. The nonparametric and the nuisance parameter approaches to consistently testing statistical models are both attempts to estimate topological measures of distance between a parametric and a nonparametric fit, and neither dominates in experiments. This topological unification allows us to greatly extend the nuisance parameter approach. How and why the nuisance parameter approach works and how it can be extended bears closely on recent developments in artificial neural networks. Statistical content is provided by viewing specification tests with nuisance parameters as tests of hypotheses about Banach-valued random elements and applying the Banach Central Limit Theorem and Law of Iterated Logarithm, leading to simple procedures that can be used as a guide to when computationally more elaborate procedures may be warranted. 1. Introduction In testing whether or not a parametric statistical model is correctly specified, there are a number of apparently distinct approaches one might take. T...

Abstract. Many nonparametric estimators and tests are naturally set in infinite dimensional contexts. Prevalence is the infinite dimensional analogue of full Lebesgue measure, shyness the analogue of being a Lebesgue null set. A prevalent set of prior distributions lead to wildly inconsistent Bayesian updating when independent and identically distributed observations happen in class of infinite spaces that includes R n and N. For any rate of convergence, no matter how slow, only a shy set of target functions can be approximated by consistent nonparametric regression schemes in a class that includes series approximations, kernels and other locally weighted regressions, splines, and artificial neural networks. When the instruments allow for the existence of an instrumental regression, the regression function only exists for a shy set of dependent variables. The instruments allow for existence in a counterintuitive dense set of cases, shyness is an open question. A prevalent set of integrated conditional moment (ICM) specification tests are consistent, a dense subset of the finitely parametrized ICM tests are consistent, prevalence is an open question.

A crucial element when undertaking monetary policies is to count on reliable projections regarding the likely effects of changes in income, interest rates, and other macroeconomic variables on monetary aggregates. Understandably, the estimation of money demand functions has been a dynamic field of econometric analysis. The frequently observed