2002s-93 ARMA Representation of Integrated and Realized … · 2009. 3. 4. · ARMA Representation...

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Montréal Décembre 2002

© 2002 Nour Meddahi. Tous droits réservés. All rights reserved. Reproduction partielle permise avec citation du document source, incluant la notice ©. Short sections may be quoted without explicit permission, if full credit, including © notice, is given to the source.

Série Scientifique Scientific Series

2002s-93

ARMA Representation of Integrated and Realized

Variances

Nour Meddahi

CIRANO

Le CIRANO est un organisme sans but lucratif constitué en vertu de la Loi des compagnies du Québec. Le financement de son infrastructure et de ses activités de recherche provient des cotisations de ses organisations-membres, d’une subvention d’infrastructure du ministère de la Recherche, de la Science et de la Technologie, de même que des subventions et mandats obtenus par ses équipes de recherche.

CIRANO is a private non-profit organization incorporated under the Québec Companies Act. Its infrastructure and research activities are funded through fees paid by member organizations, an infrastructure grant from the Ministère de la Recherche, de la Science et de la Technologie, and grants and research mandates obtained by its research teams.

Les organisations-partenaires / The Partner Organizations

•École des Hautes Études Commerciales •École Polytechnique de Montréal •Université Concordia •Université de Montréal •Université du Québec à Montréal •Université Laval •Université McGill •Ministère des Finances du Québec •MRST •Alcan inc. •AXA Canada •Banque du Canada •Banque Laurentienne du Canada •Banque Nationale du Canada •Banque Royale du Canada •Bell Canada •Bombardier •Bourse de Montréal •Développement des ressources humaines Canada (DRHC) •Fédération des caisses Desjardins du Québec •Hydro-Québec •Industrie Canada •Pratt & Whitney Canada Inc. •Raymond Chabot Grant Thornton •Ville de Montréal

ISSN 1198-8177

Les cahiers de la série scientifique (CS) visent à rendre accessibles des résultats de recherche effectuée au CIRANO afin de susciter échanges et commentaires. Ces cahiers sont écrits dans le style des publications scientifiques. Les idées et les opinions émises sont sous l’unique responsabilité des auteurs et ne représentent pas nécessairement les positions du CIRANO ou de ses partenaires. This paper presents research carried out at CIRANO and aims at encouraging discussion and comment. The observations and viewpoints expressed are the sole responsibility of the authors. They do not necessarily represent positions of CIRANO or its partners.

ARMA Representation of Integrated and Realized Variances*

Nour Meddahi†

Résumé / Abstract

Nous dérivons la représentation ARMA des variances intégrées et réalisées quand la variance instantanée est la combinaison linéaire de deux facteurs auto-régressifs, c’est-à-dire, les modèles SR-SARV(2). Cette classe de processus contient les modèles affines, diffusion GARCH, CEV, à fonctions propres, ainsi que les processus Ornstein-Uhlenbeck et positifs. Nous étudions le cas à effet de levier, et aussi le lien entre la représentation GARCH faible des rendements et la représentation ARMA de la volatilité réalisée. Finalement, nous analysons les conséquences empiriques de ces représentations ARMA. Nous trouvons qu’il est possible que certains paramètres de la représentation ARMA soient négatifs. Ainsi, la positivité de l’espérance linéaire des variances intégrées et réalisées n’est pas assurée. Nous trouvons aussi que pour certaines fréquences d’observation, les paramètres du modèle en temps continu peuvent être faiblement identifiables ou pas identifiables à partir de la représentation ARMA de variances réalisées.

This paper derives the ARMA representation of integrated and realized variances when the spot variance depends linearly on two autoregressive factors, i.e., SR-SARV(2) models. This class of processes includes affine, GARCH diffusion, CEV models, as well as the eigenfunction stochastic volatility and the positive Ornstein-Uhlenbeck models. We also study the leverage effect case, the relationship between weak GARCH representation of returns and the ARMA representation of realized variances. Finally, various empirical implications of these ARMA representations are considered. We find that it is possible that some parameters of the ARMA representation are negative. Hence, the positiveness of the expected values of integrated or realized variances is not guaranteed. We also find that for some frequencies of observations, the continuous time model parameters may be weakly or not identified through the ARMA representation of realized variances.

Mots clés : variance intégrée, variance réalisée, représentation ARMA, modèles SR-SARV, faible identification. Keywords: Integrated variance, realized variance, ARMA representation, SR-SARV models, weak identification.

* The author would like to thank Torben Andersen, Tim Bollerslev, and especially Bryan Campbell and Neil Shephard for helpful discussions and comments. The author acknowledges financial support from IFM2, MITACS, and Canada Research Chair Program (Econometrics Chair: Jean-Marie Dufour). The author thanks the Bendheim Center for Finance, Princeton University, for its hospitality during his visit and the revision of this paper, and is responsible for any remaining errors. † Département de sciences économiques, CIRANO, CIREQ, Université de Montréal, C.P. 6128, succursale Centre-ville, Montréal (Québec), H3C 3J7, Canada, phone: 514-343-2399, e-mail: nour.meddahi@umontreal.ca, webpage: http://www.mapageweb.umontreal.ca/meddahin/.

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