1. 2 EFFICIENT MARKET HYPOTHESIS n In its simplest form asserts that excess returns are...

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

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

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

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”

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

PRICE PATH

Time Price Duration

Model 1 Model 2Parameter

.2107(6.14)

.3027(18.22)

1 .0457

(2.60).0507(2.24)

2 .1731

(5.94).1578(5.19)

1 .0769

(1.00).1646(1.61)

2 .5609

(8.07).4600(5.16)

#Trans/Sec -.0440(-12.65)

-.0359(-13.40)

Spread -.0782(-15.68)

Volume/Trans -.0041(-4.58)

EMPIRICAL EVIDENCE

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

GARCH(1,1) GARCH&ECON

VARIABLE Coef Std.Err Z-Stat Coef Std.Err Z-Stat

DURS -0.008 0.004 -1.892 -0.007 0.002 -4.027

AR(1) 0.279 0.023 12.29 0.186 0.022 8.507

MA(1) -0.656 0.019 -33.86 -0.570 0.016 -35.70

VARIANCE

C 0.988 0.092 10.74 -0.111 0.047 -2.358

ARCH(1) 0.245 0.020 12.33 0.250 0.013 18.73

GARCH(1) 0.622 0.025 24.70 0.158 0.014 11.71

1/DUR 0.587 0.028 21.27

DUR/EXPDUR -0.040 0.005 -7.992

LONGVOL(-1) 0.096 0.011 8.801

1/EXPDUR

SPREAD(-1)>> 0.736 0.065 11.29

SIZE>10000 0.193 0.119 1.624

LOGLIK -112246.3 -107406.4

LB(15) 93.092 0.000 40.810 0.000

LB2(15) 30.422 0.004 169.12 0.000

EMPIRICAL EVIDENCE

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

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

Cumulative percentage quote revision after an

unexpected buy

1 3 5 7 9 11 13 15 17 19 21

1/17/91

12/24/90

Transaction Time (t)

Cumulative percentage quote revision after an unexpected buy

0 .080

Calendar time (min:sec)

1/17/91

12/24/90

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

Jeffrey R. RussellUniversity of ChicagoGraduate School of Business

Robert F. EngleUniversity of California, San Diego

http://gsbwww.uchicago.edu/fac/jeffrey.russell/research/

0 2 4 6 8 10 12 14

Time (Minutes)

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 thisadjustment 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

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 takek unique values.That is yi is a multinomial random variable.

The bivariate process (yi,ti), is called a marked point process.

A Little Notation

11 ,, iiii tytyf

where ,..., 21

i yyy and ,...,21

In the spirit of Engle (1996) we decompose the joint distribution into the product of the conditional and the marginal distribution:

We take the following conditional joint distribution of the arrival time ti and the mark yi as the general object of interest:

iiii tytqtyygtytyf 11

111 ,,,,

Engle and Russell (1998)

SPECIFYING THE PROBABILITY STRUCTURE

LET be the kx1 vectors indicating the stateobserved and the conditional probability of all k statesrespectively.

That is, takes the jth column of the kxk identity matrix ifthe jth state occurred.

A first order markov chain(1) links these with a transition probability matrix P with theproperties that

a) all elements are non-negativeb) all columns sum to unity

ttx ~ and ~

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

tj1tittijt z,exexPr)z(PP

1jjkjij

1jjijki

x~P/Plog

x~Plogx~Plog)/log(

w h i c h i m p l i e s t h a t :

]exp[1

]exp[k

R e w r i t i n g t h e k - 1 l o g f u n c t i o n s a s h ( ) t h i s c a n b e w r i t t e n i n s i m p l e

f o r m a s :

( 2 ) bAxh )(

w h e r e A i s a n u n r e s t r i c t e d ( k - 1 ) x ( k - 1 ) m a t r i x , b i s a n u n r e s t r i c t e d

( k - 1 ) x 1 v e c t o r a n d x i s a t h e ( k - 1 ) x 1 s t a t e v e c t o r .

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

ttttttttt zDhCBxAh )()( 111

)log('x);x(L tt

Even more generally, we define the Autoregressive Conditional Multinomial (ACM) model as:

jjijijti GZhCxBxAh

Where is the inverse logistic function.

Zi might contain ti, a constant term, a deterministic functionof time, or perhaps other weakly exogenous variables.

We call this an ACM(p,q,r) model.

)1()1(: KKh

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.

Price Change

.125>p if 1,0,0,0

.125p<0 if 0,1,0,0

0=p if 0,0,0,0

0<p.125- if 0,0,1,0

-.125<p if 0,0,0,1

We thereforeconsider a 5state model defined as

It is interesting to consider the sample cross correlogram ofthe state vector xi.

15 14 13 12 11

10 9 8 7 6

5 4 3 2 1 =lag

Sample cross correlations of x

up 2up 1

down 1down 2

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.

iiii tytqtyygtytyf 11111 ,,,,

Parameters are estimated using the joint distribution of arrival times and price changes.

ACM(p,q,r) specification:

)()ln( gggg

hCxBxVAh

jjijijiji

ACD(s,t) Engle and Russell (1998) specifies the conditionalprobability of the ith event arrival at time ti+by

jiji x

111lnln

Where and gj are symmetric. 1 iii tt

1where ,...,...,,| 21,21 iiiii xxttE

Expected Duration

Conditional Variance of Price Changes as a Function of Expected Duration

Simulations

We perform simulations with spreads, volume, and transactionrates 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 andtransaction rates set to their 95 percentile values, one at atime, for the initial two trades and then reset them to their median values for the remainder of the simulation.

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49

Transaction

Median High Transaction Rate Large Volume Wide Spread

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

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49

Transaction

High Transaction Rate Large Volume Wide Spread

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.

1. 2 EFFICIENT MARKET HYPOTHESIS n In its simplest form asserts that excess returns are...

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