Using Wald tests in checking GMM estimated growth models

Under the misspecified AR(1) model, GMM estimates for two sets of time series with mixed autoregressive parameters in each set will tend to select autoregressive parameters near the top of the ranges for the single estimated autoregressive parameters. Then if a Wald (Chow) test of parameter equality is performed, as is sometimes done in academic papers, the test statistic will become arbitrarily small if the two highest parameters in the two sets coincide. Thus, the Wald test will imply equality of the parameters, but most of the parameters in the first set could be far away from most of the parameters in the second set. I am not sure how Lagrange multiplier tests and likelihood ratio tests would handle things, but given their relations with the Wald test (which is most likely to reject in certain circumstances), I suspect they would have the same problem.

So here's something that works: a series of Wald tests between OLS and robust estimated parameters for the individual time series in the first set with the estimated GMM autoregressive parameter of the second set. If most of the tests reject, then the parameters are not equal. If the original test of parameter equality between the two sets has accepted, then we also have evidence of instability in the parameters.