  This paper develops a general asymptotic theory for nonparametric kernel
regression in the presence of cluster dependence. We examine nonparametric
density estimation, Nadaraya-Watson kernel regression, and local linear
estimation. Our theory accommodates growing and heterogeneous cluster sizes. We
derive asymptotic conditional bias and variance, establish uniform consistency,
and prove asymptotic normality. Our findings reveal that under heterogeneous
cluster sizes, the asymptotic variance includes a new term reflecting
within-cluster dependence, which is overlooked when cluster sizes are presumed
to be bounded. We propose valid approaches for bandwidth selection and
inference, introduce estimators of the asymptotic variance, and demonstrate
their consistency. In simulations, we verify the effectiveness of the
cluster-robust bandwidth selection and show that the derived cluster-robust
confidence interval improves the coverage ratio. We illustrate the application
of these methods using a policy-targeting dataset in development economics.
