ECON 366: ECONOMETRICS II. SPRING TERM 2005: LAB EXERCISE #10 Nonspherical Errors Continued. Brief Suggested Solutions

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1 DEPARTMENT OF ECONOMICS UNIVERSITY OF VICTORIA ECON 366: ECONOMETRICS II SPRING TERM 2005: LAB EXERCISE #10 Nonspherical Errors Continued Brief Suggested Solutions 1. In Lab 8 we considered the following model: mpg i = β 1 + β 2 sp i + β 3 hp i + β 4 wt i + e i (1) where for the i th car: mpg = average miles per gallon sp = top speed, miles per hour hp = engine horsepower wt = vehicle weight Data on 81 cars are provided in the page lab8a of the EViews workfile lab8.wf1, available from the course web page. a. Undertake White s test for homoskedasticity. The null is H 0 : σ 2 t = σ 2 vs H a : not H 0. White s test effectively compares σ 2 (X X) -1 X WX(X X) -1 (the Cov(b) under H a ) with σ 2 (X X) -1 (the Cov(b) under H 0 ). If substantially different, then suggests heteroskedasticity, if similar, then suggests homoskedasticity. Simple operational version of the test consists of: a) Estimate y=xβ+e by OLS and obtain residuals, ê t b) Form ê 2 t 2 c) Regress by OLS, ê t on a constant, and all unique (or non-redundant) variables in the set containing the regressors in X, their squares and their cross products. Let q=# of regressors in this auxiliary regression. d) Test statistic is WT=TR 2 where R 2 is from step c. The test statistic s limiting null distribution is χ 2 (q-1). EViews has this test canned : White Heteroskedasticity Test: F-statistic Probability Obs*R-squared Probability Test Equation: Dependent Variable: RESID^2 Page 1 of 8

2 Date: 03/15/05 Time: 23:15 C SP SP^ SP*HP SP*WT HP HP^ HP*WT WT WT^ R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) So, our sample value of the statistic is , with an asymptotic p-value of p=pr(χ 2 (9)>37.656)=0.000, suggesting we reject the null; i.e., in line with our prior expectation, we find evidence to suggest heteroskedasticity. b. Estimate the model using the OLS estimator of the coefficient vector, with the usual estimator of its variance-covariance matrix that assumes homoskedasticity. Repeat using White s heteroskedastic consistent estimator of the OLS estimator s variance-covariance matrix. Compare the results. The ordinary LS output is: Dependent Variable: MPG Date: 02/28/05 Time: 15:10 C SP Page 2 of 8

3 HP WT R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) The output using White s estimator of the variance-covariance matrix is: Dependent Variable: MPG Date: 03/15/05 Time: 23:26 White Heteroskedasticity-Consistent Standard Errors & Covariance C SP HP WT R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) The standard errors increase when we allow for possible heteroskedasticity, and some of them quite substantially. c. Suppose we model the variance function as: 2 i ( α + α sp + α hp + α wt ) 2 σ = (2) Estimate the auxiliary regression: 1 2 i 3 i 4 ê i = α 1 + α 2 sp i + α 3 hp i + α 4 wt i + v i (3) to obtain estimates of the α parameters; v is an error term. Use the fitted values from this auxiliary regression to estimate equation (1) by weighted i Page 3 of 8

4 least squares, which is a feasible GLS estimator here. Compare the OLS and WLS estimates of the coefficients. The output from this auxiliary regression follows: Dependent Variable: ABS(EHAT) Date: 03/16/05 Time: 08:53 C HP SP WT R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Using the option Forecast from this auxiliary regression, I have stored the fitted values, which are an estimate of σ i, in the series ehatf ; these fitted values are used as weights. Applying WLS, with these weights, we have: Page 4 of 8

5 with resultant output: Dependent Variable: MPG Date: 03/16/05 Time: 09:01 Weighting series: (1/EHATF) C SP HP WT Weighted Statistics R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Unweighted Statistics Page 5 of 8

6 R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Sum squared resid Durbin-Watson stat The table below compares the OLS and WLS coefficient estimates, from which we see substantially changes in the coefficients, and the signs of the estimates. This example illustrates a potential issue with feasible GLS it assumes that we have modeled the heteroskedasticity correctly. Such major changes as seen here would suggest this is not the case and highlights the care that must be taken when we start to model the variance function, in addition to the mean function. Coefficient OLS WLS β β β β In Lab 8 we also considered the following model: log(cu t ) = β 1 + β 2 log(i t ) + β 3 log(lme t ) + β 4 log(hs t ) + β 5 log(al t ) + e t (4) where for the t th year: cu = 12-month average US price of copper (cents per pound) i = 12-month average index of industrial production lme = 12-month average London Metal Exchange price of copper hs = number of housing starts per year (thousands of units) al = 12-month average price of aluminium (cents per pound) Annual data from 1951 through 1980 are provided in the page lab8b of the EViews workfile lab8.wf1, available from the course web page. a. Estimate model (2) by LS using the usual estimator of the LS variancecovariance matrix and the HAC estimator proposed by Newey and West (1987). Compare the standard errors. The two outputs follow: Dependent Variable: CU Date: 02/28/05 Time: 16:10 Sample: Included observations: 30 Page 6 of 8

7 C I LME HS AL R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Dependent Variable: CU Date: 03/16/05 Time: 09:21 Sample: Included observations: 30 Newey-West HAC Standard Errors & Covariance (lag truncation=3) C I LME HS AL R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) We see that some of the se s are smaller while others are larger using the Newey-West HAC estimator of the variance-covariance matrix. b. In part (a), why did we use the Newey-West estimator of the OLS coefficient estimator s variance-covariance matrix rather than White s estimator? When the errors are nonspherical with Cov(e)=W, the OLS estimator has variance-covariance matrix given by: Cov(b) = (X X) -1 X WX(X X) -1. White s estimator of this matrix explicitly allows for only heteroskedasticity, not Page 7 of 8

8 autocorrelation, evident from the formula T 1 2 = êt x tx t (T K) t= Côv(b) = T(X' X) S (X' X) with S 0 ; we only see variance type terms in this expression. The Newey-West formula, on the other hand, allows for unknown heteroskedasticity and unknown autocorrelation, as seen from the formula: 1 1 Côv(b) = T(X X) X S X(X X), with X T SX = ê j x jx j + w(j)êtêt j[xtx t j + xt jx t ] (T K (T K) j= 1 L T j= 1 t= j+ 1 which contains variance type terms and covariance type term. Note that the Newey-West formula is White s formula adjusted for covariance. Page 8 of 8