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Latent Variable Models for Stochastic Discount Factors


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  • René Garcia
  • Éric Renault



Latent variable models in finance originate both from asset pricing theory and time series analysis. These two strands of literature appeal to two different concepts of latent structures, which are both useful to reduce the dimension of a statistical model specified for a multivariate time series of asset prices. In the CAPM or APT beta pricing models, the dimension reduction is cross-sectional in nature, while in time-series state-space models, dimension is reduced longitudinally by assuming conditional independence between consecutive returns given a small number of state variables. In this chapter, we use the concept of Stochastic Discount Factor (SDF) or pricing kernel as a unifying principle to integrate these two concepts of latent variables. Beta pricing relations amount to characterize the factors as a basis of a vectorial space for the SDF. The coefficients of the SDF with respect to the factors are specified as deterministic functions of some state variables which summarize their dynamics. In beta pricing models, it is often said that only the factorial risk is compensated since the remaining idiosyncratic risk is diversifiable. Implicitly, this argument can be interpreted as a conditional cross-sectional factor structure, that is a conditional independence between contemporaneous returns of a large number of assets given a small number of factors, like in standard Factor Analysis. We provide this unifying analysis in the context of conditional equilibrium beta pricing as well as asset pricing with stochastic volatility, stochastic interest rates and other state variables. We address the general issue of econometric specifications of dynamic asset pricing models, which cover the modern literature on conditionally heteroskedastic factor models as well as equilibrium-based asset pricing models with an intertemporal specification of preferences and market fundamentals. We interpret various instantaneous causality relationships between state variables and market fundamentals as leverage effects and discuss their central role relative to the validity of standard CAPM-like stock pricing and preference-free option pricing. En finance, les modèles à variables latentes apparaissent à la fois dans les théories d'évaluation des actifs financiers et dans l'analyse de séries chronologiques. Ces deux courants de littérature font appel à deux concepts différents de structures latentes qui servent tous deux à réduire la dimension d'un modèle statistique de séries temporelles sur les prix ou les rendements de plusieurs actifs. Dans les modèles CAPM ou APT, où l'évaluation est fonction de coefficients bêtas, la réduction de dimension est de nature transversale, tandis que dans les modèles de séries chronologiques espace-état, la dimension est réduite longitudinalement en supposant l'indépendance conditionnelle entre les rendements consécutifs étant donné un petit nombre de variables d'état. Dans ce chapitre, nous utilisons le concept de facteur d'actualisation stochastique (SDF) ou noyau de valorisation comme principe unificateur en vue d'intégrer ces deux concepts de variables latentes. Les relations de valorisation avec coefficients bêtas reviennent à caractériser les facteurs comme une base d'un espace vectoriel pour le SDF. Les coefficients du SDF par rapport aux facteurs sont spécifiés comme des fonctions déterministes de certaines variables d'état qui résument leur évolution dynamique. Dans ces modèles d'évaluation à coefficients bêtas, on dit souvent que seul le risque factoriel est compensé puisque le risque résiduel idiosyncratique est diversifiable. Implicitement, cet argument peut être interprété comme une structure factorielle transversale conditionnelle, c'est-à-dire une indépendance conditionnelle entre les rendements contemporains d'un grand nombre d'actifs étant donné un petit nombre de facteurs, comme dans l'analyse factorielle standard. Nous établissons cette analyse unificatrice dans le contexte des modèles conditionnels d'équilibre à coefficients bêtas de même que dans des modèles d'évaluation des actifs financiers avec volatilité stochastique, taux d'intérêt stochastiques et autres variables d'état. Nous adressons la question générale de la spécification économétrique des modèles dynamiques d'évaluation des actifs financiers, qui regroupent la littérature moderne des modèles à facteurs conditionnellement hétéroscédastiques ainsi que les modèles d'équilibre d'évaluation des actifs financiers avec une spécification intertemporelle des préférences et des processus fondamentaux du marché. Nous interprétons diverses relations de causalité instantanées entre les variables d'état et les processus fondamentaux du marché comme des effets de levier et discutons le rôle central qu'elles jouent dans la validité des modèles de référence tels que le CAPM pour les actions ou les modèles d'évaluation sans paramètres de préférence pour les options.

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Paper provided by CIRANO in its series CIRANO Working Papers with number 99s-47.

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Date of creation: 01 Nov 1999
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Handle: RePEc:cir:cirwor:99s-47

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Keywords: Stochastic discount factors; latent variables; conditional beta pricing; conditional factor models; equilibrium asset pricing; models with latent variables; Facteurs d'actualisation stochastiques; variables latentes; évaluation des actifs financiers avec bêtas conditionnels; modèles à facteurs conditionnels; modèles d'équilibre d'évaluation des actifs financiers; modèles à variables latentes;

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Cited by:
  1. René Garcia & Richard Luger & Eric Renault, 2000. "Asymmetric Smiles, Leverage Effects and Structural Parameters," Working Papers 2000-57, Centre de Recherche en Economie et Statistique.
  2. René Garcia & Eric Ghysels & Éric Renault, 2004. "The Econometrics of Option Pricing," CIRANO Working Papers 2004s-04, CIRANO.
  3. Paul D. Gilbert & Lise Pichette, 2003. "Dynamic Factor Analysis for Measuring Money," Working Papers 03-21, Bank of Canada.


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