A Moving Linear Model Approach for Extracting Cyclical Variation from Time Series Data

We propose a methodology for decomposing time series data into multiple components, including constrained components and remaining components containing cyclical variation. Our approach employs a moving linear model and utilizes state space representation, allowing for estimation of the components using the Kalman filter. The key parameter in our model is the width of the time interval, which can be estimated using the maximum likelihood method. Notably, our approach only requires a local linear model for the constrained component, while a strict model is not necessary for the remaining component. By applying our approach iteratively, we can decompose a time series into multiple components. Furthermore, we introduce a procedure to transform the decomposed components into uncorrelated components using principal component analysis. The proposed methodology demonstrates its applicability in analyzing business cycles. To illustrate its performance, we apply it to analyze two sets of monthly time series data from Japan.