The existence theorem of approximate multibeta representation for multifactor pricing models with unobservable omitted variables: A technical note

This paper extends the approximate multibeta representation of Reisman [Reisman, H. (1988), A general approach to the APT, Econometrica, 56, 473-476, Reisman, H. (1992), Reference variables, factor structure, and the approximate multibeta representation, Journal of Finance, 47(4), 1303-1314] with insufficient information. An existence theorem is presented that if the projection error (when regressing the excess returns on a presumed information set of reference variables with linearity) follows the dependence conditions of a mixing random field, there exists an approximate multibeta representation for the risk premium. This result holds even though the linearity is an incorrect specification and/or that the included variables are not sufficiently informative for the model. In particular, the theorem includes omitted (dynamic) factor(s) which may cause unknown inter-temporal or cross-sectional memory in projection errors. An alternative model selection approach is suggested for the specification of risk premium in empirical finance.