Dynamic Panel GMM with Near Unity

Limit theory is developed for the dynamic panel GMM estimator in the presence of an autoregressive root near unity. In the unit root case, Anderson-Hsiao lagged variable instruments satisfy orthogonality conditions but are well-known to be irrelevant. For a fixed time series sample size (T) GMM is inconsistent and approaches a shifted Cauchy-distributed random variate as the cross section sample size n approaches infinity. But when T approaches infinity, either for fixed n or as n approaches infinity, GMM is square root{T} consistent and its limit distribution is a ratio of random variables that converges to twice a standard Cauchy as n approaches infinity. In this case, the usual instruments are uncorrelated with the regressor but irrelevance does not prevent consistent estimation. The same Cauchy limit theory holds sequentially and jointly as (n,T) approaches infinity with no restriction on the divergence rates of n and T. When the common autoregressive root rho = 1 + c/square root{T} the panel comprises a collection of mildly integrated time series. In this case, the GMM estimator is square root{n}n consistent for fixed T and square root{(nT)} consistent with limit distribution N(0,4) when n, T approaches infinity sequentially or jointly. These results are robust for common roots of the form rho = 1 + c/T^{gamma} for all gamma in (0,1) and joint convergence holds. Limit normality holds but the variance changes when gamma = 1. When gamma > 1 joint convergence fails and sequential limits differ with different rates of convergence. These findings reveal the fragility of conventional Gaussian GMM asymptotics to persistence in dynamic panel regressions.