Mildly Explosive Autoregression with Strong Mixing Errors

Abstract

In this paper, we consider the mildly explosive autoregression yt=ρnyt−1+ut, 1≤t≤n, where ρn=1+c/nν, c>0, ν∈(0,1), and u1,…,un are arithmetically α-mixing errors. Under some weak conditions, such as Eu1=0, E|u1|4+δ<∞ for some δ>0 and mixing coefficients α(n)=O(n−(2+8/δ)), the Cauchy limiting distribution is established for the least squares (LS) estimator ρ^n of ρn, which extends the cases of independent errors and geometrically α-mixing errors. Some simulations for ρn, such as the empirical probability of the confidence interval and the empirical density, are presented to illustrate the Cauchy limiting distribution, which have good finite sample performances. In addition, we use the Cauchy limiting distribution of the LS estimator ρ^n to illustrate real data from the NASDAQ composite index from April 2011 to April 2021.