On maximum likelihood estimators for a threshold autoregression¹

Abstract

For a stationary ergodic self-exciting threshold autoregressive model with single threshold parameter, Chan (1993) obtained the consistency and limiting distribution of the least-squares estimator for the underlying true parameters. In this paper, we derive the similar results for the maximum likelihood estimators of the same model under some regularity conditions on the error density, not necessarily Gaussian.

Introduction

In the past five decades, linear time-series models have dominated the development of the time-series analysis. But many applied fields such as electronics, oceanography, hydrology, ecology, marine engineering, medical engineering, solar astrophysics, and physics have revealed a common connecting theme: piecewise linearity. Tong (1977) mentioned the usefulness of a time-series that is piecewise linear in the past variables and in parameters. Later, Tong (1978a), Tong (1978b), Tong (1980) developed these models further in a systematic way for modeling of discrete time-series data. He argued that various phenomena such as limit cycles, jump resonance, harmonic distortion, modulation effects and chaos can be modeled by discrete time-series that are piecewise linear. He called these models the self-exciting threshold autoregressive (SETAR) models. See Tong (1983), Tong (1990) for a comprehensive introduction to general SETAR models.

This paper is concerned with the large sample behaviors of maximum likelihood estimators in a two regime SETAR model, called SETAR(2;p,p), defined as follows:

$$\text{X}{}_{\mathrm{i}}\text{=h(}\text{X}{}_{\mathrm{i-1}}\text{,}\text{θ}\text{)+ε}{}_{\mathrm{i}}\text{,}\text{i⩾1}$$

for some $\text{θ}\text{=(}\text{θ}{}_1\text{′,}\text{θ}{}_2\text{′,r,d)′∈}\text{R}{}^{\mathrm{2p+3}}\text{×{1,2,…,p}}$, where $\text{X}{}_{\mathrm{i-1}}\text{=(X}{}_{\mathrm{i-1}}\text{,…,X}{}_{\mathrm{i-p}}\text{)′}$, $\text{θ}{}_{\mathrm{j}}\text{=(θ}{}_{\mathrm{0j}}\text{,θ}{}_{\mathrm{1j}}\text{,…,θ}{}_{\mathrm{pj}}\text{)′∈}\text{R}{}^{\mathrm{p+1}}\text{,}\text{j=1,2}$ and for $\text{x}\text{∈}\text{R}{}^{\mathrm{p}}$,

$$\text{h(}\text{x}\text{,}\text{θ}\text{)=}\text{θ}{}_{01}\text{+}\text{∑}\text{k=1}\text{p}\text{θ}{}_{\mathrm{k1}}\text{x}{}_{\mathrm{k}}\text{I(x}{}_{\mathrm{d}}\text{⩽r)+}\text{θ}{}_{02}\text{+}\text{∑}\text{k=1}\text{p}\text{θ}{}_{\mathrm{k2}}\text{x}{}_{\mathrm{k}}\text{I(x}{}_{\mathrm{d}}\text{>r).}$$

The errors {ε_i} in Eq. (1)are independent and identically distributed random variables with mean zero, finite nonzero variance and ε_i is independent of X_i−1,X_i−2,…,i⩾1. The parameter r, the location of the change of the autoregressive function h, is called the threshold. The time delay d is called the delay parameter.

In this paper, we assume that the time-series in model (1) is stationary and ergodic. Detailed discussions about the stationarity and ergodicity can be found in Chan et al. (CPTW)(1985), Chan and Tong (1985).

For the case of the threshold having only finite number of possible values and assuming Gaussian errors, Tong (1983) constructed maximum likelihood estimators of the unknown parameters by using Akaike Information Criterion (Akaike, 1973). If the threshold r is known, CPTW (1985) obtained the consistency and asymptotic normality property of the least-squares estimators of the coefficient parameter $\text{θ}{}_{\mathrm{c}}\text{=(}\text{θ}{}_1\text{′,}\text{θ}{}_2\text{′)′}$ under some regularity conditions for p=1. But, in practice, the threshold parameter r is unknown and can take infinitely many values in $\text{R}$. In this case, Petruccelli (1986) proved that the conditional least-squares estimator (CLSE) of $\text{θ}$ is strongly consistent for SETAR(2;1,1) model.

The present paper is motivated by Chan (1993), who developed the strong consistency and limiting distribution of the CLSE in a SETAR(2;p,p) model (1). It derives the asymptotics of maximum likelihood estimator (MLE) of the underlying parameter $\text{θ}$ in model (1), when the errors have a density f, not necessarily Gaussian. Unlike the popular AR model, the likelihood function of SETAR(2;p,p) model is not continuous in the threshold parameter in general. Thus, the routine method of computing maximum likelihood estimator cannot be adopted. Instead, Section 2discusses the maximum likelihood method to obtain the MLE $\text{θ}\text{̂}{}_{\mathrm{n}}\text{=(}\text{θ}\text{̂}{}_{\mathrm{cn}}\text{′,}\text{r}\text{̂}{}_{\mathrm{n}}\text{,}\text{d}\text{̂}{}_{\mathrm{n}}\text{)′}$ of $\text{θ}\text{=(}\text{θ}{}_{\mathrm{c}}\text{′,r,d)′}$. Section 3describes assumptions and obtains the strong consistency of the MLE of the true parameter θ. Section 4shows the n-consistency of the threshold estimator under the assumption of discontinuity of h at r. Section 5obtains the uniform asymptotic normality of the coefficient parameter estimator $\text{θ}\text{̂}{}_{\mathrm{cn}}$ over a bounded interval and some more byproduct results. In Section 6, as a consequence of the n-consistency of $\text{r}\text{̂}{}_{\mathrm{n}}$, a suitably normalized log-likelihood sequence of processes $\text{{}\text{l}\text{̂}{}_{\mathrm{n}}\text{}}$ is shown to be approximated by a sequence of simpler processes which describe the log-likelihood under known coefficient parameter $\text{θ}{}_{\mathrm{c}}$. Through the latter processes, we obtain the limiting distribution of the standardized maximum likelihood estimator as the left endpoint of a random interval on which a superposition of independent compound Poisson processes attains a minimum.

Notation. Throughout the paper, the symbol $\text{θ}$ is the fixed unknown underlying parameter, the function f is the p.d.f. of ε₁ and F denotes the distribution function corresponding to f. The expectation under $\text{θ}$ is denoted by E.

Weak convergence is denoted by ⇒. A sequence (random) goes to zero (in probability) is denoted by o(1)(o_P(1)) while O(1)(O_P(1)) means that it is bounded (in probability). The multivariate normal distribution with mean zero and covariance matrix Γ is denoted by N(0,Γ). Let $\text{R}$ be the real line (−∞,∞), and $\text{R}\text{̄}\text{=}\text{R}\text{∪{−∞,∞}}$, then the compactness of the set $\text{R}\text{̄}$ is under the metric d(·,·) defined by d(x,y)=|arctanx−arctany|. A function ϕ satisfies the Lip(1) if $\text{∀x,}\text{y∈}\text{R}$, ∃L⩾0, such that |ϕ(x)−ϕ(y)|⩽L|x−y|. For any event A, the complement event of A is denoted by A^c and the indicator function is denoted by I(A). Throughout, the capital letter C, the symbols $\text{γ}{}_{\mathrm{i}}\text{,}\text{i=1,2,…}$ stand for absolute constants and they can have different values in different places. The notation x′y stands for the inner product of vectors x and y. For any matrix M=(m_ij), ||M||=∑_i,j|m_ij|, det(M) and adj(M) stand for the determinant and adjoint matrix of M, respectively. Vectors of dimension more than one are denoted by bold face letters. The index i in the summation varies from 1 to n unless specified otherwise.