Scoring rules assess the quality of probabilistic forecasts, by assigning a numerical score based on the predictive distribution and on the event or value that materializes. A scoring rule is proper if the forecaster maximizes the expected score for an observation drawn from the distribution F if he or she issues the probabilistic forecast F, rather than G ̸ = F. It is strictly proper if the maximum is unique. In prediction problems, proper scoring rules encourage the forecaster to make careful assessments and to be honest. In estimation problems, strictly proper scoring rules provide attractive loss and utility functions that can be tailored to the problem at hand. This article reviews and develops the theory of proper scoring rules on general probability spaces, and proposes and discusses examples thereof. Proper scoring rules derive from convex functions and relate to information measures, entropy functions, and Bregman divergences. In the case of categorical variables, we prove a rigorous version of the Savage representation. Examples of scoring rules for probabilistic forecasts in the form of predictive densities include the logarithmic, spherical, pseudospherical, and quadratic scores. The continuous ranked probability score applies to probabilistic forecasts that take the form of predictive cumulative distribution functions. It generalizes the absolute error and forms a special case of a new and very general type of score, the energy score. Like many other scoring rules, the energy score admits a kernel representation in terms of negative definite functions, with links to inequalities of Hoeffding type, in both univariate and multivariate settings. Proper scoring rules for quantile and interval forecasts are also discussed. We relate proper scoring rules to Bayes factors and to cross-validation, and propose a novel form of cross-validation known as random-fold cross-validation. A case study on probabilistic weather forecasts in the North American Pacific Northwest illustrates the importance of propriety. We note optimum score approaches to point and quantile

We propose a method for estimating VaR and related risk measures describing the tail of the conditional distribution of a heteroscedastic financial return series. Our approach combines pseudo-maximum-likelihood fitting of GARCH models to estimate the current volatility and extreme value theory (EVT) for estimating the tail of the innovation distribution of the GARCH model. We use our method to estimate conditional quantiles (VaR) and conditional expected shortfalls (the expected size of a return exceeding VaR), this being an alternative measure of tail risk with better theoretical properties than the quantile. Using backtesting of historical daily return series we show that our procedure gives better one-day estimates than methods which ignore the heavy tails of the innovations or the stochastic nature of the volatility. With the help of our fitted models we adopt a Monte Carlo approach to estimating the conditional quantiles of returns over multiple-day horizons and find that t...

A density forecast of the realization of a random variable at some future time is an estimate of the probability distribution of the possible future values of that variable. This chapter presents a selective survey of applications of density forecasting in macroeconomics and finance, and discusses some issues concerning the production, presentation, and evaluation of density forecasts. This chapter first appeared as an article with the same title in Journal of Forecasting, 19 (2000), 235-254. The helpful comments and suggestions of Frank Diebold, Stewart Hodges and two anonymous referees are gratefully acknowledged. Subsequent editorial changes have been made following suggestions from the editors of this volume. Responsibility for errors remains with the authors. 2 1. INTRODUCTION A density forecast of the realization of a random variable at some future time is an estimate of the probability distribution of the possible future values of that variable. It thus provides a complet...

Summary. Probabilistic forecasts of continuous variables take the form of predictive densities or predictive cumulative distribution functions. We propose a diagnostic approach to the evaluation of predictive performance that is based on the paradigm of maximizing the sharpness of the predictive distributions subject to calibration. Calibration refers to the statistical consistency between the distributional forecasts and the observations and is a joint property of the predictions and the events that materialize. Sharpness refers to the concentration of the predictive distributions and is a property of the forecasts only. A simple theoretical framework allows us to distinguish between probabilistic calibration, exceedance calibration and marginal calibration. We propose and study tools for checking calibration and sharpness, among them the probability integral transform histogram, marginal calibration plots, the sharpness diagram and proper scoring rules. The diagnostic approach is illustrated by an assessment and ranking of probabilistic forecasts of wind speed at the Stateline wind energy centre in the US Pacific Northwest. In combination with cross-validation or in the time series context, our proposal provides very general, nonparametric alternatives to the use of information criteria for model diagnostics and model selection.

This paper provides an empirical analysis of the role of jumps in continuous-time models of the short rate. Statistically, if jumps are present di¤usion models are misspeci…ed and I develop a test to detect jump-induced misspeci…cation. After …nding evidence for jumps, I introduce a nonparametric jump-di¤usion model and develop an estimation methodology. The results point toward a dominant statistical role for jumps in determining the dynamics of the short rate relative to di¤usive components. Estimates of jump times and sizes indicate that jumps serve an interesting economic purpose: they provide a main conduit for information about the macroeconomy to enter the term structure. Finally, I investigate the pricing implications of jumps. While jumps do not appear to have a large impact on the cross-section of bond prices, they do have important implications for interest rate derivatives.

This paper applies an extended and generalised version of the recursive modelling strategy developed in Pesaran and Timmermann (1995) to the UK stock market. The focus of the analysis is to simulate investors ' search in `real time ' for a model that can forecast stock returns. We ®nd evidence of predictability in UK stock returns which could have been exploited by investors to improve on the risk-return trade-off offered by a passive strategy in the market portfolio. Alternative interpretations of this ®nding are brie¯y discussed. Economists have long been fascinated by the sources of variations in the stock market. By the early 1970's a consensus had emerged among ®nancial economists suggesting that stock prices could be well approximated by a random walk model and that changes in stock returns were basically unpredictable. 1 Historically, the `random walk ' theory of stock prices was preceded by theories relating movements in the ®nancial markets to the business cycle. A prominent example is the interest shown by Keynes in the variation in stock returns over the business cycle. According to Skidelsky (1992) `Keynes initiated what was called an ``Active Investment Policy'', which combined investing in real assets

This paper proposes a new model for calculating VaR where the user is free to choose any probability distributions for daily changes in the market variables and parameters of the probability distributions are subject to updating schemes such as GARCH. Transformations of the probability distributions are assumed to be multivariate normal. The model is appealing in that the calculation of VaR is relatively straightforward and can make use of the RiskMetrics or a similar database. We test a version of the model using nine years of daily data on 12 different exchange rates. When the first half of the data is used to estimate the model's parameters we find that it provides a good prediction of the distribution of daily changes in the second half of the data. * Faculty of Management, University of Toronto, 105 St. George Street, Toronto, Ontario, Canada M5S 3E6. We are grateful to Tom McCurdy for comments and helpful suggestions. An earlier version of this paper was entitled "Taking account of the kurtosis in market variables when calculating value at risk" 2

Value at Risk (VaR) has emerged in recent years as a standard tool to measure and control the risk of trading portfolios. Yet, existing theoretical analyses of the optimal behavior of a trader subject to VaR limits have produced a negative viewof VaR as a risk-control tool. In particular, VaR limits have been found to induce increased risk exposure in some states and an increased probability of extreme losses. However, these conclusions are based on models that are either static or dynamically inconsistent. In this paper we formulate a dynamically consistent model of optimal portfolio choice subject to VaR limits and showthat the conclusions of earlier papers are incorrect if, consistently with common practice, the VaR is reevaluated dynamically making full use of conditioning information. In particular, we find that the risk exposure of a trader subject to a VaR limit is always lower than that of an unconstrained trader and that the probability of extreme losses is also lower. We also consider the Tail Conditional Expectation (TCE), a coherent risk measure often advocated as an alternative to VaR, and showthat in our dynamic setting it is always possible to transform a TCE limit into an equivalent VaR limit, and conversely.

Many economic and econometric applications require the integration of functions lacking a closed form antiderivative, which is therefore a task that can only be solved by numerical methods. We propose a new family of probability densities that can be used as substitutes and have the property of closed form integrability. This is especially advantageous in cases where either the complexity of a problem makes numerical function evaluations very costly, or fast information extraction is required for timevarying environments. Our approach allows generally for nonparametric maximum likelihood density estimation and may thus nd a variety of applications, two of which are illustrated briey: Estimation of Value at Risk based on approximations to the density of stock returns. Recovering risk neutral densities for the valuation of options from the option price { strike price relation. JEL Classication: C45, G13, C63; Keywords : Option Pricing, Neural Networks, Nonparametric Density Es...

It is common to have interval predictions for volatilities and other quantities governing securities prices. The purpose of this paper is to provide an exact method for converting such intervals into arbitrage based prices of nancial derivatives or industrial or contractual options. We call this procedure conservative delta hedging. The proposed approach will permit an institution's management a greater oversight of its exposure to risk.