Robust Bayes Inference for Non-identified SVARs

Transcription

1 1/29 Robust Bayes Inference for Non-identified SVARs Raffaella Giacomini (UCL) & Toru Kitagawa (UCL) 27, Dec, 2013 work in progress

2 2/29 Motivating Example Structural VAR (SVAR) is a useful tool to infer dynamic causality in empirical macroeconomics. a 11 a 12 a 13 i t a 21 a 22 a 23 Δy t = a + A 1 a 31 a 32 a 33 π t i t 1 Δy t 1 π t 1 + Examples of zero restrictions: Zero restrictions on the structural coefficients, e.g., (A1), a 12 = 0: monetary policy rule does not react to contemporaneous output. (A2), a 21 = 0: output does not respond to contemporaneous nominal interest rate. Zero restrictions on the impulse responses, e.g., (A3), y t ɛ it = 0. (A4), lim h y t+h ɛ it = 0 ɛ it ɛ yt ɛ πt

3 3/29 Motivation Identification of the impulse responses requires a sufficient number of identifying restrictions. What if the number of credible restrictions is not enough to identify the impulse responses of interest? For instance, (A1) a 12 = 0 and (A2) a 21 = 0 is credible, but the long-run money neutrality (A4) is less credible. (A1) a 12 = 0 is the only assumption that the macroeconomists/policy makers can agree upon.

4 4/29 In addition to, or as an alternative to the equality restrictions, sign restrictions on impulse responses may be imposed. (Canova & De Nicolo (02,JME), Faust (98,C-RCSPP), Uhlig (05,JME)). (A5) : i t+h ɛ it 0 for h = 0,1,..., h. (A6) : π t+h ɛ it 0 for h = 0,1,..., h.

5 4/29 In addition to, or as an alternative to the equality restrictions, sign restrictions on impulse responses may be imposed. (Canova & De Nicolo (02,JME), Faust (98,C-RCSPP), Uhlig (05,JME)). (A5) : i t+h ɛ it 0 for h = 0,1,..., h. (A6) : π t+h ɛ it 0 for h = 0,1,..., h. The insufficient number of equality restrictions and/or the sign restrictions generally lead to set-identified impulse responses.

6 5/29 Partially Identified SVARs When the available assumptions only set-identifies impulse responses, how do we run statistical inference? Bayesian inference is common in the SVAR analysis with the sign restrictions on impulse responses (e.g., Uhlig(05)). Is Bayesian inference free from the identification issue?

7 5/29 Partially Identified SVARs When the available assumptions only set-identifies impulse responses, how do we run statistical inference? Bayesian inference is common in the SVAR analysis with the sign restrictions on impulse responses (e.g., Uhlig(05)). Is Bayesian inference free from the identification issue? Some say yes; we can always obtain a proper posterior. Others say no, the posterior remains to be sensitive to a choice of prior, even asymptotically (Poirier(98,ET)), i.e., posterior consistency fails.

8 5/29 Partially Identified SVARs When the available assumptions only set-identifies impulse responses, how do we run statistical inference? Bayesian inference is common in the SVAR analysis with the sign restrictions on impulse responses (e.g., Uhlig(05)). Is Bayesian inference free from the identification issue? Some say yes; we can always obtain a proper posterior. Others say no, the posterior remains to be sensitive to a choice of prior, even asymptotically (Poirier(98,ET)), i.e., posterior consistency fails. Our view: If we cannot confidently specify a prior, we should worry about the lack of identification due to the non-diminishing posterior sensitivity. If the goal is to summarize the shape of likelihood (Sims and Zha (98,EMA)), the marginalized posterior obscure the flat likelihood.

9 6/29 Illustration: SVAR with Sign Restrictions n-variable SVAR(p): A 0 y t = a + p A l y t l + ɛ t, ɛ t (y t 1,...) N(0,I n ) l=1 Reduced form VAR(p): y t = b + p B l y t l + u t, u t (y t 1,...) N(0,Σ) l=1 where Σ=A 1 1 (A 0 0 ). Denote the reduced form parameters by φ =(B,Σ).

10 7/29 Let Σ=Σ tr Σ tr : Cholesky decomposition. For any n n orthogonal matrix Q Q, A 0 = Q Σ 1 Σ=A 1 0 (A 1 0 ). {A 0 = Q Σ 1 tr : Q Q} is the set of observationally equivalent A 0 s given Σ tr. tr satisfies

11 7/29 Let Σ=Σ tr Σ tr : Cholesky decomposition. For any n n orthogonal matrix Q Q, A 0 = Q Σ 1 Σ=A 1 0 (A 1 0 ). {A 0 = Q Σ 1 tr : Q Q} is the set of observationally equivalent A 0 s given Σ tr. VMA( ): y t = c + l=1 C l (B)A 1 0 ɛ t where IR h C h (B)A 1 0 = C h (B)Σ tr Q. tr satisfies

12 8/29 Uhlig(05) s agnostic approach to obtaining a posterior of IRs. 1. Specify a prior for reduced form parameters φ =(B,Σ). 2. Run the Bayesian reduced form VAR to obtain the posterior of φ.

13 8/29 Uhlig(05) s agnostic approach to obtaining a posterior of IRs. 1. Specify a prior for reduced form parameters φ =(B,Σ). 2. Run the Bayesian reduced form VAR to obtain the posterior of φ. 3. Draw φ s from the posterior and Q s from the uniform prior. 4. Retain the draws of IR h = C h (B)Σ tr Q that satisfy the sign restrictions.

14 Illustration: Posterior Sensitivity 3-var SVAR(2) with the sign restrictions (A5) & (A6) for h = 0,1. Data are from Aruoba & Schorfheide (11,AEJ), sample size = 163. Prior 1 (black) the Prior of Uhlig (05). Prior 2 (blue) Perturbing Prior 1 by changing the var-cov matrix. Posterior of IR(y,epsilon_i) at h=1 Density Outcome Response to Contractionary monetary Policy Shock 9/29

15 Illustration: Posterior Sensitivity 3-var SVAR(2) with the sign restrictions (A5) & (A6) for h = 0,1. Data are from Aruoba & Schorfheide (11), sample size = 163. Prior 1 (black) the Prior of Uhlig (05). Prior 2 (blue) Perturbing Prior 1 by changing the var-cov matrix. Posterior of IR(y,epsilon_i) at h=1 Density Outcome Response to Contractionary monetary Policy Shock 10/29

16 11/29 This paper Robust Bayes Analysis with multiple priors: In case that a set of credible zero/sign restrictions do not give identification, we advocate to report the range of posterior quantities when a prior of the nonidentified components in the model varies over a class (ambiguous belief). The posterior bound analysis is a useful tool to separate the information of the sample likelihood from any prior input that cannot be updated by data. Considers the benchmark prior class (Kitagawa(12)). Proposes a procedure to compute the bounds of the posterior quantities in the SVAR model.

17 Illustration: Posterior Bounds 3-var SVAR(2) with the sign restrictions (A5) & (A6) for h = 0,1. Data are from Aruoba & Schorfheide (11), sample size = 163. Prior 1 (black) the Prior of Uhlig (05). Prior 2 (blue) Perturbing Prior 1 by changing the var-cov matrix. Posterior of IR(y,epsilon_i) at h=1 Density Outcome Response to Contractionary monetary Policy Shock 12/29

18 13/29 Robust Bayes: Related Literatures Global sensitivity analysis: Berger & Berliner (86,Ann.Stat), DeRobertis & Hartigan (81,JASA), Wasserman (90,Ann.Stat), and more Posterior mean bounds: Chamberlain & Leamer (76,JRSSB), Leamer (82,EMA). Bayes vs Frequentist for non-identified SVAR: Faust (98,C-RCSPP), Gordon & Boccanfuso(01), Canova & De Nicolo (02,JME), Uhlig(05,JME), Moon, Schorfheide, & Granziela (13), Beumeister & Hamilton (13).

19 14/29 Outline 1. Framework & Identified set of IRs. 2. Constructing the benchmark prior class. 3. Derivation of the posterior bounds. 4. Real Data Illustration.

20 15/29 Recall the Notation Reduced form n-variable VAR(p) with reduced form parameters φ =(B,Σ), where Σ=A (A0 ). The set of observationally equivalent A 0 s, given Σ tr, {A 0 = Q Σ 1 tr : Q Q} where Q is an n n orthogonal matrix. IR matrix at h-th horizon is IR h C h (B)Σ tr Q.

21 16/29 Zero restrictions In terms of φ and Q =[q 1,...,q n ], Since A 0 = Q (Σ 1 tr ), A 0 [i,j]=0 (σ j ) q i = 0. Since IR h = C h (B)Σ tr Q, IR h [i,j]=0 c ih (φ) q j = 0. Since IR =(I p l=1 B l) 1 Σ tr Q, IR [i,j]=0 c i (φ) q j = 0.

22 16/29 Zero restrictions In terms of φ and Q =[q 1,...,q n ], Since A 0 = Q (Σ 1 tr ), A 0 [i,j]=0 (σ j ) q i = 0. Since IR h = C h (B)Σ tr Q, IR h [i,j]=0 c ih (φ) q j = 0. Since IR =(I p l=1 B l) 1 Σ tr Q, IR [i,j]=0 c i (φ) q j = 0. Stacking these, F 1 (φ)q 1. F(φ,Q) = 0, F F j (φ)q j (φ): f j n. j. WLOG, order the variables in such way that f 1 f 2 f n 0.

23 16/29 Zero restrictions In terms of φ and Q =[q 1,...,q n ], Since A 0 = Q (Σ 1 tr ), A 0 [i,j]=0 (σ j ) q i = 0. Since IR h = C h (B)Σ tr Q, IR h [i,j]=0 c ih (φ) q j = 0. Since IR =(I p l=1 B l) 1 Σ tr Q, IR [i,j]=0 c i (φ) q j = 0. Stacking these, F 1 (φ)q 1. F(φ,Q) = 0, F F j (φ)q j (φ): f j n. j. WLOG, order the variables in such way that f 1 f 2 f n 0. A class of SVARs to be considered is restricted to those s.t. f i n i for all i = 1,...,n.

24 17/29 Sign Restrictions Suppose s number of sign restrictions are imposed for the impulse responses to the j-th shock, S(φ)q j 0, where S(φ) is a s j n matrix. Stack the sign restrictions over multiple shocks, and represent them by S(φ,Q) 0.

25 18/29 Identified Set of IRs Let r = r(φ,q)=ir h [i,j ]=c i h(φ) q j be an impulse response of interest. Set identification of SVAR : For positive measure φ s, there are multiple Q s that satisfy F(φ,Q)=0 and S(φ,Q) 0.

26 18/29 Identified Set of IRs Let r = r(φ,q)=ir h [i,j ]=c i h(φ) q j be an impulse response of interest. Set identification of SVAR : For positive measure φ s, there are multiple Q s that satisfy F(φ,Q)=0 and S(φ,Q) 0. The set of feasible Q s; Q(φ F,S)={Q Q: F(φ,Q)=0,S(φ,Q) 0}. (1)

27 18/29 Identified Set of IRs Let r = r(φ,q)=ir h [i,j ]=c i h(φ) q j be an impulse response of interest. Set identification of SVAR : For positive measure φ s, there are multiple Q s that satisfy F(φ,Q)=0 and S(φ,Q) 0. The set of feasible Q s; Q(φ F,S)={Q Q: F(φ,Q)=0,S(φ,Q) 0}. (1) The Identified Set of r: (Moon et al (13)). IS r (φ F,S) = {r(φ,q):q Q(φ F,S)}, a set valued map of φ to R.

28 19/29 Identified Set of IRs Lemma: Assume f i n i for all i = 1,...,n, and let r = IR h [i,j ]. 1. With only zero restrictions, IS r (φ F) is a nonempty and convex interval at every φ Φ. i.e., IS r (φ F,S)=[l(φ),u(φ)]. 2. If sign restrictions are additionally imposed to the impulse responses to j -th shock, IS r (φ F,S) remains to be a convex interval whenever IS r (φ F,S) is nonempty.

29 19/29 Identified Set of IRs Lemma: Assume f i n i for all i = 1,...,n, and let r = IR h [i,j ]. 1. With only zero restrictions, IS r (φ F) is a nonempty and convex interval at every φ Φ. i.e., IS r (φ F,S)=[l(φ),u(φ)]. 2. If sign restrictions are additionally imposed to the impulse responses to j -th shock, IS r (φ F,S) remains to be a convex interval whenever IS r (φ F,S) is nonempty. Remarks: The joint restrictions of the zero restrictions and the sign restrictions are refutable. Convexity of IS r (φ F,S) simplifies computation of the posterior bounds and their asymptotic properties.

30 20/29 Source of Posterior Sensitivity Let π φq = π Q φ π φ be a prior for (φ,q). The posterior of r can be expressed as π r Y (A)= π r φ (A)dπ φ Y φ where π r φ (A)=π Q φ (r(φ,q) A).

31 20/29 Source of Posterior Sensitivity Let π φq = π Q φ π φ be a prior for (φ,q). The posterior of r can be expressed as π r Y (A)= π r φ (A)dπ φ Y φ where π r φ (A)=π Q φ (r(φ,q) A). Prior for φ is updated, while the conditional prior π Q φ is never be updated. The main source of posterior sensitivity is π Q φ. What if we make π Q φ ambiguous?

32 21/29 Benchmark Prior Class Constructing the benchmark prior class, Π φq = {π φq }. Specify a single prior for φ, π φ. Define the set of conditional priors π Q φ satisfying the imposed restrictions, Π Q φ = {π Q φ : π Q φ (Q(φ F,S)) = 1, φ} supp(π φ ). Π φq = {π Q φ π φ : π Q φ Π Q φ }

33 21/29 Benchmark Prior Class Constructing the benchmark prior class, Π φq = {π φq }. Specify a single prior for φ, π φ. Define the set of conditional priors π Q φ satisfying the imposed restrictions, Π Q φ = {π Q φ : π Q φ (Q(φ F,S)) = 1, φ} supp(π φ ). Π φq = {π Q φ π φ : π Q φ Π Q φ } The posterior class Π φq Y = {π Q φ π φ Y : π Q φ Π Q φ }

34 22/29 Posterior Bounds What are the bounds of posterior mean and the posterior probability of r, when posterior π φq Y varies over Π φq Y?

35 22/29 Posterior Bounds What are the bounds of posterior mean and the posterior probability of r, when posterior π φq Y varies over Π φq Y? Theorem (Wasserman(90, Ann.Stat), Kitagawa (12)): Assume π φ ({Q(φ F,S)= })=0. The posterior probability bounds are inf π r Y (A)=π φ Y (IS r (φ F,S) A) π φq Y Π φq Y sup π r Y (A)=1 π r Y (A c ) π φq Y Π φq Y The posterior mean bounds are [ E φ Y (l(φ)),e φ Y (u(φ)) ].

36 22/29 Posterior Bounds What are the bounds of posterior mean and the posterior probability of r, when posterior π φq Y varies over Π φq Y? Theorem (Wasserman(90, Ann.Stat), Kitagawa (12)): Assume π φ ({Q(φ F,S)= })=0. The posterior probability bounds are inf π r Y (A)=π φ Y (IS r (φ F,S) A) π φq Y Π φq Y sup π r Y (A)=1 π r Y (A c ) π φq Y Π φq Y The posterior mean bounds are [ E φ Y (l(φ)),e φ Y (u(φ)) ].

37 23/29 Computing Posterior Bounds 1. Specify π φ and run the Bayesian reduced-form VAR to get π φ Y. 2. Draw φ π φ Y, and check if Q(φ F,S) is nonempty. 3. If Q(φ F, S) is nonempty, solve max/min Q r(φ,q) s.t. Q Q = I n, F(φ,Q)=0, S(φ,Q) 0 (2) to obtain l(φ) and u(φ). 4. Repeat Step 2 and 3 to obtain many MCMC draws of (l(φ),u(φ)). 5. Report the posterior mean bounds.

38 24/29 Checking if Q(φ F, S) is nonempty 1. Draw z 1 N(0,I n ), and let e 1 be the projection residual of z 1 onto F 1 (φ). q 1 = e 1 / e Draw z j N(0,I n ), and let e j be the projection residual of z j onto (F j (φ),q 1,...,q j 1 ). q j = e j / e j. 3. Obtain a draw of Q =[q 1,...,q n ], and retain the draw if it satisfies S(φ,Q) Repeat 1-3 many times. If none of drawn Q s satisfies S(φ,Q) 0, we claim Q(φ F,S)=.

39 25/29 Posterior Lower Credible Region With the multiple posteriors, what interval estimate can we report? Posterior Lower Credible Region with credibility α, (Kitagawa (12)) C α argmin C length(c), s.t. inf π r Y (C) α. π φq Y Π φq Y The shortest interval on which all the posteriors assign probability at least α. A conservative posterior interval estimate in the presence of multiple posteriors.

40 26/29 Real Data Example A 0 i t i t ɛ it Δy t Δy = c + A(L) t ɛ + yt π t π t ɛ πt m t m t ɛ mt Quarterly data from Aruoba & Schorfheide (11), same as the empirical example of Moon et al (13), sample size = 163. Zero restrictions: (1) i t does not respond to Δy t. (2) IR [y,ɛ i ]=0. 4-var SVAR(2) with the sign restrictions i t+h m t+h ɛ it ɛ it 0, π t+h ɛ it 0, 0, for h = 0,1 (same as Moon et al(13)). Prior for φ: π(b,σ) Σ n+1 2.

41 Object of Interest: output response to contractionary monetary policy shock

42 Results Model I: Output Response Model II: Output Response impulse response impulse response horizon(quarterly) horizon(quarterly) Model III: Output Response impulse response /29

43 Results Model IV: Output Response Model V: Output Response impulse response impulse response horizon(quarterly) horizon(quarterly) Model VI: Output Response Model VII: Output Response impulse response impulse response /29

44 29/29 Conclusion We consider posterior inference for set-identified VARs from the robust Bayes viewpoint. For whom and in which occasion, the proposed method is appealing? When the goal of analysis is reporting rather than decision-making (Sims & Zha (98)). Bayesians who are anxious about their choice of prior and seek for robust inferential statement. Frequentists who are anxious about accuracy of asymptotic approximations.

45 29/29 Conclusion We consider posterior inference for set-identified VARs from the robust Bayes viewpoint. For whom and in which occasion, the proposed method is appealing? When the goal of analysis is reporting rather than decision-making (Sims & Zha (98)). Bayesians who are anxious about their choice of prior and seek for robust inferential statement. Frequentists who are anxious about accuracy of asymptotic approximations. The posterior bounds with the benchmark prior class is analytically and computationally convenient (cf. Moon et al(13)). Further analysis is needed on the frequentist properties. How to exploit non-dogmatic prior knowledge for the purpose of refining the prior class?