z-statistics (instead of t-statistics) reported in generalized method of moments (GMM) estimation?

[Francois Durant]

Hi,

I was wondering why Stata reports Z-statistics instead of t-statistics when using the command "gmm"? Would that make any sense to report the t-statistics? Thanks!
https://www.stata.com/manuals13/rgmm.pdf

[Joro Kolev]

Stata's behaviour is a bit random in this respect, because for some estimators she reports t, and F statistics , and for some z and Wald statistics.

I believe the (not very convincing to me) rationale is that some estimators have "only large sample justification", while other estimators might have a widely accepted small sample theory behind.

In some estimators you can control this behaviour as -small- option is available, e.g., -ivregress- and -sureg-, in some you cannot, e.g., I think in -gmm- the -small-option is not available.

[Francois Durant]

Thanks a lot for your response Joro. What difference does it make in the case of GMM estimation when it comes to interpret the z-stat relative the t-stat? I don't know have information about the distribution of the population (variance).

[Joro Kolev]

Thanks a lot for your response Joro. What difference does it make in the case of GMM estimation when it comes to interpret the z-stat relative the t-stat? I don't know have information about the distribution of the population (variance).

No difference in terms of interpretation. P-values have the same meaning. The only difference is that when Stata calculates z and Wald statistics, if you calculate the p-value by hand, you need to refer the statistics to the normal and Chi-square distribution respectively, instead of to the t and F distributions. However Stata does this calculation for you, so there is no difference in interpretation.

Also for more than 30 degrees of freedom the t and normal distribution are pretty much indistinguishable. So if you have more than 30 df you can simply disregard the fact that Stata calls the statistics z-statistics and think of them in your mind as t-statistics.

[Sebastian Kripfganz]

A reason why the gmm command reports p-values based on the standard normal distribution rather than the t-distribution might be that it is less easy to generally argue that the influence function is normally distributed in small samples, in particular when there are nonlinear moment functions. This normality assumption is a requirement to obtain a t-distribution for the t-statistic. When you estimate a simple linear regression model with regress instead, it might be easier to justify that the error term is normally distributed.