Asymptotic Inference for AR(1) Penal Data

TL;DR

This paper develops a comprehensive asymptotic theory for the AR(1) panel data model across various process regimes, showing how regularization ensures normal convergence of estimators with different rates depending on the process's stationarity.

Contribution

It introduces a unified asymptotic framework for AR(1) panel models covering stationary, non-stationary, and explosive processes, highlighting the impact of regularization on estimator distribution.

Findings

Estimator converges to normal distribution with rate at least O(N^{-1/3})

Variance declines at different rates depending on process regime

Explosive case estimator has standard normal limit in panel data, unlike in univariate case

Abstract

A general asymptotic theory is given for the panel data AR(1) model with time series independent in different cross sections. The theory covers the cases of stationary process, nearly non-stationary process, unit root process, mildly integrated, mildly explosive and explosive processes. It is assumed that the cross-sectional dimension and time-series dimension are respectively $N$ and $T$. The results in this paper illustrate that whichever the process is, with an appropriate regularization, the least squares estimator of the autoregressive coefficient converges to a normal distribution with rate at least $O(N^{-1/3})$. Since the variance is the key to characterize the normal distribution, it is important to discuss the variance of the least squares estimator. We will show that when the autoregressive coefficient $\rho$ satisfies $|\rho|<1$, the variance declines at the rate $O((NT)^{-1/2})$, while the rate changes to $O(N^{-1/2}T^{-1})$ when $\rho=1$ and $O(N^{-1/2}\rho^{-T+2})$ when $|\rho|>1$. $\rho=1$ is the critical point where the convergence rate changes radically. The transition process is studied by assuming $\rho$ depending on $T$ and going to $1$. An interesting phenomenon discovered in this paper is that, in the explosive case, the least squares estimator of the autoregressive coefficient has a standard normal limiting distribution in panel data case while it may not has a limiting distribution in univariate time series case.