Lecture 12 Heteroscedasticity 2

Testing for Heteroscedasticity: Example – IBM Example (continuation): The bptest in the lmtest package performs a studentized LM-BP test for the same variables used in the model (Mkt, SMB and HML). For IBM in the 3-factor FF model: > bptest(ibm_x ~ Mkt_RF + SMB + HML) #bptest only allows to test H1: fxi=model variables) studentized Breusch-Pagan test data: ibm_x ~ Mkt_RF + SMB + HML BP = 4.1385, df = 3, p-value = 0.2469 LM-BP Test: 4.1385  cannot reject H0 at 5% level (χ2[3],.05≈7.815); with a p-value = 0.2469. Note: Heteroscedasticity in financial time series is very common. In general, it is driven by squared market returns or squared past errors. 28 Testing for Heteroscedasticity: Example – DIS Example: We suspect that squared Market returns drive heteroscedasticity. We do an LM-BP (studentized) test for Disney: lr_dis <- log(x_dis[-1]/x_dis[-T]) dis_x <- lr_dis – RF fit_dis_ff3 <- lm (dis_x ~ Mkt_RF + SMB + HML) e_dis <- fit_dis_ff3$residuals e_dis2 <- e_dis^2 fit <- lm (e_dis2 ~ Mkt_RF2) Re_e2 <- summary(fit_BP)$r.squared LM_BP_test <- Re_e2 * T > LM_BP_test [1] 14.15224 > p_val <- 1 - pchisq(LM_BP_test, df = 1) > p_val [1] 0.0001685967 # Log returns for DIS # Disney excess returns # Step 1 – OLS in DGP (3-factor FF model) # Step 1 – keep residuals # Step 2 – squared residuals # Step 2 – Auxiliary regression # Step 2 – Keep R^2 from Auxiliary reg # Step 3 – Compute LM Test: R^2 * T # p-value of LM_test LM-BP Test: 14.15  reject H0 at 5% level (χ2[1],.05≈3.84); with a pvalue = .0001. 26 13 Testing for Heteroscedasticity: Example – DIS Example (continuation): We do the same test, but with SMB squared for Disney: SMB2 <- SMB^2 fit <- lm (e_dis2 ~ SMB2) Re_e2 <- summary(fit_BP)$r.squared LM_BP_test <- Re_e2 * T > LM_BP_test [1] 7.564692 > p_val <- 1 - pchisq(LM_BP_test, df = 1) # p-value of LM_test > p_val [1] 0.005952284 LM-BP Test: 7.56  reject H0 at 5% level (χ2[1],.05≈3.84); with a pvalue= .006. 27 Testing for Heteroscedasticity: White Test • Based on the difference between OLS and true OLS variances: 2 (XX – XX)= X𝚺X – 2XX = ∑ (E[ ] –  )𝒙 ′𝒙 • Empirical counterpart: (1/T) ∑ (𝑒 –𝑠 ) 𝒙 ′𝒙 • We can express each element of the 𝑘(𝑘+1) matrix as: ψ : Kolmogorov-Gabor polynomial (1/T) ∑ (𝑒 –𝑠 ) ψ ′ where ψi = (ψ ψ , ..., ψ )’ ψ = ψ ψ , 𝑝 ≥ 𝑞, 𝑝, 𝑞 = 1, 2,..., 𝑘 𝑙 = 1, 2,..., 𝑚 & 𝑚 = 𝑘*(𝑘 1 /2 • Let DT = Var [ ∑ W=[ ∑ (𝑒 –𝑠 ) ψ ′] . Then, White LM test: (𝑒 –𝑠 ) ψ ′]′ DT-1 [ ∑ (𝑒 –𝑠 ) ψ ′] χ 14 Testing for Heteroscedasticity: White Test • White heteroscedasticity test: W=[ ∑ (𝑒 –𝑠 ) ψ ′]′ DT-1 [ ∑ (𝑒 –𝑠 ) ψ ′] χ where DT = Var [ ∑ (𝑒 –𝑠 ) ψ ′] Note: W is asymptotically equivalent to a T R2 test, where R2 is calculated from a regression of 𝑒 / on the ψi’s. Testing for Heteroscedasticity: White Test • Usual calculation of the White test – Step 1. Run OLS on DGP: y = X  + . –Keep residuals, 𝑒 . – Step 2. (Auxiliary Regression). Regress 𝑒2 on all the explanatory variables (𝑥 ), their squares (𝑥 2), & all their cross products 𝑥 ∗ 𝑥 . For example, when the model contains 𝑘 = 2 explanatory variables, the test is based on: 𝑒 = β0 + β1 𝑥 , + β2 𝑥 , + β3 𝑥 , + β4 𝑥 , + β5 𝑥 , 𝑥 , + 𝑣 Let m be the number of regressors in auxiliary regression (in the above example, 𝑚 = 5). Keep R2, say 𝑅 . – Step 3. Compute the LM statistic: LM = T * 𝑅 χ . 15 Testing for Heteroscedasticity: White Test Example: White Test for 3-factor FF model residuals for IBM: HML2 <- HML^2; Mkt_HML <- Mkt_RF*HML Mkt_SMB <- Mkt_RF*SMB SMB_HML <- SMB*HML xx2 <- cbind(Mkt_RF2, SMB2, HML2, Mkt_HML, Mkt_SMB, SMB_HML) fit_ibm_W <- lm(e_ibm2 ~ xx2) # Not including original variables OK r2_e2 <- summary(fit_ibm_W)$r.squared # Keep R^2 from Auxiliary regression > r2_e2 [1] 0.0166492 lm_t <- T * r2_e2 # Compute LM test: R^2 * sample size (T) > lm_t [1] 10.93483 df_lm <- ncol(xx2) qchisq(.95, df = df_lm) LM-White Test: 10.93  cannot reject H0 at 5% level (χ2[6],.05≈12.59). 31 Testing for Heteroscedasticity: White Test Example (continuation): Now, we do a White Test for the 3 factor F-F model for DIS and GE returns. • For DIS, we get: fit_dis_W <- lm (e_dis2 ~ xx2) Re_2W <- summary(fit_dis_W)$r.squared LM_W_test <- Re_2W * T > LM_W_test [1] 25.00148  reject H0 at 5% level (χ2[6],05 ≈ 12.59). >qchisq(.95, df = df_lm) [1] 12.59159 > p_val <- 1 - pchisq(LM_W_test, df = 6) # p-value of LM_test > p_val [1] 0.0003412389 • For GE, we get: LM-White Test: 20.15 (p-value=0.0026)  reject H0 at 5% level. 32 16 Testing for Heteroscedasticity: White Test Example: We do a White Test for the residuals in the encompassing (IFE + PPP) model for changes in the USD/GBP (T=363): fit_gbp <- lm(lr_usdgbp ~ inf_dif + int_dif) e_gbp <- fit_gbp$residuals e_gbp2 <- e_gbp^2 int_dif2 <- int_dif^2 inf_dif2 <- inf_dif^2 int_inf_dif <- int_dif*inf_dif fit_W <- lm (e_gbp2 ~ int_dif2 + inf_dif2+ int_inf_dif) Re_e2W <- summary(fit_W)$r.squared LM_W_test <- Re_e2W * T p_val <- 1 - pchisq(LM_W_test, df = 3) # p-value of LM_test > LM_W_test [1] 15.46692 > p_val [1] 0.001458139  reject H0 at 5% level 33 Testing for Heteroscedasticity: Remarks • Drawbacks of the Breusch-Pagan test: - It has been shown to be sensitive to violations of the normality assumption. - Three other popular LM tests: the Glejser test; the Harvey-Godfrey test, and the Park test, are also sensitive to such violations. • Drawbacks of the White test - If a model has several regressors, the test can consume a lot of df’s. - In cases where the White test statistic is statistically significant, heteroscedasticity may not necessarily be the cause, but model specification errors. - It is general. It does not give us a clue about how to model heteroscedasticity to do FGLS. The BP test points us in a direction. 17 Testing for Heteroscedasticity: Remarks • Drawbacks of the White test (continuation) - In simulations, it does not perform well relative to others, especially, for time-varying heteroscedasticity, typical of financial time series. - The White test does not depend on normality; but the Koenker’s test is also not very sensitive to normality. In simulations, Koenker’s test seems to have more power –see, Lyon and Tsai (1996) for a Monte Carlo study of the heteroscedasticity tests presented here. . Testing for Heteroscedasticity: Remarks • General problems with heteroscedasticity tests: - The tests rely on the first four assumptions of the CLM being true. - In particular, (A2) violations. That is, if the zero conditional mean assumption, then a test for heteroskedasticity may reject the null hypothesis even if Var(y|X) is constant. - This is true if our functional form is specified incorrectly (omitted variables or specifying a log instead of a level). Recall David Hendry’s comment. • Knowing the true source (functional form) of heteroscedasticity may be difficult. A practical solution is to avoid modeling heteroscedasticity altogether and use OLS along the White heterosekdasticity-robust standard errors. 18 Estimation: WLS form of GLS • While it is always possible to estimate robust standard errors for OLS estimates, if we know the specific form of the heteroskedasticity, we can obtain more efficient estimates than OLS: GLS. • GLS basic idea: Efficient estimation through the transform the model into one that has homoskedastic errors – called WLS. • Suppose the heteroskedasticity can be modeled as: Var[𝜀|𝒙] = 2 ℎ 𝒙 • The key is to figure out what ℎ 𝒙 looks like. Suppose that we know ℎ . For example, ℎ 𝒙 = 𝑥 . (Make sure ℎ is always positive.) • Then, use to transform the model. Estimation: WLS form of GLS • Suppose that we know ℎ 𝒙 = 𝑥 . Then, use 1/√xi2 to transform the model: Var[ |𝒙] = 2 • Thus, if we divide our whole equation by ℎ 𝒙 we get a (transformed) model where the error is homoskedastic. • Assuming weights are known, we have a two-step GLS estimation: - Step 1: Use OLS, then the residuals to estimate the weights. - Step 2: Weighted least squares using the estimated weights. • Greene has a proof based on our asymptotic theory for the asymptotic equivalence of the second step to true GLS. 19 Estimation: FGLS • More typical is the situation where we do not know the form of the heteroskedasticity. In this case, we need to estimate ℎ 𝒙 . • Typically, we start by assuming a fairly flexible model, such as Var[𝜀|𝒙] = 2 exp(X) –make sure Var[𝜀 |x] > 0. But, we don’t know , it must be estimated. By our assumptions: with E(𝒗 |X) = 1. 𝜺2 = 2 exp(X) 𝒗 Then, if E(𝒗) = 1 ln(𝜺2 ) = X + u (*) where E(u) = 0 and u is independent of X. We know that 𝒆 is an estimate of 𝜺, so we can estimate (*) by OLS. Estimation: FGLS • Now, an estimate of h is obtained as ĥ = exp(ĝ), and the inverse of this is our weight. Now, we can do GLS as usual. • Summary of FGLS (1) Run the original OLS model, save the residuals, 𝒆. Get ln(𝒆2). (2) Regress ln(𝒆 2) on all of the independent variables. Get fitted values, ĝ. (3) Do WLS using 1/ exp(ĝ) as the weight. (4) Iterate to gain efficiency. • Remark: We are using WLS just for efficiency –OLS is still unbiased and consistent. Sandwich estimator gives us consistent inferences. 20 Estimation: MLE • ML estimates all the parameters simultaneously. To construct the likelihood, we assume a distribution for ε. Under normality (A5): T T T 1 1 1 ln L   ln 2  ln i2  ( yi  X i)( yi  X i) 2 2 i1 2 i1 i2   • Suppose  = exp(α + α 𝑧 , + .... + α 𝑧 , )= exp(𝒛 ’α) • Then, the first derivatives of the log likelihood wrt θ=(β, α) are: T  ln L   x i  i / i2  X '  1  i 1   ln L 1  i 2 T T 1 1 1 / i2 exp(zi ' ) zi  ( ) i2 / i4 exp(zi ' ) zi  2 i 1 2 i 1   T  z ( i 2 i / i2  1) i 1 • Then, we get the f.o.c. We get a non-linear system of equations. Estimation: MLE • We take second derivatives to calculate the information matrix : T  ln L2  x i x i ' /  i2  X '  1 X   ' i 1   ln L 1    i ' 2 T  x z ' i i i /  i2 i 1  ln L 1   i  i ' 2 T  z z ' i i 2 i /  i2 i 1 • Then, I ()  E[ 1  ln L  X '  X ] '  0  0   1 Z'Z 2  • We can estimate the model using Newton’s method: 𝜃 = 𝜃 – 𝐻 -1 𝑔 𝑔 = ∂log 𝐿 /∂𝜃′ 21 Estimation: MLE • We estimate the model using Newton’s method: 𝜃 = 𝜃 – 𝐻 -1 𝑔 𝑔 = ∂log 𝐿 /∂𝜃′ where subscript 𝑗 stands for values computed under iteration 𝑗. Since 𝐻 is block diagonal, then β = β – (X′ Σ -1 X)-1 X′ Σ 𝛼 = 𝛼 – (½Z′Z)-1 [½ ∑ -1 𝜺 𝒛 (𝜀 , / . – 1)] = 𝛼 – (Z′Z)-1 Z′𝒗, where 𝒗 = (𝜀 , / , – 1) Convergence will be achieved when 𝑔 = ∂log 𝐿 /∂𝜃 is close to zero. • We have an iterative algorithm  Iterative FGLS = MLE! Heteroscedasticity: Log Transformations • A log transformation of the data, can eliminte (or reduce) a certain type of heteroskedasticity. - Assume -  = E[𝑍 ] - Var[𝑍 ] = δ  2 (Variance proportional to the squared mean) • We log transformed the data: log(𝑍 ). Then, we use the delta method to approximate the variance of the transformed variable. Recall: Var[f(X)] using delta method: Var[ f ( X )]  f ' ( ) 2 Var[ X ] • Then, the variance of log(𝑍 ) is roughly constant: Var [log( Z t )]  (1 /  t ) 2 Var [ Z t ]   22 ARCH Models • Until the early 1980s econometrics had focused almost solely on modeling the conditional (on information set at time 𝑡, I ) mean: 𝑦 = E[𝑦 | I ] + 𝜀 , 𝜀 ~ D(0, 2) Suppose we have an AR(1) process: 𝑦 =𝛼+β𝑦 +𝜀 . Then, the conditional mean, conditioning on I E [𝑦 | I ] = 𝛼 + β 𝑦 , is: • Recall the distinction between conditional moments and unconditional ones. The unconditional mean is: E[𝑦 ] = = constant The conditional mean is time varying; the unconditional mean is not! ARCH Models • Similar idea for the variance. For the AR(1) process: 𝑦 =𝛼+β𝑦 +𝜀 . Unconditional variance: Var[𝑦 ] = E[(𝑦 – E[𝑦 ])2] = Conditional variance: Var [𝑦 | I ]=E = constant [𝑦 –E[𝑦 ]))2] = E [𝜀 ] Remark: Conditional moments are time varying; unconditional moments are not! 23 ARCH Models • The unconditional variance measures overall uncertainty. In the AR(1) example, the information set I plays no role: Var[𝑦 ] = • The conditional variance, Var [y |I ], is a better measure of uncertainty at time 𝑡. It is a function of information at time 𝑡, I . Conditional Variance 𝑦 mean Variance 𝑡 Time ARCH Models: Stylized Facts of Asset Returns - Thick tails - Mandelbrot (1963): leptokurtic (thicker than Normal) - Volatility clustering - Mandelbrot (1963): “large changes tend to be followed by large changes of either sign.” - Leverage Effects – Black (1976), Christie (1982): Tendency for changes in stock prices to be negatively correlated with changes in volatility. - Non-trading Effects, Weekend Effects – Fama (1965), French and Roll (1986) : When a market is closed information accumulates at a different rate to when it is open –for example, the weekend effect, where stock price volatility on Monday is not three times the volatility on Friday. 24