Obtaining prediction intervals for FARIMA processes using the sieve bootstrap

The sieve bootstrap (SB) prediction intervals for invertible autoregressive moving average (ARMA) processes are constructed using resamples of residuals obtained by fitting a finite degree autoregressive approximation to the time series. The advantage of this approach is that it does not require the knowledge of the orders, p and q, associated with the ARMA(p, q) model. Up until recently, the application of this method has been limited to ARMA processes whose autoregressive polynomials do not have fractional unit roots. The authors, in a 2012 publication, introduced a version of the SB suitable for fractionally integrated autoregressive moving average (FARIMA (p,d,q)) processes with 0<d<0.5 and established its asymptotic validity. Herein, we study the finite sample properties this new method and compare its performance against an older method introduced by Bisaglia and Grigoletto in 2001. The sieve bootstrap (SB) method is a numerically simpler alternative to the older method which requires the estimation of p, d, and q at every bootstrap step. Monte-Carlo simulation studies, carried out under the assumption of normal, mixture of normals, and exponential distributions for the innovations, show near nominal coverages for short-term and long-term SB prediction intervals under most situations. In addition, the sieve bootstrap method yields better coverage and narrower intervals compared to the Bisaglia–Grigoletto method in some situations, especially when the error distribution is a mixture of normals.     

Rates of convergence of autocorrelation estimates for periodically correlated autoregressive Hilbertian processes

Autoregressive Hilbertian (ARH) processes are of great importance in the analysis of functional time series data and estimation of the autocorrelation operators attracts the attention of various researchers. In this paper, we study estimators of the autocorrelation operators of periodically correlated autoregressive Hilbertian processes of order one (PCARH(1)), which is an extension of ARH(1) processes. The estimation method is based on the spectral decomposition of the covariance operator and considers two main cases: known and unknown eigenvectors. We show the consistency in the mean integrated quadratic sense of the estimators of the autocorrelation operators and present upper bounds for the corresponding rates.     

Structured priors for multivariate time series

A class of prior distributions for multivariate autoregressive models is presented. This class of priors is built taking into account the latent component structure that characterizes a collection of autoregressive processes. In particular, the state-space representation of a vector autoregressive process leads to the decomposition of each time series in the multivariate process into simple underlying components. These components may have a common structure across the series. A key feature of the proposed priors is that they allow the modeling of such common structure. This approach also takes into account the uncertainty in the number of latent processes, consequently handling model order uncertainty in the multivariate autoregressive framework. Posterior inference is achieved via standard Markov chain Monte Carlo (MCMC) methods. Issues related to inference and exploration of the posterior distribution are discussed. We illustrate the methodology analyzing two data sets: a synthetic data set with quasi-periodic latent structure, and seasonally adjusted US monthly housing data consisting of housing starts and housing sales over the period 1965 to 1974.     

Improved estimation for Poisson INAR(1) models

We consider the first-order Poisson autoregressive model proposed by McKenzie [Some simple models for discrete variate time series. Water Resour Bull. 1985;21:645–650] and Al-Osh and Alzaid [First-order integer valued autoregressive (INAR(1)) process. J Time Ser Anal. 1987;8:261–275], which may be suitable in situations where the time series data are non-negative and integer valued. We derive the second-order bias of the squared difference estimator [Weiß. Process capability analysis for serially dependent processes of Poisson counts. J Stat Comput Simul. 2012;82:383–404] for one of the parameters and show that this bias can be used to define a bias-reduced estimator. The behaviour of a modified conditional least-squares estimator is also studied. Furthermore, we access the asymptotic properties of the estimators here discussed. We present numerical evidence, based upon Monte Carlo simulation studies, showing that the here proposed bias-adjusted estimator outperforms the other estimators in small samples. We also present an application to a real data set.     

A Monte Carlo Markov chain algorithm for a class of mixture time series models

This article generalizes the Monte Carlo Markov Chain (MCMC) algorithm, based on the Gibbs weighted Chinese restaurant (gWCR) process algorithm, for a class of kernel mixture of time series models over the Dirichlet process. This class of models is an extension of Lo’s (Ann. Stat. 12:351–357, 1984) kernel mixture model for independent observations. The kernel represents a known distribution of time series conditional on past time series and both present and past latent variables. The latent variables are independent samples from a Dirichlet process, which is a random discrete (almost surely) distribution. This class of models includes an infinite mixture of autoregressive processes and an infinite mixture of generalized autoregressive conditional heteroskedasticity (GARCH) processes.     

A hierarchical model for space–time surveillance data on meningococcal disease incidence

Summary. We describe a model-based approach to analyse space–time surveillance data on meningococcal disease. Such data typically comprise a number of time series of disease counts, each representing a specific geographical area. We propose a hierarchical formulation, where latent parameters capture temporal, seasonal and spatial trends in disease incidence. We then add—for each area—a hidden Markov model to describe potential additional (autoregressive) effects of the number of cases at the previous time point. Different specifications for the functional form of this autoregressive term are compared which involve the number of cases in the same or in neighbouring areas. The two states of the Markov chain can be interpreted as representing an 'endemic' and a 'hyperendemic' state. The methodology is applied to a data set of monthly counts of the incidence of meningococcal disease in the 94 départements of France from 1985 to 1997. Inference is carried out by using Markov chain Monte Carlo simulation techniques in a fully Bayesian framework. We emphasize that a central feature of our model is the possibility of calculating—for each region and each time point—the posterior probability of being in a hyperendemic state, adjusted for global spatial and temporal trends, which we believe is of particular public health interest.     

Bootstrap-based unit root tests for higher order autoregressive models with GARCH(1, 1) errors

ABSTRACT

Bootstrap-based unit root tests are a viable alternative to asymptotic distribution-based procedures and, in some cases, are preferable because of the serious size distortions associated with the latter tests under certain situations. While several bootstrap-based unit root tests exist for autoregressive moving average processes with homoskedastic errors, only one such test is available when the innovations are conditionally heteroskedastic. The details for the exact implementation of this procedure are currently available only for the first order autoregressive processes. Monte-Carlo results are also published only for this limited case. In this paper we demonstrate how this procedure can be extended to higher order autoregressive processes through a transformed series used in augmented Dickey–Fuller unit root tests. We also investigate the finite sample properties for higher order processes through a Monte-Carlo study. Results show that the proposed tests have reasonable power and size properties.     

M‐estimation for general ARMA Processes with Infinite Variance

Abstract. General autoregressive moving average (ARMA) models extend the traditional ARMA models by removing the assumptions of causality and invertibility. The assumptions are not required under a non‐Gaussian setting for the identifiability of the model parameters in contrast to the Gaussian setting. We study M‐estimation for general ARMA processes with infinite variance, where the distribution of innovations is in the domain of attraction of a non‐Gaussian stable law. Following the approach taken by Davis et al. (1992) and Davis (1996) , we derive a functional limit theorem for random processes based on the objective function, and establish asymptotic properties of the M‐estimator. We also consider bootstrapping the M‐estimator and extend the results of Davis & Wu (1997) to the present setting so that statistical inferences are readily implemented. Simulation studies are conducted to evaluate the finite sample performance of the M‐estimation and bootstrap procedures. An empirical example of financial time series is also provided.     

Bootstrap-based bias corrections for INAR count time series

Integer-valued autoregressive (INAR) processes form a very useful class of processes suitable to model time series of counts. Several practically relevant estimators based on INAR data are known to be systematically biased away from their population values, e.g. sample autocovariances, sample autocorrelations, or the dispersion index. We propose to do bias correction for such estimators by using a recently proposed INAR-type bootstrap scheme that is tailor-made for INAR processes, and which has been proven to be asymptotically consistent under general conditions. This INAR bootstrap allows an implementation with and without parametrically specifying the innovations' distribution. To judge the potential of corresponding bias correction, we compare these bootstraps in simulations to several competitors that include the AR bootstrap and block bootstrap. Finally, we conclude with an illustrative data application.     

Joint detection for functional polynomial regression with autoregressive errors

In this article, we are concerned with detecting the true structure of a functional polynomial regression with autoregressive (AR) errors. The first issue is to detect which orders of the polynomial are significant in functional polynomial regression. The second issue is to detect which orders of the AR process in the AR errors are significant. We propose a shrinkage method to deal with the two problems: polynomial order selection and autoregressive order selection. Simulation studies demonstrate that the new method can identify the true structure. One empirical example is also presented to illustrate the usefulness of our method.     

Statistical Analysis Of Mixture Vector Autoregressive Models

In this paper, we reconsider the mixture vector autoregressive model, which was proposed in the literature for modelling non‐linear time series. We complete and extend the stationarity conditions, derive a matrix formula in closed form for the autocovariance function of the process and prove a result on stable vector autoregressive moving‐average representations of mixture vector autoregressive models. For these results, we apply techniques related to a Markovian representation of vector autoregressive moving‐average processes. Furthermore, we analyse maximum likelihood estimation of model parameters by using the expectation–maximization algorithm and propose a new iterative algorithm for getting the maximum likelihood estimates. Finally, we study the model selection problem and testing procedures. Several examples, simulation experiments and an empirical application based on monthly financial returns illustrate the proposed procedures.     

A novel and fast methodology for simultaneous multiple structural break estimation and variable selection for nonstationary time series models

A class of nonstationary time series such as locally stationary time series can be approximately modeled by piecewise stationary autoregressive (PSAR) processes. But the number and locations of the piecewise autoregressive segments, as well as the number of nonzero coefficients in each autoregressive process, are unknown. In this paper, by connecting the multiple structural break detection with a variable selection problem for a linear model with a large number of regression coefficients, a novel and fast methodology utilizing modern penalized model selection is introduced for detecting multiple structural breaks in a PSAR process. It also simultaneously performs variable selection for each autoregressive model and hence the order selection. To further its performance, an algorithm is given, which remains very fast in computation. Numerical results from simulation and a real data example show that the algorithm has excellent empirical performance.     

Simultaneous sparse model selection and coefficient estimation for heavy-tailed autoregressive processes

We propose a sparse coefficient estimation and automated model selection procedure for autoregressive processes with heavy-tailed innovations based on penalized conditional maximum likelihood. Under mild moment conditions on the innovation processes, the penalized conditional maximum likelihood estimator satisfies a strong consistency, O_P(N^?1/2) consistency, and the oracle properties, where N is the sample size. We have the freedom in choosing penalty functions based on the weak conditions on them. Two penalty functions, least absolute shrinkage and selection operator and smoothly clipped average deviation, are compared. The proposed method provides a distribution-based penalized inference to AR models, which is especially useful when the other estimation methods fail or under perform for AR processes with heavy-tailed innovations [Feigin, Resnick. Pitfalls of fitting autoregressive models for heavy-tailed time series. Extremes. 1999;1:391–422]. A simulation study confirms our theoretical results. At the end, we apply our method to a historical price data of the US Industrial Production Index for consumer goods, and obtain very promising results.     

Test of parameter changes in a class of observation-driven models for count time series

Abstract

This paper investigates the parameter-change tests for a class of observation-driven models for count time series. We propose two cumulative sum (CUSUM) test procedures for detection of changes in model parameters. Under regularity conditions, the asymptotic null distributions of the test statistics are established. In addition, the integer-valued generalized autoregressive conditional heteroskedastic (INGARCH) processes with conditional negative binomial distributions are investigated. The developed techniques are examined through simulation studies and also are illustrated using an empirical example.     

Bayesian analysis of a linear mixed model with AR(p) errors via MCMC

We develop Bayesian procedures to make inference about parameters of a statistical design with autocorrelated error terms. Modelling treatment effects can be complex in the presence of other factors such as time; for example in longitudinal data. In this paper, Markov chain Monte Carlo methods (MCMC), the Metropolis-Hastings algorithm and Gibbs sampler are used to facilitate the Bayesian analysis of real life data when the error structure can be expressed as an autoregressive model of order p. We illustrate our analysis with real data.     

Properties of generalized Levinson–Durbin–Whittle sequences

For a discrete time, second-order stationary process the Levinson–Durbin recursion is used to determine best fitting one-step-ahead linear autoregressive predictors of successively increasing order, best in the sense of minimizing the mean square error. Whittle [1963. On the fitting of multivariate autoregressions, and the approximate canonical factorization of a spectral density matrix. Biometrika 50, 129–134] generalized the recursion to the case of vector autoregressive processes. The recursion defines what is termed a Levinson–Durbin–Whittle sequence, and a generalized Levinson–Durbin–Whittle sequence is also defined. Generalized Levinson–Durbin–Whittle sequences are shown to satisfy summation formulas which generalize summation formulas satisfied by binomial coefficients. The formulas can be expressed in terms of the partial correlation sequence, and they assume simple forms for time-reversible processes. The results extend comparable formulas obtained in Shaman [2007. Generalized Levinson–Durbin sequences, binomial coefficients and autoregressive estimation. Working paper] for univariate processes.     

Bayesian Identification of Seasonal Autoregressive Models

Abstract

A very important and essential phase of time series analysis is identifying the model orders. This article develops an approximate Bayesian procedure to identify the orders of seasonal autoregressive processes. Using either a normal-gamma prior density or a noninformative prior, which is combined with an approximate conditional likelihood function, the foundation of the proposed technique is to derive the joint posterior mass function of the model orders in an easy form. Then one may inspect the posterior mass function and choose the orders with the largest posterior probability to be the suitable orders of the time series being analyzed. A simulation study, with different priors mass functions, is carried out to test the adequacy of the proposed technique and compare it with some non-Bayesian automatic criteria. The analysis of the numerical results supports the adequacy of the proposed technique in identifying the orders of the autoregressive processes.     

Time-varying autoregressive conditional duration model

The main goal of this work is to generalize the autoregressive conditional duration (ACD) model applied to times between trades to the case of time-varying parameters. The use of wavelets allows that parameters vary through time and makes possible the modeling of non-stationary processes without preliminary data transformations. The time-varying ACD model estimation was done by maximum-likelihood with standard exponential distributed errors. The properties of the estimators were assessed via bootstrap. We present a simulation exercise for a non-stationary process and an empirical application to a real series, namely the TELEMAR stock. Diagnostic and goodness of fit analysis suggest that the time-varying ACD model simultaneously modeled the dependence between durations, intra-day seasonality and volatility.     

A wavelet-based time-varying autoregressive model for non-stationary and irregular time series

In this work we propose an autoregressive model with parameters varying in time applied to irregularly spaced non-stationary time series. We expand all the functional parameters in a wavelet basis and estimate the coefficients by least squares after truncation at a suitable resolution level. We also present some simulations in order to evaluate both the estimation method and the model behavior on finite samples. Applications to silicates and nitrites irregularly observed data are provided as well.     

Marginal maximum a posteriori estimation using Markov chain Monte Carlo

Markov chain Monte Carlo (MCMC) methods, while facilitating the solution of many complex problems in Bayesian inference, are not currently well adapted to the problem of marginal maximum a posteriori (MMAP) estimation, especially when the number of parameters is large. We present here a simple and novel MCMC strategy, called State-Augmentation for Marginal Estimation (SAME), which leads to MMAP estimates for Bayesian models. We illustrate the simplicity and utility of the approach for missing data interpolation in autoregressive time series and blind deconvolution of impulsive processes.