Limit theory for random coefficient autoregressive process under possibly infinite variance error sequence

An asymptotic theory was given by Zhang and Yang (2010) for first-order random coefficient autoregressive time series yt = (ρn + φn)yt − 1 + ut, t = 1, …, n, with {ut} is a sequence of independent and identically distributed random variables with mean 0 and a finite second moment, ρn is a sequence of real numbers, and φn is a sequence of random variables. Conditional least squares estimator ρ^n$\hat{\rho }_{n}$ was shown to be asymptotically normality distributed. This model extended the moderate deviations from a unit root model proposed by Phillips and Magdalinos (2007), which just considered the case that ρn = 1 + c/kn and φn ≡ 0. In this paper, we show that the asymptotic theory in Zhang and Yang (2010) still holds when the truncated second moment of the errors l(x) = E[u211{|u1| ⩽ x}] is slowly varying function at ∞. Moreover, we propose a pivot for ρ^n$\hat{\rho }_{n}$, the limit distribution of which is proved to be standard normal.