Heteroskedasticity ECON 4550 Econometrics Memorial University of Newfoundland Adapted from Vera Tabakova’s notes

 8.1 The Nature of Heteroskedasticity  8.2 Using the Least Squares Estimator  8.3 The Generalized Least Squares Estimator  8.4 Detecting Heteroskedasticity

Figure 8.1 Heteroskedastic Errors

Food expenditure example:

Figure 8.2 Least Squares Estimated Expenditure Function and Observed Data Points

The existence of heteroskedasticity implies:  The least squares estimator is still a linear and unbiased estimator, but it is no longer best. There is another estimator with a smaller variance.  The standard errors usually computed for the least squares estimator are incorrect. Confidence intervals and hypothesis tests that use these standard errors may be misleading.

We can use a robust estimator: regress food_exp income, robust

In SHAZAM the HETCOV option on the OLS command reports the White's heteroskedasticity-consistent standard errors OLS FOOD INCOME / HETCOV Confidence interval estimate using the White standard errors CONFID INCOME / TCRIT=2.024 But the White standard errors reported by SHAZAM have numeric differences compared to our textbook results. There is a correction we could apply, but we will not worry about it for now

To obtain the best linear unbiased estimator for a model with heteroskedasticity of the type specified in equation (8.11): 1. Calculate the transformed variables given in (8.13). 2. Use least squares to estimate the transformed model given in (8.14).

The generalized least squares estimator is as a weighted least squares estimator. Minimizing the sum of squared transformed errors that is given by: When is small, the data contain more information about the regression function and the observations are weighted heavily. When is large, the data contain less information and the observations are weighted lightly.

Food example again, where was the problem coming from? regress food_exp income [aweight = 1/income]

Food example again, where was the problem coming from? Specify a weight variable (SHAZAM works with the inverse) GENR W=1/INCOME OLS FOOD INCOME / WEIGHT=W 95% confidence interval, p. 205 CONFID INCOME / TCRIT=2.024

Note that  The residual statistics reported in a WLS regression (SIGMA**2, STANDARD ERROR OF THE ESTIMATE-SIGMA, and SUM OF SQUARED ERRORS-SSE) are all based on the transformed (weighted) residuals  You should remember when making model comparisons, that the high- variance observations are systematically underweighted by this procedure  This may be a good thing, if you want to avoid having these observations dominate the model selection comparisons.  But if you want model selection to be based on how well the alternative models fit the original (untransformed) data, you must base the model selection tests on the untransformed residuals.

The steps for obtaining a feasible generalized least squares estimator for are:  1.Estimate (8.25) by least squares and compute the squares of the least squares residuals.  2.Estimate by applying least squares to the equation

3.Compute variance estimates. 4.Compute the transformed observations defined by (8.23), including if. 5.Apply least squares to (8.24), or to an extended version of (8.24) if.

For our food expenditure example: gen z = log(income) regress food_exp income predict ehat, residual gen lnehat2 = log(ehat*ehat) regress lnehat2 z * * Feasible GLS * predict sig2, xb gen wt = exp(sig2) regress food_exp income [aweight = 1/wt]

The HET command can be used for Maximum Likelihood Estimation of the model given in Equations (8.25) and (8.26), p This method is an alternative estimation method to the GLS method discussed in the text (so the results will also be different): HET FOOD INCOME (INCOME) / MODEL=MULT

Using our wage data (cps2.dta): ???

Feasible generalized least squares: 1.Obtain estimated and by applying least squares separately to the metropolitan and rural observations Apply least squares to the transformed model

* * Rural subsample regression * regress wage educ exper if metro == 0 scalar rmse_r = e(rmse) scalar df_r = e(df_r) * * Urban subsample regression * regress wage educ exper if metro == 1 scalar rmse_m = e(rmse) scalar df_m = e(df_r) * * Groupwise heteroskedastic regression using FGLS * gen rural = 1 - metro gen wt=(rmse_r^2*rural) + (rmse_m^2*metro) regress wage educ exper metro [aweight = 1/wt] STATA Commands:

Remark: To implement the generalized least squares estimators described in this Section for three alternative heteroskedastic specifications, an assumption about the form of the heteroskedasticity is required. Using least squares with White standard errors avoids the need to make an assumption about the form of heteroskedasticity, but does not realize the potential efficiency gains from generalized least squares.

8.4.1Residual Plots  Estimate the model using least squares and plot the least squares residuals.  With more than one explanatory variable, plot the least squares residuals against each explanatory variable, or against, to see if those residuals vary in a systematic way relative to the specified variable.

8.4.2The Goldfeld-Quandt Test

* * Goldfeld Quandt test * scalar GQ = rmse_m^2/rmse_r^2 scalar crit = invFtail(df_m,df_r,.05) scalar pvalue = Ftail(df_m,df_r,GQ) scalar list GQ pvalue crit

8.4.2The Goldfeld-Quandt Test For the food expenditure data You should now be able to obtain this test statistic And check whether it exceeds the critical value

8.4.2The Goldfeld-Quandt Test Sort the data by income: SORT INCOME FOOD / DESC OLS FOOD INCOME On the DIAGNOS command the CHOWONE= option reports the Goldfeld-Quandt test for heteroskedasticity (bottom of page 212) with a p-value for a one-sided test. The HET option reports the tests for heteroskedasticity reported on page 215. DIAGNOS / CHOWONE=20 HET

8.4.2The Goldfeld-Quandt Test SORT INCOME FOOD / DESC OLS FOOD INCOME On the DIAGNOS command the CHOWONE= option reports the Goldfeld-Quandt test for heteroskedasticity (bottom of page 212) with a p-value for a one-sided test. The HET option reports the tests for heteroskedasticity reported on page 215. DIAGNOS / CHOWONE=20 HET Of course, this option also computes the Chow test statistic for structural change (that is, tests the null of parameter stability in the two subsamples).

8.4.2The Goldfeld-Quandt Test SEQUENTIAL CHOW AND GOLDFELD-QUANDT TESTS N1 N2 SSE1 SSE2 CHOW PVALUE G-Q DF1 DF2 PVALUE E CHOW TEST - F DISTRIBUTION WITH DF1= 2 AND DF2= 36

8.4.2The Goldfeld-Quandt Test SHAZAM considers that the alternative hypothesis is smaller error variance in the second subset relative to the first subset. Some authors present the alternative as larger variance in the second subset. Goldfeld and Quandt recommend ordering the observations by the values of one of the explanatory variables. This can be done with the SORT command in SHAZAM. The DESC option on the SORT command should be used if it is assumed that the variance is positively related to the value of the sort variable.

8.4.3Testing the Variance Function

8.4.3a The White Test

8.4.3b Testing the Food Expenditure Example whitetst Or estat imtest, white

Further testing in SHAZAM  DIAGNOS / HET  Will yield a battery of heteroskedasticity tests using different specifications

SHAZAM for food example DIAGNOS\HET REQUIRED MEMORY IS PAR= 7 CURRENT PAR= DEPENDENT VARIABLE = FOOD 40 OBSERVATIONS REGRESSION COEFFICIENTS HETEROSKEDASTICITY TESTS CHI-SQUARE D.F. P-VALUE TEST STATISTIC E**2 ON YHAT: E**2 ON YHAT**2: E**2 ON LOG(YHAT**2): E**2 ON LAG(E**2) ARCH TEST: LOG(E**2) ON X (HARVEY) TEST: ABS(E) ON X (GLEJSER) TEST: E**2 ON X TEST: KOENKER(R2): B-P-G (SSR) : E**2 ON X X**2 (WHITE) TEST: KOENKER(R2): B-P-G (SSR) : Same in SLR

Slide 8-51 Principles of Econometrics, 3rd Edition  Breusch-Pagan test  generalized least squares  Goldfeld-Quandt test  heteroskedastic partition  heteroskedasticity  heteroskedasticity-consistent standard errors  homoskedasticity  Lagrange multiplier test  mean function  residual plot  transformed model  variance function  weighted least squares  White test

Slide 8-52 Principles of Econometrics, 3rd Edition

Slide 8-53 Principles of Econometrics, 3rd Edition (8A.1)

Slide 8-54 Principles of Econometrics, 3rd Edition

Slide 8-55 Principles of Econometrics, 3rd Edition (8A.2)

Slide 8-56 Principles of Econometrics, 3rd Edition (8A.3)

Slide 8-57 Principles of Econometrics, 3rd Edition (8B.2) (8B.1)

Slide 8-58 Principles of Econometrics, 3rd Edition (8B.4) (8B.3) (8B.5)

Slide 8-59 Principles of Econometrics, 3rd Edition (8B.6) (8B.7)

Slide 8-60 Principles of Econometrics, 3rd Edition (8B.8)