Showing papers in "Journal of Time Series Analysis in 1996"

Multivariate local polynomial regression for time series:uniform strong consistency and rates

Abstract: . Local high-order polynomial fitting is employed for the estimation of the multivariate regression function m(x1,…xd) =E{φ(Yd)φX1=x1,…,Xd=xd}, and of its partial derivatives, for stationary random processes {Yi, Xi}. The function φ may be selected to yield estimates of the conditional mean, conditional moments and conditional distributions. Uniform strong consistency over compact subsets of Rd, along with rates, are established for the regression function and its partial derivatives for strongly mixing processes.

Wavelets and time‐dependent spectral analysis

Abstract: . One of the key features of wavelet analysis is its potential use for effecting time-frequency decompositions of non-stationary signals. The relationship between wavelet analysis and time-dependent spectral analysis has so far rested mainly on heuristic reasoning:in this paper we examine the relationship in a more precise mathematical form. A crucial feature of this analysis is the need to define carefully the notion of “frequency” when applied to non-stationary signals.

A generalized fractionally integrated autoregressive moving-average process

Abstract: . This paper considers the long memory Gegenbauer autoregressive movingaverage (GARMA) process that generalizes the fractionally integrated ARMA (ARFIMA) process to allow for hyperbolic and sinusoidal decay in autocorrelations. We propose the conditional sum of squares method for estimation (which is asymptotically equivalent to the maximum likelihood estimation) and develop the asymptotic theory. Many results are in sharp contrast to those of the ARFIMA model. Simulations are conducted to assess the performance of the proposed estimators in small sample applications. Two applications to the sunspot data and the US inflation rates based on the wholesale price index are provided.

Spectral density estimation via nonlinear wavelet methods for stationary non‐gaussian time series

Abstract: . In the present paper we consider nonlinear wavelet estimators of the spectral density f of a zero mean, not necessarily Gaussian, stochastic process, which is stationary in the wide sense. It is known in the case of Gaussian regression that these estimators outperform traditional linear methods if the degree of smoothness of the regression function varies considerably over the interval of interest. Such methods are based on a nonlinear treatment of empirical coefficients that arise from an orthonormal series expansion according to a wavelet basis. The main goal of this paper is to transfer these methods to spectral density estimation. This is done by showing the asymptotic normality of certain empirical coefficients based on the tapered periodogram. Using these results we can show the risk equivalence to the Gaussian case for monotone estimators based on such empirical coefficients. The resulting estimator of f keeps all interesting properties such as high spatial adaptivity that are already known for wavelet estimators in the case of Gaussian regression. It turns out that appropriately tuned versions of this estimator attain the optimal uniform rate of convergence of their L2 risk in a wide variety of Besov smoothness classes, including classes where linear estimators (kernel, spline) are not able to attain this rate. Some simulations indicate the usefulness of the new method in cases of high spatial inhomogeneity.

Unit roots in periodic autoregressions

Abstract: text. This paper analyzes the presence and consequences of a unit root in periodic autoregressive models for univariate quarterly time series. First, we consider various representations of such models, including a new parametrization which facilitates imposing a unit root restriction. Next, we propose a class of likelihood ratio tests for a unit root, and we derive their asymptotic null distributions. Likelihood ratio tests for periodic parameter variation are also proposed. Finally, we analyze the impact on unit root inference of misspecifying a periodic process by a constant-parameter model.

Testing serial independence using the sample distribution function

Abstract: . This paper presents and discusses a nonparametric test for detecting serial dependence. We consider a Crameer-von Mises statistic based on the difference between the joint sample distribution and the product of the marginals. Exact critical values can be approximated from the asymptotic null distribution, or by resampling, randomly permuting the original series. A Monte Carlo experiment illustrates the test performance with small sample sizes. The paper also includes an application, testing the random walk hypothesis of exchange rate returns for several currencies.

Locally adaptive lag‐window spectral estimation

Abstract: . We propose a procedure for the locally optimal window width in nonparametric spectral estimation, minimizing the asymptotic mean square error at a fixed frequency Λ of a lag-window estimator. Our approach is based on an iterative plug-in scheme. Besides the estimation of a spectral density at a fixed frequency, e.g. at frequency Λ= 0, our procedure allows to perform nonparametric spectral estimation with variable window width which adapts to the smoothness of the true underlying density.

Simulation and estimation of long memory continuous time models

Abstract: . Some general properties of long memory continuous time processes are recalled or proved. Methods of simulation are studied. A comparison with the usual discrete time autoregressive fractionally integrated moving-average filter is made and illustrations are provided. Then, two methods of estimation of the parameters of such a model from a discrete sample are studied, both theoretically and empirically, with Monte Carlo experiments.

Optimal prediction of catastrophes in autoregressive moving-average processes

Abstract: . This paper presents an optimal predictor of level crossings, catastrophes, for autoregressive moving-average processes, and investigates the performance of the predictor. The optimal catastrophe predictor is the predictor that gives a minimal number of false alarms for a fixed detection probability. As a tool for evaluating, comparing and constructing the predictors a method using operating characteristics, i.e. the probability of correct alarm and the probability of detecting a catastrophe for the predictor, is used. An explicit condition for the optimal catastrophe predictor based on linear prediction of future process values is given and compared with a naive catastrophe predictor, which alarms when the predicted process values exceed a given level, and with some different approximations of the optimal predictor. Simulations of the different algorithms are presented, and the performance is shown to agree with the theoretical results. All results indicate that the optimal catastrophe predictor is far better than the naive predictor. They also show that it is possible to construct an approximate catastrophe predictor requiring fewer computations without losing too much of the optimal predictor performance. (Less)

On the robustness to small trends of estimation based on the smoothed periodogram

Abstract: . In this paper we are concerned with the robustness of inferences, carried out on a stationary process contaminated by a small trend, to this departure from stationarity. It is shown that a smoothed periodogram approach to model fining and parameter estimation is highly robust to the presence of a small trend if the underlying stationary process is short-range dependent. If the underlying process is long-range dependent the robustness properties are still good but now depend on the Hurst index of the process and deteriorate with increasing Hurst index.

Bandwidth selection in kernel smoothing of time series

Abstract: . The kernel smoothing method has been considered as a useful tool for identification and prediction in time series models. In practice this method is to be tuned by a smoothing parameter. For selection of the smoothing parameter, Hardle and Vieu (Kernel regression smoothing of time series. J. Time Ser. Anal. 13(1992), 209–32) considered a cross-validation rule and proved its asymptotic optimality. In this paper we strengthen their result for a wider use of the kernel smoothing of time series.

Model selection and order determination for time series by information between the past and the future

Abstract: . In this paper, the information between the past and the future of a Gaussian stationary sequence is calculated either by its spectral density or by its autocovariances, and is related to the problem of model fitting. It is demonstrated that the criterion of minimum mutual information is the generalization of that of maximum entropy. By employing the above information quantity, we propose a procedure, which is called LIC for simplicity, to obtain consistent estimate of the order of the Bloomfield model or the autoregressive model. In Monte Carlo studies, we illustrate the LIC procedure by several examples, and also estimate the spectral density of time series by the Bloomfield model and LIC method.

Initialization of the kalman filter with partially diffuse initial conditions

Abstract: The problem of computing estimates of the state vector when the Kalman filter is seeded with an arbitrarily large variance is considered To date the response in the literature has been the development of a number of relatively complex hybrid filters, usually involving additional quantities and equations over and above the conventional filter We show, however, that a certain square root covariance filter is capable of handling the complete range of variances (zero, positive and infinite) without modification to the filtering equations themselves and without additional computation loads Instead of the more conventional Cholesky factorization, our filter employs an alternative matrix factorization procedure based on a unit lower triangular matrix and a diagonal matrix This permits the use of a modified form of fast Givens transformations, central to the development of an efficient algorithm

The exact error in estimating the spectral density at the origin

Abstract: This paper derives expressions for the exact bias and variance of a general class of spectral density estimators at the zero frequency, building on the work of Neave (The exact error in spectrum estimates Ann Math Statist 42 (1971), 961–75) who studied the case where the mean of the series is assumed known These expressions are evaluated for 15 different windows and for a wide variety of stationary time series The exact error of the estimators is found to depend on whether the sample mean has to be estimated, and some windows are noticeably inferior at certain values of the bandwidth A response surface analysis reveals that the finite sample relationships between the bandwidth and the exact error are quite different from the ones suggested by asymptotic theory

A bayesian approach to estimating and forecasting additive nonparametric autoregressive models

Abstract: . We present a Bayesian approach for estimating nonparametrically an additive autoregressive model with the regression curve estimates cubic smoothing splines. Our approach is robust to innovation outliers; it can handle missing observations and produce multistep ahead forecasts. The computation is carried out using Markov chain Monte Carlo and requires O(nM) operations where n is the sample size and M is the number of Markov chain iterations. This makes it the first exact algorithm for spline smoothing of an additive autoregressive model which can handle large data sets. The properties of the estimates and forecasts are studied empirically using simulated and real data sets.

RELATIVE POWER OF t TYPE TESTS FOR STATIONARY AND UNIT ROOT PROCESSES

Abstract: . This paper shows numerically that the lack of power and size distortions of the Dickey-Fuller type tests for unit roots (very well documented in the unit root literature) are similar to and in many situations even smaller than the lack of power and size distortions of the standard Student t tests for stationary roots of an autoregressive model.

Asymptotic inference for non-invertible moving-average time series

Abstract: . This paper is concerned with statistical inference of nonstationary and non-invertible autoregressive moving-average (ARMA) processes. It makes use of the fact that a derived process of an ARMA(p, q) model follows an AR(q) model with an autoregressive (AR) operator equivalent to the moving-average (MA) part of the original ARMA model. Asymptotic distributions of least squares estimates of MA parameters based on a constructed derived process are obtained as corresponding analogs of a nonstationary AR process. Extensions to the nearly non-invertible models are considered and the limiting distributions are obtained as functionals of stochastic integrals of Brownian motions and Ornstein-Uhlenbeck processes. For application, a two-stage procedure is proposed for testing unit roots in the MA polynomial. Examples are given to illustrate the application.

Recursive computation of the parameters of periodic autoregressive moving-average processes

Abstract: An algorithm for recursive computation of the parameters of periodic autoregressive moving-average (ARMA) processes is given It also provides recursions for stationary multivariate ARMA processes A procedure for simultaneous estimation of the order and the parameters of a periodic ARMA process is outlined

Testing for a unit root in an AR(1) time series using irregularly observed data

Abstract: . For an AR(1) model having a unit root with nonconsecutively observed or missing data we consider the ordinary least squares estimator, the one-step Newton-Raphson estimator and an ordinary least squares type estimator which is a simple approximation of the Newton-Raphson estimator. It is shown that the limiting distributions of these estimators of the unit root are the same as those of the regression estimators as tabulated by Dickey and Fuller (Distribution of the estimators for autoregressive time series with a unit root. J. Am. Statist. Assoc. 74 (1979), 427–31) for the complete data situation. Simulation results show that our proposed unit root tests perform very well for small samples.

Spectral analysis of a stationary bivariate point process with applications to neurophysiological problems

Abstract: . In this paper we discuss the spectral analysis of a stationary bivariate point process applied to the study of a complex physiological system. An estimate of the cross-spectral density can be obtained by smoothing the modified cross-periodogram statistic. The smoothed estimate is calculated by dividing the whole length of the data into a number of disjoint subrecords. Estimates of the coherence function and the cross-intensity function follow directly from the estimate of the cross-spectral density. It is shown that the asymptotic properties of the estimate of the cross-intensity function can be improved by inserting a convergence factor in it. Examples of the estimates are illustrated by using two data sets from neurophysiological experiments and their importance is emphasized by examining the behaviour of the complex physiological system under study.

Beveridge‐nelson‐type trends for i(2) and some seasonal models

Abstract: . Beveridge and Nelson (A new approach to decomposition of economic time series into permanent and transitory components with particular attention to measurement of the business cycle. J. Monet. Econ. 7 (1981), 151–74) introduced a decomposition into trend plus irregular components for time series generated by models that are integrated of order one. The components are functions of current and past, but not future, values of the series. Therefore, these components can be viewed as estimates available to an agent at the time. Moreover, the decomposition exists whenever the generating process is stationary after first differencing. In this paper we extend the decomposition to generating processes that are integrated of order two, and to the seasonal models of Box and Jenkins (Time Series Analysis, Forecasting and Control. San Francisco:Holden Day, 1970). The analysis leads to the estimation of stochastic growth rates, as well as component series. The methodology is applied to monthly UK industrial production data.

Bias and covariance of the recursive least squares estimator with exponential forgetting in vector autoregressions

Abstract: . The recursive least squares (RLS) estimation algorithm with exponential forgetting is commonly used to estimate time-varying parameters in stochastic systems. The statistical properties of the RLS estimator are often hard to find, since they depend in a non-linear way on the time-varying characteristics. In this paper the RLS estimator with exponential forgetting factor is applied to stationary Gaussian vector autoregres-sions and the asymptotic bias and covariance function of the parameter estimates are derived.

Adequacy of asymptotic theory for goodness-of-fit criteria for spectral distributions

Abstract: . Any of the Cramer-von Mises, Anderson-Darling, and Kolmogorov-Smirnov statistics can be used to test the null hypothesis that the standardized spectral distribution of a stationary stochastic process is a specified one. The asymptotic distributions of the criteria have been characterized (Anderson, 1993). They are the same as for probability distributions if the observations are independent (all autocorrelations zero), but are different when there is dependence. In this paper simulation with 10000 replications has been used to determine the distributions of the criteria for samples of size 6, 10, 30 and 100 when the observations are independent. These empirical distributions have been compared with the asymptotic distributions in order to ascertain the sample sizes necessary for using the asymptotic tables. For practical purposes they are 30 for the Cramer-von Mises and Kolmogorov statistics and over 100 for Anderson-Darling.

Some properties of the maximum likelihood estimator in the simultaneous switching autoregressive model

Abstract: . The simultaneous switching autoregressive (SSAR) model proposed by Kunitomo and Sato (A non-linearity in economic time series and disequilibrium econometric models. In Theory and Application of Mathematical Statistics (ed. A. Takemura). Tokyo:University of Tokyo Press (in Japanese), 1994; Asymmetry in economic time series and simultaneous switching autoregressive model. Struct. Change Econ. Dyn., forthcoming (1994).) is a Markovian non-linear time series model. We investigate the finite sample as well as the asymptotic properties of the least squares estimator and the maximum likelihood (ML) estimator. Due to a specific simultaneity involved in the SSAR model, the least squares estimator is badly biased. However, the ML estimator under the assumption of Gaussian disturbances gives reasonable estimates.

Higher order moments of sample autocovariances and sample autocorrelations from an independent time series

Abstract: . Given length-n sampled time series, generated by an independent distributed process, in this paper we treat the problem of determining the greatest order, in n, that moments of the sample autocovariances and sample autocorrelations can attain. For the sample autocovariance moments, we achieve quite general results; but, for the autocorrelation moments, we restrict study to Gaussian white noise (normal, independent and identically distributed). Our main theorem relates to the cross-moments of the non-centred sample autocovariances, but we also establish a relation between these and the corresponding moments for the centred sample autocovariances.

Random sampling estimation for almost periodically correlated processes

Abstract: . In this paper we study the estimation of the spectral density functions of a continuous-time parameter almost periodically correlated process from one discrete random-time sampling. Under mixing hypotheses on the cumulant of the process, we establish the quadratic consistency of this estimator and the rate of convergence.

An outlier test for time series based on a two‐sided predictor

Abstract: . In this paper an outlier test for contaminated autoregressive processes is introduced. The test is based on a comparison of each observation with a predictor using past and future values, a so-called two-sided predictor. It is required that an upper bound for the total number of outliers is known. The asymptotic distribution of the test statistic is calculated under the null hypothesis that no outlier is present. The behaviour of the test for finite sample size is investigated by a simulation study. Moreover, the test is compared with several other outlier tests.

Spectral maximum likelihood estimation of a signal to noise ratio lying in the vicinity of zero

Abstract: . A time series model representing a decomposition into permanent plus transient components contains a deterministic component when the signal-to-noise ratio is equal to zero; otherwise, the permanent component is said to be stochastic. This distinction has important consequences in the analysis of economic phenomena. On the other hand, the absence of a stochastic permanent component in residuals from a time series regression may indicate cointegration. This paper considers the frequency domain estimation of the signal-to-noise ratio in a representative of the unobserved components model class. The sampling properties of the estimator from the resulting approximate spectral likelihood differ from those observed in the time domain and they vary substantially depending on whether the overall slope must be estimated or not. Further, it is shown that spectral estimates are T-consistent—instead of T2-consistent in the time domain. These results may explain some of the differences in estimators from frequency domain approximations to the likelihood and exact maximum likelihood estimators, and may be of use when testing for deterministic trends.

Distribution of residual autocorrelations in nonstationary autoregressive processes

Abstract: The residual autocorrelations in nonstationary autoregressive processes with autoregressive characteristic roots on the unit circle are considered Limiting distributions of the residual autocovariances and the residual autocorrelations are shown to be the same as the limiting distributions when parameters are estimated with all roots on the unit circle known The portmanteau statistic is shown to have a x2 limiting distribution The Canadian lynx data set is analysed to illustrate our theory The portmanteau test seems also useful when the characteristic roots are close to the unit circle

Testing the order of differencing in time series regression

Abstract: . In this paper we develop a test procedure for detecting overdifferencing or a moving-average unit root in time series regression models with Gaussian autoregressive moving-average errors. In addition to an intercept term the regressors consist of stable or asymptotically stationary variables and non-stationary trending variables generated by an integrated process of order 1. The test of the paper is based on the theory of locally best invariant unbiased tests. Its limiting distribution is derived under the null hypothesis and found to be non-standard but free of unknown nuisance parameters. Asymptotic critical values, which depend on the number of integrated regressors, are obtained by simulation. A limited simulation study is carried out to illustrate the finite sample properties of the test.