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Heteroskedastic linear regression

Highlights

  • Linear regression with multiplicative heteroskedastic errors
  • Flexible exponential function for the variance
  • Maximum likelihood estimator
  • Two-step GLS estimator
  • Support for Bayesian estimation
  • Robust, cluster–robust, and bootstrap standard errors
  • Complex survey designs support

What's this about?

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.

Let's see it work

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.

hetregf

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)

Heteroskedastic linear regression               Number of obs     =     10,000
ML estimation
                                                Wald chi2(10)     =  499044.20
Log likelihood =  4658.587                      Prob > chi2       =     0.0000

gpa Coef. Std. Err. z P>|z| [95% Conf. Interval]
gpa
attend .7390506 .0082289 89.81 0.000 .7229222 .755179
grade
sophomore .2361749 .0014954 157.94 0.000 .233244 .2391058
junior .7811304 .0028099 277.99 0.000 .7756231 .7866378
senior 1.27962 .003619 353.59 0.000 1.272527 1.286713
sports
yes .3872983 .0025673 150.86 0.000 .3822666 .3923301
extra
yes .3843422 .0040179 95.66 0.000 .3764673 .3922171
ap
yes .2101632 .0073612 28.55 0.000 .1957354 .224591
boy
boy -.4082246 .0015562 -262.32 0.000 -.4112748 -.4051744
pedu
college 1.599457 .0042422 377.04 0.000 1.591143 1.607772
graduate 3.070231 .0106216 289.06 0.000 3.049413 3.091049
_cons -.0445178 .008372 -5.32 0.000 -.0609266 -.0281091
lnsigma2
grade
sophomore .1260056 .0437898 2.88 0.004 .0401793 .211832
junior 1.852363 .0487709 37.98 0.000 1.756774 1.947953
senior 2.476228 .0465629 53.18 0.000 2.384967 2.56749
pedu
college -3.226682 .0324147 -99.54 0.000 -3.290214 -3.16315
graduate .5139875 .0502405 10.23 0.000 .4155179 .612457
ap
yes 1.070959 .0918929 11.65 0.000 .8908526 1.251066
extra
yes 1.064171 .053584 19.86 0.000 .9591484 1.169194
ap#extra
yes#yes .1559163 .3113822 0.50 0.617 -.4543816 .7662142
_cons -3.53778 .0361415 -97.89 0.000 -3.608616 -3.466944
LR test of lnsigma2=0: chi2(8) = 7478.32 Prob > chi2 = 0.0000

The coefficients under the heading gpa comprise 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.

Tell me more

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.


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