Asymptotic inference for AR(1) panel data
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Asymptotic inference for AR(1) panel data SHEN Jian-fei
PANG Tian-xiao
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, local to unity 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 in distribution 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 ρ satisfies |ρ| < 1, the variance declines at the rate O((N T )−1 ), while the rate changes to O(N −1 T −2 ) when ρ = 1 and O(N −1 ρ−2T +4 ) when |ρ| > 1. ρ = 1 is the critical point where the convergence rate changes radically. The transition process is studied by assuming ρ 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 the panel data case while it may not has a limiting distribution in the univariate time series case.
§1
Introduction
Dynamic models are useful in modeling time series data and have been well studied in the past few decades. One of the dynamic models is the AR(1) model which is given by yt = ρyt−1 + εt , t = 1, 2, ..., T.
(1.1)
We assume that {εt , t ≥ 1} are independent and identically distributed (i.i.d.) random variables with E[ε1 ] = 0 and E[ε21 ] = 1. Although the model (1.1) is simple, it is very useful and important in time series and econometrics literature since the model can be used to model some kinds of stationary or nonstationary time series data. The parameter ρ is the main concern in the model (1.1) since Received: 2016-08-28. Revised: 2019-01-06. MR Subject Classification: 62E20. Keywords: AR(1) model, Least squares estimator, Limiting distribution, Non-stationray, Panel data. Digital Object Identifier(DOI): https://doi.org/10.1007/s11766-020-3491-x. Supported by the National Natural Science Foundation of China (11871425), Zhejiang Provincial Natural Science Foundation of China (LY19A010022) and the Department of Education of Zhejiang Province (N20140202).
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Appl. Math. J. Chinese Univ.
Vol. 35, No. 3
whether the model is stationary is determined by the value of ρ. It is well-known that the necessary and sufficient condition for the stationarity of yt in (1.1) is |ρ| < 1 when y0 is an appropriate random variable. The least squares estimator (LSE) of ρ is given by ∑T yt yt−1 ρˆ = ∑t=1 . (1.2) T 2 t=1 yt−1 For the stationary AR(1) model, Mann and Wald (1943) proved that, if y0 = OP (1), then √ T d √ (ˆ ρ − ρ) −→ N (0
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