Nonlinear dynamics: evidence for Bucharest Stock Exchange

Nonlinear dynamics: evidence for Bucharest Stock Exchange

Dissertation paper:

Anca Svoronos(Merdescu)

Goals

To analyse a good volatility model by its ability to capture “stylized facts”

To analyse changes in models behavior with respect to temporal aggregation

To perform an empirical evidence for Bucharest Stock Exchange using its reference index BET

Introduction

The finding of nonlinear dynamics in financial time series dates back to the works of Mandelbrot and Fama in the 1960’s:

- Mandelbrot first noted in 1963 that “large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes”

- Fama developed the efficient-market hypothesis (EMH) – which asserts that financial markets are “informationally efficient”

Volatility Models

GARCH models

Engle (1982)

Bollerslev (1986)

Nelson (1991)

Glosten, Jagannathan and Runkle (1993)

Markov regime switching model

Hamilton (1989)

GARCH models

GARCH

(p,q)

TARCH

(p,q)

EGARCH

(p,q)

p

j jtj

q

i itit

ttt

hrheqiance

hreqmean

11

20:.var

:.

p

j ttjtj

q

i itit drhrheqiance1 1

211

20:.var

it

itp

j jtj

it

itq

i ith

rh

h

rheqiance

110 log

2log:.var

Markov switching model

);N(~: 2i ittt SriS

State

The model assumes the existence of an unobserved variable denoted:

;1,0iwhere

tttt SSr ][ 1010

The conditional mean and variance are defined:

;)( 10 tt SS ;)( 10 tt SS

The transition (=conditional) probabilities are :

qSSob

pSSob

tt

tt

1)01(Pr

)11(Pr

1

1

qSSob

pSSob

tt

tt

)00(Pr

1)10(Pr

1

1

The maximum likelihood will estimate the following vector containing six parameters:

),,,,,( 02

012

1 qp

, t is i.i.d N(0,1).

Data Description

Data series: BET stock index Time length: Jan 3rd, 2001 – March 4th, 2009 2131 daily returns: ]ln[ln*100 1 ttt PPr

Daily closing prices of BET index

2001 2002 2003 2004 2005 2006 2007 20080

2000

4000

6000

8000

10000

12000

Statistical properties of the returns

Non-normal distribution

-13.2 -9.9 -6.6 -3.3 0.0 3.3 6.6 9.9

0

50

100

150

200

250

300

350

Mean 0.059859

Median 0.000000

Maximum 10.09070

Minimum -13.11680

Std. Dev. 1.697205

Skewness -0.676328

Kurtosis 10.29213

Jarque-Bera 4881.677

Probability 0.000000

Observations 2130Histogram of BET returns


Heteroscedasticity

BET squared returns series

2001 2002 2003 2004 2005 2006 2007 20080

25

50

75

100

125

150

175BET return series

2001 2002 2003 2004 2005 2006 2007 2008-15

-10

-5

0

5

10

15


Autocorrelation

- High serial dependence in returns

- The Ljung-Box statistic for 20 lags is 85,75 (0.000)

Daily BET returns correlogram

-0.05

0

0.05

0.1

0.15

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

AC

PAC

Daily BET sq returns correlogram

-0.1

0

0.1

0.2

0.3

0.4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

AC

PAC

-The Ljung-Box statistic for 20 lags is 1442,6 (0.000)

-LM (1): 260,61

=> BET index returns exhibit ARCH effects

6.37

*2

01.0;20

2

RnLM


• BDS independence test (Brocht, Dechert, Scheinkman)of the null hypotheses that time series is independently and identically distributed, is a general test for identifying nonlinear dependence(m=5, ε=0,7)

Dimension BDS Statistic Std. Error z-Statistic Prob.

2 0.035426 0.002149 16.48861 0.0000

3 0.063786 0.003416 18.67311 0.0000

4 0.081721 0.004070 20.07747 0.0000

5 0.090652 0.004246 21.35227 0.0000

• The results presented above show a rejection of the independence hypothesis for all embedding dimensions m


Stationarity: Unit root tests for BET return series

ADF Test Statistic -40.43444 1% Critical Value* -3.433224 5% Critical Value -2.862696 10% Critical Value -2.567431

*MacKinnon critical values for rejection of hypothesis of a unit root.

PP Test Statistic -40.51466 1% Critical Value* -3.433224 5% Critical Value -2.862696 10% Critical Value -2.567431

*MacKinnon critical values for rejection of hypothesis of a unit root.

Models specification (daily data)

Model 1:TARCH (1, 1)

Model 2: GARCH (1,1)

Model 3: EGARCH (1,1)

Model 4: Markov Switching (MS)

dDuahauaah

urbbr

tttt

ttt

21413

2110

121

dhauaah

urbrbrbrbbr

ttt

tttttt

122

110

195144113121

dhahuah

uaah

urbrbrbrbbr

ttt

t

tt

tttttt

113112

1

110

195144113121

)log()/(2

)log(

tttt

tttt

SSSS

SSr

1010

1010

)();()(

][)(

Model Estimates

Variable Coeff T-stat Signif

1. Intercept (b1) 0.111933 3.394027 0.0007

2. AR(1) (b2) 0.140159 5.858505 0.0000

3. Constant (a0) 0.263106 11.750979 0.0000

4. ARCH (a1) 0.169324 8.292505 0.0000

5.Asymmetric coeff (a4) 0.066380** 2.297879 0.0215

6. GARCH (a3) 0.671665 32.13333 0.0000

7. Dummy(π) 26.46984 3.061015 0.0022

Q(20) st res 28.953** - 0.027

Q(20) st sq res 19.120** - 0.124

SIC 3.519069 - -

Model 1 – TGARCH(1,1)

*Denotes significance at the 1% level of significance**Denotes significance at the 5% level of significance

- Null hypothesis of BDS is not rejected at any significance level

- The standardized squared residuals are serially uncorrelated both at 5% and 1% significance level

- Volatility persistence given by is 0,874179 < 1, implying a half life volatility of about 8 days

- > 0 therefore we could stress that a leverage effect exists but testing the null hypothesis of = 0 at 1% level of significance we find that the shock is symmetric

=> a symmetric model specification should be tested

2/431 aaa

4a

4a

Dimension BDS Statistic Std. Error z-Statistic Probab

2 -0.000325 0.001966 -0.165132 0.8688

3 -0.000515 0.003116 -0.165165 0.8688

4 -0.002476 0.003701 -0.669060 0/5035

5 -0.004216 0.003848 -1,095643 0.2732

BDS test

Model Estimates


1. Intercept (b1) 0.10058 3.639956 0.0003

2. AR(1) (b2) 0.140762 5.925356 0.0000

3. AR(11) (b3) 0.033611 2.028312 0.0425

4. AR(14) (b4) 0.024985 1.444312 0.1487

5. AR(19) (b5) 0.032554 1.681572 0.0927

6. Constant (a0) 0.257882 11.59915 0.0000

7. ARCH (a1) 0.205343 11.71204 0.0000

8. GARCH (a2) 0.671857 32.11108 0.0000

9. Dummy (π) 26.55657 3.016097 0.0026

Q(20) st res 19.044** - 0.519

Q(20) st sq res 20.531** - 0.425

SIC 3.518409 - -

Model 2 – GARCH(1,1)

• Null hypothesis of BDS is accepted at any significance level for all 5 dimensions;

•The standardized squared residuals are serially uncorrelated at both significance level of 5% and 1%

• Volatility persistence is 0,8772 < 1, implying a half life volatility of about 8 days, similar to the one implied by Model 1



2 -0.000941 0.001990 -0.472662 0.6365

3 -0.001732 0.003155 -0.548993 0.5830

4 -0.003770 0.003748 -1.005762 0.3145

5 -0.005712 0.003898 -1.465476 0.1428

BDS test

0

20

40

60

80

100

120

2001 2002 2003 2004 2005 2006 2007 2008

VAR_MODEL2

Model Estimates


1. Intercept (b1) 0.049671 2.099358 0.0358

2. AR(1) (b2) 0.124025 5.44934 0.0000

3. AR(11) (b3) 0.037543 2.428243 0.0152

4. AR(14) (b4) 0.026551 1.634159 0.1022

5. AR(19) (b5) 0.015345 0.949694 0.3423

3. Constant (a0) -0.28373 -10.5953 0.0000

4. ARCH (a1) 0.53286 10.4155 0.0000

5. Asymmetric coeff (a2) -0.06249**

-2.05898 0.0395

6. GARCH (a3) 0.857612 42.09454 0.0000

7. Dummy coeff (π) 1.728375 4.90853 0.0000

Q(20) st res 18.731** 0.539

Q(20) st sq res 13.912** 0.835

SIC 3.5308

Model 3 – EGARCH(1,1)


2 -0.002040 0.001939 -1.052297 0.2927

3 -0.004246 0.003073 -1.381835 0.1670

4 -0.006847 0.003650 -1.875949 0.0607

5 -0.009025 0.003795 -2.378192 0.0174

BDS test

- Null hypothesis of BDS is being rejected by dimension m=5 and m=4 if using a significance level of 5%(1,64) and by m=5 for 1%(2,33);

- The standardized squared residuals are serially uncorrelated both at 5% and 1% significance level

- Volatility persistence given by is 0,857612 < 1, implying a half life volatility of about 8 days

- < 0 therefore we can stress a leverage effect exists although testing the null hypothesis of = 0 at 1% level of significance we find that the shock is still symmetric


3a

2a

2a

Model estimates

Model 3 – Markov Switching


1. Mean State 1(a01) 0.161229411 5.42759 0.00000006

2. Variance State 1 (σ1) 0.961946011 19.43944 0.00000000

3. Mean State 2 (a02) -0.206468087

-1.41064 0.15834962

4. Variance State 2 (σ2) 2.813250016 13.61520 0.00000000

5. Matrix of Markov transition probabilities

0.03526796 0.90738168

-

0.96473204 0.09261833

-

SIC 18576

-Both probabilities are quite small which means neither regime is too persistent – there is no evidence for “long swings” hypothesis

-We find slight asymmetry in the persistence of the regimes – upward moves are short and sharp (a01 is positive and p11 is small) and downwards moves could be gradual and drawn out (a02 negative and p22 larger)

-The ML estimates associate state 1 with a 0,16% daily increase while in state 2 the stock index falls by -0,2% with considerably more variability in state 2 than in state 1

-SIC value is significantly higher than the values estimated with GARCH models

Evidence for lower frequencies

Monthly data (99 observations)

0

2000

4000

6000

8000

10000

12000

2001 2002 2003 2004 2005 2006 2007 2008

R_D

-30

-20

-10

0

10

20

30

40

2001 2002 2003 2004 2005 2006 2007 2008

R_D_M

Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque-Bera

Prob.

2.734560 2.482586 29.,76911 -27.07954 8.433922 -0.135998 5.164572 19.43403 0.000

Monthly closing prices for BET

Autocorrelation at lag 1 0.005 Signif. 0.958

Q(20) 4.7020 Signif. 1.0000

LM(1) 0.470581

Signif. 0.9582

Q(20) for squares 12.434 Signif. 0.900

Monthly returns for BET

=> There are no significant evidence of dynamics

Model estimation

GARCH Models failed to converge (see Appendix 3)

Markov Switching models


1. Mean State 1(a01) 2.97641612 3.09876 0.00194335

2. Variance State 1 (σ1) 6.21076790 5.97818 0.00000000

3. Mean State 2 (a02) -3.44338062 -0.67761 0.49801854

4. Variance State 2 (σ2) 15.96667883

6.31980 0.00000000

5. Matrix of Markov transition probabilities

0.06728487 0.83141057

0.93271513 0.16858943

SIC 978

-Two states are again high mean/lower volatility and low mean/higher volatility

-p22 is larger than p11 which means regime 2 should be slightly more persistent – again there is no evidence for “long swings” hypothesis

-again we find asymmetry in the persistence of the regimes

-The ML estimates associate state 1 with an approx 3% monthly increase while in state 2 the stock index falls by -3,5% with considerably more variability in state 2 than in state 1

-In general, the characteristics of the regimes are still present at a monthly frequency in contrast with GARCH

Concluding remarks

If judging from the behavior of residuals, out of the GARCH models, GARCH (1,1) is the model of choice.

Compared with Markov Switching by SIC value we find GARCH(1,1) superior

Considering temporal aggregation, we find that GARCH models fail to converge while Markov Switching model still shows power

Further research:

-forecast ability of both models

Bibliography

Akgiray, V (1989) - Conditional Heteroscedasticity, Journal of Business 62: 55 - 80 Alexander, Carol (2001) – Market Models - A Guide to Financial Data Analysis, John Wiley &Sons, Ltd.; Andersen, T. G. and T. Bollerslev (1997) - Answering the Skeptics: Yes, Standard Volatility Models Do Provide Accurate Forecasts,

International Economic Review; Baillie, T. R. And DeGennaro, R.P. (1990) – Stock Returns and Volatility, Journal of Financial and Quantitative Analysis, vol.25, no.2,

June 1990; Bollerslev, Tim, Robert F. Engle and Daniel B. Nelson (1994)– ARCH Models, Handbook of Econometrics, Volume 4, Chapter 49, North

Holland; Brooks, C (2002) – Introductory econometrics for finance – Cambridge University Press 2002; Byström, H. (2001) - Managing Extreme Risks in Tranquil and Volatile Markets Using Conditional Extreme Value Theory, Department of

Economics, Lund University; DeFusco, A. R., McLeavey D. W., Pinto E. J., Runkle E. D. – Quantitative Methods for Investment Analysis, United Book Press, Inc.,

Baltimore, MD, August 2001; Engle, R.F. (1982) – Autoregressive conditional heteroskedasticity with estimates of the variance of UK inflation, Econometrica, 50, pp.

987-1008; Engle, R.F. and Victor K. Ng (1993) – Measuring and Testing the Impact of News on Volatility, The Journal of Fiance, Vol. XLVIII, No. 5; Engle, R. (2001) – Garch 101: The Use of ARCH/GARCH Models in Applied Econometrics, Journal of Economic Perspectives – Volume

15, Number 4 – Fall 2001 – Pages 157-168; Engle, R. and A. J. Patton (2001) – What good is a volatility model?, Research Paper, Quantitative Finance, Volume 1, 237-245; Engle, R. (2001) – New Frontiers for ARCH Models, prepared for Conference on Volatility Modelling and Forecasting, Perth, Australia,

September 2001; Engle, C and Hamilton, J (1990) – Long Swings in the Dollar:Are they in the Data and Do Markets know it?, American Economic Review

80:689-712; Glosten, L. R., R. Jaganathan, and D. Runkle (1993) – On the Relation between the Expected Value and the Volatility of the Normal

Excess Return on Stocks, Journal of Finance, 48, 1779-1801; Hamilton, J.D. (1994) – Time Series Analysis, Princeton University Press; Hamilton J.D. (1994) – State – Space Models, Handbook of Econometrics, Volume 4, Chapter 50, North Holland; Kanzler L.(1999) – Very fast and correctly sized estimation of the BDS statistic, Department of Economics, University of Oxford;

Kaufmann, S and Scheicher, M (1996) – Markov Regime Switching in Economic Variables:Part I. Modelling, Estimating and Testing.- Part II. A selective survey, Institute for Advanced Studies, Vienna, Economic Series, no.38, Nov. 1996

Kim, D and Kon, S (1994) – Alternative Models for the Conditional Heteroskedasticity of Stock Returns, Journal of Business 67: 563-98;

Nelson, Daniel B. (1991) – Conditional Heteroskedasticity in Asset Returns: A New Approach, Econometrica, 59, 347-370;

Pagan, A. R. and Schwert, G. W. (1990) – Alternative models for conditional stock volatility, Journal of Econometrics 45, 1990, pag. 267-290, North-Holland;

Peters, J. (2001) - Estimating and Forecasting Volatility of Stock Indices Using Asymmetric GARCH Models and (Skewed) Student-T Densities, Ecole d’Administration des Affaires, University of Liege;

Pindyck, R.S and D.L. Rubinfeld (1998) – Econometric Models and Economic Forecasts, Irwin/McGraw-Hill Poon, S.H. and C. Granger (2001) - Forecasting Financial Market Volatility - A Review, University of Lancaster,

Working paper; Rockinger, M. (1994) – Switching Regressions of Unexpected Macroeconomic Events: Explaining the French

Stock Index; Scheicher, M (1994) – Nonlinear Dynamics: Evidence for a small Stock Exchange, Department of Economics,

BWZ University of Viena, Working Paper No. 9607 Sola, M. and Timmerman, A. (1994) – Fitting the moments: A Comparison of ARCH and Regime Switching

Models for Daily Stock Returns Taylor, S.J. (1986) - Modelling Financial Time Series, John Wiley; Terasvirta, T. (1996) - Two Stylized Facts and the GARCH(1,1) Model, W.P. Series in Finance and Economics

96, Stockholm School of Economics; Van Norden, S and Schaller, H (1993) – Regime Switching in Stock Market Returns – Working paper,

November 18th, 1993;

Bibliography

Appendix 1

BDS test for TARCH (1,1)

Appendix 1BDS test for GARCH (1,1)

Appendix 1BDS test for EGARCH (1,1)

Appendix 2

0

100

200

300

400

500

600

-4 -2 0 2 4 6

Series: Standardized ResidualsSample 1/05/2001 3/04/2009Observations 2129

Mean -0.010894Median -0.070050Maximum 6.133064Minimum -5.456463Std. Dev. 1.000382Skewness 0.121775Kurtosis 5.351825

Jarque-Bera 495.9148Probability 0.000000

Residuals histogram following GARCH(1,1)

Appendix 2

0

100

200

300

400

500

600

-4 -2 0 2 4 6




Residuals histogram following GARCH(1,1)

Appendix 2

0

100

200

300

400

500

600

-4 -2 0 2 4 6




Residuals histogram following EGARCH(1,1)

Appendix 3GARCH(1,1) on monthly data

Nonlinear dynamics: evidence for Bucharest Stock Exchange

Documents

Transcript of Nonlinear dynamics: evidence for Bucharest Stock Exchange