Autori: Critchley, Frank, Marriott, Paul, Salmon, Mark
Titolo: On the differential geometry of the Wald test with nonlinear restrictions
Periodico: European University Institute of Badia Fiesolana (Fi). Department of Economics - Working papers
Anno: 1994 - Fascicolo: 45 - Pagina iniziale: 1 - Pagina finale: 35

In this paper we exploit the tools of Differential Geometry to analyse the finite sample lack of invariance of the Wald Statistic to algebraically equivalent reformulations of the null hypothesis. The Wald Statistic is shown, in general, to be an improper geometric quantity and hence is not invariant to reparameterisations of the statistical manifold in which it is being used. There is therefore little that can be done to rescue the Wald statistic from this sensitivity to the essentially arbitrary algebraic form in which the null hypothesis is expressed and the testing of nonlinear restrictions should be carried out using invariant approaches such as the Score or Likelihood Ratio procedures instead. The geometric approach also suggests an alternative invariant test based on the calculation of geodesic distances in curved manifolds. We show how this “Fisher Geodesic Statistic” may be easily calculated and applied in the case of testing nonlinear restrictions in the general linear model and also when it will coincide with the Wald Statistic. We are also able to extend the familiar inequalities relating the Wald, Score and Likelihood Ratio Statistics to the nonlinear case with the fundamental difference that the Fisher Geodesic Statistic takes the place previously occupied by the Wald statistic in the relevant inequality. The paper also provides an introduction to the methods of differential geometry and hopefully demonstrates its potential for econometricians.




Testo completo: http://hdl.handle.net/1814/524

Esportazione dati in Refworks (solo per utenti abilitati)