Statistics and Its Interface

Volume 4 (2011)

Number 2

On the least squares estimation of threshold autoregressive and moving-average models

Pages: 183 – 196

DOI: http://dx.doi.org/10.4310/SII.2011.v4.n2.a13

Authors

Dong Li (Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong)

Wai Keung Li (Department of Statistics and Actuarial Science, The University of Hong Kong, Hong Kong)

Shiqing Ling (Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong)

Abstract

This paper considers the least squares estimation and establishes its asymptotic theory for threshold autoregressive and moving-average models. Under some mild conditions, it is shown that the estimator of the threshold is $n$-consistent and after normalization it converges weakly to the smallest minimizer of a compound Poisson process, while the estimators of other coefficients are strongly consistent and asymptotically multivariate normal. This paper also provides a numerical method to tabulate the limiting distribution of the estimated threshold in practice. Simulation studies are carried out to assess the performance of the least squares estimation in finite samples.

Keywords

asymptotic normality, compound Poisson process, consistency, least squares estimation, threshold ARMA model

2010 Mathematics Subject Classification

Primary 62F12, 62M10. Secondary 60G10.

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