Bandwidth selection for continuous-time Markov processes

We propose a theoretical approach to bandwidth choice for continuous-time Markov processes. We do so in the context of stationary and nonstationary processes of the recurrent kind. The procedure consists of two steps. In the first step, by invoking local Gaussianity, we suggest an automated bandwidth selection method which maximizes the probability that the standardized data are a collection of normal draws. In the case of diffusions, for instance, this procedure selects a bandwidth which only ensures consistency of the infinitesimal variance estimator, not of the drift estimator. Additionally, the procedure does not guarantee that the rate conditions for asymptotic normality of the infinitesimal variance estimator are satisfied. In the second step, we propose tests of the hypothesis that the bandwidth(s) are either "too small" or "too big" to satisfy all necessary rate conditions for consistency and asymptotic normality. The suggested statistics rely on a randomized procedure based on the idea of conditional inference. Importantly, if the null is rejected, then the first-stage bandwidths are kept. Otherwise, the outcomes of the tests indicate whether larger or smaller bandwidths should be selected. We study scalar and multivariate diffusion processes, jump-diffusion processes, as well as processes measured with error as is the case, for instance, for stochastic volatility modelling by virtue of preliminary high-frequency spot variance estimates. The finite sample joint behavior of our proposed automated bandwidth selection method, as well as that of the associated (second-step) randomized procedure, are studied via Monte Carlo simulation.