$$\ell _{1}$$ ℓ 1 Common Trend Filtering

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.