A linear approximation to the wild bootstrap in specification testing

The specification of a nonlinear regression model E[Y|X=x] = f(x, θ) for (Y, X) ~ D for a known function f: ℝd × ⊝ → ℝ is to be tested. A possible test statistic is $${\hat T_n} = \tfrac{1}{n}\sum _{1 \leqslant i < j \leqslant n}^n{\hat U_i}{K_{ij}}{\hat U_j}$$, where $${\hat U_i}$$ denote parametrically estimated residuals, and K _ij are kernel weights. Usually the wild bootstrap algorithm (Wu, 1986) is used for for deriving the critical values. Using the structure of an U-statistic inherent to $${\hat T_n}$$, it is possible to approach its limiting distribution directly. The resulting Monte-Carlo-approximation can be viewed as a linear approximation to the wild bootstrap that consumes substantially less computer time for nonlinear models. In simulations this Monte-Carlo-approximation was demonstrated to be applicable. The theoretical foundation lies in an asymptotic consideration that differs from the usual assumptions: The kernel weights K _ij depend on a bandwidth h that is held fixed here, in contrary to the usual setting h = h _n →0. Thus the effects for n→∞ and h _n →0 are separated.