Linear regression with multiplicative heteroskedastic errors
Flexible exponential function for the variance
Maximum likelihood estimator
Two-step GLS estimator
Robust, cluster–robust, and bootstrap standard errors
Complex survey designs support
hetregress fits linear regressions in which the variance is an exponential function of covariates that you specify. It allows you to model the heteroskedasticity. When we fit models using ordinary least squares (regress), we assume that the variance of the residuals is constant. If it is not constant, regress reports biased standard errors, leading to incorrect inferences. hetregress lets you deal with the heterogeneity.
Modeling the variance as an exponential function also produces more efficient parameter estimates if the variance model is correctly specified.
hetregress implements two estimators for the variance: a maximum likelihood (ML) estimator and a two-step GLS estimator. The ML estimates are more efficient than those obtained by the GLS estimator if the mean and variance function are correctly specified and the errors are normally distributed. The two-step GLS estimates are more robust if the variance function is incorrect or the errors are nonnormal.
We model students' high school performance (grade point average or GPA) as a function of
their attendance rate (attend)
whether they are freshmen, sophomores, juniors, or seniors
their participation in sports (sports)
their participation in after school activities
whether they take advanced placement courses (ap)
whether they are boys (boy)
their parent's maximum level of educational attainment (pedu)
We could fit the model by typing
. regress gpa attend i.(grade sports extra ap boy pedu)
After fitting the model, we found evidence of heteroskedasticity using the existing postestimation command estat hettest, which did not surprise us. We suspected that the variance might increase with the student's grade level if nothing else. As students age, they become different. We had suspicions about the effects of other variables as well.
;)
So we refit the model using hetregress:
. hetregress gpa attend i.(grade sports extra ap boy pedu),
het(i.grade pedu i.ap##i.extra)
Fitting full model:
Iteration 0: Log likelihood = -8244.2526
Iteration 1: Log likelihood = -8146.4604
Iteration 2: Log likelihood = -8143.9845
Iteration 3: Log likelihood = -8143.9825
Iteration 4: Log likelihood = -8143.9825
Heteroskedastic linear regression Number of obs = 10,000
ML estimation
Wald chi2(10) = 49185.25
Log likelihood = -8143.983 Prob > chi2 = 0.0000
| gpa | Coefficient Std. err. z P>|z| [95% conf. interval] | |
| gpa | ||
| attend | .6315888 .0471474 13.40 0.000 .5391816 .7239961 | |
| grade | ||
| sophomore | -.0043576 .010086 -0.43 0.666 -.0241257 .0154105 | |
| junior | -.0161349 .01465 -1.10 0.271 -.0448484 .0125787 | |
| senior | -.0124978 .0201447 -0.62 0.535 -.0519806 .0269851 | |
| sports | ||
| yes | .7129917 .0147291 48.41 0.000 .6841232 .7418601 | |
| extra | ||
| yes | .7025737 .0152534 46.06 0.000 .6726776 .7324697 | |
| ap | ||
| yes | .3651225 .0283152 12.89 0.000 .3096258 .4206192 | |
| boy | ||
| boy | -.7186189 .008559 -83.96 0.000 -.7353942 -.7018435 | |
| pedu | ||
| college | 1.558124 .0092734 168.02 0.000 1.539948 1.576299 | |
| graduate | 2.468524 .0191345 129.01 0.000 2.431021 2.506027 | |
| _cons | .7233421 .0432877 16.71 0.000 .6384998 .8081844 | |
| lnsigma2 | ||
| grade | ||
| sophomore | .8428276 .0402258 20.95 0.000 .7639864 .9216688 | |
| junior | 1.765285 .0403254 43.78 0.000 1.686249 1.844322 | |
| senior | 2.539946 .0396568 64.05 0.000 2.46222 2.617672 | |
| pedu | ||
| college | .7894325 .0305812 25.81 0.000 .7294945 .8493705 | |
| graduate | .9831641 .0512158 19.20 0.000 .8827831 1.083545 | |
| ap | ||
| yes | .1425211 .0898203 1.59 0.113 -.0335234 .3185656 | |
| extra | ||
| yes | -.0339061 .0530107 -0.64 0.522 -.1378052 .069993 | |
| ap#extra | ||
| yes#yes | .7684617 .3065945 2.51 0.012 .1675476 1.369376 | |
| attend | .0946848 .1562184 0.61 0.544 -.2114977 .4008672 | |
| _cons | -3.057355 .1452952 -21.04 0.000 -3.342129 -2.772582 | |
The coefficients under the heading gpa compose our main model for the mean of gpa.
The coefficients under the heading lnsigma2 are the coefficients of the exponential model for the variance.
The likelihood-ratio test reported at the bottom of the table tells us that our model of the variance fits the data better than a model where the variance is constant.
Learn more about other linear models features.
You can also fit Bayesian heteroskedastic linear regression using the bayes prefix.
Read more about hetregress in the Stata Base Reference Manual.
