Thesis - Kyiv School of Economics
Financial contagion during world crisIs in 2007-2009
by
Aliaksei Malashonak
A thesis submitted in partial fulfillment of the requirements for the degree of
MA in Economics
Kyiv School of Economics
2009
Approved by
KSE Program Director
Date
Kyiv School of Economics
Abstract
Financial contagion during world crisIs in 2007-2009
by Aliaksei Malashonak
KSE Program Director: Tom Coupé
The paper is the first attempt to analyze the phenomenon of financial contagion during the financial crisis that started in 2007. Time-varying copula models of daily market returns are applied to investigate the impact of the global financial crisis on the dependence measures between seven world stock markets (USA, UK, Germany, Japan, Brazil, Russia and China) during the period of 2005-2009, based on time series of daily return rates. The model of the marginal distributions is assumed to follow a GARCH specification. Two copulas, Gaussian and Symmetrized Joe-Clayton, are employed to construct joint distributions. Such technique allows capturing excess changes in correlations and in the structure of dependence which can be interpreted as contagion.
The results indicate upward shift of constant correlation parameters and significant change of the coefficients describing the behavior of dependence parameters for the majority of country pairs, especially those including US financial market. However, the size of the effect is not homogeneous. Changes took maximum values for US, China, Russia Brazil and are lower for Japan, Germany and UK.
Overall, estimated time-varying dependence measures were found to exhibit an increase in financial markets co-movement, which supports the hypothesis of contagion during period of the crisis of 2007-2009.
Keywords: contagion, crisis, copula, dependence measure.
Table of Contents
Chapter 1. Introduction 1
Chapter 2. Literature Review 6
Chapter 3. Methodology 14
3.1 Dependence process and a copula function. 14
3.2 Conditional copula functions and is properties. 17
3.3 First step: Models of uniform distribution 19
3.4 Second step: Two models of copulas 21
3.5 Normal (Gaussian) copula dependence parameter 23
3.6 The Symmetrized Joe–Clayton copula dependence parameter. 26
Chapter 4. Data 31
Chapter 5. Results 36
5.1 General comments. 36
5.2 Main sources of contagion: US and developing countries. Optimistic expectations in Russia and Germany. 37
5.3 Dependence measures dynamics. 39
5.4 Evidence for the structure of the crisis. 43
Chapter 6. Conclusions 47
Bibliography 50
Appendix 52
List of figures
Number Page
Figure 1. Theoretical example of contagion under unchanging constant dependence parameter. 22
Figure 2. Daily returns of S&P500 and FTSE indices in 2005-2008. 33
Figure 3. S&P500 daily index in 2005-2008.. 34
Figure 4. Differences in constant correlation in Normal copula 38
Figure 5. A typical behavior of the time-varying correlation in 2008 with large positive change in level and dynamics structure of correlation parameter. Correlations in the Normal copula for US vs.UK and absolute value of S&P Index for UK. 44
Figure 6. A typical behavior of the time-varying correlation in 2008 with negative change in level and change in dynamics of correlation parameter. Correlations in the Normal copula for Germany vs. Japan and absolute value of DAX Index for Germany. 45
Figure A-1. Conditional correlations in Normal and SCJ copulas assuming the structural break in level and structure of dependence in August 2007: S&P(500) and FTSE (UK).. 52
Figure A-2. Conditional correlations in Normal and SCJ copulas assuming the structural break in level and structure of dependence: FTSE (UK) and BVSP (Brazil) in August 2007.. 53
Figure A-3. Conditional correlations in Normal and SCJ copulas assuming the structural break in level and structure of dependence: FTSE (UK) and SSE (China) in August 2007.. 54
Figure A-4 Conditional correlations in Normal and SCJ copulas assuming the structural break in level and structure of dependence: RTS (Russia) and SSE (China) in August 2007. 55
Figure A-5. Conditional correlations in SCJ copula and difference between tails assuming the structural break in level and structure of dependence: RTS (Russia) and SSE (China) in August 2007. 56
Acknowledgments
The author wishes to thank to Olesia Verchenko for thoughtful guidance and valuable comments during the research process. Special regards go to Ann for inspiration and creation of challenging environment.
Chapter 1
Introduction
A financial crisis is a situation characterized by a significant decrease in asset prices (market fall) and increase in market volatility (market risk). Financial markets which are integrated through spillover channels can transmit this instability across borders. Therefore, if we consider financial markets as investment opportunities for an investor, then information about correlation between returns of markets becomes necessary in the course of portfolio construction. A critical assumption for portfolio risk analysis is that the cross-country transmission of shocks remains constant even during crisis periods. In case the assumption of constant correlation is violated due to the instability caused by the crisis, the presence of a crisis should have direct influence on portfolio and risk management, since the change in market co-movement affects the performance of an internationally diversified portfolio without dynamic rebalancing.
Here one has to distinguish between interdependence and contagion. Markets generally are connected both during stable and unstable periods. Theoretically, long-term relationship between markets should be predetermined by spillover links between counties. The main factor which may cause increase in financial sensitivity of financial markets is the structure of the capital accounts of the countries. Financial links connect international financial systems, transmitting assets between countries. Real economic links, usually associated with international trade, reflect fundamental economic relationship among economies. Finally, social and political links can lead to clustering of shocks within political clubs or groups.
However, the excess correlation within financial systems, which can not be explained by macroeconomic fluctuations, could be observed during crisis periods. For example, the Thailand crisis of 1997, induced by massive speculative attacks on the local financial market, has spread to South Korea, Indonesia and other countries of the region.
Empirical evidence showed that market shocks can be transmitted between regions with very short lags, and the change in the structure of trade-related capital flows can not perfectly account for this excess relation, as it requires much longer period of time to be affected. Since the factors mentioned are unlikely to cause the observed excess relationship, the notion of the financial contagion was introduced by Pindyck and Rotemberg in 1990. This phenomenon is associated with the evidence of "excess co-movement" in stock and commodity prices between interdependent financial markets built on very different industry and idiosyncratic fundamentals. A more restrictive definition, allowing testing for the presence of this effect, was maintained by the World Bank: contagion occurs when cross-country correlations increase during "crisis times" relative to correlations during "tranquil times." From the researcher's point of view, each financial crisis can be considered as an opportunity to estimate contagion effects between specific countries. The latter description motivates the exploration of contagion effect during the financial crisis of 2007–2009, as the clear evidence of global instability can be observed on the world financial markets.
The most frequently used method of empirical analysis of this phenomenon, following from the definition, is based on the investigation of correlation coefficients of stock return during volatile (crisis) periods compared to relatively stable periods. There are two key points in this approach. First issue is related to defining of the timing of crisis period beginning in order to divide the historical data into crisis and control samples. Secondly, observing crises on many markets at the same time does not necessary imply that they were caused by contagion from one system to another. Instead, they could be caused by similar underlying problems that would have affected each country individually even in the absence of linkages. The natural endogeneity of cross-country time series, which was described above, complicates the estimation of excess correlation as we need to divide the connection between markets into two components: global changes in business activity and new local linkages created after beginning of unstable period.
The global financial crisis of 2007-2009 provides possibility for extensive empirical research on contagion, as it gives the chance to check for existence of excess negative spillover effects between almost all countries of the world. Dramatic movements in the US stock market could have a powerful impact on markets of very different sizes and structures worldwide. In case a significant increase in cross-market correlation coefficients after a shock to one country (or a group of countries) occurred, this could be interpreted as a presence of contagion effect.
This paper proposes the estimation of the dependency between pairs of stock markets and its dynamics during the crisis period. The response of seven stock markets to movements in daily return rates before and during the active pattern of the financial crisis of 2007-2009 is examined. This paper engages the assumption that major moves in equity markets are driven by macroeconomic variables in the long run. Therefore, lagged variables of market returns may be used to explain the trend, as they represent natural historical tendency of specific market. However, in the short run and during crisis periods excess fluctuations are mostly driven by local and global financial environment factors.
Contagion is tested within pairs of countries separately, due to the specificity of bivariate probability density functions applied for estimation. A set of time series includes developed and emerging markets such as USA, UK, Germany, Japan, Brazil, Russia and China. The selected countries represent different economic and geographical areas of the world.
The main task of this paper is to estimate the cross impact between stock markets and examine how the strength of the effect depends on volatility and direction of trend, both before and during the crisis of 2007-2009. Moreover, the paper is the first attempt to analyze financial contagion during the recent financial crisis.
Results obtained in this research can be applied for the purposes of risk management of internationally diversified portfolios. Investment portfolio construction is commonly based on the assumption that values of correlation between assets are known. Thus, non-stable correlation may induce unexpected losses for an investor. International and domestic supervisory institutions are also interested in studies on contagion effect, as the latter can change preferences of financial prudential supervision policy. For example, the crisis on one market can cause additional underestimated consequences influencing the liquidity of financial institutions on other market, thus, decreasing stability. Supervision institutions can predict such co-movement using previous data about the strength of contagion effect and take measures to minimize the crisis impact by means of controlling the capital flows in short-run.
Chapter 2
Literature Review
Increases in cross-market correlations have been documented during numerous market crises. Still, there is widespread disagreement about how contagion should be defined and empirically tested. Nevertheless, all of the previous papers on the topic have detected the evidence of contagion during the periods of high market volatility.
Different papers point toward different directions while answering the question about causes of contagion. A group of researchers claim that contagion is explained by real links, including:
• trade patterns and arrangements (Ades and Chua, 1993);
• technological factors and political instability (Chua, 1993; Easterly and Levine, 1994).
Other authors provide a financial explanation, or argue that herding behavior is the key element to understand the recent contagious episodes, relating contagion to one of the following issues:
• highly integrated capital markets, when shortages from the large country may be transmitted to small countries through trade in assets (Hoffmaister and Végh, 1994; Talvi, 1994);
• "bandwagon" effects, in which investor’s sentiment does not discriminate among different macroeconomic fundamentals across countries (Eichengreen, Rose and Wyplosz, 1995);
• regional preferences of foreign investors, implying that investors first select the larger, usually better known, countries as a place to invest (Calvo and Reinhart, 1996).
Calvo and Reinhart enriched the set of theoretical sources of contagion. They have examined “spillover” or “contagion” effect in light of the Mexican crisis in December 1994 and the effect that this event has had on other emerging market economies. The authors pointed out that institutional practices also could be a channel of spillovers. For instance, mutual funds, while expecting an increasing amount of redemptions, may sell off their holdings of equity in several emerging markets in an effort to raise cash.
Four main methods of testing are most often used in recent research on the topic: cross-market correlation analysis, GARCH analysis, long-term cointegration, and probit models.
Historically, studies examining cross-market correlation constitute the most numerous group, due to simplicity of estimation and easy interpretation of results. The method itself was developed after 1987 crisis when stock markets around the world crashed following the shock in Hong Kong. Afterwards, this method has been used for estimation of every new crisis occurring. The main idea of this method lies in testing whether the difference in correlation coefficients between highly volatile periods (crises) and stable periods is significant. The findings of changes of dependence between stable and crisis periods would suggest that the crises drive higher market dependence. This is empirically supported by the fact that while different economies follow different financial cycles, there are obvious linkages between stock markets during crisis periods (Samitas, Kenourgios and Paltalidis, 2007).
For instance, King and Wadhwani (1990) investigate why in October 1987 almost all stock markets in US, UK and Japan fell, despite difference in economic circumstances. The authors define contagion as a “mistakes-transmitting channel” which appears as the result of attempts by rational agents to infer information from price changes in other markets. Lee and Kim (1993) estimated the correlations between the stock markets for the same event across 12 largest stock indices and got similar result, which maintains the presence of contagion.
Hoffmaister and Végh (1994) and Talvi (1994) define speculative attacks or crises as large movements in exchange rates, interest rates, and international reserves, and test for heterogeneity of fundamental variables in 22 countries between 1967 and 1992. They conclude that for a number of countries there were significant differences in the behavior of key macroeconomic variables between crisis and non-crisis periods, which can be interpreted as contagion by definition.
Contagion effects also have been found by Pindyck and Rotemberg (1990, 1993). They interpret residual co-movement across stocks having very different industry and idiosyncratic fundamentals as the evidence of "excess co-movement" in stock and commodity prices.
However, Forbes and Rigobon (2002) show that unadjusted cross-market correlation coefficients are conditional on market volatility, and thus test of difference between stable and unstable periods are biased due to specific volatility dynamics structure of returns (see Methodology part for details). After correcting for this bias, the authors conclude that there is no empirical evidence of contagion during all crises analyzed (1987 US stock collapse, 1994 Mexican crisis, 1998 East Asian collapse). In other words, the hypothesis of co-integration during financial instability periods is rejected. In the authors’ opinion, interdependence existing in all states of the world acts as the cause of strong cross-market linkages.
Bartram and Wang (2004) argue upon the results obtained by Forbes and Rigobon (2002). Exploring the impact of volatility on market dependence by means of simulated and empirical time series analysis, Bartram and Wang show that “market dependence is not generally conditional on volatility regimes” and that “a bias in dependence measures occurs only for particular assumptions about the time-series dynamics”. In general, the authors argue that due to the latter fact the correction of correlation coefficients estimated during high volatility periods may be unnecessary. As a consequence, contagion can be tested using correlation coefficients approach.
Another method of testing for contagion is ARCH-GARCH variance-covariance transmission modeling, which allows estimating the effect of shocks influencing a certain market on volatility of another one. The case of 1987 US crisis was analyzed in this framework by Hamao (1990) and Chou (1994). The authors have concluded that there has been no evidence of contagion between countries, and that interdependence between markets exists constantly as stable linkage. Edwards (1998) estimated augmented GARCH model for Mexican bonds during the crisis of 1994 and detected the presence of capital transmission effect from Mexico to Chile. According to the paper, no changes in the effect have occurred during the crisis.
Long-term co-integration relationship analysis is another method used by numerous researchers. This approach is based on the assumption that cross-market relationship is constant over the observed period. As a consequence, testing for contagion becomes impossible during short crisis periods. Longin and Slonik (1995), for example, considered market returns data between 1960 and 1990 and found that correlation of the US markets with other countries increased by 0.36 over 30 years. However, the results of the research do not support hypothesis of non-stable relationship during the oil crisis in 1980s and US stock crash in late 1980s. Therefore, the method applied does not allow producing any significant result in finding the evidence of change in cross-market relations.
Several authors also include exogenous explanatory variables (for instance, news, events, and bankruptcies) into the analysis and estimate probit models to test for change in correlation coefficients. These papers also have found cross-country linkages between financial markets. For example, Baig and Goldfajn (1998) state that daily news have had a cross-country influence during 1997-1998 East Asian crisis. Forbes (1999) finds that country specific effects influenced the transmission mechanisms between individual companies during the same crisis period. Eichengreen, Rose and Wyplosz (1998) prove that crisis probability in one country depends on exogenous speculative attacks in other countries in ERM. Kaminski and Reinhart (1998) argue that conditional probability of financial downturn for the country under study increases if other countries in the region have already suffered.
Overall, all of the studies reviewed agree in that the inter-country correlations can be observed during the periods of financial crises. Still, the authors identify certain problems peculiar for the task of contagion detection. Namely, in a number of cases contagion is difficult to identify because certain time varying measures of the expected correlation are needed. During the periods of positive returns or low volatility, cross-country linkages are likely to be stable, which can not be interpreted as the contagion. Also, the papers on the topic employ different methodologies for the study of phenomenon mentioned.
Our paper follows the approach suggested by Patton (2006) and improved by other authors as the most developed method in searching for changes in the correlation of correlated markets.
Patton has empirically showed that periods of joint appreciations of the Deutschmark and Japanese yen against the U.S. dollar were followed by change in degree of correlation, compared to joint depreciation periods. The author has extended the theory of copula functions (multivariate joint distribution functions) to allow including conditioning variables in the model of the structure of exchange rates. As the result, analysis of constructed flexible conditional dependence models during ten-year period of 1991-2001 showed that the mark and yen exchange rates are more correlated when they are depreciating against the dollar than when they are appreciating.
As the findings of changes of dependence between stable and crisis periods would suggest that a crisis drives higher market dependence, some corrections have been introduced to the model. To divide correlation into constant and excess parts, we use time varying conditional copula and ARMA with extra assumption that correlation depends on previous experience observed on markets. The change in dynamics of correlation would imply that there are obvious excess linkages among stock markets during crisis periods between different economies.
Copula probability density function applied in empirical estimation can not be easily constructed for more than three time series. That is a main problem of the method, which, unfortunately, can not be solved without introduction of structural changes into theoretical model. Therefore, the presence of contagion is to be tested within each pair of countries due to peculiarity of the estimation method.
Chapter 3
Methodology
3.1 Dependence process and a copula function.
The model applied represents a theoretical system of interdependent stochastic processes, including internal trends as well as external impulses from any other processes influencing the system. Therefore, probability density function describing any specific process incorporates both independent (“natural”) components and a specific kind of correlation measure between these processes to capture the external effect. As it is shown further, it is difficult to construct a multidimensional probability density function which includes dependence measures between more than two variables.
An advantage of copula as a general procedure of formulating a multivariate distribution is that it allows including various general types of dependence into the analysis (Nelsen, 1999). By definition, copula is a multivariate joint probability function defined on the multidimensional unit cube such that every marginal distribution in this function is uniform on the interval [0, 1].
In general, the law of distribution of the specific model reflects internal behavior of stochastic variable. Now assume that the probabilities of observing a certain value of this stochastic variable are uniform random variables (i.e. the values of the cumulative distribution function represent the random variable distributed uniformly). Then, for any two stochastic variables and the corresponding cumulative densities it is possible to construct the third value as a function of these variables, such that it allows mutual compensating of the internal movements of the variables and captures interconnection between them.
By Sklar's (1959) theorem, for any random joint distribution function [pic] of n variables (which could represent the behavior of n correlated financial markets, for example), and respective marginal distribution functions [pic] there exists a copula C such that the joint distribution function is:
[pic][1].
The vector of dependence parameters [pic] included in the copula represents the complete information about relationship between marginal variables [pic]. However, the copula itself does not represent the information about the marginal distribution (as this is the dependent parameter of this function). As a result, the characteristics of variables and information about the dependence law are separated from each other. Moreover, by Sklar's theorem if marginal distributions [pic] are continuous, the copula function C is unique, as there exists only one vector of parameters. The vector [pic] usually has several parameters which relate to the strength and form of the dependence.
Therefore, it is possible to construct a bivariate function (probability density function of two variables) with multiple time-varying correlation coefficients, so that each element of the vector of dependent parameters captures different cases of extreme behavior of variables. The intuition is that low return rates unusually exhibit lower correlation between markets, while high return rates exhibit high correlation.
We construct copula (a probability function) which includes univariate marginal densities, not necessary normally distributed, of any two dependent variables and their correlation parameters. This allows separating the marginal distributions and the dependence structure, thus avoiding the common assumption of normality for both marginal distributions and their joint distribution function (Joe, 1997; Nelsen, 1999; Bartram et al., 2007).
Regression analysis consistent with multivariate normality assumption is aimed to test difference of variance of residuals between stable and crisis periods. Three assumptions are made for the purpose of analysis. Firstly, residuals are assumed to have asymmetric probability density function, due to unusual reaction of markets on short-run changes of trend and inadequateness of use of normal distribution in this case. Secondly, asymmetry parameters of the PDF are maintained to change with state shifts of markets. Thirdly, it is assumed that stable intermarket dependence can be explained by the previous dependence, and the historical average difference of cumulative probabilities of returns for the markets is a good estimator for excess co-movement.
3.2 Conditional copula functions and its properties.
Let [pic] and [pic] be returns of two markets in period t and let their conditional cumulative distribution functions (c.d.f.s) be [pic] and [pic] respectively, with [pic] denoting all previous returns, i.e. { [pic] , [pic], i>0}.
Let [pic] and [pic] be two random variables with vectors of parameters [pic]and [pic]respectively, whose marginal distributions are uniform on the interval from zero to one. Then the conditional copula density function with vector of parameters [pic], denoted by [pic], is defined by the time-varying bivariate density functions of [pic] and [pic] with specific distribution law.
Also, conditional bivariate density functions of [pic] and [pic] are given by the product of the copula density and their two marginal conditional densities, respectively denoted by [pic] and [pic].:
[pic]
where the bivariate dynamics of the returns[pic] and [pic]are determined by the three functions [pic], [pic] and [pic]
The two-stage ML (Newey and McFadden, 1994; White, 1994; Patton, 2006) provides precise estimation of structure and dynamics of the market returns. Estimates of parameters [pic] are asymptotically as efficient as one-stage ML estimates which could be obtained maximizing sum of copula likelihood functions for all observations in one step, plugging all the parameters[pic], [pic]and [pic] into the copula (Patton, 2006). However, it is difficult to apply one-stage ML for large samples in practice, because the optimal solution is the one which maximizes the sum of n likelihood functions for all the parameters simultaneously.
The first step in estimating contagion effect is the construction of univariate distribution functions [pic] and [pic] for each of market return variables. The parameters of the marginal distributions are estimated from univariate time series using ML functions:
[pic],
[pic],
where [pic], [pic] – marginal conditional densities for X and Y,
[pic] and [pic] – vectors of estimated parameters.
The results of the second step are two vectors of cumulative uniform marginal distribution computed by plugging of estimated vectors of parameters [pic] and [pic] into their conditional cumulative distribution functions [pic] and [pic].
The second step then allows estimating the dependence parameters maximizing the copula function:
[pic]
where [pic] – parameters of bivariate copula function. The second step allows to estimate the vector of parameters[pic]. Plugging the optimal values of [pic] into the copula function, we can also obtain time-varying correlation between two markets.
3.3 First step: Models of uniform distribution
The conditional density functions must be leptokurtic and have asymmetric functions of previous returns due to different investor’s reaction on different deviations from the trend (Nelson, 1991; Engle and Ng, 1993; Glosten, 1993). Previous studies have shown that the best empirical model to characterize the marginal distribution for each index is AR(2), GARCH(1, 1). Thus, we fit the appropriate ARCH specification with exogenous dummy parameter [pic] in which shows reverse of the trend and extra assumption that errors has conditional Student’s distributions:
[pic]
where [pic] – return on index i,
[pic] – conditional variance for period t ,
[pic]– dummy variable equal to 1 if there was positive return on the previous day t-1 and negative on the next t day and vice versa. This variable allows capturing asymmetry in variation of returns as we expect investor’s reaction to differ after shock and the variance to be higher.
The results of the first stage provide numerical values of uniform marginal distributions [pic] and[pic].
Following Patton (2006), we specify and estimate 12 alternative copulas (9 without time variation and 3 with time-varying property). The likelihood ratios of marginal distribution selection, which allow comparing the goodness of fit, are presented in the Appendix. As it can be seen in the results section, the findings suggest that two best models for the bivariate conditional distribution are the symmetrized Joe–Clayton copula and the normal (or Gaussian) copula with time-varying parameters.
The method used has a range of advantages compared to those applied in other papers. First, no common assumption of normality for either marginal distribution or joint distribution functions is needed. Secondly, results are invariant to strictly increasing transformations of random variables. Furthermore, the method allows accounting for asymptotic tail dependence. In addition, the parameters are allowed to change over time, and the estimations are very flexible to short-term market falls.
3.4 Second step: Two models of copulas
The absence of change of constant correlation can not be associated with non-contagion between markets. On the theoretical example below (see Figure 1) the period of decreasing correlation is followed by upward sloping trend with the average value changed not significantly. The second period clearly defines the evidence of growing contagion by definition. However, test for level change of the constant correlation does not show difference in the trends.
Therefore, to our analysis includes two procedures: testing correlations time-series for significant change of the constant correlation and testing parameters of the dynamics of dependence parameters for significant change.
[pic]
Figure 1. Theoretical example of contagion under unchanging constant dependence parameter. The increase of time-varying correlation on the second part can be interpreted as contagion. However, constant correlation parameter does not reflect reversal of the trend.
Following Patton (2006), we assume that correlation depends on the previous period market dependence and on the historical absolute differences, [pic] > 0, to capture variation in the dependence process:
[pic]
The intuition for the use of [pic]is that the smaller (larger) the difference between the realized cumulative probabilities, the higher (lower) is the dependence. This equation describes an AR(1) model in case additional assumptions are made, namely, that a linear function of the weakly mean of previous absolute differences [pic] provides a white noise innovation term. This is equivalent to the assumption that the copula parameter follows an ARMA(1,k) process. Patton proposed k=10 selected optimally by properties of copula. We take this number of periods for convenience because the difference for values up to 15 periods was found to be insignificant.
Transforming previous equations into one, we obtain the model specification which exactly follows the last assumption:
[pic],
where [pic] is the historical average difference of cumulative probabilities and [pic] is vector of parameters .
3.5 Normal (Gaussian) copula dependence parameter
Let [pic] be time-varying correlation between any two returns [pic] and[pic] at moment t. We use the values of [pic] and [pic] obtained at the first step to estimate dependence parameter [pic] of the copula function, which is conditional on [pic] and the vector[pic]. One of the appropriate kinds of copula functions is the Gaussian copula density function, which is constructed from the bivariate normal distribution using Sklar's theorem. The Gaussian copula is:
[pic].
Differentiating of C yields the copula density function:
[pic],
where [pic] is time-varying correlation between X and Y, [pic], [pic]– inverse of the p.d.f of [pic]and [pic].
This function shows the probability density of the pair [pic] assuming that two returns [pic] have joint bivariate normal distribution with time-varying correlation parameter[pic].
Any copula function is indifferent to strictly increasing transformations by its properties. Thus, apply the modified logistic transformation to keep [pic] in (−1, 1) interval without any changes in the copula:
[pic]
As a result, final specification becomes:
[pic]
Coefficient [pic] represents common dependence in correlation, and [pic] captures any variation in dependence with respect to the 10 periods mean of transformed uniform densities. In other words, [pic] reflects the dynamics of dependence under assumption that the shock on the previous day (t-1) was common for both markets. Based on the previous research, we expect this coefficient to be negative. The intuitive explanation is that the strength of any common instability decreases over time and markets return to relatively independent state.
To obtain the copula specification comparable with that of the time-varying symmetrized Joe–Clayton copula, we average [pic]over 10ags[2].
If the additional term [pic] is nonzero for some period of time, that would mean occurrence of unexpected shock on one market (low value of [pic]), and the immediate effect on the second return rate (high value of [pic]). Therefore, this term has no impact on the correlation parameter, if it is close to zero and geometrically increases with non-zero values of[pic]. We expect the coefficient on this term to be negative, as it would provide empirical evidence that markets become less correlated under the crisis. Significant difference in coefficient [pic] observed before and after crisis would imply that the dynamics of the effects has changed and market becomes more of less dependent.
3.6 The Symmetrized Joe–Clayton copula dependence parameter.
Joe (1997) constructs the copula by taking a particular Laplace transformation of Clayton’s copula. The Joe–Clayton copula is:
[pic][3],
where [pic], [pic] and [pic]
Let there exist such a limit
[pic]
By definition, copula CJC(·) has the lower tail dependence parameter if [pic] , and no lower tail dependence if [pic].
Similarly, if there exists a limit
[pic]
and if [pic] , then CJC(·) exhibits upper dependence; [pic] 0 means no such dependence.
Conditions[pic], [pic] and [pic], [pic] capture the extreme cases of random variables’ behavior. In our case, that includes the events of extremely large market movements, or shocks. The Normal copula has zero dependence parameters for correlation less than one, which means that “in the extreme case tails of the distribution of variables are independent” (Embrechts, 2001).
Any correlation parameter between two series exhibits following condition by its properties: [pic] Therefore, bivariate distribution of correlations should be perfectly symmetric:
[pic].
Still, the original JC copula has some minor asymmetry when the two tail dependence measures are equal. To solve this problem, Patton proposes the “Symmetrized Joe–Clayton” copula (SJC) which is a simple function of CJC(·) :
[pic]
To keep [pic] in the interval (−1, 1), we use modified logistic transformation:
[pic],
and estimate two modified AR(2) models:
[pic]
[pic]
Generated dependence measures in each copula should be compared to the sample period prior to autumn of 2007. Parameters of the dynamic, represented by vectors of parameters of copula functions, need to be estimated as well.
The intuition for including the average [pic] in the Normal copula now becomes clear: for very low values of [pic] and [pic], the product also takes low values, geometrically (but not linearly) increasing with the joint growth of the variables by the property of the second order polynomial on the interval between 0 and 1. Thus, we make the correlation parameter flexible enough to incorporate extreme cases (the JC has this property due to two-dimensional vector of correlation parameters).
The expected results include, first, the significant increase in constant correlation after the crisis beginning, as the markets are supposed to be more volatile during shocks. Secondly, [pic] is presumed to be positive and significantly decreasing during the crisis. This would imply that constant dependence across indices decreases under shocks. Also, [pic] is expected to be negative, significantly decreasing in time varying copula and probably insignificantly decreasing in JC copula, as we presuppose that excess effect is smaller than constant effect and the latest mean in absolute differences of cumulative probabilities decreases dependence.
Overall, we expect significant increase in conditional correlation based on estimates of time-varying copulas.
Suggested functional form may be not universal in case of presence of lags of previous returns or differences in cumulative probabilities of returns. Each group of countries probably has model specification depending on lags. This could make estimation more complicated and technically demanding.
Standard errors of the coefficients can be computed with Berndt-Hall-Hall-Hausman (BHHH) algorithm by taking first order derivatives of the log-logarithms of the copulas as a proxy for computing the information matrix. In general, for any constructed log-likelihood function [pic] coefficient at iteration step k+1 is:
[pic],
[pic],
where [pic] -- is a parameter. Expression for [pic] gives intuition for computing standard errors.
We can obtain variance-covariance matrix of the coefficients taking an inverse of the information matrix [pic], which is computed as squared vector of first order derivatives with respect to the vector of coefficients [pic] in the estimated point.
Chapter 4
Data
We use the data on daily return rates of stock indices. Return rates are computed as the difference of logarithms of daily close prices in US dollars. The data set collected from the Yahoo finance portal includes daily close prices and exchange rates excluding holidays from January 2005 to May 2009 for three emerging markets (Brazil, Russia, China) and four developed markets (USA, UK, Germany, Japan). For each country, we obtain 4 years of daily values of the stock market index.
Table 1. Summary statistics of daily returns for 7 major indices.
|Index |Mean |Std. dev. |Min |Max |Skewness |AR(1) |
| |(-0.0034) |(.070) |(-0.50) |(0.25) |(-1.08) |(-0.072) |
| |(-0.0019) |(.0268) |(-0.10) |(0.12) |(0.06) |(-0.072) |
| |(-0.0014) |(.0272) |(-0.10) |(0.13) |(0.03) |(-0.078) |
| |(-0.0022) |(0.029) |(-0.13) |(0.09) |(-0.05) |(-0.037) |
| |(-0.0028) |(0.044) |(-0.26) |(0.20) |(-0.72) |(0.1294*) |
| |(-0.0010) |(0.040) |(-0.18) |(0.17) |(-0.19) |(0.009) |
| |(-0.0012) |(.0239) |(-0.11) |
|Country |with |Constant rho |Tau-Upper |Tau-Lower |
| | |Stable |
Country |with |Rho beta |Rho alpha |Tau-U Beta |Tau-U Alpha |Tau-L Beta |Tau-L Alpha | | | |Stable |Crisis |Stable |Crisis |Stable |Crisis |Stable |Crisis |Stable |Crisis |Stable |Crisis | |US |UK |0.037 |-0.980 |1.980 |-1.79 |0.000 |-3.091 |0.000 |4.294 |-0.068 |-3.030 |-0.017 |2.892 | | |Germany |0.013 |-0.291 |0.012 |-0.22 |0.000 |-4.898 |0.000 |3.822 |-2.274 |-2.782 |4.033 |3.357 | | |Russia |-0.055 |0.091 |0.032 |1.392 |0.000 |-4.918 |0.000 |3.276 |-2.715 |-3.078 |-0.05 |4.085 | | |China |-0.068 |-0.010 |1.743 |-0.01 |0.000 |-4.193 |0.000 |-0.006 |22.152 |0.000 |6.140 |0.000 | | |Japan |0.095 |0.070 |1.428 |1.360 |0.000 |-33.145 |0.000 |-2.329 |0.000 |-50.000 |0.000 |0.387 | | |Brasil |0.049 |-0.730 |1.873 |-1.80 |0.000 |-0.026 |0.000 |-0.001 |-6.399 |-0.468 |-4.783 |0.031 | |UK |Germany |-0.258 |-0.265 |-2.590 |-0.03 |0.503 |-9.005 |3.991 |2.124 |11.097 |-2.361 |-2.38 |3.527 | | |Russia |0.033 |-1.067 |2.606 |-2.32 |1.730 |-8.012 |-2.043 |1.182 |-0.949 |4.861 |2.152 |-3.463 | | |China |-0.008 |0.263 |0.001 |0.732 |-0.001 |-3.779 |0.000 |1.696 |-8.504 |-8.701 |3.910 |0.582 | | |Japan |0.378 |-0.082 |-2.169 |-0.06 |-5.842 |8.023 |3.464 |4.618 |-11.356 |-13.650 |-4.154 |-4.856 | | |Brasil |0.031 |-0.860 |-0.006 |-2.83 |0.275 |0.398 |-0.017 |-0.430 |0.215 |-5.778 |-0.03 |-0.591 | |Germany |Russia |0.021 |-1.231 |2.553 |-2.36 |3.470 |-0.568 |-2.855 |4.238 |-6.397 |-0.893 |0.262 |4.003 | | |China |-0.392 |0.148 |-1.838 |-0.03 |-0.060 |-3.658 |0.000 |-4.541 |-8.284 |-15.518 |3.321 |0.972 | | |Japan |-0.016 |-0.323 |0.016 |-0.80 |1.154 |24.040 |-2.152 |-0.939 |-16.653 |-3.871 |-4.156 |2.910 | | |Brasil |-0.281 |-0.530 |-2.236 |-2.47 |6.699 |-0.670 |-3.189 |4.102 |-5.134 |-8.288 |2.765 |0.096 | |Russia |China |-0.155 |0.256 |1.336 |0.594 |0.000 |-2.451 |0.000 |0.041 |-3.306 |-14.669 |0.118 |-0.084 | | |Japan |-0.006 |0.002 |0.011 |0.007 |3.965 |-2.791 |0.993 |1.264 |-13.143 |-2.206 |-5.358 |4.400 | | |Brasil |-0.049 |-0.387 |0.660 |-2.48 |-4.345 |-4.269 |-7.777 |1.611 |-1.103 |-3.207 |-2.166 |0.644 | |China |Japan |0.161 |-0.429 |-0.374 |-1.96 |-0.393 |0.152 |-0.004 |-0.036 |-10.53 |8.145 |-2.25 |-4.07 | | |Brasil |0.222 |-0.021 |-0.130 |2.054 |-1.908 |2.156 |-0.003 |5.006 |4.661 |-0.829 |-0.084 |-0.038 | |Japan |Brasil |0.685 |0.046 |-2.024 |0.959 |-16.4 |-0.112 |-4.97 |-0.01 |-21.1 |-5.86 |-8.99 |0.956 | |
*Parameters selected in bold are significantly different from zero and significantly change at 95%.
5.4 Evidence for the structure of the crisis.
One interesting feature can be observed in regard to the graphs of time-varying correlations in the Normal copula: namely, there is a typical period of unusually high volatility of the correlation in the end of 2008 for all the countries indifferently to size and direction of the correlation displacement. This period, beginning in September 2008 and ending in January-February 2009, corresponds to the lowest point of the crisis. However, not all of SJC copula series of lower and upper dependence parameters represent this movement due to different structure of the probability density function (recall that SCJ copula has two parameters which occur in extreme cases).
Moreover, another common tendency can be observed in most cases of cross-dependency: time-varying correlation in Normal copula tends to increase before the beginning of the second part of the crisis (during August 2007 – August 2008) up to maximum values of the whole sample period. Such acceleration in all the cases was followed by the rapid fall and the period of high volatility of Gaussian correlation parameter in September 2008. Timing of the observed process was found to be the same for all the pairs.
Two examples are given below. Figure 5 describes the behavior of constant and time-varying correlation between S&P and FTSE indices in Normal copula and the trend of S&P during this period. Figure 6 presents the same correlations for Germany vs. Japan during the given period and absolute value of DAX Index (Germany). The difference is that the first pair of indices exhibits large increase in correlation, in contrast to Germany-Japan pair where there was no such change. Clearly, the period of high correlation in the end of 2007 – beginning of 2008 is followed by rapid fall and high volatility in the same time bounds. Similar dynamics is observed in almost all cases.
Figure 5. A typical behavior of the time-varying correlation in 2008 with large positive change in level and dynamics structure of correlation parameter. Correlations in the Normal copula for US vs.UK (upper panel) and absolute value of S&P Index for UK (lower panel).
[pic]
Figure 6. A typical behavior of the time-varying correlation in 2008 with negative change in level and change in dynamics of correlation parameter. Correlations in the Normal copula for Germany vs. Japan (upper panel) and absolute value of DAX Index for Germany (lower panel).
[pic]
The possible reason for growth of the correlation parameter during the first “latent” part of the crisis could be investor’s behavior: while observing shock on one market, investor takes measures to recombine the portfolio structure by buying or selling assets on related markets. However, investor needs a certain period of time to complete all the transactions. This period of time took approximately one year, as we can observe from the graphs.
September 2008 is the beginning of the “active” part of the crisis. Here, the influences of related regions decrease, as markets become more irrational: investors get liable to panic; their behavior becomes more stochastic in absence of predictions. This results in abnormally high volatility we can observe.
Chapter 6
Conclusions
Financial markets which are integrated through spillover channels can transmit the instability arising from financial downturns across borders. Moreover, the additional increase in interdependence between markets may be observed during crisis periods.
The examination of response of seven stock markets to movements in daily return rates before and during the active pattern of the financial crisis of 2007-2009 by means of Gaussian and Symmetrized Joe-Clayton copulas, carried out in this paper, allowed capturing excess changes in correlations and in the structure of dependence which can be interpreted as contagion.
The research findings indicate that market dependence increased for prevailing number of countries under study. The shifts in constant correlation parameters and significant change of the coefficients characterize the behavior of dependence parameters for all pairs of countries including US financial market and the majority of other pairs. However, the size of the effect was not constant. Changes took maximum values for US, China, Russia Brazil and were lower for Japan, Germany and UK.
In addition, some common tendencies have been observed for the beginning if the crisis: correlation constructed by means of normal copula is likely to be higher during the beginning of the crisis when the volatility increases moderately. Such trend results in rapid shift of correlation as soon as the crisis enters state of extremely high volatility and market trend reaches the lowest point.
In general, estimated time-varying dependence measures were found to exhibit an increase in financial markets co-movement, which supports the hypothesis of contagion during periods of the crisis of 2007-2009.
Also, as expected, the significance of parameters, indicating that the latest absolute difference of returns is consistently a relevant measure, was found when modeling market dependence.
The results provide the empirical evidence to the fact that a crisis in one market induces investors to sell their assets in other markets in order to maintain certain proportions of various countries’ assets in the portfolio. Thus, investors need to adjust their portfolios accordingly to the contagion effects, since markets display higher correlation during the periods of market crashes. Portfolio diversification across markets might be less useful than commonly anticipated, if based on correlations in relatively tranquil times.
A possible way to extend the approach taken is including into copulas more than two market returns, thus constructing a multivariate model which could prove useful for construction of internationally diversified portfolios and in risk management. Future research may also include deeper analysis of contagion dynamics with respect to crisis timing.
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Appendix
Figure A-1. Conditional correlations in Normal and SCJ copulas assuming the structural break in level and structure of dependence in August 2007: S&P(500) and FTSE (UK). Normal copula estimates (upper) and the S&P500 market trend (lower) on the left charts, SJC estimates on the right charts.
[pic][pic]
Figure A-2. Conditional correlations in Normal and SCJ copulas assuming the structural break in level and structure of dependence: FTSE (UK) and BVSP (Brazil) in August 2007. Normal copula estimates on the left chart, SJC estimates on the right charts.
[pic][pic]
Figure A-3. Conditional correlations in Normal and SCJ copulas assuming the structural break in level and structure of dependence: FTSE (UK) and SSE (China) in August 2007. Normal copula estimates on the left chart, SJC estimates on the right charts.
[pic][pic]
Figure A-4. Conditional correlations in Normal and SCJ copulas assuming the structural break in level and structure of dependence: RTS (Russia) and SSE (China) in August 2007. Differences in normal correlation are not obvious; however, SCJ shows that only upper dependence increased. Normal copula estimates on the left chart, SJC estimates on the right charts.
[pic][pic]
Figure A-5. Conditional correlations in SCJ copula and difference between tails assuming the structural break in level and structure of dependence: RTS (Russia) and SSE (China) in August 2007. Changes in dependence level are not obvious; however, differences between tails reflect the change of the dynamics. SJC conditional correlation estimates on the left charts, difference between tails on the right chart.
[pic][pic]
Table A-1. Estimates of the models of marginal distribution.
Index S_P FTSE DAX SSE RTS BVSP JPN
Mean of return -0.000 0.001** 0.001*** 0.002** 0.004*** 0.005*** 0.001
(0.001) (0.000) (0.000) (0.001) (0.001) (0.001) (0.000)
Variance
Constant -20.263 -22.613 -345.001 -11.815*** -28.734 -66.238 -12.992***
(6013.906) (.) (.) (1.006) (.) (.) (2.129)
Shift dummy S 10.059* 11.586*** 333.839*** -61.425*** 19.401*** 58.257*** 1.658
(6014.185) (0.370) (0.393) (0.361) (0.303) (0.250) (3.020)
Arch 1st lag 0.258** 0.049 0.080 0.033 0.176** 0.089 0.054
(0.085) (0.047) (0.046) (0.043) (0.068) (0.048) (0.043)
Arch 2nd lag -0.162 0.131* 0.039 0.068 -0.002 0.134* 0.047
(0.085) (0.057) (0.051) (0.049) (0.070) (0.059) (0.049)
Garch 1st lag 0.908*** 0.813*** 0.871*** 0.895*** 0.815*** 0.688*** 0.883***
(0.022) (0.033) (0.025) (0.029) (0.032) (0.041) (0.031)
DF 4 5 4 6 5 6 4
N 959 951 959 901 959 921 945
p ................
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