Common Knowledge: The Case of Linear Regression

By Lars Tyge Nielsen

Journal of Mathematical Economics 26 (1996), 285-304

Abstract

This paper reinterprets the multivariate McKelvey-Page theorem as a special case of a result about orthogonal projections in Hilbert space. When information is given by linear signals and the distribution of payoffs and signals is elliptical, it is shown how common and pooled information can be represented by information matrices, and the multivariate McKelvey-Page theorem is reinterpreted as a result about matrix algebra. Applied to a version of Grossman's (1975, 1976, 1978) securities market model with asymmetric information, the result implies that the equilibrium price is common knowledge only if all investors agree on their conditional expectations of payoffs. Combined with a result about pooling of linear signals, this observation implies that the linar rational expectations equilibrium is unique.

Keywords: Asymmetric information, common knowledge, rational expectations equilibrium.