Normal Approximation for U-Statistics with Cross-Sectional Dependence

We establish normal approximation in the Wasserstein metric for both non-degenerate and degenerate second-order U-statistics under cross-sectional dependence using Stein's method. For the non-degenerate case, our results extend recent studies on the asymptotic properties of sums of cross-sectionally dependent random variables. The degenerate case is more challenging due to the additional dependence induced by the nonlinearity of the U-statistic kernel. Through a specific implementation of Stein's method, we derive convergence rates under conditions on the mixing rate, the sparsity of the cross-sectional dependence structure, and the moments of the U-statistic kernel. Finally, we demonstrate the application of our theoretical results with a nonparametric specification test for data with cross-sectional dependence.