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The Size Problem of Kernel Based Bootstrap Tests When The Null is Nonparametric

In non- and semiparametric testing, the wild bootstrap is a standard method to determine the critical values of the test. While there exists an increasing literature on how to find a proper smoothing parameter for the nonparametric alternative, almost nothing is known how to choose a smoothing parameter in practice for the null hypothesis if this is semi or nonparametric too. Certainly, we know that at least asymptotically oversmoothing is necessary in the pre-estimation of the null model for generating the bootstrap samples, see HAardle and Marron (1990,1991). However, in practice this knowledge is of little help. Similar statements can be made about various parameter and procedures to choose in practice when performing such a test. In this article we discuss all choice questions. In particular we study the problem of bandwidth choice for the pre-estimation to generate bootstrap samples. As an alternative, we also discuss briefly the possibility of subsampling.