NEAREST-NEIGHBOUR METHODS FOR TIME SERIES ANALYSIS

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 G ₂ 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.     

SEQUENTIAL METHODS FOR DETECTING CHANGES IN THE VARIANCE OF ECONOMIC TIME SERIES

In this paper EWMA charts and CUSUM charts are introduced for detecting changes in the variance of a GARCH process. The moments of the EWMA statistics are calculated. They permit a better understanding of the underlying control procedure. In an extensive simulation study all control schemes are compared with each other. “Optimal” smoothing parameters and “optimal” reference values are tabulated. It is shown how these charts can be applied to monitor stock market returns.     

FORECASTING TIME SERIES:A COMPARATIVE ANALYSIS OF ALTERNATIVE CLASSES OF TIME SERIES MODELS

Abstract. Performance of the state dependent model developed by Priestley is evaluated relative to that of bilinear and standard linear models using two well-known time series. The results indicate the use of broader classes of time series models beyond the conventional ARMA class is likely to lead to significant reductions in forecasting error. However, there are difficult problems relating to the identification of the order of the model, estimation of the parameters, and determination of the correct nonlinear model.     

NON-LINEAR TIME SERIES ANALYSIS OF BLOWFLY POPULATION

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.     

WALSH-FOURIER ANALYSIS OF DISCRETE-VALUED TIME SERIES

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.     

STATISTICAL ANALYSIS OF ECONOMIC TIME SERIES VIA MARKOV SWITCHING MODELS

Abstract. In this paper we develop a very general Markov switching model for analysis of economic time series. Our general set-up allows us to assess the effects of a variety of model and prior assumptions on the results. The growth rates of US quarterly real gross national product are used to illustrate the proposed analysis. We find that although the evidence is not strong the analysis does not support the model in which the dynamic behavior is constant and that allowing the dynamic structure to change affects the results. We also find that the results are sensitive to the prior specification.     

OUTLIER DIAGNOSTICS IN TIME SERIES ANALYSIS

Abstract. Cook's likelihood displacement is a convenient measure of the impact of a model perturbation on parameter estimates. A commonly used model perturbation in regression is the deletion of a case , or equation. A natural model perturbation in the time series context is the deletion of an observation , or a group of observations. Diagnostics that measure the impact of individual observations on the time series estimates are explored in this paper. A diagnostic that compares the estimates of the innovation variance with and without a particular observation is studied in detail.     

REPLICATED OBSERVATIONS OF LOW ORDER AUTOREGRESSIVE TIME SERIES

Abstract. Independent replicates of first and second order autoregressive stationary time series are considered. Maximum likelihood estimates are studied with special emphasis on asymptotics when the number of replicates tends to infinity and the length of each replicate is fixed.     

ON THE SELECTION OF SUBSET AUTOREGRESSIVE TIME SERIES MODELS

Abstract. The estimation of subset autoregressive time series models has been a difficult problem because of the large number of possible alternative models involved. However, with the advent of model selection criteria based on the maximum likelihood, subset model fitting has become feasible. Using an efficient technique for evaluating the residual variance of all possible subset models, a method is proposed for the fitting of subset autoregressive models. The application of the method is illustrated by means of real and simulated data.     

A SCORE TEST FOR DETECTION OF TIME SERIES OUTLIERS

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.     

ORDER IDENTIFICATION IN MISSPECIFIED AUTOREGRESSIVE TIME SERIES MODELS

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

EFFICIENT ESTIMATION OF NONSTATIONARY TIME SERIES REGRESSION

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.     

SPECTRAL ANALYSIS FOR AMPLITUDE-MODULATED TIME SERIES

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

DISCRIMINANT ANALYSIS FOR STATIONARY VECTOR TIME SERIES

Abstract. In this paper, we shall consider the case where a stationary vector process { X_t } belongs to one of two categories described by two hypotheses π ₁ and π ₂. These hypotheses specify that { X_t } has spectral density matrices f (Λ) and g (Λ) under π ₁ and π ₂, respectively. Although Gaussianity of { X_t } is not assumed, we can formally make the Gaussian likelihood ratio (GLR) based on X (1),… X ( T ). Then an approximation I ( f : g ) of the GLR is given in terms of f (Λ) and g (Λ). If f (Λ) and g (Λ) are known, we can use I ( f : g ) as a classification statistic. It is shown that I ( f : g ) is a consistent classification criterion in the sense that the misclassification probabilities converge to zero as T →∝. When g is contiguous to f , we discuss non-Gaussian robustness of I ( f : g ). A sufficient condition for the non-Gaussian robustness will be given. Also a numerical example will be given.     

REGRESSION MODELS FOR NON-STATIONARY CATEGORICAL TIME SERIES

Abstract. Categorical time series often exhibit non-stationary behaviour, due to the influence of exogenous variables. A parsimonious and flexible class of models is proposed for the statistical analysis of such data. These models are extensions of regression models for stochastically independent observations. Statistical inference can be based on asymptotic properties of the maximum likelihood estimator and of test statistics for linear hypotheses. Weak conditions assuring these properties are stated. Some tests which are of special interest in the time series situation are treated in more detail, for example tests of stationarity or independence of parallel time series.     

DETERMINING THE DEGREE OF DIFFERENCING FOR TIME SERIES VIA THE LOG SPECTRUM

Abstract. While many time series require differencing before a model may be fitted it has been shown that 'overdifferencing' may result in a fitted model with poor long term forecasting properties. This may present real problems when the degree of differencing which is appropriate is fractional. We show that the log spectrum is a natural quantity to consider when attempting to determine the degree of differencing required and outline the distribution theory required. The ideas are shown to extend to the seasonal case and can be used to assess whether seasonal differencing is appropriate.     

ESTIMATION OF MULTIVARIATE TIME SERIES

Abstract. The algorithm proposed here is a multivariate generalization of a procedure discussed by Pearlman (1980) for calculating the exact likelihood of a univariate ARMA model. Ansley and Kohn (1983) have shown how the Kalman filter can be used to calculate the exact likelihood function when not all the observations are known. In Shea (1983) it is shown that this algorithm is much quicker than that of Ansley and Kohn (1983) for all ARMA models except an ARMA (2, 1) and a couple of low-order AR processes and therefore when we have no missing observations this algorithm should be used instead. The Fortran subroutine G13DCF in the NAG (1987) Library fits a vector ARMA model using an adaptation of this algorithm. Experience in the use of this routine suggests that having reasonably good initial estimates of the ARMA parameter matrices, and in particular the residual error covariance matrix, can not only substantially reduce the computing time but more important improve the convergence properties of the minimization procedure. We therefore propose a method of calculating initial estimates of the ARMA parameters which involves using a generalization of the concept of inverse cross covariances from the univariate to the multivariate case. Finally theory is put into practice with the fitting of a bivariate model to a couple of real-life time series.     

MARGINALS OF MULTIVARIATE FIRST-ORDER AUTOREGRESSIVE TIME SERIES MODELS

This paper is concerned with the marginal models associated with a given multivariate first-order autoregressive model. A general theory is developed to determine when reductions in the known orders of the marginal models will occur. When the auto-regressive coefficient matrix has repeated eigenvalues, there may be global reductions in the marginal models. Zeros in the eigenvectors and generalized eigenvectors of the auto-regressive coefficient matrix lead to local reductions in the marginal models. The case when the autoregressive parameter matrix has systematic zeros is also investigated.     

REVERSED RESIDUALS IN AUTOREGRESSIVE TIME SERIES ANALYSIS

Abstract. Both linear and non-linear time series can have directional features which can be used to enhance the modelling and investigation of linear or non-linear autoregressive statistical models. For this purpose, reversed p th-order residuals are introduced. Cross-correlations of residuals and squared reversed residuals allow extensions of current model identification ideas. Quadratic types of partial autocorrelation functions are introduced to assess dependence associated with non-linear models which nevertheless have linear autoregressive correlation structures. The use of these residuals and their cross-correlation functions is exemplified empirically on some deseasonalized river flow data for which a first-order autoregressive model is a satisfactory second-order fit. Parallel theoretical computations are undertaken for the non-linear first-order random coefficient autoregressive model and comparisons are made. While the data are shown to be strongly non-linear, their correlational signatures are found to be convincingly different from those of a first-order autoregressive model with random coefficients.     

NON-LINEAR TIME SERIES MODELLING AND DISTRIBUTIONAL FLEXIBILITY

Abstract. Most of the existing work in non-linear time series analysis has concentrated on generating flexible functional models by specifying non-linear specifications for the mean of a particular process, without much, if any, attention given to the distributional properties of the model. However, as Martin ( J. Time Ser. Anal. 13 (1992), 79–94) has shown, greater flexibility in perhaps a more natural way can be achieved by consideration of distributions from the generalized exponential class. This paper represents an extension of the earlier work of Martin by introducing a flexible class of non-linear time series models which can capture a wide range of empirical behaviour such as skewed, fat-tailed and even multimodal distributions. This class of models is referred to as generalized exponential non-linear time series. A maximum likelihood algorithm is given for estimating the parameters of the model and the framework is applied to estimating the distribution of the movements of the exchange rate.