Local and Global Rank Tests for Multivariate Varying-Coefficient Models

In a multivariate varying-coefficient model, the response vectors <bold>Y</bold> are regressed on known functions <bold>v</bold>(<bold>X</bold>) of some explanatory variables <bold>X</bold> and the coefficients in an unknown regression matrix <bold> <italic>θ</italic> </bold>(<bold>Z</bold>) depend on another set of explanatory variables <bold>Z</bold>. We provide statistical tests, called local and global rank tests, which allow one to estimate the rank of an unknown regression coefficient matrix <bold> <italic>θ</italic> </bold>(<bold>Z</bold>) locally at a fixed level of the variable <bold>Z</bold> or globally as the maximum rank over all levels of <bold>Z</bold> in a proper, compact subset of the support of <bold>Z</bold>, respectively. We apply our results to estimate the so-called local and global ranks in a demand system where budget shares are regressed on known functions of total expenditures and the coefficients in a regression matrix depend on prices faced by a consumer.