Predicting Returns and Volatilities with Ultra-High Frequency Data -

1. Predicting Returns and Volatilities with Ultra-High Frequency Data - Implications for the efficient market hypothesis. Robert Engle NYU and UCSD May 2001 Finnish Statsitical Society Vaasa,Finland http://weber.ucsd.edu/~mbacci/engle/

2. EFFICIENT MARKET HYPOTHESIS • In its simplest form asserts that excess returns are unpredictable - possibly even by agents with special information • Is this true for long horizons? • It is probably not true at short horizons • Microstructure theory discusses the transition to efficiency http://weber.ucsd.edu/~mbacci/engle/

3. Why Don’t Informed Traders Make Easy Profits? • Only by trading can they profit • If others watch their trades, prices will move to reduce the profit • When informed traders are buying, sellers will require higher prices until the advantage is gone. • Trades carry information about prices http://weber.ucsd.edu/~mbacci/engle/

4. 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 • When there is good news, informed traders will buy while the rest will be buyers and sellers. • When there are more buyers than sellers, there is some probability that this is due to information traders – hence prices are increased by sophisticated market makers. http://weber.ucsd.edu/~mbacci/engle/

5. CONSEQUENCES • Informed traders make temporary excess profits at the expense of uninformed traders. • The higher the proportion of informed traders, the • faster prices adjust to trades, • wider is the bid ask spread and • lower are the profits per informed trader. http://weber.ucsd.edu/~mbacci/engle/

6. Easley and O’Hara(1992) • Three possible events- Good news, Bad news and no news • Three possible actions by traders- Buy, Sell, No Trade • Same updating strategy is used http://weber.ucsd.edu/~mbacci/engle/

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8. Easley Kiefer and O’Hara • Empirically estimated these probabilities • Econometrics involves simply matching the proportions of buys, sells and non-trades to those observed. • Does not use (or need) prices, quantities or sequencing of trades http://weber.ucsd.edu/~mbacci/engle/

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13. 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 http://weber.ucsd.edu/~mbacci/engle/

14. HOW FAST IS THIS TRANSITION? • Could be decades in emerging markets • Could be seconds in big liquid markets • Speed depends on market characteristics and on the ability of the market to distinguish between informed and uninformed traders • Transparency is a factor http://weber.ucsd.edu/~mbacci/engle/

15. HOW CAN THE MARKET DETECT INFORMED TRADERS? • When traders are informed, they are more likely to be in a hurry(short durations) • When traders are informed, they prefer to trade large volumes. • When bid ask spreads are wide, it is likely that the proportion of informed traders is high as market makers protect themselves http://weber.ucsd.edu/~mbacci/engle/

16. 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/

17. APPROACH • Model the time to the next price change as a random duration • This is a model of volatility (its inverse) • Model is a point process with dependence and deterministic diurnal effects • NEW ECONOMETRICS REQUIRED http://weber.ucsd.edu/~mbacci/engle/

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19. Econometric Tools • Data are irregularly spaced in time • The timing of trades is informative • Will use Engle and Russell(1998) Autoregressive Conditional Duration (ACD) http://weber.ucsd.edu/~mbacci/engle/

20. THE CONDITIONAL INTENSITY PROCESS • The conditional intensity is the probability that the next event occurs at time t+t given past arrival times and the number of events. http://weber.ucsd.edu/~mbacci/engle/

21. THE ACD MODEL • The statistical specification is: • where xi is the duration=ti-ti-1, is the conditional duration and is an i.i.d. random variable with non-negative support http://weber.ucsd.edu/~mbacci/engle/

22. TYPES OF ACD MODELS • Specifications of the conditional duration: • Specifications of the disturbances • Exponential • Weibul • Generalized Gamma • Non-parametric http://weber.ucsd.edu/~mbacci/engle/

23. MAXIMUM LIKELIHOOD ESTIMATION • For the exponential disturbance • which is so closely related to GARCH that often theorems and software designed for GARCH can be used for ACD. It is a QML estimator. http://weber.ucsd.edu/~mbacci/engle/

24. MODELING PRICE DURATIONS • WITH IBM PRICE DURATION DATA • ESTIMATE ACD(2,2) • ADD IN PREDETERMINED VARIABLES REPRESENTING STATE OF THE MARKET • Key predictors are transactions/time, volume/transaction, spread http://weber.ucsd.edu/~mbacci/engle/

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26. 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/

27. STATISTICAL MODELS • There are two kinds of random variables: • Arrival Times of events such as trades • Characteristics of events called Marks which further describe the events • Let x denote the time between trades called durations and y be a vector of marks • Data: http://weber.ucsd.edu/~mbacci/engle/

28. A MARKED POINT PROCESS • Joint density conditional on the past: • can always be written: http://weber.ucsd.edu/~mbacci/engle/

29. MODELING VOLATILITY WITH TRANSACTION DATA • Model the change in midquote from one transaction to the next conditional on the duration. • Build GARCH model of volatility per unit of calendar time conditional on the duration. • Find that short durations and wide spreads predict higher volatilities in the future http://weber.ucsd.edu/~mbacci/engle/

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31. 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/

32. 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 http://weber.ucsd.edu/~mbacci/engle/

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35. 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 http://weber.ucsd.edu/~mbacci/engle/

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37. 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/

38. 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/ http://weber.ucsd.edu/~mbacci/engle/

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40. 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? http://weber.ucsd.edu/~mbacci/engle/

41. 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 http://weber.ucsd.edu/~mbacci/engle/

42. 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 amultinomialrandom variable. The bivariate process (yi,ti), is called a marked point process. http://weber.ucsd.edu/~mbacci/engle/

43. 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 (2000) we decompose the joint distribution into the product of the conditional and the marginal distribution: Engle and Russell (1998) http://weber.ucsd.edu/~mbacci/engle/

44. SPECIFYING THE PROBABILITY STRUCTURE • Let be a kx1 vector which has a 1 in only one place indicating the current state • Let be the conditional probability of all the states in period i. • A standard Markov chain assumes • Instead we want modifiers of P http://weber.ucsd.edu/~mbacci/engle/

45. RESTRICTIONS • For P to be a transition matrix • It must have non negative elements • All columns must sum to one • To impose these constraints, parameterize P as an inverse logistic function of its determinants http://weber.ucsd.edu/~mbacci/engle/

46. THE PARAMETERIZATION • For each time period t, express the probability of state i relative to a base state k as: • Which implies that: http://weber.ucsd.edu/~mbacci/engle/

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48. MORE GENERALLY • Let matrices have time subscripts and allow other lagged variables: • The ACM likelihood is simply a multinomial for each observation conditional on the past http://weber.ucsd.edu/~mbacci/engle/

49. THE FULL LIKELIHOOD • The sum of the ACD and ACM log likelihood is http://weber.ucsd.edu/~mbacci/engle/

50. 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. http://weber.ucsd.edu/~mbacci/engle/