Empirical likelihood based testing for regression

Abstract: Consider a random vector (X, Y ) and let m(x) = E(Y |X = x). We are interested in testing H0 : m ∈ MΘ,G = {γ(·, θ, g) : θ ∈ Θ, g ∈ G} for some known function γ, some compact set Θ ⊂ IR and some function set G of real valued functions. Specific examples of this general hypothesis include testing for a parametric regression model, a generalized linear model, a partial linear model, a single index model, but also the selection of explanatory variables can be considered as a special case of this hypothesis. To test this null hypothesis, we make use of the so-called marked empirical process introduced by [4] and studied by [16] for the particular case of parametric regression, in combination with the modern technique of empirical likelihood theory in order to obtain a powerful testing procedure. The asymptotic validity of the proposed test is established, and its finite sample performance is compared with other existing tests by means of a simulation study.