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

We construct higher order expressions for Wald and Lagrange multiplier (LM) GMM statistics that are based on 2step and continuous updating estimators (CUE). We show that the sensitivity of the limit distribution to weak and many instruments results from superfluous elements in the higher order expansion. 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 limiting distributions. Edgeworth approximations do not remove the superfluous elements. The expansion of the LM-CUE statistic, which is Kleibergen's (2003) K-statistic, does not contain the superfluous higher order elements so it is robust to weak or many instruments. An Edgeworth approximation of its finite sample distribution shows that the bootstrap reduces the size distortion. We compute power curves for tests on the autocorrelation parameter in a panel autoregressive model to illustrate the consequences of the higher order.terms and the improvement that results from applying the bootstrap