We consider conditional distribution and conditional density functionals in the space of generalized functions. The approach follows Phillips (1985, 1991, 1995) who employed generalized functions to overcome non-differentiability in order to develop expansions. We obtain the limit of the kernel estimators for weakly dependent data, even under non-differentiability of the distribution function; the limit Gaussian process is characterized as a stochastic random functional (random generalized function) on the suitable function space. An alternative simple to compute estimator based on the empirical distribution function is proposed for the generalized random functional. For test statistics based on this estimator, limit properties are established. A Monte Carlo experiment demonstrates good finite sample performance of the statistics for testing logit and probit specification in binary choice models.
The authors thank the participants of the 14th Advances in Econometrics Conference and especially Brendan Beare, and an anonymous referee, the editors Yoosoon Chang and Joon Park for helpful comments and suggestions. The authors gratefully acknowledge support of this research from the Fonds de recherche sur la société et la culture (Québec).
Tuvaandorj, P. and Zinde-Walsh, V. (2014), "Limit Theory and Inference About Conditional Distributions", Essays in Honor of Peter C. B. Phillips (Advances in Econometrics, Vol. 33), Emerald Group Publishing Limited, Bingley, pp. 397-423. https://doi.org/10.1108/S0731-905320140000033012
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