Microstructure noise in the continuous case: The pre-averaging approach

This paper presents a generalized pre-averaging approach for estimating the integrated volatility, in the presence of noise. This approach also provides consistent estimators of other powers of volatility — in particular, it gives feasible ways to consistently estimate the asymptotic variance of the estimator of the integrated volatility. We show that our approach, which possesses an intuitive transparency, can generate rate optimal estimators (with convergence rate n^−1/4$n^{-1/4}$).     

Limit theorems for bipower variation of semimartingales

This paper presents limit theorems for certain functionals of semimartingales observed at high frequency. In particular, we extend results from Jacod (2008) [5] to the case of bipower variation, showing under standard assumptions that one obtains a limiting variable, which is in general different from the case of a continuous semimartingale. In a second step a truncated version of bipower variation is constructed, which has a similar asymptotic behaviour as standard bipower variation for a continuous semimartingale and thus provides a feasible central limit theorem for the estimation of the integrated volatility even when the semimartingale exhibits jumps.     

Perfect simulation for locally continuous chains of infinite order

We establish sufficient conditions for perfect simulation of chains of infinite order on a countable alphabet. The new assumption, localized continuity, is formalized with the help of the notion of context trees, and includes the traditional continuous case, probabilistic context trees and discontinuous kernels. Since our assumptions are more refined than uniform continuity, our algorithms perfectly simulate continuous chains faster than the existing algorithms of the literature. We provide several illustrative examples.     

Volatility inference in the presence of both endogenous time and microstructure noise

In this article we consider the volatility inference in the presence of both market microstructure noise and endogenous time. Estimators of the integrated volatility in such a setting are proposed, and their asymptotic properties are studied. Our proposed estimator is compared with the existing popular volatility estimators via numerical studies. The results show that our estimator can have substantially better performance when time endogeneity exists.     

An infinite dimensional convolution theorem with applications to the efficient estimation of the integrated volatility

This paper proposes a general approach to obtain asymptotic lower bounds for the estimation of random functionals. The main result is an abstract convolution theorem in a non parametric setting, based on an associated LAMN property. This result is then applied to the estimation of the integrated volatility, or related quantities, of a diffusion process, when the diffusion coefficient depends on an independent Brownian motion.     

The effect of memory on functional large deviations of infinite moving average processes

The large deviations of an infinite moving average process with exponentially light tails are very similar to those of an i.i.d. sequence as long as the coefficients decay fast enough. If they do not, the large deviations change dramatically. We study this phenomenon in the context of functional large, moderate and huge deviation principles.     

Empirical likelihood for LAD estimators in infinite variance ARMA models

In this paper, we use an empirical likelihood method to construct confidence regions for the stationary ARMA(p,q) models with infinite variance. An empirical log-likelihood ratio is derived by the estimating equation of the self-weighted LAD estimator. It is proved that the proposed statistic has an asymptotic standard chi-squared distribution. Simulation studies show that in a small sample case, the performance of empirical likelihood method is better than that of normal approximation of the LAD estimator in terms of the coverage accuracy.     

Change detection in autoregressive time series

Autoregressive time series models of order p have p+2 parameters, the mean, the variance of the white noise and the p autoregressive parameters. Change in any of these over time is a sign of disturbance that is important to detect. The methods of this paper can test for change in any one of these p+2 parameters separately, or in any collection of them. They are available in forms that make one-sided tests possible, furthermore, they can be used to test for a temporary change. The test statistics are based on the efficient score vector. The large sample properties of the change-point estimator are also explored.     

Estimating the error distribution in nonparametric multiple regression with applications to model testing

In this paper we consider the estimation of the error distribution in a heteroscedastic nonparametric regression model with multivariate covariates. As estimator we consider the empirical distribution function of residuals, which are obtained from multivariate local polynomial fits of the regression and variance functions, respectively. Weak convergence of the empirical residual process to a Gaussian process is proved. We also consider various applications for testing model assumptions in nonparametric multiple regression. The model tests obtained are able to detect local alternatives that converge to zero at an n^−1/2-rate, independent of the covariate dimension. We consider in detail a test for additivity of the regression function.     

Kernel density estimation for spatial processes: the L1 theory

The purpose of this paper is to investigate kernel density estimators for spatial processes with linear or nonlinear structures. Sufficient conditions for such estimators to converge in L₁ are obtained under extremely general, verifiable conditions. The results hold for mixing as well as for nonmixing processes. Potential applications include testing for spatial interaction, the spatial analysis of causality structures, the definition of leading/lagging sites, the construction of clusters of comoving sites, etc.     

The rank of the covariance matrix of an evanescent field

Evanescent random fields arise as a component of the 2D Wold decomposition of homogeneous random fields. Besides their theoretical importance, evanescent random fields have a number of practical applications, such as in modeling the observed signal in the space-time adaptive processing (STAP) of airborne radar data. In this paper we derive an expression for the rank of the low-rank covariance matrix of a finite dimension sample from an evanescent random field. It is shown that the rank of this covariance matrix is completely determined by the evanescent field spectral support parameters, alone. Thus, the problem of estimating the rank lends itself to a solution that avoids the need to estimate the rank from the sample covariance matrix. We show that this result can be immediately applied to considerably simplify the estimation of the rank of the interference covariance matrix in the STAP problem.     

Limit theorems in the Fourier transform method for the estimation of multivariate volatility

In this paper, we prove some limit theorems for the Fourier estimator of multivariate volatility proposed by Malliavin and Mancino (2002, 2009) [14] and [15]. In a general framework of discrete time observations we establish the convergence of the estimator and some associated central limit theorems with explicit asymptotic variance. In particular, our results show that this estimator is consistent for synchronous data, but possibly biased for non-synchronous observations. Moreover, from our general central limit theorem, we deduce that the estimator can be efficient in the case of a synchronous regular sampling. In the non-synchronous sampling case, the expression of the asymptotic variance is in general less tractable. We study this case more precisely through the example of an alternate sampling.     

Estimating the inter-arrival time density of Markov renewal processes under structural assumptions on the transition distribution

We consider a stationary Markov renewal process whose inter-arrival time density depends multiplicatively on the distance between the past and present state of the embedded chain. This is appropriate when the jump size is governed by influences that accumulate over time. Then we can construct an estimator for the inter-arrival time density that has the parametric rate of convergence. The estimator is a local von Mises statistic. The result carries over to the corresponding semi-Markov process.     

Regular variation of order 1 nonlinear AR-ARCH models

We prove both geometric ergodicity and regular variation of the stationary distribution for a class of nonlinear stochastic recursions that includes nonlinear AR-ARCH models of order 1. The Lyapounov exponent for the model, the index of regular variation and the spectral measure for the regular variation all are characterized by a simple two-state Markov chain.     

Partial autocorrelation functions of the fractional ARIMA processes with negative degree of differencing

Let be a fractional ARIMA(p,d,q) process with partial autocorrelation function α(·). In this paper, we prove that if d∈(−1/2,0) then |α(n)|∼|d|/n as n→∞. This extends the previous result for the case 0<d<1/2.     

Representations of the optimal filter in the context of nonlinear filtering of random fields with fractional noise

The problem of nonlinear filtering of multiparameter random fields, observed in the presence of a long-range dependent spatial noise, is considered. When the observation noise is modelled by a persistent fractional Wiener sheet, several pathwise representations of the optimal filter are derived. The representations involve series of multiple stochastic integrals of different types and are particularly important since the evolution equations, satisfied by the best mean-square estimate of the signal random field, have a complicated analytical structure and fail to be proper (measure-valued) stochastic partial differential equations. Several of the above optimal filter representations involve a new family of strong martingale transforms associated to the multiparameter fractional Brownian sheet; the latter martingale family is of independent interest in fractional stochastic calculus of multiparameter random fields.     

Statistical inference for the index parameter in single-index models

In this paper, we are concerned with statistical inference for the index parameter in the single-index model . Based on the estimates obtained by the local linear method, we extend the generalized likelihood ratio test to the single-index model. We investigate the asymptotic behaviour of the proposed test and demonstrate that its limiting null distribution follows a χ²-distribution, with the scale constant and the number of degrees of freedom being independent of nuisance parameters or functions, which is called the Wilks phenomenon. A simulated example is used to illustrate the performance of the testing approach.