Bootstrap Consistency for Quadratic Forms of Sample Averages with Increasing Dimension

This paper establishes consistency of the weighted bootstrap for quadratic forms $\left( n^{-1/2} \sum_{i=1}^{n} Z_{i,n} \right)^{T}\left( n^{-1/2} \sum_{i=1}^{n} Z_{i,n} \right)$where$(Z_{i,n})_{i=1}^{n}$ are mean zero, independent ${\mathbb{R}}^d$-valued random variables and $d=d(n)$ is allowed to grow with the sample size $n$, slower than $n^{1\mathrm{/}4}$. The proof relies on an adaptation of Lindeberg interpolation technique whereby we simplify the original problem to a Gaussian approximation problem. We apply our bootstrap results to model-specification testing problems when the number of moments is allowed to grow with the sample size.