Estimating the Dynamic Mixed Hitting Time Model Using
Characteristic Function Based Moments
Yogo Purwono∗
Irwan Adi Ekaputra†
Zaafri A.Husodo‡
May 25, 2015
Abstract
We propose a characteristic function-based moment method to estimate the dynamic mixed
hitting time model which specify duration between events as the successive passage times of
components of an underlying multivariate Brownian motion relative to in itself random boundaries, while the other, correlated, Brownian components generate the marks. The proposed
estimation method afcilitates computation and overcomes problems related to the discretization error in moment conditions and to the non-tractable probability density function. An
empirical application using transaction level data on shares of a company traded on the Indonesia Stock Exchange (ISX) are illustrated. The findings suggests that durations and return
volatility have strong persistence and they are contemporanously negatively correlated. The
assessment of instantaneous volatility levels accounting for instantaneous causality between
volatilities and durations is also investigated.
Keywords: duration modeling, mixed hitting time, market microstructure, characteristic
function
JEL codes: C13, C41, G12
1
Introduction
Dynamic Mixed hitting-time (DMHT) model is a structural model of duration between events
and associated marks. In this model, durations between events are specified as the first times an
∗
Corresponding author, Graduate Program in Management, University of Indonesia, email:[email protected]
Graduate Program of Management, University of Indonesia
‡
Department of Management, University of Indonesia
†
1
underlying latent process, which is one specific component of multivariate Brownian motion,
hits random positive boundaries, and the other, correlated, Brownian components generate
the marks. The DMHT model has been developed to facilitate the analysis about asset return
movements when they are observed at endogenous random times. Observation times are
endogenous precisely because they are closely related to the price and volatility process. This
situation primarily appears in tick by tick financial data where the prices, volumes and other
marks are recorded at irregularly spaced points in time.
Endogeneity in the duration between trades has been suggested in several literatures.
Asymmetric information theory of market microstructure (e.g Easley and OHara (1987)) explains that trades are induced by market participants reacting in part to new information
hitting financial markets. Therefore, it is very likely that the time between trades is not a
random exogenous process. It ought to interact with the price formation process. So if we use
the information that is in the durations between trades on top of the information from the returns between trades, it could potentially improve the accuracy of our estimates of asset return
as well as volatility. Other theoretical models (e.g., Admati and Pfleiderer (1988) and Foster
and Viswanathan (1990)) capture the endogeneity of durations from the relationship between
volatilities and length of trading periods. These studies explain that high (low) volatilities
occur during exchange trading (nontrading) periods.
Empirical evidences about the endogeneity in the durations between trades has also been
documented in the literature. Li, et al.(2009) find evidence of endogeneity of sampling time
in high-frequency data and establish a central limit theorem for realized volatility in an endogenous time setting. Using semiparametric (GMM) approach, Renault and Werker (2011)
document some empirical evidence of instantaneous causality between durations and volatility.
Beside facilitating the analysis of price movements under endogenous observation times,
the DMHT model allows one to address problems that cannot be handled by autoregressive
conditional duration (ACD) or stochastic conditional duration (SCD) models (e.g Engle and
Russell (1998) and Bauwens and Veredas (2004)). Examples include the modeling of optimal
execution time in dynamic trading and the modeling of durations between trades of several
assets. Under the ACD/SCD approach, we are not allowed to incorporate new information
that appeared since the previous event and could be relevant for predicting the timing of
the next event. Within the DMHT model, the arrival time of such new information can be
considered a separate event leading to an update of the distribution of the remaining duration.
A major difficulty in continuous-time financial modeling is the lack of efficient tools for estimating and making an inference with discretely observed samples, especially when samples are
2
observed at endogenous random times. This is particularly striking for the DMHT model studied in this paper. The maximum likelihood (ML) estimation is usually inapplicable because
the transition density is rarely in closed-form. The frequently used simulation-based methods
are difficult to implement since the model is hard to simulate and the traditional GMM is
computationally demanding since high order derivatives need to be calculated. Fortunately,
for most of continuous time financial models are in the Levy class, and their analytical characteristic functions are obtainable. Since the characteristic function is equivalent to and contains
the same information as the probability density function, the estimation using charateristic
function can be as efficient as the ML estimator.
This paper considers characteristic function based GMM with continuum of moments to
estimate the DMHT model of the joint behavior of durations between events and asset return
in which both durations and returns are generated by an underlying multivariate Brownian
motions. Since the model that we want to estimate represent the joint process of two variables,
we propose a characteristic function-based method to jointly estimate model by using information contained in both duration between consecutive trades and the trade prices. Given the
concrete specification of duration and return proces, we derive the conditional joint characteristic function of return and duration. Following Carrasco and Florens (2000) and Carrasco et
al (2007), the estimator of the model are then derived using the characteristic function based
continuum GMM. Monte Carlo and empirical studies show that the method is computationally
less costly than the other methods and can be easily adapted to different specifications of the
model and the definition of durations. Estimation results indicate that durations and volatility
have strong persistence and they are contemporanously negatively correlated. Estimate results
also confirm some stylist facts in high frequency financial data as suggested in some market
microstructure theories.
The remainder of this paper is organized as follows. Section 2 builds the DMHT model.
Section 3 describes the characteristic function-based estimation method and provides a Monte
Carlo study. Section 4 discusses empirical results and their implications. Lastly, Section 5
concludes the paper.
3
2
Dynamic Mixed Hitting Time Model
2.1
Specification and Characterization
Let (ti )ni=0 be the increasing sequence of non-negative random times that we will call event
times for expository reasons. These times may, in general, represent any calender time, such
as the transaction (trading) times of a stock, or any event of interest, for instance, the time
at which a certain level of volume in an asset has been traded. The modeling interest is to
describe the stochastic behavior of the event time ti conditional on the information available
at some time t < ti as well as the joint stochastic behavior of event time ti and its associated
mark, such as asset price Sti . Let (Ft )t≥0 be the continuous-time filtration that represents the
time evolution of information available to the econometrician. We assume that the filtration
(Ft )t≥0 satisfies the usual condition.
The DMHT model specifies an event time as the first time a real-valued (underlying)
Wiener process (Wd,t )t≥0 , defined relatively to the filtration (Ft )t≥0 , hits a given positive
random boundary whose values depend on past observations as well as unobservable mixing
variables Mi . More precisely, if we have just seen an event at time ti , the time ti+1 , when the
next event occurs is defined by
ti+1 = inf{t > ti : Wd,t − Wd,ti + µ̄ti (t − ti ) = h̄ti }
(1)
where µ̄ti and h̄ti are the drift and the hitting boundary respectively. The dirft and hitting
boundary are assumed to be strictly positive Fti -measurable random variables. In addition,
they are also assumed to may depend on Mi . In other words, µ̄ti and h̄ti are given by
µ̄ti = µti (Mi ),
(2)
h̄ti = hti (Mi ).
(3)
The drift µ̄ti and hitting boundary h̄ti altogether capture observed heterogeneity in the
thresholds and associated hitting times. The heterogeneity in µ̄ti and h̄ti can generate all
kinds of autoregressive or log–autoregressive dynamics in order to reproduce stylized facts for
the durations of interest. In the DMHT model specified in (1), mixing variables (Mi ), for
i = 1, 2, ..., n, represent unobserved risk factor that determine duration between events and
are assumed to be i.i.d. positive random variables that are independent of Wd .
The DMHT model is introduced by Renault, van der Heijden, and Werker (2014) in or-
4
der to provides a general framework to perform inference about asset returns when they are
observed at endogenous random times, either in the univariate or multivariate context. This
modeling scheme also offers flexibility in understanding the stochastic behaviour of financial
duration data beyond the SCD/ACD specifications. In the DMHT model, the mixing component Mi potentially generates a large class of (conditional) duration distributions. Depending
on the functional form on the way the mixing variable explains µ̄ti and h̄ti and the assumed
distribution for Mi , the DMHT model can be expanded to accommodate stylized facts in intraday trading like clustering of short durations or fat tails of the duration distribution, see,
e.g., Valenti, Spagnolo, and Bonanno (2007). Although we can impose some distributions with
non negative support for Mi , the gamma distribution turns out to be sufficiently flexible to
accommodate both very short as well as very long durations (at least as shown in Whitmore
(1986)).
This general duration model nests a number of specific models, for instance, SCD and ACD
models, depending on the appropriate restrictions on the imposed parameters. For example,
the SCD duration model can be obtained by imposing the expected duration hti µti to be a
constant. Therefore, in the estimation, we can naturally impose the testable hypothesis about
the existence of SCD or ACD specification.
It is known (see, e.g., Borodin and Salminen (2002), Relation 2.2.0.1) that, when the event
times occur according to the model specified in (1), the conditional distribution of dti+1 given
Fti can be characterized by its conditional moment generating function (MGF)
s
Eti [exp(udti+1 )|Mi ] = exp µ̄ti h̄ti − µ̄ti h̄ti
2u
1 − ¯2
µ
!
(4)
ti
Here, Eti is short-hand for the conditional expectation given Fti . The conditional MGF in (4)
implies
dti+1 |Fti , Mi ∼ IG
h̄ti 2
, h̄
µ̄ti ti
where IG(α, θ) denotes the Inverse Gaussian distribution with mean α and shape parameter θ.
Following the terminology introduced by Ghysels, Gourieroux, and Jasiak (2004), the DMHT
model has a two-factor SVD like model that can be characterized by the factors (µ̄ti , h̄ti ).
The two-factor structure has an advantage in the ability to accommodate separate dynamic
patterns for the conditional mean and variance of durations. From (4), the first two moments
5
of the conditional distribution of duration are
h̄ti
,
µ̄ti
h̄t
V arti [dti+1 |Mi ] = 3i
µ̄ti
Eti [dti+1 |Mi ] =
(5)
To ensure that the first and second moment for the duration are finite, we need a strictly
positive drift in the model.
In order to apply the characteristic function based estimation method to the model, we
need to represent the conditional distribution of dti+1 given Fti and Mi in term of its conditional characteristic function (CF). For MGF MX (t) and CF ϕX (t) there is a one-to-one
correspondences, where ϕX (t) = MX (it). Based on this relationship, the conditional CF of
dti+1 given Fti and Mi is given by
s
Cµ̄t
i ,h̄ti
2.2
(u) = Eti [exp(iudti+1 )|Mi ] = exp µ̄ti h̄ti − µ̄ti h̄ti
2iu
1 − ¯2
µ
!
(6)
ti
Model Including Asset Return
The DMHT model discuss in section 2.1 covers the modeling framework for the duration
processes between transaction times only. We can extend the model to include observed
marks, for instance asset returns, at these time points as well.
Let Sti be the univariate mark, where in Section 4 they refer to the logarithm of the
prevailing mid-quote at time ti . For ease of exposition, also in this theoretical section, we
already refer to as Sti the price at time ti .
Suppose that the prices Sti are embeded in a Wiener process Ws , correlated with Wd . More
precisely, as in Renault et al (2014), we assume that for ti < t ≤ ti+1 ,
q
St − Sti = ᾱti (t − ti ) + V̄ti (Ws,ti+1 − Ws,ti )
(7)
where ᾱti = αti (Mi ) and V̄ti = Vti (Mi ) are the prevailing drift and volatility of the price
process respectively. It is assumed that Ws has correlation with the Wiener process Wd that
define durations such that
Wd,t = Z1,t
Ws,t = ρ̄s,ti Z1,t +
6
q
1 − ρ̄2s,ti Z2,t
where their coefficient correlation is ρ̄ti . For reasons of identiability, it is assumed that the
Wiener process Z1,t is independent of Z2,t . The coefficient Correlation between durations and
prices is modeled through the coefficient ρ̄ti . Hence, the concrete model for the prices Sti are
St − Sti = ᾱti (t − ti ) +
q h
q
i
V̄ti ρ̄s,ti (Z1,t − Z1,ti ) + 1 − ρ̄2s,ti (Z2,t − Z2,ti )
(8)
As with µ¯ti dan h¯ti in section 2.1, the drift ᾱti , the spot variance V̄t , and ρ̄ may all depend on
the mixing variable Mi .
Renault et al. (2014) in the Theorem 3.1 show that the distribution of the price change
Sti+1 − Sti = Rti+1 in (8) conditional on Fti , Mi , and dti+1 is Gaussian with mean
Eti [Rti+1 |dti+1 , Mi ] = ᾱti dti+1
q
+ (h̄ti − µ̄ti dti+1 ) V̄ti ρ̄ti
(9)
and variance
V arti [Rti+1 |dti+1 , Mi ] = (1 − ρ̄2s,ti )V̄ti dti+1
(10)
The theorem allows us to directly characterize the joint conditional distribution of returns
and durations in terms of its characteristic function. In other words, it paves the way for
efficient inference by GMM with a continuum of moment conditions in the spirit of Carrasco
and Florens (2000) and also Carrasco et al (2007). Because the conditional distribution of
returns Rti+1 given Fti , Mi , and dti+1 is Gaussian, then its conditional characteristic function
is given by
1
Eti [exp(ivRti+1 )|dti+1 , Mi ] = exp imti v − s2ti v 2
2
where mti and s2ti are in (9) and (10) respectively.
Equation (9) and (10) explain that both the mean and the variance of the price changes are
affine (linear in their states) in the contemporaneous duration dti+1 . Therefore, using the law
of iterated expectation, one can derive the conditional joint characteristic function of duration
dti+1 and price change Rti+1
q
Eti [exp(iudti+1 + ivRti+1 )|Mi ] = exp[iv V̄ti ρ̄,ti h̄ti ]
q
iv 2
Cµ̄t ,h̄t u + v ᾱ − v V̄ti ρ̄ti h¯ti +
(1 − ρ̄2ti )V̄ti
i
i
2
where Cµ̄t
i ,h̄ti
(11)
(.) is conditional CF in (6). In order to apply the GMM estimation with con-
7
tinuum of moment conditions, one needs to derive the conditional joint characteristic function
Eti [exp(iudti+1 + ivRti+1 )] by integrating out the mixing variable Mi . We will present explicitly this conditional joint characteristic function in Section 4 when we discuss the empirical
application.
3
Econometric Methodology
Assume we observe at random event points in time (ti ) a log-price process (Sti ), defined on some
filtered probability space (Ω, F, (Ft )t≥0 , P) that has dynamics as specified in (1). Our interest
in this paper is estimation of parameters for process Sti which is robust to the specification
of the rest of the components in the model, i.e., the drift term αt and the correlation between
log-price process and duration between events. Importantly, we will be interested in developing
an estimation method that makes full use of information contained in the duration between
events and exploits tractability of the characteristic function of the model.
Let Xt ∈ Rm (t = 1, 2, ..., T ) be an m-dimensional random vector process whose distribution
is indexed by a finite dimensional parameter θ with true value θ0 . When the process Xt
is stationary and Markovian, Feuerverger and McDunnough (1981b) show that Generalized
Method of Moments (GMM) estimation can be applied to a discrete set of the following moment
conditions by restricting the continuous index u ∈ Rm to a discrete grid u ∈ (u1 , u2 , ..., uN ),
E[y(u, Xt )(exp[iu0 Xt+1 ] − ϕ(u; θ, Xt ))] = 0
(12)
where u ∈ Rm is the Fourier transformation variable, i is imaginary number such that i2 = 1,
ϕ(u; Θ, Xt ) = E θ (exp(iu0 Xt+1 )|Xt ) is the conditional characteristic function (CCF) of Xt and
exp[iu0 Xt+1 ] is its empirical counterpart (ECF), and E θ is the expectation operator with
respect to the data generating process indexed by θ. The Estimation consists of minimizing a
norm of the sample average of the moment function y(u, Xt )(exp[iu0 Xt+1 ] − ϕ(u; θ, Xt )) with
objective function involves an optimal weighting function
−1
Z
∞
z(u, Yt ) = (2π)
−∞
∂ log f (Yt+s |Yt , θ)
exp(−iuYt+s )dYt+s ,
∂θ
(13)
that depends on the true unknown probability density function. They show that the asymptotic
variance of the resulting estimator can be made arbitrarily close to the Cramer-Rao bound by
selecting the grid for u sufficiently fine and extended. However this estimation is difficult to
8
apply because for most of continuous time models, it is impossible to have the density function
for the model.
Similar discretization approaches are used in Singleton (2001) and Chacko and Viceira
(2003). The basic idea behind their approach is to construct moment conditions by firstly
dividing the domain of the characteristic index into a finite number of grids and then approximating the instrument. However, one critical problem of their approach is when the number
of points in the grid of u is too refined or too extended in order to improve efficiency, the
associated covariance matrix of the discrete set of moment conditions becomes not invertible.
As a result, it is necessary to apply operator methods in a suitable Hilbert space to be able to
handle the estimation procedure at the limit.
Carrasco and Florens (2000) develop a characteristic function based GMM that permits to
efficiently use the whole continuum of moment conditions (CGMM) that is computationally less
demanding than the commonly used simulation-based method and traditional GMM as well
as solves the problems of singularity and instability induced by the discrete moment condition
methods. Carrasco, Chernov, Florens and Ghysels (henceforth, CCFG (2007)) extend the
scope of the CGMM procedure to Markov and weakly dependent models. Because our model
deal with the joint dynamics of return and duration between transaction, we follow CCFG
(2007) approach, but with some modification due to the nature of our process.
When the process Xt is directly observed at non random integer times, one can match the
empirical and model-implied conditional characteristic function at a given lag K using some
forms of objective function. In the CGMM procedure, when Xt is Markov instead of being
IID, CCFG (2007) propose to use the moment function based on the conditional CF (CCF):
mt (τ, θ) = (exp[iu0 Xt+1 ] − ϕ(u; θ, Xt )) exp(iv 0 Xt )
(14)
where as before ϕ(u; θ, Xt ) denote the CCF of Xt and τ = (u0 , v 0 ) ∈ R2m . CCFG (2007) show
that the instruments {exp(iv 0 Xt ), v ∈ Rm } in (8) are optimal given the Markovian structure
of the model. Moment functions defined by (8) describes a martingale difference sequence.
Under certain regularity conditions, the efficient estimator of CGMM based on CCF (which
we will call CCF–CGMM for simplicity) is given by
D
E
−1/2
−1/2
θbT (λT ) = arg min KT,λT m
b T (., θ), KT,λT m
b T (., θ)
(15)
θ
where m
b T (., θ) =
1
T
PT
t=1 mt (τ, θ)
−1/2
is the sample conterpart of moment functions, KT,λT is the
9
weighting covariance operator associated with pre-determined moment function mt (τ, θ), and
λT denotes the optimal regularization parameter for a given sample size T .
In the CGMM estimator given in (9), the objective function
D
E
−1/2
−1/2
Q = KT,λT m
b T (., θ), KT,λT m
b T (., θ)
is an inner product defined on Hilbert space L2 (π) of complex valued functions that are square
integrable, which is defined as:
2
L (π) = {f : R
m
Z
→ C;
f (u)f (u)π(u)du < ∞}
(16)
where f is the complex conjugate of f for all f ∈ C. Here π is an arbitrary finite measure on
R2m that has nothing to do with the data generating process of Xt . As long as π > 0, the
choice of π does not affect the estimation efficiency in large sample. In practice, it is customary
to set π(τ ) = exp(−τ 0 τ ) in order to be able to compute (9) using Hermitian quadratures. The
inner product on L2 (π) may be defined as
Z
hf, gi =
f (τ )g(τ )π(τ )dτ
where the overline denotes complex conjugate.
The CCF–CGMM estimator θbT (λT ), under some regularity conditions (see Carrasco and
Kotchoni, 2010), is asymptotically normal
T 1/2 θbT (λT ) − θ0 → N (0, Iθ−1
)
0
(17)
as T and λ2T T go to infinity and λT goes to zero, where Iθ−1
denotes the inverse of the Fisher
0
Information Matrix, and θ0 is the true parameter vector. Carrasco and Florens (2000) show
that the asymptotic variance of the optimal CGMM estimator is
VT
b t (., θ) =
where G
∂m
b T (.,θ)
∂θ
i
h√ T θbT (λT ) − θ0
= V ar
D
E−1
−1/2
b t (., θ) , K −1/2 E G
b t (., θ)
=
KT,λ E G
T,λ
b t,j (., θ) =
is a column vector of length l whose j th element is G
(18)
∂m
b T (.,θ)
,
∂θj
and for every two vector functions g and h, we have: hg, hii,j = hgi , hj i. The asymptotic
10
variance (12) is consistently estimated by:
D
E−1
−1/2 b
−1/2 b
\
VbT = AV
ar θb = KT,λ G
T (τ, θ), KT,λ GT (τ, θ)
b T (τ, θ) =
where G
1
T
PT
(19)
t=1 Gt (τ, θ).
As in (9), here KT is the covariance operator with the
i
kernel k(s, τ ) = E mt (s, θ), mt (τ, θ) such that
b
h
Z
Kf (τ ) =
k(s, τ )f (s)π(s)ds
(20)
Because KT is not invertible, it need to be stabilized by a regularization parameter λT ∈ [0, 1]
such that
−1
KT,λ
= KT2 + λT IT
T
−1
KT
(21)
where IT is the identity operator and T is the sample size. Hence, consistent estimate for
KT,λT can also be obtained from the sample moment conditions. See Carrasco and Florens
(2000) and Carrasco and Kotchoni (2013) for further discussion of the consistency and root–T
asymptotic normality of the CCF–CGMM estimator.
In our model, the estimation problem is more complicated as we observe the price process
Sti at endogeneous random times ti . The endogeneity of observation times are precisely because
they are probably related to the price process and its volatility. We overcome this problem by
jointly model the marks and duration between events. Hence for the estimation problem here
we can only work with the joint conditional characteristic function ϕti (u, v; Θ, Yti+1 ) (JCCF)
instead of CCF ϕt (u; θ, Xt+1 ), where Θ is now the parameters vector that we will estimate
and Yti+1 = (Rti+1 , dti+1 ) is the joint process of return and duration between events. Using
the above CCF–CGMM estimation analysis, we propose to base inference on matching the
model implied ϕti (u, v; Θ, Yti+1 ) with its sample estimate exp(iudti+1 + ivRti+1 ). If the joint
process of duration and return is stationary and Markov within a day, then exactly as above
appropriate weighting of these moment conditions will yield the Cramer-Rao efficiency bound
based on daily direct observations of (Rti+1 , dti+1 ).
More specifically, our vector of moment conditions is given by
Eti mti (τ, Θ) = (exp(iudti+1 + ivRti+1 ) − ϕti (u, v; Θ, Yti+1 )) exp(iwdti ) = 0
(22)
where Eti is short-hand for the conditional expectation given Fti and τ = (u, v, w) is the Fourier
transform. In equation (22), the set of basis functions {exp(iwdti ), w ∈ Rm } is being used as
11
instruments. These instrument are choosed by considering the efficiency of the estimation
given the Markovian structure of the model (as suggested by CCFG, 2007). Because our joint
process (Rti+1 , dti+1 ) are Markov, then the above instrument must be optimal.
Based on the moment functions
mti (τ, Θ) = (exp(iudti+1 + ivRti+1 ) − ϕti (u, v; Θ, Yti+1 )) exp(iwdti ),
then our estimator is minimum distance (GMM) with estimating equations in the vector given
c converging in probability to a positive definite matrix W :
in (22) and some weight matrix W
D
E
b N (λN ) = arg min W
cm
cm
Θ
b T (., Θ), W
b T (., θ)
Θ
where m
b N (., Θ) =
1
N
PN
ti =1 mti (τ, Θ)
(23)
is the sample conterpart of moment functions mN (τ, Θ).
c to be an estimate of the optimal weight covariance
As in CCF–CGMM analysis, we set W
operator defined by the asymptotic variance of the empirical moments to be matched, that
c = K −1/2 , while as before λN denotes the optimal regularization parameter for a given
is W
N,λN
sample size N . Note that because of the separability of data from parameters in the moment
vector mN (τ, Θ), we construct our optimal weight matrix by using only the data. This in
particular means that all moments are weighted the same way regardless of the model that is
estimated, provided the set of moments used in the estimation is kept the same of course. The
consistency and root–T asymptotic normality of our CJCF-CGMM estimator follow Carrasco
and Florens (2000) and Carrasco and Kotchoni (2013) studies.
Although in their original paper, CCFG (2007) have proved that the optimal weighting
c = K −1/2 , we also use the identity matrix as the weighting covariance
covariance operator W
N,λN
operator in order to compare the numerical instability. It is shown in Cochrane (2005) that the
use of identity matrix as the weighting matrix in GMM-based estimation, will produces more
robust and stable estimates. We conduct a Monte Carlo study to evaluate the loss of efficieny
in using other matrix such as the identity matrix as the weighting covariance operator. So far,
the Monte Carlo study shows that the loss of efficiency in using the identity matrix is tiny
and that the algorithm is more stable and faster than that based on the optimal weighting
operator. However we do not show yet the results in this paper, because we still have not been
sure about the validity of the results since the number of trial in the Monte Carlo study is
relatively little. Perhaps we present the Monte Carlo result in the next draft of paper.
12
4
Empirical Illustration
In this section, we illustrate the application of the proposed estimation method using event-byevent data for a large cap stock, and discuss their implication. Section 4.1 presents the data,
Section 4.2 explains the concrete specification of the DMHT model that we want to analyse,
Section 4.3 presents the estimation procedure, and Section 4.4 reports the parameter estimates
and discusses the results.
4.1
Data
The data used in this paper is intraday transactions of a large cap stock Telkom (ticker: TLKM)
traded in the Indonesia Stock Exchange (ISX) during March 2015, for a total of 20 trading
days. They were obtained from Thomson Reuters Tick History. The dataset contains the
following series within a day: Stock Ticker, Trading Date, Trading Time, Trading Price, Best
Bid and Ask Prices, Trading Volume, and Others. TLKM is one of the largest corporations
listed on the ISX, with a total of about 1.24B shares outstanding and a market capitalization
around US180B. .
The Thomson Reuters Tick History dataset originally contains microsecond time-stamps
transactions. Because we are interested not only in durations, but also in returns and volatility
per unit of time, we aggregate events that occur within a single second. If we include very short
(microsecond) durations in the analysis, the volatility per unit of time tends to inflate when a
nonzero return is observed. We follow the common practice (e.g., Engle and Russell, 1998 and
Bauwens and Veredas, 2004) of aggregating these zero return trades as a single transaction.
For each duration, the corresponding change in stock price (return) is recorded as well. We
occasionally see multiple trades recorded with the same timestamp. They are trades from
multiple buyers or sellers or split-transactions occurring when the volume of an order on one
side of the market is larger than the available opposing orders. We take, as in Engle (2000),
the stock price at each transaction from the log of the geometric average of the prevailing ask
and bid quotes. For simplicity, we refer to this value as the midquote. After aggregation, the
final data sets contain 35284 observed durations.
We use only single exchange to allows us discarding effects on the transaction process caused
by different market structures for various exchanges (as suggested in Engle, Ferstenberg, and
Russell, 2008). Hence, in this illustration we implicitly assume that a transaction is executed
on a given exchange just by chance and not because it has specific characteristics.
Summary statistics for both the durations dti+1 and the price changes Rti+1 for consecutive
13
Series
Duration (sec)
Return (%)
Series
Duration (sec)
Return (%)
Mean
Std dev
Min
Median
95%
99%
Max
12.72
15.99
1
3
38
72
384
-8.55E-7
9.88E-4
-1.59E-2
0
1.73E-3
1.77E-3
1.11E-2
Skewness Kurtosis ACF(1) ACF(2) ACF(3) ACF(4) ACF(5)
4.9663
64.1980
0.1729
0.1401
0.1352
0.1225
0.1155
0.1716
111.8810
-0.4253
-0.0033
-0.0044
0.0065
0.0120
Table 1: Descriptive statistics of TLKM durations (dti+1 ) and returns (Rti+1 = Sti+1 − Sti )
Figure 1: Histogram of duration between trades of TLKM during March 2015
trades are presented in Table 1, while the histograms of these durations and price changes are
plotted in Figures 1 and 2 respectively. The table and figures document well-known stylized
facts of durations, such as excess dispersion, and a large fraction of very short durations
combined with a fat right tail. Just over 30% of all durations for TLKM stock has a length
of one second. The distribution of the durations turns out to be skewed to the right and
the sample kurtosis is also larger than the normal value of three. It is interesting to know
the largest intervals between trades in this sample is around five minutes and of course the
minimal interval is one second. The average waiting time for a trade to happen is slightly less
than 15 seconds.
For the returns, we observe that returns are very much concentrated; more than half of
the TLKM trade durations do not induce a change in the midquote. The return distribution
is smoother because the tick size is smaller relative to the share price. As for temporal dependence, we see the usual strong negative autocorrelation of order one in the return series, often
associated to the price bouncing between the bid and ask. We also see decaying (persistent)
dependence in the duration series but the magnitude of the autocorrelation of order one is
smaller for the durations (17%) than for the returns (42%) .
Figures 3 and 4 display average durations and average volatility per seconds that we calcu-
14
Figure 2: Histogram of return of TLKM during March 2015
Figure 3: Diurnal pattern of durations, averaged over 30 minutes
Figure 4: Diurnal pattern of return volatility per second, averaged over 30 minutes
15
lated every 30 minutes respectively. It is widely known that transaction intensities and return
volatilities exhibit intraday seasonality. For instance, average durations between transactions
and quote changes tend to be longer around lunch time than near the opening and the closing
of the trading day. This seasonality pattern, that is usually call the diurnal pattern, holds true
for our data as well, as shown in Figures 3 and 4. These pattern also true for the adjusted
data following the correction for intraday seasonality which we described in Section 4.2.1.
4.2
Parameterization
The modeling scheme considered in Section 2 so far did not make any specific assumptions on
the distribution of the mixing variable Mi or on the way µ̄ti and h̄ti depend on Mi . In order
to apply the estimation framework to the data, we need to impose a concrete specification for
all component in DMHT model that allows us to understanding of the features of the model
more precisely. This specification is also the one that will be estimated using the TLKM data.
A fully specified DMHT model consists of assumptions about 1) the correction for intraday
seasonality, 2) the distribution of the mixing variable Mi , 3) the dynamic properties of µti and
hti , and 4) return volatility dynamics.
4.2.1
Intraday Seasonality
It is well known that there is deterministic seasonal pattern during the business day for volatility (e.g., see Andersen and Bollerslev (1997)) and trade intensity (e.g., see Engle and Russell
(1998)). For assets that are not traded around the clock, volatility is usually higher when
the markets are opening, it lowers during the middle part of the day before increasing again
after that. A similar pattern exist for durations: they are relatively shorter at the opening
and closing of the market, and relatively longer during the middle of the business day. Trade
intensity being inversely related to durations, have a smiler pattern as volatility: relatively
higher trade intensity at the beginning and end of the business day than during the middle of
the day.
To capture these diurnal patterns, we introduce two deterministic functions of the calender
time of the day, gd (t̃i ) and gd (t̃i ) for the duration and return volatility respectively. Following
Engle and Russell (1998) and functional form inspired by the intraday seasonality corrections
in Hasbrouck (1999) and Andersen, Dobrev, and Schaumburg (2008), we adjust duration and
16
return volatility using the following function respectively:
gd (t̃i ; g1 , g2 , g3 ) =
exp[−(g1 − g3 )t̃i − g3 ] + exp[−g2 (1 − t̃i )]
[exp(−g1 ) − exp(−g3 )]/(g3 − g1 ) + [1 − exp(−g2 )]/g2
(24)
1 − exp(−gS2 )
gS (t̃i ; gS1 , gS2 , gS3 , gS4 ) = 1 + gS1 exp(−gS2 t̃i ) −
gS2
1 − exp(−gS3 )
+ gS4 exp(−gS3 (1 − t̃i )) −
gS3
(25)
We assume that adjusted duration takes the form dd
ti+1 ≡ g(t̃i )dti+1 , where dti+1 is the observed
durations. The realized return volatility per unit of time will follow the same correction accordingly. The intraday seasonality correction for both durations and prices will be normalized
R1
R1
by imposing 0 g(t)dt = 1 and 0 gS (t)dt = 1, so that the correction at any point is relative
to the average duration (volatility) in the sample. The parameters of gd (t̃i ) and gd (t̃i ) will be
calibrated by using the least square error principles.
4.2.2
Mixing Distribution
For simplicity, we assume multiplicative form for hitting barrier h̄ti = hti and constant form
for the drift µ̄ti = µti . We endow the mixing variable Mi with a Gamma distribution with
parameter B and imposing E(Mi ) = 1 as an identifying restriction. Its density thus reads
fM (m) =
B B B−1
m
exp(−Bm)
Γ(B)
(26)
Depending on the value of B, the density at zero may be vanishing,
nite or diverge to infinity, which turns out to be an empirically desirable feature. While
Mi has a constant mean 1, its variance is inversely proportional to B.
4.2.3
Specification for Duration Components
Intraday financial durations exhibit significant autocorrelation that dies out only slowly. For
comparison purposes, we endow the expected durations
hti
µti
with some GARCH-type dynamics
as in Engle and Russell (1998). More precisely, we specify
ht
hti
= β0 + β1 (dti )β3 + β2 i−1
µti
µti−1
in which β0 , β1 , β2 > 0 dan β3 ∈ [0, 1].
17
(27)
In order to allow, but not to impose, an SCD specification, we model
µti = ω4 + ω1 +
ω2
hti /µti
ω6 /2
+ ω3 (dti )ω5
(28)
To test the existence of the SCD specification, we can test the following hypothesis ω1 = ω3 =
ω4 = 0 and ω6 = 1, because the SCD specification would need the constant µti hti .
4.2.4
Return Model
For the model of the returns, the drift of the price process per unit of time is assumed to be
constant αti = α and also for the coefficient correlation between the Brownian motions W and
Z (generating durations and returns, respectively) ρti = ρ . For the return volatility per unit
of time we assume that
V̄ti =
(δM
Vti
+ θM Mi )2
(29)
where δM and θM are restricted by the normalization
E
(δM
1
=1
+ θ M Mi ) 2
(30)
This normalization identifies the unconditional level of V̄ti and Vti . Finally, we use a GARCH
model that accounts for the irregularly observed data following Meddahi, Renault, and Werker
(2006),
Vti = α0 gS (t̃i ) + (1 −
dt
α1 )α2 i
Rti − αdti
p
dti
!2
dt
+ α1 α2 i Vti−1
(31)
where gS (t̃i ) is as specified in section 4.2.1.
4.3
Conditional JCF–CGMM Based Inference
The concrete model in section 4.2 is then estimated using the GMM with continuum of moment
based on the (conditional) joint characteristic function (CJCF) as presented in section 3.
However, we need to derive the CJCF that only depend on flow of information Fti in order to
apply the proposed estimation method. From the above concrete specification, applying the
law of iterated expectation to the CCF of return given contemporaneous duration and mixing
variable, we have the conditional JCF of returns and durations conditional on Mi ,
Eti [exp(iudti+1 + ivRti+1 )|Mi = mi ] =
18
s
"
exp iv
!
#
s
Vti
2ir
ρ hti mi + µti hti mi − µti hti mi 1 − 2
(δm + θm mi )2
µti
(32)
where
s
r = u + vα − v
!
Vti
iv 2
Vti
2
ρ hti mi +
(1 − ρ )
(δm + θm mi )2
2
(δm + θm mi )2
Integrating out the mixing variable Mi , one would then get a continuum of moment conditions
(indexed by (u, v) ) for the purpose of CGMM inference. The following proposition gives the
CJCF of duration and return that we will use in the estimation of TLKM data
Proposition 1 The CJCF of duration and return with concrete specification defined in section
4.2 is given by:
ψti (u, v; θ) = Eti [exp(iudti+1 + ivRti+1 )]
Z
∞
Eti [exp(iudti+1 + ivRti+1 )|Mi = mi ]
=
0
B B B−1
m
exp(−Bmi )dmi
Γ(B) i
(33)
where Eti [exp(iudti+1 + ivRti+1 )|Mi = mi ] is defined in (11), Γ(.) is the Gamma function, and
θ ∈ Θ is the set of parameters.
The estimation then will be conducted using the following moment conditions:
E[ exp(iudti+1 + ivRti+1 ) − ψti (u, v; θ) Y (., dti )] = 0
(34)
where we set Y (., dti ) = exp(iwdti ) as the instrument for the optimality of our estimation as
suggested by CCFG (2007).
Based on the moment conditions (28), our CJCF–CGMM estimator for the model using
the procedure suggested in section 3 is
D
E
b N (λN ) = arg min K −1/2 m
Θ
b N (., Θ), K −1/2 m
b N (., Θ)
Θ
(35)
where as above m
b N (., Θ) is the sample counterpart of the moment functions m(u, v, w; Θ)
which form the moment condition (28)
m(u, v, w; Θ) = exp(iudti+1 + ivRti+1 ) − ψti (u, v; θ) exp(iwdti ) = 0
The estimation and inference then just follows the procedure discusses in section 3.
19
(36)
4.4
Estimation Results
Table 2 reports estimation results including parameter estimates and their standard deviations.
Models are estimated by the Joint CCF-CGMM described in Section 3. The standard errors,
which are computed using the formula (23) combined with (25), (26), and W = K, are
presented in brackets.
θ = (β0 , β1 , β2 , β3 , ω1 , ω2 , ω3 , ω4 , ω5 , ω6 , α, α0 , α1 , α2 , ρs , ρu , δM , θM , σu2 , B)
Parameter
Estimate
(Std Errors)
Parameter
Estimate
(Std Errors)
Parameter
Estimate
(Std Errors)
β0
1.2E-13
0.0081
ω4
0.00
0.012
ρ
-0.075
0.006
β1
0.100
0.006
ω5
0.171
0.016
δM
0.000
0.005
β2
β3
ω1
0.955 0.915
0.000
0.002 0.0019 0.013
ω6
α
α0
2.300 0.000 3.7E-8
0.068 0.00 3.0E-10
θM
B
1.095 20.518
0.000 0.005
ω2
2.325
0.052
α1
0.625
0.014
ω3
0.042
0.011
α2
0.795
0.005
Table 2: Descriptive statistics of TLKM trade durations (dti+1 ) and logprice changes (Sti+1 − Sti )
The first thing to notice is all the estimates are statistically significant. A second thing
to look at is the dynamic nature in the return process when they are observed at endogeneous random times. When we look at expected duration related parameters, we find that
the conditional expected durations are non-explosive because the estimate of β1 and β2 are
both highly significant and their sum is greater than 1 (1.055), while at the same time both
beta2 < 1 and beta3 < 1. This confirm the properties of DMHT-based duration model in
Remark 4.1. of Renault et al (2013). From the estimates of the observe heterogeneity µ̄ti
parameters ω1 , ω2 , ω3 , ω4 , ω5 , ω6 , we can test if the dynamic of durations follow the SCD
(and ACD) specification. Based on discussion in the section 4.2, the null we need to test
is H0 : ω1 = ω4 = ω6 − 1 = 0. This hypotheses can be tested using a standard Wald test.
Under the null, the test statistic is asymptotically χ23 distributed. It’s 99% quantile equals
11.34. Because the sample value of the test statistic is 786, so the null is (strongly) rejected for
TLKM stock. Therefore, there is no need to further examine the additional constraint within
H0 that defines the SCD model, that is H01 : ω3 = 0. Would H0 not have been rejected, then
a test of H01 could be conducted using a Wald test. Since the ACD model is a strict subclass
of the SCD model (see Proposition 2.2 in Renault et al., 2013) and the SCD model is rejected,
then the ACD model is also rejected in favor of the more flexible DMHT model.
20
Looking at the estimated value of B, we have indication that the density function of mixing
variable Mi for TLKM stock is relatively concentrated, because the variance of Mi equals
1/B = 0.049. This density vanishes at zero. Recall that the mixing variable multiplies the
conditional expected duration, so a vanishing density at zero implies that a significant decrease
in the conditional expected duration due to the mixing variable is very unlikely. Since we only
observe durations of length at least one second in the data, it does not appear surprising to
find such result. While very small values of the mixing variable are unlikely, it can still take
on relatively large values, so as to enable the model to fit the occasional large duration found
in the data.
We have a slightly positive, but poorly identified, drift α for the parameters of the return
process, while the GARCH process that updates volatility per unit of time Vti putting almost
equal weights (α̂1 = 0.625) on both the lagged volatility and the realized squared midquote
return per second. For TLKM stock, if the duration between trades increases, the effect of
both the lagged volatility and realized squared midquote return decays geometrically with
exponent α̂2 ≈ 20.80 per second. Hence for long durations, the predicted return volatility per
unit of time is about equal to the diurnally adjusted constant term α0 gS (t∗i ) i), and this at
least indicate that there is a considerably intraday variation in volatility per unit of time.
The parameters estimate of the mixing part of the mixed variance return V̄t2i are δM = 0.000
and θM = 1.095 respectively, where the value for θM follows from the identifying constraint
(30). Hence, the return variance V̄t2i is estimated to be inversely related to the value of the
mixing variable Mi . Since Mi scales the hitting boundary, hence, a larger value of Mi implies
both a longer expected duration as well as a smaller variance, which is consistent with the
observations in for example Engle (2000) and Dufour and Engle (2000).
There is significant correlation between the Wiener process that generate price and the
Wiener process that generate duration. The correlation is estimated to be negative (ρ̂ = 0.075), so when there is a positive shock to the return volatility, then on average there is a
negative shock to the duration between trades, which means shorter durations. Therefore, our
estimation results are consistent with Easley and O’Hara (1992) arguing that a long duration
indicates the absence of new information. Factors such as characteristics of the news, market
participant and market mechanism affect each individuals trading decision. Announcement
of pubic news normally induces an increased trading intensity and a substantial up or down
movement in the price on the stock market very quickly, while private news has less impact on
the market. Informed trader tends to take advantage of the information before certain time
without moving market price to the adverse direction. There are type of investors who are
21
more willing to wait and have no tolerance of the relative high price, and type of the investor
who want to close the position soonest, such as liquidity trader. Block trade is normally
divided into orders of smaller size. The time of waiting for the order execution depends on
how much price impact the trader can take. However, we dont observe these individual level
characteristics to identify type of each trade and buyer or seller and news, what we have are
variables in a market scale, e.g. stock price, quotes, trade arrival time and volume. But
we know all factors that playing roles in affecting the trading intensity and price formation
together.
The negative correlation, although the magnitude is slightly small, in our result also justifies
the empirical support of the DMHT model to the predictions about the relation between a
contemporaneous duration and moments of the conditional return distribution as put forward
in several market micro structure theories (e.g., Diamond and Verrecchia (1987), Easley and
O’Hara (1992)). This empirical support can be confirmed through the analytic expressions of
the first two moments of the return distribution, conditional upon past information and the
contemporaneous duration derived in Renault et al. (2013):
Eti Rti+1 |dti+1
dt
i+1
V arti Rti+1 |dti+1
dti+1
√
= α + ρ V ti Eti [f (hti , µti , Mi )|dti+1 ]
= (1 − ρ2 )Vti Eti [g(Mi )|dti+1 ] +
(37)
ρ2 Vti
V arti [h(hti , µti , Mi )|dti+1 ] (38)
dti+1
where f (.), g(.), and h(.) are some different functions. From (18) and (19), it is clear that
in case the correlation parameter ρ equals zero, both the expected return per unit of time
and its variance per unit of time do not depend on the contemporaneous duration. However,
if the estimated correlation is not zero, although with small magnitude, we can state that
the correlation plays important role in explaining the above conditional moments of return
distribution.
This result in line with Renault, et al. (2013). The graphical illustration is used to analyse
the net effect of ρb on the expected return and return standard deviation per unit of time as a
function of the length of the contemporaneous duration, as specified in (18) and (19). They
show that while expected returns are slightly increasing with the contemporaneous duration,
the return volatility per unit of time are decreasing with the contemporaneous duration. Hence,
this pattern is consistent with the Easley and O’Hara (1992) and other market microstructure
models which explain that return volatility tends to decrease when there are no trading in
relatively long period.
22
4.5
Monte Carlo Studies
In this section, we implement two Monte Carlo studies using model specification described in
Section 4.2. The goal of the Monte Carlo studies are two folds. The first is to manifest the
estimation efficacy of the Conditional JCF-CGMM, and the second is to show the efficiency
gain in estimating asset return variances when jointly considering the stock price and duration
data.
Parameter
True Value
Identity Weighting Matrix
Mean
Median
RMSE
Optimal Weighting Matrix
Mean
Median
RMSE
Parameter
True Value
Identity Weighting Matrix
Mean
Median
RMSE
Optimal Weighting Matrix
Mean
Median
RMSE
Parameter
True Value
Identity Weighting Matrix
Mean
Median
RMSE
Optimal Weighting Matrix
Mean
Median
RMSE
β0
1.2E-13
β1
0.100
β2
0.955
β3
0.915
ω1
0.000
ω2
2.325
ω3
0.042
1.18E-13 0.250
1.21E-13 0.200
0.05E-13 1.225
0.957
0.955
0.085
0.920
0.918
0.057
0.001
0.000
0.0005
2.355 0.053
2.350 0.040
0.057 0.0145
1.25E-13
1.23E-13
0.04E-13
ω4
0.00
0.150
0.125
0.925
ω5
0.171
0.965
0.960
0.080
ω6
2.300
0.918
0.915
0.045
α
0.000
0.000
0.001
0.0002
α0
3.7E-8
2.350 0.048
2.345 0.045
0.060 0.0135
α1
α2
0.625 0.795
0.00
0.00
0.00
0.175
0.173
0.051
2.320
2.315
0.065
0.000
0.000
0.055
0.00
0.00
0.00
ρ
-0.075
0.173
0.171
0.045
δM
0.000
2.340
2.320
1.214
θM
1.095
0.000 3.69E-8 0.628 0.798
0.000 3.72E-8 0.624 0.792
0.000 0.063E-8 0.069 0.0241
B
20.518
-0.074
-0.075
0.0057
0.000 1.092 20.525
0.000 1.096 20.528
0.000 0.0162 0.151
-0.076
-0.074
0.005
0.000 1.097 20.522
0.000 1.094 20.517
0.000 0.0071 0.122
3.75E-8 0.635 0.799
3.73E-8 0.625 0.785
0.072E-8 0.057 0.0252
Table 3: The Results of Monte Carlo Study I
The first Monte Carlo study is based on 5000 simulation of a data set of one thousand
durations and corresponding price changes using the estimated parameter values as inputs.
Since the minimum observed duration in the data set has a length of one second, we round up
all durations between half a second and one second up to one second. Durations shorter than
23
half a second are discarded. The model is then estimated by the Conditional JCF-CGMM
using the simulated stock prices and durations, and both the identity and optimal weighting
matrices are used. We find that the optimization is less sensitive to the initial values when
using the identity weighting matrix since using the optimal weighting matrix results in 350
exploding estimates among 5000 simulations/estimations. Table 3 presents the Monte Carlo
study results, which indicate that the loss of efficiency is tiny.
In the second simulation, we simulate data from a model with endogenous durations, which
is the DMHT model as described in Section 2 and its concrete specification Section 4.2. The
same true values as in the first Monte Carlo study is used in this simulation. Each simulation
is done with 20 days worth of observations (more than 30,000 trades). For each sample, we
re-estimate two models: (1) a model allowing for endogenous durations estimated with the
simulated data and (2) the restricted version of our model where the durations are assumed
exogenous (when ρt = 0). Each time after estimating the models, The estimates of the return
variance are also calculated and compared to the true variance. The Mean Square Error (MSE)
and the Mean Absolute Error (MAE) of the estimated variance are then reported. The results
for the MSE and MAE for each of the 20 replications are reported in Table 4.
Replication
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Mean
MSE Endogenous
0.2620
0.2605
0.2618
0.2704
0.2703
0.2619
0.2509
0.3171
0.2995
0.2505
0.2717
0.2518
0.2593
0.2761
0.2362
0.2562
0.2418
0.2428
0.2575
0.2518
0.2486
MSE Exogenous
0.3021
0.3452
0.3123
0.3031
0.3150
0.2908
0.4193
0.2969
0.3511
0.3453
0.3174
0.2895
0.3095
0.2989
0.3175
0.3075
0.2875
0.2980
0.3185
0.2895
0.3209
MSE Endogenous
0.4075
0.3957
0.4092
0.4098
0.4099
0.4043
0.3969
0.4068
0.4353
0.3964
0.3934
0.3998
0.4064
0.4198
0.3852
0.3943
0.3952
0.3899
0.3862
0.3998
0.3982
MSE Exogenous
0.4373
0.4662
0.4272
0.4389
0.4388
0.4255
0.4662
0.4301
0.4319
0.4663
0.4463
0.4255
0.4195
0.4353
0.4496
0.4368
0.4465
0.4389
0.4501
0.4255
0.4403
Table 4: MSE and MAE of the estimated variances when durations are either endogenous or exogenous
24
From Table 4, we can see that, not surprisingly, for 19 out of 20 simulations the MSE or
MAE of the model that allows for endogenous durations is smaller than for the model with
exogenous durations. Averaging the results over the 15 replications, we see that the MSE
and MAE of the exogenous duration model are 29% and 11% bigger than for the endogenous
duration model.
5
Conclussion
In this paper we propose an efficient method to estimate the dynamic mixed hitting time
model which specifies durations between events as the time it takes a standard Brownian
motion with drift to hit a certain random boundary, and the corresponding values of the
marks are modeled by additional, correlated, Brownian motions. The estimation is based on
the conditional characteristic function proposed in and Carrasco et al. (2007). The technique
is particularly tractable and easy to apply when the joint conditional characteristic function
of the model is known in closed form or up to a relatively easy numerical integration.
The emprirical study using high frequency data on a stock, indicates some stylized facts
of the duration data where a duration is defined as the time span between two transactions.
These stylized facts include high persistence in the durations and fat tails, as well as a large
fraction of very short durations. More importantly, the empirical study helps characterizing
the causality relations between return volatility and durations between trades. As widely
documented in the literature, there is a negative instantaneous relation between volatility and
contemporaneous duration.
In this paper, we estimate the models for one asset or stock. We have not yet considered
the estimation of the models in terms of multivariate modeling involving different assets,
different exchanges, etc. For asset pricing, risk management and asset allocation involving
more than single assets or single exchanges, one needs to estimate the joint movements in
prices and transaction times for several assets in the portfolio. In this regard, the estimation
method for multivariate version of the DMHT models are needed to develop. Therefore, it
may be interesting to study the estimation methods and their implications using tick history
information from several assets in the portfolio.
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25
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