Emerging market volatility : a Markov-Switching Multifractal study

(1)

Master’s Thesis

Emerging Market Volatility:

a Markov-Switching Multifractal study

Daniel Tob´

on Arango

Student number: 6278310

Date of final version: July 18, 2016

Master’s programme: Econometrics

Specialisation: Econometrics

Supervisor: Prof. dr. H. P. Boswijk

Second reader: dr. S. A. Broda

Abstract

An empirical study of the volatility of emerging financial markets is performed. The performance of the Markov-Switching Multifractal model by Calvet and Fisher (2004) is evaluated on its ability to generate four properties that are found in the sample data. A comparison of performance is made between the Markov-Switching Multifractal model and two Fractionally Integrated GARCH models. Using index returns from sixteen emerging and seven developed countries, the models are estimated and a Monte Carlo study is performed. The Markov-Switching Multifractal model outperforms the other models in reproducing the data’s leptokurtic structure. Equal performance is found for the replication of the autocorrelation structure and long-term dependency and it is outperformed by both other models with respect to capturing the leverage effect.

Keywords: Markov-Switching Multifractal model (MSM), Fractionally integrated GARCH model (FIEGARCH), emerging markets, stylized facts, volatility clustering, long memory, leverage effect

(2)

Statement of Originality

This document is written by Daniel Tob´on Arango who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

(3)

Introduction

Modeling volatility is highly relevant for financial markets because it is one of the most common variables to measure the risk of financial assets. As a risk measure, it is present in valuations, risk management and many other fields. Because of the magnitude of financial markets, their riskiness is of importance for all different stakeholders like investors and governments. The worldwide impact of the most recent financial crisis has once again demonstrated the importance of understanding financial markets and the risks they may contain. With the growth of financial markets, interest in emerging financial markets has been expanding as well. Besides world factors like worldwide financial crises, emerging markets can be affected by other factors that come hand in hand with being an emerging country. Examples are liberalization policies and privatization of companies but also more grim shocks like corruption scandals. These different factors give an explanation to, for example, the empirical finding that emerging markets are more volatile than developed financial markets. The increasing investment interest in emerging markets and these findings give additional interest in understanding emerging market volatility. To understand the volatility of emerging financial markets, good performing volatility mod-els are a necessity. This thesis discusses and evaluates one of these volatility modmod-els and tries to determine whether it can explain the time-varying volatility dynamics of emerging financial markets. The model of which the performance is tested is called the Markov-Switching Multi-fractal (MSM) model by Calvet and Fisher (2004). The performance criteria in this thesis are generally based on the replication of so-called stylized facts of asset returns, being volatility clus-tering, long memory, the leverage effect and the non-normality of the asset returns’ distribution. Because Generalized Autoregressive Conditional Heteroskedasticity (GARCH) type models are among the most explored models in the existing financial econometrics literature, one of these is used as a benchmark for the MSM model. Because the replication of the stylized facts are the biggest concern of this thesis, a GARCH model that is developed to replicate these proper-ties is chosen, being the Fractionally Integrated Exponential GARCH (FIEGARCH) model by Bollerslev and Mikkelsen (1996).

The MSM model is a discrete time multifractal stochastic volatility model that models volatility by means of a product of multiple volatility components. The model as proposed by

(5)

CHAPTER 1. INTRODUCTION 2

Calvet and Fisher (2004) can be estimated by maximum likelihood but also through a general-ized method of moments (GMM) procedure (Lux, 2008). Following these methods, forecasting is possible with respectively Bayesian updating and best linear forecasts. Existing literature regarding the model is scarce but the studies show that the model has the ability to capture per-sistence in volatility, autoregressive transitions and substantial outliers. Using exchange rates, both Calvet and Fisher (2004) (ML estimation) and Lux (2008) (GMM estimation) find that their model outperforms GARCH, Markov-switching GARCH (MS-GARCH) and Fractionally Integrated GARCH (FIGARCH) in- and out-of-sample. Chuang et al. (2013) compare the per-formance of the MSM model with other volatility models for modeling the volatility of a US equity index. They find that the MSM and GARCH models outperform the other measures but that the MSM model provides better results for periods of high volatility. For crude oil prices, Wang et al. (2016) find that the MSM model fits the data significantly better than the GARCH-class models. They also conclude that the MSM model produces more accurate forecasts than the GARCH models for most loss functions. They argue that the MSM model has the ability to capture structural breaks in conditional volatility, or sudden events (like crises), relatively well. The existing MSM literature explains that the MSM model is designed to capture long range dependence, substantial outliers and that it has the ability to outperform GARCH models in periods of high volatility. These findings demonstrate the relevance of this thesis because they indicate that the MSM model might be a good fit for the relatively high volatility in emerging markets and long memory and outlier properties.

Because replicating stylized facts is the main performance criterium in this thesis, the MSM model is compared with a GARCH model that is developed to replicate these properties, the FIEGARCH model. This model has the ability to not only model the asymmetric relationship between past returns and volatility, but also allows for the long memory effect. Bollerslev and Mikkelsen (1996) show how quasi maximum likelihood (QML) estimation can be applied to esti-mate the parameters of their FIEGARCH model for modeling conditional variance. The econo-metric literature regarding fractionally integrated models consistently concludes that modeling long memory with GARCH models needs the inclusion of fractional integration. Fractionally in-tegrated versions of GARCH models like the Exponential GARCH (EGARCH) model (Nelson, 1991) therefore seem like logical extensions. Following its introduction, empirical studies of the FIEGARCH model have followed and demonstrated that the model has the ability to replicate both the leverage effect and hyperbolic decay of the autocorrelation function that appears in financial data (Bollerslev and Mikkelsen, 1996; Ruiz and Veiga, 2008; Tansuchat et al., 2009).

The stylized facts that are assessed as performance measures are three properties that follow from conditional variance; volatility clustering, long memory and the leverage effect. The other properties are related to the shape of the conditional distribution of asset returns. Volatility clustering is found in financial data if highly volatile periods are clustered together and if periods of low volatility are as well. The presence of this characteristic in the data demonstrates the conditional heteroskedasticity of time-varying volatility, it therefore motivates the use of

(6)

CHAPTER 1. INTRODUCTION 3

ARCH models. Long memory is the property that is related to the hyperbolic decay of the autocorrelation function of absolute or squared asset returns, it implies that past shocks on conditional variance die out in a hyperbolic slow rate. Financial data possesses the leverage effect if there is an negative asymmetric relation between past returns and conditional variance, meaning that negative returns have a stronger impact on future volatility than positive returns. The distributional properties that are assessed in this thesis are the skewness and kurtosis that are generated by both models. Studies have shown that the shape of the unconditional and conditional distributions of asset returns differ from the normal distribution. It is found that the distribution is asymmetric, mostly negative, meaning that losses are more regular than gains. Also, the distributions are found to have fat tails, implying that outliers are common. This thesis assesses all these properties and examines how the properties are replicated by the fitted models. A good performing model has the ability to generate the properties as they appear in the studied data.

The in-sample comparison of the estimated models result in the conclusion that the FIE-GARCH model with student−t error terms fits the empirical data slightly better than the MSM model and the FIEGARCH model with normally distributed error terms. Furthermore, the sim-ulation study to the replication of the stylized facts show that the MSM model has the ability to capture the leptokurtic structure of the return distributions relatively well, seemingly better than its benchmarks. All models are comparable in their ability to reproduce the autocorre-lation structure and the long memory effect. On the other hand it is shown that the models do not generate data that resembles the skewness from the empirical data. The leverage effect is the stylized fact where the FIEGARCH models clearly outperform the MSM model. This result is not surprising because the MSM model is not constructed to reproduce this effect. The empirical results in this thesis show a good volatility model should be able to reproduce the leverage effect as well as the other stylized facts. The in-sample outperformance by the FIEGARCH models may therefore be caused by the lack of asymmetric response from past returns on future volatility.

The remainder of this thesis is organized as follows. The second chapter gives a two-part literature study regarding the modeling of time-varying volatility and emerging market volatility. Chapter three presents both models, their specifications and estimation methods. The data that is used for the empirical study is discussed and examined in chapter four. The chapter thereafter gives the empirical results and presents a Monte-Carlo study of the model using the parameter estimates of the empirical study. Chapter six concludes, presents the discussion and recommendations.

(7)

Chapter 2

Literature Review

2.1 Modeling Volatility

The importance of knowledge of financial data’s volatility in fields like risk management and valuation of financial instruments have resulted into volatility modeling being a well represented subject in financial econometric research. Unlike other variables like stock prices and interest rates, volatility cannot be observed directly and should therefore be induced implicitly. Popular, relatively straightforward measures are historical and implied volatility but squared returns and sample variance of return series are applied as well. Throughout the years, various parametric models have been developed, aiming to model asset return’s volatility and its time varying dynamics more accurately. The models that are explored the most in the financial literature are GARCH type models, which have proven to be quite successful in this field. Other models that have been successful are the stochastic volatility models, with the standard form given by Harvey et al. (1994). The latter are found to perform better than GARCH type models (e.g. Andersen and Sørensen, 1996) but are more difficult to estimate. This section assesses different volatility models and corresponding empirical studies. The main focus is on univariate models since this thesis only considers the volatility behavior of individual stock indices and not in their possible relation or effect on one another.

Volatility dynamics in the family of GARCH models are described by the parameterization of conditional variances. These models are based on the classic ARCH model (Engle, 1982), which have the ability to model time-varying variance. Time-varying variance, or heteroskedastic variance with respect to time, leads to volatility clustering, a characteristic that is often found in asset return dynamics and can be modeled with the ARCH and GARCH models. Volatility clustering is one of the properties of asset returns that is classified as a so-called stylized fact. It basically means that high volatility periods are generally followed by high volatility periods and that periods with less volatile financial markets are followed by periods with less volatile financial markets. The first documentation that reports this finding was the study of Mandelbrot (1963). Mandelbrot (1963) stated that large changes in financial returns were followed by other large changes (of both signs) and that the same dynamics can be found in smaller changes in financial

(8)

CHAPTER 2. LITERATURE REVIEW 5

returns. This conditional heteroskedastic behavior of volatility is one of the properties that can be provided by ARCH type models. The representation of the ARCH model as proposed by Engle (1982) is given in equation (2.1).

σ_t2= ω + q X j=1 αj2t−j, (2.1) with rt= µ + t, (2.2) t∼ N (0, σt2). (2.3)

Equation (2.1) shows that an ARCH model uses q past disturbances as regressors to model time-varying variance. It also shows that large past shocks 2_t−j imply a large conditional variance σ_t2 and that this relation also holds for small past shocks. This feature leads to the replication of volatility clustering. The generalized extension of the ARCH model, the GARCH model, was introduced by Bollerslev (1986). It includes p past autoregressive terms for σ_t2, resulting into the equation below.

σ2_t = ω + q X j=1 αj2t−j+ p X j=1 βjσ2t−j. (2.4)

In these equations, rt represents the asset’s return at time t, µ is the unconditional mean of

these returns, σ_t2 the conditional variance of the asset returns at time t and t a random error

at time t. The most common GARCH model that is used for modeling financial time series is GARCH(1,1) (Bollerslev, 1986). This leads to the following, simplified model:

σ2_t = ω + α2_t−1+ βσ_t−12 . (2.5)

Sufficient conditions for the conditional variance to be positive are ω > 0, α ≥ 0 and β ≥ 0, for all t. Furthermore the condition α + β < 1 ensures that the process is weakly stationary (Bollerslev, 1986). Equations (2.4) and (2.5) demonstrate clearly that GARCH models have the ability to reproduce volatility clustering. Because shocks and conditional variance from previous periods are used as regressors for the conditional variance of future periods, high volatile periods are more likely to follow one another, and so are low volatile periods.

Another aspect that has to be taken into account when modeling the conditional variance of asset returns is the behavior of its (partial) autocorrelation function. Empirical studies have demonstrated that the sample autocorrelation function and partial autocorrelation function of conditional variance of asset returns decays slowly as the lags increase. This behavior cannot be fully captured by the ARCH model since its autocorrelation function decays exponentially and the partial autocorrelation function has a cut of point. The latter is the characteristic that requires a GARCH model, of which the partial correlation function does not have a cut off point. Equations (2.6) and (2.7) give the autocorrelation function of a GARCH(1,1) model.

ρl= (α + β)l−1ρ1 for l > 1, (2.6)

ρ1 = α +

α2β

(9)

Equation (2.7) shows that the decay of the autocorrelation function of the GARCH(1,1) model is of an exponential nature. Expression α + β in equation (2.7) is a measure for the persistence of the volatility of asset returns. As this term approaches one, shocks to volatility will be more and more persistent. Persistent shocks correspond with the empirical finding that the auto-correlation function of the conditional variance of asset returns decays in a slow hyperbolical way. This characteristic is referred to as the long memory effect, which is another stylized fact that is widely found in the volatility dynamics of asset returns. When a time series possesses long memory, it means that past shocks on asset returns die out slowly and therefore remain having a significant effect on asset returns and its volatility in the future. In order for the classic GARCH process to be (weakly) stationary and the unconditional variance to be finite, constraint α + β < 1 is imposed. Engle and Bollerslev (1986) proposed the Integrated GARCH (IGARCH) model where this inequality is set to an equality, making it an unit-root GARCH model. When restriction α + β = 1 holds, the impact of past squared shocks are persistent and remain important for ever. Motivated by the long memory effect and the statement that the degree of dependence that the IGARCH imposes is too extreme, Baillie et al. (1996) developed the FIGARCH model. This model allows the effect of past shocks to die out in a slow hyper-bolical rate, instead of remaining important forever. Fractionally integrated models and their specifications are discussed more extensively in chapter three.

Another field where the basic GARCH model is a mismatch with empirical findings is its symmetric behavior. Classic GARCH models assign an effect to the magnitude of a shock on conditional variance whereas they do not take into account the sign of the shock. Because previous studies have demonstrated that volatility reacts differently to negative shocks than to positive shocks, various models that take into account this asymmetry have been introduced. These asymmetric GARCH models allow negative and positive shocks to have separate effects on the modeled conditional variance. A stylized fact of asset returns that is directly linked to the asymmetric correlation between past shocks and conditional variance is called the leverage effect. This effect is present when negative shocks have a larger impact on conditional variance than positive shocks, implying that negative past returns increase volatility more than positive past returns. The lack of ability to reproduce this effect has been an important criticism for the classic GARCH model. It has made financial econometricians develop asymmetric models instead. Black (1976) was one of the first to find evidence for this effect. This study stated that an increase in volatility due to falling stock prices is caused by a decrease of equity value, which increases the debt to equity ratio, making the stock riskier.

An example of a widely used asymmetric GARCH model is the Exponential GARCH (EGARCH) model, proposed by Nelson (1991). This extension to the basic GARCH model allows the sign and magnitude of previous shocks to have a distinct effect on conditional vari-ance. By making this distinction, the EGARCH model accommodates an asymmetric relation between returns and volatility and can therefore model the leverage effect. This can be seen

(10)

easily from the representation of the EGARCH(1,1) model, given in equations (2.8) and (2.9).

ln(σ_t2) = ω + αg(zt−1) + β ln(σ2t−1), (2.8)

with:

g(zt) = θzt+ γ(|zt| − E[|zt|]), (2.9)

and zt = _σt_t, being the standardized residuals. Equation (2.9) demonstrates that conditional

variance is affected not only by the size of past shocks, but also its sign. The first term, θzt, allows for asymmetric responses to the sign of previous shocks whereas the second term,

γ(|zt| − E[|zt|]), ensures that shocks of different magnitudes can have different effects. If θ = 0

and γ > 0, a high (low) return compared to its mean will have a positive (negative) effect on next period’s conditional variance. For the case that θ < 0 and γ = 0, equation (2.9) shows that a positive (negative) shock has a negative (positive) effect on next period’s conditional variance. Other asymmetric GARCH models are the Glosten-Jagannathan-Runkle GARCH (GJR-GARCH) by Glosten et al. (1993), the Asymmetric GARCH (AGARCH) by Engle and Ng (1993), the Threshold GARCH (TGARCH) by Zakoian (1994) and the Quadratic GARCH (QGARCH) by Sentana (1995).

Other innovations to the classic GARCH model are the Markov-switching ARCH (MS-ARCH) model by Hamilton and Susmel (1994) and its general analogue, the MS-GARCH model of Klaassen (2002). These model types are proposed because it is argued that the classic ARCH and GARCH models do not take into account that different magnitudes of shocks may have a substantially different effect on conditional variance. The proposed Markov-switching models provide more flexibility with respect to volatility persistence. Multiple regimes with different volatility levels are defined such that volatility in the different regimes can be driven by their own GARCH effects. These different regime shifts are defined at some, but not at all frequencies. Calvet and Fisher (2004) proposed an alternative Markov-switching volatility model based on pure regime switching at all frequencies. The MSM model of Calvet and Fisher (2004) is relatively new when compared with the GARCH models in the sense that it is not studied that extensively. It assumes stochastic volatility and is found to outperform the GARCH, FIGARCH and MS-GARCH models. Because the MSM model allows for volatility regime switches at all frequencies, it has the ability to not only model time-varying volatility, but also to generate substantial outliers (volatility components with low persistence) and to replicate long memory (volatility components with high persistence). Equation (2.10) demonstrates that conditional variance is modeled as the product of discrete Markov chains in the MSM model.

σt= σ(M1,tM2,t...M¯_k,t)

1

2 (2.10)

This representation of the MSM model shows that volatility is driven by a finite number, ¯

k, volatility components with values M1,t, M2,t, ..., M¯_k,t at time t. Each volatility component

has its own switching frequency, allowing the model to generate both high persistent and low persistent components. One of the drawbacks of the model that was given by Calvet and

(11)

Fisher (2004) is the loss of estimation feasibility when the number of the models’ volatility components passes a certain level. A possible solution for allowing more volatility components is using another estimation method. Lux (2008) applies GMM estimation rather than ML estimation and shows that the former can lead to gains in forecasting accuracy for some time series. Another drawback of the MSM model is its production of symmetric relation between past returns and future volatility. Where asymmetric versions of the GARCH models were developed in order to capture the leverage effect, this has not been done for the MSM model. It is therefore expected that the MSM model does not capture this effect as well as it may capture the long memory effect and substantial outliers. The MSM model, its characterstics and its estimation procedure is explored further in chapter three.

An alternative approach of modeling volatility is with Stochastic Volatility (SV) models. The main difference between these models and GARCH type models is that volatility in the former is modeled as an unobserved component following a latent stochastic process (Taylor, 1986). Similar to the GARCH models, there have been extensions to and variations of the SV models. Among these are the Long Memory Stochastic Volatility (LMSV) model (Breidt et al., 1998; Harvey, 1998) and the Asymmetric Stochastic Volatility Model (A-SV) (Harvey and Shephard, 1996) and a combination of both, the A-LMSV model by Ruiz and Veiga (2008). SV models are sometimes found to perform better than GARCH models when modeling volatility. An important drawback of these SV models is that estimation is relatively difficult compared to the estimation of GARCH models. Ruiz and Veiga (2008) compared the A-LMSV model with the FIEGARCH model and examined their ability to capture the empirical features in the daily returns of the S&P500 Index and the DAX Index. They found that both models could replicate the long memory and leverage effects but that the former can replicate the properties for a larger scope of parameter values than the FIEGARCH model with normally distributed errors.

One advantage of SV models over GARCH models is the SV model’s ability to replicate asset returns’ distributional properties with normally distributed error terms whereas GARCH models need alternatively distributed error terms to give the same result. Assuming normally distributed error terms might not be valid because the unconditional distribution of asset re-turns generally are not normal either. Asset rere-turns are often characterized by a leptokurtic and skewed distribution rather than a normal one. These properties are consistent with respectively outliers in financial data and the finding that large upside movements are not as common as large downside drops. In order to fit the first discrepancy better, Bollerslev (1987) stated that GARCH models with error terms that have a conditional leptokurtic distribution might explain the high unconditional kurtosis better than GARCH models with error terms that are normally distributed. He proposed to model GARCH processes with standardized student−t distributed errors rather than with normally distributed error terms. It is stated that this alternative, fat tailed distribution can account for the high unconditional kurtosis and fat tails in the data. Another stylized fact related to the unconditional distribution of asset returns is its asymmetry.

(12)

In order to fit this empirical finding better, Lambert et al. (2001) propose to assume an asym-metric, leptokurtic distribution for the error terms, being the skewed standardized student−t distribution (Fern´andez and Steel, 1998). Lambert et al. (2001) apply this distribution to the Asymmetric Power ARCH (APARCH) model (Ding et al., 1993) and find that the APARCH model with skewed standardized student−t distributed error terms outperforms the same model with the normal and standardized student−t specification.

Because volatility models are very well represented in econometric research, the study of Huang (2011) assesses the performance of five popular methods that are used to quantify con-ditional variance. The methods, models and measures that were studied are the historical volatility, Monte Carlo simulation, GARCH models, quantile regression and the SV model. In this research, thirty-one emerging and developed stock indices were studied. By comparing the R2values of all measures, this study concluded that the SV model performs best for both market types. It is stated that SV models take into account asset returns’ stylized facts by explicitly allowing for both persistent and time-varying volatility terms. Also, this model generates a leptokurtic distribution, which represents fat tails and is consistent with substantial outliers in asset returns.

Chuang et al. (2013) evaluate the performance of the MSM model and compare it with implied volatility, historical volatility and the basic GARCH model. These models are applied to the S&P 500 Index and corresponding equity options. Through a regression of the realized volatility on all separate volatility models, they find that the performance of the GARCH and MSM models is similar and that both outperform the other two measures. When making the distinction between periods of global financial crisis and periods where no crisis took place, they find that the MSM model has a better forecasting performance in global financial crisis periods than the GARCH model. This result provides evidence that the MSM model can capture extreme events or unexpected shocks better than the GARCH model. Wang et al. (2016) confirm that the MSM model outperforms GARCH models (more specifically the EGARCH, GJR-GARCH, IGARCH and FIGARCH models). They find that the MSM model is a more powerful tool to capture and forecast the dynamics of crude oil return volatility than the GARCH models.

The study of Liu and Hung (2010) investigated the performance of two asymmetric GARCH models and four simple GARCH(1,1) models with different distributional specifications of the error terms. In order to capture the leverage effect, they compare the forecasting performance of the EGARCH and GJR-GARCH models. To accommodate the deficiency of the normal dis-tribution, they examine the normal, student−t, HT (heavy tailed) and the skewed generalized-t distributions for the error terms. Their empirical investigation of forecasting the daily returns volatility of the S&P500 Index indicates that the GJR-GARCH model provides the most ac-curate volatility forecasts for their sample, closely followed by the EGARCH model. These results imply that specifying an asymmetric model is preferred over specifying an sophisticated error distribution in order to capture the non-normality of the asset returns’ distribution. If

(13)

asymmetries are neglected, the symmetric GARCH model with the normally distributed error terms is preferred over the other distribution specifications. This result is in contrast with other studies and implies that asymmetric GARCH models should be assessed with both normally and a non-normally distributed errors in order to compare their performance.

Since GARCH models are the most applied models to model financial returns’ volatility, this thesis evaluates the performance of the MSM model with such a GARCH model. Comparing the performance of the MSM model with its GARCH benchmark in this thesis means that the ability to replicate the before mentioned stylized facts is used as the performance measure. Replicating the stylized facts means that the chosen GARCH model should be able to take into account not only volatility clustering but also long memory, the leverage effect and the distributional properties of asset returns. Examples of models that take these properties into account are the Fractionally Integrated Asymmetric Power (FIAPARCH) model and the FIEGARCH model. For the last stylized fact, non-normally distributed error terms might be useful to apply to the models. In a study to the external effects on emerging market volatility, Aloui (2011) applies the FIAPARCH model with student−t distributed errors. The use of this fractionally integrated model, that was introduced by Tse et al. (1998), is chosen because it seems to allow for long range volatility dependence (long memory), an asymmetric response to negative and positive shocks (leverage effect) and can determine the power of returns for which the predictable structure in the volatility pattern is strongest. Tansuchat et al. (2009) estimates long memory volatility models like the FIAPARCH model and compares their performance for modeling volatility in agricultural commodity futures returns. Because they find that the FIAPARCH model is outperformed by the FIEGARCH model, this thesis compares the performance of the MSM model with the FIEGARCH model as its benchmark. Since GARCH models with normally distributed standard errors can fall short when assessing asset returns’ leptokurtic distribution, it might be useful to model the FIEGARCH model with standard errors that can replicate this property. The performance of the FIEGARCH is therefore be explored with both standard normal and student−t distributed errors.

2.2 Emerging Markets Volatility

One of the reasons that gives the subject of emerging market volatility its importance is the general finding that asset returns in emerging markets demonstrate higher volatility than those in developed markets. The extensive study to the volatility of emerging markets of De Santis and Imrohoroglu (1997) examines the predictability of market return dynamics, the relation between market risk and expected return, price change frequency and the effect of liberalization of emerging financial markets on the volatility of stock index returns. The analysis includes the weekly market return dynamics of stock indices of four developed financial markets and fourteen emerging markets (two European, six Asian and six Latin American). Motivated by volatility clustering in the data, De Santis and Imrohoroglu (1997) employ a basic GARCH model. In this model they assume General Error Distributed (GED) errors, motivated by another stylized fact;

(14)

the inability of Gaussian errors to fully capture the leptokurtic distribution that is found in the data. They find similar results between emerging and developed markets when looking at the GARCH model’s estimates. One of these similarities is the persistence of shocks, an indication for long memory. Among the differences between both market types that are reported is a considerably higher conditional variance for emerging markets, confirming the expectation that emerging markets are more volatile than their developed peers, at all times. Another difference in conditional variance is the bigger gap between the minimum and maximum values of the generated time series of the emerging markets. Finally, they find both higher dispersion and conditional kurtosis for the emerging markets, respectively implying more frequent large changes and that outliers, or unusual shocks are more common for this market type.

In their study to events that cause large shifts in the volatility of emerging stock mar-kets, Aggarwal et al. (1999) also employed a GARCH model. In contrary with De Santis and Imrohoroglu (1997) they argue that including fat-tailed error terms does not improve the perfor-mance of the GARCH model regarding the conditional distribution’s leptokurtic behavior. This study modifies the classic GARCH model in order to capture sudden changes that are found in emerging market volatility. Their approach includes detecting sudden changes in variance and using dummy variables that are based on these sudden changes as regressors for conditional variance. For this study, they analyzed the weekly returns of indices of ten emerging countries, six indices of developed countries and four regional indices, for example the Far East Index. The authors argue that daily data might cause problems because of non-continuous trading weeks and therefore use weekly data. Again, this study finds higher unconditional variance and unconditional non-normality for the emerging markets. All time series show a high kurtosis, negative skewness for most developed markets and positive skewness for many emerging mar-kets. These results seem to be nuanced when looking at the conditional distribution that is generated by the GARCH model. Their GARCH estimate of the persistence parameter was close to unity for most time series, implying long memory. Including the dummy regressors leads to their conclusion that both local and global events cause an increase in volatility but that volatility increase occurs more often around local events. A consequence of adding these sudden change dummies is that most GARCH coefficients were reduced and no longer signif-icant. This might indicate that their innovation to the simple GARCH model means making a trade-off between modeling long memory and outliers. Finally, Aggarwal et al. (1999) state that the change dummies cannot be incorporated in forecasting models because determining shifts in volatility cannot be detected beforehand.

Edwards and Susmel (2001) also have their interest in modeling volatility such that structural shifts in the data can be captured. They state that classic GARCH models fail to capture the effect of structural shifts in financial time series that are caused by low probability events like financial crises. Their preliminary analysis of their Latin American stock returns confirmed the findings that were also reported in the above mentioned studies to emerging market volatility; high unconditional standard deviation, non-normality of the unconditional distributions and

(15)

significant autocorrelation for squared returns. The latter being their argument to employ an ARCH-type process for modeling conditional variance. Evidence for this ARCH effect was found in their estimates of the basic GARCH model, as was evidence for long memory. In order to capture both the ARCH effect and sudden changes, these authors use the MS-ARCH model that was proposed by Hamilton and Susmel (1994). In order to capture asymmetric effects, the MS-ARCH model is estimated with the asymmetric specification that was given by Glosten et al. (1993). Their univariate results provide strong evidence for state-varying volatility, meaning that it is possible to distinguish a low and high volatility state for each country. Following up on this result, the authors note that a third state might be appropriate for some countries. The same estimates also show that with the exception of one market, there is no evidence for an asymmetric effect of negative news on conditional volatility, so no evidence for the leverage effect.

Another study to the effect of local and world factors on the volatility behavior of emerging capital markets, is that of Bekaert and Harvey (1997). They state that modeling emerging mar-ket volatility brings challenges that might not need to be faced for developed financial marmar-kets. In their model they therefore allow the relative importance of local and world information shift through time as emerging markets become more or less integrated in the world market. Besides this specific challenge, they argue that standard ARCH models are unlikely to be sufficient because of the non-normality of the distribution of market returns, which is supported by the rejection of unconditional normality when testing the data of twenty emerging markets. To take into account skewness and kurtosis, the authors use three distributional specifications for the standardized residuals: the standard normal, standardized student−t and a third that accounts for both skewness and kurtosis. Also, when testing whether a symmetric or asymmetric model suits the data better, they find evidence for the last, implying the presence of a leverage effect. A joint examination of the model type and distribution assumption lead to the conclusion that an asymmetric model with normally distributed error terms gave the highest R2 and would therefore work the best for this time series.

As mentioned in the previous section, Aloui (2011) also studied emerging market volatility, with a specific interest in the volatility spillovers of Latin American stock markets. Their preliminary analysis includes the LM ARCH test (Engle, 1982) and the Ljung-Box test for the squared residuals, both revealing heteroskedasticity. Secondly, the study finds evidence for an unconditional distribution that is skewed to the left and exhibits high kurtosis. The authors also conduct various long memory tests, including the rescaled variance test, and conclude that the long memory effect is present in this data. It is argued that the presence of these characteristics means that those should be considered in the model. The estimation of their multivariate FIAPARCH model with student−t error terms replicates results into a significant asymmetry parameter, a significant fractional parameter and the conclusion that the student−t distribution is appropriate to represent the fat tails that appear in the data.

(16)

lit-CHAPTER 2. LITERATURE REVIEW 13

erature. The existing literature demonstrates that emerging markets are characterized by not only the stylized facts that are also found in developed markets but also affected by their own economic shock like liberalization policies, increase of capital flows but also their own financial crises. These idiosyncratic shocks imply that conclusions that are drawn from modeling returns and volatility of developed markets might not hold for emerging financial markets. It also shows the importance of evaluating the performance of volatility models for different emerging markets because some country events or characteristics might complicate the volatility modeling.

(17)

Chapter 3

Model

The following sections present the framework of both the MSM model by Calvet and Fisher (2004) and the FIEGARCH(1, d, 1) model by Bollerslev and Mikkelsen (1996).

3.1 The Markov-Switching Multifractal model

The volatility model that has this thesis’ primary interest is the Markov-Switching Multifrac-tal (MSM) model, introduced by Calvet and Fisher (2004). In the MSM model, volatility is modeled a multiplicative product of a finite number of volatility components. These compo-nents, also known as multipliers, have an identical marginal distribution but differ in their switching probabilities, which follow a geometric specification. These transition probabilities are heterogeneous for all components, allowing some components to be more persistent than others. These specifications allow the components to switch randomly in value over time and generate a volatility process that is highly persistent but also variable. The process results in a stochastic volatility model with a closed-form likelihood generating both substantial outliers and long memory respectively due to the inclusion of the low persistent components and high persistent components.

Equation (2.10) in the previous chapter demonstrates how the conditional variance of asset returns is driven by the various volatility components. Let log-prices be defined as pt and

log-returns represented by rt = pt− pt−1, then equation (3.1) shows how log-returns are modeled

by the MSM model.

rt= σ(M1,tM2,t...M¯_k,t)

1

2z_t, (3.1)

with σ being a positive constant, zt ∼ N (0, 1) and where the volatility components Mk,t are

persistent, non-negative and satisfy E[Mk,t] = 1. Also, it is assumed that the unconditional

mean of the log-returns equals zero. The volatility components in the model proposed by Calvet and Fisher (2004) are statistically independent at every time t, therefore parameter σ is defined as the unconditional standard deviation of return rt. Equation (3.1) gives a stochastic

volatility model with rt= σtzt, where σt= σ[g(Mt)]

1

2. Function g(M_t) is defined as the product

Qk¯

i=1Mi,t where Mtis the so-called volatility state vector with the volatility components as its

(18)

CHAPTER 3. MODEL 15

elements.

The time-varying volatility process can be established because the value of the components can switch randomly over time. At time t, the value of volatility component Mk,tis redrawn from

a distribution that is identical for all components with probability γk. The value of component

Mk,t remains the same as in the previous period (time t − 1) with probability 1 − γk. Equation

(3.2) gives the definition of transition probability γk as it was introduced by Calvet and Fisher

(2001).

γk= 1 − (1 − γ1)b

k−1

, (3.2)

with γ1 ∈ (0, 1) being the transition probability of the first component and b ∈ (1, ∞) defined

as the geometric growth rate. When these conditions hold, γk is increasing in k, implying that

the transition probability of γ¯_kis the highest and that the first volatility component is the most

persistent. The distribution from which the value of the components are drawn is binomial. The corresponding binomial random variable’s value can either be m0 ∈ [1, 2) or 2 − m0. The

full parameter vector is then defined by:

ψ = (m0, σ, b, γ¯_k) ⊂ R4₊. (3.3)

The ¯k volatility components imply that there are 2k¯ possible volatility states, given that the value volatility components are binomial random variables. The variability in the frequency of the multipliers and the high number of possible states gives this model the ability to replicate features that are found in financial data: highly persistent components model the long memory effect whereas components with a high transition probability ensure that the model generates substantial outliers.

In their paper, Calvet and Fisher (2004) give an expression for the autocorrelation function of the discrete time MSM model in levels, for squared returns it satisfies:

sup l∈I¯k ln ρ(l) ln l−δ − 1 → 0 as ¯k → ∞, (3.4)

where l equals the number of lags and δ = log_b(E[M2]/(E[M ])2). Equation (3.4) implies that ln ρ(l) ∼ ln l−δ, or ρ(l) ∼ l−δ as l → ∞, meaning that the MSM model replicates hyperbolic decay in the autocorrelation function and therefore has the ability to generate the long memory effect (Calvet and Fisher, 2004). An explicit form of the autocorrelation function for a volatility process that is generated by the binomial MSM model is given below: 1:

ρ(l) = ¯ k Y k=1 {1 2[1 − (1 − γk) l_]m 0m1+ [(1 − γk)l+ 1 2(1 − (1 − γk) l_)](1 2m 2 0+ 1 2m 2 1)} − 1 (1 2m 2 0+ 1 2m 2 1) ¯ k_{− 1} , (3.5) with m1= 2 − m0 and ρ(l) = ρ(σ2t, σt+l2 ).

1 _{The derivation of the explicit autocorrelation function of a trinomial MSM model can be found in Lee (2007),}

(19)

CHAPTER 3. MODEL 16

3.1.1 Maximum Likelihood Estimation

Calvet and Fisher (2004) describe how the parameters of the MSM model can be estimated by maximum likelihood (ML). This subsection describes the recursive procedure that results into the likelihood-function that allows this estimation method.

If the ¯k volatility components are binomial random variables, each volatility component can have either value m0 or 2 − m0. Since the volatility state equals the product of the ¯k volatility

components, the model implies d = 2k¯ possible volatility states. These states are defined as m1, ..., md ∈ R¯k

+. Because the volatility components are drawn from a discrete distribution, a

closed form likelihood function can be derived. The assumption of standard normal error terms and equation (3.1) lead to the density function of log-returns, conditional on volatility state mi_.

This is given in equation (3.6).

f (rt|Mt= mj) = 1 √ 2πσ2_mje −1 2 r2_t σ2mj. (3.6)

In order to derive the likelihood function, conditional probabilities of taking on a volatility state should be derived recursively because volatility state mj cannot be observed directly. Let Πj_t|t be this probability, which is defined in equation (3.7).

Πj_t|t= P[Mt= mj|r1, ..., rt]. (3.7)

The dynamics of the volatility components can be described by transition matrix A = (ai,j)1≤i,j≥d ∈ Rd+, where ai,j = P[Mt= mj|Mt−1 = mi] are the transition matrix’ elements. If

probabilities Πj_t|t are stacked into a row vector, Πt|t, an expression for the probability that the

volatility state takes value mj at time t, given past returns can be derived. Probability Πt|t−1

follows from the product between Πt−1|t−1 and transition matrix A, and is defined as the jth

element of the row vector in equation (3.8) (Hamilton, 1994).

Πt|t−1= Πt−1|t−1A (3.8)

The joint density distribution of rtand Mt conditional on volatility state mj is then defined by

the jth element of Hadamard product η(rt) ◦ Πt|t−1, where η(rt) represents a row vector with

the density function in equation (3.6) as its elements. Hamilton (1994) states that the density of returns, conditioned on past returns, equals the sum of the elements in the vector η(rt) ◦ Πt|t−1.

With these probabilities defined, conditional probability vector Πt|t follows from Bayes’ rule2

and is given in the equation below.

Πt|t=

η(rt) ◦ Πt|t−1

[η(rt) ◦ Πt|t−1]ι0

, (3.9)

where ι = (1, ..., 1) ∈ Rd. The sample log-likelihood function of returns now follows from the fact that the joint density of returns on time t is given by inner product η(rt) · Πt|t−1 (which

2

Bayes’ rule is given by P[A|B] = P[A∩B]

P[B] such that P[Mt= m j_|r 1, ..., rt] = P[Mt=m j_|r 1,...,rt−1]f (rt|Mt=mj) [η(rt)◦Πt|t−1]ι0

(20)

CHAPTER 3. MODEL 17

equals the denominator in equation (3.9)). This log-likelihood function is presented in equation (3.10). ln L(r1, ..., rT; ψ) = T X t=1 ln[η(rt) · Πt|t−1]. (3.10)

Calvet and Fisher (2004) state that the ML estimators that follow are consistent for a fixed ¯k and asymptotically efficient as T → ∞.

3.1.2 Model Selection

The procedure of selecting the best fitting MSM model means selecting the optimal number of volatility components ¯k. It would be too parsimonious to select the optimal number of volatility components based on the model that gives the highest log-likelihood. This thesis therefore applies a selection procedure that is two-fold. The first selection criterium follows Calvet and Fisher (2004) and is based on the Vuong test (Vuong, 1989). Because the MSM models are of a non-nested nature, a regular LR test can not be used. Instead, the Vuong test is applied for the selection of each sample’s optimal model. For each index, the Vuong test compares the log-likelihood differences across the models and tests whether the differences are significantly different from zero. If MSM(¯kM L) represents the model that gives the highest

log-likelihood and MSM(¯kX) every other model, the null-hypothesis that both models fit the

empirical data equally well is tested by the Vuong test of which the related test statistic is presented in equation (3.11). Vi(¯kX, ¯kM L) = 1 √ Ti PTi t=1[li,t(¯kX) − li,t(¯kM L)] V[li,t(¯kX) − li,t(¯kM L)] ∼ N (0, 1), (3.11)

where li,t(¯kX) and li,t(¯kM L) represent the individual log-likelihoods of index i at time t for

models MSM(¯kX) and MSM(¯kM L) respectively. The variance of the log-likelihood differences,

the denominator in equation (3.11), can be estimated consistently by its sample variance. By using the Vuong test, all models of which no evidence is found that they fit the data worse than model MSM(¯kM L) are selected. Then the second selection criterium is applied. It entails

selecting the model with the minimum number of volatility components that was left after the first selection criterium. This method is chosen because a model with a low number of volatility components simplifies the estimation procedure.

3.2 The Fractionally Integrated Exponential GARCH model

When evaluating the performance of the MSM model with respect to its ability to replicate stylized facts, it makes sense to compare it with a model that is found to perform well in the same area. A logical benchmark for a comparison is a GARCH-type model, since those have already been widely explored. The FIEGARCH model by Bollerslev and Mikkelsen (1996) is chosen as a benchmark in this thesis because of its good performance in replicating properties like volatility clustering, long memory and the leverage effect. This model is assessed with both

(21)

CHAPTER 3. MODEL 18

normally and heavy-tailed distributed standard errors because applying the latter to GARCH models tend to improve the ability to reproduce distributional characteristics of asset returns.

Bollerslev and Mikkelsen (1996) proposed the FIEGARCH model as an asymmetric alter-native of the FIGARCH model by Baillie et al. (1996). Recall from the previous chapter that the FIGARCH model allows the effect of shocks to die out in a slow hyperbolical way due to the fractionally integrated volatility process. The FIEGARCH model has the same property but also includes the asymmetric specification of the EGARCH model. Before presenting an expression for the FIGARCH(1,1) model, it is necessary to rewrite the GARCH(1,1) model in equation (2.5) into another form, presented in equation (3.12).

φ(L)(1 − L)2_t = ω + (1 − βL)vt, (3.12)

where L is the lag operator, φ(L) = (1−αL−βL)(1−L)−1, vt= 2t−σt2and the same restrictions

hold as for the model in equation (2.5). If restriction α + β = 1 is imposed to the persistence parameter, equation (3.12) represents the IGARCH(1,1) model instead. Modeling volatility with the IGARCH(1,1) model implies highly persistent shocks which remain important for all future returns. As mentioned in the previous chapter, empirical data shows that shocks do not remain important forever but their importance rather decays in a slow hyperbolical manner. In order to replicate this long memory feature, fractionally integrated volatility models have been introduced, of which the FIGARCH(1, d, 1) model represented in equation (3.13) is the most general fractionally integrated GARCH-type model.

φ(L)(1 − L)d2_t = ω + (1 − βL)vt, . (3.13)

Now, φ(L) = 1 − φL and the first difference operator (1 − L) of the IGARCH model is replaced by, (1−L)d, the fractional difference operator3. It is argued that applying these fractional orders of integration create more flexibility in modeling long-term dependency in conditional variance (Baillie et al., 1996; Bollerslev and Mikkelsen, 1996). Integrated processes of order zero and one, corresponding to the GARCH and IGARCH model, may be too restrictive because the prior does not model persistence at all and the latter models shocks with an infinite persistence. By introducing fractional order d ∈ (0, 1), identified as the decay rate of a shock to the conditional variance, the hyperbolic decay of shocks is found to be replicated sufficiently well.

Although the empirical finding of long-term dependency has been dealt with quite well by the introduction of the FIGARCH model, Bollerslev and Mikkelsen (1996) point out that there is no room for the leverage effect in this model. Bollerslev and Mikkelsen (1996) therefore introduce an asymmetric version of the FIGARCH model, the FIEGARCH model. The FIEGARCH model extends the EGARCH model that was developed by Nelson (1991) into a fractional integrated version. The FIEGARCH(1, d, 1) model now follows in equation (3.14) from the EGARCH(1,1)

3

Fractional difference operator (1 − L)d is defined by its Maclaurin series expansion, (1 − L)d = P∞

k=0

Γ(k−d) Γ(k+1)Γ(−d) =

P∞

k=0δkLk = δ(L), where Γ(·) denotes the gamma function and δ0 = 1 and

(22)

CHAPTER 3. MODEL 19

model in equation (2.8) and the FIGARCH(1, d, 1) in equation (3.13).

ln(σ_t2) = ω + 1 + αL

φ(L)(1 − L)dg(zt−1). (3.14)

Equation (3.14) demonstrates that the FIEGARCH(1, d, 1) results into EGARCH(1,1) for d = 0 and into an integrated EGARCH(1,1) for d = 1. The FIEGARCH(1, d, 1) model is covariance-stationary and invertible for d ∈ (−0.5, 0.5). Also, it is worth noticing that the model does not need to satisfy any non-negativity constraints in order to be well specified. Because of its analogy with the fractionally integrated autoregressive moving average (ARFIMA) model, the autocorrelation function of squared returns, implied by the FIEGARCH model, satisfies:

ρ(l) ∼ cl2d−1 as l → ∞, (3.15)

where l is the number lags, d is the decay rate and c is a constant. The specification in equation (3.15) shows that the autocorrelation function of the volatility proccess given by a FIEGARCH model decays hyperbolically and therefore takes into account the long memory effect.

3.2.1 Quasi Maximum Likelihood Estimation

Bollerslev and Mikkelsen (1996) describe that the parameters of the FIEGARCH model can be estimated by means of quasi maximum likelihood (QML) estimation. They define the log-likelihood function of the FIEGARCH models with normally distributed error terms as the one presented in equation (3.16). ln L(r1, ..., rT; ψN) = − 1 2T ln(2π) − 1 2 T X t=1 [ln(σ_t2) + z2_t], (3.16)

where ψN represents the parameter vector of the model with normally distributed error terms.

The log-likelihood function of the FIEGARCH model with student−t distributed error terms and corresponding parametervector ψtis defined by:

ln L(r1, ..., rT; ψt) = T ln Γ ν + 1 2 − ln Γν 2 −1 2ln (π (ν − 2)) −1 2 T X t=1 ln σ_t2+ (1 + ν) ln 1 + z 2 t ν − 2 . (3.17)

Maximizing the functions in equations (3.16) and (3.17) over their parameter vectors gives asymptotically valid estimation results. Bollerslev and Mikkelsen (1996) point out that the nor-mality assumption regarding the standardized residuals of the model with normally distributed error terms is probably violated. They therefore state that a robust covariance matrix can be estimated asymptotically consistently with matrix A( ˆψ_N)−1B( ˆψN)A( ˆψN)−1, where A( ˆψN)

and B( ˆψN) represent the hessian matrix and the outer product of the gradient respectively,

(23)

Chapter 4

Data

The empirical study in thesis includes daily index data from both emerging and developed countries. The total number of countries that are examined is twenty-three, sixteen emerging countries and seven developed. Developed markets are included to make the comparison between the time-varying volatility behavior of emerging markets and developed markets. Finding a single definition for emerging markets is hard, some countries are classified as an emerging market by one source whereas another source classifies the same countries as still developing. Following previous studies, for example Hull and McGroarty (2014), the emerging markets that are considered in this thesis are selected from the list of countries that were classified as an emerging country in the FTSE Country Classification dated September 2015. The distinction between developed and emerging markets is not the only difference that is important to assess. Differences may arise when examining countries from distinct regions. Development in for example Europe or Africa could have different effects on the volatility of its financial markets, it is therefore important to study a diversified list of countries. In order to have a diversified list of emerging markets, four countries from four different regions (Africa, Asia, Europe and Latin America) are selected. The four African countries are Egypt, Morocco, Nigeria and South Africa. China, India, Indonesia and Malaysia are selected from the list of emerging Asian countries. Hungary, Poland, Turkey and Russia are the studied European countries and Brazil, Chile, Colombia and Mexico are selected from the Latin American region. The selected developed countries are the countries that form the present Group of 7 (G7), being Canada, France, Germany, Italy, Japan, the United Kingdom and the United States. These countries are considered to be the seven major advanced economies by the International Monetary Fund. The asset class that is studied in this thesis is equity. More specifically, it studies one of each country’s main indices. The index data consists of the official daily closing prices of each index1_{. The sample period has a length of nearly eleven years, starting at the first of}

January 20052 up until the twenty-first of October 2015. All the countries that are classified as

1 _{The index data is obtained from Bloomberg. The list of indices can be found in Appendix A.}

2 _{The first of January 2005 was a Saturday so the first observation date is the first following trading day. The}

first observation dates are reported in Appendix A.

(24)

CHAPTER 4. DATA 21

emerging are assumed to have been an emerging market for the entire sample period. Because the classification was given in the year 2015 and the status of the countries changes over time, it must be noted that this assumption might not be fully justified. Nevertheless, this assumption is made because studying the historical classification of countries has shown that most countries are not classified as an emerging country for over ten years. Avoiding the assumption would result in too few observations.

4.1 Descriptive Statistics

Table 4.1 presents descriptive statistics for each country’s index returns. The first column gives the number of observations. The other columns present the unconditional mean, unconditional standard deviation, median, minimum, maximum, unconditional skewness and unconditional kurtosis of each index’ continuously compounded returns in percentages. The continuously compounded index returns, ri,t for country i at time t are defined as:

ri,t = pi,t− pi,t−1, (4.1)

where

pi,t = ln(Pi,t) (4.2)

equals the log-price of country i’s index at time t. Equations (4.3), (4.4), (4.5) and (4.6) give the definition of the sample mean, standard deviation, skewness and kurtosis respectively.

mi = 1 Ti Ti X t=1 ri,t (4.3) si = v u u t 1 Ti− 1 Ti X t=1 (ri,t− mi)2 (4.4) gi= 1 Ti PTi t=1(ri,t− mi)3 s3_i (4.5) ki= 1 Ti PTi t=1(ri,t− mi)4 s4_i (4.6)

The descriptive statistics show that the index returns have similar characteristics. The means of the daily continuously compounded returns in percentages are close to zero. Most index returns have significant negative unconditional skewness and an unconditional kurtosis that is significantly different from three, implying excess kurtosis, or fat tails. Also, most re-turns show a high unconditional standard deviation and a big spread between the minimum and maximum values. These statistics imply that the unconditional variability of the asset returns is relatively high. Chapter two discussed that one of the stylized facts is the non-normality of the unconditional distribution of asset returns. It was stated that the returns of the emerging finan-cial markets that were studied by Aggarwal et al. (1999) were positively skewed, whereas Table 4.1 shows negative skewness for most index returns. Although the results do not match, both lead to the conclusion that the skewness is significantly different from zero, which is an indica-tion that the distribuindica-tion of the returns might be non-normal. Together with an uncondiindica-tional

(25)

CHAPTER 4. DATA 22

kurtosis that significantly differs from three for all index returns, there is enough evidence to as-sume that none of the studied returns is normally distributed. Appendix B presents histograms that confirm that the index returns are not normally distributed. Comparing the density func-tions of a normal distribution with its mean and variance equal to the ones reported in Table 4.1 with the histograms clearly shows that the distributions deviate from normality.

Table 4.1: Descriptive statistics daily log-returns

Countries Obs. Mean S.D. Median Min. Max. Skewness Kurtosis

Africa Egypt 2625 0.040 1.776 0.129 -17.992 7.314 -0.924∗∗∗ 10.177∗∗∗ Morocco 2630 0.022 0.831 0.020 -5.462 4.805 -0.377∗∗∗ _8.620∗∗∗ Nigeria 2598 0.009 1.133 0.000 -9.475 11.758 0.285∗∗∗ 14.594∗∗∗ South Africa 2701 0.053 1.378 0.114 -7.959 7.707 -0.141∗∗∗ _6.635∗∗∗ Asia China 2619 0.045 1.891 0.093 -9.752 8.949 -0.439∗∗∗ _6.230∗∗∗ India 2684 0.053 1.536 0.097 -11.604 15.990 0.065 11.237∗∗∗ Indonesia 2631 0.049 1.672 0.130 -12.629 9.802 -0.513∗∗∗ 9.626∗∗∗ Malaysia 2663 0.024 0.760 0.051 -9.979 4.259 -1.145∗∗∗ 17.746∗∗∗ Europe Hungary 2700 0.014 1.656 0.031 -12.649 13.178 -0.092 9.513∗∗∗ Poland 2707 0.003 1.511 0.028 -8.443 8.155 -0.311∗∗∗ 6.260∗∗∗ Russia 2677 0.013 2.260 0.095 -21.199 20.204 -0.401∗∗∗ _14.014∗∗∗ Turkey 2718 0.040 1.859 0.078 -10.902 12.725 -0.146∗∗∗ 6.090∗∗∗ South America Brazil 2671 0.023 1.785 0.049 -12.096 13.678 -0.016 8.560∗∗∗ Chile 2694 0.028 1.044 0.057 -7.173 11.803 0.010 13.722∗∗∗ Colombia 2632 0.037 1.383 0.058 -13.254 18.126 -0.096∗ 24.523∗∗∗ Mexico 2719 0.046 1.300 0.082 -7.266 10.441 0.109∗ 9.244∗∗∗ G7 Canada 2712 0.017 1.225 0.088 -10.327 9.826 -0.663∗∗∗ 14.015∗∗∗ France 2766 0.008 1.446 0.046 -9.472 10.955 0.025 9.355∗∗∗ Germany 2748 0.032 1.397 0.106 -7.433 10.797 0.014 9.321∗∗∗ Italy 2743 -0.012 1.603 0.072 -8.599 10.874 -0.081 7.506∗∗∗ Japan 2649 0.018 1.565 0.064 -12.111 13.235 -0.533∗∗∗ 11.124∗∗∗ United Kingdom 2730 0.010 1.208 0.053 -9.266 9.384 -0.160∗∗∗ _11.215∗∗∗ United States 2719 0.019 1.268 0.073 -9.470 10.957 -0.331∗∗∗ 13.961∗∗∗ This table presents the descriptive statistics of 23 nation’s daily continuously compounded index returns in percentages. These statistics are based on the entire sample (1 January 2005 up until 21 October 2015). Obs. gives the number of trad-ing days of each country in the sample period; Mean, S.D., Median, Min., Max., Skewness and Kurtosis are respectively the mean, standard deviation, median, minimum, maximum, skewness and kurtosis of the continuously compounded daily returns. The significance levels for the unconditional skewness and kurtosis are∗∗∗,∗∗and∗for respectively the 1%, 2% and 5% significance level.

4.2 Preliminary Analysis Stylized Facts

The unconditional statistics in Table 4.1 and the histograms in Appendix B give a first indica-tion of the non-normality of the asset return distribuindica-tion. To formally test the null-hypothesis of a normal distribution, the Jarque-Bera test statistic can be applied. The Jarque-Bera test

(26)

CHAPTER 4. DATA 23

examines whether the unconditional data’s skewness and kurtosis match that of a normal dis-tribution. It tests whether the skewness differs significantly from zero and the kurtosis from three. The Jarque-Bera test statistic for country i is defined in equation (4.7)

J Bi= Ti 6 (g 2 i + (ki− 3)2 4 ) ∼ χ 2 α(2), (4.7)

where gi and ki are defined as the sample skewness and sample kurtosis in equations (4.5) and

(4.6) respectively, Ti is the number of observations and α is the significance level. The first

column of Table 4.2 reports the p-value of the Jarque-Bera test for each index at the 1% signifi-cance level. Since all p-values equal zero, the Jarque-Bera test provides enough formal evidence to reject the null-hypthesis of a normal distribution for all indices. The studied index returns for both market types meet the expectation of an asymmetric and leptokurtic distribution.

Table 4.2: Test Statistics Sample Data

Countries Jarque-Bera Autocorrelation Coefficient Hi ˆβ2,i

(p-value) ρ(1) ρ(50) Africa Egypt 0.0000 0.238∗∗∗ _0.012∗∗∗ _0.568∗∗∗ _-0.508∗∗∗ Morocco 0.0000 0.347∗∗∗ 0.115∗∗∗ 0.736∗∗∗ -0.223∗∗∗ Nigeria 0.0000 0.226∗∗∗ _0.041∗∗∗ _0.813∗∗∗ _-0.251 South Africa 0.0000 0.208∗∗∗ 0.182∗∗∗ 0.781∗∗∗ -0.149 Asia China 0.0000 0.166∗∗∗ 0.050∗∗∗ 0.951∗∗∗ -0.164 India 0.0000 0.140∗∗∗ 0.062∗∗∗ 0.841∗∗∗ -0.287 Indonesia 0.0000 0.217∗∗∗ 0.033∗∗∗ 0.737∗∗∗ -0.569∗∗∗ Malaysia 0.0000 0.122∗∗∗ 0.011∗∗∗ 0.753∗∗∗ -0.149∗∗∗ Europe Hungary 0.0000 0.338∗∗∗ 0.039∗∗∗ 0.729∗∗∗ -0.242 Poland 0.0000 0.115∗∗∗ _0.079∗∗∗ _0.804∗∗∗ _-0.131 Russia 0.0000 0.252∗∗∗ 0.046∗∗∗ 0.714∗∗∗ -0.634 Turkey 0.0000 0.109∗∗∗ _0.015∗∗∗ _0.800∗∗∗ _-0.380∗∗∗ South America Brazil 0.0000 0.174∗∗∗ _0.068∗∗∗ _0.790∗∗∗ _-0.492∗∗∗ Chile 0.0000 0.260∗∗∗ 0.010∗∗∗ 0.656∗∗∗ -0.722∗∗∗ Colombia 0.0000 0.339∗∗∗ _-0.008∗∗∗ _0.561∗∗∗ _-0.440 Mexico 0.0000 0.164∗∗∗ 0.081∗∗∗ 0.771∗∗∗ -0.391∗∗∗ G7 Canada 0.0000 0.410∗∗∗ 0.141∗∗∗ 0.727∗∗∗ -0.047 France 0.0000 0.197∗∗∗ 0.050∗∗∗ 0.768∗∗∗ -0.586∗∗∗ Germany 0.0000 0.170∗∗∗ _0.044∗∗∗ _0.790∗∗∗ _-0.556∗∗∗ Italy 0.0000 0.180∗∗∗ 0.059∗∗∗ 0.848∗∗∗ -0.545∗∗∗ Japan 0.0000 0.309∗∗∗ _0.049∗∗∗ _0.619∗∗∗ _-0.384 United Kingdom 0.0000 0.241∗∗∗ 0.056∗∗∗ 0.761∗∗∗ -0.407∗∗∗ United States 0.0000 0.213∗∗∗ _0.093∗∗∗ _0.765∗∗∗ _-0.481∗∗∗

The first column presents the p-values of the Jarque-Bera test statistic to test the normality of the sample distribution. The second and third columns show the sample autocorrelation coefficients for lags one and fifty. Column four gives the Hurst exponents, which is a measure for long memory in squared residuals. The last column gives a coefficient for the leverage effect. The significance levels are∗∗∗,∗∗ and∗for respectively the 1%, 2% and 5% significance level.

(27)

CHAPTER 4. DATA 24

Columns two and three of Table 4.2 give the sample’s squared residuals autocorrelation coefficients for lags one and fifty. The corresponding significance are based on the Ljung-Box test, which tests for autocorrelation in time series for a given number of lags. Testing squared residuals for autocorrelation indicates whether volatility clustering is present in the time series. If the null-hypothesis of no autocorrelation in squared residuals is rejected, the test implies conditional heteroskedasticity in the variance of asset returns and therefore volatility clustering. The Ljung-Box test statistic is defined by the following equation:

LBi(L) = Ti(Ti+ 2) L X l=1 ρi(l)2 Ti− l ∼ χ2_α(L), (4.8)

where L is the number of lags and ρi(l) is country i’s sample autocorrelation function of squared

residuals at lag l, defined by:

ρi(l) = 1 Ti−1

PTi−l

t=1 (e2i,t− ¯e2i)(e2i,t+l− ¯e2i)

ˆ V[e2_i]

, (4.9)

with residuals ei,t = ri,t− mi. The expressions ¯e2i and V[e2i] in equation (4.9) represent the mean

and sample variance of the squared residuals. The Ljung-Box test statistic has a chi-squared distribution. The degrees of freedom for the Ljung-Box test statistic are equal to the number of examined lags. The significance levels in Table (4.2) show that the null-hypothesis for no autocorrelation in squared residuals, for the given lags, is rejected at the 1% significance level. This result provides enough evidence to assume that all returns have time-varying volatility or conditional heteroskedastic variance and therefore exhibit volatility clustering.

In order to test whether there is long-term dependence in the squared residuals, an analysis of the Hurst exponent can be useful (Di Matteo et al., 2005; Niu and Wang, 2013; Hull and McGroarty, 2014). The Hurst exponent, introduced by Hurst (1951), gives a value to the decrease of the autocorrelation of time series as the lags increase. Determining the Hurst exponent can be done with various methods, this thesis applies the rescaled range (R/S) statistic. This method is found to have the ability to detect long-range dependency in non-Gaussian data like that of asset returns. Lo (1991) proposed a modification to the original R/S statistic by Mandelbrot and Wallis (1969) because he stated that the original statistic has a shortcoming with respect to its sensitivity to short-range dependency. The Hurst exponent follows from its relation with the R/S statistic, as defined in equation (4.10).

(R/S)i,ni = Cn

Hi

i , (4.10)

where (R/S)i,ni defines the R/S statistic for time frame ni and index i, Hi equals the Hurst

exponent of the squared residuals of index i and C is a constant. The original R/S statistic is defined as the fraction between the range and the standard deviation, Lo (1991) corrects for short-term dependency by adding weighted autocovariances to the denominator. For each set of squared residual, the time frames are defined as ni ∈ {2, .., Ti}. Each time frame ni

is formed by a sub series e2_i,1, ..., e2_i,n_i with mean ¯e2_i(ni) = _n1_i Pn_j=1i e2i,j and standard deviation

si(ni) = q ˆ V[e2i(ni)] = q 1 ni Pni j=1(e2i,j− ¯e2i(ni))2.

Emerging market volatility : a Markov-Switching Multifractal study

Master’s Thesis