Inference for local distributions at high sampling frequencies: A bootstrap approach

We study inference for the local innovations of Itô semimartingales. Specifically, we construct a resampling procedure for the empirical CDF of high-frequency innovations that have been standardized using a nonparametric estimate of its stochastic scale (volatility) and truncated to rid the effect of “large” jumps. Our locally dependent wild bootstrap (LDWB) accommodate issues related to the stochastic scale and jumps as well as account for a special block-wise dependence structure induced by sampling errors. We show that the LDWB replicates first and second-order limit theory from the usual empirical process and the stochastic scale estimate, respectively, in addition to an asymptotic bias. Moreover, we design the LDWB sufficiently general to establish asymptotic equivalence between it and a nonparametric local block bootstrap, also introduced here, up to second-order distribution theory. Finally, we introduce LDWB-aided Kolmogorov–Smirnov tests for local Gaussianity as well as local von-Mises statistics, with and without bootstrap inference, and establish their asymptotic validity using the second-order distribution theory. The finite sample performance of CLT and LDWB-aided local Gaussianity tests is assessed in a simulation study and an empirical application. Whereas the CLT test is oversized, even in large samples, the size of the LDWB tests is accurate, even in small samples. The empirical analysis verifies this pattern, in addition to providing new insights about the fine scale distributional properties of innovations to equity indices, commodities and exchange rates.