Abstract 
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[eng] Chapter 1 surveys the most important developments in volatility forecast comparison and model selection, in both the univariate and multivariate frameworks. The chapter reviews a number of evaluation methods and testing procedures for predictive accuracy based on statistical loss functions and recent contributions on the admissible form of loss functions which ensure consistency of the ordering when forecast performances are evaluated with respect to an imperfect volatility proxy.
In chapter 2 we investigate the properties of the ranking between multivariate volatility forecasts with respect to alternative statistical loss functions used to evaluate forecast performances focusing on the multivariate framework and the problems arising due to the latent nature of the conditional variance. We provide conditions on the functional form of the loss function that ensure consistency between the proxybased ranking and the true, but unobservable one. We identify a large set of loss functions that yield a consistent ranking. We illustrate our findings using artificial data and compare the ordering delivered by both consistent and inconsistent loss functions over different forecast horizons. We discuss the sensitivity of the ranking to the quality of the proxy and the degree of similarity between models. An application to three foreign exchange rates, where we compare the forecasting performance of 24 multivariate GARCH specifications over two forecast horizons, concludes the chapter.
In Chapter 3 we address the question of the selection of multivariate GARCH models in terms of forecast accuracy with a particular focus on relatively large scale problems. We consider 10 assets from NYSE and NASDAQ and compare 125 model based onestepahead conditional variance forecasts over a period of 10 years using the MCS and the SPA tests. Model performances are evaluated using four statistical loss functions which account for different types and degrees of asymmetry with respect to over/under predictions. When considering the full sample, MCS results are strongly driven by short periods of high market instability during which multivariate GARCH models appear to be rather inaccurate. Over relatively unstable periods, i.e. dotcom bubble, the set of superior models is composed of more sophisticated specifications such as orthogonal and dynamic conditional correlation, both with leverage effect in the conditional variances. However, unlike the DCC models, our results show that the orthogonal specifications tend to systematically underestimate the conditional variance. Over calm periods, simple assumptions like constant conditional correlation and symmetry in the conditional variances cannot be rejected. Finally, during the 20072008 financial crisis, accounting for nonstationarity in the conditional variance process generates superior forecasts. The SPA test suggests that, independently from the period, the best models do not provide significantly better forecasts than the DCC model with leverage effect in the conditional variances of the returns.
In Chapter 4 we derive a class of diffusion approximations based on conditional correlation models. To our knowledge, this chapter represents a first attempt to address the relationship between multivariate discrete and continuous time models, and in particular to conditional correlation models. We consider the consistent DCC (cDCC) model which is particularly appealing because it is based on a representation of the process driving the correlation which preserves the martingale difference property. We point out the existence of a degenerate diffusion limit. The degeneracy of the cDCC diffusion limit is due to the particular structure of the discrete time model in which the noise propagation system of the variances and that of the process driving the correlation are perfectly correlated. More precisely, the diffusion of the variances and that of the diagonal elements of the process driving the correlation are pairwise governed by the same Brownian motion. We also consider, as a special case, the constant conditional correlation model, which can be obtained by imposing suitable parameter restrictions to the cDCC model. In this case, we are able to recover a nondegenerate diffusion. Finally, we propose different sets of conditions regarding the speed of convergence of the parameters of the cDCC model which allow to recover other degenerate diffusion limits, characterized by a stochastic price process while variances and/or correlations remain time varying but deterministic. We also elaborate on the type of models can be obtained as Euler approximation of the different diffusions.
