ON THE DETERMINATION OF THE NUMBER OF REGIMES IN MARKOV-SWITCHING AUTOREGRESSIVE MODELS

Abstract. Dynamic models with parameters that are allowed to depend on the state of a hidden Markov chain have become a popular tool for modelling time series subject to changes in regime. An important question that arises in applications involving such models is how to determine the number of states required for the model to be an adequate characterization of the observed data. In this paper, we investigate the properties of alternative procedures that can be used to determine the state dimension of a Markov-switching autoregressive model. These include procedures that exploit the ARMA representation which Markov-switching processes admit, as well as procedures that are based on optimization of complexity-penalized likelihood measures. Our Monte Carlo analysis reveals that such procedures estimate the state dimension correctly, provided that the parameter changes are not too small and the hidden Markov chain is fairly persistent. The use of the various methods is also illustrated by means of empirical examples.     

ESTIMATION OF THE MOVING-AVERAGE REPRESENTATION OF A STATIONARY PROCESS BY AUTOREGRESSIVE MODEL FITTING

Abstract. The Hannan-Rissanen procedure for recursive order determination of an autoregressive moving-average process provides 'non-parametric' estimators of the coefficients b ( u ), say, of the moving-average representation of a stationary process by auto-regressive model fitting, and also that of the cross-covariances, c ( u ), between the process and its linear innovations. An alternative 'autoregressive' estimator of the b ( u ) is obtained by inverting the autoregressive transfer function. Some uses of these estimators are discussed, and their asymptotic distributions are derived by requiring that the order k of the fitted autoregression approaches infinity simultaneously with the length T of the observed time series. The question of bias in estimating the parameters is also examined.     

ON THE INVERTIBILITY OF MULTIVARIATE LINEAR PROCESSES

Abstract. It is shown that a multivariate linear stationary process whose coefficients are absolutely summable is invertible if and only if its spectral density is regular everywhere. This general characterization of invertibility is applied later to the case of a linear process having an autoregressive moving-average (ARMA) representation. Under the usual assumptions, it is deduced that a process Y described by an ARMA(φ, TH) model is invertible if and only if the polynomial detTH( z ) has no roots on the unit circle. Given an invertible process Y which has an ARMA representation, it is finally shown that the process YT , where YT , =ε _i =0^l S _i Y _t-i , is invertible if and only if the matrix S ( z ) =ε _{i =0}^l S _i z ⁱ is of full rank for all z of modulus 1. It follows, in particular, that any subprocess of an invertible ARMA process is also invertible.     

THE IDENTIFICATION OF SEASONAL AUTOREGRESSIVE MODELS

Abstract. In this paper we present a new approach for identifying seasonal autoregressive models and the degree of differencing required to induce stationarity in the data. The identification method is iterative and consists in systematically fitting increasing order models to the data and then verifying that the resulting residuals behave like white noise using a two-stage autoregressive order determination criterion. Once the order of the process is determined the identified structure is tested to see if it can be simplified. Simulation experiments based on different model structures with varying numbers of observations and parameter values as well as some macroeconomic data are used to evaluate the performance of the procedure.     

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 NOTE ON THE GENERATION OF INDEPENDENT REALIZATIONS OF A VECTOR AUTOREGRESSIVE MOVING-AVERAGE PROCESS

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 T ₀ 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 T ₀ 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.     

A Flexible Univariate Autoregressive Time-Series Model for Dispersed Count Data

Integer-valued time series data have an ever-increasing presence in various applications (e.g., the number of purchases made in response to a marketing strategy, or the number of employees at a business) and need to be analyzed properly. While a Poisson autoregressive (PAR) model would seem like a natural choice to model such data, it is constrained by the equi-dispersion assumption (i.e., that the variance and the mean equal). Hence, data that are over- or under-dispersed (i.e., have the variance greater or less than the mean respectively) are improperly modeled, resulting in biased estimates and inaccurate forecasts. This work instead develops a flexible integer-valued autoregressive model for count data that contain over- or under-dispersion. Using the Conway–Maxwell–Poisson (CMP) distribution and related distributions as motivation, we develop a first-order sum-of-CMP's autoregressive (SCMPAR(1)) model that will instead offer a generalizable construct that captures the PAR, and versions of what we refer to as a negative binomial AR model, and binomial AR model respectively as special cases, and serve as an overarching representation connecting these three special cases through the dispersion parameter. We illustrate the SCMPAR model's flexibility and ability to effectively model count time series data containing data dispersion through simulated and real data examples.     

ON THE RECURSIVE FITTING OF SUBSET AUTOREGRESSIONS

Abstract. In fitting a vector autoregressive process which may include lags up to and including lag K , we may wish to search for the subset vector autoregressive process of size k (where k is the number of lags with non-zero coefficient matrices, k = 1, 2, K ) which has the minimum generalized residual variance. This paper provides a recursive procedure, which is initialized by evaluating all 'forwardand'backward' autoregressions in which k = 1. The recursion then allows one to develop successively all subsets of size k = 2, k = 3 up to k = K .
The optimum subset vector autoregression is found by employing the proposed recursive procedures in conjunction with model selection criteria. This approach is used on simulated data to assess its performance and to re-examine the annual trappings of the Canadian lynx investigated by Tong (1977).     

ON THE EXISTENCE OF STATIONARY THRESHOLD AUTOREGRESSIVE MOVING-AVERAGE PROCESSES

Abstract. Conditions for the existence of causal and strictly stationary solutions of the equations defining a self-exciting threshold autoregressive moving-average (SETARMA) model are derived. For threshold autoregressive models we allow the autoregressive coefficients to be random and derive sufficient conditions for geometric ergodicity and the existence of strictly and weakly stationary solutions of the defining equations.     

THE RECURSIVE FITTING OF SUBSET VARX MODELS

Abstract. A vector time series model of the form A(L)y(t) + B(L)x(t) =ε(t) is known as a vector autoregressive model with exogenous variables (VARX model) and involves a regressand vector y(t) and a regressor vector x(t). This paper provides a method for the recursive fitting of subset VARX models. It suggests the use of ascending recursions in conjunction with an order selection criterion to choose an 'optimum' subset VARX model.     

ON THE MAXIMUM ENTROPY PROPERTY OF NONLINEAR AUTOREGRESSIONS

Abstract. The assumption of a linear autoregressive model for time series has often been justified on the basis of a maximum entropy principle. The purpose of this short note is to point out that the class of nonlinear autoregressions is also characterized by a maximum entropy property.     

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.     

Existence and Stochastic Structure of a Non-negative Integer-valued Autoregressive Process

A simple criterion is given for the existence of a generalized integer-valued autoregressive (GINAR( p )) process. We show that the GINAR( p ) process is nothing but an AR( p ) process. The spectral density gives a good insight into the stochastic structure of a GINAR( p ) model. The spectral representation of the process is explicitly given. The estimation of parameters of the process is also discussed and clarifies some results presented by Du and Li (The integer-valued autoregressive (INAR( p )) model. J. Times Ser. Anal. , 12 (1991), 129--42). Finally, we describe the number of seizures of an epileptic patient using a model of this class.     

ON THE ESTIMATION OF THE INVERSE CORRELATION FUNCTION

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.     

DETERMINING THE ORDER OF A VECTOR AUTOREGRESSION WHEN THE NUMBER OF COMPONENT SERIES IS LARGE

Abstract. We contrast the performance of several methods used for identifying the order of vector autoregressive (VAR) processes when the number K of component series is large. Through simulation experiments we show that their performance is dependent on K , the number of nonzero elements in the polynomial matrices of the VAR parameters and the permitted upper limit of the order used in testing the autoregressive structure. In addition we introduce a new quite powerful multivariate order determination criterion.     

A Stationary Spatio-Temporal GARCH Model

We introduce a lagged nearest-neighbour, stationary spatio-temporal generalized autoregressive conditional heteroskedasticity (GARCH) model on an infinite spatial grid that opens for GARCH innovations in a space-time ARMA model. This is illustrated by a real data application to a classical dataset of sea surface temperature anomalies in the Pacific Ocean. The model and its translation invariant neighbourhood system are wrapped around a torus forming a model with finite spatial domain, which we call circular spatio-temporal GARCH. Such a model could be seen as an approximation of the infinite one and simulation experiments show that the circular estimator with a straightforward bias correction performs well on such non-circular data. Since the spatial boundaries are tied together, the well-known boundary issue in spatial statistical modelling is effectively avoided. We derive stationarity conditions for these circular processes and study the spatio-temporal correlation structure through an ARMA representation. We also show that the matrices defined by a vectorized version of the model are block circulants. The maximum quasi-likelihood estimator is presented and we prove its strong consistency and asymptotic normality by generalizing results from univariate GARCH theory.     

ON THE ASYMPTOTIC EFFICIENCY OF ESTIMATORS OF THE PARAMETERS OF AN ARMA PROCESS

Abstract. In this paper we derive a lower bound on the asymptotic covariance matrix of an estimator of the parameters of an autoregressive moving average (ARMA) process when the innovations are not necessarily Gaussian.     

ON THE UNBIASEDNESS PROPERTY OF AIC FOR EXACT OR APPROXIMATING LINEAR STOCHASTIC TIME SERIES MODELS

Abstract. A rigorous analysis is given of the asymptotic bias of the log maximum likelihood as an estimate of the expected log likelihood of the maximum likelihood model, when a linear model, such as an invertible, gaussian ARMA ( p, q ) model, with or without parameter constraints, is fit to stationary, possibly non-gaussian observations. It is assumed that these data arise from a model whose spectral density function either (i) coincides with that of a member of the class of models being fit, or, that failing, (ii) can be well-approximated by invertible ARMA ( p, q ) model spectral density functions in the class, whose ARMA coefficients are parameterized separately from the innovations variance. Our analysis shows that, for the purpose of comparing maximum likelihood models from different model classes, Akaike's AIC is asymptotically unbiased, in case (i), under gaussian or separate parametrization assumptions, but is not necessarily unbiased otherwise. In case (ii), its asymptotic bias is shown to be of the order of a number less than unity raised to the power max { p, q } and so is negligible if max { p, q } is not too small. These results extend and complete the somewhat heuristic analysis given by Ogata (1980) for exact or approximating autoregressive models.