Volatility Modelling of Asset Prices using GARCH Models18949/FULLTEXT01.pdf · Volatility Modelling...

Volatility Modelling of Asset Pricesusing GARCH Models

Jens Nasstrom

Reg nr: LiTH-ISY-EX-3364-2003

February 11, 2003

Volatility Modelling of Asset Pricesusing GARCH Models

Master’s ThesisDivision of Automatic Control

Department of Electrical EngineeringLinkoping University, Sweden

Jens Nasstrom

Reg nr: LiTH-ISY-EX-3364-2003

Supervisor: O.Prof. Manfred DeistlerProf. Lennart Ljung

Examiner: Prof. Lennart Ljung

February 11, 2003

Avdelning, InstitutionDivision, Department

Institutionen för Systemteknik581 83 LINKÖPING

DatumDate2003-02-11

SpråkLanguage

RapporttypReport category

ISBN

Svenska/SwedishX Engelska/English

LicentiatavhandlingX Examensarbete ISRN LITH-ISY-EX-3364-2003

C-uppsatsD-uppsats Serietitel och serienummer

Title of series, numberingISSN

Övrig rapport____

URL för elektronisk versionhttp://www.ep.liu.se/exjobb/isy/2003/3364/

TitelTitle

Volatilitets prediktering av finansiella tillgångar - med GARCH modeller som ansats

Volatility Modelling of Asset Prices using GARCH Models

Författare Author

Jens Näsström

SammanfattningAbstractThe objective for this master thesis is to investigate the possibility to predict the risk of stocks infinancial markets. The data used for model estimation has been gathered from different branchesand different European countries. The four data series that are used in the estimation are priceseries from: Münchner Rück, Suez-Lyonnaise des Eaux, Volkswagen and OMX, a Swedish stockindex. The risk prediction is done with univariate GARCH models. GARCH models are estimatedand validated for these four data series.

Conclusions are drawn regarding different GARCH models, their numbers of lags anddistributions. The model that performs best, out-of-sample, is the APARCH model but the standardGARCH is also a good choice. The use of non-normal distributions is not clearly supported. Theresult from this master thesis could be used in option pricing, hedging strategies and portfolioselection.

NyckelordKeywordGARCH models, risk prediction, system identification and econometrics

Fur meine Eltern

Abstract

The objective for this master thesis is to investigate the possibility to predictthe risk of stocks in financial markets. The data used for model estimationhas been gathered from different branches and different European coun-tries. The four data series that are used in the estimation are price seriesfrom: Munchner Ruck, Suez-Lyonnaise des Eaux, Volkswagen and OMX, aSwedish stock index. The risk prediction is done with univariate GARCHmodels. GARCH models are estimated and validated for these four dataseries.

Conclusions are drawn regarding different GARCH models, their num-bers of lags and distributions. The model that performs best, out-of-sample,is the APARCH model but the standard GARCH is also a good choice. Theuse of non-normal distributions is not clearly supported. The result fromthis master thesis could be used in option pricing, hedging strategies andportfolio selection.

Keywords: GARCH models, risk prediction, system identification and eco-nometrics

Contents

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Presenting the companies . . . . . . . . . . . . . . . . . . . . 11.3 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Problem specification . . . . . . . . . . . . . . . . . . . . . . . 21.5 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6 Reader’s guide . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Method 52.1 Econometrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 System Identification . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 The data set . . . . . . . . . . . . . . . . . . . . . . . 62.2.2 Volatility . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.3 Selected models for estimation . . . . . . . . . . . . . 72.2.4 Criterion of fit . . . . . . . . . . . . . . . . . . . . . . 72.2.5 Validation . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Methods of estimation, toolboxes . . . . . . . . . . . . . . . . 92.3.1 The Ox GARCH package . . . . . . . . . . . . . . . . 10

3 Theory 133.1 Economic theory . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1.1 Financial data . . . . . . . . . . . . . . . . . . . . . . 133.1.2 An introduction to risk . . . . . . . . . . . . . . . . . 143.1.3 Efficient market theory . . . . . . . . . . . . . . . . . 143.1.4 Leverage effect . . . . . . . . . . . . . . . . . . . . . . 14

3.2 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.1 Correlation . . . . . . . . . . . . . . . . . . . . . . . . 153.2.2 Autocorrelation . . . . . . . . . . . . . . . . . . . . . . 153.2.3 Partial autocorrelation . . . . . . . . . . . . . . . . . . 16

3.3 Hypothesis tests . . . . . . . . . . . . . . . . . . . . . . . . . 163.3.1 Jarque-Bera test . . . . . . . . . . . . . . . . . . . . . 163.3.2 Ljung-Box test . . . . . . . . . . . . . . . . . . . . . . 173.3.3 ARCH test . . . . . . . . . . . . . . . . . . . . . . . . 17

iv Contents

3.3.4 Parameter output and t-test . . . . . . . . . . . . . . . 173.3.5 Nelson test . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4 Stochastic processes . . . . . . . . . . . . . . . . . . . . . . . 183.4.1 White noise . . . . . . . . . . . . . . . . . . . . . . . . 183.4.2 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . 193.4.3 Moving average process . . . . . . . . . . . . . . . . . 193.4.4 Autoregressive process . . . . . . . . . . . . . . . . . . 19

3.5 Computing volatility . . . . . . . . . . . . . . . . . . . . . . . 193.5.1 EWMA - Exponentially weighted moving average . . . 21

3.6 ARCH - Autoregressive conditional heteroskedasticity . . . . 213.7 Generalized ARCH - GARCH . . . . . . . . . . . . . . . . . . 22

3.7.1 GARCH(1,1) . . . . . . . . . . . . . . . . . . . . . . . 233.7.2 IGARCH . . . . . . . . . . . . . . . . . . . . . . . . . 243.7.3 EGARCH . . . . . . . . . . . . . . . . . . . . . . . . . 243.7.4 APARCH . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.8 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Volkswagen 274.1 Presenting data . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.1.1 Introducing Volkswagen . . . . . . . . . . . . . . . . . 274.2 Preestimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2.1 Data formatting . . . . . . . . . . . . . . . . . . . . . 274.2.2 Moments for the return series . . . . . . . . . . . . . . 284.2.3 Plots for Volkswagen . . . . . . . . . . . . . . . . . . . 284.2.4 Hypothesis tests . . . . . . . . . . . . . . . . . . . . . 29

4.3 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.3.1 Choosing p and q . . . . . . . . . . . . . . . . . . . . . 314.3.2 Choosing model . . . . . . . . . . . . . . . . . . . . . 324.3.3 Choosing distribution . . . . . . . . . . . . . . . . . . 324.3.4 Estimated parameters . . . . . . . . . . . . . . . . . . 33

4.4 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.4.1 Parameter time dependence . . . . . . . . . . . . . . . 344.4.2 Autocorrelation . . . . . . . . . . . . . . . . . . . . . . 344.4.3 Ljung-Box-Pierce Q-test . . . . . . . . . . . . . . . . . 354.4.4 Engle’s ARCH-test . . . . . . . . . . . . . . . . . . . . 354.4.5 Is zt normally distributed? . . . . . . . . . . . . . . . 364.4.6 Skewness of zt . . . . . . . . . . . . . . . . . . . . . . 37

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.5.1 Estimation . . . . . . . . . . . . . . . . . . . . . . . . 37

5 Munchner Ruck 395.1 Presenting data . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.1.1 Introducing Munchner Ruck . . . . . . . . . . . . . . . 395.2 Preestimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Contents v

5.2.1 Data formatting . . . . . . . . . . . . . . . . . . . . . 405.2.2 Moments for the return series . . . . . . . . . . . . . . 405.2.3 Plots for Munchner Ruck . . . . . . . . . . . . . . . . 405.2.4 Hypothesis tests . . . . . . . . . . . . . . . . . . . . . 40


5.4 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.4.1 Time dependent parameters . . . . . . . . . . . . . . . 465.4.2 Ljung-Box-Pierce Q-test . . . . . . . . . . . . . . . . . 475.4.3 Engle’s ARCH-test . . . . . . . . . . . . . . . . . . . . 475.4.4 How is zt distributed? . . . . . . . . . . . . . . . . . . 48


6 Suez-Lyonnaise des Eaux 516.1 Presenting data . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.1.1 Introducing Suez-Lyonnaise des Eaux . . . . . . . . . 516.2 Preestimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.2.1 Data formatting . . . . . . . . . . . . . . . . . . . . . 516.2.2 Moments for the return series . . . . . . . . . . . . . . 526.2.3 Correlation plots for lyoe . . . . . . . . . . . . . . . . 536.2.4 Hypothesis tests . . . . . . . . . . . . . . . . . . . . . 54

6.3 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.3.1 Choosing p and q . . . . . . . . . . . . . . . . . . . . . 566.3.2 Choosing model . . . . . . . . . . . . . . . . . . . . . 576.3.3 Choosing distribution . . . . . . . . . . . . . . . . . . 576.3.4 Estimated parameters . . . . . . . . . . . . . . . . . . 586.3.5 Are the coefficients time dependent? . . . . . . . . . . 58

6.4 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596.4.1 Parameter time dependency . . . . . . . . . . . . . . . 606.4.2 Autocorrelation . . . . . . . . . . . . . . . . . . . . . . 606.4.3 How is zt distributed? . . . . . . . . . . . . . . . . . . 616.4.4 Ljung-Box-Pierce Q-test . . . . . . . . . . . . . . . . . 616.4.5 Engle’s ARCH-test . . . . . . . . . . . . . . . . . . . . 63


7 OMX index 657.1 Presenting data . . . . . . . . . . . . . . . . . . . . . . . . . . 65

7.1.1 Introducing OMX . . . . . . . . . . . . . . . . . . . . 657.2 Preestimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

vi Contents

7.2.1 Data formatting . . . . . . . . . . . . . . . . . . . . . 657.2.2 Moments for the return series . . . . . . . . . . . . . . 657.2.3 Correlation plots for OMX . . . . . . . . . . . . . . . 667.2.4 Hypothesis tests . . . . . . . . . . . . . . . . . . . . . 66


7.4 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717.4.1 Parameter time dependence . . . . . . . . . . . . . . . 717.4.2 Autocorrelation . . . . . . . . . . . . . . . . . . . . . . 727.4.3 Ljung-Box-Pierce Q-test . . . . . . . . . . . . . . . . . 727.4.4 Engle’s ARCH-test . . . . . . . . . . . . . . . . . . . . 73


8 Conclusions 778.1 GARCH modelling discussion . . . . . . . . . . . . . . . . . . 77

8.1.1 Testing for different (p,q) . . . . . . . . . . . . . . . . 778.1.2 Testing different models . . . . . . . . . . . . . . . . . 778.1.3 Testing different distributions . . . . . . . . . . . . . . 788.1.4 Parameter instability . . . . . . . . . . . . . . . . . . . 788.1.5 User’s choice . . . . . . . . . . . . . . . . . . . . . . . 788.1.6 The OMX index . . . . . . . . . . . . . . . . . . . . . 788.1.7 Parameter values . . . . . . . . . . . . . . . . . . . . . 78

8.2 Further Studies . . . . . . . . . . . . . . . . . . . . . . . . . . 798.2.1 Different regimes . . . . . . . . . . . . . . . . . . . . . 79

8.3 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . 808.4 How good are the estimated models? . . . . . . . . . . . . . . 80

References 83

List of Figures

4.1 Price plot for Volkswagen. . . . . . . . . . . . . . . . . . . . . 294.2 Squared return for Volkswagen. . . . . . . . . . . . . . . . . . 294.3 Autocorrelation of the squared return for Volkswagen. . . . . 304.4 Partial autocorrelation of the squared return for Volkswagen. 304.5 Autocorrelation for Volkswagen with residuals after estimation. 35

5.1 Price plot for Munchner Ruck . . . . . . . . . . . . . . . . . . 415.2 Return plot for Munchner Ruck . . . . . . . . . . . . . . . . . 415.3 Squared returns for Munchner Ruck . . . . . . . . . . . . . . 425.4 Autocorrelation on the squared return for muvg. . . . . . . . 425.5 Partial autocorrelation on the squared return for muvg. . . . 43

6.1 Price plot for Suez-Lyonnaise des Eaux . . . . . . . . . . . . . 536.2 Return plot for Suez-Lyonnaise des Eaux . . . . . . . . . . . 536.3 Squared return for lyoe. . . . . . . . . . . . . . . . . . . . . . 546.4 Autocorrelation on the squared return for lyoe. . . . . . . . . 546.5 Partial autocorrelation on the squared return for lyoe. . . . . 556.6 Time dependent coefficient plot for lyoe. Window size = 200. 596.7 Time dependent coefficient plot for lyoe. Window size = 700. 596.8 Autocorrelation for lyoe after estimation. . . . . . . . . . . . 616.9 Histogram for lyoe after estimation. . . . . . . . . . . . . . . 62

7.1 Price plot for OMX. . . . . . . . . . . . . . . . . . . . . . . . 677.2 Return plot for OMX. . . . . . . . . . . . . . . . . . . . . . . 677.3 Squared return for OMX. . . . . . . . . . . . . . . . . . . . . 687.4 Autocorrelation on the squared return for OMX. . . . . . . . 687.5 Partial autocorrelation on the squared return for OMX. . . . 697.6 Autocorrelation for OMX with normalised residuals after es-

timation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737.7 Histogram for OMX after estimation. . . . . . . . . . . . . . . 74

List of Tables

2.1 GARCH features comparison. . . . . . . . . . . . . . . . . . . 11

3.1 Ljung-Box-Pierce Q-test example. . . . . . . . . . . . . . . . . 173.2 Example table for parameter test. . . . . . . . . . . . . . . . . 183.3 Time series identification with the ACF and PACF. . . . . . 20

4.1 Days with high return in the Volkswagen series. . . . . . . . . 284.2 Moments, skewness and kurtosis for the return series of Volk-

swagen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.3 Ljung-Box-Pierce Q-test for Volkswagen. . . . . . . . . . . . . 314.4 Engle’s ARCH-test for Volkswagen. . . . . . . . . . . . . . . . 314.5 Selecting p,q for Volkswagen. . . . . . . . . . . . . . . . . . . 324.6 GARCH model selection for Volkswagen. . . . . . . . . . . . . 324.7 Distribution selection for Volkswagen. . . . . . . . . . . . . . 334.8 Estimated parameters for the chosen model for Volkswagen. . 334.9 Estimated parameters for the chosen model for Volkswagen. . 344.10 Ljung-Box-Pierce Q-test, validation for Volkswagen. . . . . . 354.11 Engle’s ARCH-test, validation for Volkswagen. . . . . . . . . 364.12 Moments, skewness and kurtosis for zt of Volkswagen. . . . . 364.13 Results from the jb-test. Normality test for zt. . . . . . . . . 374.14 Skewness for zt for Volkswagen. . . . . . . . . . . . . . . . . . 37

5.1 Days with high return in the muvg series. . . . . . . . . . . . 395.2 Moments, skewness and kurtosis for the return series of muvg. 405.3 Ljung-Box-Pierce Q-test for muvg. . . . . . . . . . . . . . . . 425.4 Engle’s ARCH-test for muvg. . . . . . . . . . . . . . . . . . . 435.5 Selecting p,q for muvg. . . . . . . . . . . . . . . . . . . . . . . 445.6 GARCH model selection for muvg. . . . . . . . . . . . . . . . 455.7 APARCH model selection for muvg. . . . . . . . . . . . . . . 455.8 Distribution selection for muvg. . . . . . . . . . . . . . . . . . 465.9 Estimated parameters for the chosen model for muvg. . . . . 465.10 Estimated parameters for muvg on different data sets. . . . . 475.11 The Ljung-Box-Pierce Q-test in the validation for muvg. . . . 485.12 Engle’s ARCH-test in validation for muvg. . . . . . . . . . . . 48

x List of Tables

5.13 Moments, skewness and kurtosis for zt of muvg. . . . . . . . . 48

6.1 Days with high absolute returns in the lyoe series. . . . . . . 526.2 Moments, skewness and kurtosis for the return series of lyoe. 526.3 Ljung-Box-Pierce Q-test for lyoe. . . . . . . . . . . . . . . . . 556.4 Engle’s ARCH-test for lyoe. . . . . . . . . . . . . . . . . . . . 566.5 Selecting p,q for lyoe. . . . . . . . . . . . . . . . . . . . . . . 566.6 GARCH model selection for lyoe. . . . . . . . . . . . . . . . . 576.7 Distribution selection for lyoe. . . . . . . . . . . . . . . . . . . 576.8 Estimated parameters for the chosen model for lyoe. . . . . . 586.9 Estimated parameters for lyoe on two different data sets. . . 606.10 Estimated parameters for lyoe on two different samples of the

symmetric GARCH(1,1) model. . . . . . . . . . . . . . . . . . 616.11 Moments, skewness and kurtosis for zt of lyoe. . . . . . . . . 626.12 Ljung-Box-Pierce Q-test on validation data for lyoe. . . . . . 626.13 Engle’s ARCH-test, validation for lyoe. . . . . . . . . . . . . . 63

7.1 Days with high return in the OMX index. . . . . . . . . . . . 667.2 Moments, skewness and kurtosis for the return series of OMX. 667.3 Ljung-Box-Pierce Q-test for the OMX. . . . . . . . . . . . . . 687.4 Engle’s ARCH-test for OMX. . . . . . . . . . . . . . . . . . . 697.5 Selecting p,q for OMX. . . . . . . . . . . . . . . . . . . . . . . 707.6 GARCH model selection for OMX. . . . . . . . . . . . . . . . 707.7 Distribution selection for OMX. . . . . . . . . . . . . . . . . . 717.8 Estimated parameters for the chosen model for OMX. . . . . 727.9 Estimated parameters for the chosen model for OMX. . . . . 727.10 Ljung-Box-Pierce Q-test, validation for OMX. . . . . . . . . . 737.11 Engle’s ARCH-test, validation for OMX. . . . . . . . . . . . . 737.12 Moments, skewness and kurtosis for zt of OMX. . . . . . . . . 74

8.1 Parameter comparison. . . . . . . . . . . . . . . . . . . . . . . 808.2 Model explanation power overview for Volkswagen models. . . 818.3 Model explanation power overview for muvg models. . . . . . 818.4 Model explanation power overview for lyoe models. . . . . . . 828.5 Model explanation power overview for OMX models. . . . . . 82

Acknowledgements

First I would like to thank my supervisors and examiner Prof. ManfredDeistler and Prof. Lennart Ljung for letting me have the opportunity towrite my master thesis here in Vienna. I would also like to thank the staffat the Department of Econometrics and System Theory for making this apleasant stay.

I am also grateful to Eva Hamann and Dietmar Bauer, at the depart-ment, for interesting discussions about econometrics in general and GARCHmodels in particular.

Finally my thanks goes to my opponent Emil Tiren for useful commentand suggestions, to Anders Blomqvist and Claes Wallin for proofreading andto Thomas Schon for letting me use parts of his LATEX framework.

Vienna, December 2002

Jens Nasstrom

Notation

In this chapter symbols, operators and functions are explained. Abbrevia-tions, both technical and economical, are printed out.

Symbols

X A discrete stochastic variable.xt Stochastic process.µ Mean value of a stochastic variable.ψt The information set available at time t.

Operators and functions

L The lag operatorA(L)

∑qi=1 αiL

i

E(.) The expectation operator

xiv Notation

Abbreviations

ACF AutoCorrelation Function.AIC Akaike’s Information Criterion.AMAPE Adjusted Mean Absolute Percentage Error.APARCH Asymmetric Power ARCH.AR AutoRegressive.ARMA AutoRegressive Moving Average.ARCH AutoRegressive Conditional Heteroskedasticity.ARX AutoRegressive with eXternal input.DF Degrees of Freedom.EGARCH Exponential GARCH.EWMA Exponentially Weighted Moving Average.GARCH Generalized AutoRegressive Conditional Heteroskedasticity.IGARCH Integrated GARCH.i.i.d. Identically Independently Distributed.jb Jarque-Bera.ks Kolmogorov-Smirnov.Lbq-test Ljung-Box-Pierce Q-test.Log Log-likelihood function value.MA Moving Average.MedSE Median Squared Error.MAE Mean Absolute Error.MLE Maximum Likelihood Estimation.MSE Mean Squared Error.OLS Ordinary Least Squares.PACF Partial AutoCorrelation Function.TIC Theil Inequality Coefficient.w.s.s. Wide-Sense Stationary.

DE XETRA - Germany Stock Exchange.LYOE Suez-Lyonnaise des Eaux.MUVG Munchener Ruckversicherungs-Gesellschaft AG.PA Paris Stock Exchange.VOWG Volkswagen.Xetra Exchange Electronic Trading.

Chapter 1

Introduction

”October. This is one of the peculiarly dangerous months to speculate instocks in. The others are July, January, September, April, November, May,March, June, December, August and February.” (Twain 1894)

1.1 Background

Measuring the risk on specific assets has become increasingly importantduring the last decades. In a broad sense companies want to have goodcontrol of their risk profile. In the financial market this is of even greaterimportance. A number of large financial companies have had large lossesduring the last years. One example is the failure of Long Term CapitalManagement.

Risk can be divided into different subcategories: Strategic risks, oper-ational risks and financial risks. In this thesis we focus on risks on stockmarkets. Calculating risks is important for pricing derivatives, portfolioselection and hedging strategies.

1.2 Presenting the companies

The data used for estimation is taken from different branches and differ-ent European countries. Just to give an overview of the companies a briefpresentation is given below.

Munchner Ruck’s stock price is taken from the Germany stock marketXetra, (DE). The data was sampled from the 25th of February 1999 tothe 21st of October 2002. Munchner Ruck is one of the world’s largestreinsurance companies.

Suez-Lyonnaise des Eaux is traded at the Paris Stock Exchange (PA).The data was sampled from the 9th of June 1999 to the 17th of Septem-

2 Introduction

ber 2002. Suez-Lyonnaise des Eaux is a big water power plant com-pany.

Volkswagen’s stock price is taken from the Germany stock market (DE).The data was sampled from the 25th of February 1999 to the 21st ofOctober 2002. Volkswagen is a large producer of cars and vans.

OMX is an index that consists of 30 large Swedish companies. The datawas sampled from the 5th of January 1998 to the 21st of March 2002.The turnover in the Swedish stock market is 109 Euro/day.

The companies and the data will be further presented in the correspondingpreestimation section for each stock or index.

1.3 Objective

The objective for this thesis is to investigate to what extent it is possible topredict the risk in stocks in financial markets.

1.4 Problem specification

The specific GARCH modelling questions that are treated in this thesis are:

• Do these data sets exhibit conditional heteroskedasticity?

• What number of lags is a good model choice?

• Which GARCH model does a good job in forecasting volatility for aspecific choice of p and q?

• Do the parameters change in time?

1.5 Limitations

To be able to give reliable results it is necessary to limit the problem andfocus on a few specific topics. In this thesis we focus exclusively on univariatemodelling of volatility. The multivariate case is not treated at all. Anotherlimitation is the mean equation, which is modelled with just a constant.This is consistent with the efficient market theory, except for the fact thatinvestors also require extra return for taking risk. We are working with dailydata and are only interested in one-step-ahead prediction. The model setonly consists of different GARCH models.

Reader’s guide 3

1.6 Reader’s guide

Each chapter in the thesis is briefly presented below. The estimation chap-ters, number 4-7, can be read independently of each other. Conventions andabbreviations are presented in a special notation chapter.

Chapter 1 is the introduction chapter. It briefly presents the thesis andthe companies for which the GARCH models are estimated.

Chapter 2 is the methodology chapter. Here the model set and the estima-tion procedure is presented. A brief overview of the different toolboxesavailable is also given.

Chapter 3 presents the theory for the rest of the thesis. Statistics, stochas-tic processes and GARCH model theory are the main focuses.

Chapter 4-7 are the estimation chapters for the different data series. Thestructure for these chapters is basically the same: preestimation, es-timation and validation. This chapter can be reed independently ofeach other.

Chapter 8 is the conclusion chapter for all the data series.

The reader is assumed to have some knowledge on stochastic processes,statistic theory and system identification. General knowledge about stockmarket and economic theory for these markets is also assumed.

With this introduction we are ready to look into the system identificationmethod, concerning risk prediction, for stock markets.

Chapter 2

Method

This chapter presents the method used in this thesis. The chapter can roughlybe divided into three different parts. First econometrics and system identi-fication is presented. Second the system identification procedure is given.Finally the methodology of estimation in GARCH models is introduced.

2.1 Econometrics

Econometrics as a discipline has existed for about 100 years. It can be sepa-rated into three large subgroups: First microeconometrics dealing with, forexample, consumer behaviour: second financial econometrics dealing with fi-nancial data and finally macroeconometrics which focuses on macroeconomicphenomena such as inflation, unemployment and economic growth. (Deistler1999) This thesis is concerned with financial econometrics. We are focusingon stock markets and we are interested in prediction of the first and thesecond moments; In particular our main focus is on the second moments.

2.2 System Identification

Each chapter which contains estimation of GARCH models is presentedin the same way. First the company is presented with plots and tablescontaining interesting and important company-dependent facts. Some testsare performed to see whether or not there exists correlation in the squaredreturn and to see if there exists heteroskedasticity.

The estimation is only done in the univariate case. The procedure isto first choose the number of lags (p,q) for the symmetric GARCH with anormal distribution and then to test, for this specific (p,q), different mod-els. Finally, when the model is chosen, three different distributions for thismodel are tested. In the validation part we test to see if the correlationand heteroskedasticity has been removed from the data series and performother tests to validate the model. The system identification procedure that

6 Method

is given in this section is not the only way to do it. A general approach tosystem identification is presented by Ljung (1999).

2.2.1 The data set

Outliers are not handled in the raw estimation since it may be hard to knowwhat an outlier is in a financial market. After the estimation for the wholesample, the models are estimated for different sub-samples. By doing thiswe can get a sense of how outliers affect the model coefficients and how theparameters in the model are changing over time.

One obvious feature when estimating and predicting in the financialmarket is that we have to use the given data and cannot redo any experiment.Another limitation is the number of data samples that are used. In all ourseries we have about 1000 data samples, the data is then split into estimationpart (70%) and validation part (30%). The reason for the unequal splitis that the GARCH estimation procedure requires large sample sizes toproduce good estimates.

2.2.2 Volatility

Modelling volatility can be done in many different ways. Most volatilitymodels are from these model classes:

GARCH-methods are a way of investigating how a function of past returns,in a specific financial series, should be constructed and mapped ontothe second moment (Hull 2000). For a specific mapping, the dataseries can also be forecast with this method. This is the approach thatwill be used in this thesis and will be further explained in Section 3.5.

Stochastic volatility methods are models where the volatility follows astochastic differential equation. One way of doing this is presentedby Hull (2000):

dS

S= rdt+

√V dzs

dV = a(b− V )dt+ ξV αdzv

(2.1)

where a, b, ξ and α are constants, dzs and dzv are Wiener processesand S and r are the stock price and the risk free rate respectively. Vis the asset’s variance rate, the square of its volatility. This is a timecontinuous stochastic differential and can be seen as one possible wayof handling the problem that volatility is not constant over time. Thisapproach is not used in this thesis and will not be presented further.

Regime switching models are based on the assumption that economic andfinancial markets seem to behave differently in different periods of

2.2.3 Selected models for estimation 7

time. This is used to specify different models for different periods.Markov models can be used for switching between these different mod-els (Hamilton 1994). This approach will not be used and is not furtherexplained in this thesis.

Methods from these classes can also be combined. For example Klaasen(2002) is using regime switching models in combination with GARCHmodels.

2.2.3 Selected models for estimation

First moment modelling is not the main focus in this thesis. However areasonable model for the first moment has to be used. A misspecification inthis equation could lead to wrong conclusions about which GARCH modelto support. This is because the squared error term from the mean equationis used as the ”true” volatility when different models are measured on theirforecastability.

As described by Ljung (1999), system identification is, in many cases, aiterative procedure. It is therefore impossible to come up with the correctmodel in one try. Evenso, here a quite general model set is presented, whichis used on all the four time series.

In an efficient market, all information available up until today is alreadyincluded in the price. This would lead to the conclusion that there is noneed for any ARX/ARMAX models in the mean equation. Efficient markettheory also states that an investor is expecting the risk free rate as returnand an addition return for taking risk. (Hamilton 1994) Based on this dis-cussion the model for the return would be a constant plus a risk term and anunpredictable part εt. However, coupling of equations, between the meanequation and the second moment, is not possible in the toolbox that weuse for estimation. Therefore the model for the mean that we use, is justa constant plus the unpredictable part yt = c + εt, here yt is the return onthat specific day and c is the mean value over the values, known so far.

In the model for the second moment GARCH models are used. Themain question here is which GARCH models to use among all those thatexist. Here the choice is based on previous GARCH research (Pagan &Schwert 1990) and (Peters 2001). We will use four different GARCH models:GARCH, IGARCH, EGARCH and APARCH, with a small number of lags0, 1, 2. These are all presented in the theory chapter (see section 3.7).

2.2.4 Criterion of fit

A good performance measure of the second moment can be hard to findsince the volatility is not directly observable. One way of dealing withthis problem is to not rely on one specific measure but rather use several

8 Method

measures. In this thesis six different measures are used for evaluating theperformance of volatility forecasts from different GARCH models:

1. Mean Squared Error (MSE)

2. Median Squared Error (MedSE)

3. Mean Absolute Error (MAE)

4. Adjusted Mean Absolute Percentage Error (AMAPE)

5. Theil Inequality Coefficient (TIC)

6. Mincer-Zarnowitz R2 (R2)

The measures above need some explanation. The MSE is calculated by:

1h+ 1

S+h∑t=S

(σ2t − σ2

t )2 (2.2)

h is the number of steps ahead that we want to predict, in our case h isalways equal to 1, σ2

t is the forecast volatility, σ2t is the ”true” volatility (in

our case ε2t ) and S is the total sample size.The MedSE is:

1h+ 1

S+h∑t=S

(σmed2 − σmed2)2 (2.3)

The MAE is:

1h+ 1

S+h∑t=S

|σ2t − σ2

t | (2.4)

The AMAPE is:

1h+ 1

S+h∑t=S

∣∣∣∣ σ2t − σ2

t

σ2t + σ2

t

∣∣∣∣ (2.5)

The TIC defined by:

1h+1

∑S+ht=S (σ2

t − σ2t )

2√1

h+1

∑S+ht=S σ

2t +

√1

h+1

∑S+ht=S σ

2t

(2.6)

is a scale invariant measure that lies between 0 and 1 where 0 indicatesperfect fit.

2.2.5 Validation 9

The R2 is calculated by regressing σ2t on the ε2t , which can be formulated

with this equation

ε2t = α+ βσ2t + ut (2.7)

These measures are presented by Brooks (1997) and are implemented in theGARCH toolbox for Ox by Laurent & Peters (2002).

The first issue to address when using these measures is: What is the truevolatility? This can be done in different ways but in this thesis we will usethe squared residuals ε2t from the equation yt = c+ εt, as the true volatilitywhen measuring the fit. The reason for choosing ε2t as the true volatility iseasily seen in this equation

Eε2t = E(ztσt) = 1Eσt = σt (2.8)

zt is i.i.d. N(0, 1) and σt is the variance for εt conditioning on the informa-tion available at time t. This is further explained in Section 3.7.

Another ’true’ volatility is to use intra-day measures and then for anumber of different time periods during the day calculate the volatility,σ2t =

∑Kk=1 r

2(k),t, where r2(k),t is the return of the kth intra-day interval of

the tth day. K is the number of intervals per day (Peters 2001). This is notused since intra-day data is not used for the data series.

2.2.5 Validation

In the validation part, tests are performed to judge whether ARCH effectsand autocorrelation have been removed or not. Tests are also performedto see if the given assumptions about the model are fulfilled i.e. are thenormalised residuals distributed in the way that was assumed in the model:normal, student-t or skewed student-t.

A good way of testing a model is to see how it is performing on data notused in the estimation part. Data points (700 − 1000), depending on thedata set, have been saved for this reason. The out-of-sample measures arecomputed with one step ahead prediction (not reestimating the coefficients)and then for the next day, when new information is available, do the predic-tion again. We are of course only interested in the model performance forthe second moment (the first moment is just a constant).

2.3 Methods of estimation, toolboxes

When estimating GARCH models, some kind of computer-based softwarehas to be used. Different software has different functionality, drawbacks andfeatures. Brooks, Burke & Persand (2001) presents nine different GARCHestimating software packages and compares them. He finds that there could

10 Method

be large differences between estimated GARCH parameter values from dif-ferent toolboxes. Therefore it is important to be careful in choosing thetoolbox to use. In this thesis a toolbox1 compatible with the Ox program-ming language is used (Laurent & Peters 2002). Unfortunately this toolboxis not one of the nine which are compared by Brooks et al. (2001). Still,in the tutorial by Laurent & Peters (2002) they compare their own toolboxwith the same benchmarks that Brooks uses and come up with good results.

2.3.1 The Ox GARCH package

In this thesis the Ox GARCH package is used for estimating and forecast-ing. The package has a variety of different features. Four different distri-butions are included: normal, student-t, skewed student-t and generalizederror distribution. A number of different models are available, both for theconditional mean and the conditional variance, as well as a number of testsand forecast possibilities (Laurent & Peters 2002). In table 2.1 some of theavailable toolboxes and their features are presented. Most of these modelfeatures are not used in this thesis so the table is just to be seen as a generalcomparison. The reason for choosing the G@RCH package is on the onehand that the light version is a free version and on the other hand that ithas the GARCH model features that we want to use.

1G@RCH 2.3

2.3.1 The Ox GARCH package 11

Table 2.1. GARCH features comparison. The table is a modified version of thetable in Laurent & Peters (2002).

G@RCH PcGive Eviews S-Plus SAS StataVersion 2.3 10 4.0 6 8.2 7

Cond. meanExpl. var. + + + + + +

ARMA + + + + + +ARFIMA + - - - - -ARCH-m - + + + + +

Cond. var.Expl. var. + + + + + +

GARCH + + + + + +IGARCH + - - - + -EGARCH + + + + + +

GJR + + + + - +APARCH + - - + - +

C-GARCH - - + + - -FIGARCH + - - + - -

FIEGARCH + - - + - -FIAPARCH + - - - - -HYGARCH + - - - - -

Distr.Normal + + + + + +

Student-t + + - + + -GED + + - + - -

Skewed-t + - - - - -Double Exp. - - - + - -

EstimationML + + + + + +

QML + + + - - +

Chapter 3

Theory

This chapter contains all the theory used in this thesis. First a backgroundis given by presenting economic theory, then the necessary statistical toolsand theory are presented. Finally the theory for volatility estimation and thetheory for GARCH models are introduced.

3.1 Economic theory

This first theory section presents the economic theory, which is used in thisthesis. The main highlight from this section is the efficient market theory.

A good book that presents financial theory is Hull (2000). This book isrecommended to the reader who wants a more detailed description.

3.1.1 Financial data

In this thesis we are using a continuous approach when converting betweenstock prices and return. Let the stock price be denoted S and µ be the rateof return. If changes in stock price is denoted ∆S and changes in time aredenoted ∆t the model can be described as

∆S = µS∆t (3.1)

If we let ∆t approach zero and then solve the upcoming differential equation,the stock price is given by:

ST = S0eµT (3.2)

Equation 3.2 is the one that will be used in this thesis when convertingbetween return and stock prices. (Hull 2000)

14 Theory

3.1.2 An introduction to risk

Volatility of a stock is a measure of the uncertainty of the returns fromthat specific stock. When returns are calculated from equation (3.2), thevolatility of a stock price becomes the standard deviation of the return.

In stock markets, periods of high risk seem to be followed by periodsof high risk and for low risk periods it is the other way around. This phe-nomenon is often referred to as volatility clustering. (Hull 2000) For examplewhen a shock in a stock market occurs, the volatility is high for the subse-quent period.

3.1.3 Efficient market theory

The theory of efficient markets can be divided into three different levels:weak, semi-weak and strong efficiency. Capital markets are weak form effi-cient if it is impossible to form a trading strategy that performs better thanaverage, using the information of historical prices. This means that so calledtechnical analysis is fruitless. Semi weak efficiency means that it is impossi-ble to perform better than average, even when including exogenous variables(X), where X includes all public information. Strong efficiency means thatprivate information is also included by exogenous variables. This wouldmean, that not even with inside information would it be possible to performbetter than average. (LeRoy 1989)

3.1.4 Leverage effect

One phenomenon in equity markets is that volatility tends to increase morewhen the return decreases, than it does, when the return increases. Onereason for this is referred to as the leverage effect. As a company’s equityreduces in value the leverage for the company increases. When the stockprice for a company decreases their equity declines in value and thereforetheir leverage increases. The company has become more risky. On the otherhand, when the company’s stock price increases, the leverage is decreases.The risk of an equity is therefore dependent on the sign of the return of thestock. (Hull 2000) This will be further discussed in Section 3.7.4.

3.2 Statistics

A brief description of the statistics that will be used is presented. In allcases below, X is a discrete valued stochastic variable, k is the summationindex and px(k) is the probability that X is taking value k. A more detaileddescription is presented in Hamilton (1994).

3.2.1 Correlation 15

The first moment is the population mean and is defined as

E(X) =∑k

kpx(k) = µ (3.3)

The noncentral second moment is then defined as

E(X2) =∑k

k2px(k) = V ar(X) +E2(X) (3.4)

Noncentral moments are then in the general case defined as

E(Xr) =∑k

krpx(k) r = 1, 2, 3 . . . (3.5)

Skewness is defined as

E((X − µ)3)(V ar(X))3/2

(3.6)

A variable with positive skewness is more likely to have is values far abovethe mean value than far below. For a normal distribution the skewness iszero.Kurtosis is defined as

E((X − µ)4)(V ar(X))2

(3.7)

For a normal distribution the kurtosis is 3. A distribution with a kurtosisgreater than 3 has more probability mass in the tails, so called ”fat tails”,or leptokurtic.

3.2.1 Correlation

The population correlation between two different random variables X andY is defined by

Corr(X,Y ) ≡ Cov(X,Y )√V ar(X)V ar(Y )

(3.8)

(Hamilton 1994)

3.2.2 Autocorrelation

The jth autocorrelation is defined as the jth autocovariance divided by thevariance:

Corr(Xt, Xt−j) ≡ Cov(Xt, Xt−j)√V ar(Xt)V ar(Xt−j)

(3.9)

(Hamilton 1994)

16 Theory

3.2.3 Partial autocorrelation

The partial autocorrelation is also a useful tool in the identification of a timeseries. It is defined as the last coefficient in a linear projection of Y on them most recent values. Letting this be denoted α(m)

m , if the constant for theprocess is zero then the equation becomes:

Yt+1 = α(m)1 Yt + α

(m)2 Yt−1 + . . .+ α(m)

m Yt−m+1 (3.10)

If the process were a true AR(p) process the coefficients with lags greaterthan m would be zero. (Hamilton 1994)

3.3 Hypothesis tests

A hypothesis test is a procedure for analysing data to address questionswhether a certain criterion is fulfilled or not. This can be tested in a numberof different ways and this section presents the hypothesis test that will beused in this thesis.

All tests have a corresponding p-value. This p-value, under the assump-tion of the null hypothesis, is the probability of observing the given sampleresult.

At the significance level of 95%, which in our notation corresponds witha critical value of 0.05, then we reject the null hypothesis if the p-value islower than this critical value. A p-value greater than 0.05 corresponds toinsufficient evidence for rejecting the null hypothesis. (MathWorks 2002)

3.3.1 Jarque-Bera test

The idea behind the Jarque-Bera test is to test whether a specific distribu-tion is normal or not. The jb-value is calculated as:

jb =T − k

6

(S2 +

(K − 3)2

4

)(3.11)

where T is the number of observations, k is the number of estimated pa-rameters, S is the skewness and K is the kurtosis. In some programs theexcess kurtosis instead of the kurtosis is calculated. The kurtosis is then 3plus the excess value. The intuitive feeling about this test is that the largerthe jb-value is, the lower the probability is that the given series is drawnfrom a normal distribution. The test statistic of the Jarque-Bera test isχ2-distributed with 2 degrees of freedom under the null hypothesis, that theseries is normally distributed. (Hamann 2001)

3.3.2 Ljung-Box test 17

Table 3.1. Ljung-Box-Pierce Q-test example. P-values are given in angle brackets.With this low p-value we have to reject the hypothesis that no serial correlationexists.

Q(20) = 157.24 [2.55698e-023 ]

3.3.2 Ljung-Box test

The Ljung-Box test is performed to test whether a series has significantautocorrelation or not. The Lbq-value is calculated by:

Qk = T (T + 2)k∑i=1

r2iT − i

(3.12)

where T is the number of samples, k is the number of lags and ri the ithautocorrelation. If Qk is large then the probability that the process hasuncorrelated data decreases. The null hypothesis for the test is that thereexists no correlation and under that hypothesis, Qk is χ2 with k degrees offreedom. Table 3.1 presents an example of how this can be done (the table istaken from the estimation procedure for Volkswagen in Section 4.2.4). TheLbq-value is in this case calculated for twenty number of lags. The p-valuecorresponding to this Qk value is presented in angle brackets. (MathWorks2002)

3.3.3 ARCH test

It is fairly easy to test whether the residuals from a regression have con-ditional heteroskedasticity or not. The test is based on OLS regression,where the OLS residuals ut from the regression are saved. u2

t is thereafterregressed on a constant and its own m-lagged values. This is done for allsamples t = 1...T . This regression has a corresponding R2-value. TR2 isthen asymptotically χ2-distributed with m degrees of freedom under the nullhypothesis that ut is i.i.d. N(0, σ2). (Engle 1982)

This ARCH-test can also be performed as a test for GARCH-effects. TheARCH-test for lag (p+ q) is locally equivalent to a test for GARCH effectswith lags (p, q). (MathWorks 2002)

The null hypothesis, H0, is that no ARCH effects exists. This is testedfor lags up to T . The OX-package presents this as an F-test with the valuefrom the F-distribution and the p-value in angle brackets.

3.3.4 Parameter output and t-test

For each selected model from the estimation part, the maximum likelihoodestimates of the selected parameters are calculated. An example of this is

18 Theory

Table 3.2. Example table for parameter test. This output is from Volkswagen andmodel is an APARCH(1,1). Maximum likelihood estimation is used.

Parameter Coefficient Std.Error t-value t-probCst(M) −0.004560 0.068735 −0.06634 0.9471Cst(V) 0.654696 0.413332 1.584 0.1136

GARCH(Beta1) 0.710141 0.075228 9.440 0.0000ARCH(Alpha1) 0.133134 0.047949 2.777 0.0056

APARCH(Gamma1) −0.133827 0.120061 −1.115 0.2653APARCH(Delta) 1.957050 0.780077 2.509 0.0123

presented in table 3.2. The first column is the name of each parameter,Cst(M) stands for the constant in the mean equation. Cst(V) is the esti-mated variance that does not depend on the time t. The other parametersare specific for the chosen models. The second column contains the coeffi-cients for each parameter. In the third column, the estimated standard errorfor this coefficient value is presented. The t-value is just the coefficient valuedivided with the standard error. T-prob is the p-value given from a t-test.All these t-tests are tested against zero. Depending on whether the valuecan take negative values or not the t-test is either single or double sided.

3.3.5 Nelson test

The Nelson test is a test for time dependency of the parameters that areestimated with a maximum likelihood estimation. The null hypothesis isthat the parameters are constant and the alternative is that the parameterΘ follows a martingale process. (Hansen 1994)

3.4 Stochastic processes

This section for stochastic processes is treated in more detail by (Hamilton1994).

3.4.1 White noise

One of the basic blocks when modelling stochastic processes is the whitenoise

E(εt) = 0

E(εtεs) ={σ2 for t = s0 for t 6= s

(3.13)

3.4.2 Stationarity 19

3.4.2 Stationarity

A process Yt is said to be covariance-stationary or weakly stationary if nei-ther the mean ut nor the autocovariance γjt depend on the time t and if thegiven moments exists.

E(Yt) = µ (for all t)E((Yt − µ)(Yt−j − µ) = γj (for all t and any j)

(3.14)

3.4.3 Moving average process

A moving average process of order one MA(1) is described as

Yt = α0 + α1εt−1 + εt (3.15)

where α0 and α1 could be any real constants, and εt is a white noise processdescribed in equation 3.13. This can of course also be considered in thegeneral MA(q) case. Then the equation becomes

Yt = α0 +q∑j=1

αjεt−j + εt (3.16)

3.4.4 Autoregressive process

First we consider an AR(1) process

Yt = α0 + β1Yt−1 + εt (3.17)

where α0 and β1 can be any real constants, and εt is a white noise processdescribed in equation 3.13. An AR(1) process can also be generalized to anAR(p) process.

Yt = α0 +p∑j=1

βjYt−j + εt (3.18)

Autocorrelation and the partial autocorrelation that were introduced inthe statistical section is a great help in deciding which AR- and ARMA-models to be considered. Table 3.3 can be used in this part of the identifi-cation process. The table is presented at (www.itl.nist.gov 2002).

3.5 Computing volatility

Volatility is a measure of the uncertainty of the return for an asset.All volatility estimation used in this thesis is calculated from return

series only. This means that the trend from the price series already hasbeen removed. For further details on this procedure see section 3.1.1.

20 Theory

One obvious way to compute the volatility in a specific market on aspecific day is a linearly weighted moving average, where a specific numberof previous observations is used to calculate the volatility.

σ2n =

1p− 1

p∑i=1

(un−i − u)2 (3.19)

where u is the mean value for the process of the period where σ2n is calculated

and ui is the daily return for a specific day. σ2n is here the volatility on day

n for the variable.If something unexpected happens to a specific market, a shock of some

kind, then it would be more reasonable to assign different weights to differentperiods in time. This can be done by

σ2n =

p∑i=1

αiu2n−i (3.20)

where u is assumed to be equal to zero, which can be a good approximationif we are dealing with daily data. The problem now is of course to find theseαi. One reasonable constraint is that αi > αj when day i is more recentthan j. Other constraints should be that

p∑i=1

αi = 1 (3.21)

Table 3.3. Time series identification with the ACF and PACF.

Shape of ACF Indicated modelExponential, decaying to zero Autoregressive model.

Use the partialautocorrelation plotto identify the orderof the autoregressive model.

Alternating positive and Autoregressive model.negative, decaying to zero Use the partial autocorrelation

plot to help identify the order.One or more spikes, Moving average model.rest are essentially zero order identified by

where plot becomes zero.Decay, starting after a few lags Mixed autoregressive

and moving average model.All zero or close to zero Data is essentially random.High values at fixed intervals Include seasonal autoregressive term.No decay to zero Series is not stationary.

3.5.1 EWMA - Exponentially weighted moving average 21

and that αi > 0. This approach can be generalized in a number of differentways. One of the more popular ones, is to assign some weight to a long runvolatility. This leads to an equation like

σ2n = γV +

p∑i=1

αiu2n−i (3.22)

where V is the long run volatility and γ is the weight assigned to this volatil-ity. The constraint from equation (3.21) now looks like

γ +p∑i=1

αi = 1 (3.23)

This was first suggested by Engle (1982) and is known as the ARCH(p)model. (Hull 2000)

3.5.1 EWMA - Exponentially weighted moving average

The weights in equation 3.21 can also be calculated in a different way. Letαi = λαi−1, where λ is a constant (0 < λ < 1). This restrictions for λis consistent with equation 3.21. This weighting scheme has some goodcharacteristics. The volatility formula can now be written as

σ2n = λσ2

n−1 + (1 − λ)u2n−1 (3.24)

Here only two values have to be stored for each volatility estimate. It alsoadapts to market changes because of the functional form of the equation.Large changes in un−1 will directly effect the volatility estimate. (Hull 2000)

3.6 ARCH - Autoregressive conditional heteroske-dasticity

The ARCH-model was first presented by Engle (1982) and has since thenreceived a lot of attention. First consider an ordinary AR(p) model of thestochastic process yt.

yt = c+ α1yt−1 + . . .+ αpyt−p + ut (3.25)

where ut is white noise. The basic AR(p)-model is now extended so thatthe conditional variance of ut could change over time. One extension couldbe that u2

t itself follows an AR(m)-process.

u2t = θ0 + θ1u

2t−1 + . . .+ θmu

2t−m + wt (3.26)

22 Theory

where wt is a new white noise process and ut is the error in forecasting yt.This is the general ARCH(m)- process. (Engle 1982) For easier calculationsand for estimation, a stronger assumption about the process is added.

ut = h1/2t υt (3.27)

where υt is an i.i.d. Gaussian process with zero mean and a variance equalto one υt

iid∼ N(0, 1) and the whole model for the variance is now

εt|ψt−1 ∼ N(0, ht)

ht = α0 +q∑i=1

αiε2t−i

(3.28)

where

α0 > 0 αi > 0,i = 1, . . . , q

ψt−1 is the information available at time t − 1. Now, when the process forthe variance is defined, we add an additional equation for modelling yt. Thereturn price is modelled with a constant.

yt = c+ εt (3.29)

this means that εt is innovations from a linear regression.

3.7 Generalized ARCH - GARCH

This section is describing a generalization of the ordinary ARCH-model. Themodel structure was introduced by Bollerslev (1986). The generalization isa similar to the extension of an AR(p) to an ARMA(p,q). Formally theprocess can be written as

εt|ψt−1 ∼ N(0, ht)

ht = α0 +q∑i=1

αiε2t−i +

p∑i=1

βiht−i(3.30)

where

p integer, q integerα0 > 0, αi ≥ 0, i = 1, . . . , qβ ≥ 0, i = 1, . . . , p

thus the additional feature is that the process now also includes lagged ht−ivalues. For p = 0 the process is an ARCH(q). For p = q = 0 (an extensionallowing q = 0 if p = 0), εt is white noise. (Bollerslev 1986)

3.7.1 GARCH(1,1) 23

Theorem 1. The GARCH(p,q) process as defined in equation 3.30 is wide-sense stationary with E(εt) = 0, var(εt) = α0(1 − A(1) − B(1))−1 andcov(εt, εs) = 0 for t 6= s if and only if A(1) +B(1) < 1.

Proof. see Bollerslev (1986) page 323.

In theorem 1 A(1)=∑q

i=1 αi and B(1)=∑p

i=1 βi. In most cases the num-ber of parameters is rather small, for instance GARCH(1,1). Next we willfocus in on the model where p = q = 1, a GARCH(1,1) process.

3.7.1 GARCH(1,1)

In the case where p = q = 1 the model becomes

εt|ψt−1 ∼ N(0, ht)

ht = α0 + α1ε2t−1 + β1ht−1

(3.31)

where

α0 > 0, α1 ≥ 0, β1 ≥ 0α1 + β1 < 1

Interesting properties for εt (unconditioned) would be its moments andthey are given by:

Theorem 2. For the GARCH(1,1) process as defined in equation 3.31 anecessary and sufficient condition for existence of the 2mth moment is

µ(α1, β1,m) =m∑j=0

(m

j

)ajα

j1β

m−j1 < 1 (3.32)

where

a0 = 1, aj =j∏i=1

(2i− 1),

The 2mth moment can be described by the recursive formula

E(ε2mt ) = am

·(m−1∑n=0

a−1n E(ε2nt )αm−n

0

(m

m− n

)µ(α1, β1, n)

)

· (1 − µ(α1, β1,m))−1

(3.33)

24 Theory

Proof. Bollerslev (1986) page 325.

Due to the symmetry properties of the normal distribution, it follows thatif the 2mth moment exists, the 2(m − 1)th moment is zero. It can also beshown that the distribution of εt is heavily tailed. (Bollerslev 1986)

The existence of the 2mth moment for GARCH models with higher orderthan (1, 1) can also be interesting to investigate. The general case is notknown, but for a specific order sufficient and necessary conditions can bederived. This is done by Bollerslev (1986) page 313 and for the GARCH(1,2)this is found to be

α2 + 3α21 + 3α2

2 + β21 + 2α1β1 − 3α3

2 + 3α21α2 + 6α1α2β1 + α2β

21 < 1

(3.34)

3.7.2 IGARCH

One special case of GARCH-models is when the sum of estimated parameters(except for α0) equals one:

q∑i=1

αi +p∑j=1

βj = 1 (3.35)

This is known as integrated GARCH or IGARCH. This is similar to theEWMA-procedure if α0 is set to zero. (Hamilton 1994)

3.7.3 EGARCH

Nelson (1991) presents a model which is known as Exponential-GARCH orEGARCH. The idea is to loosen the positivity constraints from the standardGARCH but still keep the nonnegativity constraint on the volatility for theconditional variance. A suitable way of doing this is by

ln(σ2t ) = αt +

∞∑i=1

βig(zt−i) β1 ≡ 1 (3.36)

where the function g(zt) can be formulated in different ways. Nelson (1991)is suggesting

g(zt) = θzt + γ(|zt| − E|zt|) (3.37)

to handle both the sign and the magnitude of zt.A slightly different model of the EGARCH is implemented by Laurent

& Peters (2002)

ln(σ2t ) = ω + (1 − β(L))−1(1 + α(L))g(zt−i) (3.38)

3.7.4 APARCH 25

3.7.4 APARCH

One of the more general GARCH models is the APARCH model (Asymmet-ric Power ARCH) which is also implemented by Laurent & Peters (2002) inthe GARCH toolbox. The model structure is

σδt = ω +q∑i=1

αi(|εt−i| − γεt−i)δ +p∑j=1

βjσδt (3.39)

Which includes many other GARCH models as special cases.

3.8 Distributions

In this thesis three different probability distributions are used. The standardnormal distribution, the student-t and the skewed student-t. For the esti-mation of the parameters the log-likelihood functions of these distributionsare used. The reason to use different distributions (other than the normal)is that the third and fourth moment of the normal distribution could betoo restrictive. The student-t has one parameter for modelling the fourthmoment and this parameter is also estimated in the maximum likelihoodestimation. For the skewed student-t there is also an additional parameterfor having skewness not equal to zero in the distribution. This is the mostgeneral of the three distributions.

Chapter 4

Volkswagen

In this chapter, model estimation and model selection for the Volkswagenstock is performed. The procedure is following the structure that is pre-sented in the methodology chapter. Volkswagen as a company is presented,preestimation and estimation is performed and finally some conclusions aredrawn.

4.1 Presenting data

4.1.1 Introducing Volkswagen

The stock price for Volkswagen is taken from the Germany stock market(DE). The data is sampled from the 25th of February 1999 to the 21st ofOctober 2002. Particularly interesting days are presented in table 4.1. Thecertain criterion is that the squared return should be above 60 in this scaledmeasure presented in Section 4.2.1. One thing notable, is that the 11th ofSeptember does not meet the criterion but that three out of eight days withextremely high return occurred in September 2001. The dates are presentedin the form (dd.mm.yy).

4.2 Preestimation

4.2.1 Data formatting

For the data series of Volkswagen 952 data points is the whole data set. Thedata is daily data and the return is calculated by:

returnt = 100(log(pricet) − log(pricet−1)) (4.1)

The reason for multiplying by 100 is due to numerical problems in the esti-mation part. This will not affect the structure of the model since it is justa linear scaling. Data with numbers from 1-800 are used in the estimationpart and data with numbers 801-952 are used as cross validation.

28 Volkswagen

Table 4.1. Days with high return in the Volkswagen series.

number return date210 −8.4920 15.12.99540 9.6572 21.03.01666 8.2731 13.09.01670 7.8758 19.09.01672 −7.9494 21.09.01892 −9.1398 26.07.02935 −7.9203 25.09.02946 −8.8166 10.10.02

Table 4.2. Moments, skewness and kurtosis for the return series of Volkswagen.

VolkswagenFirst moment 0

Second moment 5.1369Third moment −0.4209

Fourth moment 126.4742Skewness −0.0362Kurtosis 4.7930

4.2.2 Moments for the return series

In the preestimation stage it would be interesting to get a picture of whatthe distribution of the raw return series is like. The moments, skewness andkurtosis are presented in table 4.2. For a theoretical introduction to thesemeasures see section 3.2. It is notable that the kurtosis is considerably largerthan three, which would have been the kurtosis if the return series weredrawn from a normal distribution. This behaviour is known to frequentlyoccur in financial markets.

4.2.3 Plots for Volkswagen

Before starting the estimation procedure it is interesting to see the behaviourof the specific data set presented in graphical terms. A price plot is pre-sented in figure 4.1. The squared returns of the data series are presented infigure 4.2. In the diagram we see clustering of high volatility. A graphicalpresentation of the autocorrelation is presented in figure 4.3. Here we cansee that there is a high persistence in the second moment for at least 10lags. The partial autocorrelation is presented in figure 4.4 and it also shows

4.2.4 Hypothesis tests 29

persistence in the second moment.

0 100 200 300 400 500 600 700 800 900 100030

35

40

45

50

55

60

65

70

75

Figure 4.1. Price plot for Volkswagen.

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

80

90

100

Figure 4.2. Squared return for Volkswagen.

4.2.4 Hypothesis tests

The indication from the graphical plot that there exists correlation betweendata plots is in this subsection tested with the Ljung-Box-test. The test forheteroskedasticity is performed with an ARCH test. See section 3.3 for atheoretical explanation for these tests.

30 Volkswagen

0 2 4 6 8 10 12 14 16 18 20−0.2

0

0.2

0.4

0.6

0.8

Lag

Samp

le Au

tocor

relat

ionSample Autocorrelation Function (ACF)

Figure 4.3. Autocorrelation of the squared return for Volkswagen.

0 2 4 6 8 10 12 14 16 18 20−0.2

0

0.2

0.4

0.6

0.8

Lag

Samp

le Pa

rtial A

utoco

rrelat

ions

Sample Partial Autocorrelation Function

Figure 4.4. Partial autocorrelation of the squared return for Volkswagen. Theyrepresent a measure of partial autocorrelation in the volatility.

Ljung-Box-Pierce Q-test

The computed statistical values for the certain lags are presented togetherwith the corresponding p-value in angle brackets. Result from the test ispresented in table 4.3 for Volkswagen. The null hypothesis is that no serialcorrelation exists and this hypothesis is accepted when the p-value is high.In this case we reject that hypothesis.

Estimation 31

Table 4.3. Ljung-Box-Pierce Q-test for Volkswagen. P-values are given in anglebrackets. With these low p-values we reject the hypothesis that no serial correlationexists.

Q(10) = 149.508 [4.70469e-027 ]Q(15) = 155.131 [2.30051e-025 ]Q(20) = 157.24 [2.55698e-023 ]

Table 4.4. Engle’s ARCH-test for Volkswagen on the raw return series. H0 is thatno ARCH effect exists and in this case we have p-values equal to zero which meansthat we can reject the hypothesis.

ARCH 1-5 test: F(5,789) = 14.027 [0.0000]**ARCH 1-10 test: F(10,779) = 10.222 [0.0000]**ARCH 1-15 test: F(15,769) = 7.2884 [0.0000]**

Engle’s ARCH-test

Engle’s ARCH-test performed before estimation. This test supports thatthere exists heteroskedasticity. P-values equal to zero for all of the threeseries. The result for Volkswagen is presented in table 4.4. ARCH-effectsare tested on lags 1-5, 1-10 and 1-15. With these p-values we reject thehypothesis that no ARCH-effects exist.

4.3 Estimation

This section is presenting the whole estimation procedure for the Volkswa-gen GARCH models. First p and q for the standard GARCH are chosenand then for this specific p and q different GARCH models are tested, esti-mated and finally one GARCH model with specific p and q is chosen. Forthis GARCH(p,q)-model different distributions are evaluated. Finally theparameters for the chosen GARCH models are presented.

4.3.1 Choosing p and q

Different p and q values for the standard symmetric GARCH model aretested. The selection procedure is based on each model’s ability to produceforecasts. Six different forecasts measures are used. See section 2.2.4 fora description of these measures. The Volkswagen table 4.5 shows that theGARCH(1,1) performs best in three out of six out-of-sample measures. Intwo of the measures of which the GARCH(1,1) does not score the best the

32 Volkswagen

Table 4.5. Selection for symmetric GARCH models for Volkswagen. Log-likelihoodis presented as well as Akaike’s information criterion for different p and q. Sixdifferent forecasts measures are also presented to determine which p and q thatperforms best on out-of-sample data for the symmetric GARCH.

(p,q) (0,0) (1,1) (1,2) (2,1) (2,2)Log −1726.53 −1693.55 −1693.36 −1693.15 −1692.75

Akaike 4.321326 4.243863 4.245887 4.245383 4.246866R2 0.0000 0.09341 0.08100 0.08612 0.08247

MSE 237.4908 203.1438 205.1336 204.2901 204.7869MedSE 15.7309 15.6039 15.4001 14.9537 15.4742

MAE 8.2124 8.0538 8.0759 8.0668 8.0755AMAPE 0.5570 0.5356 0.5354 0.5355 0.5352

TIC 0.7123 0.5779 0.5787 0.5784 0.5777

Table 4.6. GARCH model selection for Volkswagen. All models with p = q = 1.

GARCH EGARCH APARCH IGARCHLog −1693.545 −1701.674 −1692.588 −1706.267

Akaike 4.243863 4.269186 4.246470 4.273168R2 0.0934135 0.116252 0.115478 0.11051

MSE 203.1438 200.1063 199.5250 195.0135MedSE 15.6039 16.8086 15.5668 22.1922

MAE 8.0538 8.1821 7.9957 8.8270AMAPE 0.5356 0.5420 0.5339 0.5450

TIC 0.5779 0.5765 0.5732 0.4865

difference is in the fourth decimal. Choice for future model selection isp = 1, q = 1.

4.3.2 Choosing model

Four different GARCH models are tested. These models are described inSection 3.7. In table 4.6 the model APARCH(1,1) is performing best onout-of-sample measures. Therefore the APARCH(1,1) is used in the futuremodel selection.

4.3.3 Choosing distribution

Three different distributions are tested. A description of these models ismade in Section 3.8. The distributions are: normal, student-t and skewed

4.3.4 Estimated parameters 33

Table 4.7. Distribution selection for Volkswagen. The distribution is a Studentdistribution, with 6.0162 degrees of freedom. The distribution is a Skewed Stu-dent distribution, with a tail coefficient of 5.9079 and an asymmetry coefficient of−0.0515656.

Normal Student-t Skewed Stud.-tLog −1692.588 −1677.872 −1677.220

Akaike 4.246470 4.212179 4.213049R2 0.115478 0.113103 0.113761

MSE 199.5250 198.6129 198.7152MedSE 15.5668 16.5073 16.5735

MAE 7.9957 8.0435 8.0369AMAPE 0.5339 0.5347 0.5346

TIC 0.5732 0.5667 0.5677

Table 4.8. Estimated parameters for the chosen model for Volkswagen. The modelis an APARCH(1,1) with normal distribution.




student-t distribution. The result for the distribution comparison is pre-sented in table 4.7. Here we will use the normal distribution as our choice.

4.3.4 Estimated parameters

The best GARCH model in out-of-sample data is the APARCH(1,1). Theparameters for the chosen model is presented in table 4.8. The APARCH-(Gamma1) is the coefficient, which takes the leverage effect into account.This parameter in this case is not significant. For a more detailed discus-sion about the leverage effect, see section 3.1.4. This table is explained indetail in Section 3.3.4. One thing that is also interesting to note is thatthe APARCH(Gamma1) is shown to be time dependent in the Nelson test,section 3.3.5.

34 Volkswagen

Table 4.9. Estimated parameters for the chosen model for Volkswagen. The modelis a GARCH(1,1) with normal distribution. Data set divided in two parts (1-470)and (471-953) to see if the parameters is time dependent.

Parameter Coefficient Std.Error t-value t-probCst(M) 0.050629 0.086188 0.5874 0.5572Cst(V) 2.060946 0.690373 2.985 0.0030


Cst(M) −0.014238 0.093011 −0.1531 0.8784Cst(V) 0.208289 0.094871 2.196 0.0286


4.4 Validation

In this section the chosen GARCH model is validated. One part of the val-idation procedure has already been taking part when the model was chosenon its forecasting ability. Other tests are performed to see if the autocorrela-tion in the squared return has successfully been removed and to see whetheror not the distribution of the normalised residuals ε2t

h2t

= zt are normallydistributed. All these tests and measures will be presented in the followingsection.

4.4.1 Parameter time dependence

To check for time dependence of the parameters the data set is divided intwo parts (1-470) and (471-953). As seen in table 4.9 the estimated co-efficients are heavily dependent on the data set. The data estimation isfor the standard GARCH model with normal distribution. The reason isdue to convergence problems in the estimation of the APARCH(1,1) coef-ficients. Cst(V) is 2.06 in the first period and 0.208 in the other period.GARCH(Beta1) is 0.28 in the first data set and 0.83 in the other. It is rea-sonable that the GARCH(Beta1) parameter is changing over time. Whenbeta is low then the value of Cst(V) is going up and vice versa. It certainlylooks like the GARCH-effects in this stock are changing over time.


As seen in figure 4.5 most of the significant autocorrelation has been re-moved. Compared this result to the result from the preestimation (see fig-ure 4.3).

4.4.3 Ljung-Box-Pierce Q-test 35

0 2 4 6 8 10 12 14 16 18 20−0.2

0

0.2

0.4

0.6

0.8

Lag

Samp

le Au

tocor

relat

ion

Sample Autocorrelation Function (ACF)

Figure 4.5. Autocorrelation for Volkswagen with residuals after estimation.

Table 4.10. Ljung-Box-Pierce Q-test on the squared residuals, validation for Volk-swagen. H0 is that no significant correlation exists. H0 should be accepted whenthe probability is high. The test is performed on lags 10, 15 and 20.

Q(10) = 7.94418 [0.438942 ]Q(15) = 12.2437 [0.507767 ]Q(20) = 14.7842 [0.676728 ]

4.4.3 Ljung-Box-Pierce Q-test

The Lbq-test in the validation part is to test whether there still is a signif-icant correlation in the standardised innovations or not. The standardisedinnovations are ε2t

h2t

= zt and according to the theory for GARCH modelsand our assumptions in Section 3.7, zt is i.i.d. N(0, 1). For results on theLbq-test see table 4.10. The test is made on zt with lags 10, 15 and 20.Each row in the table represents one test with a specific lag. H0 is accepted,p-values are given within angle brackets. The theory for the test is presentedin Section 3.3.2.

4.4.4 Engle’s ARCH-test

The result from Engle’s ARCH-test is presented in table 4.11. This indicatesthat we have successfully removed the conditional heteroskedasticity that ex-isted when the test was performed on the pure return series in Section 4.2.4.

36 Volkswagen

Table 4.11. Engle’s ARCH-test, validation for Volkswagen. H0 is that no ARCHeffect exists.

ARCH 1-5 test: F(5,787) = 0.57168 [0.7218]ARCH 1-10 test: F(10,777) = 0.97848 [0.4606]ARCH 1-15 test: F(15,767) = 0.99426 [0.4590]

Table 4.12. Moments, skewness and kurtosis for zt of Volkswagen, where zt is thestandardised residuals.

VolkswagenFirst moment 0.00089

Second moment 1.00539Skewness 0.31870

Excess kurtosis 1.1851

4.4.5 Is zt normally distributed?

According to our assumptions in Section 3.7, zt is supposed to be normallydistributed. In this section we will analyse if this really is the case. Thereis a number of ways to do this. The discussion here is based on the mostcommon one, which is the Jarque-Bera test, and the calculated moments forthe distribution. For a more detailed description see section 3.3.

Moments of zt

For the calculated first two moments, skewness and excess kurtosis see ta-ble 4.12. It is quite clear that this distribution at least has first and secondmoments close to their corresponding normal values. Noticeable here is thatthe skewness has increased with a factor 10 compared with the raw returnseries. The reason for this will be treated in Section 4.4.6.

Jarque-Bera test

To test whether the zt is normally distributed or not a Jarque-Bera test isperformed. The result from the jb-test is presented in table 4.13. Theory forthe Jarque-Bera test is presented in Section 3.3.1. A p-value close to zeroindicates that we must reject the hypothesis that zt is normally distributed.

4.4.6 Skewness of zt 37

Table 4.13. Results from the jb-test. Normality test for zt.

Statistic p-valueJarque-Bera 46.917 6.4877e-011

Table 4.14. Skewness for zt for Volkswagen. With standardised residuals forthe GARCH(1,1). APARCH(1,1) not presented because of numerical convergenceproblems.

Data interval Skewness value t-test p-value(1-450) −0.13855 1.2039 0.22864

(451-952) 0.20687 1.8979 0.057715

4.4.6 Skewness of zt

For a detailed analysis of the reason for the high skewness in zt four dif-ferent estimates are made. The data is divided into two different parts,1 − 450 and 451 − 952, and then two different models are estimated. Oneis the APARCH(1,1), which is our chosen model, and one is the symmet-ric GARCH(1,1). The skewness result from this estimation is presented intable 4.14. It is quite obvious that the skewness for the raw return is de-pendent on which time part the estimation is done over. The result for theAPARCH(1,1) is not presented because of numerical convergence problemsin the estimation procedure.

4.5 Summary

4.5.1 Estimation

Different (p, q) values are tested in the symmetric case. These are then usedas fixed p and q for different GARCH-models. In this case p = q = 1appears to be the best out-of-sample choice.

Different models are tested for these specific p and q. The asymmetric-power-ARCH is the model that performs best among the models thatare tested.

Different distribution for this specific APARCH(1,1) are then tested andestimated. There is no evidence that any other distribution than thenormal should be used.

The estimated coefficient values are fairly close to the standard symmetricGARCH(1,1), with APARCH(Gamma1) close to zero and APARCH-

38 Volkswagen

(Delta) close to two.1 Still, the APARCH(1,1) outperforms the GA-RCH(1,1) in out-of-sample measures.

The standardised residuals zt are not normally distributed and this is be-cause of the skewness and the kurtosis. However the standardisedmean and the standardised variance are close to there excepted val-ues.

Convergence problems arise when the data set is too small (< 500) datasamples. Ideally more than 700 data points should be used for theestimation of GARCH parameters.

Time dependent coefficients is a problem within this data set. Even fora standard GARCH(1,1) the parameters are different when estimatedover two data samples in the same series.

The final decision for which model is up to the user. The model whichperform best on out-of-sample measures is the APARCH(1,1) but thismodel has convergence problems and the coefficients seem to be timedependent. If this model is used, then the coefficient has to be reesti-mated when new data (information) is available. The other possibilityis to use the more robust symmetric GARCH(1,1) which is not the beston out-of-sample measures but has less time dependent parameters.

1This is easily seen when these parameters are put into the APARCH-equation.

Chapter 5

Munchner Ruck

In this chapter model estimation for the Munchner Ruck stock is performed.The procedure is following the structure presented in the methodology chap-ter. The structure is the same as for Volkswagen.

5.1 Presenting data

5.1.1 Introducing Munchner Ruck

The stock price for Munchner Ruck is taken from the Germany stock market(DE). The data is sampled daily from the 25th of February 1999 to the 21stof October 2002. Especially interesting days are presented in table 5.1.There are three days with return larger than 10% over the whole period.The days before and after these large returns are also presented. The dateis in the format (dd.mm.yy). Instead of typing Munchner Ruck we will fromhere on use the abbreviation muvg.

Table 5.1. Days with high return in the muvg series.

date nr muvg22.12.99 119 0.042223.12.99 120 0.156824.12.99 121 010.09.01 567 0.012311.09.01 568 −0.170412.09.01 569 0.050826.07.02 796 0.015629.07.02 797 0.106730.07.02 798 0.0075

40 Munchner Ruck

Table 5.2. Moments, skewness and kurtosis for the return series of muvg.

muvgFirst moment 0

Second moment 6.6197Third moment 0.0867

Fourth moment 329.6006Skewness 0.0051Kurtosis 7.5215

5.2 Preestimation


The data series for muvg consists of 852 data points. The data is daily andthe return is calculated with equation (4.1). Data points (1-700) are used inthe estimation part and data points (701-832) are used for cross-validation.


The moments, skewness and kurtosis are presented in table 5.2. The dataseries has a high kurtosis which is usual for stock price return series.

5.2.3 Plots for Munchner Ruck

In this section the most important plots are presented. Before starting theestimation procedure it is interesting to see the behaviour of the specific dataset presented in graphical terms. The price and return plots are presentedin figure 5.1 and figure 5.2 respectively. The squared return and the auto-correlation is presented in figure 5.3 and figure 5.4. Here we observe thatthere is some persistence in the second moment for at least five lags. Thepartial autocorrelation is plotted in figure 5.5 and it also shows persistencein the second moment.


The feeling from the graphical plot that there exists correlation betweendata plots is in this subsection tested with the Ljung-Box-test. The testfor heteroskedasticity is performed with an ARCH-test (see section 3.3 fora theoretical explanation for these tests).


0 200 400 600 800

150

200

250

300

350

400

Münchner Rück

date

stoc

k pr

ice

Figure 5.1. Price plot for Munchner Ruck displaying daily closing prices. Thetime axis ranges from the 9th July 1999 to the 17th of September 2002.

0 200 400 600 800

−0.15

−0.05

0.00

0.05

0.10

0.15

Münchner Rück

return

Figure 5.2. Return plot for Munchner Ruck, continuously compounded. The datais daily and starts at the 9th July 1999 and ends the 17th of September 2002.


We compute the statistical value for specific lags and present them togetherwith the p-values in angle brackets; The result from the test is presented intable 5.3. The null hypothesis is that no serial correlation exists and thishypothesis is accepted when the p-value is high. In this case we have toreject that hypothesis.

42 Munchner Ruck

0 100 200 300 400 500 600 700 800 9000

50

100

150

200

250

300

Figure 5.3. Squared returns for Munchner Ruck, for daily data starts at the 9thJuly 1999 and ends the 17th of September 2002.

0 2 4 6 8 10 12 14 16 18 20−0.2

0

0.2

0.4

0.6

0.8

Lag

Samp

le Au

tocor

relat

ion

muvg

Figure 5.4. Autocorrelation for muvg on the squared return.

Table 5.3. Ljung-Box-Pierce Q-test for muvg. P-values are given in angle brackets.With these low p-values we have to reject the hypothesis that no serial correlationexists.

Q(10) = 54.0457 [6.76393e-009 ]Q(15) = 56.1805 [2.50501e-007 ]Q(20) = 65.5801 [2.49021e-007 ]

Estimation 43

0 2 4 6 8 10 12 14 16 18 20−0.2

0

0.2

0.4

0.6

0.8

Lag

Samp

le Pa

rtial A

utoco

rrelat

ions

muvg

Figure 5.5. The partial autocorrelation on the squared return for muvg, a measureof partial autocorrelation, in the volatility.

Table 5.4. Engle’s ARCH-test for muvg on the raw return series. H0 is that noARCH effect exists and in this case we have p-values equal to zero which meansthat we can reject that hypothesis.

ARCH 1-2 test: F(2,693) = 3.2265 [0.0403]*

Engle’s ARCH-test

Engle’s ARCH-test is performed before estimation. This test supports thatthere exists heteroskedasticity since the p-value is less than 0.05. The fullresult for muvg is presented in table 5.4. ARCH-effects are tested on lags1-2. With this p-value we reject the hypothesis that no ARCH effects exist,but only on the level of 95%.

5.3 Estimation

This section is presenting the whole estimation procedure for muvg GARCHmodels. First p and q for the standard GARCH model is chosen and thenfor these specific p and q different GARCH models are tested, estimatedand finally one GARCH model with a specific p and q is chosen. For thisGARCH(p,q)-model different distributions are evaluated. Finally the pa-rameters for the chosen GARCH model are presented.

44 Munchner Ruck

Table 5.5. Selection for symmetric GARCH for muvg. Log-likelihood is presentedas well as Akaike’s information criterion for different p and q. Six different mea-sures are also presented to determine which p and q that perform best on out-of-sample data.

GARCH(p,q) (0,0) (1,1) (1,2) (2,1) (2,2)Log −1595.2 −1548.66 −1548.54 −1548.41 −1548

Akaike 4.563421 4.436184 4.438688 4.438313 4.439988R2 0 0.177904 0.180966 0.18553 0.175453

MSE 405.4252 302.5713 301.3332 299.5203 303.6363MedSE 28.2512 24.8449 26.2407 27.5363 28.2426

MAE 10.9956 10.6565 10.6410 10.6349 10.7037AMAPE 0.5834 0.5355 0.5351 0.5351 0.5375

TIC 0.7169 0.4731 0.4717 0.4702 0.4752


Different p and q for the standard symmetric GARCH models are tested.The selection procedure is based on each model’s ability to produce forecasts.The six different forecasts measures are discussed in Section 2.2.4 are used.The muvg table 5.5 shows that the GARCH(2,1) performs best in three outof six out-of-sample measures. Choice for future model selection is p = 2and q = 1.


Four different GARCH models are tested. These models are theoreticallydescribed in Section 3.7. In table 5.6 the model APARCH(2,1) is performingbest on out-of-sample measures. Therefore the APARCH(2,1) is used in thefuture model selection. The EGARCH model did not converge so the Logand Akaike values are presented for the initial estimate instead.

The APARCH(1,1) has been a good choice in the other data series,therefore the APARCH(2,1) will be tested against the APARCH(1,1) toverify that it really performs best. The result is presented in table 5.7. Theresult confirms that APARCH(2,1) is our choice for further estimation.


Three different distributions are tested on APARCH(2,1): normal, student-tand skewed student-t distribution. The result for the distribution compari-son is presented in table 5.8. It is not an obvious choice which distributionto choose. The student-t and the skewed student-t are performing quiteequally in the out-of-sample measures. But looking at the Log value, we

5.3.4 Estimated parameters 45

Table 5.6. GARCH model selection for muvg. The EGARCH model did not con-verge during the estimation, due to numerical problems. All models are estimatedwith p = 2 and q = 1.


Akaike 4.438313 4.466900 4.438823 4.438140R2 0.18553 - 0.210067 0.185916

MSE 299.5203 - 289.8320 304.4905MedSE 27.5363 - 30.7050 32.5273

MAE 10.6349 - 10.6297 10.9585AMAPE 0.5351 - 0.5315 0.5344

TIC 0.4702 - 0.4541 0.4480

Table 5.7. Verification of the APARCH(2,1) model for muvg.

APARCH(1,1) APARCH (2,1)Log −1546.94 −1546.59

Akaike 4.436977 4.438823R2 0.204106 0.210067

MSE 292.2124 289.8320MedSE 32.0056 30.7050

MAE 10.6204 10.6297AMAPE 0.5317 0.5315

TIC 0.4567 0.4541

notice that it has not changed when adding an extra parameter. This is anindication that we should choose the student-t as our distribution choice.This is also the AIC choice and it will be our final model choice.


The best GARCH model in out-of-sample data is the APARCH(2,1). Theparameters for the chosen model is presented in table 5.9. The APARCH-(Gamma1) takes the leverage effect into account, but the parameter in thiscase is not significant. For a more detailed discussion about the leverageeffect see section 3.1.4. This table is explained in detail in Section 3.3.4.Three coefficients are time dependent on the level of 95%, these are: Cst(V),GARCH(Beta1) and GARCH(Beta2) (see the Nelson test in Section 3.3.5).

46 Munchner Ruck

Table 5.8. Distribution selection for muvg. The student distribution has 7.0056degrees of freedom and the skewed student distribution has a tail coefficient of7.00123 and an asymmetry coefficient of −0.00283962. All three are tested onan APARCH(2,1) model.


Akaike 4.438823 4.367206 4.370059R2 0.210067 0.219198 0.219242

MSE 289.8320 286.8509 286.8081MedSE 30.7050 28.8216 28.8743

MAE 10.6297 10.5239 10.5264AMAPE 0.5315 0.5297 0.5297

TIC 0.4541 0.4641 0.4638

Table 5.9. Estimated parameters for the chosen APARCH(2,1) model for muvg.The model is estimated with a maximum likelihood estimation.


GARCH(Beta1) 1.423870 0.300096 4.745 0.0000GARCH(Beta2) −0.483291 0.277752 −1.740 0.0823ARCH(Alpha1) 0.046485 0.024051 1.933 0.0537

APARCH(Gamma1) 0.298389 0.140086 2.130 0.0335APARCH(Delta) 1.899521 0.447514 4.245 0.0000

Student(DF) 7.005608 1.575725 4.446 0.0000

5.4 Validation

In this section the chosen GARCH model is validated. One part of the val-idation procedure has already been taking part when the model was chosenon its forecasting ability. Other tests are performed to see if the autocorre-lation in the squared return has successfully been removed. Both these testsand measures will be presented in the following sections.

5.4.1 Time dependent parameters

The data set is divided into two parts: One with data points (1-399) andthe other with data points (400-832). As one can see in table 5.10 the valuesof the parameters depend on the data set. Either large samples should beused to get accurate parameter estimation or the model has highly time

5.4.2 Ljung-Box-Pierce Q-test 47

Table 5.10. Estimated parameters for muvg on different data sets for theAPARCH(2,1) model with a skewed student-t distribution. The data set is dividedin two parts, (1-399) and (400-832) respectively.


GARCH(Beta1) 0.074981 0.079091 0.9480 0.3437GARCH(Beta2) 0.865831 0.079229 10.93 0.0000ARCH(Alpha1) 0.038896 0.015634 2.488 0.0133


Student(DF) 8.756206 3.208301 2.729 0.0066Cst(M) −0.203426 0.078598 −2.588 0.0100Cst(V) 0.063072 0.030381 2.076 0.0385

GARCH(Beta1) 1.144989 0.205460 5.573 0.0000GARCH(Beta2) −0.237451 0.185655 −1.279 0.2016ARCH(Alpha1) 0.078962 0.023409 3.373 0.0008


Student(DF) 24.694184 22.765973 1.085 0.2787

dependent coefficients. These parameter values can be compared also withthe values in table 5.9. Notice also how the APARCH(Gamma1) and theGARCH(Beta2) even have changed sign between the different periods.


The Lbq-test in the validation part tests if there still is significant correlationin the standardised innovations or not. The standardised innovations areε2th2

t= zt. For results on the Lbq-test see table 5.11. The test is made on

zt with lags 10, 15 and 20. Each row in the table represents a test with aspecific lag. H0 has to be accepted with these p-values given within anglebrackets. The theory for the test is presented in Section 3.3.2.


Result from Engle’s ARCH-test is presented in table 5.12. This result indi-cates that we have successfully removed the conditional heteroskedasticitythat existed when the test was performed on the pure return series in Sec-tion 5.2.4.

48 Munchner Ruck

Table 5.11. The Ljung-Box-Pierce Q-test on the squared residuals in the validationfor muvg. H0 is that no significant correlation exists. H0 should be accepted whenprobability is high. The test is performed on lags 10, 15, 20.

Q(10) = 5.47597 [0.602084 ]Q(15) = 7.37838 [0.83163 ]Q(20) = 10.4553 [0.883439 ]

Table 5.12. Engle’s ARCH-test in validation for muvg. H0 is that no ARCHeffect exists.

ARCH 1-5 test: F(5,686) = 0.31077 [0.9066]ARCH 1-10 test: F(10,676)= 0.54581 [0.8578]ARCH 1-15 test: F(15,666)= 0.46504 [0.9575]

5.4.4 How is zt distributed?

Moments of zt

For the calculated first two moments, skewness and excess kurtosis, seetable 5.13. Here at least the first moment and the second moment are closeto their expected values.

5.5 Summary

5.5.1 Estimation

Different (p,q) values are tested in the symmetric case. These are then usedas fixed p and q for different GARCH models. In this case p = 2, q = 1is the best out-of-sample choice.

Table 5.13. Moments, skewness and kurtosis for zt of muvg, where zt is thestandardised residuals.

muvgFirst moment 0.01913



5.5.1 Estimation 49

Different models are tested with these specific p and q. The asymmetric-power-ARCH (APARCH(2,1) performs best among the models thatare tested.

Different distributions for this specific APARCH(2,1) are then tested andestimated. The distribution we use is best on out-of-sample measurestogether with the AIC choice and our final choice is the student-tdistribution.

Numerical problems for convergence occurred in the estimation part of theEGARCH(2,1). Therefore this model has not been fully tested.

Chapter 6

Suez-Lyonnaise des Eaux

In this chapter model estimation for the volatility in Suez-Lyonnaise desEaux stock is performed. The procedure is following the structure that waspresented in the methodology chapter. The structure is the same as forVolkswagen.

6.1 Presenting data

6.1.1 Introducing Suez-Lyonnaise des Eaux

The stock price for Suez-Lyonnaise des Eaux is taken from the Paris StockExchange (PA). The data is daily from the 9th of June 1999 to the 17th ofSeptember 2002. Especially interesting days are presented in table 6.1. Theselection criterion is that the absolute return should be above 10%. The11th of September is one of those days. Days before and after interestingdays are also presented. The data is presented in the form (dd.mm.yy).During this chapter, instead of typing Suez-Lyonnaise des Eaux, we will usethe abbreviation lyoe. The stock price from date 00.06.06 is missing. Thisis handled by taking the mean of the price the day before and the day after.

6.2 Preestimation


The data series of lyoe consists of 832 data points. The return is calculatedby:

returnt = 100(ln(pricet) − ln(pricet−1) (6.1)

The reason for multiplying by 100 is due to numerical problems in the es-timation part. This will not affect the structure of the model, since this is

52 Suez-Lyonnaise des Eaux

Table 6.1. Days with high absolute returns in the lyoe series. The day before andthe day after interesting dates are also printed. Date format is (dd.mm.yy).

date nr lyoe01.02.00 148 0.022802.02.00 149 0.102603.02.00 150 0.045610.09.01 567 −0.009211.09.01 568 −0.101312.09.01 569 0.079512.07.02 786 −0.007415.07.02 787 −0.121916.07.02 788 0.076224.07.02 794 −0.050025.07.02 795 0.104326.07.02 796 0.014105.08.02 802 −0.080106.08.02 803 0.124107.08.02 804 −0.0637

Table 6.2. Moments, skewness and kurtosis for the return series of lyoe.

lyoeFirst moment 0



just a linear scaling. Data points (1-700) are used in the estimation partand data points (701-832) are used as cross validation.


In the preestimation part it would be interesting to get a picture of whatthe distribution of the raw return series is like. The moments, skewness andkurtosis is presented in table 6.2. For a theoretical introduction to thesemeasures see section 3.2. Noteworthy is the kurtosis which is considerablylarger than three.

6.2.3 Correlation plots for lyoe 53

6.2.3 Correlation plots for lyoe

Three interesting preestimation plots for the price, return and the squaredreturn are presented in figure 6.1, figure 6.2 and figure 6.3 respectively. Theautocorrelation and partial autocorrelation is presented in figure 6.4 and infigure 6.5 respectively.

0 200 400 600 800

2025

3035

40

Suez−Lyonnaise des Eaux

date

stock

price

Figure 6.1. Price plot for Suez-Lyonnaise des Eaux, showing the daily closingprices. The data series starts at the 9th July 1999 and ends at the 17th of September2002.

0 200 400 600 800

−0.10

−0.05

0.00

0.05

0.10

Suez−Lyonnaise des Eaux

return

Figure 6.2. Return plot for Suez-Lyonnaise des Eaux, continuously compounded.Daily data starts at the 9th July 1999 and end the 17th of September 2002.


0 100 200 300 400 500 600 700 800 9000

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016Lyoe

Figure 6.3. Squared return for lyoe.

0 2 4 6 8 10 12 14 16 18 20−0.2

0

0.2

0.4

0.6

0.8

Lag

Samp

le Au

tocor

relat

ion

Lyoe

Figure 6.4. Autocorrelation for lyoe. This is the autocorrelation on the squaredreturn, thus this is a measure of autocorrelation in the volatility.


The indication from the graphical plot that there exists correlation betweendata plots is in this subsection tested with the Ljung-Box-test. The test forheteroskedasticity is performed with an ARCH-test.


0 2 4 6 8 10 12 14 16 18 20−0.2

0

0.2

0.4

0.6

0.8

Lag

Samp

le Pa

rtial A

utoco

rrelat

ions

Lyoe

Figure 6.5. Partial autocorrelation for lyoe on the squared return.

Table 6.3. Ljung-Box-Pierce Q-test for lyoe. P-values are given in angle brackets.With these low p-values we have to reject the hypothesis that no serial correlationexists.

Q(10) = 98.323 [9.39552e-018 ]Q(15) = 99.1289 [2.44655e-015 ]Q(20) = 101.958 [9.68458e-014 ]


The computed statistical value for the specific lag is presented togetherwith the p-value in angle brackets. Results from the test are presented intable 6.3 for lyoe. The null hypothesis is that no serial correlation exists andthis hypothesis is accepted when the p-value is high. In this case we haveto reject that hypothesis. The correlation is tested for the squared return.

Engle’s ARCH-test

Engle’s ARCH-test performed before estimation. This test supports thatthere exists heteroskedasticity. The result for lyoe is presented in table 6.4.ARCH-effects are tested on lags 1-2. With these p-values we reject thehypothesis that no ARCH-effects exist.


Table 6.4. Engle’s ARCH-test for lyoe on the raw return series. H0 is that noARCH effect exists and in this case we have p-values equal to zero which meansthat we can reject the hypothesis.

ARCH 1-2 test: F(2,693) = 20.045 [0.0000]**

Table 6.5. Selection for symmetric GARCH for lyoe. The log-likelihood is pre-sented as well as Akaike’s information criterion for different p and q. Six differentforecasts measures are also presented to determine what p and q that perform beston out-of-sample data for the symmetric GARCH.

GARCH(p,q) (0,0) (1,1) (1,2) (2,1) (2,2)Log −1404.37 −1371.08 −1368.17 −1369.1 −1368.35

Akaike 4.018192 3.928801 3.923349 3.926012 3.926722R2 0.0000 0.203696 0.164474 0.182692 0.165933

MSE 820.9542 611.5326 620.8024 622.5953 621.2564MedSE 10.0241 8.4617 8.5447 8.6772 8.6531

MAE 14.1338 11.7340 11.7006 11.7149 11.6941AMAPE 0.6600 0.5618 0.5565 0.5594 0.5566

TIC 0.8595 0.6010 0.5937 0.6078 0.5955

6.3 Estimation

This section is presenting the whole estimation procedure for lyoe GARCHmodels. First p and q for the standard GARCH are chosen and then for thesespecific p and q different GARCH models are tested, estimated and finallyone GARCH model with a specific p and q is chosen. For this GARCH(p,q)-model different distributions are evaluated. Finally the parameters for thechosen GARCH model are presented.


Different p and q for the standard symmetric GARCH model are tested. Theselection procedure is based on each model’s ability to produce forecasts. Sixdifferent forecasts measures are used.

The lyoe table 6.5 shows that the GARCH(1,1) performs best in threeout of six out-of-sample measures. The choice for future model selection isp = 1, q = 1.

6.3.2 Choosing model 57

Table 6.6. GARCH model selection for lyoe. All models are estimated for p = 1,q = 1.


Akaike 3.928801 3.923722 3.920469 3.965381R2 0.203696 0.111235 0.260001 0.221219

MSE 611.5326 684.4780 549.2374 557.8387MedSE 8.4617 12.0363 18.0764 26.4158

MAE 11.7340 12.5614 11.4671 12.8462AMAPE 0.5618 0.5742 0.5580 0.5455

TIC 0.6010 0.6687 0.5351 0.4451

Table 6.7. Distribution selection for lyoe.


Akaike 3.920469 3.851465 3.854258R2 0.260001 0.275744 0.276033

MSE 549.2374 537.4875 536.3566MedSE 18.0764 18.5647 19.2230

MAE 11.4671 11.4685 11.4730AMAPE 0.5580 0.5521 0.5520

TIC 0.5351 0.5259 0.5244


Four different GARCH models are tested. In table 6.6 the model AP-ARCH(1,1) is performing best on out-of-sample measures and therefore theAPARCH(1,1) is used for further model selection.


Three different distributions are tested: normal, student-t and skewed stud-ent-t distribution. The result for the distribution comparison is presentedin table 6.7. The estimated degrees of freedom for the student-t distributionis 5.610786. For skewed student distribution, the tail coefficient is 5.60036and the asymmetry coefficient is −0.0116698.


Table 6.8. Estimated parameters for the chosen model for lyoe. The model is anAPARCH(1,1) estimated with a maximum likelihood estimation.




Asymmetry −0.011670 0.055067 −0.2119 0.8322Tail 5.600366 1.110430 5.043 0.0000


The best GARCH model in out-of-sample data is the APARCH(1,1). Theparameters for the chosen model is presented in table 6.8. The APAR-CH(Gamma1) is the coefficient which takes the leverage effect into ac-count. This parameter in this case is not significant. For a descriptionabout the APARCH-model see section 3.7.4. For a more detailed discus-sion about the leverage effect see section 3.1.4. This table is explained indetail in Section 3.3.4. One thing that is also interesting to note is that theAPARCH(Gamma1) and the asymmetry coefficient are shown to be timedependent in the Nelson test.

6.3.5 Are the coefficients time dependent?

When the asymmetric coefficient and the APARCH(Gamma1) are time de-pendent it is natural to ask whether or not the model can be improved if afloating window with fixed size is used.

The symmetric GARCH(1,1) coefficients are presented in figure 6.6 andfigure 6.7. The plot is from data, where data points are numbered with anoffset of 700 points for the ARCH and GARCH coefficients including ± onestandard deviation.

A closer look at figure 6.6 shows that something interesting is happeningin days (60-70). The Nelson test does not reject the hypothesis that thecoefficients are time independent, when tested over the whole data set (952data points). An interesting thing could be to estimate over this specificperiod with different window sizes to see if this can improve the forecastingability of the model. The forecast period is set to 132 days, out-of-sampledata. First all data (700 data points) are used for the estimation thenthe window size is shrunken in steps, with window sizes 700, 500, 300 and150. The result from this estimating does not support the use of smaller

Validation 59

estimation windows.

0 20 40 60 80 100 120 1400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 6.6. Time dependent coefficient plot for lyoe. Window size = 200.

0 20 40 60 80 100 120 1400.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

coef

ficien

t valu

e

Figure 6.7. Time dependent coefficient plot for lyoe. Window size = 700.

6.4 Validation

In this section the chosen GARCH model is validated. One part of the val-idation procedure has already been taking part when the model was chosenon its forecasting ability. Other tests are performed to see if the autocorre-lation in the squared return has successfully been removed. All these tests


Table 6.9. Estimated parameters for lyoe on two different data sets. Data set(1-299) has a skewed student distribution, with a tail coefficient of 4.77673 andan asymmetry coefficient of 0.190183. Data set (300-832) has a skewed studentdistribution, with a tail coefficient of 8.0258 and an asymmetry coefficient of -0.200555.




Asymmetry 0.190183 0.087422 2.175 0.0304Tail 4.776735 1.264920 3.776 0.0002

Cst(M) −0.183533 0.069957 −2.624 0.0090Cst(V) 0.172982 0.062135 2.784 0.0056



Asymmetry −0.200555 0.063555 −3.156 0.0017Tail 8.025805 2.685321 2.989 0.0029

and measures will be presented in the following sections.

6.4.1 Parameter time dependency

In table 6.9, where coefficient values for the APARCH model is displayed, itis obvious that the parameter values are changing over time. The Asymme-try coefficient is in the first period 0.19 and in the other one −0.20; in bothcases significantly unequal to zero, so either we take this model, meaningthat we have to reestimate the model when new information is available,or, if we are interested in a more robust model, we could use the standardsymmetric GARCH(1,1) with normal distribution. It is not that bad con-sidering its simplicity. Table 6.10 presents the estimates for two differentsamples. Here at least the parameters are a bit more stable.


As seen in figure 6.8 most of the significant autocorrelation has been re-moved. Compared this to the result from the preestimation (see figure 6.4).

6.4.3 How is zt distributed? 61

Table 6.10. Estimated parameters for lyoe on two different samples of the sym-metric GARCH(1,1) model. Data set one is data points (1-416) and the other dataset is (417-832), both with a normal distribution.



Cst(M) −0.069178 0.085035 −0.8135 0.4164Cst(V) 0.257725 0.116921 2.204 0.0281


0 2 4 6 8 10 12 14 16 18 20−0.2

0

0.2

0.4

0.6

0.8

Lag

Samp

le Au

tocor

relat

ion


Figure 6.8. Autocorrelation for lyoe with standardised residuals after estimation.

6.4.3 How is zt distributed?

A graphical presentation of the standardised distribution is presented infigure 6.9 and the calculated first two moments, skewness and excess kurtosisis displayed in table 6.11. The first and the second moment are close to theirexpected values.


The Lbq-test in the validation part tests if there still is or is not a significantcorrelation in the standardised innovations. Standardised innovations areε2th2

t= zt. For results on the Lbq-test, see table 6.12. The test is made on


Table 6.11. Moments, skewness and kurtosis for the standardised residuals in thedata set of lyoe.

lyoeFirst moment 0.01464

Second moment 1.00107Skewness -0.065131


−3 −2 −1 0 1 2 30

10

20

30

40

50

60

Figure 6.9. Histogram for lyoe after estimation with standardised residuals. Thehistogram is divided in 100 bins.

zt with lags 10, 15 and 20. Each row in the table represents one test witha specific lag, p-values are given within angle brackets i.e., H0 has to beaccepted. The theory for the test is presented in Section 3.3.2.

Table 6.12. Ljung-Box-Pierce Q-test on the squared residuals, validation for lyoe.H0 is that no significant correlation exists. H0 should be accepted when probabilityis high. The test is performed on lags 10, 15, 20.

Q(10) = 9.59469 [0.294633 ]Q(15) = 11.2472 [0.590121 ]Q(20) = 13.1222 [0.784262 ]

6.4.5 Engle’s ARCH-test 63

Table 6.13. Engle’s ARCH-test, validation for lyoe. H0 is that no ARCH effectexists.

ARCH 1-5 test: F(5,687) = 1.1681 [0.3233]ARCH 1-10 test: F(10,677)= 1.0342 [0.4125]ARCH 1-15 test: F(15,667)= 0.82823 [0.6463]


Result from Engle’s ARCH-test is presented in table 6.13. This indicatesthat we have successfully removed the conditional heteroskedasticity that ex-isted when the test was performed on the pure return series in Section 6.2.4.The null hypothesis is that no ARCH effect exists and we cannot reject thathypothesis.

6.5 Summary

6.5.1 Estimation

Different p,q are tested in the symmetric case. These are then used as fixedp and q for different GARCH-models. In this case p = q = 1 is thebest out-of-sample choice.

Different models are tested for this specific p and q. The asymmetric-power-ARCH is the model which performs best among the modelsthat are tested.

Different distributions for this specific APARCH(1,1) are then tested andestimated. The skewed student-t distribution is the one that performsbest on out-of-sample measures.

The estimated coefficient APARCH(Gamma1) is significantly different fr-om zero but the asymmetry coefficient in the distribution is not sig-nificant.

The autocorrelation of the standardised residuals zt exhibits no evidencethat correlation still exists. The heteroskedasticity has also success-fully been removed.

The dynamic features for the model have not been fully investigated. Eitherthe APARCH(1,1) model with skewed student-t distribution is usedand then we have to be aware of the fact that the model has to bereestimated when new information is available, or, if we want a morerobust model, we can use the standard symmetric GARCH(1,1) withnormal distribution.

Chapter 7

OMX index

In this chapter model estimation for the Swedish OMX index is performed.The procedure is following the structure presented in the methodology chap-ter. Preestimation and estimation is performed. Finally some conclusionsare drawn.

7.1 Presenting data

7.1.1 Introducing OMX

The Swedish index OMX consists of 30 large Swedish companies. The anal-ysed data is from the the 5th of January 1998 to the 21st of March 2002.The data consists of 1051 data points (price data). Daily closing data is then1050 data points (return data). Especially interesting days are presented intable 7.1. The selection criterion is that the absolute return should be above5%, smaller than the selection criterion on the stocks since the OMX indexreturns are more smoothed. The days from the 11th September to the 17thSeptember 2001 are not included in the index at all. The data is presentedin the form (dd.mm.yy).

7.2 Preestimation


For the data series of OMX, data points (1-800) are used in the estimationpart and data points (801-1050) are used as cross validation. The pricesseries is converted to a return series via equation (4.1).


In the preestimation part it would be interesting to get a picture of what thedistribution of the raw return series looks like. The moments, skewness and

66 OMX index

Table 7.1. Days with high return in the OMX index.

data return (%) date192 −6.8848 08.10.98194 11.0228 12.10.98237 −5.9640 10.12.98485 5.6546 08.12.99639 −5.2604 21.07.00704 −5.1285 23.10.00753 6.6036 04.01.01800 −7.2006 12.03.01818 5.4310 05.04.01825 5.7336 18.04.01929 5.3296 24.09.01

Table 7.2. Moments, skewness and kurtosis for the return series of OMX.

OMXFirst moment 0



kurtosis are presented in table 7.2. For a theoretical introduction to thesemeasures see section 3.2. Noticeable is the kurtosis is much larger thanthree, which would be the kurtosis if the series were normally distributed.

7.2.3 Correlation plots for OMX

The three interesting preestimation plots of the price, return and the squa-red return are presented in figure 7.1, figure 7.2 and figure 7.3 respectivily.The autocorrelation and partial autocorrelation is presented in figure 7.4and figure 7.5.


From the graphical plots we could assume that there exists correlation be-tween data points. This is in this subsection tested with the Ljung-Box-test.The test for heteroskedasticity is performed with an ARCH-test.


0 200 400 600 800 1000 1200400

600

800

1000

1200

1400

1600

Figure 7.1. Price plot for OMX with daily closing prices.

0 200 400 600 800 1000 1200−8

−6

−4

−2

0

2

4

6

8

10

12

Figure 7.2. Return plot for OMX.


The computed statistical value for a certain lag is presented together withthe p-values in angle brackets in table 7.3 for OMX. The null hypothesis isthat no serial correlation exists and this hypothesis is accepted when thep-value is high. In this case we have to reject that hypothesis.

68 OMX index

0 200 400 600 800 1000 12000

20

40

60

80

100

120

140

Figure 7.3. Squared return for OMX. The large values are because of the linearscaling in the transformation from price series to return series.

0 5 10 15 20 25 30 35 40 45 50−0.2

0

0.2

0.4

0.6

0.8

Lag

Samp

le Au

tocor

relat

ion


Figure 7.4. This is the autocorrelation of the squared return for OMX.

Table 7.3. Ljung-Box-Pierce Q-test for the OMX. P-values are given in anglebrackets. With these low p-values we reject the hypothesis that no serial correlationexists.

Q(10) = 103.985 [8.65908e-018 ]Q(15) = 131.561 [1.05139e-020 ]Q(20) = 158.337 [1.57155e-023 ]

Estimation 69

0 2 4 6 8 10 12 14 16 18 20−0.2

0

0.2

0.4

0.6

0.8

Lag

Samp

le Pa

rtial A

utoco

rrelat

ions

Sample Partial Autocorrelation Function

Figure 7.5. This is the partial autocorrelation of the squared return for OMX.

Table 7.4. Engle’s ARCH-test for OMX on the raw return series. H0 is that noARCH effect exists and in this case we have p-values equal to zero which meansthat we can reject the hypothesis that no ARCH effects exists.

ARCH 1-2 test: F(2,795) = 25.273 [0.0000]**ARCH 1-5 test: F(5,789) = 11.652 [0.0000]**

Engle’s ARCH-test

Engle’s ARCH-test performed before estimation. This test supports thatthere exists heteroskedasticity. The result for OMX is presented in table 7.4.ARCH-effects are tested on lags 1-2 and 1-5.

7.3 Estimation

This section is presenting the whole estimation procedure for OMX GARCHmodels. First p and q for the standard GARCH is chosen and then for thisspecific p and q different GARCH models are tested, estimated and finallyone GARCH model with a specific p and q is chosen. For this GARCH(p,q)-model different distributions are evaluated. Finally the parameters for thechosen GARCH models are presented.


Different p and q for the standard symmetric GARCH model is tested.The selection procedure is based on each model’s ability to produce good

70 OMX index

Table 7.5. Selection for symmetric GARCH for OMX. The log-likelihood value ispresented as well as Akaike’s information criterion for different p and q values. Sixdifferent forecast measures are also displayed to determine for which p and q theperformance is best on out-of-sample data.

(p,q) (0,0) (1,1) (1,2) (2,1) (2,2)Log −1609.06 −1546.45 −1545.88 −1543.49 −1543.357

Akaike 4.027640 3.876132 3.877211 3.871236 3.873393R2 0.0000 0.0934158 0.0783754 0.0488753 0.0549111

MSE 26.4675 24.1874 24.6836 26.3069 25.9692MedSE 6.9878 5.7996 5.8391 6.1702 6.1699

MAE 3.3548 3.3171 3.3435 3.4964 3.4665AMAPE 0.5173 0.4962 0.4971 0.5053 0.5035

TIC 0.5384 0.4580 0.4635 0.4729 0.4706

Table 7.6. GARCH model selection for OMX. The result for EGARCH(1,1) isomitted because of convergence problems.

GARCH (1,1) APARCH (1,1) IGARCH (1,1)Log −1546.45 −1538.94 −1547.44

Akaike 3.876132 3.862339 3.876093R2 0.0934158 0.113057 0.0941127

MSE 24.1874 23.9171 24.8089MedSE 5.7996 6.6722 6.8080

MAE 3.3171 3.4406 3.4731AMAPE 0.4962 0.5025 0.5021

TIC 0.4580 0.4429 0.4463

forecasts. Six different forecasts measures are used. Table 7.5 implies thatthe GARCH(1,1) performs best in six out of six out-of-sample measures.Choice for future model selection is p = 1, q = 1.


Four different GARCH models are tested. These models are theoreticallydescribed in Section 3.7. Table 7.6 does not show a clear picture of whichmodel to use. The APARCH(1,1) model is performing best is three of themeasures and the GARCH(1,1) performs best in three others. Here we usethe AIC to determine that the APARCH(1,1) model is our choice for furthermodel selection. However GARCH(1,1) is better in three of the measure soeven the GARCH(1,1) seems to do a good job.

7.3.3 Choosing distribution 71

Table 7.7. Distribution selection for OMX. The distributions are normal dis-tribution, a Student distribution, with 14.8614 degrees of freedom and a SkewedStudent distribution, with a tail coefficient of 13.1282 and an asymmetry coefficientof −0.103472.


Akaike 3.862339 3.855985 3.854248R2 0.113057 0.115848 0.116233

MSE 23.9171 23.9032 23.7695MedSE 6.6722 6.9391 6.7925

MAE 3.4406 3.4567 3.4238AMAPE 0.5025 0.5031 0.5017

TIC 0.4429 0.4407 0.4427


The three distributions are: normal, student-t and skewed student-t. Theresult for the distribution comparison is presented in table 7.7. The valuesmeasured change very little depending on the distribution. There is noneed for adding to extra parameters in the model when the measures justdiffer in the decimals. Our choice for further estimation will be the normaldistribution.


The best GARCH model in out-of-sample data is the APARCH(1,1). Theparameters for the chosen model is presented in table 7.8. For a descriptionof the APARCH-model, see section 3.7.4. This table is explained in detailin Section 3.3.4. No parameter is time dependent according to the Nelsontest. Noticeable is the APARCH(Delta) value which is lower than for theother series. All parameters have nice t-values which could be an indicationthat it is easier to find good coefficients estimates of GARCH models if theseries is a index instead of a stock.

7.4 Validation

7.4.1 Parameter time dependence

As one can see in table 7.9 the parameters are quite stable through time. Thedata is split into two different parts and then the parameters are estimated.The APARCH(1,1) model with a normal distribution is used. The reasonfor the unequal split is due to convergence problems.

72 OMX index

Table 7.8. Estimated parameters for the chosen model for OMX. The model is anAPARCH(1,1) with normal distribution. The data set consists of samples (1-800),which is the data set used in the estimation part.




Table 7.9. Estimated parameters for the chosen model for OMX. The model is anAPARCH(1,1) with normal distribution using MLE. The data set split in two parts(1-449) and (450-1050) to see if the parameters are time dependent.




Cst(M) −0.019698 0.073771 −0.2670 0.7896Cst(V) 0.128395 0.075498 1.701 0.0895




As seen in figure 7.6 most of the significant autocorrelation has been re-moved. Compared this to the result from the preestimation (see figure 7.4).


The Lbq-test in the validation part tests if there still is a significant corre-lation in the standardised innovations or not. For results on the Lbq-testsee table 7.10. The test is made on zt with lags 5, 10 and 20. Each row inthe table represents one test with a specific lag and H0 has to be accepted.

7.4.4 Engle’s ARCH-test 73

0 2 4 6 8 10 12 14 16 18 20−0.2

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0.6

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Figure 7.6. Autocorrelation for OMX with normalised residuals after estimation.

Table 7.10. Ljung-Box-Pierce Q-test on the squared normalised residuals, valida-tion for OMX. H0 is that no significant correlation exists. H0 should be acceptedwhen probability is high. The test is performed on lags 5, 10, 20. Here we acceptH0.

Q(5) = 3.40302 [0.33356 ]Q(10) = 6.24369 [0.619956 ]Q(20) = 16.4683 [0.559901 ]

Which means that we successfully have removed the autocorrelation thatexisted in the preestimation part.


Result from Engle’s ARCH-test is presented in table 7.11. This indicatesthat we have successfully removed the conditional heteroskedasticity thatexisted when the test was performed on the pure return series.

Table 7.11. Engle’s ARCH-test, validation for OMX. H0 is that no ARCH effectexists.

ARCH 1-2 test: F(2,1043)= 1.0572 [0.3478]ARCH 1-5 test: F(5,1037)= 0.64800 [0.6631]

ARCH 1-10 test: F(10,1027)= 0.64373 [0.7769]

74 OMX index

Table 7.12. Moments, skewness and kurtosis for zt of OMX, where zt is thestandardised residuals.

OMXFirst moment −0.00710



Moments of zt

The calculated first two moments, skewness and kurtosis are given in ta-ble 7.12. The distribution for zt is quite close to the normal distribution. Amore formal test is performed with the Jarque-Bera test. The Jarque-Beravalue is 8.6750 with a corresponding p-value of 0.013069 so we cannot rejectthe hypothesis that zt is normally distributed. A graphical presentation ofthis distribution is presented in figure 7.7 where the values are presented ina histogram with 50 bins.

−3 −2 −1 0 1 2 3 40

10

20

30

40

50

60

70

80

90

Figure 7.7. Histogram for OMX after estimation. This is done on standardisedresiduals. The histogram is split into 50 bins.

Summary 75

7.5 Summary

7.5.1 Estimation

Different (p,q) were tested in the symmetric case. The choice is p = q = 1based on our out-of-sample measures.

Different models were tested using the specific p and q. The asymmetric-power-ARCH is the model which performs best among the tested mod-els. The GARCH(1,1) can also be a good choice since it performsalmost as good as the APARCH(1,1) but has fewer parameters.

Different distributions for this specific APARCH(1,1) are tested and esti-mated. There is some indication for the use of skewed-t distributionbut the difference is so small that this is rejected in favour of thenormal distribution due to its simplicity.

The estimated coefficient value for the asymmetry parameter is significant.This parameter is positive which means that the risk is increasingmore when the index is moving downwards compared to how muchthe risk is increasing when the index moves upwards. This could thenbe related to the leverage effect.

The data set of an index is more homogenous than a specific stock dataset. Since all effects are more smoothed out within an index. This isseen in the stability of the parameters in the data set when it is splitand estimated as two separate parts.

Chapter 8

Conclusions

This chapter presents the general conclusions that can be drawn from theestimation chapters. A discussion about of lags, different models and dis-tributions is presented. Suggestions for further studies and some reflectionsfor the GARCH models are also given.

8.1 GARCH modelling discussion

8.1.1 Testing for different (p,q)

The whole estimation procedure for (p,q) is based on the symmetric standardGARCH with normal distribution. Five different p and q values are tested:(0,0), (1,1), (1,2), (2,1) and (2,2). p is the number of GARCH coefficientsand q is the number of ARCH coefficients. P and q equal (0,0) is just forthe naive model that no ARCH effect exists in the model. This is howevernever the case in the studied data set; (0,0) is always the worst choice amongthe models that are tested. p, q equal (1,1) is by far the most used valuesin GARCH research today and our result is also consistent with this. Inthree out of four models (1,1) is the best choice. However, in muvg the bestchoice is (2,1), adding one more beta coefficient. All these results are basedon out-of-sample measures as the main selection criterion.

8.1.2 Testing different models

Four models are tested for the specific (p,q) chosen in the first estimationpart. These are GARCH, EGARCH, APARCH and IGARCH. In all casesthe APARCH does the best job. This indicates that asymmetric coefficientand power coefficient are good to implement in the GARCH model. Ofcourse this cannot be taken as a general result. In some of the estimationsthe asymmetric coefficient is not significant and the power parameter is closeto two. This is an indication that GARCH(1,1) can be a good choice in thesecases.

78 Conclusions

8.1.3 Testing different distributions

Three different distributions are tested: normal distribution, student-t andskewed student-t. The result for the use of different distributions is notpointing in a clear direction. It is clear that different distributions doesnot affect the result to any large extent. Notice also that if one shoulduse distributions which are not normal, then at least 700 data points haveto be used in the estimation procedure to get accurate estimates for theparameters that describe the skewness and the tail of the distributions.

8.1.4 Parameter instability

One problem that arises when using models with parameters for asymmetriesand other non-linear features is that parameters depend heavily on whichpart of the data set they are estimated over, especially if the sample size issmall. Ideally more than 700 samples should be used. If models are esti-mated over smaller sample sizes, then another problem that arises is to findfeasible solutions. Especially the EGARCH has problem with convergence.

The dynamic features for the models have not been fully investigated.If we use the APARCH model then we have to be aware of the fact thatthe model has to be reestimated when new information is available. If wewant a more robust model then we use the standard symmetric GARCHwith normal distribution, because it has more stable parameter values.

8.1.5 User’s choice

The final decision of what model to use is in the end the user’s choice. TheAPARCH(1,1) is almost always a good choice for the data series that we havetested, but this model has convergence problems and the coefficients mightbe time dependent. If this model is used then the coefficients have to bereestimated when new data (information) is available. The other possibilityis to use the more robust symmetric GARCH(1,1), which is not the best onout-of-sample measures but has more time stable parameters.

8.1.6 The OMX index

The data set of the OMX index is more smoother than data for a specificstock. This is because the index is determined by the behaviour of severalstocks. This is seen in the stability of the parameters in the data set, whenit is split and estimated in two separate parts.

8.1.7 Parameter values

This section contains a comparison between parameter estimates from dif-ferent stocks. All series are best estimated with an APARCH-model, mostly

Further Studies 79

with lag values p = 1, q = 1. The parameter values are presented in ta-ble 8.1 where only the common parameters are compared. GARCH(Beta1)for muvg is not compared in this table because it has two Beta parametersfor the chosen model and a comparison for GARCH(Beta1) for this stockmodel could lead to misleading result. The values from the different distri-bution are not compared because it is only in two of the series that anotherdistribution than the normal is used. Cst(M) is close to zero as one couldexpect and the value for all different models are in the same region. Cst(V)should be positive since this is the variance of εt if we would have if p = 0,q = 0. The two series with highest Cst(V) values also have large standarddeviation. The GARCH(Beta1) parameter has values between 0.7 and 0.9,which are common values for this parameter. The ARCH(Alpha1) is in theinterval 0.05 − 0.16, all with quite low estimated standard deviations.

In three of the models the APARCH(Gamma1), which is the asymmetrymodel coefficient, has a value in the region of 0.3. The fourth model’sAPARCH(Gamma1) value is negative which means that the effect of a shockupwards affects the volatility more than an effect downwards. Bear in mindthough, that this coefficient value is not significantly unequal to zero. TheAPARCH(Delta) coefficient, which in the symmetric GARCH model is equalto two, has a value less than two in all of the models.

8.2 Further Studies

The model for the mean equation used in this thesis uses a constant for thereturn series. This is consistent with the efficient market theory, despitethe fact that investors also require extra return for taking extra risk. Oneimprovement to this model could be to do further analysis of the model forthe mean equation. This could be done with a ARMAX-model for example.

The measures in this thesis are made with εt as the ”true” volatility. Itis not obvious that this is the best choice. For example intra-day data couldbe used instead of daily data.

8.2.1 Different regimes

The problem with time dependent estimated coefficients could be an indi-cation that the stock market behaves differently (with respect to volatility)in different periods of time. One idea could be that one GARCH modelshould be used when the stock market is in a bull period and that anotherGARCH model should be used when there is a bear period. This could bemodelled with regime switching models together with GARCH models. Ithas not been done in this thesis but could be an interesting model regionfor further analysis.

80 Conclusions

Table 8.1. Parameter comparison for APARCH models. First row is Volkswagen,second is muvg, third is lyoe and fourth is OMX.

Parameter Coefficient Std.ErrorCst(M) −0.004560 0.068735Cst(M) −0.019815 0.072585Cst(M) −0.077537 0.061286Cst(M) 0.079910 0.055018Cst(V) 0.654696 0.413332Cst(V) 0.068825 0.047832Cst(V) 0.460606 0.269931Cst(V) 0.041795 0.021145

GARCH(Beta1) 0.710141 0.075228GARCH(Beta1) - -GARCH(Beta1) 0.689094 0.093886GARCH(Beta1) 0.903821 0.025832ARCH(Alpha1) 0.133134 0.047949ARCH(Alpha1) 0.046485 0.024051ARCH(Alpha1) 0.160329 0.048998ARCH(Alpha1) 0.091222 0.022479

APARCH(Gamma1) −0.133827 0.120061APARCH(Gamma1) 0.298389 0.140086APARCH(Gamma1) 0.376513 0.163371APARCH(Gamma1) 0.317723 0.125141

APARCH(Delta) 1.957050 0.780077APARCH(Delta) 1.899521 0.447514APARCH(Delta) 1.862487 0.578710APARCH(Delta) 0.944506 0.295055

8.3 Generalization

To draw general conclusions from only four data samples is of course dub-ious. Still it is fairly obvious that small p and q values, i.e. (1,1) are agood choice. It is also notable that the APARCH model seems to do a goodjob for all of the data series. The value of using different distributions isnot obvious; mostly the normal distribution does a good job in light of itssimplicity.

8.4 How good are the estimated models?

We will finish this thesis with a section that presents the explanation powerfor the chosen GARCH models that we have estimated.

How good are the estimated models? 81

Table 8.2. Model explanation power overview for Volkswagen models.

GARCH(0,0) GARCH(1,1) APARCH(1,1)Log −1726.53 −1693.55 −1692.588

Akaike 4.321326 4.243863 4.246470R2 0.0000 0.09341 0.115478

MSE 237.4908 203.1438 199.5250MedSE 15.7309 15.6039 15.5668

MAE 8.2124 8.0538 7.9957AMAPE 0.5570 0.5356 0.5339

TIC 0.7123 0.5779 0.5732

Table 8.3. Model explanation power overview for muvg models. The distributionfor the APARCH(2,1) model is a student-t.


Akaike 4.563421 4.436184 4.367206R2 0 0.177904 0.219198

MSE 405.4252 302.5713 286.8509MedSE 28.2512 24.8449 28.8216

MAE 10.9956 10.6565 10.5239AMAPE 0.5834 0.5355 0.5297

TIC 0.7169 0.4731 0.4641

In all our four stocks there exists heteroskedasticity and significant au-tocorrelation in the squared return1. Both the heteroskedasticity and theautocorrelation are satisfactorily removed. This is seen in the validationpart for the GARCH models when zt is tested for heteroskedasticity andautocorrelation.

In tables 8.2, 8.3, 8.4 and 8.5 an overview of the estimated models,with respect to their explanation power, is given. In the first column theGARCH(0,0) is presented. This is just as a benchmark that these GARCHmodel are doing a good job. In the second column the GARCH(1,1) ispresented. In the third column the chosen model for the specific stock isdisplayed. The model that has the best out-of-sample measure is marked inbold numbers. As one can see, our model choice in the third column has ahigh explanation power compared to the other two columns.

1This is calculated when the constant return value has been removed from the series.

82 Conclusions

Table 8.4. Model explanation power overview for lyoe models. The APARCH(1,1)model is model together with a skewed student-t distribution.


Akaike 4.018192 3.928801 3.854258R2 0.0000 0.203696 0.276033

MSE 820.9542 611.5326 536.3566MedSE 10.0241 8.4617 19.2230

MAE 14.1338 11.7340 11.4730AMAPE 0.6600 0.5618 0.5520

TIC 0.8595 0.6010 0.5244

Table 8.5. Model explanation power overview for OMX models. TheAPARCH(1,1) is with a normal distribution.


Akaike 4.027640 3.876132 3.862339R2 0.0000 0.0934158 0.113057

MSE 26.4675 24.1874 23.9171MedSE 6.9878 5.7996 6.6722

MAE 3.3548 3.3171 3.4406AMAPE 0.5173 0.4962 0.5025

TIC 0.5384 0.4580 0.4429

References

Bollerslev, T. (1986), ‘Generalized autoregressive conditional heteroskedas-ticity’, Journal of Econometrics 31, 307–327.

Brooks, C. (1997), ‘Linear and non-linear (non)-forecastability of high-freqency exchange rates’, Journal of Forecasting 16, 125–145.

Brooks, C., Burke, S. P. & Persand, G. (2001), ‘Benchmarks and the accu-racy of garch model estimation’, International Journal of Forecasting17, 45–56.

Deistler, M. (1999), ‘Geschichte und Perspektiven der okonometrie’, RWI-Mitteilungen 50(3), 109–127.

Engle, R. F. (1982), ‘Autoregressive conditional heteroskedasticity with es-timates of the variance of United Kingdom inflation’, Econometrica.

Hamann, E. (2001), Time series models for the West German economy,Master’s thesis, Technische Universitat Wien.

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Hansen, B. E. (1994), ‘Autoregressive conditional density estimation’, In-ternational Economic Review 35(3), 705–730.

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Klaasen, F. (2002), ‘Improving GARCH volatility forecasts with regimeswitching GARCH’, Empirical Economics 27, 363–394.

Laurent, S. & Peters, J. (2002), A tutorial for G@RCH 2.3, a completeOx Package for Estimating and Forecasting ARCH Models, http://-www.egss.ulg.ac.be/garch.

LeRoy, S. F. (1989), ‘Efficient capital markets and martingales’, Journal ofEconomic Leterature 27, 1583–1621.

84 References

Ljung, L. (1999), System Identification: Theory For the User, 2 edn,Prentice-Hall, Upper Saddle River, N.J. USA.

MathWorks (2002), Statistics Toolbox User’s Guide, 4 edn, The MathWorksInc.

Nelson, D. B. (1991), ‘Conditional heteroskedasticity in asset returns - anew approch’, Econometrica 59(2), 347–370.

Pagan, A. R. & Schwert, G. W. (1990), ‘Alternative models for conditionalstock volatility’, Journal of econometrics 45, 267–290.

Peters, J.-P. (2001), Estimating and forecasting volatility of stock indicesusing asymmetric GARCH models and (skewed) student-t densities,Technical report, University de Liege, Boulevard du Rectorat, 7 (Bat.31/ Boıte 54) 4000 Liege Belgium.

Twain, M. (1894), The Tradgedy of Pudd´nhead Wilson,, Oxford UniversityPress, New York. Chapter 13. Cited in Dangers of data mining: Thecase of calendar effects in stock returns, Journal of Econometrics 105(201).

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