Time varying correlations are often estimated with multivariate generalized autoregressive conditional heteroskedasticity (GARCH) models that are linear in squares and cross products of the data. A new class of multivariate models called dynamic conditional correlation models is proposed. These have the flexibility of univariate GARCH models coupled with parsimonious parametric models for the correlations. They are not linear but can often be estimated very simply with univariate or two-step methods based on the likelihood function. It is shown that they perform well in a variety of situations and provide sensible empirical results.

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Dynamic Conditional Correlation: A Simple Class of Multivariate Generalized Autoregressive Conditional Heteroskedasticity Models

This paper provides a general framework for integration of high-frequency intraday data into the measurement, modeling and forecasting of daily and lower frequency volatility and return distributions. Most procedures for modeling and forecasting financial asset return volatilities, correlations, and distributions rely on restrictive and complicated parametric multivariate ARCH or stochastic volatility models, which often perform poorly at intraday frequencies. Use of realized volatility constructed from high-frequency intraday returns, in contrast, permits the use of traditional time series procedures for modeling and forecasting. Building on the theory of continuous-time arbitrage-free price processes and the theory of quadratic variation, we formally develop the links between the conditional covariance matrix and the concept of realized volatility. Next, using continuously recorded observations for the Deutschemark/Dollar and Yen /Dollar spot exchange rates covering more than a decade, we find that forecasts from a simple long-memory Gaussian vector autoregression for the logarithmic daily realized volatitilies perform admirably compared to popular daily ARCH and related models. Moreover, the vector autoregressive volatility forecast, coupled with a parametric lognormal-normal mixture distribution implied by the theoretically and empirically grounded assumption of normally distributed standardized returns, gives rise to well-calibrated density forecasts of future returns, and correspondingly accurate quintile estimates. Our results hold promise for practical modeling and forecasting of the large covariance matrices relevant in asset pricing, asset allocation and financial risk management applications.

Modeling and Forecasting Realized Volatility

This paper provides a general framework for integration of high-frequency intraday data into the measurement, modeling, and forecasting of daily and lower frequency volatility and return distributions. Most procedures for modeling and forecasting financial asset return volatilities, correlations, and distributions rely on restrictive and complicated parametric multivariate ARCH or stochastic volatility models, which often perform poorly at intraday frequencies. Use of realized volatility constructed from high-frequency intraday returns, in contrast, permits the use of traditional time series procedures for modeling and forecasting. Building on the theory of continuous-time arbitrage-free price processes and the theory of quadratic variation, we formally develop the links between the conditional covariance matrix and the concept of realized volatility. Next, using continuously recorded observations for the Deutschemark / Dollar and Yen / Dollar spot exchange rates covering more than a decade, we find that forecasts from a simple long-memory Gaussian vector autoregression for the logarithmic daily realized volatilities perform admirably compared to popular daily ARCH and related models. Moreover, the vector autoregressive volatility forecast, coupled with a parametric lognormal-normal mixture distribution implied by the theoretically and empirically grounded assumption of normally distributed standardized returns, gives rise to well-calibrated density forecasts of future returns, and correspondingly accurate quantile estimates. Our results hold promise for practical modeling and forecasting of the large covariance matrices relevant in asset pricing, asset allocation and financial risk management applications.

/pdf/modeling-and-forecasting-realized-volatility-2mqo5gygak.pdf

Modeling and forecasting realized volatility

This bestselling and thoroughly classroom-tested textbook is a complete resource for finance students. A comprehensive and illustrated discussion of the most common empirical approaches in finance prepares students for using econometrics in practice, while detailed case studies help them understand how the techniques are used in relevant financial contexts. Worked examples from the latest version of the popular statistical software EViews guide students to implement their own models and interpret results. Learning outcomes, key concepts and end-of-chapter review questions (with full solutions online) highlight the main chapter takeaways and allow students to self-assess their understanding. Building on the successful data- and problem-driven approach of previous editions, this third edition has been updated with new data, extensive examples and additional introductory material on mathematics, making the book more accessible to students encountering econometrics for the first time. A companion website, with numerous student and instructor resources, completes the learning package.

Introductory Econometrics for Finance

http://finance.martinsewell.com/stylized-facts/volatility/Andersen-etal2001a.pdf

The distribution of realized stock return volatility

Existing time-varying covariance models usually impose strong restrictions on how past shocks affect the forecasted covariance matrix. In this article we compare the restrictions imposed by the four most popular multivariate GARCH models, and introduce a set of robust conditional moment tests to detect misspecification. We demonstrate that the choice of a multivariate volatility model can lead to substantially different conclusions in any application that involves forecasting dynamic covariance matrices (like estimating the optimal hedge ratio or deriving the risk minimizing portfolio). We therefore introduce a general model which nests these four models and their natural 'asymmetric' extensions. The new model is applied to study the dynamic relation between large and small firm returns. Article published by Oxford University Press on behalf of the Society for Financial Studies in its journal, The Review of Financial Studies.

Modeling Asymmetric Comovements of Asset Returns

Existing multivariate models allowing the covariance matrix to be time varying generally impose strong restrictions on how past shocks can affect the covariance matrix. Yet these restrictions are seldom compared and tested. Furthermore, few studies have examined the existence of asymmetric/leverage effects in covariance, even though there are good reasons to believe they exist and have important implications for portfolio management, derivatives pricing and hedging. In this paper, we introduce a set of robust conditional moment tests to detect misspecification in the covariance matrix, and introduce a general model which nests the four most popular multivariate GARCH models and their natural "asymmetric" extensions. The new model is applied to study the dynamic relation between large and small firm returns. Our results suggest that bad news about large firms can cause volatility in both small and large firm returns, while news about small firms, whether good or bad, has minimal effect on the covariance matrix. The conditional covariance between large firm and small firm returns also tends to increase following bad news about large firms but not bad news about small firms.

Kenneth F. Kroner

Papers

Modeling Asymmetric Comovements of Asset Returns

Modeling the Time Varying Comovement of Asset Returns