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T and fixed h da L Output: The daLASSO estimator Φ T 1 Start: Set up a grid G = γ0 × γ1 × γ2 with G = |G |. 2 for g ← 1 to G do 5 hK2 j=1 Output End da L ΦT ← : ∀g = 1, . . , G . Φ T (Λ∗T ). 6 Monte Carlo Study We use Monte Carlo to investigate the sampling properties of the PLAC-weighted adaptive LASSO estimator for VAR models. Specifically, we would like to assess its performance in terms of order identification, the parameter estimation, and subset The Doubly Adaptive LASSO for Vector Autoregressive Models 33 selection.
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For an m × n matrix A, its entrywise p-norm, denoted as A A p = vec(A) p = m n i=1 j=1 |ai j | p 1/ p p, is defined as . The Frobenius norm, which is the special case p = 2, is defined as A F = m n i=1 j=1 |ai j |2 Definition 3 (The PLAC-weighted adaptive LASSO) The PLAC-weighted adaptive da L LASSO or doubly adaptive LASSO (daLASSO) estimator φˆ T for φ ∗ is defined as φˆ da L = arg min φ ⎧ ⎨ ⎩ h y − X ⊗ IK φ 2 K K wˆ i j,k φi j,k + λT k=1 i=1 j=1 ⎫ ⎬ ⎭ , (14) The Doubly Adaptive LASSO for Vector Autoregressive Models where wˆ i j,k = 1 φ˜ i j,k γ1 h P(s) s=k γ0 γ0 h Ak = P(s) γ2 = γ0 γ0 , 27 1 φ˜ i j,k γ1 γ Ak 2 , (15) (16) s=k φ˜ i j,k is the ordinary least squares estimate or any other consistent estimate for φi j,k , P(s) γ0 = K i=1 K j=1 | Pi j (s)|γ0 1/γ0 is the entrywise γ0 -norm of the sample par- tial lag autocorrelation matrix P(s) at lag s, and γ0 > 0, γ1 ≥ 0, and γ2 ≥ 0 are some fixed constants, and h is the fixed maximum lag set initially.