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...

We use Bayesian model averaging to analyze the sample evidence on return predictability in the presence of model uncertainty. The analysis reveals in-sample and out-of-sample predictability, and shows that the out-of-sample performance of the Bayesian approach is superior to that of model selection criteria. We find that term and market premia are robust predictors. Moreover, small-cap value stocks appear more predictable than large-cap growth stocks. We also investigate the implications of model uncertainty from investment management perspectives. We show that model uncertainty is more important than estimation risk, and investors who discard model uncertainty face large utility losses.

We argue that the current framework for predictive ability testing (e.g.,West, 1996) is not necessarily useful for real-time forecast selection, i.e., for assessing which of two competing forecasting methods will perform better in the future. We propose an alternative framework for out-of-sample comparison of predictive ability which delivers more practically relevant conclusions. Our approach is based on inference about conditional expectations of forecasts and forecast errors rather than the unconditional expectations that are the focus of the existing literature. We capture important determinants of forecast performance that are neglected in the existing literature by evaluating what we call the forecasting method (the model and the parameter estimation procedure), rather than just the forecasting model. Compared to previous approaches, our tests are valid under more general data assumptions (heterogeneity rather than stationarity) and estimation methods, and they can handle comparison of both nested and non-nested models, which is not currently possible. To illustrate the usefulness of the proposed tests, we compare the forecast performance of three leading parameter-reduction methods for macroeconomic forecasting using a large number of predictors: a sequential model selection approach,

How did the Subprime Crisis, a problem in a small corner of U.S. financial markets, affect the entire global banking system? To shed light on this question we use principal components analysis to identify common factors in the movement of banks ’ credit default swap spreads. We find that fortunes of international banks rise and fall together even in normal times along with short-term global economic prospects. But the importance of common factors rose steadily to exceptional levels from the outbreak of the Subprime Crisis to past the rescue of Bear Stearns, reflecting a diffuse sense that funding and credit risk was increasing. Following the failure of Lehman Brothers, the interdependencies briefly increased to a new high, before they fell back to the pre-Lehman elevated levels – but now they more clearly reflected heightened funding and counterparty risk. After Lehman’s failure, the prospect of global recession became imminent, auguring the further deterioration of banks ’ loan portfolios. At this point the entire global financial system had become infected. 1

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We propose a theoretical framework for assessing whether a forecast model estimated over one period can provide good forecasts over a subsequent period. We formalize this idea by defining a forecast breakdown as a situation in which the out-of-sample performance of the model, judged by some loss function, is significantly worse than its in-sample performance. Our framework, which is valid under general conditions, can be used not only to detect past forecast breakdowns but also to predict future ones. We show that main causes of forecast breakdowns are instabilities in the data generating process and relate the properties of our forecast breakdown test to those of structural break tests. The empirical application finds evidence of a forecast breakdown in the Phillips ’ curve forecasts of U.S. inflation, and links it to inflation volatility and to changes in the monetary policy reaction function of the Fed.

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.

We propose methods for evaluating density forecasts. We focus primarily on methods that are applicable regardless of the particular user's loss function. We illustrate the methods with a detailed simulation example, and then we present an application to density forecasting of daily stock market returns. We discuss extensions for improving suboptimal density forecasts, multi-step-ahead density forecast evaluation, multivariate density forecast evaluation, monitoring for structural change and its relationship to density forecasting, and density forecast evaluation with known loss function.

Empirical studies using survey data on expectations have frequently observed that forecasts are biased and have concluded that agents are not rational. We establish that existing rationality tests are not robust to even small deviations from symmetric loss and hence have little ability to tell whether the forecaster is irrational or the loss function is asymmetric. We quantify the exact trade-off between forecast inefficiency and asymmetric loss leading to identical outcomes of standard rationality tests and explore new and more general methods for testing forecast rationality jointly with flexible families of loss functions that embed quadratic loss as a special case. An empirical application to survey data on forecasts of nominal output growth demonstrates the empirical significance of our results and finds that rejections of rationality may largely have been driven by the assumption of symmetric loss.

In this paper we discuss how to compare various (possibly misspecified) density forecast models using the Kullback-Leibler Information Criterion (KLIC) of a candidate density forecast model with respect to thetruedensity. TheKLIC-differential between a pair of competing models is the (predictive) log-likelihood ratio (LR) between the two models. Even though the true density is unknown, using the LR statistic amounts to comparing models with the KLIC as a loss function and thus enables us to assess which density forecast model can approximate the true density more closely. We also discuss how this KLIC is related to the KLIC based on the probability integral transform (PIT) in the framework of Diebold et al. (1998). While they are asymptotically equivalent, the PIT-based KLIC is best suited for evaluating the adequacy of each density forecast model and the original KLIC is best suited for comparing competing models. In an empirical study with the S&P500 and NASDAQ daily return series, we find strong evidence for rejecting the Normal-GARCH benchmark model, in favor of the models that can capture skewness in the conditional distribution and asymmetry and long-memory in the conditional variance.