Multivariate autoregressive time semes modeling: one scalar autoregressive model at-A-time

Multivariate (or interchangeably multichannel) autoregressive (MCAR) modeling of stationary and nonstationary time series data is achieved doing things one channel at-a-time using only scalar computations on instantaneous data. The one channel at-a-time modeling is achieved as an instantaneous response multichannel autoregressive model with orthogonal innovations variance. Conventional MCAR models are expressible as linear algebraic transformations of the instantaneous response orthogonal innovations models. By modeling multichannel time series one channel at-a-time, the problems of modeling multichannel time series are reduced to problems in the modeling of scalar autoregressive time series. The three longstanding time series modeling problems of achieving a relatively parsimonious MCAR representation, of multichannel stationary time series spectral estimation and of the modeling of nonstationary covariance time series are addressed using this paradigm.     

A note on maximum likelihood estimation for the first-order autoregressive process

The maximum likelihood estimator of the parameters of a zero-mean normal stationary first-order autoregressive process is in-vestigated. it is shown that the likelihood function is uniquely maximized at a point in the interior of the parameter space. A closed-form expression is obtained for the estimator.     

A non-stationary integer-valued autoregressive model

It is frequent to encounter a time series of counts which are small in value and show a trend having relatively large fluctuation. To handle such a non-stationary integer-valued time series with a large dispersion, we introduce a new process called integer-valued autoregressive process of order p with signed binomial thinning (INARS(p)). This INARS(p) uniquely exists and is stationary under the same stationary condition as in the AR(p) process. We provide the properties of the INARS(p) as well as the asymptotic normality of the estimates of the model parameters. This new process includes previous integer-valued autoregressive processes as special cases. To preserve integer-valued nature of the INARS(p) and to avoid difficulty in deriving the distributional properties of the forecasts, we propose a bootstrap approach for deriving forecasts and confidence intervals. We apply the INARS(p) to the frequency of new patients diagnosed with acquired immunodeficiency syndrome (AIDS) in Baltimore, Maryland, U.S. during the period of 108 months from January 1993 to December 2001.     

Structural changes in autoregressive models for binary time series

We study autoregressive models for binary time series with possible changes in their parameters. A procedure for detection and testing of a single change is suggested. The limiting behavior of the test statistic is derived. The performance of the test is analyzed under the null hypothesis as well as under different alternatives via a simulation study. Application of the method to a real data set on US recession is provided as an illustration.     

Efficient inference for parameters of unobservable periodic autoregressive time series

This paper considers estimating the model coefficients when the observed periodic autoregressive time series is contaminated by a trend. The proposed Yule–Walker estimators are obtained by a two-step procedure. In the first step, the trend is estimated by a weighted local polynomial, and the residuals are obtained by subtracting the trend estimates from the observations; in the second step, the model coefficients are estimated by the well-known Yule–Walker method via the residuals. It is shown that under certain conditions such Yule–Walker estimators are oracally efficient, i.e., they are asymptotically equivalent to those obtained from periodic autoregressive time series without a trend. An easy-to-use implementation procedure is provided. The performance of the estimators is illustrated by simulation studies and real data analysis. In particular, the simulation studies show that the proposed estimator outperforms that obtained from the residuals when the trend is estimated by kernel smoothing without taking the heteroscedasticity into consideration.     

Predicting hospital census using time series regression methods

We first describe the time series modeling problem in a general way. Then some specific assumptions and observations which are pertinent to the application of these models are made. We next propose a specific approach to the modeling problem, one which yields efficient, easily calculated estimators of all parameters (under the stated assumptions). Finally, the technique is applied to the problem of modeling the census of a particular hospital.     

Priors and component structures in autoregressive time series models

New approaches to prior specification and structuring in autoregressive time series models are introduced and developed. We focus on defining classes of prior distributions for parameters and latent variables related to latent components of an autoregressive model for an observed time series. These new priors naturally permit the incorporation of both qualitative and quantitative prior information about the number and relative importance of physically meaningful components that represent low frequency trends, quasi-periodic subprocesses and high frequency residual noise components of observed series. The class of priors also naturally incorporates uncertainty about model order and hence leads in posterior analysis to model order assessment and resulting posterior and predictive inferences that incorporate full uncertainties about model order as well as model parameters. Analysis also formally incorporates uncertainty and leads to inferences about unknown initial values of the time series, as it does for predictions of future values. Posterior analysis involves easily implemented iterative simulation methods, developed and described here. One motivating field of application is climatology, where the evaluation of latent structure, especially quasi-periodic structure, is of critical importance in connection with issues of global climatic variability. We explore the analysis of data from the southern oscillation index, one of several series that has been central in recent high profile debates in the atmospheric sciences about recent apparent trends in climatic indicators.     

A transitional Markov switching autoregressive model

ABSTRACT

This paper is concerned with properties of a transitional Markov switching autoregressive (TMSAR) model, together with its maximum-likelihood estimation and inference. We extend existing MSAR models by allowing dependence of AR parameters on hidden states at time points prior to the current time t. A stationary solution is given and expressions for the theoretical autocovariance function are derived. Two time series are analyzed and the new model outperforms two existing MSAR models in terms of maximized log-likelihood, residual correlations, and one-step-ahead forecasting performance. The new model also gives more regime changes in agreement with real events.     

On a mixture vector autoregressive model

The authors show how to extend univariate mixture autoregressive models to a multivariate time series context. Similar to the univariate case, the multivariate model consists of a mixture of stationary or nonstationary autoregressive components. The authors give the first and second order stationarity conditions for a multivariate case up to order 2. They also derive the second order stationarity condition for the univariate mixture model up to arbitrary order. They describe an EM algorithm for estimation, as well as a diagnostic checking procedure. They study the performance of their method via simulations and include a real application.     

A statistic to check model adequacy in time series

A simple statistic is suggested to examine if the assumptions on variances in a fitted time series model is valid or not. The properties of the statistic are discussed and examples are considered.     

A dynamic model for double‐bounded time series with chaotic‐driven conditional averages

In this work, we introduce a class of dynamic models for time series taking values on the unit interval. The proposed model follows a generalized linear model approach where the random component, conditioned on the past information, follows a beta distribution, while the conditional mean specification may include covariates and also an extra additive term given by the iteration of a map that can present chaotic behavior. The resulting model is very flexible and its systematic component can accommodate short‐ and long‐range dependence, periodic behavior, laminar phases, etc. We derive easily verifiable conditions for the stationarity of the proposed model, as well as conditions for the law of large numbers and a Birkhoff‐type theorem to hold. A Monte Carlo simulation study is performed to assess the finite sample behavior of the partial maximum likelihood approach for parameter estimation in the proposed model. Finally, an application to the proportion of stored hydroelectrical energy in Southern Brazil is presented.     

On nonlinear models for time series

The paper is a review of nonlinear processes used in time series analysis and presents some new original results about stationary distribution of a nonlinear autoregres-sive process of the first order. The following models are considered: nonlinear autoregessive processes, threshold AR processes, threshold MA processes, bilinear models, auto-regressive models with random parameters including double stochastic models, exponential AR models, generalized threshold models and smooth transition autoregressive models, Some tests for linearity of processes are also presented.     

Autoregressiv model identification for multivariate time series

A procedure is developed for the identification of autoregressive models for stationary invertible multivariate Gaussian time series. Model selection is based on either the AIC information criterion or on a statistic called CVR, cross-validatory residual sum of squares. An example is given to show that the forecasts generated by these models compare favorably with those generated by other common time series modeling techniques.     

Semiparametric non-linear time series model selection

Summary.  Semiparametric time series regression is often used without checking its suitability, resulting in an unnecessarily complicated model. In practice, one may encounter computational difficulties caused by the curse of dimensionality. The paper suggests that to provide more precise predictions we need to choose the most significant regressors for both the parametric and the nonparametric time series components. We develop a novel cross-validation-based model selection procedure for the simultaneous choice of both the parametric and the nonparametric time series components, and we establish some asymptotic properties of the model selection procedure proposed. In addition, we demonstrate how to implement it by using both simulated and real examples. Our empirical studies show that the procedure works well.     

C326. Ergodicity for time series and spatial processes

A statistic is presented for testing a three state observed Markov chain for independence. The test procedure is compared with the traditional X ² test. Examples are given in which the proposed test has better power than the X ² test.     

A simulation method for finite non-stationary time series

In this paper, we propose a novel simulation method which enables us to obtain a large number of simulated time series cheaply. The developed method can be applied to any non-stationary time series of finite length and it guarantees that not only the marginal distributions but also the autocorrelation structures of observed and simulated time series are the same. Extensive simulation studies have been conducted to check the performance of our method and to assess if the overall dynamics of the observed time series is preserved by the simulated realizations. The developed simulation method has also been applied to the real size data of cocoon filament, which can be reeled from a cocoon produced by a silkworm. Very good results have been achieved in all the cases considered in the paper.     

Stochastic approximation Monte Carlo Gibbs sampling for structural change inference in a Bayesian heteroscedastic time series model

We consider a Bayesian deterministically trending dynamic time series model with heteroscedastic error variance, in which there exist multiple structural changes in level, trend and error variance, but the number of change-points and the timings are unknown. For a Bayesian analysis, a truncated Poisson prior and conjugate priors are used for the number of change-points and the distributional parameters, respectively. To identify the best model and estimate the model parameters simultaneously, we propose a new method by sequentially making use of the Gibbs sampler in conjunction with stochastic approximation Monte Carlo simulations, as an adaptive Monte Carlo algorithm. The numerical results are in favor of our method in terms of the quality of estimates.     

Estimating the parameters of a BINMA Poisson model for a non-stationary bivariate time series

This article proposes a novel non-stationary BINMA time series model by extending two INMA processes where their innovation series follow the bivariate Poisson under time-varying moment assumptions. This article also demonstrates, through simulation studies, the use and superiority of the generalized quasi-likelihood (GQL) approach to estimate the regression effects, which is computationally less complicated as compared to conditional maximum likelihood estimation (CMLE) and the feasible generalized least squares (FGLS). The serial and bivariate dependence correlations are estimated by a robust method of moments.     

A threshold autoregressive model for wholesale electricity prices

Summary.  We introduce a discrete time model for electricity prices which accounts for both transitory spikes and temperature effects. The model allows for different rates of mean reversion: one for weather events, one around price jumps and another for the remainder of the process. We estimate the model by using a Markov chain Monte Carlo approach with 3 years of daily data from Allegheny County, Pennsylvania. We show that our model outperforms existing stochastic jump diffusion models for this data set. Results also demonstrate the importance of model parameters corresponding to both the temperature effect and the multilevel mean reversion rate.     

Posterior consistency for the spectral density of non-Gaussian stationary time series

Various nonparametric approaches for Bayesian spectral density estimation of stationary time series have been suggested in the literature, mostly based on the Whittle likelihood approximation. A generalization of this approximation involving a nonparametric correction of a parametric likelihood has been proposed in the literature with a proof of posterior consistency for spectral density estimation in combination with the Bernstein–Dirichlet process prior for Gaussian time series. In this article, we will extend the posterior consistency result to non-Gaussian time series by employing a general consistency theorem for dependent data and misspecified models. As a special case, posterior consistency for the spectral density under the Whittle likelihood is also extended to non-Gaussian time series. Small sample properties of this approach are illustrated with several examples of non-Gaussian time series.