Moderate deviations in a class of stable but nearly unstable processes

Abstract

We consider a stable but nearly unstable autoregressive process of any order. The bridge between stability and instability is expressed by a time-varying companion matrix An with spectral radius ρ(An)<1 satisfying ρ(An)→1. In that framework, we establish a moderate deviation principle for the empirical covariance only relying on the elements of An through 1−ρ(An) and, as a by-product, we establish a moderate deviation principle for the OLS estimator when Γ, the renormalized asymptotic variance of the process, is invertible. Finally, when Γ is singular, we also provide a compromise in the form of a moderate deviation principle for a penalized version of the estimator. Our proofs essentially rely on truncations and deviations of mn–dependent sequences, with an unbounded rate (mn).