Volatility Modelling of Asset Prices using GARCH Models

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Volatility Modelling of Asset Prices

using GARCH Models

Jens N¨asstr¨om

Reg nr: LiTH-ISY-EX-3364-2003 February 11, 2003

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Volatility Modelling of Asset Prices

using GARCH Models

Master’s Thesis

Division of Automatic Control Department of Electrical Engineering

Linköping University, Sweden Jens Näsström

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

Supervisor: O.Prof. Manfred Deistler Prof. Lennart Ljung Examiner: Prof. Lennart Ljung February 11, 2003

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Avdelning, Institution Division, Department

Institutionen för Systemteknik

581 83 LINKÖPING

Datum Date 2003-02-11 Språk Language Rapporttyp Report category ISBN Svenska/Swedish X Engelska/English Licentiatavhandling

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

C-uppsats

D-uppsats Serietitel och serienummer Title of series, numbering

ISSN Övrig rapport

____

URL för elektronisk version

http://www.ep.liu.se/exjobb/isy/2003/3364/ Titel

Title

Volatilitets prediktering av finansiella tillgångar - med GARCH modeller som ansats Volatility Modelling of Asset Prices using GARCH Models

Författare Author

Jens Näsström

Sammanfattning Abstract

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

Conclusions are drawn regarding different GARCH models, their numbers 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. The use of non-normal distributions is not clearly supported. The result from this master thesis could be used in option pricing, hedging strategies and portfolio selection.

Nyckelord Keyword

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Abstract

The objective for this master thesis is to investigate the possibility to predict the risk of stocks in financial markets. The data used for model estimation has been gathered from different branches and different European coun-tries. The four data series that are used in the estimation are price series from: M¨unchner R¨uck, Suez-Lyonnaise des Eaux, Volkswagen and OMX, a Swedish stock index. The risk prediction is done with univariate GARCH models. GARCH models are estimated and validated for these four data series.

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. The use of non-normal distributions is not clearly supported. The result from this master thesis could be used in option pricing, hedging strategies and portfolio selection.

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

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List of Figures

4.1 Price plot for Volkswagen. . . 29

4.2 Squared return for Volkswagen. . . 29

4.3 Autocorrelation of the squared return for Volkswagen. . . 30

4.4 Partial autocorrelation of the squared return for Volkswagen. 30 4.5 Autocorrelation for Volkswagen with residuals after estimation. 35 5.1 Price plot for M¨unchner R¨uck . . . 41

5.2 Return plot for M¨unchner R¨uck . . . 41

5.3 Squared returns for M¨unchner R¨uck . . . 42

5.4 Autocorrelation on the squared return for muvg. . . 42

5.5 Partial autocorrelation on the squared return for muvg. . . . 43

6.1 Price plot for Suez-Lyonnaise des Eaux . . . 53

6.2 Return plot for Suez-Lyonnaise des Eaux . . . 53

6.3 Squared return for lyoe. . . 54

6.4 Autocorrelation on the squared return for lyoe. . . 54

6.5 Partial autocorrelation on the squared return for lyoe. . . 55

6.6 Time dependent coefficient plot for lyoe. Window size = 200. 59 6.7 Time dependent coefficient plot for lyoe. Window size = 700. 59 6.8 Autocorrelation for lyoe after estimation. . . 61

6.9 Histogram for lyoe after estimation. . . 62

7.1 Price plot for OMX. . . 67

7.2 Return plot for OMX. . . 67

7.3 Squared return for OMX. . . 68

7.4 Autocorrelation on the squared return for OMX. . . 68

7.5 Partial autocorrelation on the squared return for OMX. . . . 69

7.6 Autocorrelation for OMX with normalised residuals after es-timation. . . 73

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List of Tables

2.1 GARCH features comparison. . . 11

3.1 Ljung-Box-Pierce Q-test example. . . 17

3.2 Example table for parameter test. . . 18

3.3 Time series identification with the ACF and PACF. . . 20

4.1 Days with high return in the Volkswagen series. . . 28

4.2 Moments, skewness and kurtosis for the return series of Volk-swagen. . . 28

4.3 Ljung-Box-Pierce Q-test for Volkswagen. . . 31

4.4 Engle’s ARCH-test for Volkswagen. . . 31

4.5 Selecting p,q for Volkswagen. . . 32

4.6 GARCH model selection for Volkswagen. . . 32

4.7 Distribution selection for Volkswagen. . . 33

4.8 Estimated parameters for the chosen model for Volkswagen. . 33

4.9 Estimated parameters for the chosen model for Volkswagen. . 34

4.10 Ljung-Box-Pierce Q-test, validation for Volkswagen. . . 35

4.11 Engle’s ARCH-test, validation for Volkswagen. . . 36

4.12 Moments, skewness and kurtosis for z_t of Volkswagen. . . 36

4.13 Results from the jb-test. Normality test for z_t. . . 37

4.14 Skewness for z_tfor Volkswagen. . . 37

5.1 Days with high return in the muvg series. . . 39

5.2 Moments, skewness and kurtosis for the return series of muvg. 40 5.3 Ljung-Box-Pierce Q-test for muvg. . . 42

5.4 Engle’s ARCH-test for muvg. . . 43

5.5 Selecting p,q for muvg. . . 44

5.6 GARCH model selection for muvg. . . 45

5.7 APARCH model selection for muvg. . . 45

5.8 Distribution selection for muvg. . . 46

5.9 Estimated parameters for the chosen model for muvg. . . 46

5.10 Estimated parameters for muvg on different data sets. . . 47

5.11 The Ljung-Box-Pierce Q-test in the validation for muvg. . . . 48

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x List of Tables

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

6.1 Days with high absolute returns in the lyoe series. . . 52

6.2 Moments, skewness and kurtosis for the return series of lyoe. 52 6.3 Ljung-Box-Pierce Q-test for lyoe. . . 55

6.4 Engle’s ARCH-test for lyoe. . . 56

6.5 Selecting p,q for lyoe. . . 56

6.6 GARCH model selection for lyoe. . . 57

6.7 Distribution selection for lyoe. . . 57

6.8 Estimated parameters for the chosen model for lyoe. . . 58

6.9 Estimated parameters for lyoe on two different data sets. . . 60

6.10 Estimated parameters for lyoe on two different samples of the symmetric GARCH(1,1) model. . . 61

6.11 Moments, skewness and kurtosis for z_t of lyoe. . . 62

6.12 Ljung-Box-Pierce Q-test on validation data for lyoe. . . 62

6.13 Engle’s ARCH-test, validation for lyoe. . . 63

7.1 Days with high return in the OMX index. . . 66

7.2 Moments, skewness and kurtosis for the return series of OMX. 66 7.3 Ljung-Box-Pierce Q-test for the OMX. . . 68

7.4 Engle’s ARCH-test for OMX. . . 69

7.5 Selecting p,q for OMX. . . 70

7.6 GARCH model selection for OMX. . . 70

7.7 Distribution selection for OMX. . . 71

7.8 Estimated parameters for the chosen model for OMX. . . 72

7.9 Estimated parameters for the chosen model for OMX. . . 72

7.10 Ljung-Box-Pierce Q-test, validation for OMX. . . 73

7.11 Engle’s ARCH-test, validation for OMX. . . 73

7.12 Moments, skewness and kurtosis for z_t of OMX. . . 74

8.1 Parameter comparison. . . 80

8.2 Model explanation power overview for Volkswagen models. . . 81

8.3 Model explanation power overview for muvg models. . . 81

8.4 Model explanation power overview for lyoe models. . . 82

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Acknowledgements

First I would like to thank my supervisors and examiner Prof. Manfred Deistler and Prof. Lennart Ljung for letting me have the opportunity to write my master thesis here in Vienna. I would also like to thank the staff at the Department of Econometrics and System Theory for making this a pleasant stay.

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

Finally my thanks goes to my opponent Emil Tir´en for useful comment and suggestions, to Anders Blomqvist and Claes Wallin for proofreading and to Thomas Sch¨on for letting me use parts of his LA_{TEX framework.}

Vienna, December 2002

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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.

x_t Stochastic process.

µ Mean value of a stochastic variable.

ψ_t The information set available at time t.

Operators and functions

L The lag operator

A(L) Pq_i=1α_iLi

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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 M¨unchener R¨uckversicherungs-Gesellschaft AG.

PA Paris Stock Exchange.

VOWG Volkswagen.

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Chapter 1

Introduction

”October. This is one of the peculiarly dangerous months to speculate in stocks 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 important during the last decades. In a broad sense companies want to have good control of their risk profile. In the financial market this is of even greater importance. A number of large financial companies have had large losses during the last years. One example is the failure of Long Term Capital Management.

Risk can be divided into different subcategories: Strategic risks, oper-ational risks and financial risks. In this thesis we focus on risks on stock markets. Calculating risks is important for pricing derivatives, portfolio selection 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 brief presentation is given below.

Münchner Rück’s stock price is taken from the Germany stock market Xetra, (DE). The data was sampled from the 25th of February 1999 to the 21st of October 2002. Münchner Rück is one of the world’s largest reinsurance 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

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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 of October 2002. Volkswagen is a large producer of cars and vans. OMX is an index that consists of 30 large Swedish companies. The data

was 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 corresponding preestimation section for each stock or index.

1.3 Objective

The objective for this thesis is to investigate to what extent it is possible to predict 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 a

specific 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 and focus on a few specific topics. In this thesis we focus exclusively on univariate modelling of volatility. The multivariate case is not treated at all. Another limitation is the mean equation, which is modelled with just a constant. This is consistent with the efficient market theory, except for the fact that investors also require extra return for taking risk. We are working with daily data and are only interested in one-step-ahead prediction. The model set only consists of different GARCH models.

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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 and abbreviations are presented in a special notation chapter.

Chapter 1 is the introduction chapter. It briefly presents the thesis and the 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 toolboxes available 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. The structure for these chapters is basically the same: preestimation, es-timation and validation. This chapter can be reed independently of each 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 stock market and economic theory for these markets is also assumed.

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

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Chapter 2

Method

This chapter presents the method used in this thesis. The chapter can roughly be divided into three different parts. First econometrics and system identi-fication is presented. Second the system identiidenti-fication 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, for example, consumer behaviour: second financial econometrics dealing with fi-nancial data and finally macroeconometrics which focuses on macroeconomic phenomena such as inflation, unemployment and economic growth. (Deistler 1999) This thesis is concerned with financial econometrics. We are focusing on stock markets and we are interested in prediction of the first and the second moments; In particular our main focus is on the second moments.

2.2 System Identification

Each chapter which contains estimation of GARCH models is presented in the same way. First the company is presented with plots and tables containing interesting and important company-dependent facts. Some tests are performed to see whether or not there exists correlation in the squared return and to see if there exists heteroskedasticity.

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

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6 Method

is given in this section is not the only way to do it. A general approach to system 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 know what an outlier is in a financial market. After the estimation for the whole sample, the models are estimated for different sub-samples. By doing this we can get a sense of how outliers affect the model coefficients and how the parameters in the model are changing over time.

One obvious feature when estimating and predicting in the financial market 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 our series we have about 1000 data samples, the data is then split into estimation part (70%) and validation part (30%). The reason for the unequal split is that the GARCH estimation procedure requires large sample sizes to produce good estimates.

2.2.2 Volatility

Modelling volatility can be done in many different ways. Most volatility models 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 onto the second moment (Hull 2000). For a specific mapping, the data series can also be forecast with this method. This is the approach that will be used in this thesis and will be further explained in Section 3.5. Stochastic volatility methods are models where the volatility follows a stochastic differential equation. One way of doing this is presented by Hull (2000): dS S = rdt + √ V dz_s dV = a(b− V )dt + ξVαdz_v (2.1)

where a, b, ξ and α are constants, dz_s and dz_v are Wiener processes and S and r are the stock price and the risk free rate respectively. V is the asset’s variance rate, the square of its volatility. This is a time continuous stochastic differential and can be seen as one possible way of handling the problem that volatility is not constant over time. This approach is not used in this thesis and will not be presented further. Regime switching models are based on the assumption that economic and

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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 further explained in this thesis.

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

2.2.3 Selected models for estimation

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

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

In an efficient market, all information available up until today is already included in the price. This would lead to the conclusion that there is no need for any ARX/ARMAX models in the mean equation. Efficient market theory also states that an investor is expecting the risk free rate as return and 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 an unpredictable part ε_t. However, coupling of equations, between the mean equation and the second moment, is not possible in the toolbox that we use for estimation. Therefore the model for the mean that we use, is just a constant plus the unpredictable part y_t = c + ε_t, here y_t is the return on that 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. The main question here is which GARCH models to use among all those that exist. 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 lags 0, 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 find since the volatility is not directly observable. One way of dealing with this problem is to not rely on one specific measure but rather use several

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8 Method

measures. In this thesis six different measures are used for evaluating the performance 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: 1

h + 1

S+h_X t=S

(ˆσ2_t − σ2_t)2 (2.2)

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

always equal to 1, ˆσ2_t is the forecast volatility, σ_t2 is the ”true” volatility (in our case ε2_t) and S is the total sample size.

The MedSE is:

1

h + 1

S+hX t=S

(ˆσ_med2− σ_med2)2 (2.3) The MAE is:

1 h + 1 S+h_X t=S |ˆσ2 t − σt2| (2.4)

The AMAPE is:

1 h + 1 S+h_X t=S σˆ2t − σt2 ˆ σ2_t + σ_t2 (2.5)

The TIC defined by:

1 h+1 P_S+h t=S (ˆσ2t − σ2t)2 q 1 h+1 P_S+h t=S σˆt2+ q 1 h+1 P_S+h t=S σ2t (2.6)

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

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2.2.5 Validation 9

The R2 is calculated by regressing ˆσ_t2on the ε2_t, which can be formulated with this equation

ε2_t = α + β ˆσ2_t + u_t (2.7)

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

The first issue to address when using these measures is: What is the true volatility? This can be done in different ways but in this thesis we will use the squared residuals ε2_t from the equation y_t= c + ε_t, as the true volatility when measuring the fit. The reason for choosing ε2_t as the true volatility is easily seen in this equation

Eε2_t = E(z_tσ_t) = 1Eσ_t= ˆσ_t (2.8)

z_t is i.i.d. N (0, 1) and σ_t is the variance for ε_t conditioning on the informa-tion available at time t. This is further explained in Secinforma-tion 3.7.

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

σ_t2 =PK_k=1r2_(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 not used 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 effects and autocorrelation have been removed or not. Tests are also performed to see if the given assumptions about the model are fulfilled i.e. are the normalised 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 not used in the estimation part. Data points (700− 1000), depending on the data set, have been saved for this reason. The out-of-sample measures are computed 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 for the second moment (the first moment is just a constant).

2.3 Methods of estimation, toolboxes

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

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10 Method

be large differences between estimated GARCH parameter values from dif-ferent toolboxes. Therefore it is important to be careful in choosing the toolbox to use. In this thesis a toolbox1 compatible with the Ox program-ming language is used (Laurent & Peters 2002). Unfortunately this toolbox is 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 toolbox with 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 generalized error distribution. A number of different models are available, both for the conditional mean and the conditional variance, as well as a number of tests and forecast possibilities (Laurent & Peters 2002). In table 2.1 some of the available toolboxes and their features are presented. Most of these model features are not used in this thesis so the table is just to be seen as a general comparison. The reason for choosing the G@RCH package is on the one hand that the light version is a free version and on the other hand that it has the GARCH model features that we want to use.

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2.3.1 The Ox GARCH package 11

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

G@RCH PcGive Eviews S-Plus SAS Stata

Version 2.3 10 4.0 6 8.2 7 Cond. mean Expl. 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. - - - + - -Estimation ML + + + + + + QML + + + - - +

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Chapter 3

Theory

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

3.1 Economic theory

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

A good book that presents financial theory is Hull (2000). This book is recommended 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 between stock prices and return. Let the stock price be denoted S and µ be the rate of return. If changes in stock price is denoted ∆S and changes in time are denoted ∆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:

S_T = S₀eµT (3.2)

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

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14 Theory

3.1.2 An introduction to risk

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

In stock markets, periods of high risk seem to be followed by periods of 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 example when 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 than average, using the information of historical prices. This means that so called technical analysis is fruitless. Semi weak efficiency means that it is impossi-ble to perform better than average, even when including exogenous variaimpossi-bles (X), where X includes all public information. Strong efficiency means that private information is also included by exogenous variables. This would mean, that not even with inside information would it be possible to perform better than average. (LeRoy 1989)

3.1.4 Leverage effect

One phenomenon in equity markets is that volatility tends to increase more when the return decreases, than it does, when the return increases. One reason for this is referred to as the leverage effect. As a company’s equity reduces in value the leverage for the company increases. When the stock price for a company decreases their equity declines in value and therefore their leverage increases. The company has become more risky. On the other hand, 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 the stock. (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 all cases below, X is a discrete valued stochastic variable, k is the summation index and p_x(k) is the probability that X is taking value k. A more detailed description is presented in Hamilton (1994).

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3.2.1 Correlation 15

The first moment is the population mean and is defined as

E(X) =X

k

kp_x(k) = µ (3.3)

The noncentral second moment is then defined as

E(X2) =X

k

k2p_x(k) = V ar(X) + E2(X) (3.4)

Noncentral moments are then in the general case defined as

E(Xr) =X k krp_x(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 above the mean value than far below. For a normal distribution the skewness is zero.

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 kurtosis greater 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 and

Y is defined by

Corr(X, Y )≡ p 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 the variance:

Corr(X_t, X_t−j)≡ p Cov(Xt, Xt−j)

V ar(X_t)V ar(X_t−j) (3.9) (Hamilton 1994)

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16 Theory

3.2.3 Partial autocorrelation

The partial autocorrelation is also a useful tool in the identification of a time series. It is defined as the last coefficient in a linear projection of Y on the

m most recent values. Letting this be denoted α(m)_m , if the constant for the process is zero then the equation becomes:

Y_t+1= α(m)₁ Y_t+ α₂(m)Y_t−1+ . . . + α(m)_m Y_t−m+1 (3.10)

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

3.3 Hypothesis tests

A hypothesis test is a procedure for analysing data to address questions whether a certain criterion is fulfilled or not. This can be tested in a number of different ways and this section presents the hypothesis test that will be used 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 sample result.

At the significance level of 95%, which in our notation corresponds with a critical value of 0.05, then we reject the null hypothesis if the p-value is lower than this critical value. A p-value greater than 0.05 corresponds to insufficient 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 the excess kurtosis instead of the kurtosis is calculated. The kurtosis is then 3 plus the excess value. The intuitive feeling about this test is that the larger the jb-value is, the lower the probability is that the given series is drawn from a normal distribution. The test statistic of the Jarque-Bera test is

χ2-distributed with 2 degrees of freedom under the null hypothesis, that the series is normally distributed. (Hamann 2001)

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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 correlation exists.

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 significant autocorrelation or not. The Lbq-value is calculated by:

Q_k= T (T + 2) k X i=1 r2_i T− i (3.12)

where T is the number of samples, k is the number of lags and r_i the ith autocorrelation. If Q_k is large then the probability that the process has uncorrelated data decreases. The null hypothesis for the test is that there exists no correlation and under that hypothesis, Q_k is χ2 with k degrees of freedom. Table 3.1 presents an example of how this can be done (the table is taken from the estimation procedure for Volkswagen in Section 4.2.4). The Lbq-value is in this case calculated for twenty number of lags. The p-value corresponding to this Q_k value is presented in angle brackets. (MathWorks 2002)

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 ˆu_t from the regression are saved. ˆu2_t is thereafter regressed on a constant and its own m-lagged values. This is done for all samples t = 1...T . This regression has a corresponding R2-value. T R2 is then asymptotically χ2-distributed with m degrees of freedom under the null hypothesis that u_t is i.i.d. N (0, σ2). (Engle 1982)

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

The null hypothesis, H₀, is that no ARCH effects exists. This is tested for lags up to T . The OX-package presents this as an F-test with the value from 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 likelihood estimates of the selected parameters are calculated. An example of this is

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18 Theory

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

Parameter Coefficient Std.Error t-value t-prob Cst(M) −0.004560 0.068735 −0.06634 0.9471 Cst(V) 0.654696 0.413332 1.584 0.1136 GARCH(Beta1) 0.710141 0.075228 9.440 0.0000 ARCH(Alpha1) 0.133134 0.047949 2.777 0.0056 APARCH(Gamma1) −0.133827 0.120061 −1.115 0.2653 APARCH(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 parameters are specific for the chosen models. The second column contains the coeffi-cients for each parameter. In the third column, the estimated standard error for this coefficient value is presented. The t-value is just the coefficient value divided 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 value can 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 are estimated with a maximum likelihood estimation. The null hypothesis is that the parameters are constant and the alternative is that the parameter Θ follows a martingale process. (Hansen 1994)

3.4 Stochastic processes

This section for stochastic processes is treated in more detail by (Hamilton 1994).

3.4.1 White noise

One of the basic blocks when modelling stochastic processes is the white noise E(ε_t) = 0 E(ε_tε_s) = σ2 for t = s 0 for t6= s (3.13)

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3.4.2 Stationarity 19

3.4.2 Stationarity

A process Y_t is said to be covariance-stationary or weakly stationary if nei-ther the mean u_tnor the autocovariance γ_jt depend on the time t and if the given moments exists.

E(Y_t) = µ (for all t)

E((Y_t− µ)(Y_t−j− µ) = γ_j (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

Y_t= α₀+ α₁ε_t−1+ ε_t (3.15)

where α₀ and α₁ could be any real constants, and ε_tis a white noise process described in equation 3.13. This can of course also be considered in the general MA(q) case. Then the equation becomes

Y_t= α₀+ q X j=1 α_jε_t−j+ ε_t (3.16) 3.4.4 Autoregressive process

First we consider an AR(1) process

Y_t= α₀+ β₁Y_t−1+ ε_t (3.17)

where α₀ and β₁ can be any real constants, and ε_t is a white noise process described in equation 3.13. An AR(1) process can also be generalized to an AR(p) process. Y_t= α₀+ p X j=1 β_jY_t−j+ ε_t (3.18)

Autocorrelation and the partial autocorrelation that were introduced in the 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 has been removed. For further details on this procedure see section 3.1.1.

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20 Theory

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

σ2_n= 1 p− 1 p X i=1 (u_n−i− ¯u)2 (3.19) where ¯u is the mean value for the process of the period where σ2_nis calculated and u_i is the daily return for a specific day. σ_n2 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 different periods in time. This can be done by

σ_n2 =

p

X

i=1

α_iu2_n−i (3.20)

where ¯u is assumed to be equal to zero, which can be a good approximation

if we are dealing with daily data. The problem now is of course to find these

α_i. One reasonable constraint is that α_i > α_j when day i is more recent than j. Other constraints should be that

p

X

i=1

α_i= 1 (3.21)

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

Shape of ACF Indicated model

Exponential, decaying to zero Autoregressive model. Use the partial

autocorrelation plot to identify the order

of 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.

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3.5.1 EWMA - Exponentially weighted moving average 21

and that α_i > 0. This approach can be generalized in a number of different

ways. One of the more popular ones, is to assign some weight to a long run volatility. This leads to an equation like

σ_n2 = γV +

p

X

i=1

α_iu2_n−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

X

i=1

α_i = 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. Let

α_i = λα_i−1, where λ is a constant (0 < λ < 1). This restrictions for λ is consistent with equation 3.21. This weighting scheme has some good characteristics. The volatility formula can now be written as

σ_n2 = λσ_n−12 + (1− λ)u2_n−1 (3.24) Here only two values have to be stored for each volatility estimate. It also adapts to market changes because of the functional form of the equation. Large changes in u_n−1will 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 then received a lot of attention. First consider an ordinary AR(p) model of the stochastic process y_t.

y_t= c + α₁y_t−1+ . . . + α_py_t−p+ u_t (3.25)

where u_t is white noise. The basic AR(p)-model is now extended so that the conditional variance of u_t could change over time. One extension could be that u2_t itself follows an AR(m)-process.

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22 Theory

where w_t is a new white noise process and u_t is the error in forecasting y_t. This is the general ARCH(m)- process. (Engle 1982) For easier calculations and for estimation, a stronger assumption about the process is added.

u_t= h1/2_t υ_t (3.27)

where υ_t is an i.i.d. Gaussian process with zero mean and a variance equal to one υ_tiid∼ N(0, 1) and the whole model for the variance is now

ε_t|_ψ_t−1 ∼ N(0, h_t) h_t= α₀+ q X i=1 α_iε2_t−i (3.28) where α₀> 0 α_i > 0, i = 1, . . . , q

ψ_t−1 is the information available at time t− 1. Now, when the process for the variance is defined, we add an additional equation for modelling y_t. The return price is modelled with a constant.

y_t= c + ε_t (3.29)

this means that ε_t is innovations from a linear regression.

3.7 Generalized ARCH - GARCH

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

ε_t|_ψ_t−1 ∼ N(0, h_t) h_t= α₀+ q X i=1 α_iε2_t−i+ p X i=1 β_ih_t−i (3.30) where p integer, q integer α₀> 0, α_i≥ 0, i = 1, . . . , q β≥ 0, i = 1, . . . , p

thus the additional feature is that the process now also includes lagged h_t−i values. For p = 0 the process is an ARCH(q). For p = q = 0 (an extension allowing q = 0 if p = 0), ε_tis white noise. (Bollerslev 1986)

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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 and

cov(ε_t, ε_s) = 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)=Pq_i=1α_iand B(1)=Pp_i=1β_i. In most cases the num-ber of parameters is rather small, for instance GARCH(1,1). Next we will focus 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

ε_t|_ψ_t−1 ∼ N(0, h_t)

h_t= α₀+ α₁ε2_t−1+ β₁h_t−1 (3.31)

where

α₀ > 0, α₁ ≥ 0, β₁≥ 0 α₁+ β₁ < 1

Interesting properties for ε_t (unconditioned) would be its moments and they are given by:

Theorem 2. For the GARCH(1,1) process as defined in equation 3.31 a

necessary and sufficient condition for existence of the 2mth moment is µ(α₁, β₁, m) = m X j=0 m j a_jαj₁β₁m−j < 1 (3.32) where a₀= 1, a_j = j Y i=1 (2i− 1),

The 2mth moment can be described by the recursive formula E(ε2m_t ) = a_m · m−1X n=0 a−1_n E(ε2n_t )αm−n₀ m m− n µ(α₁, β₁, n) ! · (1 − µ(α1, β₁, m))−1 (3.33)

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24 Theory

Proof. Bollerslev (1986) page 325.

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

The existence of the 2mth moment for GARCH models with higher order than (1, 1) can also be interesting to investigate. The general case is not known, but for a specific order sufficient and necessary conditions can be derived. This is done by Bollerslev (1986) page 313 and for the GARCH(1,2) this is found to be

α₂+ 3α2₁+ 3α2₂+ β₁2+ 2α₁β₁− 3α₂3+ 3α2₁α₂+ 6α₁α₂β₁+ α₂β₁2< 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 X i=1 α_i+ p X j=1 β_j = 1 (3.35)

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

3.7.3 EGARCH

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

ln(σ2_t) = α_t+

∞

X

i=1

β_ig(z_t−i) β₁≡ 1 (3.36)

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

g(z_t) = θz_t+ γ(|z_t| − E|z_t|) (3.37) to handle both the sign and the magnitude of z_t.

A slightly different model of the EGARCH is implemented by Laurent & Peters (2002)

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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) in the GARCH toolbox. The model structure is

σδ_t = ω + q X i=1 α_i(|ε_t−i| − γε_t−i)δ+ p X j=1 β_jσδ_t (3.39)

Which includes many other GARCH models as special cases.

3.8 Distributions

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

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Chapter 4

Volkswagen

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

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 of October 2002. Particularly interesting days are presented in table 4.1. The certain criterion is that the squared return should be above 60 in this scaled measure presented in Section 4.2.1. One thing notable, is that the 11th of September does not meet the criterion but that three out of eight days with extremely high return occurred in September 2001. The dates are presented in 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. The data is daily data and the return is calculated by:

return_t= 100(log(price_t)− log(price_t−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 just a linear scaling. Data with numbers from 1-800 are used in the estimation part and data with numbers 801-952 are used as cross validation.

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28 Volkswagen

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

number return date

210 −8.4920 15.12.99 540 9.6572 21.03.01 666 8.2731 13.09.01 670 7.8758 19.09.01 672 −7.9494 21.09.01 892 −9.1398 26.07.02 935 −7.9203 25.09.02 946 −8.8166 10.10.02

Table 4.2. Moments, skewness and kurtosis for the return series of Volkswagen. Volkswagen First moment 0 Second moment 5.1369 Third moment −0.4209 Fourth moment 126.4742 Skewness −0.0362 Kurtosis 4.7930

4.2.2 Moments for the return series

In the preestimation stage it would be interesting to get a picture of what the distribution of the raw return series is like. The moments, skewness and kurtosis are presented in table 4.2. For a theoretical introduction to these measures see section 3.2. It is notable that the kurtosis is considerably larger than three, which would have been the kurtosis if the return series were drawn from a normal distribution. This behaviour is known to frequently occur in financial markets.

4.2.3 Plots for Volkswagen

Before starting the estimation procedure it is interesting to see the behaviour of 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 prepre-sented in figure 4.2. In the diagram we see clustering of high volatility. A graphical presentation of the autocorrelation is presented in figure 4.3. Here we can see that there is a high persistence in the second moment for at least 10 lags. The partial autocorrelation is presented in figure 4.4 and it also shows

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4.2.4 Hypothesis tests 29

persistence in the second moment.

0 100 200 300 400 500 600 700 800 900 1000 30 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 1000 0 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 between data plots is in this subsection tested with the Ljung-Box-test. The test for heteroskedasticity is performed with an ARCH test. See section 3.3 for a theoretical explanation for these tests.

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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 Sample Autocorrelation

Sample 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

Sample Partial Autocorrelations

Sample Partial Autocorrelation Function

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

Ljung-Box-Pierce Q-test

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

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Estimation 31

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

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. H₀is that no ARCH effect exists and in this case we have p-values equal to zero which means that 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 that there exists heteroskedasticity. P-values equal to zero for all of the three series. The result for Volkswagen is presented in table 4.4. ARCH-effects are tested on lags 1-5, 1-10 and 1-15. With these p-values we reject the hypothesis 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 chosen and 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. For this GARCH(p,q)-model different distributions are evaluated. Finally the parameters 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 are tested. The selection procedure is based on each model’s ability to produce forecasts. Six different forecasts measures are used. See section 2.2.4 for a description of these measures. The Volkswagen table 4.5 shows that the GARCH(1,1) performs best in three out of six out-of-sample measures. In two of the measures of which the GARCH(1,1) does not score the best the

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32 Volkswagen

Table 4.5. Selection for symmetric GARCH models for Volkswagen. Log-likelihood is presented as well as Akaike’s information criterion for different p and q. Six different forecasts measures are also presented to determine which p and q that performs 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 4.246866 R2 0.0000 0.09341 0.08100 0.08612 0.08247 MSE 237.4908 203.1438 205.1336 204.2901 204.7869 MedSE 15.7309 15.6039 15.4001 14.9537 15.4742 MAE 8.2124 8.0538 8.0759 8.0668 8.0755 AMAPE 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 IGARCH

Log −1693.545 −1701.674 −1692.588 −1706.267 Akaike 4.243863 4.269186 4.246470 4.273168 R2 0.0934135 0.116252 0.115478 0.11051 MSE 203.1438 200.1063 199.5250 195.0135 MedSE 15.6039 16.8086 15.5668 22.1922 MAE 8.0538 8.1821 7.9957 8.8270 AMAPE 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 is

p = 1, q = 1.

4.3.2 Choosing model

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

4.3.3 Choosing distribution

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

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4.3.4 Estimated parameters 33

Table 4.7. Distribution selection for Volkswagen. The distribution is a Student distribution, 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.-t Log −1692.588 −1677.872 −1677.220 Akaike 4.246470 4.212179 4.213049 R2 0.115478 0.113103 0.113761 MSE 199.5250 198.6129 198.7152 MedSE 15.5668 16.5073 16.5735 MAE 7.9957 8.0435 8.0369 AMAPE 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 model is an APARCH(1,1) with normal distribution.

Parameter Coefficient Std.Error t-value t-prob Cst(M) −0.004560 0.068735 −0.06634 0.9471 Cst(V) 0.654696 0.413332 1.584 0.1136 GARCH(Beta1) 0.710141 0.075228 9.440 0.0000 ARCH(Alpha1) 0.133134 0.047949 2.777 0.0056 APARCH(Gamma1) −0.133827 0.120061 −1.115 0.2653 APARCH(Delta) 1.957050 0.780077 2.509 0.0123

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). The parameters 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 in detail in Section 3.3.4. One thing that is also interesting to note is that the APARCH(Gamma1) is shown to be time dependent in the Nelson test, section 3.3.5.

Volatility Modelling of Asset Prices using GARCH Models

Volatility Modelling of Asset Prices

using GARCH Models

Volatility Modelling of Asset Prices

using GARCH Models

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Abstract

Contents

List of Figures

List of Tables

Acknowledgements

Notation

Chapter 1

Introduction

1.1

Background

1.2

Presenting the companies

1.3

Objective

1.4

Problem specification

1.5

Limitations

1.6

Reader’s guide

Chapter 2

Method

2.1

Econometrics

2.2

System Identification

2.3

Methods of estimation, toolboxes

Chapter 3

Theory

3.1

Economic theory

3.2

Statistics

3.3

Hypothesis tests

3.4

Stochastic processes

3.5

Computing volatility

3.6

ARCH - Autoregressive conditional

heteroske-dasticity

3.7

Generalized ARCH - GARCH

3.8

Distributions

Chapter 4

Volkswagen

4.1

Presenting data

4.2

Preestimation

4.3

Estimation