this paper is built. First, although raw returns are clearly leptokurtic, returns standardized by realized volatilities are approximately Gaussian. Second, although the distributions of realized volatilities are clearly right-skewed, the distributions of the logarithms of realized volatilities are approximately Gaussian. Third, the long-run dynamics of realized logarithmic volatilities are well approximated by a fractionally-integrated long-memory process. Motivated by the three ABDL empirical regularities, we proceed to estimate and evaluate a multivariate model for the logarithmic realized volatilities: a fractionally-integrated Gaussian vector autoregression (VAR) . Importantly, our approach explicitly permits measurement errors in the realized volatilities. Comparing the resulting volatility forecasts to those obtained from currently popular daily volatility models and more complicated high-frequency models, we find that our simple Gaussian VAR forecasts generally produce superior forecasts. Furthermore, we show that, given the theoretically motivated and empirically plausible assumption of normally distributed returns conditional on the realized volatilities, the resulting lognormal-normal mixture forecast distribution provides conditionally well-calibrated density forecasts of returns, from which we obtain accurate estimates of conditional return quantiles. In the remainder of this paper, we proceed as follows. We begin in section 2 by formally developing the relevant quadratic variation theory within a standard frictionless arbitrage-free multivariate pricing environment. In section 3 we discuss the practical construction of realized volatilities from high-frequency foreign exchange returns. Next, in section 4 we summarize the salient distributional features of r...

It is obvious that forecasts are of great importance and widely used in economics and finance. Quite simply, good forecasts lead to good decisions. The importance of forecast evaluation and combination techniques follows immediately-- forecast users naturally have a keen interest in monitoring and improving forecast performance. More generally, forecast evaluation figures prominently in many questions in empirical economics and finance, such as: Are expectations rational? (e.g., Keane and Runkle, 1990; Bonham and Cohen, 1995) Are financial markets efficient? (e.g., Fama, 1970, 1991) Do macroeconomic shocks cause agents to revise their forecasts at all horizons, or just at short- and medium-term horizons? (e.g., Campbell and Mankiw, 1987; Cochrane, 1988) Are observed asset returns "too volatile"? (e.g., Shiller, 1979; LeRoy and Porter, 1981) Are asset returns forecastable over long horizons? (e.g., Fama and French, 1988; Mark, 1995)

helpful comments, and the National Science Foundation and the Graduate School We develop regression-based tests of hypotheses about out of sample prediction errors. Representative tests include ones for zero mean and zero correlation between a prediction error and a vector of predictors. The relevant environments are ones in which predictions depend on estimated parameters. We show that standard regression statistics generally fail to account for error introduced by estimation of these parameters. We propose computationally convenient test statistics that properly account for such error. Simulations indicate that the procedures can work well in samples of size typically available, although there sometimes are substantial size distortions.

Forecast combinations have frequently been found in empirical studies to produce better forecasts on average than methods based on the ex-ante best individual forecasting model. Moreover, simple combinations that ignore correlations between forecast errors often dominate more refined combination schemes aimed at estimating the theoretically optimal combination weights. In this chapter we analyze theoretically the factors that determine the advantages from combining forecasts (for example, the degree of correlation between forecast errors and the relative size of the individual models’ forecast error variances). Although the reasons for the success of simple combination schemes are poorly understood, we discuss several possibilities related to model misspecification, instability (non-stationarities) and estimation error in situations where thenumbersofmodelsislargerelativetothe available sample size. We discuss the role of combinations under asymmetric loss and consider combinations of point, interval and probability forecasts. Key words: Forecast combinations; pooling and trimming; shrinkage methods; model misspecification, diversification gains

We develop general model-free adjustment procedures for the calculation of unbiased volatility loss functions based on practically feasible realized volatility benchmarks. The procedures, which exploit the recent non-parametric asymptotic distributional results in Barndorff-Nielsen and Shephard (2002a) along with new results explicitly allowing for leverage effects, are both easy-to-implement and highly accurate in empirically realistic situations. On properly accounting for the measurement errors in the volatility forecast evaluations reported in Andersen, Bollerslev, Diebold and Labys (2003), the adjustments result in markedly higher estimates for the true degree of return volatility predictability.

This paper addresses that issue

This paper describes some recent advances and contributions to our understanding of economic forecasting. The framework we develop helps explain the findings of forecasting competitions and the prevalence of forecast failure. It constitutes a general theoretical background against which recent results can be judged. We compare this framework to a previous formulation, which was silent on the very issues of most concern to the forecaster. We describe a number of aspects which it illuminates, and draw out the implications for model selection. Finally, we discuss the areas where research remains needed to clarify empirical findings which lack theoretical explanations.

: We distinguish between three different ways of using real-time data to estimate forecasting equations and argue that the most frequently used approach should generally be avoided. The point is illustrated with a model that uses monthly observations of industrial production, employment, and retail sales to predict real GDP growth. When the model is estimated using our preferred method, its out-ofsample forecasting performance is clearly superior to that obtained using conventional estimation, and compares favorably with that of the Blue-Chip consensus. * Koenig and Dolmas - Federal Reserve Bank of Dallas, Piger - Federal Reserve Board and Federal Reserve Bank of Dallas. This paper had its origins in a forecasting project undertaken jointly with Ken Emery. Helpful comments and suggestions were offered by Nathan Balke, Dean Croushore, Preston Miller, John Robertson, and attendees of the November 1999 meeting of the Federal Reserve System Committee on Macroeconomics. Dean Croushore and ...

Regression-based tests of forecast probabilities of particular events of interest are constructed. The event forecast probabilities are derived from the SPF density forecasts of expected inflation and output growth. Tests of the event probabilities supplement statistically-based assessments of the forecast densities using the probability integral transform approach. The regression-based tests assess whether the forecast probabilities of particular events are equal to the true probabilities, and whether any systematic divergences between the two are related to variables in the agents ’ informa-tion set at the time the forecasts were made. Forecast encompassing tests are also used to assess the quality of the event probability forecasts.

This paper presents analytical and simulation results on the properties of two tests for forecast encompassing, allowing throughout for dependence of the forecasts on estimated regression parameters. One test, which was intended for forecasts that do not depend on regression parameters, was developed by Harvey, Leyboume and Newbold (1998). This test works relatively well when the size of the sample of forecast errors is small. A second test, which explicitly accounts for uncertainty about the regression parameters, otherwise is comparable or preferable.