The FCLT with Dependent Errors: an Helicopter Tour of the Quality of the Approximation

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9817

PERRON, Pierre

MALLET, Sylvie

The FCLT with Dependent Errors: an Helicopter

Tour of the Quality of the Approximation

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Département de sciences économiques Université de Montréal

Faculté des arts et des sciences C.P. 6128, succursale Centre-Ville Montréal (Québec) H3C 3J7 Canada http://www.sceco.umontreal.ca SCECO-information@UMontreal.CA Téléphone : (514) 343-6539 Télécopieur : (514) 343-7221

Ce cahier a également été publié par le Centre interuniversitaire de recherche en économie quantitative (CIREQ) sous le numéro 3497.

This working paper was also published by the Center for Interuniversity Research in Quantitative Economics (CIREQ), under number 3497.

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CAHIER 9817

THE FCLT WITH DEPENDENT ERRORS : AN HELICOPTER TOUR OF THE QUALITY OF THE APPROXIMATION

Pierre PERRON1 and Sylvie MALLET2

1

Centre de recherche et développement en économique (C.R.D.E.) and Département de sciences économiques, Université de Montréal, and Department of Economics, Boston University

2

Transports Canada

December 1997

_________________

This paper is drawn from the second author's M.Sc. essay at the Université de Montréal [Mallet (1995)]. The first author acknowledges financial support from the Social Sciences and Humanities Research Council (SSHRC) of Canada, the Natural Sciences and

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RÉSUMÉ

Cette note examine la pertinence de l'approximation en échantillon fini donnée par le théorème de la limite centrale fonctionnelle quand les erreurs sont dépendantes. Nous comparons la distribution des sommes partielles échelonnées de quelques donnés avec la distribution du processus de Wiener sur lequel elle converge. Notre modèle est intentionnellement très simple, compte tenu qu'il considère les donnés générées à partir d'un processus ARMA(1,1). Cependant, c'est suffisant pour obtenir des conclusions intéressantes sur les éléments particuliers qui causent des approximations inadéquates même en larges échantillons.

Mots clés : processus de Wiener, pont brownien, fonction de répartition, corrélation temporelle, approximation asymptotique

ABSTRACT

This note investigates the adequacy of the finite-sample approximation provided by the Functional Central Limit Theorem (FCLT) when the errors are allowed to be dependent. We compare the distribution of the scaled partial sums of some data with the distribution of the Wiener process to which it converges. Our setup is purposely very simple in that it considers data generated from an ARMA(1,1) process. Yet, this is sufficient to bring out interesting conclusions about the particular elements which cause the approximations to be inadequate in even quite large sample sizes.

Key words : Wiener process, Brownian bridge, distribution function, serial correlation, asymptotic approximation

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