Sampling Interval and estimated Betas : Implications for the Presence of Transitory Components in Stock Prices

9816

PERRON, Pierre

VODOUNOU, Cosme

Sampling Interval and estimated Betas :

Implications for the Presence of Transitory

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 3397.

This working paper was also published by the Center for Interuniversity Research in Quantitative Economics (CIREQ), under number 3397.

CAHIER 9816

SAMPLING INTERVAL AND ESTIMATED BETAS : IMPLICATIONS FOR THE PRESENCE OF TRANSITORY COMPONENTS IN STOCK PRICES

Pierre PERRON1 and Cosme VODOUNOU2

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

Institut National de la Statistique et de l'Analyse Économique, Bénin

December 1997

_________________

This paper is drawn from chapter 3 of Vodounou's Ph.D. Dissertation at the Université de Montréal [Vodounou (1997)]. Perron acknowledges financial support from the Social Sciences and Humanities Research Council (SSHRC) of Canada, the Natural Sciences and Engineering Research Council (NSERC) of Canada, and the Fonds pour la Formation de chercheurs et l'aide à la recherche (FCAR) of Québec.

RÉSUMÉ

Nous donnons un cadre théorique pour expliquer le fait empirique que les bêta estimés sont sensibles à l'intervalle de l'échantillon même en utilisant des rendements composés sans interruption. La composante permanente est un mouvement brownien géométrique standard et la composante transitoire est un processus stationnaire Orstein-Uhlenbeck. La représentation en temps discret des bêta dépend de l'intervalle de l'échantillon et de deux composantes appelées "bêta permanent et transitoire". Nous montrons que, si la composante transitoire n'est pas présente dans les prix des actifs, il n'existe pas d'effet d'intervalle de l'échantillon. Cependant, la présence d'une composante transitoire implique que le bêta est une fonction croissante (décroissante) de l'intervalle de l'échantillon pour actifs plus (moins) risqués. Dans notre cadre théorique, les actifs sont risqués si ses "bêta permanents" sont plus grands que ses "bêta transitoires" et vice versa pour des actifs moins risqués. Les simulations montrent que nos résultats théoriques donnent une bonne approximation pour les moyennes et écart-types des bêta estimés en petits échantillons. Nos résultats peuvent être perçus comme une évidence indirecte de la présence d'une composante transitoire dans les prix des actifs, comme proposé par Fama et French (1988) et Poterba et Summers (1988).

Mots clés : équations différentielles stochastiques, processus de Wiener, efficacité des marchés, retour à la moyenne, processus d'Ornstein-Uhlenbeck

ABSTRACT

We provide a theoretical framework to explain the empirical finding that the estimated betas are sensitive to the sampling interval even when using continuously compounded returns. We suppose that stock prices have both permanent and transitory components. The permanent component is a standard geometric Brownian motion while the transitory component is a stationary Ornstein-Uhlenbeck process. The discrete time representation of the beta depends on the sampling interval and two components labelled "permanent and transitory betas". We show that if no transitory component is present in stock prices, then no sampling interval effect occurs. However, the presence of a transitory component implies that the beta is an increasing (decreasing) function of the sampling interval for more (less) risky assets. In our framework, assets are labelled risky if their "permanent beta" is greater than their "transitory beta" and vice versa for less risky assets. Simulations show that our theoretical results provide good approximations for the means and standard deviations of estimated betas in small samples. Our results can be perceived as indirect evidence for the presence of a transitory component in stock prices, as proposed by Fama and French (1988) and Poterba and Summers (1988).

Key words : stochastic differential equations, Wiener process, market efficiency, mean-reversion, Ornstein-Uhlenbeck process

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