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YET ANOTHER ACD MODEL: THE AUTOREGRESSIVE CONDITIONAL DIRECTIONAL DURATION (ACDD) MODEL

Abstract

This paper features a new autoregressive conditional duration (ACD) model which sits within the theoretical framework provided by the recently developed observation-driven time series models by Creal et al. (2013): the generalized autoregressive score (GAS) models. The autoregressive conditional directional duration (ACDD) model itself contains three novelties. First, durations (intra-trade intervals or waiting-times) are signed, based on whether a (positive) ask-driven trade or a (negative) bid-driven trade occurred. These signed trade-durations are known as directional durations. Second, as the resultant directional durations are no longer positive and asymmetrical but are symmetrically distributed, the familiar generalized autoregressive conditional heteroskedasticity (GARCH)-like formulation of the ACD process is proposed for modeling these directional durations. Consequently, the proposed model is called the ACDD model. Third, using the alternative GARCH-like formulation, persistence or long-memory in the durations is easily addressed both via the mean and variance equations: the mean equation uses a semi-parametric fractional autoregressive (SEMIFAR) formulation and the variance equation uses a GARCH formulation. The paper demonstrates the flexibility and convenience of the generalized autoregressive score (GAS) model framework in the context of a particular ACD model specification. The model can be viewed as an alternative extension of the "asymmetric ACD model" of Bauwens and Giot (2013) which captures information related to the evolution of prices as well as the quote-durations.
YET ANOTHER ACD MODEL: THE AUTOREGRESSIVE
CONDITIONAL DIRECTIONAL
DURATION (ACDD) MODEL
NAGARATNAM JEYASREEDHARAN
Tasmanian School of Business and Economics
University of Tasmania, Australia
nj.sreedharan@utas.edu.au
DAVID E ALLEN
*
School of Mathematics and Statistics
University of Sydney and University of South Australia
profallen2007@gmail.com
JOEY WENLING YANG
UWA Business School
University of Western Australia
joeywenling.yang@uwa.edu.au
Published 29 August 2014
This paper features a new autoregressive conditional duration (ACD) model which sits
within the theoretical framework provided by the recently developed observation-driven
time series models by Creal et al. (2013): the generalized autoregressive score (GAS)
models. The autoregressive conditional directional duration (ACDD) model itself contains
three novelties. First, durations (intra-trade intervals or waiting-times) are signed, based on
whether a (positive) ask-driven trade or a (negative) bid-driven trade occurred. These
signed trade-durations are known as directional durations. Second, as the resultant direc-
tional durations are no longer positive and asymmetrical but are symmetrically distributed,
the familiar generalized autoregressive conditional heteroskedasticity (GARCH)-like for-
mulation of the ACD process is proposed for modeling these directional durations. Con-
sequently, the proposed model is called the ACDD model. Third, using the alternative
GARCH-like formulation, persistence or long-memory in the durations is easily addressed
both via the mean and variance equations: the mean equation uses a semi-parametric
fractional autoregressive (SEMIFAR) formulation and the variance equation uses a
GARCH formulation. The paper demonstrates the flexibility and convenience of the
generalized autoregressive score (GAS) model framework in the context of a particular
ACD model specification. The model can be viewed as an alternative extension of the
*
Corresponding author.
Annals of Financial Economics
Vol. 9, No. 1 (June 2014) 1450004 (20 pages)
©World Scientific Publishing Company
DOI: 10.1142/S2010495214500043
1450004-1
asymmetric ACD modelof Bauwens and Giot (2003) which captures information related
to the evolution of prices as well as the quote-durations.
Keywords: ACD model; ACDD model; directional duration; SEMIFAR; GAS models.
1. Introduction
High-frequency financial time series have become widely available during the past
decade or so. Records of all transactions and quoted prices are readily available in
pre-determined formats from many stock exchanges. An inherent feature is that
such data are irregularly spaced in time. Several approaches have been taken to
address this feature of the data.
The seminal work originated with Engle and Russell (1998), where the time
between events (trades, quotes, price changes, etc.) or durations are the quantities
being modeled. These authors proposed a class of models called the autoregressive
conditional duration (ACD), models, where conditional (expected) durations are
modeled in a fashion similar to the way conditional variances are modeled using
autoregressive conditional heteroskedasticity (ARCH) and generalized auto-
regressive conditional heteroskedasticity (GARCH) models of Engle (1982) and
Bollerslev (1986).
ACD and GARCH models share several common features, ACD models being
commonly viewed as the counterpart of GARCH models for duration data. Both
models rely on a similar economic motivation following from the clustering of
news and financial events in the markets. The ACD model captures the duration
clustering observed in high frequency data, i.e. small (large) durations being
followed by other small (large) durations in a way similar to the way the GARCH
model accounts for volatility clustering. Just as a low-order GARCH model is
often found to suffice for removing the dependence in squared returns, a low-
order ACD model is often successful in removing the temporal dependence in
durations (see Pacurar,2008). Following the GARCH literature, a number of
extensions to the original linear ACD model by Engle and Russell (1998) have
been suggested. These include the logarithmic ACD model of Bauwens and Giot
(2000), and the threshold ACD model of Zhang et al. (2001). The error dis-
tributions associated with the conditional durations has also been suggested to
have several different shapes. Examples include the exponential and Weibull
distributions as in Engle and Russell (1998), and the Burr and generalized gamma
distributions utilized by Grammig and Maurer (2000) respectively. However, a
crucial assumption for obtaining the quasi-maximum likelihood (QML) consis-
tent estimates of the ACD model and its extensions is that the conditional ex-
pectation of durations is correctly specified and that the model is linear. The QML
estimations yield consistent estimates and the inference procedures in this case
N. Jeyasreedharan, D. E. Allen & J. W. Yang
1450004-2
are straightforward to implement, but this comes at the cost of efficiency. In
practice, fully efficient maximum likelihood (ML) estimates might be preferred if
the nature of the underlying distribution is known; however, this is not likely to
be the case.
The original ACD models focus on taking into account the duration between
market events; quote or price changes, and did not include information inherent in
the evolution of the price process in the dynamics of the model. A significant
departure from this is the asymmetric ACD model of Bauwens and Giot (2003)who
follow a direction first explored by Russell and Engle (2002) in their autoregressive
conditional multinomial model which featured an ACD model fitted to the durations
plus a generalized linear model of the conditional transition probabilities of the
price process. The advantage of this type of approach is that other market-micro-
structure related information such as the traded volume and the corresponding
transaction prices, bid and ask quotes offered by the market makers, can be directly
included to enhance the precision and forecasting ability of the model.
The model developed in this paper is a variant of the asymmetric approach
explored by Bauwens and Giot (2003) and it sits within the context of recent work by
Creal et al. (2013) in their development of the generalized autoregressive score
(GAS) models provides a natural framework for our model. This new class of
observation-driven time series models adopts a mechanism to update the parameters
over time by using the scaled score of the likelihood function. This approach pro-
vides a unified and consistent framework for introducing time-varying parameters in
a wide class of nonlinear models. They suggest that their GAS model encompasses
other well-known models such as the GARCH models, ACD models, autoregressive
conditional intensity, and Poisson count models with time-varying means.
Time series models with time-varying parameters can be divided into two
classes of models: observation-driven models and parameter-driven models. In the
former approach, time variation of the parameters is introduced by letting para-
meters be functions of lagged dependent variables as well as contemporaneous and
lagged exogenous variables. Although the parameters are stochastic, they are
perfectly predictable given the past information. This simplifies likelihood evalu-
ation and observation-driven models have become popular in the applied statistics
and econometrics literature. Typical examples of these models are the GARCH
models of Engle (1982) and Bollerslev (1986), and the ACD and model of Engle
and Russell (1998). In the latter, parameter-driven models, the parameters are
stochastic processes with their own sources of error. An example of this class of
models would be stochastic volatility models, as discussed by Shephard (2005).
Creal et al. (2013) formulate their general class of observation-driven time-
varying parameter models and exploit the full density structure of the score
function. In this class of models, the time-varying parameter ftand the score
Yet Another ACD Model: The ACDD Model
1450004-3
depend on the full underlying density structure. They demonstrate that their GAS
model structure can nest both GARCH (1,1) models and ACD (1,1) models as well
as multiplicative error models (MEM).
They proceed as follows: Let N1 vector ytrepresent the dependent variable
of interest, ftthe time varying parameter vector, xta vector of exogenous variables,
(covariates), all at time t, and θa vector of static parameters. Define
Yt¼fy1,...,ytg,Ft¼ff0,f1,...,ftgand Xt¼fx1,...,xtg. The available in-
formation set available at time tconsists of fft,Ftg, where
Ft¼fYt1,Ft1,Xtg, for t¼1, ...,n:
It is assumed that ytis generated by the observation density
ytp(ytjft,Ft;θ):ð1Þ
To set the model framework in the familiar autoregressive context that provides
the context for both GARCH and ACD models assume that the mechanism for
updating the time-varying parameter ftis given by an autoregressive updating
equation:
ftþ1¼!þX
p
i¼1
Aistiþ1þX
q
j¼1
Bjftjþ1,ð2Þ
where !is a vector of constants, the coefficient matrices Aiand Bjhave the
appropriate dimensions for i¼1, ...,pand j¼1, ...,q, while stis an appropriate
function of past data. The unknown coefficients to be estimated in the expression
above are functions of θ. Clearly, both GARCH and ACD models sit within this
general GAS framework.
Our model developed in this paper presents a simple modification of the basic
ACD model. The inherent limitations in the ACD model and its extensions to date
have been a direct consequence of the positive asymmetric density assumed for the
innovations, "iin all these models; as time between successive trades are positive
(see Hautsch,2004). Distributions defined on positive support typically imply a strict
relationship between the first moment and higher-order moments and do not disen-
tangle the conditional mean and variance function. For example, under the expo-
nential distribution, all higher-order moments directly depend on the first moment.
Consequently, the corollary as derived in Engle and Russell (1998) using the
EACD(1,1) model cannot necessarily be extended to the more general ACD(p,q)
models with further proofs (see Pacurar,2008). Hence there is a certain inflexibility
and lack of published rigorous diagnostics encountered with standard ACD models.
Explicit GARCH-based ACD models circumvent these limitations for obvious reasons.
In addition, apart from being autocorrelated and having arch effects, duration
innovations also exhibit long range dependence (long memory) and non-stationarity.
N. Jeyasreedharan, D. E. Allen & J. W. Yang
1450004-4
Empirical studies based on the linear ACD model often reveal persistence in
durations as the estimated coefficients on lagged variables add up nearly to one.
Moreover, many financial duration series show a hyperbolic decay, i.e. significant
autocorrelations up to long lags. This suggests that a better fit might be obtained by
accounting for longer term dependence in durations. Indeed, the standard ACD
model imposes an exponential decay pattern on the autocorrelation function typical
for stationary and invertible ARMA processes. This may be completely inappro-
priate in the presence of long memory processes. Thus, whilst crucial for the ACD
model and its extensions the assumptions of iid innovations may be too strong and
inappropriate for describing the behavior of trade durations(see Pacurar,2008). A
further point of note is that whilst the LjungBox (LB) test statistic is assumed to
have an asymptotic χ2distribution under the null hypothesis, no formal analysis
exists that rigorously establishes this result in the context of the standard ACD
models (see Pacurar,2008).
In this paper we provide a slightly different approach to work originated by
Engle and Russell (1998). We propose an alternative definition of durations, where
positive durations depict ask-durationsand negative durations depict bid-
durations. This approach enables the innovation error density to be symmetrical.
The ensuring model is called the ACDD model.
Bid and ask durations can be important individual conveyors of market mi-
crostructure information (see Bauwens and Giot,2003;Easley and OHara,1992).
Zhang et al. (2008) demonstrate that the decomposition of the spread into two
components: the cost of buy exposure and the cost of sell exposure by taking into
account the time series characteristics of trading at the bid and ask produces richer
information about trading costs and price volatility. They test and find evidence
that the effect of volumes traded on these components is not symmetric, which is
an effect not captured in standard ACD models which do not distinguish between
trading at the bid and ask. Our model framework would facilitate the greater
exploration of these effects if warranted.
A further consideration is that recently there have been considerable advances in
algorithmic trading and in market surveillance techniques utilized by regulators.
They both utilize the analysis of microstructure patterns of buying and selling
sequences. If any patterns are found to be extractable, they will be invaluable for
smart traders. Other distinct microstructure patterns may reflect abnormal trading
behavior by market participants. These microstructure patterns can then be used to
empower market trading/surveillance agents in monitoring the markets.
The paper is organized as follows; we have set the scene in the introduction and
briefly introduced GAS models which provide a broad conceptual framework for a
wide variety of GARCH and ACD models. In Sec. 2, we briefly discuss the
standard ACD model and introduce the concept of directional durations. Section 3
Yet Another ACD Model: The ACDD Model
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introduces the semi-parametric fractional ACDD model, and the research method
and data are discussed in Secs. 4and 5. The results are discussed in Secs. 6and 7
concludes the paper.
2. The Basic ACDD Model
The time series of arrival times or durations between successive occurrences of
certain events associated with the trading process can be defined in a number of
ways. Examples include the time between successive trades, the time until a price
change occurs or until a pre-specified number of shares or level of turnover has
been traded. We define directional durations as signed durations or times between
successive trades. The signs of the durations are positive when the trade price is
above the mid-price and are negative when the trade price is below the mid-price.
The sign of the duration when the trade-price is equal to the mid-price (13.25% of
the data) is replaced with the directional sign of the previous directional duration.
The mid-price is taken to be the average of the nearest bid and ask quotes. In doing
so, we are able to differentiate between the arrival times of bid and ask-driven
trades.
The basic ACDD model relies on a linear parameterization of the conditional
duration, ψiwhich depends on ppast absolute directional durations and qpast
conditional durations, defined as:
ψi¼!þX
p
j¼1
αjjδijX
q
j¼1
βjψij,ð3Þ
where δi¼γi(titi1)are the directional durations and γi¼1 for ask durations
i.e. when the trade price is greater than the mid-price and γi¼1 for bid durations
i.e. when the trade price is lower than the mid-price, with tbeing the trade times.
To ensure positive conditional durations for all possible realizations, sufficient but
not necessary conditions are
!>0, X
p
j¼1
αj0, X
q
j¼1
βj0:
The main assumption behind ACDD model is that the standardized directional
durations,
"i¼δi
ψi
,ð4Þ
are independent and identically distributed (IID) with E("i)¼0andE("2
i)¼1.
1
1
Note that the standard ACD model assumes the standardized durations are IID with E("i)¼1 and E("2
i)¼2.
N. Jeyasreedharan, D. E. Allen & J. W. Yang
1450004-6
Equation (3) is analogous the standard ACD model with the exception of directional
durations, δi¼γi(titi1)as defined above. The significance of jδijjin the ACDD
model to is to ensure non-negative durations in the conditional duration process.
A natural choice convenient for estimation could be any family of suitable
symmetrical distributions. We adopt the generalized error distribution (GED)
family proposed by Nelson (1991) to capture the fat tails, if any, in the error terms.
Let f(",θ")be the density function for "with parameters θ". If a random variable,
"ihas a GED with mean zero and unit variance, the PDF of "iis given by:
f("i)¼exp[(1
2)j"i=λj]
λ2(þ1)=(1=),ð5Þ
where
λ¼22=(1=)
(3=)
"#
1=2
:ð6Þ
and is a positive parameter governing the thickness of the tail behavior of the
distribution. When ¼2 the above PDF reduces to the standard normal PDF;
when <2, the density has thicker tails than the normal density; when >2, the
density has thinner tails than the normal density. When the tail thickness parameter
¼1, the PDF of the GED reduces to the PDF of a double exponential distribution
(the GED nests the exponential distribution in the basic ACD model of Engle and
Russell,1998).
Based on the above PDF, the log-likelihood function of ACDD model with GED
errors can be constructed and ML and QML estimators for the ACDD parameters can
be easily derived. Furthermore, the redefinition of durations to bid- and ask-based
durations enables us to fully adopt the full range of extant GARCH formulations i.e.
meaning both the mean equation and the variance equation in the standard GARCH
model and its various extensions can be utilised for duration modeling. Various types
of GARCH models, such as EGARCH, TGARCH, PGARCH, etc. can be accessed
for analogous ACDD modeling but will not be considered here as the motivation in
this paper is to highlight and investigate the effects of embedding the bid-ask trading
dynamics into the duration processes against the standard ACD approach used in
Engle and Russell (1998). Investigations into the relevance of the other GARCH
types for ACD modeling (including nonlinear models) are left for future research.
Under the proposed ACDD formulation, the directional durations are open to
long range dependence (long memory) and non-stationarity, if any, in addition to
exhibiting autocorrelation, arch and diurnal effects (see Table 2). To address these
additional stylized characteristics and as several trend-generatingmechanisms
may be occur simultaneously, we introduce a SEMIFAR-based mean equation into
the ACDD model.
Yet Another ACD Model: The ACDD Model
1450004-7
3. The SEMIFARACDD Model
Semi-parametric fractional autoregressive (SEMIFAR) models (see Beran and
Feng,2002a,b) have been introduced for modeling different components in
the mean function of a financial time series simultaneously, such as non-parametric
trends, stochastic non-stationarity, short- and long-range dependence as well
as anti-persistence. SEMIFAR includes ARIMA and FARIMA processes
(see Hosking,1981;Granger and Joyeux,1980).
Let d¼(0:5, 0:5)be the fractional differencing parameter, m2(0, 1)be the
integer differencing parameter, Lbe the lag or backshift operator, (L)be the lag
polynomials in Lwith no common factors and all roots outside the unit circle and "i
be white noise, then the SEMIFAR model can be defined as (see Feng et al.,2007):
(L)(1L)d[(1L)myig(i)] ¼"i,ð7Þ
where i¼ti=n.
Similarly, in the SEMIFARACDD model, the mean equation is defined as
follows:
(L)(1L)d[(1L)mδig(i)] ¼ζið8Þ
with the duration equation defined by:
ψi¼!þX
p
j¼1
αjjζijX
q
j¼1
βjψij,ð9Þ
where ζiis then the SEMIFAR-filtered directional duration. To ensure positive
conditional durations for all possible realizations, sufficient but not necessary
conditions are that
!>0, X
p
j¼1
αj0, X
q
j¼1
βj0:
The main assumption behind SEMIFARACDD model is that the standardized
directional durations,
"i¼ζi
ψi
ð10Þ
are IID with E("i)¼0 and E("2
i)¼1.
4. Methodology
Based on the SEMIFARACDD model above and the asymptotic results for the
SEMIFARGARCH formulation obtained by Feng et al. (2007), the following
N. Jeyasreedharan, D. E. Allen & J. W. Yang
1450004-8
algorithm in S-PLUS is proposed for the practical implementation of the SEMI-
FARACDD model:
(a) Carry out data-driven SEMIFAR fitting using algorithm AlgB defined in Beran
and Feng (2002a) on the square-root of directional durations to obtain g
a()and
^
(L).
(b) Calculate the residuals i¼δig
a(i)and invert iusing ^
(L)into ζi
a
, the
estimates of ζi.
(c) Estimate the variance equation in ACDD model using S-PLUS/GARCH sub-
routine on the estimated residuals ζi
a
of the SEMIFAR model from (b) above.
The best SEMIFARACDD model is then determined as follows:
(a) For p¼1, pmax and q¼1, qmax estimate ACDD( p,q) and calculate its
Bayesian Information Criterion i.e. BIC(p,q);
(b) Choose the ACDD(p,q) model that minimizes the BIC. We obtain the best-fit
ACDD model, using the BIC as defined by:
BIC(p,q)¼2log (maximized likelihood)þ(log n)(pþqþ2):ð11Þ
With the trend function in the SEMIFARACDD model, it is inconvenient to
select the two Eqs. (8) and (9) at the same time. As the estimated parameter vectors
for the SEMIFAR and the ACDD models are asymptotically independent
(see Feng et al.,2007) we adopt a two-stage approach. The best-fit SEMIFAR(r)
model is chosen from r¼0, 1, 2 and the best fit ACDD(p,q) model selected from
p¼0, 1, 2 and q¼0, 1, 2, via the AIC/BIC/LL scores.
5. The Data
The dataset used in this paper is the IBM data used in the seminal paper titled
Autoregressive Conditional Duration: A New Model for Irregularly Spaced
Transaction Databy Engle and Russell (1998) and was downloaded from http://
weber.ucsd.edu/mbacci/engle. This is to enable direct comparisons to be made
with the standard ACD model using the same data. Engle and Russell (1998)give
the following account of the data set: The data were abstracted from the Trades,
Orders Reports, and Quotes (TORQ) data set constructed by Joel Hasbrouck and
NYSE. The data set contains detailed information about each transaction occurring
on the consolidated market during regular trading hours over a three-month period
beginning November 1, 1990 and ending January 31, 1991. In addition to infor-
mation about bid and ask quote movements, the volume associated with the
transactions, and the transaction prices, there is a time stamp, measured in seconds
Yet Another ACD Model: The ACDD Model
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after midnight, reflecting the time at which the transaction occurred. A plot of the
trade and quote transaction data is shown in Fig. 1.
A total of 60,328 transactions were recorded for IBM over the three-months of
trading on the consolidated market from November 1990 through January 1991.
As per the seminal paper, two days from the three months of quote and trade data
were deleted. A halt occurred on 23rd November and a more than one hour
opening delay occurred on 27th December. Following Engle and Russell (1998),
the first half hour of the trading day (i.e. trades and quotes before 10.00 am) is
omitted. This is to avoid modeling the opening of the market which is character-
ized by a call auction followed by heavy trading activity as the dynamics are likely
to be quite different over this call period. Furthermore, the call auction transactions
are not recorded at the same time each morning.
In addition, all trades and quotes after 4.00 pm were also omitted. After
omitting these two days and deleting those trade times less than 10 am and greater
the 4 pm, 51,356 observations of the original 60,328 transactions remained. Of the
IBM November 1990
12:00 12:00 12: 00
Nov 1 1990 Nov 27 1990
106 108 110 112 114 116 118 120 122 124 126
IBM December 1990
12:00 12:00 12:00
Dec 3 1990 Dec 26 1990
110 112 114 116
IBM January 1991
12:00 12:00 12: 00
Jan 2 1991 Jan 24 1991
106 108 110 112 114 116 118 120 122 124 126
Note: This is the original IBM data used by Engle and Russell (1998). It includes transactions from November
1990 through January 1991 The gray crosses depict the bid and ask quotes and the black line depicts the trade
prices.
Figure 1. IBM transaction data by Engle and Russell (1998).
N. Jeyasreedharan, D. E. Allen & J. W. Yang
1450004-10
transactions occurring at non-unique trading times, nearly all of them corre-
sponded with zero price movements. Engle and Russell (1998) suggest that these
transactions may reflect large orders that were broken up into smaller pieces. As it
is not clear that each piece should be considered a separate transaction, the zero-
second durations were considered to be a single transaction and were deleted from
the data set. After all the adjustments to the data, 46,052 observations were
collated.
In their seminal paper, Engle and Russell (1998) reported 46,091 final IBM
observations. This is probably a typo (it should have been 46,051) as their other
reported summary statistics for the same dataset was identical with the mean
duration of 28.38 s, maximum duration of 561s and standard deviation of 38.41 s
obtained from out final dataset. We ended up with 46,052 observations, the extra 1
observation is due to the way we adjusted our durations.
In Fig. 2, it can be seen that the directional durations can either be positive or
negative, whereas standard durations have strictly positive support. In addition,
IBM Nov90, Dec90 & Jan91
6:00 12:00 12:00 12:00 12:00 12:00 12:00 12:00 6:00
Nov 1 1990 Nov 27 1990 Dec 19 1990 Jan 15 1991 Jan 31 1991
100 500
Standard Durations
6:00 12:00 12:00 12:00 12:00 12:00 12:00 12:00 6:00
Nov 1 1990 Nov 27 1990 Dec 19 1990 Jan 15 1991 Jan 31 1991
-400 0 400
Directional Durations
6:00 12:00 12:00 12:00 12:00 12:00 12:00 12:00 6:00
Nov 1 1990 Nov 27 1990 Dec 19 1990 Jan 15 1991 Jan 31 1991
100 500
Abs(Directional Durations)
Note: As extracted from the original IBM data used by Engle and Russell (1998). It includes durations from
November 1990 through January 1991.
Figure 2. Standard, directional and abs(directional durations) in seconds.
Yet Another ACD Model: The ACDD Model
1450004-11
absolute values of the directional durations are equivalent to standard durations.
The directional durations as defined enable symmetrically distributed innovation
errors to be assumed.
It can be seen from Fig. 3that the autocorrelation properties of the standard
duration and the absolute directional durations are identical. However, the ACF
plot of the directional durations exhibit strong first-order AR(MA) behavior.
Herein the difference between standard and directional durations: the first-order
dependencies are fundamentally different. The inclusion of the SEMIFAR equation
as the mean equation to ACDD model ensures that the first-order dependencies are
addressed. However, the second-order dependencies are identical. Consequently,
the variance equation is analogous to that in the standard ACD model. Hence, the
mean equation captures the additional information content embedded in directional
durations. In this respect alone, the ACDD model can be deemed to be a more
adequate model than the ACD for modeling (directional) durations data.
Descriptive statistics summarizing the characteristics of unadjusted standard,
directional and absolute directional durations are shown in Table 1.
0 5 10 15 20 25 30
Standard Durations
0 5 10 15 20 25 30
sqrt(Standard Durations)
0 5 10 15 20 25 30
abs(Directional Durations)
0 5 10 15 20 25 30
0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8
sqrt(Directional Durations)
Figure 3. ACF plots of standard and directional durations.
N. Jeyasreedharan, D. E. Allen & J. W. Yang
1450004-12
It can be seen in Table 1that using directional durations increases the range of
durations, reduces their mean value, and reduces their skewness and kurtosis whilst
adding to their standard deviation.
6. Results
The seasonal adjustment to the standard durations was carried out as done by Engle
and Russell (1998) using the same scatterplot smoothing SUPSMU-subroutine in
S-PLUS. A SEMIFAR filter (the mean equation) was then applied both to the
square-root adjusted standard durations and the square-root adjusted directional
deviations.
2
This also enabled an equivalent SEMIFARACD model to be com-
pared against a similar SEMIFARACDD model (as recommended by an anon-
ymous referee).
Figure 4exhibits the adjusted square-root durations (sqrt(Adj)) for both the
standard and directional durations. As mentioned earlier, the algorithm B (AlgB) in
Beran and Feng (2002a) was used for estimating the SEMIFAR portion of the
model. The trend was estimated by local linear regression using a kernel as the
weight function. For the short memory effects, only an auto-regressive (AR)
component was considered. The SEMIFAR model is chosen from r¼0, 1, 2. The
optimal lag length obtained for the autoregressive portion of the SEMIFARACD
model was r¼0 whereas the optimal lag length obtained for the autoregressive
portion of the SEMIFARACDD model was r¼2. The LB test-statistics,
LagrangeMultiplier (LM)-statistics, rescaled range (RS)-statistics and Kwiat-
kowskiPhillipsSchmidtShin (KPSS)-statistics are listed in Table 2for the ad-
justed and square-root adjusted standard and directional durations (AdjSD, AdjDD,
sqrt(AdjSD) and sqrt(AdjDD)) before and after applying the SEMIFAR(2) filter.
The square-root adjusted standard durations (sqrt(AdjSD)) contain significant
arch effects. The square-root adjusted directional durations (sqrt(AdjDD)) contain
Table 1. Unadjusted duration descriptive statistics.
Min X1Q Median X3Q Max Mean Std Skewness Kurtosis
SD 1 5 15 36 561 28.3845 38.4121 3.5467 23.6688
DD 531 12 3 18 561 3.0724 47.6629 0.0943 15.8655
abs(DD) 1 5 15 36 561 28.3845 38.4121 3.5467 23.6688
Note: SD (standard durations), DD (directional durations) and abs(DD) (absolute directional
durations).
2
The transformation was carried out so that the ACD model can be estimated with GARCH software as per Engle
and Russell (1998).
Yet Another ACD Model: The ACDD Model
1450004-13
6:00 12: 00 12:00 12: 00 6:00
Nov 1 1990 Dec 19 1990 Jan 31 1991
-4 0 4
sqrt(AdjSD)* sqrt(AdjSD)*
Lag
ACF
0 5 10 15 20 25 30
0.0 0.4 0.80.0 0.4 0.80.0 0.4 0.80.0 0.4 0.8
6:00 12: 00 12:00 12: 00 6:00
Nov 1 1990 Dec 19 1990 Jan 31 1991
-5 -1 3
SEMIFAR-sqrt(AdjSD))* SEMIFAR-sqrt(AdjSD))*
Lag
ACF
0 5 10 15 20 25 30
6:00 12: 00 12:00 12: 00 6:00
Nov 1 1990 Dec 19 1990 Jan 31 1991
-5 -1 3
sqrt(AdjDD) sqrt(AdjDD)
Lag
ACF
0 5 10 15 20 25 30
6:00 12: 00 12:00 12: 00 6:00
Nov 1 1990 Dec 19 1990 Jan 31 1991
-5 -1 3
SEMIFAR-sqrt(AdjDD) SEMIFAR-sqrt(AdjDD)
Lag
ACF
0 5 10 15 20 25 30
Note: *The adjusted standard durations have been further transformed by a random series of 1 s and +1 s as per
Engle and Russell (1998).
Figure 4. Square-root adjusted durations and ACF plots.
Table 2. Adjusted and SEMIFAR-Adjusted duration statistics.
LB-stat LM-stat RS-stat KPSS-stat
sqrt(AdjSD)* 45.3369 2828.9225 0.8276 0.0268
p-value 0.4999 0.0000 >0.10 >0.10
sqrt(AdjDD) 4345.0041 2828.9225 2.5465 0.549
p-value 0.0000 0.0000 <0.01 <0.05
SF-sqrt(AdjSD)* 68.7266 2825.3575 0.3398 0.0025
p-value 0.0166 0.0000 >0.10 >0.10
SF-sqrt(AdjDD) 32.831 2142.5727 0.9341 0.0315
p-value 0.9279 0.0000 >0.10 >0.10
Note: *: The adjusted standard durations have been further transformed by a
random series of 1 s and þ1 s as per Engle and Russell (1998). LB
(LjungBox test), LM (Lagrange-Multiplier), RS (rescaled range), KPSS
(KwiatkowskiPhillipsSchmidtShin).
N. Jeyasreedharan, D. E. Allen & J. W. Yang
1450004-14
significant serial correlations, long memory and arch effects as expected. The SF
(0)-sqrt(AdjSD) residual standard durations exhibit serial correlations and arch
effects. However, all the SF(2)-sqrt(AdjDD) residual directional duration statistics
are insignificant with the exception of significant arch effects, indicating the
SEMIFAR filter has been efficient in capturing the serial correlation, long memory
and non-stationarity in the adjusted directional durations. Consequently, the
GARCH equation need only address the arch effects.
The ACD, SEMIFARACD, ACDD and SEMIFARACDD models are then
fitted, diagnosed and compared. The model parameters for the mean and variance
equations are listed in Table 3. The mean equation estimates are the same for both
the ACD and ACDD models are null by construction. The ACD and ACDD
variance equation parameter estimates are of the same order and significant (not
shown). The fractional differencing parameter d estimate of 0.1017 for the SF(2)-
ACDD(1,1) model indicates persistence in the bid-ask process is being captured by
the mean equation.
3
The GED parameter estimates are greater than the value of 2
(for a normal distribution) for all standardized model residuals models. However,
the GED distribtution for the SF(2)-ACDD(1,1) standardized residuals is 2.18,
indicating a closer fit to a normal distribution.
The descriptive and diagnostic statistics for the standardized residuals of various
models are displayed in Table 4. The Ljung-Box statistic for the ACD(1,1), SF(0)-
ACD(1,1) and SF(2)-ACDD(1,1) are all insignificant. However, the LB-statistic is
4053.15 for the ACDD(1,1) model, indicating high first-order dependency in the
directional durations. This high dependency is subsequently completely addressed
by the extended SEMIFAR(2)ACDD(1,1) model with some nominal but statis-
tically significant arch effects still remaining.
To remove the remaining arch effects, a best-fit SEMIFAR(2)ACDD model is
selected from p¼0, 1, 2 and q¼0, 1, 2 by means of BIC. The best fitted ACDD
3
The integer differencing parameter mis 0 for all SEMIFAR fits.
Table 3. ACD, SEMIFARACD, ACDD and SEMIFARACDD parameters.
ACD(1,1) SF(0)-ACD(1,1) ACDD(1,1) SF(2)-ACDD(1,1)
dNA 0.0095 NA 0.1017
AR(1) NA NA NA 0.1135
AR(2) NA NA NA 0.0287
A0 0.0111 0.0106 0.0102 0.0105
ARCH(1) 0.0635 0.0619 0.0617 0.0568
GARCH(1) 0.9262 0.9275 0.9282 0.9337
GED-v 2.9772 2.9336 2.9384 2.1808
Yet Another ACD Model: The ACDD Model
1450004-15
model is found to be the SEMIFAR(2)ACDD(2,1) as depicted by the lowest AIC/
BIC/LL values in Table 5.
Table 6summarises the parameters and diagnostics for variance equation of the
SEMIFAR(2)ACDD(2,1) model as selected. Though the JB-statistic still rejects
normality, the LM-statistic is no longer significant indicating insignificant arch
effects in the residuals.
In Fig. 5, the black lines depict the conditional durations from the SEMIFAR
ACDD and SEMIFARACDD models, whereas the gray crosses depict the con-
ditional durations from the ACD and ACDD models in both panels. The plots
are very similar but not identical, indicating that the alternative ACD models i.e.
the ACDD and SEMIFARACDD models are not only equivalent but are more
Table 5. AIC/BIC/LL-values for SEMIFAR(2)ACDD models.
ACDD01 ACDD02 ACDD10 ACDD11 ACDD12 ACDD20 ACDD21 ACDD22
AIC 130108 130322 129618 127419 127414 129312 127405 129113
BIC 130134 130357 129644 127454 127457 129347 127449 129166
LL 65051 65157 64806 63706 63702 64652 63698 64551
Table 6. SEMIFAR(2)ACDD(2,1) parameters.
A0 ARCH(1) ARCH(2) GARCH(1) LB-stat LM-stat JB-stat
SF(2)-ACDD(2,1) 0.0093 0.0796 0.0256 0.9376 7.1446 18.5463 256.3364
p-value 0.0000 0.0000 0.0000 0.0000 0.8479 0.1001 0.0000
Table 4. ACD/ACDD and SF-ACD/ACDD statistics on std residuals.
Mean Stdev LB-stat LM-stat JB-stat
ACD(1,1) 0.0023 1.0335 7.9554 20.6874 689.07
p-value NA NA 0.7886 0.0552 0.0000
SF(0)-ACD(1,1) 0 1.0332 9.3075 20.4316 688.4707
p-value NA NA 0.6765 0.0593 0.0000
ACDD(1,1) 0.1045 1.0282 4053.1534 20.8674 773.3262
p-value NA NA 0.0000 0.0524 0.0000
SF(2)-ACDD(1,1) 0 0.9937 6.3267 26.5903 251.478
p-value NA NA 0.8987 0.0088 0.0000
N. Jeyasreedharan, D. E. Allen & J. W. Yang
1450004-16
adequate models of the signed duration process as supported by the lower residual
statistics (as listed in Tables 46).
To highlight the subtle differences in the conditional durations as captured by
the various models. Figure 6provides two scatter plots, one for the ACD/ACDD
conditional durations and the other for the SEMIFARACD/ACDD conditional
durations As can be seen in Fig. 6the SEMIFAR models fit very different
conditional durations which must be the effects of dependencies in the mean series
when the bid-ask dynamics have been embedded.
Although there are not many differences between the conditional durations
of the ACD and ACDD models, there are significant differences between
the SEMIFAR versions of the same. The SF(2)-ACDD(1,1) model tends to
give lower conditional durations more often that the corresponding ACD(1,1)
models. One can see this tendency exhibited in panel 2 of Fig. 5,where
the black lines (depicting SF(2)-ACDD(1,1) conditional durations) are gen-
erally bounded by the gray crosses (depicting SF(0)-ACD(1,1) conditional
durations).
Conditional Durations for ACD and ACDD
12:00 12:00 12: 00 12:00 12:00 12: 00 12:00 12:00 12: 00 12:00 12: 00
Jan 2 1991 Jan 8 1991 Jan 14 1991 Jan 18 1991 Jan 24 1991 Jan 30 1991
20 40 60 80 100 120
Conditional Durations for SEMIFAR-ACD and SEMIFAR-ACDD
12:00 12:00 12: 00 12:00 12:00 12: 00 12:00 12:00 12: 00 12:00 12: 00
Jan 2 1991 Jan 8 1991 Jan 14 1991 Jan 18 1991 Jan 24 1991 J an 30 1991
20 40 60 80 100 120
Figure 5. Conditional durations plots (ACD, ACDD, SF-ACD and SF-ACDD).
Yet Another ACD Model: The ACDD Model
1450004-17
7. Conclusion
This paper modifies the standard ACD model into a SEMIFARACDD model so
that non-stationarity and long memory in durations data can be addressed and
captured using a more parsimonious parameterisation. Asymptotic results on
SEMIFARGARCH models as reported by Feng et al. (2007) are extended to the
SEMIFARACDD model. The important property that the estimates of the
SEMIFAR and ACDD parameter vectors are independent of each other, allows us to
apply the data driven SEMIFAR algorithms to estimate the trend and the SEMIFAR
parameters in the SEMIFARACDD model. The ACDD parameters from the ap-
proximated ACDD innovations are obtained by inverting the SEMIFAR residuals.
If the fitted ACDD models are adequate, then the standardized residual inno-
vations should behave as an IID sequence of random variables with the assumed
distribution. In particular, if the fitted model is adequate, both the series f"igand
f"2
igshould have no serial correlations. The AIC/BIC/LL selected SEMIFAR(2)
ACDD(2,1) model resulted in residuals that had not only small but insignificant
values of LB and LM statistics indicating strong model adequacy. The shape
parameter of the GED distribution for the standardized residuals was 2.18 with a
small JarqueBera statistic of 251.4 in addition to having mean and standard
deviation estimates as assumed (i.e. 0.0 and 0.9937 against the model assumptions
of E("i)¼0 and E("2
i)¼1).
The results indicate that the proposed SEMIFARACDD representation can be
used to capture both first-order and second-order dependencies in signed durations
ACD model (Conditional Durations)
ACDD Model (Conditional Durations)
0 20406080100120
0 20 40 60 80 100 120
ACD(1,1) vs ACDD(1,1)
SF-ACD model (Conditional Durations)
SF-ACDD Model (Conditional Durations)
0 20406080100120
0 20406080100120
SF(2)-ACD(1,1) vs SF(2)-ACDD(1,1)
Figure 6. Scatter plots of the conditional durations.
N. Jeyasreedharan, D. E. Allen & J. W. Yang
1450004-18
data. Further possible extensions to the ACDD model include leverage effects and
the full range of GARCH-type extensions that are not readily available to the
standard ACD model.
Acknowledgments
David E. Allen thanks the Australian Research Council for research funding
support and Nagaratnam Jeyasreedharan thanks the Tasmanian School of Business
and Economics, University of Tasmania for funding support to attend the
Modelling and Managing Ultra-High Frequency Data: An International Confer-
ence Perth, WA (MMUHFDIC, 2008). The authors are also grateful to Professor
Mardi Dungey, University of Tasmania and the editor and anonymous referees for
their invaluable comments and suggestions.
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N. Jeyasreedharan, D. E. Allen & J. W. Yang
1450004-20
... As stated in Jeyasreedharan et al. (2014), time series models with time-varying parameters can be divided into two classes of models: observation-driven models and parameter-driven models. In the former approach, time variation of the parameters is introduced by letting parameters be functions of lagged dependent variables as well as contemporaneous and lagged exogenous variables. ...
... Whereas in the parameter-driven models, 18 the parameters are stochastic processes with their own sources of error. Jeyasreedharan et al. (2014) develop the autoregressive conditional directional duration (ACDD) model that sits within the theoretical framework provided by the observation-driven time series models by Creal et al. (2013). This ACDD model can be seen as an alternative extension of the AACD model of Bauwens and Giot (2003). ...
... The novelty of the ACDD model is that it tries to deal with the crucial inherent limitations of the ACD model. Jeyasreedharan et al. (2014) very well point out how the standard ACD model fails to incorporate the essential features of high-frequency data. We now briefly state some of the observations of Jeyasreedharan et al. (2014). ...
Article
This paper reviews the recent literature on conditional duration modeling in high-frequency finance. These conditional duration models are associated with the time interval between trades, price, and volume changes of stocks, traded in a financial market. An earlier review by Pacurar provides an exhaustive survey of the first and some of the second generation conditional duration models. We consider almost all of the third-generation and some of the second-generation conditional duration models. Notable applications of these models and related empirical studies are discussed. The paper may be seen as an extension to Pacurar.
... It should be noted that the consistency of the QMLE hinges on the linearity (in parameter) of the conditional duration. It is unclear if the results in this paper carry over to the nonlinear models such as the log-ACD (Bauwens and Giot, 2000), the TACD (Zhang et al., 2001), the asymmetric ACD (Bauwens and Giot, 2003), the functional coefficient ACD (Fernandes et al., 2013), or the (autoregressive conditional directional duration model (ACDD), Jeyasreedharan et al., 2014). These nonlinear models are equally interesting and also attract a lot of attention. ...
... As the signed duration is not necessarily strictly positive, both asymmetric and symmetric densities can be considered in the QMLE. Jeyasreedharan et al. (2014) further considered the possible long-memory of the duration process. All in all, these extensions will improve the goodness of fit on the one hand, and they are potentially found useful in asset management and/or risk management on the other hand. ...
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