Contagion effect between financial markets : measuring the dependence structure by using GAS copulas.

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Faculty of Economics and Business

MSc Thesis Financial Econometrics

Contagion effect between financial markets: Measuring the

dependence structure by using GAS copulas

Student: Eric Lau

Student number: 0182516

Supervisor: Dr. N.P.A. van Giersbergen

2nd reader: Prof. dr. C.G.H. Diks

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Chapter 1 Introduction

Over the past ten years, investors have experienced a significant increase of interdependency between financial markets such as equities, commodities, interest rates and exchange rates. High level of interdependency usually indicates that there is a common source of risk for asset prices. An important issue in asset prices is the increase of interdependency between financial markets in time of crisis. When the economy suffers from high macro uncertainty, the asset prices become unstable and are usually largely driven by changes in the macroeconomic outlook. Studying the dependency between financial markets is a very important issue in the financial literature. In addition, gaining a proper understanding in the dependence structure of financial assets in the unstable market environment is crucial for portfolio management in terms of finding investment opportunities and improving risk management. This motivates the intention of this study to analyze the effect of changes in stock market on the oil, gold and currency markets during crises. In the financial literature this spillover effect from one market to another is referred to as financial contagion effect. Despite the extensive empirical studies on the financial contagion effect, the term 'contagion effect' remains imprecise as there is no exact definition. In the recent literature, the term contagion effect is supposed to describe unexpected events in which a financial crisis in one market brings over to a crisis in another, a process observed through downside comovements in stock prices, exchange rates, sovereign spreads and capital flows. Hence, the financial contagion effect refers to the spreading of a series of negative price shocks that can trigger financial crises. Those negative price shocks are usually due to investors' irrational behavior and not related to or cannot be explained by economic fundamentals. A very popular and widely used definition of contagion effect is from Forbes and Rigobon (2002), who defined the contagion effect as a significant increase in cross-market linkages after a shock. According to Dornbusch et al. (2000), such an increase in cross-market linkages may not be completely caused by irrational behavior on the part of investors. If one country is hit by a shock and becomes unstable, investors can be forced to withdraw funds from other countries due to liquidity constraints. The existence of financial contagion effect has been widely examined and discussed in the financial literature. However, there is no consensus on this effect.

In this study, I will analyze the contagion effect between the stock markets of U.S. and Hong Kong, and the contagion effect of each stock market on the oil, gold and U.S. dollar markets by estimating copulas and modeling the dependence measure between

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these markets. Investors in Hong Kong tend to consider the risk from the U.S. stock market when making investment decision, mainly because of the fact that the Hong Kong stock market has been heavily affected by what happened in the U.S., especially in turbulent times (He et al. 2009). Hence we may assume the existence of a certain level of interaction between the U.S. and Hong Kong stock markets. Other important markets that are strongly linked with the stock markets are the crude oil, gold and currency markets. In recent years it has become widely accepted that the oil and gold prices drive the stock market; changes in the oil and gold prices are often considered as an important factor for understanding stock price movements. In case the returns on stocks do not sufficiently compensate for risk, then the demand for alternative investments like oil and gold will increase, with the result that oil and gold are used as a hedge against the risk of stocks. Baur and Lucey (2006) investigated the behavior of gold and stock markets and come to the conclusion that gold is a hedge against the stock market. The hedging strategy that uses one asset to hedge against the risk of the other can only be effective when prices of these assets are negatively related. The linkage between the stock market and commodities which include oil and gold has been widely examined. Kling (1985) has found evidence that crude oil price increases are associated with stock market declines. Jones and Kaul (1996) indicate the existence of a stable negative relationship between oil price changes and aggregate stock returns. Moreover, Moore (1990) has found empirical evidence on a negative correlation between gold price and the stock markets. All these evidences imply a negative linkage of the stock market with the oil and gold prices. Lastly, globalized financial markets have made it easier for investors from one country to invest in the stock markets of the other countries. If the stock market in one country outperforms the stock market in another country, investors will most likely withdraw their funds from the country with a weaker stock market to the country with a stronger stock market, which will drive up the value of the currency for the country with the better stock performance. This explains the positive relation between the U.S. stock market and the U.S. dollar. The dependencies between financial markets and the market prices provide us with a large amount of information which is essential in making investment decisions. Therefore changes in the dependencies between these markets often provide valuable clues to the type of environment the U.S. stock market is a part of.

This study only focuses on the dependence structure between the above-mentioned markets during the crises and will not examine the mutually causal relationship. The contagion effect is tested within pairs of markets separately, due to the form of bivariate copulas that are applied for estimation. The definition of contagion effect by Forbes and Rigobon (2002) is applied in this study. Since I use the same definition of contagion effect by Forbes and Rigobon (2002), the dependence parameter of the copula is used to

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analyze whether it has a significant increase after a crisis, implying the existence of higher interdependence between markets. This requires the modeling of the dynamics of the copula dependence parameter. For this purpose, I will apply the Generalized Autoregressive Score model (GAS), which is proposed by Creal et al. (2012). This study examines the contagion effect in three crises: the Asian crisis in 1997, the dotcom bubble in 2000 and the financial crisis in 2008. During the Asian crisis in 1997, the contagion effect is analyzed by measuring the dependence structure between the Hong Kong stock market and other markets which include the U.S. stock market, oil and gold markets. In order to examine the contagion effect during the dotcom bubble in 2000 and the financial crisis in 2008, the dependence is measured between the U.S. stock market and other markets, which include the Hong Kong stock market, oil, gold and U.S. dollar. The main purpose of this study is to give an answer on the research question which is defined as follows: Is there evidence of a contagion effect during the economic crises in 1997, 2000 and 2008? Furthermore, this study will contribute to an understanding of the contagion effect, as there exist almost no empirical studies that have aimed at analyzing the contagion effect using the GAS specification of copulas.

This study is organized as follows. Chapter 2 provides the literature review regarding the empirical results of the contagion effect. Chapter 3 sets out the theoretical background of copula theory and methodology. Chapter 4 presents the data and estimation methods. Chapter 5 describes the empirical results and their implications are discussed. Finally, chapter 6 provides the conclusion of this study.

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Chapter 2 Literature review

A common approach to investigate the contagion effect is by estimating and modeling the dependence measure between markets. The majority of the early studies on the dependence of asset returns is based on analyzing the change in the correlation

coefficient (Longin and Solnik 1995)1_{. King and Wadhwani (1990) have analyzed the}

correlation of stock returns between London and New York stock markets during the crash of 1987. Their finding of an increase in the correlation between markets just after the crash is evidence of the contagion effect. Calvo and Reinhart (1996) study the comovements of stock indices during the Mexican crisis in 1994. They find an increase in correlation among returns in the Latin American stock markets during the unstable period of market that follows from the Mexican devaluation in 1994. On the other hand, the correlations between several Asian countries which have been mostly positive before the crisis become negative after the crisis. They point out that these findings would suggest that the contagion effect in emerging markets may be regional rather than global. However, the widely applied correlation coefficient has its own limitations. The measure depends mainly on the linearity of the relation and is not capable of describing the dependence between the variables when their joint distribution is not elliptical. In practice the returns of asset prices do not have an elliptical multivariate distribution, implying that the correlation measure is inadequate. Furthermore, the correlation coefficient does not consider conditional heteroskedasticity (Boyer et al 1999). Forbes and Rigobon (2002) show that unadjusted correlation coefficients are conditional on market volatility; therefore the test for correlation change before and after the crisis is biased due to specific volatility dynamics of returns. The authors uses the correlation coefficients that are adjusted for the bias due to heteroskedasticity and conclude there is no empirical evidence of contagion effect during the 1987 U.S. market crash, the 1994 Mexican devaluation and the 1997 Asian crisis.

In recent studies, financial contagion has been analyzed by using a methodology that goes beyond the basic correlation breakdown analysis. Several authors consider non-linear models to analyze the contagion effect. Hartman et al. (2000) and Bae et al. (2003) use approaches based on extreme value theory, assuming the existence of asymptotic dependence structure. Relying on extreme value theory, estimates for dependence between relatively large realizations of each variable can be derived. However, Poon et al. (2004) use a multivariate extreme value theory technique,

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describing a multivariate framework whereby the extreme value dependence structures

can be measured, and show that the international stock market returns2_{tend to be}

asymptotically independent. Additionally, they point out that the models based on the assumption of asymptotic dependence will give incorrect measure of the joint occurrence probability.

Copulas are an alternative and relatively new approach useful to describe the relation between the returns of asset prices. It is a flexible tool for constructing joint distributions. Using a copula to construct a joint distribution can yield a functional form that is usually more capable of capturing the dependence structure of financial asset returns than when we assume an elliptical joint distribution. Another advantage of copula is its capability in measuring asymmetric dependence between asset returns, which is an important aspect of studying the contagion effect during crises. Due to these properties, copulas have become very widely applied in finance area. In the early stage, most of the application of copulas is to model the dependence parameter unconditionally, assuming it is constant over time. Patton (2001) is the first who applied copulas for modeling time-varying dependence parameter by extending the Sklar's

theorem to the conditional case3_{. In his paper, he assumes that the dynamics of the}

copula parameter can be described by an ARMA-structured updating equation which contains an autoregressive term of copula parameter and a forcing variable.

Wen et al. (2012) follow the approach which is introduced by Patton (2001) to model the time-varying copula dependence parameter for testing the contagion effect. Using this method they measure the dependence structure between stock markets in U.S. and China and crude oil markets. As a result, they find evidence for a significant increase of dependence between stock market and crude oil market during the recent

financial crisis4_{, which supports the existence of the contagion effect in the sense of}

Forbes and Rigobon's (2002) definition. Tsui and Zhang (2010) examine the relationship between five Asian currencies during the Asian crisis in 1997, which include the Singapore Dollar, Japanese Yen, South Korean Won, Thai Baht and Indonesian Rupiah. The authors apply time-varying copulas as Patton proposed to model the possible structural breaks. They find a significant increase in the dependence after the crisis for most of the pairs of the five currencies. Rodriguez (2007) analyzes whether financial crises can be described as periods of change in the dependence structure between markets by modeling the dependence structure as a mixture of copulas. He lets the parameters vary over time according to a Markov switching model.

2_{Poon et al. (2004) use returns on five major stock market indices: S&P500, FTSE 100, DAX 30, CAC 40}

and Nikkei 225 from 1968 to 2001.

3_{See Chapter 3.1 for details}

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Using daily returns of stock indices from East Asian countries during the Asian crisis and from Latin-American countries during the Mexican crisis, the author finds evidence of changing dependence structure during the financial crisis. Furthermore, by analyzing the tail dependence in times of financial turmoil, Rodriguez (2007) points out that structural breaks in tail dependence are a potentially important dimension of the contagion phenomenon.

Since the introduction of Patton's time-varying copula, the methodology has been widely applied and extended by researchers. Regarding the extension of copulas Creal et al. (2012) have proposed a new class of dynamic copula model which is referred to as the Generalized Autoregressive Score (GAS) model. In this study, I will extend the research of Wen et al. (2012) by analyzing the contagion effect between the stock, oil, gold and currency markets using the GAS model for modeling the time-varying dependence parameter of copula. The GAS model provides a general framework for modeling time variation in a parametric model. Creal et al. (2012) use this approach to derive the GAS specification for the time-varying parameter which has a different dynamic structure compared to Patton's (2001) updating equation. The GAS updating mechanism for the time-varying copula parameter can be given by the autoregressive equation in which the parameter is updated using the scaled score vector. The use of the score vector for updating the copula parameter has a number of advantages. It defines the steepest directional derivative for improving the fit of model in terms of the likelihood at time given the current value of copula parameter, providing the natural direction for updating the copula parameter. In contrast to most of the other observation driven approaches in the literature, the score vector depends on the complete density and not only on the first and second order moments of the observations. Furthermore, the dynamic GAS copula model allows for flexibility in the choice for the way how the score is scaled for updating the copula parameter. Creal et al. (2012) carry out a simulation experiment on generated data from the symmetrized Clayton copula in order to compare the Patton's (2001) approach with the GAS model. This experiment is interesting since the symmetrized Clayton copula has both the upper and lower tail dependence measures, and it is difficult for a model with a uniform driving mechanism to capture both the upper and lower tail dependence dynamics within a single model. As the result, the authors find that the Patton's updating equation is only capable of capturing some of the variation in the dependence coefficients, while the GAS model is more capable in capturing both the upper and lower tail dependence dynamics which suggests that the GAS model performs better than Patton's (2001) approach. This property motivates the choice of the GAS model for testing the contagion effect in this study.

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Chapter 3 Copula theory

3.1 Copula: bivariate case

This chapter gives a description of the copula function theory. Suppose there are two

random variables and with the continuous marginal distribution functions ,

and their joint distribution function . A copula connects a given joint

distribution function to its marginal distributions and describes the dependence between variables, regardless of their individual distributions. Using a copula, we can isolate the dependence structure between variables. A famous result of Sklar (1959) shows that the bivariate form of copulas can be summarized as follows. Given a set of marginal distributions, a distinct copula defines specific joint densities. Hence given any

joint distribution with marginal distributions and , there is a unique

copula function in the continuous form such that:

.

In other words, equation (1) specifies the copula function as a bivariate

distribution function with marginal distributions and . The values of the

marginal distributions are uniformly distributed. This is known as the probability

integral transforms which gives us . Using for ,

the copula function of can be written as:

It is important to note that regardless of what the marginal distributions and

are, the numbers and will be uniformly distributed on . Because the

marginal distribution contains all information on the individual variable and the

joint distribution contains both univariate and multivariate information, it is

obvious from equation that the dependence information between the variables

and must be contained in the copula function . In fact, by Sklar's theorem we can

link together any two marginal distributions and any dependence structure by copula which yields a valid joint distribution function. The advantage of a copula is that we can choose an appropriate copula to model the dependence between variables without considering which type of marginal distributions.

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For the application of copulas to time series modeling the Sklar's theorem can be extended to the conditional copula (Patton 2006). The conditional copula is defined as a joint distribution of variables which are distributed conditional on an

information set , for . Using this definition, Sklar's theorem can be

extended to the time series case and allows us to separate the joint conditional distribution into its conditional marginal distributions and the conditional copula. Therefore equation can be rewritten to:

,

where and is the conditional copula of and given . By definition

we have to use the same information set in each marginal distribution and in the

copula in order to yield a valid joint conditional distribution. In financial applications, it

is common that not all information in the information set is relevant for all

variables (Patton 2006). As described later in section 3.8, the conditional marginal distribution of returns usually depends on its lag terms, but not on the lags of other

variable. Therefore, we can define as the smallest subset of , such that

. According to this definition we can construct marginal distribution

model for using only , and use for the copula to obtain a valid conditional

joint distribution:

.

Differentiating the conditional joint distribution with respect to and gives us a

form in terms of the joint density function and the marginal density

functions and :

,

where the associated copula density is the function

regarded as a function of , for .

Copulas have certain properties that are very useful to study and measure dependence between random variables. First, copula parameters are invariant under strictly increasing transformations of the underlying variables. Secondly, popular measures of concordance between variables, like Kendall's tau and Spearman's rho, can

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be expressed as a function of copula parameters. Thirdly, the asymptotic tail

dependence5_{, which is of utmost importance for studying the contagion effect, can also}

be determined by using copulas. As already described above, by using copulas we can measure the dependence structure of variables. Different copula produces different joint distribution when applied to the same marginal distributions.

In this study, the Normal, Student-t, Clayton and Symmetrized-Joe Clayton copulas are used to analyze the contagion effect. A detailed description of these copulas will be given in section 3.4. The next section will explain why the concordance measure, Kendall's , will be used as the main dependence measure in this study. Then in section 3.3, the tail dependence will be defined which is another important measure to study the contagion effect.

3.2 Measuring dependency

The dependency between two variables describes their linkage with each other. Given the dependency, if we have information about one variable we can say something about the value of the other. Dependence between two variables can be strong, weak, linear and non-linear. The terms correlation and dependency are commonly used interchangeably by financial analysts while they have different meanings. When we measure the strength of dependence between two variables, the correlation coefficient is used in most cases. In fact, the correlation coefficient which quantifies the linear relationship between two variables, is just a special case of dependency. Each variable must follow an i.i.d. process and the joint distribution of the variables has to be elliptical (such as the multivariate normal and the multivariate t-distribution). However, in practice most financial asset returns have distributions which are asymmetric and hence do not satisfy these assumptions, implying that the correlation coefficient is not an adequate measure of dependence (Embrechts, et al. 2001). To cope with the problem of measuring dependence between variables when they have no elliptical joint distribution we need an alternative to the linear correlation coefficient. Kendall's , which is a rank

correlation and hence a concordance metric6_{is the alternative that can cope with this}

problem (Embrechts, et al. 2001). Therefore Kendall's is the main dependence parameter that is applied in this study. Kendall's is a measure of dependence between two variables, which is carried out on the ranks of the data. Similar to the correlation coefficient, Kendall's takes the values between -1 and 1; it is 0 for independent

5_{Explained in section 3.3.}

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variables and it equals 1 and -1 for respectively the comonotonic 7 _and

countermonotonic case. Another property of Kendall's is that, the measure of concordance is based on the copula, implying its being invariant to increasing transformations of its arguments, Kendall's can capture non-linear dependency which cannot be measured with the linear correlation coefficient (Rodriguez (2007)). This is an additional important property underpinning the choice of using Kendall's as the main measures of dependence, because Bae et al. (2003) and Rodriguez (2007) have pointed out that the linear correlation measure is inappropriate if the contagion effect is a non-linear phenomenon. During the crisis, if panic grips investors due to excessive negative stock returns causing them to ignore the fundamentals, we would expect large negative returns to be contagious in a way that small negative returns are not. Section 3.5 will explain the relation between the Kendall's and copulas.

Another important issue with regard to the study of the contagion effect is the change of dependency in markets due to the change in general market condition. There is empirical evidence that the dependency observed in the market at the time of extremely negative market conditions (for example during financial crisis) tends to differ structurally from the dependency during normal market conditions. In most cases, financial markets usually collapse together during a bear market and increase together during a bull market. But the rate of decrease due to negative market condition is more likely to be higher than the rate of increase during the bull market; see Longin and Solnik (2001) and Ang and Chen (2002). In order to account the dependence between extreme values, the tail dependence which measures the dependence for upper and lower tails of joint distributions (the dependence in the upper-right-quadrant tail and lower-left-quadrant tail of a bivariate distribution) must be considered. A definition of the tail dependence will be given in the next section.

7_{Two random variables X and Y are called comonotonic if there is another random variable Z such that}

both X and Y are increasing or decreasing transformations of Z. If X is a monotonic decreasing transformation of Z and Y is a monotonic increasing transformation of Z, then variables X and Y are countermonotonic.

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3.3 Tail dependence: other dependence concept

Using tail dependence, which is related to a measure due to Coles, et al. (1999), we can measure the concordance in the tails, the extreme values, of the joint distribution. In the bivariate case, the th lower tail dependence is defined by:

The lower tail dependence represents the conditional probability that one variable has in its lower tail, given that the other variable takes a value in its lower tail. Because the

tail dependence parameter is a conditional probability, . A larger value for this

parameter implies a higher tail dependence. Similarly the upper tail dependence represents the conditional probability that one variable has a value in its upper tail, given that the other variable takes a value in its upper tail. The upper tail dependence is defined by:

The interpretation of the upper tail dependence coefficient is similar to the lower tail

dependence as described above. If , a copula has symmetric tail dependence. But

if , it has asymmetric tail dependence. In this study I examine two copulas with

symmetric tail dependence, which are the Normal and Student-t copulas, and other two copulas with asymmetric tail dependence, which are the Clayton and Symmetrized-Joe Clayton copulas. To give the tail dependence in a more explicit expression, we can write:

Using this form, the lower and upper tail dependence coefficients can be expressed respectively as:

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3.4 Copula classes

This section describes the theoretical concepts by defining some classes of copulas used in this study. For each class this section provides the appropriate copula density function and discusses the properties of the copula. A standard approach of using copula is to estimate the dependence over the whole dataset, which is called the constant copula. Another approach allows the dependence parameter to be estimated over every point in time of the observation period, which is called the time-varying copula. Considering time-varying dependence is important, because the conditional correlation between financial asset returns is known to vary over time (Anderson, et al 2006). Therefore it is necessary to consider the conditional copula for time series modeling as described in section 3.1. Since the main purpose of this study is to investigate how the dependence between variables varies over time, I will apply the Generalized Autoregressive Score (GAS) model of Creal, et al. (2012) which allows for time variation in the conditional copula dependence parameter. A detailed description of this "GAS" approach for time-varying copulas will be given in section 3.6.

3.4.1 Normal copula

The Normal copula is an elliptical copula, which is a type of elliptically contoured distributions. An important advantage of elliptical copula is that we can specify different levels of correlation between variables and the disadvantage is that the elliptical copula is restricted to be symmetric. In the bivariate case, the Normal copula distribution is

_,

where is the bivariate standard normal distribution function, is the univariate standard normal distribution function and is the dependence parameter which varies between -1 and 1. When there is a perfectly positive correlation, when there is no correlation, and when there is a perfectly negative correlation. The Normal copula distribution can be written alternatively:

Differentiating (12) yields the bivariate Normal copula density,

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where _and _{are the quantiles of the marginal standard}

normal distribution. Because the correlation is the only parameter of the Normal copula, this class of copula is relatively simple to calibrate. In order to estimate how the dependence parameter varies over the observation period, the GAS(1,1) model is

applied. The Normal copula is symmetric, as , and has no tail

dependence: . It is important to note that the Normal copula is usually not

appropriate for modeling the dependence structure between financial returns as the dependence structure of financial returns is usually not symmetric; they become more related when they are negatively large than when they are positively large. In other words, when two prices decrease by large amounts their dependence becomes larger than when the prices increase by large amounts. Such asymmetric tail dependence cannot be captured by the Normal copula. In order to capture the asymmetric tail dependence, other classes of copulas need to be considered. These are the Clayton and SJC copulas that will be described in subsections 3.4.3 and 3.4.4.

3.4.2 Student-t copula

Another symmetric elliptical copula is the Student-t copula. The bivariate case of Student-t copula is defined as

the corresponding bivariate Student-t copula density is

where , _and _. _{is the bivariate}

Student-t distribution with the degree of freedom parameter , and _{is the}

quantile of the marginal Student-t distribution. When the number of degrees of freedom is large (approximately around 30), the copula converges to the Normal copula. But when the number of degrees of freedom is small the behavior of the Student-t copula differs from the Normal copula; the Student-t copula has a star like shape and it contains more points in the tails than the Normal copula, which makes Student-t copula to be more capable to account for heavy-tailed distributions. Therefore, the Student-t copula

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allows for joint extreme events. By definition, the Student-t copula has the same dependence parameter as the Normal copula. The Student-t copula has symmetric tail

dependence, with , where is the CDF of the

Student-t distribution with degree of freedom.

3.4.3 Clayton copula

The Clayton copula that is introduced by Clayton (1978) has the form

Differentiating yields the Clayton copula density function,

In the Clayton copula function, the dependence is measured by the parameter with . The dependence is perfect positive if , while implies independence. The Clayton copula is an asymmetric copula which exhibits greater dependence in the negative tail than in the positive tail of the joint distribution. This copula is appropriate if asymmetry is believed to be an important issue in the dependence structure of time series, whereby the symmetric Normal and Student-t copulas may be restrictive. Moreover, the Clayton copula has asymmetric tail dependence; it has positive lower tail dependence but the upper tail dependence is zero.

In case , the lower- and upper tail dependence are respectively _and

.

3.4.4 Symmetrized-Joe Clayton (SJC) copula

SJC copula is another copula that takes the tail dependence into account. Unlike the Clayton copula, SJC copula considers the measure of both the lower- and upper tail dependence. The SJC copula which has the lower- and upper tail dependence parameters

and is given by

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and , .

In addition, and . A special feature of SJC copula is that the

dependence parameters of the copula are themselves the tail dependence coefficients.

Hence, and The dependence structure is symmetric if , it is

asymmetric if .

3.5 Relation between copulas and Kendall's

Embrechts et al (2001) demonstrate that Kendall's has a direct relationship with a

bivariate copula function by the following equation:

The integral above is the expected value of the random variable , where and

are distributed with copula function . When the copula has only one parameter then equation (19) provides a way of determining the parameter using a sample estimate of the rank correlation. By using the equation (19), it shows that the relation between Kendall's and the correlation parameter of the bivariate Normal

and Student-t copulas is given by . For the Normal and Student-t copulas,

Kendall's varies between -1 and 1. Additionally, the relation between Kendall's and

the Clayton copula parameter can be expressed in , with . Using the

updating mechanism of GAS(1, 1) model, which will be explained in the next section, the dynamics of Kendall's can be determined from the copula parameter by these definitions.

3.6 Generalized Autoregressive Score model

In order to estimate how the dependence parameter varies over the observation period, the time-varying copula approach must be considered. In this study, I use the GAS specification of Creal, et al (2012) to model the evolution of the copula parameters. This approach specifies the copula parameter that evolves as a function of the lagged copula parameter and a forcing variable which is related to the standardized score of the

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likelihood copula density. The unique feature is that this approach uses the scaled score as an updating mechanism for the observation driven part of the model. In general, the

model for a time-varying parameter has the form

In this study, I use the specification by assuming that it is capable of

capturing the dynamics of the dependence parameter. Usually the modeled is the

transformed parameter rather than the direct dependence parameter of the copula. The transformation is necessary to handle the constrained range in which the copula parameters can vary. For example, the correlation parameter of the Normal and

Student-t copulas is forced to take values inside by the function

_{. For the (tail) dependence parameters of the SJC and Clayton}

copulas, the transformation of respectively _and

are essential to ensure the parameters always remain in their domain. The

specification for the time-varying copula parameter can be described as

follows

where the scaled score is the score of the log-likelihood copula density

scaled by the scaling matrix . Similar to Koopman,

et al (2012) and Patton (2012), the square root of the inverse of the information matrix

is chosen as the scaling matrix, hence we have _{. According to}

the specification the future value of the copula parameter is a function of a

constant , the current value and the scaled score of the copula log-likelihood

Creal, et al (2012) and Koopman, et al (2012) use the Normal copula to derive the specification for the time-varying correlation parameter. The explicit

expression of the updating equation of for Normal copula is

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With respect to the Clayton and SJC copulas, no explicit expressions are available for

the scaled score in the finance literature. The main problem is the difficulty to derive a

closed-form expression for the information matrix. Therefore it is necessary to use an alternative approach to compute the information matrix numerically. To compute the information matrix numerically, I use the same method as explained in the paper of

Creal, et al (2012)8_{. Suppose the information matrix is given by}

with which is the score of the log-likelihood of copula density as defined earlier.

Additionally, is a function that depends solely on the transformed parameter . To

compute the information matrix we need first to use the simulation approach to determine the expected outer product of gradients for a range of values of copula parameter. This involves constructing a grid of values of copula parameter and hence a

grid of values of transformed parameter for a positive integer . Then

we can determine the function value by calculating the expected outer product

of gradients numerically for each of the grid points . It is noteworthy that the

determined function value is the numerical approximation of the information

matrix. Values at intermediate points can be determined by cubic spline interpolation to make certain that the first and second order derivatives of the likelihood copula are continuous. Then we need to determine the initial value of transformed parameter

by using the initial value of copula parameter . In most cases we can apply

the constant copula parameter which is estimated for the whole observation period as

the initial value . It is worth noting that this approach does not always work, especially

for the complicated GAS(1, 1) SJC copula which is bivariate dimensional. This problem

will be described in subsection 5.2.1 in more detail. With the initial value the

associated information matrix can be computed which can be used to scale the

score of log-likelihood function _{The next step involves}

obtaining the new parameter value ,at , from the specified GAS updating

equation. Thereafter, the information matrix at is computed by

interpolation. This procedure is repeated for each . Lastly, the log-likelihood copula can

be calculated for the estimated , which is used for the maximum likelihood estimation

procedure. The same numerical procedure is also applied to Student-t copula. Since the SJC copula contains two parameters that account for lower and upper tail dependence, the GAS(1, 1) updating equation results in a bivariate vector autoregressive system

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which is the most complicated among all models considered in this thesis. The details of the GAS(1, 1) updating equation for the SJC copula are described below.

According to the dynamic copula approach, the SJC copula parameters and are

modeled by transformed parameters which ensure them to remain within the domain (0, 1) by:

and .

The transformed parameters and follow the bivariate driving mechanism of

GAS(1,1) specification:

where

diagonal matrices the scaled

score vector with

and a scaling matrix which is the square root of the inverse of the information matrix. It should be noted that in this study I do not use the full information matrix, but only its diagonal elements. Since after some experimentation I find out that by using only the diagonal elements of the information matrix the dependence parameters seem to be scaled more properly. Hence the scaling matrix becomes:

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The diagonal components of information matrix

and

are numerically approximated by the above-described

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3.7 Copula estimation method

The parameters of the GAS(1, 1) copulas are estimated by Maximum likelihood estimation (MLE). In this study, I use the approach that is called the inference functions for margins (IFM) method. This is a two-step approach, where the parameters of the marginal distribution of variables are estimated first. In the second step, the parameters of the copula are estimated using the estimated marginal distributions parameters from step 1 as given. This estimation method leads to consistent estimators and is considerably simpler and more transparent than full MLE, which involves estimating all parameters of copulas and marginal distributions at the same time. The calibration algorithm of IFM method can be described by rewriting equation as

,

where is the vector of the parameters of marginal distributions and is the

vector of GAS(1, 1) copula parameters. Notice that contains the dependence parameter of the copula, which is time-varying, and the parameters of the GAS(1, 1) updating equation. From equation the log-likelihood copula can be obtained

where is the number of data points in the observation period. According to the two-step approach the log-likelihood (28) can be maximized in two two-steps:

1. The first step of the inference function for marginals is to estimate the parameters of

each marginal distribution individually using MLE. This involves estimating by

solving

The applied model for specifying the marginal distributions is explained in section

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2. The second step involves taking the estimated parameter values from the first step

as given to calibrate the GAS(1, 1) copula parameters by solving

3.8 Marginal distribution

In order to apply the copula, the marginal distributions of the variables need to be estimated first. This requires defining marginal distributions that explain the variables

as much as possible. Using the price series the returns can be determined by

. The widely applied univariate

model is used to model the dynamics of the financial returns. The model can be described by the following equations:

and

where the univariate return series is decomposed into its conditional mean , its

autoregressive lag terms and an error term . Using the Bayesian Information

Criterion (BIC), the optimal lag term for the conditional mean equation AR(p) would be found up to order . The error term is defined by (32) to be a product of the square

root of the conditional variance and the homoskedastic error term (standardized

error term). The homoskedastic error term is assumed to follow the white noise process

with its distribution . The conditional variance is defined by (33) to have a

dynamics that contains a constant , the lag-terms of the error term and the conditional

variance of respectively and , and an additional term , where

is an indicator function which equals to one if A is true and otherwise equals to 0. The last term is an extension of GARCH model that accounts for possible asymmetric response of volatility to positive and negative shocks (leverage effect). This term is necessary for modeling conditional volatility, because it has been argued in the finance

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literature that a negative shock to financial returns is likely to cause a relatively larger increase in volatility than a positive shock of the same magnitude (Brooks 2002). For simplicity, I apply the to model the conditional variance. Using the estimated model the standardized residuals can be

estimated as

.

Another important issue is that the error term of a financial time series is usually

not normally distributed. For the financial time series, the most common deviations from the normal distribution are the excess kurtosis and skewness (Bastianin 2009). Therefore, I use the flexible skewed-t distribution of Hansen (1994) to model the homoskedastic error term. The advantage of using this distribution is its capability of modeling both kurtosis and skewness. In case the true error term has skew-t

distribution, where - , the skew-t density function is given by:

- (34)

The values of , and are defined as:

where is the kurtosis parameter and is the asymmetry parameter. The parameters are restricted to and By estimating the appropriate model we can construct the marginal probabilities by using the defined probability integral transform

of the return series

, for , with the residuals from the

conditional mean equation. Hence we can construct the marginal probabilities by calculating the cumulative skewed-t distribution of the standardized residuals. The determined probability integral transform of the return series can then be used as input variables of the copula for estimation of step 2. Patton (2006) has mentioned that the modeling of the conditional copula requires modeling the true marginal distribution for each variables. In case the marginal distributions are misspecified, then the probability integral transforms will not be uniformly distributed, and consequently any copula model will be misspecified. Therefore, the testing for misspecification of marginal distribution is very important in order to apply a copula. For this purpose, I perform the Kolmogorov-Smirnov (KS) test of the distribution specification for each variable. The

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value for the test is approximated by simulation. This needs first to calculate the KS-test statistic, , for the data. Then the calculation of KS-KS-test statistic is repeated for times, but now using simulated data consisting of the number, which is equal to the sample size, of random numbers uniformly distributed over (0, 1). Let

be the values of KS-test statistic for simulations. Subsequently, by the law of

large numbers, the fraction _{approximates the p-value = for the}

KS-test.

3.9 Choosing the best copula

An important issue of modeling copulas is to choose the copula that has the best performance for the set of data. In this study, the Rivers-Vuong test (Rivers and Vuong 2002) is applied to compare the relative performances of the copula models. The null and alternative hypotheses of the Rivers-Vuong test are:

versus (35)

and

,

with . If the null hypothesis holds then the t-statistic on the

difference between the sample averages of the log-likelihoods has the standard normal distribution:

₍₃₆₎

with for and is a consistent Newey-West (1987)

HAC estimator of

An alternative approach to choose the copula that provides the best fit for the data set is to use the Akaike information criterion (AIC) or the Bayesian information criterion (BIC). These criteria give penalty for the loss of degrees of freedom from adding additional parameters. The AIC is specified as

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where is the value of maximized log-likelihood of copula and is the number of parameters in the copula. The BIC is specified as

. (38)

The copula that yields the lowest value of AIC or BIC is supposed to be the most suitable model.

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Chapter 4 4.1 Crises

This study investigates the interdependency between the financial markets during the Asian crisis in 1997, dotcom bubble in 2000 and financial crisis in 2008. A short description of each crisis is given in this section.

The Asian financial crisis initiated in Thailand, on July 2, 1997, with the collapse of the Thai baht after the Thailand's central bank was forced to float the Thai baht due to the lack of foreign currency to keep its exchange rate fixed. This was the consequence when the central bank failed in protecting its currency from speculative attack. After this action of the central bank, the Thai baht has been devaluated. At the time, Thailand had a large foreign debt that rendered the country effectively bankrupt, which in turn triggered a financial collapse that quickly spread to other countries in the region. Hong Kong, having a well developed stock market in the Asian region, was also severely affected by the crisis. In October 1997, the Hong Kong dollar, which has been pegged to the U.S. dollar at the rate of 7.8, was under the pressure of speculators because the inflation rate of Hong Kong had been higher than the U.S.'s for years. The monetary authority employed more than 1 billion USD to protect the Hong Kong dollar. At the time the Hong Kong stock market had become increasingly volatile. The severest decrease of the Hang Seng index took place on October 17, 1997, with Hang Seng index noted at 13,601. Then till October 23, 1997 it dropped with 23.34% to 10,426. In the same period, between October 17 and 27, 1997 the S&P500 index dropped 7.11% from 944.16 to 876.99.

The dotcom bubble was a period of an information technology bubble covering roughly 1997-2000, during which the equity value of the U.S. stock market increased rapidly due to the optimistic expectation of future growth in the internet sector. The bubble was a period marked by a combination of founding new internet-based companies, increasing stock prices, and overconfidence of excessive profit forecasts. The bubble burst on March 10, 2000, when the NASDAQ composite index peaked at a closing of 5,048.62. In the same period, on March 24, 2000, the S&P500 index reached its top of 1527.46. Thereafter, the index dropped with 10.06% to 1,373.86 till May 5, 2000. After a temporary recovery to 1,520.77 on September 1, 2000 the S&P500 index entered into a bear market. In Hong Kong, the Hang Seng index peaked at a closing of 18,301.69 at a later moment on March 28, 2000. Due to the burst of bubble the index has dropped 25.02% to 13,722.70 till May 26, 2000. Having the same pattern as S&P500, the Hang Seng index entered into a bear market after a temporary recovery.

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The financial crisis in 2008 resulted in the threat of collapse of many large financial institutions, the bailout of banks by governments and a slump in worldwide stock markets. It is generally thought that the financial crisis erupted on September 15, 2008 when Lehman Brothers filed for Chapter 11 protection. At the time, both S&P500 index and Hang Seng index were in the middle of a bear market. But the occurrence on September 15, 2008 had triggered excessive decrease of both stock markets. In this study, the dates October 17, 1997, March 10, 2000 and September 15, 2008 are chosen as starting dates of respectively the Asian crisis in 1997, dotcom bubble in 2000 and financial crisis in 2008.

4.2 Data description

According to Dungey and Zhumabekova (2001) the size of the crisis and non-crisis periods can affect the test for contagion. For the empirical analysis, I use the daily observations of S&P500 index, Hang Seng Index, WTI oil spot price, gold bullion per troy ounce and USD/EUR exchange rate from January 1, 1994 until December 31, 2010. The chosen observation period covers the three crises. I use the aforementioned starting dates of crises: October 17, 1997, March 10, 2000 and September 15, 2008 to divide the whole observation period into six sub-periods. The sub-periods are:

1. the pre-Asian crisis from January 1, 1994 to October 17, 1997; 2. the post-Asian crisis from October 17, 1997 to December 31, 1999; 3. the pre-dotcom bubble from January 1, 1997 to March 10, 2000; 4. the post-dotcom bubble from March 10, 2000 to December 31, 2002; 5. the pre-financial crisis from January 1, 2005 to September 15, 2008; and 6. the post-financial crisis from September 15, 2008 to December 31, 2010.

The data set, which is downloaded from Datastream consisting of totally 4435 observations (closing prices). The S&P500 index and Hang Seng index are used as a proxy for respectively the U.S. and Hong Kong stock markets. The WTI oil spot price, gold bullion price and USD/EUR exchange rate are used as a proxy for respectively the oil, gold and currency markets. Hereafter the times series are abbreviated with respectively SP500, HSI, Oil price, Gold price and USD/EUR. In Hong Kong, the Hang

Seng index is traded at the local time from 9:30 A.M. to 16:00 P.M.; while the S&P500 is

traded at the local time of the U.S. between 9:00 A.M. and 16:00 P.M. There is a time lag of

five hours between the closing time of the Hang Seng index and the opening time of the S&P500. By using the bivariate copulas, the daily returns of these markets of sequential occurrences are treated as simultaneous occurrences, allowing us to measure the daily

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dependence of the Hong Kong and U.S. stock markets. Since the Asian crisis in 1997 took place in Asia, I use the Hang Seng index to investigate the contagion effect of Asian stock market on the U.S. stock market, oil and gold markets. The dotcom bubble in 2000 and financial crisis in 2008 are initiated in the U.S., therefore the S&P500 is used to investigate the contagion effect of the U.S. stock market on the Asian stock market, and other markets.

Before estimating the marginal distributions and the copulas, it is necessary to obtain a proper understanding of the data. For the empirical analysis, the logarithmic returns of data are calculated. Table 1 summarizes the main statistics of the data set. The Jarque-Bera test strongly rejects the normality of the data. In order to examine what has been changed before and after each crisis, the main statistics of the data are also determined for each sub-period. Although the mean of returns does not differ much for all series before and after the Asian crisis in 2007, the standard deviation, skewness and kurtosis exhibit some differences after the crisis. In the post-Asian crisis period, the standard deviation and kurtosis for all series are larger than in the period before the crisis. The larger standard deviation implies that the prices become on average more volatile. The larger kurtosis implies that the returns are more fat-tailed distributed in the period after the Asian crisis. Another remarkable feature is that, except the SP500, all other return series which are negatively skewed in the pre-Asian crisis period become positively skewed in the period after the crisis. In other words, the mass of returns distribution for HSI, oil and gold prices are from being concentrated on the right to be concentrated on the left during the Asian crisis. This is in line with the fact that a crisis is associated with a sharp decrease in asset prices.

In contrast to the Asian crisis, the mean of the returns, except the gold price and USD/EUR, decreases in the period after the dotcom bubble. In general, the kurtosis decreases after the dotcom bubble and there is no obvious increase in skewness in the period after the dotcom bubble, which are the other way around compared with the Asian crisis.

Comparing the descriptive statistics of the sub-periods before and after the financial crisis in 2008, it is remarkable that the changes in the standard deviation, skewness and kurtosis are very similar to the Asian crisis. With respect to the mean of returns there is, except for Oil and Gold prices, no obvious difference between the pre- and post-crisis periods. Remarkably, the average return of Oil price of 8.2% is positively large before the financial crisis, which becomes negative of -1.7% after the crisis. Meanwhile, in comparison with the Oil price, the average return of Gold price changes in the opposite direction that increases from 5.9% before the crisis to 10.5% after the crisis. Such large increase in the average return may imply that the Gold price is not affected by crisis and may justify the use of gold as a hedge against the stock market risk.

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Table 1. Descriptive statistics of daily returns.

Note: The Jarque-Bera statistic test for the null hypothesis of normality in the sample returns distribution. *** indicates statistical significance at the 1%.

n min max mean std. Dev skewness kurtosis Jarque-Bera

Total observation period

S&P500 4435 -9.470 10.957 0.022 1.223 -0.203 11.704 14,030.70***

HSI 4435 -14.735 17.247 0.015 1.734 0.086 12.542 16,828.93***

Crud Oil 4435 -17.217 21.277 0.042 2.473 -0.048 8.201 5000.90***

Gold 4435 -7.143 7.382 0.029 1.000 -0.022 9.826 8610.02***

USD/EUR exchange rate 4435 -3.844 4.617 0.004 0.608 0.228 5.758 1443.87***

pre-Asian crisis S&P500 990 -3.131 3.077 0.071 0.706 -0.227 5.028 178.18*** HSI 990 -7.594 6.883 0.014 1.434 -0.387 6.370 493.22*** Crud Oil 990 -11.970 9.303 0.038 2.046 -0.346 6.918 653.02*** Gold 990 -2.511 2.155 -0.019 0.489 -0.147 6.094 398.39*** post-Asian crisis S&P500 576 -7.113 4.989 0.075 1.234 -0.487 7.367 480.40*** HSI 576 -14.735 17.247 0.039 2.542 0.372 10.450 1345.28*** Crud Oil 576 -13.795 15.873 0.035 2.633 0.248 8.480 726.56*** Gold 576 -3.695 7.382 -0.020 0.856 1.364 15.486 3920.42*** pre-dotcom bubble S&P500 833 -7.113 4.989 0.076 1.187 -0.462 6.787 527.43*** HSI 833 -14.735 17.247 0.034 2.304 0.295 11.189 2339.78*** Crud Oil 833 -13.795 15.873 0.024 2.446 0.168 8.489 1049.70*** Gold 833 -5.819 7.382 -0.029 0.878 1.132 16.960 6941.68***

USD/EUR exchange rate 833 -1.950 2.293 -0.033 0.561 0.469 4.153 76.723***

post-dotcom bubble

S&P500 733 -6.005 5.573 -0.064 1.439 0.239 4.376 64.81***

HSI 733 -9.285 5.434 -0.087 1.584 -0.276 5.958 276.43***

Crud Oil 733 -17.217 10.563 -0.002 2.623 -0.658 7.277 611.64***

Gold 733 -2.795 5.946 0.023 0.754 0.910 10.218 1692.44***

USD/EUR exchange rate 733 -2.266 3.321 0.011 0.654 0.262 4.255 56.474***

pre-financial crisis

S&P500 966 -4.828 4.153 -0.002 0.919 -0.357 5.859 349.61***

HSI 966 -9.051 10.184 0.032 1.424 -0.124 9.480 1692.38***

Crud Oil 966 -7.496 8.084 0.082 2.015 0.072 3.636 17.11***

Gold 966 -6.284 4.062 0.059 1.242 -0.713 5.637 361.82***

USD/EUR exchange rate 966 -2.269 1.959 0.004 0.506 -0.024 3.969 37.87***

post-financial crisis

S&P500 600 -9.470 10.957 0.001 2.014 -0.199 8.720 822.02***

HSI 600 -13.582 13.407 0.029 2.238 0.167 10.156 1282.85***

Crud Oil 600 -13.065 21.277 -0.017 3.396 0.428 7.823 599.82***

Gold 600 -7.143 6.865 0.105 1.462 0.097 6.715 346.05***

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4.3 Statistical test on contagion effect

The main purpose of this study is to test whether there is evidence of contagion effect during the crises in 1997, 2000 and 2008, based on the same definition of contagion by Forbes and Rigobon (2002) which refers to a significant increase in cross-market linkages after a shock. In order to answer this research question, I will analyze whether there are significant changes during the crises in the time-varying dependence parameter for the specified markets. For this purpose, the same approach similar to Chiang, et al (2007) is applied, by including a dummy variable in a regression model for different sub-samples to analyze the changes in dependence parameter in pre-crisis (3 years before crisis) and post-crisis (2.5 years after crisis) periods. The appropriate regression model for this task is given by:

where is the time-varying dependence parameter, which includes the Kendall's

and tail dependence parameters between the different pairs of returns series of SP500, HSI, Oil price, Gold price and USD/EUR. The lag length of the dependence

parameter is chosen by the AIC criterion. Furthermore, is a dummy variable

that takes the value of zero for the pre-crisis period and the value of one for the post-crisis period. The regression is run separately for each post-crisis in 1997, 2000 and 2008, by using the pre- and post-crisis intervals as defined in section 4.2. In case the coefficient for the dummy variable is significantly positive, there is evidence of a contagion effect between the markets of pair according to the definition of contagion by Forbes and Rigobon (2002).

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Chapter 5 Empirical Results

First of all, the model with skew-t distribution is estimated to the data. After estimating the marginal distributions, the time-varying parameters of copulas are estimated using the above described procedure. This chapter also presents the results of the Rivers-Vuong test, the AIC and BIC for determining the most appropriate model.

5.1 Marginal distribution

For estimating the marginal distributions of daily returns, the total observation period has been divided into two sub-periods consisting of observations from January 1, 1994 to December 31, 2002 and from January 1, 2003 to December 31, 2010. The first sub-period covers the Asian crisis in 1997 and the dotcom bubble in 2000, and the second period covers the financial crisis in 2008. By dividing the observation into two sub-periods, we can avoid the structural break in the marginal model which may lead to inaccurate estimation. Table 2 displays the estimated parameters for marginal distributions of daily returns of the first sub-period. The table also contains the KS-test statistics and the p-values. It is remarkable that only S&P500 and USD/EUR have no autoregressive terms in the conditional mean equation by using the BIC for selection, while the other series have AR-lag terms. Especially, HSI has four AR-terms, implying that the autocorrelation and seasonal effect in HSI are important issues. As displayed in Table 2, all p-values for the KS-test on the probability integrals of the standardized residuals are not significant at 5% significance level, which indicates that the null hypothesis of uniformly distributed probability integrals cannot be rejected. These results provide evidence that the estimated marginal distribution is well-specified for the return series in the first sub-period. Regarding the conditional variances, the leverage effect seems only to be present in stock prices, since the corresponding coefficients for SP500 and HSI of respectively 0.1372 and 0.0968 are both positive and significant at 1% significance level. The same coefficient for both Oil and Gold prices is negative and significant at 5% level, which implies that the opposite of leverage effect is present in these series. The kurtosis parameter of the skewed-t distribution is all, except for Gold price, larger than 4 and significant at 1% significance level. Finally, the asymmetry parameter is all nonzero and except for HSI and Gold price significant.

Contagion effect between financial markets : measuring the dependence structure by using GAS copulas.

Faculty of Economics and Business

MSc Thesis Financial Econometrics

Contagion effect between financial markets: Measuring the

dependence structure by using GAS copulas

Student: Eric Lau

Student number: 0182516

Supervisor: Dr. N.P.A. van Giersbergen

2nd reader: Prof. dr. C.G.H. Diks

Contents

Chapter 1

Introduction

Chapter 2

Literature review

Chapter 3

Copula theory