Some Notes on Regression With A Two-Dimensional Dependent Variable

Abstract

The purpose of these notes is to suggest procedures for dealing with dependent variables in regression models that can be represented as points in the plane. The "trick," if it should be seen as such, is to represent dependent variables in polar coordinates, in which case two-equation models can be specified in which estimation proceeds in terms of functions involving cosines, sines, and radius-vectors. Situations for which this procedure is relevant include analyses of markets in which there are duopoly suppliers. The approach allows for generalization to higher dimensions, and, perhaps most interestingly, can be applied in circumstances in which values of the dependent variable can be points in the complex plane. The procedures are illustrated using cross-sectional data on household toll-calling from a PNR & Associates BILL HARVESTING survey of the mid-1990s and data from the BLS Survey of Consumer Expenditures for the fourth quarter of 1999.