We provide a general framework for integration of high-frequency intraday data into the measurement, modeling and forecasting of daily and lower frequency return volatilities and return distributions. Most procedures for modeling and forecasting financial asset return volatilities, correlations, and distributions rely on potentially restrictive and complicated parametric multivariate ARCH or stochastic volatility models. Use of realized volatility constructed from high-frequency intraday returns, in contrast, permits the use of traditional time-series methods for modeling and forecasting. Building on the theory of continuous-time arbitrage-free price processes and the theory of quadratic variation, we develop formal links between realized volatility and the conditional covariance matrix. 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 a variety of popular daily ARCH and more complicated high-frequency 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, produces well-calibrated density forecasts of future returns, and correspondingly accurate quantile predictions. 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.
