Andrew J. Patton

Modelling Asymmetric Exchange Rate Dependence

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.

What good is a volatility model

A volatility model must be able to forecast volatility; this is the central requirement in almost all financial applications. In this paper we outline some stylized facts about volatility that should be incorporated in a model: pronounced persistence and mean-reversion, asymmetry such that the sign of an innovation also affects volatility and the possibility of exogenous or pre-determined variables influencing volatility. We use data on the Dow Jones Industrial Index to illustrate these stylized facts, and the ability of GARCH-type models to capture these features. We conclude with some challenges for future research in this area.

Copula–Based Models for Financial Time Series

This paper presents an overview of the literature on applications of copulas in the modelling of financial time series. Copulas have been used both in multivariate time series analysis, where they are used to characterize the (conditional) cross-sectional dependence between individual time series, and in univariate time series analysis, where they are used to characterize the dependence between a sequence of observations of a scalar time series process. The paper includes a broad, brief, review of the many applications of copulas in finance and economics.

A review of copula models for economic time series

This survey reviews the large and growing literature on copula-based models for economic and financial time series. Copula-based multivariate models allow the researcher to specify the models for the marginal distributions separately from the dependence structure that links these distributions to form a joint distribution. This allows for a much greater degree of flexibility in specifying and estimating the model, freeing the researcher from considering only existing multivariate distributions. The author surveys estimation and inference methods and goodness-of-fit tests for such models, as well as empirical applications of these copulas for economic and financial time series.

Evaluating Volatility and Correlation Forecasts

This chapter considers the problems of evaluation and comparison of volatility forecasts, both univariate (variance) and multivariate (covariance matrix and/or correlation). We pay explicit attention to the fact that the object of interest in these applications is unobservable, even ex post, and so the evaluation and comparison of volatility forecasts often rely on the use of a “volatility proxy”, i.e. an observable variable that is related to the latent variable of interest. We focus on methods that are robust to the presence of measurement error in the volatility proxy, and to the conditional distribution of returns.

Modelling Time-Varying Exchange Rate Dependence Using the Conditional Copula

Linear correlation is only an adequate means of describing the dependence between two random variables when they are jointly elliptically distributed. When the joint distribution of two or more variables is not elliptical the linear correlation coefficient becomes just one of many possible ways of summarising the dependence structure between the variables. In this paper we make use of a theorem due to Sklar (1959), which shows that an n-dimensional distribution function may be decomposed into its n marginal distributions, and a copula, which completely describes the dependence between the n variables. We verify that Sklars theorem may be extended to conditional distributions, and apply it to the modelling of the time-varying joint distribution of the Deutsche mark - U.S. dollar and Yen - U.S. dollar exchange rate returns. We find evidence that the conditional dependence between these exchange rates is time-varying, and that it is asymmetric: dependence is greater during appreciations of the U.S. dollar against the mark and the yen than during depreciations of the U.S. dollar. We also find strong evidence of a structural break in the conditional copula following the introduction of the euro.

Testing Forecast Optimality Under Unknown Loss

Empirical tests of forecast optimality have traditionally been conducted under the assumption of mean squared error loss or some other known loss function. In this article we establish new testable properties that hold when the forecasters loss function is unknown but testable restrictions can be imposed on the data-generating process, trading off conditions on the data-generating process against conditions on the loss function. We propose flexible estimation of the forecasters loss function in situations where the loss depends not only on the forecast error, but also on other state variables, such as the level of the target variable. We apply our results to the problem of evaluating the Federal Reserves forecasts of output growth. Forecast optimality is rejected if the Feds loss depends only on the forecast error. However, the empirical findings are consistent with forecast optimality provided that overpredictions of output growth are costlier to the Fed than underpredictions, particularly during periods of low economic growth.

Good Volatility, Bad Volatility: Signed Jumps and The Persistence of Volatility

Using estimators of the variation of positive and negative returns (realized semivariances) and high-frequency data for the S&P 500 Index and 105 individual stocks, this paper sheds new light on the predictability of equity price volatility.We showthat future volatility is more strongly related to the volatility of past negative returns than to that of positive returns and that the impact of a price jump on volatility depends on the sign of the jump, with negative (positive) jumps leading to higher (lower) future volatility. We show that models exploiting these findings lead to significantly better out-of-sample forecast performance.

Correction to “Automatic Block-Length Selection for the Dependent Bootstrap” by D. Politis and H. White

A correction on the optimal block size algorithms of Politis and White (2004) is given following a correction of Lahiris (Lahiri 1999) theoretical results by Nordman (2008).

Copula Methods for Forecasting Multivariate Time Series

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.