# Moderate deviations in a class of stable but nearly unstable processes

## TL;DR

This paper develops moderate deviation principles for nearly unstable autoregressive processes, providing insights into the behavior of empirical covariance and OLS estimators as the process approaches instability.

## Contribution

It introduces a novel moderate deviation framework for nearly unstable AR processes, including cases with singular asymptotic variance, using truncation and deviation techniques.

## Key findings

- Moderate deviation principle for empirical covariance depending on spectral radius
- Moderate deviation for OLS estimator when asymptotic variance is invertible
- Deviation results for penalized estimators in singular variance cases

## 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 $A_{n}$ with spectral radius $\rho(A_{n}) < 1$ satisfying $\rho(A_{n}) \rightarrow 1$. In that framework, we establish a moderate deviation principle for the empirical covariance only relying on the elements of $A_{n}$ through $1-\rho(A_{n})$ and, as a by-product, we establish a moderate deviation principle for the OLS estimator when $\Gamma$, the renormalized asymptotic variance of the process, is invertible. Finally, when $\Gamma$ 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 $m_{n}$--dependent sequences, with an unbounded rate $(m_{n})$.