Description
The recent advent of high frequency data provides researchers with transaction by transaction level data. Examples include scanner data from grocery stores or financial transactions data. These new data sets allow researchers to take an unprecedented look at the underlying economic structure of the markets.
Econometric Modeling of Multivariate Irregularly-Spaced High-Frequency Data
Jeffrey R. Russell
*
University of Chicago
Graduate School of Business
November 1999
Abstract: The recent advent of high frequency data provides researchers with transaction by transaction
level data. Examples include scanner data from grocery stores or financial transactions data. These new
data sets allow researchers to take an unprecedented look at the underlying economic structure of the
markets. Often the hypothesis of interest is in the context of multivariate time series data. Analysis of
multivariate high frequency data is complicated by the fact that the multiple processes are irregularly
spaced in time with different arrival rates. This has lead many investigators to work with aggregated data
which can blur the market structure and contaminate the analysis. In this paper we propose a new method
of working with the dissaggregated data and develop an econometric model for the arrival rates of
multivariate dependent point processes. We apply the model to financial transactions data and estimate
models for the bivariate point process of transaction and limit order arrival times. Since limit orders dictate
the structure of the limit order book they are a direct determinant of market liquidity. To our knowledge
little work has examined the dynamic structure of limit order submissions. Since transactions represent a
floor trader or market order demand for liquidity the bivariate model characterizes the dynamic behavior of
liquidity supply and demand. The proposed model also allows for marks, or characteristics associated with
the arrival times, to influence future arrival rates. For the stock market data analyzed we find strong
evidence of codependence in the two processes and both liquidity demand and supply are influenced by
past volume and prevailing spreads.
Key Words: ACD, High Frequency Data, Multivariate Time Series, Causality, Point Process
*
The paper benefited from comments by participants at the 1999 Econometric Society meetings. Also,
conversations with Rob Engle, Eric Ghysels, and Bruce Hansen were very helpful. Financial support from
the University of Chicago, Graduate School of Business is gratefully acknowledged.
1
Section 1: Introduction
High frequency data sets are rapidly becoming available for many types of economic and
business data including transactions data from financial markets, scanner data from grocery
stores, or records of credit card purchases to name a few. These high frequency data sets provide
the empirical investigator with an unprecedented view the underlying market microstructure.
Analysis of high frequency transaction data is complicated by the fact that transaction
data are inherently irregularly spaced in time. This is particularly troubling when the data of
interest is multivariate. For example, financial transactions data sets often contain not only
information about the transactions, but also the quote dynamics. Quote adjustments do not occur
at regularly spaced time intervals and will have a different rate than the transactions making
traditional time series analysis impossible. As a result, investigators must aggregate the time
series data to fixed time intervals so that standard econometric techniques for multivariate data
can be applied. Aggregation can blur the market structure and the choice of an optimal interval is
not obvious.
Engle and Russell (1998) propose a different approach. They propose modeling the
arrival rate of transactions data with the Autoregressive Conditional Duration (ACD) model.
Often the arrival rate is interesting in itself. Engle and Russell (1997,1998) model the time
between price changes of fixed size. This can be interpreted as a model of volatility that is
particularly well suited for irregularly spaced time series data. Engle (1996), Ghysels,
Gourieroux, and Jasiak (1998) and Russell and Engle (1999) and others propose joint modeling of
the transaction price process and the duration using the ACD model.
This paper extends the work of Engle and Russell (1998) and proposes a new
econometric model for multivariate transactions data. The data are treated as a bivariate marked
point process with dependent arrival rates. In this multivariate context it is difficult to model the
expected duration which is the foundation of the ACD model. This is the approach taken in
Engle and Lunde (1999) who propose a censored bivariate ACD model. The goal of their work is
to assess how quickly information in the transaction process impacts the prices via quote
adjustments. Therefore the primary interest is the time between transactions and subsequent
quote revisions. The transaction process is therefore specified as the "driving process". The two
processes are not treated symmetrically and some information is lost if multiple quote revisions
occur without intervening transactions. The problem is that it is difficult to model the distribution
of a duration when new information can arrive within a duration.
2
This paper takes a different approach. Rather than build models based on durations we
propose directly modeling the instantaneous arrival rates, or intensities. We propose a new model
for the instantaneous arrival rate given the multivariate filtration of arrival times and associated
marks. The model takes a particularly simple form so that the dynamics are easily interpreted
from parameter values. Basic properties of the model are established. We propose estimation by
maximum likelihood and model diagnostics are suggested.
The model is applied to financial transactions data. We examine the bivariate system of
transaction arrival times and limit order submission arrival times. Since limit orders dictate the
structure of the limit order book they are a direct determinant of market liquidity. To our
knowledge little work has examined the dynamic structure of limit order submissions. Since
transactions represent a floor trader or market order demand for liquidity the bivariate model
characterizes the dynamic behavior and relationship between the supply and demand of liquidity.
The proposed model also allows for marks, or characteristics associated with the arrival times, to
influence future arrival rates. For the stock market data analyzed we find strong evidence of
codependence in the two processes and both transactions and limit order submission are
influenced by past volume and prevailing spreads.
The paper is organized as follows. The following section introduces some notation and
introduces the conditional intensity function and compensators. Section 3 reviews the existing
statistical models for multivariate point processes. Section 4 introduces the proposed model.
Section 5 considers several extensions. Section 6 discusses the likelihood function and presents
model diagnostics. Section 7 discusses some ideas underlying market liquidity. Section 8
discusses the data and presents model estimates. Finally, section 9 concludes.
Section 2. Multivariate point processes, conditional intensity functions, and compensators.
We consider a K dimensional multivariate point process. Each point process consists of a
strictly increasing, stochastic set of arrival times. It is convenient also to introduce counting
process associated with the k
th
point process ( ) t N
k
. This is simply the number of events that
have occurred on the k
th
process by time t. The counting process is useful for indexing our arrival
times in a multivariate context since for any time t, there will likely be a different number of
events will have occurred for each process. Let ... ...
2 1 0
< < < < <
k
i
k k k
t t t t denote the arrival
times associated with the k
th
(k=1,..,K) point process. At time t the most recent arrival time will
3
be denoted by
k
t N
k
t
) (
. Associated with the ( ) t N
k
arrival time of the k
th
point process is a set of
characteristics denoted by the vector
( )
k
t N
k
Z .
( )
k
t N
k
Z is called a vector of marks with dimension
L
k
. The process ( )
( )
( )
k
t N
k
k
Z t N , is called a marked point process. For the purposes of exposition,
it is convenient to initially consider the system with no marks. Later we will return to the
possibility that some or all of the point processes are marked.
Following Cox and Isham (1980) we call ( ) t N
k
the marginal process of class k events.
We will refer to the process of points and marks taken without regard to class as
N(t). This is simply the pooled or superposed process. Throughout the discussion we assume
that the pooled process is orderly
1
in the sense that { } ) ( 1 ) , ( Pr ? ? o t t N
k
· > + for all k.
Clearly, if the pooled process is orderly then the marginal process must also be orderly.
Let F
t
denote the filtration of the pooled process. ( ) t N
k
is assumed to be adapted to F
t
.
Engle and Russell (1998) (ER) propose a model for dependent point processes based on a
particular filtration. The ACD model for process k is specified conditional on a sub-sigma field
t
k
t
F F ? consisting of the marginal history, or the internal filtration of ( ) t N
k
and the associated
marks. That is, in ER the information set consists of past arrival times and past marks of the
marginal series so that new information was assumed to arrive only at past arrival times of events
of type k.
Since the filtration in ER is unchanged between arrival times, the conditional distribution
of the waiting time until the next event is easily characterized conditional on information
available at the start of the waiting time. It was therefore natural to consider parameterizations
for the waiting time or duration until the next event. For the multivariate filtration F
t
it is very
difficult to parameterize the model in terms of the conditional distribution of waiting times. In
particular, a full characterization would require specification of the joint density of the
distribution of the next arrival time, and the complete path of information set F
t
(or covariate)
over the waiting time. A more natural approach that does not require joint modeling of the
complete path of the covariate is obtained by directly specifying a model for the instantaneous
probability that an event of type k occurs in the next instant given information available at time t.
1
This is not crucial. It is possible to relax this assumption by augmenting the dimension of the multivariate
point process by defining simultaneous occurrences to define a new class of point process. This will not be
considered in this paper.
4
The conditional probability that an event of type k occurs in the next instant is given by
the conditional intensity function. More formally, the conditional intensity function associated
with the k
th
marked point process (MPP k) is defined as
(1) ( )
( ) ( )
t
F t N t t N
F t
t
k k
t
t
k
?
> ? ? +
·
? ?
0 ) ( Pr
lim
0
?
Of course, ( )
t
k
F t ? must be non-negative. Equation (1) is simply the probability per unit time
that an event of type k occurs in the next instant. For small values of t ? ,
(2) ( ) ( ) ( ) t o F t N t t N E t F t
t
k k
t
k
? + ? ? + · ? | ) ( ) ( | ?
so that
(3) ( ) ( ) t F t t N t t N
t
k k k
? ? ? ? + | ) ( ) ( ?
will have expectation of order ( ) t o ? and as t ? goes to zero, (3) will be uncorrelated with the
past F
t
.
Next, consider adding consecutive pieces of (3) from some initial time s
0
through time s
1
(s
1
>s
0
)
as in:
(4)
( ) ( )
( )
( ) ( ) ( ) ( )
( )
? ? ? ? ? ·
?
? ? + ? ? ? + ? ? +
?
?
·
?
?
·
t
s s
j
t
k k k
t
s s
j
t
k k k
t F t j s N s N
t F t j s t j s N t j s N
0 1
0 1
1
0 1
1
0 0 0
|
| ) ) 1 ( ( ) (
?
?
Provided that ( )
t
k
F t ? is integrable taking the limit as t ? goes to zero yields:
(5) ( ) ( ) ( ) ( )
?
? ?
1
|
0 1
s
s
t
k k k
o
dt F t s N s N ?
From (3) it follows that (5) will be uncorrelated with
0
s
F . Or, equivalently,
(6) ( ) ( ) ( ) ( )
0 1
1
| s N s N dt F t E
k k
s
s
t
k
o
? ·
,
_
¸
¸
?
?
The integrated intensity function plays a central role in the theory of point processes. As
such, it has been termed the compensator, or more precisely, the F
t
-compensator
2
. The
compensator will play a central role in this paper and is denoted by:
(7) ( ) ( )
?
· ?
1
| ,
1 0
s
s
t
k k
o
dt F t s s ?
5
Of particular importance in developing a model for multivariate point process is the F
t
-
conditional survivor function. If
k
i
k
i
k
i
t t
1 ?
? · ? denotes the time interval between the i
th
and (i-1)
th
arrival time then the F
t
-conditional survivor function for the k
th
process is given by:
(8) ( ) ( )
?
? ?
+
?
> ? ·
1
|
i
t
k
i
k
i k
k
i k
F P S
Following Yashin and Arjas (1988), provided that the survivor function is absolutely continuous
with respect to Lebesgue measure then
(9) ( ) ( )
,
_
¸
¸
?
? ·
?
i
i
t
t
t
k k
i k
dt F t S
1
| exp ? ?
More important here is the implication that the integrated intensity function over the duration
between two subsequent arrival times yields a random variable with unit exponential distribution.
Hence if the survivor function is absolutely continuous then
(10) ( )
?
·
?
i
i
t
t
t
k k
t N
dt F t
1
|
~
) (
? ?
is an iid unit exponential random variable. Yashin and Arjas do not make any exogeneity
assumptions about F
t
so the result is very general. This is a stochastic version of the well know
result that any inhomogeneous Poisson process can be transformed into a homogeneous Poisson
process with arrival rate of unity by a stochastic transformation of time scale given by
( )
?
·
t
t
t
ds F s
0
? ? . Since the expected value of a unit exponential is 1, the random variable
(11) ( )
k
t N
k
t N ) ( ) (
~
1 ? ? ? ·
has mean zero with unit variance. From (6) it is clear that positive values of
k
t N ) (
? indicate that
the path of the conditional intensity function under predicted the number of events in the time
interval and negative values indicate that the conditional intensity function over predicted the
number of events in the time interval. In this sense, we may view (10) as a generalized residual
in the sense of Cox and Snell (1968).
2
A general discussion of the theory of point processes and compensators can be found in Karr (1986) or
Daley and Vere-Jones (1988).
6
The distribution of
k
t N ) (
? and its interpretation is important to the model developed in the
next section. As such, it is appropriate to ask how restrictive the continuity restriction is. Clearly
the restriction rules out the possibility of including any information in the filtration that perfectly
predicts an event occurrence of type k. While this may be the case in medical studies it is
unlikely to be a problem for financial transactions data. Medical studies have modeled the hazard
rate associated with death from a disease given some sort of treatment applied over a period of
time. One of the variables the medical studies may include is a variable taking the value 0 if the
patient dies, 4 if the patient is cured and values between 0 and 4 indicating intermediate health
quality. This type of covariate would violate the absolute continuity restrictions since a 0 would
perfectly predict the death of the patient. In this case the survivor function would have a
discontinuity at the time that the health variable takes the value 0. This type of situation should
not be a problem for our financial data since it is unlikely that any information in F
t
(the set of
past arrival times in the multivariate filtration) can perfectly predict when a transaction will
occur. We therefore assume the absolute continuity condition for the F
t
-conditional survivor
functions associated each waiting time for the multivariate point process.
3 Some Multivariate Point Process Models
Figure 1 provides a graphical view of a multivariate point process. We are interested in
modeling the conditional probability of an event of, type 1 occurring at time in the instant just
after time t given the multivariate filtration of past arrival times.
In describing the modeling strategy, it is convenient to initially consider a simple
bivariate case. Using the above notation, we consider two point processes denoted by a and b
with counting functions ( ) t N
a
and ( ) t N
b
. Then the filtration at time t is given by the complete
history of N(t), that is, all past arrival times of the pooled process
F
t=
{ }
b
t N
b b a
t N
a a
b a
t t t t t t
) (
1 0
) (
1 0
,..., , , ,..., ,
Our goal is to jointly model the arrival times of process a and process b. Models for
multivariate intensity functions have been used in a variety of disciplines. For example, Daley
(1968), Brown (1970) and others, have used multivariate point processes to study queues. These
processes may be bivariate when a model for inputs and outputs are considered. Neuronal spikes
of different types have been analyzed as a multivariate point process in Neurophysiology by
Perkel, Gerstein, and Moore (1967). Vere-Jones, Turnovsky, Erby (1964), Vere-Jones and
Davies (1966) Vere-Jones (1970) and more recently, Ogata, Akaike, and Katsura (1982) have
7
modeled earth quake occurrences as a multivariate point process. In economics, competing risks
models have been used in the analysis of unemployment spells, strikes, and recessions. These
models can be considered a type of multivariate point process provided multiple events are
observed for a single agent. Lancaster (1990) provides a very good coverage of these models.
Engle and Lunde (1999) propose a bivariate model for transactions data. Their model, however,
is a restrictive model in the sense that one process must be designated as the “driving process” of
the other so that the two processes are not treated symmetrically.
These models tend to be very specialized. Many of the models used are Markovian in
structure and therefore display only restricted dependence on the past that is likely present in
financial transactions data. Others require specification of one process as the "driving process"
not treating the two processes symmetrically. We prefer not to make a choice as to which process
drives the other.
Perhaps the most common models for multivariate point process data are natural
extensions of univariate models. The univariate renewal process can be generalized to the
multivariate setting. A simple bivariate process can be constructed based on the pair of backward
recurrence times. Let
a
i
a
i
a
i
t t
1 ?
? · ? denote the time interval between the i
th
and the (i-1)
th
arrival
times of type a. X
i
denotes the i
th
duration in the pooled process. Let
i
Y be a Bernoulli random
variable denoting the type of event associated with the i
th
event arrival of the pooled process. A
generalization of the univariate renewal is obtained from joint distribution of the i
th
arrival time of
the pooled process and its associated type if, given the filtration of past arrival times and types,
the joint distribution can be expressed as:
(12) ( ) ( ) ( )
1 1
| , | | ,
? ?
·
i i i i i t i i
Y Y h Y Y X g F Y X f
Cinlar (1969) studied the properties of these models and termed it a Semi-Markov process. Since
the marginal process follows a renewal process it is natural to call this a bivariate renewal
process. While the model is tractable from a theoretical perspective the nature of dependence is
quite limited as dictated by the Markovian structure and lack of dependence on past X
i
. Instead,
we consider the general class of self-exciting models which allow for temporal dependence in the
arrival rates and the cross arrival rates.
The conditional intensity function introduced in section 2 provides a convenient base
from which to consider parameterizations that allow for more extensive types of auto and cross
dependence than are easily characterized when focusing on the conditional distribution of the
durations between events as the Semi-Markov processes. For example, let U
a
(t) and U
b
(t) denote
the backward recurrence time to events of type a and b respectively. That is, ( )
( ) t N
a
a
t t t U ? · .
8
The backward recurrence time is grows linearly through time with discrete jumps back to the zero
occurring at each arrival time of type a. An immediate generalization of the Semi-Markov
process is conveniently expressed specifying the conditional intensity functions for a and b as
functions of on the pair of backward recurrence times U
a
(t) and U
b
(t).
(13) ( )
( )
( ) ( ) ( ) ( ) t U t U t t t F t
b a
t N t N
k
t
k
b a , , |
) (
? ? ? · ·
Now, the probability structure depends on the pair of backward recurrence times instead of the
backward recurrence time of the most recent event only. While the model is still Markovian in
nature it’s most general form should not be considered a bivariate renewal process since only in
special cases will the marginal distribution be a renewal. As discussed in Cox and Isham (1980),
this added flexibility comes with a cost, however, as analysis of even simple properties of the
marginal distributions quickly becomes infeasible for all but the simplest parameterizations.
Oakes (1976) establishes the unconditional marginal distribution as the solution to a partial
differential equation.
Hawkes (1971,1972) proposed a class of models that relaxes the Markovian structure of
the model based only on the backward recurrence times. Conditional on a bivariate filtration, the
multivariate Hawkes process for process k is expressed as:
(14) ( ) ( )
( )
( )
( )
? ? + ? ? + ·
· ·
t N
i
b
i
k
t N
i
a
i
k
t
k
b a
t t t t F t
0
2
0
1
? ? ? ?
Now the intensity depends on the backward recurrence time to all previous event arrival times
rather than just the most recent. The applicability of this type of specification for financial data is
questionable since the marginal contribution of a historic event is independent of the number of
intervening events.
Section 4: The Autoregressive Conditional Intensity (ACI) model
In this section we propose a new model for multivariate point process data. It is different
from all the models discussed above and seems particularly useful for economic and financial
transactions data. Initially, we consider a simple bivariate process. The model is most easily
formulated using the conditional intensity function. The conditional intensity function for
process k is given by:
(15) ( ) ( )
k
t N
k
t
k
F t
) (
exp | ? ? ? ·
9
where 0 >
k
? and
k
t N ) (
? is a measurable function of the bivariate filtration of all past arrival
times. Clearly, the conditional intensity function will be non-negative as required.
k
t N ) (
? is time
invariant between event arrivals of the pooled process and therefore is indexed by the counting
process associated with the pooled process. Define the vector
(16)
,
_
¸
¸
·
b
t N
a
t N
t N
) (
) (
) (
?
?
? .
Clearly, in this bivariate setting, each arrival time can be of one of two types. Let
i
y be the
indicator variable taking the value 0 if the i
th
outcome of the pooled process is of type a and 1 if
the i
th
outcome of the pooled process if of type b. We propose the following parameterization for
) (t N
? :
(17)
¹
¹
¹
'
¹
· +
· +
·
? ?
? ?
1 y if
0 y if
1 - N(t) 1 ) ( 1 ) (
1 - N(t) 1 ) ( 1 ) (
) (
t N t N b
t N t N a
t N
B a
B a
? ?
? ?
?
or, equivalently
(18) ( )
1 ) ( 1 ) ( 1 - N(t) ) ( ? ?
+ + ·
t N t N b a t N
B y a a ? ? ?
(
where ? ,
a
a , and
b
a are 2 dimensional parameter vectors B is a 2x2 matrix and
a b b
a a a ? ·
(
.
) (t N
? is an iid unit exponential random variable given by:
(19)
¹
¹
¹
'
¹
·
·
·
0 if
1 if
) (
) (
) (
) (
) (
t N
b
t N
t N
a
t N
t N
y
y
b
a
?
?
?
where
(20) ( )dt F t
a
i
a
i
t
t
t
a a
i ?
? ·
?1
1 ? ? and ( )dt F t
b
j
b
j
t
t
t
b b
j
?
? ·
?1
1 ? ?
Recall that (20) is the generalized residual discussed in section 2. So, if the most recent arrival
was of type a then
) (t N
? represents the innovation associated with the event arrival for process a.
Similarly, if the most recent arrival was of type b then
) (t N
? represents the innovation associated
with the most recent event arrival for process b.
If the event N(t)
th
arrival was of type a then
a
t N
t N
a
) (
) (
? ? · . Positive values of
a
t N ) (
?
indicate that the conditional intensity function under predicted the number of events in the time
10
interval and negative values indicate that the conditional intensity function over predicted the
number of events in the time interval.
) (t N
? is then a weighted average of its most recent value
1 ) ( ? t N
? and the error term
1 ) ( ? t N
? .
Since only one type of information can occur at each event arrival time, only a scalar
innovation appears on the right hand side of (17) at each arrival time of the pooled process. Then
a
? determines the impact of the innovation on
) (t N
? if the N(t)-1 arrival time of the pooled
process is of type a and
b
? determines the impact of the innovation on
) (t N
? if the N(t)-1 arrival
time of the pooled process is of type b. Consequently, the marginal impact of the residual on the
future arrival rate can be different depending on the type of the new information. The persistence
of the impact of
) (t N
? is determined by the eigenvalues of the matrix B. Hence the marginal
impact of the residual depends on the type of new information, but the long run decay is more
parsimoniously specified.
Since
) (t N
? is driven by the difference between the expected number of events (given the
intensity path) and the realized number of events we call the model the Autoregressive
Conditional Intensity (ACI) model. In the model in (17)
) (t N
? depends on one lag of
) (t N
? and
one lag of
) (t N
? it is natural to call this an ACI(1,1) model. Sometimes B need not be fully
saturated so that a diagonal matrix will suffice. If B is restricted to be diagonal then we refer to
the model as a Diagonal Autoregressive Conditional Duration model.
As noted in previous studies establishing even the most basic properties of multivariate point
process models is generally not feasible. The choice of
) (t N
? as the new information observed at
arrival time t
N( t)
is not only intuitively appealing, but will prove to be very useful for establishing
more general properties of the model to be discussed below.
We now present the following lemma that will be useful in establishing the dynamic
properties of the model. Appendix A establishes the following lemma.
Proposition 1: Provided that the F
t
-conditional survivor functions are absolutely continuous with
respect to Lebesgue measure,
a
i
? and
b
j
? are iid unit exponential random variables and
independent of each other for all i,j.
11
The model in (17) provides a flexible representation of the bivariate dynamics. We now take a
closer look at the dynamic properties of the model. Rearranging terms equation (18) can be
rewritten as :
(21) ( ) ( )
1 ) ( 1 ) ( ) ( ? ?
+ · ?
t N t N b a t N
y a a BL I ? ?
(
If the eigenvalues of B lie inside the unit circle equation YY can be represented in an infinite MA
representation
(22) ( ) ? + ·
?
·
? ?
?
1
) ( ) (
* 1
) (
j
j t N j t N b a
j
t N
y a a B ? ?
Appendix B establishes the following result.
Proposition 2: Consider the ACI(1,1) model given by (15) and (17). If the conditional survivor
function is absolutely continuous with respect to Lebesgue measure and the eigenvalues
of B lie inside the unit circle the bivariate process
) (t N
? will be mean reverting with
unconditional mean zero.
We conjecture that the conditions in proposition 2 are also sufficient to establish stationarity.
From (22)
) (t N
? can be viewed as an infinite MA model provided that the variance of y
N( t)
exists.
But y
N(t)
is a binary variable with finite conditional variance determined by
1 ) ( ? t N
? (see the
appendix). The only way the variance cannot exist is if it cycles or drifts off to 0 or 1. This is
unlikely however since
) (t N
? is mean reverting.
Section 5: Extensions of the ACI Model.
The model presented in (17) can be generalized in many ways. In this section we present several
directions.
5.1 Allowing the conditional intensity function to vary between event arrivals.
One important direction would be to relax the restriction that the conditional intensity
function is time invariant between event arrival times of the pooled process. A natural way to do
this is to allow the conditional intensity function to depend on the backward recurrence times
12
( )
( )
( )
( ) t U t U
b a
and where, as before,
( )
( )
) ( N
k
t - t
t
k
t U · . A particular parameterization is given
by:
(23) ( ) ( ) ( ) ( ) ( ) t U t U h F t
b a k k
t N t
k
, exp |
) (
? ? ·
where ( ) ( ) ( ) 0 , > t U t U h
b a k
Now, the function h determines how the intensity changes in the absence of an event of either
type and therefore depends only on the bivariate backward recurrence times. In fact, we can
regard h as a multivariate generalization of the baseline hazard typically used in duration models.
Clearly we have many candidates for the function of the backward recurrence
times ( ) ( ) ( ) t U t U h
b a k
, . Numerous possibilities exist. For example
) (t N
? could still be constant
between event arrivals but may have a different levels depending on which event type occurred
most recent. In this case
) (t N
? is defined as before but the constant term
k
? takes two possible
values given by:
(24) ( ) ( ) ( )
( ) ( )
( ) ( )
if
if
,
¹
¹
¹
'
¹
<
<
·
t U t U
t U t U
t U t U h
a b k
b
b a k
a b a k
?
?
Other examples include the Markov renewal process discussed above. Still further candidates
include any of semi-Markov models that have been discussed in section 3. A particular candidate
for transaction data will be discussed in the application.
5.2 Higher order dependence
Another obvious extension of the model is to allow for
) (t N
? to depend on more lags. In
the model presented in (17) the conditional intensity function depends only on the most recent
values of
) (t N
? and
) (t N
? . In general we could consider a general ACI(p,q) model where the
conditional intensity function depends on p most recent values of
) (t N
? and the q most recent
values of
) (t N
? .
13
5.3 Extending the dynamics of the model.
The model presented in (17) allows the impact of
) (t N
? to be different depending on the type
of event associated with the N(t)
th
arrival of the pooled process. The decay of the shock is
determined by B. While this model is parsimonious and easy to interpret it can also be viewed as
restrictive. Another possibility is to allow B to also depend on the most recent arrival type. Let
a b b
B B B ? ·
*
. Then
(25)
¹
¹
¹
'
¹
· +
· +
·
? ?
? ?
1 if
0 if
1 - N(t) 1 ) ( 1 ) (
1 - N(t) 1 ) ( 1 ) (
) (
y B a
y B a
t N b t N b
t N a t N a
t N
? ?
? ?
?
or equivalently,
(26) ( ) ( )
1 ) ( 1 - N(t) 1 ) ( 1 - N(t) ) ( ? ?
+ + + ·
t N b a t N b a t N
y B B y a a ? ? ?
(
(
This model allows for a richer type of interdependence between the two processes. In particular,
the model can exhibit one way causality, bi-directional causality or no causality at all. To see
this, consider the following restrictions:
(27)
1
]
1
¸
·
1
]
1
¸
·
1
]
1
¸
·
1
]
1
¸
·
b
a
b
b
a
a
a a
22
b
11
a
2
1
0
0 1
B
1 0
0
B
0
0 ?
?
?
?
(28)
( )
( )
( )
( ) ( )
( )
( ) ( )
( )
( )
( )
( ) ( )
( )
( ) ( ) t U h t U t U h
and
t U h t U t U h
b b b a b
a a b a a
,
,
·
·
In this case an occurrence of events of type a will have no impact on
b
t N ) (
? . Similarly, events of
type b will have no impact on
a
t N ) (
? . So, if h
a
is a function of only the backward recurrence time
to events of type a then the model for process a reduces to a univariate proportional hazards
model Cox (1955). That is, the conditional intensity function takes the form:
(29) ( ) ( ) ( ) ( ) t U h F t
a a
t N
t
a
a
) (
exp | ? ? ·
where ( ) ( ) t U h
a
is the baseline hazard function and ( )
) (
exp
t N
a
? simply shifts the baseline hazard
function up or down as a function of the covariates. The same is true for process b if h
b
does not
depend on the backward recurrence time to events of type a. This univariate structure is very
14
similar to the ACD parameterization of Engle and Russell (1998). However, in the ACD model is
a type of accelerated failure time model
3
where the rate at which time passes in the baseline
hazard is also accelerated or decelerated according by a factor of ( )
) (
exp
t N
a
? . For the particular
case of the exponential and Weibull version of the ACD model it can be viewed as either an
accelerated failure time or proportional hazard model. Since we have taken the exponential
function of
) (t N
a
? this is similar the Nelson form ACD model as discussed in Russell and Engle
(1999). This process will be stationary with finite variance provided that and
22 11
b a
B B lie inside
the unit circle.
This formulation should be useful for causality tests. Any deviation from the restrictions
in (26) and (27) will result in causality from one process to the other or bi-directional causality.
Clearly if h
a
depends on ( ) t U
b
then there will be causality from process a to process b and if h
b
depends on ( ) t U
a
then there will be causality from process b to process a. A simple test for
causality from one process to the other can then be based on testing the restricted model in (26)
and (27) against the unrestricted alternative of (24). A rejection of the restricted univariate model
in favor of the unrestricted model provides evidence of causality.
5.4 Deterministic Time of Day Effects
It is well documented that many types of financial data exhibit intraday deterministic
patterns. Harris (1986) was one of the first to document deterministic patterns of the variance
within day. Engle and Russell (1998) documented similar phenomenon for arrival rates of
traders. A simple way of introducing a deterministic pattern into the ACI model is given by:
(30)
( )
( )
( )
( )
( ) ( )
( )
( )
( ) ( ) ( ) t t U t U h F t
k b a k k
t N t
k
? ? ? , exp |
) (
·
?
where ( ) 0 > t
k
? is a continuous deterministic function of time. Now, the intensity can vary
deterministically as a function of time as well as stochastically. If ( ) t
k
? is left completely
unrestricted there is a lack of identification. Without loss of generality we propose normalizing
( ) t
k
? to take the value 1 at the start of the day. In our application we will specify the
deterministic component by a piecewise linear spline.
3
See Lancaster (1990) for a discussion of these models.
15
5.5 Conditioning on marks.
We now return the possibility that one or both of the point processes contains marks. In this
case there may be information associated with event arrival times. For example with a
transaction we may observe a price and volume transacted. If these marks carry information
about future arrival rates we would like to include this information in the model. Let
a
t N
a
Z
) (
and
b
t N
b
Z
) (
denote a vector of marks associated with event arrivals of types a and b respectively.
(31)
¹
¹
¹
'
¹
· + +
· + +
·
?
? ?
?
? ?
1 if
0 if
1 - N(t)
1 ) (
1 ) ( 1 ) (
1 - N(t)
1 ) (
1 ) ( 1 ) (
) (
y Z B a
y Z B a
t N
b t N t N b
t N
a t N t N a
t N
b
a
? ? ?
? ? ?
?
where
a
? and
b
? are conforming parameter vectors. So, if the most recent event arrival was of
type a then the marks from process a are used. If the most recent event was of type b then the
marks from process b are used. This will be particularly useful for transactions data where we
often have additional information such as quantities or prices. In fact, if some if appropriate
elements of a
a
and a
b
are zero it is possible for one intensity to only depend on the marks of the
second process and not on the innovations
) (t N
? .
5.6 Higher dimension systems
The ACI model is easily extended to higher dimensions.
) (t N
? can be expanded to a vectors
of dimension k and
) (t N
? will denote an iid innovation from one of k possible event arrival types
in the pooled process. Unlike many other models this extension is remarkably simple.
Furthermore, we find a bivariate model very easy to estimate so that it seems very practical to
expand the dimension beyond two.
6. The Likelihood and Model Diagnostics for an ACI(p,q)
The conditional intensity function characterizes the instantaneous probability of an event
arrival in the bivariate point process. Given the instantaneous probabilistic structure of the
bivariate point process the likelihood for the ACI model can be obtained. Following Florens and
16
Gougere (1995) it is easily expressed using the integrated intensity function described in the
previous section. For a bivariate model that requires joint estimation of both processes the
likelihood is expressed as:
(32) ( ) ( ) ( ) ( ) ( ) ( ) ? ? ? + ? ? ·
· ·
) (
1
) (
1
| | , 0 , 0 exp
t N
k
t N
j
t
b
j
b
t
a
i
a b a
a b
F t F t T T L ? ?
Simply put, the first term is the product of the F
t
-conditional survivor functions for process a and
process b associated with each duration. This term characterizes the joint probability of no events
occurring in each of those durations. The last two terms of (32) denote the joint probability of the
realized arrival times. For joint estimation of a K dimensional process, the likelihood is expanded
in the obvious way.
Early discussions of the probability structure of bivariate self-exciting processes include
Hawkes (1971) who notes that the bivariate self-exciting process is a particular Doubly Stochastic
Poisson
4
process. Thus, the likelihood can be constructed using the theory of Doubly Stochastic
process
5
. The likelihood may also be justified via the theory of compensators as discussed in
Daley and Vere-Jones (1988) Ch. 13.
Correct specification of the model implies that
k
t N
k
t N ) ( ) (
~
1 ? ? ? · should be an iid unit
exponential random variable that is uncorrelated with F
t
. . This suggests using a Ljung Box test
on the autocorrelations. A rejection of the null that
k
t N ) (
? is uncorrelated provides evidence of
mispecification. Furthermore, it
k
t N ) (
? should be distributed as a unit exponential distribution so
we can also check moment conditions associated with the unit exponential. We propose using the
test suggested in Engle and Russell (1998) for excess dispersion. In particular, a simple test of no
excess dispersion is obtained from the statistic ( ) 8 1 ˆ
2
~ ?
?
?
k
N . Under the null hypothesis of a
unit exponential the test statistic will have limiting Normal distribution. Similarly, a rejection of
the null provides evidence of model inadequacy.
7. Liquidity and Market Some Microstructure
In this section we introduce some basic concepts of liquidity. Some Market Microstructure
theory is also discussed to help motivate the empirical study in the section 6.
4
See Cox and Isham (1980) for a discussion of the Doubly Stochastic Likelihood.
5
See Cox and Isham (1980) for a good discussion of Doubly Stochastic Processes.
17
7.1 Liquidity
Market liquidity is of great interest to financial traders. Generally speaking a liquid market is
characterized by the ability to trade large amounts of volume at a price close to the efficient (or
true) price of the asset. One dimension of liquidity is often quantified by the bid ask spread. The
larger the difference in the bid and ask prices the further the transaction price will likely be from
the efficient price. This measure, however, ignores the size of the trade. Typically, the quotes are
only valid for relatively small numbers of shares. If a trader wishes to transact a large volume the
price may be quite different than prices obtained for small quantities. The price volume
relationship is conveniently expressed in figure 2.
The vertical distance along the transaction price axis denotes the bid ask spread. The
horizontal distance from the transaction price axis to the sloped line is the minimum volume that
can be transacted at the current bid and ask prices. Beyond a small number of shares larger sell
volume demands a lower price and larger buy volume demands a higher price. If an order is
larger than this small number of shares a worse price is obtained. The complete price volume plot
is called the market impact curve.
Typically this market impact curve will look quite different at different times during the day.
The spread may widen or the slope of either side of the market impact curve may change. Limit
orders represent unfilled orders awaiting transaction consisting of a desired volume and price. At
any point in time the exact nature of the market impact curve is determined by the unfilled orders
in limit order book. In this sense, limit orders can be viewed as a major contributor to market
liquidity and the market impact curve.
An agent that desires an immediate transaction will contact a floor trader. This can be done
by submitting a market order or, if the transaction is particularly difficult, by having a floor
broker "work" the transaction over a longer period of time. While this type of trade is guaranteed
to transact there is a cost. The cost will depend on the volume desired to be traded and the
current status of the market response curve.
It is therefore natural to consider a transaction as demand for liquidity and limit orders as the
supply of liquidity. In our application we will model the bivariate process of transaction arrival
times and limit order submission times. The model is used to study the dynamics the joint
dynamics of the demand and supply of liquidity how they are supply and demand are affected by
different characteristics of the market.
18
7.2 Market Microstructure and Liquidity
Market Microstructure theory typically posits two reasons an agent will decide to transact.
The first type of trader is an uninformed agent that is simply trading to adjust a portfolio. Since
the agents are uninformed and have no superior information about future prices their transactions
are often assumed to be random resulting in Poisson type arrival. The second type of trader is an
informed agent that possesses superior information. It is natural to think that the privately
informed agents would like to submit market orders to insure that their transaction occurs quickly
and therefore minimize the risk of the market learning of the private information from other
sources before they can capitalize on it. Alternatively, liquidity traders may submit either limit
orders or market orders depending on the urgency of the portfolio adjustment and other factors.
The classic models of liquidity traders are very simple and assume that they have no
strategic behavior. A slightly more sophisticated model was proposed by Admati and Pfleiderer
(1988) suggests that the uniformed by batch their trades together to minimize the probability of
trading with superiorly informed agents. If uninformed traders are concerned about trading with
better informed agents they may enter the market when the probability of transacting with better
informed agents is small. Typically, however, it is not possible to distinguish informed from
uninformed traders directly. Instead, the existence of private information must be inferred from
transaction and general market characteristics. Easley and O'Hara (1987) and Hasbrouck (1988)
find that larger volume transactions have a greater price impact than small volume trades. Early
models by Copeland and Galai (1983) and numerous papers to follow suggest that spreads should
widen if the presence of informed traders is expected.
If both uninformed and informed traders are strategic then the patterns of transaction and
limit order submission should be impacted by presence (or suspicion) that informed traders are
present. As in Easley and O'Hara (1991) informed traders will transact only when they posses
private information. Uninformed traders will submit orders limit orders when they suspect the
probability of transacting with better informed agents is small. In a rational expectations setting
this suggests that both the demand for liquidity and the supply of liquidity should be affected by
volume, spreads, and transaction rates. This hypothesis will be examined.
19
8. An Application to the Bivariate Point Process of Transactions and Limit Order
Submission
In this section of the paper we apply the ACI model to the bivariate process of transaction
arrival times and limit order submission times. We first introduce the data.
8.1 The Data
The data are abstracted from the TORQ data set. For the period November 1, 1990 to
January 31, 1991 Federal Express contains 5138 unique transaction times and 4058 unique limit
order submission times. The market operates from 9:30 am 4:00 pm local time. For details of
the operation of the market we refer the reader to Hasbrouck, Sofianos and Sosebee (1993).
Following Engle and Russell (1998) we delete arrival times occurring in the first half-
hour of the trading day since many of these trades are opening trades that operate according to a
batch auction, not the continuous market that characterizes the majority of the trades made on the
NYSE. We construct the sequence of interarrival times denoted by
r
i
r
i
r
i
t t
1 ?
? · ? and
m
j
m
j
m
j
t t
1 ?
? · ? for the trade and limit order submission processes respectively. Figure 3 presents a
histogram for the durations
r
i
? and
m
i
? . On average, transactions occur about every 248 seconds
or just over 4 minutes. Limit orders are submitted, on average once every 312 seconds or just
over 5 minutes. The standard deviations are 367 and 420 respectively. The maximum time
interval is 4139 and 4416 or a little over an hour. Naturally, both histograms are skewed to the
right with a lower bound of 1, the finest recorded interval.
It is informative to view the autocorrelation structure of the duration series
r
i
? and
m
j
? .
Table 1 contains the autocorrelations and partial autocorrelations of the two duration series.
Additionally, the series
p
s
p
s
p
s
t t
1 ?
? · ? is constructed from the pooled series containing all arrival
times of both the transaction and quote arrival times. Hence these are durations that can begin
with either a transaction or limit order time and end with either a transaction or quote time. The
duration statistics indicate that both the transaction and limit order durations are highly
autocorrelated with Ljung-box statistics of 480.1 and 668.1 respectively. These statistics are
distributed as a Chi-Squared with a 5% critical value of 24.99. The durations from the pooled
process are also highly autocorrelated.
20
8.2 Model specification and estimates
We estimate a diagonal ACI model with time varying intensity between event arrivals. We
specify the bivariate backward recurrence time function h for the k
th
process by:
(33) ( ) ( ) ( ) ( ) ( ) ( )
k k
t U t U t U t U h
b a k b a k 2 1
exp ,
? ?
? ·
Clearly, h
k
>0 as required. If 0
1
·
k
? then process k does not depend on the backward recurrence
time for process 1. So, process k may depend only on it's own backward recurrence time, only on
the backward recurrence time of the other process, or perhaps on neither. If both
0 and 0
2 1
< <
k k
? ? then the conditional intensity function will always be downward sloping with
jumps at each event arrival of the pooled process. This implies the probability of an event of type
k will decrease in the absence of new event arrivals. Alternatively, if both 0 and 0
2 1
> >
k k
? ? are
positive then the intensity function will be monotonically increasing with jumps down at each
event arrival of the pooled process. In this case the probability of an event will be increasing in
the absence of new event arrivals. If 0 and 0
2 1
> <
k k
? ? the intensity function can be non-
monotonic u-shaped or inverted u-shaped. In the absence of a event arrival in the pooled process
the intensity function will eventually be positively sloped if
k k
1 2
? ? > and eventually
negatively sloped if
k k
1 2
? ? < . Hence (33) allows for flexible variation in the intensity
function in the absence of new events and satisfies the non-negativity property.
We begin by estimating a Diagonal ACI(1,1) model with a deterministic component
given by a piecewise linear spline. Two continuous splines are specified, one for the transaction
process and one for the limit order arrivals. The nodes are at each hour with the first and last at
10:00 and 4:00 for a total of 6 linear pieces. Engle and Russell (1996) found a similar piecewise
linear spline performed well for IBM transactions data. The constant in the spline is normalized
to unity and the slope parameters for process k are denoted by ( )
k k
d d
6 1
,..., .
All the parameters are estimated jointly by conditional maximum likelihood. Initial
values for the vectors
0
? and
0
? are set to their unconditional expectation of 0. Since the scores
are recursive functions of the past similar to Bollerslev's (1986) GARCH model we use numerical
derivatives. The Bernt, Hall, Hall, Hausman (BHHH) (1974) algorithm is used to maximize the
likelihood function. The likelihood function given in (32) requires integration of the conditional
intensity function. If the function h is a constant then closed form solutions for this integral are
21
easily obtained. In the more general case of (33), however, closed form solution does not exist
and we use numerical integration for this more general case.
Estimated parameters are presented in table 1. Standard errors are obtained using the
outer product of the scores. All parameter values are significant except for some of the parameters
associated with the deterministic pattern. From the parameters for the conditional intensity
function for transaction arrivals we see that larger values of either
( )
r
t N
a
1 ?
? or
( )
m
t N
a
1 ?
? increase the
conditional intensity function. Hence if the realized arrival rate was greater (less) than the
expected arrival rate the conditional intensity function is increased (decreased) regardless of event
arrival type. Similarly, larger values of either
( )
r
t N
a
1 ?
? or
( )
m
t N
a
1 ?
? increase the conditional
intensity for limit order arrivals. All else equal, both conditional intensity functions tend to move
in the same direction. Both
a
? and
b
? are less than, but near, one. This is indicative of strong
persistence but mean reverting behavior. Finally,
r
1
? and
r
2
? are both negative and significant
indicating that the probability of a transaction monotonically declines in the absence of new event
arrivals. Similarly,
m
1
? and
m
2
? are also negative and significant indicating that the probability of
a limit order arrivals monotonically decreases in the absence of new event arrivals.
Figure 4 contains a plot the deterministic components for the two processes. They are
both U shaped with larger intensity in the morning and just prior to close. These plots are nearly
the inverted plots of those presented in Engle and Russell (1998). Engle and Russell (1998)
modeled the deterministic pattern of the expected duration which is roughly the inverse of the
intensity.
At the bottom of the table are the diagnostic statistics based on
( )
r
t N
a
1 ?
? or
( )
m
t N
a
1 ?
? .
Under the null they should be iid unit exponential and uncorrelated with F
t
. For each series, the
Ljung-Box statistic for the first 15 autocorrelations are presented at the bottom of the table. The
number of observations is different for the two series since there are different numbers of
transactions and limit order submissions. The Ljung-Box associated with the transaction process
is 16.26. The test statistic has a Chi-Squared distribution with 15 degrees of freedom. The 5%
critical value is 24.99 so we fail to reject the null. For the limit order series, the test statistic is
35.58. This results in a rejection of the null.
Next, we examine the excess dispersion test statistic. Under the null
( )
r
t N
a
1 ?
? and
( )
m
t N
a
1 ?
?
should have unit variance. The test statistics are 1.63 and -3.81 for the transaction and limit order
22
series respectively. These are asymptotically Normal and suggests that
( )
m
t N
a
1 ?
? is under
dispersed.
Since the model for limit order arrivals is rejected we estimate an ACI (2,2) model. The
results are presented in table 2. The significant coefficients on the lagged values of
( )
r
t N
a
1 ?
? and
( )
m
t N
a
1 ?
? are again positive. The sum of ( )
r r
2 1
? ? + and ( )
m m
2 1
? ? + are both equal to 0.97. This
is qualitatively similar to the findings in the ACI(1,1) model. We now return to the model
diagnostics at the bottom of the table. The Ljung-Box for the limit order series is now 26.73
which is marginal at the 5% level. The excess dispersion test statistics are basically unchanged
and we still find mild evidence of under dispersion for the limit order process. Overall, the model
fits well.
Figure 5 presents the QQ plot for
( )
r
t N
a
1 ?
? and
( )
m
t N
a
1 ?
? . Under correct specification the
QQ plots should lie on the upward sloping line. The plots indicate that while the fit is generally
good, there are too many large values of
( )
r
t N
a
1 ?
? and not enough large values of
( )
m
t N
a
1 ?
? . This is
consistent with the excess dispersion tests.
8.3 Estimates for models with marks
We now return to the more general case of a bivariate marked point process. We
estimate models where the conditional intensity functions can depend on not just the historic
bivariate arrival times but also other characteristics associated with the arrival times. By
including marks in the specification of the conditional intensity function we can see how they
impact future arrival rates of the bivariate system.
Along with the transaction arrival times we observe a transaction price, and a volume.
For each limit order the data also contains the price at which the limit order is to be executed, the
volume to be transacted, and a buy/sell indicator. Additionally, from a separate series of quotes
posted by the specialist we observe bid ask prices. These bid and ask prices can be matched up
with transaction times to obtain the prevailing bid and ask quotes at the time of the transaction.
From these marks we create the following transformations to be used in estimation. First,
we construct a series for the log of volume transacted. This series is denoted by
) (t N
r
V . Next, we
construct a measure of the volatility of price changes. ( ) ( )
1 ) ( ) ( ) (
log log
?
? ·
t N t N t N
r r r
p p ? where
i
p is the transaction price associated with the i
th
transaction.
1 ) ( ) ( ) ( ?
? · ?
t N t N t N
r r r
p p p is
23
simply the price change between subsequent transactions. Finally, using the spreads we create
the series
1 ) ( ) ( ) ( ?
? ·
t N t N t N
r r r
spd spd dspd where
i
spd is the spread, or difference between the
bid and ask prices, prevailing at the i
th
transaction time.
From the marks associated with the limit order arrivals we construct the following series.
¹
¹
¹
'
¹
·
otherwise 1
quotes. ask and bid prevailing the of midpoint the from % 1 than less is lp if 0
) (
) (
t N
t N
m
m In
So, this is an indicator variable denoting a "competitive" limit order that is not far away from the
current price.
) (t N
m
Buy is a buy sell indicator taking the value one if the order is to buy.
) (t N
m
S
is the desired number of shares to be transacted.
The marks are included linearly in the specification of
) ( t N
? as in equation 31. Parameter
estimates are given in table
6
4. The results suggest that large volume transacted tend to be
followed by higher intensity of quote submission. Large transaction volume does not appear to
be followed by higher intensity of transactions. The price change between subsequent
transactions has a positive and significant coefficient in the transaction intensity. This suggests
that the transaction intensity tends to be higher in periods of rising prices. Volatility has little
impact on the transaction rates or the limit order submission intensity. The price change between
subsequent transactions does not have a significant impact on the limit submission intensity. The
change in the spread between subsequent transaction time does not have an impact on the
intensity of transactions, however it decreases the intensity of limit order submissions. If wide
spreads are indicative of asymmetric information and price uncertainty then this suggests limit
orders are not submitted are frequently when uncertainty is greater about the value of the asset.
The proximity of the limit price to the prevailing price level is not significant for either
intensity function. The buy indicator does not have a significant for either intensity. The number
of shares associated with a limit order has a positive impact on the intensity of transactions but
has an insignificant impact on the limit order submission intensity.
In summary, we find that transaction rates and limit order submission submissions tend to
move together. Hence when demand for liquidity rises, the supply of liquidity also tends to
increase. Large transaction volume tends to be followed by more intense limit order submission.
Submission of limit orders for large volume tend to be followed by more intense transactions.
Transaction rates tend to be higher when prices are rising, but limit order submission is
6
The deterministic component parameters are not presented. They are essentially unchanged from the
previous model estimates.
24
unaffected by the direction of price movements. Finally, wide spreads tend to be followed by
increased limit order submission but we find no impact on transaction rates.
9. Conclusion:
This paper extends the work of Engle and Russell (1998) to a multivariate setting. In a
multivariate context we propose working with the conditional intensity process rather than with
the conditional distribution of durations. Modeling the instantaneous arrival rate is far easier than
modeling the conditional distribution of durations when information can arrive at any point in
time, not just at the start of a duration as considered in Engle and Russell (1998).
The model is very different from any existing model for multivariate point process and
appears to be particularly useful for analysis of financial transactions data. In particular, each
process is treated symmetrically which relieves the investigator of the task of choosing a process
to be the "driving process" as other studies have proposed. The model allows for long range
dependence typically observed in economic and financial data and is not Markovian. Finally, the
parameter values are easily interpreted in terms of an immediate impact and the persistence of an
innovation.
Extensions of the model allow the intensity function allow for rich dynamic structure.
The intensity may be constant between event arrivals or evolve as a deterministic function of the
backward recurrence times. The model can also allow for deterministic patterns in the intensity
functions. More complex dynamics allow for the possibility of performing causality tests.
Finally, we allow for the possibility that the marks, or characteristics associated with the arrival
times can also influence the intensity functions. This is particularly useful for economic
transaction data where we typically observe prices and volume at transaction times. Finally, the
model is easily expanded to higher dimensions. Our estimation suggests that this is not only
feasible, but also a practical possibility.
The model is applied to the bivariate process of transaction arrivals and limit order
submissions. Limit order submissions add to the limit order book and have a direct impact on
market liquidity and generally on the shape of the market impact curve. We are not aware of any
work that examines the dynamic process of limit order submissions. Since transactions demand
and use up liquidity the bivariate point process of transaction and limit order submission arrival
times sheds light on the dynamic relationship between liquidity supply and demand.
Estimated models suggest that the two processes tend to move together both in a
deterministic manner as well as stochastic. Hence increases in the demand for liquidity tend to be
associated with increased supply of liquidity. We also find that large transaction volume tends to
25
be followed by more intense limit order submission. Submission of limit orders for large volume
tend to be followed by more intense transactions. Transaction rates tend to be higher when prices
are rising, but limit order submission is unaffected by the direction of price movements. Finally,
wide spreads tend to be followed by increased limit order submission but we find no impact on
transaction rates.
A natural use of high frequency data is to learn about the fundamental market
microstructure. Causality testing should play a fundamental role in this learning process. It is
well know that aggregation will distort causality so it is natural to focus on transaction by
transaction data. A generalization of the model discussed in section 5.3 should be useful for
testing along these lines.
26
Appendix A
From the discussion of the martingale property of the compensator in section 2 it is clear that the
compensators for processes a and b evaluated over non-overlapping intervals must be uncorrelated. It
remains to be shown that this same is true for overlapping intervals. From the left hand side of (4) and for
small values of t ? we can express
( ) t N
a
? as:
(9)
( )
( ) ( )
( )
( )
( )
( ) ( )
? ? ? + ? ? ? + ? ? + ·
?
,
_
¸
¸
?
·
? ? ?
?
t
t t
j
t t N
a
t N a t N a
t N
t a N t a N
a a a
a
t F t j t t j t N t j t N
1
1
1 1 1
| ) ) 1 ( ( ) ( ? ?
Similarly,
(10)
( ) ( ) ( )
( )
( )
( )
( ) ( )
( )
? ? ? + ? ? ? + ? ? + ·
?
?
·
? ? ?
?
t
t t
j
t t N
b
t N b t N b t N
t b N t b N
b b b b
t F t j t t j t N t j t N
1
1
1 1 1
| ) ) 1 ( ( ) ( ? ?
Each element on the right hand side of (9) is uncorrelated with each elements on the right had side of (10)
such that t j t t j t
t N t N
a b
? + < ? +
? ? 1 ) ( 1 ) (
. Similarly, each summation element on the right hand side of
(10) is uncorrelated with all elements on the right had side of (9) such that t j t t j t
t N t N
b a
? + < ? +
? ? 1 ) ( 1 ) (
.
Since the summation is simply a linear combination of the elements it follows that
) ( t N
a
? is uncorrelated
with
) (t N
b
? .
Finally, note that the exponential distribution and its moments are determined by a single parameter.
Hence, if the
) ( t N
a
? and
) (t N
b
? are uncorrelated, they must also be independent.
Appendix B
If the eigenvalues of B lie inside the unit circle then
) ( t N
? can be expressed as:
1' ( ) ? + ·
?
·
? ?
?
1
) ( ) (
1
) (
j
j t N j t N b a
j
t N
y a a B ? ?
Under the assumptions,
) (t N
? is an iid mean zero random variable. Furthermore,
) (t N
y is a binary random
variable with conditional distribution given by:
2'
( )
( ) ( )
( )
( ) ( )
¹
¹
¹
¹
¹
'
¹
+
+
·
b
t N
b a
t N
a
b
t N
a
b
t N
b a
t N
a
b
t N
b
t N
) ( ) (
) (
) ( ) (
) (
) (
exp exp
exp
y probabilit with 0
exp exp
exp
y probabilit with 1
y
? ? ? ?
? ?
? ? ? ?
? ?
Since the distribution of
) (t N
y is determined by
1 ) ( ? t N
? ,
2 ) ( ? t N
? ,… it follows that
) (t N
? is independent of
) (t N
y . Taking expectations of both sides of 1' we conclude that the mean of
) ( t N
? exists and is equal to
zero. Since y
N(t)
is binary it's conditional variance given
) ( t N
? is bounded by 0 and .25. If the eigenvalues
of B lie inside the unit circle then
) ( t N
? is mean reverting.
27
Figure 1: Representation of a 3 Dimensional Marked Point Process
( )
1
0
1
0
, Z t
•
( )
1
1
1
1
, Z t
•
( )
1
2
1
2
, Z t
•
( )
2
0
2
0
, Z t
•
( )
2
1
2
1
, Z t
•
( )
3
0
3
0
, Z t
•
( )
3
1
3
1
, Z t
•
( )
3
2
3
2
, Z t
•
( )
3
3
3
3
, Z t
•
( )
3
3
3
3
, Z t
•
MPP 1
MPP 2
MPP 3
1
1
x
1
2
x
}}
( )
? t
F t
1
?
time
28
Figure 2: Market Impact Curve
Volume
Transaction
Price
29
Figure 3: Histogram for Transaction and Limit Order Durations
Seconds
Seconds
0
500
1000
1500
2000
2500
3000
0 500 1000 1500 2000 2500 3000 3500 4000 4500
Series: LIMIT
Sample 1 4057
Observations 4058
Mean 312.4673
Median 160.0000
Maximum 4416.000
Minimum 4.000000
Std. Dev. 420.1237
Skewness 2.957542
Kurtosis 15.15550
Jarque-Bera 30891.43
Probability 0.000000
0
1000
2000
3000
4000
0 500 1000 1500 2000 2500 3000 3500 4000
Series: TRANS
Sample 1 5137
Observations 5138
Mean 248.9017
Median 111.0000
Maximum 4139.000
Minimum 1.000000
Std. Dev. 367.7251
Skewness 3.218003
Kurtosis 18.29552
Jarque-Bera 58941.73
Probability 0.000000
30
Figure 4: Deterministic Component for Transaction and Limit Intensities
0
0.2
0.4
0.6
0.8
1
1.2
9 10 11 12 13 14 15 16 17
Time of Day
D
e
t
e
r
m
i
n
i
s
t
i
c
C
o
m
p
o
n
e
n
t
Transaction Limit Order
31
Figure 5: QQ plot of integrated Intensities
Integrated Intensity QQ Plot for Transactions
0
1
2
3
4
5
6
7
8
9
0 2 4 6 8 10
Empirical CDF
C
D
F
U
n
i
t
E
x
p
o
n
e
n
t
i
a
l
Integrated Intensity QQ Plot for Limit Orders
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6
Empirical CDF
c
d
f
U
n
i
t
E
x
p
o
n
e
n
t
i
a
l
32
Table 1. Summary Statistics for Durations
Trans. Durations Limit Durations Pooled Durations
acf pacf acf pacf acf pacf
lag 1
0.110 0.110 0.188 0.188 0.173 0.173
lag 2
0.110 0.099 0.141 0.109 0.135 0.109
lag 3
0.107 0.087 0.160 0.122 0.107 0.071
lag 4
0.116 0.090 0.131 0.076 0.105 0.066
lag 5
0.076 0.041 0.124 0.066 0.107 0.066
lag 6
0.067 0.031 0.107 0.044 0.109 0.063
lag 7
0.076 0.042 0.068 0.005 0.086 0.036
lag 8
0.057 0.021 0.064 0.011 0.098 0.051
lag 9
0.072 0.040 0.093 0.048 0.095 0.044
lag 10
0.059 0.024 0.097 0.050 0.081 0.027
lag 11
0.060 0.025 0.052 -0.002 0.081 0.029
lag 12
0.076 0.043 0.045 -0.002 0.080 0.029
lag 13
0.053 0.015 0.037 -0.006 0.064 0.011
lag 14
0.042 0.006 0.032 -0.004 0.085 0.037
lag 15
0.047 0.013 0.073 0.045 0.053 0.000
LB(15) = 480.1
Sample Size=5138
LB(15) = 668.1
Sample Size=4058
LB(15) =1413.3
Sample Size=9196
33
Table 2. Parameter Estimates for ACI(1,1)
Trade Intensity Quote Intensity
estimate t-stat estimate t-stat
r
1
?
-.0518 -3.35
m
1
?
-.0887 -3.39
r
r 1 ,
?
.04499 5.30
m
r 1 ,
?
.066 5.91
r
q 1 ,
?
.02291 2.05
m
q 1 ,
?
.1160 7.24
r
r 1 ,
?
.9752 132.12
m
q 1 ,
?
.9680 104.85
r
1
?
-.3878 -45.48
m
1
?
-.2911 -26.45
r
2
?
-.2287 -23.11
m
2
?
-.2395 -21.54
r
d
1
-.4099 -1.51
m
d
1
-.3284 -0.86
r
d
2
-.07588 -0.36
m
d
2
.1513 0.49
r
d
3
-.1360 -0.67
m
d
3
-.4996 -1.69
r
d
4
-.3043 -1.51
m
d
4
.1545 0.54
r
d
5
.5773 2.79
m
d
5
.1822 0.62
r
d
6
.1412 0.57
m
d
6
.6895 1.89
LB(15) = 16.25 LB(15) = 35.58
Mean of
r
i
? =.0117 Mean of
r
i
? =.0071
Test Stat. Ex. Disp. (
r
i
? )=0.5797 Ex Disp. (
r
i
? )=-3.81
# Events=5138 # Events=4058
34
Table 3. Parameter Estimates for ACI(2,2)
Trade Intensity Quote Intensity
estimate t-stat estimate t-stat
r
1
?
-.06321 -2.69
m
1
?
-.07757 -2.68
r
r 1 ,
?
.05460 3.06
m
r 1 ,
?
.07568 3.67
r
r 2 ,
?
.01463 0.71
m
r 2 ,
?
.0032 0.14
r
q 1 ,
?
.01493 0.62
m
q 1 ,
?
.1647 5.91
r
q 1 ,
?
.02553 1.09
m
q 1 ,
?
-.03239 -1.15
r
r 1 ,
?
.4353 1.37
m
q 1 ,
?
.6291 3.27
r
r 1 ,
?
.5345 1.72
m
q 1 ,
?
.3427 1.82
r
1
?
-.3880 -45.46
m
1
?
-.2911 -26.45
r
2
?
-.2275 -22.92
m
2
?
-.2395 -21.54
r
d
1
-.4240 -1.58
m
d
1
652 -1.02
r
d
2
-.06064 -0.296
m
d
2
.1513 0.56
r
d
3
-.1376 -0.68
m
d
3
-.4895 -1.72
r
d
4
-.3054 -1.52
m
d
4
.1778 0.64
r
d
5
.5823 2.81
m
d
5
.1660 0.58
r
d
6
.1440 0.58
m
d
6
.6984 1.96
LB(15) = 17.58 LB(15) = 26.73
Mean of
r
i
? =-.0153 Mean of
r
i
? =.0107
Test Stat. Ex. Disp. (
r
i
? )=1.56 Ex Disp. (
r
i
? )=-3.82
# Events=5138 # Events=4058
35
Table 4. Parameter Estimates for ACI(2,2)
Trade Intensity Quote Intensity
estimate t-stat estimate t-stat
r
1
?
-.06321 -2.69
m
1
?
-.07757 -2.68
r
r 1 ,
?
.05460 3.06
m
r 1 ,
?
.07568 3.67
r
r 2 ,
?
.01463 0.71
m
r 2 ,
?
.0032 0.14
r
q 1 ,
?
.01493 0.62
m
q 1 ,
?
.1647 5.91
r
q 1 ,
?
.02553 1.09
m
q 1 ,
?
-.03239 -1.15
r
q 1 ,
?
.4353 1.37
m
q 1 ,
?
.6291 3.27
r
q 1 ,
?
.5345 1.72
m
q 1 ,
?
.3427 1.82
r
1
?
-.3880 -45.46
m
1
?
-.2911 -26.45
r
2
?
-.2275 -22.92
m
2
?
-.2395 -21.54
) (t N
r
V
r
1
?
-.008947 -1.01
m
1
?
.02573 2.79
2
) (t N
r
?
r
2
?
-.001722 -.44
m
2
?
-.00107 -0.50
) (t N
r
p ?
r
3
?
21.96 4.08
m
3
?
.5297 0.081
) (t N
r
dspd
r
4
?
.00108 0.13
m
4
?
.01454 1.98
) ( t N
m
In
r
5
?
.01249 0.34
m
5
?
.02133 0.64
) (t N
m
Buy
r
6
?
.02714 1.07
m
6
?
.00717 0.33
) (t N
m
S
r
7
?
.03436 2.41
m
7
?
-.01170 -0.98
36
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doc_162305948.pdf
The recent advent of high frequency data provides researchers with transaction by transaction level data. Examples include scanner data from grocery stores or financial transactions data. These new data sets allow researchers to take an unprecedented look at the underlying economic structure of the markets.
Econometric Modeling of Multivariate Irregularly-Spaced High-Frequency Data
Jeffrey R. Russell
*
University of Chicago
Graduate School of Business
November 1999
Abstract: The recent advent of high frequency data provides researchers with transaction by transaction
level data. Examples include scanner data from grocery stores or financial transactions data. These new
data sets allow researchers to take an unprecedented look at the underlying economic structure of the
markets. Often the hypothesis of interest is in the context of multivariate time series data. Analysis of
multivariate high frequency data is complicated by the fact that the multiple processes are irregularly
spaced in time with different arrival rates. This has lead many investigators to work with aggregated data
which can blur the market structure and contaminate the analysis. In this paper we propose a new method
of working with the dissaggregated data and develop an econometric model for the arrival rates of
multivariate dependent point processes. We apply the model to financial transactions data and estimate
models for the bivariate point process of transaction and limit order arrival times. Since limit orders dictate
the structure of the limit order book they are a direct determinant of market liquidity. To our knowledge
little work has examined the dynamic structure of limit order submissions. Since transactions represent a
floor trader or market order demand for liquidity the bivariate model characterizes the dynamic behavior of
liquidity supply and demand. The proposed model also allows for marks, or characteristics associated with
the arrival times, to influence future arrival rates. For the stock market data analyzed we find strong
evidence of codependence in the two processes and both liquidity demand and supply are influenced by
past volume and prevailing spreads.
Key Words: ACD, High Frequency Data, Multivariate Time Series, Causality, Point Process
*
The paper benefited from comments by participants at the 1999 Econometric Society meetings. Also,
conversations with Rob Engle, Eric Ghysels, and Bruce Hansen were very helpful. Financial support from
the University of Chicago, Graduate School of Business is gratefully acknowledged.
1
Section 1: Introduction
High frequency data sets are rapidly becoming available for many types of economic and
business data including transactions data from financial markets, scanner data from grocery
stores, or records of credit card purchases to name a few. These high frequency data sets provide
the empirical investigator with an unprecedented view the underlying market microstructure.
Analysis of high frequency transaction data is complicated by the fact that transaction
data are inherently irregularly spaced in time. This is particularly troubling when the data of
interest is multivariate. For example, financial transactions data sets often contain not only
information about the transactions, but also the quote dynamics. Quote adjustments do not occur
at regularly spaced time intervals and will have a different rate than the transactions making
traditional time series analysis impossible. As a result, investigators must aggregate the time
series data to fixed time intervals so that standard econometric techniques for multivariate data
can be applied. Aggregation can blur the market structure and the choice of an optimal interval is
not obvious.
Engle and Russell (1998) propose a different approach. They propose modeling the
arrival rate of transactions data with the Autoregressive Conditional Duration (ACD) model.
Often the arrival rate is interesting in itself. Engle and Russell (1997,1998) model the time
between price changes of fixed size. This can be interpreted as a model of volatility that is
particularly well suited for irregularly spaced time series data. Engle (1996), Ghysels,
Gourieroux, and Jasiak (1998) and Russell and Engle (1999) and others propose joint modeling of
the transaction price process and the duration using the ACD model.
This paper extends the work of Engle and Russell (1998) and proposes a new
econometric model for multivariate transactions data. The data are treated as a bivariate marked
point process with dependent arrival rates. In this multivariate context it is difficult to model the
expected duration which is the foundation of the ACD model. This is the approach taken in
Engle and Lunde (1999) who propose a censored bivariate ACD model. The goal of their work is
to assess how quickly information in the transaction process impacts the prices via quote
adjustments. Therefore the primary interest is the time between transactions and subsequent
quote revisions. The transaction process is therefore specified as the "driving process". The two
processes are not treated symmetrically and some information is lost if multiple quote revisions
occur without intervening transactions. The problem is that it is difficult to model the distribution
of a duration when new information can arrive within a duration.
2
This paper takes a different approach. Rather than build models based on durations we
propose directly modeling the instantaneous arrival rates, or intensities. We propose a new model
for the instantaneous arrival rate given the multivariate filtration of arrival times and associated
marks. The model takes a particularly simple form so that the dynamics are easily interpreted
from parameter values. Basic properties of the model are established. We propose estimation by
maximum likelihood and model diagnostics are suggested.
The model is applied to financial transactions data. We examine the bivariate system of
transaction arrival times and limit order submission arrival times. Since limit orders dictate the
structure of the limit order book they are a direct determinant of market liquidity. To our
knowledge little work has examined the dynamic structure of limit order submissions. Since
transactions represent a floor trader or market order demand for liquidity the bivariate model
characterizes the dynamic behavior and relationship between the supply and demand of liquidity.
The proposed model also allows for marks, or characteristics associated with the arrival times, to
influence future arrival rates. For the stock market data analyzed we find strong evidence of
codependence in the two processes and both transactions and limit order submission are
influenced by past volume and prevailing spreads.
The paper is organized as follows. The following section introduces some notation and
introduces the conditional intensity function and compensators. Section 3 reviews the existing
statistical models for multivariate point processes. Section 4 introduces the proposed model.
Section 5 considers several extensions. Section 6 discusses the likelihood function and presents
model diagnostics. Section 7 discusses some ideas underlying market liquidity. Section 8
discusses the data and presents model estimates. Finally, section 9 concludes.
Section 2. Multivariate point processes, conditional intensity functions, and compensators.
We consider a K dimensional multivariate point process. Each point process consists of a
strictly increasing, stochastic set of arrival times. It is convenient also to introduce counting
process associated with the k
th
point process ( ) t N
k
. This is simply the number of events that
have occurred on the k
th
process by time t. The counting process is useful for indexing our arrival
times in a multivariate context since for any time t, there will likely be a different number of
events will have occurred for each process. Let ... ...
2 1 0
< < < < <
k
i
k k k
t t t t denote the arrival
times associated with the k
th
(k=1,..,K) point process. At time t the most recent arrival time will
3
be denoted by
k
t N
k
t
) (
. Associated with the ( ) t N
k
arrival time of the k
th
point process is a set of
characteristics denoted by the vector
( )
k
t N
k
Z .
( )
k
t N
k
Z is called a vector of marks with dimension
L
k
. The process ( )
( )
( )
k
t N
k
k
Z t N , is called a marked point process. For the purposes of exposition,
it is convenient to initially consider the system with no marks. Later we will return to the
possibility that some or all of the point processes are marked.
Following Cox and Isham (1980) we call ( ) t N
k
the marginal process of class k events.
We will refer to the process of points and marks taken without regard to class as
N(t). This is simply the pooled or superposed process. Throughout the discussion we assume
that the pooled process is orderly
1
in the sense that { } ) ( 1 ) , ( Pr ? ? o t t N
k
· > + for all k.
Clearly, if the pooled process is orderly then the marginal process must also be orderly.
Let F
t
denote the filtration of the pooled process. ( ) t N
k
is assumed to be adapted to F
t
.
Engle and Russell (1998) (ER) propose a model for dependent point processes based on a
particular filtration. The ACD model for process k is specified conditional on a sub-sigma field
t
k
t
F F ? consisting of the marginal history, or the internal filtration of ( ) t N
k
and the associated
marks. That is, in ER the information set consists of past arrival times and past marks of the
marginal series so that new information was assumed to arrive only at past arrival times of events
of type k.
Since the filtration in ER is unchanged between arrival times, the conditional distribution
of the waiting time until the next event is easily characterized conditional on information
available at the start of the waiting time. It was therefore natural to consider parameterizations
for the waiting time or duration until the next event. For the multivariate filtration F
t
it is very
difficult to parameterize the model in terms of the conditional distribution of waiting times. In
particular, a full characterization would require specification of the joint density of the
distribution of the next arrival time, and the complete path of information set F
t
(or covariate)
over the waiting time. A more natural approach that does not require joint modeling of the
complete path of the covariate is obtained by directly specifying a model for the instantaneous
probability that an event of type k occurs in the next instant given information available at time t.
1
This is not crucial. It is possible to relax this assumption by augmenting the dimension of the multivariate
point process by defining simultaneous occurrences to define a new class of point process. This will not be
considered in this paper.
4
The conditional probability that an event of type k occurs in the next instant is given by
the conditional intensity function. More formally, the conditional intensity function associated
with the k
th
marked point process (MPP k) is defined as
(1) ( )
( ) ( )
t
F t N t t N
F t
t
k k
t
t
k
?
> ? ? +
·
? ?
0 ) ( Pr
lim
0
?
Of course, ( )
t
k
F t ? must be non-negative. Equation (1) is simply the probability per unit time
that an event of type k occurs in the next instant. For small values of t ? ,
(2) ( ) ( ) ( ) t o F t N t t N E t F t
t
k k
t
k
? + ? ? + · ? | ) ( ) ( | ?
so that
(3) ( ) ( ) t F t t N t t N
t
k k k
? ? ? ? + | ) ( ) ( ?
will have expectation of order ( ) t o ? and as t ? goes to zero, (3) will be uncorrelated with the
past F
t
.
Next, consider adding consecutive pieces of (3) from some initial time s
0
through time s
1
(s
1
>s
0
)
as in:
(4)
( ) ( )
( )
( ) ( ) ( ) ( )
( )
? ? ? ? ? ·
?
? ? + ? ? ? + ? ? +
?
?
·
?
?
·
t
s s
j
t
k k k
t
s s
j
t
k k k
t F t j s N s N
t F t j s t j s N t j s N
0 1
0 1
1
0 1
1
0 0 0
|
| ) ) 1 ( ( ) (
?
?
Provided that ( )
t
k
F t ? is integrable taking the limit as t ? goes to zero yields:
(5) ( ) ( ) ( ) ( )
?
? ?
1
|
0 1
s
s
t
k k k
o
dt F t s N s N ?
From (3) it follows that (5) will be uncorrelated with
0
s
F . Or, equivalently,
(6) ( ) ( ) ( ) ( )
0 1
1
| s N s N dt F t E
k k
s
s
t
k
o
? ·
,
_
¸
¸
?
?
The integrated intensity function plays a central role in the theory of point processes. As
such, it has been termed the compensator, or more precisely, the F
t
-compensator
2
. The
compensator will play a central role in this paper and is denoted by:
(7) ( ) ( )
?
· ?
1
| ,
1 0
s
s
t
k k
o
dt F t s s ?
5
Of particular importance in developing a model for multivariate point process is the F
t
-
conditional survivor function. If
k
i
k
i
k
i
t t
1 ?
? · ? denotes the time interval between the i
th
and (i-1)
th
arrival time then the F
t
-conditional survivor function for the k
th
process is given by:
(8) ( ) ( )
?
? ?
+
?
> ? ·
1
|
i
t
k
i
k
i k
k
i k
F P S
Following Yashin and Arjas (1988), provided that the survivor function is absolutely continuous
with respect to Lebesgue measure then
(9) ( ) ( )
,
_
¸
¸
?
? ·
?
i
i
t
t
t
k k
i k
dt F t S
1
| exp ? ?
More important here is the implication that the integrated intensity function over the duration
between two subsequent arrival times yields a random variable with unit exponential distribution.
Hence if the survivor function is absolutely continuous then
(10) ( )
?
·
?
i
i
t
t
t
k k
t N
dt F t
1
|
~
) (
? ?
is an iid unit exponential random variable. Yashin and Arjas do not make any exogeneity
assumptions about F
t
so the result is very general. This is a stochastic version of the well know
result that any inhomogeneous Poisson process can be transformed into a homogeneous Poisson
process with arrival rate of unity by a stochastic transformation of time scale given by
( )
?
·
t
t
t
ds F s
0
? ? . Since the expected value of a unit exponential is 1, the random variable
(11) ( )
k
t N
k
t N ) ( ) (
~
1 ? ? ? ·
has mean zero with unit variance. From (6) it is clear that positive values of
k
t N ) (
? indicate that
the path of the conditional intensity function under predicted the number of events in the time
interval and negative values indicate that the conditional intensity function over predicted the
number of events in the time interval. In this sense, we may view (10) as a generalized residual
in the sense of Cox and Snell (1968).
2
A general discussion of the theory of point processes and compensators can be found in Karr (1986) or
Daley and Vere-Jones (1988).
6
The distribution of
k
t N ) (
? and its interpretation is important to the model developed in the
next section. As such, it is appropriate to ask how restrictive the continuity restriction is. Clearly
the restriction rules out the possibility of including any information in the filtration that perfectly
predicts an event occurrence of type k. While this may be the case in medical studies it is
unlikely to be a problem for financial transactions data. Medical studies have modeled the hazard
rate associated with death from a disease given some sort of treatment applied over a period of
time. One of the variables the medical studies may include is a variable taking the value 0 if the
patient dies, 4 if the patient is cured and values between 0 and 4 indicating intermediate health
quality. This type of covariate would violate the absolute continuity restrictions since a 0 would
perfectly predict the death of the patient. In this case the survivor function would have a
discontinuity at the time that the health variable takes the value 0. This type of situation should
not be a problem for our financial data since it is unlikely that any information in F
t
(the set of
past arrival times in the multivariate filtration) can perfectly predict when a transaction will
occur. We therefore assume the absolute continuity condition for the F
t
-conditional survivor
functions associated each waiting time for the multivariate point process.
3 Some Multivariate Point Process Models
Figure 1 provides a graphical view of a multivariate point process. We are interested in
modeling the conditional probability of an event of, type 1 occurring at time in the instant just
after time t given the multivariate filtration of past arrival times.
In describing the modeling strategy, it is convenient to initially consider a simple
bivariate case. Using the above notation, we consider two point processes denoted by a and b
with counting functions ( ) t N
a
and ( ) t N
b
. Then the filtration at time t is given by the complete
history of N(t), that is, all past arrival times of the pooled process
F
t=
{ }
b
t N
b b a
t N
a a
b a
t t t t t t
) (
1 0
) (
1 0
,..., , , ,..., ,
Our goal is to jointly model the arrival times of process a and process b. Models for
multivariate intensity functions have been used in a variety of disciplines. For example, Daley
(1968), Brown (1970) and others, have used multivariate point processes to study queues. These
processes may be bivariate when a model for inputs and outputs are considered. Neuronal spikes
of different types have been analyzed as a multivariate point process in Neurophysiology by
Perkel, Gerstein, and Moore (1967). Vere-Jones, Turnovsky, Erby (1964), Vere-Jones and
Davies (1966) Vere-Jones (1970) and more recently, Ogata, Akaike, and Katsura (1982) have
7
modeled earth quake occurrences as a multivariate point process. In economics, competing risks
models have been used in the analysis of unemployment spells, strikes, and recessions. These
models can be considered a type of multivariate point process provided multiple events are
observed for a single agent. Lancaster (1990) provides a very good coverage of these models.
Engle and Lunde (1999) propose a bivariate model for transactions data. Their model, however,
is a restrictive model in the sense that one process must be designated as the “driving process” of
the other so that the two processes are not treated symmetrically.
These models tend to be very specialized. Many of the models used are Markovian in
structure and therefore display only restricted dependence on the past that is likely present in
financial transactions data. Others require specification of one process as the "driving process"
not treating the two processes symmetrically. We prefer not to make a choice as to which process
drives the other.
Perhaps the most common models for multivariate point process data are natural
extensions of univariate models. The univariate renewal process can be generalized to the
multivariate setting. A simple bivariate process can be constructed based on the pair of backward
recurrence times. Let
a
i
a
i
a
i
t t
1 ?
? · ? denote the time interval between the i
th
and the (i-1)
th
arrival
times of type a. X
i
denotes the i
th
duration in the pooled process. Let
i
Y be a Bernoulli random
variable denoting the type of event associated with the i
th
event arrival of the pooled process. A
generalization of the univariate renewal is obtained from joint distribution of the i
th
arrival time of
the pooled process and its associated type if, given the filtration of past arrival times and types,
the joint distribution can be expressed as:
(12) ( ) ( ) ( )
1 1
| , | | ,
? ?
·
i i i i i t i i
Y Y h Y Y X g F Y X f
Cinlar (1969) studied the properties of these models and termed it a Semi-Markov process. Since
the marginal process follows a renewal process it is natural to call this a bivariate renewal
process. While the model is tractable from a theoretical perspective the nature of dependence is
quite limited as dictated by the Markovian structure and lack of dependence on past X
i
. Instead,
we consider the general class of self-exciting models which allow for temporal dependence in the
arrival rates and the cross arrival rates.
The conditional intensity function introduced in section 2 provides a convenient base
from which to consider parameterizations that allow for more extensive types of auto and cross
dependence than are easily characterized when focusing on the conditional distribution of the
durations between events as the Semi-Markov processes. For example, let U
a
(t) and U
b
(t) denote
the backward recurrence time to events of type a and b respectively. That is, ( )
( ) t N
a
a
t t t U ? · .
8
The backward recurrence time is grows linearly through time with discrete jumps back to the zero
occurring at each arrival time of type a. An immediate generalization of the Semi-Markov
process is conveniently expressed specifying the conditional intensity functions for a and b as
functions of on the pair of backward recurrence times U
a
(t) and U
b
(t).
(13) ( )
( )
( ) ( ) ( ) ( ) t U t U t t t F t
b a
t N t N
k
t
k
b a , , |
) (
? ? ? · ·
Now, the probability structure depends on the pair of backward recurrence times instead of the
backward recurrence time of the most recent event only. While the model is still Markovian in
nature it’s most general form should not be considered a bivariate renewal process since only in
special cases will the marginal distribution be a renewal. As discussed in Cox and Isham (1980),
this added flexibility comes with a cost, however, as analysis of even simple properties of the
marginal distributions quickly becomes infeasible for all but the simplest parameterizations.
Oakes (1976) establishes the unconditional marginal distribution as the solution to a partial
differential equation.
Hawkes (1971,1972) proposed a class of models that relaxes the Markovian structure of
the model based only on the backward recurrence times. Conditional on a bivariate filtration, the
multivariate Hawkes process for process k is expressed as:
(14) ( ) ( )
( )
( )
( )
? ? + ? ? + ·
· ·
t N
i
b
i
k
t N
i
a
i
k
t
k
b a
t t t t F t
0
2
0
1
? ? ? ?
Now the intensity depends on the backward recurrence time to all previous event arrival times
rather than just the most recent. The applicability of this type of specification for financial data is
questionable since the marginal contribution of a historic event is independent of the number of
intervening events.
Section 4: The Autoregressive Conditional Intensity (ACI) model
In this section we propose a new model for multivariate point process data. It is different
from all the models discussed above and seems particularly useful for economic and financial
transactions data. Initially, we consider a simple bivariate process. The model is most easily
formulated using the conditional intensity function. The conditional intensity function for
process k is given by:
(15) ( ) ( )
k
t N
k
t
k
F t
) (
exp | ? ? ? ·
9
where 0 >
k
? and
k
t N ) (
? is a measurable function of the bivariate filtration of all past arrival
times. Clearly, the conditional intensity function will be non-negative as required.
k
t N ) (
? is time
invariant between event arrivals of the pooled process and therefore is indexed by the counting
process associated with the pooled process. Define the vector
(16)
,
_
¸
¸
·
b
t N
a
t N
t N
) (
) (
) (
?
?
? .
Clearly, in this bivariate setting, each arrival time can be of one of two types. Let
i
y be the
indicator variable taking the value 0 if the i
th
outcome of the pooled process is of type a and 1 if
the i
th
outcome of the pooled process if of type b. We propose the following parameterization for
) (t N
? :
(17)
¹
¹
¹
'
¹
· +
· +
·
? ?
? ?
1 y if
0 y if
1 - N(t) 1 ) ( 1 ) (
1 - N(t) 1 ) ( 1 ) (
) (
t N t N b
t N t N a
t N
B a
B a
? ?
? ?
?
or, equivalently
(18) ( )
1 ) ( 1 ) ( 1 - N(t) ) ( ? ?
+ + ·
t N t N b a t N
B y a a ? ? ?
(
where ? ,
a
a , and
b
a are 2 dimensional parameter vectors B is a 2x2 matrix and
a b b
a a a ? ·
(
.
) (t N
? is an iid unit exponential random variable given by:
(19)
¹
¹
¹
'
¹
·
·
·
0 if
1 if
) (
) (
) (
) (
) (
t N
b
t N
t N
a
t N
t N
y
y
b
a
?
?
?
where
(20) ( )dt F t
a
i
a
i
t
t
t
a a
i ?
? ·
?1
1 ? ? and ( )dt F t
b
j
b
j
t
t
t
b b
j
?
? ·
?1
1 ? ?
Recall that (20) is the generalized residual discussed in section 2. So, if the most recent arrival
was of type a then
) (t N
? represents the innovation associated with the event arrival for process a.
Similarly, if the most recent arrival was of type b then
) (t N
? represents the innovation associated
with the most recent event arrival for process b.
If the event N(t)
th
arrival was of type a then
a
t N
t N
a
) (
) (
? ? · . Positive values of
a
t N ) (
?
indicate that the conditional intensity function under predicted the number of events in the time
10
interval and negative values indicate that the conditional intensity function over predicted the
number of events in the time interval.
) (t N
? is then a weighted average of its most recent value
1 ) ( ? t N
? and the error term
1 ) ( ? t N
? .
Since only one type of information can occur at each event arrival time, only a scalar
innovation appears on the right hand side of (17) at each arrival time of the pooled process. Then
a
? determines the impact of the innovation on
) (t N
? if the N(t)-1 arrival time of the pooled
process is of type a and
b
? determines the impact of the innovation on
) (t N
? if the N(t)-1 arrival
time of the pooled process is of type b. Consequently, the marginal impact of the residual on the
future arrival rate can be different depending on the type of the new information. The persistence
of the impact of
) (t N
? is determined by the eigenvalues of the matrix B. Hence the marginal
impact of the residual depends on the type of new information, but the long run decay is more
parsimoniously specified.
Since
) (t N
? is driven by the difference between the expected number of events (given the
intensity path) and the realized number of events we call the model the Autoregressive
Conditional Intensity (ACI) model. In the model in (17)
) (t N
? depends on one lag of
) (t N
? and
one lag of
) (t N
? it is natural to call this an ACI(1,1) model. Sometimes B need not be fully
saturated so that a diagonal matrix will suffice. If B is restricted to be diagonal then we refer to
the model as a Diagonal Autoregressive Conditional Duration model.
As noted in previous studies establishing even the most basic properties of multivariate point
process models is generally not feasible. The choice of
) (t N
? as the new information observed at
arrival time t
N( t)
is not only intuitively appealing, but will prove to be very useful for establishing
more general properties of the model to be discussed below.
We now present the following lemma that will be useful in establishing the dynamic
properties of the model. Appendix A establishes the following lemma.
Proposition 1: Provided that the F
t
-conditional survivor functions are absolutely continuous with
respect to Lebesgue measure,
a
i
? and
b
j
? are iid unit exponential random variables and
independent of each other for all i,j.
11
The model in (17) provides a flexible representation of the bivariate dynamics. We now take a
closer look at the dynamic properties of the model. Rearranging terms equation (18) can be
rewritten as :
(21) ( ) ( )
1 ) ( 1 ) ( ) ( ? ?
+ · ?
t N t N b a t N
y a a BL I ? ?
(
If the eigenvalues of B lie inside the unit circle equation YY can be represented in an infinite MA
representation
(22) ( ) ? + ·
?
·
? ?
?
1
) ( ) (
* 1
) (
j
j t N j t N b a
j
t N
y a a B ? ?
Appendix B establishes the following result.
Proposition 2: Consider the ACI(1,1) model given by (15) and (17). If the conditional survivor
function is absolutely continuous with respect to Lebesgue measure and the eigenvalues
of B lie inside the unit circle the bivariate process
) (t N
? will be mean reverting with
unconditional mean zero.
We conjecture that the conditions in proposition 2 are also sufficient to establish stationarity.
From (22)
) (t N
? can be viewed as an infinite MA model provided that the variance of y
N( t)
exists.
But y
N(t)
is a binary variable with finite conditional variance determined by
1 ) ( ? t N
? (see the
appendix). The only way the variance cannot exist is if it cycles or drifts off to 0 or 1. This is
unlikely however since
) (t N
? is mean reverting.
Section 5: Extensions of the ACI Model.
The model presented in (17) can be generalized in many ways. In this section we present several
directions.
5.1 Allowing the conditional intensity function to vary between event arrivals.
One important direction would be to relax the restriction that the conditional intensity
function is time invariant between event arrival times of the pooled process. A natural way to do
this is to allow the conditional intensity function to depend on the backward recurrence times
12
( )
( )
( )
( ) t U t U
b a
and where, as before,
( )
( )
) ( N
k
t - t
t
k
t U · . A particular parameterization is given
by:
(23) ( ) ( ) ( ) ( ) ( ) t U t U h F t
b a k k
t N t
k
, exp |
) (
? ? ·
where ( ) ( ) ( ) 0 , > t U t U h
b a k
Now, the function h determines how the intensity changes in the absence of an event of either
type and therefore depends only on the bivariate backward recurrence times. In fact, we can
regard h as a multivariate generalization of the baseline hazard typically used in duration models.
Clearly we have many candidates for the function of the backward recurrence
times ( ) ( ) ( ) t U t U h
b a k
, . Numerous possibilities exist. For example
) (t N
? could still be constant
between event arrivals but may have a different levels depending on which event type occurred
most recent. In this case
) (t N
? is defined as before but the constant term
k
? takes two possible
values given by:
(24) ( ) ( ) ( )
( ) ( )
( ) ( )
if
if
,
¹
¹
¹
'
¹
<
<
·
t U t U
t U t U
t U t U h
a b k
b
b a k
a b a k
?
?
Other examples include the Markov renewal process discussed above. Still further candidates
include any of semi-Markov models that have been discussed in section 3. A particular candidate
for transaction data will be discussed in the application.
5.2 Higher order dependence
Another obvious extension of the model is to allow for
) (t N
? to depend on more lags. In
the model presented in (17) the conditional intensity function depends only on the most recent
values of
) (t N
? and
) (t N
? . In general we could consider a general ACI(p,q) model where the
conditional intensity function depends on p most recent values of
) (t N
? and the q most recent
values of
) (t N
? .
13
5.3 Extending the dynamics of the model.
The model presented in (17) allows the impact of
) (t N
? to be different depending on the type
of event associated with the N(t)
th
arrival of the pooled process. The decay of the shock is
determined by B. While this model is parsimonious and easy to interpret it can also be viewed as
restrictive. Another possibility is to allow B to also depend on the most recent arrival type. Let
a b b
B B B ? ·
*
. Then
(25)
¹
¹
¹
'
¹
· +
· +
·
? ?
? ?
1 if
0 if
1 - N(t) 1 ) ( 1 ) (
1 - N(t) 1 ) ( 1 ) (
) (
y B a
y B a
t N b t N b
t N a t N a
t N
? ?
? ?
?
or equivalently,
(26) ( ) ( )
1 ) ( 1 - N(t) 1 ) ( 1 - N(t) ) ( ? ?
+ + + ·
t N b a t N b a t N
y B B y a a ? ? ?
(
(
This model allows for a richer type of interdependence between the two processes. In particular,
the model can exhibit one way causality, bi-directional causality or no causality at all. To see
this, consider the following restrictions:
(27)
1
]
1
¸
·
1
]
1
¸
·
1
]
1
¸
·
1
]
1
¸
·
b
a
b
b
a
a
a a
22
b
11
a
2
1
0
0 1
B
1 0
0
B
0
0 ?
?
?
?
(28)
( )
( )
( )
( ) ( )
( )
( ) ( )
( )
( )
( )
( ) ( )
( )
( ) ( ) t U h t U t U h
and
t U h t U t U h
b b b a b
a a b a a
,
,
·
·
In this case an occurrence of events of type a will have no impact on
b
t N ) (
? . Similarly, events of
type b will have no impact on
a
t N ) (
? . So, if h
a
is a function of only the backward recurrence time
to events of type a then the model for process a reduces to a univariate proportional hazards
model Cox (1955). That is, the conditional intensity function takes the form:
(29) ( ) ( ) ( ) ( ) t U h F t
a a
t N
t
a
a
) (
exp | ? ? ·
where ( ) ( ) t U h
a
is the baseline hazard function and ( )
) (
exp
t N
a
? simply shifts the baseline hazard
function up or down as a function of the covariates. The same is true for process b if h
b
does not
depend on the backward recurrence time to events of type a. This univariate structure is very
14
similar to the ACD parameterization of Engle and Russell (1998). However, in the ACD model is
a type of accelerated failure time model
3
where the rate at which time passes in the baseline
hazard is also accelerated or decelerated according by a factor of ( )
) (
exp
t N
a
? . For the particular
case of the exponential and Weibull version of the ACD model it can be viewed as either an
accelerated failure time or proportional hazard model. Since we have taken the exponential
function of
) (t N
a
? this is similar the Nelson form ACD model as discussed in Russell and Engle
(1999). This process will be stationary with finite variance provided that and
22 11
b a
B B lie inside
the unit circle.
This formulation should be useful for causality tests. Any deviation from the restrictions
in (26) and (27) will result in causality from one process to the other or bi-directional causality.
Clearly if h
a
depends on ( ) t U
b
then there will be causality from process a to process b and if h
b
depends on ( ) t U
a
then there will be causality from process b to process a. A simple test for
causality from one process to the other can then be based on testing the restricted model in (26)
and (27) against the unrestricted alternative of (24). A rejection of the restricted univariate model
in favor of the unrestricted model provides evidence of causality.
5.4 Deterministic Time of Day Effects
It is well documented that many types of financial data exhibit intraday deterministic
patterns. Harris (1986) was one of the first to document deterministic patterns of the variance
within day. Engle and Russell (1998) documented similar phenomenon for arrival rates of
traders. A simple way of introducing a deterministic pattern into the ACI model is given by:
(30)
( )
( )
( )
( )
( ) ( )
( )
( )
( ) ( ) ( ) t t U t U h F t
k b a k k
t N t
k
? ? ? , exp |
) (
·
?
where ( ) 0 > t
k
? is a continuous deterministic function of time. Now, the intensity can vary
deterministically as a function of time as well as stochastically. If ( ) t
k
? is left completely
unrestricted there is a lack of identification. Without loss of generality we propose normalizing
( ) t
k
? to take the value 1 at the start of the day. In our application we will specify the
deterministic component by a piecewise linear spline.
3
See Lancaster (1990) for a discussion of these models.
15
5.5 Conditioning on marks.
We now return the possibility that one or both of the point processes contains marks. In this
case there may be information associated with event arrival times. For example with a
transaction we may observe a price and volume transacted. If these marks carry information
about future arrival rates we would like to include this information in the model. Let
a
t N
a
Z
) (
and
b
t N
b
Z
) (
denote a vector of marks associated with event arrivals of types a and b respectively.
(31)
¹
¹
¹
'
¹
· + +
· + +
·
?
? ?
?
? ?
1 if
0 if
1 - N(t)
1 ) (
1 ) ( 1 ) (
1 - N(t)
1 ) (
1 ) ( 1 ) (
) (
y Z B a
y Z B a
t N
b t N t N b
t N
a t N t N a
t N
b
a
? ? ?
? ? ?
?
where
a
? and
b
? are conforming parameter vectors. So, if the most recent event arrival was of
type a then the marks from process a are used. If the most recent event was of type b then the
marks from process b are used. This will be particularly useful for transactions data where we
often have additional information such as quantities or prices. In fact, if some if appropriate
elements of a
a
and a
b
are zero it is possible for one intensity to only depend on the marks of the
second process and not on the innovations
) (t N
? .
5.6 Higher dimension systems
The ACI model is easily extended to higher dimensions.
) (t N
? can be expanded to a vectors
of dimension k and
) (t N
? will denote an iid innovation from one of k possible event arrival types
in the pooled process. Unlike many other models this extension is remarkably simple.
Furthermore, we find a bivariate model very easy to estimate so that it seems very practical to
expand the dimension beyond two.
6. The Likelihood and Model Diagnostics for an ACI(p,q)
The conditional intensity function characterizes the instantaneous probability of an event
arrival in the bivariate point process. Given the instantaneous probabilistic structure of the
bivariate point process the likelihood for the ACI model can be obtained. Following Florens and
16
Gougere (1995) it is easily expressed using the integrated intensity function described in the
previous section. For a bivariate model that requires joint estimation of both processes the
likelihood is expressed as:
(32) ( ) ( ) ( ) ( ) ( ) ( ) ? ? ? + ? ? ·
· ·
) (
1
) (
1
| | , 0 , 0 exp
t N
k
t N
j
t
b
j
b
t
a
i
a b a
a b
F t F t T T L ? ?
Simply put, the first term is the product of the F
t
-conditional survivor functions for process a and
process b associated with each duration. This term characterizes the joint probability of no events
occurring in each of those durations. The last two terms of (32) denote the joint probability of the
realized arrival times. For joint estimation of a K dimensional process, the likelihood is expanded
in the obvious way.
Early discussions of the probability structure of bivariate self-exciting processes include
Hawkes (1971) who notes that the bivariate self-exciting process is a particular Doubly Stochastic
Poisson
4
process. Thus, the likelihood can be constructed using the theory of Doubly Stochastic
process
5
. The likelihood may also be justified via the theory of compensators as discussed in
Daley and Vere-Jones (1988) Ch. 13.
Correct specification of the model implies that
k
t N
k
t N ) ( ) (
~
1 ? ? ? · should be an iid unit
exponential random variable that is uncorrelated with F
t
. . This suggests using a Ljung Box test
on the autocorrelations. A rejection of the null that
k
t N ) (
? is uncorrelated provides evidence of
mispecification. Furthermore, it
k
t N ) (
? should be distributed as a unit exponential distribution so
we can also check moment conditions associated with the unit exponential. We propose using the
test suggested in Engle and Russell (1998) for excess dispersion. In particular, a simple test of no
excess dispersion is obtained from the statistic ( ) 8 1 ˆ
2
~ ?
?
?
k
N . Under the null hypothesis of a
unit exponential the test statistic will have limiting Normal distribution. Similarly, a rejection of
the null provides evidence of model inadequacy.
7. Liquidity and Market Some Microstructure
In this section we introduce some basic concepts of liquidity. Some Market Microstructure
theory is also discussed to help motivate the empirical study in the section 6.
4
See Cox and Isham (1980) for a discussion of the Doubly Stochastic Likelihood.
5
See Cox and Isham (1980) for a good discussion of Doubly Stochastic Processes.
17
7.1 Liquidity
Market liquidity is of great interest to financial traders. Generally speaking a liquid market is
characterized by the ability to trade large amounts of volume at a price close to the efficient (or
true) price of the asset. One dimension of liquidity is often quantified by the bid ask spread. The
larger the difference in the bid and ask prices the further the transaction price will likely be from
the efficient price. This measure, however, ignores the size of the trade. Typically, the quotes are
only valid for relatively small numbers of shares. If a trader wishes to transact a large volume the
price may be quite different than prices obtained for small quantities. The price volume
relationship is conveniently expressed in figure 2.
The vertical distance along the transaction price axis denotes the bid ask spread. The
horizontal distance from the transaction price axis to the sloped line is the minimum volume that
can be transacted at the current bid and ask prices. Beyond a small number of shares larger sell
volume demands a lower price and larger buy volume demands a higher price. If an order is
larger than this small number of shares a worse price is obtained. The complete price volume plot
is called the market impact curve.
Typically this market impact curve will look quite different at different times during the day.
The spread may widen or the slope of either side of the market impact curve may change. Limit
orders represent unfilled orders awaiting transaction consisting of a desired volume and price. At
any point in time the exact nature of the market impact curve is determined by the unfilled orders
in limit order book. In this sense, limit orders can be viewed as a major contributor to market
liquidity and the market impact curve.
An agent that desires an immediate transaction will contact a floor trader. This can be done
by submitting a market order or, if the transaction is particularly difficult, by having a floor
broker "work" the transaction over a longer period of time. While this type of trade is guaranteed
to transact there is a cost. The cost will depend on the volume desired to be traded and the
current status of the market response curve.
It is therefore natural to consider a transaction as demand for liquidity and limit orders as the
supply of liquidity. In our application we will model the bivariate process of transaction arrival
times and limit order submission times. The model is used to study the dynamics the joint
dynamics of the demand and supply of liquidity how they are supply and demand are affected by
different characteristics of the market.
18
7.2 Market Microstructure and Liquidity
Market Microstructure theory typically posits two reasons an agent will decide to transact.
The first type of trader is an uninformed agent that is simply trading to adjust a portfolio. Since
the agents are uninformed and have no superior information about future prices their transactions
are often assumed to be random resulting in Poisson type arrival. The second type of trader is an
informed agent that possesses superior information. It is natural to think that the privately
informed agents would like to submit market orders to insure that their transaction occurs quickly
and therefore minimize the risk of the market learning of the private information from other
sources before they can capitalize on it. Alternatively, liquidity traders may submit either limit
orders or market orders depending on the urgency of the portfolio adjustment and other factors.
The classic models of liquidity traders are very simple and assume that they have no
strategic behavior. A slightly more sophisticated model was proposed by Admati and Pfleiderer
(1988) suggests that the uniformed by batch their trades together to minimize the probability of
trading with superiorly informed agents. If uninformed traders are concerned about trading with
better informed agents they may enter the market when the probability of transacting with better
informed agents is small. Typically, however, it is not possible to distinguish informed from
uninformed traders directly. Instead, the existence of private information must be inferred from
transaction and general market characteristics. Easley and O'Hara (1987) and Hasbrouck (1988)
find that larger volume transactions have a greater price impact than small volume trades. Early
models by Copeland and Galai (1983) and numerous papers to follow suggest that spreads should
widen if the presence of informed traders is expected.
If both uninformed and informed traders are strategic then the patterns of transaction and
limit order submission should be impacted by presence (or suspicion) that informed traders are
present. As in Easley and O'Hara (1991) informed traders will transact only when they posses
private information. Uninformed traders will submit orders limit orders when they suspect the
probability of transacting with better informed agents is small. In a rational expectations setting
this suggests that both the demand for liquidity and the supply of liquidity should be affected by
volume, spreads, and transaction rates. This hypothesis will be examined.
19
8. An Application to the Bivariate Point Process of Transactions and Limit Order
Submission
In this section of the paper we apply the ACI model to the bivariate process of transaction
arrival times and limit order submission times. We first introduce the data.
8.1 The Data
The data are abstracted from the TORQ data set. For the period November 1, 1990 to
January 31, 1991 Federal Express contains 5138 unique transaction times and 4058 unique limit
order submission times. The market operates from 9:30 am 4:00 pm local time. For details of
the operation of the market we refer the reader to Hasbrouck, Sofianos and Sosebee (1993).
Following Engle and Russell (1998) we delete arrival times occurring in the first half-
hour of the trading day since many of these trades are opening trades that operate according to a
batch auction, not the continuous market that characterizes the majority of the trades made on the
NYSE. We construct the sequence of interarrival times denoted by
r
i
r
i
r
i
t t
1 ?
? · ? and
m
j
m
j
m
j
t t
1 ?
? · ? for the trade and limit order submission processes respectively. Figure 3 presents a
histogram for the durations
r
i
? and
m
i
? . On average, transactions occur about every 248 seconds
or just over 4 minutes. Limit orders are submitted, on average once every 312 seconds or just
over 5 minutes. The standard deviations are 367 and 420 respectively. The maximum time
interval is 4139 and 4416 or a little over an hour. Naturally, both histograms are skewed to the
right with a lower bound of 1, the finest recorded interval.
It is informative to view the autocorrelation structure of the duration series
r
i
? and
m
j
? .
Table 1 contains the autocorrelations and partial autocorrelations of the two duration series.
Additionally, the series
p
s
p
s
p
s
t t
1 ?
? · ? is constructed from the pooled series containing all arrival
times of both the transaction and quote arrival times. Hence these are durations that can begin
with either a transaction or limit order time and end with either a transaction or quote time. The
duration statistics indicate that both the transaction and limit order durations are highly
autocorrelated with Ljung-box statistics of 480.1 and 668.1 respectively. These statistics are
distributed as a Chi-Squared with a 5% critical value of 24.99. The durations from the pooled
process are also highly autocorrelated.
20
8.2 Model specification and estimates
We estimate a diagonal ACI model with time varying intensity between event arrivals. We
specify the bivariate backward recurrence time function h for the k
th
process by:
(33) ( ) ( ) ( ) ( ) ( ) ( )
k k
t U t U t U t U h
b a k b a k 2 1
exp ,
? ?
? ·
Clearly, h
k
>0 as required. If 0
1
·
k
? then process k does not depend on the backward recurrence
time for process 1. So, process k may depend only on it's own backward recurrence time, only on
the backward recurrence time of the other process, or perhaps on neither. If both
0 and 0
2 1
< <
k k
? ? then the conditional intensity function will always be downward sloping with
jumps at each event arrival of the pooled process. This implies the probability of an event of type
k will decrease in the absence of new event arrivals. Alternatively, if both 0 and 0
2 1
> >
k k
? ? are
positive then the intensity function will be monotonically increasing with jumps down at each
event arrival of the pooled process. In this case the probability of an event will be increasing in
the absence of new event arrivals. If 0 and 0
2 1
> <
k k
? ? the intensity function can be non-
monotonic u-shaped or inverted u-shaped. In the absence of a event arrival in the pooled process
the intensity function will eventually be positively sloped if
k k
1 2
? ? > and eventually
negatively sloped if
k k
1 2
? ? < . Hence (33) allows for flexible variation in the intensity
function in the absence of new events and satisfies the non-negativity property.
We begin by estimating a Diagonal ACI(1,1) model with a deterministic component
given by a piecewise linear spline. Two continuous splines are specified, one for the transaction
process and one for the limit order arrivals. The nodes are at each hour with the first and last at
10:00 and 4:00 for a total of 6 linear pieces. Engle and Russell (1996) found a similar piecewise
linear spline performed well for IBM transactions data. The constant in the spline is normalized
to unity and the slope parameters for process k are denoted by ( )
k k
d d
6 1
,..., .
All the parameters are estimated jointly by conditional maximum likelihood. Initial
values for the vectors
0
? and
0
? are set to their unconditional expectation of 0. Since the scores
are recursive functions of the past similar to Bollerslev's (1986) GARCH model we use numerical
derivatives. The Bernt, Hall, Hall, Hausman (BHHH) (1974) algorithm is used to maximize the
likelihood function. The likelihood function given in (32) requires integration of the conditional
intensity function. If the function h is a constant then closed form solutions for this integral are
21
easily obtained. In the more general case of (33), however, closed form solution does not exist
and we use numerical integration for this more general case.
Estimated parameters are presented in table 1. Standard errors are obtained using the
outer product of the scores. All parameter values are significant except for some of the parameters
associated with the deterministic pattern. From the parameters for the conditional intensity
function for transaction arrivals we see that larger values of either
( )
r
t N
a
1 ?
? or
( )
m
t N
a
1 ?
? increase the
conditional intensity function. Hence if the realized arrival rate was greater (less) than the
expected arrival rate the conditional intensity function is increased (decreased) regardless of event
arrival type. Similarly, larger values of either
( )
r
t N
a
1 ?
? or
( )
m
t N
a
1 ?
? increase the conditional
intensity for limit order arrivals. All else equal, both conditional intensity functions tend to move
in the same direction. Both
a
? and
b
? are less than, but near, one. This is indicative of strong
persistence but mean reverting behavior. Finally,
r
1
? and
r
2
? are both negative and significant
indicating that the probability of a transaction monotonically declines in the absence of new event
arrivals. Similarly,
m
1
? and
m
2
? are also negative and significant indicating that the probability of
a limit order arrivals monotonically decreases in the absence of new event arrivals.
Figure 4 contains a plot the deterministic components for the two processes. They are
both U shaped with larger intensity in the morning and just prior to close. These plots are nearly
the inverted plots of those presented in Engle and Russell (1998). Engle and Russell (1998)
modeled the deterministic pattern of the expected duration which is roughly the inverse of the
intensity.
At the bottom of the table are the diagnostic statistics based on
( )
r
t N
a
1 ?
? or
( )
m
t N
a
1 ?
? .
Under the null they should be iid unit exponential and uncorrelated with F
t
. For each series, the
Ljung-Box statistic for the first 15 autocorrelations are presented at the bottom of the table. The
number of observations is different for the two series since there are different numbers of
transactions and limit order submissions. The Ljung-Box associated with the transaction process
is 16.26. The test statistic has a Chi-Squared distribution with 15 degrees of freedom. The 5%
critical value is 24.99 so we fail to reject the null. For the limit order series, the test statistic is
35.58. This results in a rejection of the null.
Next, we examine the excess dispersion test statistic. Under the null
( )
r
t N
a
1 ?
? and
( )
m
t N
a
1 ?
?
should have unit variance. The test statistics are 1.63 and -3.81 for the transaction and limit order
22
series respectively. These are asymptotically Normal and suggests that
( )
m
t N
a
1 ?
? is under
dispersed.
Since the model for limit order arrivals is rejected we estimate an ACI (2,2) model. The
results are presented in table 2. The significant coefficients on the lagged values of
( )
r
t N
a
1 ?
? and
( )
m
t N
a
1 ?
? are again positive. The sum of ( )
r r
2 1
? ? + and ( )
m m
2 1
? ? + are both equal to 0.97. This
is qualitatively similar to the findings in the ACI(1,1) model. We now return to the model
diagnostics at the bottom of the table. The Ljung-Box for the limit order series is now 26.73
which is marginal at the 5% level. The excess dispersion test statistics are basically unchanged
and we still find mild evidence of under dispersion for the limit order process. Overall, the model
fits well.
Figure 5 presents the QQ plot for
( )
r
t N
a
1 ?
? and
( )
m
t N
a
1 ?
? . Under correct specification the
QQ plots should lie on the upward sloping line. The plots indicate that while the fit is generally
good, there are too many large values of
( )
r
t N
a
1 ?
? and not enough large values of
( )
m
t N
a
1 ?
? . This is
consistent with the excess dispersion tests.
8.3 Estimates for models with marks
We now return to the more general case of a bivariate marked point process. We
estimate models where the conditional intensity functions can depend on not just the historic
bivariate arrival times but also other characteristics associated with the arrival times. By
including marks in the specification of the conditional intensity function we can see how they
impact future arrival rates of the bivariate system.
Along with the transaction arrival times we observe a transaction price, and a volume.
For each limit order the data also contains the price at which the limit order is to be executed, the
volume to be transacted, and a buy/sell indicator. Additionally, from a separate series of quotes
posted by the specialist we observe bid ask prices. These bid and ask prices can be matched up
with transaction times to obtain the prevailing bid and ask quotes at the time of the transaction.
From these marks we create the following transformations to be used in estimation. First,
we construct a series for the log of volume transacted. This series is denoted by
) (t N
r
V . Next, we
construct a measure of the volatility of price changes. ( ) ( )
1 ) ( ) ( ) (
log log
?
? ·
t N t N t N
r r r
p p ? where
i
p is the transaction price associated with the i
th
transaction.
1 ) ( ) ( ) ( ?
? · ?
t N t N t N
r r r
p p p is
23
simply the price change between subsequent transactions. Finally, using the spreads we create
the series
1 ) ( ) ( ) ( ?
? ·
t N t N t N
r r r
spd spd dspd where
i
spd is the spread, or difference between the
bid and ask prices, prevailing at the i
th
transaction time.
From the marks associated with the limit order arrivals we construct the following series.
¹
¹
¹
'
¹
·
otherwise 1
quotes. ask and bid prevailing the of midpoint the from % 1 than less is lp if 0
) (
) (
t N
t N
m
m In
So, this is an indicator variable denoting a "competitive" limit order that is not far away from the
current price.
) (t N
m
Buy is a buy sell indicator taking the value one if the order is to buy.
) (t N
m
S
is the desired number of shares to be transacted.
The marks are included linearly in the specification of
) ( t N
? as in equation 31. Parameter
estimates are given in table
6
4. The results suggest that large volume transacted tend to be
followed by higher intensity of quote submission. Large transaction volume does not appear to
be followed by higher intensity of transactions. The price change between subsequent
transactions has a positive and significant coefficient in the transaction intensity. This suggests
that the transaction intensity tends to be higher in periods of rising prices. Volatility has little
impact on the transaction rates or the limit order submission intensity. The price change between
subsequent transactions does not have a significant impact on the limit submission intensity. The
change in the spread between subsequent transaction time does not have an impact on the
intensity of transactions, however it decreases the intensity of limit order submissions. If wide
spreads are indicative of asymmetric information and price uncertainty then this suggests limit
orders are not submitted are frequently when uncertainty is greater about the value of the asset.
The proximity of the limit price to the prevailing price level is not significant for either
intensity function. The buy indicator does not have a significant for either intensity. The number
of shares associated with a limit order has a positive impact on the intensity of transactions but
has an insignificant impact on the limit order submission intensity.
In summary, we find that transaction rates and limit order submission submissions tend to
move together. Hence when demand for liquidity rises, the supply of liquidity also tends to
increase. Large transaction volume tends to be followed by more intense limit order submission.
Submission of limit orders for large volume tend to be followed by more intense transactions.
Transaction rates tend to be higher when prices are rising, but limit order submission is
6
The deterministic component parameters are not presented. They are essentially unchanged from the
previous model estimates.
24
unaffected by the direction of price movements. Finally, wide spreads tend to be followed by
increased limit order submission but we find no impact on transaction rates.
9. Conclusion:
This paper extends the work of Engle and Russell (1998) to a multivariate setting. In a
multivariate context we propose working with the conditional intensity process rather than with
the conditional distribution of durations. Modeling the instantaneous arrival rate is far easier than
modeling the conditional distribution of durations when information can arrive at any point in
time, not just at the start of a duration as considered in Engle and Russell (1998).
The model is very different from any existing model for multivariate point process and
appears to be particularly useful for analysis of financial transactions data. In particular, each
process is treated symmetrically which relieves the investigator of the task of choosing a process
to be the "driving process" as other studies have proposed. The model allows for long range
dependence typically observed in economic and financial data and is not Markovian. Finally, the
parameter values are easily interpreted in terms of an immediate impact and the persistence of an
innovation.
Extensions of the model allow the intensity function allow for rich dynamic structure.
The intensity may be constant between event arrivals or evolve as a deterministic function of the
backward recurrence times. The model can also allow for deterministic patterns in the intensity
functions. More complex dynamics allow for the possibility of performing causality tests.
Finally, we allow for the possibility that the marks, or characteristics associated with the arrival
times can also influence the intensity functions. This is particularly useful for economic
transaction data where we typically observe prices and volume at transaction times. Finally, the
model is easily expanded to higher dimensions. Our estimation suggests that this is not only
feasible, but also a practical possibility.
The model is applied to the bivariate process of transaction arrivals and limit order
submissions. Limit order submissions add to the limit order book and have a direct impact on
market liquidity and generally on the shape of the market impact curve. We are not aware of any
work that examines the dynamic process of limit order submissions. Since transactions demand
and use up liquidity the bivariate point process of transaction and limit order submission arrival
times sheds light on the dynamic relationship between liquidity supply and demand.
Estimated models suggest that the two processes tend to move together both in a
deterministic manner as well as stochastic. Hence increases in the demand for liquidity tend to be
associated with increased supply of liquidity. We also find that large transaction volume tends to
25
be followed by more intense limit order submission. Submission of limit orders for large volume
tend to be followed by more intense transactions. Transaction rates tend to be higher when prices
are rising, but limit order submission is unaffected by the direction of price movements. Finally,
wide spreads tend to be followed by increased limit order submission but we find no impact on
transaction rates.
A natural use of high frequency data is to learn about the fundamental market
microstructure. Causality testing should play a fundamental role in this learning process. It is
well know that aggregation will distort causality so it is natural to focus on transaction by
transaction data. A generalization of the model discussed in section 5.3 should be useful for
testing along these lines.
26
Appendix A
From the discussion of the martingale property of the compensator in section 2 it is clear that the
compensators for processes a and b evaluated over non-overlapping intervals must be uncorrelated. It
remains to be shown that this same is true for overlapping intervals. From the left hand side of (4) and for
small values of t ? we can express
( ) t N
a
? as:
(9)
( )
( ) ( )
( )
( )
( )
( ) ( )
? ? ? + ? ? ? + ? ? + ·
?
,
_
¸
¸
?
·
? ? ?
?
t
t t
j
t t N
a
t N a t N a
t N
t a N t a N
a a a
a
t F t j t t j t N t j t N
1
1
1 1 1
| ) ) 1 ( ( ) ( ? ?
Similarly,
(10)
( ) ( ) ( )
( )
( )
( )
( ) ( )
( )
? ? ? + ? ? ? + ? ? + ·
?
?
·
? ? ?
?
t
t t
j
t t N
b
t N b t N b t N
t b N t b N
b b b b
t F t j t t j t N t j t N
1
1
1 1 1
| ) ) 1 ( ( ) ( ? ?
Each element on the right hand side of (9) is uncorrelated with each elements on the right had side of (10)
such that t j t t j t
t N t N
a b
? + < ? +
? ? 1 ) ( 1 ) (
. Similarly, each summation element on the right hand side of
(10) is uncorrelated with all elements on the right had side of (9) such that t j t t j t
t N t N
b a
? + < ? +
? ? 1 ) ( 1 ) (
.
Since the summation is simply a linear combination of the elements it follows that
) ( t N
a
? is uncorrelated
with
) (t N
b
? .
Finally, note that the exponential distribution and its moments are determined by a single parameter.
Hence, if the
) ( t N
a
? and
) (t N
b
? are uncorrelated, they must also be independent.
Appendix B
If the eigenvalues of B lie inside the unit circle then
) ( t N
? can be expressed as:
1' ( ) ? + ·
?
·
? ?
?
1
) ( ) (
1
) (
j
j t N j t N b a
j
t N
y a a B ? ?
Under the assumptions,
) (t N
? is an iid mean zero random variable. Furthermore,
) (t N
y is a binary random
variable with conditional distribution given by:
2'
( )
( ) ( )
( )
( ) ( )
¹
¹
¹
¹
¹
'
¹
+
+
·
b
t N
b a
t N
a
b
t N
a
b
t N
b a
t N
a
b
t N
b
t N
) ( ) (
) (
) ( ) (
) (
) (
exp exp
exp
y probabilit with 0
exp exp
exp
y probabilit with 1
y
? ? ? ?
? ?
? ? ? ?
? ?
Since the distribution of
) (t N
y is determined by
1 ) ( ? t N
? ,
2 ) ( ? t N
? ,… it follows that
) (t N
? is independent of
) (t N
y . Taking expectations of both sides of 1' we conclude that the mean of
) ( t N
? exists and is equal to
zero. Since y
N(t)
is binary it's conditional variance given
) ( t N
? is bounded by 0 and .25. If the eigenvalues
of B lie inside the unit circle then
) ( t N
? is mean reverting.
27
Figure 1: Representation of a 3 Dimensional Marked Point Process
( )
1
0
1
0
, Z t
•
( )
1
1
1
1
, Z t
•
( )
1
2
1
2
, Z t
•
( )
2
0
2
0
, Z t
•
( )
2
1
2
1
, Z t
•
( )
3
0
3
0
, Z t
•
( )
3
1
3
1
, Z t
•
( )
3
2
3
2
, Z t
•
( )
3
3
3
3
, Z t
•
( )
3
3
3
3
, Z t
•
MPP 1
MPP 2
MPP 3
1
1
x
1
2
x
}}
( )
? t
F t
1
?
time
28
Figure 2: Market Impact Curve
Volume
Transaction
Price
29
Figure 3: Histogram for Transaction and Limit Order Durations
Seconds
Seconds
0
500
1000
1500
2000
2500
3000
0 500 1000 1500 2000 2500 3000 3500 4000 4500
Series: LIMIT
Sample 1 4057
Observations 4058
Mean 312.4673
Median 160.0000
Maximum 4416.000
Minimum 4.000000
Std. Dev. 420.1237
Skewness 2.957542
Kurtosis 15.15550
Jarque-Bera 30891.43
Probability 0.000000
0
1000
2000
3000
4000
0 500 1000 1500 2000 2500 3000 3500 4000
Series: TRANS
Sample 1 5137
Observations 5138
Mean 248.9017
Median 111.0000
Maximum 4139.000
Minimum 1.000000
Std. Dev. 367.7251
Skewness 3.218003
Kurtosis 18.29552
Jarque-Bera 58941.73
Probability 0.000000
30
Figure 4: Deterministic Component for Transaction and Limit Intensities
0
0.2
0.4
0.6
0.8
1
1.2
9 10 11 12 13 14 15 16 17
Time of Day
D
e
t
e
r
m
i
n
i
s
t
i
c
C
o
m
p
o
n
e
n
t
Transaction Limit Order
31
Figure 5: QQ plot of integrated Intensities
Integrated Intensity QQ Plot for Transactions
0
1
2
3
4
5
6
7
8
9
0 2 4 6 8 10
Empirical CDF
C
D
F
U
n
i
t
E
x
p
o
n
e
n
t
i
a
l
Integrated Intensity QQ Plot for Limit Orders
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6
Empirical CDF
c
d
f
U
n
i
t
E
x
p
o
n
e
n
t
i
a
l
32
Table 1. Summary Statistics for Durations
Trans. Durations Limit Durations Pooled Durations
acf pacf acf pacf acf pacf
lag 1
0.110 0.110 0.188 0.188 0.173 0.173
lag 2
0.110 0.099 0.141 0.109 0.135 0.109
lag 3
0.107 0.087 0.160 0.122 0.107 0.071
lag 4
0.116 0.090 0.131 0.076 0.105 0.066
lag 5
0.076 0.041 0.124 0.066 0.107 0.066
lag 6
0.067 0.031 0.107 0.044 0.109 0.063
lag 7
0.076 0.042 0.068 0.005 0.086 0.036
lag 8
0.057 0.021 0.064 0.011 0.098 0.051
lag 9
0.072 0.040 0.093 0.048 0.095 0.044
lag 10
0.059 0.024 0.097 0.050 0.081 0.027
lag 11
0.060 0.025 0.052 -0.002 0.081 0.029
lag 12
0.076 0.043 0.045 -0.002 0.080 0.029
lag 13
0.053 0.015 0.037 -0.006 0.064 0.011
lag 14
0.042 0.006 0.032 -0.004 0.085 0.037
lag 15
0.047 0.013 0.073 0.045 0.053 0.000
LB(15) = 480.1
Sample Size=5138
LB(15) = 668.1
Sample Size=4058
LB(15) =1413.3
Sample Size=9196
33
Table 2. Parameter Estimates for ACI(1,1)
Trade Intensity Quote Intensity
estimate t-stat estimate t-stat
r
1
?
-.0518 -3.35
m
1
?
-.0887 -3.39
r
r 1 ,
?
.04499 5.30
m
r 1 ,
?
.066 5.91
r
q 1 ,
?
.02291 2.05
m
q 1 ,
?
.1160 7.24
r
r 1 ,
?
.9752 132.12
m
q 1 ,
?
.9680 104.85
r
1
?
-.3878 -45.48
m
1
?
-.2911 -26.45
r
2
?
-.2287 -23.11
m
2
?
-.2395 -21.54
r
d
1
-.4099 -1.51
m
d
1
-.3284 -0.86
r
d
2
-.07588 -0.36
m
d
2
.1513 0.49
r
d
3
-.1360 -0.67
m
d
3
-.4996 -1.69
r
d
4
-.3043 -1.51
m
d
4
.1545 0.54
r
d
5
.5773 2.79
m
d
5
.1822 0.62
r
d
6
.1412 0.57
m
d
6
.6895 1.89
LB(15) = 16.25 LB(15) = 35.58
Mean of
r
i
? =.0117 Mean of
r
i
? =.0071
Test Stat. Ex. Disp. (
r
i
? )=0.5797 Ex Disp. (
r
i
? )=-3.81
# Events=5138 # Events=4058
34
Table 3. Parameter Estimates for ACI(2,2)
Trade Intensity Quote Intensity
estimate t-stat estimate t-stat
r
1
?
-.06321 -2.69
m
1
?
-.07757 -2.68
r
r 1 ,
?
.05460 3.06
m
r 1 ,
?
.07568 3.67
r
r 2 ,
?
.01463 0.71
m
r 2 ,
?
.0032 0.14
r
q 1 ,
?
.01493 0.62
m
q 1 ,
?
.1647 5.91
r
q 1 ,
?
.02553 1.09
m
q 1 ,
?
-.03239 -1.15
r
r 1 ,
?
.4353 1.37
m
q 1 ,
?
.6291 3.27
r
r 1 ,
?
.5345 1.72
m
q 1 ,
?
.3427 1.82
r
1
?
-.3880 -45.46
m
1
?
-.2911 -26.45
r
2
?
-.2275 -22.92
m
2
?
-.2395 -21.54
r
d
1
-.4240 -1.58
m
d
1
652 -1.02
r
d
2
-.06064 -0.296
m
d
2
.1513 0.56
r
d
3
-.1376 -0.68
m
d
3
-.4895 -1.72
r
d
4
-.3054 -1.52
m
d
4
.1778 0.64
r
d
5
.5823 2.81
m
d
5
.1660 0.58
r
d
6
.1440 0.58
m
d
6
.6984 1.96
LB(15) = 17.58 LB(15) = 26.73
Mean of
r
i
? =-.0153 Mean of
r
i
? =.0107
Test Stat. Ex. Disp. (
r
i
? )=1.56 Ex Disp. (
r
i
? )=-3.82
# Events=5138 # Events=4058
35
Table 4. Parameter Estimates for ACI(2,2)
Trade Intensity Quote Intensity
estimate t-stat estimate t-stat
r
1
?
-.06321 -2.69
m
1
?
-.07757 -2.68
r
r 1 ,
?
.05460 3.06
m
r 1 ,
?
.07568 3.67
r
r 2 ,
?
.01463 0.71
m
r 2 ,
?
.0032 0.14
r
q 1 ,
?
.01493 0.62
m
q 1 ,
?
.1647 5.91
r
q 1 ,
?
.02553 1.09
m
q 1 ,
?
-.03239 -1.15
r
q 1 ,
?
.4353 1.37
m
q 1 ,
?
.6291 3.27
r
q 1 ,
?
.5345 1.72
m
q 1 ,
?
.3427 1.82
r
1
?
-.3880 -45.46
m
1
?
-.2911 -26.45
r
2
?
-.2275 -22.92
m
2
?
-.2395 -21.54
) (t N
r
V
r
1
?
-.008947 -1.01
m
1
?
.02573 2.79
2
) (t N
r
?
r
2
?
-.001722 -.44
m
2
?
-.00107 -0.50
) (t N
r
p ?
r
3
?
21.96 4.08
m
3
?
.5297 0.081
) (t N
r
dspd
r
4
?
.00108 0.13
m
4
?
.01454 1.98
) ( t N
m
In
r
5
?
.01249 0.34
m
5
?
.02133 0.64
) (t N
m
Buy
r
6
?
.02714 1.07
m
6
?
.00717 0.33
) (t N
m
S
r
7
?
.03436 2.41
m
7
?
-.01170 -0.98
36
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