predicting returns and volatilities with ultra-high frequency data -
DESCRIPTION
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. EFFICIENT MARKET HYPOTHESIS. - PowerPoint PPT PresentationTRANSCRIPT
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EFFICIENT MARKET HYPOTHESIS
In its simplest form asserts that excess returns are unpredictable - possibly even by agents with special informationIs this true for long horizons?It is probably not true at short horizonsMicrostructure theory discusses the transition to efficiency
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Why Don’t Informed Traders Make Easy Profits?
Only by trading can they profitIf others watch their trades, prices will move to reduce the profitWhen 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 O’Hara(1987), Easley and O’Hara(1992), Copeland and Galai(1983) and Kyle(1985)Two indistinguishable classes of traders - informed and uninformedWhen 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 andlower are the profits per informed trader.
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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
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BEGINNING OF DAY
P(INFORMATION)=P(GOOD NEWS)=
P(AGENT IS INFORMED)=P(UNINFORMED WILL BE BUYER)=
P(UNINFORMED WILL TRADE)=
END OF DAY
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Easley Kiefer and O’Hara
Empirically estimated these probabilitiesEconometrics 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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49.9
50.0
50.1
50.2
50.3
10 20 30 40 50 60 70 80 90 100
EVA EVB
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49.9
50.0
50.1
50.2
50.3
10 20 30 40 50 60 70 80 90 100
EVA EVB
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50.00
50.05
50.10
50.15
50.20
50.25
50.30
2 4 6 8 10 12 14
ASK1ASK_EKO
ASK2ASK3
ASK4
ASKING QUOTES WITH VARIOUS FRACTIONSOF INFORMED TRADERS
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50.00
50.05
50.10
50.15
50.20
50.25
50.30
2 4 6 8 10 12 14
EVAEVANEVA2N
EVA3NEVA4NEVA5N
ASK QUOTES AFTER A SEQUENCE OF BUYSWITH INTERVENING NONTRADES
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INFORMED TRADERS
What is an informed trader? Information about true valueInformation about fundamentalsInformation about quantitiesInformation 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 marketsCould be seconds in big liquid marketsSpeed 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 EVIDENCEEngle, 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”, EconometricaDufour and Engle(2000), “Time and the Price Impact of a Trade”, Journal of Finance, forthcomingEngle 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 durationThis is a model of volatility (its inverse)Model is a point process with dependence and deterministic diurnal effectsNEW ECONOMETRICS REQUIRED
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PRICE PATH
Time Price Duration
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Econometric Tools
Data are irregularly spaced in timeThe timing of trades is informativeWill 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.( , ( ); ,..., )
( ( ) ( ) ( ), ,..., )
( )
( )lim
t N t t t
P N t t N t N t t t
t
N t
t
N t
1
0
1
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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
1 1 1 1. ,..., ,..., ;
.
i i i i i
ii i
i E x t t t t
ii x
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TYPES OF ACD MODELS
Specifications of the conditional duration:
Specifications of the disturbancesExponentialWeibulGeneralized GammaNon-parametric
iiii
jijjiji
1i1ii
z,y,x
x
x
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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.
i i
ii
xlogL
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MODELING PRICE DURATIONS
WITH IBM PRICE DURATION DATAESTIMATE ACD(2,2)ADD IN PREDETERMINED VARIABLES REPRESENTING STATE OF THE MARKETKey predictors are transactions/time, volume/transaction, spread
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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)
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EMPIRICAL EVIDENCEEngle, 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”, EconometricaDufour and Engle(2000), “Time and the Price Impact of a Trade”, Journal of Finance, forthcomingEngle 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 tradesCharacteristics of events called Marks
which further describe the events
Let x denote the time between trades called durations and y be a vector of marksData:
}N,...1i),y,x{( ii
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A MARKED POINT PROCESS
Joint density conditional on the past:
can always be written:
);y,xy,x(f~)y,x( i1i1iii1iii
F
1 1
1 1 1 1 1 2
( , , ; )
( , ; ) ( , , ; )
i i i
i i i i i ii
ii
i i
f y
y
x y
g x y y
x
xxqx
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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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GARCH(1,1) GARCH&ECON
VARIABLE Coef Std.Err Z-Stat Coef Std.Err Z-Stat
MEAN
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
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EMPIRICAL EVIDENCEEngle, 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”, EconometricaDufour and Engle(2000), “Time and the Price Impact of a Trade”, Journal of Finance, forthcomingEngle 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 Hasbrouck’s Vector Autoregressive measurement of price impact of tradesMeasure effect of time between trades on price impactUse ACD to model stochastic process of trade arrivals
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Cumulative percentage quote revision after an
unexpected buy
0
0.02
0.04
0.06
0.08
1 3 5 7 9 11 13 15 17 19 21
1/17/91
12/24/90
Transaction Time (t)
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Cumulative percentage quote revision after an unexpected buy
0
0 .02
0 .04
0 .06
0 .080
:00
02:0
5
04:1
0
06:1
5
08:2
0
10:2
5
12:3
0
14:3
5
16:4
0
18:4
5
20:
50
Calendar time (min:sec)
1/17/91
12/24/90
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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 ARRIVINGEconometric measures of information
high shares per tradeshort duration between tradessustained wide spreads
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EMPIRICAL EVIDENCEEngle, 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”, EconometricaDufour and Engle(2000), “Time and the Price Impact of a Trade”, Journal of Finance, forthcomingEngle 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. RussellUniversity of ChicagoGraduate School of Business
Robert F. EngleUniversity of California, San Diego
http://gsbwww.uchicago.edu/fac/jeffrey.russell/research/
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IBM
104.8
104.9
105
105.1
105.2
105.3
105.4
0 2 4 6 8 10 12 14
Time (Minutes)
Tra
nsa
ctio
n P
rice
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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 thisadjustment rate?
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Hausman,Lo and MacKinlay
Estimate Ordered Probit Model,JFE(1992)States are different price processes Independent variables
Time between tradesBid Ask SpreadVolumeSP500 futures returns over 5 minutesBuy-Sell indicatorLagged dependent variable
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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
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11 ,, iiii tytyf
where ,..., 21
1
ii
i yyy and ,...,21
1
ii
i ttt
In the spirit of Engle (2000) 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:
ACD
iii
iii
iiii tytqtyygtytyf 11
?
111 ,,,,
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 stateLet be the conditional probability of all the states in period i.A standard Markov chain assumes
Instead we want modifiers of P
1i iPx
1 1 1, 1( , , , )i i i i iiiP x z tt x
ix
i
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RESTRICTIONS
For P to be a transition matrixIt must have non negative elementsAll 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: , , 1log / , 1,..., 1i t k t i t iA x b for i k
1
1
exp
1 exp
ij iij k
im mm
A bP
A b
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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 .
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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
( ; ) 'log( )ACMt tL x x
1 1 1t t t t t t t t th A x B C h D z
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THE FULL LIKELIHOOD
The sum of the ACD and ACM log likelihood is
( , ; , ) 'log( ) log( ) tt t t
t
L x x
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Even more generally, we define the Autoregressive Conditional Multinomial (ACM) model as:
iji
r
jjtji
q
jjt
p
jjijijti GZhCxBxAh
1,
1,
1,
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
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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.
10-1
70
60
50
40
30
20
10
0
Price Change
Perc
ent
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.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
i
i
i
i
i
ix
We thereforeconsider a 5state model defined as
It is interesting to consider the sample cross correlogram ofthe state vector xi.
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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
up 2
up 1
dow
n 1do
wn 2
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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.
ACD
iii
ACM
iii
iiii tytqtyygtytyf 11111 ,,,,
Parameters are estimated using the joint distribution of arrival times and price changes.
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ACM(p,q,r) specification:
4321
112
2/1
)()ln( gggg
hCxBxVAh
iiiii
ji
r
jjji
q
jj
p
jjijijiji
ACD(s,t) Engle and Russell (1998) specifies the conditionalprobability of the ith event arrival at time ti+by
w
jjij
v
jjij
t
jjij
s
j ji
jiji x
1
2
111lnln
Where and gj are symmetric. 1 iii tt
iiiI
01
1where ,...,...,,| 21,21 iiiii xxttE
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0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
Expected Duration
Vo
lati
lity
Conditional Variance of Price Changes as a Function of Expected Duration
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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.
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-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49
Transaction
Do
llars
Median High Transaction Rate Large Volume Wide Spread
Price impact of 2 consecutive trades each pushing the pricedown by 1 tick.
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-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49
Transaction
Do
llars
High Transaction Rate Large Volume Wide Spread
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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.