A measure of output gap for Italy through structural time series models

The aim of this paper is to achieve a reliable estimate of the output gap for Italy through the development of several models within the class of the unobserved component time series models. These formulations imply the decomposition of output into a trend component (the 'potential output') and a cycle component (the 'output gap'). Both univariate and multivariate methods will be explored. In the former, only one measure of aggregate activity, such as GDP, is considered; in the latter, unemployment and industrial production are introduced. A comparison with alternative measures of output gap, mainly those published by international organisations, will conclude.     

Detection of patterns in noisy time series

In the context of a research project in ergonomy, myoelectric signals monitored over two to three hour periods gave rise to long noisy time series, which were smoothed using running medians. Tests developed by the authors show that the patterns displayed by the smoothed time series are not artifacts of smoothed white noise. Indeed, the smoothed series show amplitude fluctuations and short‐term correlations which are larger than those obtained by applying running medians to independent, identically distributed data. The key idea is that of reduction of data to binary signals.     

Invertibility of non-linear time series models

Sufficient conditions for invertibility of non-linear time series models are available in the literature only for a few special cases. In this paper a practical and general method for checking invertibility is presented. Briefly stated, it consists of feeding independent and identically distributed innovations into the non-linear model and then observing whether the model blows up or not. Using this idea invertibility conditions are derived for several recently proposed non-linear moving average models. Finally, the method is applied to a number of bilinear models fitted to economic time series.     

Monitoring parameter change in time series models

In this paper, we develop a monitoring procedure for an early detection of parameter changes in time series models. We design the monitoring procedure in general time series models and apply it to the changes for the autocovariances of linear processes, GARCH parameters, and underlying distributions. Simulation results are provided for illustration.     

Estimation in long memory time series models

A regression type estimator of the parameter d in fractionally differenced ARMA (p,q) processes is presented. The proposed estimator is shown to be mean square consistent. Its performance is compared with some of the existing estimators via a simulation study.     

Dynamic structural models with covariates for short-term forecasting of time series with complex seasonal patterns

This work presents a framework of dynamic structural models with covariates for short-term forecasting of time series with complex seasonal patterns. The framework is based on the multiple sources of randomness formulation. A noise model is formulated to allow the incorporation of randomness into the seasonal component and to propagate this same randomness in the coefficients of the variant trigonometric terms over time. A unique, recursive and systematic computational procedure based on the maximum likelihood estimation under the hypothesis of Gaussian errors is introduced. The referred procedure combines the Kalman filter with recursive adjustment of the covariance matrices and the selection method of harmonics number in the trigonometric terms. A key feature of this method is that it allows estimating not only the states of the system but also allows obtaining the standard errors of the estimated parameters and the prediction intervals. In addition, this work also presents a non-parametric bootstrap approach to improve the forecasting method based on Kalman filter recursions. The proposed framework is empirically explored with two real time series.     

Bootstrap-based evaluation of markov-switching time series models

This paper explores the possibility of evaluating the adequacy of Markov-switching time series models by comparing selected functionals (such as the spectral density function and moving empirical moments) obtained from the data with those of the fitted model using a bootstrap algorithm. The proposed model checking procedure is easy to implement and flexible enough to be adapted to a wide variety of models with parameters subject to Markov regime-switching. Examples with real and artificial data illustrate the potential of the methodology.     

Recursive estimation of bilinear time series models

In this paper we are concerned with the recursive estimation of bilinear models. Some methods from linear time invariant systems are adapted to suit bilinear time series models. The time-varying Kalman filter and associated parameter estimation algorithm is carried on the bilinear time series models. The methods are illustrated with examples.     

A recursive approach to parameter estimation in regression and time series models

In this paper we discuss the recursive (or on line) estimation in (i) regression and (ii) autoregressive integrated moving average (ARIMA) time series models. The adopted approach uses Kalman filtering techniques to calculate estimates recursively. This approach is used for the estimation of constant as well as time varying parameters. In the first section of the paper we consider the linear regression model. We discuss recursive estimation both for constant and time varying parameters. For constant parameters, Kalman filtering specializes to recursive least squares. In general, we allow the parameters to vary according to an autoregressive integrated moving average process and update the parameter estimates recursively. Since the stochastic model for the parameter changes will "be rarely known, simplifying assumptions have to be made. In particular we assume a random walk model for the time varying parameters and show how to determine whether the parameters are changing over time. This is illustrated with an example.     

Cluster of time series models: an example

We show that the various times series models, reported in the literature, for the Canadian lynx data form interesting clusters.     

Estimating time to detect time trends in continuous cropping

SUMMARY In long-term field trials comparing different sequences of crops and husbandry practices, the identification and understanding of trends in productivity over time is an important issue of sustainable crop production. This paper presents a statistical technique for the estimation of time trends in yield variables of a seasonal annual crop under continuous cropping. The estimation procedure incorporates the correlation structure, which is assumed to follow first-order autocorrelation in the errors that arise over time on the same plot. Because large differences in annual rainfall have a major effect on crop performance, rainfall has been allowed for in the estimation of the time trends. Expressions for the number of years (time) required to detect statistically significant time trends have been obtained. Illustrations are based on a 7-year data set of grain and straw yields from a trial in northern Syria. Although agronomic interpretation is not intended in this paper, the barley yield data indicated that a significant time trend can apparently be detected even in a suboptimal data set of 7 years' duration.     

Structural changes in autoregressive models for binary time series

We study autoregressive models for binary time series with possible changes in their parameters. A procedure for detection and testing of a single change is suggested. The limiting behavior of the test statistic is derived. The performance of the test is analyzed under the null hypothesis as well as under different alternatives via a simulation study. Application of the method to a real data set on US recession is provided as an illustration.     

Priors and component structures in autoregressive time series models

New approaches to prior specification and structuring in autoregressive time series models are introduced and developed. We focus on defining classes of prior distributions for parameters and latent variables related to latent components of an autoregressive model for an observed time series. These new priors naturally permit the incorporation of both qualitative and quantitative prior information about the number and relative importance of physically meaningful components that represent low frequency trends, quasi-periodic subprocesses and high frequency residual noise components of observed series. The class of priors also naturally incorporates uncertainty about model order and hence leads in posterior analysis to model order assessment and resulting posterior and predictive inferences that incorporate full uncertainties about model order as well as model parameters. Analysis also formally incorporates uncertainty and leads to inferences about unknown initial values of the time series, as it does for predictions of future values. Posterior analysis involves easily implemented iterative simulation methods, developed and described here. One motivating field of application is climatology, where the evaluation of latent structure, especially quasi-periodic structure, is of critical importance in connection with issues of global climatic variability. We explore the analysis of data from the southern oscillation index, one of several series that has been central in recent high profile debates in the atmospheric sciences about recent apparent trends in climatic indicators.     

On robust forecasting in dynamic vector time series models

In this article, robust estimation and prediction in multivariate autoregressive models with exogenous variables (VARX) are considered. The conditional least squares (CLS) estimators are known to be non-robust when outliers occur. To obtain robust estimators, the method introduced in Duchesne [2005. Robust and powerful serial correlation tests with new robust estimates in ARX models. J. Time Ser. Anal. 26, 49–81] and Bou Hamad and Duchesne [2005. On robust diagnostics at individual lags using RA-ARX estimators. In: Duchesne, P., Rémillard, B. (Eds.), Statistical Modeling and Analysis for Complex Data Problems. Springer, New York] is generalized for VARX models. The asymptotic distribution of the new estimators is studied and from this is obtained in particular the asymptotic covariance matrix of the robust estimators. Classical conditional prediction intervals normally rely on estimators such as the usual non-robust CLS estimators. In the presence of outliers, such as additive outliers, these classical predictions can be severely biased. More generally, the occurrence of outliers may invalidate the usual conditional prediction intervals. Consequently, the new robust methodology is used to develop robust conditional prediction intervals which take into account parameter estimation uncertainty. In a simulation study, we investigate the finite sample properties of the robust prediction intervals under several scenarios for the occurrence of the outliers, and the new intervals are compared to non-robust intervals based on classical CLS estimators.     

Stable spectral factorization with applications to the estimation of time series models

A method for generating a miniphase and inveitible spectral factor from an unstable v × v full rank polynomial matrix is proposed. The zeros inside the unit circle are reflected through the boundary |z|=1 using closed form algebraic manipulations. Also included in the procedure is a technique foi determining the stability of a polynomial operator that does not require the explicit construction of the determinant al equation. Application of the technique is illustrated and the implementation of the method in the statistical context of system estimation is discussed.     

Testing for changes in autocovariances of nonparametric time series models

In the literature on change-point analysis, much attention has been paid to detecting changes in certain marginal characteristics, such as mean, variance, and marginal distribution. For time series data with nonparametric time trend, we study the change-point problem for the autocovariance structure of the unobservable error process. To derive the asymptotic distribution of the cumulative sum test statistic, we develop substantial theory for uniform convergence of weighted partial sums and weighted quadratic forms. Our asymptotic results improve upon existing works in several important aspects. The performance of the test statistic is examined through simulations and an application to interest rates data.     

On goodness of fit for time series regression models

We Propose a Bayesian approach to chech the goodness of fit for time series regression models. The test statistics is proposed by Smith (1985) based on a sequence of random variables which are independently distributed standard normal if the model is correct. We estimate this sequence of random variables using several methods. The tests of goodness of fit are performed when either the error terms violate the Gaussian assumption, or the order is incorrect, or the model is misspecified. The methodology is illustrated using both a simulation study and three real date sets.