Time-varying Copula Research Articles

Copula contagion index and its efficiency

A Copula Contagion Index (CCI) is established to measure financial contagion, based on the time-varying copula function. Empirical studies performed on the crisis spillover of US Subprime Mortgage Crisis demonstrate the efficiency of CCI. The empirical results from event study, change-point analysis, logitistic/probit regressions and the detection of the lead–lag relations by Baba, Engle, Kraft and Kroner (BEKK)-Vector Autoregressive (VAR)-Generalized Autoregressive Conditional Heteroscedastic (GARCH) model show that, the index is efficient in detecting the financial contagions. Sensitivity analysis indicates that the index is robust. Besides, empirical results indicate that the developed markets rather than the emerging markets suffered more severely and quickly from the US subprime mortgage crisis.

Modeling Time-Varying Dependencies Between Positive-Valued High-Frequency Time Series

Multiplicative error models (MEM) became a standard tool for modeling conditional durations of intraday transactions, realized volatilities and trading volumes. The parametric estimation of the corresponding multivariate model, the so-called vector MEM (VMEM), requires a specification of the joint error term distribution, which is due to the lack of multivariate distribution functions on Rd + defined via a copula. Maximum likelihood estimation is based on the assumption of constant copula parameters and therefore, leads to invalid inference, if the dependence exhibits time variations or structural breaks. Hence, we suggest to test for time-varying dependence by calibrating a time-varying copula model and to reestimate the VMEM based on identified intervals of homogenous dependence. This paper summarizes the important aspects of (V)MEM, its estimation and a sequential test for changes in the dependence structure. The techniques are applied in an empirical example.

Copula-Based Semiparametric Models for Multivariate Time Series

In this paper, we extend copula-based univariate time series models studied in Chen & Fan (2006) to multivariate time series. Doing so, we tackle at the same time serial dependence as well as interdependence between several time series. The proposed methodology is totally different from the usual time-varying copula modeling of time series in which only the dependence between the serially independent innovations is taken into account, time series have to be modeled, and residuals are used for inference. Here, no parametric modeling of time series is necessary, so there is no need of using residuals, which facilitates the statistical analysis. We discuss issues related to parameter estimation as well as tests of goodness-of-fit. We treat in greater detail two families, specifically the meta-elliptic copulas and Archimedean copulas. The methodology is then applied to model the dependence between the Canadian/US exchange rate and value of oil futures during the last ten years.

Efficient Bayesian inference for stochastic time-varying copula models

There is strong empirical evidence that dependence in multivariate financial time series varies over time. To model this effect, a time varying copula class is developed, which is called the stochastic copula autoregressive (SCAR) model. Dependence at time t is modeled by a real-valued latent variable, which corresponds to the Fisher Z transformation of Kendall’s τ for the chosen copula family. This allows for a common scale so that a general range of copula families including the Gaussian, Clayton and Gumbel copulas can be used and compared in our modeling framework. The inclusion of latent variables makes maximum likelihood estimation computationally difficult, therefore a Bayesian approach is followed. This approach allows the computation of credibility intervals in addition to point estimates. Two Markov Chain Monte Carlo (MCMC) sampling algorithms are proposed. The first one is a naïve approach using Metropolis–Hastings within Gibbs, while the second is a more efficient coarse grid sampler. The performance of these samplers are investigated in a simulation study and are applied to data involving financial stock indices. It is shown that time varying dependence is present for this data and can be quantified by estimating the underlying time varying Kendall’s τ with point-wise credible intervals.

Speculative Market Efficiency and Hedging Effectiveness of Emerging Chinese Index Futures Market

With the daily sample of CSI300 index and CSI300 index futures prices, we investigate the market efficiency and hedging effectiveness of the Chinese index futures market. Cointegration test is employed to examine the market efficiency; OLS, the symmetric bivariate GARCH, the asymmetric bivariate GARCH and time-varying copulas are used to estimate hedging effectiveness. The result shows that CSI300 index futures prices are cointegrated with spot prices and are unbiased predictors of future spot prices. The hedging effectiveness of CSI300 index futures is about 91%, so it can help investors to avoid the systematic risk in the spot market well. Besides, the hedge ratio conducted by the OLS model is the best-performed in variance reduction, very closely followed by the time-varying copulas. Since the dynamic hedge ratios are less stable and having pronounced fluctuations, the hedgers had to adjust their futures positions. The static OLS hedge strategy might be indeed a good choice considering the high transaction cost.

Volatility Spillovers, Comovements and Contagion in Securitized Real Estate Markets

This paper analyzes the relationships between local and global securitized real estate markets, but also between securitized real estate and common stock markets. First, the volatility transmissions across markets are examined using an asymmetric t-BEKK (Baba-Engle-Kraft-Kroner) specification of their covariance matrix. Second, correlations from that model and tail dependences estimated using a time-varying copula framework are analyzed to assess whether different dynamics underlie the comovements in the whole distribution and those in the tails. Third, we investigate market contagion by testing for structural changes in the tail dependences. We use data for the U.S., the U.K. and Australia for the period 1990–2010 as a basis for our analyses. Spillover effects are found to be the largest in the U.S., both domestically and internationally. Further, comovements in tail distributions between markets appear to be quite important. We also document different dynamics between the conditional tail dependences and correlations. Finally, we find evidence of market contagion between the U.S. and the U.K. markets following the subprime crisis.

Portfolio value-at-risk estimation in energy futures markets with time-varying copula-GARCH model

This paper combines copula functions with GARCH-type models to construct the conditional joint distribution, which is used to estimate Value-at-Risk (VaR) of an equally weighted portfolio comprising crude oil futures and natural gas futures in energy market. Both constant and time-varying copulas are applied to fit the dependence structure of the two assets returns. The findings show that the constant Student t copula is a good compromise for effectively fitting the dependence structure between crude oil futures and natural gas futures. Moreover, the skewed Student t distribution has a better fit than Normal and Student t distribution to the marginal distribution of each asset. Asymmetries and excess kurtosis are found in marginal distributions as well as in dependence. We estimate VaR of the underlying portfolio to be 95% and 99%, by using the Monte Carlo simulation. Then using backtesting, we compare the out-of-sample forecasting performances of VaR estimated by different models.

Volatility Spillovers, Comovements and Contagion in Securitized Real Estate Markets

This paper analyzes the relationships between local and global securitized real estate markets, but also between securitized real estate and common stock markets. First, the volatility transmissions across markets are examined using an asymmetric t-BEKK (Baba-Engle-Kraft-Kroner) specification of their covariance matrix. Second, correlations from that model and tail dependences estimated using a time-varying copula framework are analyzed to assess whether different dynamics underlie the comovements in the whole distribution and those in the tails. Third, we investigate market contagion by testing for structural changes in the tail dependences. We use data for the U.S., the U.K. and Australia for the period 1990-2010 as a basis for our analyses. Spillover effects are found to be the largest in the U.S., both domestically and internationally. Further, comovements in tail distributions between markets appear to be quite important. We also document different dynamics between the conditional tail dependences and correlations. Finally, we find evidence of market contagion between the U.S. and the U.K. markets following the subprime crisis.

The Dependence Structure between Carbon Emission Allowances and Financial Markets - A Copula Analysis

This paper applies different copulas in order to investigate the complex dependence structure between EU emission allowance (EUA) futures returns and those of other commodities, equity and energy indices. The analysis yields important insights into the relationship between carbon, commodities and financial markets. First of all, we find a significant relationship between EUA returns and those of the other considered variables that is most appropriately modeled by a Gaussian and Student-t copula. These results contradict some earlier studies that report no statistically significant or even negative correlations between returns of emission allowances and other financial variables. Secondly, considering time-varying copulas shows that the estimated copula parameters are not constant over time. We find in particular that the dependence is stronger during the period of the financial crisis. In a Value-at-Risk (VaR) analysis, finally, we further illustrate the advantages of copula methods. In particular the Student-t copula provides an appropriate quantification of VaR at different confidence levels while other models fail to specify the risk correctly. This analysis shows that ignoring the actual nature of dependence might lead to an underestimation of the risk for portfolios combining EUAs with commodities or equity investments.

The dynamic dependence between the Chinese market and other international stock markets: A time-varying copula approach

The purpose of this paper is to study the dependence structures between the Chinese market and other major world markets, a reflection of China's increasing integration into the global economy. We used time-varying copula models to show that conditional copulas outperform both unconditional copulas and conventional GARCH models. We consistently found the Chinese market to have the highest levels of dependence, as well as the greatest variability in dependence, with markets in Japan and the Pacific. Our results provide investors interested in the Chinese market with more timely suggestions for portfolio diversification, risk management, and international asset allocation than those derived from static models.

TESTING FOR ASSET MARKET LINKAGES: A NEW APPROACH BASED ON TIME-VARYING COPULAS

This paper proposes a new approach based on time-varying copulas to test for the presence of increases in stock market interdependence (also known as shift contagion) after a financial crisis. We discuss the importance of considering simultaneously separate breaks in volatility and dependence. Without such consideration, the contagion test turns out to be biased. A sequential algorithm is proposed to tackle this problem. Applied to the recent 1997 Asian crisis, the analysis confirms that breaks in variances always precede those in the dependence parameter. Moreover, a significant ‘J-shape’ evolution of the dependence parameter is detected, supporting the idea of shift contagion.

Modeling Multivariate Interest Rates Using Time-Varying Copulas and Reducible Nonlinear Stochastic Differential Equations

We propose a new approach for modeling nonlinear multivariate interest rate processes based on time-varying copulas and reducible stochastic differential equations (SDEs). In the modeling of the marginal processes, we consider a class of nonlinear SDEs that are reducible to Ornstein–Uhlenbeck (OU) process or Cox, Ingersoll, and Ross (1985) (CIR) process. The reducibility is achieved via a nonlinear transformation function. The main advantage of this approach is that these SDEs can account for nonlinear features, observed in short-term interest rate series, while at the same time leading to exact discretization and closed-form likelihood functions. Although a rich set of specifications may be entertained, our exposition focuses on a couple of nonlinear constant elasticity volatility (CEV) processes, denoted as OU-CEV and CIR-CEV, respectively. These two processes encompass a number of existing models that have closed-form likelihood functions. The transition density, the conditional distribution function, and the steady-state density function are derived in closed form as well as the conditional and unconditional moments for both processes. In order to obtain a more flexible functional form over time, we allow the transformation function to be time varying. Results from our study of U.S. and UK short-term interest rates suggest that the new models outperform existing parametric models with closed-form likelihood functions. We also find the time-varying effects in the transformation functions statistically significant. To examine the joint behavior of interest rate series, we propose flexible nonlinear multivariate models by joining univariate nonlinear processes via appropriate copulas. We study the conditional dependence structure of the two rates using Patton (2006a) time-varying symmetrized Joe–Clayton copula. We find evidence of asymmetric dependence between the two rates, and that the level of dependence is positively related to the level of the two rates. (JEL: C13, C32, G12)

When is a Copula Constant? A Test for Changing Relationships

A copula defines the probability that observations from two time series lie below given quantiles. It is proposed that stationarity tests constructed from indicator variables be used to test against the hypothesis that the copula is changing over time. Tests associated with different quantiles may point to changes in different parts of the copula. The tests are still effective when prefiltering is carried out to correct for persistent changes in volatility. However, a “median quadrant association test” that is robust to changing volatility provides a good overall test against a time-varying copula. An empirical illustration on financial contagion is provided. (JEL: C12, G32)

Modeling exchange rate dependence dynamics at different time horizons

Despite an extensive body of research, the best way to model the dependence of exchange rates remains an open question. In this paper we present a new approach which employs a flexible time-varying copula model. It allows the conditional correlation between exchange rates to be both time-varying and modeled independently from the marginal distributions. We introduce a dynamic specification for the correlation using the Fisher transformation. Applied to Euro/US dollar and Japanese Yen/US dollar, our results reveal a significantly time-varying correlation, dependent on the past return realizations. We find that a time-varying copula with the proposed correlation specification gives better results than alternative dynamic benchmark models. The dynamic copula model outperforms at six different time horizons, ranging from hourly to daily, confirming the model specification.

Structure Linkages of the Equity, Bond and Money Markets: A Time-Varying Copula Approach

This paper studies the dependence structures between the equity, bond and money markets in Australia, the US and across the two countries. The linkages between these markets have become more important for investors during the current period of financial uncertainty. In the process of making investment decisions, investors should be aware of the structures of dependence, which show the co-movement patterns between these markets in order to reduce losses of their portfolios. This paper proposes the combination of empirical distributions and time-varying copula models to capture the structure linkages between these markets. The empirical findings have shown some important implications for a wide range of areas related to investment within these markets.

Pricing bivariate option under GARCH processes with time-varying copula

This paper develops a method for pricing bivariate contingent claims under General Autoregressive Conditionally Heteroskedastic (GARCH) process. As the association between the underlying assets may vary over time, the dynamic copula with time-varying parameter offers a better alternative to any static model for dependence structure and even to the dynamic copula model determined by dynamic dependence measure. Therefore, the proposed method proves to play an important role in pricing bivariate options. The approach is illustrated with one type of better-of-two-markets claims: call option on the better performer of Shanghai and Shenzhen Stock Composite Indexes. Results show that the option prices obtained by the time-varying copula model differ substantially from the prices implied by the static copula model and even the dynamic copula model derived from the dynamic dependence measure. Moreover, the empirical work displays the advantages of the suggested method.

Pricing bivariate option under GARCH-GH model with dynamic copula: application for Chinese market

This paper develops the method for pricing bivariate contingent claims under general autoregressive conditionally heteroskedastic (GARCH) process. In order to provide a general framework being able to accommodate skewness, leptokurtosis, fat tails as well as the time-varying volatility that are often found in financial data, generalized hyperbolic (GH) distribution is used for innovations. As the association between the underlying assets may vary over time, the dynamic copula approach is considered. Therefore, the proposed method proves to play an important role in pricing bivariate option. The approach is illustrated for Chinese market with one type of better-of-two markets claims: call option on the better performer of Shanghai Stock Composite Index and Shenzhen Stock Composite Index. Results show that the option prices obtained by the GARCH-GH model with time-varying copula differ substantially from the prices implied by the GARCH-Gaussian dynamic copula model. Moreover, the empirical work displays the advantage of the suggested method.

Portfolio Value-at-Risk with Time-Varying Copula: Evidence from Latin America

Model risk in the estimation of value-at-risk is a challenging threat for the success of any financial investments. The degree of the model risk increases when the estimation process is constructed with a portfolio in the emerging markets. The proper model should both provide flexible joint distributions by splitting the marginality from the dependencies among the financial assets within the portfolio and also capture the non-linear behaviours and extremes in the returns arising from the special features of the emerging markets. In this paper, we use time-varying copula to estimate the value-at-risk of the portfolio comprised of the Bovespa and the IPC Mexico in equal and constant weights. The performance comparison of the copula model to the EWMA portfolio model made by the Christoffersen back-test shows that the copula model captures the extremes most successfully. The copula model, by estimating the portfolio value-at-risk with the least violation number in the back-tests, provides the investors to allocate the minimum regulatory capital requirement in accordance with the Basel II Accord.

The Euro and European financial market dependence

A time-varying copula model is used to investigate the impact of the introduction of the Euro on the dependence between 17 European stock markets during the period 1994–2003. The model is implemented with a GJR-GARCH-MA- t model for the marginal distributions and the Gaussian copula for the joint distribution, which allows capturing time-varying, non-linear relationships. The results show that, within the Euro area, market dependence increased after the introduction of the common currency only for large equity markets, such as in France, Germany, Italy, the Netherlands and Spain. Structural break tests indicate that the increase in financial market dependence started around the beginning of 1998 when Euro membership was determined and the relevant information was announced. The UK and Sweden, but not other European countries outside the Euro area, are found to exhibit an increase in equity market co-movement, which is consistent with the interpretation that these countries may be expected to join the Euro in the future.

The Euro and European Financial Market Dependence

A time-varying copula model is used to investigate the impact of the introduction of the Euro on the dependence between seventeen European stock markets during the period 1994-2003. The model is implemented with a GJR-GARCH-MA-t model for the marginal distributions and the Gaussian copula for the joint distribution, which allows capturing time-varying, non-linear relationships. The results show that, within the Euro area, market dependence increased after the introduction of the common currency only for large equity markets, such as in France, Germany, Italy, the Netherlands and Spain. Structural break tests indicate that the increase in financial market dependence started around the beginning of 1998 when Euro membership was determined and the relevant information was announced. The UK and Sweden, but not other European countries outside the Euro area, are found to exhibit an increase in equity market co-movement, which is consistent with the interpretation that these countries may be expected to join the Euro in the future.