1
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
2
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
3
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.
4
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
5
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
6
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
7
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)
8
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.
9
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
10
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/en
gle/
11
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
12
PRICE PATH
Time
Price Duration
13
Model 1
Model 2
Parameter

.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)
-.0440
(-12.65)
-.0359
(-13.40)
#Trans/Sec
Spread
Volume/Trans
-.0782
(-15.68)
-.0041
(-4.58)
14
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/en
gle/
15
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
16
GARCH(1,1)
VARIABLE
Coef
Std.Err Z-Stat
GARCH&ECON
Coef
Std.Err Z-Stat
MEAN
DURS
-0.008
AR(1)
0.279
MA(1)
-0.656
0.004 -1.892 -0.007
0.002 -4.027
0.023
0.022
12.29
0.186
8.507
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
0.587
0.028
21.27
1/DUR
DUR/EXPDUR
-0.040
0.005 -7.992
LONGVOL(-1)
0.096
0.011
8.801
SPREAD(-1)>>
0.736
0.065
11.29
SIZE>10000
0.193
0.119
1.624
1/EXPDUR
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
17
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/en
gle/
18
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
19
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/en
gle/
20
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
21
Cumulative percentage quote revision after an
unexpected buy
0.08
0.06
0.04
0.02
1/17/91
12/24/90
0
1
3
5
7
9
11
13
15
17
19
21
Transaction Time (t)
22
Cumulative percentagequote revisionafter an
unexpected buy
0.08
1/17/91
0.06
0.04
12/24/90
0.02
20:50
18:45
16:40
14:35
12:30
10:25
08:20
06:15
04:10
02:05
0:00
0
Calendar time(min:sec)
23
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
24
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/en
gle/
25
Jeffrey R. Russell
Robert F. Engle
University of Chicago
University of California, San Diego
Graduate School of Business
http://gsbwww.uchicago.edu/fac/jeffrey.russell/research/
26
IBM
Transaction Price
105.4
105.3
105.2
105.1
105
104.9
104.8
0
2
4
6
8
10
12
14
Time (Minutes)
27
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?
28
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
29
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.
30
We take the following conditional joint distribution of the
arrival time ti and the mark yi as the general object of interest:

f yi , ti y i 1 ,t i 1

where y i 1   yi 1 , yi  2 ,...  and t i 1  ti 1 , ti  2 ,...
In the spirit of Engle (1996) we decompose the joint distribution
into the product of the conditional and the marginal distribution:

 


f yi , ti y i 1 ,t i 1  g yi y i 1 ,t i  q ti y i 1 ,t i 1

 
 


?
ACD
Engle and Russell (1998)
31
SPECIFYING THE PROBABILITY STRUCTURE
LET ~
xt and ~t be the kx1 vectors indicating the state
observed and the conditional probability of all k states
respectively.
~
th
That is, xt takes the j column of the kxk identity matrix if
th
the j state occurred.
A first order markov chain
~
~
(1)  t  Pxt 1
links these with a transition probability matrix P with the
properties that
a) all elements are non-negative
b) all columns sum to unity
32
WITH COVARIATES

TRANSITION MATRIX P BECOMES

  
Pt  Pij ( z t )  Pr xt  ei xt 1  e j , z t

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
33
 k ~ 
 k

log(  i /  k )  log   Pij x j   log   Pkj ~
xj 
 j 1

 j 1






 log Pij / Pkj ~x j
k
j 1

k 1
 Aij x j  bi
j 1
which implies that:
Pij 
exp[ Aij  bi ]
k 1
1   exp[ Alj  bl ]
l 1
34
Rewriting the k-1 log functions as h() this can be written in simple
form as:
(2)
h( )  Ax  b
where A is an unrestricted (k-1)x(k-1) matrix, b is an unrestricted
(k-1)x1 vector and x is a the (k-1)x1 state vector.
35
MORE GENERALLY

Let matrices have time subscripts and
allow other lagged variables:
h( t )  At xt 1  Bt t 1  Ct h( t 1 )  Dt zt

The likelihood is simply a multinomial
for each observation conditional on the
past
L( x; )   xt ' log(  t )
36
Even more generally, we define the Autoregressive Conditional
Multinomial (ACM) model as:
h i    At , j xi  j   i  j    Bt , j xi  j   Ct , j h i  j   GZi
p
q
r
j 1
j 1
j 1
Where h : ( K 1)  ( K 1) 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.
37
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.
70
60
P ercent
50
40
30
20
10
0
-1
0
1
P ric e C ha ng e
38
We therefore
consider a 5
state model
defined as
1,0,0,0 if p < -.125
i

0,1,0,0 if - .125  p < 0
i



xi  0,0,0,0 if p i = 0


0,0,1,0 if 0 < p i  .125


0,0,0,1 if p i > .125

It is interesting to consider the sample cross correlogram of
the state vector xi.
39
Sample cross correlations of x
lag = 1






  
   
  

  
3
    
   


   


    
    
   


   


    
7
8
9
10
    
   


   








    
   


   








    
   


   








14
15
up 2
up 1
down 1
down 2
up 2
up 1
down 1
down 2
2
6
    
   


   














  
   
  

  
11
12
13
    
   


   


    
    
   


   


    
    
   


   


    
4






  
   
  

  
    
   


   


    
5
    
   


   


    
    
   


   


    
40
Parameters are estimated using the joint distribution of arrival
times and price changes.

 


f yi , ti y i 1 ,t i 1  g yi y i 1 ,t i  q ti y i 1 ,t i 1

 
 


ACM
ACD
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.
41
ACM(p,q,r) specification:
h i      A V
p
1/ 2
j i j
j 2
x
i j
  i  j    B j xi  j   C j h i  j  
q
r
j 1
j 1
ln(  i ) g1   i g 2  ( i  i ) g 3   i g 4
Where  i  ti  ti 1 and gj are symmetric.
ACD(s,t) Engle and Russell (1998) specifies the conditional
probability of the ith event arrival at time ti+ by
 
  I i 1   0   where  i  E  | ti 1 , ti  2, ..., xi 1 , xi  2 ,...
 i  i 
1
v
w
 i j t
ln  i       j
   j ln  i  j     j xi  j    j i2 j
j 1
j 1
j 1
 i  j j 1
s
42
Conditional Variance of Price Changes as a Function of Expected Duration
0.008
0.007
0.006
0.004
0.003
0.002
0.001
4.
9
4.
6
4.
3
4
3.
7
3.
4
3.
1
2.
8
2.
5
2.
2
1.
9
1.
6
1.
3
1
0.
7
0.
4
0
0.
1
Volatility
0.005
Expected Duration
43
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.
44
Price impact of 2 consecutive trades each pushing the price
down by 1 tick.
0
1
4
7
10
13
16
19
22
25
28
31
34
37
40
43
46
49
-0.05
Dollars
-0.1
-0.15
-0.2
-0.25
-0.3
Transaction
Median
High Transaction Rate
Large Volume
Wide Spread
45
0
1
4
7
10
13
16
19
22
25
28
31
34
37
40
43
46
49
-0.01
Dollars
-0.02
-0.03
-0.04
-0.05
-0.06
Transaction
High Transaction Rate
Large Volume
Wide Spread
46
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.
47