ics 9: 583–591.Swanson, N. R. and H. White (1995), A Model-Selection Approach to Assessing the Information inthe Term-Structure Using Linear Models and Neural Networks. Journal of Business and EconomicStatistics 13: 265–275.Taylor, S. J. (1986), Modelling Financial Time Series; John Wiley: Chichester.Tenti, P. (1996), Forecasting Foreign Exchange Rates Using Recurrent Neural Networks. AppliedArtiﬁcial Intelligence 10: 567–581.Theil, H. (1966), Applied Economic Forecasting; North-Holland: Amsterdam.Trippi, R. and E. Turban (eds) (1993), Neural Networks in Finance and Investing – Using ArtiﬁcialIntelligence to Improve Real-World Performance; Probus: Chicago.West, K. D. and D. Cho (1995), The Predictive Ability of Several Models of Exchange Rate Volatility. Journal of Econometrics 69: 367–391.Xu, X. and S. J. Taylor (1996), Conditional Volatility and the Informational Efﬁciency of thePHLX Currency Options Market. In Forecasting Financial Markets, Dunis, C. (ed.); John Wiley:Chichester.Zhang, G., B. E. Patuwo and M. Y. Hu (1998), Forecasting with Artiﬁcial Neural Networks: TheState of the Art. International Journal of Forecasting 14: 35–62.
Analysis of Financial Time Series. Ruey S. TsayCopyright  2002 John Wiley & Sons, Inc.ISBN: 0-471-41544-8
CHAPTER 5
High-Frequency Data Analysisand Market Microstructure
High-frequency data are observations taken at fine time intervals. In finance, theyoften mean observations taken daily or at a finer time scale. These data have becomeavailable primarily due to advances in data acquisition and processing techniques,and they have attracted much attention because they are important in empiricalstudy of market microstructure. The ultimate high-frequency data in finance are thetransaction-by-transaction or trade-by-trade data in security markets. Here time isoften measured in seconds. The Trades and Quotes (TAQ) database of the New YorkStock Exchange (NYSE) contains all equity transactions reported on the Consolidated Tape from 1992 to present, which includes transactions on NYSE, AMEX,NASDAQ, and the regional exchanges. The Berkeley Options Data Base providessimilar data for options transactions from August 1976 to December 1996. Transactions data for many other securities and markets, both domestic and foreign,are continuously collected and processed. Wood (2000) provides some historicalperspective of high-frequency financial study.High-frequency financial data are important in studying a variety of issues relatedto trading process and market microstructure. They can be used to compare the efficiency of different trading systems in price discovery (e.g., the open out-cry systemof NYSE and the computer trading system of NASDAQ). They can also be used tostudy the dynamics of bid and ask quotes of a particular stock (e.g., Hasbrouck, 1999;Zhang, Russell, and Tsay, 2001b). In an order-driven stock market (e.g., the TaiwanStock Exchange), high-frequency data can be used to study the order dynamic and,more interesting, to investigate the question “who provides the market liquidity.”Cho, Russell, Tiao, and Tsay (2000) use intraday 5-minute returns of more than 340stocks traded in the Taiwan Stock Exchange to study the impact of daily stock pricelimits and find significant evidence of magnet effects toward the price ceiling.However, high-frequency data have some unique characteristics that do not appearin lower frequencies. Analysis of these data thus introduces new challenges to financial economists and statisticians. In this chapter, we study these special characteristics, consider methods for analyzing high-frequency data, and discuss implications175
176
HIGH - FREQUENCY DATA
of the results obtained. In particular, we discuss nonsynchronous trading, bid-askspread, duration models, price movements that are in multiples of tick size, andbivariate models for price changes and time durations between transactions associated with price changes. The models discussed are also applicable to other scientificareas such as telecommunications and environmental studies.
5.1 NONSYNCHRONOUS TRADINGWe begin with nonsynchronous trading. Stock tradings such as those on the NYSEdo not occur in a synchronous manner; different stocks have different trading frequencies, and even for a single stock the trading intensity varies from hour to hourand from day to day. Yet we often analyze a return series in a fixed time interval suchas daily, weekly, or monthly. For daily series, price of a stock is its closing price,which is the last transaction price of the stock in a trading day. The actual time of thelast transaction of the stock varies from day to day. As such we incorrectly assumedaily returns as an equally-spaced time series with a 24-hour interval. It turns outthat such an assumption can lead to erroneous conclusions about the predictabilityof stock returns even if the true return series are serially independent.For daily stock returns, nonsynchronous trading can introduce (a) lag-1 crosscorrelation between stock returns, (b) lag-1 serial correlation in a portfolio return,and (c) in some situations negative serial correlations of the return series of a singlestock. Consider stocks A and B. Assume that the two stocks are independent andstock A is traded more frequently than stock B. For special news affecting the marketthat arrives near the closing hour on one day, stock A is more likely than B to showthe effect of the news on the same day simply because A is traded more frequently.The effect of the news on B will eventually appear, but it may be delayed until thefollowing trading day. If this situation indeed happens, return of stock A appearsto lead that of stock B. Consequently, the return series may show a significant lag1 cross-correlation from A to B even though the two stocks are independent. Fora portfolio that holds stocks A and B, the prior cross-correlation would become asignificant lag-1 serial correlation.In a more complicated manner, nonsynchronous trading can also induce erroneousnegative serial correlations for a single stock. There are several models available inthe literature to study this phenomenon; see Campbell, Lo, and MacKinlay (1997)and the references therein. Here we adopt a simplified version of the model proposed in Lo and MacKinlay (1990). Let rt be the continuously compounded returnof a security at the time index t. For simplicity, assume that {rt } is a sequence ofindependent and identically distributed random variables with mean E(rt ) = µ andvariance Var(rt ) = σ 2 . For each time period, the probability that the security is nottraded is π, which is time-invariant and independent of rt . Let rto be the observedreturn. When there is no trade at time index t, we have rto = 0 because there isno information available. Yet when there is a trade at time index t, we define rto asthe cumulative return from the previous trade (i.e., rto = rt + rt−1 + · · · + rt−kt ,where kt is the largest non-negative integer such that no trade occurred in the periods
177
NONSYNCHRONOUS TRADING
t − kt , t − kt + 1, . . . , t − 1). Mathematically, the relationship between rt and rto is
rto =
0rtrt + rt−1r + r+rt
t−1
...ki=0 rt−i...
with probability πwith probability (1 − π)2with probability (1 − π)2 πwith probability (1 − π)2 π 2...
t−2
(5.1)
with probability (1 − π)2 π k−1...
These probabilities are easy to understand. For example, rto = rt if and only if thereare trades at both t and t − 1, rto = rt + rt−1 if and only if there are trades at t andt − 2, but no trade at t − 1, and rto = rt + rt−1 + rt−2 if and only if there are tradesat t and t − 3, but no trades at t − 1 and t − 2, and so on. As expected, the totalprobability is 1 given byπ + (1 − π)2 [1 + π + π 2 + · · ·] = π + (1 − π)2
1= π + 1 − π = 1.1−π
We are ready to consider the moment equations of the observed return series {rto }.First, the expectation of rto isE(rto ) = (1 − π)2 E(rt ) + (1 − π)2 π E(rt + rt−1 ) + · · ·= (1 − π)2 µ + (1 − π)2 π2µ + (1 − π)2 π 2 3µ + · · ·= (1 − π)2 µ[1 + 2π + 3π 2 + 4π 3 + · · ·]= (1 − π)2 µ
1= µ.(1 − π)2
(5.2)
1In the prior derivation, we use the result 1 + 2π + 3π 2 + 4π 3 + · · · = (1−π)2 . Next,
for the variance of rto , we use Var(rto ) = E[(rto )2 ] − [E(rto )]2 andE(rto )2 = (1 − π)2 E[(rt )2 ] + (1 − π)2 π E[(rt + rt−1 )2 ] + · · ·
= (1 − π)2 [(σ 2 + µ2 ) + π(2σ 2 + 4µ2 ) + π 2 (3σ 2 + 9µ2 ) + ·]
(5.3)
= (1 − π) {σ [1 + 2π + 3π + · · ·] + µ [1 + 4π + 9π + · · ·]}2= σ 2 + µ2−1 .1−π
(5.4)
2
2
2
2
2
(5.5)
In Eq. (5.3), we useE
rt−ii=0

2
k

k
= Var
rt−ii=0
2
k
+ E
rt−ii=0
= (k + 1)σ 2 + [(k + 1)µ]2
178
HIGH - FREQUENCY DATA
under the serial independence assumption of rt . Using techniques similar to that ofEq. (5.2), we can show that the first term of Eq. (5.4) reduces to σ 2 . For the secondterm of Eq. (5.4), we use the identity1 + 4π + 9π 2 + 16π 3 + · · · =
21−,3(1 − π)(1 − π)2
which can be obtained as follows: LetH = 1 + 4π + 9π 2 + 16π 3 + · · ·
and
G = 1 + 3π + 5π 2 + 7π 3 + · · · .
Then (1 − π)H = G and(1 − π)G = 1 + 2π + 2π 2 + 2π 3 + · · ·= 2(1 + π + π 2 + · · ·) − 1 =
2− 1.(1 − π)
Consequently, from Eqs. (5.2) and (5.5), we have22πµ2o22− 1 − µ2 = σ 2 +.Var(rt ) = σ + µ1−π1−π
(5.6)
o ) =Consider next the lag-1 autocovariance of {rto }. Here we use Cov(rto , rt−1oooE(rtort−1) − E(rt0 )E(rt−1) = E(rtort−1) − µ2 . The question then reduces to findingooooE(rt rt−1 ). Notice that rt rt−1 is zero if there is no trade at t, no trade at t − 1, or notrade at both t and t − 1. Therefore, we have0with probability 2π − π 2rt rt−1with probability (1 − π)3rt (rt−1 + rt−2 )with probability (1 − π)3 πr (r3 2t t−1 + rt−2 + rt−3 ) with probability (1 − π) πo(5.7)rto rt−1= ......kwith probability (1 − π)3 π k−1rt ( i=1 rt−i )......
Again the total probability is unity. To understand the prior result, notice thatorto rt−1= rt rt−1 if and only if there are three consecutive trades at t − 2, t − 1,and t. Using Eq. (5.7) and the fact that E(rt rt− j ) = E(rt )E(rt− j ) = µ2 for j > 0,we haveo) = (1 − π)3 {E(rt rt−1 ) + π E[rt (rt−1 + rt−2 )]E(rtort−13
+ π 2 E rt
rt−i
+ · · ·}
i=1
= (1 − π)3 µ2 [1 + 2π + 3π 2 + · · ·] = (1 − π)µ2 .
BID - ASK SPREAD
179
The lag-1 autocovariance of {rto } is thenoCov(rto , rt−1) = −πµ2 .
(5.8)
Provided that µ is not zero, the nonsynchronous trading induces a negative lag-1autocorrelation in rto given byρ1 (rto ) =
−(1 − π)πµ2.(1 − π)σ 2 + 2πµ2
In general, we can extend the prior result and show thato2 jCov(rto , rt−j ) = −µ π ,
j ≥ 1.
The magnitude of the lag-1 ACF depends on the choices of µ, π, and σ and canbe substantial. Thus, when µ = 0, the nonsynchronous trading induces negativeautocorrelations in an observed security return series.The previous discussion can be generalized to the return series of a portfolio thatconsists of N securities; see Campbell, Lo, and MacKinlay (1997, Chapter 3). Inthe time series literature, effects of nonsynchronous trading on the return of a singlesecurity are equivalent to that of random temporal aggregation on a time series, withthe trading probability π governing the mechanism of aggregation.
5.2 BID-ASK SPREADIn some stock exchanges (e.g., NYSE) market makers play an important role in facilitating trades. They provide market liquidity by standing ready to buy or sell whenever the public wishes to sell or buy. By market liquidity, we mean the ability to buyor sell significant quantities of a security quickly, anonymously, and with little priceimpact. In return for providing liquidity, market makers are granted monopoly rightsby the exchange to post different prices for purchases and sales of a security. Theybuy at the bid price Pb and sell at a higher ask price Pa . (For the public, Pb is thesale price and Pa is the purchase price.) The difference Pa − Pb is call the bid-askspread, which is the primary source of compensation for market makers. Typically,the bid-ask spread is small—namely, one or two ticks.The existence of bid-ask spread, although small in magnitude, has several important consequences in time series properties of asset returns. We briefly discuss thebid-ask bounce—namely, the bid-ask spread introduces negative lag-1 serial correlation in an asset return. Consider the simple model of Roll (1984). The observedmarket price Pt of an asset is assumed to satisfySPt = Pt∗ + It ,2
(5.9)
180
HIGH - FREQUENCY DATA
where S = Pa − Pb is the bid-ask spread, Pt∗ is the time-t fundamental value of theasset in a frictionless market, and {It } is a sequence of independent binary randomvariables with equal probabilities (i.e., It = 1 with probability 0.5 and = −1 withprobability 0.5). The It can be interpreted as an order-type indicator, with 1 signifying buyer-initiated transaction and −1 seller-initiated transaction. Alternatively, themodel can be written asPt = Pt∗ +
+S/2 with probability 0.5,−S/2 with probability 0.5.
If there is no change in Pt∗ , then the observed process of price changes isSPt = (It − It−1 ) .2
(5.10)
Under the assumption of It in Eq. (5.9), E(It ) = 0 and Var(It ) = 1, and we haveE( Pt ) = 0 andVar( Pt ) = S 2 /2
(5.11)
Cov( Pt ,
Pt−1 ) = −S 2 /4
(5.12)
Cov( Pt ,
Pt− j ) = 0,
(5.13)
Therefore, the autocorrelation function ofρ j ( Pt ) =
j > 1.
Pt is
−0.50
if j = 1,if j > 1.
(5.14)
The bid-ask spread thus introduces a negative lag-1 serial correlation in the seriesof observed price changes. This is referred to as the bid-ask bounce in the financeliterature. Intuitively, the bounce can be seen as follows. Assume that the fundamental price Pt∗ is equal to (Pa + Pb )/2. Then Pt assumes the value Pa or Pb . If thepreviously observed price is Pa (the higher value), then the current observed price iseither unchanged or lower at Pb . Thus, Pt is either 0 or −S. However, if the previous observed price is Pb (the lower value), then Pt is either 0 or S. The negativelag-1 correlation in Pt becomes apparent. The bid-ask spread does not introduceany serial correlation beyond lag 1, however.A more realistic formulation is to assume that Pt∗ follows a random walk so that∗Pt∗ = Pt∗ − Pt−1= t , which forms a sequence of independent and identicallydistributed random variables with mean zero and variance σ 2 . In addition, { t } isindependent of {It }. In this case, Var( Pt ) = σ 2 + S 2 /2, but Cov( Pt , Pt− j )remains unchanged. Therefore,ρ1 ( Pt ) =
−S 2 /4S 2 /2 + σ 2
≤ 0.
EMPIRICAL CHARACTERISTICS
181
The magnitude of lag-1 autocorrelation of Pt is reduced, but the negative effectremains when S = Pa − Pb > 0. In finance, it might be of interest to study thecomponents of the bid-ask spread. Interested readers are referred to Campbell, Lo,and MacKinlay (1997) and the references therein.The effect of bid-ask spread continues to exist in portfolio returns and in multivariate financial time series. Consider the bivariate case. Denote the bivariate order-typeindicator by It = (I1t , I2t ) , where I1t is for the first security and I2t for the secondsecurity. If I1t and I2t are contemporaneously correlated, then the bid-ask spreadscan introduce negative lag-1 cross-correlations.
5.3 EMPIRICAL CHARACTERISTICS OF TRANSACTIONS DATALet ti be the calendar time, measured in seconds from midnight, at which the i-thtransaction of an asset takes place. Associated with the transaction are several variables such as the transaction price, the transaction volume, the prevailing bid and askquotes, and so on. The collection of ti and the associated measurements are referredto as the transactions data. These data have several important characteristics that donot exist when the observations are aggregated over time. Some of the characteristicsare given next.1. Unequally spaced time intervals: Transactions such as stock tradings on anexchange do not occur at equally spaced time intervals. As such the observedtransaction prices of an asset do not form an equally spaced time series. Thetime duration between trades becomes important and might contain usefulinformation about market microstructure (e.g., trading intensity).2. Discrete-valued prices: The price change of an asset from one transaction tothe next only occurs in multiples of tick size. In the NYSE, the tick size wasone eighth of a dollar before June 24, 1997, and was one sixteenth of a dollarbefore January 29, 2001. All NYSE and AMEX stocks started to trade in decimals on January 29, 2001. Therefore, the price is a discrete-valued variable intransactions data. In some markets, price change may also be subject to limitconstraints set by regulators.3. Existence of a daily periodic or diurnal pattern: Under the normal trading conditions, transaction activity can exhibit periodic pattern. For instance, in theNYSE, transactions are heavier at the beginning and closing of the tradinghours and thinner during the lunch hours, resulting in a “U-shape” transaction intensity. Consequently, time durations between transactions also exhibita daily cyclical pattern.4. Multiple transactions within a single second: It is possible that multiple transactions, even with different prices, occur at the same time. This is partly dueto the fact that time is measured in seconds that may be too long a time scalein periods of heavy tradings.
182
HIGH - FREQUENCY DATA
Table 5.1. Frequencies of Price Change in Multiples of Tick Size for IBM Stock fromNovember 1, 1990 to January 31, 1991.Number (tick)
≤ −3
−2
−1
0
1
2
≥3
Percentage
0.66
1.33
14.53
67.06
14.53
1.27
0.63
To demonstrate these characteristics, we consider first the IBM transactions datafrom November 1, 1990 to January 31, 1991. These data are from the Trades, OrdersReports, and Quotes (TORQ) dataset; see Hasbrouck (1992). There are 63 tradingdays and 60,328 transactions. To simplify the discussion, we ignore the price changesbetween trading days and focus on the transactions that occurred in the normal trading hours from 9:30 am to 4:00 pm Eastern Time. It is well known that overnightstock returns differ substantially from intraday returns; see Stoll and Whaley (1990)and the references therein. Table 5.1 gives the frequencies in percentages of pricechange measured in the tick size of $1/8 = $0.125. From the table, we make thefollowing observations:1. About two-thirds of the intraday transactions were without price change.2. The price changed in one tick approximately 29% of the intraday transactions.3. Only 2.6% of the transactions were associated with two-tick price changes.4. Only about 1.3% of the transactions resulted in price changes of three ticks ormore.5. The distribution of positive and negative price changes was approximatelysymmetric.Consider next the number of transactions in a 5-minute time interval. Denote theseries by xt . That is, x1 is the number of IBM transactions from 9:30 am to 9:35 amon November 1, 1990 Eastern time, x2 is the number of transactions from 9:35 am to9:40 am, and so on. The time gaps between trading days are ignored. Figure 5.1(a)shows the time plot of xt , and Figure 5.1(b) the sample ACF of xt for lags 1 to 260. Ofparticular interest is the cyclical pattern of the ACF with a periodicity of 78, whichis the number of 5-minute intervals in a trading day. The number of transactionsthus exhibits a daily pattern. To further illustrate the daily trading pattern, Figure 5.2shows the average number of transactions within 5-minute time intervals over the 63days. There are 78 such averages. The plot exhibits a “smiling” or “U” shape, indicating heavier tradings at the opening and closing of the market and thinner tradingsduring the lunch hours.Since we focus on transactions that occurred in the normal trading hours of atrading day, there are 59,838 time intervals in the data. These intervals are called theintraday durations between trades. For IBM stock, there were 6531 zero time intervals. That is, during the normal trading hours of the 63 trading days from November 1, 1990 to January 31, 1991, multiple transactions in a second occurred 6531times, which is about 10.91%. Among these multiple transactions, 1002 of them had
n(trades)0 20 40 60 80 1200
1000
200030005-minute time intervals
4000
5000
ACF0.0 0.2 0.4 0.6 0.8 1.0
Series : x
0
50
100
150
200
250
Lag
10
average15
20
25
Figure 5.1. IBM intraday transactions data from 11/01/90 to 1/31/91: (a) the number of transactions in 5-minute time intervals, and (b) the sample ACF of the series in part(a).
0
10
20
3040505-minute time intervals
60
70
80
Figure 5.2. Time plot of the average number of transactions in 5-minute time intervals. Thereare 78 observations, averaging over the 63 trading days from 11/01/90 to 1/31/91 for IBMstock.183
184
HIGH - FREQUENCY DATA
Table 5.2. Two-Way Classification of Price Movements in Consecutive Intraday Tradesfor IBM Stock. The Price Movements Are Classified Into “Up,” “Unchanged,” and“Down.” The Data Span is From 11/01/90 to 1/31/91.ith trade(i − 1)th trade
“+”
“0”
“−”
Margin
“+”“0”“−”
44148674580
5498297794841
39485473410
9887401199831
Margin
9888
40118
9831
59837
different prices, which is about 1.67% of the total number of intraday transactions.Therefore, multiple transactions (i.e., zero durations) may become an issue in statistical modeling of the time durations between trades.Table 5.2 provides a two-way classification of price movements. Here price movements are classified into “up,” “unchanged,” and “down.” We denote them by “+,”“0,” and “−,” respectively. The table shows the price movements between two consecutive trades (i.e., from the [i − 1]th to the ith transaction) in the sample. From thetable, trade-by-trade data show that1. consecutive price increases or decreases are relatively rare, which are about441/59837 = 0.74% and 410/59837 = 0.69%, respectively;2. there is a slight edge to move from “up” to “unchanged” than to “down”; seerow 1 of the table;3. there is a high tendency for price to remain “unchanged”;4. the probabilities of moving from “down” to “up” or “unchanged” are about thesame. See row 3.The first observation mentioned before is a clear demonstration of bid-ask bounce,showing price reversals in intraday transactions data. To confirm this phenomenon,we consider a directional series Di for price movements, where Di assumes the value+1, 0, −1 for “up,” “unchanged,” and “down” price movement, respectively, for theith transaction. The ACF of {Di } has a single spike at lag 1 with value −0.389, whichis highly significant for a sample size of 59,837 and confirms the price reversal inconsecutive trades.As a second illustration, we consider the transactions data of IBM stock inDecember 1999 obtained from the TAQ database. The normal trading hours are from9:30 am to 4:00 pm Eastern time, except for December 31 when the market closedat 13:00 pm. Comparing with the 1990–1991 data, two important changes haveoccurred. First, the number of intraday tradings has increased sixfold. There were134,120 intraday tradings in December 1999 alone. The increased trading intensityalso increased the chance of multiple transactions within a second. The percentageof trades with zero time duration doubled to 22.98%. At the extreme, there were
185
EMPIRICAL CHARACTERISTICS
0
2000
number of trades400060008000
10000
after-hourregular
day
Figure 5.3. IBM transactions data for December 1999. The plot shows the number of transactions in each trading day with the after-hours portion denoting the number of trades withtime stamp after 4:00 pm.
42 transactions within a given second that happened twice on December 3, 1999.Second, the tick size of price movement was $1/16 = $0.0625 instead of $1/8.The change in tick size should reduce the bid-ask spread. Figure 5.3 shows the dailynumber of transactions in the new sample. Figure 5.4(a) shows the time plot of timedurations between trades, measured in seconds, and Figure 5.4(b) is the time plot ofprice changes in consecutive intraday trades, measured in multiples of the tick sizeof $1/16. As expected, Figures 5.3 and 5.4(a) show clearly the inverse relationshipbetween the daily number of transactions and the time interval between trades. Figure 5.4(b) shows two unusual price movements for IBM stock on December 3, 1999.They were a drop of 63 ticks followed by an immediate jump of 64 ticks and a dropof 68 ticks followed immediately by a jump of 68 ticks. Unusual price movementslike these occurred infrequently in intraday transactions.Focusing on trades recorded within the regular trading hours, we have 61,149trades out of 133,475 with no price change. This is about 45.8% and substantiallylower than that between November 1990 and January 1991. It seems that reducingthe tick size increased the chance of a price change. Table 5.3 gives the