Priors from General Equilibrium Models for VARs by Marco del Negro - - PowerPoint PPT Presentation

Priors from General Equilibrium Models for VARs by Marco del Negro and Frank Schorfheide

Presenter: Keith O’Hara March 10, 2014

Presenter: Keith O’Hara (∼/) DSGE-VARs March 10, 2014 1 / 30

The Main Idea of the Paper

Use the implied moments of a DSGE model as the prior for a Bayesian VAR (a ‘DSGE-VAR(λ)’).

◮ This is similar to a ‘dummy observation’ approach.

We can view the policy functions of a DSGE model, St = FSt−1 + Gεt, as a VAR(1) with tight cross-equation restrictions. The parameter λ ∈ (0, ∞) controls how ‘close’ the DSGE-VAR matches the DSGE model dynamics; it corresponds to the ratio of ‘dummy observations’ to actual data. As λ → ∞, we approach the DSGE model.

Presenter: Keith O’Hara (∼/) DSGE-VARs March 10, 2014 2 / 30

Solving and Estimating a DSGE Model

Presenter: Keith O’Hara (∼/) DSGE-VARs March 10, 2014 3 / 30

Solving a DSGE Model

We can express a log-linearized DSGE model as a system of linear expectational difference equations: AθSt = BθEt[St+1] + CθSt−1 + Dθεt which we ‘solve’ to get a first-order VAR for the state’s transition: St = FSt−1 + Gεt. F solves the following matrix quadratic, which implies a solution for G: 0 = BθF2 − AθF + Cθ G = (Aθ − BθF)−1 Dθ We solve the matrix polynomial as a generalized eigenvalue problem.

Presenter: Keith O’Hara (∼/) DSGE-VARs March 10, 2014 4 / 30

The Log-Linearized Model

For the log-linearized model in this paper, this corresponds to xt = Etxt+1 − τ −1(Rt − Et[πt+1]) + (1 − ρg)gt + ρz 1 τ zt πt = γ r ∗ Et[πt+1] + κ(xt − gt) Rt = ρRRt−1 + (1 − ρR) (ψ1πt + ψ2xt) + νt with shock processes: zt = ρzzt−1 + σzεz,t gt = ρggt−1 + σgεg,t νt = σRεR,t So St = [xt, πt, Rt, zt, gt, νt]⊤.

Presenter: Keith O’Hara (∼/) DSGE-VARs March 10, 2014 5 / 30

Solving a DSGE Model

Three cases to consider: (1) In the case of as many explosive eigenvalues (λe) as forward-looking equations, we have a unique solution to the problem. (2) If there are more stable eigenvalues (λs) than forward-looking equations, there are many stable solutions for F, one for each block-partition of λs. Here, equilibrium is indeterminate, and we face the issue of equilibrium selection. (3) If there are less stable eigenvalues than forward-looking equations, then there are no non-explosive solutions, as there is no block-partition of λs such that all λ are stable.

Presenter: Keith O’Hara (∼/) DSGE-VARs March 10, 2014 6 / 30

Estimating a DSGE Model

With our solution St = FSt−1 + Gεt, we have the state equation of a filtering problem. Assuming Gaussian disturbances, which, coupled with our linear problem, implies a Kalman filter approach with measurement equation Yt = H⊤St + εY,t We proceed in three short steps, repeated for all t in {1, . . . , T}:

◮ predict the state at time t + 1 given information available at time t; ◮ update the state with new Yt+1 information; and ◮ calculate the likelihood at t + 1 based on forecast errors of Yt+1 and

the covariance matrix of these forecasts.

Classical ML and Bayesian estimation procedures are standard, with the latter being particularly popular; probably due to identification.

Presenter: Keith O’Hara (∼/) DSGE-VARs March 10, 2014 7 / 30

DSGE-VAR Details: Setup and Prior

Presenter: Keith O’Hara (∼/) DSGE-VARs March 10, 2014 8 / 30

VAR Model

The standard VAR(p) model is denoted by yt = Φ0 + Φ1yt−1 + · · · + Φpyt−p + ut, ut|y t−1 ∼ N(0, Σu) Let k = 1 + n × p. In stacked form, we have Y = XΦ + U with likelihood function p(Y |Φ,Σu) ∝ |Σu|−T/2× exp

• −1

2tr

• Σ−1

u (Y ⊤Y − Φ⊤X ⊤Y − Y ⊤XΦ + Φ⊤X ⊤XΦ)

• Presenter: Keith O’Hara (∼/)

DSGE-VARs March 10, 2014 9 / 30

The Prior

We look at the prior in terms of a ‘dummy observation’ approach. The prior is of the form p(Φ, Σu|θ) = c−1(θ)|Σu|− λT+n+1

2

× exp

• −1

2tr

• λTΣ−1

u (Γ∗ yy(θ) − Φ⊤Γ∗ xy(θ) − Γ∗ yx(θ)Φ + Φ⊤Γ∗ xx(θ)Φ)

• where Γ∗

yy(θ) := Eθ[yty ⊤ t ], etc, are the population moments.

The form of the normalizing term c−1(θ) is a little complicated.

Presenter: Keith O’Hara (∼/) DSGE-VARs March 10, 2014 10 / 30

The Prior

For a model solution of the form St = FSt−1 + Gεt Yt = H⊤St we first compute the steady state covariance matrix of the state by solving the discrete Lyapunov equation Ωss = FΩssF⊤ + GQG⊤ using a doubling algorithm, where Eθ[StS⊤

t ] = Ωss.

Then we compute the Γ∗ matrices with Γ∗

yy(θ) = H⊤ΩssH

Γ∗

yxh(θ) = H⊤FhΩssH

Presenter: Keith O’Hara (∼/) DSGE-VARs March 10, 2014 11 / 30

Computational Aside: Solving for Ωss

For completeness... we solve Ωss = FΩssF⊤ + GQG⊤ by iteration. Let Q := GQG⊤. Set Ωss(0) = Q and B(0) = F. Then, for i = 1, 2, . . . Ωss(i + 1) = Ωss(i) + B(i)Ωss(i)B(i)⊤ B(i + 1) = B(i)B(i)⊤ Continue until the difference between Ωss(i + 1) and Ωss(i) is ‘small’. Note: Ωss is a symmetric positive-definite matrix, so the relevant matrix norm here is the largest singular value (from a SVD). Could also use the vec-Kronecker trick: vec(ABC) = (C ⊤ ⊗ A)vec(B).

Presenter: Keith O’Hara (∼/) DSGE-VARs March 10, 2014 12 / 30

The Prior

Let Φ∗(θ) = [Γ∗

xx(θ)]−1Γ∗ xy(θ)

Σ∗

u(θ) = Γ∗ yy(θ) − Γ∗ yx(θ)[Γ∗ xx(θ)]−1Γ∗ xy(θ)

◮ Interpretation: If the data were generated by the DSGE model at hand,

Φ∗(θ) is the coefficient matrix of the VAR(p) that minimizes the

• ne-step-ahead QFE loss.

Given a θ, the prior distribution is of the usual IW-N form: Σu|θ ∼ IW (λTΣ∗

u(θ), λT − k, n)

Φ|Σu, θ ∼ N

• Φ∗(θ), Σu ⊗ (λTΓ∗

xx(θ))−1

The joint prior is then given by p(Φ, Σu, θ) = p(Φ, Σu|θ)p(θ)

Presenter: Keith O’Hara (∼/) DSGE-VARs March 10, 2014 13 / 30

DSGE-VAR Posterior

Presenter: Keith O’Hara (∼/) DSGE-VARs March 10, 2014 14 / 30

The Posterior: Block 1

The joint posterior distribution is factorized similarly: p(Φ, Σu, θ|Y ) = p(Φ, Σu|Y , θ)p(θ|Y ) The ML estimates are

• Φ(θ) =
• λTΓ∗

xx(θ) + X ⊤X

−1 [λTΓ∗

xy + X ⊤Y ]

• Σu(θ) =

1 (λ + 1)T

• (λTΓ∗

yy(θ) + Y ⊤Y )

• −

1 (λ + 1)T ×

• (λTΓ∗

yx(θ) + Y ⊤X)(λTΓ∗ xx(θ) + X ⊤X)−1(λTΓ∗ xy(θ) + X ⊤Y )

• The prior and likelihood are conjugate, so

Σu|Y , θ ∼ IW

• (λ + 1)T

Σu(θ), (1 + λ)T − k, n

• Φ|Y , Σu, θ ∼ N
• Φ(θ), Σu ⊗ (λTΓ∗

xx(θ) + X ⊤X)−1

Presenter: Keith O’Hara (∼/) DSGE-VARs March 10, 2014 15 / 30

The Posterior: Block 2

The posterior distribution of the DSGE parameters is p(θ|Y ) ∝ p(Y |θ)p(θ) where the marginal likelihood is p(Y |θ) =

• p(Y |Φ, Σu)p(Φ, Σu)d(Φ, Σu)

(1) The authors show (in the appendix) that the closed form for (1) is p(Y |θ) = p(Y |Φ, Σ)p(Φ, Σ|θ) p(Φ, Σ|Y ) = |λTΓ∗

xx(θ) + X ⊤X|− n

2 |(λ + 1)T

Σu(θ)|− (λ+1)T−k

2

|λTΓ∗

xx(θ)|− n

2 |λTΣ∗

u(θ)|− λT−k

2

× (2π)−nT/22

n((λ+1)T−k) 2

n

i=1 Γ[((λ + 1)T − k + 1 − i)/2]

2

n(λT−k) 2

n

i=1 Γ[(λT − k + 1 − i)/2]

Presenter: Keith O’Hara (∼/) DSGE-VARs March 10, 2014 16 / 30

Sampling Algorithm

Our previous discussion implies that a Metropolis-within-Gibbs MCMC algorithm would be appropriate. Given some value for θ, we sample Σu from Σu|Y , θ ∼ IW

• (λ + 1)T

Σu(θ), (1 + λ)T − k, n

• Then, given θ and Σu, sample

Φ|Y , Σu, θ ∼ N

• Φ(θ), Σu ⊗ (λTΓ∗

xx(θ) + X ⊤X)−1

Given Φ and Σu, we evaluate a new θ draw using a Random Walk Metropolis MCMC algorithm.

Presenter: Keith O’Hara (∼/) DSGE-VARs March 10, 2014 17 / 30

Random Walk Metropolis Sampling Algorithm

Given some initial θ (perhaps the posterior mode), draw a proposal θ(∗) from a jumping distribution, N(θ(h−1), c · Σm) where Σm is the inverse of the Hessian computed at the posterior mode and c is a scaling factor. Compute the acceptation ratio, ν = p(Y |θ(∗))p(θ(∗)) p(Y |θ(h−1))p(θ(h−1)) Finally, we accept or reject the proposal according to θ(h) =

• θ(∗)

P = min{ν, 1} θ(h−1) else Given this θ(h), draw a new Σu, and so on.

Presenter: Keith O’Hara (∼/) DSGE-VARs March 10, 2014 18 / 30

Choosing λ

How to choose λ? Construct a grid Λ and select the λ that maximizes the marginal data density pλ(Y ) =

• pλ(Y |θ)p(θ)dθ

That is, ˆ λ = argmax

λ∈Λ

pλ(Y ) ˆ λ is roughly 0.6, but forecasting exercises show pλ(Y ) to be relatively flat between 0.5 and 2.

Presenter: Keith O’Hara (∼/) DSGE-VARs March 10, 2014 19 / 30

The Log-Linearized Model

Presenter: Keith O’Hara (∼/) DSGE-VARs March 10, 2014 20 / 30

The Log-Linearized Model

Three main equations: xt = Etxt+1 − τ −1(Rt − Et[πt+1]) + (1 − ρg)gt + ρz 1 τ zt πt = γ r ∗ Et[πt+1] + κ(xt − gt) Rt = ρRRt−1 + (1 − ρR) (ψ1πt + ψ2xt) + σRεR,t Shock processes: zt = ρzzt−1 + σzεz,t gt = ρggt−1 + σgεg,t Measurement equations: ∆ ln Xt = ln γ + ∆xt + zt ∆ ln Pt = ln π∗ + πt ln Ra

t = 4 [(ln r ∗ + ln π∗) + Rt]

Presenter: Keith O’Hara (∼/) DSGE-VARs March 10, 2014 21 / 30

Priors

Prior Θ Meaning Density P1 P2 ln γ Technology Scaling N 0.500 0.250 ln π∗ SS Inflation N 1.000 0.500 ln r ∗ γ/β G 0.500 0.250 κ NKPC Slope G 0.300 0.150 τ

• Coef. RRA

G 2.000 0.500 ψ1 MP Inflation G 1.500 0.250 ψ2 MP Output Gap G 0.125 0.100 ρR Interest Smoothing B 0.500 0.200 ρg AR Gov. Spending B 0.800 0.100 ρz AR Technology B 0.300 0.100 σR SD Int. IG 0.251 0.139 σg SD Gov. IG 0.630 0.323 σz SD Tech. IG 0.875 0.430

Presenter: Keith O’Hara (∼/) DSGE-VARs March 10, 2014 22 / 30

Posterior CI for Different λ

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RMSFE versus unrestricted VAR(4)

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RMSFE versus unrestricted VAR(4) and BVAR(4)

Presenter: Keith O’Hara (∼/) DSGE-VARs March 10, 2014 25 / 30

Impulse Response Functions

Not entirely obvious how to relate the structural shocks from a DSGE model (εt) to the one-step ahead forecast errors of a VAR (ut). We’re used to seeing IRFs of the form...

Presenter: Keith O’Hara (∼/) DSGE-VARs March 10, 2014 26 / 30

IRFs from VAR(4)

• 0.2

0.0 0.2 0.4 0.6 0.8 5 10 15 20

Horizon Shock from GDP to GDP

• 0.2

0.0 0.2 0.4 0.6 5 10 15 20

Horizon Shock from GDP to INF

0.0 0.2 0.4 0.6 5 10 15 20

Horizon Shock from GDP to INT

• 0.2
• 0.1

0.0 0.1 0.2 0.3 5 10 15 20

Horizon Shock from INF to GDP

0.0 0.2 0.4 0.6 0.8 1.0 5 10 15 20

Horizon Shock from INF to INF

0.0 0.1 0.2 0.3 0.4 0.5 0.6 5 10 15 20

Horizon Shock from INF to INT

• 0.3
• 0.2
• 0.1

0.0 0.1 5 10 15 20

Horizon Shock from INT to GDP

0.0 0.2 0.4 0.6 5 10 15 20

Horizon Shock from INT to INF

0.0 0.5 1.0 5 10 15 20

Horizon Shock from INT to INT

Presenter: Keith O’Hara (∼/) DSGE-VARs March 10, 2014 27 / 30

Impulse Response Functions

Let Σtr = Chol(Σu). We have ut = ΣtrΩǫt (2) where Ω⊤Ω = I. ∂yt ∂εt

• VAR

= ΣtrΩ For a DSGE model, there is a unique A(θ) that determines the contemporaneous effect of ε on yt. ∂yt ∂εt

• DSGE

= A0(θ) = Σ∗

tr(θ)Ω∗(θ)

Use a QR decomposition to get A0. Keep Σtr in (2) but replace Ω with Ω∗(θ).

Presenter: Keith O’Hara (∼/) DSGE-VARs March 10, 2014 28 / 30

IRFs

Presenter: Keith O’Hara (∼/) DSGE-VARs March 10, 2014 29 / 30

Conclusion

Good forecasting performance. Perhaps better to compare forecasts from a DSGE/DSGE-VAR against a steady state BVAR. Mattias Villani has some nice work on this. DSGE-VAR extended to higher-order expansions?

Presenter: Keith O’Hara (∼/) DSGE-VARs March 10, 2014 30 / 30