Generalized Resampling Scheme With Application to Spectral Density Matrix in Almost Periodically Correlated Class of Time Series

The aim of this article is to introduce new resampling scheme for nonstationary time series, called generalized resampling scheme (GRS). The proposed procedure is a generalization of well known in the literature subsampling procedure and is simply related to existing block bootstrap techniques. To document the usefulness of GRS, we consider the example of model with almost periodic phenomena in mean and variance function, where the consistency of the proposed procedure was examined. Finally, we prove the consistency of GRS for the spectral density matrix for nonstationary, multivariate almost periodically correlated time series. We consider both zero mean and non‐zero mean case. The consistency holds under general assumptions concerning moment and α‐mixing conditions for multivariate almost periodically correlated time series. Proving the consistency in this case poses a difficulty since the estimator of the spectral density matrix can be interpreted as a sum of random matrixes whose dependence grow with the sample size.     

Exponential inequalities for Mann’s stochastic algorithm

Abstract

In this article, we investigate the problem of approximating the fixed point for some function using a Mann iterative process with random errors. After establishing some exponential inequalities, we prove the complete convergence of Mann’s algorithm toward the fixed point and deduce a confidence interval for this one. In addition, we establish the convergence rate of Mann’s algorithm. Several numerical examples are sketched to illustrate the performance of the proposed algorithm.     

Random effects mixture models for clustering electrical load series

For purposes such as rate setting and long‐term capacity planning, electrical utility companies are interested in dividing their customers into homogeneous groups or clusters in terms of the customers’ electricity demand profiles. Such demand profiles are typically represented by load series, long time series of daily or even hourly rates of energy consumption of individual customers. The high dimension and time series nature inherent in the load series render existing methods of clustering analysis ineffective. To handle the high dimension and to take advantage of the time‐series nature of load series, we introduce a class of mixture models for time series, the random effects mixture models, which are particularly useful for clustering the load series. The random effects mixture models are based on a hierarchical model for individual components. They employ highly flexible antedependence models to effectively capture the time‐series characteristics of the covariance of the load series. We present details on the construction of such mixture models and discuss a special Expectation‐maximization (EM) algorithm for their computation. We also apply these models to cluster the data set which had motivated this research, a set of 923 load series from BC Hydro, a crown utility company in British Columbia, Canada.     

Exponential inequalities for Mann’s iterative scheme with functional random errors

In this article, we deal with an iteration method for approximating a fixed point of a contraction mapping using Mann’s algorithm under functional random errors. We first show its almost complete convergence to the fixed point by mean of an exponential inequality and then we specify the induced rate of convergence. We finally build a confidence domain for the fixed point. Moreover, some numerical examples are considered.     

RESIDUAL AUTOCOVARIANCES AND UNIT ROOT TESTS BASED ON INSTRUMENTAL VARIABLE ESTIMATORS FROM TIME SERIES REGRESSION MODELS

Abstract. Hall (Testing for a unit root in the presence of moving average errors. Biometrika 76 (1989), 49–56; Joint hypothesis tests for a random walk based on instrumental variable estimators. J. Time Ser. Anal. 13 (1992), 29–45), Pantula and Hall (Testing for unit roots in autoregressive moving average models:an instrumental variable approach. J. Econometrics 48 (1991), 325–53) and Lee and Schmidt (Unit root tests based on instrumental variable estimation. Int. Econ. Rev. 39 (1994), 449–62) proposed instrumental variable (IV) based tests for a unit root in an ARMA(p+ 1, q) time series. To perform the tests it is essentially necessary to know (p, q) but in many cases this information is unknown. In practice a natural solution to this problem is to estimate (p, q) from the data using a strategy based on the residual autocovariances from the IV regression. In this paper we examine the properties of these residual autocovariances under various assumptions about the true nature of the time series. This analysis allows us to propose a model selection procedure which has desirable asymptotic and finite sample properties whether the time series is stationary or possesses a unit root. A sideproduct of our analysis is that we extend Box and Pierce's (Distribution of residual autocorrelations in autoregressive integrated moving average time series models. J. Am. Statist. Assoc. 65 (1970), 1509–26) analysis of the least squares residual autocorrelations to the residual autocovariances from IV regressions.     

A NOTE ON TESTING FOR SEASONAL COINTEGRATION USING PRINCIPAL COMPONENTS IN THE FREQUENCY DOMAIN

Abstract. This paper deals with testing for cointegration at any frequency with a focus on the bounds tests proposed by Joyeux (Tests for seasonal cointegration using principal components. J. Time Ser. Anal. 13 (1992), 109–18). It is shown that this class of tests has asymptotic size equal to one because the author does not take into account non-contemporaneous cointegration at frequencies other than zero and π. The consequences of this size distortion with finite samples are investigated by a Monte Carlo experiment. Bounds tests for contemporaneous cointegration are also proposed. Finally, an empirical example of testing for seasonal cointegration in monthly time series is presented and discussed.     

Data-Efficient Quickest Change Detection with On–Off Observation Control

Abstract

In this article we extend Shiryaev's quickest change detection formulation by also accounting for the cost of observations used before the change point. The observation cost is captured through the average number of observations used in the detection process before the change occurs. The objective is to select an on–off observation control policy that decides whether or not to take a given observation, along with the stopping time at which the change is declared, to minimize the average detection delay, subject to constraints on both the probability of false alarm and the observation cost. By considering a Lagrangian relaxation of the constraint problem and using dynamic programming arguments, we obtain an a posteriori probability-based two-threshold algorithm that is a generalized version of the classical Shiryaev algorithm. We provide an asymptotic analysis of the two-threshold algorithm and show that the algorithm is asymptotically optimal—that is, the performance of the two-threshold algorithm approaches that of the Shiryaev algorithm—for a fixed observation cost, as the probability of false alarm goes to zero. We also show, using simulations, that the two-threshold algorithm has good observation cost-delay trade-off curves and provides significant reduction in observation cost compared to the naïve approach of fractional sampling, where samples are skipped randomly. Our analysis reveals that, for practical choices of constraints, the two thresholds can be set independent of each other: one based on the constraint of false alarm and another based on the observation cost constraint alone.     

ESTIMATION OF THE MEMORY PARAMETER FOR NONSTATIONARY OR NONINVERTIBLE FRACTIONALLY INTEGRATED PROCESSES

Abstract. We consider the asymptotic characteristics of the periodogram ordinates of a fractionally integrated process having memory parameter d≥ 0.5, for which the process is nonstationary, or d≤ -.5, for which the process is noninvertible. Series having d outside the range (-.5,.5) may arise in practice when a raw series is modeled without preliminary consideration of the stationarity and invertibility of the series or when a wrong decision is made concerning the stationarity and invertibility of the series. We find that the periodogram of a nonstationary or noninvertible fractionally integrated process at the jth Fourier frequency ωj= 2πj/n, where n is the sample size, suffers from an asymptotic relative bias which depends on j. We also examine the impact of periodogram bias on the regression estimator of d proposed by Geweke and Porter-Hudak (1983) in finite samples. The results indicate that the bias in the periodogram ordinates can strongly affect the GPH estimator even when the number of Fourier frequencies used in the regression is allowed to depend on the length of the series. We find that data tapering and elimination of the first periodogram ordinate in the regression can reduce this bias, at the cost of an increase in variance for nonstationary series. Additionally, we find for nonstationary series that the GPH estimator is more nearly invariant to first-differencing when a data taper is used.     

Bayesian Model Uncertainty In Smooth Transition Autoregressions

Abstract. In this paper, we propose a fully Bayesian approach to the special class of nonlinear time‐series models called the logistic smooth transition autoregressive (LSTAR) model. Initially, a Gibbs sampler is proposed for the LSTAR where the lag length, k, is kept fixed. Then, uncertainty about k is taken into account and a novel reversible jump Markov Chain Monte Carlo (RJMCMC) algorithm is proposed. We compared our RJMCMC algorithm with well‐known information criteria, such as the Akaikes? information criteria, the Bayesian information criteria (BIC) and the deviance information criteria. Our methodology is extensively studied against simulated and real‐time series.     

Testing for nonlinear deterministic components when the order of integration is unknown

We consider testing for the presence of nonlinearities in the deterministic component of a time series, approximating the potential nonlinear behaviour using a Fourier function expansion. In contrast to procedures that are currently available, we develop tests that are robust to the order of integration, in the sense that they are asymptotically correctly sized regardless of whether the stochastic component of the series is stationary or contains a unit root. The tests we propose take the form of Wald statistics based on cumulated series, together with a correction factor to line up the asymptotic critical values across the I(0) and I(1) environments. The local asymptotic power and finite sample properties of the tests are evaluated using various different correction factors. We envisage that the testing procedure we recommend should be very useful to applied researchers wishing to draw robust inference regarding the presence of nonlinear deterministic components in a series.     

Further Results on Pseudo-Maximum Likelihood Estimation and Testing in the Constant Elasticity of Variance Continuous Time Model

Constant elasticity volatility processes have been shown to be useful, for example, to encompass a number of existing models that have closed-form likelihood functions. In this article, we extend the existing literature in two directions: first we find explicit closed form solutions of the pseudo maximum likelihood estimators (MLEs) by discretizing the diffusion function and we provide their asymptotic theory in the context of the constant elasticity of variance (CEV) model characterized by a general CEV parameter ρ ≥ 0. Second we obtain bias expansions for those pseudo MLEs also in terms of ρ ≥ 0. We provide a general framework since only the cases with ρ = 0 and ρ = 0.5 have been considered in the literature so far. When the time series is not positive almost surely, we need to impose the restriction that ρ is a non-negative integer.     

Distributions for residual autocovariances in parsimonious periodic vector autoregressive models with applications

The asymptotic distribution of the residual autocovariance matrices in the class of periodic vector autoregressive time series models with structured parameterization is derived. Diagnostic checking with portmanteau test statistics represents a useful application of the result. Under the assumption that the periodic white noise process of the periodic vector autoregressive time series model is composed of independent random variables, we demonstrate that the finite sample distributions of the Hosking‐Li‐McLeod portmanteau test statistics can be approximated by those of weighted sums of independent chi‐square random variables. The quantiles of the asymptotic distribution can be computed using the Imhof algorithm or other exact methods. Thus, using the (single) chi‐square distribution for these test statistics appears inadequate in general, although it is often recommended in practice for diagnostic methods of that kind. A simulation study provides empirical evidence.     

Asymptotic optimality of the wald sequential test

We give a generalization to the case of m hypotheses of a theorem of Lai and derive an asymptotic optimality property of the Wald sequential test for general, possibly dependent, random variables with respect to the r^th moment of observation time, under some r-quick convergence condition. We also extend the definition of this convergence and give an application to sequential analysis.     

A FAST FRACTIONAL DIFFERENCE ALGORITHM

We provide a fast algorithm for calculating the fractional difference of a time series. In standard implementations, the calculation speed (number of arithmetic operations) is of order T², where T is the length of the time series. Our algorithm allows calculation speed of order TlogT. For moderate and large sample sizes, the difference in computation time is substantial. Copyright © 2014 John Wiley & Sons, Ltd.     

Time series analysis based on running Mann‐Whitney Z Statistics

A time series analysis method based on the calculation of Mann–Whitney U statistics is described. This method samples data rankings over running time windows, converts those samples to Mann–Whitney U statistics, and then normalizes the U statistics to Z statistics using Monte‐Carlo generated null parameters. Based on the Z statistics’ magnitudes this algorithm can identify time windows containing significant incidences of low or high data rankings, where the window length is determined by the sample size. By repeating this process with sampling windows of varying duration ranking regimes of arbitrary onset and duration can be objectively identified in a time series. The simplicity of the procedure's output – a time series’ most significant non‐overlapping ranking sequences – makes it possible to graphically identify common temporal breakpoints and patterns of variability in the analyses of multiple time series. This approach is demonstrated using United States annual temperature data during 1896–2008.     

A GENERALIZED VARIANCE RATIO TEST OF ARIMA (p, 1, q) MODEL SPECIFICATION

Abstract. The variance ratio test is often used as a check of the hypothesis that a time series is generated by a random walk. A natural extension of the test is developed to cover the case where the assumed model is ARIMA(p, 1, q), with unknown parameters. Small sample properties of the generalized test are investigated, and the test is applied to a frequently analysed data set on US quarterly real gross national product. In effect, we are testing for low frequency misspecification in assumed autoregressive moving-average (ARMA) models for a differenced series.     

Determining the order of the functional autoregressive model

We propose a multistage testing procedure to determine the order p of a functional autoregressive process, FAR (p). At its core is the representation of the FAR(p) process as a fully functional linear model with dependent regressors. Estimating the kernel function in this linear model allows us to construct a test statistic which has, approximately, a chi–square distribution with the number of degrees of freedom determined by the number of functional principal components used to represent the data. The asymptotic justification relies on the concept of L^p–m–approximability which quantifies the temporal dependence of functional time series. The procedure enjoys very good finite sample properties, as confirmed by a simulation study and applications to functional time series derived from credit card transactions and Eurodollar futures data.     

Improved prediction limits for a general class of Gaussian models

In this article we consider the problem of prediction for a general class of Gaussian models, which includes, among others, autoregressive moving average time‐series models, linear Gaussian state space models and Gaussian Markov random fields. Using an idea presented in Sjöstedt‐De Luna and Young (2003) , in the context of spatial statistics, we discuss a method for obtaining prediction limits for a future random variable of interest, taking into account the uncertainty introduced by estimating the unknown parameters. The proposed prediction limits can be viewed as a modification of the estimative prediction limit, with unconditional, and eventually conditional, coverage error of smaller asymptotic order. The modifying term has a quite simple form and it involves the bias and the mean square error of the plug‐in estimators for the conditional expectation and the conditional variance of the future observation. Applications of the results to Gaussian time‐series models are presented.     

A Plug‐in Bandwidth Selection Procedure for Long‐Run Covariance Estimation with Stationary Functional Time Series

In several arenas of application, it is becoming increasingly common to consider time series of curves or functions. Many inferential procedures employed in the analysis of such data involve the long‐run covariance function or operator, which is analogous to the long‐run covariance matrix familiar to finite‐dimensional time‐series analysis and econometrics. This function may be naturally estimated using a smoothed periodogram type estimator evaluated at frequency zero that relies on the choice of a bandwidth parameter. Motivated by a number of prior contributions in the finite‐dimensional setting, in particular Newey and West ( 1994 ), we propose a bandwidth selection method that aims to minimize the estimator's asymptotic mean‐squared normed error (AMSNE) in L²[0,1]². As the AMSNE depends on unknown population quantities including the long‐run covariance function itself, estimates for these are plugged in in an initial step after which the estimated AMSNE can be minimized to produce an empirical optimal bandwidth. We show that the bandwidth produced in this way is asymptotically consistent with the AMSNE optimal bandwidth, with quantifiable rates, under mild stationarity and moment conditions. These results and the efficacy of the proposed methodology are evaluated by means of a comprehensive simulation study, from which we can offer practical advice on how to select the bandwidth parameter in this setting.     

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.