ALTERNATIVE MEASURES OF THE EXPLANATORY POWER OF GENERAL MULTIVARIATE REGRESSION MODELS

Abstract

In this paper R ²-type measures of the explanatory power of multivariate linear and categorical probit models proposed in the literature are reviewed and their deficiencies discussed. It is argued that a measure of the explanatory power should take into account the components which are explicitly modelled when a regression model is estimated while it should be indifferent to components not explicitly modelled. Based on this view three different measures for multivariate probit models are proposed. Results of a simulation study are presented, designed to compare two measures in various situations, to evaluate the BC _a bootstrap technique for testing the hypothesis that the corresponding measure is zero, and to calculate approximate confidence intervals. The BC _a bootstrap technique turned out to work quite well for a wide range of situations, but may lead to misleading results if the true values of the corresponding measure are close to zero.

Keywords:

• Pseudo-R ²
• Multivariate linear model
• Multivariate probit model
• Panel model
• Simulation study
• Bootstrap confidence intervals

We would like to thank an anonymous referee and the editor for helpful comments and suggestions

Notes

¹In fact, Hooper (1959) considers a model where all of the variables are centered. How- ever, the results remain unchanged regardless of whether the variables are centered or not (cf. Mardia, Kent and Bibby, 1995).

²In applications the true values are generally not known. However, as described in Spiess and Keller (1999), a global strategy (see, e.g., Dennis and Schnabel, 1983) can be used to allow for less optimal starting values. Additionally, if for given starting values the algorithm does not converge, several different starting values should be tried.