Updating the forecast function of ARIMA models and the link with DLMs

This paper shows that the whole forecast function of ARIMA time series models, and not just the eventual forecast function, may be updated each time an observation is received. The paper also shows that the coefficients in the updating equations for the forecast function may be expressed in exactly the same form as the Kalman filter updating equations for canonical time series DLMs. Moreover, the adaptive factors in the updating equations are shown to be a simple function of the ARIMA model parameters. Copyright © 1999 John Wiley & Sons, Ltd.     

Convergence and the Constant Dynamic Linear Model

It is well known that, as calculated using the Kalman filter recurrence relationships, the posterior parameter variance and the adaptive vector of observable constant dynamic linear models converge to limiting values. However, most proofs are tortuous, some have subtle errors and some relate only to specific cases. An elegant probabilistic convergence proof demonstrates that the limit is independent of the initial parametric prior. The result is extended to a class of multivariate dynamic linear models. Finally the proof is shown to apply to many non-observable constant DLMs. © 1997 John Wiley & Sons, Ltd.     

Variance intervention

Variance intervention is a simple state-space approach to handling sharp discontinuities of level or slope in the states or parameters of models for non-stationary time-series. It derives from earlier procedures used in the 1960s for the design of self-adaptive, state variable feedback control systems. In the alternative state-space forecasting context considered in the present paper, it is particularly useful when applied to structural time series models. The paper compares the variance intervention procedure with the related ‘subjective intervention’ approach proposed by West and Harrison in a recent issue of the Journal of Forecasting, and demonstrates it efficacy by application to various time-series data, including those used by West and Harrison.     

Temporal Aggregation in Dynamic Linear Models

One important aspect concerning the analysis and forecasting of time series that is sometimes neglected is the relationship between a model and the sampling interval, in particular, when the observation is cumulative over the sampling period. This paper intends to study the temporal aggregation in Bayesian dynamic linear models (DLM). Suppose that a time series Y_t is observed at time units t and the observations of the process are aggregated over r units of time, defining a new time series Z_k=Σ^r_i=1Y_rk+i. The relevant factors explaining the variation of Z_k can, and in general will, be different, depending on how the sampling interval r is chosen. It is shown that if Y_t follows certain dynamic linear models, then the aggregated series can also be described by possibly different DLM. In the examples, the industrial production of Brazil is analysed under various aggregation periods and the results are compared. © 1997 John Wiley & Sons, Ltd.     

Variance estimation for multivariate dynamic linear models

The problem of estimating unknown observational variances in multivariate dynamic linear models is considered. Conjugate procedures are possible for univariate models and also for special very restrictive common components models but they are not generally applicable. However, for clarity of operation and in order to avoid numerical integration, it is desirable to have conjugacy or approximate conjugacy. Such an approximate procedure is proposed based upon a simple analytic approximation. It is exact for the sub-class of conjugate models and improves on a previous procedure based upon the Robust filter.     

A Dynamic Model Selection Procedure to Forecast Using Multi-Process Models

Multi-process models are particularly useful when observations appear extreme relative to their forecasts, because they allow for explanations of any behaviour of a time series, considering more generating sources simultaneously. In this paper, the multi-process approach is extended by developing a dynamic procedure to assess the weights of the various sources, alias the prior probabilities of the rival models, that compete in the collection to make forecasts. The new criterion helps the forecasting system to learn about the most plausible scenarios for the time series, considering all the combinations of consecutive models to be a function of the magnitude of the one-step-ahead forecast error. Throughout the paper, the different treatments of outliers and structural changes are highlighted using the concepts of robustness and sensitivity. Finally, the dynamic selection procedure is tested on the CP6 dataset, showing an effective improvement in the overall predictive ability of multi-process models whenever anomalous observations occur. © 1997 John Wiley & Sons, Ltd.     

Structural Time-series Modelling of Monetary Aggregates: A Case Study for Eleven European Countries

Monetary aggregates for eleven European countries are analysed using the structural time-series methodology, paying special attention to unit root issues. Estimation of the parameters of the models is carried out by applying the asymptotic least squares (ALS) procedure. A comparison with the maximum likelihood estimates obtained via the Kalman filter shows that ALS is an alternative to Kalman filter estimation. The empirical results show that for only a small number of series the four variance parameters of the basic structural model are strictly positive. For the majority of the series the variance of the irregular component is equal to 0.©1997 John Wiley & Sons, Ltd.     

A dynamic linear model approach for disaggregating time series data

An approach is proposed for obtaining estimates of the basic (disaggregated) series, x_i, when only an aggregate series, y_t, of k period non-overlapping sums of x_i's is available. The approach is based on casting the problem in a dynamic linear model form. Then estimates of x_i can be obtained by application of the Kalman filtering techniques. An ad hoc procedure is introduced for deriving a model form for the unobserved basic series from the observed model of the aggregates. An application of this approach to a set of real data is given.