Forecast Evaluation in the Presence of Unobserved Volatility

Abstract

A number of volatility forecasting studies have led to the perception that the ARCH- and Stochastic Volatility-type models provide poor out-of-sample forecasts of volatility. This is primarily based on the use of traditional forecast evaluation criteria concerning the accuracy and the unbiasedness of forecasts. In this paper we provide an analytical assessment of volatility forecasting performance. We use the volatility and log volatility framework to prove how the inherent noise in the approximation of the true- and unobservable-volatility by the squared return, results in a misleading forecast evaluation, inflating the observed mean squared forecast error and invalidating the Diebold–Mariano statistic. We analytically characterize this noise and explicitly quantify its effects assuming normal errors. We extend our results using more general error structures such as the Compound Normal and the Gram–Charlier classes of distributions. We argue that evaluation problems are likely to be exacerbated by non-normality of the shocks and that non-linear and utility-based criteria can be more suitable for the evaluation of volatility forecasts.     

Functional variable selection via Gram–Schmidt orthogonalization for multiple functional linear regression

ABSTRACT

Functional linear model is of great practical importance, as exemplified by applications in high-throughput studies such as meteorological and biomedical research. In this paper, we propose a new functional variable selection procedure, called functional variable selection via Gram–Schmidt (FGS) orthogonalization, for a functional linear model with a scalar response and multiple functional predictors. Instead of the regularization methods, FGS takes into account the similarity between the functional predictors in a data-driven way and utilizes the technique of Gram–Schmidt orthogonalization to remove the irrelevant predictors. FGS can successfully discriminate between the relevant and the irrelevant functional predictors to achieve a high true positive ratio without including many irrelevant predictors, and yield explainable models, which offers a new perspective for the variable selection method in the functional linear model. Simulation studies are carried out to evaluate the finite sample performance of the proposed method, and also a weather data set is analysed.     

Large sample power of absolute moment tests

Sample kurtosis is a member of the large class of absolute moment tests of normality. We compare kurtosis to other absolute moment tests to determine which are the most powerful at detecting long‐tailed symmetric departures from normality for large samples. The large sample power of the tests is calculated using Geary's (1947) approximations of the moments of the test statistics. Using the system of Gram-Charlier symmetric distributions as alternatives, the most power is obtained using a moment in the range 2.5 ‐ 3.5.