Exact Moments for Autor1egressive and Random walk Models for a Zero or Stationary Initial Value
研究了一阶自回归模型在零初始值和随机游走情形下最大似然估计量的偏误、均方误差、偏度和峰度，给出了精确的小样本表达式，用于校准统计推断。
For a first-order autoregressive AR(1) model with zero initial value, x i = ax i−1 ,_, + e i , 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 – α 2 )). Our expressions are exact for small samples computed by relatively reliable Gaussian quadrature methods, rather than approximate ones in powers of n −l or α 2 .