Master Thesis Modeling tail distributions with regime switching GARCH models. huh. Ebbe Filt Petersen huh huh huh... huh Number of pages: 71

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1 Master Thesis Modeling tail distributions with regime switching GARCH models. huh huh Mentor: David Skovmand huh Ebbe Filt Petersen huh huh huh... huh Number of pages: 71 Cand.merc.(mat) Copenhagen Business School juli

2 Abstract For many years researchers and also the financial industry have been trying to replicate financial data and market movements in order to generate profits, for risk management activities or various other reasons. Different models have been developed for this purpose, where one of the most famous is the GARCH model. The GARCH model has proven to be a good model for modeling financial time series. However, due to an often high persistence found in GARCH models, resulting in an overestimation of the conditional variance, the model has also been criticized. For that reason it would make sense to expand the GARCH model in a way that could account for this drawback, but still maintain the qualities of GARCH. The purpose of this master thesis is to expand the GARCH model in order to reduce the drawbacks of GARCH, resulting in a better model for describing the tail distribution of financial return data. This thesis expands the GARCH model to a regime switching GARCH, switching between two different volatility regimes. The regime switching MS-GARCH is expected to reduce the high persistence and long convergence of GARCH, by switching between two regimes when sudden changes in the volatility is observed in the return series. Initially a normal distributed MS-GARCH is introduced. This model is later expanded to include t-distributed innovations. The MS-tGARCH allows for a very flexible distribution structure, by modeling different t-distributions in each regime. For this reason it was the model, expected to model the return distribution best. The analysis involves a total of four GARCH type models, namely GARCH, tgarch, MS-GARCH and MS-tGARCH. The data that the models is replicating is daily returns of the Danish OMXC20 index in the period from to The empirical evidence to test the hypothesis was found through a one-period and a multi-period VaR analysis. The analyses showed that the MS-tGARCH outperformed all other models and was the best model for describing the tail distribution of the OMXC20, when forecasting VaR more than one period head. In the one-period VaR analysis the single regime tgarch was the best performing model and introduction did not improve the model. The overall conclusion of this study was that using the t-distribution in a regime switching framework, results in the overall best model for modeling the return tail distribution, amongst the models tested. 2

3 Acknowledgements First of all, I would like to thank my mentor David Skovmand for leading me in the right direction and helping me pick out this interesting topic for my master thesis. Further, I owe the biggest possible thanks to my friend Christian Duffau Rasmussen for always being willing to the discuss the problems that occurred and reoccured throughout this process. Also a big thanks to my colleague Dr. Jon Simpson for helping me set up virtual machines around the world, powerful enough to run various simulation experiments, which otherwise would have been impossible. Last but not least, a massive thanks to my lovely girlfriend Anne Karina Asbjørn for being the best support at all times. 3

4 Indhold 1 Problem formulation Main Problem Formulation Theoretical Questions Empirical Questions Method and Structure Theoretical Analysis Empirical Analysis Financial Time Series 11 4 Autoregressive Conditional Heteroscedasticity Models ARCH GARCH Stationarity Persistence Kurtosis Generalized Regime-Switching Models MS-GARCH The Conditional Means Specification of Switching Probabilities The Conditional Variance Specification of the Innovation Distribution Construction of the likelihood The Likelihood Dynamics QMLE, Stationarity and Consistency Transformation to the MS-tGARCH(1,1) Estimation & Results 47 4

5 6.1 Data Diagnostics Analysis Value at Risk One-period Value at Risk Kupiec and Wilks Test OMXC20 results Multi-Period Value at Risk Conclusion 72 9 References Appendix Appendix

6 1 Problem formulation 1.1 Main Problem Formulation The main objective of this master thesis is to investigate whether the introduction of different volatility regimes in the GARCH model, results in a model that describes the lower tail distribution of stock market returns better. 1.2 Theoretical Questions - Why is GARCH a good model for modeling financial data? What strengths and weaknesses does it have? - What is the fundamental difference between a GARCH and a regime switching GARCH model? - How can the MS-GARCH theoretically reduce the high persistence often found in GARCH models. - What advantages does the use of the t-distribution, compared to the normal distribution, have in a regime switching framework. 1.3 Empirical Questions - How is the MS-GARCH estimated by Maximum Likelihood and what problems arise in regards to regime dependence in the conditional variance? - How can the potential improvement by introducing different volatility regimes in the standard GARCH be tested? - Does the MS-GARCH solve the drawbacks of the GARCH? - Does the MS-GARCH describe the different quantiles of the return distribution better than GARCH? 6

7 2 Method and Structure Throughout this section, the overall choice of method, test and assumptions will be presented. The study seeks to answer the main problem formulation and sub questions by the use of a theoretical analysis followed by an empirical analysis. It is the structure of the chosen method that binds these two analyses together. The overall structure of this study is a combination of a deductive- and inductive structure. A deductive structure means that specific theory is used and tested by the chosen method on a limited set of data 1. On the other side, method, theory and empirics will also be elaborated and discussed throughout the study in both the theoretical and empirical section, which is the inductive part of the thesis. The purpose of this study and its contribution to the literature is to test whether extending the GARCH model to include two different volatility regimes will improve the models ability describe the tail distribution of the daily OMXC20 returns. 2.1 Theoretical Analysis A large amount of time series models have the purpose of estimating variance. Most famous is the GARCH model introduced by Bollerslev (1986). GARCH has been tested and proven to be a solid model for this purpose, which is why it has been chosen as the cornerstone in this thesis. The first part of the theoretical analysis is an argumentation for why GARCH is a good model for financial data. The argumentation consists of a description of some characteristics about financial data, followed by an introduction to GARCH, describing how GARCH posses some of the same characteristics. Further, advantages and disadvantages by estimating GARCH models on stock market return data will be presented and discussed. Trying to meet some of the disadvantages of GARCH, an extension is introduced. The extension nests in the theory of Markov Switching Models introduced by Hamilton (1889). Applying this theory to GARCH results in a Markov-Switching GARCH. The second part of the theoretical analysis will mainly consist of introducing the MS-GARCH model discuss theoretical advantages over GARCH. The primary issue when estimating MS-GARCH mo- 1 Ankersborg, Vibeke (2011); Specialeprocessen: Tag magten over dit speciale 7

8 dels is the regime path dependence that occur due to dependence between conditional variances. This issue will be discussed and solved by taking the approach of Gray (1996), where the expected conditional variance is used instead of the actual variance to eliminate the path dependence. Combining all elements of MS-GARCH results in the likelihood function, used when estimating the model with MLE. The MS-GARCH was introduced to allow a more flexible variance structure of GARCH. However, MS-GARCH limits the distribution to be normal in both regimes. To allow for an even more flexible variance structure, MS-GARCH is transformed to the MS-tGARCH, which models each regime with t-distributions. When ν then t(ν, µ, σ) N(µ, σ), which in theory allows MS-tGARCH to model the returns with a normal distribution in one regime and a t-distribution in the other. It is expected that the more flexible distribution structure of the MS-tGARCH makes it the best model for modeling the tail distribution of the data investigated in this study. 2.2 Empirical Analysis The main purpose of the empirical analysis is to find empirical evidence that can answer the problem formulation. The empirical analysis consists of estimating the MS-GARCH and MS-tGARCH on Danish stock market data. The data will be described in section (6.1). Due to the complexity of MS-GARCH and MS-tGARCH it has been necessary to hardcode the models in order to estimate them, since no pre-programmed versions of these models are available. For this purpose the program R has been used. The MS-extension of GARCH has been chosen to get a more flexible variance structure, which from a theoretical point of view should result in a model, capable of describing the stock market return tail distribution more precisely than the single regime GARCH. As benchmark for the regime switching models, a single regime GARCH and tgarch is also estimated on the data. The MS-GARCH, extends the GARCH to include two different volatility regimes, where switches between the two regimes are driven by a hidden Markov model. In both regimes the variance is modeled with a GARCH(1,1) process. The innovations in the MS-GARCH are assumed to be normal, whereas a comparison with GARCH, 8

9 will strictly be a test of whether the introduction of regimes to the GARCH variance structure has improved the models ability to model the tail of the return distribution. As part of this empirical analysis a tgarch is further extended to the MS-tGARCH to investigate the effect of introducing regimes in a t-distributed GARCH. The MS-tGARCH assumes t-distributed innovations in both regimes, which allows for a more flexible structure of the conditional distribution, since the MS-tGARCH allows the degree of freedom parameter ν i to vary over regimes. In total, four models will thus be estimated, MS-GARCH, MS-tGARCH and their respective benchmarks. The four models require the following parameters to be estimated during the analysis: Model Parameters to be estimated GARCH (ω, α, β) tgarch (ω, α, β, ν) MS-GARCH (ω 1, ω 2, α 1, α 2, β 1, β 2, µ 1, µ 2, p 11, p 22 ) MS-tGARCH (ω 1, ω 2, α 1, α 2, β 1, β 2, µ 1, µ 2, p 11, p 22, ν 1, ν 2 ) A diagnostics analysis of the estimated models is presented in section (6.2). This analysis consists of testing significance of the parameters with a Wald test. Significant autocorrelation in the squared model residuals are investigated via plots of the autocorrelation function. To test the overall choice of model and distribution qq-plots are presented and both Kolmogorov-Smirnov and Tail-Sensitive confidence bands are applied. The main analysis in this study is presented in section (7). Section (7) includes a one-period and a multi-period Value at Risk analysis. The multi-period ahead forecast of the return distribution is not known in a GARCH setup, whereas this distribution is estimated by simulation of the four models. Simulating distributions of MS-GARCH models has to the best of knowledge not been done before, whereas this aspect further contributes to the literature. The results of both VaR analyses are tested with the Kupiec and Wilks test. It is results of both VaR analyses that will be answering the problem formulation. Making this extensive analysis on Danish stock market data, will be this studys contribution to the literature. 9

10 Many studies have shown that the t-distribution is a better match on financial data relative to the normal distribution. 2 As a byproduct of answering the problem formulation, using both normal- and t-distributed models, this study will also be answering whether the t-distribution is a better match for the Danish data relative to the normal distribution. 2 Haas, Markus & Pigorsch, Christian (2007) - Financial Economics, Fat-tailed Distributions 10

11 3 Financial Time Series The study of financial time series has been a topic for many years. Many researchers have studied the financial markets to understand the way they move and later tried to replicate their movements with different stochastic models. Recently the financial industry has adapted the theoretical approach to understand the financial markets in order to generate profit by forecasting market movements. Some Hedge Fund investment strategies depends solely on modeling financial asset prices and the parameters expected to influence those prices. Baseline is that modeling financial time series has become a very relevant and interesting field of study. However, due to the characteristics and complexity of the financial markets, modeling these time series is a complex problem. The difficulty is mainly due to the existing statistical empirical findings, often referred to as stylized facts, which are difficult to replicate artificially with the use of stochastic models 3. Let the price of a financial asset at time t be denoted by p t. The price return, y t can then be calculated as y t = log( p t p t 1 ) (1) The price itself and the price return is closely related since the return determines the price and vise versa, but the two time series are very different. c c20.r Time Time Figur 1: The index value of the danish OMXC20 index (left) and daily returns (right) in the period from to Francq, Christian & Zakoian, Jean-Michel (2010); GARCH Models 11

12 Figure (1) displays the daily index value and return for the danish stock market index, OMXC20, in the period from to As seen, the evolution of the index value is not far from the picture of random walk, whereas the return series could be related to a stationary process with finite variance. The fluctuations in the return series vary in magnitude over time, where very large fluctuations is observed around the recent great stock market crash in However, the fluctuations seem to be concentrated around a long term mean around zero. In one of the classic articles that describes financial data it is stated:...large changes tend to be followed by large changes - of either sign - and small changes tend to be followed by small changes. 4 The statement of Mandelbrot (1963) does not seem to be far away from the truth when looking at the OMXC20 returns in figure (1). It is observed that the period around the recent financial crises displays a much higher variance than in the rest of the return series. Also around the dot-com crisis in the beginning of the new millennium are clusters of higher volatility found. In both volatility regimes the mean remains close to zero. The assumption that volatility is dependent on past periods volatility does not seem unrealistic. Looking at the squared returns and hence only looking at the magnitude of the fluctuations, a measure of the return volatility is the outcome. c20.r^ Time Figur 2: Squared daily returns of the OMXC20 in the period from to Mandelbrot, Benoit (1963); The Variation of Certain Speculative Prices 12

13 Figure (2) displays the squared OMXC20 returns. It is seen that these periods of higher volatility reoccur over this 14 year time horizon. However there is no obvious system in the sudden change of variance level. Observing figure (2) it does not seem unreasonable to believe that financial time series is subject to the phenomenon of volatility clustering and is therefore incompatible with a marginal distribution that has constant variance, also referred to as a homoscedastic marginal distribution. 5 A more realistic assumption is that the marginal distribution has a time varying and non-constant conditional variance, which is referred to as a heteroscedastic marginal distribution. The most commonly used model for modeling homoscedastic processes is the ARMA model by Wold (1938) whereas the ARCH and GARCH models by Engle (1982) and Bollerslev (1986) respectively are the most recognized models for modeling heteroscedastic processes. The phrase volatility clustering is used for a time series having serial dependence in the variance structure. So a process that displays volatility clustering should have significant autocorrelation in the squared returns. Figure (3) below shows the autocorrelation function for the return series and the squared return series for the OMXC20, for lags {0, 1,..., 30} ACF ACF Lag Lag Figur 3: ACF of OMXC20 returns (left) and of OMXC20 squared returns (right) in the period form to , together with a 95% confidence band. As seen from the left hand side graph in figure (3) the return series itself shows no obvious 5 Francq, Christian & Zakoian, Jean-Michel (2010); GARCH Models 13

14 sign of autocorrelation which makes it comparable with a white noise process. But examining the autocorrelation function of the squared returns a very significant autocorrelation in the data is observed. So when looking past the direction of the return and hence only focusing on the magnitude of the returns, a serial dependence occurs. This means that the model chosen to replicate return data, must have a heteroscedastic variance structure, which makes it capable of capturing the serial dependence in the return variance. Throughout the literature of financial data modeling the normal distribution is often used. The reason for the popularity of this assumption might nest in the fact that the normal distributed returns is an implication of the random walk theory of stock prices by Fama (1965) 6,7. However, many studies have subsequently shown that daily returns have densities with much fatter tails than the normal distribution and are more sharply centered around a zero mean 8. If a return distribution displays these two features, the returns are said to be leptokurtic distributed. In more technical terms, a probability distribution is said to be leptokurtic when it has kurtosis larger than the one of the normal distribution, which is three. 9 The assumption of normality in daily returns tend to be more reasonable when the time horizon is long relative to shorter time horizons. 10 In figure (4) below is shown a histogram of the daily OMXC20 returns together with the density curve of a normal distribution and a t-distribution with ν = 5, which illustrates that the actual return distribution has fatter tails than what the normal distribution is able to capture. 6 Aparicio, Felipe & Estrada, Javier (1997); Empirical Distributions of Stock Returns: Scandivian Securities Markets, Fama, Eugine (1965); The Behavior of Stock.Market Prices 8 Haas, Markus & Pigorsch, Christian (2007) - Financial Economics, Fat-tailed Distributions 9 Shumway, Robert H. & Stoffer, David S. (2011); Time Series Analysis and Its Applications 10 Haas, Markus & Pigorsch, Christian (2007) - Financial Economics, Fat-tailed Distributions 14

15 Density N(0,1) t(5) C20 returns Figur 4: Distribution histogram of OMXC20 returns in the period from to together with density curves of a normal distribution and a t-distribution with 5 degrees of freedom. Figure (4) indicate that the actual distribution of the OMXC20 returns is leptokurtic. The kurtosis is a measure of the how leptokurtic a distribution is. For the OMXC20 returns an excess kurtosis of around 5.5 is calculated in R. A distribution that has this positive excess kurtosis is the t-distribution, which is also very commonly used throughout the literature of modeling financial time series. 11. The variance structure and the distribution are two of the stylized facts about financial time series that makes them very hard to replicate artificially. A model that have the purpose of modeling this type of data must be capable of capturing both aspects. Clearly the classic ARMA model is not capable of capturing these characteristics, mainly due to the assumption of a constant variance over time. But as mentioned, the ARCH and GARCH type models assume a non-constant variance and have therefore been very popular in modeling financial return series and their variance structure. In the following section the ARCH and GARCH models will be introduced and their strengths and weaknesses discussed. 11 Klaassen, Franc (2002); Improving GARCH Valotility Forecasts with Regime-Switching GARCH 15

16 4 Autoregressive Conditional Heteroscedasticity Models The family of Autoregressive Conditional Heteroscedasticity models all have roots in the classic ARCH model by Engle (1982). Since then, many expansions of Engles model have been developed and presented in numerous scientific articles. The most commonly used is the popular generalized ARCH, called GARCH, by Bollerslev (1986). The ARCH family of models basically consists of three different elements, which are all mentioned in the name ARCH. The AR is related to the autoregressive structure of the models. The C is related to the variance, where the C stands for conditional since it is the conditional variance that is modeled. The H represent the heteroscedasticity present in the model, which means that the variance is time varying. Throughout this section the ARCH model family will be discussed in terms of their strengths and weaknesses. This include an analysis of the model parameters and their influence on persistence and stationarity. The ARCH model will be briefly introduced followed by a more adequate review of the GARCH. 4.1 ARCH The ARCH model was first introduced by Engle (1982) to account for the complex and specific behavior of financial time series. The model has its roots in the widely used general model y t = σ t z t (2) where σ t is the volatility of y t which represents the return process defined in equation (1). The innovations z t are initially assumed to be a process of i.i.d. normally distributed random variables with zero mean and unit variance, where σ t is restricted to be strictly positive, i.e. σ t > 0 for all values of t. With this very general model introduced, the definition of an ARCH process lies right at hand. The process y t, defined in (2) is referred to as an 16

17 ARCH(1) if the first two moments, the mean and variance, exist and satisfy 12 E[y t ỹ t 1 ] = 0 σ 2 t = V ar(y t ỹ t 1 ) = ω + αy 2 t 1 (3) where ỹ t 1 = {y t 1, y t 2,...} which is all observable information gathered by the data generating process y t until time t 1. The model parameters ω and α are constants to be estimated. The restrictions on the parameters are introduced further below. Clearly the return series is dependent on its own history through the conditional variance. The specific model defined in (1) and (3) is called an ARCH(1) due to one period lag dependence. The dynamics of ARCH in terms of the dependence between the variance σ t and the return process y t is left for section (4.2) where the GARCH model is introduced. Investigating the ARCH model a bit further it is noted that y t is conditionally normal distributed with zero mean and conditional variance as in equation (3), 13 y t ỹ t 1 N(0, ω + αyt 1) 2 (4) This is a consequence of the assumption of normal distributed innovations z t. Note that any distribution can be chosen for the innovations. By applying the law of iterated expectations and using the information in equation (4) it can be shown that the process y t is a martingale difference, i.e. a process with zero mean, conditioned on all observable information. E[y t ] = EE[y t ỹ t 1 ] = EE[y t y t 1 ] = 0 (5) Knowing that y t is a martingale difference it can be shown that it is also a sequence of uncorrelated variables. This is seen by again applying of the law of iterated expectations and using the properties of y t. cov[y t+h y t ] = E[y t+h y t ] E[y t+h ] E[y t ] = E[y t+h y t ] = EE[y t+h y t ỹ t+h 1 ] (6) = E[y t E[y t+h ỹ t+h 1 ]] (7) = 0 (8) 12 Shumway, Robert H. & Stoffer, David S. (2011); Time Series Analysis and Its Applications 13 Shumway, Robert H. & Stoffer, David S. (2011); Time Series Analysis and Its Applications 17

18 Where y t becomes deterministic under the conditioning of ỹ t+h 1, given h > 0. Equation (5) further ensures the zero. In conclusion then the return process y t is an uncorrelated sequence, with zero mean and time varying conditional variance given in equation (3). That the ARCH model posses these properties gives a first indication that it might be a good model for replicating stock market returns, since returns display the same features, as described in the section (3). A plot of the autocorrelation function for a simulated ARCH return process and the square of the process is displayed in figure (5) below. The simulation is made in R and is based on equation (1) and (3). Returns Squared Returns ACF ACF Lag Lag Figur 5: Acf of simulated ARCH returns and squared returns Investigating the autocorrelation function for the return process it is seen that it is more or less perfectly uncorrelated. But looking at the acf for the squared return process on the right hand side of the figure (5) it is observed that a significant autocorrelation is present in the data. By using the property of y t being a matingale difference it is seen that var[y t ] = E[yt 2 ] E[y t ] 2 = E[yt 2 ] 0 So the variance of y t equals its own second moment. By squaring the returns and hence only looking at the magnitude of the returns, a significant autocorrelation is observed and the ARCH model thus has serial dependence in the variance, which was one of the stylized facts about real life financial times series as discussed in section (3). 18

19 The main issue with the simple ARCH model is that it only uses last periods return to estimate the current periods conditional variance. This means that a large shock to the return at time t only has effect in period t + 1, whereas an ARCH(1) does not capture nearly enough of the persistence in the conditional variance as observed in real life data. 14 So to model the kind of persistence that is actually observed in financial data it is necessary to include a lot more lags in the conditional variance, than just one. This can be done by expanding the ARCH(1) to a ARCH(q), with conditional variance given by σ 2 t = ω + α 1 y 2 t 1 + α 2 y 2 t α q y 2 t q = ω + q α i yt i 2 i=1 By using the ARCH(q) model it is possible to capture as many lags as needed, but this however gives a lot more parameters to estimate i.e. ARCH models do not have very flexible variance structures 15. The lack of ability to capture the persistence in the conditional variance has given birth to the generalized-arch, GARCH, first introduced by Bollerslev (1986). 4.2 GARCH Due to the little flexibility of the lag structure in ARCH models, as described in the previous section, a constant variance model such as the ARMA will often be preferred over the ARCH 16. To avoid these lag structure issues Bollerslev (1986) introduced the GARCH model. Compared to ARCH, the GARCH model, has a lot more flexible lag structure, making it capable of capturing the same autoregressive effects as the ARCH model, but with a lot less parameters to be estimated. This extra flexibility compared to ARCH comes from the fact that GARCH is based upon a ARCH( ). Throughout this section various aspects of the GARCH model will be presented and described. This section seeks to describe advantages and drawbacks of the model. 14 Reider, Rob (2009); Volatility Forecasting I: GARCH Models 15 Bollerslev, Tim (1986); Generalized Autoregressive Conditional Heteroskedasticity 16 Engle, Robert F. (1982); Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of U.K. Inflation 19

20 The GARCH model maintains the general model for financial time series in equation (2), whereas the difference between ARCH and GARCH is to be found in the definition of the conditional variance. Where the conditional variance in ARCH models only depend on past squared returns the GARCH model incorporate lagged conditional variances as well. The GARCH(1,1) model is thus the combination of (2) with conditional variance defined as σ 2 t = ω + αy 2 t 1 + βσ 2 t 1 (9) From equation (9) it is seen that the conditional variance is indeed heteroscedastic and determined by past squared returns and past conditional variances. Assuming normal distributed innovations, the distribution of y t conditioned on ỹ t 1 is given by y t ỹ t 1 N(0, ω + αyt βσt 1) 2 (10) This dependence and hence the dynamics for the GARCH(1,1) variance process is seen if a sudden shock, is observed in the return process, y t. If a shock is observed in period t then the conditional variance forecast in period t + 1 will be large, dependent on the value of α. In period t + 2 the shock in period t will not have any direct effect on the conditional variance. However, the shock in period t will still persist through the conditional variance in period t + 2 and in future conditional variances. To see how α and β effect the GARCH process, two different GARCH(1,1) processes are shown in figure (6). test.garch test.garch Time Time Figur 6: Simulated GARCH(1,1) processes with different parameter values. Both processes are simulated in R. Left: {ω, α, β} = {1, 0.1, 0.85}, Rright: {ω, α, β} = {1, 0.8, 0.15} 20

21 Figure (6) displays the dependency structure of past returns and past conditional variances. The left hand side plot, with small α (0.10) and large β (0.85) responds weakly to last periods return due to the small α value. Due to the high β value the persistence in the shocks decay very slowly, which is why volatility clustering is not particularly pronounced in this case. The right hand side plot has a larger α (0.8) and smaller β (0.15) and the clustering of volatility is observed. The return responds aggressively to last periods return resulting in large sudden fluctuations when a large absolute value of z t occurs. With the smaller β however, the shocks are much less persistent and hence faster at converging back to the unconditional variance, due to the smaller dependence on last periods conditional variance. The unconditional variance will be introduced later in this section. Expanding the GARCH(1,1) to a GARCH(p, q), in similar ways as in the ARCH case, the conditional variance take the following form σ 2 t = ω + p α i yt i 2 + i=1 q β i σt i 2 (11) i=1 Where the GARCH(p, q) boils down to an ARCH(p) if the parameter q is set to zero. The parameters {ω, α i, β i } are restricted by ω > 0 and α i, β i 0, to ensure strict positivity of the conditional variance. From equation (11) it can be shown that the GARCH(p, q) nests an ARCH process of infinite order. 17 Since the studies in this thesis only concerns GARCH(1,1) processes, the GARCH(p, q) will not be gone into more details with. With this introduction to GARCH models it is now time to take a closer look at some of the theoretical properties that defines the GARCH model Stationarity One of the most important subjects to discuss when estimating GARCH models is the stationarity of the process. Shumway & Stoffer (2011) define a strict stationary process 17 Bollerslev, Tim (1986); Generalized Autoregressive Conditional Heteroskedasticity 21

22 X t, as one for which it holds that P {x t1 c 1,..., x tk c k } = P {x t1 +h c 1,..., x tk +h c k } (12) for all k = 1, 2,... and all time points t 1, t 2,..., t k and all numbers c 1, c 2,..., c k and all time shifts h = 0, ±1, ±2,.... The interpretation behind strict stationarity is that for any subset of X t then the multivariate distribution function must be identical to the corresponding shifted subset, time shifted with h. From strict stationarity follows a few convenient results. If the mean function, µ t, is defined for the strict stationary process X t then µ t = µ s for all t and s and hence the mean must be constant i.e. µ t = µ. With constant mean it further follows, given that the second order moment of X t is well defined, that the autocovariance function satisfies γ(x s, x t ) = E[(x s µ)(x t µ)] = γ(x s+h, x t+h ) (13) The interpretation of this result is that for a strict stationary process then the autocovariance function does not depend on the actual time in the process, but only on the time difference between the times s and t. Since strict stationarity can be difficult to fulfill Shumway & Stoffer (2011) define a second and milder form of stationarity. They define a weak stationary process X t, as one for which it holds that (i) the mean value function, µ t is constant and does not depend on time (ii) the autocovariance function, γ(x s, x t ), depends on s and t only through their difference s t It is obvious that a strict stationary process with finite variance is also weak stationary. It should be noted that this is a one way implication. However in the case of GARCH processes with α + β > 1 then the process is not weak stationary, since the process will have infinite variance, but with α + β > 1 not too far from 1 then the GARCH process can still be strict stationary Francq, Christian & Zakoian, Jean-Michel (2010); GARCH Models 22

23 In the following the strict and weak stationary conditions for GARCH processes will be introduced. Consider the GARCH process defined by equation (2) and (9). Francq and Zakoian (2010) defines the strict stationary zone of such process when the GARCH parameters satisfy E[log( αz 2 t + β )] < 0 (14) Further Franq and Zakoian (2010) defines the weak stationary zone of such process given ω > 0 when α + β < 1 (15) It is noticed that equation (14) implies β < 1. If a GARCH is weak stationary it is seen that strict stationarity follows. By application of Jensens inequality to (14) it is seen E[log(αz 2 t + β)] log(e[αz 2 t + β]) = log(α + β) < 0 (16) So given the commonly seen condition for stationary of GARCH processes defined in equation (15) the, in some sense, less strict condition in (14) will be fulfilled as well. The two stationary zones are shown in figure (7) below. Beta Alpha Strict stationary zone Weak stationary zone Figur 7: The area under the blue line contain all parameter combinations (α, β) where the weak stationary criteria defined in equation (15) are met. The area under red line contain all the parameter combinations for which the strict stationary criteria in equation (14) are met. 23

24 That a GARCH process, not satisfying the weak stationary condition, has infinite variance, is seen by looking the l-step ahead forecast of the conditional variance. But first the unconditional variance must be introduced. Assume that a GARCH process y t is weak stationary, then, by using that y t is a martingale difference and a sequence of uncorrelated variables as was showed in equation (5) and (6), the unconditional variance of the process y t can be derived as V ar(y t ) = E[yt 2 ] E[y t ] 2 = E[σt 2 ] = E[ω + αyt βσt 1] 2 = ω + αe[yt 1] 2 + βe[yt 1] 2 = ω + (α + β)e[yt 1] 2 Where the weak stationarity of y t ensures strict stationary by Jensens Inequality and hence the distribution of y t is identical to distribution of y t 1 from what it follows that V ar(y t ) = V ar(y t 1 ) = E[y 2 t 1]. The unconditional can now be written as V ar(y t ) = ω + (α + β)e[y 2 t 1] V ar(y t ) (α + β)v ar(y t 1 ) = ω V ar(y t ) = ω 1 (α + β) (17) The unconditional variance is denoted by σ 2. Using the definition of σ 2 from equation (17) and following Reider (2009) the l-step ahead forecast can be derived as ˆσ 2 t+l = σ 2 + (α + β) l (σ 2 t σ 2 ) (18) The l-step ahead forecast tells a lot about the behavior of GARCH processes. First it is made clear from equation (18) that a GARCH process that does not satisfy the weak stationary conditions has infinite variance. This is seen by looking at the limit of equation (18), lim l ˆσ t+l 2 = lim l σ 2 +(α+β) l (σt 2 σ 2 ) = since the term (α+β) l because the process was assumed not to be weak stationary. It is very important to remember that even if the GARCH process has infinite variance it can still be strict stationary as seen from figure (7), just with infinite variance. When discussing persistence and convergence 24

25 of GARCH processes in the next section, both the unconditional variance and the l-step ahead forecast is of high importance Persistence In this section the persistence of GARCH processes will be discussed in terms of the parameters which influence the persistence, but also in terms of convergence of the conditional variance. Start by expressing the conditional variance in terms of the unconditional variance defined in equation (17) σt 2 = ω + αyt βσt 1 2 = (1 α β)e[σ 2 ] + αyt βσt 1 2 (19) By expressing the conditional variance this way it is seen that it is a weighted average between the unconditional variance, the one lag squared returns and the one lag conditional variance, weighted with the parameters {(1 α β), α, β} respectively. Trivially the three weightings sum to one. As α + β approaches 1 the persistence in the conditional variance increases and becomes stronger, due to the increasing weights on yt 1 2 and σt 1. 2 In the case where α + β = 1 the GARCH process is referred to as an integrated GARCH, in which the unconditional variance is not defined as a consequence of (17). In the text below figure (6) it was described how two simulated GARCH(1,1) processes behaved in terms of persistence, given the choice of parameters. It was also mentioned that after a shock to the return, dependent on the parameter values, the process would either be high or low persistent and hence be either slow or fast at converging back to the unconditional variance. From equation (18) it is clear that the conditional variance will converge to the unconditional variance with time if the process is weak stationary. Given weak stationarity, the term (α + β) l (σt 2 σ 2 ) 0 and hence σt+l 2 σ2 as l. Further it is noticed that the convergence will be slower as α + β approaches one. Where the IGARCH in theory never will converge towards the unconditional mean, which does not exist in that case. This relationship between parameter values and convergence speed is shown by creating a small simulation experiment. Three GARCH processes are simulated with α = 0.05 and 25

26 β = {0.2, 0.8, 0.9} respectively. The estimations of the conditional variances have been made in R. The three GARCH processes are given an artificial shock at time t = 30 and hence it should be possible to see the convergence speed differ in the three cases. The experiment is shown in figure (8) below beta = 0.9 beta = 0.8 beta = Time Figur 8: The figure show a simulation experiment of convergence and persistence in three GARCH(1,1) processes. At time t = 30 the three processes are given an artificial shock and the convergence is afterwards observed. As seen the process with β = 0.9 has very slow convergence whereas the process with β = 0.2 converges very fast. Further is it observed that all processes converges back to their respective unconditional variances, The unconditional variance is calculated from equation (17) and plotted as well. Just as the theory predicts the experiment shows that the higher β value and hence the higher parameter sum, since α is kept constant in all simulations, the slower convergence back to the unconditional variance. The topic of persistence in GARCH models has been discussed throughout the literature and one main criticism of the GARCH model is that when estimated on real life data then α+β tends to be very close to one 19, making the conditional variance estimate highly persistent as 19 Nelson, Daniel B. (1991); Conditional Heteroskedasticity in Asset Returns: A New Approach 26

27 seen from figure (8). With high persistence and hence long convergence time it could imply that GARCH models tend to overestimate the actual variance found in financial data after a sudden shock to the return process 20. A study of the autocorrelation function will make a good indication of whether this phenomenon is present in real life data. The small study consists of deriving the actual autocorrelation function for daily squared S&P500 returns and comparing it with the autocorrelation function of an estimated GARCH process. First, Franq & Zakoian (2010) defines the autocorrelation function of a GARCH with finite fourth moment as ρ y 2(h) := Corr(y 2 t, y 2 t+h) = ρ y 2(1)(α + β) h 1 (20) for h 1, being the number of lags and where ρ y 2(1) = α(1 β(α + β)) 1 (α + β) 2 + α 2 Both the actual and the estimated autocorrelation functions are plotted in figure (9) below for h = 200 and h = 50 respectively. The estimated parameter values are {α, β} = {0.079, 0.914} which could be a sign of high persistence. All calculations are made in R. ACF GARCH estimate ACF GARCH estimate Lag Lag Figur 9: Autocovariance function of daily squared S&P500 returns and GARCH(1,1) estimated autocovarience function. The GARCH(1,1) model has estimated the parameters (α, β) = (0.079, 0.914). 20 Benavides, Guillermo (2007); GARCH processes and Value at Risk: An Empirical Analysis for Mexican Interest Rates Futures 27

28 In this case the over estimation of persistence, and hence perhaps also of the conditional variance, is clear. The squared S&P500 returns displays an autocorrelation function that is lower than the estimated one of the GARCH process. In figure (10) below is shown a small sample of the estimated conditional variance where the overestimation is observed. squared returns squared returns Time Time Figur 10: Sample of squared S&P500 returns and GARCH(1,1) estimate of the conditional variance. It is seen that the estimated conditional variance overestimates the actual variance after a shock to the squared returns. Figure (10) shows actual variance of the S&P500 returns for a small sample period. The phenomenon of over estimating is especially clear in the period around time t = 60. It is seen that when the volatility of the S&P500 drops to low levels after a sudden shock, the estimated conditional variance has a hard time following the pace of shift in volatility level. This is an important drawback of using GARCH to model financial data. Sudden shifts in volatility levels are often observed in financial time series and hence it should motivate an expansion of the GARCH that enables the model to react faster to these sudden changes Kurtosis As discussed in section (3), one of the stylized facts about financial time series is the distribution of the returns. The distribution of stock market returns has in many occasions been shown to be leptokurtic 21. As briefly mentioned, a distribution is leptokurtic if it is 21 Haas, Markus & Pigorsch, Christian (2007) - Financial Economics, Fat-tailed Distributions 28

29 more peaked in the center and has fatter tails than the normal distribution, given identical mean and variance. The kurtosis measure is exactly a measure of these two properties and hence it could be interesting to examine this measure a bit further in a GARCH context. Following Rahbek (2009) the necessary condition for existence of the moment of order four, which is the kurtosis, in a GARCH(1,1) model is β 2 + 2αβ + 3α 2 < 1 (21) Given that the fourth moment of the process y t excist then Posedel (2005) defines the fourth moment as E[yt 4 ] = E[zt 4 ] E[σt 4 ] = 3ω(1 + α + β)[(1 α β)(1 β 2 2αβ 3α 2 )] 1 With the marginal kurtosis given by the fourth moment over the squared unconditional variance and the unconditional variance of GARCH given in equation (17), the kurtosis of GARCH equals 22 κ = E[y4 t ] + α + β)(1 α β) = 3(1 (E[yt 2 ]) 2 1 (β 2 + 2αβ + 3α 2 ) (22) Given that the condition in equation (21) is met, which ensure that the kurtosis exists, and the GARCH process is weak stationary, both the nominator and denominator of (22) is positive, whereas the kurtosis will always be larger than three. From what it follows that GARCH processes are leptokurtic, since the kurtosis of the normal distribution is three. Further it could be interesting to see for which values of α and β the GARCH kurtosis actually exists. In the figure (11) below is plotted possible values of the α parameter as a function of α +β for which the criteria in equation (21) is satisfied. The normal distribution with κ = 3 is plotted together with two t-distributions with 7 and 5 degrees of freedom with kurtosis 5 and 9 respectively. The possible sets of parameters lies below the three different lines. The thick black line illustrate the the trivial condition on α that α α + β, since β 0, due to the existence of a positive variance. 22 Shumway, Robert H. & Stoffer, David S. (2011); Time Series Analysis and Its Applications 29

30 alpha t(5) t(7) Gaussian alpha+beta Figur 11: Parameter spaces for which the GARCH kurtosis is defined, for t-distributions with 5 and 7 degrees of freedom and the standard normal distribution. It is observed that the parameter space for which the kurtosis is defined decreases as the tails of the distributions enlarges. Given a high β value, which is commonly seen in GARCH estimates, the α parameter is restricted to quite small values if the process is to have finite fourth moment. Moreover, the restriction on α get stricter the more heavy tailed the distribution gets. Throughout this section some strengths and weaknesses about GARCH models have been introduced. The ability to capture serial dependence in the squared returns and the leptokurtic features of GARCH makes it an attractive model for modeling returns. However, the often high values, of the estimated persistence parameter β, can result in overestimation of the conditional variance due to the high persistence and hence slow convergence as seen in figure (8) and (10). In section (5) the Markov Switching GARCH model is presented and it is argued why this model might be able to avoid some the unfortunate features of GARCH while still maintaining the strengths and essence of the popular model. 30

31 5 Generalized Regime-Switching Models The GARCH model is very commonly used when modelling the volatility of time series. However, if the return experiences a sudden shock, which leads to a short period with high volatility then the GARCH tends to overestimate the persistence in a period after the shock, 23 which further leads to an overestimation of the conditional variance as explained in section (4.2.2). Lamoureux & Lastrapes (1990) show that the high persistence often estimated by GARCH models might nest in structural changes in the conditional variance process. This means for example, if the variance is high and homoscedastic in one period but low and homoscedastic otherwise, then persistence of such two periods already result in volatility persistence. A GARCH, which is not able to distinguish between such two periods and hence not able to capture the persistence of such periods, will put all persistence in the persistence of individual shocks. 24 One possible way to deal with this issue is to model the variance with two different models depending on whether the current period experiences high or low volatility. That will result in an introduction of shifts in the variance structure, allowing the variance to take on different structures. This is possible by merging the GARCH with a Markov Switching model, first introduced by Hamilton (1989). The basic idea behind the Markov Switching GARCH is to reduce the long GARCH persistence by switching from one variance structure to another. To get an understanding of the MS-GARCH model, a small experiment is set up. A data vector is created by simulating normal distributed i.i.d variables, with different variance. The simulation is made in R. In the left hand side plot of figure (12) is displayed the data series, which should imitate a financial return series with clear shifts in volatility level. The idea behind the experiment is entirely the illustrate the regime switching effect. 23 Lamoureux, Christopher G. & Lastrapes, William D. (1990); Persistence in Variance, Structural Change, and the GARCH Model 24 Klaassen, Franc (2002); Improving GARCH Valotility Forecasts with Regime-Switching GARCH 31