Discussion of Robust Bayes Inference for non-identied SVARs", by March Giacomini 2014 and1 Kitagaw / 14

Transcription

1 Discussion of Robust Bayes Inference for non-identied SVARs", by Giacomini and Kitagawa Sophocles Mavroeidis 1 1 Oxford University March 2014 Discussion of Robust Bayes Inference for non-identied SVARs", by March Giacomini 2014 and1 Kitagaw / 14

2 Outline My understanding of the paper. What I like about the paper. What I don't like about the paper. Conclusion. Discussion of Robust Bayes Inference for non-identied SVARs", by March Giacomini 2014 and2 Kitagaw / 14

3 The main idea using a simple example Bivariate SVAR(0) Reduced form A 0 Y t = ε t iidn (0, I 2 ). Y t = u t iidn (0, Σ). Under-identication: 3 RF parameters in Σ, call them φ = (σ 1, σ 2, ρ) ; 4 in A 0. Choleski decomposition of Σ = Σ tr Σ 0 tr. Write A 0 = Q 0 Σ 1 tr, where Q = Σ tr is identied, θ is not. cos θ sin θ sin θ cos θ, θ 2 [ π, π]. Discussion of Robust Bayes Inference for non-identied SVARs", by March Giacomini 2014 and3 Kitagaw / 14

4 Inference on IRFs Data is completely uninformative about θ 2 [ π, π]. Impulse responses: IR 0 σ1 0 = Σ tr Q = p cos θ σ 2 ρ σ 2 1 ρ 2 sin θ = σ 2 p 1 σ 1 cos θ ρ 2 sin θ + ρ cos θ sin θ cos θ σ 1 sin θ σ 2 p 1 ρ 2 cos θ ρ sin θ From this we can obtain identied sets (IS) on each IR, e.g., for r = IR 0 [1, 2] = Y 1 / ε 2, we have Nice and convex! (Lemma 1) IS r (φ) = [ σ 1, σ 1 ].! Discussion of Robust Bayes Inference for non-identied SVARs", by March Giacomini 2014 and4 Kitagaw / 14

5 Bayes inference Existing practice (Uhlig, 2005): 1 postulate `agnostic' prior on Q, here θ Uniform[ π, π]. 2 draw φ from its posterior π φjy. 3 keep Q or θ such that sign restrictions at φ hold (here all θ no restrictions) 4 collect posterior draws of Σ tr Q. Is this really `agnostic'? Discussion of Robust Bayes Inference for non-identied SVARs", by March Giacomini 2014 and5 Kitagaw / 14

6 Example Σ 1 iw (Σ 0, T ), Σ 0 = , T = 100. Q uniform on O (2). IR[11] IR[12] IR[21] IR[22] Discussion of Robust Bayes Inference for non-identied SVARs", by March Giacomini 2014 and6 Kitagaw / 14

7 Example Σ 1 iw (Σ 0, T ), Σ 0 = , T = 100. q 11 uniform on [ 1, 1]. 0.5 IR[11] IR[12] IR[21] IR[22] Discussion of Robust Bayes Inference for non-identied SVARs", by March Giacomini 2014 and7 Kitagaw / 14

8 Example with sign restriction IR[11]>=0 Q uniform on O (2). 1.5 IR[11] 0.75 IR[12] IR[21] IR[22] Pr [IR > 0jY ] = Discussion of Robust Bayes Inference for non-identied SVARs", by March Giacomini 2014 and8 Kitagaw / 14

9 Example with sign restriction IR[11]>=0 q 11 uniform on [ 1, 1] IR[11] IR[12] IR[21] IR[22] Pr [IR > 0jY ] = Discussion of Robust Bayes Inference for non-identied SVARs", by March Giacomini 2014 and9 Kitagaw / 14

10 Truly agnostic (in my opinion) Bayes inference The Giacomini-Kitagawa way: 1 Obtain posterior of φ, π φjy. (this is all we can ever hope to learn from the data). 2 Consider all possible priors on Q such that the restrictions hold Π Qjφ (here all distributions π θ supported on [ π, π] with sign restriction support is π 2, π 2 ). 3 Derive bounds for the (posterior) probability of any event of interest. 4 Derive agnostic credible sets for any parameter of interest. In our example, GK produce Pr [IR 12 > 0jY ] 2 [0, 1] and Pr [IR 21 > 0jY ] 2 [0, 1] quite sensibly, in my opinion. Discussion of Robust Bayes Inference for non-identied SVARs", March by Giacomini 2014 and10 Kitagaw / 14

11 Robust evaluation of restrictions If you want to know whether the restrictions are consistent with the data, use this: Pr (identied set is nonempty). With sign restrictions S (φ, Q) 0, this is Pr max S (φ, Q) 0. Q This only depends on π φjy, which is the only thing we can learn from the likelihood/data..... unlike Pr (S (φ, Q) 0) which depends on the prior on Q. Discussion of Robust Bayes Inference for non-identied SVARs", March by Giacomini 2014 and11 Kitagaw / 14

12 What I like about the paper 1 Very neat way of conveying our ignorance about things we can never hope to learn from the data! 2 Lemma 1 (convexity of identied sets) is very useful in practice (though not essential for point 1). 3 Intuitive way of checking the consistency of sign restrictions with data robust to prior! 4 Clear algorithm for implementing the procedure. 5 Real life examples that show how it works in practice. Discussion of Robust Bayes Inference for non-identied SVARs", March by Giacomini 2014 and12 Kitagaw / 14

13 What I don't like about the paper Not much: Needs some editing. Term under-exactly-identied sounds awkward. Discussion of Robust Bayes Inference for non-identied SVARs", March by Giacomini 2014 and13 Kitagaw / 14

14 Conclusion Fantastic paper! Discussion of Robust Bayes Inference for non-identied SVARs", March by Giacomini 2014 and14 Kitagaw / 14