Measuring the Effects of a Shock to Monetary Policy - Humboldt ...

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Bayesian FAVARs with Agnostic Identification 27 multi move version of the Gibbs sampler one has to prepare the model further which is done step by step in the following. The multi move Gibbs Sampling, alternately samples the parameters θ and the factors Ft given the data. We use the multi move version of the Gibbs sampler because this approach allows us as, a first step to estimate the unobserved common components, namely the factors via the Kalman filtering technique conditional on the given hyperparameters and as a second step calculate the hyperparameters of the model given the factors via the Gibbs sampler in the respective blocking 28 . For the state space representation we define X ′ t = (X ′ t, Y ′ t ) , e ′ t = (e ′ t, 0) ′ and F ′ t = (F ′ t, Y ′ t ). For the case that Φ(L) is of order one, the model can be rewritten as: with Λ = ⎡ ⎢ ⎣ Λf Λy 0 I Xt = ΛFt + et Ft = Φ(L)Ft−1 + vt ⎤ ⎥ ⎦ , R = ⎡ ⎢ ⎣ R 0 0 0 But in most applications one can expect the order to be d > 1, so is the case in the dataset I analyze. The dataset I analyze is in monthly frequency therefore I chose a lag order of 12 for Φ(L). The FAVAR equation has to be transformed into a first- order Markov process, in order to be able to draw the factors via Bayesian Kalman filtering. For that we define Φ(L) = Φ1L+Φ2L 2 +...+ΦdL d , ¯ Ft = (F ′ t, F ′ t−1, ..., F ′ t−1−d) ′ and ¯vt = (vt, 0, ..., 0) ′ . The lag polynomial of the FAVAR equation in the first-order representation changes to: 28 Please note that we always also condition on the data due to notational convenience it is left out but is implicitly assumed and not further explicitly written ⎤ ⎥ ⎦ . (9) (10)
28 Bayesian FAVARs with Agnostic Identification ⎡ ⎢ Φ1 Φ2 . . . Φd−1 Φd ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ IK+M 0 . . . 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ¯Φ = ⎢ 0 IK+M . . . 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ . . . . . . . . . . . . . . . ⎥ ⎣ ⎦ 0 0 . . . IK+M 0 Now we have to transform the VCV of the FAVAR disturbances with 0’s in a straight- forward way to adjust the dimensions of the state equation which results in the following matrix: ⎡ ⎤ ⎢ Q ⎢ 0 ¯Q = ⎢ . . . ⎣ 0 0 . . . . . . . . . . . . 0 ⎥ 0 ⎥ . . . ⎥ ⎦ 0 0 . . . 0 where the 0 ′ s and ¯ Q have dimension [(K + M) × (K + M)] , and [d(K + M) × d(K + M)] respectively. This results in the final observation equation of the following form: And finally the last extensions Xt = ¯ Λ ¯ Ft + et ⎤ (11) (12) (13) ¯Λ = [Λ 0 . . . 0] (14) The final state-space representation prepared to fit the estimation procedure are: ¯Ft = ¯ Φ ¯ Ft−1 + ¯vt (15)
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