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1. INTRODUCTION
For the stable first-order autoregressive (AR(1)) model, extensive literature on the finite-sample properties of the ordinary least squares (OLS) estimator of the autoregressive coefficient has developed; see Hurwicz (1950), White (1957, 1961), Shenton and Johnson (1965), Copas (1966), Phillips (1977), Sawa (1978), Grubb and Symons (1987), and Kiviet and Phillips (1993), to list just a few. In these studies, the error term was assumed to be normally distributed to enable the authors to derive analytical results. Peters (1989) considered the nonnormal case and derived the exact moments of the OLS estimator under an Edgeworth-Gram-Charlier-type nonnormal specification for the error term. Abadir and Lucas (2004) analyzed the mean squared error (MSE) in the nonnormal case for Ornstein-Uhlenbeck processes (hence nearly integrated AR models). Recently, Bao and Ullah (2006a) found that the second-order bias of the OLS estimator is robust to nonnormality. Under normality, they were able to derive the analytical approximate MSE result explicitly in terms of model parameters.
The purpose of this note is twofold. First, I will investigate whether the robust bias result still holds if one moves to a higher order, namely, O (T-2 ), where T is the sample size. Second, I will derive the analytical MSE result under a general error distribution. Consequently, one can study explicitly the effects of nonnormality on the sampling properties of the OLS estimator. In the next section, I consider the AR(1) model without an intercept and the model with an intercept, named the pure and intercept models, respectively. I show that for both models, nonnormality affects the approximate bias and MSE, up to O (T-2 ), of the OLS estimator through the skewness and kurtosis coefficients of the nonnormal density. Setting the nonnormal parameters equal to zero yields the normal results developed in the literature. The nonnormality coefficients do affect the bias of the OLS estimator through order O (T-2 ) terms. Section 3 concludes, and technical details are collected in the Appendix.
2. MAIN RESULTS
Consider the pure model
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and the intercept model
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where, throughout, the nonnormal independent and identically distributed (i.i.d.) error term [varepsilon]t in (1) and (2) is assumed to have the...