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Quite so. If you have only just enough instruments to identify the model, there are none available to test overidentifying restrictions, as there is no overid. The J statistic is by definition zero for all exactly identified models. On the other hand, if you have found some variables which you believe are appropriate instruments, transformations of them (powers, lags, interactions) should also be valid instruments, so it is not clear why you are stuck with exact ID. 


---------------------------------------------------------------------------From   Michael Hanson <> 
Subject   Re: st: ivreg2: No validity tests if just-identified? 
Date   Wed, 15 Apr 2009 21:59:21 -0400 

On Apr 15, 2009, at 7:46 PM, Jennifer Beardsley wrote:

Dear Stata-list,

I worked quite carefully through the various options in ivreg2 and ivregress for testing (a) instrument validity (orthogonality etc) and (b) instrument strength (correlatedness with endogenous regressors). After doing so, I seem to be arriving at the conclusion that there is no way to test (a) if the model is just- identified, i.e. if I have the same number of excluded instruments as I have endogenous regressors). For example, the Sargan overid statistic, the C-statistic, the LR IV redundancy test statistic, etc. all don't get produced unless the model is overidentified. Is that true?! If yes, I would need to rely on persuasion using economic intuition to make my case that the instruments are valid, and there are no statistical tools to use? 

In contrast, there do seem to be ready statistics I can draw on to examine instrument weakness/strength. 

Thanks for any response,


The short answer to your question is "yes", as most the tests you reference are tests of over-identifying restrictions. Think of it this way: you cannot test assumptions required to just identify a model, as -- by definition -- these assumptions are necessary in order to merely proceed with estimation. But, if you have more instruments than you need to just identify a model, you *can* test certain properties given the "extra" (I'm speaking loosely here) instruments. That's what many of the tests you cite above do, in so many words. 

I'm not sure what you are looking for, but it sounds like you have some concern about the first-stage fit of your instruments. A useful rule of thumb is to look at the F-statistic for the first-stage regression in 2SLS: if it is larger than about 10, then you are unlikely to suffer from problems that arise with weak instruments. See the discussion in ch. 12 of Stock & Watson (2007); -estat firststage- after -ivregress- (in Stata 10) gives more precise critical values from Stock & Yogo (2005). I believe recent versions of -ivreg2- also report these statistics if you use Stata 9. 

To your concern about "economic persuasion": one would use instrumental variables if OLS is expected to yield biased results -- that is, if E[X'u] ~= 0. Of course, this moment condition is not testable: you must use economic theory and/or intuition ("persuasion") to argue for the need for an IV estimator in the first place. Similarly, an IV estimator requires the moment conditions E [Z'u] = 0 and E[Z'X] ~= 0. The second of these is "verifiable" through weak instrument tests as discussed above. The first cannot be tested, but the existence of "extra" moments beyond those needed to just identify the model allows one to test for over-identifying restrictions. (Again, speaking loosely.) 

Hope that is helpful. In light of your questions, you might find it worthwhile to review textbook treatments of IV -- such as ch. 15 in Wooldridge (2006) or ch. 12 in Stock & Watson (2007). Both do a good job, in my opinion, of developing intuition for the IV estimator. For further discussion of the problems associated with weak instruments, you might start with Stock, Wright & Yogo (2002). There are many other folks on this list with expertise in this area who may chime in (such as Kit, who I see has already gave a pithy reply while I was composing this more prolix one). 



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