PROFILE SUMMARIES FOR ARIMA TIME SERIES MODEL PARAMETERS

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.     

REGRESSION OF SPECTRAL ESTIMATORS WITH FRACTIONALLY INTEGRATED TIME SERIES

Abstract. Assuming a normal distribution we supplement the proof of periodogram regression suggested by Geweke and Porter-Hudak ( J. Time Ser. Anal. 4 (1983) 221–38) in order to estimate and test the difference parameter of fractionally integrated autoregressive moving-average models. The procedure proposed by Kashyap and Eom ( J. Time Ser. Anal. 9 (1988) 35–41) arises as a special case and is found to be correct if the true parameter value is negative. Regression of the smoothed periodogram yields estimators for the difference parameter with much faster vanishing variance; no asymptotic distribution can be derived, however. In computer experiments we find that the smoothed periodogram regression may be superior to pure periodogram regression when we have to discriminate between autoregression and fractional integration     

NONPARAMETRIC ESTIMATORS FOR TIME SERIES

Abstract. Kernel multivariate probability density and regression estimators are applied to a univariate strictly stationary time series X _r We consider estimators of the joint probability density of X _t at different t -values, of conditional probability densities, and of the conditional expectation of functionals of X _v given past behaviour. The methods seem of particular relevance in light of recent interest in non-Gaussian time series models. Under a strong mixing condition multivariate central limit theorems for estimators at distinct points are established, the asymptotic distributions being of the same nature as those which would derive from independent multivariate observations.     

ESTIMATION FOR THE FIRST-ORDER DIAGONAL BILINEAR TIME SERIES MODEL

Abstract. The problem of estimation of the parameter b in the simple diagonal bilinear model { X _t }, X_t = e_t + be _{t -1} X_{t -1}, is considered, where { e_t } is Gaussian white noise with zero mean and possibly unknown variance ². The asymptotic normality of the moment estimator of b is established for the two cases when ² is known and ² is unknown. It is noted that the limit distribution of the least-squares cannot easily be derived analytically. A bootstrap comparison of the sampling distributions of the least-squares and moment estimates shows that both are asymptotically normal with the least-squares estimate being the more efficient.     

MULTIPLE BILINEAR TIME SERIES MODELS

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.     

A CONDITIONAL LEAST SQUARES APPROACH TO BILINEAR TIME SERIES ESTIMATION

Abstract. In this paper a conditional least squares (CLS) procedure for estimating bilinear time series models is introduced. This method is applied to a special superdiagonal bilinear model which includes the classical linear autoregressive moving-average model as a particular case and it is proven that the limiting distribution of the CLS estimates is Gaussian and that the law of the iterated logarithm holds.     

GEOMETRIC ERGODICITY OF A DOUBLY STOCHASTIC TIME SERIES MODEL

Abstract. We demonstrate that a large class of doubly stochastic time series models are geometrically ergodic, and hence admit second-order stationary solutions.
We also establish a version of the strong law of large numbers, the law of the interated logorithm and the central limit theorem for the stochastic processes under consideration.     

A TEST FOR LINEARITY OF STATIONARY TIME SERIES

Abstract. A standard assumption that is often made in time series analysis is that the series conforms to a linear model. The object of this paper is to describe statistical tests for testing this assumption. The tests are constructed from the bispectral density function, and depend on the application of Hotelling T ². These tests are illustrated with two real time series and four simulated time series. Some guidelines about the choice of the parameters are also included.     

ON THE ERGODICITY OF BILINEAR TIME SERIES MODELS

Abstract. Existence, strict stationarity and ergodicity of Bilinear Time Series Models for a given input White Noise and parameter values are studied in detail in this paper. The use of ergodicity in the estimation of parameters is also hinted at in this article.     

SOME PROPERTIES OF CONDITIONAL QUASI-LIKELIHOOD FUNCTIONS FOR TIME SERIES MODEL FITTING

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.     

NONLINEAR TRANSFORMATIONS OF INTEGRATED TIME SERIES:A RECONSIDERATION

Abstract. In this paper I reconsider two of the questions raised by Granger and Hallman (Nonlinear transformations of integrated time series. J. Time Ser. Anal. 12 (1991), 207–24):(i) If X_t is I(1) and Z_t=h(X_t), is Z_t also I(1)? (ii) Can X_t and h(X_t) be cointegrated? The distinction between I(1) and I(0) processes is replaced by the distinction between long memory and short memory processes, where for short memory I mean strong mixing. By exploiting the fact that random walks (with positive trend component) are martingales (submartingales) and are also first-order Markov, I show that (a) unbounded convex (concave) and strictly monotonic transformations of random walks are always long memory processes, (b) polynomial, strictly convex (concave) transformations of random walks display a unit root component, but the first differences of such transformations need not be short memory, and (c) X_t and h(X_t), with h an unbounded convex (concave) or strictly monotonic function, can never be cointegrated.     

ON THE EXISTENCE OF SOME BILINEAR TIME SERIES MODELS

Abstract. The existence of a multivariate strictly stationary stochastic process conforming to a certain bilinear time series model is discussed.     

COMPARISON OF TIME AND CROSS-SECTIONAL AGGREGATION UNDER A TIME SERIES RANDOM COMPONENT MODEL

Abstract. Small-area estimation under a stationary time series random component model is considered. Cross-sectional aggregation and varying degrees of time aggregation are treated as competing prediction methods. An estimated mean-squared prediction error criterion is used to compare these methods. Some exact and asymptotic properties of this criterion are developed, a consistent estimator of the associated asymptotic variance is presented and simultaneous approximate confidence intervals for the mean-squared prediction errors are discussed. Time aggregation of a single series is considered as a special case. In addition, an extension to the assessment of mean-squared prediction errors of synthetic small-area predictors is outlined.     

ON THE THIRD-ORDER MOMENT STRUCTURE AND BISPECTRAL ANALYSIS OF SOME BILINEAR TIME SERIES

For the bilinear time series model X_t=βX_t-k^e_t-t+^e_t, 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.     

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.     

SPECTRAL ESTIMATION AND DECONVOLUTION FOR A LINEAR TIME SERIES MODEL

Abstract. A vector linear time series model is observed as the sum of a convolution of an unknown signal and an additive noise process. The main objective is the estimation or deconvolution of the signal when the spectra of the signal and noise processes are unknown. We prove the strong consistency of a class of nonparametric spectral estimators derived by maximizing a particular Gaussian likelihood function. We also study the mean square convergence of the finite-sample deconvolution estimators as a function of the sample length T , the filter length M and the spectral bandwidth B_T = L_T/T .     

ON LINEAR PROCESSES WITH GIVEN MOMENTS

Abstract. Let X _t = c ₀ Y _t + c ₁ Y _{t -1}+… be a linear process with known coefficients c _k , where Y _t is a strict white noise. Let m ₁, …, m _2r be given numbers. A method is presented to determine whether there exists a distribution of Y _t such that EX ^k _t = m _k for k = 1, …, 2 r . In the positive case, such a distribution of Y _t is described. Some explicit formulas for AR(1) and AR(2) models are derived. The results can be used for simulating a process with given moments of its stationary distribution. The procedure also enables proof that some stationary distributions cannot belong to the given linear process.     

BAYES STOPPING RULES IN A CHANGE-POINT MODEL WITH A RANDOM HAZARD RATE

In the Bayes sequential change-point problem, an assumption of a fully known prior distribution of a change-point is usually impracticable. At every moment, one often knows only the discrete hazard function, that is, the probability of a change occurring before the next observation is collected, given that it has not occurred so far. In the randomized model, the observed or predicted values of the hazard function are assumed to form a Markov chain. Under these assumptions, the optimal change-point detection stopping rules are derived for two popular loss functions introduced in Shiryaev (1978) and Ritov (1990). Derivations are based on the theory of optimal stopping of Markov sequences.     

A NONSTATIONARY TIME SERIES MODEL AND ITS FITTING BY A RECURSIVE FILTER

Abstract. The use of the state space representation for the analysis of nonstationary time series is proposed. For the fitting of the models, the use of a modified AIC based on the likelihood of the innovation process is proposed. A square root filter/smoother algorithm for the evaluation of the likelihood and state estimation is discussed.     

ON THE SQUARED RESIDUAL AUTOCORRELATIONS IN NON-LINEAR TIME SERIES WITH CONDITIONAL HETEROSKEDASTICITY

Abstract. Time series with a changing conditional variance have been found useful in many applications. Residual autocorrelations from traditional autoregressive moving-average models have been found useful in model diagnostic checking. By analogy, squared residual autocorrelations from fitted conditional heteroskedastic time series models would be useful in checking the adequacy of such models. In this paper, a general class of squared residual autocorrelations is defined and their asymptotic distribution is obtained. The result leads to some useful diagnostic tools for statisticians using conditional heteroskedastic time series models. Some simulation results and an illustrative example are also reported.