A new test procedure of independence in copula models via chi-square-divergence

We introduce a new test procedure of independence in the framework of parametric copulas with unknown marginals. The method is based essentially on the dual representation of $\chi^2$-divergence on signed finite measures. The asymptotic properties of the proposed estimate and the test statistic are studied under the null and alternative hypotheses, with simple and standard limit distributions both when the parameter is an interior point or not.Comment: 23 pages (2 figures). Submitted to publicatio

On Empirical Likelihood in Semiparametric Two-Sample Density Ratio Models

We consider estimation and test problems for some semiparametric two-sample density ratio models. The profile empirical likelihood (EL) poses an irregularity problem under the null hypothesis that the laws of the two samples are equal. We show that a “dual” form of the profile EL is well defined even under the null hypothesis. A statistical test, based on the dual form of the EL ratio statistic (ELRS), is then proposed. We give an interpretation for the dual form of the ELRS through φ-divergences and “duality” technique. The asymptotic properties of the test statistic are presented both under the null and the alternative hypotheses, and an approximation to the power function is deduced

Inhomogeneous Dependency Modelling with Time Varying Copulae

Measuring dependence in a multivariate time series is tantamount to modelling its dynamic structure in space and time. In the context of a multivariate normally distributed time series, the evolution of the covariance (or correlation) matrix over time describes this dynamic. A wide variety of applications, though, requires a modelling framework different from the multivariate normal. In risk management the non-normal behaviour of most financial time series calls for nonlinear (i.e. non-gaussian) dependency. The correct modelling of non-gaussian dependencies is therefore a key issue in the analysis of multivariate time series. In this paper we use copulae functions with adaptively estimated time varying parameters for modelling the distribution of returns, free from the usual normality assumptions. Further, we apply copulae to estimation of Value-at-Risk (VaR) of a portfolio and show its better performance over the RiskMetrics approach, a widely used methodology for VaR estimation.Value-at-Risk, time varying copula, adaptive estimation, nonparametric estimation.

Building and using semiparametric tolerance regions for parametric multinomial models

We introduce a semiparametric ``tubular neighborhood'' of a parametric model in the multinomial setting. It consists of all multinomial distributions lying in a distance-based neighborhood of the parametric model of interest. Fitting such a tubular model allows one to use a parametric model while treating it as an approximation to the true distribution. In this paper, the Kullback--Leibler distance is used to build the tubular region. Based on this idea one can define the distance between the true multinomial distribution and the parametric model to be the index of fit. The paper develops a likelihood ratio test procedure for testing the magnitude of the index. A semiparametric bootstrap method is implemented to better approximate the distribution of the LRT statistic. The approximation permits more accurate construction of a lower confidence limit for the model fitting index.Comment: Published in at http://dx.doi.org/10.1214/08-AOS603 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

SFB 649 Discussion Paper 2006-075 Inhomogeneous Dependency Modelling with Time Varying

Measuring dependence in a multivariate time series is tantamount to modelling its dynamic structure in space and time. In the context of a multivariate normally distributed time series, the evolution of the covariance (or correlation) matrix over time describes this dynamic. A wide variety of applications, though, requires a modelling framework different from the multivariate normal. In risk management the non-normal behaviour of most financial time series calls for nonlinear (i.e. non-gaussian) dependency. The correct modelling of non-gaussian dependencies is therefore a key issue in the analysis of multivariate time series. In this paper we use copulae functions with adaptively estimated time varying parameters for modelling the distribution of returns, free from the usual normality assumptions. Further, we apply copulae to estimation of Value-at-Risk (VaR) of a portfolio and show its better performance over the RiskMetrics approach, a widely used methodology for VaR estimation. JEL classification: C 1

Laws and Limits of Econometrics

We start by discussing some general weaknesses and limitations of the econometric approach. A template from sociology is used to formulate six laws that characterize mainstream activities of econometrics and the scientific limits of those activities, we discuss some proximity theorems that quantify by means of explicit bounds how close we can get to the generating mechanism of the data and the optimal forecasts of next period observations using a finite number of observations. The magnitude of the bound depends on the characteristics of the model and the trajectory of the observed data. The results show that trends are more elusive to model than stationary processes in the sense that the proximity bounds are larger. By contrast, the bounds are of smaller order for models that are unidentified or nearly unidentified, so that lack or near lack of identification may not be as fatal to the use of a model in practice as some recent results on inference suggest, we look at one possible future of econometrics that involves the use of advanced econometric methods interactively by way of a web browser. With these methods users may access a suite of econometric methods and data sets online. They may also upload data to remote servers and by simple web browser selections initiate the implementation of advanced econometric software algorithms, returning the results online and by file and graphics downloads.Activities and limitations of econometrics, automated modeling, nearly unidentified models, nonstationarity, online econometrics, policy analysis, prediction, quantitative bounds, trends, unit roots, weak instruments