Approximations to the Distribution of the Sample Autocorrelation Coefficient in the Presence of Unit Roots

Abstract

In this work, we start with the first order autoregressive model [AR (1)]                                             yt = \(\rho\)yt-1 + \(\varepsilon\)t,  t = 1,2, ... , n, where y0 = 0, the variables \(\varepsilon\)1, ... , \(\varepsilon\)n are independent and identically distributed (iid) N(0, \(\sigma\)2), \(|\rho|\) \(\le\) 1 and we will study approximations to the distribution of the autocorrelation coefficient as an estimator of \(\rho\). First, we develop the general case for all \(\rho\) provided that \(|\rho|\) \(\le\) 1 and then we focus our attention on the non stationary case under the presence of unit roots that is when \(|\rho|\) = 1. The approximations are obtained using the expansions developed by Francis Ysidro Edgeworth in 1904 to approximate the distributions of estimators. We introduce a comparison of the results from Monte Carlo simulations and our approximations. Several authors have formed that Edgeworth approximations are insufficiently accurate in practical situations. Indeed, Edgeworth approximations to the density function can produce negative values on the tails of distributions. However, the advantage of these approximations is that they generate analytical results (specifically, formulas) that are simple to handle and interpret, making them ideal for comparison with other methods. Our findings are similar to theirs.