Description
From a passing airplane one can see the rush hour traffic snaking home far below. For some, it is enough to know that the residents will all get home at some point. Alternatively, from a tall building in the center of the city one can observe individuals in transit from work to home.
Analysis of High Frequency Financial Data
Robert F. Engle
New York University and University of California, San Diego
Je?rey R. Russell
University of Chicago, Graduate School of Business
December 21, 2004
Contents
1 Introduction 2
1.1 Data Characteristics . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Irregular Temporal Spacing . . . . . . . . . . . . . . . 3
1.1.2 Discreteness . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.3 Diurnal Patterns . . . . . . . . . . . . . . . . . . . . . 5
1.1.4 Temporal Dependence . . . . . . . . . . . . . . . . . . 7
1.2 Types of economic data . . . . . . . . . . . . . . . . . . . . . 10
1.3 Economic Questions . . . . . . . . . . . . . . . . . . . . . . . 11
2 Econometric Framework 13
2.1 Examples of Point Processes . . . . . . . . . . . . . . . . . . . 16
2.1.1 The ACD model . . . . . . . . . . . . . . . . . . . . . 18
2.1.2 Thinning point processes . . . . . . . . . . . . . . . . 26
2.2 Modeling in Tick Time - the Marks . . . . . . . . . . . . . . . 28
2.2.1 VAR Models for Prices and Trades in Tick Time . . . 28
2.2.2 Volatility Models in Tick Time . . . . . . . . . . . . . 32
2.3 Models for discrete prices . . . . . . . . . . . . . . . . . . . . 34
2.4 Calendar Time Conversion . . . . . . . . . . . . . . . . . . . . 40
2.4.1 Bivariate Relationships . . . . . . . . . . . . . . . . . 42
3 Conclusion 45
A EACD(3,3) parameter estimates using EVIEWS GARCH
module. 51
B VAR parameter estimates 51
1
1 Introduction
From a passing airplane one can see the rush hour tra?c snaking home far
below. For some, it is enough to know that the residents will all get home
at some point. Alternatively, from a tall building in the center of the city
one can observe individuals in transit from work to home. Why one road is
moving more quickly than another can be observed. Roads near the coastal
waters might be immersed in a thick blanket of fog forcing the cars to travel
slowly due to poor visibility while roads in the highlands, above the fog,
move quickly. Tra?c slows as it gets funneled through a narrow pass while
other roads with alternate routes make good time. If a critical bridge is
washed out by rain then some travelers may not make it home at all that
night.
Like the view from the airplane above, classic asset pricing research
assumes only that prices eventually reach their equilibrium value, the route
taken and speed of achieving equilibrium is not speci?ed. How does the
price actually adjust from one level to another? How long will it take?
Will the equilibrium be reached at all? How do market characteristics
such as transparency, the ability of traders to view others actions, or the
presence of several markets trading the same asset a?ect the answers to
these questions? Market microstructure studies the mechanism by which
prices adjust to re?ect new information.
Answers to these questions require studying the details of price adjust-
ment. From the passing plane in the sky, the resolution is insu?cient and
the view from the market ?oor, like the view from the street below the build-
ing, provides a very good description of the actions of some individuals, but
lacks perspective. With high frequency ?nancial data we stand atop the
tall building, poised to empirically address such questions.
1.1 Data Characteristics
With these new data sets come new challenges associated with their analysis.
Modern data sets may contain tens of thousands of transactions or posted
quotes in a single day time stamped to the nearest second. The analysis of
these data are complicated by irregular temproal spacing, diurnal patterns,
price discretness, and complex often very long lived dependence.
2
1.1.1 Irregular Temporal Spacing
Perhaps most important is that virtually all transactions data are inherently
irregularly spaced in time. Figure 1 plots one two hours of transaction
prices for an arbitrary day in March 2001. The stock used is the US stock
Airgas which will be the subject of several examples throughout the paper.
The horizontal axis is the time of day and the vertical axis is the price.
Each diamond denotes a transaction. The irregular spacing of the data
is immediately evident as some transactions appear to occur only seconds
apart while others, for example between 10:30 and 11:00 may be ?ve or ten
minutes apart.
Since most econometric models are speci?ed for ?xed intervals this poses
an immediate complication. A choice must be made regarding the time
intervals over which to analyze the data. If ?xed intervals are chosen then
some sort of interpolation rule must be used when no transaction occurs
exactly at the end of the interval. Alternatively if stochastic intervals are
used then the spacing of the data will likely need to be taken into account.
The irregular spacing of the data becomes even more complex when dealing
with multiple series each with its own transaction rate. Here, interpolation
can introduce spurious correlations due to non-syncronous trading.
1.1.2 Discreteness
All economic data is discrete. When viewed over long time horizons
the variance of the process is usually quite large relative to the magnitude
of the minimum movement. For transaction by transaction data, however,
this is not the case and for many data sets the transaction price changes
take only a handful of values called ticks. Institutional rules restrict prices
to fall on a pre-speci?ed set of values. Price changes must fall on multiples
of the smalles allowable price change called a tick. In a market for an
actively traded stock it is generally not common for the price to move a
large number of ticks from one transaction to another. In open outcry
markets the small price changes are indirectly imposed by discouraging the
specialist from making radical price changes from one transaction to the
next and for other markets, such as the Taiwan stock exchange these price
restrictions are directly imposed in the form of price change limits from one
3
8.6
8.61
8.62
8.63
8.64
8.65
8.66
8.67
8.68
8.69
8.7
10:00 10:30 11:00 11:30 12:00
Time
P
r
i
c
e
Figure 1: Plot of a small sample of transaction prices for the Airgas stock
transaction to the next (say 2 ticks). The result is that price chages often
fall on a very small number of possible outcomes.
US stocks have recently undergone a transition from trading in 1/8ths of
a dollar to decimalization. This transition was initially tested for 7 NYSE
stocks in August of 2000 and was completed for the NYSE listed stocks
on January 29th, 2001. NASDAQ began testing with 14 stocks on March
12, 2001 and completed the transition April 9, 2001. In June of 97 NYSE
permitted 1/16th prices.
As an example, Figure 2 presents a histogram of the Airgas data trans-
action price changes after deleting the overnight and opening transactions.
The sample used here contains 10 months of data spanning from March 1,
2001 through December 31, 2001. The horizontal axis is measured in cents.
52% of the transaction prices are unchanged from the previous price. Over
70% of the transaction prices fall on one of three values; no change, up one
cent or down one cent. Over 90% of the values lie between -5 and +5 cents.
Since the bid and ask prices are also restricted to the same minimum adjust-
ment the bid, ask, and the midpoint of the bid ask prices will exhibit similar
4
0
10
20
30
40
50
60
Cents
P
e
r
c
e
n
t
5 10 0 -5 -10 15 -15
Figure 2: Histogram of price transaction changes for Airgas stock
discreteness. Of course data prior to decimalization is even more extreme.
For these data sets it is not uncommon to ?nd over 98% of the data taking
just one of 5 values. This discreteness will have an impact on measuring
volatility, dependence, or any characteristic of prices that is small relative
to the tick size.
This discreteness also induces a high degree of kurtosis in the data. For
example, for the Airgas data the sample kurtosis is 66. Such large kurtosis
is typical of high frequency data.
1.1.3 Diurnal Patterns
Intraday ?nancial data typically contain very strong diurnal or periodic
patterns. For most stock markets volatility, the frequency of trades, volume,
and spreads all typically exhibit a U-shaped pattern over the course of the
day. For an ealry reference see McInish and Wood (1992). Volatility is
systematically higher near the open and generally just prior to the close.
Volume and spreads have a similar pattern. The time between trades, or
durations, tend to be shortest near the open and just prior to the close.
This was ?rst documented in Engle and Russell (1998).
5
0
20
40
60
80
100
120
140
160
180
10:00 11:00 12:00 13:00 14:00 15:00 16:00
Time of Day
C
e
n
t
s
/
S
e
c
o
n
d
s
Duration
Std Dev.
Figure 3: Diurnal pattern for durations and standard deviation of mid-quote
price changes.
Figure 3 presents the diurnal patterns estimated for the ARG data. The
diurnal patterns were estimated by ?tting a piecewise linear spline to the
duration between trades and the squared midquote price changes. The ver-
tical axis is measured in seconds for the duration and cents for the standard
deviation of price changes.
Diurnal patterns are also typically present in the foreign exchange mar-
ket although here there is no opening and closing of the market. These
markets operate 24 hours a day seven days a week. Here the pattern is typ-
ically driven by ”active” periods of the day. See Andersen and Bollerslev
(1997) for patterns in foreign exchange volaitlity. For example, prior to the
induction of the Euro, US dollar exchange rates with European countries
typically exhibit the highest volatility during the overlap of time that both
the US markets and the European markets were active. This occurred in
the late afternoon GMT when it is morning in the US and late afternoon in
Europe.
6
-0.45
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Lag in transaction time
A
C
FMid quote
Trans. Price
Figure 4: Autocorrelation function for the mid quote and transaction price
changes
1.1.4 Temporal Dependence
Unlike their lower frequency counterparts, high frequency ?nancial returns
data typically display strong dependence. The dependence is largely the
result of price discreteness and the fact that there is often a spread between
the price paid by buyer and seller initiated trades. This is typically referred
to as bid-ask bounce and is responsible for the large ?rst order negative
autocorrelation. Bid ask bounce will be discussed in more detail in section
2.3. Other factors leading to dependence in price changes include traders
breaking large orders up into a sequence of smaller orders in hopes of trans-
acting at a better price overall. These sequences of buys or sells can lead to
a sequence of transactions that move the price in the same direction. Hence
at longer horizons we sometimes ?nd positive autocorrelations. Figure 4
contains a plot of the autocorrelation function for changes in the transaction
and midpoint prices from one transaction to the next for the Airgass stock
using the 10 months of data. Again, overnight price changes have been
deleted.
7
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
1 12 23 34 45 56 67 78 89 100 111 122 133 144 155 166 177 188 199
Figure 5: Autocorrelations for the squared midquote price changes
Similar to lower frequency returns, high frequency data tends to exhibit
volatility clustering. Large price changes tend to follow large price changes
and vice-versa. The ACF for the absolute value of the transaction price
change for ARG is shown in ?gure 5. Since the diurnal pattern will likely
in?uence the autocorrelation function it is ?rst removed by dividing the
price change by the square root of its variance by time of day. The variance
by time of day was estimated with linear splines. The usual long set of
positive autocorrelations is present.
The transaction rates also exhibit strong temporal dependence. Figure
6 presents a plot of the autocorrelations for the durations between trades
after removing the deterministic component discussed above. Figure 7
presents the autocorrelations for the log of volume. Both series exhibit
long sets of positive autocorrelation spanning many transactions. These
autocorrelations indicate clustering of durations and volume respectively.
Under temporal aggregation the dependence in the price changes tends
to decrease. However, even at intervals of a half hour or longer negative
?rst order autocorrelation often remains.
1.2 Types of economic data
8
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
1 12 23 34 45 56 67 78 89 100 111 122 133 144 155 166 177 188 199
Figure 6: Autocorrelations for durations
-0.05
0
0.05
0.1
0.15
0.2
1 12 23 34 45 56 67 78 89 100 111 122 133 144 155 166 177 188 199
Figure 7: Autocorrelations for the log of volume
9
We begin this section with a general discussion of the types of high fre-
quency data currently available. With the advancement and integration
of computers in ?nancial markets data sets containing detailed information
about market transactions are now commonplace. Many ?nancial assets
are traded in centralized markets which potentially contain the most de-
tailed information. The NYSE and the Paris Bourse are two such markets
commonly analyzed. These centralized markets might be order driven (like
the Paris Bourse) where a computer uses algorithms to match market par-
ticipants or they might be open outcry (like the NYSE) where there is a
centralized trading ?oor that market participants must trade through. In
either case, the data sets constructed from these markets typically contain
detailed information about the transactions and quotes for each asset traded
on the market. The exact time of a transaction usually down to the second,
and the price and quantity transacted are common data. Similarly, the
bid and ask quotes are also generally available along with a time stamp for
when the quotes became active. The trades and quotes (TAQ) data set
distributed by the NYSE is an example of such a data set.
The currency exchange market is a commonly analyzed decentralized
market. Here the market participants are banks that communicate and
arrange transactions on a one on one basis with no central recording institu-
tion. Quotes are typically fed through reuters for customers with a reuters
account to view continuous updates of quotes. Olsen and Associates has
been downloading and storing this quote data and made it available for aca-
demic research. The most comprehensive data sets contain all quotes that
pass through the Reuters screens and an associated time stamp. As the
transactions do not pass through a centralized system there is no compre-
hensive source for transaction data.
The de?nition of the quote can vary across markets with important im-
plications. Foreign exchange quotes are not binding. Quotes from the
NYSE are valid for a ?xed (typically small) quantity or depth. Quotes in
electronic markets can come in various forms. For example, quotes for the
Paris bourse are derived from the limit order book and represent the best ask
and bid prices in the book. The depth here is determined by the quantity of
volume at the best prices. Alternatively, the Taiwan stock exchange, which
is an electronic batch auction market, posts a ”reference price” derived from
the past transaction and is only a benchmark from which to gauge where
the next transaction price may fall. These di?erence are important not
only from an economic perspective but also determine the reliability of the
data. Non-binding quotes are much more likely to contain large errors than
binding ones.
10
Many empirical studies have focused on the stock and currency exchange
high frequency data. However, data also exists for other markets, most
notably the options and futures markets. These data sets treat each contract
as a separate asset reporting time quotes and transactions just as for the
stocks. The Berkeley Options data base is a common source for options
data.
Other specialized data sets are available that contain much more detailed
information. Perhaps the most well known data set is the TORQ data set
put together by Joel Hasbrouck and the NYSE. This data set is not very
comprehensive in that it contains only 144 stocks traded on US markets
covering three months in the early 1990’s. However, it contains detailed
information regarding the nature of the transactions including the order type
(limit order, market order etc.) as well as detailed information about the
submission of orders. The limit order information provides a widow into the
limit order book of the specialist although it cannot be exactly replicated.
The Paris Bourse data typically contain detailed information about the limit
order book near the current price.
1.3 Economic Questions
Market microstructure economics focuses on how prices adjust to new in-
formation and how the trading mechanism a?ects asset prices. In a perfect
world, new information would be immediately disseminated and interpreted
by all market participants. In this full information setting prices would
immediately adjust to a new equilibrium value determined by the agents
preferences and the content of the information. This setting, however, is
not likely to hold in practice. Not all relevant information is known by
all market participants at the same time. Furthermore, information that
becomes available is not processed at the same speed by all market partic-
ipants implying variable lag time between a news announcement and the
agents realization of price implications. Much of modern microstructure
theory is therefore driven by models of asymmetric information.
In the simplest form, there is a subset of the agents that are endowed
with superior knowledge regarding the value of an asset. These agents
are referred to as privately informed or simply informed agents. Agents
without superior information are referred to as noise or liquidity traders
11
and are assumed to be indistinguishable from the informed agents. Ques-
tions regarding the means by which the asset price transitions to re?ect the
information of the privately informed agents can be couched in this con-
text. Early theoretical papers utilizing this framework include Glosten and
Milgrom (1985), Easley and O’Hara (1992), Copland and Galai (1983) and
Kyle (1985). A very comprehensive review of this literature can be found
in O’Hara (1995).
The premise of these models is that market makers optimally update bid
and ask prices to re?ect all public information and remaining uncertainty.
For the NYSE it is the specialist that plays the role of market maker. Even
in markets without a designated specialist bid and ask quotes are generally
inferred either explicitly or implicitly from the buy and sell limit orders
closest to the current price. Informed and uniformed traders are assumed
to be indistinguishable when arriving to trade so the di?erence between the
bid and the ask prices can be viewed as compensation for the risk associated
with trading against potentially better informed agents. Informed traders
will make pro?table transactions at the expense of the uninformed.
In a rational expectations setting market makers learn about private
information by observing the actions of traders. Informed traders only
transact when they have private information and would like to trade larger
quantities to capitalize on their information before it becomes public. The
practical implications is that characteristics of transactions carry informa-
tion. An overview of the predictions of these models is that prices adjust
more quickly to re?ect private information when the proportion of unin-
formed traders is higher. Volume is higher, transaction rates are higher
when the proportion of uninformed traders is higher. The bid ask spread
is therefore predicted to be increasing in volume and transaction rates.
It is unlikely that aggregated daily data will be very useful in empiri-
cally evaluating market microstructure e?ects on asset prices. Using high
frequency data we seek to empirically asses the e?ects of market microstruc-
ture on asset price dynamics. Although private information is inherently
unobservable the impact of transaction characteristics on subsequent price
revisions can be measured and quanti?ed. We can asses which trades are
likely to move the price or what characteristics of the market should we ex-
pect to see large rapid price changes. A related and very practical issue is
liquidity loosely de?ned as the ability to trade large volume with little or no
price impact. How much will it cost to transact and how is this cost related
to transaction characteristics? An empirical understanding of the answers
to these questions might lead to optimal order submission strategies.
12
Answers to these questions necessarily involve studying the relationship
between transaction prices and market characteristics. The bid ask spread
is often considered a measure of liquidity since it measures the cost of pur-
chasing and then immediately reselling. The wider the spread the higher the
cost of transacting. Of course spreads vary both cross sectionally across dif-
ferent stocks as well as temporally. Using high frequency data the variation
in the spread can be linked to the variation in market characteristics.
The price impact of a trade can be measured by relating quote revisions
to characteristics of a trade. Again with high frequency transactions data
the magnitude and possibly the direction of price adjustments can be linked
to characteristics of the market. Volatility models for high frequency data
are clearly relevant here.
A privately informed agent may have the ability to exploit that informa-
tion in multiple markets. In some cases a single asset is traded in multiple
markets. A natural question is in which market does price discovery take
place? Do price movements in one market precede price movements in
the other market. If the two series are cointegrated these questions can
be addressed in the context of error correction models. Another possibly
for informed agents to have multiple outlets to exploit their information is
the derivative market. From an empirical perspective this can be reduced
to causality questions. Do price movements in one market precede price
movements in the other? Do trades in one market tend to have a greater
price impact across both markets?
Implementing the econometric analysis, however, is complicated by the
data features discussed in the previous section. This chapter provides a
review of the techniques and issues encountered in the analysis of high fre-
quency data. The chapter is organized as follows.
2 Econometric Framework
One of the most salient features of high frequency transactions data is that
transactions do not occur at regularly spaced time intervals. In the statis-
tics literature this type of process has been referred to as a point process
and a large body of literature has been produced studying and applying
models of point processes. Examples of applications include the study of
?ring of neurons or the study of earthquake occurrences. More formally, let
t
1
, t
2
, ..., t
i
, ... denote a sequence of strictly increasing random variables cor-
responding to event arrival times such as transactions. Jointly, these arrival
times are referred to as a point process. It is convenient to introduce the
13
counting function N(t) which is simply the number of event arrivals that
have occurred at or prior to time t. This will be a step function with unit
increments at each arrival time.
Often, there will be additional information associated with the arrival
times. In the study of earthquake occurrences there might be additional
information about the magnitude of the earthquake associated with each
arrival time. Similarly, for the ?nancial transactions data there is often a
plethora of information associated with the transaction arrival times includ-
ing price, volume, bid and ask quotes, depth, and more. If there is additional
information associated with the arrival times then the process is referred to
as a marked point process. Hence, if the marks associated with the i
th
ar-
rival time are denoted by an M-dimensional vector y
i
then the information
associated with the i
th
event is summarized by its arrival time and the value
of the marks [t
i
, y
i
].
Depending on the economic question at hand, either the arrival time,
the marks, or both may be of interest. Often economic hypothesis can be
couched in the framework of conditional expectations of future values. We
denote the ?ltration of arrival times and marks at the time of the i
th
event
arrival by
b
t
i
= {t
i
, t
i?1
, ..., t
0
} and b y
i
= {y
i
, y
i?1
, ..., y
0
} respectively. The
probability structure for the dynamics associated with a stationary, marked
point process is can be completely characterized and conveniently expressed
as the joint distribution of marks and arrival times given the ?ltration of
past arrival times and marks:
f
¡
t
N(t)+1
, y
N(t)+1
|
b
t
N(t)
, b y
N(t)
¢
(1)
While this distribution provides a complete description of the dynamics of
a marked point process it is rarely speci?ed in practice. Often the question
of economic interest can be expressed in one of four ways. When will the
next event happen? What value should we expect for the mark at the next
arrival time? What value should we expect for the mark after a ?xed time
interval? Or, how long should we expect to wait for a particular type of
event to occur?
The answers to the ?rst two questions are immediately obtained from (1).
Alternatively, if the contemporaneous relationship between y
i
and t
i
is not
of interest, then the analysis may be greatly simpli?ed by restricting focus
to the marginalized distributions provided that the marks and arrival times
are weakly exogneous. If the waiting time until the next event regardless of
the value of the marks at termination is of interest, then the marginalized
distribution given by
14
f
t
¡
t
N(t)+1
|
b
t
N(t)
, b y
N(t)
¢
=
Z
f
¡
t
N(t)+1
, y|
b
t
N(t)
, b y
N(t)
¢
dy (2)
may be analyze. This is simply a point process where the arrival times may
depend on the past arrival times and the past marks. We will refer to this
as a model for the event arrival times, or simply a point process. Examples
here include models for the arrival of traders.
Alternatively, many economic hypotheses involve the dynamics of the
marks such as models for the spread, or prices. In this case, one may be
interested in modeling or forecasting the value for the next mark, regardless
of when it occurs, given the ?ltration of the joint process. This is given by
f
y
¡
y
N(t)+1
|
b
t
N(t)
, b y
N(t)
¢
=
Z
f
¡
t, y
N(t)+1
|
b
t
N(t)
, b y
N(t)
¢
dt (3)
Here, the information set is updated at each event arrival time and we refer
to such models as event time or tick time models of the marks. Of course,
multiple step forecasts from 2 would require, in general, a model for the
marks and multiple step forecasts for the mark in 3 would generally require
a model for the durations.
Yet another alternative approach is to model the value of the mark to
be at some future time t + ? (? > 0) given the ?ltration at time t. That is
g
¡
y
N(t+?)
|
b
t
N(t)
, b y
N(t)
¢
(4)
Here the conditional distribution associated with the mark over a ?xed time
interval is the object of interest. Theoretically, speci?cation of (1) implies
a distribution for (4), only in very special cases,however, will this exist in
closed form. Since the distribution of the mark is speci?ed over discrete
?xed calendar time intervals we refer to this type of analysis as ?xed interval
analysis. A ?nal approach taken in the literature is to study the distribution
of the length of time it will take for a particular type of event, de?ned by
the mark, to occur. For example, one might want to know how long will it
take for the price to move by more than a speci?ed amount, or how long will
it take for a set amount of volume to be transacted. This can be expressed
as
g
¡
t + ?
min
|
b
t
N(t)
, b y
N(t)
¢
(5)
where if E
t
de?nes some event associated with the marks, then ?
min
=
min
?>0
y
N(t+?)
? E
t
. Again, only in special cases can this distribution be de-
rived analytically from (1). t
min
is called the hitting time in the stochastic
15
process literature. Since the marks are associated with arrival times, the
?rst crossing times will simply be a subset of the original set of arrival times.
In the point process literature the subset of points is called a thinned point
process.
This section proceeds to discuss each of the above approaches. We begin
with a discussion and examples of point processes. Next, we consider tick
time models. We then consider ?xed interval analysis by ?rst discussing
methods of converting to ?xed time intervals and then give examples of
various approaches used in the literature.
2.1 Examples of Point Processes
It is convenient to begin this section with a discussion of point processes
with no marks. A point process is referred to as a simple point process if,
as a time interval goes to zero, the probability of multiple events occurring
over that time interval can be made an arbitrarily small fraction of the
probability of a single event occurring. In this case, characterization of the
instantaneous probability of a single event dictates the global behavior of
the process. A convenient way of characterizing a simple point process,
therefore is by the instantaneous arrival rate of the intensity function given
by:
?(t) = lim
?t?0
Pr (N(t +?t > N (t))
?t
(6)
Perhaps the most well known simple point process is the homogenous Pois-
son process. For a homogeneous Poisson process the probability of an
event arrival is constant. A homogenous Poisson process can therefore be
described by a single parameter where ?(t) =?. For many types of point
process the assumption of a constant arrival rate is not likely realistic. In-
deed, for ?nancial data we tend to observe bursts of trading activity followed
by lulls. This feature becomes apparent when looking at the series of the
time between transactions, or durations. Figure 6 presents the autocorrela-
tions associated with the intertrade durations of Airgas. The plot indicates
strong temporal dependence in the durations between transaction events.
Clearly the homogenous Poisson model is not suitable for such data.
For a point process with no marks, Snyder and Miller (1991) conve-
niently classify point processes into two categories, those that evolve with
after-e?ects and those that do not. A point process on [t
0
, ?] is said to
16
evolve without after e?ects if for any t > t
0
the realization of events on
[t, ?) does not depend on the sequence of events in the interval [t
0
, t). A
point process is said to be conditionally orderly at time t ? t
0
if for a suf-
?ciently short interval of time and conditional on any event P de?ned by
the realization of the process on [t
0
, t) the probability of two or more events
occurring is in?nitessimal relative to the probability of one event occurring.
Our discussion here focuses on point processes that evolve with after-e?ects
and are conditionally orderly A point process that evolves with after-e?ects
can be conveniently described using the conditional intensity function which
speci?es the instantaneous probability of an event arrival conditional upon
?ltration of event arrival times. That is, the conditional intensity is given
by
?(t|N(t), t
i?1
, t
i?2
, ..., t
0
) = lim
?t?0
P
¡
N(t +?t) > N(t)|N(t), t
N(t)
, t
N(t)?1
, ..., t
0
¢
?t
(7)
The conditional intensity function associated with any single waiting time
has traditionally been called a hazard function in the econometrics literature.
Here, however, the intensity function is de?ned as a function of t across
multiple events, unlike much of the literature in macroeconomics that tends
to focus on large cross sections with a single spell.
Perhaps the simplest example of a point process that evolves with afteref-
fects is a ?rst order homogeneous point process where ?
¡
t|N(t), t
N(t)?1
, t
N(t)?2
, ..., t
0
¢
=
?
¡
t|N(t), t
N(t)
¢
and the durations between events x
i
= t
i
? t
i?1
form a se-
quence of independent random variables. If in addition, the durations are
identically distributed then the process is referred to as a renewal process.
More generally, for an m
th
order self exciting point process the conditional
intensity depends on N(t) and the m most recent event arrivals.
As discussed in Snyder and Miller (1975), for example, the conditional
intensity function, the conditional survivor function, and the durations or
”waiting times” between events each completely describe a conditionally
orderly point process. Letting p
i
be a family of conditional probability
density functions for arrival time t
i
, the log likelihood can be expressed in
17
terms of the conditional density or intensity as
L =
N(T)
X
i=1
log p
i
(t
i
|t
0
, ...., t
i?1
) (8)
L =
N(T)
X
i=1
log ?(t
i
|t
0
, ...., t
i?1
) ?
T
Z
t
0
?(u|N(t), t
0
, ...., t
i?1
) (9)
Equation (??) is referred to as a self exciting point process. It was orig-
inally proposed by Hawkes (1971) and by Rubin (1972) and are sometimes
called Hawkes self exciting processes. Numerous parameterizations have
been proposed in the statistics literature.
2.1.1 The ACD model
Engle and Russell [?] propose the Autoregressive Conditional Duration (ACD)
which is particularly well suited for high frequency ?nancial data. This pa-
rameterization is most easily expressed in terms of the waiting times between
events. Let x
i
= t
i
?t
i?1
be the interval of time between event arrivals which
will be called the duration. The distribution of the duration is speci?ed di-
rectly conditional on the past durations. The ACD model is then speci?ed
by two conditions. Let ?
i
be the expectation of the duration given the past
arrival times which is given by
E (x
i
|x
i?1
, x
i?2
, ..., x
1
) = ?
i
(x
i?1
, x
i?2
, ..., x
1
; ) = ?
i
(10)
Furthermore, let
x
i
= ?
i
?
i
(11)
where ?
i
˜i.i.d. with density p(²; ?) with non-negative support, and ? and ?
are variation free. The baseline intensity, or baseline hazard, is given by
?
0
=
p(²; ?)
S(²; ?)
(12)
where S
0
(²; ?) =
?
R
²
p(u; ?)du is the survivor function. The intensity function
for an ACD model is then given by
?(t|N(t), t
i?1
, t
i?2
, ..., t
0
) = ?
0
Ã
t ?t
N(t)?1
?
N(t)
!
1
?
N(t)
(13)
18
Since ?
i
enters the baseline hazard this type of model is referred to as
an accelerated failure time model in the duration literature. The rate at
which time progresses through the hazard function is dependent upon ?
i
and
therefore can be viewed in the context of time deformation models. During
some periods the pace of the market is more rapid than other periods.
The ?exibility of the ACD model stems from the variety of choices for
parameterizations of the conditional mean in (10) and the i.i.d. density
p(²; ?). Engle and Russell (1998) suggest and apply linear parameterizations
for the expectation given by
?
i
= ? +
p
X
j=1
?
j
x
i?j
+
q
X
j=1
?
j
?
i?j
(14)
Since the conditional expectation of the duration depends on p lags of the
duration and q lags of the expected duration this is termed an ACD(p, q)
model. Popular choices for the density p(²; ?) include the exponential and
the Weibull distributions suggested in Engle and Russell (1998). These
models are termed the Exponential ACD (EACD) and Weibull ACD (WACD)
models respectively. The exponential distribution has the property that
the baseline hazard is monotonic. The Weibull distribution relaxes this as-
sumption and allows for a hump-shaped baseline intensity. An appropriate
choice of the distribution, and hence the baseline intensity will depend on
the characteristics of the data at hand. Other choices include the Gamma
distribution Lunde (1998) and Zhang Russell and Tsay (2001) or the Burr
distribution suggested in Grammig and Maurer (2000). These distributions
allow for even greater ?exibility in the baseline hazard. Given a choice for
(10) and p(²; ?) the likelihood function is constructed from (8).
For each choice of p(²; ?) from (12) and (13) there is an implied intensity
function. Since the exponential distribution implies a constant hazard, the
intensity function takes a particularly simple form given by
?(t|N(t), t
i?1
, t
i?2
, ..., t
0
) =
1
?
N(t)
(15)
and for the Weibull distribution the intensity is slightly more complicated
?(t|N(t), t
i?1
, t
i?2
, ..., t
0
) = ?
_
_
?
³
1 +
1
?
´
?
N(t)
_
_
?
¡
t ?t
N(t)
¢
??1
(16)
which reduces to (15) when ? = 1.
19
The ACD(p, q) speci?cation in (14) appears very similar to a ARCH(p, q)
models of Engle (1982) and Bollerslev (1986) and indeed the two models
share many of the same properties. From (11) and (14) it follows that the
durations x
i
follow an ARMA(max(p, q), q). Let ?
i
? x
i
? ?
i
which is a
martingale di?erence by construction then
x
i
= ? +
max(p,q)
X
j=1
¡
?
j
+ ?
j
¢
x
i?j
?
q
X
j=1
?
j
?
i?j
+ ?
i
If ?(L) and ?(L) denote polynomials in the lag operator of orders p and q
respectively then the persistence of the model can be measured by ?(1) +
? (1) . For most duration data this sum is very close to (but less than) one
indicating strong persistence but stationarity. It also becomes clear from
this representation that restrictions must be placed on parameter values to
ensure non-negative durations. These restrictions impose that the in?nite
AR representation implied by inverting the MA component must contain
non-negative coe?cients for all lags. These conditions are identical to the
conditions derived in Nelson and Cao (1992) to ensure non-negativity of
GARCH models. For example, for the ACD(1,1) model this reduces to
? ? 0, ? ? 0, ? ? 0. Similarly,
The most basic application of the ACD model to ?nancial transactions
data is to model the arrival times of trades. In this case it denotes the
arrival of the i
th
transaction and x
i
denotes the time between the i
th
and
(i-1)
th
transactions. Engle and Russell (1998) propose using an ACD(2,2)
model with Weibull errors to model the arrival times of IBM transactions.
Like volatility, the arrival rate of transactions on the NYSE can have a
strong diurnal (intraday) pattern. Volatility tends to be relatively high just
after the open and just prior to the close; that is, they have volatility for
stocks tends to exhibit a U shaped diurnal pattern. Similarly, Engle and
Russell (1998) document that the durations between trades have a diurnal
pattern with high activity just after the open and just prior to the close;
that is, the durations exhibit an inverse U shaped diurnal pattern. Let
?
N(t)+1
= E
¡
x
N(t)+1
|t
N(t)
¢
denote the expectation of the duration given
time of day alone. Engle and Russell (1998) suggest including an additional
term on the right hand side of (11) to account for a diurnal pattern so that
the i
th
duration is given by:
x
i
= ?
i
?
i
?
i
(17)
Now, ?
i
is the expectation of the duration after partialing out the determin-
istic pattern and is interpreted as the fraction above or below the average
20
value for that time of day. The expected (non-standardized) duration is
now given by ?
i
?
i
. It is natural to refer to ?
i
as the deterministic compo-
nent and ?
i
as the stochastic component. Engle and Russell (1998) suggest
using cubic splines to model the deterministic pattern.
The parameters of the two components as well as any parameters as-
sociated with ?
i
can be estimated jointly by maximizing (8) or, a two step
procedure can be implemented in which ?rst the terms of the deterministic
pattern are estimated and in a second stage the remaining parameters are
estimated. The two step procedure can be implemented by ?rst running an
OLS regression of durations on a cubic spline. Let
b
?
i
denote the prediction
for the i
th
duration obtained from the OLS regression. Then let e x
i
=
x
i
?
i
denote the normalized duration. This standardized series should be free of
any diurnal pattern and should have a mean near unity. An ACD model
can then be estimated by MLE using the normalized durations e x
i
in place
of x
i
in (8). While this is not e?cient the two step procedure will provide
consistent estimates under correct speci?cation. (3) plots the estimated
diurnal pattern for ARG. This plot was constructed by regressing the du-
ration on a linear spline for the time of day at the start of the duration.
We ?nd the typical inverted U shaped pattern with durations longest in the
middle of the day and shortest near the open and close.
The similarity between the ACD model . Indeed the link is close as
detailed in the following corollary proven in Engle and Russell (1998).
Corollary 1 QMLE results for the EACD(1,1) model
If
1. E
i?1
(x
i
) = ?
i
= ? + ?x
i?1
+ ? ?
i?1
,
2. ²
i
=
x
i
?
i
is
(a) strictly stationary
(b) nondegenerate
(c) has bounded conditional second moments
(d) sup
i
E
i?1
[ln(? + ?²
i
)] < 0
3. ?
0
? (?, ?, ?) is in the interior of ?
21
4. L(?) = ?
N(T)
P
i=1
³
log (?
i
) +
x
i
?
i
´
Then the maximizer of L will be consistent and asymptotically normal
with a covariance matrix given by the familiar robust standard errors from
Lee-Hansen (1994).
This result is a direct corollary from the Lee and Hansen (1994 and
Lumsdaine (1996) proofs for the class of GARCH(1,1) models. The theorem
is powerful since under the conditions of the theorem we can estimate an
ACD model assuming an exponential distribution and even if the assumption
is false we still obtain consistent estimates although the standard errors need
to be adjusted as in White (1982). Furthermore, the corollary establishes
that we can use standard GARCH software to perform QML estimation of
ACD models. This is accomplished by setting the dependent variable equal
to the square root of the duration and imposing a conditional mean equation
of zero. The resulting parameter values provide consistent estimates of the
parameters used to forecast the expected duration.
Additionally, an estimate of the conditional distribution can be ob-
tained non-parametrically by considering the residuals b²
i
=
x
i
b
?
i
where
b
?
i
=
E
i?1
³
x
i
|
b
?
´
. Under correct speci?cation the standardized durations b²
i
should
be i.i.d. and the distribution can be estimated using non-parametric meth-
ods such as kernal smoothing. Alternatively, it is often more informative to
consider the baseline hazard. Given an estimate of the density the baseline
hazard is obtained from (12). Engle and Russell (1998) therefore propose a
semiparametric estimation procedure where in the ?rst step QMLE is per-
formed using the exponential distribution and in a second state the density
of ² is estimated nonparametrically. This is referred to as a semiparametric
ACD model.
ACD Model Diagnostics The properties of the standardized duration
also provide a means to asses the goodness of ?t of the estimated model.
For example, the correlation structure, or other types of dependence can be
tested. Engle and Russell (1998) suggest simply examining the Ljung-Box
statistic although other types of nonlinear dependence can be examined.
Engle and Russell (1998) suggest examining autocorrelations associated
with nonlinear transformations of the residuals b²
i
, for example, squares or
square roots. An alternative test of nonlinearity advocated in Engle and
Russell (1998) is to divide the diurnally adjusted durations into bins. Then
regress b²
i
on a constant and indicators for the magnitude of the previous
22
duration. One indicator must be omitted to avoid perfect multicollinearity.
If the b²
i
are indeed i.i.d. then there should be no predictability implied
from this regression. Often these tests suggests that the linear speci?cation
tends to over predict the duration following extremely short or extremely
long durations. This suggests that a model where the expectation is more
sensitive following short durations and less sensitive following long durations
may work well.
Engle and Russell (1998) suggest examining autocorrelations associated
with nonlinear transformations of the residuals b²
i
, for example, squares or
square roots. An alternative test of nonlinearity advocated in Engle and
Russell (1998) is to divide the diurnally adjusted durations into bins. Then
regress (b²
i
? 1) on indicators for the magnitude of the previous duration.
If the b²
i
are indeed i.i.d. then there should be no predictability implied
from this regression. Often these tests suggests that the linear speci?cation
tends to over predict the duration following extremely short or extremely
long durations. This suggests that a model where the expectation is more
sensitive following short durations and less sensitive following long durations
may work well.
Tests of the distributional assumptions of ² can also be examined. A
general test is based on the fact that the integrated intensity over the dura-
tion
u
i
=
t
i
Z
s=t
i?1
?(s|N(s), t
i?1
, t
i?2
, ..., t
0
)ds (18)
will be distributed as a unit exponential as discussed in Russell (1999).
Often this takes a very simple form. For example, substituting the expo-
nential intensity (15) into (18) simply yields the residual b u
i
= b²
i
=
x
i
b
?
i
b
?
i
.
Similarly, for the substituting the Weibull intensity (16) into (15) yields
b u
i
=
Ã
?
³
1+
1
?
´
x
i
b
?
i
b
?
i
!
?
. The variance of u
i
should be unity leading Engle and
Russell (1998) to suggest the test statistic
p
N(T)
(b ?
u
?1)
?
8
(19)
which should have a limiting standard normal distribution. This is a formal
test for remaining excess dispersion often observed in duration data. Fur-
thermore, since the survivor function for an exponential random variable
U is simply exp(?u) a plot of the the negative of the log of the empirical
23
survivor function should be linearly related to u
i
with a slope of unity hence
providing a graphical measure of ?t.
Nonlinear ACD models The tests for nonlinearity discussed above often
suggest nonlinearity. Zhang Russell and Tsay (2000) propose a nonlinear
threshold ACD model with this feature in mind. Here the dynamics of the
conditional mean are given by
?
i
=
_
_
_
?
1
+ ?
1
x
i?1
+ ?
1
?
i?1
if x
i?1
? a
1
?
2
+ ?
2
x
i?1
+ ?
2
?
i?1
if a
1
< x
i?1
? a
2
?
3
+ ?
3
x
i?1
+ ?
3
?
i?1
if a
2
< x
i?1
where a
1
and a
2
are parameters to be estimated. Hence the dynamics of
the expected duration depend on the magnitude of the previous duration.
Indeed, using the same IBM data as analyzed in Engle and Russell (1998)
they ?nd ?
1
> ?
2
> ?
3
as expected from the nonlinear test results. Esti-
mation is performed using a combination of a grid search across a
1
and a
2
and maximum likelihood for all pairs of a
1
and a
2
. The MLE is the pair
of a
1
and a
2
and the corresponding parameters of the ACD model for each
regime that produces the highest maximized likelihood.
Another nonlinear ACD model applied in Engle and Lunde (1999), Rus-
sell and Engle (2002) is the Nelson Form ACD model. The properties of
the Nelson form ACD are developed in Bauwens and Giot (2000). We refer
to this as the Nelson form ACD model because it is in the spirit of Nelson
(1991) EGARCH model and we want to minimize confusion with the version
of the ACD model that uses the exponential distribution for ?. Here the
log of the expected duration follows a linear speci?cation.
ln(?
i
) = ? +
p
X
j=1
?
j
²
i?j
+
q
X
j=1
?
j
ln(?
i?j
)
This formulation is particularly convenient when other market variables are
included in the ACD model since non-negativity of the expected duration is
directly imposed.
An interesting approach to nonlinearity is taken in Fernandes and Gram-
mig (2002) who propose a class of nonlinear ACD models. The parametriza-
tion is constructed using the Box Cox transformation of the expected du-
ration and a ?exible non-linear function of ²
i?j
that allows the expected
duration to respond in a distinct manner to small and large shocks. The
model nests many of the common ACD models and is shown to work well
for a variety of duration data sets.
24
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
1 10 19 28 37 46 55 64 73 82 91 100 109 118 127 136 145 154 163 172 181 190 199
Figure 8: Autocorrelations for ACD duration residuals b²
i
=
x
i
b
?
i
ACD example Appendix (A) contains parameter estimates for an EACD(3,2)
model estimated using the GARCH module of EVIEWS. The durations
were ?rst adjusted by dividing by the time of day e?ect estimated by linear
splines in ?gure 3. The sum of the ? and ? coe?cients is in excess of .999
but less than one indicating strong persistence but the impact of shocks dies
o? after a su?cient period of time. A plot of the autocorrelations of the
residuals is presented in ?gure 8. The autocorrelations are no longer all
positive and appear insigni?cant. A formal test for the null that the ?rst
15 auto correlations are zero yields a 12.15 with a p-value of 67%.
A test for remaining excess dispersion in (19) yields a test statistic of
?
32365(1.41?1)
?
8
= 26.07. There is evidence of excess dispersion indicating
that it is unlikely that ? is exponential. However, under the conditions
of the corollary the parameter estimates can be viewed as QML estimates.
A plot of the nonparametric hazard is given in ?gure 9. The estimate
was obtained using a nearest neighbor estimator. The hazard is nearly
monotonically decreasing indicating that the longer it has been since the
last transaction the less likely it is for a transaction to occur in the next
instant. This is a common ?nding for transactions data.
25
0
0.5
1
1.5
2
2.5
3
3.5
4
0 1 2 3 4 5 6 7 8 9
Figure 9: Non-parametric estimate of baseline intensity.
2.1.2 Thinning point processes
The above discussion focuses on modeling the arrival of transaction events.
For some economic questions it may not be necessary to examine the arrival
of all transactions. Rather, we might want to focus on a subset of all trans-
actions which have special meaning. For example, when foreign exchange
data are examined many of the posted quotes appear to be simply noisy
repeats of the previous posted quotes. Alternatively, for stock data, often
times many transactions are recorded with the exact same transaction price.
Another example might be the time until a set amount of volume has been
transacted - a measure related to liquidity.
Depending on the question at hand, we might be able to focus on the
subset of event arrival times for which the marks take special importance -
such as a price movement or attained a speci?ed cumulative volume. In this
case, the durations would correspond to the time interval between marks of
special importance. If the timing of these special events is important, then
econometrically this corresponds to focusing on the distribution of ?
min
in
(5). The sequence of arrival times corresponding to the times at which the
marks take special values is called a thinned point process since the new set
of arrival times will contain fewer events than the original series.
Engle and Russell (1998) refer to the series of durations between events
26
constructed by thinning the events with respect to price and volume as price
based and volume based durations. They suggest that the ACD model
might be a good candidate for modeling for these thinned series. In Engle
and Russell (1997) a Weibull ACD model is applied to a thinned series of
quotes arrival times for the Dollar Deutchemark exchange rate series. More
formally if t
i
denotes the arrival times of the original series of quotes then let
?
0
= t
0
Next, let N
?
(t)denote the counting function for the thinned process
de?ned by t
N
?
(t)+1
= t +?
N
?
(t)+1
where ?
N
?
(t)+1
= min
?>0
¯
¯
p
N(t+?)
?p
N
?
(t)
¯
¯
>
c and N(t
0
) = N
?
(t
0
) = 0. So, the sequence of durations ?
i
corresponds
to the price durations de?ned by price movements greater than a threshold
value c. An ACD Weibull model appears to provide a nice ?t for the thinned
series.
The authors suggest that this provides a convenient way of characteriz-
ing volatility when the data are irregularly spaced. The intuition is that
instead of modeling the price change per unit time, as is typically done
for volatility models constructed using regularly spaced data, the model for
price durations models the time per unit price change. In fact, assuming
that the price process locally follows a geometric Brownian motion leads to
implied measures of volatility using ?rst crossing time theory.
Engle and Lange (2001) combine the use of price durations discussed
above with the cumulative signed volume transacted over the price duration
to measure liquidity. For each of the US stocks analyzed, the volume quan-
tity associated with each transaction is given a positive sign if it is buyer
initiated and negative sign if it is seller initiated using the rule proposed
by Lee and Ready (1991). This signed volume is then cumulated for each
price duration. The cumulative signed volume, referred to as VNET, is the
total net volume that can be transacted before inducing a price move hence
providing a time varying measure of the depth of the market. For each price
duration several other measures are also constructed including the cumula-
tive (unsigned) volume and number of transactions. Regressing VNET on
these market variables suggest that market depth is lower following periods
high transaction rates, and high volatility. While VNET increases with past
volume it does so less than proportionally indicating that order imbalance
as a fraction of overall (unsigned) volume decreases with overall volume.
Jointly these results suggest that market depth tends to be lower during
periods of high transaction rates, high volatility, and high volume. In an
asymmetric information environment this is indicative of informed trading
transpiring during period of high transaction rates and high volume.
27
2.2 Modeling in Tick Time - the Marks
Often, economic hypothesis of interest are cast in terms of the marks asso-
ciated with the arrival times. For example, many hypothesis in the asym-
metric information literature focusing on the mechanism by which private
information becomes impounded asset prices. In a rational expectations
environment, the specialist will learn about a traders private information
from the characteristics of their transactions. Hence, many asymmetric
information models of ?nancial participants have implications about how
price adjustments should depend on the characteristics of trades such as
volume or frequency of transactions. By the very nature of market micro
structure ?eld, these theories often need to be examined at the transaction
by transaction frequency This section of the paper examines transaction by
transaction analysis of the marks. We refer to this approach generally as
tick time analysis.
2.2.1 VAR Models for Prices and Trades in Tick Time
Various approaches to tick time modeling of the marks have been considered
in the literature. The approaches are primarily driven by the economic
question at hand as well as assumptions about the role of the timing of
trades. Many times the hypothesis of interest can be expressed as how
the prices adjust given characteristics of past order ?ow. In this case, it is
not necessary to analyze the joint distribution in (1) but only the marginal
distribution of the mark given in (3).
Perhaps the simplest approach in application is to assume that timing
of past of transactions has no impact on the distribution of the marks, that
is f
y
¡
y
i+1
|
b
t
i
, b y
i
¢
= f
y
(y
i+1
|, b y
i
) . This is the approach taken in Hasbrouck
(1991) where the price impact of a trade on future transaction prices is
examined. Hasbrouck focuses on the midpoint of the bid and ask quotes as
a measure of the price of the asset. We refer to this as the midprice and
would appear to be a good approximation to the value of the asset given
the information available. Hasbrouck is interested in testing and identifying
how buyer and seller initiated trades di?er in their impact on the expectation
of the future price.
Let ?m
i
denote the change in the midprice from the (i-1)
th
to the i
th
transactions or m
i
? m
i?1
. The bid and ask prices used to construct the
midprice are those prevailing just prior to the transaction time t
i
. Let w
i
denote the signed volume of a transaction taking a positive value if the i
th
trade is buyer initiated and a negative value if it is seller initiated. The
28
direction of trade is inferred using the Lee and Ready rule discussed in sec-
tion . Hasbrouck persuasively argues that market frictions induce temporal
correlations in both the price and volume series. Regulations require the
specialist to operate an ”orderly” market meaning that the price should not
?uctuate dramatically from one trade to the next. Hence in the face of
a large price move the specialist will have to take intervening transactions
at intermediate prices to smooth the price transition. Volume might also
be autocorrelated as a result of the common practice of breaking up large
orders into multiple small orders to achieve at a better overall price than
had the order been executed in one large transaction. Finally, since neither
price nor direction of trades can be viewed as exogenous the series must be
analyzed jointly to get a full picture of the series dynamics. Hasbrouck
analyzes the bivariate system using the following VAR:
?m
i
=
J
X
j=1
a
j
?m
i?j
+
J
X
j=0
b
j
w
i?j
+ v
1i
(20)
w
i
=
J
X
j=1
c
j
?m
i?j
+
J
X
j=1
d
j
w
i?j
+ v
2i
Notice that the signed volume appears contemporaneously on the right
hand side of the quote update equation. The quote revision equation is
therefore speci?ed conditional on the contemporaneous trade. In reality, it
is likely that the contemporaneous quote revision will in?uence the decision
to transact as the marginal trader might be enticed to transact when a new
limit order arrives improving the price. In fact, our application suggests
some evidence that this may be the case for the Airgas stock. We estimate
a VAR is for price changes and a trade direction indicator variable taking
the value 1 if the trade is deemed buyer initiated and -1 if the trade is
deemed seller initiated using the Lee and Reedy rule. Volume e?ects are
not considered here. As expected, the b
j
coe?cients tend to be postive
meaning that buys tend to lead to increasing quote revisinos and sells tend
to lead to decreasing quote revisions. The market frictions suggest that
the full impact of a trade may not be instantaneous, but rather occur over
a longer period of time. To gauge this e?ect the VAR can be expressed
as an in?nite vector moving average model. The coe?cients then form the
impulse response function.
The cumulants of the impulse response functions then provide a measure
of the total price impact of a trade. Since the model operates in transaction
29
0
0.002
0.004
0.006
0.008
0.01
0.012
0 5 10 15 20
Transactions
P
r
i
c
e
I
m
p
a
c
t
Figure 10: Cummulative price impact of an unexpected buy
time these price impacts are therefore also measured in transaction time.
The asymptote associated with the cummulative impulse response function
is then de?ned as the total price impact of a trade. Since the data are
indexed in tick time the price impact is measured in units of transactions.
The results indicate that it can take several transactions before the full
price impact of a transaction is realized. Figure 1 below presents a price
impact plot for the stock Airgas. Similar to Hasbrouck’s ?ndings using
the earlier data sets we ?nd that the price impact can take many periods
to be fully realized and that the function is concave. The price impact is
in the expected direction - buys increase the price and sells decreased the
priced. Looking at a cross section of stocks, Hasbrouck constructs measures
for the information asymmetry by taking the ratio of the price impact of
a 90th percentile volume trade over the average price. This measure is
decreasing with market capitalization suggesting that ?rms with smaller
market capitalization have larger information asymmetries.
Engle and Dufour also analyze the price impact of trades, but relax the
30
assumption that the timing of trades has no impact on the marginal distri-
bution of price changes. Easley and O’Hara (1992) propose a model with
informed and uninformed traders. On any given day private information
may or may not exist. Informed and uninformed traders are assumed to
arrive in a random fashion. Informed traders only transact when private
information is present so on days with no private information all transac-
tions are by the uninformed. Days with high transaction rates are therefore
viewed as days with more informed trading. Admatti and Pfeiderer (1988)
suggest that in the presence of short sales constraints the timing of trades
should also carry information. Here, long durations imply bad news and
suggest falling prices. The important thread here is that the timing of
trades should carry informative. This leads Dufour and Engle (2000) to
consider expanding Hasbrouck’s VAR structure to allow the durations to
impact price updates. The duration between trades is treated as a pre-
determined variable that in?uences the informativeness of past trades on
future quote revisions. This is done with by allowing the b
j
parameters in
(20) to be time varying parameters. In particular,
b
j
= ?
j
+
K
X
k=1
?
k
D
j,i?k
+ ?
i
ln(x
i?j
)
where D
j,i?k
are dummy variables for the time of day and x
i
is the duration.
Since the b
j
dictate the impact of past trades on quote revisions it is clear
that these e?ects will be time varying whenever the coe?cients d
k
or ? are
non-zero. The model therefore extends the basic VAR of Hasbrouck by
allowing the impact of trades to depend on the time of day as well the
trading frequency as measured by the elapsed time between trades. A
similar adjustment is made to the coe?cients d
j
in the trade equation.
The modi?ed VAR is speci?ed conditional on the durations and may
therefore be estimated directly. Impulse response function, however, will
require complete speci?cation of the trivariate system of trades quotes and
arrival times. Dufour and Engle propose using the ACD model for the
arrival times.
Parameters are estimated for 18 stocks. As in the simple Hasbrouck
VAR the impact of past transactions on quote revisions tends to be positive
meaning that buys tend to lead to increasing quote revisions and sells lead to
decreasing quote revisions. The magnitude of the d
k
indicate some degree of
time of day e?ects in the impact of trades. Trades near the open tend to be
more informative, or have a larger price impact than trades at other times
during the day although this e?ect is not uniform across all 18 stocks. The
31
coe?cients on the durations tend to be negative indicating that the longer
it has been since the last trade, the smaller the price impact will be. The
cumulative impulse response from an unanticipated order can be examined
only now these functions will depend on the state of the market as dictated
by transaction rates measured by the durations. The result is that the
price impact curves will shift up when the transaction occurs with a short
duration and shift down when transactions occur with long durations.
The VAR approach to modeling tick data is particularly appealing due
to its ease of use. Furthermore, information about the spacing of the data
can be included in these models as suggested in Dufour and Engle (2000).
These VAR models can, of course, be expanded to include other variables
such as the bid ask spread or measures of volatility.
2.2.2 Volatility Models in Tick Time
The VARs of the previous section where proved useful in quantifying the
price impact of trades. As such, they focus on the predictable change in
the quotes given characteristics of a transaction. Alternatively, we might
want to ask how characteristics of a transaction a?ect our uncertainty about
the quote updates. Volatility models provide a means of quantifying our
uncertainty.
The class of GARCH models of Engle (1982) and Bollerslev (1986) have
proven to be a trusted work horse in modeling ?nancial data at the daily
frequency. Irregular spacing of transaction by transaction data seems par-
ticularly important for volatility modeling of transaction by transaction data
since volatility is generally measured over ?xed time intervals. Furthermore,
it is very unlikely that, all else equal, the volatility of the asset price over
a one hour intertrade duration should be the same as the volatility over a
5 second intertrade duration. Engle (2000) proposes adapting the GARCH
model for application to irregularly spaced transaction by transaction data.
Let the return from the i ?1
th
to the i
th
transaction be denoted by r
i
.
De?ne the conditional variance per transaction as
V
i?1
(r
i
|x
i
) = h
i
(21)
where this variance is de?ned conditional on the contemporaneous duration
as well as past price changes. The variance of interest, however, is the
variance per unit time. This is related to the variance per transaction as
32
V
i?1
µ
r
i
?
x
i
|x
i
¶
= ?
2
i
(22)
so that the relationship between the two variances is h
i
= x
i
?
2
i
.
The volatility per unit time is then modeled as a GARCH process. En-
gle proposes an ARMA(1,1) model for the series
r
i
?
x
i
. Let e
i
denote the
innovation to this series. If the durations are not informative about the
variance per unit time then the GARCH(1,1) model for irregularly spaced
data is simply
?
2
i
=
.
? +
.
?e
2
i?1
+
.
??
2
i?1
(23)
where we have placed dots above the GARCH parameters to di?erentiate
these from the parameters of the ACD model with similar notation. Engle
terms this model the UHF-GARCH model or Ultra High-Frequency GARCH
model.
A more general model is inspired by the theoretical models of Easley
and O’Hara (1992) and Admatti and Pleiderer (1985) discussed in section
(2.2.1) above. These models suggest that the timing of transactions is re-
lated to the likelihood of asymmetric trading and hence more uncertainty.
Engle therefore proposes augmenting the GARCH(1,1) models with addi-
tional information about the contemporaneous duration and perhaps other
characteristics of the market that might be thought to carry information
about uncertainty such as spreads and past volume.
While the model speci?es the volatility per unit time it is still operates
in transaction time updating the volatility on a time scale determined by
transaction arrivals. If calendar time forecasts of volatility are of interest
then a model for the arrival times must be speci?ed and estimated. Toward
this end Engle proposes using an ACD model for the arrival times. If the
arrival times are deemed exogenous then the ACD model and the GARCH
model can be estimated separately although this estimation may be inef-
?cient. In particular, under the exogeneity assumption the ACD model
could be estimated ?rst and then the volatility model could be speci?ed
conditional on the contemporaneous duration and expected duration in a
second step using canned GARCH software that admits additional explana-
tory variables. This is the approach taken in Engle (2000) where estimation
is performed via (Q)MLE. Engle considers the following speci?cation
?
2
i
=
.
? +
.
?e
2
i?1
+
.
??
2
i?1
+ ?
1
x
?1
i
+ ?
2
x
i
?
i
+ ?
3
?
?1
i
+ ?
4
?
i?1
33
where ?
i
is the expected duration obtained from an ACD model and ?
i?1
characterizes the long run volatility via exponential smoothing of the squared
return per unit time.
An alternative approach to modeling volatility of irregularly spaced data
was simultaneously and independently developed in Ghysels and Jasiak
(1998). Here the authors suggest using temporal aggregation to handle
the spacing of the data. GARCH models are not closed under temporal
aggregation so the authors propose working with the weak GARCH class of
models proposed by Drost and Nijman (1993). For the weak GARCH class
of models Drost and Nijman derive the implied low frequency weak GARCH
model implied by a higher frequency weak GARCH model. Ghysels and
Jasiak propose a GARCH model with time varying parameters driven by
the expected spacing of the data. This approach is complicated, however,
by the fact that the temporal aggregation results apply to aggregation from
one ?xed interval to another, exogenously speci?ed, ?xed interval. The
spacing of the transactions data is not ?xed and it is unlikely that transac-
tion arrival times are exogenous. Nevertheless, the authors show that the
proposed model is an exact discretization of a time deformed di?usion with
ACD as the directing process. The authors propose using GMM to estimate
the model.
2.3 Models for discrete prices
Discrete prices in ?nancial markets pose an additional complication in the
analysis of ?nancial data. For the US markets the graduation to decimal-
ization is now complete, but we still ?nd price changes clustering on just
a handful of values. This discreteness can have an important in?uence on
analysis of prices. Early analysis of discrete prices focused on the notion
of a ”true” or e?cient price. The e?cient price is de?ned as the expected
value of the asset given all currently available public information. The fo-
cus of these early studies, therefore, was on the relationship between the
e?cient price and observed discrete prices. In particular, much empha-
sis was placed on how inference about the e?cient price is in?uenced by
measurement errors induced by discreteness.
Let P
t
denote the observed price at time t and let P
e
t
denote the e?cient
or ”true” price of the asset at time t. Early models for discrete prices can
generally be described in the following setting:
34
P
e
t
= P
e
t?1
+v
t
P
t
= round (P
e
t
+ c
t
Q
t
, d)
v
t
˜N
¡
0, ?
2
t
¢
(24)
where d ? 0 is the tick size and round is a function rounding the argument
to the nearest tick. Q
t
is an unobserved i.i.d. indicator for whether the
trade was buyer or seller initiated taking the value 1 for buyer initiated and
-1 for seller initiated trades and probability given by 1/2. The parameter
c
t
? 0 denotes the cost of market making. It includes both tangible costs
of market making as well as compensation for risk.
With c = 0 and ?
2
t
= ?
2
we obtain the model of Gottlieb and Kalay
(1985)
1
. When d = 0 (no rounding) we obtain the model of Roll (1984).
Harris (1990) considers the full model in (24). In this case, we can write
?P
t
= c (Q
t
?Q
t?1
) + ?
t
??
t?1
+ v
t
(25)
where ?
t
= P
e
t
? P
t
is the rounding error. The variance of the observed
price series is therefore given by
E (?P
t
)
2
= ?
2
+ 2c
2
+ E
¡
?
t+1
??
t
¢
2
(26)
. Hence the variance of the observed transaction price will exceed the vari-
ance by an amount that depends on the cost of market making and the
discrete rounding errors. Furthermore, the ?rst order serial correlation is
given by
E (?P
t
?P
t?1
) = ?c
2
+ E
¡
?
t+1
??
t
¢ ¡
?
t
??
t?1
¢
(27)
which is shown to be negative in Harris (1990). The ?rst order serial corre-
lation will be larger when the cost of market making is larger and depends
on the discreteness rounding errors. The Harris model goes a long way in
describing key features of price discreteness and the implications regarding
inference on the e?cient price dynamics but it is still very simplistic in sev-
eral dimensions since it assumes that both the volatility of the e?cient price
and the cost of market making is constant. As new information hits the
market the volatility of the e?cient price will change. Since part of the cost
of making market is the risk of holding the asset the cost of making market
will also be time varying.
1
For simplicity we have neglected a drift term in the e?cient price equation and the
possiblity of dividend payments considered in Gottlieb and Kalay (1985).
35
Hasbrouck (1999) builds on the previous discrete price literature by re-
laxing these two assumptions - both the volatility of the e?cient price and
the cost of making market are time varying. He also proposes working with
bid and ask prices as opposed to the transaction prices circumventing the
need to sign trades as buyer or seller initiated. P
t
of 24 is therefore replaced
by two prices P
a
t
and P
b
t
, the bid and ask prices respectively where
P
a
t
= ceiling (P
e
t
+ c
a
t
, d)
P
b
t
= floor
¡
P
e
t
?c
b
t
, d
¢
(28)
c
a
t
> 0 and c
b
t
> 0 are the cost of exposure on the ask and bid side respec-
tively. These costs are the economic cost to the specialist including both
the ?xed cost of operation and the expected cost incurred as a result of the
obligation to trade a ?xed quantity at these prices with potentially better
informed traders. The ceiling function rounds up to the nearest discrete
tick and the ?oor function rounds down to the nearest tick recognizing the
market maker would not set quotes that would fail to cover the cost.
The dynamics of this model are not suited for the type of analysis pre-
sented by Harris due to it’s more complex dynamic structure. Instead,
Hasbrouck focuses on characterizing the dynamics of the both cost of expo-
sure and, second, estimating models for the volatility of the e?cient price
given only observations of the perturbed discrete bid and ask prices. Has-
brouck proposes using a GARCH model for the dynamics of the e?cient
price volatility ?
2
t
. The dynamics of the cost of exposure is assumed to be
of an autoregressive form.
ln(c
a
t
) = µ
t
+ ?
¡
ln(c
a
t
) ?µ
t?1
¢
+ ?
?
t
(29)
ln
³
c
b
t
´
= µ
t
+ ?
³
ln
³
c
b
t
´
?µ
t?1
´
+ ?
?
t
where µ
t
is a common deterministic function of time of day and ? is the
common autoregressive parameter. ?
?
t
and ?
?
t
are assumed to be i.i.d. and
independent of the e?cient price innovation v
t
.
Estimation of the model is complicated by the fact that the e?cient price
is inherently unobserved - only the discrete bid and ask quotes are observed.
Hasbrouck (1999) proposes using a non-Gaussian, non-linear state space of
Kitagawa (1987) and Hasbrouck (1999b) and Manrique and Sheppard (1997)
propose using MCMC methods which treat the price at any given date as
an unknown parameter. We refer the reader to this literature for further
details on the estimation.
36
The early work with no time varying parameters focused on the impact
of discrete rounding errors on inference regarding the e?cient price. Has-
brouck also treats the object of interest as the dynamics of the e?cient price
and demonstrates a methodology to study the second moment dynamics of
the e?cient price while accounting for the discrete rounding errors. In ad-
dition, the model allows for asymmetric cost of exposure. In some states of
the world the specialist may set quotes more conservatively on one side of
the market than the other.
More recently Zhang Russell and Tsay (2000) appeal to the asymmetric
information microstructure theory that suggests that transaction character-
istics should in?uence the market makers perception of the exposure risk.
They include in the cost of exposure dynamics (29) measures of the order
imbalance and overall volume and ?nd that the cost of exposure is a?ected
by order imbalance. In particular, unexpectedly large buyer initiated vol-
ume tends to increase the cost of exposure on the ask side and decrease the
cost of exposure on the bid side with analogous results for unexpected seller
initiated volume. These e?ects are mitigated, however, the larger the total
volume transacted.
If the structural parameters are not of primary interest then an alter-
native is to directly model transaction prices with a reduced form model
for discrete valued random variables. This is the approach taken in Haus-
man Lo and MacKinlay (1992). They propose modeling the transaction
by transaction price changes with an ordered Probit model. In doing so,
the structural models linking the unobserved e?cient price to the observed
transaction price is replaced by a reduced from Probit link. The model
is applied to transaction by transaction price dynamics. Let k denote the
number of discrete values that the price changes can take which is assumed
to be ?nite. Let s
i
denote a vector of length k taking the j
th
column of
the kxk identity matrix if the j
th
state occurs on the i
th
transaction. Let
?
i
denote a k dimensional vector with j
th
where ?
i
= E (s
i
|I
i?1
) where I
i
is the information set associated with the i
th
transaction. Clearly, the j
th
element of x
i
denotes the conditional probability of the j
th
state occurring.
At the heart of the Probit model lies the assumption that the observed dis-
crete transaction price changes can be represented as a transformation of a
continuous latent price given by ?P
?
i
˜N
¡
µ
i
, ?
2
i
¢
where µ
i
and ?
2
i
are the
mean and variance of the latent price given I
i?1
. The Hausman Lo and
MacKinlay model assumes that the j
th
element of ?
i
is given by
?
j
˜F
P
?
i
(c
j?1
) ?F
P
?
i
(c
j
) (30)
where F
P
?
i
is the cdf associated with the price changes ?P
?
i
and c
j
, j =
37
1, k ?1 are time invariant parameters.
Since bid ask bounce induces dependence in the price changes it is nat-
ural to allow the conditional mean of price changes to depend on past price
changes. Hausman Lo and Mackinlay are particularly interested in testing
asymmetric information theories regarding the information content in a se-
quence of trades. In particular, they study how transaction prices respond
to a sequence of buyer initiated trades versus a sequence of seller initiated
trades. By conditioning on recent buys and sells the authors ?nd evidence
that persistent selling predicts falling prices and persistent buying predicts
rising prices. The authors also suggest that the conditional variance may
depend on the contemporaneous duration so that the variance associated
with a price change over a long duration may not be the same as the vari-
ance of a price change associated with a relatively short duration. Indeed,
they ?nd that long durations lead to higher variance per transaction.
Russell and Engle (2002) are also interested in reduced form models for
discrete prices that explicitly account for the irregular spacing of the data.
They propose joint modeling the arrival times and the price changes as a
marked point process. The joint likelihood for the arrival times and the price
changes is decomposed into the product of the conditional distribution of the
price change given the duration and the marginal distribution of the arrival
times which are assumed to be given by an ACD model. More speci?cally
if x
i
denotes the i
th
intertransaction duration and b z
i
= (z
i
, z
i?1
, z
i?2
...) then
f (x
i+1
, ?p
i+1
|b x
i
, ?b p
i
) = ?(?p
i+1
|b x
i
, ?b p
i+1
) ?(x
i
|b x
i
, ?b p
i
)
where ? denotes the distribution of price changes given the past price
changes and durations as well as the contemporaneous duration. ? de-
notes the distribution of price changes given the past price changes and
returns. Engle and Russell propose using the ACD model for the durations
and propose the Autoregressive Conditional Multinomial (ACM) model for
the conditional distribution of the discrete price changes. A simple model
for the price dynamics might assume a VARMA model for the state vector
s
i
. Since the state vector is simply a vector of ones and zeros its expecta-
tion should be bounded between zero and one. Russell and Engle use the
logistic transformation to directly impose this condition. Using the logistic
link function the VARMA model is expressed in terms of the log odds. Let
h
j
denote a k ?1 vector with j
th
element given by ln
¡
?
j
/?
k
¢
. Let e s
i
and
e ?
i
denote k ?1 dimensional vectors consisting of the ?rst k-1 elements of s
i
and ?
i
. Hence the k
th
element has been omitted and is referred to as the
base state. Then the ACM(u, v) model with duration dependence is given
38
by:
h
i
= c +
u
X
m=1
A
m
(e s
i?m
? e ?
i
) +
v
X
m=1
B
m
h
i?m
+
w
X
m=1
?
m
ln(x
i?m+1
) (31)
where A
m
and B
m
are (k ?1) x(k ?1) parameter matrices and ? and ?
m
are k ? 1 parameter vector. Given the linear structure of the log odds
VARMA the choice of the the base state is arbitrary. The ?rst k ? 1
probabilities are obtained by applying the logistic link function
?
i
=
1
1 + ?
0
exp(h
i
)
exp(h
i
) (32)
where ?
0
is a k?1 vector of ones and exp(h
i
) should be interpreted as applying
the exponential function element by element. The omitted state is obtained
by imposing the condition that the probabilities sum to 1. The ACD(p, q)
speci?cation for the durations allows feedback from the price dynamics into
the duration dynamics as follows:
ln(?
i
) = ?+
p
X
m=1
?
m
x
i?m
?
i?m
+
q
X
m=1
?
m
ln
¡
?
i?m
¢
+
r
X
m=1
¡
?
m
?p
i?m
+ ?
m
?p
2
i?m
¢
For the stocks analyzed, the longer the contemporaneous duration the lower
the expected price change and large price changes tend to be followed by
short durations.
Another reduced form model for discrete prices is proposed by Rydberg
and Sheppard (1999). The model decomposes the discrete price changes
into the trivariate process ?P
i
= Z
i
D
i
M
i
where Z
i
is an indicator for
the i
th
transaction price change being non-zero and is referred to as ac-
tivity. Conditional on a price move (Z
i
6= 0) D
i
is takes the value of 1
or -1 denoting an upward or downward price change respectively. Given a
non-zero price change and its direction M
i
is the magnitude of the price
change given it is non-zero and the direction. The authors suggest de-
composing the distribution of price changes given an information set I
i
Pr (?P
i
|I
i?1
) = Pr (Z
i
|I
i?1
) Pr (D
i
|Z
i
, I
i?1
) Pr (M
i
|Z
i
, D
i
, I
i?1
) . The au-
thors propose modeling the binary variables Z
i
and D
i
following an autore-
gressive logistic process ?rst proposed by Cox (1958). A simple version for
the activity variable is given by:
ln
µ
Pr (Z
i
= 1)
1 ?Pr (Z
i
= 1)
¶
= c +
u
X
m=1
Z
i?m
(33)
39
The direction indicator variable is modeled in a similar fashion. Finally, the
magnitude of the price change is modeled by a distribution for count data.
Hence it is positively valued over integers. The integers here are measured
in units of ticks, or the smallest possible price change.
The Russell Engle ACM approach and the Rydberg Sheppard compo-
nents model are very similar in spirit both implementing an autoregressive
structure. While this decomposition breaks the estimation down into a se-
quence of simpler problems it comes with a cost. In order to estimate the
model sequentially the ?rst model for the activity can not be a function of
lagged values of Pr (D
i
|I
i?1
) or Pr(M
i
|I
i?1
). Similarly the model for the
direction cannot depend on the past probability associated with the magni-
tude. The importance of this restriction surly depends on the application at
hand. A second advantage of the Rydberg Sheppard model easily accounts
for a large number of states (possibly in?nite).
2.4 Calendar Time Conversion
Most ?nancial econometric analyses are carried out in ?xed time units.
These time intervals for many years were months or weeks or days but now
time intervals of hours, ?ve minutes or seconds are being used for econo-
metric model building. Once the data are converted from their natural
irregular spacing to regular spaced observations, econometric analysis typ-
ically proceeds without considering the original form of the data. Models
are constructed for volatility, price impact, correlation, extreme values, and
many other ?nancial constructs. In this section we discuss the most com-
mon approaches used to convert irregularly spaced data to equally spaced
observations and in the next sections we will examine the implications of
this conversion.
Suppose the data on prices arrive at times {t
i
; i = 1, ..., N(T)} so
that there are N observations occurring over time (0, T). These times could
be times at which transactions occur and the price could be either the trans-
action price or the prevailing midquote at that time. An alternative formu-
lation would have these times as the times at which quotes are posted and
then the prices are naturally considered to be midquotes. Let the log of the
price at time t
i
be denoted p
?
i
.
The task is to construct data on prices at each ?xed interval of time.
Denoting the discrete time intervals by integers of t = 1, . . . , T, a task is to
estimate p
t
. The most common speci?cation is to use the most recent price
at the end of the time interval as the observation for the interval. Thus:
40
p
t
= p
?
i
where t
i
? t < t
i+1
(34)
For example, Huang and Stoll (1994) use this scheme where p is the pre-
vailing midquote at the time of the last trade. Andersen Bollerslev Diebold
and Ebens (2001) use the last trade price.
Various alternative schemes have been used. One could interpolate
the price path from some or all of the p
?
observations and then record the
value at time t. For example smoothing splines could be ?t through all the
data points. A particularly simple example of this uses the weighted average
of the last price in one interval and the ?rst price in the next interval:
e p
t
=
¡
?p
?
i
+ (1 ??) p
?
i+1
¢
where t
i
? t < t
i+1
, and ? =
t ?t
i
t
i+1
?t
i
(35)
Andersen, Bollerslev, Diebold and Labys (2001,2002) use this procedure
with midquotes to get 5 minute and 30 minute calendar time data. The
advantage of this formulation is supposed to be its reduced sensitivity to
measurement error in prices. Clearly this comes at a cost of using future
information. The merits of such a scheme must be evaluated in the context
of a particular data generating process and statistical question.
A third possibility is adopted by Hasbrouck(2002). Since the time
of a trade is recorded only to the nearest second, then if t is measured in
seconds, there is at most one observation per time period. The calendar
price is either set to this price or it is set to the previous period price. This
version follows equation 34 but does not represent an approximation in the
same sense.
Returns are de?ned as the ?rst di?erence of the series.
y
?
i
= p
?
i
?p
?
i?1
, y
t
= p
t
?p
t?1
, and e y
i
= e p
i
? e p
i?1
(36)
thus y
?
de?nes returns over irregular intervals while y de?nes returns
over calendar intervals. In the case of small calendar intervals, there will
be many zero returns in y. In this case, there are some simpli?cations.
Means and variances are preserved if there never is more than one trade per
calendar interval
N(T)
X
i=1
y
?
i
=
T
X
t=1
y
t
and
N(T)
X
i=1
(y
?
i
)
2
=
T
X
t=1
y
2
t
(37)
41
If the calendar time intervals are larger, then means will still be preserved
but not variances as there may be more than one price in a calendar interval.
N(T)
X
i=1
y
?
i
=
T
X
t=1
y
t
and
N(T)
X
i=1
_
_
_
_
_
_
_
_
X
multiple
trades
y
?
i
_
_
_
_
_
_
_
_
2
=
T
X
t=1
y
2
t
(38)
However, if the prices are Martingales, then the expectation of the cross
products is zero and the expected value and probability limit of the calendar
time and event time variances is the same.
When prices are interpolated, these relations no longer hold. In this
scheme there would be many cases of multiple observations in a calendar
interval. The mean will approximately be the same in both sets of returns,
however the variances will not. The sum of squared transformed returns is
given by:
N(T)
X
i=1
[e y
i
]
2
=
N(T)
X
i=1
£
?
i
p
?
i
+ (1 ??
i
) p
?
i?1
??
j
p
?
j
?(1 ??
j
) p
?
j?1
¤
2
(39)
=
N(T)
X
i=1
£
?
i
¡
p
?
i
?p
?
i?1
¢
+
¡
p
?
i
?p
?
j
¢
+ (1 ??
j
)
¡
p
?
j
?p
?
j?1
¢¤
2
where i and j are the events just after the end points of the calendar
intervals. In the right hand side of expression 39, the returns will all be
uncorrelated if the y
?
are Martingale di?erences, hence the expected vari-
ance and the probability limit of the variance estimators will be less than
the variance of the process. Furthermore, the returns will be positively au-
tocorrelated because they are formulated in terms of future prices. This is
easily seen in equation 39 because the change in price around the interval
endpoints is included in both adjacent returns.
2.4.1 Bivariate Relationships
We now consider two correlated assets with Martingale prices. One of
these asset prices is only observed at random time periods while the other
is continuously observable. In this case the stochastic process of the infre-
quently observed process is de?ned on times {t
i
; i = 1, ..., N(T)}. Let the
42
log price of the second asset be q
t
, and let its return, measured respectively
in the ?rst asset trade time and in calendar time, be:
z
?
t
i
? z
?
i
= q
t
i
?q
t
i?1
, z
t
= q
t
?q
t?1
The return on the ?rst asset is given by
y
?
i
= ?z
?
i
+ ²
?
i
(40)
where the innovation is a Martingale di?erence sequence, independent
of z, with potential heteroskedasticity since each observation may have a
di?erent time span. Since this model is formulated in transaction time, it
is natural to estimate the unknown parameter beta with transaction time
data. It is straightforward to show that least squares will be consistent.
Calendar time data on y can be constructed from 40 and 36. Con-
sider the most disaggregated calendar time interval and let d
t
be a dummy
variable for the time periods in which a price is observed on the ?rst asset.
Then a useful expression for y
t
is
y
t
= d
t
µ
(?z
t
+ ²
t
) + (1 ?d
t?1
) (?z
t?1
+ ²
t?1
)
+(1 ?d
t?2
) (?z
t?2
+ ²
t?2
) + ...
¶
(41)
With this data and the comparable calendar data on z, we are in a
position to estimate beta by ordinary least squares in calendar time. The
estimator is simply
b
? =
T
P
t=1
z
t
y
t
T
P
t=1
z
2
t
(42)
which has an interesting probability limit under simplifying assumptions.
Theorem 1
If
a) (z
t
, ²
t
) are independent Martingale di?erence sequences with ?-
nite variance
b) d
t
˜ independent Bernoulli with parameter ?
Then
plim
b
? = ?? (43)
Proof:
Substituting and taking probability limits:
plim
b
? =
1
?
2
z
E [z
t
d
t
((?z
t
+ ²
t
) + (1 ?d
t?1
) (?z
t?1
+²
t?1
) + ...)] (44)
43
Writing the expectation of independent variables as the product of their
expectation gives
plim
b
? =
1
?
2
z
·
E (d
t
) E
¡
?z
2
t
+ z
t
²
t
¢
+
E (d
t
) E (z
t
) E (1 ?d
t?1
) E (?z
t?1
+ z²
t?1
) + ...
¸
= ??
QED.
The striking result is that the regression coe?cient is heavily down-
ward biased purely because of the non-trading bias. If trading is infrequent,
then the regression coe?cient will be close to zero.
In this setting, researchers will often regress on many lags of z. Suppose
the regression computed incorporates k lags of z.
y
t
= ?
0
z
t
+ ?
1
z
t?1
+ ... + ?
k
z
t?k
+ ²
t
The result is given by Theorem II.
THEOREM II:
Under the assumptions of Theorem I the regression in (11) has a prob-
ability limit
plim
b
? = ? (1 ??)
j?1
?
and the sum of these coe?cients approaches beta as k gets large.
PROOF:
Because z is a Martingale di?erence sequence, the matrix of regressors
approaches a diagonal matrix with the variance of z on the diagonals. Each
row of the z0y matrix has dummy variables d
t
(1 ?d
t?1
) (1 ?d
t?2
) ... (1 ?d
t?j
)
multiplying the square of z. The result follows from independence.
QED.
The regression coe?cients decline from the contemporaneous one
but ultimately summing up to the total impact of asset one on two. The
result is however misleading because it appears that the price movements
in asset two predict future movements in asset one. There appears to be
causality or price discovery between these assets merely because of the ran-
dom trade times.
Similar results can be found in more general contexts including dy-
namic structure in the price observations and dependence with the z’s. Con-
tinuing research will investigate the extent of the dependence and how results
change with the economic structure.
44
3 Conclusion
The introduction of widely available ultra high frequency data sets over
the past decade has spurred interest in empirical market microstructure.
The black box determining equilibrium prices in ?nancial markets has been
opened up. Intraday transaction by transaction dynamics of asset prices,
volume, and spreads are available for analysis. These vast data sets present
new and interesting challenges to econometricians.
Since transactions data are inherently irregularly spaced we view the
process as a marked point process. The arrival times form the points and
the characteristics of the trades form the marks. We ?rst discuss models
for the timing of events when the arrival rate may be time varying. Since
the introduction of the ACD model of Engle and Russell (1998) numerous
other models for the timing of event arrivals have been proposed and applied
to ?nancial data. The models have been applicable to transaction arrival
times or, if some arrival times are thought to be more informative than
others the point process can be ”thinned” to contain only those arrival
times with special information. Examples include volume based durations
which correspond to the time it takes for a speci?ed amount of volume to
be transacted. Another example is price durations which correspond to the
time it takes for the price to move a speci?ed amount. These models can
be thought of as models of volatility where the volatility is intuitively the
inverse of our usual measures of volatility - namely the time it takes for the
price to move a speci?ed amount.
Models for the marks are also discussed. Often the focus is on the
transaction price dynamics or joint modeling of transaction prices and vol-
ume. If the spacing of the data is ignored then the modeling problem can
be reduced to standard econometric modeling procedures of VARs, simple
linear regression, or GARCH models. Models that address the inherent
discreteness in transaction by transaction prices are also discussed.
Alternatively, if the spacing of the data is thought to carry information
then the simple approaches may be mispeci?ed. Choices then include con-
ditioning the marks on the arrival times as in Hausman Lo and Mackinlay,
or, if forecasting is of interest joint modeling of the arrival times. The latter
approach is considered in Engle (2000), Russell and Engle (2002), Rydberg
and Sheppard (1999), or Ghysels (1999) among others.
Finally, while arti?cially descretizing the time intervals at which prices
(or other marks) is a common practice in the literature, it does not come
without cost. Di?erent descretizing schemes trade of bias associated with
45
temporally aggregating with variance. Averaging reduces the variability but
blurs the timing of events. We also show, in a stylized model, that causal
relationships can be arti?cially induced by descretizing the data. Care
should be taken in interpreting results from this type of analysis.
46
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50
Appendix
A EACD(3,3) parameter estimates using EVIEWS
GARCH module.
Coefficient Robust Std. Err.
? 0.004244 0.000855
?
1
0.070261 0.007157
?
2
0.038710 0.012901
?
3
-0.055966 0.008640
?
1
0.835806 0.125428
?
2
0.107894 0.118311
where ?
i
= ? +
3
P
j=1
?
j
x
i?j
+
2
P
j=1
?
j
?
i?j
Model diagnostics
B VAR parameter estimates
Price Equation Trade Equation
Variable Coefficient Std. Error Coefficient Std. Error
c -0.006553 0.000284 0.509648 0.004785
w
i
0.014230 0.000430
w
i-1
0.000891 0.000493 0.298146 0.005557
w
i-2
-0.000175 0.000493 0.059228 0.005797
w
i-3
-0.000533 0.000493 0.036385 0.005803
w
i-4
0.000176 0.000493 0.026645 0.005798
w
i-5
-0.001295 0.000425 0.035205 0.005558
?m
i-1
-0.262310 0.005734 0.250909 0.071635
?m
i-2
-0.121951 0.005934 0.108735 0.081696
?m
i-3
-0.054038 0.005968 -0.000260 0.084009
?m
i-4
-0.026460 0.005934 -0.022889 0.081695
?m
i-5
-0.011011 0.005734 -0.220448 0.071634
51
doc_107925136.pdf
From a passing airplane one can see the rush hour traffic snaking home far below. For some, it is enough to know that the residents will all get home at some point. Alternatively, from a tall building in the center of the city one can observe individuals in transit from work to home.
Analysis of High Frequency Financial Data
Robert F. Engle
New York University and University of California, San Diego
Je?rey R. Russell
University of Chicago, Graduate School of Business
December 21, 2004
Contents
1 Introduction 2
1.1 Data Characteristics . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Irregular Temporal Spacing . . . . . . . . . . . . . . . 3
1.1.2 Discreteness . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.3 Diurnal Patterns . . . . . . . . . . . . . . . . . . . . . 5
1.1.4 Temporal Dependence . . . . . . . . . . . . . . . . . . 7
1.2 Types of economic data . . . . . . . . . . . . . . . . . . . . . 10
1.3 Economic Questions . . . . . . . . . . . . . . . . . . . . . . . 11
2 Econometric Framework 13
2.1 Examples of Point Processes . . . . . . . . . . . . . . . . . . . 16
2.1.1 The ACD model . . . . . . . . . . . . . . . . . . . . . 18
2.1.2 Thinning point processes . . . . . . . . . . . . . . . . 26
2.2 Modeling in Tick Time - the Marks . . . . . . . . . . . . . . . 28
2.2.1 VAR Models for Prices and Trades in Tick Time . . . 28
2.2.2 Volatility Models in Tick Time . . . . . . . . . . . . . 32
2.3 Models for discrete prices . . . . . . . . . . . . . . . . . . . . 34
2.4 Calendar Time Conversion . . . . . . . . . . . . . . . . . . . . 40
2.4.1 Bivariate Relationships . . . . . . . . . . . . . . . . . 42
3 Conclusion 45
A EACD(3,3) parameter estimates using EVIEWS GARCH
module. 51
B VAR parameter estimates 51
1
1 Introduction
From a passing airplane one can see the rush hour tra?c snaking home far
below. For some, it is enough to know that the residents will all get home
at some point. Alternatively, from a tall building in the center of the city
one can observe individuals in transit from work to home. Why one road is
moving more quickly than another can be observed. Roads near the coastal
waters might be immersed in a thick blanket of fog forcing the cars to travel
slowly due to poor visibility while roads in the highlands, above the fog,
move quickly. Tra?c slows as it gets funneled through a narrow pass while
other roads with alternate routes make good time. If a critical bridge is
washed out by rain then some travelers may not make it home at all that
night.
Like the view from the airplane above, classic asset pricing research
assumes only that prices eventually reach their equilibrium value, the route
taken and speed of achieving equilibrium is not speci?ed. How does the
price actually adjust from one level to another? How long will it take?
Will the equilibrium be reached at all? How do market characteristics
such as transparency, the ability of traders to view others actions, or the
presence of several markets trading the same asset a?ect the answers to
these questions? Market microstructure studies the mechanism by which
prices adjust to re?ect new information.
Answers to these questions require studying the details of price adjust-
ment. From the passing plane in the sky, the resolution is insu?cient and
the view from the market ?oor, like the view from the street below the build-
ing, provides a very good description of the actions of some individuals, but
lacks perspective. With high frequency ?nancial data we stand atop the
tall building, poised to empirically address such questions.
1.1 Data Characteristics
With these new data sets come new challenges associated with their analysis.
Modern data sets may contain tens of thousands of transactions or posted
quotes in a single day time stamped to the nearest second. The analysis of
these data are complicated by irregular temproal spacing, diurnal patterns,
price discretness, and complex often very long lived dependence.
2
1.1.1 Irregular Temporal Spacing
Perhaps most important is that virtually all transactions data are inherently
irregularly spaced in time. Figure 1 plots one two hours of transaction
prices for an arbitrary day in March 2001. The stock used is the US stock
Airgas which will be the subject of several examples throughout the paper.
The horizontal axis is the time of day and the vertical axis is the price.
Each diamond denotes a transaction. The irregular spacing of the data
is immediately evident as some transactions appear to occur only seconds
apart while others, for example between 10:30 and 11:00 may be ?ve or ten
minutes apart.
Since most econometric models are speci?ed for ?xed intervals this poses
an immediate complication. A choice must be made regarding the time
intervals over which to analyze the data. If ?xed intervals are chosen then
some sort of interpolation rule must be used when no transaction occurs
exactly at the end of the interval. Alternatively if stochastic intervals are
used then the spacing of the data will likely need to be taken into account.
The irregular spacing of the data becomes even more complex when dealing
with multiple series each with its own transaction rate. Here, interpolation
can introduce spurious correlations due to non-syncronous trading.
1.1.2 Discreteness
All economic data is discrete. When viewed over long time horizons
the variance of the process is usually quite large relative to the magnitude
of the minimum movement. For transaction by transaction data, however,
this is not the case and for many data sets the transaction price changes
take only a handful of values called ticks. Institutional rules restrict prices
to fall on a pre-speci?ed set of values. Price changes must fall on multiples
of the smalles allowable price change called a tick. In a market for an
actively traded stock it is generally not common for the price to move a
large number of ticks from one transaction to another. In open outcry
markets the small price changes are indirectly imposed by discouraging the
specialist from making radical price changes from one transaction to the
next and for other markets, such as the Taiwan stock exchange these price
restrictions are directly imposed in the form of price change limits from one
3
8.6
8.61
8.62
8.63
8.64
8.65
8.66
8.67
8.68
8.69
8.7
10:00 10:30 11:00 11:30 12:00
Time
P
r
i
c
e
Figure 1: Plot of a small sample of transaction prices for the Airgas stock
transaction to the next (say 2 ticks). The result is that price chages often
fall on a very small number of possible outcomes.
US stocks have recently undergone a transition from trading in 1/8ths of
a dollar to decimalization. This transition was initially tested for 7 NYSE
stocks in August of 2000 and was completed for the NYSE listed stocks
on January 29th, 2001. NASDAQ began testing with 14 stocks on March
12, 2001 and completed the transition April 9, 2001. In June of 97 NYSE
permitted 1/16th prices.
As an example, Figure 2 presents a histogram of the Airgas data trans-
action price changes after deleting the overnight and opening transactions.
The sample used here contains 10 months of data spanning from March 1,
2001 through December 31, 2001. The horizontal axis is measured in cents.
52% of the transaction prices are unchanged from the previous price. Over
70% of the transaction prices fall on one of three values; no change, up one
cent or down one cent. Over 90% of the values lie between -5 and +5 cents.
Since the bid and ask prices are also restricted to the same minimum adjust-
ment the bid, ask, and the midpoint of the bid ask prices will exhibit similar
4
0
10
20
30
40
50
60
Cents
P
e
r
c
e
n
t
5 10 0 -5 -10 15 -15
Figure 2: Histogram of price transaction changes for Airgas stock
discreteness. Of course data prior to decimalization is even more extreme.
For these data sets it is not uncommon to ?nd over 98% of the data taking
just one of 5 values. This discreteness will have an impact on measuring
volatility, dependence, or any characteristic of prices that is small relative
to the tick size.
This discreteness also induces a high degree of kurtosis in the data. For
example, for the Airgas data the sample kurtosis is 66. Such large kurtosis
is typical of high frequency data.
1.1.3 Diurnal Patterns
Intraday ?nancial data typically contain very strong diurnal or periodic
patterns. For most stock markets volatility, the frequency of trades, volume,
and spreads all typically exhibit a U-shaped pattern over the course of the
day. For an ealry reference see McInish and Wood (1992). Volatility is
systematically higher near the open and generally just prior to the close.
Volume and spreads have a similar pattern. The time between trades, or
durations, tend to be shortest near the open and just prior to the close.
This was ?rst documented in Engle and Russell (1998).
5
0
20
40
60
80
100
120
140
160
180
10:00 11:00 12:00 13:00 14:00 15:00 16:00
Time of Day
C
e
n
t
s
/
S
e
c
o
n
d
s
Duration
Std Dev.
Figure 3: Diurnal pattern for durations and standard deviation of mid-quote
price changes.
Figure 3 presents the diurnal patterns estimated for the ARG data. The
diurnal patterns were estimated by ?tting a piecewise linear spline to the
duration between trades and the squared midquote price changes. The ver-
tical axis is measured in seconds for the duration and cents for the standard
deviation of price changes.
Diurnal patterns are also typically present in the foreign exchange mar-
ket although here there is no opening and closing of the market. These
markets operate 24 hours a day seven days a week. Here the pattern is typ-
ically driven by ”active” periods of the day. See Andersen and Bollerslev
(1997) for patterns in foreign exchange volaitlity. For example, prior to the
induction of the Euro, US dollar exchange rates with European countries
typically exhibit the highest volatility during the overlap of time that both
the US markets and the European markets were active. This occurred in
the late afternoon GMT when it is morning in the US and late afternoon in
Europe.
6
-0.45
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Lag in transaction time
A
C
FMid quote
Trans. Price
Figure 4: Autocorrelation function for the mid quote and transaction price
changes
1.1.4 Temporal Dependence
Unlike their lower frequency counterparts, high frequency ?nancial returns
data typically display strong dependence. The dependence is largely the
result of price discreteness and the fact that there is often a spread between
the price paid by buyer and seller initiated trades. This is typically referred
to as bid-ask bounce and is responsible for the large ?rst order negative
autocorrelation. Bid ask bounce will be discussed in more detail in section
2.3. Other factors leading to dependence in price changes include traders
breaking large orders up into a sequence of smaller orders in hopes of trans-
acting at a better price overall. These sequences of buys or sells can lead to
a sequence of transactions that move the price in the same direction. Hence
at longer horizons we sometimes ?nd positive autocorrelations. Figure 4
contains a plot of the autocorrelation function for changes in the transaction
and midpoint prices from one transaction to the next for the Airgass stock
using the 10 months of data. Again, overnight price changes have been
deleted.
7
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
1 12 23 34 45 56 67 78 89 100 111 122 133 144 155 166 177 188 199
Figure 5: Autocorrelations for the squared midquote price changes
Similar to lower frequency returns, high frequency data tends to exhibit
volatility clustering. Large price changes tend to follow large price changes
and vice-versa. The ACF for the absolute value of the transaction price
change for ARG is shown in ?gure 5. Since the diurnal pattern will likely
in?uence the autocorrelation function it is ?rst removed by dividing the
price change by the square root of its variance by time of day. The variance
by time of day was estimated with linear splines. The usual long set of
positive autocorrelations is present.
The transaction rates also exhibit strong temporal dependence. Figure
6 presents a plot of the autocorrelations for the durations between trades
after removing the deterministic component discussed above. Figure 7
presents the autocorrelations for the log of volume. Both series exhibit
long sets of positive autocorrelation spanning many transactions. These
autocorrelations indicate clustering of durations and volume respectively.
Under temporal aggregation the dependence in the price changes tends
to decrease. However, even at intervals of a half hour or longer negative
?rst order autocorrelation often remains.
1.2 Types of economic data
8
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
1 12 23 34 45 56 67 78 89 100 111 122 133 144 155 166 177 188 199
Figure 6: Autocorrelations for durations
-0.05
0
0.05
0.1
0.15
0.2
1 12 23 34 45 56 67 78 89 100 111 122 133 144 155 166 177 188 199
Figure 7: Autocorrelations for the log of volume
9
We begin this section with a general discussion of the types of high fre-
quency data currently available. With the advancement and integration
of computers in ?nancial markets data sets containing detailed information
about market transactions are now commonplace. Many ?nancial assets
are traded in centralized markets which potentially contain the most de-
tailed information. The NYSE and the Paris Bourse are two such markets
commonly analyzed. These centralized markets might be order driven (like
the Paris Bourse) where a computer uses algorithms to match market par-
ticipants or they might be open outcry (like the NYSE) where there is a
centralized trading ?oor that market participants must trade through. In
either case, the data sets constructed from these markets typically contain
detailed information about the transactions and quotes for each asset traded
on the market. The exact time of a transaction usually down to the second,
and the price and quantity transacted are common data. Similarly, the
bid and ask quotes are also generally available along with a time stamp for
when the quotes became active. The trades and quotes (TAQ) data set
distributed by the NYSE is an example of such a data set.
The currency exchange market is a commonly analyzed decentralized
market. Here the market participants are banks that communicate and
arrange transactions on a one on one basis with no central recording institu-
tion. Quotes are typically fed through reuters for customers with a reuters
account to view continuous updates of quotes. Olsen and Associates has
been downloading and storing this quote data and made it available for aca-
demic research. The most comprehensive data sets contain all quotes that
pass through the Reuters screens and an associated time stamp. As the
transactions do not pass through a centralized system there is no compre-
hensive source for transaction data.
The de?nition of the quote can vary across markets with important im-
plications. Foreign exchange quotes are not binding. Quotes from the
NYSE are valid for a ?xed (typically small) quantity or depth. Quotes in
electronic markets can come in various forms. For example, quotes for the
Paris bourse are derived from the limit order book and represent the best ask
and bid prices in the book. The depth here is determined by the quantity of
volume at the best prices. Alternatively, the Taiwan stock exchange, which
is an electronic batch auction market, posts a ”reference price” derived from
the past transaction and is only a benchmark from which to gauge where
the next transaction price may fall. These di?erence are important not
only from an economic perspective but also determine the reliability of the
data. Non-binding quotes are much more likely to contain large errors than
binding ones.
10
Many empirical studies have focused on the stock and currency exchange
high frequency data. However, data also exists for other markets, most
notably the options and futures markets. These data sets treat each contract
as a separate asset reporting time quotes and transactions just as for the
stocks. The Berkeley Options data base is a common source for options
data.
Other specialized data sets are available that contain much more detailed
information. Perhaps the most well known data set is the TORQ data set
put together by Joel Hasbrouck and the NYSE. This data set is not very
comprehensive in that it contains only 144 stocks traded on US markets
covering three months in the early 1990’s. However, it contains detailed
information regarding the nature of the transactions including the order type
(limit order, market order etc.) as well as detailed information about the
submission of orders. The limit order information provides a widow into the
limit order book of the specialist although it cannot be exactly replicated.
The Paris Bourse data typically contain detailed information about the limit
order book near the current price.
1.3 Economic Questions
Market microstructure economics focuses on how prices adjust to new in-
formation and how the trading mechanism a?ects asset prices. In a perfect
world, new information would be immediately disseminated and interpreted
by all market participants. In this full information setting prices would
immediately adjust to a new equilibrium value determined by the agents
preferences and the content of the information. This setting, however, is
not likely to hold in practice. Not all relevant information is known by
all market participants at the same time. Furthermore, information that
becomes available is not processed at the same speed by all market partic-
ipants implying variable lag time between a news announcement and the
agents realization of price implications. Much of modern microstructure
theory is therefore driven by models of asymmetric information.
In the simplest form, there is a subset of the agents that are endowed
with superior knowledge regarding the value of an asset. These agents
are referred to as privately informed or simply informed agents. Agents
without superior information are referred to as noise or liquidity traders
11
and are assumed to be indistinguishable from the informed agents. Ques-
tions regarding the means by which the asset price transitions to re?ect the
information of the privately informed agents can be couched in this con-
text. Early theoretical papers utilizing this framework include Glosten and
Milgrom (1985), Easley and O’Hara (1992), Copland and Galai (1983) and
Kyle (1985). A very comprehensive review of this literature can be found
in O’Hara (1995).
The premise of these models is that market makers optimally update bid
and ask prices to re?ect all public information and remaining uncertainty.
For the NYSE it is the specialist that plays the role of market maker. Even
in markets without a designated specialist bid and ask quotes are generally
inferred either explicitly or implicitly from the buy and sell limit orders
closest to the current price. Informed and uniformed traders are assumed
to be indistinguishable when arriving to trade so the di?erence between the
bid and the ask prices can be viewed as compensation for the risk associated
with trading against potentially better informed agents. Informed traders
will make pro?table transactions at the expense of the uninformed.
In a rational expectations setting market makers learn about private
information by observing the actions of traders. Informed traders only
transact when they have private information and would like to trade larger
quantities to capitalize on their information before it becomes public. The
practical implications is that characteristics of transactions carry informa-
tion. An overview of the predictions of these models is that prices adjust
more quickly to re?ect private information when the proportion of unin-
formed traders is higher. Volume is higher, transaction rates are higher
when the proportion of uninformed traders is higher. The bid ask spread
is therefore predicted to be increasing in volume and transaction rates.
It is unlikely that aggregated daily data will be very useful in empiri-
cally evaluating market microstructure e?ects on asset prices. Using high
frequency data we seek to empirically asses the e?ects of market microstruc-
ture on asset price dynamics. Although private information is inherently
unobservable the impact of transaction characteristics on subsequent price
revisions can be measured and quanti?ed. We can asses which trades are
likely to move the price or what characteristics of the market should we ex-
pect to see large rapid price changes. A related and very practical issue is
liquidity loosely de?ned as the ability to trade large volume with little or no
price impact. How much will it cost to transact and how is this cost related
to transaction characteristics? An empirical understanding of the answers
to these questions might lead to optimal order submission strategies.
12
Answers to these questions necessarily involve studying the relationship
between transaction prices and market characteristics. The bid ask spread
is often considered a measure of liquidity since it measures the cost of pur-
chasing and then immediately reselling. The wider the spread the higher the
cost of transacting. Of course spreads vary both cross sectionally across dif-
ferent stocks as well as temporally. Using high frequency data the variation
in the spread can be linked to the variation in market characteristics.
The price impact of a trade can be measured by relating quote revisions
to characteristics of a trade. Again with high frequency transactions data
the magnitude and possibly the direction of price adjustments can be linked
to characteristics of the market. Volatility models for high frequency data
are clearly relevant here.
A privately informed agent may have the ability to exploit that informa-
tion in multiple markets. In some cases a single asset is traded in multiple
markets. A natural question is in which market does price discovery take
place? Do price movements in one market precede price movements in
the other market. If the two series are cointegrated these questions can
be addressed in the context of error correction models. Another possibly
for informed agents to have multiple outlets to exploit their information is
the derivative market. From an empirical perspective this can be reduced
to causality questions. Do price movements in one market precede price
movements in the other? Do trades in one market tend to have a greater
price impact across both markets?
Implementing the econometric analysis, however, is complicated by the
data features discussed in the previous section. This chapter provides a
review of the techniques and issues encountered in the analysis of high fre-
quency data. The chapter is organized as follows.
2 Econometric Framework
One of the most salient features of high frequency transactions data is that
transactions do not occur at regularly spaced time intervals. In the statis-
tics literature this type of process has been referred to as a point process
and a large body of literature has been produced studying and applying
models of point processes. Examples of applications include the study of
?ring of neurons or the study of earthquake occurrences. More formally, let
t
1
, t
2
, ..., t
i
, ... denote a sequence of strictly increasing random variables cor-
responding to event arrival times such as transactions. Jointly, these arrival
times are referred to as a point process. It is convenient to introduce the
13
counting function N(t) which is simply the number of event arrivals that
have occurred at or prior to time t. This will be a step function with unit
increments at each arrival time.
Often, there will be additional information associated with the arrival
times. In the study of earthquake occurrences there might be additional
information about the magnitude of the earthquake associated with each
arrival time. Similarly, for the ?nancial transactions data there is often a
plethora of information associated with the transaction arrival times includ-
ing price, volume, bid and ask quotes, depth, and more. If there is additional
information associated with the arrival times then the process is referred to
as a marked point process. Hence, if the marks associated with the i
th
ar-
rival time are denoted by an M-dimensional vector y
i
then the information
associated with the i
th
event is summarized by its arrival time and the value
of the marks [t
i
, y
i
].
Depending on the economic question at hand, either the arrival time,
the marks, or both may be of interest. Often economic hypothesis can be
couched in the framework of conditional expectations of future values. We
denote the ?ltration of arrival times and marks at the time of the i
th
event
arrival by
b
t
i
= {t
i
, t
i?1
, ..., t
0
} and b y
i
= {y
i
, y
i?1
, ..., y
0
} respectively. The
probability structure for the dynamics associated with a stationary, marked
point process is can be completely characterized and conveniently expressed
as the joint distribution of marks and arrival times given the ?ltration of
past arrival times and marks:
f
¡
t
N(t)+1
, y
N(t)+1
|
b
t
N(t)
, b y
N(t)
¢
(1)
While this distribution provides a complete description of the dynamics of
a marked point process it is rarely speci?ed in practice. Often the question
of economic interest can be expressed in one of four ways. When will the
next event happen? What value should we expect for the mark at the next
arrival time? What value should we expect for the mark after a ?xed time
interval? Or, how long should we expect to wait for a particular type of
event to occur?
The answers to the ?rst two questions are immediately obtained from (1).
Alternatively, if the contemporaneous relationship between y
i
and t
i
is not
of interest, then the analysis may be greatly simpli?ed by restricting focus
to the marginalized distributions provided that the marks and arrival times
are weakly exogneous. If the waiting time until the next event regardless of
the value of the marks at termination is of interest, then the marginalized
distribution given by
14
f
t
¡
t
N(t)+1
|
b
t
N(t)
, b y
N(t)
¢
=
Z
f
¡
t
N(t)+1
, y|
b
t
N(t)
, b y
N(t)
¢
dy (2)
may be analyze. This is simply a point process where the arrival times may
depend on the past arrival times and the past marks. We will refer to this
as a model for the event arrival times, or simply a point process. Examples
here include models for the arrival of traders.
Alternatively, many economic hypotheses involve the dynamics of the
marks such as models for the spread, or prices. In this case, one may be
interested in modeling or forecasting the value for the next mark, regardless
of when it occurs, given the ?ltration of the joint process. This is given by
f
y
¡
y
N(t)+1
|
b
t
N(t)
, b y
N(t)
¢
=
Z
f
¡
t, y
N(t)+1
|
b
t
N(t)
, b y
N(t)
¢
dt (3)
Here, the information set is updated at each event arrival time and we refer
to such models as event time or tick time models of the marks. Of course,
multiple step forecasts from 2 would require, in general, a model for the
marks and multiple step forecasts for the mark in 3 would generally require
a model for the durations.
Yet another alternative approach is to model the value of the mark to
be at some future time t + ? (? > 0) given the ?ltration at time t. That is
g
¡
y
N(t+?)
|
b
t
N(t)
, b y
N(t)
¢
(4)
Here the conditional distribution associated with the mark over a ?xed time
interval is the object of interest. Theoretically, speci?cation of (1) implies
a distribution for (4), only in very special cases,however, will this exist in
closed form. Since the distribution of the mark is speci?ed over discrete
?xed calendar time intervals we refer to this type of analysis as ?xed interval
analysis. A ?nal approach taken in the literature is to study the distribution
of the length of time it will take for a particular type of event, de?ned by
the mark, to occur. For example, one might want to know how long will it
take for the price to move by more than a speci?ed amount, or how long will
it take for a set amount of volume to be transacted. This can be expressed
as
g
¡
t + ?
min
|
b
t
N(t)
, b y
N(t)
¢
(5)
where if E
t
de?nes some event associated with the marks, then ?
min
=
min
?>0
y
N(t+?)
? E
t
. Again, only in special cases can this distribution be de-
rived analytically from (1). t
min
is called the hitting time in the stochastic
15
process literature. Since the marks are associated with arrival times, the
?rst crossing times will simply be a subset of the original set of arrival times.
In the point process literature the subset of points is called a thinned point
process.
This section proceeds to discuss each of the above approaches. We begin
with a discussion and examples of point processes. Next, we consider tick
time models. We then consider ?xed interval analysis by ?rst discussing
methods of converting to ?xed time intervals and then give examples of
various approaches used in the literature.
2.1 Examples of Point Processes
It is convenient to begin this section with a discussion of point processes
with no marks. A point process is referred to as a simple point process if,
as a time interval goes to zero, the probability of multiple events occurring
over that time interval can be made an arbitrarily small fraction of the
probability of a single event occurring. In this case, characterization of the
instantaneous probability of a single event dictates the global behavior of
the process. A convenient way of characterizing a simple point process,
therefore is by the instantaneous arrival rate of the intensity function given
by:
?(t) = lim
?t?0
Pr (N(t +?t > N (t))
?t
(6)
Perhaps the most well known simple point process is the homogenous Pois-
son process. For a homogeneous Poisson process the probability of an
event arrival is constant. A homogenous Poisson process can therefore be
described by a single parameter where ?(t) =?. For many types of point
process the assumption of a constant arrival rate is not likely realistic. In-
deed, for ?nancial data we tend to observe bursts of trading activity followed
by lulls. This feature becomes apparent when looking at the series of the
time between transactions, or durations. Figure 6 presents the autocorrela-
tions associated with the intertrade durations of Airgas. The plot indicates
strong temporal dependence in the durations between transaction events.
Clearly the homogenous Poisson model is not suitable for such data.
For a point process with no marks, Snyder and Miller (1991) conve-
niently classify point processes into two categories, those that evolve with
after-e?ects and those that do not. A point process on [t
0
, ?] is said to
16
evolve without after e?ects if for any t > t
0
the realization of events on
[t, ?) does not depend on the sequence of events in the interval [t
0
, t). A
point process is said to be conditionally orderly at time t ? t
0
if for a suf-
?ciently short interval of time and conditional on any event P de?ned by
the realization of the process on [t
0
, t) the probability of two or more events
occurring is in?nitessimal relative to the probability of one event occurring.
Our discussion here focuses on point processes that evolve with after-e?ects
and are conditionally orderly A point process that evolves with after-e?ects
can be conveniently described using the conditional intensity function which
speci?es the instantaneous probability of an event arrival conditional upon
?ltration of event arrival times. That is, the conditional intensity is given
by
?(t|N(t), t
i?1
, t
i?2
, ..., t
0
) = lim
?t?0
P
¡
N(t +?t) > N(t)|N(t), t
N(t)
, t
N(t)?1
, ..., t
0
¢
?t
(7)
The conditional intensity function associated with any single waiting time
has traditionally been called a hazard function in the econometrics literature.
Here, however, the intensity function is de?ned as a function of t across
multiple events, unlike much of the literature in macroeconomics that tends
to focus on large cross sections with a single spell.
Perhaps the simplest example of a point process that evolves with afteref-
fects is a ?rst order homogeneous point process where ?
¡
t|N(t), t
N(t)?1
, t
N(t)?2
, ..., t
0
¢
=
?
¡
t|N(t), t
N(t)
¢
and the durations between events x
i
= t
i
? t
i?1
form a se-
quence of independent random variables. If in addition, the durations are
identically distributed then the process is referred to as a renewal process.
More generally, for an m
th
order self exciting point process the conditional
intensity depends on N(t) and the m most recent event arrivals.
As discussed in Snyder and Miller (1975), for example, the conditional
intensity function, the conditional survivor function, and the durations or
”waiting times” between events each completely describe a conditionally
orderly point process. Letting p
i
be a family of conditional probability
density functions for arrival time t
i
, the log likelihood can be expressed in
17
terms of the conditional density or intensity as
L =
N(T)
X
i=1
log p
i
(t
i
|t
0
, ...., t
i?1
) (8)
L =
N(T)
X
i=1
log ?(t
i
|t
0
, ...., t
i?1
) ?
T
Z
t
0
?(u|N(t), t
0
, ...., t
i?1
) (9)
Equation (??) is referred to as a self exciting point process. It was orig-
inally proposed by Hawkes (1971) and by Rubin (1972) and are sometimes
called Hawkes self exciting processes. Numerous parameterizations have
been proposed in the statistics literature.
2.1.1 The ACD model
Engle and Russell [?] propose the Autoregressive Conditional Duration (ACD)
which is particularly well suited for high frequency ?nancial data. This pa-
rameterization is most easily expressed in terms of the waiting times between
events. Let x
i
= t
i
?t
i?1
be the interval of time between event arrivals which
will be called the duration. The distribution of the duration is speci?ed di-
rectly conditional on the past durations. The ACD model is then speci?ed
by two conditions. Let ?
i
be the expectation of the duration given the past
arrival times which is given by
E (x
i
|x
i?1
, x
i?2
, ..., x
1
) = ?
i
(x
i?1
, x
i?2
, ..., x
1
; ) = ?
i
(10)
Furthermore, let
x
i
= ?
i
?
i
(11)
where ?
i
˜i.i.d. with density p(²; ?) with non-negative support, and ? and ?
are variation free. The baseline intensity, or baseline hazard, is given by
?
0
=
p(²; ?)
S(²; ?)
(12)
where S
0
(²; ?) =
?
R
²
p(u; ?)du is the survivor function. The intensity function
for an ACD model is then given by
?(t|N(t), t
i?1
, t
i?2
, ..., t
0
) = ?
0
Ã
t ?t
N(t)?1
?
N(t)
!
1
?
N(t)
(13)
18
Since ?
i
enters the baseline hazard this type of model is referred to as
an accelerated failure time model in the duration literature. The rate at
which time progresses through the hazard function is dependent upon ?
i
and
therefore can be viewed in the context of time deformation models. During
some periods the pace of the market is more rapid than other periods.
The ?exibility of the ACD model stems from the variety of choices for
parameterizations of the conditional mean in (10) and the i.i.d. density
p(²; ?). Engle and Russell (1998) suggest and apply linear parameterizations
for the expectation given by
?
i
= ? +
p
X
j=1
?
j
x
i?j
+
q
X
j=1
?
j
?
i?j
(14)
Since the conditional expectation of the duration depends on p lags of the
duration and q lags of the expected duration this is termed an ACD(p, q)
model. Popular choices for the density p(²; ?) include the exponential and
the Weibull distributions suggested in Engle and Russell (1998). These
models are termed the Exponential ACD (EACD) and Weibull ACD (WACD)
models respectively. The exponential distribution has the property that
the baseline hazard is monotonic. The Weibull distribution relaxes this as-
sumption and allows for a hump-shaped baseline intensity. An appropriate
choice of the distribution, and hence the baseline intensity will depend on
the characteristics of the data at hand. Other choices include the Gamma
distribution Lunde (1998) and Zhang Russell and Tsay (2001) or the Burr
distribution suggested in Grammig and Maurer (2000). These distributions
allow for even greater ?exibility in the baseline hazard. Given a choice for
(10) and p(²; ?) the likelihood function is constructed from (8).
For each choice of p(²; ?) from (12) and (13) there is an implied intensity
function. Since the exponential distribution implies a constant hazard, the
intensity function takes a particularly simple form given by
?(t|N(t), t
i?1
, t
i?2
, ..., t
0
) =
1
?
N(t)
(15)
and for the Weibull distribution the intensity is slightly more complicated
?(t|N(t), t
i?1
, t
i?2
, ..., t
0
) = ?
_
_
?
³
1 +
1
?
´
?
N(t)
_
_
?
¡
t ?t
N(t)
¢
??1
(16)
which reduces to (15) when ? = 1.
19
The ACD(p, q) speci?cation in (14) appears very similar to a ARCH(p, q)
models of Engle (1982) and Bollerslev (1986) and indeed the two models
share many of the same properties. From (11) and (14) it follows that the
durations x
i
follow an ARMA(max(p, q), q). Let ?
i
? x
i
? ?
i
which is a
martingale di?erence by construction then
x
i
= ? +
max(p,q)
X
j=1
¡
?
j
+ ?
j
¢
x
i?j
?
q
X
j=1
?
j
?
i?j
+ ?
i
If ?(L) and ?(L) denote polynomials in the lag operator of orders p and q
respectively then the persistence of the model can be measured by ?(1) +
? (1) . For most duration data this sum is very close to (but less than) one
indicating strong persistence but stationarity. It also becomes clear from
this representation that restrictions must be placed on parameter values to
ensure non-negative durations. These restrictions impose that the in?nite
AR representation implied by inverting the MA component must contain
non-negative coe?cients for all lags. These conditions are identical to the
conditions derived in Nelson and Cao (1992) to ensure non-negativity of
GARCH models. For example, for the ACD(1,1) model this reduces to
? ? 0, ? ? 0, ? ? 0. Similarly,
The most basic application of the ACD model to ?nancial transactions
data is to model the arrival times of trades. In this case it denotes the
arrival of the i
th
transaction and x
i
denotes the time between the i
th
and
(i-1)
th
transactions. Engle and Russell (1998) propose using an ACD(2,2)
model with Weibull errors to model the arrival times of IBM transactions.
Like volatility, the arrival rate of transactions on the NYSE can have a
strong diurnal (intraday) pattern. Volatility tends to be relatively high just
after the open and just prior to the close; that is, they have volatility for
stocks tends to exhibit a U shaped diurnal pattern. Similarly, Engle and
Russell (1998) document that the durations between trades have a diurnal
pattern with high activity just after the open and just prior to the close;
that is, the durations exhibit an inverse U shaped diurnal pattern. Let
?
N(t)+1
= E
¡
x
N(t)+1
|t
N(t)
¢
denote the expectation of the duration given
time of day alone. Engle and Russell (1998) suggest including an additional
term on the right hand side of (11) to account for a diurnal pattern so that
the i
th
duration is given by:
x
i
= ?
i
?
i
?
i
(17)
Now, ?
i
is the expectation of the duration after partialing out the determin-
istic pattern and is interpreted as the fraction above or below the average
20
value for that time of day. The expected (non-standardized) duration is
now given by ?
i
?
i
. It is natural to refer to ?
i
as the deterministic compo-
nent and ?
i
as the stochastic component. Engle and Russell (1998) suggest
using cubic splines to model the deterministic pattern.
The parameters of the two components as well as any parameters as-
sociated with ?
i
can be estimated jointly by maximizing (8) or, a two step
procedure can be implemented in which ?rst the terms of the deterministic
pattern are estimated and in a second stage the remaining parameters are
estimated. The two step procedure can be implemented by ?rst running an
OLS regression of durations on a cubic spline. Let
b
?
i
denote the prediction
for the i
th
duration obtained from the OLS regression. Then let e x
i
=
x
i
?
i
denote the normalized duration. This standardized series should be free of
any diurnal pattern and should have a mean near unity. An ACD model
can then be estimated by MLE using the normalized durations e x
i
in place
of x
i
in (8). While this is not e?cient the two step procedure will provide
consistent estimates under correct speci?cation. (3) plots the estimated
diurnal pattern for ARG. This plot was constructed by regressing the du-
ration on a linear spline for the time of day at the start of the duration.
We ?nd the typical inverted U shaped pattern with durations longest in the
middle of the day and shortest near the open and close.
The similarity between the ACD model . Indeed the link is close as
detailed in the following corollary proven in Engle and Russell (1998).
Corollary 1 QMLE results for the EACD(1,1) model
If
1. E
i?1
(x
i
) = ?
i
= ? + ?x
i?1
+ ? ?
i?1
,
2. ²
i
=
x
i
?
i
is
(a) strictly stationary
(b) nondegenerate
(c) has bounded conditional second moments
(d) sup
i
E
i?1
[ln(? + ?²
i
)] < 0
3. ?
0
? (?, ?, ?) is in the interior of ?
21
4. L(?) = ?
N(T)
P
i=1
³
log (?
i
) +
x
i
?
i
´
Then the maximizer of L will be consistent and asymptotically normal
with a covariance matrix given by the familiar robust standard errors from
Lee-Hansen (1994).
This result is a direct corollary from the Lee and Hansen (1994 and
Lumsdaine (1996) proofs for the class of GARCH(1,1) models. The theorem
is powerful since under the conditions of the theorem we can estimate an
ACD model assuming an exponential distribution and even if the assumption
is false we still obtain consistent estimates although the standard errors need
to be adjusted as in White (1982). Furthermore, the corollary establishes
that we can use standard GARCH software to perform QML estimation of
ACD models. This is accomplished by setting the dependent variable equal
to the square root of the duration and imposing a conditional mean equation
of zero. The resulting parameter values provide consistent estimates of the
parameters used to forecast the expected duration.
Additionally, an estimate of the conditional distribution can be ob-
tained non-parametrically by considering the residuals b²
i
=
x
i
b
?
i
where
b
?
i
=
E
i?1
³
x
i
|
b
?
´
. Under correct speci?cation the standardized durations b²
i
should
be i.i.d. and the distribution can be estimated using non-parametric meth-
ods such as kernal smoothing. Alternatively, it is often more informative to
consider the baseline hazard. Given an estimate of the density the baseline
hazard is obtained from (12). Engle and Russell (1998) therefore propose a
semiparametric estimation procedure where in the ?rst step QMLE is per-
formed using the exponential distribution and in a second state the density
of ² is estimated nonparametrically. This is referred to as a semiparametric
ACD model.
ACD Model Diagnostics The properties of the standardized duration
also provide a means to asses the goodness of ?t of the estimated model.
For example, the correlation structure, or other types of dependence can be
tested. Engle and Russell (1998) suggest simply examining the Ljung-Box
statistic although other types of nonlinear dependence can be examined.
Engle and Russell (1998) suggest examining autocorrelations associated
with nonlinear transformations of the residuals b²
i
, for example, squares or
square roots. An alternative test of nonlinearity advocated in Engle and
Russell (1998) is to divide the diurnally adjusted durations into bins. Then
regress b²
i
on a constant and indicators for the magnitude of the previous
22
duration. One indicator must be omitted to avoid perfect multicollinearity.
If the b²
i
are indeed i.i.d. then there should be no predictability implied
from this regression. Often these tests suggests that the linear speci?cation
tends to over predict the duration following extremely short or extremely
long durations. This suggests that a model where the expectation is more
sensitive following short durations and less sensitive following long durations
may work well.
Engle and Russell (1998) suggest examining autocorrelations associated
with nonlinear transformations of the residuals b²
i
, for example, squares or
square roots. An alternative test of nonlinearity advocated in Engle and
Russell (1998) is to divide the diurnally adjusted durations into bins. Then
regress (b²
i
? 1) on indicators for the magnitude of the previous duration.
If the b²
i
are indeed i.i.d. then there should be no predictability implied
from this regression. Often these tests suggests that the linear speci?cation
tends to over predict the duration following extremely short or extremely
long durations. This suggests that a model where the expectation is more
sensitive following short durations and less sensitive following long durations
may work well.
Tests of the distributional assumptions of ² can also be examined. A
general test is based on the fact that the integrated intensity over the dura-
tion
u
i
=
t
i
Z
s=t
i?1
?(s|N(s), t
i?1
, t
i?2
, ..., t
0
)ds (18)
will be distributed as a unit exponential as discussed in Russell (1999).
Often this takes a very simple form. For example, substituting the expo-
nential intensity (15) into (18) simply yields the residual b u
i
= b²
i
=
x
i
b
?
i
b
?
i
.
Similarly, for the substituting the Weibull intensity (16) into (15) yields
b u
i
=
Ã
?
³
1+
1
?
´
x
i
b
?
i
b
?
i
!
?
. The variance of u
i
should be unity leading Engle and
Russell (1998) to suggest the test statistic
p
N(T)
(b ?
u
?1)
?
8
(19)
which should have a limiting standard normal distribution. This is a formal
test for remaining excess dispersion often observed in duration data. Fur-
thermore, since the survivor function for an exponential random variable
U is simply exp(?u) a plot of the the negative of the log of the empirical
23
survivor function should be linearly related to u
i
with a slope of unity hence
providing a graphical measure of ?t.
Nonlinear ACD models The tests for nonlinearity discussed above often
suggest nonlinearity. Zhang Russell and Tsay (2000) propose a nonlinear
threshold ACD model with this feature in mind. Here the dynamics of the
conditional mean are given by
?
i
=
_
_
_
?
1
+ ?
1
x
i?1
+ ?
1
?
i?1
if x
i?1
? a
1
?
2
+ ?
2
x
i?1
+ ?
2
?
i?1
if a
1
< x
i?1
? a
2
?
3
+ ?
3
x
i?1
+ ?
3
?
i?1
if a
2
< x
i?1
where a
1
and a
2
are parameters to be estimated. Hence the dynamics of
the expected duration depend on the magnitude of the previous duration.
Indeed, using the same IBM data as analyzed in Engle and Russell (1998)
they ?nd ?
1
> ?
2
> ?
3
as expected from the nonlinear test results. Esti-
mation is performed using a combination of a grid search across a
1
and a
2
and maximum likelihood for all pairs of a
1
and a
2
. The MLE is the pair
of a
1
and a
2
and the corresponding parameters of the ACD model for each
regime that produces the highest maximized likelihood.
Another nonlinear ACD model applied in Engle and Lunde (1999), Rus-
sell and Engle (2002) is the Nelson Form ACD model. The properties of
the Nelson form ACD are developed in Bauwens and Giot (2000). We refer
to this as the Nelson form ACD model because it is in the spirit of Nelson
(1991) EGARCH model and we want to minimize confusion with the version
of the ACD model that uses the exponential distribution for ?. Here the
log of the expected duration follows a linear speci?cation.
ln(?
i
) = ? +
p
X
j=1
?
j
²
i?j
+
q
X
j=1
?
j
ln(?
i?j
)
This formulation is particularly convenient when other market variables are
included in the ACD model since non-negativity of the expected duration is
directly imposed.
An interesting approach to nonlinearity is taken in Fernandes and Gram-
mig (2002) who propose a class of nonlinear ACD models. The parametriza-
tion is constructed using the Box Cox transformation of the expected du-
ration and a ?exible non-linear function of ²
i?j
that allows the expected
duration to respond in a distinct manner to small and large shocks. The
model nests many of the common ACD models and is shown to work well
for a variety of duration data sets.
24
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
1 10 19 28 37 46 55 64 73 82 91 100 109 118 127 136 145 154 163 172 181 190 199
Figure 8: Autocorrelations for ACD duration residuals b²
i
=
x
i
b
?
i
ACD example Appendix (A) contains parameter estimates for an EACD(3,2)
model estimated using the GARCH module of EVIEWS. The durations
were ?rst adjusted by dividing by the time of day e?ect estimated by linear
splines in ?gure 3. The sum of the ? and ? coe?cients is in excess of .999
but less than one indicating strong persistence but the impact of shocks dies
o? after a su?cient period of time. A plot of the autocorrelations of the
residuals is presented in ?gure 8. The autocorrelations are no longer all
positive and appear insigni?cant. A formal test for the null that the ?rst
15 auto correlations are zero yields a 12.15 with a p-value of 67%.
A test for remaining excess dispersion in (19) yields a test statistic of
?
32365(1.41?1)
?
8
= 26.07. There is evidence of excess dispersion indicating
that it is unlikely that ? is exponential. However, under the conditions
of the corollary the parameter estimates can be viewed as QML estimates.
A plot of the nonparametric hazard is given in ?gure 9. The estimate
was obtained using a nearest neighbor estimator. The hazard is nearly
monotonically decreasing indicating that the longer it has been since the
last transaction the less likely it is for a transaction to occur in the next
instant. This is a common ?nding for transactions data.
25
0
0.5
1
1.5
2
2.5
3
3.5
4
0 1 2 3 4 5 6 7 8 9
Figure 9: Non-parametric estimate of baseline intensity.
2.1.2 Thinning point processes
The above discussion focuses on modeling the arrival of transaction events.
For some economic questions it may not be necessary to examine the arrival
of all transactions. Rather, we might want to focus on a subset of all trans-
actions which have special meaning. For example, when foreign exchange
data are examined many of the posted quotes appear to be simply noisy
repeats of the previous posted quotes. Alternatively, for stock data, often
times many transactions are recorded with the exact same transaction price.
Another example might be the time until a set amount of volume has been
transacted - a measure related to liquidity.
Depending on the question at hand, we might be able to focus on the
subset of event arrival times for which the marks take special importance -
such as a price movement or attained a speci?ed cumulative volume. In this
case, the durations would correspond to the time interval between marks of
special importance. If the timing of these special events is important, then
econometrically this corresponds to focusing on the distribution of ?
min
in
(5). The sequence of arrival times corresponding to the times at which the
marks take special values is called a thinned point process since the new set
of arrival times will contain fewer events than the original series.
Engle and Russell (1998) refer to the series of durations between events
26
constructed by thinning the events with respect to price and volume as price
based and volume based durations. They suggest that the ACD model
might be a good candidate for modeling for these thinned series. In Engle
and Russell (1997) a Weibull ACD model is applied to a thinned series of
quotes arrival times for the Dollar Deutchemark exchange rate series. More
formally if t
i
denotes the arrival times of the original series of quotes then let
?
0
= t
0
Next, let N
?
(t)denote the counting function for the thinned process
de?ned by t
N
?
(t)+1
= t +?
N
?
(t)+1
where ?
N
?
(t)+1
= min
?>0
¯
¯
p
N(t+?)
?p
N
?
(t)
¯
¯
>
c and N(t
0
) = N
?
(t
0
) = 0. So, the sequence of durations ?
i
corresponds
to the price durations de?ned by price movements greater than a threshold
value c. An ACD Weibull model appears to provide a nice ?t for the thinned
series.
The authors suggest that this provides a convenient way of characteriz-
ing volatility when the data are irregularly spaced. The intuition is that
instead of modeling the price change per unit time, as is typically done
for volatility models constructed using regularly spaced data, the model for
price durations models the time per unit price change. In fact, assuming
that the price process locally follows a geometric Brownian motion leads to
implied measures of volatility using ?rst crossing time theory.
Engle and Lange (2001) combine the use of price durations discussed
above with the cumulative signed volume transacted over the price duration
to measure liquidity. For each of the US stocks analyzed, the volume quan-
tity associated with each transaction is given a positive sign if it is buyer
initiated and negative sign if it is seller initiated using the rule proposed
by Lee and Ready (1991). This signed volume is then cumulated for each
price duration. The cumulative signed volume, referred to as VNET, is the
total net volume that can be transacted before inducing a price move hence
providing a time varying measure of the depth of the market. For each price
duration several other measures are also constructed including the cumula-
tive (unsigned) volume and number of transactions. Regressing VNET on
these market variables suggest that market depth is lower following periods
high transaction rates, and high volatility. While VNET increases with past
volume it does so less than proportionally indicating that order imbalance
as a fraction of overall (unsigned) volume decreases with overall volume.
Jointly these results suggest that market depth tends to be lower during
periods of high transaction rates, high volatility, and high volume. In an
asymmetric information environment this is indicative of informed trading
transpiring during period of high transaction rates and high volume.
27
2.2 Modeling in Tick Time - the Marks
Often, economic hypothesis of interest are cast in terms of the marks asso-
ciated with the arrival times. For example, many hypothesis in the asym-
metric information literature focusing on the mechanism by which private
information becomes impounded asset prices. In a rational expectations
environment, the specialist will learn about a traders private information
from the characteristics of their transactions. Hence, many asymmetric
information models of ?nancial participants have implications about how
price adjustments should depend on the characteristics of trades such as
volume or frequency of transactions. By the very nature of market micro
structure ?eld, these theories often need to be examined at the transaction
by transaction frequency This section of the paper examines transaction by
transaction analysis of the marks. We refer to this approach generally as
tick time analysis.
2.2.1 VAR Models for Prices and Trades in Tick Time
Various approaches to tick time modeling of the marks have been considered
in the literature. The approaches are primarily driven by the economic
question at hand as well as assumptions about the role of the timing of
trades. Many times the hypothesis of interest can be expressed as how
the prices adjust given characteristics of past order ?ow. In this case, it is
not necessary to analyze the joint distribution in (1) but only the marginal
distribution of the mark given in (3).
Perhaps the simplest approach in application is to assume that timing
of past of transactions has no impact on the distribution of the marks, that
is f
y
¡
y
i+1
|
b
t
i
, b y
i
¢
= f
y
(y
i+1
|, b y
i
) . This is the approach taken in Hasbrouck
(1991) where the price impact of a trade on future transaction prices is
examined. Hasbrouck focuses on the midpoint of the bid and ask quotes as
a measure of the price of the asset. We refer to this as the midprice and
would appear to be a good approximation to the value of the asset given
the information available. Hasbrouck is interested in testing and identifying
how buyer and seller initiated trades di?er in their impact on the expectation
of the future price.
Let ?m
i
denote the change in the midprice from the (i-1)
th
to the i
th
transactions or m
i
? m
i?1
. The bid and ask prices used to construct the
midprice are those prevailing just prior to the transaction time t
i
. Let w
i
denote the signed volume of a transaction taking a positive value if the i
th
trade is buyer initiated and a negative value if it is seller initiated. The
28
direction of trade is inferred using the Lee and Ready rule discussed in sec-
tion . Hasbrouck persuasively argues that market frictions induce temporal
correlations in both the price and volume series. Regulations require the
specialist to operate an ”orderly” market meaning that the price should not
?uctuate dramatically from one trade to the next. Hence in the face of
a large price move the specialist will have to take intervening transactions
at intermediate prices to smooth the price transition. Volume might also
be autocorrelated as a result of the common practice of breaking up large
orders into multiple small orders to achieve at a better overall price than
had the order been executed in one large transaction. Finally, since neither
price nor direction of trades can be viewed as exogenous the series must be
analyzed jointly to get a full picture of the series dynamics. Hasbrouck
analyzes the bivariate system using the following VAR:
?m
i
=
J
X
j=1
a
j
?m
i?j
+
J
X
j=0
b
j
w
i?j
+ v
1i
(20)
w
i
=
J
X
j=1
c
j
?m
i?j
+
J
X
j=1
d
j
w
i?j
+ v
2i
Notice that the signed volume appears contemporaneously on the right
hand side of the quote update equation. The quote revision equation is
therefore speci?ed conditional on the contemporaneous trade. In reality, it
is likely that the contemporaneous quote revision will in?uence the decision
to transact as the marginal trader might be enticed to transact when a new
limit order arrives improving the price. In fact, our application suggests
some evidence that this may be the case for the Airgas stock. We estimate
a VAR is for price changes and a trade direction indicator variable taking
the value 1 if the trade is deemed buyer initiated and -1 if the trade is
deemed seller initiated using the Lee and Reedy rule. Volume e?ects are
not considered here. As expected, the b
j
coe?cients tend to be postive
meaning that buys tend to lead to increasing quote revisinos and sells tend
to lead to decreasing quote revisions. The market frictions suggest that
the full impact of a trade may not be instantaneous, but rather occur over
a longer period of time. To gauge this e?ect the VAR can be expressed
as an in?nite vector moving average model. The coe?cients then form the
impulse response function.
The cumulants of the impulse response functions then provide a measure
of the total price impact of a trade. Since the model operates in transaction
29
0
0.002
0.004
0.006
0.008
0.01
0.012
0 5 10 15 20
Transactions
P
r
i
c
e
I
m
p
a
c
t
Figure 10: Cummulative price impact of an unexpected buy
time these price impacts are therefore also measured in transaction time.
The asymptote associated with the cummulative impulse response function
is then de?ned as the total price impact of a trade. Since the data are
indexed in tick time the price impact is measured in units of transactions.
The results indicate that it can take several transactions before the full
price impact of a transaction is realized. Figure 1 below presents a price
impact plot for the stock Airgas. Similar to Hasbrouck’s ?ndings using
the earlier data sets we ?nd that the price impact can take many periods
to be fully realized and that the function is concave. The price impact is
in the expected direction - buys increase the price and sells decreased the
priced. Looking at a cross section of stocks, Hasbrouck constructs measures
for the information asymmetry by taking the ratio of the price impact of
a 90th percentile volume trade over the average price. This measure is
decreasing with market capitalization suggesting that ?rms with smaller
market capitalization have larger information asymmetries.
Engle and Dufour also analyze the price impact of trades, but relax the
30
assumption that the timing of trades has no impact on the marginal distri-
bution of price changes. Easley and O’Hara (1992) propose a model with
informed and uninformed traders. On any given day private information
may or may not exist. Informed and uninformed traders are assumed to
arrive in a random fashion. Informed traders only transact when private
information is present so on days with no private information all transac-
tions are by the uninformed. Days with high transaction rates are therefore
viewed as days with more informed trading. Admatti and Pfeiderer (1988)
suggest that in the presence of short sales constraints the timing of trades
should also carry information. Here, long durations imply bad news and
suggest falling prices. The important thread here is that the timing of
trades should carry informative. This leads Dufour and Engle (2000) to
consider expanding Hasbrouck’s VAR structure to allow the durations to
impact price updates. The duration between trades is treated as a pre-
determined variable that in?uences the informativeness of past trades on
future quote revisions. This is done with by allowing the b
j
parameters in
(20) to be time varying parameters. In particular,
b
j
= ?
j
+
K
X
k=1
?
k
D
j,i?k
+ ?
i
ln(x
i?j
)
where D
j,i?k
are dummy variables for the time of day and x
i
is the duration.
Since the b
j
dictate the impact of past trades on quote revisions it is clear
that these e?ects will be time varying whenever the coe?cients d
k
or ? are
non-zero. The model therefore extends the basic VAR of Hasbrouck by
allowing the impact of trades to depend on the time of day as well the
trading frequency as measured by the elapsed time between trades. A
similar adjustment is made to the coe?cients d
j
in the trade equation.
The modi?ed VAR is speci?ed conditional on the durations and may
therefore be estimated directly. Impulse response function, however, will
require complete speci?cation of the trivariate system of trades quotes and
arrival times. Dufour and Engle propose using the ACD model for the
arrival times.
Parameters are estimated for 18 stocks. As in the simple Hasbrouck
VAR the impact of past transactions on quote revisions tends to be positive
meaning that buys tend to lead to increasing quote revisions and sells lead to
decreasing quote revisions. The magnitude of the d
k
indicate some degree of
time of day e?ects in the impact of trades. Trades near the open tend to be
more informative, or have a larger price impact than trades at other times
during the day although this e?ect is not uniform across all 18 stocks. The
31
coe?cients on the durations tend to be negative indicating that the longer
it has been since the last trade, the smaller the price impact will be. The
cumulative impulse response from an unanticipated order can be examined
only now these functions will depend on the state of the market as dictated
by transaction rates measured by the durations. The result is that the
price impact curves will shift up when the transaction occurs with a short
duration and shift down when transactions occur with long durations.
The VAR approach to modeling tick data is particularly appealing due
to its ease of use. Furthermore, information about the spacing of the data
can be included in these models as suggested in Dufour and Engle (2000).
These VAR models can, of course, be expanded to include other variables
such as the bid ask spread or measures of volatility.
2.2.2 Volatility Models in Tick Time
The VARs of the previous section where proved useful in quantifying the
price impact of trades. As such, they focus on the predictable change in
the quotes given characteristics of a transaction. Alternatively, we might
want to ask how characteristics of a transaction a?ect our uncertainty about
the quote updates. Volatility models provide a means of quantifying our
uncertainty.
The class of GARCH models of Engle (1982) and Bollerslev (1986) have
proven to be a trusted work horse in modeling ?nancial data at the daily
frequency. Irregular spacing of transaction by transaction data seems par-
ticularly important for volatility modeling of transaction by transaction data
since volatility is generally measured over ?xed time intervals. Furthermore,
it is very unlikely that, all else equal, the volatility of the asset price over
a one hour intertrade duration should be the same as the volatility over a
5 second intertrade duration. Engle (2000) proposes adapting the GARCH
model for application to irregularly spaced transaction by transaction data.
Let the return from the i ?1
th
to the i
th
transaction be denoted by r
i
.
De?ne the conditional variance per transaction as
V
i?1
(r
i
|x
i
) = h
i
(21)
where this variance is de?ned conditional on the contemporaneous duration
as well as past price changes. The variance of interest, however, is the
variance per unit time. This is related to the variance per transaction as
32
V
i?1
µ
r
i
?
x
i
|x
i
¶
= ?
2
i
(22)
so that the relationship between the two variances is h
i
= x
i
?
2
i
.
The volatility per unit time is then modeled as a GARCH process. En-
gle proposes an ARMA(1,1) model for the series
r
i
?
x
i
. Let e
i
denote the
innovation to this series. If the durations are not informative about the
variance per unit time then the GARCH(1,1) model for irregularly spaced
data is simply
?
2
i
=
.
? +
.
?e
2
i?1
+
.
??
2
i?1
(23)
where we have placed dots above the GARCH parameters to di?erentiate
these from the parameters of the ACD model with similar notation. Engle
terms this model the UHF-GARCH model or Ultra High-Frequency GARCH
model.
A more general model is inspired by the theoretical models of Easley
and O’Hara (1992) and Admatti and Pleiderer (1985) discussed in section
(2.2.1) above. These models suggest that the timing of transactions is re-
lated to the likelihood of asymmetric trading and hence more uncertainty.
Engle therefore proposes augmenting the GARCH(1,1) models with addi-
tional information about the contemporaneous duration and perhaps other
characteristics of the market that might be thought to carry information
about uncertainty such as spreads and past volume.
While the model speci?es the volatility per unit time it is still operates
in transaction time updating the volatility on a time scale determined by
transaction arrivals. If calendar time forecasts of volatility are of interest
then a model for the arrival times must be speci?ed and estimated. Toward
this end Engle proposes using an ACD model for the arrival times. If the
arrival times are deemed exogenous then the ACD model and the GARCH
model can be estimated separately although this estimation may be inef-
?cient. In particular, under the exogeneity assumption the ACD model
could be estimated ?rst and then the volatility model could be speci?ed
conditional on the contemporaneous duration and expected duration in a
second step using canned GARCH software that admits additional explana-
tory variables. This is the approach taken in Engle (2000) where estimation
is performed via (Q)MLE. Engle considers the following speci?cation
?
2
i
=
.
? +
.
?e
2
i?1
+
.
??
2
i?1
+ ?
1
x
?1
i
+ ?
2
x
i
?
i
+ ?
3
?
?1
i
+ ?
4
?
i?1
33
where ?
i
is the expected duration obtained from an ACD model and ?
i?1
characterizes the long run volatility via exponential smoothing of the squared
return per unit time.
An alternative approach to modeling volatility of irregularly spaced data
was simultaneously and independently developed in Ghysels and Jasiak
(1998). Here the authors suggest using temporal aggregation to handle
the spacing of the data. GARCH models are not closed under temporal
aggregation so the authors propose working with the weak GARCH class of
models proposed by Drost and Nijman (1993). For the weak GARCH class
of models Drost and Nijman derive the implied low frequency weak GARCH
model implied by a higher frequency weak GARCH model. Ghysels and
Jasiak propose a GARCH model with time varying parameters driven by
the expected spacing of the data. This approach is complicated, however,
by the fact that the temporal aggregation results apply to aggregation from
one ?xed interval to another, exogenously speci?ed, ?xed interval. The
spacing of the transactions data is not ?xed and it is unlikely that transac-
tion arrival times are exogenous. Nevertheless, the authors show that the
proposed model is an exact discretization of a time deformed di?usion with
ACD as the directing process. The authors propose using GMM to estimate
the model.
2.3 Models for discrete prices
Discrete prices in ?nancial markets pose an additional complication in the
analysis of ?nancial data. For the US markets the graduation to decimal-
ization is now complete, but we still ?nd price changes clustering on just
a handful of values. This discreteness can have an important in?uence on
analysis of prices. Early analysis of discrete prices focused on the notion
of a ”true” or e?cient price. The e?cient price is de?ned as the expected
value of the asset given all currently available public information. The fo-
cus of these early studies, therefore, was on the relationship between the
e?cient price and observed discrete prices. In particular, much empha-
sis was placed on how inference about the e?cient price is in?uenced by
measurement errors induced by discreteness.
Let P
t
denote the observed price at time t and let P
e
t
denote the e?cient
or ”true” price of the asset at time t. Early models for discrete prices can
generally be described in the following setting:
34
P
e
t
= P
e
t?1
+v
t
P
t
= round (P
e
t
+ c
t
Q
t
, d)
v
t
˜N
¡
0, ?
2
t
¢
(24)
where d ? 0 is the tick size and round is a function rounding the argument
to the nearest tick. Q
t
is an unobserved i.i.d. indicator for whether the
trade was buyer or seller initiated taking the value 1 for buyer initiated and
-1 for seller initiated trades and probability given by 1/2. The parameter
c
t
? 0 denotes the cost of market making. It includes both tangible costs
of market making as well as compensation for risk.
With c = 0 and ?
2
t
= ?
2
we obtain the model of Gottlieb and Kalay
(1985)
1
. When d = 0 (no rounding) we obtain the model of Roll (1984).
Harris (1990) considers the full model in (24). In this case, we can write
?P
t
= c (Q
t
?Q
t?1
) + ?
t
??
t?1
+ v
t
(25)
where ?
t
= P
e
t
? P
t
is the rounding error. The variance of the observed
price series is therefore given by
E (?P
t
)
2
= ?
2
+ 2c
2
+ E
¡
?
t+1
??
t
¢
2
(26)
. Hence the variance of the observed transaction price will exceed the vari-
ance by an amount that depends on the cost of market making and the
discrete rounding errors. Furthermore, the ?rst order serial correlation is
given by
E (?P
t
?P
t?1
) = ?c
2
+ E
¡
?
t+1
??
t
¢ ¡
?
t
??
t?1
¢
(27)
which is shown to be negative in Harris (1990). The ?rst order serial corre-
lation will be larger when the cost of market making is larger and depends
on the discreteness rounding errors. The Harris model goes a long way in
describing key features of price discreteness and the implications regarding
inference on the e?cient price dynamics but it is still very simplistic in sev-
eral dimensions since it assumes that both the volatility of the e?cient price
and the cost of market making is constant. As new information hits the
market the volatility of the e?cient price will change. Since part of the cost
of making market is the risk of holding the asset the cost of making market
will also be time varying.
1
For simplicity we have neglected a drift term in the e?cient price equation and the
possiblity of dividend payments considered in Gottlieb and Kalay (1985).
35
Hasbrouck (1999) builds on the previous discrete price literature by re-
laxing these two assumptions - both the volatility of the e?cient price and
the cost of making market are time varying. He also proposes working with
bid and ask prices as opposed to the transaction prices circumventing the
need to sign trades as buyer or seller initiated. P
t
of 24 is therefore replaced
by two prices P
a
t
and P
b
t
, the bid and ask prices respectively where
P
a
t
= ceiling (P
e
t
+ c
a
t
, d)
P
b
t
= floor
¡
P
e
t
?c
b
t
, d
¢
(28)
c
a
t
> 0 and c
b
t
> 0 are the cost of exposure on the ask and bid side respec-
tively. These costs are the economic cost to the specialist including both
the ?xed cost of operation and the expected cost incurred as a result of the
obligation to trade a ?xed quantity at these prices with potentially better
informed traders. The ceiling function rounds up to the nearest discrete
tick and the ?oor function rounds down to the nearest tick recognizing the
market maker would not set quotes that would fail to cover the cost.
The dynamics of this model are not suited for the type of analysis pre-
sented by Harris due to it’s more complex dynamic structure. Instead,
Hasbrouck focuses on characterizing the dynamics of the both cost of expo-
sure and, second, estimating models for the volatility of the e?cient price
given only observations of the perturbed discrete bid and ask prices. Has-
brouck proposes using a GARCH model for the dynamics of the e?cient
price volatility ?
2
t
. The dynamics of the cost of exposure is assumed to be
of an autoregressive form.
ln(c
a
t
) = µ
t
+ ?
¡
ln(c
a
t
) ?µ
t?1
¢
+ ?
?
t
(29)
ln
³
c
b
t
´
= µ
t
+ ?
³
ln
³
c
b
t
´
?µ
t?1
´
+ ?
?
t
where µ
t
is a common deterministic function of time of day and ? is the
common autoregressive parameter. ?
?
t
and ?
?
t
are assumed to be i.i.d. and
independent of the e?cient price innovation v
t
.
Estimation of the model is complicated by the fact that the e?cient price
is inherently unobserved - only the discrete bid and ask quotes are observed.
Hasbrouck (1999) proposes using a non-Gaussian, non-linear state space of
Kitagawa (1987) and Hasbrouck (1999b) and Manrique and Sheppard (1997)
propose using MCMC methods which treat the price at any given date as
an unknown parameter. We refer the reader to this literature for further
details on the estimation.
36
The early work with no time varying parameters focused on the impact
of discrete rounding errors on inference regarding the e?cient price. Has-
brouck also treats the object of interest as the dynamics of the e?cient price
and demonstrates a methodology to study the second moment dynamics of
the e?cient price while accounting for the discrete rounding errors. In ad-
dition, the model allows for asymmetric cost of exposure. In some states of
the world the specialist may set quotes more conservatively on one side of
the market than the other.
More recently Zhang Russell and Tsay (2000) appeal to the asymmetric
information microstructure theory that suggests that transaction character-
istics should in?uence the market makers perception of the exposure risk.
They include in the cost of exposure dynamics (29) measures of the order
imbalance and overall volume and ?nd that the cost of exposure is a?ected
by order imbalance. In particular, unexpectedly large buyer initiated vol-
ume tends to increase the cost of exposure on the ask side and decrease the
cost of exposure on the bid side with analogous results for unexpected seller
initiated volume. These e?ects are mitigated, however, the larger the total
volume transacted.
If the structural parameters are not of primary interest then an alter-
native is to directly model transaction prices with a reduced form model
for discrete valued random variables. This is the approach taken in Haus-
man Lo and MacKinlay (1992). They propose modeling the transaction
by transaction price changes with an ordered Probit model. In doing so,
the structural models linking the unobserved e?cient price to the observed
transaction price is replaced by a reduced from Probit link. The model
is applied to transaction by transaction price dynamics. Let k denote the
number of discrete values that the price changes can take which is assumed
to be ?nite. Let s
i
denote a vector of length k taking the j
th
column of
the kxk identity matrix if the j
th
state occurs on the i
th
transaction. Let
?
i
denote a k dimensional vector with j
th
where ?
i
= E (s
i
|I
i?1
) where I
i
is the information set associated with the i
th
transaction. Clearly, the j
th
element of x
i
denotes the conditional probability of the j
th
state occurring.
At the heart of the Probit model lies the assumption that the observed dis-
crete transaction price changes can be represented as a transformation of a
continuous latent price given by ?P
?
i
˜N
¡
µ
i
, ?
2
i
¢
where µ
i
and ?
2
i
are the
mean and variance of the latent price given I
i?1
. The Hausman Lo and
MacKinlay model assumes that the j
th
element of ?
i
is given by
?
j
˜F
P
?
i
(c
j?1
) ?F
P
?
i
(c
j
) (30)
where F
P
?
i
is the cdf associated with the price changes ?P
?
i
and c
j
, j =
37
1, k ?1 are time invariant parameters.
Since bid ask bounce induces dependence in the price changes it is nat-
ural to allow the conditional mean of price changes to depend on past price
changes. Hausman Lo and Mackinlay are particularly interested in testing
asymmetric information theories regarding the information content in a se-
quence of trades. In particular, they study how transaction prices respond
to a sequence of buyer initiated trades versus a sequence of seller initiated
trades. By conditioning on recent buys and sells the authors ?nd evidence
that persistent selling predicts falling prices and persistent buying predicts
rising prices. The authors also suggest that the conditional variance may
depend on the contemporaneous duration so that the variance associated
with a price change over a long duration may not be the same as the vari-
ance of a price change associated with a relatively short duration. Indeed,
they ?nd that long durations lead to higher variance per transaction.
Russell and Engle (2002) are also interested in reduced form models for
discrete prices that explicitly account for the irregular spacing of the data.
They propose joint modeling the arrival times and the price changes as a
marked point process. The joint likelihood for the arrival times and the price
changes is decomposed into the product of the conditional distribution of the
price change given the duration and the marginal distribution of the arrival
times which are assumed to be given by an ACD model. More speci?cally
if x
i
denotes the i
th
intertransaction duration and b z
i
= (z
i
, z
i?1
, z
i?2
...) then
f (x
i+1
, ?p
i+1
|b x
i
, ?b p
i
) = ?(?p
i+1
|b x
i
, ?b p
i+1
) ?(x
i
|b x
i
, ?b p
i
)
where ? denotes the distribution of price changes given the past price
changes and durations as well as the contemporaneous duration. ? de-
notes the distribution of price changes given the past price changes and
returns. Engle and Russell propose using the ACD model for the durations
and propose the Autoregressive Conditional Multinomial (ACM) model for
the conditional distribution of the discrete price changes. A simple model
for the price dynamics might assume a VARMA model for the state vector
s
i
. Since the state vector is simply a vector of ones and zeros its expecta-
tion should be bounded between zero and one. Russell and Engle use the
logistic transformation to directly impose this condition. Using the logistic
link function the VARMA model is expressed in terms of the log odds. Let
h
j
denote a k ?1 vector with j
th
element given by ln
¡
?
j
/?
k
¢
. Let e s
i
and
e ?
i
denote k ?1 dimensional vectors consisting of the ?rst k-1 elements of s
i
and ?
i
. Hence the k
th
element has been omitted and is referred to as the
base state. Then the ACM(u, v) model with duration dependence is given
38
by:
h
i
= c +
u
X
m=1
A
m
(e s
i?m
? e ?
i
) +
v
X
m=1
B
m
h
i?m
+
w
X
m=1
?
m
ln(x
i?m+1
) (31)
where A
m
and B
m
are (k ?1) x(k ?1) parameter matrices and ? and ?
m
are k ? 1 parameter vector. Given the linear structure of the log odds
VARMA the choice of the the base state is arbitrary. The ?rst k ? 1
probabilities are obtained by applying the logistic link function
?
i
=
1
1 + ?
0
exp(h
i
)
exp(h
i
) (32)
where ?
0
is a k?1 vector of ones and exp(h
i
) should be interpreted as applying
the exponential function element by element. The omitted state is obtained
by imposing the condition that the probabilities sum to 1. The ACD(p, q)
speci?cation for the durations allows feedback from the price dynamics into
the duration dynamics as follows:
ln(?
i
) = ?+
p
X
m=1
?
m
x
i?m
?
i?m
+
q
X
m=1
?
m
ln
¡
?
i?m
¢
+
r
X
m=1
¡
?
m
?p
i?m
+ ?
m
?p
2
i?m
¢
For the stocks analyzed, the longer the contemporaneous duration the lower
the expected price change and large price changes tend to be followed by
short durations.
Another reduced form model for discrete prices is proposed by Rydberg
and Sheppard (1999). The model decomposes the discrete price changes
into the trivariate process ?P
i
= Z
i
D
i
M
i
where Z
i
is an indicator for
the i
th
transaction price change being non-zero and is referred to as ac-
tivity. Conditional on a price move (Z
i
6= 0) D
i
is takes the value of 1
or -1 denoting an upward or downward price change respectively. Given a
non-zero price change and its direction M
i
is the magnitude of the price
change given it is non-zero and the direction. The authors suggest de-
composing the distribution of price changes given an information set I
i
Pr (?P
i
|I
i?1
) = Pr (Z
i
|I
i?1
) Pr (D
i
|Z
i
, I
i?1
) Pr (M
i
|Z
i
, D
i
, I
i?1
) . The au-
thors propose modeling the binary variables Z
i
and D
i
following an autore-
gressive logistic process ?rst proposed by Cox (1958). A simple version for
the activity variable is given by:
ln
µ
Pr (Z
i
= 1)
1 ?Pr (Z
i
= 1)
¶
= c +
u
X
m=1
Z
i?m
(33)
39
The direction indicator variable is modeled in a similar fashion. Finally, the
magnitude of the price change is modeled by a distribution for count data.
Hence it is positively valued over integers. The integers here are measured
in units of ticks, or the smallest possible price change.
The Russell Engle ACM approach and the Rydberg Sheppard compo-
nents model are very similar in spirit both implementing an autoregressive
structure. While this decomposition breaks the estimation down into a se-
quence of simpler problems it comes with a cost. In order to estimate the
model sequentially the ?rst model for the activity can not be a function of
lagged values of Pr (D
i
|I
i?1
) or Pr(M
i
|I
i?1
). Similarly the model for the
direction cannot depend on the past probability associated with the magni-
tude. The importance of this restriction surly depends on the application at
hand. A second advantage of the Rydberg Sheppard model easily accounts
for a large number of states (possibly in?nite).
2.4 Calendar Time Conversion
Most ?nancial econometric analyses are carried out in ?xed time units.
These time intervals for many years were months or weeks or days but now
time intervals of hours, ?ve minutes or seconds are being used for econo-
metric model building. Once the data are converted from their natural
irregular spacing to regular spaced observations, econometric analysis typ-
ically proceeds without considering the original form of the data. Models
are constructed for volatility, price impact, correlation, extreme values, and
many other ?nancial constructs. In this section we discuss the most com-
mon approaches used to convert irregularly spaced data to equally spaced
observations and in the next sections we will examine the implications of
this conversion.
Suppose the data on prices arrive at times {t
i
; i = 1, ..., N(T)} so
that there are N observations occurring over time (0, T). These times could
be times at which transactions occur and the price could be either the trans-
action price or the prevailing midquote at that time. An alternative formu-
lation would have these times as the times at which quotes are posted and
then the prices are naturally considered to be midquotes. Let the log of the
price at time t
i
be denoted p
?
i
.
The task is to construct data on prices at each ?xed interval of time.
Denoting the discrete time intervals by integers of t = 1, . . . , T, a task is to
estimate p
t
. The most common speci?cation is to use the most recent price
at the end of the time interval as the observation for the interval. Thus:
40
p
t
= p
?
i
where t
i
? t < t
i+1
(34)
For example, Huang and Stoll (1994) use this scheme where p is the pre-
vailing midquote at the time of the last trade. Andersen Bollerslev Diebold
and Ebens (2001) use the last trade price.
Various alternative schemes have been used. One could interpolate
the price path from some or all of the p
?
observations and then record the
value at time t. For example smoothing splines could be ?t through all the
data points. A particularly simple example of this uses the weighted average
of the last price in one interval and the ?rst price in the next interval:
e p
t
=
¡
?p
?
i
+ (1 ??) p
?
i+1
¢
where t
i
? t < t
i+1
, and ? =
t ?t
i
t
i+1
?t
i
(35)
Andersen, Bollerslev, Diebold and Labys (2001,2002) use this procedure
with midquotes to get 5 minute and 30 minute calendar time data. The
advantage of this formulation is supposed to be its reduced sensitivity to
measurement error in prices. Clearly this comes at a cost of using future
information. The merits of such a scheme must be evaluated in the context
of a particular data generating process and statistical question.
A third possibility is adopted by Hasbrouck(2002). Since the time
of a trade is recorded only to the nearest second, then if t is measured in
seconds, there is at most one observation per time period. The calendar
price is either set to this price or it is set to the previous period price. This
version follows equation 34 but does not represent an approximation in the
same sense.
Returns are de?ned as the ?rst di?erence of the series.
y
?
i
= p
?
i
?p
?
i?1
, y
t
= p
t
?p
t?1
, and e y
i
= e p
i
? e p
i?1
(36)
thus y
?
de?nes returns over irregular intervals while y de?nes returns
over calendar intervals. In the case of small calendar intervals, there will
be many zero returns in y. In this case, there are some simpli?cations.
Means and variances are preserved if there never is more than one trade per
calendar interval
N(T)
X
i=1
y
?
i
=
T
X
t=1
y
t
and
N(T)
X
i=1
(y
?
i
)
2
=
T
X
t=1
y
2
t
(37)
41
If the calendar time intervals are larger, then means will still be preserved
but not variances as there may be more than one price in a calendar interval.
N(T)
X
i=1
y
?
i
=
T
X
t=1
y
t
and
N(T)
X
i=1
_
_
_
_
_
_
_
_
X
multiple
trades
y
?
i
_
_
_
_
_
_
_
_
2
=
T
X
t=1
y
2
t
(38)
However, if the prices are Martingales, then the expectation of the cross
products is zero and the expected value and probability limit of the calendar
time and event time variances is the same.
When prices are interpolated, these relations no longer hold. In this
scheme there would be many cases of multiple observations in a calendar
interval. The mean will approximately be the same in both sets of returns,
however the variances will not. The sum of squared transformed returns is
given by:
N(T)
X
i=1
[e y
i
]
2
=
N(T)
X
i=1
£
?
i
p
?
i
+ (1 ??
i
) p
?
i?1
??
j
p
?
j
?(1 ??
j
) p
?
j?1
¤
2
(39)
=
N(T)
X
i=1
£
?
i
¡
p
?
i
?p
?
i?1
¢
+
¡
p
?
i
?p
?
j
¢
+ (1 ??
j
)
¡
p
?
j
?p
?
j?1
¢¤
2
where i and j are the events just after the end points of the calendar
intervals. In the right hand side of expression 39, the returns will all be
uncorrelated if the y
?
are Martingale di?erences, hence the expected vari-
ance and the probability limit of the variance estimators will be less than
the variance of the process. Furthermore, the returns will be positively au-
tocorrelated because they are formulated in terms of future prices. This is
easily seen in equation 39 because the change in price around the interval
endpoints is included in both adjacent returns.
2.4.1 Bivariate Relationships
We now consider two correlated assets with Martingale prices. One of
these asset prices is only observed at random time periods while the other
is continuously observable. In this case the stochastic process of the infre-
quently observed process is de?ned on times {t
i
; i = 1, ..., N(T)}. Let the
42
log price of the second asset be q
t
, and let its return, measured respectively
in the ?rst asset trade time and in calendar time, be:
z
?
t
i
? z
?
i
= q
t
i
?q
t
i?1
, z
t
= q
t
?q
t?1
The return on the ?rst asset is given by
y
?
i
= ?z
?
i
+ ²
?
i
(40)
where the innovation is a Martingale di?erence sequence, independent
of z, with potential heteroskedasticity since each observation may have a
di?erent time span. Since this model is formulated in transaction time, it
is natural to estimate the unknown parameter beta with transaction time
data. It is straightforward to show that least squares will be consistent.
Calendar time data on y can be constructed from 40 and 36. Con-
sider the most disaggregated calendar time interval and let d
t
be a dummy
variable for the time periods in which a price is observed on the ?rst asset.
Then a useful expression for y
t
is
y
t
= d
t
µ
(?z
t
+ ²
t
) + (1 ?d
t?1
) (?z
t?1
+ ²
t?1
)
+(1 ?d
t?2
) (?z
t?2
+ ²
t?2
) + ...
¶
(41)
With this data and the comparable calendar data on z, we are in a
position to estimate beta by ordinary least squares in calendar time. The
estimator is simply
b
? =
T
P
t=1
z
t
y
t
T
P
t=1
z
2
t
(42)
which has an interesting probability limit under simplifying assumptions.
Theorem 1
If
a) (z
t
, ²
t
) are independent Martingale di?erence sequences with ?-
nite variance
b) d
t
˜ independent Bernoulli with parameter ?
Then
plim
b
? = ?? (43)
Proof:
Substituting and taking probability limits:
plim
b
? =
1
?
2
z
E [z
t
d
t
((?z
t
+ ²
t
) + (1 ?d
t?1
) (?z
t?1
+²
t?1
) + ...)] (44)
43
Writing the expectation of independent variables as the product of their
expectation gives
plim
b
? =
1
?
2
z
·
E (d
t
) E
¡
?z
2
t
+ z
t
²
t
¢
+
E (d
t
) E (z
t
) E (1 ?d
t?1
) E (?z
t?1
+ z²
t?1
) + ...
¸
= ??
QED.
The striking result is that the regression coe?cient is heavily down-
ward biased purely because of the non-trading bias. If trading is infrequent,
then the regression coe?cient will be close to zero.
In this setting, researchers will often regress on many lags of z. Suppose
the regression computed incorporates k lags of z.
y
t
= ?
0
z
t
+ ?
1
z
t?1
+ ... + ?
k
z
t?k
+ ²
t
The result is given by Theorem II.
THEOREM II:
Under the assumptions of Theorem I the regression in (11) has a prob-
ability limit
plim
b
? = ? (1 ??)
j?1
?
and the sum of these coe?cients approaches beta as k gets large.
PROOF:
Because z is a Martingale di?erence sequence, the matrix of regressors
approaches a diagonal matrix with the variance of z on the diagonals. Each
row of the z0y matrix has dummy variables d
t
(1 ?d
t?1
) (1 ?d
t?2
) ... (1 ?d
t?j
)
multiplying the square of z. The result follows from independence.
QED.
The regression coe?cients decline from the contemporaneous one
but ultimately summing up to the total impact of asset one on two. The
result is however misleading because it appears that the price movements
in asset two predict future movements in asset one. There appears to be
causality or price discovery between these assets merely because of the ran-
dom trade times.
Similar results can be found in more general contexts including dy-
namic structure in the price observations and dependence with the z’s. Con-
tinuing research will investigate the extent of the dependence and how results
change with the economic structure.
44
3 Conclusion
The introduction of widely available ultra high frequency data sets over
the past decade has spurred interest in empirical market microstructure.
The black box determining equilibrium prices in ?nancial markets has been
opened up. Intraday transaction by transaction dynamics of asset prices,
volume, and spreads are available for analysis. These vast data sets present
new and interesting challenges to econometricians.
Since transactions data are inherently irregularly spaced we view the
process as a marked point process. The arrival times form the points and
the characteristics of the trades form the marks. We ?rst discuss models
for the timing of events when the arrival rate may be time varying. Since
the introduction of the ACD model of Engle and Russell (1998) numerous
other models for the timing of event arrivals have been proposed and applied
to ?nancial data. The models have been applicable to transaction arrival
times or, if some arrival times are thought to be more informative than
others the point process can be ”thinned” to contain only those arrival
times with special information. Examples include volume based durations
which correspond to the time it takes for a speci?ed amount of volume to
be transacted. Another example is price durations which correspond to the
time it takes for the price to move a speci?ed amount. These models can
be thought of as models of volatility where the volatility is intuitively the
inverse of our usual measures of volatility - namely the time it takes for the
price to move a speci?ed amount.
Models for the marks are also discussed. Often the focus is on the
transaction price dynamics or joint modeling of transaction prices and vol-
ume. If the spacing of the data is ignored then the modeling problem can
be reduced to standard econometric modeling procedures of VARs, simple
linear regression, or GARCH models. Models that address the inherent
discreteness in transaction by transaction prices are also discussed.
Alternatively, if the spacing of the data is thought to carry information
then the simple approaches may be mispeci?ed. Choices then include con-
ditioning the marks on the arrival times as in Hausman Lo and Mackinlay,
or, if forecasting is of interest joint modeling of the arrival times. The latter
approach is considered in Engle (2000), Russell and Engle (2002), Rydberg
and Sheppard (1999), or Ghysels (1999) among others.
Finally, while arti?cially descretizing the time intervals at which prices
(or other marks) is a common practice in the literature, it does not come
without cost. Di?erent descretizing schemes trade of bias associated with
45
temporally aggregating with variance. Averaging reduces the variability but
blurs the timing of events. We also show, in a stylized model, that causal
relationships can be arti?cially induced by descretizing the data. Care
should be taken in interpreting results from this type of analysis.
46
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50
Appendix
A EACD(3,3) parameter estimates using EVIEWS
GARCH module.
Coefficient Robust Std. Err.
? 0.004244 0.000855
?
1
0.070261 0.007157
?
2
0.038710 0.012901
?
3
-0.055966 0.008640
?
1
0.835806 0.125428
?
2
0.107894 0.118311
where ?
i
= ? +
3
P
j=1
?
j
x
i?j
+
2
P
j=1
?
j
?
i?j
Model diagnostics
B VAR parameter estimates
Price Equation Trade Equation
Variable Coefficient Std. Error Coefficient Std. Error
c -0.006553 0.000284 0.509648 0.004785
w
i
0.014230 0.000430
w
i-1
0.000891 0.000493 0.298146 0.005557
w
i-2
-0.000175 0.000493 0.059228 0.005797
w
i-3
-0.000533 0.000493 0.036385 0.005803
w
i-4
0.000176 0.000493 0.026645 0.005798
w
i-5
-0.001295 0.000425 0.035205 0.005558
?m
i-1
-0.262310 0.005734 0.250909 0.071635
?m
i-2
-0.121951 0.005934 0.108735 0.081696
?m
i-3
-0.054038 0.005968 -0.000260 0.084009
?m
i-4
-0.026460 0.005934 -0.022889 0.081695
?m
i-5
-0.011011 0.005734 -0.220448 0.071634
51
doc_107925136.pdf