We base our conditions in Assumption A on those employed in KPS (2019), augmented by conditions on the existence of the LRV of $εit2$⁠. The kernel conditions given in Assumption B are similar to those employed by Sun et al. (2008) and Sun (2014) and are satisfied by the Bartlett–Newey–West kernel used in the body of the chapter.

Assumption A.

Assumption B.

The identical distribution assumption in A(i) and A(ii) is convenient in what follows but can no doubt be relaxed under stronger uniform moment conditions that assure the validity of laws of large numbers and central limit theory. Some changes in the formulae for the variances and LRVs in those cases would be needed. Assumption B is similar to the kernel conditions in Sun et al. (2008) and Sun (2014), assures a nonnegative LRV estimator, and is sufficient to validate first order and some higher order asymptotics, although the latter are not considered here. The conditions in Assumption B are satisfied by the Bartlett–Newey–West estimator employed in the LRV estimators (17)–(19) used in the text.