The asymptotic distribution of the autoregressive parameter in a misspecified AR(1) model under GMM estimation

I have written many times on this blog about estimation using the generalised method of moments of the equation

y(i,t)=a(i)*y(i,t-1)+error

where the i indicates the ith group and t indicates time. This equation is important in growth theory estimation. The misspecification is that a(i) is assumed to be a constant across groups, but really varies. Under GMM System, GMM Difference, and OLS estimators, the estimated a coefficient tends to be near the top of the real range of the a(i)s.

The distribution of the estimated coefficient around the limiting value for a is normal with mean a and a standard error taking the same form as in a correctly specified model. The proof follows similarly to the original proof for the correctly specified model. I am reasonably sure this works - I've followed things through - but haven't worked everything out in detail, so there might be some small issues left.