Robust Standard Errors for Nonlinear Models

André Richter wrote to me from Germany, commenting on the reporting of robust standard errors in the context of nonlinear models such as Logit and Probit. He said he 'd been led to believe that this doesn't make much sense. I told him that I agree, and that this is another of my "pet peeves"!

Yes, I do get grumpy about some of the things I see so-called "applied econometricians" doing all of the time. For instance, see my Gripe of the Day post back in 2011. Sometimes I feel as if I could produce a post with that title almost every day!

Anyway, let's get back to André's point.

The following facts are widely known (e.g., check any recent edition of Greene's text) and it's hard to believe that anyone could get through a grad. level course in econometrics and not be aware of them:

• In the case of a linear regression model, heteroskedastic errors render the OLS estimator, b, of the coefficient vector, β, inefficient. However, this estimator is still unbiased and weakly consistent. 
• In this same linear model, and still using OLS, the usual estimator of the covariance matrix of b is an inconsistent estimators of the true covariance matrix of b. Consequently, if the standard errors of the elements of b are computed in the usual way, they will inconsistent estimators of the true standard deviations of the elements of b.
• For this reason,we often use White's "heteroskedasticity consistent" estimator for the covariance matrix of b, if the presence of heteroskedastic errors is suspected.
• This covariance estimator is still consistent, even if the errors are actually homoskedastic.
• In the case of the linear regression model, this makes sense. Whether the errors are homoskedastic or heteroskedastic, both the OLS coefficient estimators and White's standard errors are consistent.

However, in the case of a model that is nonlinear in the parameters:

• The MLE of the parameter vector is biased and inconsistent if the errors are heteroskedastic (unless the likelihood function is modified to correctly take into account the precise form of heteroskedasticity).
• This stands in stark contrast to the situation above, for the linear model.
• The MLE of the asymptotic covariance matrix of the MLE of the parameter vector is also inconsistent, as in the case of the linear model.
• Obvious examples of this are Logit and Probit models, which are nonlinear in the parameters, and are usually estimated by MLE.

I've made this point in at least one previous post. The results relating to nonlinear models are really well-known, and this is why it's extremely important to test for model mis-specification (such as heteroskedasticity) when estimating models such as Logit, Probit, Tobit, etc. Then, if need be, the model can be modified to take the heteroskedasticity into account before we estimate the parameters. For more information on such tests, and the associated references, see this page on my professional website.

Unfortunately, it's unusual to see "applied econometricians" pay any attention to this! They tend to just do one of two things. They either

1. use Logit or Probit, but report the "heteroskedasticity-consistent" standard errors that their favourite econometrics package conveniently (but misleading) computes for them. This involves a covariance estimator along the lines of White's "sandwich estimator". Or, they
2. estimate a "linear probability model" (i.e., just use OLS, even though the dependent variable is a binary dummy variable, and report the "het.-consistent standard errors".

If they follow approach 2, these folks defend themselves by saying that "you get essentially the same estimated marginal effects if you use OLS as opposed to Probit or Logit." I've said my piece about this attitude previously (here, here, here, and here), and I won't go over it again here.

My concern right now is with approach 1 above.

The "robust" standard errors are being reported to cover the possibility that the model's errors may be heteroskedastic. But if that's the case, the parameter estimates are inconsistent. What use is a consistent standard error when the point estimate is inconsistent? Not much!!

This point is laid out pretty clearly in Greene (2012, pp. 692-693), for example. Here's what he has to say:

"...the probit (Q-) maximum likelihood estimator is not consistent in the presence of any form of heteroscedasticity, unmeasured heterogeneity, omitted variables (even if they are orthogonal to the included ones), nonlinearity of the form of the index, or an error in the distributional assumption [ with some narrow exceptions as described by Ruud (198)]. Thus, in almost any case, the sandwich estimator provides an appropriate asymptotic covariance matrix for an estimator that is biased in an unknown direction." (My underlining; DG.) "White raises this issue explicitly, although it seems to receive very little attention in the literature.".........."His very useful result is that if the QMLE converges to a probability limit, then the sandwich estimator can, under certain circumstances, be used to estimate the asymptotic covariance matrix of that estimator. But there is no guarantee the the QMLE will converge to anything interesting or useful. Simply computing a robust covariance matrix for an otherwise inconsistent estimator does not give it redemption. Consequently, the virtue of a robust covariance matrix in this setting is unclear."

Back on July 2006, on the R Help feed, Robert Duval had this to say:

"This discussion leads to another point which is more subtle, but more important...

You can always get Huber-White (a.k.a robust) estimators of the standard errors even in non-linear models like the logistic regression. However, if you believe your errors do not satisfy the standard assumptions of the model, then you should not be running that model as this might lead to biased parameter estimates.

For instance, in the linear regression model you have consistent parameter estimates independently of whether the errors are heteroskedastic or not. However, in the case of non-linear models it is usually the case that heteroskedasticity will lead to biased parameter estimates (unless you fix it explicitly somehow).

Stata is famous for providing Huber-White std. errors in most of their regression estimates, whether linear or non-linear. But this is nonsensical in the non-linear models since in these cases you would be consistently estimating the standard errors of inconsistent parameters.

This point and potential solutions to this problem is nicely discussed in Wooldrige's Econometric Analysis of Cross Section and Panel Data."

Amen to that!

Regrettably, it's not just Stata that encourages questionable practices in this respect. These same options are also available in EViews, for example.

Reference

Greene, W. H., 2012. Econometric Analysis. Prentice Hall, Upper Saddle River, NJ.

© 2013, David E. Giles