Expansions of GMM statistics that indicate their properties under weak and/or many instruments and the bootstrap

We show that the sensitivity of the limit distribution of commonly used GMM statistics to weak and many instruments results from superfluous elements in the higher order expansion of these statistics. When the instruments are strong and their number is small, these elements are of higher order and result in higher order biases. When instruments are weak and/or their number is large, they are, however, of zero-th order and influence the limit distributions of GMM statistics. Edgeworth approximations do not remove these superfluous elements. Expansions of GMM statistics that are robust to weak or many instruments do not possess these superfluous elements. Their robustness is therefore the result of improved higher order properties. This renders an additional reason for usage of these statistics. An Edgeworth expansion of the robust statistics can be constructed so the approximation of their finite sample distribution can be further improved upon by use of the bootstrap. We illustrate the finite sample performance of GMM statistics by constructing power curves for tests on the autocorrelation parameter in a panel autoregressive model using both asymptotic and bootstrap critical values.