percentagesof trades associated with a price change. The price movements remain approximatelysymmetric with respect to zero. Large price movements in intraday tradings are stillrelatively rare.Remark: The record keeping of high-frequency data is often not as good as thatof observations taken at lower frequencies. Data cleaning becomes a necessity in
186
HIGH - FREQUENCY DATA
0
20
duration40 60
80
(a) Intraday duration
20000
40000
6000080000sequence
100000
120000
0
20000
40000
6000080000sequence
100000
120000
-60
change-20 0 20 40 60
0
Figure 5.4. IBM transactions data for December 1999. Part (a) is the time plot of time durations between trades and part (b) is the time plot of price changes in consecutive trades measured in multiples of the tick size of $1/16. Only data in the normal trading hours are included.
high-frequency data analysis. For transactions data, missing observations may happen in many ways, and the accuracy of the exact transaction time might be questionable for some trades. For example, recorded trading times may be beyond 4:00 pmEastern time even before the opening of after-hours tradings. How to handle theseobservations deserves a careful study. A proper method of data cleaning requires a
Table 5.3. Percentages of Intraday Transactions Associated with a Price Change for IBMStock Traded in December 1999. The Percentage of Transactions without Price ChangeIs 45.8% and the Total Number of Transactions Recorded within the Regular TradingHours Is 133,475. The Size Is Measured in Multiples of Tick Size $1/16.(a) Upward movementssizepercentage
118.03
25.80
31.79
40.66
50.25
60.15
70.09
>70.32
0.17
0.10
0.31
(b) Downward movementspercentage
18.24
5.57
1.79
0.71
0.24
187
MODELS FOR PRICE CHANGES
deep understanding of the way by which the market operates. As such, it is importantto specify clearly and precisely the methods used in data cleaning. These methodsmust be taken into consideration in making inference.Again, let ti be the calendar time, measured in seconds from the midnight, whenthe ith transaction took place. Let Pti be the transaction price. The price changefrom the (i − 1)th to the ith trade is yi ≡ Pti = Pti − Pti−1 and the time durationis ti = ti − ti−1 . Here it is understood that the subscript i in ti and yi denotes thetime sequence of transactions, not the calendar time. In what follows, we considermodels for yi and ti both individually and jointly.
5.4 MODELS FOR PRICE CHANGESThe discreteness and concentration on “no change” make it difficult to model theintraday price changes. Campbell, Lo, and MacKinlay (1997) discuss several econometric models that have been proposed in the literature. Here we mention two models that have the advantage of employing explanatory variables to study the intradayprice movements. The first model is the ordered probit model used by Hauseman,Lo, and MacKinlay (1992) to study the price movements in transactions data. Thesecond model has been considered recently by McCulloch and Tsay (2000) and is asimplified version of the model proposed by Rydberg and Shephard (1998); see alsoGhysels (2000).5.4.1 Ordered Probit ModelLet yi∗ be the unobservable price change of the asset under study (i.e., yi∗ = Pt∗i −Pt∗i−1 ), where Pt∗ is the virtual price of the asset at time t. The ordered probit modelassumes that yi∗ is a continuous random variable and follows the modelyi∗ = xi β + i ,
(5.15)
where xi is a p-dimensional row vector of explanatory variables available at timeti−1 , β is a k × 1 parameter vector, E( i | xi ) = 0, Var( i | xi ) = σi2 , andCov( i , j ) = 0 for i = j. The conditional variance σi2 is assumed to be a positive function of the explanatory variable wi —that is,σi2 = g(wi ),
(5.16)
where g(.) is a positive function. For financial transactions data, wi may contain thetime interval ti − ti−1 and some conditional heteroscedastic variables. Typically, onealso assumes that the conditional distribution of i given xi and wi is Gaussian.Suppose that the observed price change yi may assume k possible values. In theory, k can be infinity, but countable. In practice, k is finite and may involve combin-
188
HIGH - FREQUENCY DATA
ing several categories into a single value. For example, we have k = 7 in Table 5.1,where the first value “−3 ticks” means that the price change is −3 ticks or lower. Wedenote the k possible values as {s1 , . . . , sk }. The ordered probit model postulates therelationship between yi and yi∗ asyi = s j
if α j−1 < yi∗ ≤ α j ,
j = 1, . . . , k,
(5.17)
where α j s are real numbers satisfying −∞ = α0 < α1 < · · · < αk−1 < αk = ∞.Under the assumption of conditional Gaussian distribution, we haveP(yi = s j | xi , wi ) = P(α j−1 < xi β + i ≤ α j | xi , wi )if j = 1 P(xi β + i ≤ α1 | xi , wi )= P(α j−1 < xi β + i ≤ α j | xi , wi ) if j = 2, . . . , k − 1P(αk−1 < xi β + i | xi , wi )if j = k α1 −xi βif j = 1 σi (wi ) α−x βα −x β=  σji (wii ) −  jσ−1i (wi )iif j = 2, . . . , k − 11 −  αk−1 −xi βif j = k,σi (wi )(5.18)where (x) is the cumulative distribution function of the standard normal randomvariable evaluated at x, and we write σi (wi ) to denote that σi2 is a positive functionof wi . From the definition, an ordered probit model is driven by an unobservablecontinuous random variable. The observed values, which have a natural ordering,can be regarded as categories representing the underlying process.The ordered probit model contains parameters β, αi (i = 1, . . . , k − 1), and thosein the conditional variance function σi (wi ) in Eq. (5.16). These parameters can beestimated by the maximum likelihood or Markov Chain Monte Carlo methods.Example 5.1. Hauseman, Lo, and MacKinlay (1992) apply the ordered probit model to the 1988 transactions data of more than 100 stocks. Here we only reporttheir result for IBM. There are 206,794 trades. The sample mean (standard deviation) of price change yi , time duration ti , and bid-ask spread are −0.0010(0.753),27.21(34.13), and 1.9470(1.4625), respectively. The bid-ask spread is measured inticks. The model used has nine categories for price movement, and the functionalspecifications arexi β = β1 ti∗ +
3
3
βv+1 yi−v +v=1
3
βv+4 SP5i−v +v=1
βv+7 IBSi−vv=1
3
+
βv+10 [Tλ (Vi−v ) × IBSi−v ]v=1
(5.19)
189
MODELS FOR PRICE CHANGES
σi2 (wi ) = 1.0 + γ12 ti∗ + γ22 ABi−1 ,
(5.20)
where Tλ (V ) = (V λ − 1)/λ is the Box-Cox (1964) transformation of V with λ ∈[0, 1] and the explanatory variables are defined by the following:•
ti∗ = (ti − ti−1 )/100 is a rescaled time duration between the (i − 1)th and ithtrades with time measured in seconds.
• ABi−1 is the bid-ask spread prevailing at time ti−1 in ticks.•
yi−v (v = 1, 2, 3) is the lagged value of price change at ti−v in ticks. Withk = 9, the possible values of price changes are {−4, −3, −2, −1, 0, 1, 2, 3, 4}in ticks.
• Vi−v (v = 1, 2, 3) is the lagged value of dollar volume at the (i − v)th transac-
tion, defined as the price of the (i − v)th transaction in dollars times the numberof shares traded (denominated in hundreds of shares). That is, the dollar volumeis in hundreds of dollars.
• SP5i−v (v = 1, 2, 3) is the 5-minute continuously compounded returns of the
Standard and Poor’s 500 index futures price for the contract maturing in theclosest month beyond the month in which transaction (i − v) occurred, wherethe return is computed with the futures price recorded one minute before thenearest round minute prior to ti−v and the price recorded 5 minutes before this.• IBSi−v (v = 1, 2, 3) is an indicator variable defined by
 1IBSi−v =0−1
abif Pi−v > (Pi−v+ Pi−v)/2ab)/2if Pi−v = (Pi−v + Pi−vabif Pi−v < (Pi−v + Pi−v )/2,
where P ja and P jb are the ask and bid price at time t j .The parameter estimates and their t ratios are given in Table 5.4. All the t ratiosare large except one, indicating that the estimates are highly significant. Such high tratios are not surprising as the sample size is large. For the heavily traded IBM stock,the estimation results suggest the following conclusions:1. The boundary partitions are not equally spaced, but are almost symmetric withrespect to zero.2. The transaction duration ti affects both the conditional mean and conditionalvariance of yi in Eqs. (5.19) and (5.20).3. The coefficients of lagged price changes are negative and highly significant,indicating price reversals.4. As expected, the bid-ask spread at time ti−1 significantly affects the conditional variance.
190
HIGH - FREQUENCY DATA
Table 5.4. Parameter Estimates of the Ordered-Probit Model in Eq. (5.19) and Eq. (5.20)for the 1988 Transaction Data of IBM, Where t Denotes the t Ratio.(a) Boundary partitions of the probit modelPar.
α1
α2
α3
α4
α5
α6
α7
α8
Est.t
−4.67−145.7
−4.16−157.8
−3.11−171.6
−1.34−155.5
1.33154.9
3.13167.8
4.21152.2
4.73138.9
(b) Equation parameters of the probit modelPar.
γ1
γ2
β1 :
ti∗
β2 : y−1
β3
β4
β5
β6
Est.t
0.4015.6
0.5271.1
−0.12−11.4
−1.01−135.6
−0.53−85.0
−0.21−47.2
1.1254.2
−0.26−12.1
Par.
β7
β8
β9 :
β10
β11
β12
β13
Est.t
0.010.26
−1.14−63.6
−0.37−21.6
−0.17−10.3
0.1247.4
0.0518.6
0.027.7
5.4.2 A Decomposition ModelAn alternative approach to modeling price change is to decompose it into three components and use conditional specifications for the components; see Rydberg andShephard (1998). The three components are an indicator for price change, the direction of price movement if there is a change, and the size of price change if a changeoccurs. Specifically, the price change at the ith transaction can be written asyi ≡ Pti − Pti−1 = Ai Di Si ,
(5.21)
where Ai is a binary variable defined asAi =
10
if there is a price change at the ith tradeif price remains the same at the ith trade.
(5.22)
Di is also a discrete variable signifying the direction of the price change if a changeoccurs—that is,Di | (Ai = 1) =
1 if price increases at the ith trade−1 if price drops at the ith trade,
(5.23)
where Di | (Ai = 1) means that Di is defined under the condition of Ai = 1, and Siis size of the price change in ticks if there is a change at the ith trade and Si = 0 ifthere is no price change at the ith trade. When there is a price change, Si is a positiveinteger-valued random variable.Note that Di is not needed when Ai = 0, and there is a natural ordering in thedecomposition. Di is well defined only when Ai = 1 and Si is meaningful when
191
MODELS FOR PRICE CHANGES
Ai = 1 and Di is given. Model specification under the decomposition makes use ofthe ordering.Let Fi be the information set available at the ith transaction. Examples of elementsin Fi are ti− j , Ai− j , Di− j , and Si− j for j ≥ 0. The evolution of price change undermodel (5.21) can then be partitioned asP(yi | Fi−1 ) = P(Ai Di Si | Fi−1 )= P(Si | Di , Ai , Fi−1 )P(Di | Ai , Fi−1 )P(Ai | Fi−1 ).
(5.24)
Since Ai is a binary variable, it suffices to consider the evolution of the probabilitypi = P(Ai = 1) over time. We assume thatln
pi1 − pi
= xi β
pi =
or
exi β,1 + exi β
(5.25)
where xi is a finite-dimensional vector consisting of elements of Fi−1 and β is aparameter vector. Conditioned on Ai = 1, Di is also a binary variable, and we usethe following model for δi = P(Di = 1 | Ai = 1),
δiln1 − δi
= zi γ
or δi =
ezi γ,1 + ezi γ
(5.26)
where zi is a finite-dimensional vector consisting of elements of Fi−1 and γ isa parameter vector. To allow for asymmetry between positive and negative pricechanges, we assume thatSi | (Di , Ai = 1) ∼ 1 +
g(λu,i )g(λd,i )
if Di = 1, Ai = 1if Di = −1, Ai = 1,
(5.27)
where g(λ) is a geometric distribution with parameter λ and the parameters λ j,ievolve over time as
λ j,iln1 − λ j,i
= wi θ j
or λ j,i =
ewi θ j1 + ewi θ j
,
j = u, d,
(5.28)
where wi is again a finite-dimensional explanatory variables in Fi−1 and θ j is aparameter vector.In Eq. (5.27), the probability mass function of a random variable x, which followsthe geometric distribution g(λ), isp(x = m) = λ(1 − λ)m ,
m = 0, 1, 2, . . . .
We added 1 to the geometric distribution so that the price change, if it occurs, is atleast 1 tick. In Eq. (5.28), we take the logistic transformation to ensure that λ j,i ∈[0, 1].
192
HIGH - FREQUENCY DATA
The previous specification classifies the ith trade, or transaction, into one of threecategories:1. no price change: Ai = 0 and the associated probability is (1 − pi );2. a price increase: Ai = 1, Di = 1, and the associated probability is pi δi . Thesize of the price increase is governed by 1 + g(λu,i ).3. a price drop: Ai = 1, Di = −1, and the associated probability is pi (1 − δi ).The size of the price drop is governed by 1 + g(λd,i ).Let Ii ( j) for j = 1, 2, 3 be the indicator variables of the prior three categories. Thatis, Ii ( j) = 1 if the jth category occurs and Ii ( j) = 0 otherwise. The log likelihoodfunction of Eq. (5.24) becomesln[P(yi | Fi−1 )] = Ii (1) ln[(1 − pi )] + Ii (2)[ln( pi ) + ln(δi )+ ln(λu,i ) + (Si − 1) ln(1 − λu,i )]+ Ii (3)[ln( pi ) + ln(1 − δi ) + ln(λd,i ) + (Si − 1) ln(1 − λd,i )],and the overall log likelihood function isn
ln[P(y1 , . . . , yn | F0 )] =
ln P(yi | Fi−1 )],
(5.29)
i=1
which is a function of parameters β, γ, θu , and θd .Example 5.2. We illustrate the decomposition model by analyzing the intraday transactions of IBM stock from November 1, 1990 to January 31, 1991. Therewere 63 trading days and 59,838 intraday transactions in the normal trading hours.The explanatory variables used are1. Ai−1 : The action indicator of the previous trade (i.e., the [i − 1]th trade withina trading day).2. Di−1 : The direction indicator of the previous trade.3. Si−1 : The size of the previous trade.4. Vi−1 : The volume of the previous trade, divided by 1000.5. ti−1 : Time duration from the (i − 2)th to (i − 1)th trade.6. B Ai : The bid-ask spread prevailing at the time of transaction.Because we use lag-1 explanatory variables, the actual sample size is 59,775. It turnsout that Vi−1 , ti−1 and B Ai are not statistically significant for the model entertained. Thus, only the first three explanatory variables are used. The model employedis
193
MODELS FOR PRICE CHANGES

piln= β0 + β1 Ai−11 − piδiln= γ0 + γ1 Di−11 − δiλu,iln= θu,0 + θu,1 Si−11 − λu,iλd,iln= θd,0 + θd,1 Si−1 .1 − λd,i
(5.30)
The parameter estimates, using the log-likelihood function in Eq. (5.29), are givenin Table 5.5. The estimated simple model shows some dynamic dependence in theprice change. In particular, the trade-by-trade price changes of IBM stock exhibitsome appealing features:1. The probability of a price change depends on the previous price change.Specifically, we haveP(Ai = 1 | Ai−1 = 0) = 0.258,
P(Ai = 1 | Ai−1 = 1) = 0.476.
The result indicates that a price change may occur in clusters and, as expected,most transactions are without price change. When no price change occurredat the (i − 1)th trade, then only about one out of four trades in the subsequent transaction has a price change. When there is a price change at the(i − 1)th transaction, the probability of a price change in the ith trade increasesto about 0.5.2. The direction of price change is governed by0.483P(Di = 1 | Fi−1 , Ai ) = 0.0850.904
if Di−1 = 0 (i.e., Ai−1 = 0)if Di−1 = 1, Ai = 1if Di−1 = −1, Ai = 1.
This result says that (a) if no price change occurred at the (i − 1)th trade, thenthe chances for a price increase or decrease at the ith trade are about even; and(b) the probabilities of consecutive price increases or decreases are very low.The probability of a price increase at the ith trade given that a price change
Table 5.5. Parameter Estimates of the ADS Model in Eq. (5.30) for IBM Intraday Transactions: 11/01/90 to 1/31/91.Parameter
β0
β1
γ0
γ1
θu,0
θu,1
θd,0
θd,1
EstimateStd.Err.
−1.0570.104
0.9620.044
−0.0670.023
−2.3070.056
2.2350.029
−0.6700.050
2.0850.187
−0.5090.139
194
HIGH - FREQUENCY DATA
occurs at the ith trade and there was a price increase at the (i − 1)th trade isonly 8.6%. However, the probability of a price increase is about 90% giventhat a price change occurs at the ith trade and there was a price decrease at the(i − 1)th trade. Consequently, this result shows the effect of bid-ask bounceand supports price reversals in high-frequency trading.3. There is weak evidence suggesting that big price changes have a higher probability to be followed by another big price change. Consider the size of a priceincrease. We haveSi | (Di = 1) ∼ 1 + g(λu,i ),
λu,i = 2.235 − 0.670Si−1 .
Using the probability mass function of a geometric distribution, we obtain thatthe probability of a price increase by one tick is 0.827 at the ith trade if thetransaction results in a price increase and Si−1 = 1. The probability reduces to0.709 if Si−1 = 2 and to 0.556 if Si−1 = 3. Consequently, the probability ofa large Si is proportional to Si−1 given that there is a price increase at the ithtrade.A difference between the ADS and ordered probit models is that the ADS modeldoes not require any truncation or grouping in the size of a price change.
5.5 DURATION MODELSDuration models are concerned with time intervals between trades. Longer durations indicate lack of trading activities, which in turn signify a period of no newinformation. The dynamic behavior of durations, thus, contains useful informationabout intraday market activities. Using concepts similar to the ARCH models forvolatility, Engle and Russell (1998) propose an autoregressive conditional duration(ACD) model to describe the evolution of time durations for (heavily traded) stocks.Zhang, Russell, and Tsay (2001) extend the ACD model to account for nonlinearityand structural breaks in the data. In this section, we introduce some simple durationmodels. As mentioned before, intraday transactions exhibit some diurnal pattern.Therefore, we focus on the adjusted time durationti∗ =
ti / f (ti ),
(5.31)
where f (ti ) is a deterministic function consisting of the cyclical component of ti .Obviously, f (ti ) depends on the underlying asset and the systematic behavior of themarket. In practice, there are many ways to estimate f (ti ), but no single methoddominates the others in terms of statistical properties. A common approach is to usesmoothing spline. Here we use simple quadratic functions and indicator variables totake care of the deterministic component of daily trading activities.
195
DURATION MODELS
For the IBM data employed in the illustration of ADS models, we assume7
f (ti ) = exp[d(ti )],
d(ti ) = β0 +
β j f j (ti ),
(5.32)
j=1
where
ti − 43200f 1 (ti ) = −14400f 2 (ti ) = −
ti − 483009300
2,
2,
 ti − 38700 2f 3 (ti ) = −75000 2− ti − 486009000f 4 (ti ) =0
if ti < 43200otherwise,if ti ≥ 43200otherwise,
f 5 (ti ) and f 6 (ti ) are indicator variables for the first and second 5 minutes of marketopening [i.e., f 5 (.) = 1 if and only if ti is between 9:30 am and 9:35 am Eastern(c)
-1.0
-1.0
-0.6
-0.6
-0.2
-0.2
(a)
0
100
200minutes
300
400
0
100
300
400
300
400
(d)
-1.0
-2.0
-0.6
-1.0
-0.2
0.0
(b)
200minutes
0
100
200minutes
300
400
0
100
200minutes
Figure 5.5. Quadratic functions used to remove the deterministic component of IBM intradaytrading durations: (a)–(d) are the functions f 1 (.) to f 4 (.) of Eq. (5.32), respectively.
196
HIGH - FREQUENCY DATA
Time], and f 7 (ti ) is the indicator for the last 30 minutes of daily trading [i.e., f 7 (ti ) =1 if and only if the trade occurred between 3:30 pm and 4:00 pm Eastern Time].Figure 5.5 shows the plot of fi (.) for i = 1, . . . , 4, where the time scales in thex-axis is in minutes. Note that f 3 (43,200) = f 4 (43,200), where 43,200 correspondsto 12:00 noon.The coefficients β j of Eq. (5.32) are obtained by the least squares method of thelinear regression7
ln( ti ) = β0 +
β j f j (ti ) + i .j=1
The fitted model isln(ti ) = 2.555 + 0.159 f 1 (ti ) + 0.270 f 2 (ti ) + 0.384 f 3 (ti )+ 0.061 f 4 (ti ) − 0.611 f 5 (ti ) − 0.157 f 6 (ti ) + 0.073 f 7 (ti ).Figure 5.6 shows the time plot of average durations in 5-minute time intervals overthe 63 trading days before and after adjusting for the deterministic component. Part(b)
ave-dur2.62.0
10
2.2
2.4
20
ave-dur
2.8
30
3.0
40
3.2
(a)
0
10 20 30 40 50 605-minute intervals
70
80
0
10 20 30 40 50 605-minute intervals
70
80
Figure 5.6. IBM transactions data from 11/01/90 to 1/31/91: (a) The average durations in 5minute time intervals, and (b) the average durations in 5-minute time intervals after adjustingfor the deterministic component.
197
DURATION MODELS
(a) is the average durations of(b) is the average durations ofis largely removed.
ti and, as expected, it exhibits a diurnal pattern. Partti∗ (i.e., after the adjustment), and the diurnal pattern
5.5.1 The ACD ModelThe autoregressive conditional duration (ACD) model uses the idea of GARCH models to study the dynamic structure of the adjusted duration ti∗ of Eq. (5.31). For easein notation, we define xi = ti∗ .Let ψi = E(xi | Fi−1 ) be the conditional expectation of the adjusted durationbetween the (i − 1)th and ith trades, where Fi−1 is the information set available atthe (i − 1)th trade. In other words, ψi is the expected adjusted duration given Fi−1 .The basic ACD model is defined asxi = ψi i ,
(5.33)
where { i } is a sequence of independent and identically distributed non-negative random variables such that E( i ) = 1. In Engle and Russell (1998), i follows a standardexponential or a standardized Weibull distribution, and ψi assumes the formr
ψi = ω +
s
γ j xi− j +j=1
ω j ψi− j .
(5.34)
j=1
Such a model is referred to as an ACD(r, s) model. When the distribution of i isexponential, the resulting model is called an EACD(r, s) model. Similarly, if i follows a Weibull distribution, the model is a WACD(r, s) model. If necessary, readersare referred to Appendix A for a quick review of exponential and Weibull distributions.Similar to GARCH models, the process ηi = xi − ψi is a Martingale differencesequence [i.e., E(ηi | Fi−1 ) = 0], and the ACD(r, s) model can be written asmax(r,s)
xi = ω +
s
(γ j + ω j )xi− j −j=1
ω j ηi− j + η j ,
(5.35)
j=1
which is in