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 y t = s t ρ y t − 1 + ε t , t = 1 , … , n , where s t is an i.i.d. binary variable with p = P ( s t = 1 ) , independent of ε t i.i.d. with mean zero and finite variance. We say that the process y t is persistent if the autoregressive coefficient ρ ˆ n of y t on y t − 1 is close to one. We take p < 1 < p ρ 2 which implies 1 < ρ and that y t is stationary with infinite variance. Under this assumption we prove the curious result that ρ ˆ n → P ρ − 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 = 1 n y t − 1 2 and ∑ t = 1 n y t y t − 1 and hence the limit of ρ ˆ n .