Exact Moments for Autor1egressive and Random walk Models for a Zero or Stationary Initial Value

Abstract

For a first-order autoregressive AR(1) model with zero initial value, xi= axi−1,_, + ei, we provide the bias, mean squared error, skewness, and kurtosis of the maximum likelihood estimator â. Brownian motion approximations by Phillips (1977, Econometrica 45, 463–485; 1978, Biometrika 65, 91–98; 1987, Econometrica 55, 277–301), Phillips and Perron (1988, Biometrika 75, 335–346), and Perron (1991, Econometric Theory 7, 236–252; 1991, Econometrica 59, 211–236), among others, yield an elegant unified theory but do not yield convenient formulas for calibration of skewness and kurtosis. In addition to the usual stationary case |α| < 1, we include the unstable |α| = 1 case of the random walk model. For the |α| < 1 case, we give new exact results for White's (1961, Biometrika 48, 85–94) model B, where the initial value x0 is a normal random variable N(0,σ2/(l – α2)). Our expressions are exact for small samples computed by relatively reliable Gaussian quadrature methods, rather than approximate ones in powers of n−lor α2.