Re: st: How to save a hessian matrix for post-estimation analysis?

[kirin_guess@yahoo.com.tw]

Dear Gordon, 
 
Thanks for your reply. What I’m doing is compiling a command for score tests. According to the David A. FREEDMAN’s paper (2007: 291), it is possible that the observed negative hessian of the restricted model can generate negative variance, which makes the test results inconsistent. 
 
When the negative hessian is not positive definite, Stata will compute the variance-covariance matrix by the generalised inversion, so there won’t be negative variances. However, that makes me wonder whether the generalised inverse matrix is valid for the score test. Of course, we cannot run either the score test or the wald test, if the variance of the tested variable is set to be zero because of the generalised inversion. The problem is the validity of the testing results for the other variables with non-zero variances. 
 
Your reply is useful to me, and I will make my command to show a warning message, when the variance-covariance matrix is degenerate. Moreover, if it’s convenient, could you please suggest me some key papers that discuss the robustness of the generalised inverse matrix when it is used for obtaining the variance-covariance matrix? 

Thank you very much indeed.
 
 
Best Regards,
  
Chi-lin Tsai
 
 
 
 

============================
From: Gordon Hughes <G.A.Hughes@ed.ac.uk>
To: statalist@hsphsun2.harvard.edu 
Sent: Thursday, 23 August 2012, 18:19
Subject: Re: st: How to save a hessian matrix for post-estimation analysis?

Sorry about the delay in responding.  Unfortunately, the real answer to your question is that it all depends on what you are trying to do.  I think that the best advice is to try and remove the source of the degeneracy - i.e. remove collinear variables or whatever.  If there is some reason why this can't be done (I can't think of an obvious example), then the generalised inverse is pretty robust so that the non-missing diagonal elements might be used for tests on the relevant coefficients.

I would give one warning.  Degeneracy in the variance-covariance matrix is commonly a sign that the maximisation has failed - i.e. the likelihood function does not have a single local maximum and the gradient procedure has headed off to a region that is infeasible or degenerate for some reason.  Any inferences based on estimates of this kind are worthless.  This brings us back full circle to the question of why you have degenerate variance-covariance matrix.

Gordon Hughes
g.a.hughes@ed.ac.uk

 
 
 
 
============================

Dear Gordon,

Thanks for the answer. It seems that there is no easy way to use the official command, e.g. "logit", "mprobit", etc., and save the original hessian.

By the way, I wonder that, if the negative hessian is singular or not positive definite, is there any drawback to using the generalised inverse hessian? I know that some standard errors in the output will be labelled with a dot, which means we can't do the statistical tests for those predictors. How about the other standard errors appearing as usual? We can use them to do the test, but will them cause invalid results?

Thanks for the help.

Best Regards,

Chi-lin Tsai


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