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, S t = F S t − 1 + G ε t , 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: A θ S t = B θ E t [ S t +1 ] + C θ S t − 1 + D θ ε t which we ‘solve’ to get a first-order VAR for the state’s transition: S t = F S t − 1 + G ε t . F solves the following matrix quadratic, which implies a solution for G : 0 = B θ F 2 − A θ F + C θ G = ( A θ − B θ F ) − 1 D θ 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 1 x t = E t x t +1 − τ − 1 ( R t − E t [ π t +1 ]) + (1 − ρ g ) g t + ρ z τ z t π t = γ r ∗ E t [ π t +1 ] + κ ( x t − g t ) R t = ρ R R t − 1 + (1 − ρ R ) ( ψ 1 π t + ψ 2 x t ) + ν t with shock processes: z t = ρ z z t − 1 + σ z ε z , t g t = ρ g g t − 1 + σ g ε g , t ν t = σ R ε R , t So S t = [ x t , π t , R t , z t , g t , ν t ] ⊤ . 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 ( λ e ) as forward-looking equations, we have a unique solution to the problem. (2) If there are more stable eigenvalues ( λ s ) than forward-looking equations, there are many stable solutions for F , one for each block-partition of λ s . 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 λ s such that all λ are stable. Presenter: Keith O’Hara ( ∼ /) DSGE-VARs March 10, 2014 6 / 30
Estimating a DSGE Model With our solution S t = F S t − 1 + G ε t , 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 Y t = H ⊤ S t + ε Y , 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 Y t +1 information; and ◮ calculate the likelihood at t + 1 based on forecast errors of Y t +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 u t | y t − 1 ∼ N ( 0 , Σ u ) y t = Φ 0 + Φ 1 y t − 1 + · · · + Φ p y t − p + u t , Let k = 1 + n × p . In stacked form, we have Y = X Φ + U with likelihood function p ( Y | Φ , Σ u ) ∝ | Σ u | − T / 2 × � �� � − 1 Σ − 1 u ( Y ⊤ Y − Φ ⊤ X ⊤ Y − Y ⊤ X Φ + Φ ⊤ X ⊤ X Φ) exp 2tr 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 , Σ u | θ ) = c − 1 ( θ ) | Σ u | − λ T + n +1 × 2 � �� � − 1 λ T Σ − 1 u (Γ ∗ yy ( θ ) − Φ ⊤ Γ ∗ xy ( θ ) − Γ ∗ yx ( θ )Φ + Φ ⊤ Γ ∗ exp 2tr xx ( θ )Φ) where Γ ∗ yy ( θ ) := E θ [ y t y ⊤ 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 S t = F S t − 1 + G ε t Y t = H ⊤ S t we first compute the steady state covariance matrix of the state by solving the discrete Lyapunov equation Ω ss = F Ω ss F ⊤ + G Q G ⊤ using a doubling algorithm, where E θ [ S t S ⊤ t ] = Ω ss . Then we compute the Γ ∗ matrices with Γ ∗ yy ( θ ) = H ⊤ Ω ss H Γ ∗ yx h ( θ ) = H ⊤ F h Ω ss H Presenter: Keith O’Hara ( ∼ /) DSGE-VARs March 10, 2014 11 / 30
Computational Aside: Solving for Ω ss For completeness... we solve Ω ss = F Ω ss F ⊤ + G Q G ⊤ by iteration. Let Q := G Q G ⊤ . Set Ω ss (0) = Q and B (0) = F . Then, for i = 1 , 2 , . . . Ω ss ( i + 1) = Ω ss ( i ) + B ( i )Ω ss ( i ) B ( i ) ⊤ B ( i + 1) = B ( i ) B ( i ) ⊤ Continue until the difference between Ω ss ( i + 1) and Ω ss ( i ) is ‘small’. Note: Ω ss 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 xx ( θ )] − 1 Γ ∗ Φ ∗ ( θ ) = [Γ ∗ xy ( θ ) xx ( θ )] − 1 Γ ∗ Σ ∗ u ( θ ) = Γ ∗ yy ( θ ) − Γ ∗ yx ( θ )[Γ ∗ 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 one-step-ahead QFE loss. Given a θ , the prior distribution is of the usual IW-N form: Σ u | θ ∼ IW ( λ T Σ ∗ u ( θ ) , λ T − k , n ) � xx ( θ )) − 1 � Φ ∗ ( θ ) , Σ u ⊗ ( λ T Γ ∗ Φ | Σ u , θ ∼ N The joint prior is then given by p (Φ , Σ u , θ ) = p (Φ , Σ u | 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 , Σ u , θ | Y ) = p (Φ , Σ u | Y , θ ) p ( θ | Y ) The ML estimates are � � − 1 � λ T Γ ∗ xx ( θ ) + X ⊤ X [ λ T Γ ∗ xy + X ⊤ Y ] Φ( θ ) = � � 1 1 � ( λ T Γ ∗ yy ( θ ) + Y ⊤ Y ) − ( λ + 1) T × Σ u ( θ ) = ( λ + 1) T � � xx ( θ ) + X ⊤ X ) − 1 ( λ T Γ ∗ ( λ T Γ ∗ yx ( θ ) + Y ⊤ X )( λ T Γ ∗ xy ( θ ) + X ⊤ Y ) The prior and likelihood are conjugate, so � � ( λ + 1) T � Σ u | Y , θ ∼ IW Σ u ( θ ) , (1 + λ ) T − k , n � xx ( θ ) + X ⊤ X ) − 1 � � Φ( θ ) , Σ u ⊗ ( λ T Γ ∗ Φ | Y , Σ u , θ ∼ N 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 , Σ u ) p (Φ , Σ u ) d (Φ , Σ u ) (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 ) Σ u ( θ ) | − ( λ +1) T − k xx ( θ ) + X ⊤ X | − n 2 | ( λ + 1) T � = | λ T Γ ∗ 2 xx ( θ ) | − n u ( θ ) | − λ T − k 2 | λ T Σ ∗ | λ T Γ ∗ 2 � n n (( λ +1) T − k ) × (2 π ) − nT / 2 2 i =1 Γ[(( λ + 1) T − k + 1 − i ) / 2] 2 � n n ( λ T − k ) i =1 Γ[( λ T − k + 1 − i ) / 2] 2 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 Σ u from � � ( λ + 1) T � Σ u | Y , θ ∼ IW Σ u ( θ ) , (1 + λ ) T − k , n Then, given θ and Σ u , sample � xx ( θ ) + X ⊤ X ) − 1 � � Φ( θ ) , Σ u ⊗ ( λ T Γ ∗ Φ | Y , Σ u , θ ∼ N Given Φ and Σ u , 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 · Σ m ) where Σ m is the inverse of the Hessian computed at the posterior mode and c is a scaling factor. Compute the acceptation ratio, p ( Y | θ ( ∗ ) ) p ( θ ( ∗ ) ) ν = p ( Y | θ ( h − 1) ) p ( θ ( h − 1) ) Finally, we accept or reject the proposal according to � θ ( ∗ ) P = min { ν, 1 } θ ( h ) = θ ( h − 1) else Given this θ ( h ) , draw a new Σ u , and so on. Presenter: Keith O’Hara ( ∼ /) DSGE-VARs March 10, 2014 18 / 30
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