CFA Level 2 - Quantitative Analysis Session 3 - Reading 11

CFA Level 2 - Quantitative Analysis Session 3 - Reading 11 (Practice Questions, Sample Questions) 1. Thomas Manx is attempting to determine the correlation between the number of times a stock quote is requested on his firm’s website and the number of trades his firm actually processes. He has examined samples from several days trading and quotes and has determined that the covariance between these two variables is 88.6, the standard deviation of the number of quotes is 18, and the standard deviation of the number of trades processed is 14. Based on Manx’s sample, what is the correlation between the number of quotes requested and the number of trades processed? A) 0.78. B) 0.18. C) 0.35. Explanation: Correlation = Cov (X,Y) / (Std. Dev. X)(Std. Dev. Y) Correlation = 88.6 / (18)(14) = 0.35 2. Which of the following statements regarding scatter plots is most accurate? Scatter plots: A) illustrate the relationship between two variables. B) illustrate the scatterings of a single variable. C) are used to examine the third moment of a distribution (skewness). Explanation: A scatter plot is a collection of points on a graph where each point represents the values of two variables. They are used to examine the relationship between two variables. 3. If the correlation between two variables is −1.0, the scatter plot would appear along a: A) straight line running from southwest to northeast. B) a curved line running from southwest to northeast. C) straight line running from northwest to southeast. Explanation: If the correlation is −1.0, then higher values of the y-variable will be associated with lower values of the x-variable. The points would lie on a straight line running from northwest to southeast. 4. Determine and interpret the correlation coefficient for the two variables X and Y. The standard deviation of X is 0.05, the standard deviation of Y is 0.08, and their covariance is −0.003. A) −0.75 and the two variables are negatively associated. B) +0.75 and the two variables are positively associated. C) −1.33 and the two variables are negatively associated. Explanation: The correlation coefficient is the covariance divided by the product of the two standard deviations, i.e. −0.003 / (0.08 × 0.05). 5. Unlike the coefficient of determination, the coefficient of correlation: A) indicates whether the slope of the regression line is positive or negative. B) indicates the percentage of variation explained by a regression model. C) measures the strength of association between the two variables more exactly. Explanation: In a simple linear regression the coefficient of determination (R2) is the squared correlation coefficient, so it is positive even when the correlation is negative. 6. In order to have a negative correlation between two variables, which of the following is most accurate? A) The covariance must be negative. B) Either the covariance or one of the standard deviations must be negative. C) The covariance can never be negative. Explanation: In order for the correlation between two variables to be negative, the covariance must be negative. (Standard deviations are always positive.) 7. Which of the following statements regarding a correlation coefficient of 0.60 for two variables Y and X is most accurate? This correlation: A) is significantly different from zero. B) indicates a positive causal relation between the two variables. C) indicates a positive covariance between the two variables. Explanation: A test of significance requires the sample size, so we cannot conclude anything about significance. There is some positive relation between the two variables, but one may or may not cause the other. 8. Which model does not lend itself to correlation coefficient analysis? A) Y = X + 2. B) Y = X3. C) X = Y × 2. Explanation: The correlation coefficient is a measure of linear association. All of the functions except for Y = X3 are linear functions. 9. Rafael Garza, CFA, is considering the purchase of ABC stock for a client’s portfolio. His analysis includes calculating the covariance between the returns of ABC stock and the equity market index. Which of the following statements regarding Garza’s analysis is most accurate? A) The actual value of the covariance is not very meaningful because the measurement is very sensitive to the scale of the two variables. B) The covariance measures the strength of the linear relationship between two variables. C) A covariance of +1 indicates a perfect positive covariance between the two variables. Explanation: Covariance is a statistical measure of the linear relationship of two random variables, but the actual value is not meaningful because the measure is extremely sensitive to the scale of the two variables. Covariance can range from negative to positive infinity. 10. Consider the case when the Y variable is in U.S. dollars and the X variable is in U.S. dollars. The 'units' of the covariance between Y and X are: A) a range of values from −1 to +1. B) U.S. dollars. C) squared U.S. dollars. Explanation: The covariance is in terms of the product of the units of Y and X. It is defined as the average value of the product of the deviations of observations of two variables from their means. The correlation coefficient is a standardized version of the covariance, ranges from −1 to +1, and is much easier to interpret than the covariance. 11. Which of the following statements about covariance and correlation is least accurate? A)The covariance and correlation are always the same sign, positive or negative. B) A zero covariance implies a zero correlation. C) There is no relation between the sign of the covariance and the correlation. Explanation: The correlation is the ratio of the covariance to the product of the standard deviations of the two variables. Therefore, the covariance and the correlation have the same sign. 12. Which of the following statements regarding the coefficient of determination is least accurate? The coefficient of determination: A) may range from −1 to +1. B) cannot decrease as independent variables are added to the model. C) is the percentage of the total variation in the dependent variable that is explained by the independent variable. Explanation: In a simple regression, the coefficient of determination is calculated as the correlation coefficient squared and ranges from 0 to +1. 13. A sample covariance for the common stock of the Earth Company and the S&P 500 is −9.50. Which of the following statements regarding the estimated covariance of the two variables is most accurate? A) The relationship between the two variables is not easily predicted by the calculated covariance. B) The two variables will have a slight tendency to move together. C) The two variables will have a strong tendency to move in opposite directions. Explanation: The actual value of the covariance for two variables is not very meaningful because its measurement is extremely sensitive to the scale of the two variables, ranging from negative to positive infinity. Covariance can, however be converted into the correlation coefficient, which is more straightforward to interpret. 14. Which term is least likely to apply to a regression model? A) Coefficient of variation. B) Goodness of fit. C) Coefficient of determination. Explanation: Goodness of fit and coefficient of determination are different names for the same concept. The coefficient of variation is not directly part of a regression model. 15. sample covariance of two random variables is most commonly utilized to: A) identify and measure strong nonlinear relationships between the two variables. B) estimate the “pure” measure of the tendency of two variables to move together over a period of time. C) calculate the correlation coefficient, which is a measure of the strength of their linear relationship. Explanation: Since the actual value of a sample covariance can range from negative to positive infinity depending on the scale of the two variables, it is most commonly used to calculate a more useful measure, the correlation coefficient. 16. For the case of simple linear regression with one independent variable, which of the following statements about the correlation coefficient is least accurate? A) If the regression line is flat and the observations are dispersed uniformly about the line, the correlation coefficient will be +1. B) If the correlation coefficient is negative, it indicates that the regression line has a negative slope coefficient. C) The correlation coefficient can vary between −1 and +1. Explanation: Correlation analysis is a statistical technique used to measure the strength of the relationship between two variables. The measure of this relationship is called the coefficient of correlation. If the regression line is flat and the observations are dispersed uniformly about the line,there is no linear relationship between the two variables and the correlation coefficient will be zero. Both of the other choices are CORRECT. 17. The Y variable is regressed against the X variable resulting in a regression line that is horizontal with the plot of the paired observations widely dispersed about the regression line. Based on this information, which statement is most likely accurate? A) The R2 of this regression is close to 100%. B) The correlation between X and Y is close to zero. C) X is perfectly positively correlated to Y. Explanation: Perfect correlation means that all the observations fall on the regression line. An R2 of 100% means perfect correlation. When there is no correlation, the regression line is horizontal. 18. Which of the following statements about linear regression is least accurate? A) The correlation coefficient, ρ, of two assets x and y = (covariance x,y) × standard deviation x × standard deviation y. B) The independent variable is uncorrelated with the residuals (or disturbance term). C) R2 = RSS / SST. Explanation: The correlation coefficient, ρ, of two assets x and y = (covariance x,y) divided by (standard deviation x × standard deviation y). The other statements are true. For the examination, memorize the assumptions underlying linear regression! 19. A simple linear regression equation had a coefficient of determination (R2) of 0.8. What is the correlation coefficient between the dependent and independent variables and what is the covariance between the two variables if the variance of the independent variable is 4 and the variance of the dependent variable is 9? Correlation coefficient Covariance A) 0.91 4.80 B) 0.89 5.34 C) 0.89 4.80 Explanation: The correlation coefficient is the square root of the R2, r = 0.89. To calculate the covariance multiply the correlation coefficient by the product of the standard deviations of the two variables: COV = 0.89 × √4 × √9 = 5.34 20. Ron James, CFA, computed the correlation coefficient for historical oil prices and the occurrence of a leap year and has identified a statistically significant relationship. Specifically, the price of oil declined every fourth calendar year, all other factors held constant. James has most likely identified which of the following conditions in correlation analysis? A) Positive correlation. B) Spurious correlation. C) Outliers. Explanation: Spurious correlation occurs when the analysis erroneously indicates a linear relationship between two variables when none exists. There is no economic explanation for this relationship; therefore this would be classified as spurious correlation. 21. One major limitation of the correlation analysis of two random variables is when two variables are highly correlated, but no economic relationship exists. This condition most likely indicates the presence of: A) outliers. B) nonlinear relationships. C) spurious correlation. Explanation: Spurious correlation occurs when the analysis erroneously indicates a relationship between two variables when none exists. 22. One of the limitations of correlation analysis of two random variables is the presence of outliers, which can lead to which of the following erroneous assumptions? A) The presence of a nonlinear relationship between the two variables, when in fact, there is a linear relationship. B) The absence of a relationship between the two variables, when in fact, there is a linear relationship. C) The presence of a nonlinear relationship between the two variables, when in fact, there is no relationship whatsoever between the two variables. Explanation: Outliers represent a few extreme values for sample observations in a correlation analysis. They can either provide statistical evidence that a significant relationship exists, when there is none, or provide evidence that no relationship exists when one does. 23. We are examining the relationship between the number of cold calls a broker makes and the number of accounts the firm as a whole opens. We have determined that the correlation coefficient is equal to 0.70, based on a sample of 16 observations. Is the relationship statistically significant at a 10% level of significance, why or why not? The relationship is: A) significant; the t-statistic exceeds the critical value by 3.67. B) not significant; the critical value exceeds the t-statistic by 1.91. C) significant; the t-statistic exceeds the critical value by 1.91. Explanation: The calculated test statistic is t-distributed with n – 2 degrees of freedom: t = r√(n – 2) / √(1 – r2) = 2.6192 / 0.7141 = 3.6678 From a table, the critical value = 1.76 24. A study of 40 men finds that their job satisfaction and marital satisfaction scores have a correlation coefficient of 0.52. At 5% level of significance, is the correlation coefficient significantly different from 0? A) No, t = 1.68. B) No, t = 2.02. C) Yes, t = 3.76. Explanation: H0: r = 0 vs. Ha: r ≠ 0 t = [r √(n – 2)] / √(1 – r2)

="[(0.52" √(38)] √(1 – 0.522)="3.76" tc (α = 0.05 and degrees of freedom = 38) = 2.021 t > tc hence we reject H0. 25. Suppose the covariance between Y and X is 0.03 and that the variance of Y is 0.04 and the variance of X is 0.12. The sample size is 30. Using a 5% level of significance, which of the following is most accurate? The null hypothesis of: A) no correlation is rejected. B) significant correlation is rejected. C) no correlation is not rejected. Explanation: The correlation coefficient is r = 0.03 / (√0.04 * √0.12) = 0.03 / (0.2000 * 0.3464) = 0.4330. The test statistic is t = (0.4330 × √28) / √(1 − 0.1875) = 2.2912 / 0.9014 = 2.54. The critical t-values are ± 2.048. Therefore, we reject the null hypothesis of no correlation. 26. Consider a sample of 60 observations on variables X and Y in which the correlation is 0.42. If the level of significance is 5%, we: A) cannot test the significance of the correlation with this information. B) conclude that there is no significant correlation between X and Y. C) conclude that there is statistically significant correlation between X and Y. Explanation: The calculated t is t = (0.42 × √58) / √(1-0.42^2) = 3.5246 and the critical t is approximately 2.000. Therefore, we reject the null hypothesis of no correlation. 27. Consider a sample of 32 observations on variables X and Y in which the correlation is 0.30. If the level of significance is 5%, we: A) conclude that there is significant correlation between X and Y. B) conclude that there is no significant correlation between X and Y. C) cannot test the significance of the correlation with this information. Explanation: The calculated t = (0.30 × √30) / √(1 − 0.09) = 1.72251 and the critical t values are ± 2.042. Therefore, we fail to reject the null hypothesis of no correlation. 28. Suppose the covariance between Y and X is 10, the variance of Y is 25, and the variance of X is 64. The sample size is 30. Using a 5% level of significance, which of the following statements is most accurate? The null hypothesis of: A) no correlation is rejected. B) significant correlation is rejected. C) no correlation cannot be rejected. Explanation: The correlation coefficient is r = 10 / (5 × 8) = 0.25. The test statistic is t = (0.25 × √28) / √(1 − 0.0625) = 1.3663. The critical t-values are ± 2.048. Therefore, we cannot reject the null hypothesis of no correlation. 29. The purpose of regression is to: A) get the largest R2 possible. B) explain the variation in the dependent variable. C) explain the variation in the independent variable. Explanation: The goal of a regression is to explain the variation in the dependent variable. 30. The capital asset pricing model is given by: Ri =Rf + Beta ( Rm -Rf) where Rm = expected return on the market, Rf = risk-free market and Ri = expected return on a specific firm. The dependent variable in this model is: A) Ri. B) Rm - Rf. C) Rf. Explanation: The dependent variable is the variable whose variation is explained by the other variables. Here, the variation in Ri is explained by the variation in the other variables, Rf and Rm. 31. The independent variable in a regression equation is called all of the following EXCEPT: A) predicted variable. B) predicting variable. C) explanatory variable. Explanation: The dependent variable is the predicted variable. 32. Joe Harris is interested in why the returns on equity differ from one company to another. He chose several company-specific variables to explain the return on equity, including financial leverage and capital expenditures. In his model: A) return on equity is the independent variable, and financial leverage and capital expenditures are dependent variables B) return on equity is the dependent variable, and financial leverage and capital expenditures are independent variables. C) return on equity, financial leverage, and capital expenditures are all independent variables. Explanation: The dependent variable is return on equity. This is what he wants to explain. The variables he uses to do the explaining (i.e., the independent variables) are financial leverage and capital expenditures. 33. Sera Smith, a research analyst, had a hunch that there was a relationship between the percentage change in a firm’s number of salespeople and the percentage change in the firm’s sales during the following period. Smith ran a regression analysis on a sample of 50 firms, which resulted in a slope of 0.72, an intercept of +0.01, and an R2 value of 0.65. Based on this analysis, if a firm made no changes in the number of sales people, what percentage change in the firm’s sales during the following period does the regression model predict? A) +1.00%. B) +0.72%. C) +0.65%. Explanation: The slope of the regression represents the linear relationship between the independent variable (the percent change in sales people) and the dependent variable, while the intercept represents the predicted value of the dependent variable if the independent variable is equal to zero. In this case, the percentage change in sales is equal to: 0.72(0) + 0.01 = +0.01. 34. Paul Frank is an analyst for the retail industry. He is examining the role of television viewing by teenagers on the sales of accessory stores. He gathered data and estimated the following regression of sales (in millions of dollars) on the number of hours watched by teenagers (in hours per week): Sales t = 1.05 + 1.6 TVt Which of the following is the most accurate interpretation of the estimated results? If TV watching: A) goes up by one hour per week, sales of accessories increase by $1.60. B) goes up by one hour per week, sales of accessories increase by $1.6 million. C) changes, no change in sales is expected. Explanation: The interpretation of the slope coefficient is the change in the dependent variable (sales in millions of dollars) for a given one-unit change in the independent variable (TV hours per week). The intercept of 1.05 means that 1.05 million dollars worth of accessories is expected to be sold even if TV watching is zero. 35. In the estimated regression equation Y = 0.78 - 1.5 X, which of the following is least accurate when interpreting the slope coefficient? A) If the value of X is zero, the value of Y will be -1.5. B) The dependent variable increases by 1.5 units if X decreases by 1 unit. C) The dependent variable declines by -1.5 units if X increases by 1 unit. Explanation: The slope represents the change in the dependent variable for a one-unit change in the independent variable. If the value of X is zero, the value of Y will be equal to the intercept, in this case, 0.78. 36. The most appropriate test statistic to test statistical significance of a regression slope coefficient with 45 observations and 2 independent variables is a: A) one-tail t-statistic with 43 degrees of freedom. B) two-tail t-statistic with 42 degrees of freedom. C) one-tail t-statistic with 42 degrees of freedom. Explanation: df = n − k − 1 = 45 − 2 − 1 37. Consider the regression results from the regression of Y against X for 50 observations: Y = 5.0 - 1.5 X The standard error of the estimate is 0.40 and the standard error of the coefficient is 0.45. The predicted value of Y if X is 10 is: A) 10. B) 20. C) -10. Explanation: The predicted value of Y is: Y = 5.0 – [1.5 (10)] = 5.0 – 15 = -10 38. Consider the regression results from the regression of Y against X for 50 observations: Y = 5.0 + 1.5 X The standard error of the coefficient is 0.50 and the standard error of the forecast is 0.52. The 95% confidence interval for the predicted value of Y if X is 10 is: A) {19.480 < Y < 20.052}. B) {18.980 < Y < 21.019}. C) {18.954 < Y < 21.046}. Explanation: The predicted value of Y is: Y = 5.0 + [1.5 (10)] = 5.0 + 15 = 20. The confidence interval is 20 ± 2.011 (0.52) or {18.954 < Y < 21.046}. 39. A dependent variable is regressed against a single independent variable across 100 observations. The mean squared error is 2.807, and the mean regression sum of squares is 117.9. What is the correlation coefficient between the two variables? A) 0.55. B) 0.30. C) 0.99. Explanation: The correlation coefficient is the square root of the R2, which can be found by dividing the regression sum of squares by the total sum of squares. The regression sum of squares is the mean regression sum of squares multiplied by the number of independent variables, which is 1, so the regression sum of squares is equal to 117.9. The residual sum of squares is the mean squared error multiplied by the denominator degrees of freedom, which is the number of observations minus the number of independent variables, minus 1, which is equal to 100 − 1 − 1 = 98. The residual sum of squares is then 2.807 × 98 = 275.1. The total sum of squares is the sum of the regression sum of squares and the residual sum of squares, which is 117.9 + 275.1 = 393.0. The R2 = 117.9 / 393.0 = 0.3, so the correlation is the square root of 0.3 = 0.55. 40. Regression analysis has a number of assumptions. Violations of these assumptions include which of the following? A) Independent variables that are not normally distributed. B) A zero mean of the residuals. C) Residuals that are not normally distributed. Explanation: The assumptions include a normally distributed residual with a constant variance and a mean of zero. 41. Limitations of regression analysis include all of the following EXCEPT: A) parameter instability. B) regression results do not indicate anything about economic significance. C) outliers may affect the estimated regression line. Explanation: The estimated coefficients tell us something about economic significance – they tell us the expected or average change in the dependent variable for a given change in the independent variable. 42. Wanda Brunner, CFA, is working on a regression analysis based on publicly available macroeconomic time-series data. The most important limitation of regression analysis in this instance is: A) the error term of one observation is not correlated with that of another observation. B) limited usefulness in identifying profitable investment strategies. C) low confidence intervals. Explanation: Regression analysis based on publicly available data is of limited usefulness if other market participants are also aware of and make use of this evidence.