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 → ∞. But when T → ∞, either for fixed n or as n → ∞, GMM is √T consistent and its limit distribution is a ratio of random variables that converges to twice a standard Cauchy as n → ∞. 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) → ∞ with no restriction on the divergence rates of n and T. When the common autoregressive root ρ = 1 c/√T the panel comprises a collection of mildly integrated time series. In this case, the GMM estimator is √n consistent for fixed T and √(nT) consistent with limit distribution N(0,4) when n,T → ∞ sequentially or jointly. These results are robust for common roots of the form ρ = 1 c/T^{γ} for all γ ∈ (0,1) and joint convergence holds. Limit normality holds but the variance changes when γ = 1. When γ > 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
