Why GMM selects high parameters in misspecifications

My earlier posts have described how within group OLS, difference GMM, and system GMM estimators select parameters from the high end of the range b(i) when they are estimating a model like

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

(y(i,t) is data for group i at time t, a(i) is a constant for each group, and b is a constant within and across groups)

when the true generating process is

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

(b(i) is a constant within groups).

It's a feature of GMM, particularly where the weighting matrix is the identity matrix. Within groups OLS is an instance of GMM, so is covered by the same theory.

If we look at the main moment conditions for any of the estimators, they are

E(y*error)=0 or E((change in y)*error)=0 or E(y*(change in error))=0.

There's a slight difference for system and difference GMM when dealing with homoscedastic errors.

Take the first moment condition, applicable to OLS. GMM sets equal to zero sums of terms of the form

y(i,t-1) * (y(i,t) - b*y(i,t-1) - c(i)).

We can replace y(i,t) with its true generating process to get

y(i,t-1) * (b(i)*y(i,t-1) - b*y(i,t-1) + error).

When the b(i) are positive, y(i, t) will in the long run almost certainly be larger when b(i) is larger. It follows that b will tend to the highest of the b(i), since this parameter has more weight in the equation. The expected tendency is monotonic upward as sample sizes increase.

The original GMM recommended optimal weighting matrix may help to correct for the tendency, but is often not used in the growth literature, and the misspecification means that no parameter is more accurate than another although the impression it gives may be more useful. I think that Wald test estimation between two subsets of the data may also be distorted - high standard errors combined with selection of high parameter values from the ranges may give an artificial impression that two different regions of the world have the same generating processes.