[Paper Review] Moderate deviations principle for empirical covariance from a unit root

Yu Miao, Yanling Wang+1 more|arXiv (Cornell University)|Jul 17, 2012

Monetary Policy and Economic Impact15 references3 citations

TL;DR

This paper establishes a moderate deviations principle (MDP) for the empirical covariance in a linear autoregressive model with a unit root, where the autoregressive parameter θₙ → 1. Using martingale approximations and exponential moment bounds under moment conditions on the noise, the authors derive MDPs for both the empirical covariance and the least squares and Yule-Walker estimators of θₙ, extending prior results beyond the Gaussian or log-Sobolev assumptions.

ABSTRACT

In the present paper, we consider the linear autoregressive model in $ r$, $$ X_{k,n}=θ_n X_{k,n-1}+ξ_k, k=0,1,...,n, n\ge 1$$ where $θ_n\in [0,1)$ is unknown, $(ξ_k)_{k\in\zz}$ is a sequence of centered i.i.d. r.v. valued in $ r$ representing the noise. When $θ_n o 1$, the moderate deviations principle for empirical covariance is discussed and as statistical applications we provide the moderate deviation estimates of the least square and the Yule-Walker estimators of the parameter $θ_n$.

Motivation & Objective

• To establish a moderate deviations principle (MDP) for the empirical covariance in a linear autoregressive model when the autoregressive parameter θₙ approaches 1.
• To extend existing large and moderate deviation results beyond the Gaussian or log-Sobolev assumptions on the noise distribution.
• To provide moderate deviation estimates for the least squares and Yule-Walker estimators of θₙ under the unit root asymptotic regime.
• To analyze the asymptotic behavior of the empirical covariance and parameter estimators in the moderate deviation regime, where deviations scale as bₙ / √n with bₙ → ∞ and bₙ / n → 0.

Proposed method

• Derives a moderate deviations principle for the empirical covariance Cₗ,n* using a martingale approximation technique to handle the dependent structure of the time series.
• Applies the Gärtner-Ellis theorem via exponential moment bounds on truncated increments of the empirical covariance process.
• Imposes moment conditions on the noise ξₖ, specifically E|ξₖ|²⁺ᵞ < ∞ for some γ > 0, to control tail behavior and ensure convergence of exponential moments.
• Uses truncation and coupling arguments to control the difference between truncated and original processes, showing their contribution vanishes in the moderate deviation scaling.
• Establishes convergence of the normalized cumulant generating function to a quadratic form, confirming the MDP for the empirical covariance.
• Applies the MDP for the empirical covariance to derive moderate deviation estimates for the least squares and Yule-Walker estimators of θₙ via continuous mapping and delta-method type arguments.

Experimental results

Research questions

• RQ1 What is the moderate deviations behavior of the empirical covariance in a unit root autoregressive model when θₙ → 1?
• RQ2 How do the least squares and Yule-Walker estimators of θₙ behave under moderate deviations when the true parameter approaches unity?
• RQ3 Can the moderate deviations principle be established for the empirical covariance without assuming Gaussianity or log-Sobolev conditions on the noise?
• RQ4 What moment conditions on the noise ξₖ are sufficient to ensure the validity of the moderate deviations principle for the empirical covariance and parameter estimators?
• RQ5 How does the asymptotic behavior of the empirical covariance and estimators change in the moderate deviation regime compared to the central limit theorem regime?

Key findings

• The empirical covariance Cₗ,n* satisfies a moderate deviations principle with speed bₙ² and rate function I(x) = x² / (2σ²), where σ² is the asymptotic variance of the covariance estimator.
• The least squares estimator ̂θₙ satisfies a moderate deviations principle under the same scaling, with the rate function derived from the MDP of the empirical covariance.
• The Yule-Walker estimator ̃θₙ also satisfies a moderate deviations principle, with the same speed bₙ², under the same moment conditions.
• The proof relies on truncation and moment bounds: under E|ξₖ|²⁺ᵞ < ∞ for some γ > 0, the exponential moments of the truncated increments converge to zero at the required rate.
• The contribution of the truncated tails vanishes in the moderate deviation scaling, as shown by the convergence of the tail probability terms to zero under the condition limₙ→∞ (bₙ (mp)¹⁺ᵞ) / √n = 0.
• The result generalizes prior work by removing the need for log-Sobolev or Gaussian integrability assumptions, extending the MDP to a broader class of noise distributions.