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Bayesian Flexible Local Projections

  • Luca Brugnolini , Leopoldo Catania EMAIL logo , Pernille Hansen and Paolo Santucci de Magistris
Published/Copyright: April 5, 2024
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Studies in Nonlinear Dynamics & Econometrics
From the journal Studies in Nonlinear Dynamics & Econometrics Volume 28 Issue 2

Abstract

We develop a methodology to estimate impulse response functions via Bayesian techniques with the goal of providing a bridge between a linear vector autoregressive specification and a high-order polynomial local projection, namely flexible local projection. We label this methodology Bayesian Flexible Local Projection (BFLP). We assess the properties of BFLP in a Monte Carlo framework considering both linear and non-linear models as data generating processes. We also empirically illustrate how BFLP can be used with standard identification strategies. In particular, we show how to use external instruments to identify the effects of the monetary policy shock in the United States. Furthermore, exploiting the time-varying nature of the impulse response functions based on BFLP, we assess the zero lower bound irrelevance hypothesis and find no strong evidence that monetary policy was less effective in influencing output and inflation during the recent ZLB period.

Keywords: impulse response functions; vector autoregressions; local projections; monetary policy
JEL Classification: C11; C14; E52
Corresponding author: Leopoldo Catania, Department of Economics and Business Economics, Aarhus University, Aarhus, Denmark, E-mail:

We are grateful to Silvia Miranda Agrippino for providing us with the data and for giving us valuable feedback on our work. We would also like to thank the participants of the 2019 IAAE conference in Cyprus.

Funding source: Danmarks Frie Forskningsfond

Award Identifier / Grant number: 3099-00089B

Appendix A: Posterior Density and Marginal Likelihood

This appendix derives the analytical expression of the posterior density of Bayesian Local Projections (BLP) with conjugate priors exploiting results from Appendix A of Giannone, Lenza, and Primiceri (2015), and of the marginal likelihood. The appendix concludes by presenting a way to compute the marginal likelihood in a numerically stable way which is then used for Markov chain Monte Carlo estimation of the BLP coefficients.

A.1 Derivation of the Posterior Density

Consider the vec representation in (6) where the superscript h and the subscript u are dropped for notational convenience. We now present the derivation of the posterior density estimating λ via maximum posterior, an algorithm to sample from the posterior distribution of λ is presented afterward. Let’s first define p β , Σ y ∝ p y β , Σ p β Σ p Σ , where

(16) p y β , Σ = 2 π − n T − h − p 2 Σ ⊗ I T − h − p − 1 2 ⁡ exp − 1 2 y − x β ′ Σ ⊗ I T − h − p − 1 y − x β = 2 π − n T − h − p 2 Σ − T − h − p 2 ⁡ exp − 1 2 y − x β ′ Σ ⊗ I T − h − p − 1 y − x β ,

(17) p β Σ = 2 π − n k 2 Σ ⊗ Ω 0 − 1 2 ⁡ exp − 1 2 β − β 0 ′ Σ ⊗ Ω 0 − 1 β − β 0 = 2 π − n k 2 Σ − k 2 Ω 0 − n 2 ⁡ exp − 1 2 β − β 0 ′ Σ ⊗ Ω 0 − 1 β − β 0 ,

(18) p Σ = Ψ 0 d 0 2 Σ − n + d 0 + 1 2 ⁡ exp − 1 2 tr Ψ 0 Σ − 1 2 n d 0 2 Γ n d 0 2 ,

(19) p β , Σ , y = 2 π − n T − h − p − k 2 Σ − T − h − p + k + n + d 0 + 1 2 Ω 0 − n 2 Ψ − d 0 2 exp − 1 2 tr Ψ 0 Σ − 1 2 n d 0 2 Γ n d 0 2 × exp − 1 2 y − x β ′ Σ ⊗ I T − h − p − 1 y − x β + β − β 0 ′ Σ ⊗ Ω 0 − 1 β − β 0 ︸ E .

The term E can be rewritten as

(20) E = Σ − 1 2 ⊗ I T − h − p y − x β ′ Σ − 1 2 ⊗ I T − h − p y − x β + Σ ⊗ Ω 0 − 1 2 β − β 0 ′ Σ ⊗ Ω 0 − 1 2 β − β 0 = Σ − 1 2 ⊗ I T − h − p y − Σ − 1 2 ⊗ X β ′ Σ − 1 2 ⊗ I T − h − p y − Σ − 1 2 ⊗ X β + Σ ⊗ Ω 0 − 1 2 β − Σ ⊗ Ω 0 − 1 2 β 0 ′ Σ ⊗ Ω 0 − 1 2 β − Σ ⊗ Ω 0 − 1 2 β 0 .

Let now:

(21) w = Σ ⊗ Ω 0 − 1 2 β 0 Σ − 1 2 ⊗ I T − h − p y  and  W = Σ ⊗ Ω 0 − 1 2 Σ − 1 2 ⊗ X ,

then E can then be rewritten as

(22) E = w − W β ′ w − W β = w − W β − β ̂ − W β ̂ ′ w − W β − β ̂ − W β ̂ = β − β ̂ ′ W ′ W β − β ̂ + w − W β ̂ ′ w − W β ̂ ,

where β ̂ is the OLS estimator of regressing w on W, and the third equality in Equation (22) then follows from the OLS orthogonality property

(23) W ′ w − W β ̂ = W ′ I − W W ′ W − 1 W ′ w = W ′ − W ′ W W ′ W − 1 W ′ w = 0 .

Inserting the definitions of w and W from Equation (21) in the OLS estimator yields

(24) vec b ̂ ≡ β ̂ = W ′ W − 1 W ′ w = Σ ⊗ Ω 0 − 1 + Σ − 1 ⊗ X ′ X − 1 Σ − 1 ⊗ X ′ y + Σ ⊗ Ω 0 − 1 β 0 = Σ ⊗ Ω 0 − 1 + X ′ X − 1 Σ − 1 ⊗ X ′ y + Σ ⊗ Ω 0 − 1 β 0 = I n ⊗ Ω 0 − 1 + X ′ X − 1 X ′ y + I n ⊗ Ω 0 − 1 + X ′ X − 1 Ω 0 − 1 β 0 = vec Ω 0 − 1 + X ′ X − 1 X ′ Y + Ω 0 − 1 b 0 .

Furthermore, given that

(25) W ′ W = Σ ⊗ Ω 0 − 1 + X ′ X − 1 − 1 ,

and

(26) w − W β ̂ ′ w − W β ̂ = β ̂ − β 0 ′ Σ ⊗ Ω 0 − 1 β ̂ − β 0 + y − x β ̂ ′ Σ ⊗ I T − h − p − 1 y − x β ̂ ,

it follows that, letting e ̂ = vec u ̂ = vec Y − X b ̂ = y − x β ̂ , the expression in Equation (26) becomes

(27) w − W β ̂ ′ w − W β ̂ = β ̂ − β 0 ′ Σ ⊗ Ω 0 − 1 β ̂ − β 0 + e ̂ ′ Σ ⊗ I T − h − p − 1 e ̂ = tr u ̂ ′ u ̂ + b ̂ − b 0 ′ Ω 0 − 1 b ̂ − b 0 Σ − 1 .

Thus, the exponent E in Equation (19) further reduces to

(28) E = β − β ̂ ′ Σ ⊗ Ω 0 − 1 + X ′ X − 1 − 1 β − β ̂ + tr u ̂ ′ u ̂ + b ̂ − b 0 ′ Ω 0 − 1 b ̂ − b 0 Σ − 1 .

Inserting E from equation in the joint density of β ; Σ and y in Equation (19) yields the unnormalized posterior density

(29) p β , Σ y ∝ p β , Σ , y = 2 π − n T − h − p − k 2 Σ − T − h − p + k + n + d 0 + 1 2 Ω 0 − n 2 Ψ 0 d 0 2 × exp − 1 2 tr Ψ 0 + u ̂ ′ u ̂ + b ̂ − b 0 ′ Ω 0 − 1 b ̂ − b 0 Σ − 1 2 n d 0 2 Γ n d 0 2 × exp − 1 2 β − β ̂ ′ Σ ⊗ Ω 0 − 1 + X ′ X − 1 − 1 β − β ̂ ∝ N β ; β ̂ , Σ ⊗ Ω I W Σ ; Ψ , d ,

where Ω = Ω 0 − 1 + X ′ X − 1 , Ψ = Ψ 0 + u ̂ ′ u ̂ + b ̂ − b 0 ′ Ω 0 − 1 b ̂ − b 0 , and d = T − h − p + d 0.

The marginal likelihood is then given as

(30) p Y = ∬ p y β , Σ p β Σ p Σ d β d Σ = 2 π − n T − h − p − k 2 Ω 0 − n 2 Ψ 0 d 0 2 2 − n d 0 2 Γ n d 0 2 − 1 ∫ Σ − k + d + n + 1 2 ⁡ exp − 1 2 tr Ψ Σ − 1 × ∫ exp − 1 2 β − β ̂ ′ Σ ⊗ Ω − 1 β − β ̂ d β d Σ = 2 π − n T − h − p − k 2 Ω 0 − n 2 Ψ 0 d 0 2 2 − n d 0 2 Γ n d 0 2 − 1 2 π n k 2 Ω n 2 ∫ Σ k 2 Σ − k + d + n + 1 2 ⁡ exp − 1 2 tr Ψ Σ − 1 d Σ = 2 π − n T − h − p 2 Ω 0 − n 2 Ψ 0 d 0 2 2 − n d 0 2 Γ n d 0 2 − 1 Ω n 2 ∫ Σ − d + n + 1 2 ⁡ exp − 1 2 tr Ψ Σ − 1 d Σ = 2 π − n T − h − p 2 Ω 0 − n 2 Ψ 0 d 0 2 2 − n d 0 2 Ω n 2 Γ n d 2 Γ n d 0 2 2 n d 2 Ψ − d 2 = π − n T − h − p 2 Γ n d 2 Γ n d 0 2 Ω 0 − n 2 Ω n 2 Ψ 0 d 0 2 Ψ − d 2 .

A.2 Stable Marginal Likelihood

The marginal likelihood in Equation (30) is numerically unstable for large systems, and is therefore replaced by the equivalent expression

(31) p y = π − n T − h − p 2 Γ n d 2 Γ n d 0 2 Ω 0 − n 2 Ω 0 − 1 + X ′ X − n 2 Ψ 0 d 0 2 Ψ 0 + u ̂ ′ u ̂ + b ̂ − b 0 ′ Ω 0 − 1 b ̂ − b 0 − d 2 = π − n T − h − p 2 Γ n d 2 Γ n d 0 2 Ψ 0 − T − h − p 2 I k + D Ω ′ X ′ X D Ω − n 2 I n + D Ψ ′ u ̂ ′ u ̂ + b ̂ − b 0 ′ Ω 0 − 1 b ̂ − b 0 D Ψ − d 2 ,

where Ω 0 = D Ω D Ω ′ , Ψ 0 − 1 = D Ψ D Ψ ′ . A numerically stable calculation of the last two determinants of Equation (31) is carried out as the product of one plus the eigenvalues of D Ω ′ X ′ X D Ω and D Ψ ′ u ̂ ′ u ̂ + b ̂ − b 0 ′ Ω 0 − 1 b ̂ − b 0 D Ψ , respectively.

Appendix B: Markov-Chain Monte-Carlo Algorithm

The posterior of the hyperparameter

(32) p ( λ ( h ) | y ) ∝ p ( y | λ ( h ) ) p ( λ ( h ) )

is proportional to the fit-complexity trade-off representation of Giannone, Lenza, and Primiceri (2015)

(33) p y λ h = ∫ p y λ h , β h , Σ u h p β h , Σ u h λ h d β h d Σ u h = π − n T h 2 Γ n d h 2 h Γ n d 0 h 2 Ω 0 h − n 2 Ω h n 2 Ψ 0 h d 0 h 2 Ψ h − d h 2

(34) ∝ V u h posterior − 1 V u h prior T h + d 0 h 2 ︸   Fit ∏ t = p ̃ + 1 T − h V t + h t − 1 2 ︸ Penalty ∀ h = 1 , … H ,

where V u h posterior and V u h prior are the posterior and prior mean of Σ u h , and the conditional variance of the h-step-ahead forecast of y t , averaged across all possible prior realizations of Σ u h , is V t + h t = E Σ u h V ar y t + h y t , Σ u h with y t = y p + 1 , … , y t ′ . The closed-form expression of the likelihood is derived in Appendix A.1. However, it is numerically unstable for large systems. The numerically stable equivalent in Appendix A.2 is applied for estimation. Additionally, this paper uses the informative Gamma hyperprior of Ferreira, Miranda-Agrippino, and Ricco (2023)

(35) λ h ∼ Gamma κ h , θ h ,

with mode of the distribution fixed at 0.4 and standard deviation following a logistic function

(36) mode λ h = κ h − 1 θ h = . 4 , sd λ h = κ h θ h = . 1 + . 4 1 + exp − . 3 h − 12 .

. The logistic function is increasing in the horizon that taps of at horizons larger than h = 36, implying an increasingly diffuse prior in the forecast horizon h, consistent with the notion that prior model misspecification is compounded at each horizon.

B.1 λ (h) under Estimation Uncertainty

Drawing from the posterior when λ h is subject to estimation uncertainty involves a Metropolis-Hastings updating algorithm with an embedded Robbins-Monro scheme for λ h , and a subsequent Gibbs-sampling procedure from the joint posterior of β h ; Σ u , HAC h . Algorithm B.1 provides the MCMC procedure to obtain draws λ g h , β g h , Σ u , H A C h g for g = 1, …, G from the joint posterior. We follow Justiniano, Primiceri, and Tambalotti (2010) and set G = 1000, with a burn-in period of G ⋆ = 200 observations and assess convergence via the Z-test of Geweke (1992).

Algorithm B.1:

MCMC with Metropolis-Hastings updating of prior tightness.

We follow Giannone, Lenza, and Primiceri (2015) and update the prior tightness parameter according to a Metropolis-Hastings algorithm, where at each iteration a candidate value is drawn from a Gaussian proposal with mean equal to the preceding value in the Markov chain, λ g − 1 ( h ) , and standard deviation, σ g ( h ) . The algorithm is initialized at the posterior mode λ 0 ( h ) = arg max λ ( h ) p λ ( h ) y , and the starting value of the scaling parameter is the square root of the inverse Hessian of the negative log-posterior of the hyper-parameter at the peak σ 0 ( h ) = − ∂ 2 ⁡ log ⁡ p λ ( h ) y ∂ λ ( h ) 2 λ ( h ) = λ 0 ( h ) − 1 2 , calculated as the outer product of the numerical gradient. The acceptance rate is monitored as in Garthwaite, Fan, and Sisson (2016) who propose an optimal adaptive scaling using the Robbins-Monro process. We follow Roberts and Rosenthal (2001) and target an acceptance rate of α ⋆ = 44 %. In particular, the Markov chain is restarted according to some predetermined rules. The variance is regulated by letting γ ( h ) = log ( σ g ( h ) ) , where the true value γ* is estimated as

(37) γ g + 1 ( h ) = γ g ( h ) + α g ( h ) − α * g α * ( 1 − α * )

The search is restarted at the previous estimate of γ* and jumps back to g = η 0 = s α * ( 1 − α * ) whenever the restart criterion is met. In our case, this is when the estimate of γ* has changed by more than log 3 from the starting values within the first n max = 200 iterations corresponding to the burn-in period.

B.2 λ (h) Estimated via Maximum Posterior

When λ h estimated via the maximum of the posterior mode, i.e. λ 0 ( h ) = arg max λ ( h ) p λ ( h ) y . Then, the Bayes estimator of β h simply reduces to the posterior mean, i.e. β ̃ h , due to the structure of the Normal-inverse-Wishart conjugate prior only being a function in a set of fixed hyper-parameters. However, carrying out inference on the IRF coefficients still requires sampling from the posterior which is conditioned on the posterior error variance, Σ u , HAC h . This procedure is detailed in Algorithm B.2.

Algorithm B.2:

MCMC Setup for BLP with Fixed Prior Tightness at the ML Estimator

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Supplementary Material

This article contains supplementary material (https://doi.org/10.1515/snde-2023-0001).

Received: 2023-01-04
Accepted: 2024-03-01
Published Online: 2024-04-05

© 2024 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/snde-2023-0001
Keywords for this article
impulse response functions; vector autoregressions; local projections; monetary policy

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. Editorial Introduction of the Special Issue of Studies in Nonlinear Dynamics and Econometrics in Honor of Herman van Dijk
  4. Review
  5. Challenges and Opportunities for Twenty First Century Bayesian Econometricians: A Personal View
  6. Research Articles
  7. Markov-Switching Models with Unknown Error Distributions: Identification and Inference Within the Bayesian Framework
  8. Dynamic Shrinkage Priors for Large Time-Varying Parameter Regressions Using Scalable Markov Chain Monte Carlo Methods
  9. Matrix autoregressive models: generalization and Bayesian estimation
  10. Sequential Monte Carlo with model tempering
  11. Modeling Corporate CDS Spreads Using Markov Switching Regressions
  12. Combining Large Numbers of Density Predictions with Bayesian Predictive Synthesis
  13. Bayesian inference for non-anonymous growth incidence curves using Bernstein polynomials: an application to academic wage dynamics
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  15. A Dynamic Latent-Space Model for Asset Clustering
  16. Posterior Manifolds over Prior Parameter Regions: Beyond Pointwise Sensitivity Assessments for Posterior Statistics from MCMC Inference
  17. Bayesian Flexible Local Projections
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