Testing whether or not an observed sample follows a specific, hypothesised distribution is one of the fundamental problems in statistics. The applications within Economics and Econometrics are myriad and include basic specification testing for the adequacy of fitted models, tests on predictive densities and of predictive ability as well as, in the two or many sample cases, tests for stochastic dominance and also in cumulative prospect theory. Traditionally such tests have been built on the empirical distribution function (the proportion of observed values falling below a given value) and in particular the statistics applied are either the Kolmogorov-Smirnov or Cramer-von Mises test statistics. Corradi and Swanson's (2006) chapter in the Handbook of Econometrics provides a thorough review of the application of these procedures in the context of evaluating predictive densities, for example.
In this Nottingham School of Economics working Marsh employs a different approach for the fundamental problem. Rather than utilising the empirical distribution the procedures are based upon a consistent non-parametric estimator of the density. Because this density estimator (developed in Barron and Sheu (1991)) is obtained via non-parametric likelihood the resulting test then takes the form of a non-parametric likelihood ratio test. The optimality properties such tests enjoy in the simpler parametric context carry over to the non-parametric case. The results of this paper complement those of Marsh (2007 and 2010) in that here the null hypothesis (of correct specification) is imposed via moment restrictions on the non-parametric density estimator.
It is as a consequence of this approach that the resulting tests are said to be 'Hausman-type', in that the resulting likelihood ratio is then formed as the difference between an estimator which is consistent (but efficient) only under the null hypothesis with one which is consistent (but inefficient) under the null and also the alternative. The test is proven to have a limiting standard normal distribution under the null and to be consistent under any fixed alternative. Numerical experiments in the paper demonstrate the flexibility and fit of the non-parametric density estimator as well as competitive small sample performance of the test itself.
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GC Discussion Paper 15/03, Hausman type tests for nonparametric likelihood by Patrick Marsh
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Posted on Monday 7th September 2015