Predicting Returns and Volatilities with Ultra-High Frequency Data - Implications for the efficient market hypothesis. Robert Engle NYU and UCSD May 2000 Santa Fe

EFFICIENT MARKET HYPOTHESIS In its simplest form asserts that excess returns are unpredictable - possibly even by agents with special information Even if this is true for long horizons, it might not be true at short horizons Microstructure theory discusses the transition to efficiency OUTLINE: 1.Simplest definition of Efficient Market Hypothesis - excess returns are unpredictable based upon some information set 2. Microstructure evidence Some autoregressions 3. Interpretation -

TRANSITION TO EFFICIENCY Glosten-Milgrom(1985), Easley and O’Hara(1987), Easley and O’Hara(1992), Copeland and Galai(1983) and Kyle(1985) Two indistinguishable classes of traders - informed and uninformed Bid and Ask prices are optimally updated by market maker until information is incorporated in prices

CONSEQUENCES Informed traders make excess profits at the expense of uninformed traders. The higher the proportion of informed traders, the faster prices adjust to trades, the wider is the bid ask spread and the lower are the profits per informed trader. In real settings with choice over volumes and speed of trading, informed traders partly reveal their identity, reducing profits.

INFORMED TRADERS What is an informed trader? Information about true value Information about fundamentals Information about quantities Information about who is informed Temporary profits from trading but ultimately will be incorporated into prices

HOW FAST IS THIS TRANSITION? Difficult to estimate Data Problems Discreteness of dependent variable Bid Ask bounce in transaction prices Irregular timing of measurements Measuring independent variables Cannot observe private information trading Must infer information events

SIMPLE STATISTICS First order autoregression of transaction prices (50K observations on IBM) has coefficient of -.4 with t-stat of -101, R2=.16 No implication for trading since cannot buy at the bid price or sell at the ask Same autoregression for midquote has coefficient -.26 with t-stat -62 and R2=.07

TIME SERIES PROPERTIES Both are primarily MA(1) - bid ask bounce for transactions but why for midquotes? Test for autocorrelation after MA(1): Transaction prices LB(15)=52 (>>25) Midquotes LB(15)=1106 (>>>>25)

THEORY The higher the proportion of information traders, the faster prices adjust in trade time When there is information, there is typically a higher proportion of information traders When there is information, traders are in a hurry so trades are close together When there is information, prices adjust very fast in calendar time.

MEASURING INFORMATION When traders are in a hurry, they are more likely to be informed (short durations) When trades are large they are more likely to be informative (except perhaps for block trades) When bid ask spreads are wide, it is likely that the proportion of informed traders is high

EMPIRICAL EVIDENCE Engle, Robert and Jeff Russell,(1998) “Autoregressive Conditional Duration: A New Model for Irregularly Spaced Data, Econometrica Engle, Robert,(2000), “The Econometrics of Ultra-High Frequency Data”, Econometrica Dufour and Engle(2000), “Time and the Price Impact of a Trade”, Journal of Finance, forthcoming Engle and Lunde, “Trades and Quotes - A Bivariate Point Process” Russell and Engle, “Econometric analysis of discrete-valued, irregularly-spaced, financial transactions data” http://weber.ucsd.edu/~mbacci/engle/

APPROACH Model the time to the next price change as a random duration (ACD Model) This is a model of volatility (its inverse) ACD(2,2) with economic predetermined variables Key predictors are transactions/time, volume/transaction, spread

EMPIRICAL EVIDENCE Engle, Robert and Jeff Russell,(1998) “Autoregressive Conditional Duration: A New Model for Irregularly Spaced Data, Econometrica Engle, Robert,(2000), “The Econometrics of Ultra-High Frequency Data”, Econometrica Dufour and Engle(2000), “Time and the Price Impact of a Trade”, Journal of Finance, forthcoming Engle and Lunde, “Trades and Quotes - A Bivariate Point Process” Russell and Engle, “Econometric analysis of discrete-valued, irregularly-spaced, financial transactions data” http://weber.ucsd.edu/~mbacci/engle/

MODELING VOLATILITY WITH TRANSACTION DATA Model the change in midquote from one transaction to the next Build GARCH model of volatility per unit of calendar time Find that short durations and wide spreads predict higher volatilities in the future

EMPIRICAL EVIDENCE Engle, Robert and Jeff Russell,(1998) “Autoregressive Conditional Duration: A New Model for Irregularly Spaced Data, Econometrica Engle, Robert,(2000), “The Econometrics of Ultra-High Frequency Data”, Econometrica Dufour and Engle(2000), “Time and the Price Impact of a Trade”, Journal of Finance, forthcoming Engle and Lunde, “Trades and Quotes - A Bivariate Point Process” Russell and Engle, “Econometric analysis of discrete-valued, irregularly-spaced, financial transactions data” http://weber.ucsd.edu/~mbacci/engle/

APPROACH Measure the time between a trade and a new price quote Predict this based on economic variables correcting for censoring by intervening trades Find that information variables predict quicker price revisions

EMPIRICAL EVIDENCE Engle, Robert and Jeff Russell,(1998) “Autoregressive Conditional Duration: A New Model for Irregularly Spaced Data, Econometrica Engle, Robert,(2000), “The Econometrics of Ultra-High Frequency Data”, Econometrica Dufour and Engle(2000), “Time and the Price Impact of a Trade”, Journal of Finance, forthcoming Engle and Lunde, “Trades and Quotes - A Bivariate Point Process” Russell and Engle, “Econometric analysis of discrete-valued, irregularly-spaced, financial transactions data” http://weber.ucsd.edu/~mbacci/engle/

APPROACH Extend Hasbrouck’s Vector Autoregressive measurement of price impact of trades Measure effect of time between trades on price impact Use ACD to model stochastic process of trade arrivals

SUMMARY The price impacts, the spreads, the speed of quote revisions, and the volatility all respond to information variables TRANSITION IS FASTER WHEN THERE IS INFORMATION ARRIVING Econometric measures of information high shares per trade short duration between trades sustained wide spreads

EMPIRICAL EVIDENCE Engle, Robert and Jeff Russell,(1998) “Autoregressive Conditional Duration: A New Model for Irregularly Spaced Data, Econometrica Engle, Robert,(2000), “The Econometrics of Ultra-High Frequency Data”, Econometrica Dufour and Engle(2000), “Time and the Price Impact of a Trade”, Journal of Finance, forthcoming Engle and Lunde, “Trades and Quotes - A Bivariate Point Process” Russell and Engle, “Econometric analysis of discrete-valued, irregularly-spaced, financial transactions data” http://weber.ucsd.edu/~mbacci/engle/

Graduate School of Business Robert F. Engle Jeffrey R. Russell University of Chicago Graduate School of Business Robert F. Engle University of California, San Diego http://gsbwww.uchicago.edu/fac/jeffrey.russell/research/

Goal: Develop an econometric model for discrete-valued, irregularly-spaced time series data. Method: Propose a class of models for the joint distribution of the arrival times of the data and the associated price changes. Questions: Are returns predictable in the short or long run? How long is the long run? What factors influence this adjustment rate?

Hausman,Lo and MacKinlay Estimate Ordered Probit Model,JFE(1992) States are different price processes Independent variables Time between trades Bid Ask Spread Volume SP500 futures returns over 5 minutes Buy-Sell indicator Lagged dependent variable

A Little Notation Let ti be the arrival time of the ith transaction where t0<t1<t2… A sequence of strictly increasing random variables is called a simple point process. N(t) denotes the associated counting process. Let pi denote the price associated with the ith transaction and let yi=pi-pi-1 denote the price change associated with the ith transaction. Since the price changes are discrete we define yi to take k unique values. That is yi is a multinomial random variable. The bivariate process (yi,ti), is called a marked point process.

We take the following conditional joint distribution of the arrival time ti and the mark yi as the general object of interest: In the spirit of Engle (1996) we decompose the joint distribution into the product of the conditional and the marginal distribution: Engle and Russell (1998)

WITH COVARIATES TRANSITION MATRIX P BECOMES where ei is the ith column of identity matrix. TO INSURE THAT THIS IS A TRANSITION MATRIX FOR ALL POSSIBLE VALUES OF THE COVARIATES, USE INVERSE LOGISTIC TRANSFORMATION

MORE GENERALLY Let matrices have time subscripts and allow other lagged variables: The likelihood is simply a multinomial for each observation conditional on the past

Even more generally, we define the Autoregressive Conditional Multinomial (ACM) model as: Where is the inverse logistic function. Zi might contain ti, a constant term, a deterministic function of time, or perhaps other weakly exogenous variables. We call this an ACM(p,q,r) model.

The data: 58,944 transactions of IBM stock over the 3 months of Nov. 1990 - Jan. 1991 on the consolidated market. (TORQ) 98.6% of the price changes took one of 5 different values.

We therefore consider a 5 state model defined as It is interesting to consider the sample cross correlogram of the state vector xi.

Sample cross correlations of x up 2 up 1 down 1 down 2 up 2 up 1 down 1 down 2

Parameters are estimated using the joint distribution of arrival times and price changes. Initially, we consider simple parameterizations in which the information set for the joint likelihood consists of the filtration of past arrival times and past price changes.

ACM(p,q,r) specification: Where and gj are symmetric. ACD(s,t) Engle and Russell (1998) specifies the conditional probability of the ith event arrival at time ti+t by where

Simulations We perform simulations with spreads, volume, and transaction rates all set to their median value and examine the long run price impact of two consecutive trades that push the price down 1 ticks each. We then perform simulations with spreads, volume and transaction rates set to their 95 percentile values, one at a time, for the initial two trades and then reset them to their median values for the remainder of the simulation.

Price impact of 2 consecutive trades each pushing the price down by 1 tick.

Conclusions 1. Both the realized and the expected duration impact the distribution of the price changes for the data studied. 2. Transaction rates tend to be lower when price are falling. 3. Transaction rates tend to be higher when volatility is higher. 4. Simulations suggest that the long run price impact of a trade can be very sensitive to the volume but is less sensitive to the spread and the transaction rates.