Empirical Likelihood for a Long Range Dependent Process Subordinated to a Gaussian Process

This article develops empirical likelihood methodology for a class of long range dependent processes driven by a stationary Gaussian process. We consider population parameters that are defined by estimating equations in the time domain. It is shown that the standard block empirical likelihood (BEL) method, with a suitable scaling, has a non‐standard limit distribution based on a multiple Wiener–Itô integral. Unlike the short memory time series case, the scaling constant involves unknown population quantities that may be difficult to estimate. Alternative versions of the empirical likelihood method, involving the expansive BEL (EBEL) methods are considered. It is shown that the EBEL renditions do not require an explicit scaling and, therefore, remove this undesirable feature of the standard BEL. However, the limit law involves the long memory parameter, which may be estimated from the data. Results from a moderately large simulation study on finite sample properties of tests and confidence intervals based on different empirical likelihood methods are also reported.     

NON-PARAMETRIC ESTIMATION WITH STRONGLY DEPENDENT MULTIVARIATE TIME SERIES

Smooth non-parametric kernel density and regression estimators are studied when the data are strongly dependent. In particular, we derive central (and non-central) limit theorems for the kernel density estimator of a multivariate Gaussian process and an infinite-order moving average of an independent identically distributed process, as well as the estimator's consistency for other types of data, such as non-linear functions of a Gaussian process. We find that the kernel density estimator at two different points, under certain conditions, is not only perfectly correlated but may converge to the same random variable. Also, central (and non-central) limit theorems of the non-parametric kernel regression estimator are studied. One important and surprising characteristic found is that its asymptotic variance does not depend on the point at which the regression function is estimated and also that its asymptotic properties are the same whether or not regressors are strongly dependent. Finally, a Monte Carlo experiment is reported to assess the behaviour of the estimators in finite samples.     

ON M‐Estimation Under Long‐Range Dependence in Volatility

Abstract. We consider M‐estimation of a location parameter for processes with zero autocorrelations but long‐range dependence in volatility. The observed process is the product of i.i.d. Gaussian observations and a long‐memory Gaussian process. For nonlinear estimators, the rate of convergence depends on the type of the ψ‐function. For skew‐symmetric ψ‐functions, a central limit theorem with ‐rate of convergence holds, under suitable regularity assumptions. This is not true in general for M‐estimators where the ψ‐function is not skewsymmetric.     

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.     

Identification of Persistent Cycles in Non‐Gaussian Long‐Memory Time Series

Abstract. Asymptotic distribution is derived for the least squares estimates (LSE) in the unstable AR(p) process driven by a non‐Gaussian long‐memory disturbance. The characteristic polynomial of the autoregressive process is assumed to have pairs of complex roots on the unit circle. In order to describe the limiting distribution of the LSE, two limit theorems involving long‐memory processes are established in this article. The first theorem gives the limiting distribution of the weighted sum, is a non‐Gaussian long‐memory moving‐average process and (c_n,k,1 ≤ k ≤ n) is a given sequence of weights; the second theorem is a functional central limit theorem for the sine and cosine Fourier transforms      

On the limit theory of the Gaussian SQMLE in the EGARCH(1,1) model

We derive the limit theory of the Gaussian stable quasi maximum likelihood estimator for the stationary EGARCH(1,1) model when the squared innovation process has marginals with regularly varying tails. We derive regularly varying rates and limiting stable distributions. We perform Monte Carlo experiments to assess the extent of the parameter space corresponding to the invertibility condition, and the quality of the asymptotic approximation.     

Flexible and Robust Mixed Poisson INGARCH Models

In this article, we propose a general class of INteger‐valued Generalized AutoRegressive Conditional Heteroskedastic (INGARCH) models based on a flexible family of mixed Poisson (MP) distributions. Our proposed class of count time series models contains the negative binomial (NB) INGARCH process as particular case and open the possibility to introduce new models such as the Poisson‐inverse Gaussian (PIG) and Poisson generalized hyperbolic secant processes. In particular, the PIG INGARCH model is an interesting and robust alternative to the NB model. We explore first‐order and second‐order stationary properties of our MPINGARCH models and provide expressions for the autocorrelation function and mean and variance marginals. Conditions to ensure strict stationarity and ergodicity properties for our class of INGARCH models are established. We propose an Expectation‐Maximization algorithm to estimate the parameters and obtain the associated information matrix. Further, we discuss two additional estimation methods. Monte Carlo simulation studies are considered to evaluate the finite‐sample performance of the proposed estimators. We illustrate the flexibility and robustness of the MPINGARCH models through two real‐data applications about number of cases of Escherichia coli and Campylobacter infections. This article contains a Supporting Information.     

Process design and control optimization: A simultaneous approach by multi‐parametric programming

We present a framework for the application of design and control optimization via multi‐parametric programming through four case studies. We develop design dependent multi‐parametric model predictive controllers that are able to provide the optimal control actions as functions of the system state and the design of the process at hand, via our recently introduced PAROC framework (Pistikopoulos et al, Chem Eng Sci. 2015;136:115–138). The process and the design dependent explicit controllers undergo a mixed integer dynamic optimization (MIDO) step for the determination of the optimal design. The result of the MIDO is the optimal design of the process under optimal operation. We demonstrate the framework through case studies of a tank, a continuously stirred tank reactor, a binary distillation column and a residential cogeneration unit. © 2017 American Institute of Chemical Engineers AIChE J, 2017     

Block Bootstrap for the Empirical Process of Long‐Range Dependent Data

We consider the bootstrapped empirical process of long‐range dependent data. It is shown that this process converges to a semi‐degenerate limit, where the random part of this limit is always Gaussian. Thus the bootstrap might fail when the original empirical process accomplishes a noncentral limit theorem. However, even in this case our results can be used to estimate a nuisance parameter that appears in the limit of many nonparametric tests under long memory. Moreover, we develop a new resampling procedure for goodness‐of‐fit tests and a test for monotonicity of transformations.     

Zero‐Modified Geometric INAR(1) Process for Modelling Count Time Series with Deflation or Inflation of Zeros

In this article, we propose a first‐order integer‐valued autoregressive [INAR(1)] process for dealing with count time series with deflation or inflation of zeros. The proposed process has zero‐modified geometric marginals and contains the geometric INAR(1) process as a particular case. The proposed model is also capable of capturing underdispersion and overdispersion, which sometimes are caused by deflation or inflation of zeros. We explore several statistical and mathematical properties of the process, discuss point estimation of the parameters and find the asymptotic distribution of the proposed estimators. We also propose a test based on our model for checking if the count time series considered is deflated or inflated of zeros. Two empirical illustrations are presented in order to show the potential for practice of our zero‐modified geometric INAR(1) process. This article contains a Supporting Information.     

Combining Cumulative Sum Change‐Point Detection Tests for Assessing the Stationarity of Univariate Time Series

We derive tests of stationarity for univariate time series by combining change‐point tests sensitive to changes in the contemporary distribution with tests sensitive to changes in the serial dependence. The proposed approach relies on a general procedure for combining dependent tests based on resampling. After proving the asymptotic validity of the combining procedure under the conjunction of null hypotheses and investigating its consistency, we study rank‐based tests of stationarity by combining cumulative sum change‐point tests based on the contemporary empirical distribution function and on the empirical autocopula at a given lag. Extensions based on tests solely focusing on second‐order characteristics are proposed next. The finite‐sample behaviors of all the derived statistical procedures for assessing stationarity are investigated in large‐scale Monte Carlo experiments, and illustrations on two real datasets are provided. Extensions to multi‐variate time series are briefly discussed as well.     

TWO‐STEP ESTIMATION OF A MULTI‐VARIATE LÉVY PROCESS

Based on the concept of a Lévy copula to describe the dependence structure of a multi‐variate Lévy process, we present a new estimation procedure. We consider a parametric model for the marginal Lévy processes as well as for the Lévy copula and estimate the parameters by a two‐step procedure. We first estimate the parameters of the marginal processes and then estimate in a second step only the dependence structure parameter. For infinite Lévy measures, we truncate the small jumps and base our statistical analysis on the large jumps of the model. Prominent example will be a bivariate stable Lévy process, which allows for analytic calculations and, hence, for a comparison of different methods. We prove asymptotic normality of the parameter estimates from the two‐step procedure, and in particular, we derive the Godambe information matrix, whose inverse is the covariance matrix of the normal limit law. A simulation study investigates the loss of efficiency because of the two‐step procedure and the truncation.     

Improved Prediction Limits For AR(p) and ARCH(p) Processes

Abstract. A new simulation‐based prediction limit that improves on any given estimative d‐step‐ahead prediction limit for a Markov process is described. This improved prediction limit can be found with almost no algebraic manipulations. Nonetheless, it has the same asymptotic coverage properties as the Barndorff‐Nielsen and Cox [Inference and Asymptotics (1994) Chapman and Hall, London] and Vidoni [Journal of Time Series Analysis Vol. 25, pp. 137–154.] (2004) improved prediction limits. The new simulation‐based prediction limit is ideally suited to those Markov process models for which the algebraic manipulations required for the latter improved prediction limits are very complicated. We illustrate the new method by applying it in the context of one‐step‐ahead prediction for a zero‐mean Gaussian AR(2) process and an ARCH(2) process.     

A CENTRAL LIMIT THEOREM OF FOURIER TRANSFORMS OF STRONGLY DEPENDENT STATIONARY PROCESSES

Abstract. We consider a limiting distribution of the finite Fourier transforms of observations drawn from a strongly dependent stationary process. It is proved that the finite Fourier transforms at different frequencies are asymptotically independent and normally distributed. Our result can apply to a fractional autoregressive integrated moving-average process and a fractional Gaussian noise, two examples of strongly dependent stationary processes.     

Multi‐variate time‐series simulation

In this article we present a method for simulating a multi‐variate time series via a vector auto regressive moving average (p, q) process. We also carried out two simulation studies to check the performance of the method and applied the methodology to a real sea condition time series. All results show that the method works very well in practice.     

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.     

A CONDITIONAL LEAST SQUARES APPROACH TO BILINEAR TIME SERIES ESTIMATION

Abstract. In this paper a conditional least squares (CLS) procedure for estimating bilinear time series models is introduced. This method is applied to a special superdiagonal bilinear model which includes the classical linear autoregressive moving-average model as a particular case and it is proven that the limiting distribution of the CLS estimates is Gaussian and that the law of the iterated logarithm holds.     

The Periodogram of fractional processes1

Abstract. We analyse asymptotic properties of the discrete Fourier transform and the periodogram of time series obtained through (truncated) linear filtering of stationary processes. The class of filters contains the fractional differencing operator and its coefficients decay at an algebraic rate, implying long‐range‐dependent properties for the filtered processes when the degree of integration α is positive. These include fractional time series which are nonstationary for any value of the memory parameter (α ≠ 0) and possibly nonstationary trending (α ≥ 0.5). We consider both fractional differencing or integration of weakly dependent and long‐memory stationary time series. The results obtained for the moments of the Fourier transform and the periodogram at Fourier frequencies in a degenerating band around the origin are weaker compared with the stationary nontruncated case for α > 0, but sufficient for the analysis of parametric and semiparametric memory estimates. They are applied to the study of the properties of the log‐periodogram regression estimate of the memory parameter α for Gaussian processes, for which asymptotic normality could not be showed using previous results. However, only consistency can be showed for the trending cases, 0.5 ≤ α < 1. Several detrending and initialization mechanisms are studied and only local conditions on spectral densities of stationary input series and transfer functions of filters are assumed.     

Dynamic spatial Bayesian models for radioactivity deposition

Dynamic spatial Bayesian (DSB) models are proposed for the analytical modelling of radioactivity deposition after a nuclear accident. The proposed models are extensions of the multi‐variate time‐series dynamic linear models of West and Harrison (1997) to Markov random field processes. They combine the outputs from a long‐range atmospheric dispersal model with measured data (and prior information) to provide improved deposition prediction in space and time. Two versions of a Gaussian DSB model were applied to the radioactivity deposition in Bavaria over a 15 days period during the Chernobyl nuclear accident. One version had fixed functional forms for its spatial variances and covariances while the other allowed those to adapt and ‘learn’ from data in the conjugate Bayesian paradigm. There were two main sources of information for radioactivity deposition in our application: radioactivity measurements at a sparse set of 13 monitoring stations, and the numerical deposition evaluation of the atmospheric dispersal K‐model for the points of a 64 × 64 regular grid. We have analysed the temporal predictions (one‐step‐ahead forecasting) of those DSB models to show that the dispersal K‐model tended in general to underestimate the deposition levels at all times while the DSB models corrected for that although with different degrees of adjustment.     

Local Whittle estimation of multi‐variate fractionally integrated processes

This article derives a semi‐parametric estimator of multi‐variate fractionally integrated processes covering both stationary and non‐stationary values of d. We utilize the notion of the extended discrete Fourier transform and periodogram to extend the multi‐variate local Whittle estimator of Shimotsu (2007) to cover non‐stationary values of d. Consistency and asymptotic normality is shown for d ∈ (?1/2,∞). A simulation study illustrates the performance of the proposed estimator for relevant sample sizes. Empirical justification of the proposed estimator is shown through an empirical analysis of log spot exchange rates. We find that the log spot exchange rates of Germany, United Kingdom, Japan, Canada, France, Italy and Switzerland against the US Dollar for the period January 1974 until December 2001 are well decribed as I(1) processes.