Evaluating dimensionality in spatial voting models

Abstract

Spatial models of voting are widely used in Economics and Political Science. Spatial theory is an attractive framework to analyze choice because Euclidean geometry is easy to visualize and the language of politics is full of spatial references. Empirical tests have generally supported the theory but the estimation methods employed to produce the spatial representations of voters have raised serious statistical issues which have not been fully resolved. One of these issues is determining the number of dimensions. This is a difficult problem because the number of estimated parameters increases with the number of dimensions. We show a solution for this problem in this paper. We assume a uniform distribution of voters through an N-dimensional unit hypersphere with perfect spatial voting and equally salient dimensions. We then solve for the projection of this perfect voting onto one dimension and the resultant classification error. We derive the probability density function of the classification errors which can be used to calculate the likelihood that a sample of votes was drawn from an N-dimensional hypersphere. Our Monte-Carlo investigation into the properties of this likelihood function shows that it can be used reliably for low N with small sample sizes.