A Little Magic with the Cauchy Distribution

The standard Cauchy distribution is completely characterized by theproperty that it has no atmos and is distributionally equivalent under the involution X → –1/X, i.e., X ≡ –1/X. Since maximum likelihood is invariant to the choice of normalization rule in structural equation estimation this property establishes that the LIML estimator is standard Cauchy in the leading case of a canonical structural equation. This is a proof by identifying characteristics and is a major improvement over the usual apparatus of change of variable methods and reductions by multiple integration.

The new approach has applications in many other contexts. A second example considered in the paper is the unidentified ARMA with degenerate common factors. Such models commonly arise from overfitting or overdifferencing. They have eluded conventional asymptotic methods for many years. Yet they are resolved quite simply by the present approach, which yields both an exact finite sample theory and the relevant asymptotics.