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.     

A Bayesian estimation of lag lengths in distributed lag models

Dynamic regression models are widely used because they express and model the behaviour of a system over time. In this article, two dynamic regression models, the distributed lag (DL) model and the autoregressive distributed lag model, are evaluated focusing on their lag lengths. From a classical statistics point of view, there are various methods to determine the number of lags, but none of them are the best in all situations. This is a serious issue since wrong choices will provide bad estimates for the effects of the regressors on the response variable. We present an alternative for the aforementioned problems by considering a Bayesian approach. The posterior distributions of the numbers of lags are derived under an improper prior for the model parameters. The fractional Bayes factor technique [A. O'Hagan, Fractional Bayes factors for model comparison (with discussion), J. R. Statist. Soc. B 57 (1995), pp. 99–138] is used to handle the indeterminacy in the likelihood function caused by the improper prior. The zero-one loss function is used to penalize wrong decisions. A naive method using the specified maximum number of DLs is also presented. The proposed and the naive methods are verified using simulation data. The results are promising for the method we proposed. An illustrative example with a real data set is provided.     

Transformed symmetric generalized autoregressive moving average models

Cordeiro and Andrade [Transformed generalized linear models. J Stat Plan Inference. 2009;139:2970–2987] incorporated the idea of transforming the response variable to the generalized autoregressive moving average (GARMA) model, introduced by Benjamin et al. [Generalized autoregressive moving average models. J Am Stat Assoc. 2003;98:214–223], thus developing the transformed generalized autoregressive moving average (TGARMA) model. The goal of this article is to develop the TGARMA model for symmetric continuous conditional distributions with a possible nonlinear structure for the mean that enables the fitting of a wide range of models to several time series data types. We derive an iterative process for estimating the parameters of the new model by maximum likelihood and obtain a simple formula to estimate the parameter that defines the transformation of the response variable. Furthermore, we determine the moments of the original dependent variable which generalize previous published results. We illustrate the theory by means of real data sets and evaluate the results developed through simulation studies.     

Autoregressive Moving Average Infinite Hidden Markov-Switching Models

Markov-switching models are usually specified under the assumption that all the parameters change when a regime switch occurs. Relaxing this hypothesis and being able to detect which parameters evolve over time is relevant for interpreting the changes in the dynamics of the series, for specifying models parsimoniously, and may be helpful in forecasting. We propose the class of sticky infinite hidden Markov-switching autoregressive moving average models, in which we disentangle the break dynamics of the mean and the variance parameters. In this class, the number of regimes is possibly infinite and is determined when estimating the model, thus avoiding the need to set this number by a model choice criterion. We develop a new Markov chain Monte Carlo estimation method that solves the path dependence issue due to the moving average component. Empirical results on macroeconomic series illustrate that the proposed class of models dominates the model with fixed parameters in terms of point and density forecasts.     

Modelling spatio-temporal air pollution data from a mobile monitoring station

ABSTRACT

Environmental data is typically indexed in space and time. This work deals with modelling spatio-temporal air quality data, when multiple measurements are available for each space-time point. Typically this situation arises when different measurements referring to several response variables are observed in each space-time point, for example, different pollutants or size resolved data on particular matter. Nonetheless, such a kind of data also arises when using a mobile monitoring station moving along a path for a certain period of time. In this case, each spatio-temporal point has a number of measurements referring to the response variable observed several times over different locations in a close neighbourhood of the space-time point. We deal with this type of data within a hierarchical Bayesian framework, in which observed measurements are modelled in the first stage of the hierarchy, while the unobserved spatio-temporal process is considered in the following stages. The final model is very flexible and includes autoregressive terms in time, different structures for the variance-covariance matrix of the errors, and can manage covariates available at different space-time resolutions. This approach is motivated by the availability of data on urban pollution dynamics: fast measures of gases and size resolved particulate matter have been collected using an Optical Particle Counter located on a cabin of a public conveyance that moves on a monorail on a line transect of a town. Urban microclimate information is also available and included in the model. Simulation studies are conducted to evaluate the performance of the proposed model over existing alternatives that do not model data over the first stage of the hierarchy.     

Inference for the random coefficients bifurcating autoregressive model for cell lineage studies

Cell lineage data consist of observations on quantitative characteristics of the descendants of an initial cell. The bifurcating autoregressive model has been previously used to model dependencies in cell lineage data by considering each line of descent to be a first order autoregressive process and allowing the environmental effects of sisters to be correlated. Here the basic bifurcating autoregressive model is modified to include random coefficients which allows for the relationship between mother and daughter cells to depend on environmental factors. Maximum likelihood inference under the assumption of multivariate normality is considered and the method is illustrated on several data sets.     

Non-penalty shrinkage estimation of random effect models for longitudinal data with AR(1) errors

In this paper, we consider the non-penalty shrinkage estimation method of random effect models with autoregressive errors for longitudinal data when there are many covariates and some of them may not be active for the response variable. In observational studies, subjects are followed over equally or unequally spaced visits to determine the continuous response and whether the response is associated with the risk factors/covariates. Measurements from the same subject are usually more similar to each other and thus are correlated with each other but not with observations of other subjects. To analyse this data, we consider a linear model that contains both random effects across subjects and within-subject errors that follows autoregressive structure of order 1 (AR(1)). Considering the subject-specific random effect as a nuisance parameter, we use two competing models, one includes all the covariates and the other restricts the coefficients based on the auxiliary information. We consider the non-penalty shrinkage estimation strategy that shrinks the unrestricted estimator in the direction of the restricted estimator. We discuss the asymptotic properties of the shrinkage estimators using the notion of asymptotic biases and risks. A Monte Carlo simulation study is conducted to examine the relative performance of the shrinkage estimators with the unrestricted estimator when the shrinkage dimension exceeds two. We also numerically compare the performance of the shrinkage estimators to that of the LASSO estimator. A longitudinal CD4 cell count data set will be used to illustrate the usefulness of shrinkage and LASSO estimators.     

Classical model selection via simulated annealing

Summary. The classical approach to statistical analysis is usually based upon finding values for model parameters that maximize the likelihood function. Model choice in this context is often also based on the likelihood function, but with the addition of a penalty term for the number of parameters. Though models may be compared pairwise by using likelihood ratio tests for example, various criteria such as the Akaike information criterion have been proposed as alternatives when multiple models need to be compared. In practical terms, the classical approach to model selection usually involves maximizing the likelihood function associated with each competing model and then calculating the corresponding criteria value(s). However, when large numbers of models are possible, this quickly becomes infeasible unless a method that simultaneously maximizes over both parameter and model space is available. We propose an extension to the traditional simulated annealing algorithm that allows for moves that not only change parameter values but also move between competing models. This transdimensional simulated annealing algorithm can therefore be used to locate models and parameters that minimize criteria such as the Akaike information criterion, but within a single algorithm, removing the need for large numbers of simulations to be run. We discuss the implementation of the transdimensional simulated annealing algorithm and use simulation studies to examine its performance in realistically complex modelling situations. We illustrate our ideas with a pedagogic example based on the analysis of an autoregressive time series and two more detailed examples: one on variable selection for logistic regression and the other on model selection for the analysis of integrated recapture–recovery data.     

Comparison of three-dimensional profiles over time

In this paper, we describe an analysis for data collected on a three-dimensional spatial lattice with treatments applied at the horizontal lattice points. Spatial correlation is accounted for using a conditional autoregressive model. Observations are defined as neighbours only if they are at the same depth. This allows the corresponding variance components to vary by depth. We use the Markov chain Monte Carlo method with block updating, together with Krylov subspace methods, for efficient estimation of the model. The method is applicable to both regular and irregular horizontal lattices and hence to data collected at any set of horizontal sites for a set of depths or heights, for example, water column or soil profile data. The model for the three-dimensional data is applied to agricultural trial data for five separate days taken roughly six months apart in order to determine possible relationships over time. The purpose of the trial is to determine a form of cropping that leads to less moist soils in the root zone and beyond. We estimate moisture for each date, depth and treatment accounting for spatial correlation and determine relationships of these and other parameters over time.     

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.     

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.     

Empirical likelihood estimation for linear regression models with AR(p) error terms with numerical examples

Linear regression models are useful statistical tools to analyze data sets in different fields. There are several methods to estimate the parameters of a linear regression model. These methods usually perform under normally distributed and uncorrelated errors. If error terms are correlated the Conditional Maximum Likelihood (CML) estimation method under normality assumption is often used to estimate the parameters of interest. The CML estimation method is required a distributional assumption on error terms. However, in practice, such distributional assumptions on error terms may not be plausible. In this paper, we propose to estimate the parameters of a linear regression model with autoregressive error term using Empirical Likelihood (EL) method, which is a distribution free estimation method. A small simulation study is provided to evaluate the performance of the proposed estimation method over the CML method. The results of the simulation study show that the proposed estimators based on EL method are remarkably better than the estimators obtained from CML method in terms of mean squared errors (MSE) and bias in almost all the simulation configurations. These findings are also confirmed by the results of the numerical and real data examples.     

Bayesian LASSO-Regularized quantile regression for linear regression models with autoregressive errors

Quantile regression (QR) is a natural alternative for depicting the impact of covariates on the conditional distributions of a outcome variable instead of the mean. In this paper, we investigate Bayesian regularized QR for the linear models with autoregressive errors. LASSO-penalized type priors are forced on regression coefficients and autoregressive parameters of the model. Gibbs sampler algorithm is employed to draw the full posterior distributions of unknown parameters. Finally, the proposed procedures are illustrated by some simulation studies and applied to a real data analysis of the electricity consumption.     

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.     

Specification Test for Spatial Autoregressive Models

This article considers a simple test for the correct specification of linear spatial autoregressive models, assuming that the choice of the weight matrix W_n is true. We derive the limiting distributions of the test under the null hypothesis of correct specification and a sequence of local alternatives. We show that the test is free of nuisance parameters asymptotically under the null and prove the consistency of our test. To improve the finite sample performance of our test, we also propose a residual-based wild bootstrap and justify its asymptotic validity. We conduct a small set of Monte Carlo simulations to investigate the finite sample properties of our tests. Finally, we apply the test to two empirical datasets: the vote cast and the economic growth rate. We reject the linear spatial autoregressive model in the vote cast example but fail to reject it in the economic growth rate example. Supplementary materials for this article are available online.     

基于大数据的碳价预测

为了研究大数据是否能够帮助我们预测碳排放权价格,本文讨论了结构化数据和非结构化信息对预测碳价所起的作用。结构化数据选取了国际碳现货价格、碳期货价格和汇率,非结构化信息选择百度搜索指数和媒体指数。考虑到当解释变量很多时,平等对待每一个解释变量是不合理的,所以提出了网络结构自回归分布滞后（ADL）模型,在参数估计和变量选择的同时兼顾了解释变量之间的网络关系。实证分析表明,网络结构ADL模型明显优于其他模型,可以获得较高的预测准确性,更适合基于大数据的预测。     

Maximum likelihood estimation for directional conditionally autoregressive models

A spatial process observed over a lattice or a set of irregular regions is usually modeled using a conditionally autoregressive (CAR) model. The neighborhoods within a CAR model are generally formed using only the inter-distances or boundaries between the regions. To accommodate directional spatial variation, a new class of spatial models is proposed using different weights given to neighbors in different directions. The proposed model generalizes the usual CAR model by accounting for spatial anisotropy. Maximum likelihood estimators are derived and shown to be consistent under some regularity conditions. Simulation studies are presented to evaluate the finite sample performance of the new model as compared to the CAR model. Finally, the method is illustrated using a data set on the crime rates of Columbus, OH and on the elevated blood lead levels of children under the age of 72 months observed in Virginia in the year of 2000.     

A First-Order Spatial Integer-Valued Autoregressive SINAR(1, 1) Model

Binomial thinning operator has a major role in modeling one-dimensional integer-valued autoregressive time series models. The purpose of this article is to extend the use of such operator to define a new stationary first-order spatial non negative, integer-valued autoregressive SINAR(1, 1) model. We study some properties of this model like the mean, variance and autocorrelation function. Yule-Walker estimator of the model parameters is also obtained. Some numerical results of the model are presented and, moreover, this model is applied to a real data set.     

Production Frontiers and Panel Data

Value at risk (VaR) is the standard measure of market risk used by financial institutions. Interpreting the VaR as the quantile of future portfolio values conditional on current information, the conditional autoregressive value at risk (CAViaR) model specifies the evolution of the quantile over time using an autoregressive process and estimates the parameters with regression quantiles. Utilizing the criterion that each period the probability of exceeding the VaR must be independent of all the past information, we introduce a new test of model adequacy, the dynamic quantile test. Applications to real data provide empirical support to this methodology.     

Influential Observations in Time Series

This article studies how to identify influential observations in univariate autoregressive integrated moving average time series models and how to measure their effects on the estimated parameters of the model. The sensitivity of the parameters to the presence of either additive or innovational outliers is analyzed, and influence statistics based on the Mahalanobis distance are presented. The statistic linked to additive outliers is shown to be very useful for indicating the robustness of the fitted model to the given data set. Its application is illustrated using a relevant set of historical data.