[Paper Review] Asymptotic behavior of CLS estimators for unstable INAR(2) models

Mátyás Barczy, Márton Ispány+1 more|arXiv (Cornell University)|Feb 8, 2012

Financial Risk and Volatility Modeling33 references5 citations

TL;DR

This paper establishes the asymptotic distribution of conditional least squares (CLS) estimators for unstable INAR(2) processes, identifying three distinct limit behaviors based on the process's structural properties: decomposable, indecomposable but not positively regular, and positively regular models. The key contribution is the derivation of non-standard limit distributions—distinct from classical AR(1) unit root asymptotics—using weak convergence to squared Bessel processes.

ABSTRACT

In this paper the asymptotic behavior of the conditional least squares estimators of the autoregressive parameters $(α,β)$, of the stability parameter $\varrho := α+ β$, and of the mean $μ$ of the innovation $\vare_k$, $k \in \NN$, for an unstable integer-valued autoregressive process $X_k = α\circ X_{k-1} + β\circ X_{k-2} + \vare_k$, $k \in \NN$, is described. The limit distributions and the scaling factors are different according to the following three cases: (i) decomposable, (ii) indecomposable but not positively regular, and (iii) positively regular models.

Motivation & Objective

• To characterize the asymptotic behavior of conditional least squares (CLS) estimators for autoregressive parameters, stability parameter ϱ = α + β, and innovation mean μ in unstable INAR(2) processes.
• To identify distinct limit distributions for CLS estimators under three structural regimes: decomposable, indecomposable but not positively regular, and positively regular models.
• To extend existing asymptotic theory for stable INAR(p) models to the unstable (unit root) case, particularly for p=2.
• To establish weak convergence of scaled random step processes to a squared Bessel process (Cox–Ingersoll–Ross process), a continuous-time branching process with immigration.
• To provide a foundation for asymptotic inference in critical multitype branching processes and unstable integer-valued time series models.

Proposed method

• Uses conditional least squares (CLS) estimation for the autoregressive parameters (α, β), stability parameter ϱ = α + β, and innovation mean μ in an INAR(2) model: X_k = α∘X_{k-1} + β∘X_{k-2} + ε_k.
• Applies weak convergence of scaled random step processes formed from the INAR(2) process to a squared Bessel process (Cox–Ingersoll–Ross process) under the unstable regime.
• Employs martingale difference arrays and functional central limit theorem techniques to derive weak convergence of the estimator processes.
• Classifies the limit behavior based on the spectral properties of the autoregressive polynomial: decomposable, indecomposable but not positively regular, or positively regular.
• Applies a functional limit theorem (Theorem C.1) for stochastic processes with time-changed martingale increments to prove convergence to a diffusion process.
• Uses the convergence of empirical covariance and variance statistics to derive the limiting distributions of the CLS estimators under different structural assumptions.

Experimental results

Research questions

• RQ1 What is the asymptotic distribution of the CLS estimator for the autoregressive parameters (α, β) in an unstable INAR(2) process?
• RQ2 How does the limit distribution of the CLS estimator for the stability parameter ϱ = α + β differ across the three structural regimes: decomposable, indecomposable but not positively regular, and positively regular?
• RQ3 What is the limiting behavior of the CLS estimator for the innovation mean μ in an unstable INAR(2) model?
• RQ4 How do the scaling factors and limit distributions for CLS estimators in unstable INAR(2) models compare to those in classical unstable AR(1) models?
• RQ5 Can the weak convergence of the scaled path of the INAR(2) process to a squared Bessel process be rigorously established under the unstable condition?

Key findings

• For decomposable unstable INAR(2) models, the CLS estimators of (α, β) and ϱ converge to a non-degenerate limit distribution that is non-normal and depends on the joint behavior of the process.
• In the indecomposable but not positively regular case, the limit distribution of the CLS estimator for ϱ is non-standard and involves a ratio of stochastic integrals with respect to a Bessel process.
• For positively regular models, the CLS estimator of the stability parameter ϱ converges to a distribution involving a ratio of stochastic integrals of the form ∫₀¹ W_t dW_t / ∫₀¹ W_t² dt, analogous to the Dickey–Fuller statistic but in a discrete-time integer-valued setting.
• The CLS estimator of the innovation mean μ is asymptotically normal with a scaling factor of √n, consistent with standard asymptotic theory.
• The limit process of the scaled random step function formed from the INAR(2) process is a squared Bessel process (Cox–Ingersoll–Ross process), confirming the weak convergence result.
• The limit distributions are fundamentally different from those in classical unstable AR(1) models, particularly in the decomposable and indecomposable cases, due to the discrete nature and thin-tailed innovations of the INAR(2) process.