J stat problem, GMM I have recently performed a GMM estimations, my problem is that all the J-stats are 0.0000. It means that the IV are overrefined right or the model is not well specified. I used one-period lags of the independent variables as interments. Should I just report that the GMM model is misspecified or even if the J stat is 0.0000, i can make some inferences about the coefficients.?
 A: If the $J$-stat is 0, then you have a ridiculously good model. If the $p$ value is 0 you have a ridiculously bad model. Say you're estimating moment restrictions 
$$ \operatorname{E}[\,g(X_i,\theta)\,]=0$$
with $X$ being the data and $\theta$ being the true parameter value. You have an estimate $\hat{\theta}$, and an estimate of the inverse variance covariance matrix of moments $\hat W$. The $J$-stat is:
$$ N\,\, \bigg(\frac{1}{N}\sum_{i=1}^N g(X_i,\hat\theta)\bigg)' \hat{W} \bigg(\frac{1}{N}\sum_{t=1}^N g(X_i,\hat\theta)\bigg)\ $$
If your model correctly describes the data (and your estimation routine is working okay) then $\frac{1}{N}\sum_{t=1}^Ng(X_i,\hat\theta)$ will be very close to 0, and this statistic will be extremely low. It will in fact be asymptotically $\chi^2$ distributed (degrees of freedom = #moments - #parameters) under the null of a correctly identified model. For a 5% significance compare this statistic to the 0.95 quantile of the distribution - so you need to be careful what this $p$ value corresponds to. Higher $J$-stats should lead to higher rejections. I'm assuming that you're using a package for this which calculates the $p$-value for you, in which case everything should be correct and low $p$-values will mean a rejected model. I also assume your $p$-value is 0, because I have never seen a $J$-stat that low, although who knows?
What does it mean if your $J$ stat is extremely far away from zero? You have a bad model, but again what does that mean? The $J$-stat is a test of over-identifying restrictions - your model places enough restrictions that you can check to make sure that they are all consistent. If your model were completely correct, it should be possible to satisfy all the moment restrictions simultaneously, even though there are more equations there than unknowns. You could imagine a situation where you have five correct moments, and add one incorrect moment restriction into a model. The $J$-stat will reject the model. That tells you the last restriction is inconsistent with the parameters necessary to satisfy the others. If this is the case you should not use the parameters. With a more complicated model they may still be of interest for telling you something about the model and moments themselves, what kind of restrictions they place on your parameters and why the moments might be impossible to consistently satisfy. It still answers the question "what parameters give us the best this model can do to explain the data?" But you should not really carry these estimates over to other applications.
In an IV model, your moment restrictions are really exclusion restrictions, $E[z_i^\prime(y_i-x_i\beta)]=0$ (plus the model itself, $x_i\beta$). A $J$ test can never tell you whether every one of your individual IV exclusion restrictions are correct or incorrect - there's no really good way to determine that. Instead, if you don't reject the null it tells you that given some of your exclusion restrictions/model are correct, it seems the rest of the restrictions can be satisfied by the same parameter values. In other words, it can't tell you about the validity of your instruments, just about their consistency. You should check to see whether there are particular instruments whose exclusion restrictions are giving you problems, but don't rely on a $J$-test to validate your underlying exogeneity assumptions.
