Appendix - The great recession and its effects on monetary policy

with n = 1, . . . , N number of countries, t = 1, . . . , T points in time and

^π_n,t ∼ N(0, e^h^t)

h_t = h_t−1 +^h_t

^h_t ∼ N(0, σ_h²)

^u_n,t ∼ N(0, σ_n,u² )

^ρ^π ∼ T N(−ρ^π_t−1,1−ρ^π_t−1; 0, σ_ρ²π)

^λ ∼ T N(−1−λt−1,0−λt−1; 0, σ_λ²)

^β ∼ T N(−β_t−1,1−β_t−1; 0, σ_β²)

(2.4)

Moreover, we impose that the unemployment-gaps evolve as a stationary AR(2) process, restricting ρ^u_n,1 + ρ^u_n,2 < 1, ρ^u_n,2 − ρ^u_n,1 < 1 and |ρ^u_n,2| < 1.

Additionally, we assume that λ_t, ρ^π_t and β_t lie in the intervals (−1,0), (0,1) and (0,1), respectively. The prior for initial conditions of the state equations are

τ₁^π,EU ∼ N(τ₀^π,EU, ω_τ²^π) τ_n,1^u ∼ N(τ_n,0^u , ω_τ²u) ρ^π₁ ∼ T N(0,1;ρ^π₀, ω²_ρ^π) λ₁ ∼ T N(−1,0;λ₀, ω_λ²) β1 ∼ T N(0,1;β0, ω_β²) h₁ ∼ T N(h₀, ω_h²)

whereby τ₀^π,EU, τ_n,0^u , ρ^π₀, λ₀, β₀, h₀, ω²_τπ, ω_τ²u, ω²_ρπ,ω_λ²,ω_β² and ω_h² are known constants. The specific choice of initial conditions is shown in Table 2.2. For the model parameters we choose the following priors

σ_u,n² ∼ IG(v_u, S_u) σ_h² ∼ IG(v_h, S_h) σ_τ²π ∼ IG(v_τπ, S_τπ) σ_τ²u,n ∼ IG(v_τ^u_,n, S_τ^u_,n) σ_ρ²^π ∼ IG(v_ρ^π, S_ρ^π) σ_λ² ∼ IG(v_λ, S_λ) σ_β² ∼ IG(v_β, S_β)

IG denotes the inverse-Gamma distribution. The initial values and priors are shown in Table 2.2. The prior for the degrees of freedom for the parameters is small as v = 10, implying a large variance and therewith a relatively non-informative prior. The scale parameters are set in way as to reflect the desired smoothness desired smoothness of tvp parameters and trends in terms of expected value of the respective variances. For example, S_τ^π = 0.9

with E(σ_τ^π) = 0.1 then the prior for E(σ_τ^π) implies a relatively smooth transition of τ^π. With a high probability τ^π changes between −0.01 and 0.01 from one period to another. Since the inflation trend is common across countries but the unemployment trend is country-specific we employ distinct scale parameters for the unemployment trends, reflecting differences across country-specific NAIRUs due to structural differences across labour markets.

Table 2.2: Initial conditions and priors Initial conditions:

τ₀^π τ₀^u h0

OE BG BD ES FN FR IR IT NL PT

1.9 4.2 8.3 5.4 9.6 10.6 9.2 5.4 9.6 4.7 5.8 1 ρ^π₀ λ₀ β₀ ρ^u_1,0 ρ^u_2,0 ω_τ²π ω_τ²u ω²_h ω_ρ²π ω²_λ ω_β² ω²_u 0.7 −0.4 0.001 1.6 −0.7 0.01 0.1 0.2 0.005 0.005 10⁻⁸ 0.2

Priors

S_τπ S_τu S_h

OE BG BD ES FN FR IR IT NL PT

0.9 0.9 0.9 0.9 3.6 1.8 0.9 4.5 0.9 1.8 0.9 1.8 S_u S_ρπ S_λ S_β Forσ² of τ^π, τ^u,h, ρ^π,λ and β we set v

4.5 0.081 0.081 4.5e⁻⁶ 10

The table indicates initial conditions and priors of the benchmark model estimation.

2.7.2 MCMC sampling

We adapt the algorithm introduced by Chan et al. (2016) and sequentially draw from

1. p(τ^π,EU|π, u, τ^u, ρ^π, λ, β, h, θ, IV)

2. p(τ^u|π, u, τ^π,EU, ρ^π, λ, β, h, θ, IV)

3. p(ρ^π|π, u, τ^π,EU, τ^u, λ, β, h, θ, IV)

4. p(λ|π, u, τ^π,EU, τ^u, ρ^π, β, h, θ, IV)

5. p(β|π, u, τ^π,EU, τ^u, ρ^π, λ, h, θ, IV)

6. p(h|π, u, τ^π,EU, τ^u, ρ^π, λ, β, θ, IV)

7. p(θ|π, u, τ^π,EU, τ^u, ρ^π, λ, β, h, IV)

with θ = (σ_u, σ_τ^π,EU, σ_τ^u, σ_h, σ_ρ^π, σ_λ, σ_β, ρ^u) and IV being the initial values for the respective parameters

Conditional distribution of τ^π,EU

To obtain the conditional distribution of τ^π,EU we rewrite the inflation equa-tion in the following way:

K_ππ = µ_π +K_πX₀τ^π,EU +^π, ^π ∼ N(0,Ω_π) (2.5) whereby π is N T ×1, τ^π,EU is T ×1, ^π is N T ×1, Ω_π is

diag(σ²_1,1, . . . , σ²_N,1, . . . , σ_N,T² ) and

K_π = N T ×N T







I_N 0 0 0

−ρ^π₂I_N I_N 0 0 0 −ρ^π₂I_N I_N 0 ... . . ...

0 0 0 −ρ^π₂I_N







IN is an identity matrix of N × N. Since |K| = 1, Kπ is invertible for all values of ρ^π

µ_π = N T ×1







ρ^π₁(π_1,0 −τ^π,EU) +λ₁(u_1,1 −τ_1,1^u ) +β₁π₁^oil ...

ρ^π₁(π_N,0 −τ^π,EU) +λ₁(u_N,1 −τ_N,1^u ) +β₁π₁^oil λ₂(u_1,2 −τ_1,2^u ) +β₂π₂^oil

...

λ₂(u_N,2 −τ_N,2^u ) +β₂π₂^oil ...

λ_T(u_N,T −τ_N,T^u ) + β_Tπ_T^oil







X₀ = N T ×T







ι . . . 0 0 ι . . . ...

0 0 ι . . . ...

... . . ...

0 . . . ι







ι as a column vector of N × 1 ones. Note that (X₀⁰X₀) is an invertible

N T ×N T matrix.

(M π|u, τ^u, ρ^π, ρ^u, λ, β, h, θ) ∼N(M K_π⁻¹µ_π +τ^πEU, M⁰K_π⁰⁻¹Ω_πK⁻¹M) (2.6) whereby M = (X₀⁰X₀)⁻¹X₀⁰. Then the prior density of M π is given by

log p(M π|u, τ^u, ρ^π, ρ^u, λ, β, h, θ) ∝

−¹₂jh− ¹₂(M π−M K_π⁻¹µ_π−τ^π,EU)⁰(M⁰K_π⁰⁻¹Ω_πK⁻¹M)⁻¹ (M π−M K_π⁻¹µπ−τ^π,EU)

(2.7)

with j being a N T × 1 columns of ones. The state equation of τ^π,EU is defined as

Hτ^π,EU = α_π +^{τ π} (2.8)

with

απ = T ×1





 τ₀^π,EU

0 ...







H = T ×T







1 0 0 . . . 0

−1 1 0 . . . ...

0 −1 1 . . . ...

0 0 −1 1 . . . ...

... . . ...

0 0 . . . −1 1







so that

(τ^π,EU|σ_τ,π² ) ∼ N(H⁻¹α_π,(H⁰Ω⁻¹_{τ π}H)⁻¹) (2.9) with Ω_{τ π} = diag(ω_τ²π, σ_τ²π, . . . , σ²_τπ). The prior density of τ^π,EU is given by

log p(τ^π,EU|σ_τ,π² ) ∝

−¹₂(τ^π,EU −H⁻¹απ)⁰H⁰Ω⁻¹_{τ π}H(τ^π,EU −H⁻¹απ) (2.10)

Combining (2.7) and (2.10)

log p(τ^π,EU|M π, u, τ^u, ρ^π, λ, β, h, θ) ∝

−¹₂(τ^π −τˆ^π)⁰D_τ,π⁻¹(τ^π −τˆ^π) (2.11) with

τ^π,EU = Dτ,π((M⁰K_π⁰⁻¹ΩπK_π⁻¹M)⁻¹(M π−M K_π⁻¹µπ) +H⁰Ω⁻¹_{τ π}απ) D_τ,π = ((M⁰K_π⁰⁻¹Ω_πK_π⁻¹M)⁻¹ +H⁰Ω⁻¹_τ H)⁻¹

We sample N(ˆτ^π,EU, D_τ,π) by using the precision-based-algorithm developed by Chan and Jeliazkov (2009). This implies that we sample τˆ^π,EU by apply-ing the Cholesky factorisation to D_τ,π that is a block-banded matrix so that C⁰C = Dτ,π. Then we solve for τˆ^π,EU by backward and forward substitu-tion, sample u ∝ N(0, I), solve for Cx = u and get a draw of τ^π,EU by τ^π,EU = ˆτ^π,EU +x with τ^π,EU ∝N(ˆτ^π,EU, Dτ,π).

Conditional distribution of τ^u

Next, we derive the conditional distribution of τ^u. Therefore, we rewrite the Phillips curve equation as

z = Λτ^u +^π, ^π ∼N(0,Ω_π) (2.12)

with

z =







(π1,1 −τ₁^π,EU)−ρ^π_1,1(π1,0 −τ₀^π,EU)−λ1u1,1 −β1π₁^oil ...

(πN,1 −τ₁^π,EU)−ρ^π_N,1(πN,0 −τ₀^π,EU)−λ1uN,1 −β1π₁^oil ...

(πN,T −τ_T^π,EU)−ρ^π_N,T(πN,T−1 −τ_T^π,EU₋₁ )−λTuN,T −βTπ^oil_T







Λ = diag(−λ_1,1, . . . ,−λ_N,1, . . . ,−λ_N,T)

τ^u = [τ_1,1^u , . . . , τ_N,1^u , . . . , τ_N,T^u ]⁰ The prior density of π is then given by

log p(π|u, τ^u, τ^π, ρ^π, λ, β, h, θ) ∝ (z−Λτ^u)⁰Ω⁻¹_π (z−Λτ_u) (2.13) The second measurement equation for τ^u stems from the unemployment-gap formulation.

K_uu = µ_u+K_uτ^u+^u, ^u ∼ N(0,Ω_u) (2.14) with Ω_u = I_T ⊗σ_u², σ_u² = [ω_u², σ_1,u² , . . . , σ_N,u² ]⁰ and

µu =







ρ^u_1,1(u1,0 −τ_1,0^u ) +ρ^u_1,2(u1,−1 −τ_1,−1^u ) ...

ρ^u_N,1(uN,0 −τ_N,0^u ) +ρ^u_N,2(u_N,−1 −τ_N,−1^u ) ...

ρ^u_1,2(u1,0 −τ_1,0^u ) ...

ρ^u_N,2(uN,0 −τ_N,0^u ) ...







Ku =







I_N 0 . . . 0

−ρ^u₁I_N I_N . . . .

−ρ^u₂I_N −ρ^u₁I_N I_N . . . ...

0 −ρ^u₂I_N −ρ^u₁I_N I_N . . . ...

... . . ...

0 . . . −ρ^u₂I_N −ρ^u₁I_N I_N







whereby ρ^u₁ and ρ^u₂ are column vectors with 1×N. The prior density of u is then given by

log p(u|τ^u, θ) ∝

−¹₂(u−K_u⁻¹µu −τ^u)⁰K_u⁰Ω⁻¹_u Ku(u−K_u⁻¹µu−τ^u) (2.15)

The state equation τ^u takes on the form

τ^u = H⁻¹α_u +^τ_t^u (2.16)

with α_u = (τ₀^u, . . . ,0)⁰ and Ω_{τ u} = diag(ω_{τ u}² , σ_{τ u}² , . . . , σ_{τ u}² ) log p(τ^u|σ_{τ u}² ) ∝

−¹₂(τ^u−H⁻¹αu)⁰H⁰Ω⁻¹_{τ u}H(τ^u−H⁻¹αu)

(2.17)

Next, combining (2.13), (2.15) and (2.17) yields log p(τ^u|π, u, ρ^π, τ^π, λ, β, h, θ) ∝

−¹₂(z −Λτ^u)⁰Ω⁻¹_π (z −Λτ^u)

−¹₂(u−K_u⁻¹µ_u −τ^u)⁰K_u⁰Ω⁻¹_u K_u(u−K_u⁻¹µ_u−τ^u)

−¹₂(τ^u−H⁻¹α_u)⁰H⁰Ω⁻¹_{τ u}H(τ^u−H⁻¹α_u)

= −¹₂(τ^u −τˆ^u)⁰D_τ^u(τ^u−τˆ^u)

with

(2.18)

τ^u = D_τ^u(Λ⁰Ω⁻¹_π z+K_u⁰Ω⁻¹_u K_u(u−K_u⁻¹µ_u) +H⁰Ω⁻¹_{τ u}α_u) Dτ^u = (Λ⁰Ω⁻¹_π Λ +K_u⁰Ω⁻¹_u Ku+H⁰Ω⁻¹_{τ u}H)⁻¹

(2.19) As before we sample the distribution by using the precision-based algorithm.

Conditional distribution of ρ^π The measurement equation for ρ_π is

π^∗ + Λu^∗ +βtπ_t^oil = XπX0ρ^π+^π (2.20)

wherebyπ^∗ = π−X₀τ^π,EU,X_π = diag(π₀^∗, . . . , π_N,T^∗ ₋₁),ρ^π = [ρ^π₀, ρ^π₁, . . . , ρ^π_T]⁰ and u^∗ = u−τ^u. Then it follows that

(M X_π⁻¹π^∗ +M X_π⁻¹Λu^∗) ∼ N(ρ^π, M⁰X_π⁰⁻¹ΩπX_π⁻¹M) (2.21)

log p(M X_π⁻¹π^∗ +M X_π⁻¹Λu^∗|τ^π, τ^u, ρ^π, λ, β, h, θ) ∝

−¹₂j_Th− ¹₂(M X_π⁻¹π^∗ +M X_π⁻¹Λu^∗ −ρ^π)⁰ (M X_π⁻¹Ω_πX_π⁰⁻¹M⁰)⁻¹

(M X_π⁻¹π^∗ +M X_π⁻¹Λu^∗ −ρ^π)

(2.22)

The state equation of ρ_π is given by

Hρ^π = ^ρ^π , ρ^π ∼ N(0, H⁰⁻¹Ω_ρπH⁻¹) (2.23)

with

log p(ρ^π|σ²_ρπ) ∝

−¹₂(ρ^π⁰H⁰Ω⁻¹_ρπHρ^π) +g_ρ^π_,σ²

ρπ

(2.24)

Combining (2.22) and (2.24) yields

log p(ρ^π|π, u, τ^π, τ^u, λ, β, h, θ) ∝

−¹₂j_Th− ¹₂(M X_π⁻¹π^∗ +M X_π⁻¹Λu^∗ −ρ^π)⁰(M X_π⁻¹Ω_πX_π⁰⁻¹M⁰)⁻¹ (M X_π⁻¹π^∗ +M X_π⁻¹Λu^∗ −ρ^π)

−¹₂(ρ^π⁰H⁰Ω⁻¹_ρπHρ^π) +g_ρ^π_,σ²

ρπ

∝ −¹₂(ρ^π −ρˆ^π)⁰D⁻¹_ρ^π (ρ^π −ρˆ^π) +g_ρ^π_,σ²

ρπ

(2.25)

with

g_ρ^π_,σ²

ρπ = −PT

t=2(Φ(^1−ρ

π t−1

σ_ρπ² −Φ(^−ρ_σ2^π ρπ

))

ρˆ^π = D_ρ^π((M⁰X_π⁰⁻¹Ω_πX_π⁻¹M)⁻¹M X_π⁻¹(π^∗ + Λu^∗)) D_ρ^π = ((M⁰X_π⁰⁻¹Ω_πX_π⁻¹M)⁻¹ +H⁰Ω⁻¹_ρπH)⁻¹

As it can be seen in EQ (2.25) the conditional density for ρ^π is truncated-normal. We follow Chan et al. (2016) and apply an independence chain Metropolis-Hastings step, whereby the candidate draws resulting from the precision-based method are accepted or rejected by an acceptance-rejection Metropolis-Hastings step.

Conditional distribution of λ

The measurement equation of λ takes on the following form

π_λ = X_uX₀λ+ ^π (2.26)

with Xu = diag(u^∗_1,0, . . . , u^∗_N,T₋₁) and πλ = [π^∗_1,1−ρ^π₁π^∗_1,0−β1π₁^oil, . . . , π_N,1^∗ − ρ^π₁π_N,0^∗ −β₁π₁^oil, . . . , π_N,T^∗ −ρ^π_Tπ_N,T^∗ ₋₁ −β_Tπ_T^oil]⁰. Then it follows that

M X_u⁻¹w ∼ N(λ, M⁰X_u⁰⁻¹Ω_πX_u⁻¹M) (2.27) with

log p(M X_u⁻¹π_λ|τ^π, τ^u, ρ^π, λ, β, h, θ) ∝

−¹₂j_Th− ¹₂(M X_u⁻¹π_λ−λ)⁰(M X_u⁻¹Ω_πX_u⁰⁻¹M⁰)⁻¹ (M X_u⁻¹π_λ −λ)

(2.28)

The state equation of λ is given by

Hλ = ^λ_t, ^λ_t ∼ N(0,Ω_λ) (2.29) with

log p(λ|σ_λ²) ∝

−¹₂(λ)⁰H⁰Ω⁻¹_λ H(λ) +g_λ(λ, σ_λ²)

(2.30)

Combining (2.28) and (2.30) yields log p(λ|π, u, τ^π, τ^u, ρ^π, β, h, θ) ∝

−¹₂(λ−λ)ˆ ⁰D⁻¹_λ (λ−λ) +ˆ g_λ

(2.31)

with

g_λ(λ, σ²_λ) = −PT

t=2(Φ(^−λ_σ^t−1

λ −Φ(^−1−λ_σ

λ ))

λˆ = D_λ((M X_u⁻¹Ω_πX_u⁰⁻¹M⁰)⁻¹M X_u⁻¹π_λ) D_λ = ((M X_u⁻¹Ω_πX_u⁰⁻¹M⁰)⁻¹ +H⁰Ω⁻¹_λ H)⁻¹

Similarly to the sampling ofρ^π, we include an acceptance-rejection Metropolis-Hastings (ARMH) step additional to the precision-based algorithm as the conditional density is of non-standard form.

Conditional distribution of β

We apply a similar derivation strategy as before. Then the measurement equation of β takes on the following form

π_oil^∗ = X0Xoilβ +^π (2.32) with X_oil = diag(π^oil₁ , . . . , π_T^oil), β = [β₀, β₁, . . . , β_T]⁰ and π^∗_oil = [π_1,1 − τ₁^π,EU − ρ^π₁(π_1,0 − τ₀^π,EU) − λ₁u^∗_1,1, . . . , π_N,1 − τ₁^π,EU − ρ^π₁(π_N,0 − τ₀^π,EU) −

λ₁u^∗_N,1, . . . , π_N,T−τ_T^π,EU−ρ^π_T(π_N,T₋₁−τ_T^π,EU−1)−λ_Tu^∗_N,T]⁰. Then it follows that

log p(X_oil⁻¹M π_oil^∗ |τ^π, τ^u, ρ^π, λ, h, θ) ∝

−¹₂jTh− ¹₂(X_oil⁻¹M π_oil^∗ −β)⁰(X_oil⁰⁻¹M⁰ΩoilM X_oil⁻¹)⁻¹ (X_oil⁻¹M π_oil^∗ −β)

(2.33)

The state equation of β is given by

Hβ = ^β_t, ^β_t ∼N(0,Ω_β) (2.34) with

log p(β|σ_β²) ∝

−¹₂(β)⁰H⁰Ω⁻¹_β H(β) +g_β(β, σ²_β)

(2.35)

Combining (2.33) and (2.35) yields log p(β|π, u, τ^π, τ^u, ρ^π, β, h, θ) ∝

−¹₂(β −β)ˆ ⁰D⁻¹_β (β−β) +ˆ g_β

(2.36)

with

g_β,σ²

β = −PT

t=2(Φ(^1−β_σ2^t−1 β

−Φ(^−β_σ2 β

)) βˆ = D_β(X_oil⁰ M⁰⁻¹Ω⁻¹_β π_oil^∗ )

D_β = (X_oil⁰ M⁰⁻¹Ω⁻¹_β M⁻¹X_oil +H⁰Ω⁻¹_β H)⁻¹

As before, we include an acceptance-rejection Metropolis-Hastings (ARMH) step additional to the precision-based algorithm as the conditional density is of non-standard form.

Sampling h and θ

For sampling h and the remaining parameters summarized by θ, we stick to the algorithm developed by Chan and Strachan (2012) that is also used in Chan et al. (2016). Thereby, we draw ρ^u from a bivariate truncated normal distribution, employing an ARMH step. Moreover, we draw all remaining variances in separate blocks from inverse-Gamma distributions. We refer the reader to Chan and Strachan (2012) and Chan et al. (2016) for further technical details.⁵

2.7.3 Specifications of other models

We now report details on the additional unobserved component models pre-sented in the model comparison and forecasting exercise in the paper. The algorithm underlying these models are in principle variants of the algorithm presented in the previous section and are very close to those of Chan et al.

(2013) and Chan et al. (2016). For all model variants that differ with respect to the time variation of parameters, we employ the settings as presented in Table 2.2 and switch off the respective state equation(s). For the PC vari-ants that do not include oil prices we set the same starting values and priors for the panel PC model as in Table 2.2 and for the aggregate PC model as described below. Thus, in the remainder of this section we focus on the univariate unobserved component models and the aggregate Phillips curve model. Turning first to the univariate models, the UCSV, panel UCSV and UC model take on the following forms:

5 It should be noted that we do not boundτ^π,EU norτ^u as in Chan et al. (2016).

UCSV

π_t −τ_t^π = ρ^π_t(π_t−1 −τ_t−1^π ) +^π_t τ_t^π = τ_t−1^π +^τ,π_t

ρ^π_t = ρ^π_t−1 +^ρ,π_t ^π_t ∼ N(0, e^h^t) h_t = h_t−1 +^h_t ^h_t ∼ N(0, σ_h²)

^ρ^π ∼ T N(−ρ^π_t−1,1−ρ^π_t−1; 0, σ_ρ²π) panel UCSV

π_n,t−τ_t^π,EU = ρ^π_t(π_n,t−1 −τ_t−1^π,EU) +^π_n,t τ_t^π,EU = τ_t−1^π,EU +^τ,π_t

ρ^π_t = ρ^π_t−1 +^ρ,π_t ^π_t ∼ N(0, e^h^t) h_t = h_t−1 +^h_t ^h_t ∼ N(0, σ_h²)

^ρ^π ∼ T N(−ρ^π_t−1,1−ρ^π_t−1; 0, σ_ρ²π) UC

u_t−τ_t^u = ρ^u₁(u_t−1 −τ_t−1^u ) + ρ^u₂(u_t−2 −τ_t−2^u ) +^u_t τ_t^u = τ_t−1^u +^τ_t^u

^u_t ∼ N(0, σ_u²)

For comparability across models, we employ similar priors and starting val-ues as in our benchmark specification. Thus, for panel and aggregate, UCSV models we set τ₀^π = τ₀^π,EU = 1.9, ρ^π₀ = 0.7,h₀ = 1, ω_τ,π² = 0.01, ω_ρ,π² = 0.005, ω_h² = 0.2. We specify the model parameters as inverse-Gamma distributions

so that for σ_τ²

π, σ_ρ,π² and σ_h² we have σ² ∼ IG(v, S). We set v_h = v_τ = v_ρ,π = 10, S_h = 1.8, S_τ = 0.9 and S_ρ,π = 0.81. The starting values for the UC model are τ₀^u = [9,9], ρ^u_1,0 = 1.6, ρ^u_2,0 = −0.7, ω_u² = 0.2 and ω²_τu = 0.01.

We again assume that σ_u² and σ_τ²u follow an inverse-Gamma distribution and set v_τ^u = v_u = 10 and S_τ^u = 3.6. For the aforementioned models ρ^π and ρ^u stem from truncated normal distributions as detailed in the benchmark model specification.

The multivariate, aggregate Phillips curve model is specified below. Model parameters are again specified as inverse-Gamma distributions and we apply the same starting values as well as priors as in our baseline model (see Table 2.2), except for τ₀^u and S_τ^u, which we set to [9; 9] and 3.6, respectively.

ag. PC

π_t −τ_t^π = ρ^π_t(π_t−1 −τ_t−1^π,EU) +λ_t(u_t −τ_t^u) +β_tπ^oil +^π_t (u_t −τ_t^u) = ρ^u₁(u_t−1 −τ_t−1^u ) +ρ^u_n,2(u_t−2 −τ_t−2^u ) + ^u_t

τ_t^π,EU = τ_t−1^π,EU +^τ,π_t

τ_t^u = τ_t−1^u +^τ,u_t ^h_t ∼N(0, σ²_h) ρ^π_t = ρ^π_t−1 +^ρ,π_t ^u_t ∼ N(0, σ_u²)

λ_t = λ_t−1 +^λ_t ^ρ^π ∼T N(−ρ^π_t−1,1−ρ^π_t−1; 0, σ²_ρ^π) ^π_n,t ∼N(0, e^h^t) ^λ ∼T N(−1−λ_t−1,0−λ_t−1; 0, σ_λ²)

h_t = h_t−1 +^h_t ^β ∼ T N(−β_t−1,1−β_t−1; 0, σ_β²)

2.7.4 Prior predictive analysis

To emphasize the sensibility of our prior settings, we perform a prior pre-dictive analysis. Therefore, we draw from the prior distribution using the starting values and priors shown in Table 2.2 and simulate with the state equations as to generate artificial data series for inflation and unemployment.

We repeat this exercise 10⁴ times. We compute the mean, the median, the 16%- and 84%-percentile as well as the variance of each draw of the artifi-cial series. Then we evaluate the observed data with the cumulative density functions from the artificial data series. Table 2.3 presents the prior cdfs evaluate at the observed data for the distinct features. It can be seen that the baseline model explains well the observed data.

Table 2.3: Prior cdfs for observed data of inflation and unemployment inflation unemployment

mean 0.50 0.68

median 0.49 0.40

16% 0.50 0.31

84% 0.50 0.94

variance 0.50 0.89

The table presents the results of the prior predictive analysis of the benchmark model.

2.7.5 Forecasting results on EMU unemployment rates

Turning to the evaluation of unemployment forecasts, aggregate Phillips curves and the UC models offer a substantial improvement in forecasting, compared to the AR and RW models. Panel model variants, however, again perform best from horizon 12 onwards. This implies that variations of the panel PC model only change the forecasting performance marginally across

models (ratios show some variations from the third decimal point onwards).

Using cumulated oil price inflation improves the unemployment forecast for the medium term. Thus, our proposed panel Phillips curve specification re-veals a better forecast accuracy of EU10 inflation and unemployment (from the medium term onwards) than a variety of other UCSV and univariate time-series models.

inEurope:cross-sectionalidentificationoflong-runinflationexpectations horizon

model 1 2 4 6 9 12 15 18 21 24 27 30 33 36

ARIMA(1,0,1) 1 1 1 1 1 1 1 1 1 1 1 1 1 1

ARIMA(2,0,1) 1.04 0.96 0.70 0.73 0.87 0.98 1.08 1.14 1.18 1.18 1.16 1.12 1.09 1.07

ARIMA(2,0,2) 0.98 0.87 0.76 0.77 0.87 0.95 1.01 1.07 1.09 1.10 1.10 1.07 1.05 1.03

ARIMA(1,0,2) 1.00 0.89 0.91 0.91 0.94 0.96 0.97 0.98 0.99 0.99 0.99 0.99 0.99 0.99

RW 1.01 1.01 1.03 1.02 1.02 1.01 1.01 1.00 1.00 1.00 1.00 1.00 1.00 1.00

panel PC 0.31 0.49 0.74 0.84 0.90 0.92 0.94 0.95 0.94 0.94 0.95 0.95 0.94 0.95 panel PC excl. oil 0.31 0.49 0.74 0.83 0.90 0.92 0.93 0.94 0.93 0.94 0.94 0.94 0.94 0.94

ag. PC 0.65 0.88 1.16 1.31 1.38 1.37 1.34 1.28 1.24 1.21 1.19 1.17 1.15 1.15

ag. PC excl. oil 0.41 0.73 1.19 1.41 1.52 1.51 1.47 1.41 1.35 1.30 1.27 1.24 1.21 1.20

UC 0.52 0.66 1.00 1.28 1.51 1.59 1.61 1.56 1.51 1.47 1.43 1.40 1.37 1.36

panel PC const. λ 0.31 0.49 0.73 0.83 0.90 0.92 0.94 0.95 0.94 0.95 0.95 0.95 0.94 0.95 panel PC const. ρ 0.31 0.50 0.75 0.85 0.92 0.94 0.95 0.96 0.95 0.96 0.96 0.96 0.95 0.96 panel PC const. o 0.31 0.49 0.74 0.83 0.90 0.92 0.93 0.94 0.94 0.94 0.95 0.95 0.94 0.95 panel PC const. 0.31 0.50 0.75 0.85 0.92 0.95 0.96 0.96 0.95 0.95 0.96 0.95 0.95 0.95 panel PC excl. sv 0.30 0.49 0.73 0.83 0.90 0.93 0.94 0.95 0.95 0.96 0.96 0.96 0.96 0.96 panel PC cum. oil 0.30 0.49 0.73 0.83 0.89 0.91 0.93 0.94 0.93 0.94 0.94 0.94 0.94 0.94

panel PC cs ρ 0.31 0.50 0.74 0.84 0.91 0.93 0.94 0.95 0.95 0.95 0.96 0.96 0.96 0.96

This table displays the root mean squared forecast errors (RMSFEs) of respective model relative to the RMSFEs of an ARIMA(1,0,1) model that stem from a pseudo-out-of-sample unemployment rate forecast for the time span January 2009 to April 2017. Values lower than 1 indicate that the RMSFEs of the respective model are lower than those of the ARIMA(1,0,1) model. The results from the benchmark model are highlighted in bold. The shading indicates the respective forecasting performance, whereby green (light) shadings emphasize better forecasting performance.

125

in ASEAN-5 economies since the Asian Financial Crisis

This chapter is joint work with Juan-Angel Garcia Morales and has been written as part of the Inter-national Monetary Fund Funded Internship Program. The content of this chapter contains abstracts of Dany-Knedlik, G. and Garcia-Morales, J. A. (2018): Monetary Policy and Inflation Dynamics in ASEAN Economies. IMF Working Papers 18/147, International Monetary Fund and Dany-Knedlik, G. and Garcia-Morales, J. A. (2018 forthcoming): Forecasting Inflation in Asia: What has changed since the Asian Financial Crisis?. IMF Working Paper. International Monetary Fund. The views expressed in this chapter are those of the authors and do not necessarily represent the views of the IMF, its Executive Board, or IMF management. We thank Ana Corbacho, Giovanni Ganelli, Jaime Guajardo, Minsuk Kim, Changyong Rhee, Umang Rawat, Aubrey Poon, Gregor von Schweinitz, Jan Christopher Scherer, Axel Lindner and Katja Heinisch for very valuable comments on earlier drafts.

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