Statistical Inference in Copula Models

In this paper, we provide an exposition of statistical inference procedures on the copula associated with a multivariate distribution from which we have a random sample, without any knowledge of the marginal distributions. According to the \mathscr{G}-sufficiency criterion, the inference should be based on vectors of coordinatewise ranks only. The empirical copula and its variants are nonparametric estimators of copula, and depend on the data only through their ranks. Their asymptotic properties are briefly reviewed. Then estimation of functionals of the copula such as rank correlation coefficients are discussed. Semiparametric estimation of the dependence parameter of copula is also considered with a special focus on rank approximate Z-estimators which includes the well-known pseudo-likelihood estimators. After a short review of general goodness-of-fit testing on copulas, we conclude the paper with a detailed examination of some resampling schemes based on the empirical and related copulas, which are indispensable for carrying out the aforementioned inference procedures in practice.

Key words: Copula, \mathscr{G}-sufficiency, empirical copula, semiparametric estimation, goodness-of-fit testing, bootstrap.

Properties of Divergence for Semiparametric Copula Models

A semiparametric copula model is a statistical model in which the copula is assumed to be parametric and the marginal distribution is arbitrary. In this paper, we investigate the divergence of the model. In particular, we establish a relationship between the rank divergence induced from the marginal distribution of the multivariate rank statistic and the profile divergence defined by infimum of the Kullback–Leibler divergence with respect to the nuisance parameter. We also obtain formulas for piecewise uniform and Gaussian copulas.

Key words: Composite transformation model, copula, divergence, holonomic gradient method, information geometry, optimal transport.

Properties of Skew-t Copulas and Their Statistical Estimation
—Application to Asset Returns—

The skew-t copula is the copula that is implicit in a multivariate skew-t distribution. There are various types of multivariate skew-t distributions, depending on the way that skewness is incorporated into the multivariate Student-t distribution. First, we summarize the representative multivariate skew-t distributions and show the procedures for applying the maximum likelihood estimation to the three types of skew-t copulas. Next, we refer to empirical studies for the applications of skew-t copulas and show estimation results for three indices from TOPIX33 Sector Indices to indicate the effectiveness of skew-t copulas in representing asset returns. Finally, we conclude by describing future research tasks.

Key words: Copula, multivariate skew-t distribution, tail dependence.

Realized Stochastic Volatility Model
—Extensions and Application to Japanese Stock Index—

Realized volatility (RV), which is the sum of squared intraday returns over a certain interval (such as a day), has widely been used as an estimator of the financial volatility. In the real market, however, the presence of non-trading hours and market microstructure noise in transaction prices may create bias in the RV. Taking account of this bias, several studies propose modeling daily returns and RV simultaneously. The resultant model, based on the stochastic volatility (SV) model, is called the realized stochastic volatility (RSV) model. Because the likelihood of the RSV model, as well as the SV model, is difficult to evaluate analytically, the Bayesian estimation method via the Markov chain Monte Carlo (MCMC) is often used. In this article, we explain the RSV model, its Bayesian estimation method via MCMC, and several extensions of the RSV model. Further, we apply the models to the Nikkei 225 stock index and explain the estimation results.

Key words: Markov chain Monte Carlo, realized volatility, stochastic volatility model.

A Copula Model with Stochastic Tail Dependence: Statistical Inference and Applications to Quantitative Finance

We survey model structures of stochastic copulas in which the dependence structures stochastically vary, as well as statistical estimation methods for these dependence structures. Because dependence structures in stochastic copulas are described by state equations that incorporate latent variables, likelihood evaluation requires numerical calculation. In this survey, we summarize these methodologies and discuss the application of stochastic copulas to multivariate dependence structures through vine copulas. We also introduce some applications to the field of finance, including time-varying copula models with leverage and copula models with time-varying dependence parameters. As an example of the latter models, we report a currency hedging model for time-varying dependence parameters in copula models.

Key words: Stochastic dependence structure, stochastic copula, vine copula.

An Extension of a CDO Pricing Model Using a Copula toward a Risk Evaluation Model

Many financial risk evaluation models were developed before the worldwide financial crisis, but none of them could predict the crisis. This is because all of these models are purely statistical, meaning that they are based on analyses of historical data and therefore cannot predict crises that have never happened. In this article, we consider the CDO pricing model used in the paper that proposed the implied copula, and propose extending it to a risk evaluation model. In the pricing model, assuming that the default times are conditionally independent, the non-parametric distribution of conditional default probabilities are estimated from the market prices of CDO tranches, and the estimated distributions in previous studies show that the default probabilities will increase dramatically with some small probabilities. It is our model that will apply such estimated results to evaluating the risk of a portfolio. By using not only the historical data but also the market prices of CDOs and derivatives, market participants' potential fears of future catastrophic loss can be reflected in the risk evaluation. In this article, we show how to extend the pricing model toward a risk evaluation model according to some known theoretical results, and in particular, discuss in detail the importance of the change of measure and its mathematical description. The important tool of our model is the factor that has a large influence on conditional default probabilities for all entities. We show some numerical results that can hardly be obtained from the existing models, and that are consistent with the remarkable features seen in the financial crisis.

Key words: Financial risk management, statistical model, implied copula, conditional independence, physical probability, risk-neutral probability.

Copula-based Continuous Event History Analysis

Scholars are interested not just in when an event happens but also in what kind of event happens. Moreover, the latter can depend on the former. If an event variable is nominal, the dependent competing risks approach is available. The model of the present paper, copula-based continuous event history analysis (CCEHA), takes advantage of copulas to model dependence between time and continuous event variables. Monte Carlo simulation illustrates that separate estimation of time and event models creates bias in the estimate of the event model when time and the event are not independent of each other (and some durations are censored). By contrast, CCEHA does not have this problem. We reanalyzed a dataset on election timing and outcome in the postwar U.K. and found an asymmetric positive dependence: later elections helped the governing party, whereas earlier elections did not necessarily hurt it.

Key words: Survival analysis, duration analysis, timing, dependence, election, United Kingdom.

Survival Analysis Using Copulas
—Meta-analysis with Correlated Endpoints—

With the rapid development this decade of open databases for biomedical researchers, we have gained access to complex and yet accurate patient-level information. For cancer patients in particular, these databases record information on individual patients, including overall survival time, time-to-tumor progression, tumor size, and gene expressions. In addition, there has been significant development of meta-analytical methodologies for analyzing data from different sources. To fully utilize such complex survival data, it is insufficient to apply the classical tools used in survival data analysis, such as Cox regression. In this paper, we review copula-based statistical methods for analyzing two survival time variables, namely overall survival time and time-to-tumor progression. We also review the joint frailty-copula model for individual patient data (IPD) meta-analysis methods, which account for the heterogeneity of patients from different sources. Regarding the construction of an appropriate likelihood function for a given dataset, we explain the importance of acknowledging the semi-competing risks relationship between overall survival time and time-to-tumor progression. Finally, we introduce a dynamic prediction method for overall survival time according to gene expressions and tumor progression, which may contribute to the development of personalized medicine.

Key words: Clinical trial, competing risk, Cox proportional hazards model, dynamic prediction, gene expression, personalized medicine.

Verification of the Effectiveness of Sensitivity Analysisas a Variable Selection in Support Vector Regression
—Analysis of Factors Affecting Prefectural All-cause Mortality Rates—

The sensitivity analysis method as a variable selection in support vector regression has been applied to the search of factors affecting prefectural all-cause mortality rates, and its effectiveness has been verified. In Japan, a multi-death society will come after an aging society with a declining birthrate, and various social problems such as shortage of doctors, end-of-life care, lonely death, death place refugees, vacant houses are feared. In order for local governments to take countermeasures against mortality, it is important to clarify the factors that have a significant effect on mortality among many factors and estimate their relative impact.All-cause age-adjusted mortality rates of 47 prefectures were used as an objective variable, and 56 indices in lifestyle habit, medical care/welfare, and society/economyfields were employed as potential explanatory variables. Factors related to the mortality rates were searched by applying a support vector regression technique to these data, and their sensitivities to the mortality rates were estimated by employinga sensitivity analysis method. Eleven kinds of factors which reproduce the observed mortalities of 47 prefectures with an accuracy of statistical significance level were obtained. It is found thatrates of social workers not examined in previous studies as well as those of smoking habit and of elderly singles highly contribute to the observed mortality rates.Countermeasures for decreasing the mortality rate of Aomori prefecture showing the highest rate in Japan were proposed on the basis of the affecting factors obtained in this study. From these results, the effectiveness of the sensitivity analysis method as a variable selection in support vector regression has been demonstrated.

Key words: Support vector regression, variable selection, sensitivity analysis, all-cause mortality rates, affecting factors.