A Multivariate Periodic Unobserved Components Time Series Analysis for Sectoral U.S. Employment

Authored by: Siem Jan Koopman , Marius Ooms , Irma Hindrayanto

Economic Time Series

Print publication date:  March  2012
Online publication date:  March  2012

Print ISBN: 9781439846575
eBook ISBN: 9781439846582
Adobe ISBN:


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An unobserved components time series (UC) model is built of a number of stochastic linear processes that typically capture trend, seasonal, cycle, and remaining stationary dynamic features in an observed time series. The basic theory and methodology are laid out in Harvey (1989). Each stochastic component can be represented as an autoregressive integrated moving average (ARIMA) process. We can therefore consider the UC model as a special case of the RegComponents framework of Bell (2004). The typical parameters that need to be estimated in a UC model are the variances of the innovations driving the components and a selection of other coefficients associated with the ARIMA processes. For a seasonal time series, the UC model usually contains a stochastic seasonal component that is able to capture the time-varying seasonal effects. However, other dynamic features in the time series may also be subject to seasonal effects. For example, in the case of an economic time series, cyclical effects may be relatively more apparent or have a bigger impact in a particular season (winter) compared to another season (summer). In such cases, we can let the coefficients associated with a particular component be dependent on the season. We define this extension of the UC model as a periodic UC (PUC) model, which is explored in Koopman and Ooms (2002, 2006). In this chapter, we extend the PUC approach into a multivariate framework where several observed time series are modeled simultaneously.

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