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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 model”of 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 Ljung–Box (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-durations”and 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 O’Hara,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

1450004-5

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

αjjδijjþX

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 γi¼1 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-generating”mechanisms

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 SEMIFAR–ACDD 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 SEMIFAR–ACDD 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

αjjζijjþX

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 SEMIFAR–ACDD 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 SEMIFAR–ACDD model above and the asymptotic results for the

SEMIFAR–GARCH 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-

FAR–ACDD 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 SEMIFAR–ACDD model is then determined as follows:

(a) For p¼1, pmax and q¼1, qmax estimate ACDD( p,q) and calculate it’s

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)¼2log (maximized likelihood)þ(log n)(pþqþ2):ð11Þ

With the trend function in the SEMIFAR–ACDD 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 Data”by 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

1450004-9

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 SEMIFAR–ACD model to be com-

pared against a similar SEMIFAR–ACDD 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 SEMIFAR–ACD

model was r¼0 whereas the optimal lag length obtained for the autoregressive

portion of the SEMIFAR–ACDD model was r¼2. The LB test-statistics,

Lagrange–Multiplier (LM)-statistics, rescaled range (RS)-statistics and Kwiat-

kowski–Phillips–Schmidt–Shin (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

(Ljung–Box test), LM (Lagrange-Multiplier), RS (rescaled range), KPSS

(Kwiatkowski–Phillips–Schmidt–Shin).

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, SEMIFAR–ACD, ACDD and SEMIFAR–ACDD 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, SEMIFAR–ACD, ACDD and SEMIFAR–ACDD 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 SEMIFAR–ACDD 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 SEMIFAR–ACDD 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 4–6).

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 SEMIFAR–ACD/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 SEMIFAR–ACDD model so

that non-stationarity and long memory in durations data can be addressed and

captured using a more parsimonious parameterisation. Asymptotic results on

SEMIFAR–GARCH models as reported by Feng et al. (2007) are extended to the

SEMIFAR–ACDD 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 SEMIFAR–ACDD 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 Jarque–Bera 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 SEMIFAR–ACDD 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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