Group LASSO for structural break time series

Consider a structural break autoregressive (SBAR) process (formula presented), where Y_t-1 = (1, Y_t-1, …, Y_t-p)^T, β_j = (β_j0, …, β_jp)^T ∈ ℝ^p+1,j = 1, …, m + 1, {t₁, …, t_m} are change-points, 1 = t₀ < t₁ <・・・ <t_m+1 = n + 1, σ(・) is a measurable function on ℝ^q, and {ε_t} are white noise with unit variance. In practice, the number of change-points m is usually assumed to be known and small, because a large m would involve a huge amount of computational burden for parameters estimation. By reformulating the problem in a variable selection context, the group least absolute shrinkage and selection operator (LASSO) is proposed to estimate an SBAR model when m is unknown. It is shown that both m and the locations of the change-points {t₁, …, t_m} can be consistently estimated from the data, and the computation can be efficiently performed. An improved practical version that incorporates group LASSO and the stepwise regression variable selection technique are discussed. Simulation studies are conducted to assess the finite sample performance. Supplementary materials for this article are available online.