Abstract
The ell _{1} trend filtering enables us to estimate a continuous piecewise linear trend of univariate time series. This filter and its variants have subsequently been applied in various fields, including astronomy, climatology, economics, electronics, environmental science, finance, and geophysics. Although the ell _{1} trend filtering can estimate a continuous piecewise linear trend of univariate time series, it cannot estimate a common continuous piecewise linear trend of multiple time series. This paper develops a statistical procedure that enables us to estimate it, which is a multivariate extension of the ell _{1} trend filtering. We provide an algorithm for estimating it and a clue to specify the tuning parameter of the procedure, both required for its application. We also (i) numerically illustrate how well the algorithm works, (ii) provide an empirical illustration, and (iii) introduce a generalization of our novel method.
Highlights
The ‘1 trend filtering, which was developed by Steidl et al (2006), Steidl (2006), Kim et al (2009), Tibshirani (2014), and Guntuboyina et al (2020), enables us to extract a continuous piecewise linear trend of univariate time series
The filter and its variants have been subsequently applied in various fields, including astronomy (Politsch et al 2020), climatology (Khodadadi and McDonald 2019), economics (Yamada and Jin 2013; Yamada and Yoon 2014; Winkelried 2016; Yamada 2017; Klein 2018), electronics (Suo et al 2019), environmental science (Brantley et al 2019), finance (Mitra and Rohit 2018), and geophysics (Wu et al 2018)
It is possible to say that the ‘1 trend filtering is a method to obtain the trend considered by Perron (1989) and Rappoport and Reichlin (1989)
Summary
The ‘1 trend filtering, which was developed by Steidl et al (2006), Steidl (2006), Kim et al (2009), Tibshirani (2014), and Guntuboyina et al (2020), enables us to extract a continuous piecewise linear trend of univariate time series. Figure 1 illustrates a continuous piecewise linear trend. HP filtering has been used to extract the cyclical component of a univariate time series For other such methods, see, e.g., Pollock (2016) and Michaelides et al (2018). By extending the ‘1 trend filtering, we develop a novel method to estimate xt and ai from yi;t. The ‘1 trend filtering is defined by XT min ðyi;t À xi;tÞ2 þ w jD2xi;tj; ð6Þ xi;1;...;xi;T 2R t1⁄41 t1⁄43 where w [ 0 is a tuning parameter In matrix notation, it is expressed as min xi 2RT kyi. We extend the ‘1 trend filtering so that we may estimate a common continuous piecewise linear trend of multiple time series, y1;t; .
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