# Hessian of profile likelihood used for standard error estimation

This question is motivated by this one. I looked up two sources and this is what I found.

A. van der Vaart, Assymptotic Statistics:

It is rarely possible to compute a profile likelihood explicitly, but its numerical evaluation is often feasible. Then the profile likelihood may serve to reduce the dimension of the likelihood function. Profile likelihood functions are often used in the same way as (ordinary) likelihood functions of parametric models. Apart from taking their points of maximum as estimators $\hat\theta$, the second derivative at $\hat\theta$ is used as an estimate of minus the inverse of the asymptotic covariance matrix of e. Recent research appears to validate this practice.

J. Wooldridge, Econometric Analysis of Cross Section and Panel Data (the same in both editions):

As a device for studying asymptotic properties, the concentrated objective function is of limited value because $g(W,\beta)$ generally depends on all of $W$, in which case the objective function cannot be written as the sum of independent, identically distributed summands. One setting where equation (12.89) is a sum of i.i.d. functions occurs when we concentrate out individual-specific effects from certain nonlinear panel data models. In addition, the concentrated objective function can be useful for establishing the equivalence of seemingly different estimation approaches.

Wooldridge discusses the problem in wider context of M-estimators, so it applies to maximum likelihood estimators also.

So we get two different answers for the same question. The devil in my opinion is in the details. For some models we can use hessian of profile likelihood safely for some models not. Are there any general results which give conditions when can we do that (or cannot)?

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These passages do not seem to address the same question at all: the first concerns numerical calculation for a given dataset whereas the second concerns "studying asymptotic properties." Use of the Hessian is typically a purely mathematical consideration with typically straightforward answers: see our related discussion. – whuber Aug 24 '11 at 13:43
van der Vaart says that Hessian is used for calculation of asymptotic covariance matrix. Since Wooldridge talks that concentrated objective function cannot be used for study of asymptotic properties, this implies that its hessian (numerical) cannot be used for estimating standard errors. I did not forget our discussion, so I take this passage with the grain of salt. However neither van der Vaart nor Wooldridge given any references. Before doing the extensive research I just wanted to check maybe this is something well known. – mpiktas Aug 24 '11 at 14:01
Excellent point: somehow I overlooked the "asymptotic" in the van der Vaart quotation. There still may be no contradiction, however: Wooldridge merely says that the obvious simple justification (iid summands) is not available for demonstrating that van der Vaart's approach works; Wooldridge does not say it doesn't work ;-). – whuber Aug 24 '11 at 14:04
@whuber, yes but he doesn't say that it work either :) I am aware that there might be no contradiction, I only want to know whether there are some definite results. – mpiktas Aug 24 '11 at 14:08
See On Profile Likelihood (S. A. Murphy and A. W. van der Vaart), jstor.org/pss/2669386 – whuber Aug 24 '11 at 15:00

For some models we can use hessian of profile likelihood safely for some models not

Unfortunately, that is true for now and unlikley to change.

The clearest discussion that I am aware of is The rules of conditional inference: Is there a universal definition of nonformation? B Jørgensen - Statistical Methods & Applications, 1994.

And for some of the issues specific to adressing failures of profile likelhood Stafford, J. E. (1996). A robust adjustment of the profile likelihood, Annals of Statistics, 24, 336-52.

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A fast answer: This is discussed in chapter three of O E Barndorff-Nielsen & D R Cox: Inference and asymptotics, Chapman & Hall, page 90, equation 3.31, which they ascribe to Patefield. They conclude that for a scalar parameter this is valid (they do not analyse other cases).

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