Selection of non-autoregressive parameters in GMM under misspecification

Here's an observation about the behaviour of the GMM estimates in a misspecified AR(1) model. The estimated model is

y(t) = a.y(t-1) + b.x(t) + error

while the real generating process is

y(t) = a(i).y(t-1) + b(i).x(t) + error.

The a(i) and b(i) vary across groups. I showed on 14th July that if the b(i)=b=0 for all i, then GMM selects an a estimate near the top of the range of the a(i). This post discusses what happens if the bs are not zero.

My procedure for analysis is that one takes the standard formula for an OLS estimate of the a and b parameters, then plugs in the misspecified formulae, and rearranges the formulae a wee bit to make the expressions look like weighted averages. If the b(i)s are all the same, then no matter what the behaviour of the as, the estimate for b is unbiased. If the b(i)s vary, then the estimator tends to select the parameters from the series with the highest xs, and the lowest autoregressive parameters.

I haven't worked this analysis through the GMM System and Difference estimators, but suspect that the behaviour will be the same, as the OLS is a variant of GMM. Cross-country growth theory empirics often use these GMM estimators, so the estimates will tend to be near the top of the range across countries for the lagged autoregressive parameter, and the non-autoregressive parameters will tend to come from low autoregressive parameter countries.

The next couple of sentences are more conjectural. Given that the majority of growth tends to be explained outside the augmented classical determinants of capital and education accumulation, countries whose growth is substantially explained by these determinants may be among the low growth countries with low autoregressive parameters. Thus, the non-autoregressive parameters may be near the top of the range too, and growth is likely to be overstated by the estimates.