequal class distribution in the dataset and it is necessary to measure the equilibrium of precision and recall. Confusion Matrix C demonstrates the most favorable value for the F1 score (0.92), which is higher than the F1 scores of Confusion Matrix A (0.91) and Confusion Matrix B (0.91). Although Dataset XYZ has an unequal class distribution between positive sentiment and negative sentiment sentences, Azarov is most concerned with the cost of false positives, or Type I errors, and not with finding the equilibrium between precision and recall. 30. B is correct. Accuracy is the percentage of correctly predicted classes out of total predictions and is calculated as (TP + TN)/(TP + FP + TN + FN). In order to obtain the values for true positive (TP), true negative (TN), false positive (FP), and false negative (FN), predicted sentiment for the positive (Class “1”) and the negative (Class “0”) classes are determined based on whether each individual target p-value is greater than or less than the threshold p-value of 0.65. If an individual target p-value is greater than the threshold p-value of 0.65, the predicted sentiment for that instance is positive (Class “1”). If an individual target p-value is less than the threshold p-value of 0.65, the predicted sentiment for that instance is negative (Class “0”). Actual sentiment and predicted sentiment are then classified as follows: Actual Sentiment Predicted Sentiment Classification 1 1 TP 0 1 FP 1 0 FN 362 Learning Module 7 Big Data Projects Actual Sentiment Predicted Sentiment Classification 0 0 TN Exhibit 2, with added “Predicted Sentiment” and “Classification” columns, is presented below: 10 Sample Results of Test Data for Dataset XYZ Sentence # Actual Sentiment Target p-Value Predicted Sentiment Classification 1 1 0.75 1 TP 2 0 0.45 0 TN 3 1 0.64 0 FN 4 1 0.81 1 TP 5 0 0.43 0 TN 6 1 0.78 1 TP 7 0 0.59 0 TN 8 1 0.60 0 FN 9 0 0.67 1 FP 10 0 0.54 0 TN Based on the classification data obtained from Exhibit 2, a confusion matrix can be generated: Confusion Matrix for Dataset XYZ Sample Test Data with Threshold p-Value = 0.65 Actual Training Labels Predicted Results Class “1” Class “0” Class “1” TP = 3 FP = 1 Class “0” FN = 2 TN = 4 Using the data in the confusion matrix above, the accuracy metric is computed as follows: Accuracy = (TP + TN)/(TP + FP + TN + FN). Accuracy = (3 + 4)/(3 + 1 + 4 + 2) = 0.70. A is incorrect because 0.67 is the F1 score, not accuracy metric, for the sample of the test set for Dataset XYZ, based on Exhibit 2. To calculate the F1 score, the precision (P) and the recall (R) ratios must first be calculated. Precision and recall for the sample of the test set for Dataset XYZ, based on Exhibit 2, are calculated as follows: Precision (P) = TP/(TP + FP) = 3/(3 + 1) = 0.75. Recall (R) = TP/(TP + FN) = 3/(3 + 2) = 0.60. The F1 score is calculated as follows: Solutions F1 score = (2 × P × R)/(P + R) = (2 × 0.75 × 0.60)/(0.75 + 0.60) = 0.667, or 0.67. C is incorrect because 0.75 is the precision ratio, not the accuracy metric, for the sample of the test set for Dataset XYZ, based on Exhibit 2. The precision score is calculated as follows: Precision (P) = TP/(TP + FP) = 3/(3 + 1) = 0.75. 31. A is correct. Only Remark 1 is correct. Method selection is the first task of ML model training and is governed by the following factors: (1) supervised or unsupervised learning, (2) the type of data, and (3) the size of data. The second and third tasks of model training, respectively, are performance evaluation and tuning. Remark 2 is incorrect because model fitting errors (bias error and variance error) are used in tuning, not performance evaluation. The techniques used in performance evaluation, which measure the goodness of fit for validation of the model, include (1) error analysis, (2) receiver operating characteristic (ROC) plots, and (3) root mean squared error (RMSE) calculations. B and C are incorrect because Remark 2 is incorrect. Model fitting errors (bias error and variance error) are used in tuning, not performance evaluation. The techniques used in performance evaluation, which measure the goodness of fit for validation of the model, include (1) error analysis, (2) receiver operating characteristic plots, and (3) root mean squared error calculations. 363 444 Learning Module 1 Currency Exchange Rates: Understanding Equilibrium Value PRACTICE PROBLEMS The following information relates to questions 1-5 Ed Smith is a new trainee in the foreign exchange (FX) services department of a major global bank. Smith’s focus is to assist senior FX trader Feliz Mehmet, CFA. Mehmet mentions that an Indian corporate client exporting to the United Kingdom wants to estimate the potential hedging cost for a sale closing in one year. Smith is to determine the premium/discount for an annual (360-day) forward contract using the exchange rate data presented in Exhibit 1. Exhibit 1: Select Currency Data for GBP and INR Spot (INR/GBP) 79.5093 Annual (360-day) MRR (GBP) 5.43% Annual (360-day) MRR (INR) 7.52% Mehmet is also looking at two possible trades to determine their profit potential. The first trade involves a possible triangular arbitrage trade using the Swiss, US, and Brazilian currencies, to be executed based on a dealer’s bid/offer rate quote of 0.2355/0.2358 in CHF/BRL and the interbank spot rate quotes presented in Exhibit 2. Exhibit 2: Interbank Market Quotes Currency Pair Bid/Offer CHF/USD 0.9799/0.9801 BRL/USD 4.1699/4.1701 Mehmet is also considering a carry trade involving the USD and the EUR. He anticipates it will generate a higher return than buying a one-year domestic note at the current market quote due to low US interest rates and his predictions of exchange rates in one year. To help Mehmet assess the carry trade, Smith provides Mehmet with selected current market data and his one-year forecasts in Exhibit 3. Exhibit 3: Spot Rates and Interest Rates for Proposed Carry Trade Today’s One-Year MRR Currency Pair (Price/ Base) Spot Rate Today Projected Spot Rate in One Year USD 0.80% CAD/USD 1.3200 1.3151 CAD 1.71% EUR/CAD 0.6506 0.6567 EUR 2.20% Practice Problems Finally, Mehmet asks Smith to assist with a trade involving a US multinational customer operating in Europe and Japan. The customer is a very cost-conscious industrial company with an AA credit rating and strives to execute its currency trades at the most favorable bid–offer spread. Because its Japanese subsidiary is about to close on a major European acquisition in three business days, the client wants to lock in a trade involving the Japanese yen and the euro as early as possible the next morning, preferably by 8:05 a.m. New York time. At lunch, Smith and other FX trainees discuss how best to analyze currency market volatility from ongoing financial crises. The group agrees that a theoretical explanation of exchange rate movements, such as the framework of the international parity conditions, should be applicable across all trading environments. They note such analysis should enable traders to anticipate future spot exchange rates. But they disagree on which parity condition best predicts exchange rates, voicing several different assessments. Smith concludes the discussion on parity conditions by stating to the trainees, I believe that in the current environment both covered and uncovered interest rate parity conditions are in effect. 1. Based on Exhibit 1, the forward premium (discount) for a 360-day INR/GBP forward contract is closest to: A. –1.546. B. 1.546. C. 1.576. 2. Based on Exhibit 2, the most appropriate recommendation regarding the triangular arbitrage trade is to: A. decline the trade, because no arbitrage profits are possible. B. execute the trade, buy BRL in the interbank market, and sell BRL to the dealer. C. execute the trade, buy BRL from the dealer, and sell BRL in the interbank market. 3. Based on Exhibit 3, the potential all-in USD return on the carry trade is closest to: A. 0.83%. B. 1.23%. C. 1.63%. 4. The factor least likely to lead to a narrow bid–offer spread for the industrial company’s needed currency trade is the: A. timing of its trade. B. company’s credit rating. C. pair of currencies involved. 5. If Smith’s statement on parity conditions is correct, future spot exchange rates are most likely to be forecast by: A. current spot rates. 445 446 Learning Module 1 Currency Exchange Rates: Understanding Equilibrium Value B. forward exchange rates. C. inflation rate differentials. The following information relates to questions 6-13 Anna Goldsworthy is the chief financial officer of a manufacturing firm headquartered in the United Kingdom. She is responsible for overseeing exposure to price risk in both the commodity and currency markets. Goldsworthy is settling her end-of-quarter transactions and creating reports. Her intern, Scott Underwood, assists her in this process. The firm hedges input costs using forward contracts that are priced in US dollars (USD) and Mexican pesos (MXN). Processed goods are packaged for sale under licensing agreements with firms in foreign markets. Goldsworthy is expecting to receive a customer payment of JPY 225,000,000 (Japanese yen) that she wants to convert to pounds sterling (GBP). Underwood gathers the exchange rates from Dealer A in Exhibit 1. Exhibit 1: Dealer A’s Spot Exchange Rates Spot Exchange Rates Currency Pair (Price/Base) Bid Offer Midpoint JPY/GBP 129.65 129.69 129.67 MXN/USD 20.140 20.160 20.150 GBP/EUR 0.9467 0.9471 0.9469 USD/EUR 1.1648 1.1652 1.1650 USD/GBP 1.2301 1.2305 1.2303 The firm must also buy USD to pay a major supplier. Goldsworthy calls Dealer A with specific details of the transaction and asks to verify the USD/GBP quote. Dealer A calls her back later with a revised USD/GBP bid–offer quote of 1.2299/1.2307. Goldsworthy must purchase MXN 27,000,000 to pay an invoice at the end of the quarter. In addition to the quotes from Dealer A, Underwood contacts Dealer B, who provides a bid–offer price of GBP/MXN 0.0403/0.0406. To check whether the dealer quotes are reflective of an efficient market, Underwood examines whether the prices allow for an arbitrage profit. In three months, the firm will receive EUR 5,000,000 from another customer. Six months ago, the firm sold EUR 5,000,000 against the GBP using a nine-month forward contract at an all-in price of GBP/EUR 0.9526. To mark the position to market, Underwood collects the GBP/EUR forward rates in Exhibit 2. Exhibit 2: GBP/EUR Forward Rates Maturity One month Forward Points 4.40/4.55 Practice Problems 447 Maturity Forward Points Three months 14.0/15.0 Six months 29.0/30.0 Goldsworthy also asks for the current 90-day MRRs for the major currencies. Selected three-month MRRs (annualized) are shown in Exhibit 3. Goldsworthy studies Exhibit 3 and says, “We have the spot rate and the 90-day forward rate for GBP/EUR. As long as we have the GBP 90-day MRR, we will be able to calculate the implied EUR 90-day MRR.” Exhibit 3: 90-Day MRR Currency Annualized Rate GBP 0.5800% JPY 0.0893% USD 0.3300% After reading a draft report, Underwood notes, “We do not hedge the incoming Japanese yen cash flow. Your report asks for a forecast of the JPY/GBP exchange rate in 90 days. We know the JPY/GBP spot exchange rate.” He asks, “Does the information we have collected tell us what the JPY/GBP exchange rate will be in 90 days?” Goldsworthy replies, “The JPY/GBP exchange rate in 90 days would be a valuable piece of information to know. An international parity condition can be used to provide an estimate of the future spot rate.” 6. Using the quotes in Exhibit 1, the amount received by Goldsworthy from converting JPY 225,000,000 will be closest to: A. GBP 1,734,906. B. GBP 1,735,174. C. GBP 1,735,442. 7. Using Exhibit 1, which of the following would be the best reason for the revised USD/GBP dealer quote of 1.2299/1.2307? A. A request for a much larger transaction B. A drop in volatility in the USD/GBP market C. A request to trade when both New York and London trading centers are open 8. Using the quotes from Dealer A and B, the triangular arbitrage profit on a transaction of MXN 27,000,000 would be closest to: A. GBP 0. B. GBP 5,400. C. GBP 10,800. 448 Learning Module 1 Currency Exchange Rates: Understanding Equilibrium Value 9. Based on Exhibits 1, 2, and 3, the mark-to-market gain for Goldsworthy’s forward position is closest to: A. GBP 19,971. B. GBP 20,500. C. GBP 21,968. 10. Based on Exhibit 2, Underwood should conclude that three-month EUR MRR is: A. below three-month GBP MRR. B. equal to three-month GBP MRR. C. above three-month GBP MRR. 11. Based on the exchange rate midpoint in Exhibit 1 and the rates in Exhibit 3, the 90-day forward premium (discount) for the USD/GBP would be closest to: A. –0.0040. B. –0.0010. C. +0.0010. 12. Using Exhibits 1, 2, and 3, which international parity condition would Goldsworthy most likely use to calculate the EUR MRR? A. Real interest rate parity B. Covered interest rate parity C. Uncovered interest rate parity 13. The international parity condition Goldsworthy will use to provide the estimate of the future JPY/GBP spot rate is most likely: A. covered interest rate parity. B. uncovered interest rate parity. C. relative purchasing power parity. The following information relates to questions 14-20 Connor Wagener, a student at the University of Canterbury in New Zealand, has been asked to prepare a presentation on foreign exchange rates for his international business course. Wagener has a basic understanding of exchange rates but would like a practitioner’s perspective, and he has arranged an interview with currency trader Hannah McFadden. During the interview, Wagener asks McFadden, Could you explain what drives exchange rates? I’m curious as to why our New Zealand dollar was affected by the European debt crisis in 2011 and what other factors impact it. In response, McFadden begins with a general discussion of exchange rates. She Practice Problems 449 notes that international parity conditions illustrate how exchange rates are linked to expected inflation, interest rate differences, and forward exchange rates as well as current and expected future spot rates. McFadden makes the following statement: Statement 1: “Fortunately, the international parity condition most relevant for FX carry trades does not always hold.” McFadden continues her discussion: FX carry traders go long (i.e., buy) high-yield currencies and fund their position by shorting—that is, borrowing in—low-yield currencies. Unfortunately, crashes in currency values can occur which create financial crises as traders unwind their positions. For example, in 2008, the New Zealand dollar was negatively impacted when highly leveraged carry trades were unwound. In addition to investors, consumers and business owners can also affect currency exchange rates through their impact on their country’s balance of payments. For example, if New Zealand consumers purchase more goods from China than New Zealand businesses sell to China, New Zealand will run a trade account deficit with China. McFadden further explains, Statement 2: “A trade surplus will tend to cause the currency of the country in surplus to appreciate, while a deficit will cause currency depreciation. Exchange rate changes will result in immediate adjustments in the prices of traded goods as well as in the demand for imports and exports. These changes will immediately correct the trade imbalance.” McFadden next addresses the influence of monetary and fiscal policy on exchange rates: Countries also exert significant influence on exchange rates both through the initial mix of their fiscal and monetary policies and by subsequent adjustments to those policies. Various models have been developed to identify how these policies affect exchange rates. The Mundell–Fleming model addresses how changes in both fiscal and monetary policies affect interest rates and ultimately exchange rates in the short term. McFadden describes monetary models by stating, Statement 3: “Monetary models of exchange rate determination focus on the effects of inflation, price level changes, and risk premium adjustments.” McFadden continues her discussion: So far, we’ve touched on balance of payments and monetary policy. The portfolio balance model addresses the impacts of sustained fiscal policy on exchange rates. I must take a client call but will return shortly. In the meantime, here is some relevant literature on the models I mentioned along with a couple of questions for you to consider: Question 1: Assume an emerging market (EM) country has restrictive monetary and fiscal policies under low capital mobility conditions. Are these policies likely to lead to currency appreciation or currency depreciation or to have no impact? Question 2: Assume a developed market (DM) country has an expansive fiscal policy under high capital mobility conditions. Why is its currency most likely to depreciate in the long run under an integrated Mundell–Fleming and portfolio balance approach? 450 Learning Module 1 Currency Exchange Rates: Understanding Equilibrium Value Upon her return, Wagener and McFadden review the questions. McFadden notes that capital flows can have a significant impact on exchange rates and have contributed to currency crises in both EM and DM countries. She explains that central banks, such as the Reserve Bank of New Zealand, use FX market intervention as a tool to manage exchange rates. McFadden states, Statement 4: “Some studies have found that EM central banks tend to be more effective in using exchange rate intervention than DM central banks, primarily because of one important factor.” McFadden continues her discussion: Statement 5: “I mentioned that capital inflows could cause a currency crisis, leaving fund managers with significant losses. In the period leading up to a currency crisis, I would predict that an affected country’s: Prediction 1: foreign exchange reserves will increase. Prediction 2: broad money growth will increase. Prediction 3: exchange rate will be substantially higher than its mean level during tranquil periods.” After the interview, McFadden agrees to meet the following week to discuss more recent events on the New Zealand dollar. 14. The international parity condition McFadden is referring to in Statement 1 is: A. purchasing power parity. B. covered interest rate parity. C. uncovered interest rate parity. 15. In Statement 2, McFadden is most likely failing to consider the: A. initial gap between the country’s imports and exports. B. effect of an initial trade deficit on a country’s exchange rates. C. lag in the response of import and export demand to price changes. 16. The least appropriate factor used to describe the type of models mentioned in Statement 3 is: A. inflation. B. price level changes. C. risk premium adjustments. 17. The best response to Question 1 is that the policies will: A. have no impact. B. lead to currency appreciation. C. lead to currency depreciation. 18. The most likely response to Question 2 is a(n): A. increase in the price level. Practice Problems B. decrease in risk premiums. C. increase in government debt. 19. The factor that McFadden is most likely referring to in Statement 4 is: A. FX reserve levels. B. domestic demand. C. the level of capital flows. 20. Which of McFadden’s predictions in Statement 5 is least likely to be correct? A. Prediction 1 B. Prediction 2 C. Prediction 3 451 452 Learning Module 1 Currency Exchange Rates: Understanding Equilibrium Value SOLUTIONS 1. C is correct. The equation to calculate the forward premium (discount) is: ​[_ ​ 360 ​]​ _ Actual ​​Ff/d ​ ​− ​Sf/d ​ ​ = ​Sf/d ​ ​ ​ Actual ​ ​(​i​f​− i​ ​d​)​. (1 + ​i​d​[_ ​ 360 ​]​) Sf/d is the spot rate with GBP the base currency or d and INR the foreign currency or f. Sf/d per Exhibit 1 is 79.5093, if is equal to 7.52%, and id is equal to 5.43%. With GBP as the base currency (i.e., the “domestic” currency) in the INR/GBP quote, substituting in the relevant base currency values from Exhibit 1 yields the following: ​[_ ​360 ]​ ​ ____________ ( ) F ​ ​− ​Sf/d ​ f/d ​ ​ = 79.5093​ ​   360 ​ ​ 0.0752 − 0.0543 ​. (1 + 0.0543​[_ ​360 ​]​) 360 ​​            1 ​Ff/d ​ ​− S​ f/d ​ ​ = 79.5093​(_ ​1.0543 ​)(​ 0.0752 − 0.0543)​. ​Ff/d ​ ​− S​ f/d ​ ​ = 1.576. 2. B is correct. The dealer is posting a bid rate to buy BRL at a price that is too high. This overpricing is determined by calculating the interbank implied cross rate for the CHF/BRL using the intuitive equation-based approach: CHF/BRL = CHF/USD × (BRL/USD)–1, or CHF/BRL = CHF/USD × USD/BRL. Inverting the BRL/USD given the quotes in Exhibit 2 determines the USD/BRL bid–offer rates of 0.23980/0.23982. (The bid of 0.23980 is the inverse of the BRL/ USD offer, calculated as 1/4.1702; the offer of 0.23982 is the inverse of the BRL/ USD bid, calculated as 1/4.1698.) Multiplying the CHF/USD and USD/BRL bid– offer rates then leads to the interbank implied CHF/BRL cross rate: Bid: 0.9799 × 0.23980 = 0.2349. Offer: 0.9801 × 0.23982 = 0.23505. Since the dealer is willing to buy BRL at 0.2355 but BRL can be purchased from the interbank market at 0.23505, there is an arbitrage opportunity to buy BRL in the interbank market and sell BRL to the dealer for a profit of 0.0045 CHF (0.2355 – 0.23505) per BRL transacted. 3. A is correct. The carry trade involves borrowing in a lower-yielding currency to invest in a higher-yielding one and netting any profit after allowing for borrowing costs and exchange rate movements. The relevant trade is to borrow USD and lend in EUR. To calculate the all-in USD return from a one-year EUR MRR deposit, first determine the current and one-year-later USD/EUR exchange rates. Because one USD buys CAD 1.3200 today and one CAD buys EUR 0.6506 today, today’s EUR/USD rate is the product of these two numbers: 1.3200 × 0.6506 = 0.8588. The projected rate one year later is 1.3151 × 0.6567 = 0.8636. Accordingly, measured in dollars, the investment return for the unhedged EUR MRR deposit is equal to Solutions 453 (1.3200 × 0.6506) × (1 + 0.022) × [1/(1.3151 × 0.6567)] – 1 = 0.8588 × (1.022)(1/0.8636) – 1 = 1.01632 – 1 = 1.632%. However, the borrowing costs must be charged against this gross return to fund the carry trade investment (one-year USD MRR was 0.80%). The net return on the carry trade is therefore 1.632% – 0.80% = 0.832%. 4. B is correct. While credit ratings can affect spreads, the trade involves spot settlement (i.e., two business days after the trade date), so the spread quoted to this highly rated (AA) firm is not likely to be much tighter than the spread that would be quoted to a somewhat lower-rated (but still high-quality) firm. The relationship between the bank and the client, the size of the trade, the time of day the trade is initiated, the currencies involved, and the level of market volatility are likely to be more significant factors in determining the spread for this trade. 5. B is correct. By rearranging the terms of the equation defining covered interest rate parity and assuming that uncovered interest rate parity is in effect, the forward exchange rate is equal to the expected future spot exchange rate—Ff/d = e ​—with the expected percentage change in the spot rate equal to the interest S​f/d rate differential. Thus, the forward exchange rate is an unbiased forecast of the future spot exchange rate. 6. A is correct. Goldsworthy has been given a bid–offer spread. Because she is buying the base currency—in this case, GBP—she must pay the offer price of JPY 129.69 per GBP. JPY 225,000,000 ____________ 129.69 JPY/GBP = GBP 1,734,906. 7. A is correct. Posted quotes are typically for transactions in 1 million units of the base currency. Larger transactions may be harder for the dealer to sell in the interbank market and would likely require the dealer to quote a wider spread (lower bid price and higher offer price). 8. A is correct. Using quotes from Dealer A, she can find USD MXN MXN _ _ ​ ​_ GBP ​ = ​USD ​× ​GBP ​. The bid from Dealer A for MXN/GBP is effectively ​ ​ ​ ​= ( ​_ ​USD ) ​ ​ ​× ( ​_ ​GBP ​)​ ​(_      ​​ GBP ) bid bid bid = 20.140 × 1.2301 = 24.7742. MXN MXN USD The offer from Dealer A is ​ ​ ​ ​= ( ​_ ​USD ) ​ ​ ​× ( ​_ ​GBP ) ​​ ​_ ( ) offer offer​      ​​ GBP offer = 20.160 × 1.2305 = 24.8069. MXN MXN USD To compare with Dealer B’s quote, she must take the inverse of MXN/GBP, so she has an offer to sell MXN at a rate of 1/24.7742 = GBP 0.0404 and a bid to purchase MXN at a rate of 1/24.8069 = GBP 0.0403. Dealer A is effectively quoting MXN/GBP at 0.0403/0.0404. Although she can effectively buy MXN more cheaply from Dealer A (GBP 0.0404 from Dealer A, versus GBP 0.0406 from Dealer B), she cannot resell them to Dealer B for a higher price than GBP 0.0403. There is no profit from triangular arbitrage. 9. A is correct. Marking her nine-month contract to market six months later requires buying GBP/EUR three months forward. The GBP/EUR spot rate 454 Learning Module 1 Currency Exchange Rates: Understanding Equilibrium Value is 0.9467/0.9471, and the three-month forward points are 14.0/15.0. The three-month forward rate to use is 0.9471 + (15/10000) = 0.9486. Goldsworthy sold EUR 5,000,000 at 0.9526 and bought at 0.9486. The net cash flow at the settlement date will equal EUR 5,000,000 × (0.9526 – 0.9486) GBP/EUR = GBP 20,000. This cash flow will occur in three months, so we discount at the three-month GBP MRR of 58 bps: GBP 20,000 ____________    ​​   90 ​ = GBP 19,971.04. _ 1 + 0.0058​(​360 ​)​ 10. A is correct. The positive forward points for the GBP/EUR pair shown in Exhibit 2 indicate that the EUR trades at a forward premium at all maturities, including three months. Covered interest rate parity, 1 + ​i​f​[_ ​ 360 ]​ ​ _ Actual ​​Ff/d ​ ​ = ​Sf/d ​ ​ ​ Actual ​ ​, (1 + ​i​d​[_ ​ 360 ]​ ​) suggests a forward rate greater than the spot rate requires a non-domestic risk-free rate (in this case, the GBP MRR) greater than the domestic risk-free rate (EUR MRR). When covered interest rate parity is violated, traders can step in and conduct arbitrage. 11. B is correct. Using covered interest rate parity, the forward rate is 1 + ​i​f​[_ ​ 360 ]​ ​ Actual ​Ff/d ​ ​ = ​Sf/d ​ ​ ​ _ Actual ​ (1 + ​i​d​[_ ​ 360 ]​ ​)     90 _ 1 + 0.0033​[​360 ​] ____________ = 1.2303​    ​ 90 ​ ​ = 1.2295. (1 + 0.0058​[_ ​360 ]​ ​) Because the domestic MRR is higher than the non-domestic MRR, the forward rate will be less than the spot rate, giving a forward discount of ​ ​ = 1.2295 − 1.2303 = − 0.0008.​ ​​Ff/d ​ ​− ​Sf/d 12. B is correct. The covered interest rate parity condition, 1 + ​i​f​[_ ​ 360 ]​ ​ Actual ​ ​ _ ​ ​​Ff/d ​ ​ = ​Sf/d Actual ​ ​,​ (Equation 1) (1 + ​i​d​[_ ​ 360 ]​ ​) specifies the forward exchange rate that must hold to prevent arbitrage given the spot exchange rate and the risk-free rates in both countries. If the forward and spot exchange rates, as well as one of the risk-free rates, are known, the other risk-free rate can be calculated. 13. B is correct. According to uncovered interest rate parity, ​%Δ ​Sf/d ​e ​ = ​i​f​− ​i​d​,​ (Equation 2) the expected change in the spot exchange rate should reflect the interest rate spread between the two countries, which can be found in Exhibit 3. Given the spot exchange rate (from Exhibit 1) and the expected future change, she should be able to estimate the future spot exchange rate. 14. C is correct. The carry trade strategy is dependent on the fact that uncovered interest rate parity does not hold in the short or medium term. If uncovered interest rate parity held, it would mean that investors would receive identical returns from either an unhedged foreign currency investment or a domestic currency Solutions investment because the appreciation/depreciation of the exchange rate would offset the yield differential. However, during periods of low volatility, evidence shows that high-yield currencies do not depreciate enough and low-yield currencies do not appreciate enough to offset the yield differential. 15. C is correct. McFadden states that exchange rates will immediately correct the trade imbalance. She is describing the flow supply/demand channel, which assumes that trade imbalances will be corrected as the deficit country’s currency depreciates, causing its exports to become more competitive and its imports to become more expensive. Studies indicate that there can be long lags between exchange rate changes, changes in the prices of traded goods, and changes in the trade balance. In the short run, exchange rates tend to be more responsive to investment and financing decisions. 16. C is correct. Risk premiums are more closely associated with the portfolio balance approach. The portfolio balance approach addresses the impact of a country’s net foreign asset/liability position. Under the portfolio balance approach, investors are assumed to hold a diversified portfolio of assets including foreign and domestic bonds. Investors will hold a country’s bonds as long as they are compensated appropriately. Compensation may come in the form of higher interest rates and/or higher risk premiums. 17. B is correct. The currency is likely to appreciate. The emerging market country has both a restrictive monetary policy and a restrictive fiscal policy under conditions of low capital mobility. Low capital mobility indicates that interest rate changes induced by monetary and fiscal policy will not cause large changes in capital flows. Implementation of restrictive policies should result in an improvement in the trade balance, which will result in currency appreciation. 18. C is correct. Expansionary fiscal policies result in currency depreciation in the long run. Under a portfolio balance approach, the assumption is that investors hold a mix of domestic and foreign assets including bonds. Fiscal stimulus policies result in budget deficits, which are often financed by debt. As the debt level rises, investors become concerned as to how the ongoing deficit will be financed. The country’s central bank may need to create more money in order to purchase the debt, which would cause the currency to depreciate. Or the government could adopt a more restrictive fiscal policy, which would also depreciate the currency. 19. A is correct. EM countries are better able to influence their exchange rates because their reserve levels as a ratio of average daily FX turnover are generally much greater than those of DM countries. This means that EM central banks are in a better position to affect currency supply and demand than DM countries, where the ratio is negligible. EM policymakers use their foreign exchange reserves as a kind of insurance to defend their currencies, as needed. 20. A is correct. Prediction 1 is least likely to be correct. Foreign exchange reserves tend to decline precipitously, not increase, as a currency crisis approaches. Broad money growth tends to rise in the period leading up to a currency crisis, and the exchange rate is substantially higher than its mean level during tranquil periods. 455 530 Learning Module 2 Economic Growth PRACTICE PROBLEMS The following information relates to questions 1-9 Victor Klymchuk, the chief economist at ECONO Consulting (EC), is reviewing the long-term GDP growth of three countries. Klymchuk is interested in forecasting the long-term change in stock market value for each country. Exhibit 1 presents current country characteristics and historical information on select economic variables for the three countries. Exhibit 1: Select Country Factors and Historical Economic Data 2009–2019 Country Factors Country A Country B Country C ■ High level of savings and investment ■ Highly educated workforce ■ Low tariffs on foreign imports ■ Limited natural resources ■ Developed financial markets ■ Moderate levels of disposable income ■ Significant foreign direct and indirect investments ■ Significant natural resources ■ Politically unstable ■ Limited property rights ■ Poor public education and health ■ Significant natural resources Growth in Hours Worked (%) Growth in Labor Productivity (%) Growth in TFP (%) Growth in GDP (%) 0.9 2.4 0.6 3.3 −0.3 1.6 0.8 1.3 1.8 0.8 −0.3 2.6 Klymchuk instructs an associate economist at EC to assist him in forecasting the change in stock market value for each country. Klymchuk reminds the associate of the following: Practice Problems Statement 1: 531 “Over short time horizons, percentage changes in GDP, the ratio of earnings to GDP, and the price-to-earnings ratio are important factors for describing the relationship between economic growth and stock prices. However, I am interested in a long-term stock market forecast.” A client is considering investing in the sovereign debt of Country A and Country B and asks Klymchuk his opinion of each country’s credit risk. Klymchuk tells the client the following: Statement 2: “Over the next 10 years, I forecast higher potential GDP growth for Country A and lower potential GDP growth for Country B. The capital per worker is similar and very high for both countries, but per capita output is greater for Country A.” The client tells Klymchuk that Country A will offer 50-year bonds and that he believes the bonds could be a good long-term investment given the higher potential GDP growth. Klymchuk responds to the client as follows: Statement 3: “After the next 10 years, I think the sustainable rate of economic growth for Country A will be affected by a growing share of its population over the age of 65, a declining percentage under age 16, and minimal immigration.” The client is surprised to learn that Country C, a wealthy, oil-rich country with significant reserves, is experiencing sluggish economic growth and asks Klymchuk for an explanation. Klymchuk responds by stating: Statement 4: “Although countries with greater access to natural resources are often wealthier, the relationship between resource abundance and economic growth is not clear. My analysis shows that the presence of a dominant natural resource (oil) in Country C is constraining growth. Interestingly, Country A has few natural resources but is experiencing a strong rate of increase in per capita GDP growth.” Klymchuk knows that growth in per capita income cannot be sustained by pure capital deepening. He asks the associate economist to determine how important capital deepening is as a source of economic growth for each country. Klymchuk instructs the associate to use the data provided in Exhibit 1. Klymchuk and his associate debate the concept of convergence. The associate economist believes that developing countries, irrespective of their particular characteristics, will eventually equal developed countries in per capita output. Klymchuk responds as follows: Statement 5: “Poor countries will only converge to the income levels of the richest countries if they make appropriate institutional changes.” 1. Based on the country factors provided in Exhibit 1, the country most likely to be considered a developing country is: A. Country A. B. Country B. C. Country C. 532 Learning Module 2 Economic Growth 2. Based on Exhibit 1, capital deepening as a source of growth was most important for: A. Country A. B. Country B. C. Country C. 3. Based on Statement 1, over the requested forecast horizon, the factor that will most likely drive stock market performance is the percentage change in: A. GDP. B. the earnings to GDP ratio. C. the price-to-earnings ratio. 4. Based solely on the predictions in Statement 2, over the next decade Country B’s sovereign credit risk will most likely: A. increase. B. decrease. C. not change. 5. Based on Statement 2, the difference in per capita output between Country A and Country B most likely results from differences in: A. capital deepening. B. capital per worker. C. total factor productivity. 6. Based on Statement 3, after the next 10 years, the growth rate of potential GDP for Country A will most likely be: A. lower. B. higher. C. unchanged. 7. Based on Statement 4 and Exhibit 1, the sluggish economic growth in Country C is least likely to be explained by: A. limited labor force growth. B. export driven currency appreciation. C. poorly developed economic institutions. 8. Based on Statement 4, the higher rate of per capita income growth in Country A is least likely explained by the: A. rate of investment. B. growth of its population. C. application of information technology. Practice Problems 533 9. The type of convergence described by Klymchuk in Statement 5 is best described as: A. club convergence. B. absolute convergence. C. conditional convergence. The following information relates to questions 10-15 At an international finance and economics conference in Bamako, Mali, Jose Amaral of Brazil and Lucinda Mantri of India are discussing how to spur their countries’ economic growth. Amaral believes that growth can be bolstered by removing institutional impediments and suggests several possibilities for Brazil: launching a rural literacy program, clarifying property rights laws, and implementing a new dividend tax on foreign investors. Mantri responds that for India, capital deepening will be more effective, and she has proposed the following ideas: building a group of auto and textile factories in the southern states, developing a north–south and east–west highway network, and sponsoring a patent initiative. In response, Amaral says to Mantri: “Based on endogenous growth theory, one of those proposals is more likely to raise total factor productivity than result in pure capital deepening.” Although Mantri recognizes that India lacks the significant natural resources that Brazil has, she states that India can overcome this challenge by bolstering long-term growth through three channels: Channel 1 Deepening the capital base Channel 2 Making investments in technology Channel 3 Maintaining a low rupee exchange rate Each country’s basic economic statistics were presented at the conference. Selected data for Brazil and India are presented in Exhibit 1. Adama Kanté, a fund manager based in Mali, is planning to increase the fund’s allocation to international equities and, after some preliminary analysis, has determined the new allocation will be to Brazilian or Indian equities. After reviewing the data in Exhibit 1, Kanté decides that the allocation will be to Indian equities. Exhibit 1: Economic Statistics for Brazil and India Economic Statistic GDP per capita, 2018 Brazil India $14,360 $6,999 GDP per capita growth, 2000–2018 1.3% 5.8% GDP growth, 2005–2018 2.0% 7.2% - Growth resulting from labor productivity component 1.3% 6.3% 534 Learning Module 2 Economic Growth Economic Statistic Brazil India - Growth resulting from capital deepening component 1.9% 4.5% Kanté is concerned about the low standard of living in Mali and its large informal sector. To improve per capita GDP, Kanté is considering five specific strategies: Strategy 1 Lower the country’s tax rate. Strategy 2 Introduce policies that encourage the return of highly educated Malian emigrants. Strategy 3 Build daycare centers to increase women’s participation in the workforce. Strategy 4 Impose high tariffs on imports to protect the country’s nascent industries. Strategy 5 Use economic development bank loans to improve the country’s transport and manufacturing infrastructure. 10. Which of Amaral’s initiatives is least likely to achieve his stated growth objective? A. Dividend tax B. Rural literacy C. Property rights 11. Which proposal for India is Amaral most likely referring to in his response to Mantri? A. Patent initiative B. Highway network C. Auto and textile factories 12. The channel that is least likely to help India overcome its challenge of lacking significant natural resources is: A. Channel 1. B. Channel 2. C. Channel 3. 13. Based on Exhibit 1, which Indian economic statistic least likely supports Kanté’s international equity allocation preference? A. GDP per capita B. Growth resulting from labor productivity C. Growth resulting from capital deepening 14. The strategy that is least likely to improve per capita GDP in Mali is: A. Strategy 1. B. Strategy 2. C. Strategy 3. Practice Problems 535 15. Which of the following strategies being considered by Kanté is most likely to undermine or delay convergence with developed economies? A. Strategy 2 B. Strategy 4 C. Strategy 5 The following information relates to questions 16-21 Hans Schmidt, CFA, is a portfolio manager with a boutique investment firm that specializes in sovereign credit analysis. Schmidt’s supervisor asks him to develop estimates for GDP growth for three countries. Information on the three countries is provided in Exhibit 1. Exhibit 1: Select Economic Data for Countries A, B, and C Country Economy Capital per Worker A Developed High B Developed High C Developing Low After gathering additional data on the three countries, Schmidt shares his findings with colleague, Sean O’Leary. After reviewing the data, O’Leary notes the following observations: Observation 1: The stock market of Country A has appreciated considerably over the past several years. Also, the ratio of corporate profits to GDP for Country A has been trending upward over the past several years and is now well above its historical average. Observation 2: The government of Country C is working hard to bridge the gap between its standard of living and that of developed countries. Currently, the rate of potential GDP growth in Country C is high. Schmidt knows that a large part of the analysis of sovereign credit is to develop a thorough understanding of the potential GDP growth rate for a particular country and the region in which the country is located. Schmidt is also doing research on Country D for a client of the firm. Selected economic facts on Country D are provided in Exhibit 2. Exhibit 2: Select Economic Facts for Country D ■ Slow GDP Growth ■ Abundant Natural Resources ■ Developed Economic Institutions Prior to wrapping up his research, Schmidt schedules a final meeting with O’Leary to see if he can provide any other pertinent information. O’Leary makes the following statements to Schmidt: 536 Learning Module 2 Economic Growth Statement 1: Many countries that have the same population growth rate, savings rate, and production function will have growth rates that converge over time. Statement 2: Convergence between countries can occur more quickly if economies are open and there is free trade and international borrowing and lending; however, there is no permanent increase in the rate of growth in an economy from a more open trade policy. 16. Based on Exhibit 1, the factor that would most likely have the greatest positive impact on the per capita GDP growth of Country A is: A. free trade. B. technology. C. saving and investment. 17. Based on Observation 1, in the long run the ratio of profits to GDP in Country A is most likely to: A. remain near its current level. B. increase from its current level. C. decrease from its current level. 18. Based on Observation 2, Country C is most likely to have: A. relatively low real asset returns. B. a relatively low real interest rate. C. a relatively high real interest rate. 19. Based on Exhibit 2, the least likely reason for the current pace of GDP growth in Country D is: A. a persistently strong currency. B. strong manufacturing exports. C. strong natural resource exports. 20. The type of convergence described by O’Leary in Statement 1 is best described as: A. club convergence. B. absolute convergence. C. conditional convergence. 21. Which of the following growth models is most consistent with O’Leary’s Statement 2? A. Classical B. Endogenous C. Neoclassical Solutions SOLUTIONS 1. C is correct. Country C is the most likely to be a developing economy. Political instability, limited property rights, and poor public education and health are all factors that limit economic growth and thereby contribute to a relatively low standard of living. 2. A is correct. The associate economist can measure the effect of pure capital deepening by measuring the difference of the growth rates of labor productivity and total factor productivity. The larger the difference, the more important capital deepening is as a source of economic growth. From 2000–2010, Country A’s labor productivity grew by 2.4% per year, of which 0.6% came from TFP growth and 1.8% from capital deepening (2.4% − 0.6% = 1.8%). 3. A is correct. In the long run, the GDP growth rate is the most important driver of stock market performance. Therefore, the associate economist should focus on the drivers of long-run potential GDP growth. The ratio of earnings to GDP cannot increase indefinitely because that would imply that profit would eventually absorb all of GDP. This ratio cannot shrink forever, either, because unprofitable companies will go out of business. Thus, the annualized growth rate of the earnings to GDP ratio must be approximately zero over long time horizons, and this ratio should not be a dominant factor in forecasting long-term stock market performance. Similarly, the price-to-earnings ratio cannot grow or contract at a finite rate forever because investors will not pay an excessive price for each dollar of earnings, nor will they give away earnings for free. Therefore the rate of change in the price-to-earnings ratio must be approximately zero over long time horizons and should not be a dominant factor in the forecast of long-term stock market performance. 4. A is correct. Credit rating agencies consider the growth rate of potential GDP when evaluating the credit risk of sovereign debt. The chief economist’s expectation for lower potential GDP growth for Country B over the next decade increases the perceived credit risk of its sovereign bonds. 5. C is correct. The higher per capita output for Country A is most likely the result of differences in the cumulative impact of technological progress embodied in total factor productivity. Technological progress raises a country’s productive capacity and causes an upward shift in the entire production function, resulting in higher output per worker for a given level of capital per worker. 6. A is correct. Demographic factors can positively or negatively contribute to a country’s sustainable rate of economic growth. After the next 10 years, Country A is expected to experience a growing share of the population over the age of 65 and a declining percentage of the population under the age of 16. All else the same, this implies slower growth of the labor force and hence slower growth of potential GDP. Immigration could offset these demographic challenges. However, Statement 3 indicates that Country A is expected to experience minimal immigration. 7. A is correct. Country C is an example of a country endowed with an abundant natural resource yet experiencing slow economic growth. Although labor force growth is an important source of economic growth, it is the least likely explanation of the sluggish economic growth in Country C. As shown in Exhibit 1, growth in total hours worked has accounted for most of Country C’s growth. Furthermore, export driven currency appreciation and poorly developed economic institutions are both likely causes of sluggish growth in countries with abundant 537 538 Learning Module 2 Economic Growth natural resources. 8. B is correct. Population growth can increase the growth rate of the overall economy but does not affect the rate of increase in per capita GDP. Therefore, population growth does not explain Country A’s higher rate of per capita income growth. An increase in labor force participation could, however, raise the growth rate of per capita GDP. 9. A is correct. Klymchuk is referring to the concept of club convergence. The basic premise is that lower-income members of the club are converging to the income levels of the richest countries. This implies that the countries with the lowest per capita income in the club grow at the fastest rate. Countries outside the club, however, continue to fall behind. 10. A is correct. Amaral’s initiative to implement a new dividend tax is likely to impede inflows of equity capital by making equity investment in Brazil less attractive for foreign investors. Capital flows, or lack thereof, have a major impact on economic growth because, in an open economy, world savings can finance domestic investment. As a potential source of funds, foreign investment breaks the vicious cycle of low income, low domestic savings, and low investment. 11. A is correct. Mantri’s proposal to sponsor a patent initiative, which is likely to result in technology investment and improvement, is likely to cause a proportional upward shift in the entire production function, allowing the economy to produce higher output per worker for a given level of capital per worker. Technological progress also increases the marginal product of capital relative to its marginal cost. 12. C is correct. Maintaining a low currency exchange rate is a policy aimed at maintaining demand for the country’s exports. It would have little direct impact on the potential growth rate of aggregate supply. It might boost long-term capacity growth indirectly, however, by encouraging adoption of leading edge technology. Nonetheless, it would not be expected to be as powerful as capital deepening and/or investment in technology. 13. A is correct. Kanté’s decision to invest in equities in India is supported by the country’s strong economic growth. For global investors, economic growth is important because equity composite valuations depend to a great extent on both the level of economic output (GDP per capita and GDP overall) and the rate of economic growth. Relative to Brazil, India’s growth rate in per capita GDP has been much higher, and furthermore, the growth rate in GDP has also been much higher than that of Brazil. In contrast to the growth rate, the relatively low level of GDP per capita in India is less likely to indicate attractive equity investment opportunities. Low per capita GDP suggests that India may lack sufficient industrial and financial infrastructure to support some types of industries. It also indicates that domestic purchasing power is relatively limited, decreasing the potential for higher-margin, domestically oriented businesses. 14. A is correct. With Mali’s low standard of living (i.e., GDP per capita and large informal workforce), the tax rate is unlikely to be an impediment to growth, so lowering the tax rate is not likely to be a major contributor to growth. 15. B is correct. The strategy for Mali to impose high tariffs (trade restrictions) on imports is likely to undermine rather than enhance growth and therefore is not supportive of convergence with developed economies. Freer trade (fewer trade restrictions) tends to enhance growth by, for example, inducing a shift of resources into industries in which the country has a comparative thereby increasing Solutions overall productivity; forcing less efficient domestic companies to exit and more efficient ones to innovate; allowing domestic producers to more fully exploit economies of scale by selling to a larger market; and enabling less advanced sectors of an economy to catch up with more-advanced countries or sectors through knowledge spillovers. 16. B is correct. Country A is a developed country with a high level of capital per worker. Technological progress and/or more intensive use of existing technology can help developed countries increase productivity and thereby increase per capita GDP. Most developed countries have reasonably low trade barriers; thus, somewhat freer trade is likely to have only an incremental, and probably transitory, impact on per capita GDP growth. Also, because the country already has a high capital-to-labor ratio, increased saving/investment is unlikely to increase the growth rate substantially unless it embodies improved technology. 17. C is correct. The ratio of profits to GDP for Country A has been trending upward over the past several years and is now well above its historical average. The ratio of profits to GDP cannot rise forever. At some point, stagnant labor income would make workers unwilling to work without an increase in wages and would also undermine demand, making further expansion of profit margins unsustainable. Thus, the ratio of profits to GDP will likely decline in the long run toward its historical average. 18. C is correct. A high growth rate of potential GDP would cause real incomes to rise more rapidly and also translate into higher real interest rates and higher expected/required real asset returns. The real interest rate is essentially the real return that consumers/savers demand in exchange for postponing consumption. Faster growth in potential GDP means that consumers expect their real income to rise more rapidly. This implies that an extra unit of future income/consumption is less valuable than it would be if income were expected to grow more slowly. All else the same, the real interest rate will have to be relatively high in order to induce the savings required to fund required/desired capital accumulation. 19. B is correct. Country D is a country with abundant resources and has developed the economic institutions necessary for growth, yet the country is experiencing slow economic growth. It is likely that Country D is experiencing the Dutch Disease, whereby currency appreciation driven by strong export demand for natural resources makes other segments of the economy, in particular manufacturing, globally uncompetitive. Strong manufacturing exports indicate that Country D is globally competitive and likely to have adopted leading edge technology. Thus, it is unlikely that the slow growth reflects inability to maintain productivity growth. Similarly, strong exports suggest adequate demand for its products. Thus, strong exports are unlikely to be the cause of slow growth. 20. C is correct. Conditional convergence means that convergence is conditional on the countries having the same savings rate, population growth rate, and production function. If these conditions hold, the neoclassical model implies convergence to the same level of per capita output as well as the same steady-state growth rate. 21. C is correct. According to the neoclassical model, convergence should occur more quickly if economies are open and there is free trade and international borrowing and lending. Opening up the economy should increase the rate at which the capital-to-labor ratio converges among countries. In the neoclassical Solow model, however, after the reallocation of world savings, there is no permanent increase in the rate of growth in an economy. Both the developed and developing countries eventually grow at the same steady-state rate. 539 570 Learning Module 3 Economics of Regulation PRACTICE PROBLEMS The following information relates to questions 1-7 Cate Stephenson is an analyst in the economics research division of an international securities firm. She is conducting research on the regulatory environment in certain European countries. Stephenson begins with an analysis of a hypothetical country, Genovia. Genovia has recently introduced a new accounting statute. In Genovia, there is an independent regulator—“Le régulateur.” Le régulateur is not a self-regulating organization (SRO). There is also an SRO—“L’organisation.” L’organisation is not an independent regulator. In her research report, Stephenson makes the following statements: Statement 1 Le régulateur has been given legal authority by the government to enforce the new statute. Statement 2 L’organisation issues administrative regulations related to the new statute using government funding. Statement 3 L’organisation has member companies that accept the authorization of L’organisation to set and enforce rules and standards. Stephenson and her supervisor discuss the intended and unintended effects of implementing the new statute, and Stephenson makes two comments. Comment 1 It is likely that some unintended consequences will be identified in regulatory filings prior to implementation of the new legislation. Comment 2 Indirect costs arise because of unintended consequences and may result in high unanticipated costs. Stephenson reads a report titled “International Trade,” which has three sections about Genovia’s policies and regulations. ■ The first section of the report discusses policies that legislators may implement to accomplish Genovia’s objective of promoting free trade on industrial goods. ■ The second section of the report covers corporate domicile. Stephenson learns that regulators in Genovia recently amended regulations to encourage foreign businesses to move their corporate domicile to Genovia. ■ The third section of the report reviews the regulation of commerce. Genovia’s goal is to establish an environment that encourages foreign businesses to increase trade with domestic businesses. Stephenson considers two features of Genovia’s regulation of commerce. Feature 1 Recent court decisions have upheld financial liability and bankruptcy laws. Feature 2 A legal structure that governs contracts and each party’s rights is in place. Practice Problems 571 Stephenson then reviews two initiatives by Genovia to improve domestic policies and regulations. ■ The first initiative by Genovia is its passage of conflict-of-interest regulations. Regulators implement regulatory restrictions and regulatory mandates that apply to employees of securities firms. One of Stephenson’s research colleagues writes reports on a company in which he owns shares. ■ The second initiative by Genovia is to reduce pollution and promote renewable electricity generation. Two years ago, the government implemented taxes on fossil fuels and subsidies on hydropower and other renewables. Stephenson reviews the changes in sources of electricity production since the policies were introduced, shown in Exhibit 1. Exhibit 1: Genovia’s Domestic Electricity Generation Production Sector Year 0 Year 1 Year 2 Fossil fuels 462 446 426 Hydropower 186 231 273 Other renewables 97 120 154 Total 745 797 853 Note: Amounts are in terawatt hours (TWh). 1. Which of Stephenson’s statements regarding Le régulateur and L’organisation is correct? A. Only Statement 1 is correct. B. Only Statement 2 is correct. C. Both Statement 1 and Statement 2 are correct. 2. Is Stephenson’s Statement 3 correct? A. Yes B. No, because L’organisation is given the authority to enforce regulations by a government agency C. No, because pressure from its member companies prevents L’organisation from enforcing its rules and standards 3. Which of Stephenson’s comments to her supervisor is most likely correct? A. Only Comment 1 is correct. B. Only Comment 2 is correct. C. Both Comment 1 and Comment 2 are correct. 4. Which of the following policies would best address Genovia’s objective of promoting free trade on industrial goods? A. Imposing tariffs on foreign-produced goods 572 Learning Module 3 Economics of Regulation B. Allowing a floating currency C. Providing subsidies to domestic companies 5. By amending regulations to encourage foreign businesses to change their corporate domicile, regulators are engaging in regulatory: A. capture. B. arbitrage. C. competition. 6. Which feature discussed in the third section of “International Trade” will most likely help Genovia achieve its goal of encouraging foreign businesses to increase trade with domestic businesses? A. Only Feature 1 B. Only Feature 2 C. Both Feature 1 and Feature 2 7. Based on Exhibit 1, which government policy has been most effective in helping Genovia achieve its second initiative? A. Tax on fossil fuels B. Subsidy on hydropower C. Subsidy on other renewables The following information relates to questions 8-11 Tiu Asset Management (TAM), a hypothetical financial services firm, recently hired Jonna Yun. A member of TAM’s global equity portfolio team, Yun is assigned the task of analyzing the effects of regulation on the financial services sector of a particular country. In her first report to the team, Yun makes the following statements: Statement 1 The country’s regulator, a government agency, concerned about systemic risk, is calling for an accelerated adoption of centralized derivatives settlement (as opposed to bilateral settlement between two counterparties)—a more stringent rule—ahead of other major countries that are considering a similar move. Statement 2 Regulators use various tools to intervene in the financial services sector. Statement 3 Regulations may bring benefits to the economy, but they may also have unanticipated costs. Practice Problems Statement 4 573 The country’s regulatory authorities are considering a regulation that is similar to Regulation Q in the United States, which imposed a ceiling on interest rates paid by banks for certain bank deposits. 8. What is the most likely basis for the concerns noted in Statement 1? A. Externalities B. Regulatory arbitrage C. Informational friction 9. The tools least likely to be used by regulators to intervene in financial markets owing to informational frictions are: A. blackout periods. B. capital requirements. C. insider-trading restrictions. 10. Which of the following is most likely an unanticipated effect of regulation? A. Hiring compliance lawyers B. Setting legal standards for contracts C. Establishing employers’ rights and responsibilities 11. After Regulation Q was imposed, the demand for money market funds most likely: A. increased. B. decreased. C. remained unchanged. 574 Learning Module 3 Economics of Regulation SOLUTIONS 1. A is correct. Le régulateur, as an independent regulator but not an SRO, has legal authority from the Genovia government to regulate. Therefore, Le régulateur both enacts and enforces regulations related to the new accounting statute in Genovia. 2. A is correct. L’organisation is an SRO but not an independent regulator, so it is a private entity that is not affiliated with Genovia’s government. SROs that are not independent regulators receive authority from their members, who agree to comply with the organization’s rules and standards and its enforcement thereof. 3. C is correct. Comment 1 is correct because regulatory filings, in response to proposed regulations, often identify at least some of the unintended consequences prior to the implementation of the regulation. Comment 2 is correct because the cost of unintended consequences, including both indirect costs and unanticipated implementation costs, can be high. 4. B is correct. A floating currency allows international trade in Genovia to be market based. International disputes about whether a country is manipulating or fixing its currency price often center on issues related to competitiveness. 5. C is correct. Regulatory competition describes actions by regulators to encourage behaviors. Regulators may compete to provide a regulatory environment designed to attract certain entities (regulatory competition). By amending regulations, Genovia’s regulators seek to encourage foreign companies to change their corporate domicile. 6. C is correct. Genovia needs unambiguous laws concerning financial liability and bankruptcy to encourage foreign businesses to enter into contracts, particularly those that are long term and may involve sunk costs. The court decisions help Genovia achieve its goal. Also, clearly defined rules governing contracts, their interpretation, and each party’s legal rights under a contract are necessary. Thus, both features help Genovia achieve its goal. 7. C is correct. At the end of Year 2, the compound annual growth rate (CAGR) for each sector is calculated as follows: (Year 2/Year 0)0.5 – 1. Fossil fuels: (426/462)0.5 – 1 = –4% Hydropower: (273/186)0.5 – 1 = 21% Other renewables: (154/97)0.5 – 1 = 26% The CAGR indicates that the 26% increase in production from the subsidy on other renewables has been more effective than the 4% decrease in production from the tax on fossil fuels or the 21% increase in production from the subsidy on hydropower. Thus, the subsidy on other renewables of 26% is the highest, indicating that this policy has been the most effective in helping Genovia achieve its second initiative. 8. B is correct. Firms based in the country are likely to be concerned because of the earlier timing of the application of new (more stringent) regulations in the country than in other large countries. With more stringent regulations, some business may flow to less stringent regulatory environments or jurisdictions. 9. A is correct. Blackout periods are established by companies in response to Solutions concerns about insider trading. Thus, blackout periods are not a tool used by regulators to intervene in the financial services sector. Capital requirements are used by government regulators to reduce systemic risk and financial contagion. Insider-trading restrictions are used by regulators concerned about insiders using their greater knowledge to the disadvantage of others; insider-trading restrictions respond to informational frictions. 10. A is correct. The hiring of more lawyers to deal with compliance is an example of an “unintended” implementation cost. Establishing legal standards for contracts and employers’ rights and responsibilities are objectives (intended consequences) of some regulation. 11. A is correct. Regulation Q set a ceiling on the interest rates paid by banks for various types of deposits, which resulted in investors’ shifting funds to money market funds. 575