Moderate deviations in a class of stable but nearly unstable processes

2019/05/07 by Frédéric Proïa, Proïa, Frédéric
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Probability and Risk Models #Random Matrices and Applications #Statistical Methods and Inference #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.1905.02618

openalex publication_date 2019/05/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ρ(A_n) < 1 satisfying ρ(A_n) → 1. In that framework, we establish a moderate deviation principle for the empirical covariance only relying on the elements of A_n through 1-ρ(A_n) 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 m_n--dependent sequences, with an unbounded rate (m_n).

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