Inference for Continuous Semimartingales Observed at High Frequency: A General Approach

The econometric literature of high frequency data usually relies on moment estimators which are derived from assuming local constancy of volatility and related quantities. We here show that this first order approximation is not always valid if used näıvely. We find that such approximations require an ex post adjustment involving asymptotic likelihood ratios. These are given. Several examples (powers of volatility, leverage effect, ANOVA) are provided. The first order approximations in this study can be over the period of one observation, or over blocks of successive observations. The theory relies heavily on the interplay between stable convergence and measure change, and on asymptotic expansions for martingales. Practically, the procedure permits (1) the definition of estimators of hard to reach quantities, such as the leverage effect, (2) the improvement in efficiency in classical estimators, and (3) easy analysis. More conceptually, we show that the approximation induces a measure change similar to that occurring in options pricing theory. In particular, localization over one observation induces a measure change related to the leverage effect, while localization over a block of observations creates an effect that connects to the volatility of volatility. Another conceptual gain is the relationship to Hermite polynomials. The three measure changes mentioned relate, respectively, to the first, third, and second such polynomial.