Nonparametric testing of an exclusion restriction

Following a framework proposed in Bickel, Ritov, and Stoker (2001) we propose and analyze the behavior of a broad family of tests for H: E(Y | U, V) = E(Y | U) when we observe (Uⁱ, Vⁱ, Yⁱ) ∈ i.i.d., i = 1, …, n. INTRODUCTION The practice of statistical testing plays several roles in empirical research. These roles range from the careful assessment of the evidence against specific scientific hypotheses to the judgment of whether an estimated model displays decent goodness of fit to the empirical data. The paradigmatic situation we consider is one where the investigator views some departures from the hypothesized model as being of primary importance with others of interest if sufficiently gross but otherwise secondary. For instance consider a signal hypothesized to be constant. Low frequency departures from a constant value might be considered of interest, even if of low amplitude, and high-frequency departures as less important, unless they are of high amplitude. Bickel, Ritov, and Stoker (2001) follow this point of view by proposing a general approach to testing semiparametric hypotheses within a nonparametric model in the context of observing n i.i.d. observations. They proposed that tests should be tailored in such a way that on the n^−1/2 scale, power can be concentrated in a few selected directions with some power reserved at the same scale in all other directions.