Nonparametric Inferences
on Conditional Quantile Processes
First draft: 31 July 2005
This version: 15 January 2007
Submitted for publication
Abstract
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This
paper is concerned with tests of restrictions on the sample path of
conditional quantile processes. These
tests are tantamount to assessments of lack of fit for models of conditional
quantile functions or more generally as tests of how certain covariates
affect the distribution of an outcome variable of interest. This paper extends tests of the generalized
likelihood ratio (GLR) type as introduced by Fan, Zhang & Zhang (2001) to
nonparametric inference problems regarding conditional quantile
processes. As such, the tests proposed
here present viable alternatives to existing methods based on the Khmaladze
(1981, 1988) martingale transformation.
The range of inference problems that may be addressed by the methods
proposed here is wide, and includes tests of nonparametric nulls against
nonparametric alternatives as well as tests of parametric specifications
against nonparametric alternatives. In
particular, it is shown that a class of GLR statistics based on nonparametric
additive quantile regressions have pivotal asymptotic distributions given by
the suprema of squares of Bessel processes, as in Hawkins (1987) and Andrews
(1993). The tests proposed here are
also shown to be asymptotically rate-optimal for tests of nonparametric
hypotheses according to the formulations of Ingster (1993) and of Spokoiny
(1996). |
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KEYWORDS: Quantile regression, nonparametric
inference, minimax rate, additive models, local polynomials, generalized likelihood ratio |
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