Per A. Mykland

A Tale of Two Time Scales: Determining Integrated Volatility With Noisy High-Frequency Data

It is a common practice in finance to estimate volatility from the sum of frequently sampled squared returns. However, market microstructure poses challenges to this estimation approach, as evidenced by recent empirical studies in finance. The present work attempts to lay out theoretical grounds that reconcile continuous-time modeling and discrete-time samples. We propose an estimation approach that takes advantage of the rich sources in tick-by-tick data while preserving the continuous-time assumption on the underlying returns. Under our framework, it becomes clear why and where the “usual” volatility estimator fails when the returns are sampled at the highest frequencies. If the noise is asymptotically small, our work provides a way of finding the optimal sampling frequency. A better approach, the “two-scales estimator,” works for any size of the noise.

Regeneration in Markov Chain Samplers

Abstract Markov chain sampling has recently received considerable attention, in particular in the context of Bayesian computation and maximum likelihood estimation. This article discusses the use of Markov chain splitting, originally developed for the theoretical analysis of general state-space Markov chains, to introduce regeneration into Markov chain samplers. This allows the use of regenerative methods for analyzing the output of these samplers and can provide a useful diagnostic of sampler performance. The approach is applied to several samplers, including certain Metropolis samplers that can be used on their own or in hybrid samplers, and is illustrated in several examples.

INFERENCE FOR CONTINUOUS SEMIMARTINGALES OBSERVED AT HIGH FREQUENCY

The econometric literature of high frequency data often relies on moment estimators which are derived from assuming local constancy of volatility and related quantities. We here study this local-constancy approximation as a general approach to estimation in such data. We show that the technique yields asymptotic properties (consistency, normality) that are correct subject to an ex post adjustment involving asymptotic likelihood ratios. These adjustments are derived and documented. Several examples of estimation are provided: powers of volatility, leverage effect, and integrated betas. The first order approximations based on local constancy can be over the period of one observation or over blocks of successive observations. It has the advantage of gaining in transparency in defining and analyzing estimators. The theory relies heavily on the interplay between stable convergence and measure change, and on asymptotic expansions for martingales. Copyright 2009 The Econometric Society.

Looking at Markov samplers through cusum path plots: a simple diagnostic idea

In this paper, we propose to monitor a Markov chain sampler using the cusum path plot of a chosen one-dimensional summary statistic. We argue that the cusum path plot can bring out, more effectively than the sequential plot, those aspects of a Markov sampler which tell the user how quickly or slowly the sampler is moving around in its sample space, in the direction of the summary statistic. The proposal is then illustrated in four examples which represent situations where the cusum path plot works well and not well. Moreover, a rigorous analysis is given for one of the examples. We conclude that the cusum path plot is an effective tool for convergence diagnostics of a Markov sampler and for comparing different Markov samplers.

Nonlinear Experiments: Optimal Design and Inference Based on Likelihood

Abstract Nonlinear experiments involve response and regressors that are connected through a nonlinear regression-type structure. Examples of nonlinear models include standard nonlinear regression, logistic regression, probit regression. Poisson regression, gamma regression, inverse Gaussian regression, and so on. The Fisher information associated with a nonlinear experiment is typically a complex nonlinear function of the unknown parameter of interest. As a result, we face an awkward situation. Designing an efficient experiment will require knowledge of the parameter, but the purpose of the experiment is to generate data to yield parameter estimates! Our principal objective here is to investigate proper designing of nonlinear experiments that will let us construct efficient estimates of parameters. We focus our attention on a very general nonlinear setup that includes many models commonly encountered in practice. The experiments considered have two fundamental stages: a static design in the initial stage,...

Are Volatility Estimators Robust with Respect to Modeling Assumptions

We consider microstructure as an arbitrary contamination of the underlying latent securities price, through a Markov kernel Q. Special cases include additive error, rounding, and combinations thereof. Our main result is that, subject to smoothness conditions, the two scales realized volatility (TSRV) is robust to the form of contamination Q. To push the limits of our result, we show what happens for some models involving rounding (which is not, of course, smooth) and see in this situation how the robustness deteriorates with decreasing smoothness. Our conclusion is that under reasonable smoothness, one does not need to consider too closely how the microstructure is formed, while if severe non-smoothness is suspected, one needs to pay attention to the precise structure and also to what use the estimator of volatility will be put.

Realized Volatility When Sampling Times are Possibly Endogenous

When estimating integrated volatilities based on high-frequency data, simplifying assumptions are usually imposed on the relationship between the observation times and the price process. In this paper, we establish a central limit theorem for the realized volatility in a general endogenous time setting. We also establish a central limit theorem for the tricity under the hypothesis that there is no endogeneity, based on which we propose a test and document that this endogeneity is present in financial data.

Estimators of diffusions with randomly spaced discrete observations: A general theory

We provide a general method to analyze the asymptotic properties of a variety of estimators of continuous time diffusion processes when the data are not only discretely sampled in time but the time separating successive observations may possibly be random. We introduce a new operator, the generalized infinitesimal generator, to obtain Taylor expansions of the asymptotic moments of the estimators. As a special case, our results apply to the situation where the data are discretely sampled at a fixed nonrandom time interval. We include as specific examples estimators based on maximum-likelihood and discrete approximations such as the Euler scheme.

The Estimation of Leverage Effect with High Frequency Data

The leverage effect has become an extensively studied phenomenon that describes the (usually) negative relation between stock returns and their volatility. Although this characteristic of stock returns is well acknowledged, most studies of the phenomenon are based on cross-sectional calibration with parametric models. On the statistical side, most previous works are conducted over daily or longer return horizons, and few of them have carefully studied its estimation, especially with high-frequency data. However, estimation of the leverage effect is important because sensible inference is possible only when the leverage effect is estimated reliably. In this article, we provide nonparametric estimation for a class of stochastic measures of leverage effect. To construct estimators with good statistical properties, we introduce a new stochastic leverage effect parameter. The estimators and their statistical properties are provided in cases both with and without microstructure noise, under the stochastic volatility model. In asymptotics, the consistency and limiting distribution of the estimators are derived and corroborated by simulation results. For consistency, a previously unknown bias correction factor is added to the estimators. Applications of the estimators are also explored. This estimator provides the opportunity to study high-frequency regression, which leads to the prediction of volatility using not only previous volatility but also the leverage effect. The estimator also reveals a theoretical connection between skewness and the leverage effect, which further leads to the prediction of skewness. Furthermore, adopting the ideas similar to the estimation of the leverage effect, it is easy to extend the methods to study other important aspects of stock returns, such as volatility of volatility.

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 naively. 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.