bluuebunny/crossref_metadata_2025 huggingface dataset:
A NOTE ON THE THRESHOLD AR(1) MODEL WITH CAUCHY INNOVATIONS
Authors: Anděl Jiři , Bartoň Tomáŝ
Published: May 2008

Abstract. A threshold autoregressive process of the first order with one threshold r and with Cauchy innovations is investigated in the paper. An explicit formula for the stationary density of such process is derived for the special case that r = 0 and that the autoregressive parameters have the same absolute value.
ASYMPTOTIC PROPERTIES OF SOME PRELIMINARY ESTIMATORS FOR AUTOREGRESSIVE MOVING AVERAGE TIME SERIES MODELS
Authors: Saikkonen Pentti
Published: May 2008

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.
WHY DO NONINVERTIBLE ESTIMATED MOVING AVERAGES OCCUR?*
Authors: Anderson T. W. , Takemura Akimichi
Published: May 2008

Abstract. The positive probability that an estimated moving average process is noninvertible is studied for maximum likelihood estimation of a university process. Upper and lower bounds for the probability in the first‐order case are obtained as well as limits when the sample size tends to infinity. Higher order moving average models and autoregressive moving average models are also treated.
A PROCEDURE FOR OBTAINING M‐ESTIMATES IN REGRESSION MODELS WITH SERIALLY DEPENDENT ERRORS
Authors: Coursey Don , Nyquist Hans
Published: May 2008

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.
TEMPORAL AGGREGATION IN THE ARIMA PROCESS
Authors: Stram Daniel O. , Wei William W. S.
Published: May 2008

Abstract. The effect of temporal aggregation on ARIMA models is investigated. The paper discusses the change of model form resulting from aggregation. For the IMA model it is noted that reduction of model order may occur, due to aggregation, which takes an arbitrarily high order IMA (d, q) process to an IMA (d, 0) process for the aggregates. For the AR process, we derive the exact order for the aggregate model and show that aggregation of an AR (p) series does not necessarily produce an ARMA (p, q) aggregate series as has been suggested in the literature. In particular, the AR order of AR (p) and ARIMA (p, d, q) can be reduced upon aggregation. The paper gives the general conditions under which this reduction of order may occur.
AN AUTOMATIC NON‐PARAMETRIC SPECTRUM ESTIMATOR
Authors: Cameron M. A.
Published: May 2008

Abstract.An estimator of the spectral density of a stationary process is obtained by approximating it with a step function. The positions of the level changes are determined using partitioning algorithms. The algorithms are stopped by using criteria such as AIC. In examples, the resulting estimates are shown to be good representations of the true spectra in most circumstances, even when the series are fairly short. In particular the estimates highlight spikes in the spectral density that are caused by periodicities.
EXACT LEAST SQUARES MULTI‐STEP PREDICTION FROM NONLINEAR AUTOREGRESSIVE MODELS
Authors: Pemberton John.
Published: May 2008

Abstract.We obtain an integral equation recurrence relation for the optimal least squares predictor of a nonlinear autoregressive time series model. Numerical solutions are given for a first order threshold model up to three steps ahead.
WALSH‐FOURIER ANALYSIS OF DISCRETE‐VALUED TIME SERIES
Authors: Stoffer David S.
Published: May 2008

Abstract.An approach to the analyses of discrete‐valued time series is discussed. The analyses are accomplished in the spectral domain using the Walsh‐Fourier transform which is based on Walsh functions. This approach will enable an investigator of discrete systems to analyse the data in terms of square waveforms and sequency rather than sine waves and frequency.

We develop a general signal‐plus‐noise type model for discrete‐valued time series in which Walsh‐Fourier spectral analysis is of interest. We consider the problems of detecting whether a common signal exists in repeated measures on discrete‐valued time series and in discrete‐valued processes collected in an experimental design. We show that these models may depend on unknown regression parameters and we develop consistent estimates of these parameters based on the finite Walsh‐Fourier transform. Applications to certain Markov models are given; however, the methods presented also apply to non‐Markov cases.
A METHOD FOR GENERATING INDEPENDENT REALIZATIONS OF A MULTIVARIATE NORMAL STATIONARY AND INVERTIBLE ARMA(p, q) PROCESS
Authors: Barone Piero
Published: May 2008

Abstract. A method for generating finite independent realizations of a normal multivariate stationary ARMA(p, q) process is proposed. It is based on an AR(1) representation of an ARMA(p, q) process allowing for an exact generation of the initial values of the simulation algorithm. Input facilities are supplied in order to assure stationarity and invertibility of the considered process.
MULTIPLE BILINEAR TIME SERIES MODELS
Authors: Stensholt Boonchai K. , Tjøstheim Dag
Published: May 2008

Abstract.A definition of multiple bilinear time series models is given. Sufficient conditions are obtained for the existence of strictly stationary solutions conforming to the model, and a brief discussion of the first and second order structure is included.
NEAREST‐NEIGHBOUR METHODS FOR TIME SERIES ANALYSIS
Authors: Yakowitz S.
Published: May 2008

Abstract. The nearest‐neighbour method, because of its intuitively appealing nature and competitive theoretical properties, deserves consideration in time‐series applications akin to attention it has received lately in the i.i.d. case. Here it is shown that as a nonparametric regression device, like the kernel method, under the G2 mixing assumption, it converges in quadratic mean at the Stone‐optimal rate. In the closing sections, our methodology is extended to a broader pattern‐recognition context, and applied to hydrologic data.
A TEST FOR NON‐LINEARITY OF PREDICTION IN TIME SERIES
Authors: O'Brien C.
Published: May 2008

Abstract.A test for non‐linearity of prediction in time series is suggested; the test makes use of the partial correlation between a series and its square, eliminating linear terms. The distribution of partial correlation estimates is considered both theoretically and by simulation. Extensions of the test to include partial correlations with other nonlinear functions of the observed data are suggested. The nature of optimum predictors for some particular non‐linear time series is described, and the performance of the test when used with simulated non‐linear series is investigated.
ON THE THIRD‐ORDER MOMENT STRUCTURE AND BISPECTRAL ANALYSIS OF SOME BILINEAR TIME SERIES
Authors: Gabr M. M.
Published: May 2008

For the bilinear time series modelXt=βXt‐ket‐t+et, k > l, k = l andk<lformulae for the third‐order theoretical moments and an expression for the bispectral density function are obtained. These results can be used to distinguish between bilinear models and white noise and, in general, linear models. Furthermore, they give an indication of the type combination (k, l) in the above model. The modulus of the bispectral density function of the above bilinear time series model for different combinations of (k, l) and values ofβare computed and the properties are studied.
TIME DELAY ESTIMATION
Authors: Hannan E. J. , Thomson P. J.
Published: May 2008

The problem considered is that of estimating the delays that arise when a waveform propagates across an array. Problems that arise in previous estimation techniques when the signal‐to‐noise ratio is low are reduced by introducing an improved weighting of frequencies into the formulae and by using finite parameter models to estimate this weight function. The virtues of the various procedures are checked by simulations.
PREDICTION ERROR OF MULTIVARIATE TIME SERIES WITH MIS‐SPECIFIED MODELS
Authors: Lewis Richard A. , Reinsel Gregory C.
Published: May 2008

We consider the situation in which an incorrectly specified autoregressive moving‐average model is used to predict future values of a stationary multivariate time series. The use of an incorrect model for prediction results in an increase in mean‐square prediction error over that of the optimal predictor, and an expression for this increase is first given for fixed values of the parameters in the incorrect model. For the case in which the incorrect model is an autoregression, we also take into account parameter estimation error by first deriving the asymptotic distribution and limiting moment properties of the least‐squares estimator of the parameters in the mis‐specified model. An asymptotic approximation to the increase in mean‐square prediction error is then obtained. Numerical examples are provided to demonstrate the accuracy of the asymptotic approximation in finite samples. Our results are consistent with those obtained in the univariate case, indicating that fitted autoregressions of high order can yield substantially sub‐optimal forecasts.
AN EXACT TEST FOR A STOCHASTIC COEFFICIENT IN A TIME SERIES REGRESSION MODEL
Authors: Shively Thomas S.
Published: May 2008

An exact small‐sample test is developed for testing the hypothesis that a regression coefficient is constant against the alternative that it is generated by a random walk process. The test is mean‐ and scale‐invariant and approximates the most powerful invariant test against any specific alternative. It thus outperforms tests previously given in the literature. Computationally efficient algorithms are given to compute the test statistic and its distribution using a modified version of the Kalman filter.
ESTIMATION FOR NON‐LINEAR TIME SERIES MODELS USING ESTIMATING EQUATIONS
Authors: Thavaneswaran A. , Abraham B.
Published: May 2008

Godambe's (1985) theorem on optimal estimating equations for stochastic processes is applied to non‐linear time series estimation problems. Examples are considered from the usual classes of non‐linear time series models. A recursive estimation procedure based on optimal estimating equations is provided. It is also shown that pre‐filtered estimates can be used to obtain the optimal estimate from a non‐linear state‐space model.
TESTING SEPARATE TIME SERIES MODELS
Authors: McAleer Michael , McKenzie C. R. , Hall A. D.
Published: May 2008

Abstract.We develop simple procedures for testing the adequacy of separate time series models. The test statistics may be calculated using auxiliary regressions that are very similar to those used for calculating Lagrange multiplier test statistics. While the separate tests are designed to yield high power against separate alternatives, they are also powerful as diagnostic checks against a range of inappropriate alternatives. The small‐sample properties of the separate and Lagrange multiplier tests are compared on the basis of a Monte Carlo experiment. In these experiments it is found that the separate tests are frequently more powerful than the Lagrange multiplier tests, even for alternatives against which the latter are asymptotically optimal.
STATIONARITY OF THE SOLUTION OF Xt= AtXt‐1+εtAND ANALYSIS OF NON‐GAUSSIAN DEPENDENT RANDOM VARIABLES
Authors: POURAHMADI MOHSEN
Published: May 2008

Abstract.We give general and concrete conditions in terms of the coefficient (stochastic) process {At} so that the (doubly) stochastic difference equation Xt= AtXt‐1+εthas a second‐order strictly stationary solution. It turns out that by choosing {At} and the “innovation” process {εt} properly, a host of stationary processes with non‐Gaussian marginals and long‐range dependence can be generated using this difference equation. Examples of such nowGaussian marginals include exponential, mixed exponential, gamma, geometric, etc. When {At} is a binary time series, the conditional least‐squares estimator of the parameters of this model is the same as those of the parameters of a Galton‐Watson branching process with immigration.
A NOTE ON AIC ORDER DETERMINATION FOR MULTIVARIATE AUTOREGRESSIONS
Authors: Quinn B. G.
Published: May 2008

Abstract.Formulae are obtained for the limiting probability of overestimating the order of a multivariate autoregression when using AIC‐type procedures.
ON THE EXISTENCE OF THE STATIONARY AND ERGODIC NEAR(p) MODEL
Authors: Chan Kung‐sik
Published: May 2008

Abstract.The NEAR(2) model proposed by Lawrance and Lewis in 1985 is generalized to the NEAR(p) model. A necessary and sufficient condition for the existence of an ‘innovation’ sequence and a stationary ergodic process satisfying the NEAR(p) model is derived. It is shown that the ‘innovation’ sequence is distributed as a mixture of exponentials.
PROPERTIES OF PREDICTORS FOR MULTIVARIATE AUTOREGRESSIVE MODELS WITH ESTIMATED PARAMETERS
Authors: Samaranayake V. A. , Hasza David P.
Published: May 2008

Abstract.Thek‐dimensionalpth‐order autoregressive processes {Yt} that are either stationary or have one unstable or explosive root are considered. The properties of thes‐periods‐ahead predictorŶn+s, obtained by replacing the unknown parameters in the expression for the best linear predictor YTn+sby 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 ofŶn+sis derived through terms of ordern‐1for normal stationary processes when the parameters are estimated from the realization to be predicted. In addition, small‐sample properties ofŶn+sare investigated.
YULE‐WALKER TYPE DIFFERENCE EQUATIONS FOR HIGHER‐ORDER MOMENTS AND CUMULANTS FOR BILINEAR TIME SERIES MODELS
Authors: Sesay S. A. O. , Rao T. Subba
Published: May 2008

Abstract.In this paper we obtain difference equations for the third‐ and fourth‐order lagged moments and cumulants when the time series {Xt} satisfies a bilinear model and is stationary up to fourth order. These equations are similar to the well‐known Yule‐Walker equations which are available for linear time series models.
A SIMPLE CONDITION FOR THE EXISTENCE OF SOME STATIONARY BILINEAR TIME SERIES
Authors: Liu Jian
Published: May 2008

Abstract.A sufficient condition is derived for the existence of a strictly stationary solution of some bilinear time series which may have infinite variance innovations. This condition is equivalent to the condition that a polynomial of degreerhas no zeros within the unit circle. In the special case when the innovations have finite variance, the computational effort involved in checking this condition is significantly reduced compared with checking the stationarity conditions given by Bhaskara Raoet al.and Liu and Brockwell which requires a knowledge of the maximum eigenvalue in the absolute value of anr2xr2matrix.
CONTRIBUTIONS TO EVOLUTIONARY SPECTRAL THEORY
Authors: Mélard Guy , Schutter Annie Herteleer‐de
Published: May 2008

Abstract.The purpose of this paper is to discuss several fundamental issues in the theory of time‐dependent spectra for univariate and multivariate non‐stationary processes. The general framework is provided by Priestley's evolutionary spectral theory which is based on a family of stochastic integral representations. A particular spectral density function can be obtained from the Wold—Cramér decomposition, as illustrated by several examples. It is shown why the coherence is time invariant in the evolutionary theory and how the theory can be generalized so that the coherence becomes time dependent. Statistical estimation of the spectrum is also considered. An improved upper bound for the bias due to non‐stationarity is obtained which does not rely on the characteristic width of the process. The results obtained in the paper are illustrated using time series simulated from an evolving bivariate autoregressive moving‐average process of order (1, 1) with a highly time‐varying coherence.
THE ESTIMATION OF THE ORDER OF AN AUTOREGRESSION USING RECURSIVE RESIDUALS AND CROSS‐VALIDATION
Authors: Kavalieris L.
Published: May 2008

Abstract.Several criteria for the estimation of the order of an autoregressive representation of a stationary time series are examined. There need not be a true finite‐order autoregression model for the data, so that the purpose of model identification is to obtain an adequate representation of the data. It is proved that minimizing the sum of squares of recursive residuals (the ‘predictive minimizing description length’) is equivalent to minimizing BIC. The equivalence between the cross‐validation and Akaike information criterion methods of autoregressive modelling is also established.
FAST LINEAR ESTIMATION METHODS FOR VECTOR AUTOREGRESSIVE MOVING‐AVERAGE MODELS
Authors: Koreisha Sergio , Pukkila Tarmo
Published: May 2008

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.
ON THE EXISTENCE OF A GENERAL MULTIPLE BILINEAR TIME SERIES
Authors: Liu Jian
Published: May 2008

Abstract.A sufficient condition is derived for the existence of a strictly stationary solution of the general multiple bilinear time series equations (without assuming subdiagonality). The condition is shown to reduce to the condition of Stensholt and Tjostheim in the special case which they consider. Under this condition a solution is constructed which is shown to be casual in the sense we define, strictly stationary and ergodic. It is moreover the unique causal solution and the unique stationary solution of the defining equations. In the special case when the defining equations contain no non‐linear terms, i.e. the multiple autoregressive moving‐average (ARMA) model. the condition given here reduces to the well‐known sufficient condition for the existence of a casual stationary solution.
A SIMULATION METHOD FOR NON‐NORMAL RANDOM PROCESSES
Authors: Xinjian Gu , Yiyun Huang
Published: May 2008

Abstract. In 1980 Davies, Spedding, and Watson presented a simulation method for autoregressive moving‐average (ARMA) processes with non‐normal residuals using Johnson transformations. In this paper we present another simulation method by which the desired non‐normal series are directly obtained from normal series with the required correlation. The new method has some advantages over that of Davies et al.
ALARM CHARACTERISTICS FOR A FLOOD WARNING SYSTEM WITH DETERMINISTIC COMPONENTS
Authors: Beckman Stig‐Inge , Holst Jan , Lindgren Georg
Published: May 2008

Abstract. A method for evaluating a predictor‐based alarm system is studied in this paper. The predictor is composed of a ‘deterministic’ component reflecting external information and a statistically based component for the error between the measurements and the external predictor. The aim of the predictor study is twofold:it is a means of interpreting the connections between the alarm and the catastrophe, and it can be used to select suitable alarm levels. As an application, the performance of a water‐level predictor as part of a flood warning system has been evaluated. The result of this analysis shows that an alarm system which operates when the predictor reaches a certain level will tend to give either too many alarms or alarms that are out of phase with the catastrophe.
A CLASS OF MODELS FOR NON‐NORMAL TIME SERIES
Authors: Janacek G. J. , Swift A. L.
Published: May 2008

Abstract.We propose a model for non‐normal times series in which we regard the series as an instantaneous transformation of an underlying ‘generating’ series which is normal. We describe a procedure which simultaneously estimates both the transformation to normality and the time series structure of the underlying series. This model has several advantages; in particular several alternative types of forecast can easily be calculated. The relative merits of these forecasts are considered.
NUMERICAL EVALUATION OF DISTRIBUTIONS IN NON‐LINEAR AUTOREGRESSION
Authors: Moeanaddin R. , Tong Howell
Published: May 2008

Abstract. We use the Chapman‐Kolmogorov formula as a recursive relation for computing the m‐step‐ahead conditional density of a non‐linear autoregressive model. We approximate the stationary marginal probability density function of the model by the m‐step‐ahead conditional density for sufficiently large m. An advantage of our method is its simple implementation; only one NAG subroutine is needed. We have also studied the advantage of incorporating the matrix‐squaring procedure.
SUBSET THRESHOLD AUTOREGRESSION WITH APPLICATIONS
Authors: Thanoon B. Y.
Published: May 2008

Abstract.In this paper we define subset threshold autoregressive models and suggest a simple algorithm for fitting them. The suggested algorithm is applied to simulated as well as real data. Two well‐known time series are studied:the Canadian lynx data and Wolf's sunspot numbers. The fitted models are compared with the threshold models of Tong and Lim and with the subset bilinear models of Gabr and Subba Rao. Some statistical properties of these models are also studied. An examination of forecasting performance is included.
CROSS‐VALIDATORY CHOICE OF A SPECTRUM ESTIMATE AND ITS CONNECTIONS WITH AIC
Authors: Hurvich Clifford M. , Beltrato Kaizô I.
Published: May 2008

Abstract. This paper concerns the use of a generalized version of the cross‐validated log likelihood criterion (CVLL) for selecting a spectrum estimator from an arbitrary class of candidate estimators. It is shown that CVLL is asymptotically equivalent to the expected Kullback‐Leibler information of the candidate estimator. The Akaike information criterion (AIC) is also asymptotically equivalent to Kullback‐Leibler information, but the applicability of AIC is limited to parametric estimators. Thus CVLL can be viewed as a cross‐validatory generalization of AIC. Monte Carlo results show that CVLL is able to provide an effective choice from a class of candidates which simultaneously includes autoregressive and classical smoothed periodogram estimators. To save computation time, CVLL can be evaluated only for the classical estimators while the computationally more efficient AIC is evaluated for the parametric estimators. The criterion values are all directly comparable in this case. As an additional computation‐saving device, a non‐cross‐validatory version of CVLL for classical estimators is proposed and studied.
DIFFERENTIAL GEOMETRY OF ARMA MODELS
Authors: Ravishanker Nalini , Melnick Edward L. , Tsai Chih‐Ling
Published: May 2008

Abstract.A general approach for the development of a statistical inference on autoregressive moving‐average (ARMA) models is presented based on geometric arguments. ARMA models are characterized as members of the curved exponential family. Geometric properties of ARMA models are computed and used to suggest parameter transformations that satisfy predetermined properties. In particular, the effect on the asymptotic bias of the maximum likelihood estimator of model parameters is illustrated. Hypothesis testing of parameters is discussed through the application of a modified form of the likelihood ratio test statistic.
STATIONARY PROCESSES WITH A FINITE NUMBER OF NON‐ZERO CANONICAL CORRELATIONS BETWEEN FUTURE AND PAST
Authors: Vaninskii K. L. , Yaglom A. M.
Published: May 2008

Abstract. The spectral densities f(γ) are determined for stationary random processes X(t) with continuous time which have the property that the number of non‐zero canonical correlations between the past X(t) (t≤ 0) (more accurately the present and the past) and the future X(t) (t≥θ) is finite (equal to N) at any θ≥τ for some r > 0. A method for finding all the corresponding canonical correlations P1, …, PN and the canonical variables X1‐, …, XN‐ and X1+, …, XN+ is given. Similar results related to processes X(t) with discrete (integral) time are briefly considered.
THE INTEGER‐VALUED AUTOREGRESSIVE (INAR(p)) MODEL
Authors: Jin‐Guan Du , Yuan Li
Published: May 2008

Abstract.The integer‐valued autoregressive (INAR) model with lagpdependence is discussed. The existence and ergodic property of the INAR model are proved. It is shown that the correlation structure of the INAR model is similar to that of the continuous‐valued autoregressive (AR) process, and the stationary conditions of INAR and AR processes are also the same.
PROFILE SUMMARIES FOR ARIMA TIME SERIES MODEL PARAMETERS
Authors: Lam Raymond L. H. , Watts Donald G.
Published: May 2008

Abstract. Autoregressive intergrated moving average (ARIMA) times series models are nonlinear in the parameters and so summarizing the inferential results for such models can be difficult. A common approach is to present parameter joint and marginal inference regions based on the linear approximation, and although such approximate regions are easy to calculate, it is not known generally whether they approximate the true regions adequately. In this paper we present exact approaches for summarizing the inferential results for time series model parameters, called profile t and profile trace plots, which are based on the work of Bates and Watts. Calculations for the profile plots are simple and can be used to determine exact regions, and so can be used to assess the accuracy of linear approximation regions. In addition to developing the profile plots for time series models, the main finding of the paper is that, for ARIMA model parameters, linear approximation regions are very satisfactory except when a parameter estimate is within about two standard errors of the stationarity or invertibility region boundary.
STATIONARITY AND CENTRAL LIMIT THEOREM ASSOCIATED WITH BILINEAR TIME SERIES MODELS
Authors: Chanda Kamal C.
Published: May 2008

Abstract. Consider the general bilinear times series model

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where {Xt; t= 0, L1, …} is a p‐variate process, C(p x (s+ 1)), A(p x p). Bt(p x p) (1 ≤j≤q) are arbitrary matrices of constants, εT=[εt,…εt‐q+1] and {εt; t=0, ±1, …} is a strictly stationary ergodic sequence of random variables. We investigate a set of minimal regularity conditions (on C, A, Bj and {εt}) under which we can establish the existence and causality of Xt and the asymptotic normality of the sample mean derived from {Xt}.
A LINEAR ESTIMATION PROCEDURE FOR THE PARAMETERS OF AUTOREGRESSIVE MOVING‐AVERAGE PROCESSES
Authors: Chiu Shean‐Tsong
Published: May 2008

Abstract.A linear estimation procedure for the parameters of autoregressive moving‐average processes is proposed. The basic idea is to write the spectrum for the moving‐average part as a linear function of a properly selected set of parameters and to use Chiu's weighted least‐squares procedure to reduce the problem to a weighted linear least‐squares problem. The proposed procedure finds estimates by solving systems of linear equations and does not need optimization programs. An one‐step estimate is also suggested. It is shown that the estimates are asymptotically equal to the commonly used ‘approximate’ maximum likelihood estimate described in the paper. For Gaussian processes, the estimates obtained by the proposed procedures are asymptotically efficient.
GRAPHICAL METHODS FOR DETERMINING THE PRESENCE OF PERIODIC CORRELATION
Authors: Hurd Harry L. , Gerr Neil L.
Published: May 2008

Abstract. Methods for determining whether an observed time series is, or contains, a periodically correlated sequence are presented. They are based on the fact that the support of the spectral measure for a harmonizable periodically correlated stochastic sequence is contained in a set of equally spaced lines parallel to the main diagonal in the bifrequency plane. We show how a coherence statistic devised by N. R. Goodman can be used to test for the presence of spectral support (correlation) along diagonal lines, thus providing a test for the presence of periodic correlation. Plots of this coherence statistic as a function of two frequency variables visually reveal the presence of diagonal support lines. Two additional tests for the presence of diagonal support lines are constructed by collapsing the two‐dimensional coherence plots into one‐dimensional plots of difference frequency.
CONSISTENT ESTIMATION OF THE ASYMPTOTIC COVARIANCE STRUCTURE OF MULTIVARIATE SERIAL CORRELATIONS
Authors: Melard Guy , Paesmans Marianne , Roy Roch
Published: May 2008

Abstract.A method is proposed for estimating, in a consistent way, the asymptotic covariance structure of serial correlations for a multivariate second‐order stationary process. To obtain a consistent estimator of this structure, which is also of the non‐negative definite type, results relative to the scalar case are generalized. The method consists in weighting appropriately the elements of the sample autocorrelation matrices in a generalization of Bartlett's formula so that the estimator converges in probability. Several useful applications of the results of the paper are mentioned.
JOINT HYPOTHESIS TESTS FOR A RANDOM WALK BASED ON INSTRUMENTAL VARIABLE ESTIMATORS
Authors: Hall Alastair
Published: May 2008

Abstract. In this paper we propose test statistics for the null hypothesis of a random walk or a random walk with drift for the case in which the innovations to the series are a moving‐average process. The statistics are based on the instrumental variable estimators proposed by Hall and by Pantula and Hall and are shown to have the limiting distributions tabulated by Dickey and Fuller.
ADAPTIVE SEMIPARAMETRIC ESTIMATION IN THE PRESENCE OF AUTOCORRELATION OF UNKNOWN FORM
Authors: Hidalgo F. Javier
Published: May 2008

Abstract.In a time series regression model the residual autoregression function is an unknown, possibly non‐linear, function. It is estimated by non‐parametric kernel regression. The resulting least‐squares estimate of the regression function is shown to be adapative, in the sense of having the same asymptotic distribution, to first order, as estimates based on knowledge of the autoregression function. Also, a Monte Carlo experiment about the behaviour of the estimator is described.
THRESHOLD TIME SERIES MODELS AS MULTIMODAL DISTRIBUTION JUMP PROCESSES
Authors: Martin Vance L.
Published: May 2008

Abstract.Recent contributions by Tong and others in modelling time series exhibiting threshold points have generally been based on approximating non‐linear processes by piecewise linear time series models. In this paper we provide an alternative framework in which to model time series displaying jump behaviour by using a multimodal conditional distribution to capture the jump process. Each subordinate model of the distribution is determined by an autoregressive process, and jump behaviour occurs when the relative heights of the modes of the distribution change whilst the threshold points are identified by the antimodes of the distribution. This class of models is referred to as multipredictor autoregressive time series (MATS).
ON THE EXISTENCE OF STATIONARY THRESHOLD AUTOREGRESSIVE MOVING‐AVERAGE PROCESSES
Authors: Brockwell Peter J. , Liu Jian , Tweedie Richard L.
Published: May 2008

Abstract. Conditions for the existence of causal and strictly stationary solutions of the equations defining a self‐exciting threshold autoregressive moving‐average (SETARMA) model are derived. For threshold autoregressive models we allow the autoregressive coefficients to be random and derive sufficient conditions for geometric ergodicity and the existence of strictly and weakly stationary solutions of the defining equations.
ON THE STABILITY OF A THRESHOLD AR(1) WITHOUT INTERCEPTS
Authors: Lim K. S.
Published: May 2008

Abstract. Binary polynomials are used to determine the stability of a threshold AR(1) without intercepts.
KERNEL REGRESSION SMOOTHING OF TIME SERIES
Authors: Härdle Wolfgang , Vieu Philippe
Published: May 2008

Abstract. A class of non‐parametric regression smoothers for times series is defined by the kernel method. The kernel approach allows flexible modelling of a time series without reference to a specific parametric class. The technique is applicable to detection of non‐linear dependences in time series and to prediction in smooth regression models with serially correlated observations.

In practice these estimators are to be tuned by a smoothing parameter. A data‐driven selector for this smoothing parameter is presented that asymptotically minimizes a squared error measure. We prove asymptotic optimality of this selector. We illustrate the technique with a simulated example and by constructing a smooth prediction curve for the variation of gold prices. In both cases the non‐parametric method proves to be useful in uncovering non‐linear structure.
REPARAMETRIZATION ASPECTS OF NUMERICAL BAYESIAN METHODOLOGY FOR AUTOREGRESSIVE MOVING‐AVERAGE MODELS
Authors: Marriott J. M. , Smith A. F. M.
Published: May 2008

Abstract.Within the context of likelihood and Bayes approaches to inference in autoregressive moving‐average (ARMA) time series models, previous ideas on parameter transformation and numerical integration for implementing Bayesian procedures are reviewed. Some novel transformation ideas are introduced and their role in an efficient numerical integration approach is examined. Some comparisons of the effectivesness of different numerical integration strategies are made.
VECTOR AUTOREGRESSIVE MODELS WITH UNIT ROOTS AND REDUCED RANK STRUCTURE:ESTIMATION. LIKELIHOOD RATIO TEST, AND FORECASTING
Authors: Reinsel Gregory C. , Ahn Sung K.
Published: May 2008

Abstract.The nonstationary multivariate autoregressive (AR) model Φ (L)Yt=εtis considered for anm‐dimensional process {Yt}, where it is assumed that det {Φ(L)}= 0 hasd<munit 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 numberdof 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.
DETECTING SINUSOIDS IN NON‐GAUSSIAN NOISE
Authors: Lii K.‐S. , Tsou T.‐H.
Published: May 2008

Abstract.Spectral analysis is a well‐established procedure for detecting harmonic signals in a noisy environment. Much research has been done on methods that use second‐order statistics (i.e. the autocovariance function and power spectrum) such as Whittle's test, Bartlett's test, Hannan's test and the PriestleyP(Λ) test. When the noise is non‐Gaussian, statistics of order greater than two can provide more information to detect the periodicities in noisy data. We direct our main attention to the third (fourth) order cumulant and bispectral (trispectral) methods. New test statistics are derived and are shown to be more powerful than other methods based on second‐order statistics under a mixed spectrum condition. The asymptotic power functions of the new test statistics and other tests are studied. Some Monte Carlo simulations are used to evaluate the performance of the new methods with moderate sample sizes.
TESTING FOR WHITE NOISE AGAINST MULTIMODAL SPECTRAL ALTERNATIVES
Authors: Reschenhofer E. , Bomze I. M.
Published: May 2008

Abstract.The test for white noise proposed by Reschenhofer (Biometrika76 (1989) 629–32) is adaptive in that the form of its test statistic depends on the data. It proceeds (i) by using the real part of the finite Fourier transform of the data for the selection of the form of the test statistic and (ii) by applying the selected test statistic to the imaginary part.

In this paper it is shown that independently of using information contained in the real part to construct a test statistic for the imaginary part, information contained in the imaginary part can be used to construct a test statistic for the real part. Because of the independence of these tests, they can easily be combined.

Then a further test is proposed which again is adaptive but where both the real and imaginary parts contribute simultaneously to a single test statistic.
REDUCTION OF THE ASYMPTOTIC BIAS OF AUTOREGRESSIVE AND SPECTRAL ESTIMATORS BY TAPERING
Authors: Zhang H.‐C.
Published: May 2008

Abstract. The asymptotic bias to terms of order T‐1, where T is the observed series length, is studied for estimators of the coefficients and disturbance variance in an AR(p) model. Reduction of the asymptotic bias by tapering is established and, if the tapering function is defined appropriately to depend on T, not only is the asymptotic bias reduced, but the asymptotic distribution of the estimators is not altered. In addition, the asymptotic biases of other time series parameter estimators constructed from the sample covariance function, such as several types of spectral estimators, can also be reduced by tapering.
ASYMPTOTIC RESULTS FOR PERIODIC AUTOREGRESSIVE MOVING‐AVERAGE PROCESSES
Authors: Anderson P. L. , Vecchia A. V.
Published: May 2008

Abstract.This paper is concerned with the derivation of asymptotic distributions for the sample autocovariance and sample autocorrelation functions of periodic autoregressive moving‐average processes, which are useful in modelling periodically stationary time series. In an effort to obtain a parsimonious model representing a periodically stationary time series, the asymptotic properties of the discrete Fourier transform of the estimated periodic autocovariance and autocorrelation functions are presented. Application of the asymptotic results to some specific models indicates their usefulness for model identification analysis.
DETERMINING THE ORDER OF A VECTOR AUTOREGRESSION WHEN THE NUMBER OF COMPONENT SERIES IS LARGE
Authors: Koreisha Sergio G. , Pukkila Tarmo
Published: May 2008

Abstract.We contrast the performance of several methods used for identifying the order of vector autoregressive (VAR) processes when the numberKof component series is large. Through simulation experiments we show that their performance is dependent onK, the number of nonzero elements in the polynomial matrices of the VAR parameters and the permitted upper limit of the order used in testing the autoregressive structure. In addition we introduce a new quite powerful multivariate order determination criterion.
ESTIMATION OF THE PREDICTION ERROR VARIANCE AND AN R2MEASURE BY AUTOREGRESSIVE MODEL FITTING
Authors: Bhansali R. J.
Published: May 2008

Abstract.For predicting the future values of a stationary process {xt} (t= 0, pL 1, pL 2,…) on the basis of its past, two key parameters are the varianceV(h),h≥ 1, of theh‐step prediction error andZ(h) ={R(0) ‐V(h)}/R(0), the corresponding measure, in anR2sense, of the predictability of the process from its past, whereR(0) denotes the process variance. The estimation ofV(h) andZ(h) from a realization ofTconsecutive observations of {xt} is considered, without requiring that the process follows a finite parameter model. Three different autoregressive estimators are examined and are shown to be asymptotically equivalent in the sense that asT∝ they have the same asymptotic normal distribution. The question of bias in estimating these parameters is also examined and a bias correction is proposed. Finite sample behaviour is investigated by a simulation study.
EXPECTATION‐MAXIMIZATION ALGORITHMS AND THE ESTIMATION OF TIME SERIES MODELS IN THE PRESENCE OF OUTLIERS
Authors: Abraham Bovas , Chuang Alice
Published: May 2008

Abstract.The expectation‐maximization algorithm is reviewed briefly. The algorithm is applied to time series situations where outliers may be present. An approximation of the algorithm is considered to reduce the computational complexity. Examples are given to illustrate the application of this algorithm.
ON THE PARTIAL SUMS OF RESIDUALS IN AUTOREGRESSIVE AND MOVING AVERAGE MODELS
Authors: Bai Jushan
Published: May 2008

Abstract.The limiting process of partial sums of residuals in stationary and invertible autoregressive moving‐average models is studied. It is shown that the partial sums converge to a standard Brownian motion under the assumptions that estimators of unknown parameters are root‐nconsistent and that innovations are independent and identically distributed random variables with zero mean and finite variance or, more generally, are martingale differences with moment restrictions specified in Theorem 1. Applications for goodness‐of‐fit and change‐point problems are considered. The use of residuals for constructing nonparametric density estimation is discussed.
THE EFFECT OF AGGREGATION ON PREDICTION IN AUTOREGRESSIVE INTEGRATED MOVING‐AVERAGE MODELS
Authors: Hotta L. K. , Neto J. Cardosc
Published: May 2008

Abstract.Letxtbe a time series generated by an autoregressive integrated moving‐average process ARIMA(p, d, q). The non‐overlapping aggregate series also follows an ARIMA process. Thus, the prediction of the aggregated observations could be done by either the disaggregate model or the aggregate model. We derive the efficiency of the predictors for two important disaggregate models, ARIMA(0, 1, 1) and ARIMA(0, 2, 2), when the models are assumed known. When the models are not known we estimate the efficiency through simulation with the models being selected using Akaike's information criterion.
YULE‐WALKER ESTIMATES FOR CONTINUOUS‐TIME AUTOREGRESSIVE MODELS
Authors: Hyndman Rob J.
Published: May 2008

Abstract.I consider continuous‐time autoregressive processes of orderpand 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 THE INVERTIBILITY OF MULTIVARIATE LINEAR PROCESSES
Authors: Nsiri Saïd , Roy Roch
Published: May 2008

Abstract.It is shown that a multivariate linear stationary process whose coefficients are absolutely summable is invertible if and only if its spectral density is regular everywhere. This general characterization of invertibility is applied later to the case of a linear process having an autoregressive moving‐average (ARMA) representation. Under the usual assumptions, it is deduced that a processYdescribed by an ARMA(φ, TH) model is invertible if and only if the polynomial detTH(z) has no roots on the unit circle. Given an invertible processYwhich has an ARMA representation, it is finally shown that the processYT, whereYT, =εi=0lSiYt‐i, is invertible if and only if the matrixS(z) =εi=0lSiziis of full rank for allzof modulus 1. It follows, in particular, that any subprocess of an invertible ARMA process is also invertible.
TESTS FOR FRACTIONAL INTEGRATION:A MONTE CARLO INVESTIGATION
Authors: Cheung Yin‐Wong
Published: May 2008

Abstract. The performance of the Geweke‐Porter‐Hudak (GPH) test, the modified rescaled range (MRR) test and two Lagrange multiplier (LM) type tests for fractional integration in small samples is examined using Monte Carlo methods. Both the GPH and MRR tests are found to be robust to moderate autoregressive moving‐average components, autoregressive conditional heteroskedasticity effects and shifts in the variance. However, these two tests are sensitive to large autoregressive moving‐average components and shifts in the mean. It is also found that the LM tests are sensitive to deviations from the null hypothesis. As an illustration, the GPH test is applied to two economic data series.
NON‐PARAMETRIC APPROACH IN TIME SERIES ANALYSIS
Authors: Taniguchi Masanobu , Kondo Masao
Published: May 2008

Abstract.Suppose that {Xt} is a Gaussian stationary process with spectral densityf(Λ). In this paper we consider the testing problem, whereK(Λ) is an appropriate function andcis a given constant. This test setting is unexpectedly wide and can be applied to many problems in time series. For this problem we propose a test based onK{fn(Λ)}dΛwherefn(Λ) is a non‐parametric spectral estimator off(Λ), and we evaluate the asymptotic power under a sequence of non‐parametric contiguous alternatives. We compare the asymptotic power of our test with the other and show some good properties of our test. It is also shown that our testing problem can be applied to testing for independence. Finally some numerical studies are given for a sequence of exponential spectral alternatives. They confirm the theoretical results and the goodness of our test.
SPECTRAL ANALYSIS FOR AMPLITUDE‐MODULATED TIME SERIES
Authors: Toloi Clélia M. C. , Morettin Pedro A.
Published: May 2008

Abstract.Let {X(n)} be a non‐observed strictly stationary process, {a(n)} a sequence independent of {X(n)} andY(n) =a(n)X(n) the observed process. This work deals with the estimation of the spectral density functionfx(Λ) of the process of interest, {X(n)}, using observations of the modulated process {Y(n)}. We obtain estimators offx(Λ) for three types of modulating functions:deterministic, random independent and random correlated.