Bound and convergence of the non-constant dynamic linear model

This paper extends the limiting results of West and Harrison (1997, section 5.5) about the convergence of the variances of time series dynamic linear models (TSDLMs) when both, the variances of the observation and evolution errors of the model, are time-varying with steady limits. Analytical results are derived and an illustrative example is provided.     

Bayesian identification of double seasonal autoregressive time series models

ABSTRACT

Seasonal autoregressive (SAR) models have been modified and extended to model high frequency time series characterized by exhibiting double seasonal patterns. Some researchers have introduced Bayesian inference for double seasonal autoregressive (DSAR) models; however, none has tackled the problem of Bayesian identification of DSAR models. Therefore, in order to fill this gap, we present a Bayesian methodology to identify the order of DSAR models. Assuming the model errors are normally distributed and using three priors, i.e. natural conjugate, g, and Jeffreys’ priors, on the model parameters, we derive the joint posterior mass function of the model order in a closed-form. Accordingly, the posterior mass function can be investigated and the best order of DSAR model is chosen as a value with the highest posterior probability for the time series being analyzed. We evaluate the proposed Bayesian methodology using simulation study, and we then apply it to real-world hourly internet amount of traffic dataset.     

Multivariate Seasonal Adjustment,Economic Identities,and Seasonal Taxonomy

This article extends the methodology for multivariate seasonal adjustment by exploring the statistical modeling of seasonality jointly across multiple time series, using latent dynamic factor models fitted using maximum likelihood estimation. Signal extraction methods for the series then allow us to calculate a model-based seasonal adjustment. We emphasize several facets of our analysis: (i) we quantify the efficiency gain in multivariate signal extraction versus univariate approaches; (ii) we address the problem of the preservation of economic identities; (iii) we describe a foray into seasonal taxonomy via the device of seasonal co-integration rank. These contributions are developed through two empirical studies of aggregate U.S. retail trade series and U.S. regional housing starts. Our analysis identifies different seasonal subcomponents that are able to capture the transition from prerecession to postrecession seasonal patterns. We also address the topic of indirect seasonal adjustment by analyzing the regional aggregate series. Supplementary materials for this article are available online.     

Forecasting Economic Time Series With Structural and Box-Jenkins Models: A Case Study

The basic structural model is a univariate time series model consisting of a slowly changing trend component, a slowly changing seasonal component, and a random irregular component. It is part of a class of models that have a number of advantages over the seasonal ARIMA models adopted by Box and Jenkins (1976). This article reports the results of an exercise in which the basic structural model was estimated for six U.K. macroeconomic time series and the forecasting performance compared with that of ARIMA models previously fitted by Prothero and Wallis (1976).     

Gaussian variational approximation with sparse precision matrices

We consider the problem of learning a Gaussian variational approximation to the posterior distribution for a high-dimensional parameter, where we impose sparsity in the precision matrix to reflect appropriate conditional independence structure in the model. Incorporating sparsity in the precision matrix allows the Gaussian variational distribution to be both flexible and parsimonious, and the sparsity is achieved through parameterization in terms of the Cholesky factor. Efficient stochastic gradient methods that make appropriate use of gradient information for the target distribution are developed for the optimization. We consider alternative estimators of the stochastic gradients, which have lower variation and are more stable. Our approach is illustrated using generalized linear mixed models and state-space models for time series.     

Identification of seasonal arima models using a filtering method

This paper proposes an identification method of ARIMA models for seasonal time series using an intermediary model and a filtering method. This method is found to be useful when conventional methods, such as using sample ACF and PACF, fail to reveal a clear-cut model. This filtering identification method is also found to be particularly effective when a seasonal time series is subjected to calendar variations, moving-holiday effects, and interventions.     

Bayesian Inferences and Forecasts With Multiple Autoregressive Moving Average Models

The main objective of this paper is to develop convenient Bayesian techniques for estimation and forecasting which can be used to analyze multiple (multivariate) autoregressive moving average processes. Based on the conditional likelihood function and the least squares estimates of the residuals, the marginal posterior distribution of the coefficients of the model is approximated by a matrix t distribution, the marginal posterior distribution of the precision matrix is approximated by a Wishart distribution, and the predictive distribution is approximated by a multivariate t distribution. Some numerical examples are given to demonstrate the idea of using the proposed techniques to analyze different types of multiple ARMA models.     

Multivariate stochastic volatility with Bayesian dynamic linear models

This paper develops a Bayesian procedure for estimation and forecasting of the volatility of multivariate time series. The foundation of this work is the matrix-variate dynamic linear model, for the volatility of which we adopt a multiplicative stochastic evolution, using Wishart and singular multivariate beta distributions. A diagonal matrix of discount factors is employed in order to discount the variances element by element and therefore allowing a flexible and pragmatic variance modelling approach. Diagnostic tests and sequential model monitoring are discussed in some detail. The proposed estimation theory is applied to a four-dimensional time series, comprising spot prices of aluminium, copper, lead and zinc of the London metal exchange. The empirical findings suggest that the proposed Bayesian procedure can be effectively applied to financial data, overcoming many of the disadvantages of existing volatility models.     

Dynamic structural models with covariates for short-term forecasting of time series with complex seasonal patterns

This work presents a framework of dynamic structural models with covariates for short-term forecasting of time series with complex seasonal patterns. The framework is based on the multiple sources of randomness formulation. A noise model is formulated to allow the incorporation of randomness into the seasonal component and to propagate this same randomness in the coefficients of the variant trigonometric terms over time. A unique, recursive and systematic computational procedure based on the maximum likelihood estimation under the hypothesis of Gaussian errors is introduced. The referred procedure combines the Kalman filter with recursive adjustment of the covariance matrices and the selection method of harmonics number in the trigonometric terms. A key feature of this method is that it allows estimating not only the states of the system but also allows obtaining the standard errors of the estimated parameters and the prediction intervals. In addition, this work also presents a non-parametric bootstrap approach to improve the forecasting method based on Kalman filter recursions. The proposed framework is empirically explored with two real time series.     

Seasonal Extraction in the Presence of Feedback

Time series seasonal extraction techniques are quite often applied in the context of a policy aimed at controlling the nonseasonal components of a time series. Monetary policies targeting the nonseasonal components of monetary aggregates are an example. Such policies can be studied as a quadratic optimal control model in which observations are contaminated by seasonal noise. Optimal extraction filters in such models do not correspond to univariate time series seasonal extraction filters. The linear quadratic control model components are nonorthogonal due to the presence of control feedback. This article presents the Kalman filter as a conceptual and computational device used to extract seasonal noise in the presence of feedback.     

Extended complex error correction models for seasonal cointegration

In this paper, we extend the complex error correction model (ECM) of [Cubadda, G. (2001). Complex reduced rank models for seasonally cointegrated time series. Oxford Bulletin of Economics and Statistics, 63, 497–511] to models with two types of deterministic terms: (i) restricted seasonal dummies and constant; (ii) restricted seasonal dummies and unrestricted constant. These types of deterministic terms are most frequently adopted in the analysis of seasonal cointegration by many practitioners and researchers, because the other type–where all seasonal dummies and constant terms are unrestricted–may yield oscillating trends. We obtain the limiting distribution of the likelihood ratio (LR) test for the seasonal cointegrating (CI) rank in the extended models. We also provide asymptotic and finite critical values for the test.     

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.     

Parallel algorithms for Markov chain Monte Carlo methods in latent spatial Gaussian models

Markov chain Monte Carlo (MCMC) implementations of Bayesian inference for latent spatial Gaussian models are very computationally intensive, and restrictions on storage and computation time are limiting their application to large problems. Here we propose various parallel MCMC algorithms for such models. The algorithms' performance is discussed with respect to a simulation study, which demonstrates the increase in speed with which the algorithms explore the posterior distribution as a function of the number of processors. We also discuss how feasible problem size is increased by use of these algorithms.     

A Benchmarking Approach to Forecast Combination

This article is concerned with the development of a statistical model-based approach to optimally combine forecasts derived from an extrapolative model, such as an autoregressive integrated moving average (ARIMA) time series model, with forecasts of a particular characteristic of the same series obtained from independent sources. The methods derived combine the strengths of all forecasting approaches considered in the combination scheme. The implications of the general theory are investigated in the context of some commonly encountered seasonal ARIMA models. An empirical example to illustrate the method is included.     

Multichannel electroencephalographic analyses via dynamic regression models with time-varying lag–lead structure

Multiple time series of scalp electrical potential activity are generated routinely in electroencephalographic (EEG) studies. Such recordings provide important non-invasive data about brain function in human neuropsychiatric disorders. Analyses of EEG traces aim to isolate characteristics of their spatiotemporal dynamics that may be useful in diagnosis, or may improve the understanding of the underlying neurophysiology or may improve treatment through identifying predictors and indicators of clinical outcomes. We discuss the development and application of non-stationary time series models for multiple EEG series generated from individual subjects in a clinical neuropsychiatric setting. The subjects are depressed patients experiencing generalized tonic–clonic seizures elicited by electroconvulsive therapy (ECT) as antidepressant treatment. Two varieties of models—dynamic latent factor models and dynamic regression models—are introduced and studied. We discuss model motivation and form, and aspects of statistical analysis including parameter identifiability, posterior inference and implementation of these models via Markov chain Monte Carlo techniques. In an application to the analysis of a typical set of 19 EEG series recorded during an ECT seizure at different locations over a patient's scalp, these models reveal time-varying features across the series that are strongly related to the placement of the electrodes. We illustrate various model outputs, the exploration of such time-varying spatial structure and its relevance in the ECT study, and in basic EEG research in general.     

Estimation across multiple models with application to Bayesian computing and software development

Statistical models are sometimes incorporated into computer software for making predictions about future observations. When the computer model consists of a single statistical model this corresponds to estimation of a function of the model parameters. This paper is concerned with the case that the computer model implements multiple, individually-estimated statistical sub-models. This case frequently arises, for example, in models for medical decision making that derive parameter information from multiple clinical studies. We develop a method for calculating the posterior mean of a function of the parameter vectors of multiple statistical models that is easy to implement in computer software, has high asymptotic accuracy, and has a computational cost linear in the total number of model parameters. The formula is then used to derive a general result about posterior estimation across multiple models. The utility of the results is illustrated by application to clinical software that estimates the risk of fatal coronary disease in people with diabetes.     

Applied state space modelling of non-Gaussian time series using integration-based Kalman filtering

The main topic of the paper is on-line filtering for non-Gaussian dynamic (state space) models by approximate computation of the first two posterior moments using efficient numerical integration. Based on approximating the prior of the state vector by a normal density, we prove that the posterior moments of the state vector are related to the posterior moments of the linear predictor in a simple way. For the linear predictor Gauss-Hermite integration is carried out with automatic reparametrization based on an approximate posterior mode filter. We illustrate how further topics in applied state space modelling, such as estimating hyperparameters, computing model likelihoods and predictive residuals, are managed by integration-based Kalman-filtering. The methodology derived in the paper is applied to on-line monitoring of ecological time series and filtering for small count data.     

Oracle GMM estimation for misspecified models via thresholding

We propose a thresholding generalized method of moments (GMM) estimator for misspecified time series moment condition models. This estimator has the following oracle property: its asymptotic behavior is the same as of any efficient GMM estimator obtained under the a priori information that the true model were known. We propose data adaptive selection methods for thresholding parameter using multiple testing procedures. We determine the limiting null distributions of classical parameter tests and show the consistency of the corresponding block-bootstrap tests used in conjunction with thresholding GMM inference. We present the results of a simulation study for a misspecified instrumental variable regression model and for a vector autoregressive model with measurement error. We illustrate an application of the proposed methodology to data analysis of a real-world dataset.     

Generalized linear models with random effects in the two-parameter exponential family

In this paper, we develop a new class of double generalized linear models, introducing a random-effect component in the link function describing the linear predictor related to the precision parameter. This is a useful procedure to take into account extra variability and also to make the model more robust. The Bayesian paradigm is adopted to make inference in this class of models. Samples of the joint posterior distribution are drawn using standard Monte Carlo Markov Chain procedures. Finally, we illustrate this algorithm by considering simulated and real data sets.     

Skewness of maximum likelihood estimators in dispersion models

We introduce the dispersion models with a regression structure to extend the generalized linear models, the exponential family nonlinear models (Cordeiro and Paula, 1989) and the proper dispersion models (Jørgensen, 1997a). We provide a matrix expression for the skewness of the maximum likelihood estimators of the regression parameters in dispersion models. The formula is suitable for computer implementation and can be applied for several important submodels discussed in the literature. Expressions for the skewness of the maximum likelihood estimators of the precision and dispersion parameters are also derived. In particular, our results extend previous formulas obtained by Cordeiro and Cordeiro (2001) and Cavalcanti et al. (2009). A simulation study is performed to show the practice importance of our results.