The ℓ1 trend filter, which is similar to the popular Hodrick–Prescott (HP) filter, seems to be very promising because it enables us to estimate a piecewise linear trend without specifying the location and number of kink points a priori. Such a trend may be regarded as a result of occasional permanent shocks to the growth rate. Similarly to the HP filter, the value of the tuning parameter needs to be selected in applying this filter. This paper proposes a method for selecting the tuning parameter of the ℓ1 trend filter and its generalization.
We are very grateful to two anonymous referees and the editor for their valuable suggestions and comments. The usual caveat applies. Hiroshi Yamada’s work was partly supported by JSPS KAKENHI Grant Numbers 22530272, 15K13010.
Proofs of (8) and (9)
As in Osborne, Presnell, and Turlach (2000) and Kim et al. (2009), there exists v=[v1, …, vT−2]′ such that (a): 2(y−x** )=ϕ**D′v, where, for t=1, …, T−2, vt=1 if ηt>0, vt=−1 if ηt<0, and vt∈[−1, 1] if ηt=0. Here, [η1, …, ηT−2]′=Dx** . As Dx** ≠0, we see (b): ||v||∞=1. Combining (a) and (b), we obtain (8). In addition, from the definition of v, we see (c): (Dx** )′v=||Dx** ||1. Combining (a) and (c), we obtain (9).
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