Asymptotic Normality of the Least-Squares Estimates for Higher Order Autoregressive Integrated Processes with Some Applications

Abstract

Using the asymptotic normality of the least-squares estimates for the autoregressive (AR) process with real, positive unit roots and at least one stable root, we consider the asymptotic distributions of the Wald and t ratio tests on AR coefficients. In addition, we propose a method of constructing confidence intervals for the sum of AR coefficients possibly in the presence of a unit root. Using simulation methods, we compare the finite-sample cumulative distributions of the t ratios for individual autoregressive coefficients with those of standard normal distributions, and investigate the finite-sample performance of our confidence intervals and t ratios. Our simulation results show that the t ratios for nonstationary processes converge to a standard normal distribution more slowly than those for stationary processes. Further, the confidence intervals are shown to work reasonably well in moderately large samples, but they display unsatisfactory performance at small sample sizes.