Moment-Condition Contamination and Pseudo-True Values of Difference-GMM and System-GMM under Persistent AR(1) Measurement Error

Abstract This paper studies Difference-GMM and System-GMM estimation of the autoregressive parameter of a dynamic panel when the dependent variable is observed with persistent AR(1) measurement error. Our contribution is a closed-form derivation of how the error contaminates the moment conditions of both estimators. For the lag-$s$ lagged-level instrument of the differenced equation the contamination is $(\phi-1)(\phi-\rho)\phi^{s-2}\sigma_e^2$, and for the lag-$s$ lagged-difference instrument of the level equation it is $(1-\phi)(\phi-\rho)\phi^{s-1}\sigma_e^2$, where $\rho$ is the autoregressive parameter and $\phi$ and $\sigma_e^2$ are the persistence and variance of the error. Both vanish when the error is exactly as persistent as the latent process, because the composite disturbance then collapses to serially uncorrelated noise, and, except at the shallowest lag, when the error is serially independent. We also derive the pseudo-true value of each estimator: for a single instrument it is a weighted average of $\rho$ and $\phi$, with the weight on $\phi$ equal to the noise share of the differenced variance at that instrument's lag. A Monte Carlo study shows that the derived expressions reproduce the simulated biases, that the bias does not vanish as the cross-section or the panel length grows, that the two estimators are biased by almost the same amount under classical error, and that dropping the shallowest instruments removes most of the classical-error bias but not the bias caused by persistent error.