The asymptotic distribution of the sum of weighted squared residuals in binary choice models

It is claimed by some authors that the distribution of the sum of weighted squared residuals, used as a goodness of fit measure in binary choice models, behaves for large n as a x 2 n – k–1 distribution. This claim seems to be based on a false analogy with the well–known Pearson x 2 statistic for frequency tables with a fixed number of cells and cell sizes tending to infinity. We derive the asymptotic (normal) distribution and show that the approximation by the x 2 distribution in general will not be valid. A new x 2 test is proposed based on the asymptotic normality of the measure.