Poisson Regression: Z-Statistic, LR Test, and Overdispersion

[mornington]

Hi all,

I have come across a few problems while estimating a Poisson regression model. I have chosen the simplest Poisson method in eviews, with Huber-White robust standard errors.

1)
While checking for the significance of a binary regressor (gender dummy), the z-statistic and the associated p-value suggest that the coefficient is insignificant. On the other hand, a Likelihood Redundant Variable Test with a single restriction, gender (i.e., under the null, the coefficient on gender is zero), suggests that the coefficient is now significant.

Questions: Which of these is an appropriate result? Is this anomaly a relic of the fact that the two tests are governed by different distributions under the null, or is the Eviews method of executing the Likelihood Redundant Variable test simply not valid for a Poisson regression? This might be of some help: A friend suggested that Eviews might not be calculating consistent estimates for the standard errors for the Z-statistic, or that the LR test is not robust to misspecification.

2)
The standard Poisson regression assumes that the conditional population mean of the dependent variable is equal to its conditional variance. If, on the other hand, we were to relax this assumption along the lines of Generalised Linear Models (GLM), so that the conditional variance is now proportional to (rather than equal to) the conditional mean [a case of over-dispersion], the standard errors of the "equality" model must now be multiplied by the same factor of proportionality to arrive at the standard errors in our new model.

Questions: Is the simplest Poisson regression option in Eviews assuming equality of the conditional mean and conditional variance, or proportionality (as in GLM assumption)? If the former is true, do I need to manually inflate my standard errors in case of over-dispersion, or does Eviews do it automatically? Lastly, do these conclusions change if I use the Huber-White rubust standard errors?

Many thanks.
T

[EViews Glenn]

Two questions:

I have chosen the simplest Poisson method in eviews, with Huber-White robust standard errors.

While checking for the significance of a binary regressor (gender dummy), the z-statistic and the associated p-value suggest that the coefficient is insignificant. On the other hand, a Likelihood Redundant Variable Test with a single restriction, gender (i.e., under the null, the coefficient on gender is zero), suggests that the coefficient is now significant.

This one confused me a bit until I actually read the posting carefully. The key is in the first line of the posting. I read the simplest possible Poisson model part, but skipped over the Huber-White part.

If you used the default Poisson estimation settings, EViews would compute the standard errors using ML. In this case, you would expect to see the same results for your Wald and LR test statistics.

In your case, you are computing your standard errors using the Huber-White robust standard errors. These SEs allow for conditional variances that not only differ from the conditional mean, but also permit departures from the information matrix equality. Thus, a Wald test using these results might differ from those from the LR test statistic which still imposes the mean-variance equality or the QLR test statistic which allows for overdispersion. Hence the potential for different results. I won't comment on which I think is right since there are power considerations to bear in mind, but I will note that the Wald tests are valid under a less restrictive set of assumptions.

The standard Poisson regression assumes that the conditional population mean of the dependent variable is equal to its conditional variance. If, on the other hand, we were to relax this assumption along the lines of Generalised Linear Models (GLM), so that the conditional variance is now proportional to (rather than equal to) the conditional mean [a case of over-dispersion], the standard errors of the "equality" model must now be multiplied by the same factor of proportionality to arrive at the standard errors in our new model.

There is a GLM standard errors option that specifically allows for over-dispersion alone. The Huber/White standard errors are a bit more general than that.

[mornington]

Thanks Glenn! I did browse through the e-views manual, and your advice seems to sit well with what's mentioned there. I think I'm a bit clearer on this now.

I have one more question though. The question is : If I run the regression under the GLM assumption of over-dispersion (GLM robust errors), is the LR test-statistic still calculated assuming equality between the the conditional variance of the regressand and the its conditional expectation? Should I simply stick to the wald test?

Thanks once again

[EViews Glenn]

The LR statistic does maintain the mean-variance assumption.

As you note, the Wald does.

But all is not lost. EViews doesn't adjust the LR statistics for overdispersion automatically (it's on our lengthy list of things to look at). Part of the reason it hasn't been done is that it's not difficult to compute the QLR statistic as in Wooldridge Our manual does a mediocre job of discussing this issue, but Wooldridge's Handbook chapter and even his undergraduate text discuss this. Just take the LR statistic, and scale it by the estimate of sigma^2 obtained under the unrestricted model, and have at it...

[mornington]

As you note, the Wald does.

I didn't understand that


From what I have understood, I should scale down the reported LR statistic from a Redundant Variable Test by dividing by the square of the standard error of the regression that is reported, and compare that value manually with the the critical values in the Chi square table.

Is that right ?

Is the reported S.E. of the regression correct ? or would i have to manually estimate that ?



Thanks again !

[EViews Glenn]

The S.E. of the regression is simply the standard estimator of the root of the variance of the residuals. I'm not certain what you mean by correct here.

For the scaling of the test statistic, you do need an estimate of the variance factor that relates the variance of the data to the restricted ML variance (it's the overdispersion or underdispersion factor). This value is provided in the top of your GLM output. When you select GLM standard errors, your coefficient variance matrix is multiplied by this number. Thus, Wald statistics that you compute from the equation and the z-stats in your output will be GLM variance robust. As you note the LR tests that we do automatically are not robust , but can be made so by dividing, not as you suggest by the S.E. of the regression, but rather by the estimate of the variance matrix factor. So your prescription for proceeding is correct with the exception of using the factor instead of the S.E.