Testing index sufficiency with a predicted
index
Andreas Dzemski∗
November 20, 2015
This paper tackles the problem of testing the null hypothesis of index sufficiency
H0 : E[Y |X] = E[Y |r0 (X)]
when the index rule r0 is unknown and has to be estimated at a first stage.
I extend a testing approach by Delgado and Manteiga 2001 to allow for
predicted variables in the conditioning set of the null model. The class of
permissible estimators of r0 is characterized in terms of assumptions about
their precision and complexity, and comprises a wide range of parametric,
semiparametric and fully nonparametric estimators. I provide a stochastic
expansion that describes how the estimation of the index rule affects the
asymptotic distribution of the test statistic. This expansion holds uniformly
over the class of permissible first-stage estimators. As demonstrated for kernelbased estimators, the first-stage estimation typically affects the asymptotic
distribution of the test statistic. I suggest a multiplier bootstrap procedure
which explicitly accounts for the first-stage estimation error. A rejection rule
based on bootstrapped critical values guarantees that the test is correctly sized.
In contrast to the case of an observed index, employing a higher-order kernel
is not sufficient to eliminate the bias from kernel smoothing. An alternative
procedure that uses an estimator of the bias to center the test statistic works
under relatively weak assumptions about the first-stage estimator.
JEL codes: C12, C14, C52
Keywords: significance test, generated regressors, U-statistic, multiplier
bootstrap
∗
School of Business, Economics and Law, University of Gothenburg.
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