Q: Heteroscedasticity:
"I'm fitting an OLS regression model, which assumes that all cases have equal error variance (homscedasticity). I suspect my data may violate this assumption. Is this a problem; if so what can I do about it?"

A: When heteroscedasticity is mild, OLS standard errors behave quite well (Long and Ervin 2000). However, when heteroscedasticity is severe, ignoring it may bias your standard errors and p values. The direction of the bias depends on the pattern of heteroscedasticity: p values may be too large or too small.

Sometimes heteroscedasticity can be removed by a data transformation, such as logging the dependent variable. This may also improve the approximation to normality. Be careful, however, that your transformation doesn't make the results hard to interpret.

Sometimes the form of the heteroscedasticity is clear and can be modeled. More commonly, though, heteroscedasticity is a nuisance that cannot be modeled because its source is not well understood. In this case, the classic correction for heteroscedasticity is the HC0 estimator proposed by Huber (1967) and White (1980). But although this estimator is correct in large samples, it is no better than OLS in small samples. MacKinnon and White (1985) discussed three improvements, HC1, HC2, and HC3. An evaluation by Long and Ervin (2000) suggests that HC3 is the best, especially in small samples.

It is possible to correct for heteroscedasticity using popular software:

1. In Stata, use the HC3 option in the REG command, e.g.,

   reg y x, hc3

   In small samples, this is better than the ROBUST option, which implements HC1.
2. In SAS, there are several options:
   1. In the REG procedure, the ACOV option implements the HC0 correction, e.g.:

      proc reg;
      model y = x / acov;
      run;

      However, ACOV only corrects the covariance matrix; it does not correct the standard errors. To get the corrected standard errors, you have to take the square roots of the diagonal elements in the covariance matrix. :(
   2. In the MIXED procedure, the EMPIRICAL option in the MIXED procedure implements the HC0 correction, e.g.:

      proc mixed empirical;
      model y = x ;
      run;

      Unlike the ACOV option in PROC REG, the EMPIRICAL option corrects the standard errors as well as the covariance matrix. The EMPIRICAL option also works more complicated mixed models, not just vanilla regressions.
   3. Finally,Andrew Hayes has written a SAS macro that gives HC0 standard errors directly, and implements the improved small-sample corrections HC1, HC2, and HC3.
3. SPSS has not implemented any heteroscedasticity correction, but again Hayes has written an SPSS macro that implements the HC0, HC1, HC2, and HC3 corrections.
4. LIMDEP implements the HC0 correction. In addition, Bob Kaufman has written the following LIMDEP code for implementing the HC3 correction:

Title; Calculate MacKinnon and White's HC3 estimator of OLS Standard Errors $
Namelist; iv=one,list of predictors; dv=name of dependent variable$
Regress; lhs=dv; rhs=iv; res=resy$
Matrix; xpxinv=; bols=b$
Create; resysq=resy^2$
Create; hii=qfr(iv,xpxinv); hc3= resysq/ (1-hii) $
Matrix; varhc3=xpxinv*iv'[hc3]iv*xpxinv; stat(bols,varhc3)$

Now, how do you know if you should correct for heteroscedasticity? There are a number of tests for heteroscedasticity, so it seems natural to conduct a test, then use a correction if the test suggest heteroscedasticity. The trouble with this is that the tests often fail to detect heteroscedasticity, leading you to neglect the correction when it is actually needed. In simulations, Long and Ervin (2000) found that this possibility was quite serious. As a result, they recommended that "a test for heteroscedasticity should not be used to determine whether [an HC estimator] should be used." It is better to use an HC estimator whenever heteroscedasticity is suspected.

References

Greene, W.F. (1997). Econometric Analysis (3rd edition). New York: Prentice-Hall.

Hayes, A. F. & Cai, L. (in review). "Heteroscedasticity-robust moderated multiple regression using heteroscedasticity-consistent standard error estimates." Manuscript submitted for publication.

Huber, P.J. 1967. "The behavior of maximum likelihood estimates under non-standard conditions." Proceeding of the Fifth Berkeley Symposium on Mathematical Statistics and Probability 1: 221-233.

Long, J.S and L.H. Ervin, 2000, "Using Heteroscedasticity Consistent Standard Errors in the Linear Regression Model." The American Statistician 54:217-224.

MacKinnon, J.G. and H. White. 1985. "Some heteroskedasticity consistent covariance matrix estimators with improved finite sample properties." Journal of Econometrics, 29, 53-57.

White, Halbert. 1980. "A heteroskedastic-consistent covariance matrix estimator and a direct test of heteroskedasticity." Econometrica 48:817-838.