Showing papers in "Journal of Time Series Analysis in 1988"
TL;DR: In this paper, the correlation structure for the squares from the generalized autoregressive conditional heteroskedastic (GARCH) process is presented, and the behavior of the correlations for the square mimics the usual correlations of an appropriately defined ARMA process, although the admissible regions for the correlations are somewhat more restrictive.
Abstract: . The correlation structure for the squares from the generalized autoregressive conditional heteroskedastic (GARCH) process is presented. It is shown that the behaviour of the correlations for the squares mimics the usual correlations of an appropriately defined ARMA process, although the admissible regions for the correlations are somewhat more restrictive. Simulation experiments are used to study the applicability of the theoretical results for order identification and diagnostic checking. Finally, an empirical example is given for the IBM stock market price series from Box and Jenkins (1976).
258 citations
TL;DR: In this paper, the relative efficiency of the two estimators for time series whose spectrum behaves like a power at the origin (e.g., fractional Gaussian noise and fractional ARIMA) was investigated.
Abstract: . When estimating the unknown mean of a stationary time series, the best linear unbiased estimator is often a significantly better estimator than the ordinary least squares estimates Xn. The relative efficiency of these two estimators is investigated for time series whose spectrum behaves like a power at the origin (e.g., fractional Gaussian noise and fractional ARIMA).
92 citations
TL;DR: Godambe's theorem on optimal estimating equations for stochastic processes is applied to non-linear time series estimation problems in this article, and a recursive estimation procedure based on the theorem is provided.
Abstract: 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.
73 citations
TL;DR: In this article, an unbiased and consistent estimator is proposed based on a least-squares method in the frequency domain, and the Cramer-Rao lower bound is derived.
Abstract: This study deals with the parameter estimation in long-memory time series models. An unbiased and consistent estimator is proposed. The proposed estimator is based on a least-squares method in the frequency domain, and it is computationally simple. Also, the Cramer–Rao lower bound is derived. The mean-square error of the proposed estimator is order of O(1/N), where N is the number of samples. The accuracy of the estimates is verified using synthetic long-memory time series data.
72 citations
TL;DR: In this article, the problem of modeling time series driven by non-Gaussian innovations is considered and the asymptotic normality of the maximum likelihood estimator is established under some general conditions.
Abstract: . The problem of modelling time series driven by non-Gaussian innovations is considered. The asymptotic normality of the maximum likelihood estimator is established under some general conditions. The distribution of the residual autocorrelations is also obtained. This gives rise to a potentially useful goodness-of-fit statistic. Applications of the results to two important cases are discussed. Two real examples are considered.
63 citations
TL;DR: In this article, 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, which is mean-and scale-invariant and approximates the most powerful invariant test against any specific alternative.
Abstract: 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.
42 citations
TL;DR: In this paper, extended results associated with the predictors of long-memory time series models are considered. But these direct methods of obtaining predictors for fractionally differenced autoregressive integrated moving average (ARIMA) processes have advantages from the theoretical point of view.
Abstract: . This paper considers some extended results associated with the predictors of long-memory time series models. These direct methods of obtaining predictors of fractionally differenced autoregressive integrated moving-average (ARIMA) processes have advantages from the theoretical point of view.
36 citations
TL;DR: In this paper, the authors give general and concrete conditions in terms of the coefficient (stochastic) process {At} so that the (doubly) stochastic difference equation Xt= AtXt-1+et has a second-order strictly stationary solution.
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+et has a second-order strictly stationary solution. It turns out that by choosing {At} and the “innovation” process {et} 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.
35 citations
TL;DR: For the bilinear time series model Xt=βXt-ket-t+et, k > l, k = l and k < l formulae for the third-order theoretical moments and an expression for the bispectral density function are obtained as discussed by the authors.
Abstract: For the bilinear time series model Xt=βXt-ket-t+et, k > l, k = l and k < l formulae 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 of β are computed and the properties are studied.
35 citations
TL;DR: In this paper, the authors consider non-linear time series modeling via a case study and discuss several important issues concerning nonlinear time-series models and data analysis, including data analysis.
Abstract: . In recent years there has been a growing interest in studying non-linear time series and various non-linear models have been proposed in the literature. In this paper, I consider non-linear time series modelling via a case study. Several important issues concerning non-linear time series models and data analysis emerge from the study.
33 citations
TL;DR: In this article, a procedure based on the well-known score-test is discussed for detection of outliers and distinguishing between the outlier types, and the importance levels of the tests are also obtained and illustrated with simulated examples.
Abstract: . Two characterizations, the aberrant observation and innovation models, for outliers in time series are considered. A procedure based on the well-known score-test is discussed for detection of outliers and distinguishing between the outlier types. Significance levels of the tests are also obtained and the method is illustrated with simulated examples.
TL;DR: In this article, the difference equations for the third and fourth-order lagged moments and cumulants were obtained for linear time series with a bilinear model and stationary up to fourth order.
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.
TL;DR: It is shown that although these processes are not stationary with regard to an additive binary operation, they are stationary with respect to a multiplicative binary operation and this property is exploited in such a way as to guarantee essentially the same structure as is available for stationary processes.
Abstract: . In this paper a class of nonstationary processes, referred to as multiplicative stationary processes, is investigated. It is shown that although these processes are not stationary with regard to an additive binary operation, i.e. in the classical sense, they are stationary with respect to a multiplicative binary operation. This property is then exploited in such a way as to guarantee essentially the same structure as is available for stationary processes. In particular, suitable definitions for the autocorrelation, power spectrum and linear processes are given. In addition, the Euler process is introduced as the nonstationary or multiplicative stationary dual of the classical autoregressive processes. Some ergodic theorems are also obtained and numerous examples are given.
TL;DR: In this paper, a class of linear filters are considered that when applied to a constant generate a wide class of deterministic trends in mean, which may correspond to fractional integrated series.
Abstract: . Using the theory of divergent series, a class of linear filters are considered that when applied to a constant generate a wide class of deterministic trends in mean. If these filters are applied to a white noise, series are produced that have changing variances and may correspond to fractional integrated series. Some other trend-generating mechanisms are also considered and methods of characterizing different trends are discussed.
TL;DR: In this article, a multiple time series regression model with trending regressors has residuals that are believed to be not only serially dependent but nonstationary, and the residuals can be decomposed as a stationary autoregressive process of known order multiplied by an unknown time-varying scale factor.
Abstract: . A multiple time series regression model with trending regressors has residuals that are believed to be not only serially dependent but nonstationary. Assuming the residuals can be decomposed as a stationary autoregressive process of known order multiplied by an unknown time-varying scale factor, we propose estimators of the regression coefficients and show them to be as efficient as estimators based on known scale factors. Our estimators have features in common with adaptive estimators proposed by Carroll (1982) and Hannan (1963) for different regression problems, involving respectively independent residuals with heteroskedasticity of unknown type, and stationary residuals with unknown serial dependence structure.
TL;DR: In this paper, an autoregressive moving-average model is used to predict future values of a stationary multivariate time series and an asymptotic approximation to the increase in mean-square prediction error is obtained.
Abstract: 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.
TL;DR: In this paper, the limiting probability of overestimating the order of a multivariate autoregression when using AIC-type procedures was investigated and shown to be bounded by a factor of 1.
Abstract: . Formulae are obtained for the limiting probability of overestimating the order of a multivariate autoregression when using AIC-type procedures.
TL;DR: In this article, two new methods for estimating the inverse covariance and inverse correlation functions of a time series are proposed, one based on an orthogonality property, the other suggested by interpolation considerations.
Abstract: Two new methods for estimating the inverse covariance and inverse correlation functions of a time series are proposed. One of them is based on an orthogonality property, the other is suggested by interpolation considerations. The two methods are shown to be asymptotically equivalent, and their asymptotic distribution is derived. The asymptotic distribution turns out to be the same as that of the autoregressive estimates of the inverse correlations. The problem of choosing an estimation method in practice is discussed.
TL;DR: In this paper, the authors developed simple procedures for testing the adequacy of separate time series models using auxiliary regressions that are very similar to those used for calculating Lagrange multiplier test statistics.
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.
TL;DR: In this paper, it is shown that the sample autocovariance of a periodically correlated process converges to a limit which reveals the same periodicity as the process, and a strongly consistent estimate of hidden period is proposed.
Abstract: . It is shown that the sample autocovariance of a periodically correlated process converges to a limit which reveals the same periodicity as the process. A theorem is proved relating to the rate of almost sure convergence, which is uniform in the lag up to some orders of observation length. Based on the limiting property, a strongly consistent estimate of hidden period is proposed.
TL;DR: In this article, an alternative procedure is developed to detect the hidden frequencies in linear processes which differs from the procedure proposed earlier by the author, and the advantage of the new procedure is that it enables us to detect a hidden frequency in the noise when the spectral density possesses high peaks.
Abstract: . An alternative procedure is developed to detect the hidden frequencies in linear processes which differs from the procedure proposed earlier by the author. The advantage of the new procedure is that it enables us to detect a hidden frequency in the noise when the spectral density possesses high peaks. In such cases the earlier procedure often fails in practice. We also prove the convergence of a spectral estimate designed to reduce the influence of hidden frequencies.
TL;DR: In this paper, Coates and Diggle introduced a test procedure for the comparison of the spectral densities of two stationary processes and extended this test to a situation in which replicated observations are available for each process.
Abstract: . Coates and Diggle (1986) introduced a test procedure for the comparison of the spectral densities of two stationary processes. We extend this test to a situation in which replicated observations are available for each process.
TL;DR: In this paper, the k-dimensional pth-order autoregressive processes with one unstable or explosive root were considered and the s-periods-ahead predictor was shown to be asymptotically equivalent to the optimal predictor except in the explosive case.
Abstract: The k-dimensional pth-order autoregressive processes {Yt} that are either stationary or have one unstable or explosive root are considered The properties of the s-periods-ahead predictor Ŷn+s, obtained by replacing the unknown parameters in the expression for the best linear predictor YTn+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 Ŷn+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 Ŷn+s are investigated
TL;DR: In this article, the problem of estimating the delays that arise when a waveform propagates across an array is considered, and 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.
Abstract: 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.
TL;DR: In this paper, the NEAR(2) model was generalized to the PEAR(p) model, and a necessary and sufficient condition for the existence of an innovation sequence and a stationary ergodic process satisfying the NEARP model was derived.
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.
TL;DR: A direct method of computing the initial state covariance matrix T0 which, unless the number of time series is large, is usually faster than using the doubling algorithm of Anderson and Moore.
Abstract: . Barone has described a method for generating independent realizations of a vector autoregressive moving-average (ARMA) process which involves recasting the ARMA model in state space form. We discuss a direct method of computing the initial state covariance matrix T0 which, unless the number of time series is large, is usually faster than using the doubling algorithm of Anderson and Moore. Our numerical comparisons are particularly valuable because T0 must also be computed when calculating the likelihood function. A number of other computational refinements are described. In particular, we advocate the use of Choleski factorizations rather than spectral decompositions. For a pure moving-average process computational savings can be achieved by working directly with the ARMA model rather than with its state space representation.
TL;DR: In this paper, the authors consider fitting a parametric model to a time series and obtain the maximum likelihood estimates of unknown parameters included in the model by regarding the time series as a Gaussian process satisfying the model.
Abstract: . We consider fitting a parametric model to a time series and obtain the maximum likelihood estimates of unknown parameters included in the model by regarding the time series as a Gaussian process satisfying the model. We evaluate the asymptotic value of the conditional quasi-likelihood function when the number of observations tends to infinity. We show what properties of the time series we can find by examining the behaviour of the conditional quasi-likelihood function, even when the time series does not necessarily satisfy the model and is not necessarily Gaussian.
TL;DR: In this article, a quick algorithm for obtaining estimates of autoregressive parameters for moving-average models is presented, which can be used for model selection by providing a criterion and a two-way table of certain partial covariances.
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.
TL;DR: In this article, a structural, stationary version of the well-known state-space model is used to model covariance-stationary stochastic processes and a rank condition for local parameter identifiability is given.
Abstract: In this paper a structural, stationary version of the well-known state-space model is used to model covariance-stationary stochastic processes. The identifiability of the model parameters is discussed and a rank condition for local parameter identifiability is given. Ljung's results on prediction-error estimation are used to establish strong consistency and asymptotic efficiency of the non-linear ML-estimates obtained from dependent observations. It turns out that the model can be identified by using simultaneously the steady-state Kalman filter for the unobservable state vector and the prediction-error estimation method for the model parameters.
TL;DR: In this article, an expression for the asymptotic mean square error in predicting more than one step ahead from a p-variate autoregressive model with random coefficients is derived.
Abstract: In this paper, an expression for the asymptotic mean square error in predicting more than one step ahead from a p-variate autoregressive model with random coefficients is derived. Two cases are investigated: (i) when the parameters are known, and (ii) when the parameters are estimated.