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Bootstrap Consistency for Quadratic Forms of Sample Averages with Increasing Dimension

Abstract

This paper establishes consistency of the weighted bootstrap for quadratic forms (n1/2i=1nZi,n)T(n1/2i=1nZi,n)\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 (Zi,n)i=1n(Z_{i,n})_{i=1}^{n} are mean zero, independent Rd\mathbb{R}^{d}-valued random variables and d=d(n)d=d(n) is allowed to grow with the sample size nn, slower than n1/4n^{1/4}. 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.

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