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On Loss Functions and Ranking Forecasting Performances of Multivariate Volatility Models

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  • Sébastien Laurent
  • Jeroen Rombouts

    ()

  • Francesco Violente

Abstract

A large number of parameterizations have been proposed to model conditional variance dynamics in a multivariate framework. However, little is known about the ranking of multivariate volatility models in terms of their forecasting ability. The ranking of multivariate volatility models is inherently problematic because it requires the use of a proxy for the unobservable volatility matrix and this substitution may severely affect the ranking. We address this issue by investigating the properties of the ranking with respect to alternative statistical loss functions used to evaluate model performances. We provide conditions on the functional form of the loss function that ensure the proxy-based ranking to be consistent for the true one - i.e., the ranking that would be obtained if the true variance matrix was observable. We identify a large set of loss functions that yield a consistent ranking. In a simulation study, we sample data from a continuous time multivariate diffusion process and compare the ordering delivered by both consistent and inconsistent loss functions. We further 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 16 multivariate GARCH specifications, is provided. Un grand nombre de méthodes de paramétrage ont été proposées dans le but de modéliser la dynamique de la variance conditionnelle dans un cadre multivarié. Toutefois, on connaît peu de choses sur le classement des modèles de volatilité multivariés, du point de vue de leur capacité à permettre de faire des prédictions. Le classement des modèles de volatilité multivariés est forcément problématique du fait qu’il requiert l’utilisation d’une valeur substitutive pour la matrice de la volatilité non observable et cette substitution peut influencer sérieusement le classement. Nous abordons ce problème en examinant les propriétés du classement en relation avec les fonctions de perte statistiques alternatives utilisées pour évaluer la performance des modèles. Nous présentons des conditions liées à la forme fonctionnelle de la fonction de perte qui garantissent que le classement fondé sur une valeur de substitution est constant par rapport au classement réel, c’est-à-dire à celui qui serait obtenu si la matrice de variance réelle était observable. Nous établissons un vaste ensemble de fonctions de perte qui produisent un classement constant. Dans le cadre d’une étude par simulation, nous fournissons un échantillon de données à partir d’un processus de diffusion multivarié en temps continu et comparons l’ordre généré par les fonctions de perte constantes et inconstantes. Nous approfondissons la question de la sensibilité du classement à la qualité de la substitution et le degré de ressemblance entre les modèles. Une application à trois taux de change est proposée et, dans ce contexte, nous comparons l’efficacité de prédiction de 16 paramètres du modèle GARCH multivarié (approche d’hétéroscédasticité conditionnelle autorégressive généralisée).

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Bibliographic Info

Paper provided by CIRANO in its series CIRANO Working Papers with number 2009s-45.

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Date of creation: 01 Nov 2009
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Handle: RePEc:cir:cirwor:2009s-45

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Keywords: Volatility; multivariate GARCH; matrix norm; loss function; model confidence set; Volatilité; modèle GARCH multivarié; norme matricielle; fonction de perte; ensemble de modèles de confiance.;

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