Previous posts have looked at the GMM estimation of the AR(1) process
y(i,t)=a(i)*y(i,t-1) + zero mean error
where i is a group indicator and t is time, and the estimation assumes that a(i) is a constant across groups. I showed that GMM estimators tend to estimate a value for a near the top of the a(i) range.
Testing for equality of subgroup a parameters using the Chow test is therefore misleading in that what is compared is two parameters near the top of each subgroup range. The Sargan and Hausman test may identify the misspecification, as they examine residual patterns, but as all groups are pooled in their testing (from memory) they may not be very powerful as some groups will exhibit positive serial autocorrelation and others will exhibit negative serial autocorrelation in their residuals, as previously shown on this site.
Here is a test which should work. Perform the GMM estimation, then estimate an AR(1) on the residuals for each group by OLS. Under the null of a(i) constancy, the residual AR(1) parameters should be asymptotically zero mean and normally distributed with an unbiased estimated correlation. Normalise to N(0,1), then sum their squares to get a chi squared distribution on i degrees of freedom. Reject a(i) constancy if the test statistic is too large at a set level of chi squared significance.