Moving average threshold heterogeneous autoregressive (MAT‐HAR) models

• Author: Xiaojing Cai,Kaiji Motegi,Haifeng Xu,Shigeyuki Hamori
• Date: 01 November 2020
• DOI: http://doi.org/10.1002/for.2671
• Published date: 01 November 2020

Received: 18 September 2019 Accepted: 12 January 2020

DOI: 10.1002/for.2671

RESEARCH ARTICLE

Moving average threshold heterogeneous autoregressive

(MAT-HAR) models

Kaiji Motegi1Xiaojing Cai2Shigeyuki Hamori1Haifeng Xu3

1Graduate School of Economics, Kobe

University, Kobe,Japan

2Graduate School of Humanities and

Social Sciences, Okayama University,

Okayama, Japan

3Department of Statistics, School of

Economics, Xiamen University and Wang

Yanan Institute for Studies in Economics

(WISE), Xiamen University, China

Correspondence

Haifeng Xu, Department of Statistics,

School of Economics, and Wang Yanan

Institute for Studies in Economics (WISE),

Xiamen University, Xiamen 361005,

China.

Email: xhf1984@hotmail.co.jp

Funding information

Japan Society for the Promotion of

Science, Grant/AwardNumber: (A)

17H00983; National Natural Science

Foundation of China, Grant/Award

Number: 71801184; Natural Science

Foundation of Fujian Province,

Grant/AwardNumber: 2018J01114

Abstract

We propose moving average threshold heterogeneous autoregressive

(MAT-HAR) models as a novel combination of heterogeneous autoregres-

sion (HAR) and threshold autoregression (TAR). The MAT-HAR has multiple

groups of lags of a target series, and a threshold term can appear in each group.

The threshold is a moving average of lagged target series, which guarantees

time-varying thresholds and simple estimation via least squares. We show

via Monte Carlo simulations that the MAT-HAR has sharp in-sample and

out-of-sample performance. An empirical application on the industrial produc-

tion of Japan suggests that significant threshold effects exist, and the MAT-HAR

has a higher forecast accuracy than the HAR.

KEYWORDS

heterogeneousautoregression (HAR), model selection, out-of-sample forecast, threshold autoregres-

sion (TAR), time series analysis

1INTRODUCTION

Heterogeneous autoregressive (HAR) models are a popu-

lar tool in time series analysis because of their parsimo-

nious specifications and intuitively reasonable parametric

constraints that are in line with sampling frequencies.

The original HAR model was proposed by Corsi (2009)

and has been extensively applied to predict economic and

financial time series, especially the realized volatility of

asset returns. An advantage of the HAR model and its

variants is that estimation and inference are relatively sim-

ple. Another advantage of HAR is that, due to its tight

parametrization, sufficiently many lags of a target series

can be included in the model. A potential issue of the

existing HAR models is that threshold effects are not con-

sidered.1

It is often plausible to assume that financial markets and

macroeconomy have several regimes that switch stochas-

tically over time (e.g., recession and expansion periods).

Economic time series may well have different proper-

ties across regimes, which motivates threshold models.

Tong (1978) is a seminal paper that proposed the thresh-

old autoregressive (TAR) model, and there are many

well-known extensions including the smooth-transition

TAR (STAR) and self-exciting TAR (SETAR) models. An

1See, for example, Andersen, Bollerslev, and Diebold (2007), Corsi,

Audrino, and Renò (2012), and Ghyselsand Marcellino (2018, ch. 14) for

extensive discussions on HAR.

Journal of Forecasting. 2020;39:1035–1042. wileyonlinelibrary.com/journal/for © 2020 John Wiley & Sons, Ltd. 1035