Transcript Volume, Volatility and Stock Return on the Romanian Stock

Volume, Volatility and Stock Return
on the Romanian Stock Market
Dissertation paper
MSc Student:
Valentin STANESCU
Supervisor:
Professor Moisa ALTAR
Previous research about volume
Lamorieux and Laplace (1991)
 Gallant, Rossi and Tauchen (1992)
 Karpoff (1987)

•

Rogalsky (1978), Smirlock and Starks (1988), Jain
and Joh (1988) and Antoniewicz (1992)
•

contemporaneous stock price-volume relation
traditional Granger causality tests
Baek and Brock (1992), Hiemstra and Jones
(1993,1994)
•
nonlinear Granger tests
The Data

Estimation and training:
•

953 observations 16/6/1997 until 2/08/2001
test data:
•
200 from 3/08/2001 until 1/7/2002
Eliminated non trading days
 Volume = no. of shares
 Price = closing price
 Volume precedes price

Modeling the series

Unit root in price => return. Volume is I(0).
8.8
16
8.4
14
8.0
12
7.6
7.2
10
6.8
8
6.4
6
6.0
4
5.6
5.2
2
250
500
750
250
LOGINCHID
500
750
LOGVOL
.3
6
.2
4
2
.1
0
.0
-2
-.1
-4
-.2
-6
-8
-.3
250
500
LOGRET
750
250
500
DTRLOGVOL
750
Detrended volume
Dependent Variable: LOGVOL
Method: Least Squares
Date: 06/ 23/03 Time: 08:46
Sample: 1 953
Included observations: 953
Variable
Coefficient
C
12.51883
@TREND
-0.004228
Std. Error
0.097349
0.000177
t-Statistic
128.5976
-23.87742
Long term analysis is not a goal of the paper
Short term trend might contain relevant information
Prob.
0.0000
0.0000
Unit root tests: ADF, price
Null Hypothesis: LOGINCHID has a unit root
Exogenous: Constant
Lag Length: 0 (Automatic based on SIC, MAXLAG=21)
Augmented Dickey-Fuller test statistic
Test critical values:
1% level
5% level
10% level
t-Statistic
-2.091756
-3.436998
-2.864364
-2.568326
Prob.*
0.2482
Null Hypothesis: LOGINCHID has a unit root
Exogenous: Constant, Linear Trend
Lag Length: 0 (Automatic based on SIC, MAXLAG=21)
Augmented Dickey-Fuller test statistic
Test critical values:
1% level
5% level
10% level
t-Statistic
-1.386975
-3.967718
-3.414541
-3.129413
Prob.*
0.8644
Unit root tests: PP, price
Null Hypothesis: LOGINCHID has a unit root
Exogenous: Constant
Bandwidth: 6 (Newey-W est using Bartlett kernel)
Phillips-Perron test statistic
Test critical values:
1% level
5% level
10% level
Adj. t-Stat
-2.091832
-3.436998
-2.864364
-2.568326
Prob.*
0.2482
Adj. t-Stat
-1.375348
-3.967718
-3.414541
-3.129413
Prob.*
0.8677
Null Hypothesis: LOGINCHID has a unit root
Exogenous: Constant, Linear Trend
Bandwidth: 7 (Newey-W est using Bartlett kernel)
Phillips-Perron test statistic
Test critical values:
1% level
5% level
10% level
Unit root tests: return
ADF test for return
Null Hypothesis: LOGRE T has a unit root
Exogenous: None
Lag Length: 0 (Automatic based on SIC, MA XLA G=21)
Augmented Dickey-Fuller test statistic
Test critical values:
1% level
5% level
10% level
t-Statistic
-28.82061
-2.567396
-1.941156
-1.616475
Prob.*
0.0000
Adj. t-Stat
-28.75681
-2.567396
-1.941156
-1.616475
Prob.*
0.0000
PP test for return
Null Hypothesis: LOGRE T has a unit root
Exogenous: None
Bandwidth: 7 (Newey-W est using Bartlett kernel)
Phillips-Perron test statistic
Test critical values:
1% level
5% level
10% level
Unit root tests: volume
ADF test for detrended logvolume with no intercept or trend
Null Hypothesis: DTRLOGVOL has a unit root
Exogenous: None
Lag Length: 3 (Automatic based on SIC, MA XLA G=21)
Augmented Dickey-Fuller test statistic
Test critical values:
1% level
5% level
10% level
t-Statistic
-9.333024
-2.567401
-1.941157
-1.616475
Prob.*
0.0000
Adj. t-Stat
-25.50840
-2.567393
-1.941156
-1.616476
Prob.*
0.0000
PP test for detrended logvolume with no intercept or trend
Null Hypothesis: DTRLOGVOL has a unit root
Exogenous: None
Bandwidth: 18 (Newey-West using Bartlett kernel)
Phillips-Perron test statistic
Test critical values:
1% level
5% level
10% level
GARCH equation for the return
Dependent Variable: LOGRE T
Method: ML - ARCH (Marquardt)
Date: 06/ 28/03 Time: 13:08
Sample(adjusted): 2 953
Included observations: 952 after adjusting endpoints
Convergence ac hieved after 16 iterations
Bollerslev-Wooldrige robust standard errors & covariance
Variance backcast: ON
Coefficient
Std.
z-Statistic
Variance Equation
C
0.00013
4.28E-05
3.068983
ARCH(1)
0.14537
0.035324
4.115530
GARCH(1)
0.80495
0.044306
18.16810
R-squared
-0.000878
Mean dependent
Adjusted R-squared
-0.002987
S.D.
dependent
S.E. of regression
0.046581
Akaike
info
Sum squared resid
2.059142
Schwarz criterion
Log likelihood
1634.855
Durbin-Watson
Note the persistence in volatility
Prob.
0.0021
0.0000
0.0000
-0.001378
0.046512
-3.428268
-3.412957
1.864309
Zooming in...
3
2
Notice how the volume spikes up
when the volatility increases
1
0
-1
-2
-3
100
110
120
130
140
150
DTRLOGVOL
.15
.10
.05
.00
Sometimes the reaction of the
volume follows the increase of the
volatility (continuous line) but
sometimes it precedes the turbulent
period (dotted line).
-.05
-.10
Is there a link between the two?
-.15
-.20
100
110
120
130
LOGRET
140
150
Linear Granger tests, volume vs
variance and vs return
Pairwise Granger Causality Tests
Date: 06/ 28/03 Time: 14:06
Sample: 1 953
Lags: 2
Null Hypothesis:
Obs
DTRLOGVOL does not Granger Cause
950
V11LOGRET does not Granger Cause DTRLOGVOL
Pairwise Granger Causality Tests
Date: 06/ 28/03 Time: 14:30
Sample: 1 953
Lags: 2
Null Hypothesis:
LOGRET does
not
Granger
Cause
DTRLOGVOL does not Granger Cause LOGRET
Obs
950
F-Statistic
20.6786
1.12091
F-Statistic
0.83012
2.07032
Probability
1.6E-09
0.32642
Probability
0.43631
0.12672
Volume causes the variance but there is no linear relation to
the return
Explanations for causality

the sequential information arrival models
•

tax and non-tax related motives for trading
•

Lakonishok and Schmidt (1989)
mixture of distributions models
•

Copenland (1976), Jennings, Starks and Fellingham
(1981)
Clark (1973) and Epps and Epps (1976)
noise trader models
•
•
not based on fundamentals
stock returns are positively autocorrelated in the short
run, but negatively autocorrelated in the long run
VAR of Volume and Variance
Vector Autoregression Estimates
Date: 06/23/03 Time: 09:28
Sample(adjusted): 6 953
Included observations: 948 after adjusting
Endpoints
Standard errors in ( ) & t-statistics in [ ]
R-squared
Adj. R-squared
Sum sq. resids
S.E. equation
F-statistic
Log likelihood
Akaike AIC
Schwarz SC
Mean dependent
S.D. dependent
Determinant Residual Covariance
Log Likelihood (d. f. adjusted)
Akaike Information Criteria
Schwarz Criteria
V11LOGRE T
0.818853
0.817310
0.000465
0.000704
530.5791
5540. 649
-11.67015
-11.62406
0.002289
0.001647
DTRLOGVOL
0.205484
0.198715
1707. 575
1.348519
30.35649
-1624.091
3.445340
3.491425
-0.002682
1.506481
9.00E-07
3908. 447
-8.207694
-8.115523
Lags of variance and volume explain 80% of the variance
Volume in variance equation
Dependent Variable: LOGRET
Method: ML - ARCH (Marquardt)
Date: 06/ 23/03 Time: 09:27
Sample(adjusted): 2 953
Included observations: 952 after adjusting endpoints
Convergence ac hieved after 28 iterations
Bollerslev-Wooldrige robust standard errors & covariance
Variance backcast: ON
Coefficient
Std.
z-Statistic
Variance Equation
C
0.001051
0.00028
3.716772
ARCH(1)
0.237641
0.05659
4.199324
GARCH(1)
0.257225
0.14688
1.751144
DTRLOGVOL
0.000216
4.75E4.543745
R-squared
-0.000878
Mean dependent
Adjusted R-squared
-0.004045
S.D. dependent
S.E. of regression
0.046606
Akaike
info
Sum squared resid
2.059142
Schwarz
Log likelihood
1647.243
Durbin-Watson
There is no more persistence in volatility
Prob.
0.0002
0.0000
0.0799
0.0000
-0.001378
0.046512
-3.452191
-3.431777
1.864309
Model for volume
Dependent Variable: DTRLOGVOL
Method: ML - ARCH (Marquardt)
Date: 06/ 27/03 Time: 21:34
Sample(adjusted): 5 953
Included observations: 949 after adjusting endpoints
Convergence ac hieved after 20 iterations
Bollerslev-Wooldrige robust standard errors & covariance
Variance backcast: ON
Coefficient
Std.
z-Statistic
DTRLOGVOL(-1)
0.277545
0.0340
8.141118
DTRLOGVOL(-2)
0.136964
0.0346
3.948569
DTRLOGVOL(-3)
0.128357
0.0327
3.922292
DTRLOGVOL(-4)
0.123724
0.0324
3.813235
Variance Equation
C
0.004705
0.0005
8.617856
ARCH(1)
-0.011571
0.0061
-1.886693
(RES ID<0)*A RCH(1)
0.010752
0.0047
2.254126
GARCH(1)
1.005476
0.0046
214.8090
R-squared
0.202899
Mean dependent
Adjusted R-squared
0.196970
S.D.
dependent
S.E. of regression
1.349367
Akaike
info
Sum squared resid
1713.365
Schwarz criterion
Log likelihood
-1546.996
Durbin-Watson
Prob.
0.0000
0.0001
0.0001
0.0001
0.0000
0.0592
0.0242
0.0000
-0.002110
1.505789
3.277124
3.318055
2.020404
Dummies for volume
Dependent Variable: DTRLOGVOL
Method: Least Squares
Date: 07/ 01/03 Time: 14:56
Sample: 1 953
Included observations: 953
White Heteroskedasticity-Consistent Standard Errors & Covariance
Variable
Coefficient
Std.
t-Statistic
C
0.284290
0.0743
3.824710
M5
-0.418717
0.1701
-2.461389
M7
-0.738126
0.1683
-4.384672
M8
-0.579498
0.1744
-3.322586
M9
-0.553598
0.2007
-2.757699
M10
-0.295297
0.1451
-2.033973
M11
-0.665681
0.1423
-4.675975
R-squared
0.037817
Mean dependent
Adjusted R-squared
0.031714
S.D.
dependent
S.E. of regression
1.478980
Akaike
info
Sum squared resid
2069.263
Schwarz criterion
Log likelihood
-1721.695
F-statistic
Durbin-Watson stat
1.285220
Prob(F-statistic)
Prob.
0.0001
0.0140
0.0000
0.0009
0.0059
0.0422
0.0000
3.79E1.5030
3.6279
3.6635
6.1968
0.0000
Significant coefficients but small and irrelevant R squared.
Implications





return variance was slowly adjusted because of the
persistence, now it is volume dependent
mean return is still set to zero because of a lack of a better
prediction
the volume has a AR mean equation which leads to a
predictable value, unlike the return’s
return variance is forecasted instead of adapted
Applications
• Risk management
• Option strategies
• Delta hedged portfolio
• Other strategies involving the volatility
Granger causality
General: F X I   F X I
 Time series (linear):

t
t 1
t
t 1
 Yt LyLy

X t  AL X t  BL Yt  U X ,t
Yt  C L X t  DL Yt  U Y , y
t  1,2,...,

Non-linear:
•
•
X t  Yt L  X t M   t
is not detected by a linear Granger test
let: X tm  X t , X t 1,..., X t m1 
X tLxLx   X t Lx , X t Lx 1,..., X t 1 
Yt LyLy  Yt Ly , Yt Ly 1,...,Yt 1 
Non-linear causality

Testable implication:

Pr X

Pr X tm  X sm  e | X tLx Lx  X sLx Lx  e, Yt LyLy  YsLy Ly  e 

m
t
 X sm  e | X tLx Lx  X sLx Lx  e
we note:


C1m  Lx, Ly, e   Pr X tmLxLx  X smLxLx  e, Yt LyLy  YsLy
 Ly  e

C 3m  Lx, e   Pr  X
X
C 4Lx, e   Pr  X
X
 e
C 2Lx, Ly, e   Pr X tLx Lx  X sLx Lx  e, Yt LyLy  YsLy
 Ly  e
m  Lx
t  Lx
Lx
t  Lx
Lx
s  Lx
m  Lx
s  Lx
e



Statistic

Statistic:
 C1m  Lx, Ly, e, n C3m  Lx, e, n 
  N 0,  2 m, Lx, Ly, e
n 

C 4Lx, e, n 
 C 2Lx, Ly, e, n

Estimated:
C1m  Lx, Ly, e, n  

 
2
m  Lx
m  Lx
Ly
Ly
I
x
,
x
,
e

I
y
,
y

t  Lx
s  Lx
t  Ly
s  Ly , e


n n  1 s t s

 


2
Lx
Lx
Ly
Ly
C 2Lx, Ly, e, n  
I
x
,
x
,
e

I
y
,
y

t  Lx
s  Lx
t  Ly
s  Ly , e


n n  1 s t s
2
C 3m  Lx, e, n  
I xtmLxLx , xsmLxLx , e

nn  1 s t  s
C 4Lx, e, n  

2
I xtLx Lx , xsLx Lx , e

nn  1 s t  s



Estimation

2
ˆ
ˆ









m
,
Lx
,
Ly
,
e
,
n

d
n

n
d
n
 Variance:
•
d(n) = { 1/C2(Lx,Ly,e,n) , -C1(m+Lx,Ly,e,n)/C22(Lx,Ly,e,n) , 1/C4(Lx,e,n) , C3(m+Lx,e,n)/C42(Lx,e,n) }

K n 


1
ˆ n Aˆ
ˆ
ˆ n 




ˆ i , j n  4   wk n
A
n

A
n

A
 i ,t
j ,t  k 1
j ,t  k 1
j ,t

k 1
 2n  k  1 t

1 

Aˆ1,t n 
  I xtmLxLx , xsmLxLx , e  I ytLy Ly , ysLy Ly , e   C1m  Lx, Ly, e, n
n  1  s t

1 

Aˆ 2,t n 
  I xtLx Lx , xsLx Lx , e  I ytLy Ly , ysLy Ly , e   C 2Lx, Ly, e, n
n  1  s t



 
 


1 

Aˆ3,t n 
  I xtmLxLx , xsmLxLx , e   C3m  Lx, e, n
n  1  s t



1 

Aˆ 4,t n 
  I xtLx Lx , xsLx Lx , e   C 4Lx, e, n
n  1  s t



Nonlinear Granger test





Linear VAR residuals of simple GARCH filtered returns
and of volume factor GARCH filtered returns, scaled to
share a common standard deviation of 1 and thus a
common scale factor e.
Heuristic approach to find “the needle in the hay stack”.
10% statistically significant causality from volume to
return in both cases for T=952, m=1, Lx=8, Ly=6 and
e=0.3
Insignificant results for a relation running from returns to
volume
Problems:
•
•
how to determine the remaining relations?
is the relation always present?
Fuzzy logic and neural networks

Classic algebra
1, if  x  A
1A x   
0, if  x  A

1A B  1A 1B
1A B  min1,1A  1B 
1A   1  1A
com plem ent
Patched function
•
•
•
subsets
noise resistant
triangles are probabilities
Explicit rules
 Internal significance test

The internal mechanism

The fuzzy logic neural network does not extract all the possible rules
and assign probabilities to them, instead it tries to
•




increase or decrease the degree of a fuzzy variable, the number of sets, the
location of the set separator, the links between the subnetworks and the
variables and the probabilities.
The degree of a variable is the number of sets it can belong to
The number of sets is adjusted and the center of all the sets is moved to
minimize their cumulated distance to the observations
For each required output a network is created and they are trained
simultaneously
By changing the connections between the inputs and the subnetworks
of a network an input may be found to be irrelevant to a certain output.
The information set and targets

The information set considers the parameters where the
nonlinear Granger test indicated causality:
• eight lags of the return, absolute return, and squared
return
• current and six lags of volume
• alternate set that included the price

Targets:
•
•
•
current return, absolute return, squared return
buy decision, buy/sell decision
volume (adjusted information set)
Results

Failure...to model:
•
•
•
current return
buy decision
buy/sell decision
Weird results for price => discarded
 Success:

•
•
•

absolute returns
squared returns
volume
All the variables had a degree of two showing that
they belonged to just two subsets.
Squared return

Relevant information:
•

Rules extracted
•
•

volume and 1 lag squared return, divided in two subsets
if volume is small then squared return will almost
surely be small
if volume is large then squared return will almost surely
be large
This confirms the econometric results in which
volume was linked to volatility as in Lamourieux
and Laplace (1991) and Clark’s models.
Absolute returns

Relevant information:
•

Rules extracted:
if volume is small then absolute return will be almost surely small
• if volume is large then absolute return will be small with 32% probability
or large with 68% probability
• if the 1 lag absolute return is small then the current absolute return will be
small with 72% probability or large with 28% probability
• if the 1 lag absolute return is large then the current absolute return will
almost surely be large
Rule 1 and the second part of rule 2 show a positive correlation between
volume and absolute return but the first part indicates the possibility of a
negative correlation. This result is consistent with the sequential information
arrival models of Copeland (1976) and Jennings, Starks and Fellingham
(1981).
•

1 lag absolute return and volume, both with two subsets
Volume


Relevant information: two lags of volume
Rules extracted:
•
•
•
•

if the first lagged volume is small then
surely be small
if the first lagged volume is large then
surely be large
if the second lag is small then current
probability or large with 23% probability
if the second lag is large then current
probability or large with 88% probability
the current volume will almost
the current volume will almost
value will be small with 77%
value will be small with 12%
These rules only confirm the autoregressive in mean, heteroskedastic,
with persistence in variance estimated TARCH model, but only the
first two lags were deemed relevant. It may be that the volume depends
on other factors like the speed of information flow from Clark’s model.
Evaluating the network


Just for the absolute and squared returns, and volume
The network performs its own internal tests when deciding
whether to drop a rule, a subnetwork or make other
adjustments. If the new configuration is better at
explaining the data then the components added were
significant and those removed had little explanatory power.
When evaluating the network on the last 200 observations
these internal mechanisms showed that the network
performed well on data not presented to it in the training
stages.
Conclusions and remarks




In line with the empirical results obtained in the literature on the stock
price-volume relation by indicating the presence of a nonlinear Granger
causality between closing prices and volume.
Suggests that researchers should consider nonlinear theoretical
mechanisms and empirical regularities when devising and evaluating
models of the joint dynamics of stock prices and trading volume. Or the
fact that they could be related through different functions for each subset.
No guidance regarding the source or form of the nonlinear dependence.
The relation between return, variance and volume was partially modeled
but the process governing the volume could not be modeled better than
using a GARCH model. This could be an interesting goal for future
research.
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