volume, volatility and stock return on the romanian stock market dissertation paper

Volume, Volatility and Stock Returnon 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)

• contemporaneous stock price-volume relation Rogalsky (1978), Smirlock and Starks (1988), Jain

and Joh (1988) and Antoniewicz (1992) • 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).

5.2

5.6

6.0

6.4

6.8

7.2

7.6

8.0

8.4

8.8

250 500 750

LOGINCHID

-.3

-.2

-.1

.0

.1

.2

.3

250 500 750

LOGRET

2

4

6

8

10

12

14

16

250 500 750

LOGVOL

-8

-6

-4

-2

0

2

4

6

250 500 750

DTRLOGVOL

Detrended volume

Dependent Variable: LOGVOLMethod: Least SquaresDate: 06/23/03 Time: 08:46Sample: 1 953Included observations: 953

Variable Coefficient Std. Error t-Statistic Prob.C 12.51883 0.097349 128.5976 0.0000

@TREND -0.004228 0.000177 -23.87742 0.0000

Long term analysis is not a goal of the paper

Short term trend might contain relevant information

Unit root tests: ADF, price

Null Hypothesis: LOGINCHID has a unit rootExogenous: ConstantLag Length: 0 (Automatic based on SIC, MAXLAG=21)

t-Statistic Prob.*Augmented Dickey-Fuller test statistic -2.091756 0.2482Test critical values: 1% level -3.436998

5% level -2.86436410% level -2.568326

Null Hypothesis: LOGINCHID has a unit rootExogenous: Constant, Linear TrendLag Length: 0 (Automatic based on SIC, MAXLAG=21)


5% level -3.41454110% level -3.129413

Unit root tests: PP, price

Null Hypothesis: LOGINCHID has a unit rootExogenous: ConstantBandwidth: 6 (Newey-West using Bartlett kernel)

Adj. t-Stat Prob.*Phillips-Perron test statistic -2.091832 0.2482Test critical values: 1% level -3.436998

5% level -2.86436410% level -2.568326

Null Hypothesis: LOGINCHID has a unit rootExogenous: Constant, Linear TrendBandwidth: 7 (Newey-West using Bartlett kernel)


5% level -3.41454110% level -3.129413

Unit root tests: return

ADF test for return

Null Hypothesis: LOGRET has a unit rootExogenous: NoneLag Length: 0 (Automatic based on SIC, MAXLAG=21)


5% level -1.94115610% level -1.616475

PP test for return

Null Hypothesis: LOGRET has a unit rootExogenous: NoneBandwidth: 7 (Newey-West using Bartlett kernel)


5% level -1.94115610% level -1.616475

Unit root tests: volumeADF test for detrended logvolume with no intercept or trend

Null Hypothesis: DTRLOGVOL has a unit rootExogenous: NoneLag Length: 3 (Automatic based on SIC, MAXLAG=21)


5% level -1.94115710% level -1.616475

PP test for detrended logvolume with no intercept or trend

Null Hypothesis: DTRLOGVOL has a unit rootExogenous: NoneBandwidth: 18 (Newey-West using Bartlett kernel)


5% level -1.94115610% level -1.616476

GARCH equation for the return

Dependent Variable: LOGRETMethod: ML - ARCH (Marquardt)Date: 06/28/03 Time: 13:08Sample(adjusted): 2 953Included observations: 952 after adjusting endpointsConvergence achieved after 16 iterationsBollerslev-Wooldrige robust standard errors & covarianceVariance backcast: ON

Coefficient Std. z-Statistic Prob. Variance Equation

C 0.00013 4.28E-05 3.068983 0.0021ARCH(1) 0.14537 0.035324 4.115530 0.0000

GARCH(1) 0.80495 0.044306 18.16810 0.0000R-squared -0.000878 Mean dependent -0.001378Adjusted R-squared -0.002987 S.D. dependent 0.046512S.E. of regression 0.046581 Akaike info -3.428268Sum squared resid 2.059142 Schwarz criterion -3.412957Log likelihood 1634.855 Durbin-Watson 1.864309

Note the persistence in volatility

Zooming in...

-3

-2

-1

0

1

2

3

100 110 120 130 140 150

DTRLOGVOL

-.20

-.15

-.10

-.05

.00

.05

.10

.15

100 110 120 130 140 150

LOGRET

Notice how the volume spikes up when the volatility increases

Sometimes the reaction of the volume follows the increase of the volatility (continuous line) but sometimes it precedes the turbulent period (dotted line).

Is there a link between the two?

Linear Granger tests, volume vs variance and vs return

Pairwise Granger Causality TestsDate: 06/28/03 Time: 14:06Sample: 1 953Lags: 2 Null Hypothesis: Obs F-Statistic Probability DTRLOGVOL does not Granger Cause 950 20.6786 1.6E-09 V11LOGRET does not Granger Cause DTRLOGVOL 1.12091 0.32642

Pairwise Granger Causality TestsDate: 06/28/03 Time: 14:30Sample: 1 953Lags: 2 Null Hypothesis: Obs F-Statistic Probability LOGRET does not Granger Cause 950 0.83012 0.43631 DTRLOGVOL does not Granger Cause LOGRET 2.07032 0.12672

Volume causes the variance but there is no linear relation to the return

Explanations for causality

the sequential information arrival models• Copenland (1976), Jennings, Starks and Fellingham

(1981) tax and non-tax related motives for trading

• Lakonishok and Schmidt (1989) mixture of distributions models

• 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 [ ]

V11LOGRET DTRLOGVOL R-squared 0.818853 0.205484 Adj. R-squared 0.817310 0.198715 Sum sq. resids 0.000465 1707.575 S.E. equation 0.000704 1.348519 F-statistic 530.5791 30.35649 Log likelihood 5540.649 -1624.091 Akaike AIC -11.67015 3.445340 Schwarz SC -11.62406 3.491425 Mean dependent 0.002289 -0.002682 S.D. dependent 0.001647 1.506481 Determinant Residual Covariance 9.00E-07 Log Likelihood (d. f. adjusted) 3908.447 Akaike Information Criteria -8.207694 Schwarz Criteria -8.115523

Lags of variance and volume explain 80% of the variance

Volume in variance equation

Dependent Variable: LOGRETMethod: ML - ARCH (Marquardt)Date: 06/23/03 Time: 09:27Sample(adjusted): 2 953Included observations: 952 after adjusting endpointsConvergence achieved after 28 iterationsBollerslev-Wooldrige robust standard errors & covarianceVariance backcast: ON

Coefficient Std. z-Statistic Prob. Variance Equation

C 0.001051 0.00028 3.716772 0.0002ARCH(1) 0.237641 0.05659 4.199324 0.0000

GARCH(1) 0.257225 0.14688 1.751144 0.0799DTRLOGVOL 0.000216 4.75E- 4.543745 0.0000

R-squared -0.000878 Mean dependent -0.001378Adjusted R-squared -0.004045 S.D. dependent 0.046512S.E. of regression 0.046606 Akaike info -3.452191Sum squared resid 2.059142 Schwarz -3.431777Log likelihood 1647.243 Durbin-Watson 1.864309

There is no more persistence in volatility

Model for volume

Dependent Variable: DTRLOGVOLMethod: ML - ARCH (Marquardt)Date: 06/27/03 Time: 21:34Sample(adjusted): 5 953Included observations: 949 after adjusting endpointsConvergence achieved after 20 iterationsBollerslev-Wooldrige robust standard errors & covarianceVariance backcast: ON

Coefficient Std. z-Statistic Prob.DTRLOGVOL(-1) 0.277545 0.0340 8.141118 0.0000DTRLOGVOL(-2) 0.136964 0.0346 3.948569 0.0001DTRLOGVOL(-3) 0.128357 0.0327 3.922292 0.0001DTRLOGVOL(-4) 0.123724 0.0324 3.813235 0.0001

Variance EquationC 0.004705 0.0005 8.617856 0.0000

ARCH(1) -0.011571 0.0061 -1.886693 0.0592(RESID<0)*ARCH(1) 0.010752 0.0047 2.254126 0.0242

GARCH(1) 1.005476 0.0046 214.8090 0.0000R-squared 0.202899 Mean dependent -0.002110Adjusted R-squared 0.196970 S.D. dependent 1.505789S.E. of regression 1.349367 Akaike info 3.277124Sum squared resid 1713.365 Schwarz criterion 3.318055Log likelihood -1546.996 Durbin-Watson 2.020404

Dummies for volume

Dependent Variable: DTRLOGVOLMethod: Least SquaresDate: 07/01/03 Time: 14:56Sample: 1 953Included observations: 953White Heteroskedasticity-Consistent Standard Errors & Covariance

Variable Coefficient Std. t-Statistic Prob.C 0.284290 0.0743 3.824710 0.0001

M5 -0.418717 0.1701 -2.461389 0.0140M7 -0.738126 0.1683 -4.384672 0.0000M8 -0.579498 0.1744 -3.322586 0.0009M9 -0.553598 0.2007 -2.757699 0.0059

M10 -0.295297 0.1451 -2.033973 0.0422M11 -0.665681 0.1423 -4.675975 0.0000

R-squared 0.037817 Mean dependent 3.79E-Adjusted R-squared 0.031714 S.D. dependent 1.5030S.E. of regression 1.478980 Akaike info 3.6279Sum squared resid 2069.263 Schwarz criterion 3.6635Log likelihood -1721.695 F-statistic 6.1968Durbin-Watson stat 1.285220 Prob(F-statistic) 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: Time series (linear):

Non-linear:• is not detected by a linear Granger test• let:

LyLyttttt YIXFIXF 11

,...,2,1

,

,

t

UYLDXLCY

UYLBXLAX

yYttt

tXttt

tMtLtt XYX

11,...,, mtttmt XXXX

11,...,, tLxtLxtLx

Lxt XXXX

11,...,, tLytLytLy

Lyt YYYY

Non-linear causality

Testable implication:

we note:

eXXeXX

eYYeXXeXX

LxLxs

LxLxt

ms

mt

LyLys

LyLyt

LxLxs

LxLxt

ms

mt

|Pr

,|Pr

eYYeXXeLyLxmC LyLys

LyLyt

LxmLxs

LxmLxt

,Pr,,1

eYYeXXeLyLxC LyLys

LyLyt

LxLxs

LxLxt ,Pr,,2

eXXeLxmC LxmLxs

LxmLxt

Pr,3

eXXeLxC LxLxs

LxLxt Pr,4

Statistic

Statistic:

Estimated:

eLyLxmN

neLxC

neLxmC

neLyLxC

neLyLxmCn ,,,,0

,,4

,,3

,,,2

,,,1 2

eyyIexxInn

neLyLxmC LyLys

LyLyt

s st

LxmLxs

LxmLxt ,,,,

1

2,,,1

eyyIexxInn

neLyLxC LyLys

LyLyt

s st

LxLxs

LxLxt ,,,,

1

2,,,2

s st

LxmLxs

LxmLxt exxI

nnneLxmC ,,

1

2,,3

s st

LxLxs

LxLxt exxI

nnneLxC ,,

1

2,,4

Estimation

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) }

ndnndneLyLxm ˆ,,,,ˆ 2

nK

k ttjktjktjtikji nAnAnAnA

knnwn

1,1,1,,,

ˆˆˆˆ12

14ˆ

neLyLxmCeyyIexxIn

nAts

LyLys

LyLyt

LxmLxs

LxmLxtt ,,,1,,,,

1

1ˆ,1

neLyLxCeyyIexxIn

nAts

LyLys

LyLyt

LxLxs

LxLxtt ,,,2,,,,

1

1ˆ,2

neLxmCexxIn

nAts

LxmLxs

LxmLxtt ,,3,,

1

1ˆ,3

neLxCexxIn

nAts

LxLxs

LxLxtt ,,4,,

1

1ˆ,4

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

Patched function• subsets• noise resistant• triangles are probabilities

Explicit rules Internal significance test

Axif

AxifxA ,0

,11

AA

BABA

BABA

complement 111

11,1min1

111

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:• volume and 1 lag squared return, divided in two subsets

Rules extracted• 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:• 1 lag absolute return and volume, both with two subsets

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).

Volume

Relevant information: two lags of volume Rules extracted:

• if the first lagged volume is small then the current volume will almost surely be small

• if the first lagged volume is large then the current volume will almost surely be large

• if the second lag is small then current value will be small with 77% probability or large with 23% probability

• if the second lag is large then current value will be small with 12% probability or large with 88% probability

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.

References Andersen, T. (1992), Return volatility and trading volume in financial markets: An

information flow interpretation of stochastic volatility, Working paper, Northwestern University

Antoniewicz, R. (1992), A causal relationship between stock returns and volume, Working paper, Federal Reserve Board

Baek, E. and W.Brock (1992), A general test for nonlinear Granger causality: Bivariate model, Working paper, Iowa State University and University of Wisconsin, Madison

Berndt, E., B. Hall, and J. Hausman (1974), Estimation and inference in nonlinear structural models, Annals of Economic and Social Measurement 3, 653-665

Brock, W., D. Hsieh, and B. LeBaron (1991), A Test of Nonlinear Dynamics, Chaos, and Instability: Statistical Theory and Economic Evidence, MIT Press, Cambridge.

Campbell, J., S. Grossman, and J. Wang (1993), Trading volume and serial correlation in stock returns, Quarterly Journal of Economics 108, 905-939

Clark, P. (1973), A subordinated stochastic process model with finite variances for speculative prices, Econometrica 41, 135-155

Copeland T. (1976), A model of asset trading under the assumption of sequential information arrival, Journal of Finance 31, 135-155

Dikerson, J, and B. Kosko (1996), Fuzzy function approximation with ellipsoidal rules, IEEE Transations on systems, man, and cybernetics – part B: Cybernetics, vol 26, no. 4

Duffe, G. (1992), Trading volume and return reversals, Working paper, Federal Reserve Board

References Engle, R. (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of

the United Kingdom inflation, Econometrica 50, 987-1007 Epps, T., and M. Epps (1976), The stochastic dependence of security price changes and transaction

volumes: Implications for the mixture of distributions hypothesis, Econometrica 44, 305-321 Fama, E., and K. French (1988), Permanent and temporary components of stock prices, Journal of

Political Economy 96, 246-273 Gallant, R., P. Rossi, and G. Tauchen (1992a), Stock prices and volume, Review of Financial

Studies 5, 199-242 (1992b), Nonlinear dynamic structures, Econometrica 61, 871-907 Hiemstra, C., and J. Jones (1992), Detection and description of linear and nonlinear dependence in

daily Dow Jones stock returns and NYSE trading volume, Working paper, University of Strathclyde and Securities and Exchange Commission

Hinich, M. and D. Patterson, (1985), Evidence of nonlinearity in stock returns, Journal of Business and Economic Statistics 3, 69-77

Hsieh, D. (1991), Chaos and nonlinear dynamics: Application to financial markets, Journal of Finance 46, 1839-1877

Jain, P. and G. Joh (1988), The dependence between hourly prices and trading volume, Journal of Financial and Quantitative Analysis 23, 269-283

References

Jennings, R., L. Starks, and J. Fellingham (1981), An equilibrium model of asset trading with sequential information arrival, Journal of Finance 36, 143-161

Karpoff, J. (1987), The relation between price changes and trading volume: A survey, Journal of Financial and Quantitative Analysis 22, 109-126

Kosko, B. (1988), Bidirectional Associative Memory, IEEE Transactions on systems, man and cybernetics, vol. 18, no. 1

Kosko, B. (1990a), Unsupervised learning in noise, IEEE Transactions on neural networks, vol. 1, no. 1

Kosko, B., and S. Mitaim (1997), Adaptive stochastic resonance, IEEE Transactions on neural networks

Lakonishok, J., and S. Smidt (1989), Past price changes and current trading volume, The Journal of Portfolio Management 15, 18-24

Lamoureux, C., and W. Lastrapes (1990), Heteroskedasticity in stock return data: Volume versus GARCH effects, Journal of Finance 45, 221-229

Scheinkman, J., and B. LeBaron (1989), Nonlinear dynamics and stock returns, Journal of Business 62, 311-337

White, H. (1980), A heteroscedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity, Econometrica 48, 817-838

volume, volatility and stock return on the romanian stock market dissertation paper

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