Smoothing a Time Series by Segments of the Data Range

Abstract

We consider the problem of estimating a trend with different amounts of smoothness for segments of a time series subjected to different variability regimes. We propose using an unobserved components model to consider the existence of at least two data segments. We first fix some desired percentages of smoothness for the trend segments and deduce the corresponding smoothing parameters involved. Once the size of each segment is chosen, the smoothing formulas here derived produce trend estimates for all segments with the desired smoothness as well as their corresponding estimated variances. Empirical examples from demography and economics illustrate our proposal.

Keywords:

• Comparability
• Generalized least squares
• Graduation
• Hodrick–Prescott filter
• Smoothing parameter.

Mathematics Subject Classification:

• 65D10
• 37M10.