Biases in common estimation methods

The GMM System estimator and other estimators of the AR(1) model have been tested many times in the academic literature in clinical situations where data is generated from Monte Carlo models which look like y(t)=a + b.y(t-1) + group constant + error. They are also frequently used in testing complex applied models such as y(t) = a + b.y(t-1) + other covariates + group constant + error.

I was interested in testing the performance of the estimators in Monte Carlo situations, but facing a complexity of data similar to that of the applications. So I generated data for a hundred countries, based on a model

income per person (t) = a + b.income(t-1) + c.education(t) + d.investment(t) + e.population growth(t-1 to t) + f.changes in the number of telephones per person (t-1 to t) + error.

The various variables are given common proxies: ln(gdp per capita), years of education, and investment/GDP.

The starting points for the countries were those actually observed over the last forty years, that is, I used as the starting data for the Monte Carlo simulations the actual country data for the first year that sufficient data was available for any estimation to be viable. The synthetic data was generated subsequently by Monte Carlo using the parameters and standard errors from the within groups estimator of the base data and updated on a period by period basis, so that the right hand side variables could themselves be updated from their own Monte Carlo sequences based on the parameters of OLS estimates (for example from investment(t) = g + h.investment(t-1) + i.income(t-1) + error).

The final Monte Carlo data was then put through the estimation methods. For the GMM estimators, both the theoretically optimal instrument set (which Sargan tests on GMM SYS comprehensively rejected) and optimal tests (according to Sargan) were tried. The results below are representative.

As you can see, the GMM DIF has good estimates compared with the generating model, and the within groups OLS is pretty good too. The GMM SYS is less good, though the signs are correct. I am not sure why.