Least squares estimation in a simple random coefficient autoregressive model

The question we discuss is whether a simple random coefficient autoregressive model with infinite variance can create the long swings, or persistence, which are observed in many macroeconomic variables. The model is defined by yt=stρyt−1+εt,t=1,…,n, where st is an i.i.d. binary variable with p=P(st=1), independent of εt i.i.d. with mean zero and finite variance. We say that the process yt is persistent if the autoregressive coefficient ρˆn of yt on yt−1 is close to one. We take p<1<pρ2 which implies 1<ρ and that yt is stationary with infinite variance. Under this assumption we prove the curious result that ρˆn→Pρ−1. The proof applies the notion of a tail index of sums of positive random variables with infinite variance to find the order of magnitude of ∑t=1nyt−12 and ∑t=1nytyt−1 and hence the limit of ρˆn.