Supervisor: Professor Moisa Altar

Modelling and Forecasting Stock Index Volatility

–a comparison between GARCH models and the Stochastic Volatility model–

Supervisor: Professor Moisa Altar

Table of Contents

Competing volatility models Data description Model estimates and forecasting

performances Concluding remarks

The Stylized Facts The distribution of financial time series has

heavier tails than the normal distribution

Highly correlated values for the squared returns

Changes in the returns tend to cluster

Why model and forecast volatility?

investment

security valuation

risk management

policy issues

Competing Volatility Models

ARCH/GARCH class of models Engle (1982) Bollerslev (1986) Nelson (1991) Glosten, Jaganathan, and Runkle

(1993) Stochastic Volatility (Variance) model

Taylor (1986)

The GARCH model

p

1j jtj

q

1i

2iti0t

ttt

hrh:.eqiancevar

hr:.eqmean

Parameter constraints: ensuring variance to be positive

stationarity condition:

1j0

,1i0

,0

i

i

0

p

j j

q

i i 111

Error distribution1. Normal

The density function:

Implied kurtosis: k=3

The log-likelihood function:

t

t

t

t hhf

2

2

1exp

2

1

T

t t

ttNormal h

hL1

2

ln2ln2

1

2. Student-t Bollerslev (1987)


Implied kurtosis:


2,

2

svar;

s12

s21f t

t21

t2t

21

21t

t

4,

4

23

k

T

t t

ttStudent TL

12

22

21ln1ln

2

12ln

2

1

2ln

2

1ln

3. Generalized Error Distribution (GED) Nelson (1991)


Implied kurtosis:


3

21

;1

2

21

exp

f

2

1

t

t

2351

k

T

t

tGEDL

1

1ln2ln

1

2

1ln

The SV model

2vtt1tt

tt

tttt

0,N~v,vhh:.eqvolatility

)h2

1exp(

)1,0(N~,r:.eqmean

Parameter constraints: stationarity condition:

Linearized form:

1||

ttt

tttttt

vhh

hhry

1

22 27.1)ln()ln(

2

,02 tt VarE

Forecast Evaluation Measures Root Mean Square Error (RMSE)

Mean Absolute Error (MAE)

Theil-U Statistics

LINEX loss function

I

iiiI

RMSE1

222 )ˆ(1

I

iiiI

MAE1

22ˆ1

I

i ii

I

i iiUTheil

1

2221

1

222

)(

)ˆ(

I

iiiii aa

ILINEX

1

2222 1)ˆ())ˆ(exp(1

Data Description

data series: BET-C stock index

time length: April 17, 1998 - April 21, 2003

1255 daily returns

Pt – daily closing value of BET-C

Software: Eviews, Ox

Descriptive statistics for BET-C return seriesMean Median Maximu

mMinimum Std.

Dev.Skewnes

sKurtosis Jarque-

BeraProb.

0.000102

-0.0000519

0.1038602

-0.0975698

0.0153105

0.106634 9.423705 2160.141 0.000

1ttt PlnPlnr

400

500

600

700

800

900

1000

1100

1200

1300

250 500 750 1000 1250

BETC

Daily closing prices of BET-C index

Tested Hypotheses 1. Normality

Histogram of the BET-C returns BET-C return quantile plotted

against the Normal quantile

0

100

200

300

400

500

-10 -5 0 5 10

Series: R100Sample 2 1257Observations 1256

Mean 0.010406Median -0.004572Maximum 10.38602Minimum -9.756982Std. Dev. 1.531051Skewness 0.106367Kurtosis 9.430919

Jarque-Bera 2166.704Probability 0.000000-4

-3

-2

-1

0

1

2

3

4

-.10 -.05 .00 .05 .10 .15

R

Norm

al Q

uantil

e

2.Homoscedasticity

-.12

-.08

-.04

.00

.04

.08

.12

250 500 750 1000 1250

RETURN

.000

.002

.004

.006

.008

.010

.012

250 500 750 1000 1250

SQUARED_RETURN

BET-C return series

BET-C squared return series

3. Stationarity

Unit root tests for BET-C return series

ADF Test Statistic -13.53269 1% Critical Value* -3.4384

5% Critical Value -2.8643


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

PP Test Statistic -28.07887 1% Critical Value* -3.4384



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

4. Serial independence

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

1 4 7 10 13 16 19 22 25 28 31 34

AC

PAC

Autocorrelation coefficients for returns (lags 1 to 36)

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.31 5 9 13

17

21

25

29

33

AC

PAC

Autocorrelation coefficients for squared returns (lags 1 to 36)

Model estimates and forecasting performances

Constant Y(-1) R-squared

Mean equation with intercept -0.000355 0.276034 0.076278

t-statistic (probability that the coefficient equals 0)

-0.768264 (0.4425)

9.087175(0.000)

-

Mean equation without intercept - 0.276769 0.075733

t-statistic (probability that the coefficient equals 0)

- 9.117758(0.000)

-

Mean equation specification

GARCH models

Methodology: - two sets: 1004 observations for model estimation 252 observations for out-of-sample forecast evaluation

Lagnumber

Correlogram of residuals

Correlogram ofsquared residuals

Q-stat Prob Q-stat Prob

1 0.0085 0.927 103.60 0.0005 3.3598 0.645 162.76 0.000

10 5.7904 0.833 165.21 0.00015 8.0496 0.922 167.21 0.000 0

40

80

120

160

200

-0.05 0.00 0.05

Series: ResidualsSample 3 1004Observations 1002

Mean -0.000355Median -0.000463Maximum 0.093143Minimum -0.077582Std. Dev. 0.014613Skewness -0.022081Kurtosis 8.209193

Jarque-Bera 1132.997Probability 0.000000

ARCH Test:

F-statistic 114.8229 Probability 0.000000

Obs*R-squared 103.1921 Probability 0.000000

Residual tests

White Heteroskedasticity Test:



ARCH-LM test and White Heteroscedasticity Test

Autocorrelation tests

Normality test

GARCH (1,1) – Normal Distribution – QML parameter estimatesCoefficient Std.Error t-value Probability

AR (1)

0.302055 0.045561 6.630 0.0000

Constant (V) 0.0000472947 0.141153 3.351 0.0008

ARCH(Alpha1) 0.320832 0.065118 4.927 0.0000

GARCH(Beta1) 0.483147 0.102838 4.698 0.0000

GARCH (1,1) – Student-T Distribution – QML parameter estimates

Coefficient Std.Error t-value Probability

AR(1)

0.280817 0.037364 7.516 0.0000

Constant(V)

0.0000527251 0.144746

3.643 0.0003

ARCH(Alpha1) 0.350230 0.067874

5.160 0.0000

GARCH(Beta1) 0.439533 0.091994

4.778 0.0000

Student(DF)

4.512539 0.656110

6.878 0.0000

Diagnostic test based on the news impact curve (EGARCH vs. GARCH) Test ProbSign Bias t-Test 0.41479 0.67830 Negative Size Bias t-Test 0.66864 0.50373Positive Size Bias t-Test 0.02906 0.97682Joint Test for the Three Effects 0.47585 0.92416

Diagnostic test based on the news impact curve (EGARCH vs. GARCH) Test ProbSign Bias t-Test 0.38456 0.70056Negative Size Bias t-Test 0.81038 0.41772Positive Size Bias t-Test 0.21808 0.82736Joint Test for the Three Effects 0.73189 0.86568

SV– QML parameter estimates

Coefficient Std. Error z-Statistic Probability

C(1) -1.269102 0.450023 -2.820081 0.0048

C(2) 0.858869 0.050340 17.06149 0.0000

C(3) -1.486221 0.456019 -3.259119 0.0011

GARCH (1,1) –GED Distribution – QML parameter estimates

Coefficient Std.Error t-value Probability

AR(1)

0.285181 0.057321 4.975 0.0000

Constant(V)

0.0000496321

0.130000 3.818 0.0001

ARCH(Alpha1) 0.333678 0.062854 5.309 0.0000

GARCH(Beta1) 0.450807 0.091152 4.946 0.0000

Student(DF)

1.172517 0.081401 14.40 0.0000Diagnostic test based on the news impact curve (EGARCH vs. GARCH) Test ProbSign Bias t-Test 0.47340 0.63592Negative Size Bias t-Test 0.82446 0.40968Positive Size Bias t-Test 0.14047 0.88829Joint Test for the Three Effects 0.74931 0.86155

SV model To estimate the SV model, the return series was first filtered in order to eliminate the first order autocorrelation of the returns

In-sample model evaluationa) Residual tests Autocorrelation of the residuals

Lag GARCH(1,1) Nomal GARCH(1,1) Student-T GARCH(1,1) GED SV

Q-stat. p-value Q-stat. p-value Q-stat. p-value Q-stat. p-value1 1.131 0.287 2.289 0.130 2.014 0.156 0.506 0.4775 3.286 0.511 4.755 0.313 4.408 0354 2.802 0.59110 5.654 0.774 7.046 0.632 6.720 0.667 6.237 0.71615 8.679 0.851 10.144 0.752 9.796 0.777 7.571 0.910

Lag GARCH(1,1) Nomal GARCH(1,1) Student-T GARCH(1,1) GED SVQ-stat. p-value Q-stat. p-value Q-stat. p-value Q-stat. p-value

1 0.127 1 0.204 1 0.186 1 0.589 0.4435 3.198 0.362 3.606 0.307 3.499 0.321 2.681 0.61310 6.033 0.644 6.235 0.621 6.180 0.627 6.539 0.68515 6.782 0.913 6.936 0.905 6.895 0.907 8.824 0.842

Autocorrelation of the squared residuals

Kurtosis explanationUnexplained

kurtosisGARCH (1,1) Normal 4.28

GARCH (1,1) Student-t -7.21GARCH (1,1) GED 2.56SV -2.05

b) In-sample forecast evaluation

RMSE MAE THEIL-U1

GARCH 11 Normal 0.0000196062 0.000257336 0.646352

GARCH 11 T 0.0000195026 0.000256516 0.639539

GARCH 11 GED 0.0000194814 0.000253146 0.638149

SV 0.0000186253 0.000231101 0.583293

LINEX a=-20 a=-10 a= 10 a= 20

GARCH 11 Normal 7,70895E-09 1,92751E-09 1,92806E-09 7,71335E-09

GARCH 11 T 7,62777E-09 1,9072E-09 1,90773E-09 7,63198E-09

GARCH 11 GED 7,61114E-09 1,90305E-09 1,90359E-09 7,61545E-09

SV 6,95655E-09 1,73942E-09 1,73999E-09 6,96113E-09

1 Benchmark model - Random Walk

Out-of-sample Forecast Evaluation

Forecast methodology - rolling sample window: 1004 observations - at each step, the n-step ahead forecast is stored - n=1, 5, 10 Benchmark: realized volatility = squared returns

.000

.002

.004

.006

.008

.010

.012

1050 1100 1150 1200 1250

RR

Forecast output

0

0,0005

0,001

0,0015

0,002

0,0025

0,003

0,0035

0,004

0,0045

0,005

1

19 37 55 73 91

109

127

145

163

181

199

217

235

253

1day

5days

10 days

0

0,001

0,002

0,003

0,004

0,005

0,006

1

19 37 55 73 91

109

127

145

163

181

199

217

235

253

1day

5days

10 days

0

0,0005

0,001

0,0015

0,002

0,0025

0,003

0,0035

0,004

0,0045

0,005

1 20 39 58 77 96 115

134

153

172

191

210

229

248

1day

5days

10 days

0

0,0001

0,0002

0,0003

0,0004

0,0005

0,0006

0,0007

0,0008

1 19 37 55 73 91 109

127

145

163

181

199

217

235

253

1 day

5 days

10 days

a) GARCH (1,1) Normal c) GARCH (1,1) GED

b) GARCH (1,1) Student-t d) SV

Evaluation Measures

1-step ahead forecast evaluationRMSE MAE THEIL-U1

GARCH 11 Normal 0,000035300 0,00022591 0,583721

GARCH 11 T 0,000035111 0,000204242 0,580597

GARCH 11 GED 0,000035760 0,000203486 0,591337

SV 0,000048823 0,000253071 0,807336

LINEX a=-20 a=-10 a= 10 a= 20


GARCH 11 T 6,23593E-09 1,55923E-09 1,55971E-09 6,2398E-09

GARCH 11 GED 6,46868E-09 1,61743E-09 1,61795E-09 6,47286E-09

SV 1,2055E-08 3,01454E-09 3,01612E-09 1,20676E-08


5-step ahead forecast evaluation

RMSE MAE THEIL-U1

GARCH 11 Normal 0.0000512767 0.0003042315 0.847915

GARCH 11 T 0.0000512001 0.0003077174 0.846648

GARCH 11 GED 0.0000511668 0.0002983467 0.846097

SV 0.0000511653 0.0002851430 0.846073


LINEX a=-20 a=-10 a= 10 a= 20

GARCH 11 Normal 1.3297E-08 3.325E-09 3.3268E-09 1.33108E-08

GARCH 11 T 1.3257E-08 3.315E-09 3.3169E-09 1.32711E-08

GARCH 11 GED 1.3241E-08 3.311E-09 3.3126E-09 1.32539E-08

SV 1.3239E-08 3.310E-09 3.3125E-09 1.32534E-08

10-step ahead forecast evaluationRMSE MAE THEIL-U1

GARCH 11 Normal 0.0000513675 0.0003060239 0.849416

GARCH 11 T 0.0000513716 0.0003107481 0.849484

GARCH 11 GED 0.0000513779 0.000300542 0.849588

SV 0.0000514735 0.0002870131 0.851169

LINEX a=-20 a=-10 a= 10 a= 20


GARCH 11 T 1,33467E-08 3,33753E-09 3,33925E-09 1,33604E-08

GARCH 11 GED 1,33499E-08 3,33834E-09 3,34007E-09 1,33637E-08

SV 1,33996E-08 3,35077E-09 3,35251E-09 1,34135E-08


Comparison between the statistical features of the two sample periods

In-sample Out-of-sample

Number of observations 1004 252

Mean -0.000468 0.002371

Median -0.000378 0.001137

Maximum 0.093332 0.103860

Minimum -0.097570 -0.065731

Standard Deviation 0.015209 0.015531

Skewness -0.116772 0.925148

Kurtosis 8.666434 11.94869

Jarque-Bera 1344.146 880.2563

Probability 0 0

Concluding remarks

In-sample analysis: a) residual tests: all models may be appropriate; b) evaluation measures: SV model is the best

performer;

Out-of-sample analysis: - for a 1-day forecast horizon GARCH models

outperform SV; - for the 5-day and 10-day forecast horizon, model

performances seem to converge; - the best model changes with forecast horizon and

with forecast evaluation measure; - there is no clear winner;

Concluding remarks

Sample construction problems;

Further research: - allowing for switching regimes; - allowing for leptokurtotic distributions in

the SV - a better proxy for realized volatility;

Bibliography 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; Armstrong, J.S. (1995) - On the Selection of Error Measures for Comparisons Among Forecasting

Methods, Journal of Forecasting; Armstrong, J.S (1978) – Forecasting with Econometric Methods: Folklore versus Fact, Journal of

Business, 51 (4), 1978, 549-564; Bluhm, H.H.W. and J. Yu (2000) - Forecasting volatility: Evidence from the German stock market,

Working paper, University of Auckland; Bollerslev, Tim, Robert F. Engle and Daniel B. Nelson (1994)– ARCH Models, Handbook of

Econometrics, Volume 4, Chapter 49, North Holland; Byström, H. (2001) - Managing Extreme Risks in Tranquil and Volatile Markets Using Conditional

Extreme Value Theory, Department of Economics, Lund University; Christodoulakis, G.A. and Stephen E. Satchell (2002) – Forecasting Using Log Volatility Models, Cass

Business School, Research Paper; Christoffersen, P. F and F. X. Diebold. (1997) - How Relevant is Volatility Forecasting for Financial Risk

Management?, The Wharton School, University of Pennsylvania; 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; 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;

Hol, E. and S. J. Koopman (2000) - Forecasting the Variability of Stock Index Returns with Stochastic Volatility Models and Implied Volatility, Tinbergen Institute Discussion Paper;

Koopman, S.J. and Eugenie Hol Uspenski (2001) –The Stochastic volatility in Mean model: Empirical evidence from international stock markets,

Liesenfeld, R. and R.C. Jung (2000) Stochastic Volatility Models: Conditional Normality versus Heavy-Tailed Distributions, Journal of Applied Econometrics, 15, 137-160;

Lopez, J.A.(1999) – Evaluating the Predictive Accuracy of Volatility Models, Economic Research Deparment, Federal Reserve Bank of San Francisco;

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

Ozaki, T. and P.J. Thomson (1998) – Transformation and Seasonal Adjustment, Technical Report, Institute of Statistics and Operations Research, New Zealand

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;

Peters, J. and S. Laurent (2002) – A Tutorial for G@RCH 2.3, a Complete Ox Package for Estimating and Forecasting ARCH Models;

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; Ruiz, E. (1994) - Quasi-Maximum Likelihood Estimation of Stochastic Volatility Models, Journal of

Econometrics, 63, 289-306; Ruiz, Esther, Angeles Carnero and Daniel Pena (2001) – Is Stochastic Volatility More Flexible than Garch?

, Universidad Carlos III de Madrid, Statistics and Econometrics Series, Working Paper 01-08; Sandmann, G. and S.J. Koopman (1997)– Maximum Likelihood Estimation of Stochastic Volatility Models,

Financial Markets Group, London School of Economics, Discussion Paper 248; Shephard, H. (1993) – Fitting Nonlinear Time-series Models with Applications to stochastic Variance

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Applied Financial Economics, 8, 477-485

Appendix – GARCH mean equation

Dependent Variable: Y

Method: Least Squares

Date: 06/23/03 Time: 00:45

Sample(adjusted): 3 1004

Included observations: 1002 after adjusting endpoints

Variable Coefficient Std. Error t-Statistic Prob.

C -0.000355 0.000462 -0.768264 0.4425

Y(-1) 0.276034 0.030376 9.087175 0.0000

R-squared 0.076278 Mean dependent var -0.000487

Adjusted R-squared 0.075354 S.D. dependent var 0.015204

S.E. of regression 0.014620 Akaike info criterion -5.610880

Sum squared resid 0.213740 Schwarz criterion -5.601080

Log likelihood 2813.051 F-statistic 82.57675

Durbin-Watson stat 2.002722 Prob(F-statistic) 0.000000

1. The AR(1) model with intercept

2.The AR(1) model without intercept

Dependent Variable: Y


Date: 06/23/03 Time: 00:46




Y(-1) 0.276769 0.030355 9.117758 0.0000

R-squared 0.075733 Mean dependent var -0.000487




Log likelihood 2812.755 Durbin-Watson stat 2.003016

Appendix – Residual TestsDate: 06/23/03 Time: 00:48

Sample: 3 1004

Included observations: 1002

Autocorrelation Partial Correlation AC PAC Q-Stat Prob

.| | .| | 1 -0.003 -0.003 0.0085 0.927

.| | .| | 2 -0.011 -0.011 0.1228 0.940

.| | .| | 3 0.041 0.041 1.8102 0.613

.| | .| | 4 0.004 0.004 1.8256 0.768

.| | .| | 5 0.039 0.040 3.3598 0.645

.| | .| | 6 0.030 0.028 4.2395 0.644

.| | .| | 7 0.013 0.014 4.4124 0.731

.| | .| | 8 0.027 0.025 5.1482 0.742

.| | .| | 9 -0.025 -0.027 5.7834 0.761

.| | .| | 10 -0.003 -0.005 5.7904 0.833

.| | .| | 11 0.034 0.029 6.9812 0.801

.| | .| | 12 0.008 0.008 7.0442 0.855

.| | .| | 13 0.030 0.029 7.9561 0.846

.| | .| | 14 -0.007 -0.009 8.0088 0.889

.| | .| | 15 0.006 0.007 8.0496 0.922

.| | .| | 16 -0.049 -0.055 10.543 0.837

.| | .| | 17 0.021 0.020 10.994 0.857

.| | .| | 18 -0.002 -0.008 10.998 0.894

.| | .| | 19 0.007 0.009 11.051 0.922

.| | .| | 20 0.023 0.023 11.599 0.929

Correlogram of Residuals

Date: 06/23/03 Time: 00:49

Sample: 3 1004


Autocorrelation Partial Correlation AC PAC Q-Stat Prob

.|** | .|** | 1 0.321 0.321 103.60 0.000

.|* | .|* | 2 0.194 0.101 141.44 0.000

.|* | .| | 3 0.125 0.041 157.05 0.000

.|* | .| | 4 0.075 0.010 162.73 0.000

.| | .| | 5 0.005 -0.043 162.76 0.000

.| | .| | 6 0.008 0.005 162.82 0.000

.| | .| | 7 0.042 0.045 164.59 0.000

.| | .| | 8 0.024 0.003 165.18 0.000

.| | .| | 9 0.005 -0.012 165.21 0.000

.| | .| | 10 -0.027 -0.040 165.97 0.000

.| | .| | 11 -0.004 0.012 165.98 0.000

.| | .| | 12 -0.009 0.000 166.06 0.000

.| | .| | 13 -0.028 -0.022 166.84 0.000

.| | .| | 14 -0.011 0.005 166.96 0.000

.| | .| | 15 -0.016 -0.012 167.21 0.000

.| | .| | 16 0.007 0.020 167.26 0.000

.| | .| | 17 -0.019 -0.020 167.61 0.000

.| | .| | 18 -0.004 0.005 167.62 0.000

.| | .| | 19 0.000 0.003 167.62 0.000

.| | .| | 20 -0.017 -0.019 167.91 0.000

Correlogram of Squared Residuals

ARCH Test:



Test Equation:

Dependent Variable: RESID^2


Date: 06/23/03 Time: 00:52




C 0.000145 1.83E-05 7.903650 0.0000

RESID^2(-1) 0.321081 0.029964 10.71555 0.0000

R-squared 0.103089 Mean dependent var 0.000213






ARCH-LM test

White Heteroskedasticity Test:



Test Equation:

Dependent Variable: RESID^2


Date: 06/23/03 Time: 00:53

Sample: 3 1004



C 0.000144 1.82E-05 7.933013 0.0000

Y(-1) -0.000222 0.001125 -0.197479 0.8435

Y(-1)^2 0.299471 0.026700 11.21598 0.0000

R-squared 0.112508 Mean dependent var 0.000213






White Heteroskedasticity Test

Documents

Supervisor: Professor Moisa Altar