WHY DO NONINVERTIBLE ESTIMATED MOVING AVERAGES OCCUR?*

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.     

EXACT MAXIMUM LIKELIHOOD ESTIMATION IN AUTOREGRESSIVE PROCESSES

Abstract. The purpose of this paper is to complement the theory of exact maximum likelihood estimation in pure autoregressive processes by differentiating the exact Gaussian likelihood function with respect to the model parameters and obtaining a set of likelihood equations very similar in form to the Yule—Walker equations. The main contribution of this paper is a very simple expression for the derivatives and the resulting likelihood equations in terms of the components of a (p+ 1) x (p+ 1) function of the data, the model parameters (s?², φ) and the autocovariances at lags 0 through p. We propose an iterative algorithm for solving the likelihood equations by alternately solving two linear systems, first for (s?², φ) given current estimates of the autocovariances, then for updated estimates of the autocovariances given current estimates of (s?², φ). The number of operations per iteration is independent of the series length since the algorithm uses the data only through the value of the (p+ 1) x (p+ 1) sufficient statistic.     

Maximum likelihood estimation of higher‐order integer‐valued autoregressive processes

Abstract. In this article, we extend the earlier work of Freeland and McCabe [Journal of time Series Analysis (2004) Vol. 25, pp. 701–722] and develop a general framework for maximum likelihood (ML) analysis of higher‐order integer‐valued autoregressive processes. Our exposition includes the case where the innovation sequence has a Poisson distribution and the thinning is binomial. A recursive representation of the transition probability of the model is proposed. Based on this transition probability, we derive expressions for the score function and the Fisher information matrix, which form the basis for ML estimation and inference. Similar to the results in Freeland and McCabe (2004) , we show that the score function and the Fisher information matrix can be neatly represented as conditional expectations. Using the INAR(2) specification with binomial thinning and Poisson innovations, we examine both the asymptotic efficiency and finite sample properties of the ML estimator in relation to the widely used conditional least squares (CLS) and Yule–Walker (YW) estimators. We conclude that, if the Poisson assumption can be justified, there are substantial gains to be had from using ML especially when the thinning parameters are large.     

ESTIMATION OF THE MULTIVARIATE AUTOREGRESSIVE MOVING AVERAGE HAVING PARAMETER RESTRICTIONS AND AN APPLICATION TO ROTATIONAL SAMPLING

Abstract. The vector autoregressive moving average model with nonlinear parametric restrictions is considered. A simple and easy-to-compute Newton-Raphson estimator is proposed that approximates the restricted maximum likelihood estimator which takes full advantage of the information contained in the restrictions. In the case when there are no parametric restrictions, our Newton-Raphson estimator is equivalent to the estimator proposed by Reinsel et al. (Maximum likelihood estimators in the multivariate autoregressive moving-average model from a generalized least squares view point. J. Time Ser. Anal. 13 (1992), 133–45). The Newton-Raphson estimation procedure also extends to the vector ARMAX model. Application of our Newton-Raphson estimation method in rotational sampling problems is discussed. Simulation results are presented for two different restricted models to illustrate the estimation procedure and compare its performance with that of two alternative procedures that ignore the parametric restrictions.     

A Bayesian Approach to Event Prediction

Abstract.  In a series of papers, Lindgren (1975a, 1985) and de Maré (1980) set the principles of optimal alarm systems and obtained the basic results. Application of these ideas to linear discrete time-series models was carried out by Svensson et al. (1996) . In this paper, we suggest a Bayesian predictive approach to event prediction and optimal alarm systems for discrete time series. There are two novelties in this approach: first, the variation in the model parameters is incorporated in the analysis; second, this method allows 'on-line prediction' in the sense that, as we observe the process, our posterior probabilities and predictions are updated at each time point.     

ON THE STRUCTURE OF THE LIKELIHOOD FUNCTION OF AUTOREGRESSIVE AND MOVING AVERAGE MODELS

Abstract. In finite order normal moving average models the maximum likelihood estimates always exist. For finite order normal autoregressive models sufficient conditions for the existence of maximum likelihood estimates is given. Some cases not satisfying the conditions are studied.     

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.     

APPROXIMATE DISTRIBUTION OF PARAMETER ESTIMATORS FOR FIRST-ORDER AUTOREGRESSIVE MODELS

Abstract. Formulae for the exact bias and mean square error for the least squares for forward-backward least squares estimators are obtained based on the explicit expressions for the moment-generating and characteristic functions of quadratic form in the first-order autoregressive process. Asymptotic expressions for their cumulants and the maximum likelihood estimator are given. Approximations of the distributions of the above estimators are proposed based on the Ornstein-Ulenbeck process. A simple computational procedure for the exact distribution is developed, and some numerical comparisons are given which support the overall good accuracy of the approximation and confirm that the maximum likelihood estimator performs better than the others.     

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.     

VALIDITY OF EDGEWORTH EXPANSIONS FOR STATISTICS OF TIME SERIES

Abstract. In this paper, we discuss the validity of the multivariate Edgeworth expansion of distribution functions of statistics which need not be standardized sums of independent and identically distributed vectors. We apply this result to statistics of time series. In particular, we shall give the asymptotic expansion of the distribution of the maximum likelihood estimator of a parameter of a circular autoregresive moving average process.     

Convolution‐closed models for count time series with applications

There has recently been an upsurge of interest in time series models for count data. Many papers focus on the model with first‐order (Markov) dependence and Poisson innovations. Our paper considers practical models that can capture higher‐order dependence based on the work of Joe (1996). In this framework we are able to model both equidispersed and overdispersed marginal distributions of data. The latter is approached using generalized Poisson innovations. Central to the models is the use of the property of closure under convolution of certain families of random variables. The models can be thought of as stationary Markov chains of finite order. Parameter estimation is undertaken by maximum likelihood, inference procedures are considered and means of assessing model adequacy employed. Applications to two new data sets are provided.     

On the Evaluation of the Information Matrix for Multiplicative Seasonal Time‐Series Models

Abstract. This paper gives a procedure for evaluating the Fisher information matrix for a general multiplicative seasonal autoregressive moving average time‐series model. The method is based on the well‐known integral specification of Whittle [Ark. Mat. Fys. Astr. (1953) vol. 2. pp. 423–434] and leads to a system of linear equations, which is independent of the seasonal period and has a closed solution. It is shown to be much simpler, in general, than the method of Klein and Mélard [Journal of Time Series Analysis (1990) vol. 11, pp. 231–237], which depends on the seasonal period. It is also shown that the nonseasonal method of McLeod [Biometrika (1984) vol. 71, pp. 207–211] has the same basic features as that of Klein and Mélard. Explicit solutions are obtained for the simpler nonseasonal and seasonal models in common use, a feature which has not been attempted with the Klein–Mélard or the McLeod approaches. Several illustrations of these results are discussed in detail.     

Generalized least squares estimation for cointegration parameters under conditional heteroskedasticity

In the presence of generalized conditional heteroscedasticity (GARCH) in the residuals of a vector error correction model (VECM), maximum likelihood (ML) estimation of the cointegration parameters has been shown to be efficient. On the other hand, full ML estimation of VECMs with GARCH residuals is computationally difficult and may not be feasible for larger models. Moreover, ML estimation of VECMs with independently identically distributed residuals is known to have potentially poor small sample properties and this problem also persists when there are GARCH residuals. A further disadvantage of the ML estimator is its sensitivity to misspecification of the GARCH process. We propose a feasible generalized least squares estimator which addresses all these problems. It is easy to compute and has superior small sample properties in the presence of GARCH residuals.     

Poisson QMLE of Count Time Series Models

Regularity conditions are given for the consistency of the Poisson quasi‐maximum likelihood estimator of the conditional mean parameter of a count time series model. The asymptotic distribution of the estimator is studied when the parameter belongs to the interior of the parameter space and when it lies at the boundary. Tests for the significance of the parameters and for constant conditional mean are deduced. Applications to specific integer‐valued autoregressive (INAR) and integer‐valued generalized autoregressive conditional heteroscedasticity (INGARCH) models are considered. Numerical illustrations, Monte Carlo simulations and real data series are provided.