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.     

Analysis of autoregressive models with symmetric stable innovations

This paper develops algorithms for fitting autoregressive models with symmetric stable innovations using auto-covariation function. A recursive algorithm is proposed for generalized Yule-Walker estimation of autoregressive coefficients and partial auto-covariation function. It also introduces a new information criterion, useful for consistent order selection. Applications of the proposed methods are illustrated using observations simulated from autoregressive models with symmetric stable innovations as well as by analysing a set of real data.     

The evaluation of exact maximum likelihood estimates for varma models

This paper describes an algorithm for the evaluation of the exact likelihood function in order to obtain estimates of the coefficients of vector autoregressive moving average (VARMA) models. The use of the algorithm is illustrated by a Monte Carlo experiment and an application to the analysis of a set of bivariate animal population data. Fanally it is shown how to extend the algorithm, in a simple manner, to obtain exact maximum likelihood estimates of the coefficients of vector autoregressive moving average models with included exogenous variables.     

Risk efficient estimation of fully dependent random coefficient autoregressive models of general order

We consider a stochastic dynamic model with autoregressive progression. The drift coefficients of the autoregressive model are random where the randomness in the coefficients can have any dependence structure. We propose a two-step sequential estimator and study the asymptotic behavior of few important properties. Paradigm of sequential estimation has its own advantage in reducing sample size and plugging estimates of nuisance parameters while inferring about the main parameters. Our proposed estimator is asymptotically optimal as the predictive risk of the proposed estimator attains the risk of the oracle that assumes known nuisance parameters. Extensive simulation confirms our results.     

AN EXAMINATION OF ESTIMATED RESIDUALS IN A REGRESSION WITH AN INFINITE ORDER PARAMETRIC MODEL

We consider a linear regression with the error term that obeys an autoregressive model of infinite order and estimate parameters of the models. The parameters of the autoregressive model should be estimated based on estimated residuals obtained by means of the method of ordinary least squares, because the errors are unobservable. The consistency of the coefficients, variance and spectral density of the model obeyed by the error term is shown. Further, we estimate the coefficients of the linear regression by means of the method of estimated generalized least squares. We also show the consistency of the estimator.

     

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.     

Parameter estimation for generalized random coefficient autoregressive processes

A generalized random coefficient autoregressive (GRCA) process is introduced in which the random coefficients are permitted to be correlated with the error process. The ordinary random coefficient autoregressive process, the Markovian bilinear model and its generalization, and the random coefficient exponential autoregressive process, among others, are seen to be special cases of the GRCA process. Conditional least squares, and weighted least-squares estimators of the mean of the random coefficient vector are derived and their limit distributions are studied. Estimators of the variance-covariance parameters are also discussed. A simulation study is presented which shows that the weighted least-squares estimator dominates the unweighted least-squares estimator.     

Estimation of a multiple-threshold AR(p) model

The present paper deals with the multiple-threshold p-order autoregressive model which has been introduced by Tong and Lim [H. Tong, K.S. Lim, Threshold autoregression, limit cycles and cyclical data, J. R. Stat. Soc. Ser. B 42 (1980) 245–292] in nonlinear system modelling. Under some conditions on the coefficients of the model which ensure the stationarity, the existence of moments and the strong mixing property of this process and under other mild assumptions, we establish the asymptotic properties (consistency and asymptotic normality) of the minimum Hellinger distance estimates of the autoregressive coefficients of the model.     

Properties of generalized Levinson–Durbin–Whittle sequences

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

Bayesian 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.     

Time-varying coefficient models with ARMA–GARCH structures for longitudinal data analysis

Time-varying coefficient models with autoregressive and moving-average–generalized autoregressive conditional heteroscedasticity structure are proposed for examining the time-varying effects of risk factors in longitudinal studies. Compared with existing models in the literature, the proposed models give explicit patterns for the time-varying coefficients. Maximum likelihood and marginal likelihood (based on a Laplace approximation) are used to estimate the parameters in the proposed models. Simulation studies are conducted to evaluate the performance of these two estimation methods, which is measured in terms of the Kullback–Leibler divergence and the root mean square error. The marginal likelihood approach leads to the more accurate parameter estimates, although it is more computationally intensive. The proposed models are applied to the Framingham Heart Study to investigate the time-varying effects of covariates on coronary heart disease incidence. The Bayesian information criterion is used for specifying the time series structures of the coefficients of the risk factors.     

Bayes Estimates for the Price and Income Elasticities of Alcoholic Beverages in Finland From 1955 to 1980

The Lindley–Smith theory of Bayes estimates for multiple regression equations with exchangeability between the regression coefficients of the individual equations is extended to the case in which a first-order autoregressive process generates the regression coefficients. The ensuing formulas are applied to study monthly Finnish consumption of alcohol. The point of this application is that exchangeability between the regression coefficients is less than we can assess beforehand when there is a natural ordering, in this case according to chronological time, of the equations. Still, the general task of the Lindley–Smith estimators, to consider the combined data when estimating individual regression coefficients, is a relevant one.     

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.     

Bayesian analysis of threshold autoregressions

A nonasymptotic Bayesian approach is developed for analysis of data from threshold autoregressive processes with two regimes. Using the conditional likelihood function, the marginal posterior distribution for each of the parameters is derived along with posterior means and variances. A test for linear functions of the autoregressive coefficients is presented. The approach presented uses a posterior p-value averaged over the values of the threshold. The one-step ahead predictive distribution is derived along with the predictive mean and variance. In addition, equivalent results are derived conditional upon a value of the threshold. A numerical example is presented to illustrate the approach.     

Asymptotic Results for Spatial ARMA Models

Causal quadrantal-type spatial ARMA(p, q) models with independent and identically distributed innovations are considered. In order to select the orders (p, q) of these models and estimate their autoregressive parameters, estimators of the autoregressive coefficients, derived from the extended Yule–Walker equations are defined. Consistency and asymptotic normality are obtained for these estimators. Then, spatial ARMA model identification is considered and simulation study is given.     

Least squares estimation of the parameters of a stochastic difference equation with polynomial regression component

In this paper, strong consistency of least squares estimates of the coefficients of stochastic difference equations with polynomial regression components, has been established, using martingale arguments, under autoregressive, partially explosive and purely explosive situations. The asymptotic normality of these estimates have also been discussed in this paper.     

Kullback-Leibler divergence to evaluate posterior sensitivity to different priors for autoregressive time series models

In this paper we use the Kullback-Leibler divergence to measure the distance between the posteriors of the autoregressive (AR) model coefficients, aiming to evaluate mathematically the sensitivity of the coefficients posterior to different types of priors, i.e. Jeffreys’, g, and natural conjugate priors. In addition, we evaluate the impact of the posteriors distance in Bayesian estimates of mean and variance of the model coefficients by generating a large number of Monte Carlo simulations from the posteriors. Simulation study results show that the coefficients posterior is sensitive to prior distributions, and the posteriors distance has more influence on Bayesian estimates of variance than those of mean of the model coefficients. Same results are obtained from the application to real-world time series datasets.     

Generalizing the Bayesian Vector Autoregression Approach for Regional Interindustry Employment Forecasting

The Bayesian vector autoregression (BVAR) employment-forecasting approach is generalized using data for the state of Georgia. This study advances previous regional BVAR approaches by (a) incorporating regional input-output coefficients instead of national coefficients, (b) using the coefficients both to specify the prior means in one model and to weight the variances of a Minnesota-type prior in a second model, and (c) including final-demand effects and links to national and world economies. Out-of-sample forecasts produced by the generalized BVAR models are compared to forecasts produced from an autoregressive model, an unconstrained VAR model, and a Minnesota BVAR model.     

Lag order selection for an optimal autoregressive covariance matrix estimator

A good parametric spectral estimator requires an accurate estimate of the sum of AR coefficients, however a criterion which minimizes the innovation variance not necessarily yields the best spectral estimate. This paper develops an alternative information criterion considering the bias in the sum of the parameters for the autoregressive estimator of the spectral density at frequency zero.     

A study on geographically weighted spatial autoregression models with spatial autoregressive disturbances

Abstract

Spatial heterogeneity and correlation are both considered in the geographical weighted spatial autoregressive model. At present, this kind of model has aroused the attention of some scholars. For the estimation of the model, the existing research is based on the assumption that the error terms are independent and identically distributed. In this article we use a computationally simple procedure for estimating the model with spatially autoregressive disturbance terms, both the estimates of constant coefficients and variable coefficients are obtained. Finally, we give the large sample properties of the estimators under some ordinary conditions. In addition, application study of the estimation methods involved will be further explored in a separate study.