Testing (Infinitely) Many Zero Restrictions
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We present a max-test statistic for testing (possibly infinitely) many zero parameter restrictions in a general parametric regression framework. The test statistic is based on estimating the key parameters one at a time in many models of possibly vastly smaller dimension than the original model, and choosing the largest in absolute value from these individually estimated parameters. Under mild conditions the parsimonious models identify whether the original parameter of interest is or is not zero. The potentially much lower dimension of the many models used ensures greater estimator accuracy, and using only the largest in a sequence of weighted estimators reduces test statistic complexity and therefore estimation error, ensuring sharper size and greater power in practice. Weights allow for standardization in order to control for estimator dispersion. We use a simulation method or parametric bootstrap for p-value computation without directly requiring the max-statistic's limit distribution. This is critically useful when asymptotically infinitely many parameters are estimated. Existing extreme theory does not exist for the maximum of a sequence of a strongly dependent or nonstationary sequence that arise from estimating many regression models. A simulation experiment shows the max-test dominates a conventional bootstrapped test.
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