Local and Global Rank Tests for Multivariate Varying-Coefficient Models

Abstract

In a multivariate varying-coefficient model, the response vectors Y are regressed on known functions v(X) of some explanatory variables X and the coefficients in an unknown regression matrix θ(Z) depend on another set of explanatory variables Z. We provide statistical tests, called local and global rank tests, which allow one to estimate the rank of an unknown regression coefficient matrix θ(Z) locally at a fixed level of the variable Z or globally as the maximum rank over all levels of Z in a proper, compact subset of the support of Z, 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.

Stationary subspace analysis of nonstationary covariance processes: Eigenstructure description and testing

“…The proof of the proposition can be found in Appendix A.2, and follows the approach taken in Donald et al (2011).…”

“…The proof of Proposition 4.2 follows the path taken by Donald et al (2011); see, in particular, its technical appendix online. We shall try to minimize repetitions by indicating only the key needed assertions.…”

“…Some of the developments will be somewhat simpler since many of the considered matrices are symmetric. But we shall also need several new auxiliary results to account for the key difference from Donald et al (2011) in that smoothing through a kernel is carried out here in time t while this was over the values of a random variable in Donald et al (2011). Some of the auxiliary results for the proof of Proposition 4.2 can be found in Appendix A.3 and the reader interested in proofs might want to look at them first, before going through the arguments in this section.…”

“…According to one possible definition, a function f is real analytic if its Taylor series converges to the function f in a neighborhood of each point. As noted for similar assumptions G4, G5 in Donald et al (2011), analyticity is assumed to have smoothness of the eigenvectors of analytic matrices involving A 2 (u). It is well known that smoothness of a matrix is not sufficient to have smooth eigenvectors (see, e.g., Kato (1976), Bunse-Gerstner et al (1991)).…”

Multivariate varying‐coefficient models for an ecological system

“…Donald et al (2005) consider rank tests for multivariate varying-coefficient models applied to economic demand systems with errors assumed independent throughout time.…”