Robust estimation for the covariance matrix of multi‐variate time series

In this article, we study the robust estimation for the covariance matrix of stationary multi‐variate time series. As a robust estimator, we propose to use a minimum density power divergence estimator (MDPDE) proposed by Basu et al. (1998) . Particularly, the MDPDE is designed to perform properly when the time series is Gaussian. As a special case, we consider the robust estimator for the autocovariance function of univariate stationary time series. It is shown that the MDPDE is strongly consistent and asymptotically normal under regularity conditions. Simulation results are provided for illustration.     

Subsampling inference for the autocovariances and autocorrelations of long‐memory heavy‐ tailed linear time series

We provide a self‐normalization for the sample autocovariances and autocorrelations of a linear, long‐memory time series with innovations that have either finite fourth moment or are heavy‐tailed with tail index 2 < α < 4. In the asymptotic distribution of the sample autocovariance there are three rates of convergence that depend on the interplay between the memory parameter d and α, and which consequently lead to three different limit distributions; for the sample autocorrelation the limit distribution only depends on d. We introduce a self‐normalized sample autocovariance statistic, which is computable without knowledge of α or d (or their relationship), and which converges to a non‐degenerate distribution. We also treat self‐normalization of the autocorrelations. The sampling distributions can then be approximated non‐parametrically by subsampling, as the corresponding asymptotic distribution is still parameter‐dependent. The subsampling‐based confidence intervals for the process autocovariances and autocorrelations are shown to have satisfactory empirical coverage rates in a simulation study. The impact of subsampling block size on the coverage is assessed. The methodology is further applied to the log‐squared returns of Merck stock.     

Density estimation with locally identically distributed data and with locally stationary data

Abstract. We consider multivariate density estimation when the assumptions of identically distributed data or stationary data are relaxed to the assumptions of locally identically distributed data or locally stationary data. We assume that the distribution of the data is changing continuously as function of time. To estimate densities non‐parametrically with these local regularity conditions, we need time localization in addition to the usual space localization. We define a time‐localized kernel estimator that estimates the density non‐parametrically at any given point of time. The consistency of the time‐localized kernel estimator is proved and the rates of convergence of the estimator are derived under conditions on the β‐and α‐mixing coefficients. Both the time‐series setting and spatial setting are covered.     

DETERMINING THE NUMBER OF REGIMES IN MARKOV SWITCHING VAR AND VMA MODELS

We give stable finite‐order vector autoregressive moving average (p ^* ,q ^* ) representations for M‐state Markov switching second‐order stationary time series whose autocovariances satisfy a certain matrix relation. The upper bounds for p ^* and q ^* are elementary functions of the dimension K of the process, the number M of regimes, the autoregressive and moving‐average orders of the initial model. If there is no cancellation, the bounds become equalities, and this solves the identification problem. Our classes of time series include every M‐state Markov switching multi‐variate moving‐average models and autoregressive models in which the regime variable is uncorrelated with the observable. Our results include, as particular cases, those obtained by Krolzig (1997) and improve the bounds given by Zhang and Stine (2001) and Francq and Zakoïan (2001) for our classes of dynamic models. A Monte Carlo experiment and an application on foreign exchange rates complete the article. Copyright © 2013 John Wiley & Sons, Ltd.     

Stationary bootstrapping for non‐parametric estimator of nonlinear autoregressive model

We consider stationary bootstrap approximation of the non‐parametric kernel estimator in a general kth‐order nonlinear autoregressive model under the conditions ensuring that the nonlinear autoregressive process is a geometrically Harris ergodic stationary Markov process. We show that the stationary bootstrap procedure properly estimates the distribution of the non‐parametric kernel estimator. A simulation study is provided to illustrate the theory and to construct confidence intervals, which compares the proposed method favorably with some other bootstrap methods.     

Integration of CARMA processes and spot volatility modelling

Continuous‐time autoregressive moving average (CARMA) processes with a non‐negative kernel and driven by a non‐decreasing Lévy process constitute a useful and very general class of stationary, non‐negative continuous‐time processes which have been used, in particular for the modelling of stochastic volatility. In the celebrated stochastic volatility model of Barndorff‐Nielsen and Shephard (2001) , the spot (or instantaneous) volatility at time t, V(t), is represented by a stationary Lévy‐driven Ornstein‐Uhlenbeck process. This has the shortcoming that its autocorrelation function is necessarily a decreasing exponential function, limiting its ability to generate integrated volatility sequences, , with autocorrelation functions resembling those of observed realized volatility sequences. (A realized volatility sequence is a sequence of estimated integrals of spot volatility over successive intervals of fixed length, typically 1 day.) If instead of the stationary Ornstein–Uhlenbeck process, we use a CARMA process to represent spot volatility, we can overcome the restriction to exponentially decaying autocorrelation function and obtain a more realistic model for the dependence observed in realized volatility. In this article, we show how to use realized volatility data to estimate parameters of a CARMA model for spot volatility and apply the analysis to a daily realized volatility sequence for the Deutsche Mark/ US dollar exchange rate.     

High‐frequency sampling and kernel estimation for continuous‐time moving average processes

Interest in continuous‐time processes has increased rapidly in recent years, largely because of high‐frequency data available in many applications. We develop a method for estimating the kernel function g of a second‐order stationary Lévy‐driven continuous‐time moving average (CMA) process Y based on observations of the discrete‐time process Y^Δ obtained by sampling Y at Δ, 2Δ, …, nΔ for small Δ. We approximate g by g^Δ based on the Wold representation and prove its pointwise convergence to g as Δ → 0 for continuous‐time autoregressive moving average (CARMA) processes. Two non‐parametric estimators of g^Δ, on the basis of the innovations algorithm and the Durbin–Levinson algorithm, are proposed to estimate g. For a Gaussian CARMA process, we give conditions on the sample size n and the grid spacing Δ(n) under which the innovations estimator is consistent and asymptotically normal as n → ∞. The estimators can be calculated from sampled observations of any CMA process, and simulations suggest that they perform well even outside the class of CARMA processes. We illustrate their performance for simulated data and apply them to the Brookhaven turbulent wind speed data. Finally, we extend results of Brockwell et al. (2012) for sampled CARMA processes to a much wider class of CMA processes.     

On processes with hyperbolically decaying autocorrelations

We discuss some relations between autocorrelations (ACFs) and partial autocorrelations (PACFs) of weakly stationary processes. First, we construct an extension of a process ARIMA(0,d,0) for d ∈ (?∞, 0), which enjoys non‐summable partial autocorrelations and autocorrelations decaying as rapidly as ρ_n ? n^?1+2d. Such a situation is impossible if the absolute sum of autocorrelations is sufficiently small. We show that then the PACF is less than the ACF up to a multiplicative constant. Our second result complements a similar result of Baxter (1962).     

Asymptotic Distribution of the Bias Corrected Least Squares Estimators in Measurement Error Linear Regression Models Under Long Memory

This article derives the consistency and asymptotic distribution of the bias corrected least squares estimators (LSEs) of the regression parameters in linear regression models when covariates have measurement error (ME) and errors and covariates form mutually independent long memory moving average processes. In the structural ME linear regression model, the nature of the asymptotic distribution of suitably standardized bias corrected LSEs depends on the range of the values of where d _X ,d _u , and d _ε are the LM parameters of the covariate, ME and regression error processes respectively. This limiting distribution is Gaussian when and non‐Gaussian in the case . In the former case some consistent estimators of the asymptotic variances of these estimators and a log(n)‐consistent estimator of an underlying LM parameter are also provided. They are useful in the construction of the large sample confidence intervals for regression parameters. The article also discusses the asymptotic distribution of these estimators in some functional ME linear regression models, where the unobservable covariate is non‐random. In these models, the limiting distribution of the bias corrected LSEs is always a Gaussian distribution determined by the range of the values of d _ε ? d _u .     

Monitoring of emulsion polymerisations: A study of reaction kinetics in the presence of secondary nucleation

A non‐linear state estimator that provides on‐line information on residual monomer concentration and radical concentration is experimentally validated under dynamic conditions, and in particular in the presence of secondary particle nucleation. The proposed estimator uses the heat of reaction as an input. Practical methods for determining the initial values of the parameters in the energy balance are also proposed and tested on laboratory and pilot scale installations. It is shown that this method provides a robust tool for the on‐line monitoring of batch (with and without shot additions of monomer, surfactant and initiator) and semi‐batch emulsion polymerisation reactors. The observer was shown to converge rapidly, even under conditions where the number of particles per litre of emulsion (N_p) changed rapidly.     

Multi‐variate stochastic volatility modelling using Wishart autoregressive processes

A new multi‐variate stochastic volatility estimation procedure for financial time series is proposed. A Wishart autoregressive process is considered for the volatility precision covariance matrix, for the estimation of which a two step procedure is adopted. The first step is the conditional inference on the autoregressive parameters and the second step is the unconditional inference, based on a Newton‐Raphson iterative algorithm. The proposed methodology, which is mostly Bayesian, is suitable for medium dimensional data and it bridges the gap between closed‐form estimation and simulation‐based estimation algorithms. An example, consisting of foreign exchange rates data, illustrates the proposed methodology.     

NON‐PARAMETRIC ESTIMATION OF HIGH‐FREQUENCY SPOT VOLATILITY FOR BROWNIAN SEMIMARTINGALE WITH JUMPS

The availability of high‐frequency financial data has led to substantial improvements in our understanding of financial volatility. Most existing literature focuses on estimating the integrated volatility over a fixed period. This article proposes a non‐parametric threshold kernel method to estimate the time‐dependent spot volatility and jumps when the underlying price process is governed by Brownian semimartingale with finite activity jumps. The threshold kernel estimator combines the threshold estimation for integrated volatility and the kernel filtering approach for spot volatility when the price process is driven only by diffusions without jumps. The estimator proposed is consistent and asymptotically normal and has the same rate of convergence as the estimator studied by Kristensen (2010) in a setting without jumps. The Monte Carlo simulation study shows that the proposed estimator exhibits excellent performance over a wide range of jump sizes and for different sampling frequencies. An empirical example is given to illustrate the potential applications of the proposed method.     

Consistent estimation of the memory parameter for nonlinear time series

Abstract. For linear processes, semiparametric estimation of the memory parameter, based on the log‐periodogram and local Whittle estimators, has been exhaustively examined and their properties well established. However, except for some specific cases, little is known about the estimation of the memory parameter for nonlinear processes. The purpose of this paper is to provide the general conditions under which the local Whittle estimator of the memory parameter of a stationary process is consistent and to examine its rate of convergence. We show that these conditions are satisfied for linear processes and a wide class of nonlinear models, among others, signal plus noise processes, nonlinear transforms of a Gaussian process ξ_t and exponential generalized autoregressive, conditionally heteroscedastic (EGARCH) models. Special cases where the estimator satisfies the central limit theorem are discussed. The finite‐sample performance of the estimator is investigated in a small Monte Carlo study.     

Local Likelihood for non‐parametric ARCH(1) models

Abstract. We propose a non‐parametric local likelihood estimator for the log‐transformed autoregressive conditional heteroscedastic (ARCH) (1) model. Our non‐parametric estimator is constructed within the likelihood framework for non‐Gaussian observations: it is different from standard kernel regression smoothing, where the innovations are assumed to be normally distributed. We derive consistency and asymptotic normality for our estimators and show, by a simulation experiment and some real‐data examples, that the local likelihood estimator has better predictive potential than classical local regression. A possible extension of the estimation procedure to more general multiplicative ARCH(p) models with p > 1 predictor variables is also described.     

MULTIVARIATE LIMIT THEOREMS IN THE CONTEXT OF LONG‐RANGE DEPENDENCE

We study the limit law of a vector made up of normalized sums of functions of long‐range dependent stationary Gaussian series. Depending on the memory parameter of the Gaussian series and on the Hermite ranks of the functions, the resulting limit law may be (a) a multi‐variate Gaussian process involving dependent Brownian motion marginals, (b) a multi‐variate process involving dependent Hermite processes as marginals or (c) a combination. We treat cases (a) and (b) in general and case (c) when the Hermite components involve ranks 1 and 2. We include a conjecture about case (c) when the Hermite ranks are arbitrary, although the conjecture can be resolved in some special cases.     

Synthesis of zwitterionic stationary phase based on hydrophilic non‐porous poly(glycidymethacrylate‐co‐ethylenedimethacrylate) beads and their application for fast separation of proteins

A new hydrophilic strong/strong type zwitterionic stationary phase for high performance liquid chromatography (HPLC) was synthesized by chemical modification of 3.0 μm non‐porous monodisperse poly(glycidylmethacrylate‐co‐ethylenedimethacrylate)(P_GMA/EDMA) beads in the following steps. First, the beads were reacted with hydrochloride to obtain chlorizated beads; second, chlorizated beads were reacted with dimethylamine to obtain ammoniated beads; third, ammoniated beads were reacted with 1,3‐propanesultone to obtain non‐porous hydrophilic zwitterionic stationary phase. The stationary phase was evaluated in detail to determine its ion‐exchange properties, separability, reproducibility, hydrophilicity, and the effect of column loading and pH on the separation and retention of proteins. The highest dynamic protein loading capacity of the synthesized zwitterionic packing for bovin serum albumin and Lys were 18.3 and 27.4 mg g^?1, respectively. The zwitterionic stationary phase was capable of separating two acidic and three basic proteins simultaneously in less than 2.5 min by the flow‐rates of 3.0 mL min^?1. The zwitterionic resin was also used for rapid separation and purification of recombinant human interferon‐r (rhIFN‐r) and human granulocyte colony‐stimulation factor (hG‐CSF) from the crude extract solution. The satisfactory results were obtained. © 2009 Wiley Periodicals, Inc. J Appl Polym Sci, 2009     

Testing non‐parametric hypotheses for stationary processes by estimating minimal distances

In this article, new tests for non‐parametric hypotheses in stationary processes are proposed. Our approach is based on an estimate of the L²‐distance between the spectral density matrix and its best approximation under the null hypothesis. We explain the main idea in the problem of testing for a constant spectral density matrix and in the problem of comparing the spectral densities of several correlated stationary time series. The method is based on direct estimation of integrals of the spectral density matrix and does not require the specification of smoothing parameters. We show that the limit distribution of the proposed test statistic is normal and investigate the finite sample properties of the resulting tests by means of a small simulation study.     

A Robbins–Monro Algorithm for Non‐Parametric Estimation of NAR Process with Markov Switching: Consistency

We approach the problem of non‐parametric estimation for autoregressive Markov switching processes. In this context, the Nadaraya–Watson‐type regression functions estimator is interpreted as a solution of a local weighted least‐square problem, which does not admit a closed‐form solution in the case of hidden Markov switching. We introduce a non‐parametric recursive algorithm to approximate the estimator. Our algorithm restores the missing data by means of a Monte Carlo step and estimates the regression function via a Robbins–Monro step. We prove that non‐parametric autoregressive models with Markov switching are identifiable when the hidden Markov process has a finite state space. Consistency of the estimator is proved using the strong α‐mixing property of the model. Finally, we present some simulations illustrating the performances of our non‐parametric estimation procedure.     

Properties of the Sieve Bootstrap for Fractionally Integrated and Non‐Invertible Processes

Abstract. In this article, we investigate the consequences of applying the sieve bootstrap under regularity conditions that are sufficiently general to encompass both fractionally integrated and non‐invertible processes. The sieve bootstrap is obtained by approximating the data‐generating process by an autoregression, whose order h increases with the sample size T. The sieve bootstrap may be particularly useful in the analysis of fractionally integrated processes since the statistics of interest can often be non‐pivotal with distributions that depend on the fractional index d. The validity of the sieve bootstrap is established for |d|<1/2 and it is shown that when the sieve bootstrap is used to approximate the distribution of a general class of statistics then the error rate will be of an order smaller than , β>0. Practical implementation of the sieve bootstrap is considered and the results are illustrated using a canonical example.     

Least Squares Bias in Time Series with Moderate Deviations from a Unit Root

This articles derives the approximate bias of the least squares estimator of the autoregressive coefficient in discrete autoregressive time series where the autoregressive coefficient is given by α_T=1+c/k_T, with k_T being a deterministic sequence increasing to infinity at a rate slower than T, such that k_T=o(T) as T→∞. The cases in which c<0, c=0 and c>0 are considered correspond to (moderately) stationary, non‐stationary and (moderately) explosive series respectively. The result is used to derive the limiting distribution of the indirect inference method for such processes with moderate deviations from a unit root and for explosive series with a fixed coefficient, which does not depend on the sample size. Second, the result demonstrates why the jackknife estimator cannot be constructed for explosive time series for values of the autoregressive parameter close to unity in view of the discontinuity of the bias function, which the article derives. Finally, the expression is used to construct a bias‐corrected estimator, and simulations are carried out to assess the three estimators' bias reduction capabilities.