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Vector moving average threshold heterogeneousautoregressive (VMAT-HAR) model∗
Kaiji Motegi† Shigeyuki Hamori‡
Kobe University Kobe University
March 2, 2020
Abstract
The existing vector heterogeneous autoregression (VHAR) does not allow forthreshold effects. The threshold autoregressions are well established in the lit-erature, but the presence of an unknown threshold complicates inference. Toresolve this dilemma, we propose the vector moving average threshold (VMAT)HAR model. Observed moving averages of lagged target series are used asthresholds, which guarantees time-varying thresholds and the least squares es-timation. We show via simulations that the proposed model performs well insmall samples. We analyze daily realized volatilities of the stock price indicesof Hong Kong and Shanghai, detecting significant threshold effects and mutualGranger 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
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), Bubak, Kocenda, and Zikes (2011), Busch,
Christensen, and Nielsen (2011), Chiriac and Voev (2011), Soucek and Todorova
(2013), Patton and Sheppard (2015), Cech and Barunık (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 Reno (2012) and Ghysels and Marcellino (2018, Ch. 14) forcomprehensive discussions on HAR.
2 Cubadda, Guardabascio, and Hecq (2017) proposed the VHAR index (VHARI) model, which isa VHAR model with suitable parametric restrictions, in order to detect the presence of commonalitiesin 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.
2
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-
ios.
As an empirical application, we fit the proposed model to recent daily log RVs
3
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)
where
y(k)t = (y
(k)1t , . . . , y
(k)Dt )
⊤ =
∑tτ=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., Bubak, Kocenda, and Zikes, 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 threshold effects in economic time series
is well documented. A naıve way to add threshold terms to the VHAR model would
4
be as follows.
yt = A(0) +K∑k=1
{A(k)y
(k)t−1 +Ψ(k) × I
(k)t−1
(µ(k)
)× y
(k)t−1
}+ ut, (3)
where µ(k) = (µ(k)1 , . . . , µ
(k)D )⊤ is a vector of unknown thresholds,
I(k)t (µk) =
1(y(k)1t ≥ µ
(k)1
). . . 0
.... . .
...
0 . . . 1(y(k)Dt ≥ µ
(k)D
) ,
and 1(A) is the indicator function that equals 1 if event A occurs and 0 otherwise.
In model (3), threshold terms are added to each of the K sampling frequencies.
There are two issues with model (3). First, the presence of DK unknown thresholds,
(µ(1), . . . ,µ(K)), causes a large adverse impact on inference. Indeed, a simplified model
with D = 1 or K = 1 is not necessarily easy to handle since a numerical search for
multiple unknown thresholds is required.
Second, the thresholds are fixed over time, which could be an unrealistic assump-
tion. In economic applications, adjacent lags of a target variable typically contain
more important information than remote lags. In-sample and out-of-sample per-
formance of the model might well be improved if we specify the thresholds to be
the moving averages of the lagged target variable instead of mere constants. Salisu,
Gupta, and Ogbonna (2019) and Motegi, Cai, Hamori, and Xu (2020) demonstrated
that is indeed the case in univariate frameworks. Time-varying threshold models are
not fully explored in the multivariate time series literature, and the present paper fills
this gap.
3 Vector MAT-HAR models
To resolve the dilemma between VHAR and TAR, we propose the vector moving
average threshold heterogeneous autoregressive (VMAT-HAR) model as follows.
yt = A(0) +K∑k=1
{A(k)y
(k)t−1 +Ψ(k)I
(k)t−1y
(k)t−1
}+ ut, (4)
5
where ut = (u1t, . . . , uDt)⊤ is a strictly stationary martingale difference sequence with
respect to the increasing σ-field Ft = σ(yt,yt−1, . . . ), Σu = E(utu⊤t ) is a positive
definite error covariance matrix,
A(0) =
a(0)1
...
a(0)D
, A(k) =
a(k)11 . . . a
(k)1D
.... . .
...
a(k)D1 . . . a
(k)DD
, Ψ(k) =
ψ
(k)11 . . . ψ
(k)1D
.... . .
...
ψ(k)D1 . . . ψ
(k)DD
,
I(k)t =
1(y(k)1t ≥ µ
(k)1t
). . . 0
.... . .
...
0 . . . 1(y(k)Dt ≥ µ
(k)Dt
) , (5)
and
µ(k)dt =
∑tτ=max{t−ℓT ,1} y
(k)dτ
t+ 1−max{t− ℓT , 1}, d ∈ {1, . . . , D}. (6)
The moving average threshold, µ(k)dt , is specified as the sample mean of {y(k)d,t−ℓT
, . . . , y(k)dt }
if t ≥ ℓT+1, or {y(k)d1 , . . . , y(k)dt } otherwise. It measures the average level of y
(k)d in recent
periods. If y(k)dt exceeds the recent average (i.e., y
(k)dt ≥ µ
(k)dt ), then the (d, d)-element
of I(k)t becomes 1 and it affects the persistence of yt through Ψ(k).
The lag length ℓT is chosen by the researcher, and can depend on the sample size
T . A trade-off between small and large values of ℓT is that a small value would make
µ(k)dt volatile and hence hard to interpret while a large value would possibly make y
(k)dt
and µ(k)dt far from each other.4 It is beyond the scope of this paper to find an optimal
choice of ℓT . A suggested rule of thumb is to use ℓT = δ√T with some δ > 0. In
the present paper, we use δ = 1 for Monte Carlo simulations and empirical analysis,
obtaining reasonable results.
To further understand the structure of VMAT-HAR, pick the dth equation of (4):
ydt = a(0)d +
K∑k=1
{D∑i=1
a(k)di y
(k)it−1 +
D∑i=1
ψ(k)di 1
(y(k)it−1 ≥ µ
(k)it−1
)y(k)it−1
}+ udt. (7)
Equation (7) reveals several key features of the VMAT-HAR model. First, (7) reduces
4 This is a bias-variance trade-off that arises in various topics of econometrics such as a bandwidthselection in variance estimation (e.g., Newey and West, 1987, 1994) and a selection of block size inbootstraps (e.g., Shao, 2010, 2011).
6
to the univariate MAT-HAR model for y1 if D = 1:
y1t = a(0)1 +
K∑k=1
{a(k)11 y
(k)1t−1 + ψ
(k)11 1
(y(k)1t−1 ≥ µ
(k)1t−1
)y(k)1t−1
}+ u1t (8)
(see Motegi, Cai, Hamori, and Xu, 2020, Eq. (5)). Hence, the proposed model (4) is a
well-defined extension of the univariate MAT-HAR model. The extra feature brought
by the multivariate extension is that yd depends on the lags of all D variables with
time-varying threshold effects.
Second, (7) reduces to the dth equation of the VHAR model (1) if ψ(k)di = 0 for all
i ∈ {1, . . . , D} and k ∈ {1, . . . , K}. Hence, the proposed model (4) is a well-defined
extension of the VHAR model with threshold effects being allowed. Further, the zero
restrictions (i.e., no threshold effects) can easily be tested via Wald tests as described
later in Section 4.
Third and importantly, the time-varying threshold term 1(y(k)it−1 ≥ µ
(k)it−1) can be
computed directly from data for all i ∈ {1, . . . , D} and k ∈ {1, . . . , K}. It is thereforestraightforward to estimate the entire parameters (A(0),A(1), . . . ,A(K),Ψ(1), . . . ,Ψ(K))
via the least squares, a remarkable advantage from a practical point of view. A specific
procedure of the least squares estimation is described in Section 4.
Fourth, the proposed model is parsimoniously specified by virtue of the HAR
structure. For the bivariate case with K = 3 sampling frequencies, for instance, the
number of parameters in (7) is only 2KD + 1 = 13. Hence, parameter proliferation
should not be an issue as far as the target series {yt} are sampled at a sufficiently
high frequency (e.g., daily, weekly, or even monthly data should be workable). This
claim shall be verified via the Monte Carlo simulation in Section 5.
In summary, the VMAT-HAR model (4) is a natural extension of the univariate
MAT-HAR model (8) with the useful feature of observable, time-varying, and intu-
itively reasonable thresholds being carried over. Besides, VMAT-HAR is a natural
extension of VHAR (1) with the parsimonious parameterization being preserved.5
5 It is beyond the scope of this paper to derive a stationarity condition of the VMAT-HAR model.If Ψ(k) = 0, then the stationarity condition is well documented since VHAR is essentially equivalentto VAR(mK) with parametric restrictions. In the general case with Ψ(k) = 0, the stationaritycondition becomes hard to derive. Hence, this paper simply assumes that the VMAT-HAR model isstrictly stationary.
7
4 Statistical inference
By virtue of the observable thresholds (6), the statistical inference of VMAT-HAR
follows straightforwardly from the well-known VAR theory (see, e.g., Hamilton, 1994,
Lutkepohl, 2006). We describe the least squares estimation in Section 4.1, asymptotic
Wald tests in Section 4.2, and bootstrapped Wald tests in Section 4.3.
4.1 Estimation
To rewrite model (4) in a matrix form, define
Y = (y1, . . . ,yT ), U = (u1, . . . ,uT ),
X t =(1, {y(1)
t }⊤, . . . , {y(K)t }⊤, {I(1)
t y(1)t }⊤, . . . , {I(K)
t y(K)t }⊤
)⊤,
X = (X0, . . . ,XT−1) ,
B =(A(0),A(1), . . . ,A(K),Ψ(1), . . . ,Ψ(K)
).
Then, model (4) is rewritten compactly as
Y = BX +U .
The multivariate least squares estimator for B is given by
B = Y X⊤(XX⊤)−1. (9)
Let β = vec(B) and β = vec(B), where vec(·) is the column-wise vectorization
operator. Under certain regularity conditions, it follows that
βp→ β as T → ∞ (10)
and √T (β − β)
d→ N (0, Γ−1 ⊗Σu) as T → ∞, (11)
where Γ = plimT→∞ (T−1XX⊤), Σu = E(utu⊤t ), and ⊗ is the Kronecker product.6
6 It is beyond the scope of this paper to provide the specific regularity conditions for the con-sistency (10) and the asymptotic normality (11); they are numerically verified via Monte Carlosimulations in Section 5.
8
4.2 Asymptotic Wald tests
4.2.1 Linear and zero restrictions
Consider linear parametric restrictions:
H0 : Rβ = r, (12)
where R is an n × (2KD2 + D) selection matrix of full row rank n; r is an n × 1
vector; n is the number of parametric restrictions. The alternative hypothesis is
simply H1 : Rβ = r. The Wald test statistic is defined as
W = T (Rβ − r)⊤{R(Γ
−1⊗ Σu)R
⊤}−1
(Rβ − r), (13)
where Γ = T−1XX⊤ and
Σu =1
T − (2KD + 1)(Y − BX)(Y − BX)⊤ (14)
(2KD + 1 parameters are present in each of the D equations). The consistency
(10), the asymptotic normality (11), and the convergences in probability Γp→ Γ and
Σup→ Σu imply that
Wd→ χ2
n under H0 and Wp→ ∞ under H1,
where χ2n is the chi-squared distribution with degrees of freedom n. The asymptotic
p-value is given by p = 1−Fn(W ), where Fn(·) is the cumulative distribution function
of χ2n. Reject H0 at the 100α% level if p < α, and accept H0 otherwise.
The linear parametric restrictions (12) include zero restrictions as a special case.
A particularly important type of zero restrictions is no threshold effects expressed as
H th0 : Ψ(k) = 0D×D for all k ∈ {1, . . . , K}.
Under H th0 , the VMAT-HAR model (4) reduces to the VHAR model (1). H th
0 can
simply be tested via the Wald test with R = (0n×(KD2+D), In) and r = 0n×1, where
n = KD2.
9
4.2.2 Granger causality tests
Another important type of zero restrictions is Granger non-causality. In particular,
if target series are volatility measures as in the empirical application of this paper,
then the Granger causality is associated with causation in variance (Cheung and Ng,
1996). To elaborate Granger causality tests in VMAT-HAR, we focus on the bivariate
case D = 2. When D = 2, Granger (non-)causality at prediction horizon h = 1 is
equivalent to Granger (non-)causality at any horizon h ≥ 1, and hence the single-
horizon test suffices. When D ≥ 3, there can be causal chains and Granger causality
at each horizon h ≥ 1 needs to be tested separately. The multi-horizon Granger
causality tests were developed by Lutkepohl (1993), Dufour and Renault (1998), and
Dufour, Pelletier, and Renault (2006), and elaborated further by Hill (2007), Al-
Sadoon (2014), and Ghysels, Hill, and Motegi (2016), among others. The procedure
of the multi-horizon Granger causality tests is well established, but many researchers
restrict their attention to the bivariate case due to its simplicity (e.g., Gotz, Hecq,
and Smeekes, 2016, Ghysels, Hill, and Motegi, 2020).
When D = 2, the VMAT-HAR model (4) becomesy1ty2t
=
a(0)1
a(0)2
+ K∑k=1
a(k)11 a
(k)12
a(k)21 a
(k)22
y(k)1t−1
y(k)2t−1
+
ψ(k)11 ψ
(k)12
ψ(k)21 ψ
(k)22
I(k)1t−1y
(k)1t−1
I(k)2t−1y
(k)2t−1
+
u1tu2t
. (15)We first consider Granger causality from y2 to y1. For d ∈ {1, 2}, define Fdt =
σ(ydt, ydt−1, . . . ), the σ-field spanned by {ydt, ydt−1, . . . }. Fdt is called the information
set of yd up to time t. Define Ft = σ(y1t, y1t−1, . . . , y2t, y2t−1, . . . ) and call it the
information set up to time t. By definition, y2 does not Granger cause y1 if
E(y1,t+1 | F1t) = E(y1,t+1 | Ft). (16)
Given (15), y2 does not Granger cause y1 if and only if
HGC0 : a
(k)12 = ψ
(k)12 = 0, k ∈ {1, . . . , K}. (17)
The equivalence between Granger non-causality in (16) and the zero restrictions in
(17) is a straightforward application of the classical result in the literature (e.g.,
Dufour and Renault, 1998). A key insight is that y(k)dt and I
(k)dt are elements of Fdt and
Ft, but not elements of Fjt with j = d; recall the constructions (2), (5), and (6).
10
Under HGC0 , the first equation of (15) reduces to
y1t = a(0)1 +
K∑k=1
{a(k)11 y
(k)1t−1 + ψ
(k)11 I
(k)1t−1y
(k)1t−1
}+ u1t,
which is identical to the univariate MAT-HAR model on y1 (Motegi, Cai, Hamori,
and Xu, 2020, Eq. (5)). Thus, rejecting Granger non-causality is equivalent to re-
jecting the univariate MAT-HAR model in favor of the bivariate version. Conversely,
accepting Granger non-causality is evidence for the univariate MAT-HAR model.
HGC0 can be tested via the Wald test with suitably chosen (R, r). Specifically,
R = (ei1 , ei2 , . . . , ei2K )⊤ and r = 02K×1, where ig = 5+4(g−1) with g ∈ {1, . . . , 2K}
and ei is the 2(4K + 1) × 1 vector whose ith element is 1 and all other elements are
0. This construction follows from the fact that (a(1)12 , . . . , a
(K)12 , ψ
(1)12 , . . . , ψ
(K)12 ) appear
at the 5th, 9th, . . . , {5+ 4(2K − 1)}th elements of the 2(4K +1)× 1 parameter vector
β = vec(B), respectively.
Similarly, Granger causality from y1 to y2 can be tested by testing a(k)21 = ψ
(k)21 = 0
for all k ∈ {1, . . . , K}. Specifically, R = (ei1 , . . . , ei2K )⊤ and r = 02K×1, where
ig = 4g with g ∈ {1, . . . , 2K}.
4.3 Bootstrapped Wald tests
In the previous section, the Wald test is performed via the asymptotic p-value asso-
ciated with the χ2 distribution. The asymptotic χ2 test may cause size distortions
when the sample size T is not large enough (Dufour, Pelletier, and Renault, 2006,
Ghysels, Hill, and Motegi, 2016, 2020). When H0 implies zero restrictions, the para-
metric bootstrap of Dufour, Pelletier, and Renault (2006) can readily be employed to
control the size of the Wald test. The procedure is as follows.
Step 1 Estimate the VMAT-HAR model (4) and compute the least squares
estimator B in (9), the error covariance matrix estimator Σu in (14), and
the Wald test statistic W in (13).
Step 2 Generate u∗1, . . . ,u
∗T
i.i.d.∼ N (0D×1, Σu). Generate {y∗t}Tt=1 from the
VMAT-HAR process (4), where {u∗t}Tt=1 is used as the error term and
B with H0 being imposed is used as the parameters.
Step 3 Fit the model (4) to {y∗t}Tt=1 and compute a bootstrapped Wald test
statistic W ∗ according to (13).
11
Step 4 Repeat Steps 2-3 S times, resulting in a set of bootstrapped Wald test
statistics {W ∗s }Ss=1. The bootstrapped p-value is given by
p∗ =1
S
S∑s=1
1(W ∗s ≥ W ).
Reject H0 at the 100α% level if p∗ < α, and accept H0 otherwise.
5 Monte Carlo simulation
In this section, Monte Carlo simulations are performed to investigate the finite sample
performance of the VMAT-HAR model. In particular, we inspect the empirical size
and power of the asymptotic and bootstrapped Wald tests with respect to the two
important zero restrictions: no threshold effects and Granger non-causality.
5.1 Simulation design
The data generating process (DGP) is the bivariate VMAT-HAR:y1ty2t
=
a(0)1,0
a(0)2,0
+K∑k=1
a(k)11,0 a
(k)12,0
a(k)21,0 a
(k)22,0
y(k)1t−1
y(k)2t−1
+
ψ(k)11,0 ψ
(k)12,0
ψ(k)21,0 ψ
(k)22,0
I(k)1t−1y
(k)1t−1
I(k)2t−1y
(k)2t−1
+
ϵ1tϵ2t
or compactly
yt = A(0)0 +
K∑k=1
{A
(k)0 y
(k)t−1 +Ψ
(k)0 I
(k)t−1y
(k)t−1
}+ ϵt,
where K = 3 and ϵti.i.d.∼ N (02×1, I2). The constants are set as a
(0)1,0 = a
(0)2,0 = −1. The
individual persistence parameters are set as a(1)11,0 = a
(1)22,0 = 0.3, a
(2)11,0 = a
(2)22,0 = 0.2,
and a(3)11,0 = a
(3)22,0 = 0.1. Assume that a
(k)21,0 = ψ
(k)21,0 = ψ
(k)22,0 = 0 for k ∈ {1, 2, 3}. This
implies that y2 follows the univariate HAR process whose finite sample property is
well documented in the literature. Consider two cases for the remaining parameters:
DGP-1 a(1)12,0 = 0.15, a
(2)12,0 = 0.1, a
(3)12,0 = 0.05, and ψ
(k)11,0 = ψ
(k)12,0 = 0 for
k ∈ {1, 2, 3}. In this case, y2 Granger-causes y1 and a threshold effect does
not exist.
DGP-2 a(k)12,0 = ψ
(k)12,0 = 0 for k ∈ {1, 2, 3}, ψ(1)
11,0 = 0.25, ψ(2)11,0 = 0.2, and
ψ(3)11,0 = 0.1. In this case, y2 does not Granger-cause y1 and a threshold
12
effect exists.
Execute the lease squares to the simulated data and then test the following null
hypotheses:
H th0 : Ψ(k) = 02×2 for k ∈ {1, 2, 3} (i.e., no threshold effects),
HGC0 : a
(k)12,0 = ψ
(k)12,0 = 0 for k ∈ {1, 2, 3} (i.e., Granger non-causality from y2 to y1).
Under DGP-1, H th0 is true and HGC
0 is false. Under DGP-2, H th0 is false and HGC
0 is
true. Each null hypothesis is tested via the Wald test, where the p-value is computed
via the asymptotic χ2 distribution or the parametric bootstrap of Dufour, Pelletier,
and Renault (2006) with S = 500 bootstrap iterations. We compute rejection fre-
quencies of the Wald tests across J = 1000 Monte Carlo samples, where the nominal
sizes are α ∈ {0.01, 0.05, 0.10}. The rejection frequency is interpreted as empirical
size when H0 is true and empirical power when H0 is false.
In terms of the sampling frequencies (m1,m2,m3) and the sample size T , we
consider two realistic scenarios:
Financial scenario: (m1,m2,m3) = (1, 5, 22) and T ∈ {250, 500, 750}. This
set-up matches daily, weekly, and monthly levels with approximately 1, 2,
or 3 years of sample period.
Macroeconomic scenario: (m1,m2,m3) = (1, 3, 12) and T ∈ {120, 240, 360}.This set-up matches monthly, quarterly, and yearly levels with 10, 20, or
30 years of sample period.
The financial scenario with T = 750 days matches the empirical application in Section
6. The macroeconomic scenario matches the empirical application of Motegi, Cai,
Hamori, and Xu (2020), although they used the univariate MAT-HAR model. The
macroeconomic scenario is more challenging for inference than the financial scenario
since the sample size tends to be smaller. For each of the two scenarios, the lag length
for computing moving average thresholds µ(k)dt is ℓT = ⌊
√T ⌋.
5.2 Simulation results
See Table 1 for the rejection frequencies. Focus on the financial scenario first. The
empirical size of the asymptotic Wald test is fairly close to the nominal size α when
T = 250, and almost identical to α when T ≥ 500. See HGC0 : y2 ↛ y1 under
13
GDP-2 with α = 0.05, for instance. The empirical size is {0.078, 0.048, 0.051} for
T ∈ {250, 500, 750}, respectively. These results indicate that the asymptotic χ2 con-
vergence operates well in small samples by virtue of the parsimonious specification.
The empirical power of the asymptotic Wald test is reasonably high and ap-
proaches 1 as the sample size T grows. SeeH th0 : Ψ(k) = 0 under GDP-2 with α = 0.05,
for example. The empirical power is {0.284, 0.661, 0.931} for T ∈ {250, 500, 750}, re-spectively. It is evidence for the consistency of the asymptotic Wald test.
When the bootstrapped Wald test is used, we achieve perfectly accurate empirical
size for any T ∈ {250, 500, 750}. See HGC0 : y2 ↛ y1 under GDP-2 with α = 0.05, for
instance. The empirical size is {0.052, 0.041, 0.046} for T ∈ {250, 500, 750}, respec-tively. It indicates that the parametric bootstrap controls for size successfully. The
empirical power of the bootstrapped test is almost as high as that of the asymptotic
test, highlighting the use of the bootstrap. In summary, the VMAT-HAR model and
the asymptotic or bootstrapped Wald tests perform strikingly well in the financial
scenarios.
Now focus on the macroeconomic scenario. The empirical size of the asymptotic
Wald test is nearly identical to the nominal size when T ≥ 240, but slight size distor-
tions arise when the sample size is only T = 120. See HGC0 : y2 ↛ y1 under GDP-2
with T = 120, for instance. The empirical sizes are {0.030, 0.089, 0.149}, where the
nominal sizes are {0.010, 0.050, 0.100}, respectively.Once the parametric bootstrap is used, the over-rejection problem under T = 120
is resolved successfully. The empirical sizes of the bootstrapped test forHGC0 : y2 ↛ y1
under GDP-2 are {0.012, 0.056, 0.118}, which are virtually identical to the nominal
sizes α ∈ {0.010, 0.050, 0.100}.The empirical power of the Wald test is reasonably high and approaches 1 as the
sample size T increases, whether the asymptotic or bootstrapped test is used. See
HGC0 : y2 ↛ y1 under GDP-1 with α = 0.05, for example. For T ∈ {120, 240, 360}, the
empirical power is {0.349, 0.661, 0.845} for the asymptotic test and {0.268, 0.611, 0.827}for the bootstrapped test. In summary, the VMAT-HAR model and the bootstrapped
Wald test perform remarkably well in the macroeconomic scenarios. Existing multi-
variate time series models often perform poorly in small samples due to parameter
proliferation, while the VMAT-HAR model keeps sharp performance by virtue of its
parsimonious specification.
14
6 Empirical application
6.1 Data and preliminary analysis
We analyze two daily series of five-minute realized volatilities (RVs) from January
23, 2017 through January 22, 2020 (T = 730 days). The data are publicly available
at Realized Library of the Oxford-Man Institute of Quantitative Finance.7 The first
target series, {y1t}, is the log RV of the Hang Seng Index of Hong Kong (HSI). The
second target series, {y2t}, is the log RV of 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. Besides, a practical advantage of selecting HSI
and SSEC is that there is no time difference between the two markets, which makes
it easier to perform the daily-level analysis.8
Formulate a bivariate VMAT-HAR model (15) with K = 3 sampling frequencies.
Set (m1,m2,m3) = (1, 5, 22) in order to capture the daily, weekly, and monthly fluc-
tuations of the log RVs. Use ℓT = ⌊√T ⌋ = 27 days of lags for computing moving
average thresholds µ(k)dt in (6). See Figure 1 for the time series plots of y
(k)dt and µ
(k)dt for
each index d ∈ {1, 2} and frequency k ∈ {1, 2, 3}. It is evident that the time-varying
thresholds trace the log RVs fairly well for each index and frequency.
The Hong Kong protests starting around March 2019 seem to have added volatility
to HSI in the first few months, but the excess volatility seems to have vanished rapidly
(Figure 1). It is not clear from the figure whether the protests affected the volatility
of SSEC at all. It suggests that the financial conditions of Hong Kong and Shanghai
are not always in parallel with their political situations.
See Table 2 for sample statistics of the daily log RVs, y(1)dt . The volatility of
HSI fluctuates more extensively than SSEC, judging from the fact that the former
has the larger standard deviation and range (i.e., the distance between the minimum
and maximum). The skewness is positive and moderately large for both indices,
suggesting the presence of sudden volatility increase. The kurtosis is close to 3 for
both indices, suggesting the absence of extreme outliers. For both indices, we observe
7 Our RV measure is a simple sum of squared five-minute returns. Other related measures such asbi-power variation are often analyzed in the literature (see, e.g., Ghysels, Santa-Clara, and Valkanov,2006, Andersen, Bollerslev, and Diebold, 2007, Patton and Sheppard, 2015). We leave it as a futuretask to apply the VMAT-HAR model to those various RV measures.
8 In a similar vein, other interesting targets would include Canada/Mexico/U.S.,China/Japan/South Korea, and France/Germany/U.K., among others. Investigating these casesis left as future empirical projects.
15
mixed results on the normality tests; the null hypothesis is not rejected at the 5%
level by the Kolmogorov-Smirnov test but rejected by the Anderson-Darling test.
Finally, the Phillips-Perron unit root test with the Bartlett kernel and the Newey-
West automatic bandwidth selection is performed, where only an intercept is included
in the test equation. The resulting one-sided p-values of MacKinnon (1996) are 0.000
for both indices, an overwhelming rejection of the unit root hypothesis. Hence, the
daily log RVs are most likely stationary.
6.2 Empirical results
Table 3 reports the least squares estimates and the associated asymptotic t-statistics
for all parameters in (15). a(1)11 , a
(2)11 , a
(1)22 , and a
(2)22 take significantly positive values,
which indicates that both indices have strong persistence at the daily and weekly lev-
els. ψ(k)11 and ψ
(k)22 are not significantly different from 0 for any k ∈ {1, 2, 3}, suggesting
that the threshold effects in the univariate sense are absent.
The cross-sectional effects of SSEC on HSI, namely {a(k)12 , ψ(k)12 }3k=1, have mixed
signs, magnitudes, and statistical significance. It suggests that the way SSEC affects
HSI is relatively complex. Interestingly, ψ(2)12 = −0.019 and the associated t-statistic
is −3.197, suggesting the presence of the cross-sectional threshold effect at the weekly
level. We observe similar patterns for the cross-sectional effects of HSI on SSEC, but
the statistical significance is generally weaker than the converse case.
See Table 4 for results of the asymptotic and bootstrapped Wald tests. The
null hypotheses are (i) no threshold effects Ψ(k) = 02×2, (ii) Granger non-causality
from SSEC to HSI a(k)12 = ψ
(k)12 = 0, and (iii) Granger non-causality from HSI to
SSEC a(k)21 = ψ
(k)21 = 0, where k ∈ {1, 2, 3}. For the bootstrapped test, the parametric
bootstrap of Dufour, Pelletier, and Renault (2006) with S = 1000 bootstrap iterations
is used.
As expected from the simulation results in Section 5, the asymptotic and boot-
strapped p-values are almost identical to each other for all cases (Table 4). For the
sake of brevity, we discuss the asymptotic p-values only hereafter. The null hypothesis
of no threshold effects is rejected at the 5% level with the p-value being 0.045. This
rejection is evidence in favor of the VMAT-HAR model against the VHAR model. In
view of Table 3, this rejection likely stems from the statistically significant ψ(2)12 .
The null hypothesis of Granger non-causality from SSEC to HSI is rejected at
the 5% level with the p-value being 0.035. Although the signs, magnitudes, and
significance of {a(k)12 , ψ(k)12 }3k=1 are mixed, their overall significance is strong enough to
16
reject the joint zero hypothesis. This result indicates that the log RV of SSEC provides
incremental predictive ability on the log RV of HSI, a practically useful implication.
It is also evidence in favor of the bivariate VMAT-HAR model against the univariate
MAT-HAR model.
The null hypothesis of Granger non-causality from HSI to SSEC is not rejected
at the 5% level but rejected at the 10% level with the p-value being 0.071. It suggests
that Granger causality from HSI to SSEC is somewhat weaker than Granger causality
from SSEC and HSI, which is an intuitively reasonable result in view of the large
presence of Shanghai in the global stock markets. This result is also consistent with
Table 3, where the statistical significance of {a(k)12 , ψ(k)12 }3k=1 is generally stronger than
that of {a(k)21 , ψ(k)21 }3k=1.
7 Conclusion
The HAR model proposed by Corsi (2009) and Andersen, Bollerslev, and Diebold
(2007) has been adopted extensively to model and predict RVs of financial markets.
The VHAR model was proposed by Bubak, Kocenda, and Zikes (2011) among others,
and has been popularly fitted to realized variances or covariances. A potential draw-
back of the existing VHAR model is that the possibility of threshold effects is ruled
out.
The TAR model of Tong (1978) and its variants are a well-known approach for
capturing threshold effects, but the presence of an unknown threshold complicates
statistical inference, even in the univariate setting. Further, the constant threshold
could be an unrealistic assumption. Because of these issues, the TAR structure has
not been incorporated in the existing VHAR model.
To resolve this dilemma, the present paper has proposed the vector moving av-
erage threshold heterogeneous autoregressive (VMAT-HAR) model. It is essentially
a multivariate extension of the univariate MAT-HAR model of Motegi, Cai, Hamori,
and Xu (2020). Observed moving averages of lagged target series are used as thresh-
olds, which guarantees time-varying thresholds and the least squares estimation. 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 has established the statistical procedure of the VMAT-HAR model.
We have shown via Monte Carlo simulations that the VMAT-HAR model performs
well in small samples. As an empirical application, we fitted VMAT-HAR to daily
17
log RVs of the stock price indices of Hong Kong and Shanghai. Significant threshold
effects and mutual Granger causality are detected, an evidence in favor of VMAT-HAR
against the existing VHAR and MAT-HAR models.
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21
Table 1: Rejection frequencies of the asymptotic and bootstrapped Wald tests
Financial scenario: (m1,m2,m3) = (1, 5, 22)
Empirical size Empirical power
DGP-1 DGP-2 DGP-1 DGP-2
Hth0 : Ψ(k) = 0 HGC
0 : y2 ↛ y1 HGC0 : y2 ↛ y1 Hth
0 : Ψ(k) = 0
(Hth0 is true) (HGC
0 is true) (HGC0 is false) (Hth
0 is false)
Method T 1%, 5%, 10% 1%, 5%, 10% 1%, 5%, 10% 1%, 5%, 10%
Asymptotic 250 .028, .062, .130 .020, .078, .118 .330, .572, .694 .129, .284, .400
500 .008, .054, .101 .010, .048, .101 .768, .902, .939 .431, .661, .773
750 .013, .053, .107 .017, .051, .093 .994, .998, 1.000 .834, .931, .960
Bootstrap 250 .008, .042, .099 .012, .052, .107 .267, .523, .649 .091, .238, .350
500 .011, .048, .099 .010, .041, .080 .706, .878, .938 .414, .645, .748
750 .006, .043, .088 .014, .046, .089 .932, .981, .988 .800, .929, .979
Macroeconomic scenario: (m1,m2,m3) = (1, 3, 12)
Empirical size Empirical power
DGP-1 DGP-2 DGP-1 DGP-2
Hth0 : Ψ(k) = 0 HGC
0 : y2 ↛ y1 HGC0 : y2 ↛ y1 Hth
0 : Ψ(k) = 0
(Hth0 is true) (HGC
0 is true) (HGC0 is false) (Hth
0 is false)
Method T 1%, 5%, 10% 1%, 5%, 10% 1%, 5%, 10% 1%, 5%, 10%
Asymptotic 120 .020, .067, .136 .030, .089, .149 .171, .349, .472 .077, .192, .290
240 .016, .062, .126 .008, .057, .110 .435, .661, .768 .226, .459, .570
360 .011, .057, .114 .015, .059, .116 .688, .845, .901 .530, .734, .809
Bootstrap 120 .007, .053, .105 .012, .056, .118 .108, .268, .401 .048, .141, .249
240 .008, .064, .118 .011, .049, .107 .365, .611, .726 .214, .408, .532
360 .009, .051, .104 .009, .057, .107 .604, .827, .892 .427, .672, 772
The DGP is bivariate VMAT-HAR yt = A(0)0 +
∑Kk=1{A
(k)0 y
(k)t−1 + Ψ
(k)0 I
(k)t−1y
(k)t−1} + ϵt, where K = 3 and
ϵti.i.d.∼ N (02×1, I2). Under DGP-1, y2 Granger-causes y1 and a threshold effect does not exist. Under DGP-2,
y2 does not Granger-cause y1 and a threshold effect exists. In the financial scenario, (m1,m2,m3) = (1, 5, 22)
and T ∈ {250, 500, 750}. In the macroeconomic scenario, (m1,m2,m3) = (1, 3, 12) and T ∈ {120, 240, 360}.The null hypotheses of no threshold effects, Hth
0 , and Granger non-causality from y2 to y1, HGC0 , are tested
separately via the Wald tests, where the p-value is computed via the asymptotic χ2 distribution or the
parametric bootstrap of Dufour, Pelletier, and Renault (2006) with S = 500 bootstrap iterations. This
table reports the rejection frequencies across J = 1000 Monte Carlo samples, where the nominal sizes are
α ∈ {0.01, 0.05, 0.10}.22
Table 2: Sample statistics of the daily log realized volatilities of HSI and SSEC
# Obs. Mean Median Stdev Min Max Skew Kurt p-KS p-AD p-PP
HSI 730 −10.01 −10.10 0.764 −11.97 −7.251 0.429 3.071 0.069 0.001 0.000
SSEC 730 −10.17 −10.21 0.625 −11.66 −8.096 0.364 3.166 0.245 0.001 0.000
This table presents the sample statistics of the daily log realized volatilities, {y(1)dt }Tt=1, of Hang Seng
Index of Hong Kong (HSI) and SSE Composite Index of Shanghai (SSEC). The sample period is
January 23, 2017 through January 22, 2020 (T = 730 days). p-KS, p-AD, and p-PP signify p-values
of the Kolmogorov-Smirnov normality test, the Anderson-Darling normality test, and the Phillips-
Perron unit root test, respectively. The Phillips-Perron test is performed with the Bartlett kernel
and the Newey-West automatic bandwidth selection, where only an intercept is included in the test
equation. p-PP is the one-sided p-value of MacKinnon (1996).
23
Tab
le3:
Least
squares
estimates
andt-statistics
forthebivariate
VMAT-H
AR
model
onHSIan
dSSEC
EquationforHangSengIndex
ofHongKong(H
SI)
a(0
)1
a(1
)11
a(1
)12
a(2
)11
a(2
)12
a(3
)11
a(3
)12
ψ(1
)11
ψ(1
)12
ψ(2
)11
ψ(2
)12
ψ(3
)11
ψ(3
)12
Estim
ate
−1.131
0.321
0.130
0.568
−0.327
−0.001
0.197
0.007
0.007
0.006
−0.019
−0.006
0.005
(t-statistic)
(−2.415)
(5.345)
(1.948)
(5.627)
(−2.804)
(−0.010)
(1.735)
(1.218)
(1.104)
(0.941)
(−3.197)
(−1.268)
(1.111)
EquationforSSE
Composite
Index
ofShanghai(SSEC)
a(0
)2
a(1
)21
a(1
)22
a(2
)21
a(2
)22
a(3
)21
a(3
)22
ψ(1
)21
ψ(1
)22
ψ(2
)21
ψ(2
)22
ψ(3
)21
ψ(3
)22
Estim
ate
−1.350
0.007
0.256
0.209
0.434
−0.193
0.150
0.007
0.002
0.006
0.004
−0.007
−0.000
(t-statistic)
(−3.054)
(0.125)
(4.072)
(2.196)
(3.934)
(−2.013)
(1.400)
(1.287)
(0.426)
(0.934)
(0.636)
(−1.584)
(−0.112)
Thebivariate
VMAT-H
AR
model
yt=
A(0
)+
∑ K k=1{A
(k)y(k
)t−
1+
Ψ(k
)I(k
)t−
1y(k
)t−
1}+
utwithK
=3and(m
1,m
2,m
3)=
(1,5,22)is
fitted
tothe
daily
logrealized
volatilitiesof
Han
gSengIndex
ofHon
gKong(y
1t)andSSE
Composite
Index
ofShanghai(y
2t).
Thesample
periodis
January
23,2017
through
Jan
uary22,2020
(T=
730day
s).Thethreshold
term
sare
computedwithℓ T
=⌊√T⌋=
27day
soflags.
This
table
reportsthe
leastsquares
estimates
andtheassociated
asymptotict-statisticsforallparameters.
24
Table 4: Asymptotic and bootstrapped p-values of the Wald tests on HSI and SSEC
H0 : Ψ(k) = 02×2 H0 : a
(k)12 = ψ
(k)12 = 0 H0 : a
(k)21 = ψ
(k)21 = 0
(No threshold effects) (SSEC ↛ HSI) (HSI ↛ SSEC)
Asymptotic 0.045 0.035 0.071
Bootstrap 0.045 0.038 0.077
The bivariate VMAT-HAR model yt = A(0) +∑K
k=1{A(k)y
(k)t−1 +Ψ(k)I
(k)t−1y
(k)t−1} + ut with K = 3
and (m1,m2,m3) = (1, 5, 22) is fitted to the daily log realized volatilities of Hang Seng Index of
Hong Kong (HSI, y1t) and SSE Composite Index of Shanghai (SSEC, y2t). The sample period is
January 23, 2017 through January 22, 2020 (T = 730 days). The threshold terms are computed with
ℓT = ⌊√T ⌋ = 27 days of lags. The null hypotheses considered are Ψ(k) = 02×2 (i.e., no threshold
effects), a(k)12 = ψ
(k)12 = 0 (i.e., Granger non-causality from SSEC to HSI), and a
(k)21 = ψ
(k)21 = 0
(i.e., Granger non-causality from HSI to SSEC), where k ∈ {1, 2, 3}. Each null hypothesis is tested
separately by the Wald test, where the p-value is computed via the asymptotic χ2 distribution or the
parametric bootstrap of Dufour, Pelletier, and Renault (2006) with S = 1000 bootstrap iterations.
The p-values are reported in the table.
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Figure 1: Time series plots of log realized volatilities and moving average thresholds
2018 2019 2020-12
-11
-10
-9
-8
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(a) Daily HSI {y(1)1t }
2018 2019 2020-12
-11
-10
-9
-8
-7
(b) Daily SSEC {y(1)2t }
2018 2019 2020-12
-11
-10
-9
-8
-7
(c) Weekly HSI {y(2)1t }
2018 2019 2020-12
-11
-10
-9
-8
-7
(d) Weekly SSEC {y(2)2t }
2018 2019 2020-12
-11
-10
-9
-8
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(e) Monthly HSI {y(3)1t }
2018 2019 2020-12
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-9
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(f) Monthly SSEC {y(3)2t }
The sample period is January 23, 2017 through January 22, 2020 (T = 730 days). Daily, weekly,
and monthly log realized volatilities of Hang Seng Index of Hong Kong (HSI) and SSE Composite
Index of Shanghai (SSEC), denoted as {y(k)dt }Tt=1, are plotted with the blue, solid lines. For each
index d ∈ {1, 2} and frequency k ∈ {1, 2, 3}, the moving average threshold {µ(k)dt }Tt=1 is plotted with
the red, dotted lines, where ℓT = ⌊√T ⌋ = 27 days of lags are used.
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