Inferring the rank of a matrix

This paper considers methods of inference concerning the rank of matrix a π - ξ based on an asymptotically normal estimate of π and some identifiable specification for ξ. One such specification is ξ = 0, in which case one is interested in the rank of π. We first propose, and examine the properties of, a test of the hypothesis that the rank is of a given size against the alternative that the rank is larger. We then look at the problem of estimating the rank of this matrix using model selection procedures and sequential hypothesis testing. Conditions for consistency of such procedures are obtained. Some economic applications are discussed and the various methods are compared in a Monte Carlo experiment.