RE vs FE test after xtoverid

[Paul Gerrans]

Hi
I have a panel dataset and would like to test the validity of a FE vs RE specification. I use xtoverid to estimate a RE specification with an endogenous regressor and I need to rely on multiple instruments. In one estimation I have one endogenous continuous variable and in a second I have two endogenous continuous variables. The general form of each is:
Y = X1 (X2= Z1 Z2 Z3), re vce(cluster id)
Y = X1 (X2 X3 = Z1 Z2 Z3), re vce(cluster id)

While my primary interest is the endogenous regressor I prefer the RE estimation because l’d like to estimate the role of time invariant regressors (notably gender). However I would like to check on the statistical justification of RE or FE. As I allow for non-homoskedastic errors I understand this means I cannot rely on the Hausman test to guide the FE vs RE choice.

There have been postings on this topic such as
http://www.stata.com/statalist/archi.../msg01183.html
and elsewhere Jeff Wooldridge illustrates an augmented regression approach in section 11.2 of “Econometric Analysis of Cross Section and Panel Data” which is summarised at http://www.iza.org/conference_files/..._endog_iza.pdf (from slide 18).

In this latter case the time averaged instrument is included in an augmented REIV estimation.

My questions are:
1. Does this approach extend to the multiple instrument case and/or multiple endogenous regressors case?
2. Or, is there an alternative routine for estimating an appropriate test for RE vs FE after xtoverid given non-homoskedastic errors, and multiple instruments/ multiple endogenous regressors?
Thanks
Paul

[Paul Gerrans]

Just to correct myself, I meant "RE vs FE test after xtivreg" as the topic. I'm aware that xtoverid has a Hausman test for RE vs FE after xtreg.
Thanks
Paul

[Mark Schaffer]

Short answer: (1) yes, it extends to K>1 and/or L>1; (2) xtoverid implements the test described by Jeff Wooldridge, so if you ask xtoverid for a robust version of the test statistic, that's what you'll get.

[Paul Gerrans]

Thanks Mark for the quick response.

So for (1) does that mean, in the case of 3 instruments, include the time average of Z1 Z2 Z3 in an augmented REIV estimation and the t-est on each will inform the RE vs FE choice?

To clarify for (2), that test obtained from using xtoverid after xtivreg will be of the endogenous variables being instrumented not a FE vs RE test. Is that the case?

Thanks Paul

[Mark Schaffer]

Ah, sorry, I misunderstood the question. Let me back up, and ignore what I wrote above.

In the straight FE vs. RE case with no endogenous regressors, the RE estimation is overidentified in the sense that, for every regressor, it uses both the within and between orthogonality conditions. The FE estimation is just-identified, because it uses just the within orthogonality conditions. (I think this is briefly explained in the xtoverid help file.) So we can do an overid test by including in our estimation the time averages (aka "Mundlak fixed effects") of the regressors (which relate to the between orthogonality conditions that the RE estimator uses).

When you've got endogenous regressors, the situation is different. In this case, xtoverid reports an overid test based on the estimator that you're using. For the FE case, it just reports a Sargan-Hansen J stat for the FE (LSDV with IV) estimator that relates to the excluded instruments (Zs in your example).

For the RE case, you've got a choice of RE estimators and what xtoverid reports relates to which estimator you use. The raw instruments Z are each transformed twice, once using the within transformation (so you get a "within Z" IV) and once using the between transformation (so you get a "between Z"). The EC2SLS RE estimator uses these instruments separately, and xtoverid reports the J stat for this overidentified estimation. The G2SLS RE estimator combines these two transformed Zs into a single IV using the RE GLS transformation (so you get a "GLS Z"), and xtoverid reports the J stat for this overidentified estimation.

If you try xtoverid after your estimator using the ec2sls option, you'll see you get more degrees of freedom for the J stat than if you use the default g2sls option. The above is why.

[Paul Gerrans]

Thanks very much Mark, I understand that. With the endogenous regressors, as you point out I can test overidentifying restrictions which in the context of xtivreg and then xtoverid will be providing information about the endogenous regressors, not RE vs FE. In the case of the RE estimators this will vary depending on the specific RE estimator employed.

In the case of using xtivreg then, is there a RE vs FE test that can be estimated in a case with one endogenous regressor and in a separate case with two endogenous regressors?

Thanks
Paul

[Mark Schaffer]

Which RE estimator, and which orthogonality conditions do you want to test? The orthogonality conditions associated with the Zs, the Xs, or both?

[Paul Gerrans]

Hi Mark. My understanding would be that I would ultimately be needing to test both - that is, the RE specification with instruments vs the FE with instruments. I have been using the G2SLS RE estimator.
Thanks
Paul

[Mark Schaffer]

I look at this a bit. Maybe I'm missing something obvious, but I can't see an easy way to test the additional orthogonality conditions associated with both the Xs (included exogenous) and the Zs (excluded exogenous) without doing something like rolling your own.

[Paul Gerrans]

Okay, thanks very much Mark. I'll look to "roll my own"!

[Mark Schaffer]

Paul - an option, if you want to give a try, is to use a nearly-working-version I have of xtivreg2 that handles random effects and lets you specify your orthogonality conditions with some flexiblity. I haven't written a help file, but I have some examples in a do file that might be good enough. If you're interested, contact me off-list and I'll email it to you.