Consistency of Kernel Estimators of Heteroscedastic and Autocorrelated Covariance Matrices

Conditions are derived for the consistency of kernel estimators of the covariance matrix of a sum of vectors of dependent heterogeneous random variables, which match those of the currently best-known conditions for the central limit theorem, as required for a unified theory of asymptotic inference. These include finite moments of order no more than 2 + for > 0, trending variances, and variables which are near-epoch dependent on a mixing process, but not necessarily mixing. The results are also proved for the case of sample-dependent bandwidths.