Asymptotically unbiased estimation of auto-covariances and auto-correlations with long panel data∗

Many economic variables are correlated over time. It is important to determine whether this observed correlation comes from time invariant unobserved heterogeneity among individuals or from temporal persistency of a shock. This paper examines how to estimate the auto-covariances and auto-correlations of individual dynamics separately from unobserved heterogeneity. When both cross-sectional and time-series sample sizes tend to infinity, we show that the within-group auto-covariances are consistent for the auto-covariances of individual dynamics, but that they are severely biased when the length of the time series is short. The biases have the leading term that converges to the long-run variance of the individual dynamics. This paper develops methods to estimate the long-run variance in panel data settings, and methods to alleviate the biases of the within-group auto-covariances based on the proposed long-run variance estimators. Monte Carlo simulations reveal that the procedures developed in this paper effectively reduce the biases of the estimators in small samples.