Projection Inference for Set-Identified Svars

Transcription

1 Projection Inference for Set-Identified Svars Bulat Gafarov (PSU), Matthias Meier (University of Bonn), and José-Luis Montiel-Olea (Columbia) September 21, / 38

2 Introduction: Set-id. SVARs SVAR: Theoretical restrictions R imposed on a VAR. (Sims [1980, 1986]) Y t = A 1 Y t A p Y t p + η t, Σ = E[η t η t] Goal of the restrictions: (A 1,... A p, Σ) R IRF k,i,j (response of variable i to a j-th structural shock at horizon k) Map R can be 1-to-1 (point id.) or 1-to-many (set id.). (set i.d. SVARs have become popular in applied macro work) Common practice: set-identify SVARs with / = restrictions. (Faust [1998]; Canova and De Nicolo [2002]; Uhlig[2005]) 2 / 38

3 Motivation Most empirical studies report Bayesian credible sets for IRF k,i,j (Bayesian Inference depends on the specification of prior beliefs) Practical concern: prior beliefs are not dominated by the data (results are sensitive to the choice of priors even if T ) Theoretical Critique: Coverage and Robust credibility 0. (as T ; Moon & Schorfheide [2012], Kitagawa [2012]) Recent work on non-bayesian Inference for set-id. SVARs. (MSG [2013]-Freq. Inference; GK [2014]-Robust Bayes) Is there a simple way to conduct inference in set-id. SVARs? (that pleases both a frequentist and a robust Bayesian, and that is general and computationally feasible?) 3 / 38

4 Description of the Inference Problem IRF k,i,j I R k,i,j (µ) [ v k,i,j (µ), v k,i,j (µ) ], µ (A, Σ). 4 / 38

5 This paper Studies the properties of projection inference for set i.d. SVARs. (Scheffé [1953]; Dufour [1990]; Dufour and Taamouti [2005]) We collect IRF k,i,j s in a 1 α Wald Ellipsoid for µ (A, Σ). (that is, we project a nominal 1 α Wald Ellipsoid) Strategy: focus on the endpoints of the identified set for IRF k,i,j. (the maximum and minimum response, v k,i,j (µ), v k,i,j (µ)) [ inf µ CS T (1 α) v k,i,j(µ), ] sup v k,i,j (µ) µ CS T (1 α) Our projection region has coverage and RB credibility 1 α. (for any vector of IRFs! Thus providing simultaneous inference) 5 / 38

6 Pros: Pros & Cons Generality: can handle the typical application in applied work (+/0 restrictions on IRFs, long-run restrictions, elasticity bounds) Feasibility: solve two nonlinear optimization problems per IRF k,i,j (we use state-of-the-art solution algorithms for these problems) Cons: Projection is conservative for a frequentist and a Robust Bayesian (coverage and robust credibility are strictly above 1-α.) We calibrate projection to remove the excess of Robust Cred. (= 1 α and not > 1 α. Calibration based on KMS[2016]) 6 / 38

7 Outline 1. Model and Main Definitions 2. Assumptions and Results 3. Implementation and Illustrative Example 4. Conclusion 7 / 38

8 1. Model and Main Definitions 8 / 38

9 SVAR(p) Structural VAR for the n-dimensional vector Y t : Y t = A 1 Y t A p Y t p + Bε t, Σ BB Vector of reduced form parameters is: µ = (vec(a 1, A 2,..., A p ), vech(σ) ) R d Coefficients of the Structural Impulse Response Function: IRF H = {IRF kh,i h,j h (A, B)} H h=1, IRF k h,i h,j h (A, B) = e i h C kh (A) B jh. }{{} 1 n Interested in simulatenous inference about λ H IRF H. (Inoue and Kilian [2013,2016] and Lütkepohl et. al [2016]) 9 / 38

10 Restrictions R(µ) on B Identified set for λ H : } IH {λ R (µ) R H λh = IRF kh,i h,j h s.t. BB = Σ, B R(µ), h ±/0 restrictions on IRFs: e i C k (A)B j 0 (e.g. Sims [1980], Uhlig [2005]) ±/0 long-run restrictions: e i (I n A(1)) 1 B j 0 (e.g. Blanchard, Quah [1989], Gali [1999]) Elasticity bounds: (e i B j )/(e i B j ) [c, d] (e.g. Kilian, Murphy [2012], Baumeister, Hamilton [2015]) 10 / 38

11 Bounds on the Identified Set: Max and Min Response The endpoints of the identified set for each IRF k,i,j : v k,i,j (µ) sup IRF k,i,j (A, B) B v k,i,j (µ) inf B IRF k,i,j(a, B) s.t. BB = Σ, B R(µ) s.t. BB = Σ, B R(µ) Nonlinear, possibly nondifferentiable transformations of µ. Obviously... I R H (µ) H h=1 [ ] v kh,i h,j h (µ), v kh,i h,j h (µ). No need to assume the i.d. set is connected. 11 / 38

12 Projection region for λ H Let CS T (1 α; µ) be the (typical) Wald ellipsoid for µ. Let CS T (1 α; IRF k,i,j ) be the interval defined by: [ ] inf sup v k,i,j (µ) v k,i,j(µ), µ CS T (1 α;µ) µ CS T (1 α;µ) The projection region for λ H = {IRF kh,i h,j h (A, B)} H h=1 is: CS T (1 α; λ H ) CS T (1 α; IRF k1,i 1,j 1 )... CS T (1 α; IRF kh,i H,j H ) We now present the properties of CS T (1 α; λ H ) as T 12 / 38

13 2. Assumptions and Results 1 to 4 13 / 38

14 Result 1: Frequentist Coverage Let P be a DGP for the data. Parameterized by (A, B, F ). We want projection to be valid over a class P of DGPs: A1: Suppose the class of DGPs P is such that ( ) lim inf inf P µ(p) CS T (1 α; µ) 1 α. T P P R1: Under Assumption A1: lim inf T inf inf P P P λ H IH R(µ(P)) ( λ H CS T (1 α; λ H ) ) 1 α. 14 / 38

15 Proof: Straightforward Projection Argument Suppose that H = 1. For any λ I R k,i,j (µ(p)) : ( [ P λ inf ( [ P v k,i,j (µ(p)), v k,i,j (µ(p)) sup v k,i,j(µ), µ CS T (1 α) µ CS T (1 α) inf ]) v k,i,j (µ) sup v k,i,j(µ), µ CS T (1 α) µ CS T (1 α) ( as I R k,i,j (µ(p)) [v k,i,j(µ(p)), v k,i,j (µ(p))] ) ]) v k,i,j (µ) ( ) P µ(p) CS T (1 α). 15 / 38

16 Robust Bayes Framework Let P be a prior for the structural parameters (A, B). (F is now a fixed known distribution; we use N (0, I n )) Represent the prior P in terms of (P µ, P Q µ ), Q Σ 1/2 B. (Orthogonal reduced-form parameterization Arias et. al [2014]) Let P(P µ) denote the class of priors such that µ P µ. The robust credibility of CS T (1 α, λ H ) is defined as: ) inf P ( λ H (A, B) CS T (1 α; λ H ) Y T P P(Pµ ) 16 / 38

17 Result 2: Robust Bayesian credibility We can view robust credibility as a random variable (as it depends on the data Y T ) A2: Suppose that P is such that whenever Y T f (Y T µ 0 ): P (µ(a, B) CS T (1 α; µ) Y T ) = 1 α + o p (Y T µ 0 ). This is implied by the Bernstein von-mises Theorem for µ. R2: Under Assumption 2: ) inf P ( λ H CS T (1 α; λ H ) Y T 1 α + o p (Y T µ 0 ) P P(Pµ ) Proof: Another embarrassingly simple projection argument! 17 / 38

18 Calibrated Projection Yes: We know that projection inference is conservative! (both in terms of frequentist coverage and a robust credibility) In theory, it is conceptually simple to remove projection bias (project a smaller Wald ellipsoid as suggested by KMS[2016]) In practice, removing the excess of robust Bayesian credibility is much easier than removing the excess of frequentist coverage. We suggest an algorithm to calibrate robust credibility. The algorithm also removes the excess of frequentist coverage (provided the bounds of i.d. set are differentiable) 18 / 38

19 Result 3: Calibrated Robust Credibility Our calibration algorithm is based on the following result. Suppose there is a nominal level 1 α (Y T ) s.t: ( Pµ H [ h=1 v kh,i h,j h (µ), v kh,i h,j h (µ) ] CS T (1 α (Y T ), λ H ) Y T ) = 1 α R3: Then, for every data realization: ) inf P ( λ H (A, B) CS T (1 α (Y T ); λ H ) Y T = 1 α, P P(Pµ ) Proof: slightly more involved. See Appendix A / 38

20 Calibration Algorithm Take M draws (µ m) from the posterior distribution of µ (or from its asymptotic approximation based on BvM) For each h = 1,... H and each m = 1,..., M evaluate: [v kh,i h,j h (µ m), v kh,i h,j h (µ m)] (we use nonlinear numerical solvers to evaluate the bounds) Fix a confidence level 1 α s < 1 α. Count how often: [v kh,i h,j h (µ m), v kh,i h,j h (µ m)] CS T (1 α s ; λ kh,i h,j h ) for all h = 1,... H. If RBC is not in between [1 α η, 1 α + η], change α s. (η is a tolerance level for the excess of robust credibility) 20 / 38

21 Result 4: Coverage of Calibrated Projection Question: Suppose we have found α (Y T ). Is it true that: ( P µ0 [ v h (µ 0 ), v h (µ 0 ) ] CS T (1 α (Y T ); λ h ), h = 1,..., H ) 1 α? Answer: Yes. The following regularity conditions suffice: v h (µ 0 ), v h (µ 0 ) are differentiable at µ 0 for each h = 1,..., H. T ( µ µ 0 ) d N(0, Ω), Ω T p Ω, and BvM holds. Proof: Under differentiability CS T (1 α (Y T ); λ h ) is approximately: [ v h ( µ T ) r T σ h(µ 0 ), v h ( µ T ) + r T σ ] h(µ 0 ) T T 21 / 38

22 3. Implementation and illustrative example 22 / 38

23 Projection as an Optimization Problem To implement projection we need to solve the program: sup v k,i,j (µ) µ CS T (1 α;µ) The confidence set for the reduced-form parameters is taken as: { } CS T (1 α) µ R d T ( µt µ) Ω 1 T ( µ T µ) χ 2 1 α,d Thus, the program of interest becomes: ( sup µ CS T (1 α) sup B I R k,i,j (µ) e i C k (A)B j Which is a twice differentiable, non-convex, nonlinear program: sup e i C k (A)B j s.t. B Ik,i,j R (µ) and µ CS T (1 α) µ,b ) 23 / 38

24 Solution Algorithm State-of-the-art nonlinear, non-convex, large-scale problems? Large literature on local and global optimization algorithms (Local: AL, SQP, IP; Global: Multistart, GSearch, GAs) Since the problem is non-convex, local solvers are not enough. (we use a two-phase solution algorithm: local+global) SQP/IP most powerful local algorithm for nonlinear prog. (Nocedal and Wright [2006], p. 253; implemented in fmincon) For the global stage we use Multistart, GlobalSearch, ga (take the local solution as an input for global algorithm) 24 / 38

25 Example: Demand-Supply SVAR BH(2015), ECMA Effect of a structural shock on labor demand over wages/emp? (2-SVAR, 6 lags (1-AIC, 1-BIC), Q1-70/Q2-14: w t, η t.) ( ) wt η t ( ) ( ) ( ) wt 1 wt 6 ɛ d = A A η 6 + B t t 1 η t 6 ɛ s. t Sign restrictions set-identify structural demand and supply shocks: ( ) [ ] b1 b B 3 + satisfies. b 2 b Elasticities of supply and demand: α b 2 /b 1 β b 4 /b / 38

26 Table: Additional Identifying Restrictions Motivation BH(2015) This paper Empirical studies α max{.6 +.6t 3, 0}.27 α 2 report α [.27, 2] Empirical studies β min{.6 +.6t 3, 0} 2.5 β.15 β [ 2.5,.15] γ = 0 is too strong γ N (0, V ) 2V γ 2V γ e 2(I n A 1 A 2... A 6 ) 1 B 1 26 / 38

27 cumulative % change in wage cumulative % change in employment 68% Projection Region and 68% Credible Set Quarters after shock Quarters after shock Expansionary Demand Shock (BH priors) 27 / 38

28 cumulative % change in wage cumulative % change in employment 68% Projection Region and 68% Credible Set Quarters after shock Quarters after shock Expansionary Demand Shock (Uhlig s priors) 28 / 38

29 cumulative % change in wage cumulative % change in employment 68% Projection Region and 68% Credible Set Quarters after shock Quarters after shock Expansionary Supply Shock (BH priors) 29 / 38

30 cumulative % change in wage cumulative % change in employment 68% Projection Region and 68% Credible Set Quarters after shock Quarters after shock Expansionary Supply Shock (Uhlig s priors) 30 / 38

31 Robustness or Conservativeness? Projection is informative about the effects of ɛ d on η t... But not very informative about the rest of the dynamic effects. This could be a consequence of the robustness of projection, Or a consequence of its conservativeness (> 1 α). To separate these effects, we report the calibrated projection. (represented as dotted line in the following figures) 31 / 38

32 cumulative % change in wage cumulative % change in employment 68% Projection Region and 68% Calibrated Projection Quarters after shock Quarters after shock Expansionary Demand Shock Boxes: horizon-by-horizon 68% robust Bayesian credible set Whiskers: minimum/maximum IRF (100,000 draws) 32 / 38

33 cumulative % change in wage cumulative % change in employment 68% Projection Region and 68% Calibrated Projection Quarters after shock Quarters after shock Expansionary Supply Shock Boxes: horizon-by-horizon 68% robust Bayesian credible set Whiskers: minimum/maximum IRF (100,000 draws) 33 / 38

34 Comments Credible sets differ substantially depending on the prior beliefs. (compare BH with Uhlig priors) Prior-free, projected region: qualitatively different inference. (only the employment response to demand shock significant) SQP/IP: 12 min; Uhlig: 38 min; BH: 66min; Global: 9hrs. (see Table III, p. 20 in the paper) Global algorithms do not improve the local solution. (see Appendix B, p. 48 in the paper) Calibration of RCS takes around 3 minutes for M = 1, 000. (and around 5 hours with M = 100, 000 and 50 parallel workers) 34 / 38

35 4. Conclusion 35 / 38

36 Main Messages from our Paper We studied the properties of projection inference for SVARs. (delivers frequentist and Robust Bayes interpretation) We have emphasized the generality of projection inference. (can handle typical applications in applied macro work) We thought seriously about computational feasibility. (implementation requires solving two mathematical programs) We showed how to calibrate the RB credibility of projection. (which also calibrates coverage under some reg. conditions) 36 / 38

37 Bottom Line We think that projection is a simple way to conduct inference in set-id SVARs. (it has both frequentist coverage and Robust Bayes credibility) (it is also general, feasible, and delivers simultaneous inference) 37 / 38

38 Thanks very much for listening! 38 / 38