NON-NEGATIVE AUTOREGRESSIVE MODELS

Abstract. Consider a stationary non-negative autoregressive (AR) model given x _t = b ₁ x _t -1, +…+ b _p x _t-p + e _t , where the e _t are independent identically distributed non-negative variables and b ₁, …, b _p are non-negative parameters, and all the roots of the equation 1 – b ₁ u –…– b _p u ^p = 0 are outside the unit circle. The stationary solution of the above AR model is called a stationary non-negative AR process. Let x ₁, x ₂, … x _n be an example of a stationary non-negative AR process. Under very general conditions strongly consistent estimators of the AR parameters b ₁, b ₂, …, b _p have been studied. In this paper a new procedure is proposed to estimate not only b ₁, b ₂, …, b _p but also b _o which is the essential lower bound of the variable e _t . We shall show that the new estimators obtained using the new procedure are consistent estimators of b _o, b ₁, …, b _p under the weakest condition which guarantees that the stationary non-negative AR model has a stationary non-degenerative solution.     

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.     

NON-NEGATIVE AUTOREGRESSIVE PROCESSES

Abstract. Consider a stationary autoregressive process given by X _t = b ₁ X _{t -1}+…+ b _p X _t-p + Y _t , where the Y _t are independent identically distributed positive variables and b ₁,…, b _p are non-negative parameters. Let the variables X ₁,…, X _n be given. If p = 1 then it is known that b ₁*= min( X _t / X _{t -1}) is a strongly consistent estimator for b ₁ under very general conditions. In this paper the case p = 2 is analysed in detail. It is proved that min( X _t / X _{t -1})→ b ₁ almost surely (a.s.) and min( X _t / X _{t -2})→ b ₂+ b ₁² a.s. as n → 8. The convergence is very slow. Denote by b ₁* and b ₂* values of b ₁ and b ₂ respectively which maximize b ₂+ b ₂ under the conditions X _t - b ₁ X _{t -1}- b ₂ X _{t -2}≥ 0 for t = 3,…, n . We prove that b ₁* b ₁ and b ₂* b ₂ a.s. Simulations show that b ₁* and b ₂* are better than the least-squares estimators of the autoregressive coefficients when the distribution of Y _t is exponential.     

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.     

FAST LINEAR ESTIMATION METHODS FOR VECTOR AUTOREGRESSIVE MOVING-AVERAGE MODELS

Abstract. Three linear methods for estimating parameter values of vector auto-regressive moving-average (VARMA) models which are in general at least an order of magnitude faster than maximum likelihood estimation are developed in this paper. Simulation results for different model structures with varying numbers of component series and observations suggest that the accuracy of these procedures is in most cases comparable with maximum likelihood estimation. Procedures for estimating parameter standard error are also discussed and used for identification of nonzero elements in the VARMA polynomial structures. These methods can also be used to establish the order of the VARMA structure. We note, however, that the primary purpose of these estimates is to generate initial estimates for the nonzero parameters in order to reduce subsequent computational time of more efficient estimation procedures such as exact maximum likelihood.     

LARGE SAMPLE ANALYSIS OF AUTOREGRESSIVE MOVING-AVERAGE MODELS WITH ERRORS IN VARIABLES

Abstract. We consider estimation of parameters of an unobservable ARMA(p, q) process {U_t; t= 1,2,…} based on a set of n observables, X₁, …, X_n, where X_t=U_t, +ε_t, 1 ≤t≤n, it being assumed that {ε_t} is independent of {U_t}. We examine the asymptotic properties of these ARMA estimators under a set of weak regularity conditions on {ε_t}.     

THE ESTIMATION OF PARAMETERS FOR AUTOREGRESSIVE MOVING AVERAGE MODELS

Abstract. This paper provides some new and improved versions of an earlier procedure for the estimation of parameters for autoregressive moving average models suggested by the author (1979). Some numerical examples of the application of the procedure are also given.     

A GENERALIZED LEAST-SQUARES APPROACH FOR ESTIMATION OF AUTOREGRESSIVE MOVING-AVERAGE MODELS

Abstract. In this paper we present a generalized least-squares approach for estimating autoregressive moving-average (ARMA) models. Simulation results based on different model structures with varying numbers of observations are used to contrast the performance of our procedure with that of maximum likelihood estimates. Existing software packages can be utilized to derive these estimates.     

ESTIMATION OF AUTOREGRESSIVE PARAMETERS AND ORDER SELECTION FOR ARMA MODELS

Abstract. A quick algorithm for obtaining estimates of autoregressive parameters for autoregressive moving-average model is presented. The algorithm is recursive in the orders, and can be used for model selection by providing a criterion and a two-way table of certain partial covariances. Consistency and asymptotic normality of the estimates are shown.     

ASYMPTOTIC PROPERTIES OF SOME PRELIMINARY ESTIMATORS FOR AUTOREGRESSIVE MOVING AVERAGE TIME SERIES MODELS

Abstract. Some simple preliminary estimators for the coefficients of mixed autoregressive moving average time series models are considered. As the first step the estimators require the fitting of a long autoregression to the data. The first two methods of the paper are non-iterative and generally inefficient. The estimators are Yule-Walker type modifications of the least squares estimators of the coefficients in auxiliary linear regression models derived, respectively, for the coefficients of the long autoregression and for the coefficients of the corresponding long moving average approximation of the model. Both of these estimators are shown to be strongly consistent and their asymptotic distributions are derived. The asymptotic distributions are used in studying the loss in efficiency and in constructing the third estimator of the paper which is an asymptotically efficient two-step estimator. A numerical illustration of the third estimator with real data is given.     

ESTIMATION AND BLIND DECONVOLUTION OF AUTOREGRESSIVE SYSTEMS WITH NONSTATIONARY BINARY INPUTS

Abstract. The problem of parameter estimation and blind deconvolution of auto-regressive (AR) systems with independent nonstationary binary inputs is considered. The estimation procedure consists of applying a moving-average filter (equalizer) to the observed data and adjusting the parameters of the filter so as to minimize a criterion that measures the binariness of its output. The output sequence itself serves as an estimate of the unobservable binary input of the AR system. Without assuming stationarity of the inputs, it is shown that the proposed method produces a consistent estimator of the AR system not only in the sense of converging to the true parameter as the sample size increases, but also in the sense of attaining the true parameter of the AR system for a sufficiently large sample size. For noisy data, the estimation criterion is modified on the basis of an asymptotic analysis of the effect of the noise. It is shown that the modified criterion is also consistent (in the usual sense) and its variability depends upon the filtered noise. Some simulation results are presented to demonstrate the performance of the proposed method for parameter estimation as well as for blind deconvolution.     

SOME PROPERTIES OF AUTOREGRESSIVE ESTIMATES FOR PROCESSES WITH MIXED SPECTRA

Abstract. A formal justification for the use of the method of autoregressive spectral estimation for time series consisting of a sinusoidal signal in additive noise is given in this paper. The analytical properties of the autoregressive approximation to the generalized spectral density of the process are presented, and the operational characteristics of the statistical estimation procedure are discussed. In particular, strong convergence of the autoregressive parameters and the autoregressive transfer function approximation is shown.     

A PROCEDURE FOR OBTAINING M-ESTIMATES IN REGRESSION MODELS WITH SERIALLY DEPENDENT ERRORS

Abstract. Applications where error terms in a regression model display both non-normal and serially dependent behavior are considered. For the estimation of the parameters, an iterative Cochrane-Orcutt type M-estimator is proposed. A proof of convergence of the iterative procedure is given. In a simulation experiment, where the least absolute error criterion is applied, the performance of the estimator is tested and the theoretical convergence properties illustrated. In particular, the existence of multiple stationary points in an iterative process is discussed.     

ESTIMATION OF AUTOREGRESSIVE MOVING-AVERAGE ORDER GIVEN AN INFINITE NUMBER OF MODELS AND APPROXIMATION OF SPECTRAL DENSITIES

Abstract. A modification of the minimum Akaike information criterion (AIC) procedure (and of related procedures like the Bayesian information criterion (BIC)) for order estimation in autoregressive moving-average (ARMA) models is introduced. This procedure has the advantage that consistency for the order estimators obtained via this procedure can be established without restricting attention to only a finite number of models. The behaviour of these newly introduced order estimators is also analysed for the case when the data-generating process is not an ARMA process (transfer function/spectral density approximation). Furthermore, the behaviour of the order estimators obtained via minimization of BIC (or of related criteria) is investigated for a non-ARMA data-generating process.     

RESIDUAL AUTOCOVARIANCES AND UNIT ROOT TESTS BASED ON INSTRUMENTAL VARIABLE ESTIMATORS FROM TIME SERIES REGRESSION MODELS

Abstract. Hall (Testing for a unit root in the presence of moving average errors. Biometrika 76 (1989), 49–56; Joint hypothesis tests for a random walk based on instrumental variable estimators. J. Time Ser. Anal. 13 (1992), 29–45), Pantula and Hall (Testing for unit roots in autoregressive moving average models:an instrumental variable approach. J. Econometrics 48 (1991), 325–53) and Lee and Schmidt (Unit root tests based on instrumental variable estimation. Int. Econ. Rev. 39 (1994), 449–62) proposed instrumental variable (IV) based tests for a unit root in an ARMA(p+ 1, q) time series. To perform the tests it is essentially necessary to know (p, q) but in many cases this information is unknown. In practice a natural solution to this problem is to estimate (p, q) from the data using a strategy based on the residual autocovariances from the IV regression. In this paper we examine the properties of these residual autocovariances under various assumptions about the true nature of the time series. This analysis allows us to propose a model selection procedure which has desirable asymptotic and finite sample properties whether the time series is stationary or possesses a unit root. A sideproduct of our analysis is that we extend Box and Pierce's (Distribution of residual autocorrelations in autoregressive integrated moving average time series models. J. Am. Statist. Assoc. 65 (1970), 1509–26) analysis of the least squares residual autocorrelations to the residual autocovariances from IV regressions.     

VECTOR AUTOREGRESSIVE MODELS WITH UNIT ROOTS AND REDUCED RANK STRUCTURE:ESTIMATION. LIKELIHOOD RATIO TEST, AND FORECASTING

Abstract. The nonstationary multivariate autoregressive (AR) model Φ ( L ) Y _t =ε _t is considered for an m -dimensional process { Y _t }, where it is assumed that det {Φ( L )}= 0 has d < m unit roots and all other roots are outside the unit circle, and also that rank {Φ(1)}= r ( r = m – d ). Limiting distribution results obtained by Ahn and Reinsel for the least-squares and the Gaussian reduced rank (unit roots imposed) estimators for this AR model are extended to a model where the AR parameters possess additional structure such as nested reduced rank, and based on these results the asymptotic distribution of the likelihood ratio test statistic for testing the number d of unit roots is obtained. An analysis of three US monthly interest rate series is presented to illustrate the testing and estimation procedures. A small simulation study is also performed to examine the finite-sample properties of the likelihood ratio test and the prediction performance of models which impose different numbers of unit roots.     

CONTROL AND ESTIMATION FOR PERTURBED NONLINEAR PROCESSES WITH DEADTIME

Feedback linearization techniques are used to deal with the nonlinear controller designs which have attracted many researchers' attention in recent years. The approach has been applied successfully to solve a number of practical nonlinear control problems, but typically requires on-line full state measurement which is usually not the case in real chemical process industries. In this paper, we address the problem of synthesizing nonlinear state feedback controllers for time-delay nonlinear systems which are perturbed by disturbances. On-line estimation of the unmeasurable disturbances and unavailable state variables is introduced to facilitate the implementation of coordinate transformations and state feedback and prediction. Two kinds of dynamic compensators are then proposed to handle the process deadtime. Finally numerical simulations in a CSTR example demonstrate the promising performance of the overall nonlinear control structure in disturbance rejection.     

试差矩阵熬及其在复杂反应动力学研究中的应用

本文提出了一种研究高维拟一级复杂反应动力学的新方法——试差矩阵法.该方法充分运用有效的数学方法和计算技术,以减少实验工作量,并能大大减少待定参数个数,提高估值精度.研究表明,该方法具有良好的方法统计特性,能耐受较大的实验误差。本文还提出了新的八碳芳烃临氢异构化和六碳组份重整体系的反应网络,应用试差矩阵法成功地研究了这两个体系的双曲型动力学模型,求取了各模型参数,模型拟合值与实验值能很好地吻合.     

ESTIMATION OF THE ORDER OF A MOVING AVERAGE MODEL FROM AUTOREGRESSIVE AND WINDOW ESTIMATES OF THE INVERSE CORRELATION FUNCTION

Abstract. The autoregressive and window estimates of the inverse correlation function are used for estimating the order of a finite moving average process by using criteria similar to the FPEα criterion of Bhansali and Downham (1977). The asymptotic distribution of the estimates is derived. Their finite sample behaviour is examined by means of a simulation study.     

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.