Narrow Results

Hyperbolic cotangent function is proposed as a generator of new Archimedean copula family and several properties are revealed. To show performance in real data analysis, application to modeling dependence between monthly temperature extremes as well as between flood peak and volume is given.

In this paper, we emphasize and elaborate on two important and relatively new aspects in uncertainty analysis in order to increase the credibility of empirical results in statistics in general, and in econometrics in particular, namely, the problem of partial identification, and the use of random set statistics. We elaborate on the current interests in partially identified models, exemplified by econometric structures involving copulas. We spell out the rationale and the statistical methods based upon random set theory for analyzing partial identification problem towards credible econometrics.

It has been empirically verified that the strength of dependence in stock markets usually rises with volatility. In this paper we exploit this stylized fact combined with local maximum likelihood estimation of copula models to analyze the dynamic joint behavior of series of financial log returns. Explanatory variables based on the estimated GARCH volatilities are considered as potential regressors for explaining the dynamics in the copula parameters. The proposed model can assess and discriminate how much of the strength of dependence is due just to the time-varying volatility. The final local-parametric estimates may be used to compute risk measures, to simulate portfolio behavior, and so on. We illustrate our methods using two American indexes. Results indicate that volatility does affect the strength of dependence. The in-sample Value-at-Risk based on the dynamic model outperforms those based on the empirical estimates.

Using a large set of daily US and Japanese stock returns, we test in detail the relevance of Student models, and of more general elliptical models, for describing the joint distribution of returns. We find that while Student copulas provide a good approximation for strongly correlated pairs of stocks, systematic discrepancies appear as the linear correlation between stocks decreases, that rule out all elliptical models. Intuitively, the failure of elliptical models can be traced to the inadequacy of the assumption of a single volatility mode for all stocks. We suggest several ideas of methodological interest to efficiently visualise and compare different copulas. We identify the rescaled difference with the Gaussian copula and the central value of the copula as strongly discriminating observables. We insist on the need to shun away from formal choices of copulas with no financial interpretation.

The estimation of the multiplier parameter of portfolio insurance strategies is crucial for its implementation because it determines the risk exposure to the performance-seeking asset (PSA) at each point in time. Studies that address the estimation of the multiplier’s upper bound have been limited to strategies that use as the safe asset a short-term bank account, in which case the co-movements of the safe and the PSA become irrelevant. However, in several relevant applications, portfolio insurance strategies use stochastic reference assets different from cash, such as the control of active-risk relative to a benchmark, or insuring a minimum level of retirement income. We find that the implications of taking into account the assets’ co-movements in the multiplier estimation can be crucial. In Monte Carlo simulations the multiplier doubles in size across scenarios, and the strategy using the proposed approach presents stochastic dominance over the strategy that ignores the asset dependency structure.

This paper investigates the linkages of Chinese yuan to other currencies before and after the yuan devaluation on 11 August 2015. Linear regression analysis shows that only a few of the 14 currencies considered are significantly affected by the devaluation. However, the devaluation of Chinese yuan has been associated with larger fluctuations in these currencies and the occurrence of extreme positive and negative returns. The regression method may under estimate the tail dependence between currencies, as financial data are usually non-normally distributed, especially when extreme event occurs. We apply the Archimedean copulas to capture the presence of lower and upper tail dependence between the exchange rate returns of Chinese yuan and the selected currencies, and found dependencies not revealed by the linear regression analysis. The extreme returns after the Chinese yuan devaluation have resulted in higher dependence with the selected currencies. While the dependence structure was dominated by risks due to unusual currency gains before the devaluation, the market responses to large losses and gains have become more symmetric after the devaluation.

Copulas, as dependence measures of random variables, have wide applications in distribution theory, medical research, multivariate survival analysis, risk management, and other fields. The basic properties of copulas have been studied extensively in the literature. However, their geometric and topological properties, which are very important for properly characterizing the dependence patterns of random variables, have not as yet caught statisticians’ attention. In this paper, we shall study the geometric structures of copulas and the local dependence pattern of random variables. Important classes of copulas, such as the polynomial copulas, the piecewise linear and quadratic copulas are also investigated.

There are more and more recent copula models aimed at describing the behavior of multivariate data sets. However, no effective methods are known for checking the validity of these models, especially for the case of higher dimensions.

Our approach is based on the multivariate probability integral transformation of the joint distribution, which reduces the multivariate problem to one dimension. We compare the above goodness of fit tests to those, which are based on the copula density function. We present the background of the methods as well as simulations for their power.

In the last Chapter, we saw the important topic of risk management and the method of conditional heteroskedasticity in modelling dynamically changing volatilities. We pursue the theme of financial risk management further in this chapter by investigating other approaches including nonlinear methods of estimation. In particular we try to understand the paramount concept of systemic risk which has been at the heart of global financial market crisis using a statistical tool called copula.