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  • Vector moving average threshold heterogeneous autoregressive (VMAT-HAR) model∗

    Kaiji Motegi† Shigeyuki Hamori‡

    Kobe University Kobe University

    March 2, 2020


    The existing vector heterogeneous autoregression (VHAR) does not allow for threshold effects. The threshold autoregressions are well established in the lit- erature, but the presence of an unknown threshold complicates inference. To resolve this dilemma, we propose the vector moving average threshold (VMAT) HAR model. Observed moving averages of lagged target series are used as thresholds, which guarantees time-varying thresholds and the least squares es- timation. We show via simulations that the proposed model performs well in small samples. We analyze daily realized volatilities of the stock price indices of Hong Kong and Shanghai, detecting significant threshold effects and mutual Granger causality.

    JEL codes: C32, C51, C58.

    Keywords: Granger causality test, multivariate time series analysis, realized volatil-

    ity, threshold autoregression (TAR), vector heterogeneous autoregression (VHAR).

    ∗The second author is grateful for the financial support of JSPS KAKENHI Grant Number (A) 17H00983.

    †Corresponding author. Graduate School of Economics, Kobe University. Address: 2-1 Rokkodai- cho, Nada, Kobe, Hyogo 657-8501 Japan. E-mail: [email protected]

    ‡Graduate School of Economics, Kobe University. E-mail: [email protected]


  • 1 Introduction

    The heterogeneous autoregressive (HAR) model proposed by Corsi (2009) and Ander-

    sen, Bollerslev, and Diebold (2007) has been adopted extensively to model and predict

    realized volatilities (RVs) of financial markets. The HAR model is essentially an AR

    model with large enough lag length, where parametric restrictions are imposed from

    a viewpoint of sampling frequencies. The large lag length captures strong persistence

    in RVs, and the intuitively reasonable parametric restrictions address parameter pro-

    liferation. Besides, the HAR model can easily be estimated via the least squares. Due

    to its sharp performance and practical applicability, the HAR model has been applied

    and extended in various directions.1

    Multivariate versions of HAR are proposed and applied to realized variances or

    covariances by Bauer and Vorkink (2011), Bubák, Kočenda, and Žikeš (2011), Busch,

    Christensen, and Nielsen (2011), Chiriac and Voev (2011), Souček and Todorova

    (2013), Patton and Sheppard (2015), Čech and Baruńık (2017), and Caloia, Cipollini,

    and Muzzioli (2018). The specification of the vector HAR (VHAR) model is a natural

    extension of the univariate HAR, and the least squares estimation is still feasible.

    The multivariate extension leads to higher prediction accuracy and richer economic

    implications, since dynamic feedback effects among multiple RVs can be captured.2

    A potential drawback of the existing VHAR model is that the possibility of thresh-

    old effects is ruled out. It is often plausible to assume that financial markets and

    macroeconomy have several regimes that switch stochastically over time (e.g., reces-

    sion and expansion periods). Economic time series may well have different properties

    across regimes, which motivates threshold models. Tong (1978) is a seminal paper

    that proposed the threshold autoregressive (TAR) model, and there are many well-

    known extensions including the smooth-transition TAR (STAR) and self-exciting TAR

    (SETAR) models. A large amount of literature documents the existence of threshold

    effects in economic and financial time series.3

    A practical challenge of the TAR-type models is that the presence of an unknown

    threshold complicates statistical inference, even in the univariate setting. A related

    problem is that the number of lags included in the model needs to be small in order

    1 See, e.g., Corsi, Audrino, and Renò (2012) and Ghysels and Marcellino (2018, Ch. 14) for comprehensive discussions on HAR.

    2 Cubadda, Guardabascio, and Hecq (2017) proposed the VHAR index (VHARI) model, which is a VHAR model with suitable parametric restrictions, in order to detect the presence of commonalities in a set of RV measures. Also see Cubadda, Hecq, and Riccardo (2019).

    3 See, e.g., Chen, So, and Liu (2011), Hansen (2011), Tong (2015), Elliott and Timmermann (2016, Ch. 8), and Ghysels and Marcellino (2018, Ch. 9) for extensive discussions on TAR.


  • to keep inference simple. Consequently, multivariate threshold models have not been

    fully explored (see, e.g., Tsay, 1998, Huang, Hwang, and Peng, 2005, Huang, Yang,

    and Hwang, 2009, for early contributions to the literature). Further, the constant

    threshold could be an unrealistic assumption since a threshold may vary over time,

    depending on the state of the market or economy. Because of these issues, the TAR

    structure has not been incorporated in the existing VHAR model.

    Recently, Motegi, Cai, Hamori, and Xu (2020) proposed the moving average

    threshold HAR (MAT-HAR) model, which can be thought of as a univariate HAR

    model that allows for time-varying threshold effects. As in the standard HAR, the

    MAT-HAR model has multiple groups of lags of a target series, where the groups are

    constructed from a viewpoint of sampling frequencies. An observed moving average

    of lagged target series is included as a threshold for each group, which guarantees

    time-varying thresholds and simple estimation via the least squares. Motegi, Cai,

    Hamori, and Xu (2020) show via Monte Carlo simulations and a macroeconomic ap-

    plication that the MAT-HAR model achieves the sharper in-sample and out-of-sample

    performance than the benchmark HAR model. Salisu, Gupta, and Ogbonna (2019)

    applied the MAT-HAR model to the monthly RV of the U.S. stock market, finding

    an improved forecast performance relative to the conventional HAR models.

    Inspired by Motegi, Cai, Hamori, and Xu (2020), the present paper proposes

    the vector MAT-HAR (VMAT-HAR) model to incorporate time-varying threshold

    effects in the VHAR model. As in the univariate MAT-HAR, a vector of thresholds

    at each sampling frequency is specified as an observed moving average of lagged target

    variables. The thresholds can be calculated directly from data, and hence the entire

    model can be estimated via the least squares, a considerable advantage for applied

    researchers. To the authors’ best knowledge, the VMAT-HAR is the only multivariate

    model where multiple time-varying thresholds exist and the least squares is feasible.

    This paper establishes the statistical procedure of the VMAT-HAR model. Specif-

    ically, the least squares estimation and asymptotic and bootstrapped Wald tests with

    respect to linear parametric restrictions are described. In particular, the Wald tests

    for no threshold effects or Granger non-causality are discussed in detail so that the

    VMAT-HAR model can formally be compared with the benchmark VHAR or MAT-

    HAR model. We show via Monte Carlo simulations that the proposed model exhibits

    sharp performance in small samples under both financial and macroeconomic scenar-


    As an empirical application, we fit the proposed model to recent daily log RVs


  • of the Hang Seng Index of Hong Kong (HSI) and SSE Composite Index of Shanghai

    (SSEC). Given the increasing political tension between Hong Kong and Mainland

    China since the 2019-20 Hong Kong protests, it is of broad interest to investigate how

    their stock markets are linked with each other. The null hypothesis of no threshold

    effects is rejected, an evidence in favor of VMAT-HAR against VHAR. Further, mutual

    Granger causality between HSI and SSEC is detected, an evidence in favor of VMAT-

    HAR against MAT-HAR.

    The rest of this paper is organized as follows. In Section 2, the notation and basic

    framework are introduced. In Section 3, the VMAT-HAR model is proposed. In Sec-

    tion 4, statistical inference under VMAT-HAR is described. In Section 5, the Monte

    Carlo simulations are performed. In Section 6, the empirical analysis is presented. In

    Section 7, brief concluding remarks are provided.

    2 Set-up and motivation

    Let yt = (y1t, . . . , yDt) ⊤ be D-dimensional target variables. The vector heterogeneous

    autoregressive (VHAR) model is specified as

    yt = A (0) +

    K∑ k=1

    A(k)y (k) t−1 + ut, t ∈ {1, . . . , T}, (1)


    y (k) t = (y

    (k) 1t , . . . , y

    (k) Dt )

    ⊤ =

    ∑t τ=max{t+1−mk,1} yτ

    t+ 1−max{t+ 1−mk, 1} , k ∈ {1, . . . , K}, (2)

    with m1 = 1 and hence y (1) t = yt (see, e.g., Bubák, Kočenda, and Žikeš, 2011).

    mk signifies the ratio of sampling frequencies. Typical choices include (m1,m2,m3) =

    (1, 5, 22) (i.e., day, week, and month) for financial applications including the empirical

    analysis of the present paper, and (m1,m2,m3) = (1, 3, 12) (i.e., month, quarter, and

    year) for macroeconomic applications as in Motegi, Cai, Hamori, and Xu (2020).

    A potential drawback of the VHAR model (1) is that the possibility of threshold

    effects is ruled out. The behavior of yt might well be different when it is above or

    below a certain threshold. The existence of thresh