Journal of Econometrics 205 (2018) 140–155 Contents lists available at ScienceDirect Journal of Econometrics journal homepage: www.elsevier.com/locate/jeconom Exact Bayesian moment based inference for the distribution of the small-time movements of an Itô semimartingale A. Ronald Gallant a, *, George Tauchen b a Department of Economics, Penn State University, University Park PA 16802, United States b Department of Economics, Duke University, Durham NC 27708, United States article info Article history: Available online 24 March 2018 JEL classification: C51 C52 G12 Keywords: High-frequency data Activity index Efficient method of moments Semimartingale Specification test Spot variance Stochastic volatility abstract We modify the Gallant and Tauchen (1996) efficient method of moments (EMM) method to perform exact Bayesian inference, where exact means no reliance on asymptotic ap- proximations. We use this modification to evaluate the empirical plausibility of recent predictions from high frequency financial theory regarding the small-time movements of an Itô semimartingale. The theory indicates that the probability distribution of the small moves should be locally stable around the origin. It makes no predictions regarding large rare jumps, which get filtered out. Our exact Bayesian procedure imposes support conditions on parameters as implied by this theory. The empirical application uses S&P Index options extending over a wide range of moneyness, including deep out of the money puts. The evidence is consistent with a locally stable distribution valid over most of the support of the observed data while mildly failing in the extreme tails, about which the theory makes no prediction. We undertake diagnostic checks on all aspects of the proce- dure. In particular, we evaluate the distributional assumptions regarding a semi-pivotal statistic, and we test by Monte Carlo that the posterior distribution is properly centered with short credibility intervals. Taken together, our results suggest a more important role than previously thought for pure jump-like models with diminished, if not absent, diffusive component. © 2018 Elsevier B.V. All rights reserved. 1. Introduction 1.1. Overview This paper has two closely connected objectives. The first is to gain further understanding of the distributional aspects of high frequency small-time moves of financial prices. In particular, we aim to evaluate the empirical plausibility of some recent sharp theoretical predictions regarding the probability distribution governing the small moves. As explained further below, this effort directly puts us into a situation where we have a parametric probability model for the data that can be easily simulated, but the density itself is not available in convenient closed form. Of course, this aspect of the paper puts us into the classical indirect estimation context, e.g., simulated method of moments, etc. Of the various available techniques for this context, the EMM approach as reviewed at length in Gallant and Tauchen (2010) has certain optimality properties. However, EMM was developed long before new computing techniques made Bayesian inference feasible for more complicated problems than the classical implausibly simplistic setups. Since Bayes is arguably preferred for parametric * Corresponding author. E-mail addresses: aronaldg@gmail.com (A. Ronald Gallant), george.tauchen@duke.edu (G. Tauchen). https://doi.org/10.1016/j.jeconom.2018.03.008 0304-4076/© 2018 Elsevier B.V. All rights reserved.