A review of copula models for economic time series
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paper_title: Copulas and Temporal Dependence

paper_content:

An emerging literature in time series econometrics concerns the modeling of potentially nonlinear temporal dependence in stationary Markov chains using copula functions. We obtain sucient conditions for a geometric rate of mixing in models of this kind. Geometric beta-mixing is established under a rather strong sucient condition that rules out asymmetry and tail dependence in the copula function. Geometric rho-mixing is obtained under a weaker condition that permits both asymmetry and tail dependence. We verify one or both of these conditions for a range of parametric copula functions that are popular in applied work.
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paper_title: Pair-Copula Constructions of Multiple Dependence

paper_content:

Building on the work of Bedford, Cooke and Joe, we show how multivariate data, which exhibit complex patterns of dependence in the tails, can be modelled using a cascade of pair-copulae, acting on two variables at a time. We use the pair-copula decomposition of a general multivariate distribution and propose a method for performing inference. The model construction is hierarchical in nature, the various levels corresponding to the incorporation of more variables in the conditioning sets, using pair-copulae as simple building blocks. Pair-copula decomposed models also represent a very flexible way to construct higher-dimensional copulae. We apply the methodology to a financial data set. Our approach represents the first step towards the development of an unsupervised algorithm that explores the space of possible pair-copula models, that also can be applied to huge data sets automatically.
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paper_title: Copula Methods for Forecasting Multivariate Time Series

paper_content:

Copula-based models provide a great deal of flexibility in modeling multivariate distributions, allowing the researcher to specify the models for the marginal distributions separately from the dependence structure (copula) that links them to form a joint distribution. In addition to flexibility, this often also facilitates estimation of the model in stages, reducing the computational burden. This chapter reviews the growing literature on copula-based models for economic and financial time series data, and discusses in detail methods for estimation, inference, goodness-of-fit testing, and model selection that are useful when working with these models. A representative data set of two daily equity index returns is used to illustrate all of the main results.
---
paper_title: Goodness-of-Fit Tests for Copulas of Multivariate Time Series

paper_content:

In this paper, we study the asymptotic behavior of the sequential empirical process and the sequential empirical copula process, both constructed from residuals of multivariate stochastic volatility models. Applications for the detection of structural changes and specification tests of the distribution of innovations are discussed. It is also shown that if the stochastic volatility matrices are diagonal, which is the case if the univariate time series are estimated separately instead of being jointly estimated, then the empirical copula process behaves as if the innovations were observed; a remarkable property. As a by-product, one also obtains the asymptotic behavior of rank-based measures of dependence applied to residuals of these time series models.
---
paper_title: Efficient Estimation of Copula-Based Semiparametric Markov Models

paper_content:

This paper considers efficient estimation of copula-based semiparametric strictly stationary Markov models. These models are characterized by nonparametric invariant (one-dimensional marginal) distributions and parametric bivariate copula functions; where the copulas capture temporal dependence and tail dependence of the processes. The Markov processes generated via tail dependent copulas may look highly persistent and are useful for financial and economic applications. We first show that Markov processes generated via Clayton, Gumbel and Student's $t$ copulas and their survival copulas are all geometrically ergodic. We then propose a sieve maximum likelihood estimation (MLE) for the copula parameter, the invariant distribution and the conditional quantiles. We show that the sieve MLEs of any smooth functionals are root-$n$ consistent, asymptotically normal and efficient; and that their sieve likelihood ratio statistics are asymptotically chi-square distributed. We present Monte Carlo studies to compare the finite sample performance of the sieve MLE, the two-step estimator of Chen and Fan (2006), the correctly specified parametric MLE and the incorrectly specified parametric MLE. The simulation results indicate that our sieve MLEs perform very well; having much smaller biases and smaller variances than the two-step estimator for Markov models generated via Clayton, Gumbel and other tail dependent copulas.
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paper_title: Inferences on the association parameter in copula models for bivariate survival data

paper_content:

We investigate two-stage parametric and two-stage semi-parametric estimation procedures for the association parameter in copula models for bivariate survival data where censoring in either or both components is allowed. We derive asymptotic properties of the estimators and compare their performance by simulations. Both parametric and semi-parametric estimators of the association parameter are efficient at independence, and the parameter estimates in the margins have high efficiency and are robust to misspecification of dependency structures. In addition, we propose a consistent variance estimator for the semi-parametric estimator of the association parameter. We apply the proposed methods to an AIDS data set for illustration.
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paper_title: Time dependent copulas

paper_content:

For the study of dynamic dependence structures, the authors introduce the concept of a pseudo-copula, which extends Patton's definition of a conditional copula. They state the equivalent of Sklar's theorem for pseudo-copulas. They establish the asymptotic normality of nonparametric estimators of pseudo-copulas under strong mixing assumptions, and discuss applications to specification tests. They complement the theory with a small simulation study on the power of the proposed tests.
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paper_title: A Primer on Copulas for Count Data

paper_content:

The authors review various facts about copulas linking discrete distributions. They show how the possibility of ties that results from atoms in the probability distribution invalidates various familiar relations that lie at the root of copula theory in the continuous case. They highlight some of the dangers and limitations of an undiscriminating transposition of modeling and inference practices from the continuous setting into the discrete one.
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paper_title: Dynamic Copula Methods in Finance

paper_content:

The latest tools and techniques for pricing and risk management ::: ::: This book introduces readers to the use of copula functions to represent the dynamics of financial assets and risk factors, integrated temporal and cross-section applications. The first part of the book will briefly introduce the standard the theory of copula functions, before examining the link between copulas and Markov processes. It will then introduce new techniques to design Markov processes that are suited to represent the dynamics of market risk factors and their co-movement, providing techniques to both estimate and simulate such dynamics. The second part of the book will show readers how to apply these methods to the evaluation of pricing of multivariate derivative contracts in the equity and credit markets. It will then move on to explore the applications of joint temporal and cross-section aggregation to the problem of risk integration.
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paper_title: Modelling Longitudinal Data using a Pair-Copula Decomposition of Serial Dependence

paper_content:

Copulas have proven to be very successful tools for the flexible modeling of cross-sectional dependence. In this paper we express the dependence structure of continuous-valued time series data using a sequence of bivariate copulas. This corresponds to a type of decomposition recently called a “vine” in the graphical models literature, where each copula is entitled a “pair-copula.” We propose a Bayesian approach for the estimation of this dependence structure for longitudinal data. Bayesian selection ideas are used to identify any independence pair-copulas, with the end result being a parsimonious representation of a time-inhomogeneous Markov process of varying order. Estimates are Bayesian model averages over the distribution of the lag structure of the Markov process. Using a simulation study we show that the selection approach is reliable and can improve the estimates of both conditional and unconditional pairwise dependencies substantially. We also show that a vine with selection outperforms a Gaussian...
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paper_title: Estimation of copula-based semiparametric time series models

paper_content:

This paper studies the estimation of a class of copula-based semiparametric stationary Markov models. These models are characterized by nonparametric invariant (or marginal) distributions and parametric copula functions that capture the temporal dependence of the processes; the implied transition distributions are all semiparametric. Models in this class are easy to simulate, and can be expressed as semiparametric regression transformation models. One advantage of this copula approach is to separate out the temporal dependence(such as tail dependence) from the marginal behavior (such as fat tailedness) of a time series. We present conditions under which processes generated by models in this class are $\beta $-mixing; naturally, these conditions depend only on the copula specification. Simple estimators of the marginal distribution and the copula parameter are provided, and their asymptotic properties are established under easily verifiable conditions. Estimators of important features of the transition distribution such as the (nonlinear) conditional moments and conditional quantiles are easily obtained from estimators of the marginal distribution and the copula parameter; their $\sqrt{n}-$ consistency and asymptotic normality can be obtained using the Delta method. In addition, the semiparametric conditional quantile estimators are automatically monotonic across quantiles.
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paper_title: MODELLING ASYMMETRIC EXCHANGE RATE DEPENDENCE*

paper_content:

We test for asymmetry in a model of the dependence between the Deutsche mark and the yen, in the sense that a different degree of correlation is exhibited during joint appreciations against the U.S. dollar versus during joint depreciations. We consider an extension of the theory of copulas to allow for conditioning variables, and employ it to construct flexible models of the conditional dependence structure of these exchange rates. We find evidence that the mark-dollar and yen-dollar exchange rates are more correlated when they are depreciating against the dollar than when they are appreciating. Copyright 2006 by the Economics Department Of The University Of Pennsylvania And Osaka University Institute Of Social And Economic Research Association.
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paper_title: Everything you always wanted to know about copula modeling but were afraid to ask

paper_content:

This paper presents an introduction to inference for copula models, based on rank methods. By working out in detail a small, fictitious numerical example, the writers exhibit the various steps involved in investigating the dependence between two random variables and in modeling it using copulas. Simple graphical tools and numerical techniques are presented for selecting an appropriate model, estimating its parameters, and checking its goodness-of-fit. A larger, realistic application of the methodology to hydrological data is then presented.
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paper_title: A semiparametric estimation procedure of dependence parameters in multivariate families of distributions

paper_content:

SUMMARY This paper investigates the properties of a semiparametric method for estimating the dependence parameters in a family of multivariate distributions. The proposed estimator, obtained as a solution of a pseudo-likelihood equation, is shown to be consistent, asymptotically normal and fully efficient at independence. A natural estimator of its asymptotic variance is proved to be consistent. Comparisons are made with alternative semiparametric estimators in the special case of Clayton's model for association in bivariate data.
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paper_title: Multivariate models and dependence concepts

paper_content:

Introduction. Aspects of Interpretation. Technical Considerations. Statistical Analysis. Special Methods for Joint Responses. Some Examples. Strategical Aspects. More Specialized Topics. Appendices.
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paper_title: Goodness-of-Fit Tests for Copulas of Multivariate Time Series

paper_content:

In this paper, we study the asymptotic behavior of the sequential empirical process and the sequential empirical copula process, both constructed from residuals of multivariate stochastic volatility models. Applications for the detection of structural changes and specification tests of the distribution of innovations are discussed. It is also shown that if the stochastic volatility matrices are diagonal, which is the case if the univariate time series are estimated separately instead of being jointly estimated, then the empirical copula process behaves as if the innovations were observed; a remarkable property. As a by-product, one also obtains the asymptotic behavior of rank-based measures of dependence applied to residuals of these time series models.
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paper_title: On the Out-of-Sample Importance of Skewness and Asymmetric Dependence for Asset Allocation

paper_content:

Recent studies in the empirical finance literature have reported evidence of two types of asymmetries in the joint distribution of stock returns. The first is skewness in the distribution of individual stock returns. The second is an asymmetry in the dependence between stocks: stock returns appear to be more highly correlated during market downturns than during market upturns. In this article we examine the economic and statistical significance of these asymmetries for asset allocation decisions in an out-of-sample setting. We consider the problem of a constant relative risk aversion (CRRA) investor allocating wealth between the risk-free asset, a small-cap portfolio, and a large-cap portfolio. We use models that can capture time-varying moments up to the fourth order, and we use copula theory to construct models of the time-varying dependence structure that allow for different dependence during bear markets than bull markets. The importance of these two asymmetries for asset allocation is assessed by comparing the performance of a portfolio based on a normal distribution model with a portfolio based on a more flexible distribution model. For investors with no short-sales constraints, we find that knowledge of higher moments and asymmetric dependence leads to gains that are economically significant and statistically significant in some cases. For short sales-constrained investors the gains are limited. Copyright 2004, Oxford University Press.
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paper_title: Estimation of copula-based semiparametric time series models

paper_content:

This paper studies the estimation of a class of copula-based semiparametric stationary Markov models. These models are characterized by nonparametric invariant (or marginal) distributions and parametric copula functions that capture the temporal dependence of the processes; the implied transition distributions are all semiparametric. Models in this class are easy to simulate, and can be expressed as semiparametric regression transformation models. One advantage of this copula approach is to separate out the temporal dependence(such as tail dependence) from the marginal behavior (such as fat tailedness) of a time series. We present conditions under which processes generated by models in this class are $\beta $-mixing; naturally, these conditions depend only on the copula specification. Simple estimators of the marginal distribution and the copula parameter are provided, and their asymptotic properties are established under easily verifiable conditions. Estimators of important features of the transition distribution such as the (nonlinear) conditional moments and conditional quantiles are easily obtained from estimators of the marginal distribution and the copula parameter; their $\sqrt{n}-$ consistency and asymptotic normality can be obtained using the Delta method. In addition, the semiparametric conditional quantile estimators are automatically monotonic across quantiles.
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paper_title: Test of independence and randomness based on the empirical copula process

paper_content:

Deheuvels (1981a) described a decomposition of the empirical copula process into a finite number of asymptotically mutually independent sub-processes whose joint limiting distribution is tractable under the hypothesis that a multivariate distribution is equal to the product of its margins. It is proved here that this result can be extended to the serial case and that the limiting processes have the same joint distribution as in the non-serial setting. As a consequences, linear rank statistics have the same asymptotic distribution in both contexts. It is also shown how these facts can be exploited to construct simple statistics for detecting dependence graphically and testing it formally. Simulation are used to explore the finite-sample behavior of these statistics, which are found to be powerful against varions types of alternatives.
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paper_title: Kendall's Tau for Serial Dependence

paper_content:

The authors show how Kendall's tau can be adapted to test against serial dependence in a univariate time series context. They provide formulas for the mean and variance of circular and noncircular versions of this statistic, and they prove its asymptotic normality under the hypothesis of independence. They present also a Monte Carlo study comparing the power and size of a test based on Kendall's tau with the power and size of competing procedures based on alternative parametric and nonparametric measures of serial dependence. In particular, their simulations indicate that Kendall's tau outperforms Spearman's rho in detecting first-order autoregressive dependence, despite the fact that these two statistics are asymptotically equivalent under the null hypothesis, as well as under local alternatives.
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paper_title: Goodness-of-Fit Tests for Copulas of Multivariate Time Series

paper_content:

In this paper, we study the asymptotic behavior of the sequential empirical process and the sequential empirical copula process, both constructed from residuals of multivariate stochastic volatility models. Applications for the detection of structural changes and specification tests of the distribution of innovations are discussed. It is also shown that if the stochastic volatility matrices are diagonal, which is the case if the univariate time series are estimated separately instead of being jointly estimated, then the empirical copula process behaves as if the innovations were observed; a remarkable property. As a by-product, one also obtains the asymptotic behavior of rank-based measures of dependence applied to residuals of these time series models.
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paper_title: A multivariate version of Hoeffding's Phi-Square

paper_content:

A multivariate measure of association is proposed, which extends the bivariate copula-based measure Phi-Square introduced by Hoeffding [22]. We discuss its analytical properties and calculate its explicit value for some copulas of simple form; a simulation procedure to approximate its value is provided otherwise. A nonparametric estimator for multivariate Phi-Square is derived and its asymptotic behavior is established based on the weak convergence of the empirical copula process both in the case of independent observations and dependent observations from strictly stationary strong mixing sequences. The asymptotic variance of the estimator can be estimated by means of nonparametric bootstrap methods. For illustration, the theoretical results are applied to financial asset return data.
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paper_title: Goodness-of-Fit Tests for Copulas of Multivariate Time Series

paper_content:

In this paper, we study the asymptotic behavior of the sequential empirical process and the sequential empirical copula process, both constructed from residuals of multivariate stochastic volatility models. Applications for the detection of structural changes and specification tests of the distribution of innovations are discussed. It is also shown that if the stochastic volatility matrices are diagonal, which is the case if the univariate time series are estimated separately instead of being jointly estimated, then the empirical copula process behaves as if the innovations were observed; a remarkable property. As a by-product, one also obtains the asymptotic behavior of rank-based measures of dependence applied to residuals of these time series models.
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paper_title: A multivariate version of Hoeffding's Phi-Square

paper_content:

A multivariate measure of association is proposed, which extends the bivariate copula-based measure Phi-Square introduced by Hoeffding [22]. We discuss its analytical properties and calculate its explicit value for some copulas of simple form; a simulation procedure to approximate its value is provided otherwise. A nonparametric estimator for multivariate Phi-Square is derived and its asymptotic behavior is established based on the weak convergence of the empirical copula process both in the case of independent observations and dependent observations from strictly stationary strong mixing sequences. The asymptotic variance of the estimator can be estimated by means of nonparametric bootstrap methods. For illustration, the theoretical results are applied to financial asset return data.
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paper_title: __________________ Volatility and Correlation Forecasting *

paper_content:

Volatility has been one of the most active and successful areas of research in time series econometrics and economic forecasting in recent decades. This chapter provides a selective survey of the most important theoretical developments and empirical insights to emerge from this burgeoning literature, with a distinct focus on forecasting applications. Volatility is inherently latent, and Section 1 begins with a brief intuitive account of various key volatility concepts. Section 2 then discusses a series of different economic situations in which volatility plays a crucial role, ranging from the use of volatility forecasts in portfolio allocation to density forecasting in risk management. Sections 3-5 present a variety of alternative procedures for univariate volatility modeling and forecasting based on the GARCH, stochastic volatility and realized volatility paradigms, respectively. Section 6 extends the discussion to the multivariate problem of forecasting conditional covariances and correlations, and Section 7 discusses volatility forecast evaluation methods in both univariate and multivariate cases. Section 8 concludes briefly.
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paper_title: Copula Methods for Forecasting Multivariate Time Series

paper_content:

Copula-based models provide a great deal of flexibility in modeling multivariate distributions, allowing the researcher to specify the models for the marginal distributions separately from the dependence structure (copula) that links them to form a joint distribution. In addition to flexibility, this often also facilitates estimation of the model in stages, reducing the computational burden. This chapter reviews the growing literature on copula-based models for economic and financial time series data, and discusses in detail methods for estimation, inference, goodness-of-fit testing, and model selection that are useful when working with these models. A representative data set of two daily equity index returns is used to illustrate all of the main results.
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paper_title: The Stationary Bootstrap

paper_content:

Abstract This article introduces a resampling procedure called the stationary bootstrap as a means of calculating standard errors of estimators and constructing confidence regions for parameters based on weakly dependent stationary observations. Previously, a technique based on resampling blocks of consecutive observations was introduced to construct confidence intervals for a parameter of the m-dimensional joint distribution of m consecutive observations, where m is fixed. This procedure has been generalized by constructing a “blocks of blocks” resampling scheme that yields asymptotically valid procedures even for a multivariate parameter of the whole (i.e., infinite-dimensional) joint distribution of the stationary sequence of observations. These methods share the construction of resampling blocks of observations to form a pseudo-time series, so that the statistic of interest may be recalculated based on the resampled data set. But in the context of applying this method to stationary data, it is natural...
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paper_title: Goodness-of-Fit Tests for Copulas of Multivariate Time Series

paper_content:

In this paper, we study the asymptotic behavior of the sequential empirical process and the sequential empirical copula process, both constructed from residuals of multivariate stochastic volatility models. Applications for the detection of structural changes and specification tests of the distribution of innovations are discussed. It is also shown that if the stochastic volatility matrices are diagonal, which is the case if the univariate time series are estimated separately instead of being jointly estimated, then the empirical copula process behaves as if the innovations were observed; a remarkable property. As a by-product, one also obtains the asymptotic behavior of rank-based measures of dependence applied to residuals of these time series models.
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paper_title: Autoregressive Conditional Density Estimation

paper_content:

R. F. Engle's autoregressive conditional heteroskedastic model is extended to permit parametric specifications for conditional dependence beyond the mean and variance. The suggestion is to model the conditional density with a small number of parameters, and then model these parameters as functions of the conditioning information. This method is applied to two data sets. The first application is to the monthly excess holding yield on U.S. Treasury securities, where the conditional density used is a Student's t distribution. The second application is to the U.S. Dollar/Swiss Franc exchange rate, using a new skewed Student t conditional distribution. Copyright 1994 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.
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paper_title: Copula Methods for Forecasting Multivariate Time Series

paper_content:

Copula-based models provide a great deal of flexibility in modeling multivariate distributions, allowing the researcher to specify the models for the marginal distributions separately from the dependence structure (copula) that links them to form a joint distribution. In addition to flexibility, this often also facilitates estimation of the model in stages, reducing the computational burden. This chapter reviews the growing literature on copula-based models for economic and financial time series data, and discusses in detail methods for estimation, inference, goodness-of-fit testing, and model selection that are useful when working with these models. A representative data set of two daily equity index returns is used to illustrate all of the main results.
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paper_title: Moment tests for density forecast evaluation in the presence of parameter estimation uncertainty

paper_content:

Density forecast (DF) possesses appealing properties when it is correctly specified for the true conditional distribution. Although a number of parametric specification tests have been introduced for the DF evaluation (DFE) in the parameter-free context, econometric DF models are typically parameter‐dependent. In this paper, we first use a generalized probability integral transformation‐based moment test to unify these existing tests, and then apply the Newey–Tauchen method (the West–McCracken method) to correct this unified test as a generalized full‐sample (out‐of‐sample) test in the parameter‐dependent context. Unlike the corrected tests, the uncorrected tests could be substantially undersized (oversized) when they are directly applied to the full‐sample (out‐of‐sample) DFE in the presence of parameter estimation uncertainty. We also use a simulation to show the usefulness of the corrected tests in rectifying the size distortion problem, and apply the corrected tests to an empirical study of stock index returns. Copyright (C) 2010 John Wiley & Sons, Ltd.
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paper_title: Estimation, Inference and Specification Analysis.

paper_content:

This book examines the consequences of misspecifications ranging from the fundamental to the nonexistent for the interpretation of likelihood-based methods of statistical estimation and interference. Professor White first explores the underlying motivation for maximum-likelihood estimation, treats the interpretation of the maximum-likelihood estimator (MLE) for misspecified probability models, and gives the conditions under which parameters of interest can be consistently estimated despite misspecification, and the consequences of misspecification, for hypothesis testing in estimating the asymptotic covariance matrix of the parameters. Although the theory presented in the book is motivated by econometric problems, its applicability is by no means restricted to economics. Subject to defined limitations, the theory applies to any scientific context in which statistical analysis is conducted using approximate models.
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paper_title: Asymptotic efficiency of the two-stage estimation method for copula-based models

paper_content:

For multivariate copula-based models for which maximum likelihood is computationally difficult, a two-stage estimation procedure has been proposed previously; the first stage involves maximum likelihood from univariate margins, and the second stage involves maximum likelihood of the dependence parameters with the univariate parameters held fixed from the first stage. Using the theory of inference functions, a partitioned matrix in a form amenable to analysis is obtained for the asymptotic covariance matrix of the two-stage estimator. The asymptotic relative efficiency of the two-stage estimation procedure compared with maximum likelihood estimation is studied. Analysis of the limiting cases of the independence copula and Frechet upper bound help to determine common patterns in the efficiency as the dependence in the model increases. For the Frechet upper bound, the two-stage estimation procedure can sometimes be equivalent to maximum likelihood estimation for the univariate parameters. Numerical results are shown for some models, including multivariate ordinal probit and bivariate extreme value distributions, to indicate the typical level of asymptotic efficiency for discrete and continuous data.
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paper_title: Estimation of multivariate models for time series of possibly different lenghts

paper_content:

We consider the problem of estimating parametric multivariate density models when unequal amounts of data are available on each variable. We focus in particular on the case that the unknown parameter vector may be partitioned into elements relating only to a marginal distribution and elements relating to the copula. In such a case we propose using a multi-stage maximum likelihood estimator (MSMLE) based on all available data rather than the usual one-stage maximum likelihood estimator (1SMLE) based only on the overlapping data. We provide conditions under which the MSMLE is not less asymptotically efficient than the 1SMLE, and we examine the small sample efficiency of the estimators via simulations. The analysis in this paper is motivated by a model of the joint distribution of daily Japanese yen-US dollar and euro-US dollar exchange rates. We find significant evidence of time variation in the conditional copula of these exchange rates, and evidence of greater dependence during extreme events than under the normal distribution. Copyright © 2006 John Wiley & Sons, Ltd.
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paper_title: Multivariate models and dependence concepts

paper_content:

Introduction. Aspects of Interpretation. Technical Considerations. Statistical Analysis. Special Methods for Joint Responses. Some Examples. Strategical Aspects. More Specialized Topics. Appendices.
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paper_title: Pseudo maximum likelihood methods : theory

paper_content:

Estimators obtained by maximizing a likelihood function are studied in the case where the true p.d.f. does not necessarily belong to the family chosen for the likelihood function. When such a procedure is applied to the estimation of the parameters of the first order moments, it is possible to prove a necessary and sufficient condition for its consistency. Asymptotic normality is shown as well as the existence of a lower bound for the asymptotic covariance matrix. It is also seen that this bound can be reached if consistent estimates are available for the parameters of the second order moments. Finally, a necessary and sufficient condition for the consistency of the pseudo maximum likelihood estimation of the first and second moments is given. IN THIS PAPER we are interested in the properties of the estimators obtained by maximizing a likelihood function associated with a family of probability distributions, which does not necessarily contain the true distribution. This kind of method will be called the pseudo maximum likelihood method.' More precisely, we determine the families of pseudo likelihood functions providing consistent and asymptotically normal estimators of parameters involved in the true distribution. After some preliminaries in Section 2, we define in Section 3 the linear exponential families and we show in Section 4 that this kind of family gives consistent and asymptotically normal estimators of the parameters appearing in the first order moment of the true distribution; moreover, it is shown that, conversely, any family with this property is necessarily linear exponential. In Section 5, we propose a generalization of the previous method adapted to the case in which it is possible to estimate consistently the second order moments of the true distribution. Finally, in Section 6 we define the quadratic exponential families and we show that these families are the only ones providing consistent estimators of the parameters appearing in the first and the second moments. Some technical proofs are gathered in five appendices and the application of the method of scoring is discussed in another one.
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paper_title: Efficient Estimation of Semiparametric Multivariate Copula Models

paper_content:

We propose a sieve maximum likelihood estimation procedure for a broad class of semiparametric multivariate distributions. A joint distribution in this class is characterized by a parametric copula function evaluated at nonparametric marginal distributions. This class of distributions has gained popularity in diverse fields due to its flexibility in separately modeling the dependence structure and the marginal behaviors of a multivariate random variable, and its circumvention of the “curse of dimensionality” associated with purely nonparametric multivariate distributions. We show that the plug-in sieve maximum likelihood estimators (MLEs) of all smooth functionals, including the finite-dimensional copula parameters and the unknown marginal distributions, are semiparametrically efficient, and that their asymptotic variances can be estimated consistently. Moreover, prior restrictions on the marginal distributions can be easily incorporated into the sieve maximum likelihood estimation procedure to achieve fu...
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paper_title: Goodness-of-Fit Tests for Copulas of Multivariate Time Series

paper_content:

In this paper, we study the asymptotic behavior of the sequential empirical process and the sequential empirical copula process, both constructed from residuals of multivariate stochastic volatility models. Applications for the detection of structural changes and specification tests of the distribution of innovations are discussed. It is also shown that if the stochastic volatility matrices are diagonal, which is the case if the univariate time series are estimated separately instead of being jointly estimated, then the empirical copula process behaves as if the innovations were observed; a remarkable property. As a by-product, one also obtains the asymptotic behavior of rank-based measures of dependence applied to residuals of these time series models.
---
paper_title: Inferences on the association parameter in copula models for bivariate survival data

paper_content:

We investigate two-stage parametric and two-stage semi-parametric estimation procedures for the association parameter in copula models for bivariate survival data where censoring in either or both components is allowed. We derive asymptotic properties of the estimators and compare their performance by simulations. Both parametric and semi-parametric estimators of the association parameter are efficient at independence, and the parameter estimates in the margins have high efficiency and are robust to misspecification of dependency structures. In addition, we propose a consistent variance estimator for the semi-parametric estimator of the association parameter. We apply the proposed methods to an AIDS data set for illustration.
---
paper_title: Estimation of copula-based semiparametric time series models

paper_content:

This paper studies the estimation of a class of copula-based semiparametric stationary Markov models. These models are characterized by nonparametric invariant (or marginal) distributions and parametric copula functions that capture the temporal dependence of the processes; the implied transition distributions are all semiparametric. Models in this class are easy to simulate, and can be expressed as semiparametric regression transformation models. One advantage of this copula approach is to separate out the temporal dependence(such as tail dependence) from the marginal behavior (such as fat tailedness) of a time series. We present conditions under which processes generated by models in this class are $\beta $-mixing; naturally, these conditions depend only on the copula specification. Simple estimators of the marginal distribution and the copula parameter are provided, and their asymptotic properties are established under easily verifiable conditions. Estimators of important features of the transition distribution such as the (nonlinear) conditional moments and conditional quantiles are easily obtained from estimators of the marginal distribution and the copula parameter; their $\sqrt{n}-$ consistency and asymptotic normality can be obtained using the Delta method. In addition, the semiparametric conditional quantile estimators are automatically monotonic across quantiles.
---
paper_title: A semiparametric estimation procedure of dependence parameters in multivariate families of distributions

paper_content:

SUMMARY This paper investigates the properties of a semiparametric method for estimating the dependence parameters in a family of multivariate distributions. The proposed estimator, obtained as a solution of a pseudo-likelihood equation, is shown to be consistent, asymptotically normal and fully efficient at independence. A natural estimator of its asymptotic variance is proved to be consistent. Comparisons are made with alternative semiparametric estimators in the special case of Clayton's model for association in bivariate data.
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paper_title: STATISTICAL INFERENCE FOR MULTIVARIATE RESIDUAL COPULA OF GARCH MODELS

paper_content:

Recently a flexible class of semiparametric copula-based multivariate GARCH models has been proposed to quantify multivariate risks, in which uni- variate GARCH models are used to capture the dynamics of individual financial series, and parametric copulas are used to model the contemporaneous dependence among GARCH residuals with nonparametric marginals. In this paper we address two questions regarding statistical inference for this class of models. (1) Under what mild sufficient conditions is the asymptotic distribution of the pseudo max- imum likelihood estimator (MLE) of the residual copula parameter of Chen and Fan (2006a) justified? (2) How do we test the correct specification of a parametric copula for the GARCH residuals? In order to answer both questions rigorously, we establish a new weighted approximation for the empirical distributions of the GARCH residuals, which is of interest in its own right. Simulation studies and data examples are provided to examine the finite sample performance of the pseudo MLE of the residual copula parameter and the proposed goodness-of-fit test.
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paper_title: Statistical Inference Procedures for Bivariate Archimedean Copulas

paper_content:

Abstract A bivariate distribution function H(x, y) with marginals F(x) and G(y) is said to be generated by an Archimedean copula if it can be expressed in the form H(x, y) = ϕ–1[ϕ{F(x)} + ϕ{G(y)}] for some convex, decreasing function ϕ defined on [0, 1] in such a way that ϕ(1) = 0. Many well-known systems of bivariate distributions belong to this class, including those of Gumbel, Ali-Mikhail-Haq-Thelot, Clayton, Frank, and Hougaard. Frailty models also fall under that general prescription. This article examines the problem of selecting an Archimedean copula providing a suitable representation of the dependence structure between two variates X and Y in the light of a random sample (X 1, Y 1), …, (X n , Y n ). The key to the estimation procedure is a one-dimensional empirical distribution function that can be constructed whether the uniform representation of X and Y is Archimedean or not, and independently of their marginals. This semiparametric estimator, based on a decomposition of Kendall's tau statistic...
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paper_title: Rank-based inference for bivariate extreme-value copulas

paper_content:

Consider a continuous random pair (X, Y ) whose dependence is characterized by an extreme-value copula with Pickands dependence function A. When the marginal distributions of X and Y are known, several consistent estimators of A are available. Most of them are variants of the estimators due to Pickands [Bull. Inst. Internat. Statist. 49 (1981) 859–878] and Cap´era`a, Foug`eres and Genest [Biometrika 84 (1997) 567–577]. In this paper, rank-based versions of these estimators are proposed for the more common case where the margins of X and Y are unknown. Results on the limit behavior of a class of weighted bivariate empirical processes are used to show the consistency and asymptotic normality of these rank-based estimators. Their finite- and large-sample performance is then compared to that of their known-margin analogues, as well as with endpoint-corrected versions thereof. Explicit formulas and consistent estimates for their asymptotic variances are also suggested
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paper_title: THE BERNSTEIN COPULA AND ITS APPLICATIONS TO MODELING AND APPROXIMATIONS OF MULTIVARIATE DISTRIBUTIONS

paper_content:

We define the Bernstein copula and study its statistical properties in terms of both distributions and densities. We also develop a theory of approximation for multivariate distributions in terms of Bernstein copulas. Rates of consistency when the Bernstein copula density is estimated empirically are given. In order of magnitude, this estimator has variance equal to the square root of the variance of common nonparametric estimators, e.g., kernel smoothers, but it is biased as a histogram estimator. We would thank Mark Salmon for interesting us in the copula function and Peter Phillips, an associate editor, and the referees for many valuable comments. All remaining errors are our sole responsibility.
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paper_title: ESTIMATORS BASED ON KENDALL'S TAU IN MULTIVARIATE COPULA MODELS

paper_content:

Summary ::: The estimation of a real-valued dependence parameter in a multivariate copula model is considered. Rank-based procedures are often used in this context to guard against possible misspecification of the marginal distributions. A standard approach consists of maximizing the pseudo-likelihood. Here, we investigate alternative estimators based on the inversion of two multivariate extensions of Kendall's tau developed by Kendall and Babington Smith, and by Joe. The former, which amounts to the average value of tau over all pairs of variables, is often referred to as the coefficient of agreement. Existing results concerning the finite- and large-sample properties of this coefficient are summarized, and new, parallel findings are provided for the multivariate version of tau due to Joe, along with illustrations. The performance of the estimators resulting from the inversion of these two versions of Kendall's tau is compared in the context of copula models through simulations.
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paper_title: Goodness-of-Fit Tests for Copulas of Multivariate Time Series

paper_content:

In this paper, we study the asymptotic behavior of the sequential empirical process and the sequential empirical copula process, both constructed from residuals of multivariate stochastic volatility models. Applications for the detection of structural changes and specification tests of the distribution of innovations are discussed. It is also shown that if the stochastic volatility matrices are diagonal, which is the case if the univariate time series are estimated separately instead of being jointly estimated, then the empirical copula process behaves as if the innovations were observed; a remarkable property. As a by-product, one also obtains the asymptotic behavior of rank-based measures of dependence applied to residuals of these time series models.
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paper_title: Frank's family of bivariate distributions

paper_content:

SUMMARY This paper examines the properties of a new class of bivariate distributions whose members are stochastically ordered and likelihood ratio dependent. The proposed class can be used to construct bivariate families of distributions whose marginals are arbitrary and which include the Frechet bounds as well as the distribution corresponding to independent variables. Three nonparametric estimators of the association parameter are suggested and Monte Carlo experiments are used to compare their small-sample behaviour to that of the maximum likelihood estimate.
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paper_title: Simulated Method of Moments Estimation for Copula-Based Multivariate Models

paper_content:

This article considers the estimation of the parameters of a copula via a simulated method of moments (MM) type approach. This approach is attractive when the likelihood of the copula model is not known in closed form, or when the researcher has a set of dependence measures or other functionals of the copula that are of particular interest. The proposed approach naturally also nests MM and generalized method of moments estimators. Drawing on results for simulation-based estimation and on recent work in empirical copula process theory, we show the consistency and asymptotic normality of the proposed estimator, and obtain a simple test of overidentifying restrictions as a specification test. The results apply to both iid and time series data. We analyze the finite-sample behavior of these estimators in an extensive simulation study. We apply the model to a group of seven financial stock returns and find evidence of statistically significant tail dependence, and mild evidence that the dependence between these assets is stronger in crashes than booms. Supplementary materials for this article are available online.
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paper_title: Modelling Longitudinal Data using a Pair-Copula Decomposition of Serial Dependence

paper_content:

Copulas have proven to be very successful tools for the flexible modeling of cross-sectional dependence. In this paper we express the dependence structure of continuous-valued time series data using a sequence of bivariate copulas. This corresponds to a type of decomposition recently called a “vine” in the graphical models literature, where each copula is entitled a “pair-copula.” We propose a Bayesian approach for the estimation of this dependence structure for longitudinal data. Bayesian selection ideas are used to identify any independence pair-copulas, with the end result being a parsimonious representation of a time-inhomogeneous Markov process of varying order. Estimates are Bayesian model averages over the distribution of the lag structure of the Markov process. Using a simulation study we show that the selection approach is reliable and can improve the estimates of both conditional and unconditional pairwise dependencies substantially. We also show that a vine with selection outperforms a Gaussian...
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paper_title: Efficient Bayesian inference for Gaussian copula regression models

paper_content:

A Gaussian copula regression model gives a tractable way of handling a multivariate regression when some of the marginal distributions are non-Gaussian. Our paper presents a general Bayesian approach for estimating a Gaussian copula model that can handle any combination of discrete and continuous marginals, and generalises Gaussian graphical models to the Gaussian copula framework. Posterior inference is carried out using a novel and efficient simulation method. The methods in the paper are applied to simulated and real data. Copyright 2006, Oxford University Press.
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paper_title: Modelling dependence using skew t copulas: Bayesian inference and applications

paper_content:

We construct a copula from the skew t distribution of Sahu, Dey & Branco (2003). This copula can capture asymmetric and extreme dependence between variables, and is one of the few copulas that can do so and still be used in high dimensions effectively. However, it is difficult to estimate the copula model by maximum likelihood when the multivariate dimension is high, or when some or all of the marginal distributions are discrete-valued, or when the parameters in the marginal distributions and copula are estimated jointly. We therefore propose a Bayesian approach that overcomes all these problems. The computations are undertaken using a Markov chain Monte Carlo simulation method which exploits the conditionally Gaussian representation of the skew t distribution. We employ the approach in two contemporary econometric studies. The first is the modeling of regional spot prices in the Australian electricity market. Here, we observe complex non-Gaussian margins and nonlinear inter-regional dependence. Accurate characterization of this dependence is important for the study of market integration and risk management purposes. The second is the modeling of ordinal exposure measures for 15 major websites. Dependence between websites is important when measuring the impact of multi-site advertising campaigns. In both cases the skew t copula substantially out-performs symmetric elliptical copula alternatives, demonstrating that the skew t copula is a powerful modeling tool when coupled with Bayesian inference.
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paper_title: Bayesian Approaches to Copula Modelling

paper_content:

Copula models have become one of the most widely used tools in the applied modelling of multivariate data. Similarly, Bayesian methods are increasingly used to obtain efficient likelihood-based inference. However, to date, there has been only limited use of Bayesian approaches in the formulation and estimation of copula models. This article aims to address this shortcoming in two ways. First, to introduce copula models and aspects of copula theory that are especially relevant for a Bayesian analysis. Second, to outline Bayesian approaches to formulating and estimating copula models, and their advantages over alternative methods. Copulas covered include Archimedean, copulas constructed by inversion, and vine copulas; along with their interpretation as transformations. A number of parameterisations of a correlation matrix of a Gaussian copula are considered, along with hierarchical priors that allow for Bayesian selection and model averaging for each parameterisation. Markov chain Monte Carlo sampling schemes for fitting Gaussian and D-vine copulas, with and without selection, are given in detail. The relationship between the prior for the parameters of a D-vine, and the prior for a correlation matrix of a Gaussian copula, is discussed. Last, it is shown how to compute Bayesian inference when the data are discrete-valued using data augmentation. This approach generalises popular Bayesian methods for the estimation of models for multivariate binary and other ordinal data to more general copula models. Bayesian data augmentation has substantial advantages over other methods of estimation for this class of models.
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paper_title: Copula Methods for Forecasting Multivariate Time Series

paper_content:

Copula-based models provide a great deal of flexibility in modeling multivariate distributions, allowing the researcher to specify the models for the marginal distributions separately from the dependence structure (copula) that links them to form a joint distribution. In addition to flexibility, this often also facilitates estimation of the model in stages, reducing the computational burden. This chapter reviews the growing literature on copula-based models for economic and financial time series data, and discusses in detail methods for estimation, inference, goodness-of-fit testing, and model selection that are useful when working with these models. A representative data set of two daily equity index returns is used to illustrate all of the main results.
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paper_title: Goodness-of-Fit Tests for Copulas of Multivariate Time Series

paper_content:

In this paper, we study the asymptotic behavior of the sequential empirical process and the sequential empirical copula process, both constructed from residuals of multivariate stochastic volatility models. Applications for the detection of structural changes and specification tests of the distribution of innovations are discussed. It is also shown that if the stochastic volatility matrices are diagonal, which is the case if the univariate time series are estimated separately instead of being jointly estimated, then the empirical copula process behaves as if the innovations were observed; a remarkable property. As a by-product, one also obtains the asymptotic behavior of rank-based measures of dependence applied to residuals of these time series models.
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paper_title: Autoregressive Conditional Density Estimation

paper_content:

R. F. Engle's autoregressive conditional heteroskedastic model is extended to permit parametric specifications for conditional dependence beyond the mean and variance. The suggestion is to model the conditional density with a small number of parameters, and then model these parameters as functions of the conditioning information. This method is applied to two data sets. The first application is to the monthly excess holding yield on U.S. Treasury securities, where the conditional density used is a Student's t distribution. The second application is to the U.S. Dollar/Swiss Franc exchange rate, using a new skewed Student t conditional distribution. Copyright 1994 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.
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paper_title: Estimation of copula-based semiparametric time series models

paper_content:

This paper studies the estimation of a class of copula-based semiparametric stationary Markov models. These models are characterized by nonparametric invariant (or marginal) distributions and parametric copula functions that capture the temporal dependence of the processes; the implied transition distributions are all semiparametric. Models in this class are easy to simulate, and can be expressed as semiparametric regression transformation models. One advantage of this copula approach is to separate out the temporal dependence(such as tail dependence) from the marginal behavior (such as fat tailedness) of a time series. We present conditions under which processes generated by models in this class are $\beta $-mixing; naturally, these conditions depend only on the copula specification. Simple estimators of the marginal distribution and the copula parameter are provided, and their asymptotic properties are established under easily verifiable conditions. Estimators of important features of the transition distribution such as the (nonlinear) conditional moments and conditional quantiles are easily obtained from estimators of the marginal distribution and the copula parameter; their $\sqrt{n}-$ consistency and asymptotic normality can be obtained using the Delta method. In addition, the semiparametric conditional quantile estimators are automatically monotonic across quantiles.
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paper_title: Copulas and Temporal Dependence

paper_content:

An emerging literature in time series econometrics concerns the modeling of potentially nonlinear temporal dependence in stationary Markov chains using copula functions. We obtain sucient conditions for a geometric rate of mixing in models of this kind. Geometric beta-mixing is established under a rather strong sucient condition that rules out asymmetry and tail dependence in the copula function. Geometric rho-mixing is obtained under a weaker condition that permits both asymmetry and tail dependence. We verify one or both of these conditions for a range of parametric copula functions that are popular in applied work.
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paper_title: Dynamic Copula Quantile Regressions and Tail Area Dynamic Dependence in Forex Markets

paper_content:

We introduce a general approach to nonlinear quantile regression modelling based on the copula function that defines the dependency structure between the variables of interest. Hence, we extend Koenker and Bassett's (1978. Regression quantiles. Econometrica, 46, no. 1: 33-50.) original statement of the quantile regression problem by determining a distribution for the dependent variable Y conditional on the regressors X, and hence the specification of the quantile regression functions. The approach exploits the fact that the joint distribution function can be split into two parts: the marginals and the dependence function (or copula). We then deduce the form of the (invariably nonlinear) conditional quantile relationship implied by the copula. This can be achieved with arbitrary distributions assumed for the marginals. Some properties of the copula-based quantiles or c-quantiles are derived. Finally, we examine the conditional quantile dependency in the foreign exchange market and compare our quantile approach with standard tail area dependency measures.
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paper_title: Statistical properties of parametric estimators for Markov chain vectors based on copula models

paper_content:

To estimate and measure risks, two key classes of dependence relationship must be identified: temporal dependence and contemporaneous dependence. In this paper, we propose a parametric estimation model that uses a three-stage pseudo maximum likelihood estimation (3SPMLE), and we investigate the consistency and asymptotic normality of parametric estimators. The proposed model combines the concept of a copula and the methods of parametric estimators of two-stage pseudo maximum likelihood estimation (2SPMLE). The selection of a copula model that best captures the dependence structure is a critical problem. To solve this problem, we propose a model selection method that is based on the parametric pseudo-likelihood ratio under the 3SPMLE for stationary Markov vector-type models.
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paper_title: ARCHIMEDEAN COPULAS AND TEMPORAL DEPENDENCE

paper_content:

We study the dependence properties of stationary Markov chains generated by Archimedean copulas. Under some simple regularity conditions, we show that regular variation of the Archimedean generator at zero and one implies geometric orgodicityof the associated Markov chain. We verify our assumptions for a range of Archimedean copulas used in applications.
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paper_title: Time irreversible copula-based Markov Models

paper_content:

Economic and financial time series frequently exhibit time irreversible dynamics. For instance, there is considerable evidence of asymmetric fluctuations in many macroeconomic and financial variables, and certain game theoretic models of price determination predict asymmetric cycles in price series. In this paper we make two primary contributions to the econometric literature on time reversibility. First, we propose a new test of time reversibility, applicable to stationary Markov chains. Compared to existing tests, our test has the advantage of being consistent against arbitrary violations of reversibility. Second, we explain how a circulation density function may be used to characterize the nature of time irreversibility when it is present. We propose a copula-based estimator of the circulation density, and verify that it is well behaved asymptotically under suitable regularity conditions. We illustrate the use of our time reversibility test and circulation density estimator by applying them to five years of Canadian gasoline price markup data.
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paper_title: Efficient Estimation of Copula-Based Semiparametric Markov Models

paper_content:

This paper considers efficient estimation of copula-based semiparametric strictly stationary Markov models. These models are characterized by nonparametric invariant (one-dimensional marginal) distributions and parametric bivariate copula functions; where the copulas capture temporal dependence and tail dependence of the processes. The Markov processes generated via tail dependent copulas may look highly persistent and are useful for financial and economic applications. We first show that Markov processes generated via Clayton, Gumbel and Student's $t$ copulas and their survival copulas are all geometrically ergodic. We then propose a sieve maximum likelihood estimation (MLE) for the copula parameter, the invariant distribution and the conditional quantiles. We show that the sieve MLEs of any smooth functionals are root-$n$ consistent, asymptotically normal and efficient; and that their sieve likelihood ratio statistics are asymptotically chi-square distributed. We present Monte Carlo studies to compare the finite sample performance of the sieve MLE, the two-step estimator of Chen and Fan (2006), the correctly specified parametric MLE and the incorrectly specified parametric MLE. The simulation results indicate that our sieve MLEs perform very well; having much smaller biases and smaller variances than the two-step estimator for Markov models generated via Clayton, Gumbel and other tail dependent copulas.
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paper_title: Copula credibility for aggregate loss models

paper_content:

Abstract This paper develops credibility predictors of aggregate losses using a longitudinal data framework. For a model of aggregate losses, the interest is in predicting both the claims number process as well as the claims amount process. In a longitudinal data framework, one encounters data from a cross-section of risk classes with a history of insurance claims available for each risk class. Further, explanatory variables for each risk class over time are available to help explain and predict both the claims number and claims amount process. For the marginal claims distributions, this paper uses generalized linear models, an extension of linear regression, to describe cross-sectional characteristics. Elliptical copulas are used to model the dependencies over time, extending prior work that used multivariate t -copulas. The claims number process is represented using a Poisson regression model that is conditioned on a sequence of latent variables. These latent variables drive the serial dependencies among claims numbers; their joint distribution is represented using an elliptical copula. In this way, the paper provides a unified treatment of both the continuous claims amount and discrete claims number processes. The paper presents an illustrative example of Massachusetts automobile claims. Estimates of the latent claims process parameters are derived and simulated predictions are provided.
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paper_title: Copula-Based Nonlinear Quantile Autoregression

paper_content:

Parametric copulas are shown to be attractive devices for specifying quantile autoregressive models for nonlinear time-series. Estimation of local, quantile-specific copula-based time series models offers some salient advantages over classical global parametric approaches. Consistency and asymptotic normality of the proposed quantile estimators are established under mild conditions, allowing for global misspecification of parametric copulas and marginals, and without assuming any mixing rate condition. These results lead to a general framework for inference and model specification testing of extreme conditional value-at-risk for financial time series data.
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paper_title: Heavy-tailed longitudinal data modeling using copulas

paper_content:

In this paper, we consider "heavy-tailed" data, that is, data where extreme values are likely to occur. Heavy-tailed data have been analyzed using flexible distributions such as the generalized beta of the second kind, the generalized gamma and the Burr. These distributions allow us to handle data with either positive or negative skewness, as well as heavy tails. Moreover, it has been shown that they can also accommodate cross-sectional regression models by allowing functions of explanatory variables to serve as distribution parameters. The objective of this paper is to extend this literature to accommodate longitudinal data, where one observes repeated observations of cross-sectional data. Specifically, we use copulas to model the dependencies over time, and heavy-tailed regression models to represent the marginal distributions. We also introduce model exploration techniques to help us with the initial choice of the copula and a goodness-of-fit test of elliptical copulas for model validation. In a longitudinal data context, we argue that elliptical copulas will be typically preferred to the Archimedean copulas. To illustrate our methods, Wisconsin nursing homes utilization data from 1995 to 2001 are analyzed. These data exhibit long tails and negative skewness and so help us to motivate the need for our new techniques. We find that time and the nursing home facility size as measured through the number of beds and square footage are important predictors of future utilization. Moreover, using our parametric model, we provide not only point predictions but also an entire predictive distribution.
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paper_title: Estimation of copula-based semiparametric time series models

paper_content:

This paper studies the estimation of a class of copula-based semiparametric stationary Markov models. These models are characterized by nonparametric invariant (or marginal) distributions and parametric copula functions that capture the temporal dependence of the processes; the implied transition distributions are all semiparametric. Models in this class are easy to simulate, and can be expressed as semiparametric regression transformation models. One advantage of this copula approach is to separate out the temporal dependence(such as tail dependence) from the marginal behavior (such as fat tailedness) of a time series. We present conditions under which processes generated by models in this class are $\beta $-mixing; naturally, these conditions depend only on the copula specification. Simple estimators of the marginal distribution and the copula parameter are provided, and their asymptotic properties are established under easily verifiable conditions. Estimators of important features of the transition distribution such as the (nonlinear) conditional moments and conditional quantiles are easily obtained from estimators of the marginal distribution and the copula parameter; their $\sqrt{n}-$ consistency and asymptotic normality can be obtained using the Delta method. In addition, the semiparametric conditional quantile estimators are automatically monotonic across quantiles.
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paper_title: Modeling statistical dependence of Markov chains via copula models

paper_content:

Conditional probability distributions have been commonly used in modeling Markov chains. In this paper we consider an alternative approach based on copulas to investigate Markov-type dependence structures. Based on the realization of a single Markov chain, we estimate the parameters using one- and two-stage estimation procedures. We derive asymptotic properties of the marginal and copula parameter estimators and compare performance of the estimation procedures based on Monte Carlo simulations. At low and moderate dependence structures the two-stage estimation has comparable performance as the maximum likelihood estimation. In addition we propose a parametric pseudo-likelihood ratio test for copula model selection under the two-stage procedure. We apply the proposed methods to an environmental data set.
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paper_title: Modelling dependence using skew t copulas: Bayesian inference and applications

paper_content:

We construct a copula from the skew t distribution of Sahu, Dey & Branco (2003). This copula can capture asymmetric and extreme dependence between variables, and is one of the few copulas that can do so and still be used in high dimensions effectively. However, it is difficult to estimate the copula model by maximum likelihood when the multivariate dimension is high, or when some or all of the marginal distributions are discrete-valued, or when the parameters in the marginal distributions and copula are estimated jointly. We therefore propose a Bayesian approach that overcomes all these problems. The computations are undertaken using a Markov chain Monte Carlo simulation method which exploits the conditionally Gaussian representation of the skew t distribution. We employ the approach in two contemporary econometric studies. The first is the modeling of regional spot prices in the Australian electricity market. Here, we observe complex non-Gaussian margins and nonlinear inter-regional dependence. Accurate characterization of this dependence is important for the study of market integration and risk management purposes. The second is the modeling of ordinal exposure measures for 15 major websites. Dependence between websites is important when measuring the impact of multi-site advertising campaigns. In both cases the skew t copula substantially out-performs symmetric elliptical copula alternatives, demonstrating that the skew t copula is a powerful modeling tool when coupled with Bayesian inference.
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paper_title: Duration time-series models with proportional hazard

paper_content:

The analysis of liquidity in financial markets is generally performed by means of the dynamics of the observed intertrade durations (possibly weighted by price or volume). Various dynamic models for duration data have been considered in the literature, such as the Autoregressive Conditional Duration (ACD) model. These models are often excessively constrained, introducing, for example, a deterministic link between conditional expectation and variance in the case of the ACD model. Moreover, the stationarity properties and the patterns of the stationary distributions are often unknown. The aim of this article is to solve these difficulties by considering a duration time series satisfying the proportional hazard property. We describe in detail this class of dynamic models, discuss its various representations and provide the ergodicity conditions. The proportional hazard copula can be specified either parametrically, or nonparametrically. We discuss estimation methods in both contexts, and explain why they are efficient, that is, why they reach the parametric (respectively, nonparametric) efficiency bound. Copyright 2007 The Authors Journal compilation 2007 Blackwell Publishing Ltd.
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paper_title: Copula-based semiparametric models for multivariate time series

paper_content:

The authors extend to multivariate contexts the copula-based univariate time series modeling approach of Chen & Fan [X. Chen, Y. Fan, Estimation of copula-based semiparametric time series models, J. Econometrics 130 (2006) 307-335; X. Chen, Y. Fan, Estimation and model selection of semiparametric copula-based multivariate dynamic models under copula misspecification, J. Econometrics 135 (2006) 125-154]. In so doing, they tackle simultaneously serial dependence and interdependence between time series. Their technique differs from the usual approach to time series copula modeling in which the series are first modeled individually and copulas are used to model the dependence between their innovations. The authors discuss parameter estimation and goodness-of-fit testing for their model, with emphasis on meta-elliptical and Archimedean copulas. The method is illustrated with data on the Canadian/US exchange rate and the value of oil futures over a ten-year period.
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paper_title: Multivariate models and dependence concepts

paper_content:

Introduction. Aspects of Interpretation. Technical Considerations. Statistical Analysis. Special Methods for Joint Responses. Some Examples. Strategical Aspects. More Specialized Topics. Appendices.
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paper_title: Goodness-of-Fit Tests for Copulas of Multivariate Time Series

paper_content:

In this paper, we study the asymptotic behavior of the sequential empirical process and the sequential empirical copula process, both constructed from residuals of multivariate stochastic volatility models. Applications for the detection of structural changes and specification tests of the distribution of innovations are discussed. It is also shown that if the stochastic volatility matrices are diagonal, which is the case if the univariate time series are estimated separately instead of being jointly estimated, then the empirical copula process behaves as if the innovations were observed; a remarkable property. As a by-product, one also obtains the asymptotic behavior of rank-based measures of dependence applied to residuals of these time series models.
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paper_title: Copula goodness-of-fit testing: an overview and power comparison

paper_content:

Several copula goodness-of-fit approaches are examined, three of which are proposed in this paper. Results are presented from an extensive Monte Carlo study, where we examine the effect of dimension, sample size and strength of dependence on the nominal level and power of the different approaches. While no approach is always the best, some stand out and conclusions and recommendations are made. A novel study of p-value variation due to permutation order, for approaches based on Rosenblatt's transformation is also carried out. Results show significant variation due to permutation order for some of the approaches based on this transform. However, when approaching rejection regions, the additional variation is negligible.
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paper_title: Moment tests for density forecast evaluation in the presence of parameter estimation uncertainty

paper_content:

Density forecast (DF) possesses appealing properties when it is correctly specified for the true conditional distribution. Although a number of parametric specification tests have been introduced for the DF evaluation (DFE) in the parameter-free context, econometric DF models are typically parameter‐dependent. In this paper, we first use a generalized probability integral transformation‐based moment test to unify these existing tests, and then apply the Newey–Tauchen method (the West–McCracken method) to correct this unified test as a generalized full‐sample (out‐of‐sample) test in the parameter‐dependent context. Unlike the corrected tests, the uncorrected tests could be substantially undersized (oversized) when they are directly applied to the full‐sample (out‐of‐sample) DFE in the presence of parameter estimation uncertainty. We also use a simulation to show the usefulness of the corrected tests in rectifying the size distortion problem, and apply the corrected tests to an empirical study of stock index returns. Copyright (C) 2010 John Wiley & Sons, Ltd.
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paper_title: Local Power Analyses of Goodness-of-fit Tests for Copulas

paper_content:

The asymptotic behaviour of several goodness-of-fit statistics for copula families is obtained under contiguous alternatives. Many comparisons between a Cramer-von Mises functional of the empirical copula process and new moment-based goodness-of-fit statistics are made by considering their associated asymptotic local power curves. It is shown that the choice of the estimator for the unknown parameter can have a significant influence on the power of the Cramer-von Mises test and that some of the moment-based statistics can provide simple and efficient goodness-of-fit methods. Copyright (c) 2009 Board of the Foundation of the Scandinavian Journal of Statistics.
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paper_title: Goodness-of-fit tests for copulas: A review and a power study

paper_content:

Many proposals have been made recently for goodness-of-fit testing of copula models. After reviewing them briefly, the authors concentrate on "blanket tests", i.e.,Â those whose implementation requires neither an arbitrary categorization of the data nor any strategic choice of smoothing parameter, weight function, kernel, window, etc. The authors present a critical review of these procedures and suggest new ones. They describe and interpret the results of a large Monte Carlo experiment designed to assess the effect of the sample size and the strength of dependence on the level and power of the blanket tests for various combinations of copula models under the null hypothesis and the alternative. To circumvent problems in the determination of the limiting distribution of the test statistics under composite null hypotheses, they recommend the use of a double parametric bootstrap procedure, whose implementation is detailed. They conclude with a number of practical recommendations.
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paper_title: Multivariate Density Forecast Evaluation and Calibration In Financial Risk Management: High-Frequency Returns on Foreign Exchange

paper_content:

We provide a framework for evaluating and improving multivariate density forecasts. Among other things, the multivariate framework lets us evaluate the adequacy of density forecasts involving cross-variable interactions, such as time-varying conditional correlations. We also provide conditions under which a technique of density forecast "calibration" can be used to improve deficient density forecasts, and we show how the calibration method can be used to generate good density forecasts from econometric models, even when the conditional density is unknown. Finally, motivated by recent advances in financial risk management, we provide a detailed application to multivariate high-frequency exchange rate density forecasts. © 2000 by the President and Fellows of Harvard College and the Massachusetts Institute of Technology
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paper_title: Goodness-of-Fit Tests for Copulas of Multivariate Time Series

paper_content:

In this paper, we study the asymptotic behavior of the sequential empirical process and the sequential empirical copula process, both constructed from residuals of multivariate stochastic volatility models. Applications for the detection of structural changes and specification tests of the distribution of innovations are discussed. It is also shown that if the stochastic volatility matrices are diagonal, which is the case if the univariate time series are estimated separately instead of being jointly estimated, then the empirical copula process behaves as if the innovations were observed; a remarkable property. As a by-product, one also obtains the asymptotic behavior of rank-based measures of dependence applied to residuals of these time series models.
---
paper_title: Estimation of copula-based semiparametric time series models

paper_content:

This paper studies the estimation of a class of copula-based semiparametric stationary Markov models. These models are characterized by nonparametric invariant (or marginal) distributions and parametric copula functions that capture the temporal dependence of the processes; the implied transition distributions are all semiparametric. Models in this class are easy to simulate, and can be expressed as semiparametric regression transformation models. One advantage of this copula approach is to separate out the temporal dependence(such as tail dependence) from the marginal behavior (such as fat tailedness) of a time series. We present conditions under which processes generated by models in this class are $\beta $-mixing; naturally, these conditions depend only on the copula specification. Simple estimators of the marginal distribution and the copula parameter are provided, and their asymptotic properties are established under easily verifiable conditions. Estimators of important features of the transition distribution such as the (nonlinear) conditional moments and conditional quantiles are easily obtained from estimators of the marginal distribution and the copula parameter; their $\sqrt{n}-$ consistency and asymptotic normality can be obtained using the Delta method. In addition, the semiparametric conditional quantile estimators are automatically monotonic across quantiles.
---
paper_title: A New Approach to Measuring Financial Contagion

paper_content:

This article proposes a new approach to evaluate contagion in financial markets. Our measure of contagion captures the coincidence of extreme return shocks across countries within a region and across regions. We characterize the extent of contagion, its economic significance, and its determinants using a multinomial logistic regression model. Applying our approach to daily returns of emerging markets during the 1990s, we find that contagion is predictable and depends on regional interest rates, exchange rate changes, and conditional stock return volatility. Evidence that contagion is stronger for extreme negative returns than for extreme positive returns is mixed. Copyright 2003, Oxford University Press.
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paper_title: Extreme Correlation of International Equity Markets

paper_content:

Testing the hypothesis that international equity market correlation increases in volatile times is a difficult exercise and misleading results have often been reported in the past because of a spurious relationship between correlation and volatility. This paper focuses on extreme correlation, that is to say the correlation between returns in either the negative or positive tail of the multivariate distribution. Using "extreme value theory" to model the multivariate distribution tails, we derive the distribution of extreme correlation for a wide class of return distributions. Using monthly data on the five largest stock markets from 1958 to 1996, we reject the null hypothesis of multivariate normality for the negative tail, but not for the positive tail. We also find that correlation is not related to market volatility per se but to the market trend. Correlation increases in bear markets, but not in bull markets.
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paper_title: Asymmetric correlations of equity portfolios

paper_content:

Correlations between US stocks and the aggregate US market are much greater for downside moves, especially for extreme downside moves, than for upside moves. We develop a new statistic for measuring, comparing and testing asymmetries in conditional correlations. Conditional on the downside, correlations in the data differ from the conditional correlations implied by a normal distribution by 11.6%. We find that conditional asymmetric correlations are fundamentally different from other measures of asymmetry like skewness and co-skewness. We find that small stocks, value stocks and past loser stocks have more asymmetric movements. Controlling for size, we find that stocks with lower betas exhibit greater correlation asymmetries and we find no relationship between leverage and correlation asymmetries. Correlation asymmetries in the data reject the null of multivariate normal distributions at daily, weekly and monthly frequencies. However, several empirical models with greater flexibility, particularly regime-switching models, perform much better at capturing correlation asymmetries.
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paper_title: Forecasting International Equity Correlations

paper_content:

An important component of asset allocation decisions is the future correlation structure of equity returns. Other studies have found that correlations change through time. Examination of the changing cross-country correlations in the G-7 countries provides clues as to why they change. Equity cross-correlations are related to the coherence between business cycles in the respective countries. Correlations are higher during recessions than during growth periods. Correlations are low when two countries' business cycles are out of phase. A semicorrelation metric differentiates equity comovements in bull and bear markets and provides a method for forecasting multiperiod equity correlations. Two applications are investigated: out-of-sample global portfolio allocation and derivative instruments.
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paper_title: Quantile Prediction

paper_content:

This chapter is concerned with the problem of quantile prediction (or forecasting). There are numerous applications in economics and finance where quantiles are of interest. We primarily focus on methods that are relevant for dynamic time series data. The chapter is organized around two key questions: first, how to measure and forecast the conditional quantiles of some series of interest given the information currently available and second, how to assess the accuracy of alternative conditional quantile predictors.
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paper_title: Extreme VaR scenarios in higher dimensions

paper_content:

For a sequence of random variables X1, ..., Xn, the dependence scenario yielding the worst possible Value-at-Risk at a given level α for X1+...+Xn is known for n=2. In this paper we investigate this problem for higher dimensions. We provide a geometric interpretation highlighting the dependence structures which imply the worst possible scenario. For a portfolio (X1,..., Xn) with given uniform marginals, we give an analytical solution sustaining the main result of Ruschendorf (Adv. Appl. Probab. 14(3):623–632, 1982). In general, our approach allows for numerical computations.
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paper_title: Value at Risk trade-off and capital allocation with copulas

paper_content:

type="main" xml:lang="en"> This paper uses copula functions to evaluate tail probabilities and market risk trade-offs at a given confidence level, dropping the joint normality assumption on returns. Copulas enable one to represent distribution functions separating the marginal distributions from the association structure. We present an application to two stock market indices: for each market we recover the marginal probability distribution. We then calibrate copula functions and recover the joint distribution. The estimated copulas directly give the joint probabilities of extreme losses. Their level curves measure the trade-off between losses over different desks. This trade-off can be exploited for capital allocation and is shown to depend on fat tails. (J.E.L.: C14, G19, G29).
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paper_title: Using copulae to bound the value-at-risk for functions of dependent risks

paper_content:

The theory of copulae is known to provide a useful tool for modelling dependence in integrated risk management. In the present paper we review and extend some of the more recent results for finding distributional bounds for functions of dependent risks. As an example, the main emphasis is put on Value-at-Risk as a risk measure.
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paper_title: Option prices and risk-neutral densities for currency cross rates

paper_content:

The theoretical relationship between the risk‐neutral density (RND) of the euro/ pound cross rate and the bivariate RND of the dollar/euro and the dollar/pound rates is derived; the required bivariate RND is defined by the dollar‐rate marginal RNDs and a copula function. The cross‐rate RND can be used by banks, international businesses, and central bankers to assess market expectations, to measure risks, and to value options, without relying on over‐the‐counter markets, which may be either non‐existent or illiquid. Empirical comparisons are made between cross‐rate RNDs estimated from several data sets. Five one‐parameter copula functions are evaluated and it is found that the Gaussian copula is the only one‐parameter copula function that is ranked highly in all of the comparisons we have made. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:324–360, 2010
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paper_title: CDO pricing with nested Archimedean copulas

paper_content:

Companies in the same industry sector are usually more correlated than firms in different sectors, as they are similarly affected by macroeconomic effects, political decisions, and consumer trends. Despite the many stock return models taking this fact into account, there are only a few credit default models that take it into consideration. In this paper we present a default model based on nested Archimedean copulas that is able to capture hierarchical dependence structures among the obligors in a credit portfolio. Nested Archimedean copulas have a surprisingly simple and intuitive interpretation. The dependence among all companies in the same sector is described by an inner copula and the sectors are then coupled via an outer copula. Consequently, our model implies a larger default correlation for companies in the same industry sector than for companies in different sectors. A calibration to CDO tranche spreads of the European iTraxx portfolio is performed to demonstrate the fitting capability of the model. This portfolio consists of CDS on 125 companies from six different industry sectors and is therefore an excellent portfolio for a comparison of our generalized model with a traditional copula model of the same family that does not take different sectors into account.
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paper_title: Pricing multivariate currency options with copulas

paper_content:

Multivariate options are widely used when there is a need to hedge against a number of risks simultaneously; such as when there is an exposure to several currencies or the need to provide cover against an index such as the FTSE100, or indeed any portfolio of assets. In the case of a basket option the payoff depends on the value of the entire portfolio or basket of assets where the basket is some weighted average of the underlying assets. The principal reason for using basket options is that they are cheaper to use for portfolio insurance than a corresponding portfolio of plain vanilla options on the individual assets. This cost saving depends on the correlation structure between the assets; the lower the correlation between currency pairs in a currency portfolio for instance, the greater the cost saving.
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paper_title: Correlated default with incomplete information

paper_content:

We propose a model of correlated multi-firm default with incomplete information. While public bond investors observe issuers' assets and defaults, we suppose that they are not informed about the threshold asset level at which a firm is liquidated. Bond investors form instead a prior on these thresholds. Stochastic dependence between default events is induced through correlated asset values and correlated default thresholds. The former results from dependence of firms on common macroeconomic factors, while the latter corresponds to direct inter-firm linkages. Having addressed this issuer interdependence, the predictions of our model are consistent with empirically well documented facts, in particular the clustering of defaults. We characterize joint conditional default probabilities as assessed by the imperfectly informed secondary market. The representation of dependence via (conditional) copulas is emphasized. We propose the default time copula as a consistent default correlation measure, which overcomes the limitations of existing covariance based measures. A case study is examined, where issuers' assets follow geometric Brownian motions and bond investors' threshold prior is uniform.
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paper_title: Nonparametric Pricing of Multivariate Contingent Claims

paper_content:

In this paper, I derive and implement a nonparametric, arbitrage-free technique for multivariate contingent claim (MVCC) pricing. Using results from the method of copulas, I show that the multivariate risk-neutral density can be written as a product of marginal risk-neutral densities and a risk-neutral dependence function. I then develop a pricing technique using nonparametrically estimated marginal risk-neutral densities (based on options data) and a nonparametric dependence function (based on historical return data). By using nonparametric estimation, I avoid the pricing biases that result from incorrect parametric assumptions such as lognormality. ; I apply this technique to estimate the joint risk-neutral density of euro-dollar and yen-dollar returns. I compare the nonparametric risk-neutral density with density based on a lognormal dependence function and nonparametric marginals. The nonparametric euro-yen risk-neutral density has greater volatility, skewness, and kurtosis than the density based on a lognormal dependence function. In a comparison of pricing accuracy for euro-yen futures options, I find that the nonparametric model is superior to the lognormal model.
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paper_title: Quanto Pricing with Copulas

paper_content:

“Valuing a derivative that depends on more than one underlying asset, like a currency quanto, which pays off in one currency based on the exchange rate between two others, requires knowledge of the correlation structure among them. Pricing is not difficult if one knows the joint probability distribution. But calibrating a pricing model to the market is much harder, because it requires an implied distribution that is simultaneously consistent with quanto prices and also with the volatility smiles from plain vanilla options in the underlying currencies. Recent work on copulas offers a start. Univariate distributions for the individual underlying assets (the “marginals”) can be obtained from plain vanilla options and then combined, using a copula function, into a joint distribution with whatever dependence structure is desired. But things are still not easy, because the common copula functions used in the finance literature do not have enough flexibility to match the actual implied distributions. In this article, Bennett and Kennedy offer a neat and easily implemented solution, in the form of a procedure for modifying the normal copula to produce realistic dependence structures. The ability to match a quanto volatility smiles is significantly enhanced, relative to the current standard procedure in the market.”
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paper_title: Measuring Corporate Default Risk

paper_content:

This book, based on the author's Clarendon Lectures in Finance, examines the empirical behaviour of corporate default risk. A new and unified statistical methodology for default prediction, based on stochastic intensity modeling, is explained and implemented with data on U.S. public corporations since 1980. Special attention is given to the measurement of correlation of default risk across firms. The underlying work was developed in a series of collaborations over roughly the past decade with Sanjiv Das, Andreas Eckner, Guillaume Horel, Nikunj Kapadia, Leandro Saita, and Ke Wang. Where possible, the content based on methodology has been separated from the substantive empirical findings, in order to provide access to the latter for those less focused on the mathematical foundations. A key finding is that corporate defaults are more clustered in time than would be suggested by their exposure to observable common or correlated risk factors. The methodology allows for hidden sources of default correlation, which are particularly important to include when estimating the likelihood that a portfolio of corporate loans will suffer large default losses. The data also reveal that a substantial amount of power for predicting the default of a corporation can be obtained from the firm's "distance to default," a volatility-adjusted measure of leverage that is the basis of the theoretical models of corporate debt pricing of Black, Scholes, and Merton. The findings are particularly relevant in the aftermath of the financial crisis, which revealed a lack of attention to the proper modelling of correlation of default risk across firms.
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paper_title: On the Out-of-Sample Importance of Skewness and Asymmetric Dependence for Asset Allocation

paper_content:

Recent studies in the empirical finance literature have reported evidence of two types of asymmetries in the joint distribution of stock returns. The first is skewness in the distribution of individual stock returns. The second is an asymmetry in the dependence between stocks: stock returns appear to be more highly correlated during market downturns than during market upturns. In this article we examine the economic and statistical significance of these asymmetries for asset allocation decisions in an out-of-sample setting. We consider the problem of a constant relative risk aversion (CRRA) investor allocating wealth between the risk-free asset, a small-cap portfolio, and a large-cap portfolio. We use models that can capture time-varying moments up to the fourth order, and we use copula theory to construct models of the time-varying dependence structure that allow for different dependence during bear markets than bull markets. The importance of these two asymmetries for asset allocation is assessed by comparing the performance of a portfolio based on a normal distribution model with a portfolio based on a more flexible distribution model. For investors with no short-sales constraints, we find that knowledge of higher moments and asymmetric dependence leads to gains that are economically significant and statistically significant in some cases. For short sales-constrained investors the gains are limited. Copyright 2004, Oxford University Press.
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paper_title: Dependence Structure and Extreme Comovements in International Equity and Bond Markets

paper_content:

Common negative extreme variations in returns are prevalent in international equity markets. This has been widely documented with statistical tools such as exceedance correlation, extreme value theory, and Gaussian bivariate GARCH or regime-switching models. We point to limits of these tools to characterize extreme dependence and propose an alternative regime-switching copula model that includes one normal regime in which dependence is symmetric and a second regime characterized by asymmetric dependence. We apply this model to international equity and bond markets, to allow for inter-market movements. Empirically, we find that dependence between international assets of the same type is strong in both regimes, especially in the asymmetric one, but weak between equities and bonds, even in the same country.
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paper_title: Dynamic stochastic copula models: estimation, inference and applications

paper_content:

We propose a new dynamic copula model where the parameter characterizing dependence follows an autoregressive process. As this model class includes the Gaussian copula with stochastic correlation process, it can be viewed as a generalization of multivariate stochastic volatility models. Despite the complexity of the model, the decoupling of marginals and dependence parameters facilitates estimation. We propose estimation in two steps, where ﬁrst the parameters of the marginal distributions are estimated, and then those of the copula. Parameters of the latent processes (volatilities and dependence) are estimated using efficient importance sampling (EIS). We discuss goodness-of-ﬁt tests and ways to forecast the dependence parameter. For two bivariate stock index series, we show that the proposed model outperforms standard competing models.
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paper_title: Tracking a changing copula

paper_content:

A copula models the relationships between variables independently of their marginal distributions. When the variables are time series, the copula may change over time. Recursive procedures based on indicator variables are proposed for tracking these changes over time. Estimation of the unknown parameters is by maximum likelihood. When the marginal distributions change, pre-filtering is necessary before constructing the indicator variables on which the recursions are based. This entails estimating time-varying quantiles and a simple method based on time-varying histograms is proposed. The techniques are applied to the Hong Kong and Korean stock market indices. Some interesting and unexpected movements are detected, particularly after the attack on the Hong Kong dollar in 1997.
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paper_title: Inhomogeneous Dependence Modeling with Time-Varying Copulae

paper_content:

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.
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paper_title: Is the Potential for International Diversification Disappearing

paper_content:

Quantifying the evolution of security co-movements is critical for asset pricing and portfolio allocation, and so we investigate patterns and trends in correlations and tail dependence over time using weekly returns for developed markets (DMs) and emerging markets (EMs) during the period 1973-2009. We use the DECO, DCC, and BEKK correlation models, and develop a novel dynamic t-copula to allow for dynamic tail dependence. We show that it is possible to characterize co-movements for many countries simultaneously. Correlations have significantly trended upward for both DMs and EMs, but correlations between EMs are lower than between DMs. Further, our evidence clearly contradicts the decoupling hypothesis. The tail dependence has also increased for both EMs and DMs, but its level is still very low for EMs as compared to DMs. Thus, while our correlation analysis suggests that the diversification potential of EMs has reduced over time, the tail dependence analysis suggests that EMs offer diversification benefits during large market moves.
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paper_title: Generalized Autoregressive Conditional Heteroskedasticity

paper_content:

The present paper proposes a generalization of the canonical AutoRegressive Conditional Heteroskedasticity (ARCH) model by extending the conditional variance equation toward past conditional variances. The stationarity conditions and autocorrelation structure of the Generalized AutoRegressive Conditional Heteroskedastic (GARCH) model are derived. Using an empirical example of uncertainty of the inflation rate the paper demonstrates that the GARCH model provides a better fit and a more plausible learning mechanism than the ARCH model.
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paper_title: When is a Copula Constant? A Test for Changing Relationships

paper_content:

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, with the lower quantiles being of particular interest in financial applications concerned with risk. Tests located at the median provide an overall test of a changing relationship. The properties of various tests are compared and it is shown that they are still effective if pre-filtering is carried out to correct for changing volatility or, more generally, changing quantiles. Applying the tests to daily stock return indices in Korea and Thailand over the period 1995-9 indicates that the relationship between them is not constant over time.
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paper_title: Change analysis of dynamic copula for measuring dependence in multivariate financial data

paper_content:

This paper proposes a new approach to measure dependencies in multivariate financial data. Data in finance and insurance often cover a long time period. Therefore, the economic factors may induce some changes within the dependence structure. Recently, two methods have been proposed using copulas to analyse such changes. The first approach investigates changes within the parameters of the copula. The second determines the sequence of copulas using moving windows. In this paper we take into account the non-stationarity of the data and analyse the impact of (1) time-varying parameters for a copula family, and (2) the sequence of copulas, on the computations of the VaR and ES measures. We propose tests based on conditional copulas and the goodness-of-fit to decide the type of change, and further give the corresponding change analysis. We illustrate our approach using the Standard & Poor 500 and Nasdaq indices in order to compute risk measures using the two previous methods.
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paper_title: Goodness-of-Fit Tests for Copulas of Multivariate Time Series

paper_content:

In this paper, we study the asymptotic behavior of the sequential empirical process and the sequential empirical copula process, both constructed from residuals of multivariate stochastic volatility models. Applications for the detection of structural changes and specification tests of the distribution of innovations are discussed. It is also shown that if the stochastic volatility matrices are diagonal, which is the case if the univariate time series are estimated separately instead of being jointly estimated, then the empirical copula process behaves as if the innovations were observed; a remarkable property. As a by-product, one also obtains the asymptotic behavior of rank-based measures of dependence applied to residuals of these time series models.
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paper_title: Generalized Autoregressive Score Models with Applications ∗

paper_content:

We propose a class of observation driven time series models referred to as Generalized Autoregressive Score (GAS) models. The mechanism to update the parameters over time is the scaled score of the likelihood function. This new approach provides a unified and consistent framework for introducing time-varying parameters in a wide class of non-linear models. The GAS model encompasses other well-known models such as the generalized autoregressive conditional heteroskedasticity, the autoregressive conditional duration, the autoregressive conditional intensity, and Poisson count models with time-varying mean. In addition, our approach can lead to new formulations of observation driven models. We illustrate our framework by introducing new model specifications for time-varying copula functions and for multivariate point processes with time-varying parameters. We study the models in detail and provide simulation and empirical evidence.
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paper_title: Time Variation in Asset Return Dependence: Strength or Structure? ∗

paper_content:

The dependence between asset returns varies. Its strength can become stronger or weaker. Also, its structure can change, for example, when asymmetries related to bull and bear markets become more or less pronounced. To analyze these different types of variations, we develop a model that separately accommodates these changes. It combines a mixture of structurally different copulas with time variation. Our model shows both types of changes in the dependence between several equity market returns. Ignoring them leads to biases in risk measures. An underestimation of Value-at-Risk by maximum 15% occurs exactly when most harmful, during crisis periods.
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paper_title: Stochastic Volatility: Selected Readings

paper_content:

Stochastic volatility is the main concept used in the fields of financial economics and mathematical finance to deal with time-varying volatility in financial markets. This book brings together some of the main papers that have influenced the field of the econometrics of stochastic volatility, and shows that the development of this subject has been highly multidisciplinary, with results drawn from financial economics, probability theory, and econometrics, blending to produce methods and models that have aided our understanding of the realistic pricing of options, efficient asset allocation, and accurate risk assessment. A lengthy introduction by the editor connects the papers with the literature. Contributors to this volume - Contributors: Torben Andersen, Northwestern University; Ole E. Barndorff-Nielsen, University of Aarhus; Tim Bollerslev, Duke University; Mikhail Chernov; Siddhartha Chib, Washington University in St. Louis; Peter Clark, University of California, Davis; Fabienne Comte, Universite Rene Descartes- Paris 5; Frank Diebold, University of Pennsylvania; Dean Foster, University of Pennsylvania; A Ronald Gallant, Duke University; Eric Ghysels, University of North Carolina - Chapel Hill; Andrew Harvey, University of Cambridge; Steven Heston, University of Maryland; David Hsieh, Duke University; John Hull, University of Toronto; Eric Jacquier, H.E.C. MONTREAL; Sangjoon Kim, RBS Securities Japan Limited; Paul Labys, Charles River Associates; Angelo Melino, University of Toronto; Daniel Nelson; Marc Nerlove, University of Maryland; Nicholas Polson, University of Chicago; Eric Renault, University of Montreal; Peter Rossi, University of Chicago; Esther Ruiz, Universidad Carlos III de Madrid; Barr Rosenberg, AXA Rosenberg Investment Management; Neil Shephard, Nuffield College, University of Oxford; Stephen Taylor, Lancaster University; George Tauchen, Duke University; Stuart Turnbull, University of Houston; Alan White, University of Toronto.
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paper_title: Modeling exchange rate dependence dynamics at different time horizons

paper_content:

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.
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paper_title: On the Out-of-Sample Importance of Skewness and Asymmetric Dependence for Asset Allocation

paper_content:

Recent studies in the empirical finance literature have reported evidence of two types of asymmetries in the joint distribution of stock returns. The first is skewness in the distribution of individual stock returns. The second is an asymmetry in the dependence between stocks: stock returns appear to be more highly correlated during market downturns than during market upturns. In this article we examine the economic and statistical significance of these asymmetries for asset allocation decisions in an out-of-sample setting. We consider the problem of a constant relative risk aversion (CRRA) investor allocating wealth between the risk-free asset, a small-cap portfolio, and a large-cap portfolio. We use models that can capture time-varying moments up to the fourth order, and we use copula theory to construct models of the time-varying dependence structure that allow for different dependence during bear markets than bull markets. The importance of these two asymmetries for asset allocation is assessed by comparing the performance of a portfolio based on a normal distribution model with a portfolio based on a more flexible distribution model. For investors with no short-sales constraints, we find that knowledge of higher moments and asymmetric dependence leads to gains that are economically significant and statistically significant in some cases. For short sales-constrained investors the gains are limited. Copyright 2004, Oxford University Press.
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paper_title: Tails of correlation mixtures of elliptical copulas

paper_content:

Correlation mixtures of elliptical copulas arise when the correlation parameter is driven itself by a latent random process. For such copulas, both penultimate and asymptotic tail dependence are much larger than for ordinary elliptical copulas with the same unconditional correlation. Furthermore, for Gaussian and Student t-copulas, tail dependence at sub-asymptotic levels is generally larger than in the limit, which can have serious consequences for estimation and evaluation of extreme risk. Finally, although correlation mixtures of Gaussian copulas inherit the property of asymptotic independence, at the same time they fall in the newly defined category of near asymptotic dependence. The consequences of these findings for modeling are assessed by means of a simulation study and a case study involving financial time series.
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paper_title: The Copula-GARCH model of conditional dependencies: An international stock market application

paper_content:

Abstract Modeling the dependency between stock market returns is a difficult task when returns follow a complicated dynamics. When returns are non-normal, it is often simply impossible to specify the multivariate distribution relating two or more return series. In this context, we propose a new methodology based on copula functions, which consists in estimating first the univariate distributions and then the joining distribution. In such a context, the dependency parameter can easily be rendered conditional and time varying. We apply this methodology to the daily returns of four major stock markets. Our results suggest that conditional dependency depends on past realizations for European market pairs only. For these markets, dependency is found to be more widely affected when returns move in the same direction than when they move in opposite directions. Modeling the dynamics of the dependency parameter also suggests that dependency is higher and more persistent between European stock markets.
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paper_title: Time-varying joint distribution through copulas

paper_content:

The analysis of temporal dependence in multivariate time series is considered. The dependence structure between the marginal series is modelled through the use of copulas which, unlike the correlation matrix, give a complete description of the joint distribution. The parameters of the copula function vary through time, following certain evolution equations depending on their previous values and the historical data. The marginal time series follow standard univariate GARCH models. Full Bayesian inference is developed where the whole set of model parameters is estimated simultaneously. This represents an essential difference from previous approaches in the literature where the marginal and the copula parameters are estimated separately in two consecutive steps. Moreover, a Bayesian procedure is proposed for the estimation of several measures of risk, such as the variance, Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) of a portfolio of assets, providing point estimates and predictive intervals. The proposed copula model enables to capture the dependence structure between the individual assets which strongly influences these risk measures. Finally, the problem of optimal portfolio selection based on the estimation of mean-variance, mean-VaR and mean-CVaR efficient frontiers is also addressed. The proposed approach is illustrated with simulated and real financial time series.
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paper_title: MODELLING ASYMMETRIC EXCHANGE RATE DEPENDENCE*

paper_content:

We test for asymmetry in a model of the dependence between the Deutsche mark and the yen, in the sense that a different degree of correlation is exhibited during joint appreciations against the U.S. dollar versus during joint depreciations. We consider an extension of the theory of copulas to allow for conditioning variables, and employ it to construct flexible models of the conditional dependence structure of these exchange rates. We find evidence that the mark-dollar and yen-dollar exchange rates are more correlated when they are depreciating against the dollar than when they are appreciating. Copyright 2006 by the Economics Department Of The University Of Pennsylvania And Osaka University Institute Of Social And Economic Research Association.
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paper_title: Dependence Structure and Extreme Comovements in International Equity and Bond Markets

paper_content:

Common negative extreme variations in returns are prevalent in international equity markets. This has been widely documented with statistical tools such as exceedance correlation, extreme value theory, and Gaussian bivariate GARCH or regime-switching models. We point to limits of these tools to characterize extreme dependence and propose an alternative regime-switching copula model that includes one normal regime in which dependence is symmetric and a second regime characterized by asymmetric dependence. We apply this model to international equity and bond markets, to allow for inter-market movements. Empirically, we find that dependence between international assets of the same type is strong in both regimes, especially in the asymmetric one, but weak between equities and bonds, even in the same country.
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paper_title: Modeling International Financial Returns with a Multivariate Regime-switching Copula

paper_content:

In order to capture observed asymmetric dependence in international financial returns, we construct a multivariate regime-switching model of copulas. We model dependence with one Gaussian and one canonical vine copula regime. Canonical vines are constructed from bivariate conditional copulas and provide a very flexible way of characterizing dependence in multivariate settings. We apply the model to returns from the G5 and Latin American regions, and document three main findings. First, we discover that models with canonical vines generally dominate alternative dependence structures. Second, the choice of copula is important for risk management, since it modifies the Value-at-Risk (VaR) of international portfolios and produces a better out-of-sample performance. Third, ignoring asymmetric dependence and regime-switching in portfolio selection leads to significant costs for an investor. Copyright The Author 2009. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org., Oxford University Press.
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paper_title: New Evidence of Asymmetric Dependence Structures in International Equity Markets

paper_content:

A number of recent studies finds two asymmetries in dependence structures in international equity markets; specifically, dependence tends to be high in both highly volatile markets and in bear markets. In this paper, a further investigation of asymmetric dependence structures in international equity markets is performed by using the Markov switching model and copula theory. Combining these two theories enables me to model dependence structures with sufficient flexibility. Using this flexible framework, I indeed find that there are two distinct regimes in the U. S.-U. K. market. I also show that for the U. S.-U. K. market the bear regime is better described by an asymmetric copula with lower tail dependence with clear rejection of the Markov switching multivariate normal model. In addition, I show that ignorance of this further asymmetry in bear markets is very costly for risk management. Lastly, I conduct a similar analysis for other G7 countries, where I find other cases in which the use of a Markov switching multivariate normal model would be inappropriate.
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paper_title: CDO pricing with nested Archimedean copulas

paper_content:

Companies in the same industry sector are usually more correlated than firms in different sectors, as they are similarly affected by macroeconomic effects, political decisions, and consumer trends. Despite the many stock return models taking this fact into account, there are only a few credit default models that take it into consideration. In this paper we present a default model based on nested Archimedean copulas that is able to capture hierarchical dependence structures among the obligors in a credit portfolio. Nested Archimedean copulas have a surprisingly simple and intuitive interpretation. The dependence among all companies in the same sector is described by an inner copula and the sectors are then coupled via an outer copula. Consequently, our model implies a larger default correlation for companies in the same industry sector than for companies in different sectors. A calibration to CDO tranche spreads of the European iTraxx portfolio is performed to demonstrate the fitting capability of the model. This portfolio consists of CDS on 125 companies from six different industry sectors and is therefore an excellent portfolio for a comparison of our generalized model with a traditional copula model of the same family that does not take different sectors into account.
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paper_title: Pair-Copula Constructions of Multiple Dependence

paper_content:

Building on the work of Bedford, Cooke and Joe, we show how multivariate data, which exhibit complex patterns of dependence in the tails, can be modelled using a cascade of pair-copulae, acting on two variables at a time. We use the pair-copula decomposition of a general multivariate distribution and propose a method for performing inference. The model construction is hierarchical in nature, the various levels corresponding to the incorporation of more variables in the conditioning sets, using pair-copulae as simple building blocks. Pair-copula decomposed models also represent a very flexible way to construct higher-dimensional copulae. We apply the methodology to a financial data set. Our approach represents the first step towards the development of an unsupervised algorithm that explores the space of possible pair-copula models, that also can be applied to huge data sets automatically.
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paper_title: Is the Potential for International Diversification Disappearing

paper_content:

Quantifying the evolution of security co-movements is critical for asset pricing and portfolio allocation, and so we investigate patterns and trends in correlations and tail dependence over time using weekly returns for developed markets (DMs) and emerging markets (EMs) during the period 1973-2009. We use the DECO, DCC, and BEKK correlation models, and develop a novel dynamic t-copula to allow for dynamic tail dependence. We show that it is possible to characterize co-movements for many countries simultaneously. Correlations have significantly trended upward for both DMs and EMs, but correlations between EMs are lower than between DMs. Further, our evidence clearly contradicts the decoupling hypothesis. The tail dependence has also increased for both EMs and DMs, but its level is still very low for EMs as compared to DMs. Thus, while our correlation analysis suggests that the diversification potential of EMs has reduced over time, the tail dependence analysis suggests that EMs offer diversification benefits during large market moves.
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paper_title: Likelihood inference for Archimedean copulas in high dimensions under known margins

paper_content:

Explicit functional forms for the generator derivatives of well-known one-parameter Archimedean copulas are derived. These derivatives are essential for likelihood inference as they appear in the copula density, conditional distribution functions, and the Kendall distribution function. They are also required for several asymmetric extensions of Archimedean copulas such as Khoudraji-transformed Archimedean copulas. Availability of the generator derivatives in a form that permits fast and accurate computation makes maximum-likelihood estimation for Archimedean copulas feasible, even in large dimensions. It is shown, by large scale simulation of the performance of maximum likelihood estimators under known margins, that the root mean squared error actually decreases with both dimension and sample size at a similar rate. Confidence intervals for the parameter vector are derived under known margins. Moreover, extensions to multi-parameter Archimedean families are given. All presented methods are implemented in the R package nacopula and can thus be studied in detail.
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paper_title: Beyond simplified pair-copula constructions

paper_content:

Pair-copula constructions (PCCs) offer great flexibility in modeling multivariate dependence. For inference purposes, however, conditional pair-copulas are often assumed to depend on the conditioning variables only indirectly through the conditional margins. The authors show here that this assumption can be misleading. To assess its validity in trivariate PCCs, they propose a visual tool based on a local likelihood estimator of the conditional copula parameter which does not rely on the simplifying assumption. They establish the consistency of the estimator and assess its performance in finite samples via Monte Carlo simulations. They also provide a real data application.
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paper_title: Constructing hierarchical Archimedean copulas with Lévy subordinators

paper_content:

A probabilistic interpretation for hierarchical Archimedean copulas based on Levy subordinators is given. Independent exponential random variables are divided by group-specific Levy subordinators which are evaluated at a common random time. The resulting random vector has a hierarchical Archimedean survival copula. This approach suggests an efficient sampling algorithm and allows one to easily construct several new parametric families of hierarchical Archimedean copulas.
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paper_title: Modelling dependence using skew t copulas: Bayesian inference and applications

paper_content:

We construct a copula from the skew t distribution of Sahu, Dey & Branco (2003). This copula can capture asymmetric and extreme dependence between variables, and is one of the few copulas that can do so and still be used in high dimensions effectively. However, it is difficult to estimate the copula model by maximum likelihood when the multivariate dimension is high, or when some or all of the marginal distributions are discrete-valued, or when the parameters in the marginal distributions and copula are estimated jointly. We therefore propose a Bayesian approach that overcomes all these problems. The computations are undertaken using a Markov chain Monte Carlo simulation method which exploits the conditionally Gaussian representation of the skew t distribution. We employ the approach in two contemporary econometric studies. The first is the modeling of regional spot prices in the Australian electricity market. Here, we observe complex non-Gaussian margins and nonlinear inter-regional dependence. Accurate characterization of this dependence is important for the study of market integration and risk management purposes. The second is the modeling of ordinal exposure measures for 15 major websites. Dependence between websites is important when measuring the impact of multi-site advertising campaigns. In both cases the skew t copula substantially out-performs symmetric elliptical copula alternatives, demonstrating that the skew t copula is a powerful modeling tool when coupled with Bayesian inference.
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paper_title: The t copula and related copulas

paper_content:

Summary The t copula and its properties are described with a focus on issues related to the dependence of extreme values. The Gaussian mixture representation of a multivariate t distribution is used as a starting point to construct two new copulas, the skewed t copula and the grouped t copula, which allow more heterogeneity in the modelling of dependent observations. Extreme value considerations are used to derive two further new copulas: the t extreme value copula is the limiting copula of componentwise maxima of t distributed random vectors; the t lower tail copula is the limiting copula of bivariate observations from a t distribution that are conditioned to lie below some joint threshold that is progressively lowered. Both these copulas may be approximated for practical purposes by simpler, better-known copulas, these being the Gumbel and Clayton copulas respectively.
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paper_title: Modelling Dependence in High Dimensions with Factor Copulas

paper_content:

his paper presents flexible new models for the dependence structure, or copula, of economic variables based on a latent factor structure. The proposed models are particularly attractive for relatively high dimensional applications, involving fifty or more variables, and can be combined with semiparametric marginal distributions to obtain flexible multivariate distributions. Factor copulas generally lack a closed-form density, but we obtain analytical results for the implied tail dependence using extreme value theory, and we verify that simulation-based estimation using rank statistics is reliable even in high dimensions. We consider "scree" plots to aid the choice of the number of factors in the model. The model is applied to daily returns on all 100 constituents of the S&P 100 index, and we find significant evidence of tail dependence, heterogeneous dependence, and asymmetric dependence, with dependence being stronger in crashes than in booms. We also show that factor copula models provide superior estimates of some measures of systemic risk.
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paper_title: Dependence Structures for Multivariate High-Frequency Data in Finance

paper_content:

Stylized facts for univariate high-frequency data in finance are well known. They include scaling behaviour, volatility clustering, heavy tails and seasonalities. The multivariate problem, however, has scarcely been addressed up to now. In this paper, bivariate series of high-frequency FX spot data for major FX markets are investigated. First, as an indispensable prerequisite for further analysis, the problem of simultaneous deseasonalization of high-frequency data is addressed. In the following sections we analyse in detail the dependence structure as a function of the timescale. Particular emphasis is put on the tail behaviour, which is investigated by means of copulas.
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paper_title: A Longer Look at the Asymmetric Dependence between Hedge Funds and the Equity Market

paper_content:

This paper reexamines, at a range of investment horizons, the asymmetric dependence between hedge fund returns and market returns. Given the current availability of hedge fund data, the joint distribution of longer-horizon returns is extracted from the dynamics of monthly returns using the filtered historical simulation; we then apply the method based on copula theory to uncover the dependence structure therein. While the direction of asymmetry remains unchanged, the magnitude of asymmetry is attenuated considerably as the investment horizon increases. Similar horizon effects also occur on the tail dependence. Our findings suggest that nonlinearity in hedge fund exposure to market risk is more short term in nature, and that hedge funds provide higher benefits of diversification, the longer the horizon.
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paper_title: Multivariate autoregressive modeling of time series count data using copulas

paper_content:

Abstract We introduce the Multivariate Autoregressive Conditional Double Poisson model to deal with discreteness, overdispersion and both auto and cross-correlation, arising with multivariate counts. We model counts with a double Poisson and assume that conditionally on past observations the means follow a Vector Autoregression. We resort to copulas to introduce contemporaneous correlation. We apply it to the study of sector and stock-specific news related to the comovements in the number of trades per unit of time of the most important US department stores traded on the NYSE. We show that the market leaders inside a specific sector are related to their size measured by their market capitalization.
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paper_title: Margins and Hedge Fund Contagion

paper_content:

Funding risk measures the extent to which a fund can borrow money by posting collateral. Using a novel measure of funding risk based on futures margins, we are able to empirically identify the mechanism by which changes in funding risk affect the likelihood of contagion. An increase in margins of the order of magnitude observed during the subprime crisis increases the probability of contagion among certain types of funds by up to 34%. Our analysis shows that some types of hedge funds are more vulnerable to contagion than others. Our results also suggest that policies that limit the magnitude of changes in margins over short periods of time may reduce the likelihood of contagion among hedge funds.
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paper_title: Higher Education Funding Reforms in England: the Distributional Effects and the Shifting Balance of Costs

paper_content:

This article undertakes a quantitative analysis of substantial reforms to the system of higher education (HE) finance in England, first announced in 2004 and revised in 2007. The reforms introduced deferred fees for HE, payable by graduates through the tax system via income-contingent repayments on loans subsidised by the government. The article uses lifetime earnings simulated by the authors to consider the likely distributional consequences of the reforms for graduates. It also considers the costs of the reforms for taxpayers, and how the reforms are likely to shift the balance of funding for HE between the public and private sectors.
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paper_title: Modeling exchange rate dependence dynamics at different time horizons

paper_content:

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.
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paper_title: Common Factors in Conditional Distributions for Bivariate Time Series

paper_content:

A definition for a common factor for bivariate time series is suggested by considering the decomposition of the conditional density into the product of the marginals and the copula, with the conditioning variable being a common factor if it does not directly enter the copula. We show the links between this definition and the idea of a common factor as a dominant feature in standard linear representations. An application using a business cycle indicator as the common factor in the relationship between U.S. income and consumption found that both series held the factor in their marginals but not in the copula.
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paper_title: Are Market Neutral Hedge Funds Really Market Neutral ?

paper_content:

One can consider the concept of market neutrality for hedge funds as having breadth and depth: breadth reflects the number of market risks to which a fund is neutral, while depth reflects the completeness of the neutrality of the fund to market risks. We focus on market neutrality depth, and propose five different neutrality concepts. Mean neutrality nests the standard correlation-based definition of neutrality. Variance neutrality, Value-at-Risk neutrality and tail neutrality all relate to the neutrality of the risk of the hedge fund to market risks. Finally, complete neutrality. corresponds to independence of the fund to market risks. We suggest statistical tests for each neutrality concept, and apply the tests to a combined database of monthly market neutral hedge fund returns from the HFR and TASS hedge fund databases. We find that around one-quarter of these funds exhibit some significant exposure to market risk.
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paper_title: The Euro and European Financial Market Dependence

paper_content:

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.
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Title: A review of copula models for economic time series

Section 1: Introduction
Description 1: This section introduces the topic, providing background on copula models and their relevance for economic and financial time series. It also outlines the structure and goals of the survey.

Section 2: Estimation and inference
Description 2: This section describes various estimation and inference methods for copula-based models that have been proposed in the literature, including fully parametric and semiparametric models.

Section 3: Dependence summary statistics
Description 3: This section discusses the computation of simple measures of dependence before specifying and estimating a copula model, along with methods for conducting inference on these measures.

Section 4: Time-varying dependence
Description 4: This section explores evidence and tests for time-varying dependence in economic time series, and reviews models for time-varying conditional copulas.

Section 5: Empirical illustration
Description 5: This section presents a case study of estimating copula-based models using daily returns on two equity indices to illustrate the practical application of the methods discussed.

Section 6: Estimating copula-based multivariate time series models
Description 6: This section covers the process of estimating copula-based models for multivariate time series, including both parametric and semiparametric approaches.

Section 7: Fully parametric
Description 7: This section details the fully parametric estimation method for copula models, highlighting the use of maximum likelihood estimation and discussing its computational complexity.

Section 8: Semiparametric
Description 8: This section examines semiparametric copula-based models that use nonparametric methods for the marginal distributions and parametric methods for the copula, including canonical maximum likelihood estimation.

Section 9: Nonparametric
Description 9: This section looks at fully nonparametric estimation of copulas, particularly in the context of i.i.d. and time series data.

Section 10: Other estimation methods
Description 10: This section reviews alternative estimation methods for copula models, such as method of moments, generalized method of moments, minimum distance estimation, and Bayesian estimation.

Section 11: Empirical illustration, continued
Description 11: This section provides an additional example of estimating both parametric and semiparametric copula models, presenting results and discussing different types of standard errors.

Section 12: Estimating copula-based univariate time series models
Description 12: This section focuses on models for univariate time series using copulas, including maximum likelihood estimation and conditions for asymptotic normality of the copula parameter.

Section 13: Goodness-of-fit tests
Description 13: This section addresses goodness-of-fit tests for copula models, discussing in-sample tests such as Kolmogorov-Smirnov and Cramér-von Mises, and specific considerations for parametric and semiparametric models.

Section 14: Applications of copula methods for economic time series
Description 14: This section surveys various applications of copula methods in economic and financial research, including risk management, derivative contracts, portfolio decisions, and high-dimensional copula models.

Section 15: Conclusion
Description 15: This section summarizes the key points discussed in the survey, reflecting on the utility of copula models for economic time series and highlighting areas for future research.