STABLE ALGORITHMS FOR THE STATE SPACE MODEL

Abstract. Numerically stable algorithms are developed for filtering, likelihood evaluation, generalized least squares computation and smoothing where data are generated by a state space model. The algorithms handle diffuse initial states in a numerically safe way. Singular innovation covariance matrices, such as those which arise in series with missing values, are dealt with. The algorithms generalize stable algorithms for ordinary least-squares computations.     

STATE SPACE MODELS WITH DIFFUSE INITIAL CONDITIONS

Abstract. A state space model with diffuse initial conditions is considered. A simple and direct proof of the algorithm for computing the likelihood function and minimum mean square estimators of the state is given     

A FLEXIBLE STATE SPACE MODEL AND ITS APPLICATIONS

The standard state space model treats observations as imprecise measurement of the Markovian states. Our flexible model handles the states and observations symmetrically, which are simultaneously determined by past observations and up to first‐lagged states. The only distinction between the states and observations is the observability. When it is applied to the autoregressive moving average, dynamic factor and stochastic volatility models, the state space form is both parsimonious and intuitive, for low‐dimension states are constructed simply by stacking all the relevant but unobserved components in the structural model.     

ORDERS AND INITIAL VALUES OF NON-STATIONARY MULTIVARIATE ARMA MODELS

Abstract. Akaike's stepwise canonical correlations procedure identifies each autoregressive (AR) order (assuming a lesser moving-average (MA) order) and gives initial parameter estimates of a stationary multivariate autoregressive moving-average model. We show that a similar procedure is valid for the non-stationary model when this can be approximated by a large-order AR-only model which has consistent least-squares estimators.
The procedure can also be adapted for all MA orders by fixing the overall maximum of any MA order less its corresponding AR order. We perform the canonical correlations procedure on a succession of values of this maximum and compare the results to obtain one or more prospective model structures.     

FILTERING AND SMOOTHING IN STATE SPACE MODELS WITH PARTIALLY DIFFUSE INITIAL CONDITIONS

Abstract. Ansley and Kohn ( Annals of Statistics , 1985) generalized the Kalman filter to handle state space models with partially diffuse initial conditions and used this filter to compute the marginal likelihood of the observations efficiently. In this paper we simplify the algorithm and make it numerically more accurate and operationally more efficient. Based on this filtering algorithm we obtain a corresponding smoothing algorithm for the state vector.     

DATA AUGMENTATION AND DYNAMIC LINEAR MODELS

Abstract. We define a subclass of dynamic linear models with unknown hyperpara-meter called d -inverse-gamma models. We then approximate the marginal probability density functions of the hyperparameter and the state vector by the data augmentation algorithm of Tanner and Wong. We prove that the regularity conditions for convergence hold. For practical implementation a forward-filtering-backward-sampling algorithm is suggested, and the relation to Gibbs sampling is discussed in detail.     

BOOTSTRAPPING STATIONARY AUTOREGRESSIVE MOVING-AVERAGE MODELS

Abstract. In this paper we develop an asymptotic theory for application of the bootstrap to stationary stochastic processes of autoregressive moving-average (ARMA) type, with known order ( p, q ). We give a proof of the asymptotic validity of the bootstrap proposal applied to M estimators for the unknown parameter vector of the process. For this purpose we derive an asymptotic expansion for M estimators in ARMA models and construct an estimate for the unknown distribution function of the residuals which in principle are not observable. A small simulation study is also included.     

PARTIAL AUTOCORRELATION PROPERTIES FOR NON-STATIONARY AUTOREGRESSIVE MOVING-AVERAGE MODELS

Abstract. We are primarily interested in relating the partial autocorrelation behaviour of an autoregressive integrated moving-average process of order ( p, d, q ), { Z _i } say, with those of its D -differenced processes {(1 - B ) ^D Z _i } ( D = 1, …, d ). To this end, we evaluate the early partial correlations corresponding to serial correlations which initially follow a slow linear decline from unity. These partials, to a first approximation, take a small constant negative value from lag 2 onwards. We also demonstrate a relationship between the theoretical partials π _k and π _k (Δ) for a once-integrated process { Z _i } and its first-differenced process {(1 - B ) Z _i } respectively. These results carry over to cases where the non-stationary zeros are at -1, and differencing is replaced by the corresponding 'simplifying' transformation implicit in the operator 1 + B.     

LINEAR INTERPOLATORS AND THE INVERSE CORRELATION FUNCTION OF NON-STATIONARY TIME SERIES

Abstract. The inverse correlation function of a stationary time series was introduced by Cleveland (The inverse autocorrelations of a time series and their applications. Technometrics 14 (1972), 277–93). In this paper inverse correlations are defined for non-stationary time series {x_t, integer t} such that y_t= (1 —B^s)^dx_t is second-order stationary. The linear interpolator and the inverse process of {x_t} are also defined:their weights are shown to be time invariant and proportional to the inverse correlations. The interpolation method for the estimation of the inverse correlation function of a stationary time series is extended to the non-stationary series {x_t} and the asymptotic properties of the estimates are found to be similar to those in the stationary case.     

LEAVE-K-OUT DIAGNOSTICS IN STATE-SPACE MODELS

Abstract. The paper derives an algorithm for computing leave- k -out diagnostics for the detection of patches of outliers for stationary and nonstationary state-space models with regression effects. The algorithm is based on a reverse run of the Kalman filter on the smoothing errors and is both efficient and easy to implement. The US index of industrial production for textiles is used to illustrate the application of the algorithm.     

ESTIMATION OF THE NON-STATIONARY FACTOR IN ARUMA MODELS

Abstract. Two methods for the estimation of the non-stationary factor in ARUMA models are given. Both methods yield strongly consistent estimators and the roots of the corresponding filters lie on the unit circle.     

STATE TRANSITION SPECIFICATION IN STATE-SPACE MODELS

Abstract. This note investigates an approach to state transition specification in statespace models. The approach generalizes the procedure whereby first or higher order differences in the state are modelled as white noise.     

ORDER IDENTIFICATION STATISTICS IN STATIONARY AUTOREGRESSIVE MOVING-AVERAGE MODELS:VECTOR AUTOCORRELATIONS AND THE BOOTSTRAP

Abstract. In this paper we consider the vector autocorrelation approach for identifying ARMA ( p, q ) models and use a bootstrap procedure in order to evaluate the distribution of the corresponding sample statistics by means of a resampling scheme for the residuals when p and q are unknown. The asymptotic validity of the bootstrap procedure applied to the vector autocorrelation estimates is established. Some simulations and examples demonstrating the appropriateness of the proposed bootstrap procedure in comparison with large-sample Gaussian approximations are included.     

DYNAMIC STATE-SPACE MODELS

Abstract. In many cases, multiple time series can be viewed as realizations of the same underlying process and such data usually accumulate in time. The historic time-series data provide important information for our current prediction. In this paper, we extend the traditional state-space model to a general dynamic scheme, in which estimation and prediction across time series and within a time series are handled by a unified O ( N ) sequential procedure. Under this framework, the information from historic data serves as the prior for the current time series and the estimation and prediction of a time series can incorporate the information from other time series as well as its own history. The solution is to construct sequentially a new state-space model for the next time series conditional on the past time series. Because we achieve the general dynamic estimation and prediction through constructing new conditional state-space models, existing estimation procedures for state-space models can be adapted into this framework with minimal modifications. An application to infant growth curves is used as illustration.     

STATIONARY TANGENT: THE DISCRETE AND NON-SMOOTH CASE

Abstract. In Keich (2000 ),we define a stationary tangent process, or a locally optimal stationary approximation, to a real non-stationary smooth Gaussian process. This paper extends the idea by constructing a discrete tangent – a `locally' optimal stationary approximation – for a discrete time, real Gaussian process. Analogously to the smooth case, our construction relies on a generalization of the recursion formula for the orthogonal polynomials of the spectral distribution function. More precisely, we use a generalization of the Schur parameters to identify the stationary tangent. By way of discretizing, we later demonstrate how this tangent can be used to obtain `good' local stationary approximations to non-smooth continuous time, real Gaussian processes. Further, we demonstrate how, analogously to the curvatures in the smooth case, the Schur parameters can be used to determine the order of stationarity of a non-smooth process.     

DETERMINING THE NUMBER OF REGIMES IN MARKOV SWITCHING VAR AND VMA MODELS

We give stable finite‐order vector autoregressive moving average (p ^* ,q ^* ) representations for M‐state Markov switching second‐order stationary time series whose autocovariances satisfy a certain matrix relation. The upper bounds for p ^* and q ^* are elementary functions of the dimension K of the process, the number M of regimes, the autoregressive and moving‐average orders of the initial model. If there is no cancellation, the bounds become equalities, and this solves the identification problem. Our classes of time series include every M‐state Markov switching multi‐variate moving‐average models and autoregressive models in which the regime variable is uncorrelated with the observable. Our results include, as particular cases, those obtained by Krolzig (1997) and improve the bounds given by Zhang and Stine (2001) and Francq and Zakoïan (2001) for our classes of dynamic models. A Monte Carlo experiment and an application on foreign exchange rates complete the article. Copyright © 2013 John Wiley & Sons, Ltd.     

FINITE SAMPLE PREDICTION AND OVERDIFFERENCING

Abstract. If the process generating a time series contains a deterministic component the differencing operations carried out to achieve stationarity may lead to an ARMA model which is strictly noninvertible. This is known as overdifferencing but it is shown here that overdifferencing need not have serious implications for prediction provided that a finite sample prediction procedure is used. The proposed method is based on the Kalman filter and it allows both the optimal predictors and their mean square errors to be computed.     

AN INNOVATION STATE SPACE APPROACH FOR TIME SERIES FORECASTING

Abstract. An innovation state space modelling approach is presented in which the structures and parameters of a model are determined by an identification algorithm proposed by Tse and Weinert ( IEEE Trans. Automat. Contr. 120 (1975), 603–13) and the singular value decomposition technique. This approach is applied to two typical data series to illustrate its use, and its forecasting accuracy is compared with other time series approaches.     

A STUDY OF THE APPLICATION OF STATE-DEPENDENT MODELS IN NON-LINEAR TIME SERIES ANALYSIS

Abstract. The theory of state-dependent models was developed by Priestley (1980), and a few simple applications were given in Priestley (1981). In this paper, an extensive study of the application of state-dependent models to a wide variety of non-linear time series data is carried out. Both real and simulated data are used in the study, and the problems encountered are highlighted. The method is demonstrated to be successful in practice in many cases, and suggestions for the further development of the algorithm are also given.     

广义线性模型与非线性模型参数的统计推断

概述了广义线性模型(含半参数广义线性模型)与非线性模型研究的新进展,着重讨论了它们参数估计的统计推断。近年来,广义线性模型与非线性模型的研究发展迅速,有着广泛的应用前景。