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 → ∞. 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.
Phillips, Peter C.B., "Dynamic Panel GMM with Near Unity" (2014). Cowles Foundation Discussion Papers. 2376.