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

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)

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 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.

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.