On Estimating the Integrated Co-Volatility Using Noisy High-Frequency Data with Jumps

In this article, we consider the estimation of covariation of two asset prices which contain jumps and microstructure noise, based on high-frequency data. We propose a realized covariance estimator, which combines pre-averaging method to remove the microstructure noise and the threshold method to reduce the jumps effect. The asymptotic properties, such as consistency and asymptotic normality, are investigated. The estimator allows very general structure of jumps, for example, infinity activity or even infinity variation. Simulation is also included to illustrate the performance of the proposed procedure.     

The Relationship between the Volatility of Returns and the Number of Jumps in Financial Markets

We propose a methodology to employ high frequency financial data to obtain estimates of volatility of log-prices which are not affected by microstructure noise and Lévy jumps. We introduce the “number of jumps” as a variable to explain and predict volatility and show that the number of jumps in SPY prices is an important variable to explain the daily volatility of the SPY log-returns, has more explanatory power than other variables (e.g., high and low, open and close), and has a similar explanatory power to that of the VIX. Finally, the number of jumps is very useful to forecast volatility and contains information that is not impounded in the VIX.     

Rate efficient estimation of realized Laplace transform of volatility with microstructure noise

In this paper, we consider the problem of estimating the Laplace transform of volatility within a fixed time interval [0,T] using high‐frequency sampling, where we assume that the discretized observations of the latent process are contaminated by microstructure noise. We use the pre‐averaging approach to deal with the effect of microstructure noise. Under the high‐frequency scenario, we obtain a consistent estimator whose convergence rate is , which is known as the optimal convergence rate of the estimation of integrated volatility functionals under the presence of microstructure noise. The related central limit theorem is established. The simulation studies justify the finite‐sample performance of the proposed estimator.     

Estimation of the integrated volatility using noisy high-frequency data with jumps and endogeneity

In this paper, we investigate a new estimator of the integrated volatility of Itô semimartingales in the presence of both market microstructure noise and jumps when sampling times are endogenous. In the first step, our estimation wipes off the effects of the microstructure noise, and in the second step our estimator shrinks the effects of jumps. We provide consistency of the estimator when the jumps have finite variation and infinite variation and establish a central limit theorem for the estimator in a general endogenous time setting when the jumps only have finite variation. Simulation illustrates the performance of the proposed estimator.     

An empirical analysis of the volatility of the Japanese stock price index: a non-parametric approach

This paper presents an empirical analysis of stochastic features of volatility in the Japanese stock price index, or TOPIX, using high-frequency data sampled every 5 min. The process of TOPIX is modeled by a stochastic differential equation with the time-homogeneous drift and diffusion coefficients. To avoid the risk of misspecification for the volatility function, which is defined by the squared diffusion coefficient, the local polynomial model is applied to the data, and then produced the estimates of the volatility function together with their confidence intervals. The result of the estimation suggests that the volatility function shows similar patterns for one period, but drastically changes for another.     

The failure of meta-analytic asymptotics for the seemingly efficient estimator of the common risk difference

We consider the case of a multicenter trial in which the center specific sample sizes are potentially small. Under homogeneity, the conventional procedure is to pool information using a weighted estimator where the weights used are inverse estimated center-specific variances. Whereas this procedure is efficient for conventional asymptotics (e. g. center-specific sample sizes become large, number of center fixed), it is commonly believed that the efficiency of this estimator holds true also for meta-analytic asymptotics (e.g. center-specific sample size bounded, potentially small, and number of centers large). In this contribution we demonstrate that this estimator fails to be efficient. In fact, it shows a persistent bias with increasing number of centers showing that it isnot meta-consistent. In addition, we show that the Cochran and Mantel-Haenszel weighted estimators are meta-consistent and, in more generality, provide conditions on the weights such that the associated weighted estimator is meta-consistent.     

The limiting distribution of the least-squares estimator in nearly integrated seasonal models

We consider the least-squares estimator of the autoregressive parameter in a nearly integrated seasonal model. Building on the study by Chan (1989), who obtained the limiting distribution, we derive a closed-form expression for the appropriate limiting joint moment generating function. We use this function to tabulate percentage points of the asymptotic distribution for various seasonal periods via numerical integration. The results are extended by deriving a stochastic asymptotic expansion to order O_p(T^-l), whose percentage points are also obtained by numerically integrating the appropriate limiting joint moment generating function. The adequacy of the approximation to the finite-sample distribution is discussed.     

Bayesian estimation and the application of long memory stochastic volatility models

A new sampling-based Bayesian approach to the long memory stochastic volatility (LMSV) process is presented; the method is motivated by the GPH-estimator in fractionally integrated autoregressive moving average (ARFIMA) processes, which was originally proposed by J. Geweke and S. Porter-Hudak [The estimation and application of long memory time series models, Journal of Time Series Analysis, 4 (1983) 221–238]. In this work, we perform an estimation of the memory parameter in the Bayesian framework; an estimator is obtained by maximizing the posterior density of the memory parameter. Finally, we compare the GPH-estimator and the Bayes-estimator by means of a simulation study and our new approach is illustrated using several stock market indices; the new estimator is proved to be relatively stable for the various choices of frequencies used in the regression.     

Wavelet semi-parametric inference for long memory in volatility in the presence of a trend

Risk of investing in a financial asset is quantified by functionals of squared returns. Discrete time stochastic volatility (SV) models impose a convenient and practically relevant time series dependence structure on the log-squared returns. Different long-term risk characteristics are postulated by short-memory SV and long-memory SV models. It is therefore important to test which of these two alternatives is suitable for a specific asset. Most standard tests are confounded by deterministic trends. This paper introduces a new, wavelet-based, test of the null hypothesis of short versus long memory in volatility which is robust to deterministic trends. In finite samples, the test performs better than currently available tests which are based on the Fourier transform.     

Nonparametric estimation of jump characteristics under market microstructure noise

This article proposes a simple nonparametric method to estimate the jump characteristics in asset price with noisy high-frequency data. We combine the pre-averaging approach and the threshold technique to identify the jumps, and then propose the pre-averaging threshold estimators for the number and sizes of jumps occurred. We further present the asymptotic properties of the proposed estimators. The Monte Carlo simulation shows that the estimators are robust to microstructure noise and work very well especially when the data frequency is ultra-high. Finally, an empirical example further demonstrates the power of the proposed method.     

Power Variations and Testing for Co‐Jumps: The Small Noise Approach

In this paper, we study the effects of noise on bipower variation, realized volatility (RV) and testing for co‐jumps in high‐frequency data under the small noise framework. We first establish asymptotic properties of bipower variation in this framework. In the presence of the small noise, RV is asymptotically biased, and the additional asymptotic conditional variance term appears in its limit distribution. We also propose consistent estimators for the asymptotic variances of RV. Second, we derive the asymptotic distribution of the test statistic proposed in (Ann. Stat. 37, 1792‐1838) under the presence of small noise for testing the presence of co‐jumps in a two‐dimensional Itô semimartingale. In contrast to the setting in (Ann. Stat. 37, 1792‐1838), we show that the additional asymptotic variance terms appear and propose consistent estimators for the asymptotic variances in order to make the test feasible. Simulation experiments show that our asymptotic results give reasonable approximations in the finite sample cases.     

An estimator for the cumulative co‐volatility of asynchronously observed semimartingales with jumps

In this paper, we consider two semimartingales sampled at stopping times in an asynchronous manner. We are interested in estimating their cumulative co‐volatility separately from the sum of their co‐jumps. For this purpose, we combine the Hayashi–Yoshida method (to deal with the asynchronicity) with the threshold technique (to separate the jumps) and consider a class of statistics called the truncated Hayashi–Yoshida estimator. We prove the consistency and the asymptotic mixed normality of the truncated Hayashi–Yoshida estimator under some mild conditions allowing the presence of infinite activity jumps.     

A nonparametric specification test for the volatility functions of diffusion processes

This paper develops a new test for the parametric volatility function of a diffusion model based on nonparametric estimation techniques. The proposed test imposes no restriction on the functional form of the drift function and has an asymptotically standard normal distribution under the null hypothesis of correct specification. It is consistent against any fixed alternatives and has nontrivial asymptotic power against a class of local alternatives with proper rates. Monte Carlo simulations show that the test performs well in finite samples and generally has better power performance than the nonparametric test of Li (2007 Li, F. (2007). Testing the parametric specification of the diffusion function in a diffusion process. Econometric Theory 23(2):221–250.[Crossref], [Web of Science ®] , [Google Scholar]) and the stochastic process-based tests of Dette and Podolskij (2008 Dette, H., Podolskij, M. (2008). Testing the parametric form of the volatility in continuous time diffusion models–a stochastic process approach. Journal of Econometrics 143(1):56–73.[Crossref], [Web of Science ®] , [Google Scholar]). When applying the test to high frequency data of EUR/USD exchange rate, the empirical results show that the commonly used volatility functions fit more poorly when the data frequency becomes higher, and the general volatility functions fit relatively better than the constant volatility function.     

Robust estimation of a high-dimensional integrated covariance matrix

In this article, we consider a robust method of estimating a realized covariance matrix calculated as the sum of cross products of intraday high-frequency returns. According to recent articles in financial econometrics, the realized covariance matrix is essentially contaminated with market microstructure noise. Although techniques for removing noise from the matrix have been studied since the early 2000s, they have primarily investigated a low-dimensional covariance matrix with statistically significant sample sizes. We focus on noise-robust covariance estimation under converse circumstances, that is, a high-dimensional covariance matrix possibly with a small sample size. For the estimation, we utilize a statistical hypothesis test based on the characteristic that the largest eigenvalue of the covariance matrix asymptotically follows a Tracy–Widom distribution. The null hypothesis assumes that log returns are not pure noises. If a sample eigenvalue is larger than the relevant critical value, then we fail to reject the null hypothesis. The simulation results show that the estimator studied here performs better than others as measured by mean squared error. The empirical analysis shows that our proposed estimator can be adopted to forecast future covariance matrices using real data.     

The robustness of the quasilikelihood estimator

The quasilikelihood estimator is widely used in data analysis where a likelihood is not available. We illustrate that with a given variance function it is not only conservative, in minimizing a maximum risk, but also robust against a possible misspecification of either the likelihood or cumulants of the model. In examples it is compared with estimators based on maximum likelihood and quadratic estimating functions.     

Estimation of the Stochastic Volatility Model by the Empirical Characteristic Function Method

The stochastic volatility model has no closed form for its likelihood and hence the maximum likelihood estimation method is difficult to implement. However, it can be shown that the model has a known characteristic function. As a consequence, the model is estimable via the empirical characteristic function. In this paper, the characteristic function of the model is derived and the estimation procedure is discussed. An application is considered for daily returns of Australian/New Zealand dollar exchange rate. Model checking suggests that the stochastic volatility model together with the empirical characteristic function estimates fit the data well.     

Faster Convergence to the Estimation of Quadratic Variation with Microstructure Noise

The continuous quadratic variation of asset return plays a critical role for high-frequency trading. However, the microstructure noise could bias the estimation of the continuous quadratic variation. Zhang et al. (2005 Zhang, L., Mykland, P., Ait-Sahalia, Y. (2005). A tale of two time scales: determining integrated volatility with noisy high-frequency data. J. Amer. Statist. Assoc. 100(472):1394–1411.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) proposed a batch estimator for the continuous quadratic variation of high-frequency data in the presence of microstructure noise. It gives the estimates after all the data arrive. This article proposes a recursive version of their estimator that outputs variation estimates as the data arrive. Our estimator gives excellent estimates well before all the data arrive. Both real high-frequency futures data and simulation data confirm the performance of the recursive estimator.     

Asymptotic properties of the maximum-likelihood estimator in zero-inflated binomial regression

The zero-inflated binomial (ZIB) regression model was proposed to account for excess zeros in binomial regression. Since then, the model has been applied in various fields, such as ecology and epidemiology. In these applications, maximum-likelihood estimation (MLE) is used to derive parameter estimates. However, theoretical properties of the MLE in ZIB regression have not yet been rigorously established. The current paper fills this gap and thus provides a rigorous basis for applying the model. Consistency and asymptotic normality of the MLE in ZIB regression are proved. A consistent estimator of the asymptotic variance–covariance matrix of the MLE is also provided. Finite-sample behavior of the estimator is assessed via simulations. Finally, an analysis of a data set in the field of health economics illustrates the paper.     

A large-scale monte carlo study of the buckley-james estimator with censored data

Estimation in the presence of censoring is an important problem. In the linear model, the Buckley-James method proceeds iteratively by estimating the censored values than re-estimating the regression coeffi- cients. A large-scale Monte Carlo simulation technique has been developed to test the performance of the Buckley-James (denoted B-J) estimator. One hundred and seventy two randomly generated data sets, each with three thousand replications, based on four failure distributions, four censoring patterns, three sample sizes and four censoring rates have been investigated, and the results are presented. It is found that, except for Type I1 censoring, the B-J estimator is essentially unbiased, even when the data sets with small sample sizes are subjected to a high censoring rate. The variance formula suggested by Buckley and James (1979) is shown to be sensitive to the failure distribution. If the censoring rate is kept constant along the covariate line, the sample variance of the estimator appears to be insensitive to the censoring pattern with a selected failure distribution. Oscillation of the convergence values associated with the B-J estimator is illustrated and thoroughly discussed.