An Exact Multivariate Model-Based Structural Decomposition

We propose a simple and structured procedure for decomposing a vector of time series into trend, cycle, seasonal, and irregular components. Contrary to common practice, we do not assume these components to be orthogonal conditional on their past. However, the state–space representation employed ensures that their estimates converge to values with null variances and covariances. Null variances are very important, as they ensure that the components do not change when the sample increases. This lack of “revisions” is the most important feature of our method, in comparison with most alternative procedures. On the other hand, null covariances provide a solid statistical foundation for the decomposition, as it ensures that a given component can be analyzed and interpreted independently of any other component(s). Other convenient properties of our method derive from the use of a state–space approach. First, defining the problem in state–space avoids dependence on particular model specifications, so the same procedure can be applied to a wide class of data representations, including ARIMA, VARMAX, univariate transfer functions, and structural time series models. Also, state–space methods deal easily with nonstandard situations, such as samples with missing values or constraints upon the structural components. Practical application of the procedure is illustrated with both simulated and real data. A MATLAB toolbox for time series modeling and decomposition is available via the Internet.