Estimation of Models with Heteroskedastic errorsWith heteroskedastic errors the OLS estimator still gives an unbiased estimator. That is, the proof that the OLS estimator is unbiased does not use the heteroskedasticity assumption. The heteroskedasticity affects the results in two ways:
A number of solution approaches can be considered as follows. 1. The observed heteroskedasticity in the residuals may be an indication of model misspecification such as incorrect functional form. For example, a loglog model may reduce heteroskedasticity compared to a linear model. Gujarati [1995, p.386] comments that the log transformation compresses the scales in which the variables are measured. 2.
If the model specification is considered adequate then it may be useful
to focus on just correcting the second problem above.
The 3. To obtain an efficient estimator an estimation method is weighted least squares (WLS). This is a special case of generalized least squares (GLS). The application of this method requires specifying a functional form for the error variance.
Note: Another estimation approach that has application
to the estimation of models with heteroskedastic errors is
maximum likelihood estimation. This is available in SHAZAM with the
[SHAZAM Guide home] Computing HeteroskedasticityConsistent Standard ErrorsThe ExampleThis example uses the Griffiths, Hill and Judge
data set on household expenditure
that was analyzed in the section on
testing for heteroskedasticity.
The SHAZAM commands (filename:
The SHAZAM output can be viewed. The OLS standard errors are reported in the following output:
When the
Both regressions report identical OLS estimated coefficients. But the second regression reports larger standard errors. So hypothesis testing that relies on the results of the first regression may give misleading results.
[SHAZAM Guide home] SHAZAM output_SAMPLE 1 40 _READ (GHJ.txt) FOOD INCOME UNIT 88 IS NOW ASSIGNED TO: GHJ.txt 2 VARIABLES AND 40 OBSERVATIONS STARTING AT OBS 1 _OLS FOOD INCOME OLS ESTIMATION 40 OBSERVATIONS DEPENDENT VARIABLE = FOOD ...NOTE..SAMPLE RANGE SET TO: 1, 40 RSQUARE = .3171 RSQUARE ADJUSTED = .2991 VARIANCE OF THE ESTIMATESIGMA**2 = 46.853 STANDARD ERROR OF THE ESTIMATESIGMA = 6.8449 SUM OF SQUARED ERRORSSSE= 1780.4 MEAN OF DEPENDENT VARIABLE = 23.595 LOG OF THE LIKELIHOOD FUNCTION = 132.672 VARIABLE ESTIMATED STANDARD TRATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 38 DF PVALUE CORR. COEFFICIENT AT MEANS INCOME .23225 .5529E01 4.200 .000 .563 .5631 .6871 CONSTANT 7.3832 4.008 1.842 .073 .286 .0000 .3129 _* Test for heteroskedasticity _DIAGNOS / HET DEPENDENT VARIABLE = FOOD 40 OBSERVATIONS REGRESSION COEFFICIENTS 0.232253330328 7.38321754308 HETEROSKEDASTICITY TESTS CHISQUARE D.F. PVALUE TEST STATISTIC E**2 ON YHAT: 12.042 1 0.00052 E**2 ON YHAT**2: 13.309 1 0.00026 E**2 ON LOG(YHAT**2): 10.381 1 0.00127 E**2 ON LAG(E**2) ARCH TEST: 2.565 1 0.10926 LOG(E**2) ON X (HARVEY) TEST: 4.358 1 0.03683 ABS(E) ON X (GLEJSER) TEST: 11.611 1 0.00066 E**2 ON X TEST: KOENKER(R2): 12.042 1 0.00052 BPG (SSR) : 11.283 1 0.00078 E**2 ON X X**2 (WHITE) TEST: KOENKER(R2): 14.582 2 0.00068 BPG (SSR) : 13.662 2 0.00108 _STOP _* Get the heteroskedasticity corrected standard errors _OLS FOOD INCOME / HETCOV OLS ESTIMATION 40 OBSERVATIONS DEPENDENT VARIABLE = FOOD ...NOTE..SAMPLE RANGE SET TO: 1, 40 USING HETEROSKEDASTICITYCONSISTENT COVARIANCE MATRIX RSQUARE = .3171 RSQUARE ADJUSTED = .2991 VARIANCE OF THE ESTIMATESIGMA**2 = 46.853 STANDARD ERROR OF THE ESTIMATESIGMA = 6.8449 SUM OF SQUARED ERRORSSSE= 1780.4 MEAN OF DEPENDENT VARIABLE = 23.595 LOG OF THE LIKELIHOOD FUNCTION = 132.672 VARIABLE ESTIMATED STANDARD TRATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 38 DF PVALUE CORR. COEFFICIENT AT MEANS INCOME .23225 .6911E01 3.361 .002 .479 .5631 .6871 CONSTANT 7.3832 4.292 1.720 .094 .269 .0000 .3129 _STOP
