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The Vector AutoRegressive Moving Average (VARMA) model is fundamental to the theory of multivariate time series; however, identifiability issues have led practitioners to abandon it in favor of the simpler but more restrictive Vector AutoRegressive (VAR) model. We narrow this gap with a new optimization-based approach to VARMA identification built upon the principle of parsimony. Among all equivalent data-generating models, we use convex optimization to seek the parameterization that is "simplest" in a certain sense. A user-specified strongly convex penalty is used to measure model simplicity, and that same penalty is then used to define an estimator that can be efficiently computed. We establish consistency of our estimators in a double-asymptotic regime. Our non-asymptotic error bound analysis accommodates both model specification and parameter estimation steps, a feature that is crucial for studying large-scale VARMA algorithms. Our analysis also provides new results on penalized estimation of infinite-order VAR, and elastic net regression under a singular covariance structure of regressors, which may be of independent interest. We illustrate the advantage of our method over VAR alternatives on three real data examples.

A bootstrap procedure for constructing prediction bands for a stationary functional time series is proposed. The procedure exploits a general vector autoregressive representation of the time-reversed series of Fourier coefficients appearing in the Karhunen-Loève representation of the functional process. It generates backward-in-time functional replicates that adequately mimic the dependence structure of the underlying process in a model-free way and have the same conditionally fixed curves at the end of each functional pseudo-time series. The bootstrap prediction error distribution is then calculated as the difference between the model-free, bootstrap-generated future functional observations and the functional forecasts obtained from the model used for prediction. This allows the estimated prediction error distribution to account for the innovation and estimation errors associated with prediction and the possible errors due to model misspecification. We establish the asymptotic validity of the bootstrap procedure in estimating the conditional prediction error distribution of interest, and we also show that the procedure enables the construction of prediction bands that achieve (asymptotically) the desired coverage. Prediction bands based on a consistent estimation of the conditional distribution of the studentized prediction error process also are introduced. Such bands allow for taking more appropriately into account the local uncertainty of the prediction. Through a simulation study and the analysis of two data sets, we demonstrate the capabilities and the good finite-sample performance of the proposed method.

We use ten common macroeconomic variables to test for the predictability of the quarterly growth rate of house price index (HPI) in the United States during 1975:Q1–2018:Q2. We extend the instrumen...

Small-angle X-ray scattering (SAXS) is a technique that yields low-resolution structural information of biological macromolecules by exposing a large ensemble of molecules in solution to a powerful...

Many researchers in biology and medicine have focused on trying to understand biological rhythms and their potential impact on disease. A common biological rhythm is circadian, where the cycle repeats itself every 24 hours. However, a disturbance of the circadian pattern may be indicative of future disease. In this article, we develop new statistical methodology for assessing the degree of disturbance or irregularity in a circadian pattern for count sequences that are observed over time in a population of individuals. We develop a latent variable Poisson modeling approach with both circadian and stochastic short-term trend (autoregressive latent process) components that allow for individual variation in the degree of each component. A parameterization is proposed for modeling covariate dependence on the proportion of these two model components across individuals. In addition, we incorporate covariate dependence in the overall mean, the magnitude of the trend, and the phase-shift of the circadian pattern. Innovative Markov chain Monte Carlo sampling is used to carry out Bayesian posterior computation. Several variations of the proposed models are considered and compared using the deviance information criterion. We illustrate this methodology with longitudinal physical activity count data measured in a longitudinal cohort of adolescents.

Maximum likelihood estimation of a censored autoregressive model with exogenous variables (CARX) requires computing the conditional likelihood of blocks of data of variable dimensions. As the random block dimension generally increases with the censoring rate, maximum likelihood estimation becomes quickly numerically intractable with increasing censoring. We introduce a new estimation approach using the complete-incomplete data framework with the complete data comprising the observations were there no censoring. We introduce a system of unbiased estimating equations motivated by the complete-data score vector, for estimating a CARX model. The proposed quasi-likelihood method reduces to maximum likelihood estimation when there is no censoring, and it is computationally efficient. We derive the consistency and asymptotic normality of the quasi-likelihood estimator, under mild regularity conditions. We illustrate the efficacy of the proposed method by simulations and a real application on phosphorus concentration in river water.

We present an approach for modeling areal spatial covariance in observed genetic allele data by considering the stationary (limiting) distribution of a spatio-temporal Markov random walk model for gene flow. This stationary distribution corresponds to an intrinsic simultaneous autoregressive (SAR) model for spatial correlation, and provides a principled approach to specifying areal spatial models when a spatio-temporal generating process can be assumed. We apply the approach to a study of spatial genetic variation of trout in a stream network in Connecticut, USA.

Practical Bayesian nonparametric methods have been developed across a wide variety of contexts. Here, we develop a novel statistical model that generalizes standard mixed models for longitudinal data that include flexible mean functions as well as combined compound symmetry (CS) and autoregressive (AR) covariance structures. AR structure is often specified through the use of a Gaussian process (GP) with covariance functions that allow longitudinal data to be more correlated if they are observed closer in time than if they are observed farther apart. We allow for AR structure by considering a broader class of models that incorporates a Dirichlet Process Mixture (DPM) over the covariance parameters of the GP. We are able to take advantage of modern Bayesian statistical methods in making full predictive inferences and about characteristics of longitudinal profiles and their differences across covariate combinations. We also take advantage of the generality of our model, which provides for estimation of a variety of covariance structures. We observe that models that fail to incorporate CS or AR structure can result in very poor estimation of a covariance or correlation matrix. In our illustration using hormone data observed on women through the menopausal transition, biology dictates the use of a generalized family of sigmoid functions as a model for time trends across subpopulation categories.

The Whittle likelihood estimation (WLE) has played a fundamental role in the development of both theory and computation of time series analysis. However, WLE is only applicable to models whose theoretical spectral density function (SDF) is known up to the parameters in the models. In this article, we propose a residual-based WLE, called extended WLE (XWLE), which can estimate models with their SDFs only partially available, including many popular time series models with correlated residuals. Asymptotic properties of XWLE are established. In particular, XWLE is asymptotically equivalent to WLE in estimating linear ARMA models, and is also capable of estimating nonlinear AR models with MA residuals and even with exogenous variables. The finite-sample performances of XWLE are checked by simulated examples and real data analysis.

The primary contributions of this article are rigorously developed novel statistical methods for detecting change points in multivariate time series. We extend the class of score type change point statistics considered in 2007 by Huskova, Praskova, and Steinebach to the vector autoregressive (VAR) case and the epidemic change alternative. Our proposed procedures do not require the observed time series to actually follow the VAR model. Instead, following the strategy implicitly employed by practitioners, our approach takes model misspecification into account so that our detection procedure uses the model background merely for feature extraction. We derive the asymptotic distributions of our test statistics and show that our procedure has asymptotic power of 1. The proposed test statistics require the estimation of the inverse of the long-run covariance matrix which is particularly difficult in higher-dimensional settings (i.e., where the dimension of the time series and the dimension of the parameter vecto...

Consider a structural break autoregressive (SBAR) process where j = 1, …, m + 1, {t1, …, tm} are change-points, 1 = t0 < t1 < ⋅⋅⋅ < tm + 1 = n + 1, σ( · ) is a measurable function on , and {ϵt} are white noise with unit variance. In practice, the number of change-points m is usually assumed to be known and small, because a large m would involve a huge amount of computational burden for parameters estimation. By reformulating the problem in a variable selection context, the group least absolute shrinkage and selection operator (LASSO) is proposed to estimate an SBAR model when m is unknown. It is shown that both m and the locations of the change-points {t1, …, tm} can be consistently estimated from the data, and the computation can be efficiently performed. An improved practical version that incorporates group LASSO and the stepwise regression variable selection technique are discussed. Simulation studies are conducted to assess the finite sample performance. Supplementary materials for this article are av...

Let observations y1, …, yn be generated from a first-order autoregressive (AR) model with positive errors. In both the stationary and unit root cases, we derive moment bounds and limiting distributions of an extreme value estimator, , of the AR coefficient. These results enable us to provide asymptotic expressions for the mean squared error (MSE) of and the mean squared prediction error (MSPE) of the corresponding predictor, , of yn + 1. Based on these expressions, we compare the relative performance of () and the least-squares predictor (estimator) from the MSPE (MSE) point of view. Our comparison reveals that the better predictor (estimator) is determined not only by whether a unit root exists, but also by the behavior of the underlying error distribution near the origin, and hence is difficult to identify in practice. To circumvent this difficulty, we suggest choosing the predictor (estimator) with the smaller accumulated prediction error and show that the predictor (estimator) chosen in this way is as...

In many applications, time series exhibit nonstationary behavior that might reasonably be modeled as a time-varying autoregressive (AR) process. In the context of such a model, we discuss the problem of testing for modality of the variance function. We propose a test of modality that is local and, when used iteratively, can be used to identify the total number of modes in a given series. This problem is closely related to peak detection and identification, which has applications in many fields. We propose a test that, under appropriate assumptions, is asymptotically distribution free under the null hypothesis, even though nonparametric estimation of the AR parameter functions is involved. Simulation studies and applications to real datasets illustrate the behavior of the test.

Of the various renewable energy resources, wind power is widely recognized as one of the most promising. The management of wind farms and electricity systems can benefit greatly from the availability of estimates of the probability distribution of wind power generation. However, most research has focused on point forecasting of wind power. In this article, we develop an approach to producing density forecasts for the wind power generated at individual wind farms. Our interest is in intraday data and prediction from 1 to 72 hours ahead. We model wind power in terms of wind speed and wind direction. In this framework, there are two key uncertainties. First, there is the inherent uncertainty in wind speed and direction, and we model this using a bivariate vector autoregressive moving average-generalized autoregressive conditional heteroscedastic (VARMA-GARCH) model, with a Student t error distribution, in the Cartesian space of wind speed and direction. Second, there is the stochastic nature of the relationship of wind power to wind speed (described by the power curve), and to wind direction. We model this using conditional kernel density (CKD) estimation, which enables a nonparametric modeling of the conditional density of wind power. Using Monte Carlo simulation of the VARMA-GARCH model and CKD estimation, density forecasts of wind speed and direction are converted to wind power density forecasts. Our work is novel in several respects: previous wind power studies have not modeled a stochastic power curve; to accommodate time evolution in the power curve, we incorporate a time decay factor within the CKD method; and the CKD method is conditional on a density, rather than a single value. The new approach is evaluated using datasets from four Greek wind farms.

Temperature control for a large data center is both important and expensive. On the one hand, many of the components produce a great deal of heat, and on the other hand, many of the components require temperatures below a fairly low threshold for reliable operation. A statistical framework is proposed within which the behavior of a large cooling system can be modeled and forecast under both steady state and perturbations. This framework is based upon an extension of multivariate Gaussian autoregressive hidden Markov models (HMMs). The estimated parameters of the fitted model provide useful summaries of the overall behavior of and relationships within the cooling system. Predictions under system perturbations are useful for assessing potential changes and improvements to be made to the system. Many data centers have far more cooling capacity than necessary under sensible circumstances, thus resulting in energy inefficiencies. Using this model, predictions for system behavior after a particular component of the cooling system is shut down or reduced in cooling power can be generated. Steady-state predictions are also useful for facility monitors. System traces outside control boundaries flag a change in behavior to examine. The proposed model is fit to data from a group of air conditioners within an enterprise data center from the IT industry. The fitted model is examined, and a particular unit is found to be underutilized. Predictions generated for the system under the removal of that unit appear very reasonable. Steady-state system behavior also is predicted well.

A class of hierarchical Bayesian models is proposed for adverse birth outcomes such as preterm birth, which are assumed to follow a conditional binomial distribution. The log-odds of an adverse outcome in a particular county, logit(pi), follow a linear model that includes observed covariates and normally-distributed random effects. Spatial dependence between neighboring regions is allowed for by including an intrinsically autoregressive (IAR) prior or an IAR convolution prior in the linear predictor. Temporal dependence is incorporated by including a temporal IAR term also. It is shown that the variance parameters underlying these random effects (IAR, convolution, convolution plus temporal IAR) are identifiable. The Deviance Information Criterion (DIC) is considered as a way to compare spatial hierarchical models. Simulations are performed to test whether the DIC can identify whether binomial outcomes come from a hierarchical model that includes different combinations of random and fixed effects. Having v...

High-dimensional regression problems which reveal dynamic behavior are typically analyzed by time propagation of a few number of factors. The inference on the whole system is then based on the low-dimensional time series analysis. Such highdimensional problems occur frequently in many different fields of science. In this paper we address the problem of inference when the factors and factor loadings are estimated by semiparametric methods. This more flexible modelling approach poses an important question: Is it justified, from inferential point of view, to base statistical inference on the estimated times series factors? We show that the difference of the inference based on the estimated time series and 'true' unobserved time series is asymptotically negligible. Our results justify fitting vector autoregressive processes to the estimated factors, which allows one to study the dynamics of the whole high-dimensional system with a low-dimensional representation. We illustrate the theory with a simulation study. Also, we apply the method to a study of the dynamic behavior of implied volatilities and discuss other possible applications in finance and economics.

The class of functional signal plus noise (FSN) models is introduced that provides a new, general method for modeling and forecasting time series of economic functions. The underlying, continuous economic function (or “signal”) is a natural cubic spline whose dynamic evolution is driven by a cointegrated vector autoregression for the ordinates (or “y-values”) at the knots of the spline. The natural cubic spline provides flexible cross-sectional fit and results in a linear state-space model. This FSN model achieves dimension reduction, provides a coherent description of the observed yield curve and its dynamics as the cross-sectional dimension N becomes large, and can be feasibly estimated and used for forecasting when N is large. The integration and cointegration properties of the model are derived. The FSN models are then applied to forecasting 36-dimensional yield curves for U.S. Treasury bonds at the 1-month-ahead horizon. The method consistently outperforms the dynamic Nelson–Siegel and random walk fo...

The assessment of air pollution regulatory programs designed to improve ground level ozone concentrations is a topic of considerable interest to environmental managers. To aid this assessment, it is necessary to model the space-time behavior of ozone for predicting summaries of ozone across spatial domains of interest and for the detection of long-term trends at monitoring sites. These trends, adjusted for the effects of meteorological variables, are needed for determining the effectiveness of pollution control programs in terms of their magnitude and uncertainties across space. This paper proposes a space-time model for daily 8-hour maximum ozone levels to provide input to regulatory activities: detection, evaluation, and analysis of spatial patterns of ozone summaries and temporal trends. The model is applied to analyzing data from the state of Ohio which has been chosen because it contains a mix of urban, suburban, and rural ozone monitoring sites in several large cities separated by large rural areas. The proposed space-time model is auto-regressive and incorporates the most important meteorological variables observed at a collection of ozone monitoring sites as well as at several weather stations where ozone levels have not been observed. This problem of misalignment of ozone and meteorological data is overcome by spatial modeling of the latter. In so doing we adopt an approach based on the successive daily increments in meteorological variables. With regard to modeling, the increment (or change-in-meteorology) process proves more attractive than working directly with the meteorology process, without sacrificing any desired inference. The full model is specified within a Bayesian framework and is fitted using MCMC techniques. Hence, full inference with regard to model unknowns is available as well as for predictions in time and space, evaluation of annual summaries and assessment of trends.

The problem of testing noncorrelation between two multivariate time series is considered. Assuming that the global process admits a joint vector autoregressive (VAR) representation, noncorrelation between the two component series is equivalent to the hypothesis that all off-diagonal blocks in the matrix coefficients and the innovation covariance of the joint VAR representation are zero. We establish an adequate local asymptotic normality (LAN) property for this VAR model in the vicinity of noncorrelation. This LAN structure allows construction of optimal pseudo-Gaussian tests—that is, tests that are locally and asymptotically optimal under Gaussian innovations, but remain valid under non-Gaussian ones—for the null hypothesis of noncorrelation and for comparing their local asymptotic powers with those of the heuristic tests (Haugh–El Himdi–Roy and Koch–Yang–Hallin–Saidi) proposed in the literature.

Attachment loss, the extent of a tooth's root (in millimeters) that is no longer attached to surrounding bone by periodontal ligament, is often used to measure the current state of a patient's periodontal disease and monitor disease progression. Attachment loss data can be analyzed using a conditionally autoregressive (CAR) prior distribution that smooths fitted values toward neighboring values. However, it may be desirable to have more than one class of neighbor relation in the spatial structure, so the different classes of neighbor relations can induce different degrees of smoothing. For example, we may wish to allow smoothing of neighbor pairs bridging the gap between teeth to differ from smoothing of pairs that do not bridge such gaps. Adequately modeling the spatial structure may improve the monitoring of periodontal disease progression. This article develops a two-neighbor-relation CAR model to handle this situation and presents associated theory to help explain the sometimes unusual posterior distr...

Novel periodic extensions of dynamic long-memory regression models with autoregressive conditional heteroscedastic errors are considered for the analysis of daily electricity spot prices. The parameters of the model with mean and variance specifications are estimated simultaneously by the method of approximate maximum likelihood. The methods are implemented for time series of 1,200-4,400 daily price observations in four European power markets. Apart from persistence, heteroscedasticity, and extreme observations in prices, a novel empirical finding is the importance of day-of-the-week periodicity in the autocovariance function of electricity spot prices. In particular, the very persistent daily log prices from the Nord Pool power exchange of Norway are effectively modeled by our framework, which is also extended with explanatory variables to capture supply-and-demand effects. The daily log prices of the other three electricity markets - EEX in Germany, Powernext in France, and APX in The Netherlands - are less persistent, but periodicity is also highly significant. The dynamic behavior differs from market to market and depends primarily on the method of power generation: hydro power, power generated from fossil fuels, or nuclear power. The article improves on existing models in capturing the memory characteristics, which are important in derivative pricing and real option analysis. © 2007 American Statistical Association.

We consider the problem of prediction for stationary and nonstationary univariate time series using a modification suggested by the usual exponentially weighted moving average method. The modification leads to a class of general exponential predictors that can improve on the usual finite approximations to an infinite autoregressive process. We provide the theoretical justifications and suggest a class of predictors that covers modified and finite autoregressive fits as special cases. Two examples involving sample data show how the method is competitive with autoregressive integrated moving average (ARIMA) when applied to a U.S. energy use series and improves on ARIMA when applied to a global temperature series. A simulation indicates that considerable improvements are possible for infinite autoregressive (ARIMA) processes exhibiting certain special patterns of long-range dependence.

This article considers the problem of modeling a class of nonstationary time series using piecewise autoregressive (AR) processes. The number and locations of the piecewise AR segments, as well as the orders of the respective AR processes, are assumed unknown. The minimum description length principle is applied to compare various segmented AR fits to the data. The goal is to find the “best” combination of the number of segments, the lengths of the segments, and the orders of the piecewise AR processes. Such a “best” combination is implicitly defined as the optimizer of an objective function, and a genetic algorithm is implemented to solve this difficult optimization problem. Numerical results from simulation experiments and real data analyses show that the procedure has excellent empirical properties. The segmentation of multivariate time series is also considered. Assuming that the true underlying model is a segmented autoregression, this procedure is shown to be consistent for estimating the location of...

Twenty years ago, Kay proposed an iterative filtering algorithm (IFA) for jointly estimating the frequencies of multiple complex sinusoids from noisy observations. IFA is based on the fact that the noiseless signal is an autoregressive (AR) process, so the frequency estimation problem can be reformulated as the problem of estimating the AR coefficients. By iterating the cycle of AR coefficient estimation and AR filtering, IFA provides a computationally simple procedure yet capable of accurate frequency estimation especially at low signal-to-noise ratio (SNR). However, the convergence of IFA has not been established beyond simulation and a very special case of a single frequency and infinite sample size. This article provides a statistical analysis of the algorithm and makes several important contributions. It shows that the poles of the AR filter must be reduced by an extra shrinkage parameter to accommodate poor initial values and avoid being trapped into false solutions. It also shows that the AR estima...

Here we present a novel method for modeling stationary time series. Our approach is to construct the model with a specified marginal family and build the dependence structure around it. We show that the resulting time series is linear with a simple autocorrelation structure. We construct models that parallel existing structures, namely state-space models, autoregressive conditional heteroscedasticity (ARCH) models, and generalized ARCH models. We use Bayesian techniques to estimate the resulting models. We also demonstrate that the models perform well compared with competing methods for the applications considered, count models and volatility models.

The theory developed for bootstrapping unit root tests in an autoregressive (AR) context has been concerned mainly with the large-sample behavior of the methods proposed under the assumption that the null hypothesis is true. No results exist for the relative performance and the power behavior of the bootstrap methods under the alternative. This article studies the properties of different AR bootstrap schemes of the unit root hypothesis, including a new proposal based on unrestricted residuals. It shows that bootstrap procedures based on differencing the observed series suffer from power problems as compared with bootstrap procedures based on unrestricted residuals. Whereas for finite-order AR processes differencing leads to just a loss of power, for infinite-order autoregressions such a differencing makes the application of sieve AR bootstrap schemes inappropriate if the alternative is true. The superiority of the new bootstrap proposal is shown, and some numerical examples illustrate our theoretical findings. © 2005 American Statistical Association. 100 470 545 553 Cited By :23

Proactive management of web server farms requires accurate prediction of workload. An exemplary measure of workload is the amount of service requests per unit time. As a time series, the workload exhibits not only short-term random fluctuations, but also prominent periodic (daily) patterns that evolve randomly from one period to another. A hierarchical framework with multiple time scales is proposed to model such time series. This framework leads to an adaptive procedure that provides both long-term (in days) and short-term (in minutes) predictions with simultaneous confidence bands that accommodate not only serial correlation, but also heavy tailedness, heteroscedasticity, and nonstationarity of the data.

Time series data are often subject to measurement error, usually the result of needing to estimate the variable of interest. Although it is often reasonable to assume that the measurement error is additive (i.e., the estimator is conditionally unbiased for the missing true value), the measurement error variances often vary as a result of changes in the population/process over time and/or changes in sampling effort. In this article we address estimation of the parameters in linear autoregressive models in the presence of additive and uncorrelated measurement errors, allowing heteroscedasticity in the measurement error variances. We establish the asymptotic properties of naive estimators that ignore measurement error and propose an estimator based on correcting the Yule–Walker estimating equations. We also examine a pseudo-likelihood method based on normality assumptions and computed using the Kalman filter. We review other techniques that have been proposed, including two that require no information about ...

A popular model for assessing dependence on time is the time series model composed of a linear trend plus a zero mean autoregressive (AR) process. Although considerable effort has been devoted developing tests for linear trend in the presence of serial correlation, the testing procedures used in practice are less than satisfactory for portions of the parameter space for the AR coefficient. This is because the variance of the feasible generalized least squares (FGLS) estimator of the trend coefficient is heavily dependent on the parameters of the AR process. A test based on the Gauss–Newton procedure is shown to have more uniform behavior over the parameter space than tests based on FGLS. The test based on Gauss–Newton procedure also has good power.

A class of generalized autoregressive moving average (GARMA) models is developed that extends the univariate Gaussian ARMA time series model to a flexible observation-driven model for non-Gaussian time series data. The dependent variable is assumed to have a conditional exponential family distribution given the past history of the process. The model estimation is carried out using an iteratively reweighted least squares algorithm. Properties of the model, including stationarity and marginal moments, are either derived explicitly or investigated using Monte Carlo simulation. The relationship of the GARMA model to other models is shown, including the autoregressive models of Zeger and Qaqish, the moving average models of Li, and the reparameterized generalized autoregressive conditional heteroscedastic GARCH model (providing the formula for its fourth marginal moment not previously derived). The model is demonstrated by the application of the GARMA model with a negative binomial conditional distribution to ...

We propose a mixture autoregressive conditional heteroscedastic (MAR-ARCH) model for modeling nonlinear time series. The models consist of a mixture of K autoregressive components with autoregressive conditional heteroscedasticity; that is, the conditional mean of the process variable follows a mixture AR (MAR) process, whereas the conditional variance of the process variable follows a mixture ARCH process. In addition to the advantage of better description of the conditional distributions from the MAR model, the MARARCH model allows a more flexible squared autocorrelation structure. The stationarity conditions, autocorrelation function, and squared autocorrelation function are derived. Construction of multiple step predictive distributions is discussed. The estimation can be easily done through a simple EM algorithm, and the model selection problem is addressed. The shape-changing feature of the conditional distributions makes these models capable of modeling time series with multimodal conditional distr...

Whittle pseudo-maximum likelihood estimates of parameters for stationary time series have been found to be consistent and asymptotically normal in the presence of long-range dependence. Generalizing the definition of the memory parameter d, we extend these results to include possibly nonstationary (.5 $\leq d

This article considers maximum likelihood (ML) and restricted maximum likelihood (REML) estimation of time series regression models with autoregressive AR(p) noise. Approximate biases of the ML and REML estimators of the AR parameters, based on their approximate representations, are derived. In addition, a bias result for the ML estimator (MLE) of the error variance is established. Numerical results are presented to illustrate the biases of the MLE and REML estimator for the AR parameters, and simulation results are provided to assess the adequacy of our approximations. The impact of bias of the AR estimates on testing of linear trend in a regression trend model is also investigated. For a time series of short or moderate sample length, the REML estimator is generally much less biased than the MLE. Consequently, the REML approach leads to more accurate inferences for the regression parameters.

Functional magnetic resonance imaging (fMRI) is a new technique for studying the workings of the active human brain. During an fMRI experiment, a sequence of magnetic resonance images is acquired while the subject performs specific behavioral tasks. Changes in the measured signal can be used to identify and characterize the brain activity resulting from task performance. The data obtained from an fMRI experiment are a realization of a complex spatiotemporal process with many sources of variation, both biological and technological. This article describes a nonlinear Bayesian hierarchical model for fMRI data and presents inferential methods that enable investigators to directly target their scientific questions of interest, many of which are inaccessible to current methods. The article describes optimization and posterior sampling techniques to fit the model, both of which must be applied many thousands of times for a single dataset. The model is used to analyze data from a psychological experiment...

Optimal rank-based procedures have been derived for testing arbitrary linear restrictions on the parameters of autoregressive moving average (ARMA) models with unspecified innovation densities. The finite-sample performances of these procedures are investigated here in the context of AR order identification and compared to those of classical (partial correlograms and Lagrange multipliers) methods. The results achieved by rank-based methods are quite comparable, in the Gaussian case, to those achieved by the traditional ones, which, under Gaussian assumptions, are asymptotically optimal. However, under non-Gaussian innovation densities, especially heavy-tailed or nonsymmetric, or when outliers are present, the percentages of correct order selection based on rank methods are strikingly better than those resulting from traditional approaches, even in the case of very short (n = 25) series. These empirical findings confirm the often ignored theoretical fact that the Gaussian case, in the ARMA context...

This article presents an autoregressive random coefficient model with overdispersed negative multinomial marginal distributions for the analysis of heterogeneity and serial dependencies in multivariate longitudinal count data. The model structure consists of four components that take into account (a) individual difference effects, (b) random time effects, (c) multiple event categories, and (d) autodependencies. The last component is based on a stochastic integer-valued autoregressive process proposed by McKenzie. The model is applied to analyze count data from a panel diary study about the relationship between personality factors and emotion experiences. It is shown that there are large and stable individual personality differences in the incidence and duration of self-reported emotional experiences. Theoretical and clinical implications of this result are discussed.

We explore and illustrate the use of time series decomposition methods for evaluating and comparing latent structure in nonstationary electroencephalographic (EEG) traces obtained from depressed patients during brain seizures induced as part of electroconvulsive therapy (ECT). Analysis of the patterns of change over time in the frequency structure of such EEG data provides insight into the neurophysiological mechanisms of action of this effective but poorly understood antidepressant treatment, and allows clinicians to modify ECT treatments to optimize therapeutic benefits while minimizing associated side effects. Our work has introduced new methods of time-frequency analysis of EEG series that identify the complete pattern of time evolution of frequency structure over the course of a seizure, and usefully assist in these scientific and clinical studies. New methods of decomposition of flexible dynamic models provide time domain decompositions of individual EEG series into collections of latent co...

Threshold autoregressive models in which the process is piecewise linear in the threshold space have received much attention in recent years. In this article I use predictive residuals to construct a test statistic for detecting threshold nonlinearity in a vector time series and propose a procedure for building a multivariate threshold model. The thresholds and the model are selected jointly based on the Akaike information criterion. The finite-sample performance of the proposed test is studied by simulation. The modeling procedure is then used to study arbitrage in security markets and results in a threshold cointegration between logarithms of future contracts and spot prices of a security after adjusting for the cost of carrying the contracts. In this particular application, thresholds are determined in part by the transaction costs. I also apply the proposed procedure to U.S. monthly interest rates and two river flow series of Iceland.

Efficient means of modeling aberrant behavior in times series are developed. Our methods are based on state-space forms and allow test statistics for various interventions to be computed from a single run of the Kalman filter smoother. The approach encompasses existing detection methodologies. Departures commonly observed in practice, such as outlying values, level shifts, and switches, are readily dealt with. New diagnostic statistics are proposed. Implications for structural models, autoregressive integrated moving average models, and models with explanatory variables are given.

This article considers the power properties of the McCabe and Tremayne (MT) test for the difference stationarity of a time series. The limiting distribution of the MT test is derived under sequences of locally heteroscedastic and locally explosive autoregressive (AR) alternatives. The limiting distribution of Dickey–Fuller (DF) statistics is also considered under a sequence of locally heteroscedastic alternatives. Whereas the MT test possesses asymptotic power against both forms of nonstationary local alternative, the DF tests set up to test against explosive AR alternatives display little or no ability to reject difference stationarity under local heteroscedastic integration.

This article develops a predictive model for National Football League (NFL) game scores using data from the period 1988–1993. The parameters of primary interest—measures of team strength—are expected to vary over time. Our model accounts for this source of variability by modeling football outcomes using a state-space model that assumes team strength parameters follow a first-order autoregressive process. Two sources of variation in team strengths are addressed in our model; week-to-week changes in team strength due to injuries and other random factors, and season-to-season changes resulting from changes in personnel and other longer-term factors. Our model also incorporates a home-field advantage while allowing for the possibility that the magnitude of the advantage may vary across teams. The aim of the analysis is to obtain plausible inferences concerning team strengths and other model parameters, and to predict future game outcomes. Iterative simulation is used to obtain samples from the joint ...

This article introduces a nonparametric method to estimate the covariance matrix for the stationary part of the signal (hidden in data), to enable benchmarking via signal extraction. Some discussions and simulations are carried out to compare the proposed benchmarking method to the regression method development by Cholette and Dagum and the signal extraction method developed by Hillmer and Trabelsi suggesting autoregression integrated moving average (ARIMA) models for the signal. The results show that the nonparametric method is feasible, robust, and almost as efficient as the signal extraction method when the true model for the signal is known.

This article considers fractionally integrated autoregressive moving-average time series models with conditional heteroscedasticity, which combines the popular generalized autoregressive conditional heteroscedastic (GARCH) and the fractional (ARMA) models. The fractional differencing parameter d can be greater than 1/2, thus incorporating the important unit root case. Some sufficient conditions for stationarity, ergodicity, and existence of higher-order moments are derived. An algorithm for approximate maximum likelihood (ML) estimation is presented. The asymptotic properties of ML estimators, which include consistency and asymptotic normality, are discussed. The large-sample distributions of the residual autocorrelations and the square-residual autocorrelations are obtained, and two portmanteau test statistics are established for checking model adequacy. In particular, non-stationary FARIMA(p, d, q)-GARCH(r, s) models are also considered. Some simulation results are reported. As an illustration,...

We analyze a time series of 20 years of daily sea surface temperatures measured off the California coast. The temperatures exhibit quite complicated features, such as effects on many different time scales, nonlinear effects, and long-range dependence. We show how a time series version of the multivariate adaptive regression splines (MARS) algorithm, TSMARS, can be used to obtain univariate adaptive spline threshold autoregressive models that capture many of the physical characteristics of the temperatures and are useful for short- and long-term prediction. We also discuss practical modeling issues, such as handling cycles, long-range dependence, and concurrent predictor time series using TSMARS. Models for the temperatures are evaluated using out-of-sample forecast comparisons, residual diagnostics, model skeletons, and sample functions of simulated series. We show that a categorical seasonal indicator variable can be used to model nonlinear structure in the data that is changing with time of yea...

A data-determined method for testing structural models of the errors in vector autoregressions is discussed. The method can easily be combined with prior economic knowledge and a subjective analysis of data characteristics to yield valuable information concerning model selection and specification. In one dimension, it turns out that standard t statistics can be used to test the various overidentifying restrictions that are implied by a model. In another dimension, the method compares a priori knowledge of a structural model for the errors with the properties exhibited by the data. Thus this method may help to ensure that orderings of the errors for impulse response and forecast error variance decomposition analyses are sensible, given the data. Two economic examples are used to illustrate the method.

For the partially nonstationary vector autoregressive model of Ann and Reinsel, I further assume that the first differenced series has scalar components of lower order and study estimation of these models along with asymptotic properties of the estimators. It is shown that Gaussian reduced rank estimation can be easily carried out by simple modification of the Ahn and Reinsel's method. The asymptotic distribution for the estimator of the nonstationary parameter is a locally asymptotically mixed normal, and for that of the stationary parameter is asymptotically a normal. Testing hypothesis of the assumed structure of scalar components, including serial correlation common feature, is briefly discussed. A numerical example is provided to illustrate the methods.