TRANSFORMED POLYNOMIALS FOR NONLINEAR AUTOREGRESSIVE MODELS OF THE CONDITIONAL MEAN

This article proposes a flexible set of transformed polynomial functions for modelling the conditional mean of autoregressive processes. These functions enjoy the same approximation theoretic properties of polynomials and, at the same time, ensure that the process is strictly stationary, is ergodic, has fading memory and has bounded unconditional moments. The consistency and asymptotic normality of the least‐squares estimator is easily obtained as a result. A Monte Carlo study provides evidence of good finite sample properties. Applications in empirical time‐series modelling, structural economics and structural engineering problems show the usefulness of transformed polynomials in a wide range of settings.     

BATCH-TO-BATCH OPTIMAL CONTROL OF BATCH PROCESSES BASED ON RECURSIVELY UPDATED NONLINEAR PARTIAL LEAST SQUARES MODELS

A batch-to-batch optimal control approach for batch processes based on batch-wise updated nonlinear partial least squares (NLPLS) models is presented in this article. To overcome the difficulty in developing mechanistic models for batch/semi-batch processes, a NLPLS model is developed to predict the final product quality from the batch control profile. Mismatch between the NLPLS model and the actual plant often exists due to low-quality training data or variations in process operating conditions. Thus, the optimal control profile calculated from a fixed NLPLS model may not be optimal when applied to the actual plant. To address this problem, a recursive nonlinear PLS (RNPLS) algorithm is proposed to update the NLPLS model using the information newly obtained after each batch run. The proposed algorithm is computationally efficient in that it updates the model using the current model parameters and data from the current batch. Then the new optimal control profile is recalculated from the updated model and implemented on the next batch. The procedure is repeated from batch to batch and, usually after several batches, the control profile will converge to the optimal one. The effectiveness of this method is demonstrated on a simulated batch polymerization process. Simulation results show that the proposed method achieves good performance, and the optimization with the proposed NLPLS model is more effective and stable than that with a batch-wise updated linear PLS model.     

BATCH-TO-BATCH OPTIMAL CONTROL OF BATCH PROCESSES BASED ON RECURSIVELY UPDATED NONLINEAR PARTIAL LEAST SQUARES MODELS

A batch-to-batch optimal control approach for batch processes based on batch-wise updated nonlinear partial least squares (NLPLS) models is presented in this article. To overcome the difficulty in developing mechanistic models for batch/semi-batch processes, a NLPLS model is developed to predict the final product quality from the batch control profile. Mismatch between the NLPLS model and the actual plant often exists due to low-quality training data or variations in process operating conditions. Thus, the optimal control profile calculated from a fixed NLPLS model may not be optimal when applied to the actual plant. To address this problem, a recursive nonlinear PLS (RNPLS) algorithm is proposed to update the NLPLS model using the information newly obtained after each batch run. The proposed algorithm is computationally efficient in that it updates the model using the current model parameters and data from the current batch. Then the new optimal control profile is recalculated from the updated model and implemented on the next batch. The procedure is repeated from batch to batch and, usually after several batches, the control profile will converge to the optimal one. The effectiveness of this method is demonstrated on a simulated batch polymerization process. Simulation results show that the proposed method achieves good performance, and the optimization with the proposed NLPLS model is more effective and stable than that with a batch-wise updated linear PLS model.     

BAYESIAN THRESHOLD AUTOREGRESSIVE MODELS FOR NONLINEAR TIME SERIES

Abstract. This paper provides a Bayesian approach to statistical inference in the threshold autoregressive model for time series. The exact posterior distribution of the delay and threshold parameters is derived, as is the multi-step-ahead predictive density. The proposed methods are applied to the Wolfe's sunspot and Canadian lynx data sets.     

BAYESIAN INFERENCE OF THRESHOLD AUTOREGRESSIVE MODELS

Abstract. The study of non-linear time series has attracted much attention in recent years. Among the models proposed, the threshold autoregressive (TAR) model and bilinear model are perhaps the most popular ones in the literature. However, the TAR model has not been widely used in practice due to the difficulty in identifying the threshold variable and in estimating the associated threshold value. The main focal point of this paper is a Bayesian analysis of the TAR model with two regimes. The desired marginal posterior densities of the threshold value and other parameters are obtained via the Gibbs sampler. This approach avoids sophisticated analytical and numerical multiple integration. It also provides an estimate of the threshold value directly without resorting to a subjective choice from various scatterplots. We illustrate the proposed methodology by using simulation experiments and analysis of a real data set.     

A NOTE ON USING THRESHOLD AUTOREGRESSIVE MODELS FOR MULTI-STEP-AHEAD PREDICTION OF CYCLICAL DATA

Abstract. We give a brief account of how the class of threshold autoregressive time series models may be used to make short, medium and long range predictions of cyclical data.     

LEAST SQUARES ESTIMATES AND ORDER DETERMINATION PROCEDURES FOR AUTOREGRESSIVE PROCESSES WITH A TIME DEPENDENT VARIANCE

Abstract. We study nonstationary autoregressive processes, where the variance of the generating white noise process is allowed to depend on time. It is shown that ordinary least squares estimates are strongly consistent and with a proper scaling factor asymptotically normal, but, as can be expected, they are not efficient. Furthermore, AIC type order determination criteria, used as if the underlying process is stationary, are consistent, whereas identification of order in terms of the partial autocorrelation function may lead one astray.     

GENERALIZED LEAST SQUARES ESTIMATION OF ARMA MODELS

Abstract. Two multistage methods for estimating scalar ARMA models are investigated. Both estimate innovations using an autoregression; these are used to obtain initial ARMA parameter estimates by regression and finally the initial estimates are refined by generalized least squares or nonlinear optimization to achieve efficiency. We provide a proof of the generalized least squares procedure.     

非负约束总体最小二乘问题

在第一部分中，提出了非负约束总体最小二乘问题，即利用总体最小二乘思想处理问题Ａｘ＝ｂｘ≥０｛；在第二部分中给出了非负约束最小二乘问题解集的结构和扰动分析，并对在实际计算中要用到的一个参数ｐ提出了可行的方法     

ORDER IDENTIFICATION IN MISSPECIFIED AUTOREGRESSIVE TIME SERIES MODELS

Abstract. In this paper we examine the properties of Akaike's (Fitting autoregressive models for prediction, Ann. Inst. Statist. Math. 21 (1969), 243–47) and Hannan and Quinn's (The determination of the order of an autoregression, J. R. Statist. Soc., Ser. B 41 (1979), 190–95) information criteria in stationary autoregressive time series models when the true order of the process is greater than the maximum considered by the analyst. The limiting distributions of the estimated orders are derived and the implications of these results for model building are considered.     

YULE-WALKER ESTIMATES FOR CONTINUOUS-TIME AUTOREGRESSIVE MODELS

Abstract. I consider continuous-time autoregressive processes of order p and develop estimators of the model parameters based on Yule-Walker type equations. For continuously recorded data, it is shown that these estimators are least squares estimators and have the same asymptotic distribution as maximum likelihood estimators.
In practice, though, data can only be observed discretely. For discrete data, I consider approximations to the continuous-time estimators. It is shown that some of these discrete-time estimators are asymptotically biased. Alternative estimators based on the autocovariance function are suggested. These are asymptotically unbiased and are a fast alternative to the maximum likelihood estimators described by Jones. They may also be used as starting values for maximum likelihood estimation.     

ON ESTIMATING THRESHOLDS IN AUTOREGRESSIVE MODELS

Abstract. The problem of estimating the threshold parameter, i.e., the change point, of a threshold autoregressive model is studied. By introducing smoothness into the model, sampling properties of the conditional least-squares estimate may be obtained. Artificial and real data are used for illustrations.     

UNIFIED INTERVAL ESTIMATION FOR RANDOM COEFFICIENT AUTOREGRESSIVE MODELS

The consistency of the quasi‐maximum likelihood estimator for random coefficient autoregressive models requires that the coefficient be a non‐degenerate random variable. In this article, we propose empirical likelihood methods based on weighted‐score equations to construct a confidence interval for the coefficient. We do not need to distinguish whether the coefficient is random or deterministic and whether the process is stationary or non‐stationary, and we present two classes of equations depending on whether a constant trend is included in the model. A simulation study confirms the good finite‐sample behaviour of our resulting empirical likelihood‐based confidence intervals. We also apply our methods to study US macroeconomic data.     

ON THE RELATIONSHIP BETWEEN GENERALIZED LEAST SQUARES AND GAUSSIAN ESTIMATION OF VECTOR ARMA MODELS

Abstract. This paper investigates theoretical aspects of the relationship between the generalized least squares and Gaussian estimation schemes for vector autoregressive moving-average models. The asymptotic convergence of the generalized least squares estimator to the Gaussian estimator is established and an alternative numerical method for implementing the generalized least squares scheme is proposed. Finally, some simulation results are presented to illustrate the theory.     

ON THE CONSISTENCY OF LEAST SQUARES ESTIMATORS FOR A THRESHOLD AR(1) MODEL

Abstract. For the SETAR (2; 1,1) model

where {a_t(i)} are i.i.d. random variables with mean 0 and variance σ²(i), i = 1,2, and {a_t(l)} is independent of {a_t(2)}, we consider estimators of φ₁, φ ₂ and r which minimize weighted sums of the sum of squares functions for σ²(1) and σ²(2). These include as a special case the usual least squares estimators. It is shown that the usual least squares estimators of φ₁, φ₂ and r are consistent. If σ²(1) ≠σ²(2) conditions on the weights are found under which the estimators of r and φ₁ or φ₂ are not consistent.     

PROPERTIES OF PREDICTORS FOR MULTIVARIATE AUTOREGRESSIVE MODELS WITH ESTIMATED PARAMETERS

Abstract. The k -dimensional p th-order autoregressive processes { Y _t} that are either stationary or have one unstable or explosive root are considered. The properties of the s -periods-ahead predictor Ŷ _n+s, obtained by replacing the unknown parameters in the expression for the best linear predictor YT_n+s by their least-squares estimators, is shown to be asymptotically equivalent to the optimal predictor except in the explosive case. An expression for the mean squared error of Ŷ _n+s is derived through terms of order n ^-1 for normal stationary processes when the parameters are estimated from the realization to be predicted. In addition, small-sample properties of Ŷ _n+s are investigated.     

BOOTSTRAP FOR RANDOM COEFFICIENT AUTOREGRESSIVE MODELS

In this paper, we deal with autoregressive processes with random coefficients. We propose a least‐squares estimator for the fourth‐order moments of both the innovation and disturbance noises and state its consistency. The main theme of the paper is the development of bootstrap procedures for the autoregressive parameter. We show how to obtain approximative residuals for the process even though the standard method for autoregressive processes does not work in this context since one then would obtain convoluted residuals of the innovation and disturbance noises. These ideas lead to a modification of the classical residual bootstrap for autoregressive processes. The consistency of the bootstrap procedure is established. Further, the estimators proposed in the first part are used to form two wild bootstrap modifications. Finally, the performances of the three bootstrap procedures are explored by a simulation study and compared with each other.     

THE ESTIMATION OF RANDOM COEFFICIENT AUTOREGRESSIVE MODELS. I

Abstract. This paper is concerned with autoregressive models in which the coefficients are assumed to be not constant but subject to random perturbations so that we are considering a class of random coefficient autoregressive models. By means of a two stage regression procedure estimates of the unknown parameters of these models are obtained. The estimates are shown to be strongly consistent and to satisfy a central limit theorem. A number of Monte Carlo experiments was carried out to illustrate the estimation procedure and their results are reported.