Online Appendix To: Bayesian Doubly Adaptive Elastic-Net Lasso For VAR Shrinkage

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1 Online Appendix To: Bayesian Doubly Adaptive Elastic-Net Lasso For VAR Shrinkage Deborah Gefang Department of Economics University of Lancaster April 7, 203 I would like to thank Gary Koop, Esther Ruiz and two anonymous referees for their constructive comments. I would also like to thank the conference participants of CFE, ESEM202, and RCEF202 for helpful discussions. Any remaining errors are my own responsibility.

2 Technical Details for Models Nested in DAE- Lasso This section presents the priors, posteriors, and full conditional Gibbs schemes for Lasso, adaptive Lasso, e-net Lasso, and adaptive e-net Lasso.. Lasso VAR Shrinkage Following Song and Bickel (20), we define Lasso estimator for a VAR as: N 2 k ˆβ L = arg min β {[y (I n X)β] [y (I n X)β] + λ β j } () Correspondingly, the conditional multivariate mixture prior for β takes the following form: j= N 2 k π(β Σ, Γ, λ ) j= { 0 2πfj (Γ)) exp[ 2f j (Γ) β2 j ]d(f j (Γ))} { M 2 exp( 2 Γ M Γ)} 2 (2) where Γ = [γ, γ 2,..., γ N 2 k], M = Σ I Nk, and f j (Γ) is a function of Γ and Λ to be defined later. In this mixture prior, the terms associated with the L penalty are conditional on Σ through f j (Γ). In equation (2), the variances of β a and β b for a b are related through M. However, β a and β b themselves are independent of each other. We need to find an appropriate f j (Γ) which provides us tractable posteriors. The last term in equation (2) takes the form of a multivariate Normal distribution Γ N(0, M). For ease of exposition, we first write the 2

3 N 2 k N 2 k covariance matrix M as following: M = M,... M,j M,j+... M,N 2 k M j,... M j,j M j,j+... M j,n 2 k M j+,... M j+,j M j+,j+... M j+,n 2 k (3) M N 2 k,... M N 2 k,j M N 2 k,j+... M N 2 k,n 2 k M j+,j+... M j+,n 2 k Let H j = (M j,j+,..., M j,n 2 k) M N 2 k,j+... M N 2 k,n 2 k We next construct independent variables τ j for j =, 2,..., N 2 k using standard textbook techniques (e.g. Anderson, 2003; Muirhead 982). τ = γ + H (γ 2, γ 3,..., γ N 2 k) (4) τ 2 = γ 2 + H 2 (γ 3, γ 4,..., γ N 2 k) (5)... τ N 2 K = γ N 2 k + H N 2 k γ N 2 k (6) τ N 2 K = γ N 2 k (7) 3

4 The joint density of τ, τ 2,..., τ N 2 k is N(τ 0, σ 2 γ )N(τ 2 0, σ 2 γ 2 )...N(τ N 2 k 0, σ 2 γ N 2 k ) (8) where σ 2 γ j = M j,j H j (M j,j+,..., M j,n 2 k), with σ 2 γ N 2 k = M N 2 k,n 2 k. Note that it is computationally feasible to derive σ 2 γ j when M is sparse. The Jacobian of transforming Γ N(0, M) to (8) is. Defining η j = τ j /λ, we can write (8) as N(η 0, σ 2 γ )N(η 2 0, σ 2 γ 2 )...N(η N 2 k 0, σ 2 γ N 2 k ) (9) Let f j (Γ) = 2(ηj 2 ), the scale mixture prior is: N 2 k π(β Σ, Γ, λ ) { exp[ β2 j 0 2π(2ηj 2)) 2(2ηj 2)]d(2η2 j ) j= λ2 2σγ 2 exp[ j 2 (σγ 2 j )/λ 2 ]} (0) The last two terms in (0) constitute a scale mixture of Normals (with an exponential mixing density), which can be expressed as the univariate Laplace distribution λ 2 exp( λ β σγ 2 j σ 2 j ). γj Equation (0) shows that the conditional prior for β j is N(0, ), and 2ηj 2 the conditional prior for β is β Γ, Σ, Λ, Λ 2 N(0, D Γ) () where DΓ = diag([,,..., ]). 2η 2 2η2 2 2η 2 N 2 k 4

5 Priors for Σ and λ 2 can be elicited following standard practice in VAR and Lasso literature. In this paper, we set Wishart prior for Σ and Gamma prior for λ 2 : Σ W (S, ν), λ 2 G(µ λ 2, ν λ 2 ). The full conditional posterior for β is β N(β, V β ), where V β = [(I N X) )(Σ I Nk )(I N X)+(D Γ ) ], and β = V β [(I N X) (Σ I Nk )y]. The Full conditional posterior for Σ is W (S, ν), with S = (Y XB) (Y XB) + 2Q Q + S and ν = T + 2Nk + ν, with vec(q) = Γ. The Full conditional posterior for λ 2 is G( µ λ, ν λ ), where ν λ = ν λ + 2N 2 k ν λ µ λ and µ λ = ν λ +2µ λ τ 2 Finally the full conditional posterior of is j /σγ 2 j 2ηj 2 λ Inverse Gaussian: IG( 2, λ2 βj 2 ). Γ can not be directly drawn from the σ2 γ σ 2 j γ j posteriors. But it can be recovered in each Gibbs iteration using the draws of and Σ. Conditional on arbitrary starting values, the Gibbs sampler contains the following six steps:. draw β Σ, Λ, Γ from N(β, V β ); 2. draw Σ β, Λ, Γ from W (S, ν) 3. draw λ 2 Σ, β, Γ from G( µ λ, ν λ ) 4. draw λ β, Σ, Λ 2ηj 2 from IG( 2, λ2 βj 2 ) for j =, 2,...N 2 k. σ2 γ σ 2 j γ j 5. calculate Γ based on draws of Σ and in the current iteration. We adopt the same form of the inverse-gaussian density used in Park and Casella (2008). 5

6 .2 Adaptive Lasso VAR Shrinkage We define the adaptive Lasso estimator for a VAR as: N 2 k ˆβ AL = arg min β {[y (I n X)β] [y (I n X)β] + λ,j β j } (2) Correspondingly, the conditional multivariate mixture prior for β takes the following form: j= N 2 k π(β Σ, Γ, Λ ) j= { 0 2πfj (Γ)) exp[ 2f j (Γ) β2 j ]d(f j (Γ))} { M 2 exp( 2 Γ M Γ)} 2 (3) where Γ = [γ, γ 2,..., γ N 2 k], M = Σ I Nk, and f j (Γ) is a function of Γ and Λ to be defined later. In this mixture prior, the terms associated with the L penalty are conditional on Σ through f j (Γ). In equation (3), the variances of β a and β b for a b are related through M. However, β a and β b themselves are independent of each other. We need to find an appropriate f j (Γ) which provides us tractable posteriors. The last term in equation (3) takes the form of a multivariate Normal distribution Γ N(0, M). For ease of exposition, we first write the N 2 k N 2 k covariance matrix M as following: 6

7 M = M,... M,j M,j+... M,N 2 k M j,... M j,j M j,j+... M j,n 2 k M j+,... M j+,j M j+,j+... M j+,n 2 k (4) M N 2 k,... M N 2 k,j M N 2 k,j+... M N 2 k,n 2 k M j+,j+... M j+,n 2 k Let H j = (M j,j+,..., M j,n 2 k) M N 2 k,j+... M N 2 k,n 2 k We next construct independent variables τ j for j =, 2,..., N 2 k using standard textbook techniques (e.g. Anderson, 2003; Muirhead 982). τ = γ + H (γ 2, γ 3,..., γ N 2 k) (5) τ 2 = γ 2 + H 2 (γ 3, γ 4,..., γ N 2 k) (6)... τ N 2 K = γ N 2 k + H N 2 k γ N 2 k (7) τ N 2 K = γ N 2 k (8) The joint density of τ, τ 2,..., τ N 2 k is N(τ 0, σ 2 γ )N(τ 2 0, σ 2 γ 2 )...N(τ N 2 k 0, σ 2 γ N 2 k ) (9) 7

8 where σ 2 γ j = M j,j H j (M j,j+,..., M j,n 2 k), with σ 2 γ N 2 k = M N 2 k,n 2 k. Note that it is computationally feasible to derive σ 2 γ j when M is sparse. The Jacobian of transforming Γ N(0, M) to (9) is. Defining η j = τ j /λ,j, we can write (9) as N(η 0, σ 2 γ, )N(η 2 0, σ 2 γ 2,2 )...N(η N 2 k 0, σ 2 γ N 2 k,n 2 k ) (20) Let f j (Γ) = 2(ηj 2 ), the scale mixture prior is: N 2 k π(β Σ, Γ, Λ ) { exp[ β2 j 0 2π(2ηj 2)) 2(2ηj 2)]d(2η2 j ) j= λ2,j 2σγ 2 exp[ j 2 (σγ 2 j )/λ 2 ]},j (2) Equation (2) shows that the conditional prior for β j is N(0, ), and 2ηj 2 the conditional prior for β is β Γ, Σ, Λ, Λ 2 N(0, D Γ) (22) where DΓ = diag([,,..., ]). 2η 2 2η2 2 2η 2 N 2 k Priors for Σ and λ 2,j can be elicited following standard practice in VAR and Lasso literature. In this paper, we set Wishart prior for Σ and Gamma prior for λ 2,j : Σ W (S, ν), λ 2,j G(µ λ 2,j, ν λ 2,j ). The full conditional posterior for β is β N(β, V β ), where V β = [(I N X) )(Σ I Nk )(I N X)+(D Γ ) ], and β = V β [(I N X) (Σ I Nk )y]. The Full conditional posterior for Σ is W (S, ν), with S = (Y XB) (Y XB) + 2Q Q + S and ν = T + 2Nk + ν, with vec(q) = Γ. The 8

9 Full conditional posterior for λ 2,j is G( µ λ,j, ν λ,j ), where ν λ,j = ν λ,j + 2 and µ λ,j = ν λ,j σj 2µ λ,j 2τj 2µ. Finally the full conditional posterior of is +ν λ λ,j σγ 2,j j 2ηj 2 λ Inverse Gaussian: IG( 2,j, λ2,j ). Γ can not be directly drawn from the βj 2σ2 γ σ 2 j γ j posteriors. But it can be recovered in each Gibbs iteration using the draws of and Σ. Conditional on arbitrary starting values, the Gibbs sampler contains the following six steps:. draw β Σ, Λ, Γ from N(β, V β ); 2. draw Σ β, Λ, Γ from W (S, ν) 3. draw λ 2,j β, Σ, Λ, j, Γ from G( µ λ,j, ν λ,j ) for j =, 2,...N 2 k 4. draw λ β, Σ, Λ 2ηj 2 from IG( 2,j, λ2,j ) for j =, 2,...N 2 k. βj 2σ2 γ σ 2 j γ j 5. calculate Γ based on draws of Σ and in the current iteration..3 E-net Lasso VAR Shrinkage We define the e-net Lasso estimator for a VAR as: N 2 k N 2 k ˆβ EL = arg min β {[y (I n X)β] [y (I n X)β] + λ β j + λ 2 βj 2 } j= j= (23) Correspondingly, the conditional multivariate mixture prior for β takes 9

10 the following form: N 2 k λ2 π(β Σ, Γ, λ, λ 2 ) { exp( λ 2 2π j= 0 2 β2 j ) 2πfj (Γ)) exp[ 2f j (Γ) β2 j ]d(f j (Γ))} { M 2 exp( 2 Γ M Γ)} 2 (24) where Γ = [γ, γ 2,..., γ N 2 k], M = Σ I Nk, and f j (Γ) is a function of Γ and Λ to be defined later. In this mixture prior, the terms associated with the L penalty are conditional on Σ through f j (Γ). In equation (24), the variances of β a and β b for a b are related through M. However, β a and β b themselves are independent of each other. We need to find an appropriate f j (Γ) which provides us tractable posteriors. The last term in equation (24) takes the form of a multivariate Normal distribution Γ N(0, M). For ease of exposition, we first write the N 2 k N 2 k covariance matrix M as following: M = M,... M,j M,j+... M,N 2 k M j,... M j,j M j,j+... M j,n 2 k M j+,... M j+,j M j+,j+... M j+,n 2 k (25) M N 2 k,... M N 2 k,j M N 2 k,j+... M N 2 k,n 2 k 0

11 M j+,j+... M j+,n 2 k Let H j = (M j,j+,..., M j,n 2 k) M N 2 k,j+... M N 2 k,n 2 k We next construct independent variables τ j for j =, 2,..., N 2 k using standard textbook techniques (e.g. Anderson, 2003; Muirhead 982). τ = γ + H (γ 2, γ 3,..., γ N 2 k) (26) τ 2 = γ 2 + H 2 (γ 3, γ 4,..., γ N 2 k) (27)... τ N 2 K = γ N 2 k + H N 2 k γ N 2 k (28) τ N 2 K = γ N 2 k (29) The joint density of τ, τ 2,..., τ N 2 k is N(τ 0, σ 2 γ )N(τ 2 0, σ 2 γ 2 )...N(τ N 2 k 0, σ 2 γ N 2 k ) (30) where σ 2 γ j = M j,j H j (M j,j+,..., M j,n 2 k), with σ 2 γ N 2 k = M N 2 k,n 2 k. Note that it is computationally feasible to derive σ 2 γ j when M is sparse. The Jacobian of transforming Γ N(0, M) to (30) is. Defining η j = τ j /λ, we can write (30) as N(η 0, σ 2 γ )N(η 2 0, σ 2 γ 2 )...N(η N 2 k 0, σ 2 γ N 2 k ) (3)

12 Let f j (Γ) = 2(ηj 2 ), the scale mixture prior is: N 2 k π(β Σ, Γ, λ, λ 2 ) j= λ2 { exp( λ 2 2π 2 β2 j ) 0 exp[ β2 j 2π(2ηj 2)) 2(2ηj 2)]d(2η2 j ) (32) 2σγ 2 exp[ j 2 (σγ 2 j )/λ 2 ]} λ2 where η j = τ j /λ. The last two terms in (32) constitute a scale mixture of Normals (with an exponential mixing density), which can be expressed as the univariate Laplace distribution λ 2 σ 2 γ j exp( λ σ 2 γj β j ). Equation (32) shows that the conditional prior for β j is N(0, and the conditional prior for β is 2λ 2 η 2 j +), β Γ, Σ, Λ, Λ 2 N(0, D Γ) (33) where D Γ = diag([ 2η 2 2λ 2 η 2+, 2η2 2 2η 2λ 2 η2 2 +,..., 2 N 2 k 2λ 2 η 2 N 2 k +]). Priors for Σ and λ 2 can be elicited following standard practice in VAR and Lasso literature. In this paper, we set Wishart prior for Σ and Gamma priors for λ 2 and λ 2: Σ W (S, ν), λ 2 G(µ λ 2, ν λ 2 ), λ 2 G(µ λ2, ν λ2 ). The full conditional posterior for β is β N(β, V β ), where V β = [(I N X) )(Σ I Nk )(I N X)+(D Γ ) ], and β = V β [(I N X) (Σ I Nk )y]. The Full conditional posterior for Σ is W (S, ν), with S = (Y XB) (Y XB) + 2Q Q + S and ν = T + 2Nk + ν, with vec(q) = Γ. The 2

13 Full conditional posterior for λ 2 is G( µ λ, ν λ ), where ν λ = ν λ + 2N 2 k and µ λ = ν λ µ λ ν λ +2µ λ τ 2 j /σ 2 γ j. The Full conditional posterior for λ 2 is G( µ λ2, ν λ2 ), µ λ2 ν λ2 where ν λ2 = ν λ2 + N 2 k and µ λ2 = ν λ2 +µ λ2 β 2. Finally the full conditional j λ posterior of is Inverse Gaussian: IG( 2 2ηj 2, λ2 βj 2 ). Γ can not be directly σ2 γ σ 2 j γ j drawn from the posteriors. But it can be recovered in each Gibbs iteration using the draws of and Σ. 2ηj 2 Conditional on arbitrary starting values, the Gibbs sampler contains the following six steps:. draw β Σ, Λ, Λ 2, Γ from N(β, V β ); 2. draw Σ β, Λ, Λ 2, Γ from W (S, ν) 3. draw λ 2 β, Σ, Λ 2, Γ from G( µ λ, ν λ ) 4. draw λ 2 β, Σ, Λ, Γ from G( µ λ2, ν λ2 ) 5. draw λ β, Σ, Λ 2ηj 2, Λ 2 from IG( 2, λ2 βj 2 ) for j =, 2,...N 2 k. σ2 γ σ 2 j γ j 6. calculate Γ based on draws of Σ and in the current iteration..4 Adaptive E-net Lasso VAR Shrinkage In line with Zou and Zhang (2009), we define the adaptive e-net Lasso estimator for a VAR as following: N 2 k N 2 k ˆβ AEL = arg min β {[y (I n X)β] [y (I n X)β] + λ,j β j + λ 2 βj 2 } j= j= (34) 3

14 Correspondingly, the conditional multivariate mixture prior for β takes the following form: N 2 k λ2 π(β Σ, Γ, Λ, λ 2 ) { exp( λ 2 2π j= 0 2 β2 j ) 2πfj (Γ)) exp[ 2f j (Γ) β2 j ]d(f j (Γ))} { M 2 exp( 2 Γ M Γ)} 2 (35) where Γ = [γ, γ 2,..., γ N 2 k], M = Σ I Nk, and f j (Γ) is a function of Γ and Λ to be defined later. In this mixture prior, the terms associated with the L penalty are conditional on Σ through f j (Γ). We need to find an appropriate f j (Γ) which provides us tractable posteriors. The last term in equation (35) takes the form of a multivariate Normal distribution Γ N(0, M). For ease of exposition, we first write the N 2 k N 2 k covariance matrix M as following: M = M,... M,j M,j+... M,N 2 k M j,... M j,j M j,j+... M j,n 2 k M j+,... M j+,j M j+,j+... M j+,n 2 k (36) M N 2 k,... M N 2 k,j M N 2 k,j+... M N 2 k,n 2 k 4

15 M j+,j+... M j+,n 2 k Let H j = (M j,j+,..., M j,n 2 k) M N 2 k,j+... M N 2 k,n 2 k We next construct independent variables τ j for j =, 2,..., N 2 k using standard textbook techniques (e.g. Anderson, 2003; Muirhead 982). τ = γ + H (γ 2, γ 3,..., γ N 2 k) (37) τ 2 = γ 2 + H 2 (γ 3, γ 4,..., γ N 2 k) (38)... τ N 2 K = γ N 2 k + H N 2 k γ N 2 k (39) τ N 2 K = γ N 2 k (40) The joint density of τ, τ 2,..., τ N 2 k is N(τ 0, σ 2 γ )N(τ 2 0, σ 2 γ 2 )...N(τ N 2 k 0, σ 2 γ N 2 k ) (4) where σ 2 γ j = M j,j H j (M j,j+,..., M j,n 2 k), with σ 2 γ N 2 k = M N 2 k,n 2 k. Note that it is computationally feasible to derive σ 2 γ j when M is sparse. The Jacobian of transforming Γ N(0, M) to (4) is. Defining η j = τ j /λ,j, we can write (4) as N(η 0, σ 2 γ, )N(η 2 0, σ 2 γ 2,2 )...N(η N 2 k 0, σ 2 γ N 2 k,n 2 k ) (42) 5

16 Let f j (Γ) = 2(ηj 2 ). The scale mixture prior in (35) can be rewritten as: N 2 k π(β Σ, Γ, Λ, λ 2 ) j= λ2 { exp( λ 2 2π 2 β2 j ) 0 exp[ β2 j 2π(2ηj 2)) 2(2ηj 2)]d(2η2 j ) (43) λ2,j 2σγ 2 exp[ j 2 (σγ 2 j )/λ 2 ]},j The last two terms in (43) constitute a scale mixture of Normals (with an exponential mixing density), which can be expressed as the univariate Laplace distribution λ,j 2 σ 2 γ j exp( λ,j σ 2 γj β j ). Equation (43) shows that the conditional prior for β j is N(0, and the conditional prior for β is 2λ 2 η 2 j +), β Γ, Σ, Λ, Λ 2 N(0, D Γ) (44) where D Γ = diag([ 2η 2 2λ 2 η 2+, 2η2 2 2η 2λ 2 η2 2 +,..., 2 N 2 k 2λ 2 η 2 N 2 k +]). Priors for Σ and λ 2,j can be elicited following standard practice in VAR and Lasso literature. In this paper, we set Wishart prior for Σ and Gamma priors for λ 2,j and λ 2,j: Σ W (S, ν), λ 2,j G(µ λ 2,j, ν λ 2,j ), λ 2,j G(µ λ2, ν λ2 ). The full conditional posterior for β is β N(β, V β ), where V β = [(I N X) )(Σ I Nk )(I N X)+(D Γ ) ], and β = V β [(I N X) (Σ I Nk )y]. The Full conditional posterior for Σ is W (S, ν), with S = (Y XB) (Y XB) + 2Q Q + S and ν = T + 2Nk + ν, with vec(q) = Γ. The 6

17 Full conditional posterior for λ 2,j is G( µ λ,j, ν λ,j ), where ν λ,j = ν λ,j +2 and µ λ,j = ν λ,j σ 2 j µ λ,j 2τ 2 j µ λ,j +ν λ,j σ 2 γ j. The Full conditional posterior for λ 2 is G( µ λ2, ν λ2 ), µ λ2 ν λ2 where ν λ2 = ν λ2 + N 2 k and µ λ2 = ν λ2 +µ λ2 β 2. Finally the full conditional j λ posterior of is Inverse Gaussian: IG( 2,j, λ2,j ). Γ can not be directly 2ηj 2 βj 2σ2 γ σ 2 j γ j drawn from the posteriors. But it can be recovered in each Gibbs iteration using the draws of and Σ. 2ηj 2 Conditional on arbitrary starting values, the Gibbs sampler contains the following six steps:. draw β Σ, Λ, Λ 2, Γ from N(β, V β ); 2. draw Σ β, Λ, Λ 2, Γ from W (S, ν) 3. draw λ 2,j β, Σ, Λ, j, Λ 2, Γ from G( µ λ,j, ν λ,j ) for j =, 2,...N 2 k 4. draw λ 2 β, Σ, Λ, Γ from G( µ λ2, ν λ2 ) 5. draw λ β, Σ, Λ 2ηj 2, Λ 2 from IG( 2,j, λ2,j ) for j =, 2,...N 2 k. βj 2σ2 γ σ 2 j γ j 6. calculate Γ based on draws of Σ and in the current iteration. 2 Detailed Forecast Evaluation Results Tables -4 report the DAELasso forecasts results along with Lasso, adaptive Lasso, e-net Lasso, adaptive e-net Lasso, and those of the factor models and the seven popular Bayesian shrinkage priors in Koop (20). In line with Koop (20), we present MSFE relative to the random walk and log predictive likelihood for GDP, CPI and FFR. The results for DAELasso 7

18 and four other Lasso types of shrinkage methods are reported at the top of each table, followed by those of the methods reported in Koop (20). Koop (20) considers three variants of the Minnesota prior. The first is the natural conjugate prior used in Banbura et al (200), which is labelled Minn. Prior as in BGR. The second is the traditional Minnesota prior of Litterman (986), which is labelled Minn. Prior Σ diagonal. The third is the traditional Minnesota prior except that the upper left 3 3 block of Σ is not assumed to be daigonal, which is labelled Minn. Prior Σ not diagonal. Koop (20) also evaluates the performances of four types of SSVS priors. The first is the same as George et al (2008), which is labelled SSVS Non-conj. semi-automatic. The second is a combination of the non-conjugate SSVS prior and Minnesota prior with variables selected in a data based fashion, which is labelled SSVS Non-conj. plus Minn. Prior. The Third is a conjugate SSVS prior, which is labelled SSVS Conjugate Semi-automatic. The fourth is a combination of the conjugate SSVS prior and Minnesota prior, which is labelled SSVS Conjugate plus Minn. Prior. Finally the results for factor-augmented VAR models with one and four lagged factors are labelled as Factor model p= and Factor model p=4, respectively. We refer to Koop (20) for a lucid description of these priors. 8

19 DAELasso adaptive e-net Lasso e-net Lasso adaptive Lasso Lasso Minn. Prior as in BGR Minn. Prior Σ diagonal Minn. Prior Σ not diagonal SSVS Conjugate semi-automatic SSVS Conjugate plus Minn. Prior SSVS Non-conj. semi-automatic SSVS Non-conj. plus Minn. Prior Factor model p= Factor model p=4 Notes: Table : Rolling Forecasting for h = MSFEs as proportion of random walk MSFEs. Sum of log predictive likelihoods in parentheses. GDP CPI FFR ( ) ( ) ( -2.7 ) ( ) ( ) ( ) ( ) ( -2.6 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( -90.5) ( ) ( ) ( ) ( ) ( -8.7 ) ( -92. ) ( ) ( ) ( ) ( ) ( ) ( -9.4 ) ( -22. ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 9

20 DAELasso adaptive e-net Lasso e-net Lasso adaptive Lasso Lasso Minn. Prior as in BGR Minn. Prior Σ diagonal Minn. Prior Σ not diagonal SSVS Conjugate semi-automatic SSVS Conjugate plus Minn. Prior SSVS Non-conj. semi-automatic SSVS Non-conj. plus Minn. Prior Factor model p= Factor model p=4 Notes: Table 2: Rolling Forecasting for h = 4 MSFEs as proportion of random walk MSFEs. Sum of log predictive likelihoods in parentheses. GDP CPI FFR ( ) ( ) ( ) ( ) ( ) ( -29.9) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( -27. ) ( ) ( ) ( -2. ) ( ) ( ) ( ) ( ) ( -22. ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( -98. ) ( ) ( ) ( ) ( ) ( ) ( ) 20

21 DAELasso adaptive e-net Lasso e-net Lasso adaptive Lasso Lasso Minn. Prior as in BGR Minn. Prior Σ diagonal Minn. Prior Σ not diagonal SSVS Conjugate semi-automatic Table 3: Recursive Forecasting for h = SSVS Conjugate plus Minn. Prior SSVS Non-conj. semi-automatic SSVS Non-conj. plus Minn. Prior Factor model p= Factor model p=4 Notes: MSFEs as proportion of random walk MSFEs. Sum of log predictive likelihoods in parentheses. GDP CPI FFR ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( -22. ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( -84. ) ( ) ( ) ( -9.2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( -29. ) ( ) 2

22 DAELasso adaptive e-net Lasso e-net Lasso adaptive Lasso Lasso Minn. Prior as in BGR Minn. Prior Σ diagonal Minn. Prior Σ not diagonal SSVS Conjugate semi-automatic Table 4: Recursive Forecasting for h = 4 SSVS Conjugate plus Minn. Prior SSVS Non-conj. semi-automatic SSVS Non-conj. plus Minn. Prior Factor model p= Factor model p=4 Notes: MSFEs as proportion of random walk MSFEs. Sum of log predictive likelihoods in parentheses. GDP CPI FFR ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( -26. ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( -2.6 ) ( ) ( ) ( ) ( ) 22