Predicting Returns and Volatilities with Ultra-High Frequency Data - Implications for the efficient market hypothesis.

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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
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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

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Why Dont 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

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TRANSITION TO EFFICIENCY

• Glosten-Milgrom(1985), Easley and OHara(1987),
  Easley and OHara(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.

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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.

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Easley and OHara(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

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Easley Kiefer and OHara

• 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

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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

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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

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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

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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

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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

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Econometric Tools

• Data are irregularly spaced in time
• The timing of trades is informative
• Will use Engle and Russell(1998) Autoregressive
  Conditional Duration (ACD)

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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.

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THE ACD MODEL

• The statistical specification is
• where xi is the durationti-ti-1, is the
  conditional duration and is an i.i.d. random
  variable with non-negative support

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TYPES OF ACD MODELS

• Specifications of the conditional duration
• Specifications of the disturbances
• Exponential
• Weibul
• Generalized Gamma
• Non-parametric

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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.

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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

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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

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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

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A MARKED POINT PROCESS

• Joint density conditional on the past
• can always be written

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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

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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

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APPROACH

• Extend Hasbroucks 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

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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

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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

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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/res
earch/
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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?
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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

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A Little Notation
Let ti be the arrival time of the ith transaction
where t0ltt1ltt2 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
yipi-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.
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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)
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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

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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

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THE PARAMETERIZATION

• For each time period t, express the probability
  of state i relative to a base state k as
• Which implies that

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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

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THE FULL LIKELIHOOD

• The sum of the ACD and ACM log likelihood is

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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.
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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.
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We therefore consider a 5 state model defined as
It is interesting to consider the sample cross
correlogram of the state vector xi.
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Sample cross correlations of x
up 2 up 1 down 1 down 2
up 2 up 1 down 1 down 2
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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.
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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 tit by
where
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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.
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Price impact of 2 consecutive trades each pushing
the price down by 1 tick.
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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.