A probabilistic forecast is the estimated probability with which a future event will occur. One interesting feature of such forecasts is their calibration, or the match between the predicted probabilities and the actual outcome probabilities. Calibration has been evaluated in the past by...

Using the tail index of returns on U.S. equities as a summary measure of extreme behavior, we examine changes in the equity markets surrounding the development of program trading for portfolio insurance, the crash of 1987, and the subsequent introduction of circuit breakers and other changes in...

Many processes can be represented in a simple form as infinite-order linear series. In such cases, an approximate model is often derived as a truncation of the infinite-order process, for estimation on the finite sample. The literature contains a number of asymptotic distributional results for...

Financial returns typically display heavy tails and some degree of skewness, and conditional variance models with these features often outperform more limited models. The difference in performance may be especially important in estimating quantities that depend on tail features, including risk...

For a general stationary ARMA(<italic>p,q</italic>) process <italic>u</italic> we derive the <italic>exact</italic> form of the orthogonalizing matrix <italic>R</italic> such that <italic>R</italic>′<italic>R</italic> = Σ⁻¹, where Σ = <italic>E</italic>(<italic>uu</italic>′) is the covariance matrix of <italic>u</italic>, generalizing the known formulae for <italic>AR</italic>(<italic>p</italic>) processes. In a linear regression model with an ARMA(<italic>p,q</italic>) error process,...