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Re: st: hetroscedasticity test after probit

From   Maarten Buis <>
Subject   Re: st: hetroscedasticity test after probit
Date   Thu, 5 Jul 2012 10:56:13 +0200

On Thu, Jul 5, 2012 at 10:32 AM, Yuval Arbel wrote:
> Prakash, returning to your original question, I see no point at all to
> check for hetroscedasticity after -probit- simply because this problem
> is inherent in the family of models with binary dependent variables.
> Take, for example, the so called LPM (linear probability model), where
> the dependent variable is derived from Binomial distribution (which
> is by itself an approximation to the Normal distribution, from which
> the probit model is derived). Every elementary Econometric textbook
> (e.g. Jan Kmenta, Elements of Econometrics, 1997, pp. 548-549), will
> show you a very simple proof revealing the fact that the LPM is
> inherently heteroscdastic, where the variance of the random
> disturbance term equals yhat(1-yhat) and yhat is the vector of
> predicted values

The problem is subtly but completely different with models like
-probit- and -logit-. These models already use the variance function
yhat(1-yhat), so there is no need to further adjust for that.

However, heteroskedasticity is still a huge and unsolvable problem in
these models. Think of it this way: your dependent variable is a
probability. A probabiltiy embodies uncertainty, and that uncertainty
comes from all variables we have not included in our model. In one
sense this makes it very easy to deal with heteroskedasticity: We just
define our dependent variable of interest to be the probability given
the control variabels in our model. The results of your model give an
accurate description of what you have found in your data. However, we
often want to give parameters a counterfactual interpretation (e.g.
"if the men suddenly became women, then the probabiltiy changes by x
percentage points"). Such a counterfactual interpretation is only
correct if we can assume that there is no heteroscedasticity. Several
solutions have been proposed and I trust none of them: they are just
too sensitive. If you really want to do something about it, than I
you'll really need to do some reading. Since these models are so
sensitive, you really need to know what you are doing. A good entry
point for that literature is (Williams 2009). But my position is that
that problem is basically unsolvable, so not worth worrying about.

Hope that helps,

Williams, R. 2009. Using heterogenous choice models to compare logit
and probit coefficients across groups. Sociological Methods & Research
37: 531--559.

Maarten L. Buis
Institut fuer Soziologie
Universitaet Tuebingen
Wilhelmstrasse 36
72074 Tuebingen
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