On the maximum and minimum response to an impulse in Svars

Transcription

1 On the maximum and minimum response to an impulse in Svars Bulat Gafarov (PSU), Matthias Meier (UBonn), and José-Luis Montiel-Olea (NYU) September 23, / 58

2 Introduction Structural VAR: Theoretical Restrictions 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] 2 / 58

3 Introduction Structural VAR: Theoretical Restrictions 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: I k,ij : (A 1,... A p, Σ) IRF k,ij. (response of variable i at horizon k to a structural impulse j) 2 / 58

4 Introduction Structural VAR: Theoretical Restrictions 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: I k,ij : (A 1,... A p, Σ) IRF k,ij. (response of variable i at horizon k to a structural impulse j) Restrictions = point identification or set identification. (i.e., I k,ij is one-to-one or one-to-many) 2 / 58

5 Introduction Structural VAR: Theoretical Restrictions 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: I k,ij : (A 1,... A p, Σ) IRF k,ij. (response of variable i at horizon k to a structural impulse j) Restrictions = point identification or set identification. (i.e., I k,ij is one-to-one or one-to-many) We study SVARs that are set-identified with ±/0 restrictions. (Faust [1998], Uhlig[2005]) 2 / 58

6 Introduction Structural VAR: Theoretical Restrictions 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: I k,ij : (A 1,... A p, Σ) IRF k,ij. (response of variable i at horizon k to a structural impulse j) Restrictions = point identification or set identification. (i.e., I k,ij is one-to-one or one-to-many) We study SVARs that are set-identified with ±/0 restrictions. (Faust [1998], Uhlig[2005]) Our context: Prior-free inference in sign-restricted SVARs. (Giacomini & Kitagawa [2015]-RBayes; MSG [2013]-M. Inequ.) 2 / 58

7 This paper Uniformly consistent in level, frequentist confidence interval for: 3 / 58

8 This paper Uniformly consistent in level, frequentist confidence interval for: 1. The k-th coefficient of the structural impulse-response fn: IRF k,ij R 3 / 58

9 This paper Uniformly consistent in level, frequentist confidence interval for: 1. The k-th coefficient of the structural impulse-response fn: IRF k,ij R 2. The (i, j)-th structural impulse response function, up to k = h: (IRF 0,ij, IRF 1,ij,... IRF h,ij ) R h+1 3 / 58

10 This paper Uniformly consistent in level, frequentist confidence interval for: 1. The k-th coefficient of the structural impulse-response fn: IRF k,ij R 2. The (i, j)-th structural impulse response function, up to k = h: (IRF 0,ij, IRF 1,ij,... IRF h,ij ) R h+1 Strategy: focus on the bounds of the identified set for IRF k,ij. (the maximum and minimum response, v k,ij (A, Σ), v k,ij (A, Σ)) 3 / 58

11 This paper Uniformly consistent in level, frequentist confidence interval for: 1. The k-th coefficient of the structural impulse-response fn: IRF k,ij R 2. The (i, j)-th structural impulse response function, up to k = h: (IRF 0,ij, IRF 1,ij,... IRF h,ij ) R h+1 Strategy: focus on the bounds of the identified set for IRF k,ij. (the maximum and minimum response, v k,ij (A, Σ), v k,ij (A, Σ)) Message: Frequentist inference is conceptually straightforward. (and as general as Uhlig s current Bayesian Approach) 3 / 58

12 This paper Uniformly consistent in level, frequentist confidence interval for: 1. The k-th coefficient of the structural impulse-response fn: IRF k,ij R 2. The (i, j)-th structural impulse response function, up to k = h: (IRF 0,ij, IRF 1,ij,... IRF h,ij ) R h+1 Strategy: focus on the bounds of the identified set for IRF k,ij. (the maximum and minimum response, v k,ij (A, Σ), v k,ij (A, Σ)) Message: Frequentist inference is conceptually straightforward. (and as general as Uhlig s current Bayesian Approach) Proposal: Projection Inference 3 / 58

13 This paper Uniformly consistent in level, frequentist confidence interval for: 1. The k-th coefficient of the structural impulse-response fn: IRF k,ij R 2. The (i, j)-th structural impulse response function, up to k = h: (IRF 0,ij, IRF 1,ij,... IRF h,ij ) R h+1 Strategy: focus on the bounds of the identified set for IRF k,ij. (the maximum and minimum response, v k,ij (A, Σ), v k,ij (A, Σ)) Message: Frequentist inference is conceptually straightforward. (and as general as Uhlig s current Bayesian Approach) [ ] CS for θ (A, Σ) = min v k,ij(θ), max v k,ij(θ) θ CS θ CS 3 / 58

14 Details max θ CS v k,ij (θ) is a nonlinear mathematical program. (with gradients that can be computed analytically) 4 / 58

15 Details max θ CS v k,ij (θ) is a nonlinear mathematical program. (with gradients that can be computed analytically) Implementing projection requires us to solve such program. (No numerical inversion of tests, no sampling from rotations) 4 / 58

16 Details max θ CS v k,ij (θ) is a nonlinear mathematical program. (with gradients that can be computed analytically) Implementing projection requires us to solve such program. (No numerical inversion of tests, no sampling from rotations) Projection is conservative (> 1-α), but can be calibrated. (the degrees of freedom in CS for (A, Σ) can be adjusted) 4 / 58

17 Details max θ CS v k,ij (θ) is a nonlinear mathematical program. (with gradients that can be computed analytically) Implementing projection requires us to solve such program. (No numerical inversion of tests, no sampling from rotations) Projection is conservative (> 1-α), but can be calibrated. (the degrees of freedom in CS for (A, Σ) can be adjusted) MC output provides a lower bound on the d.o.f. adjustment. (this point is illustrated with economic applications) 4 / 58

18 Details max θ CS v k,ij (θ) is a nonlinear mathematical program. (with gradients that can be computed analytically) Implementing projection requires us to solve such program. (No numerical inversion of tests, no sampling from rotations) Projection is conservative (> 1-α), but can be calibrated. (the degrees of freedom in CS for (A, Σ) can be adjusted) MC output provides a lower bound on the d.o.f. adjustment. (this point is illustrated with economic applications) +/0 on only one shock, delta-method inference is available. (we emphasized this in the previous version of the paper) 4 / 58

19 Brief Overview of Related Work a) Set Identified SVARs Bayesian Approaches Faust (1998); Uhlig (2005); Baumeister and Hamilton (2014); b) Set Identified SVARs Non-Bayesian Approaches Moon, Schorfheide, Granziera (2013); Giacomini, Kitagawa (2015) c) Transformations of set-identified parameters Bugni, Canay, Shi (2015); Kaido, Molinari, Stoye (2015) 5 / 58

20 Brief Overview of Related Work a) Set Identified SVARs Bayesian Approaches Faust (1998); Uhlig (2005); Baumeister and Hamilton (2014); F-U: Numerical Bayes; BH: Analytic, Full-Bayesian Analysis. b) Set Identified SVARs Non-Bayesian Approaches Moon, Schorfheide, Granziera (2013); Giacomini, Kitagawa (2015) c) Transformations of set-identified parameters Bugni, Canay, Shi (2015); Kaido, Molinari, Stoye (2015) 5 / 58

21 Brief Overview of Related Work a) Set Identified SVARs Bayesian Approaches Faust (1998); Uhlig (2005); Baumeister and Hamilton (2014); b) Set Identified SVARs Non-Bayesian Approaches Moon, Schorfheide, Granziera (2013); Giacomini, Kitagawa (2015) c) Transformations of set-identified parameters Bugni, Canay, Shi (2015); Kaido, Molinari, Stoye (2015) 5 / 58

22 Brief Overview of Related Work a) Set Identified SVARs Bayesian Approaches Faust (1998); Uhlig (2005); Baumeister and Hamilton (2014); b) Set Identified SVARs Non-Bayesian Approaches Moon, Schorfheide, Granziera (2013); Giacomini, Kitagawa (2015) MSG: Frequentist Bonferroni-M.I.; GK: Robust Bayesian c) Transformations of set-identified parameters Bugni, Canay, Shi (2015); Kaido, Molinari, Stoye (2015) 5 / 58

23 Brief Overview of Related Work a) Set Identified SVARs Bayesian Approaches Faust (1998); Uhlig (2005); Baumeister and Hamilton (2014); b) Set Identified SVARs Non-Bayesian Approaches Moon, Schorfheide, Granziera (2013); Giacomini, Kitagawa (2015) c) Transformations of set-identified parameters Bugni, Canay, Shi (2015); Kaido, Molinari, Stoye (2015) 5 / 58

24 Brief Overview of Related Work a) Set Identified SVARs Bayesian Approaches Faust (1998); Uhlig (2005); Baumeister and Hamilton (2014); b) Set Identified SVARs Non-Bayesian Approaches Moon, Schorfheide, Granziera (2013); Giacomini, Kitagawa (2015) c) Transformations of set-identified parameters Bugni, Canay, Shi (2015); Kaido, Molinari, Stoye (2015) SVARs could be a nice application of these general papers! 5 / 58

25 Brief Overview of Related Work a) Set Identified SVARs Bayesian Approaches Faust (1998); Uhlig (2005); Baumeister and Hamilton (2014); b) Set Identified SVARs Non-Bayesian Approaches Moon, Schorfheide, Granziera (2013); Giacomini, Kitagawa (2015) c) Transformations of set-identified parameters Bugni, Canay, Shi (2015); Kaido, Molinari, Stoye (2015) SVARs could be a nice application of these general papers! (Projection inference involved in MI models, but not in SVARs.) 5 / 58

26 Outline 1. Notation and Main Assumptions 2. Projection Inference and Implementation in SVARs 3. Calibrated Projection Inference: Idea 4. Illustrative Example: Unconventional MP 5. Conclusion 6 / 58

27 Outline 1. Notation and Main Assumptions 2. Projection Inference and Implementation in SVARs 3. Calibrated Projection Inference: Idea 4. Illustrative Example: Unconventional MP 5. Conclusion 6 / 58

28 Outline 1. Notation and Main Assumptions 2. Projection Inference and Implementation in SVARs 3. Calibrated Projection Inference: Idea 4. Illustrative Example: Unconventional MP 5. Conclusion 6 / 58

29 Outline 1. Notation and Main Assumptions 2. Projection Inference and Implementation in SVARs 3. Calibrated Projection Inference: Idea 4. Illustrative Example: Unconventional MP 5. Conclusion 6 / 58

30 Outline 1. Notation and Main Assumptions 2. Projection Inference and Implementation in SVARs 3. Calibrated Projection Inference: Idea 4. Illustrative Example: Unconventional MP 5. Conclusion 6 / 58

31 1. Notation and Main Assumptions 7 / 58

32 Gaussian SVAR(p) Vector Autoregression for the n-dimensional vector Y t : Y t = A 1 Y t A p Y t p + η t, A (A 1, A 2,... A p ) and Σ E[η t η t]. 8 / 58

33 Gaussian SVAR(p) Vector Autoregression for the n-dimensional vector Y t : Y t = A 1 Y t A p Y t p + η t, A (A 1, A 2,... A p ) and Σ E[η t η t]. θ = (vec(a), vech(σ) ) 8 / 58

34 Gaussian SVAR(p) Vector Autoregression for the n-dimensional vector Y t : Y t = A 1 Y t A p Y t p + η t, A (A 1, A 2,... A p ) and Σ E[η t η t]. Structural Model for the vector of Forecast Errors η t : η t = Hε t, H R n n, ε t N n (0, I n ), i.i.d. 8 / 58

35 Gaussian SVAR(p) Vector Autoregression for the n-dimensional vector Y t : Y t = A 1 Y t A p Y t p + η t, A (A 1, A 2,... A p ) and Σ HH Structural Model for the vector of Forecast Errors η t : η t = Hε t, H R n n, ε t N n (0, I n ), i.i.d. 8 / 58

36 Structural IRF Structural Impulse Response Functions (variable i, shock j) 9 / 58

37 Structural IRF Structural Impulse Response Functions (variable i, shock j) Y i,t+k ε jt IRF k,ij (A, H) = e i C k (A) }{{} 1 n H j 9 / 58

38 Structural IRF Structural Impulse Response Functions (variable i, shock j) Y i,t+k ε jt IRF k,ij (A, H) = e i C k (A) }{{} 1 n Linear combination of the j-th column of H. H j 9 / 58

39 Structural IRF Structural Impulse Response Functions (variable i, shock j) Y i,t+k ε jt IRF k,ij (A, H) = e i C k (A) }{{} 1 n Linear combination of the j-th column of H. Moving-Average Representation of the SVAR Y t = C k (A)Hε t k, k=0 C k (A) is a nonlinear transformation of A H j 9 / 58

40 Structural IRF Structural Impulse Response Functions (variable i, shock j) Y i,t+k ε jt (e i is the i-th column of I n ) IRF k,ij (A, H) = e i C k (A) }{{} 1 n H j Moving-Average Representation of the SVAR Y t = C k (A)Hε t k, k=0 C k (A) is a nonlinear transformation of A 9 / 58

41 (A, Σ) IRF k,ij is one-to-many The reduced-form parameters are compatible with any θ = (A, Σ) IRF k,ij e i C k (A)H j such that H j satisfies HH = Σ 10 / 58

42 (A, Σ) IRF k,ij is one-to-many The reduced-form parameters are compatible with any θ = (A, Σ) IRF k,ij e i C k (A)H j such that H j satisfies HH = Σ The set is usually refined with other restrictions on H (the most common are zero/sign restrictions) 10 / 58

43 Identified set for IRF k,ij I k,ij (A, Σ) { } v R v = e i C k (A)H j, HH = Σ, ±/0 rest. on H 11 / 58

44 Identified set for IRF k,ij } I k,ij (A, Σ) {v R v = e i C k (A)H j, HH = Σ, e i C k (A)H j 0 11 / 58

45 Identified set for IRF k,ij I k,ij (A, Σ) { } v R v = e i C k (A)H j, HH = Σ, e i (H ) 1 e j 0 11 / 58

46 The maximum response in the population subject to v k,ij (A, Σ) max H R n n e i C k (A)H j HH = Σ and 12 / 58

47 The maximum response in the population subject to v k,ij (A, Σ) max H R n n e i C k (A)H j HH = Σ and zero restrictions: Z (A, Σ)H j = 0, Z(A, Σ) R n mz 12 / 58

48 The maximum response in the population subject to v k,ij (A, Σ) max H R n n e i C k (A)H j HH = Σ and zero restrictions: Z (A, Σ)H j = 0, Z(A, Σ) R n mz sign restrictions: S (A, Σ)H j 0, S(A, Σ) R n ms 12 / 58

49 Obviously... I k,ij (A, Σ) [ ] v k,ij (A, Σ), v k,ij (A, Σ) 13 / 58

50 And this is almost all we need to know to conduct inference on IRF k,ij 14 / 58

51 2. Projection Inference and its implementation 15 / 58

52 2.1 Main Idea 16 / 58

53 What is it that we are looking for? Want to construct CS T : Data Subs(R) such that: ) lim inf inf inf P θ (IRF k,ij CS T 1 α T θ Θ IRF k,ij I k,ij (θ) 17 / 58

54 What is it that we are looking for? Want to construct CS T : Data Subs(R) such that: ) lim inf inf inf P θ (IRF k,ij CS T 1 α T θ Θ IRF k,ij I k,ij (θ) And we know that (uniformly over Θ): T ( θt θ) d N dθ (0, Ω(θ)) 17 / 58

55 What is it that we are looking for? Want to construct CS T : Data Subs(R) such that: ) lim inf inf inf P θ (IRF k,ij CS T 1 α T θ Θ IRF k,ij I k,ij (θ) And we know that (uniformly over Θ): T ( θt θ) d N dθ (0, Ω(θ)) Thus, a CS is available for the reduced-form parameters: { } CS( θ T ; 1 α) θ Θ T ( θ T θ) Ω 1 ( θ T θ) χ 2 d θ (1 α) 17 / 58

56 What is it that we are looking for? Want to construct CS T : Data Subs(R) such that: ) lim inf inf inf P θ (IRF k,ij CS T 1 α T θ Θ IRF k,ij I k,ij (θ) And we know that (uniformly over Θ): T ( θt θ) d N dθ (0, Ω(θ)) Thus, a CS is available for the reduced-form parameters: { } CS( θ T ; 1 α) θ Θ T ( θ T θ) Ω 1 ( θ T θ) χ 2 d θ (1 α) And this CS is uniformly valid of level (1 α): ( ) lim inf inf P θ θ CS( θ T ; 1 α) 1 α T θ Θ 17 / 58

57 Projection Confidence Set for IRF k,ij A conceptually straightforward CS for IRF k,ij uses projection: [ ] CS P T min v k,ij (θ), max v k,ij (θ) θ CS( θ T,1 α) θ CS( θ T,1 α) 18 / 58

58 Projection Confidence Set for IRF k,ij A conceptually straightforward CS for IRF k,ij uses projection: [ ] CS P T min v k,ij (θ), max v k,ij (θ) θ CS( θ T,1 α) θ CS( θ T,1 α) Build a confidence set for θ, and evaluate the max/min. 18 / 58

59 Projection Confidence Set for IRF k,ij A conceptually straightforward CS for IRF k,ij uses projection: [ ] CS P T min v k,ij (θ), max v k,ij (θ) θ CS( θ T,1 α) θ CS( θ T,1 α) Build a confidence set for θ, and evaluate the max/min. Proof: Note that for any IRF k,ij I k,ij (θ): P θ (IRF k,ij CS P T ) ) P θ (v k,ij (θ), v k,ij (θ) CS P T ) P θ (θ CS P T ( θ T, 1 α) 18 / 58

60 Projection Confidence Set for IRF k,ij A conceptually straightforward CS for IRF k,ij uses projection: [ ] CS P T min v k,ij (θ), max v k,ij (θ) θ CS( θ T,1 α) θ CS( θ T,1 α) Build a confidence set for θ, and evaluate the max/min. Proof: Note that for any IRF k,ij I k,ij (θ): P θ (IRF k,ij CS P T ) ) P θ (v k,ij (θ), v k,ij (θ) CS P T ) P θ (θ CS P T ( θ T, 1 α) = lim inf T inf θ Θ inf IRFk,ij I k,ij (θ) P θ ( IRF k,ij CS T ) 18 / 58

61 Projection Confidence Set for IRF k,ij A conceptually straightforward CS for IRF k,ij uses projection: [ ] CS P T min v k,ij (θ), max v k,ij (θ) θ CS( θ T,1 α) θ CS( θ T,1 α) Build a confidence set for θ, and evaluate the max/min. Proof: Note that for any IRF k,ij I k,ij (θ): P θ (IRF k,ij CS P T ) ) P θ (v k,ij (θ), v k,ij (θ) CS P T ) P θ (θ CS P T ( θ T, 1 α) ) lim inf T inf θ Θ P θ (θ CS P T ( θ T, 1 α) = 1 α 18 / 58

62 v k,ij ( ), v k,ij ( ) are only required to be measurable fns. of θ! 19 / 58

63 2.2 Implementation 20 / 58

64 The upper end of the projection CS subject to max A,Σ v k,ij(a, Σ) 21 / 58

65 The upper end of the projection CS subject to max A,Σ v k,ij(a, Σ) θ { } θ T ( θ T θ) Ω 1 ( θ T θ) χ 2 d θ (1 α) θ = (vec(a), vech(σ) ) R d θ 21 / 58

66 Since the maximum response is given by subject to v k,ij (A, Σ) max H R n n e i C k (A)H j 22 / 58

67 Since the maximum response is given by subject to and v k,ij (A, Σ) max H R n n e i C k (A)H j HH = Σ zero restrictions: Z (A, Σ)H j = 0, Z(A, Σ) R n mz sign restrictions: S (A, Σ)H j 0, S(A, Σ) R n ms 22 / 58

68 The upper end of projection CS is the value of a NLP subject to and max A,Σ,H R n n e i C k (A)H j HH = Σ zero restrictions: Z (A, Σ)H j = 0, Z(A, Σ) R n mz sign restrictions: S (A, Σ)H j 0, S(A, Σ) R n ms CS: T ( θ T θ) Ω 1 ( θ T θ) χ 2 d θ (1 α) 23 / 58

69 We solve this program using fmincon 24 / 58

70 fmincon finds local minimum even when d θ is large, say 100. (as long as the CS for θ contains only stationary matrices) 25 / 58

71 2.3 A Simple (Small-Scale) Toy Example 26 / 58

72 Demand-Supply SVAR HB(2015): two shocks (d θ = 7) Effect of a structural shock on labor demand over wages/emp? (2-SVAR, 1 lag (AIC, BIC), Q1-1970/Q2-2014: ln w t, ln e t.) SR1: demand shock increases w t and emp t SR2: supply shock increases w t, but decreases emp t w t : comprnfb; e t : payems (St. Louis Fed). 27 / 58

73 % change % change 95% Projection (blue) vs. 95% Bayesian Credible Set (gray) 5 WAGE 5 EMP Months after shock Months after shock 19 seconds vs seconds IntelCore i7) 28 / 58

74 % change % change 95% Projection (blue) vs. 95% Bayesian Credible Set (gray) 5 WAGE 5 EMP Months after shock Months after shock 19 seconds vs seconds IntelCore i7) Point estimator:.01 seconds 29 / 58

75 Frequentist CS are larger than Bayesian Credible Sets (no surprise) (Moon and Schorfheide [ECMA; 2012]) 30 / 58

76 Are they unnecessarily large? 31 / 58

77 3. Calibrated Projection 32 / 58

78 The well-known problem of Projection Projection inference could be conservative, in the sense that: ( ) lim inf inf inf P θ IRF k,ij CS P T > 1 α. T θ Θ IRF k,ij I k,ij (θ) 33 / 58

79 The well-known problem of Projection Projection inference could be conservative, in the sense that: ( ) lim inf inf inf P θ IRF k,ij CS P T > 1 α. T θ Θ IRF k,ij I k,ij (θ) In fact, when the max/min are differentiable CS P T is approx: [ χ 2 d θ (1 α) χ 2 d θ (1 α) ] v k,ij ( θ T ) σ, v k,ij ( θ T ) + σ T T 33 / 58

80 The well-known problem of Projection Projection inference could be conservative, in the sense that: ( ) lim inf inf inf P θ IRF k,ij CS P T > 1 α. T θ Θ IRF k,ij I k,ij (θ) In fact, when the max/min are differentiable CS P T is approx: [ χ 2 d θ (1 α) χ 2 d θ (1 α) ] v k,ij ( θ T ) σ, v k,ij ( θ T ) + σ T T But in this case, we would like to use something like: [ v k,ij ( θ T ) 1.64 σ, v k,ij ( θ T ) ] σ T T 33 / 58

81 The well-known problem of Projection Projection inference could be conservative, in the sense that: ( ) lim inf inf inf P θ IRF k,ij CS P T > 1 α. T θ Θ IRF k,ij I k,ij (θ) In fact, when the max/min are differentiable CS P T is approx: [ χ 2 d θ (1 α) χ 2 d θ (1 α) ] v k,ij ( θ T ) σ, v k,ij ( θ T ) + σ T T But in this case, we would like to use something like: [ v k,ij ( θ T ) 1.64 σ, v k,ij ( θ T ) ] σ T T Thus, when d θ = 186: 14 times std. errors vs / 58

82 Calibrated Projection: Thought Experiment Asymptotically θ T is approximately N dθ (θ, Ω(θ)/T ). 34 / 58

83 Calibrated Projection: Thought Experiment Asymptotically θ T is approximately N dθ (θ, Ω(θ)/T ). Hence, we can use this limiting DGP to compute: C k (d) inf θ Θ inf IRF k,ij I k,ij (θ) P θ ( IRF k,ij CS P T ( θ T, d) ) 34 / 58

84 Calibrated Projection: Thought Experiment Asymptotically θ T is approximately N dθ (θ, Ω(θ)/T ). Hence, we can use this limiting DGP to compute: C k (d) inf θ Θ inf IRF k,ij I k,ij (θ) P θ ( IRF k,ij CS P T ( θ T, d) Which depends on the degrees of freedom used in CS for θ. ) 34 / 58

85 Calibrated Projection: Thought Experiment Asymptotically θ T is approximately N dθ (θ, Ω(θ)/T ). Hence, we can use this limiting DGP to compute: C k (d) inf θ Θ inf IRF k,ij I k,ij (θ) P θ ( IRF k,ij CS P T ( θ T, d) Which depends on the degrees of freedom used in CS for θ. ) Brute Force: Calibrate the parameter d until C k (d) = 1 α. (theoretically, all we need to show is continuity of C k (d)) 34 / 58

86 Calibrated Projection: Thought Experiment Asymptotically θ T is approximately N dθ (θ, Ω(θ)/T ). Hence, we can use this limiting DGP to compute: C k (d) inf θ Θ inf IRF k,ij I k,ij (θ) P θ ( IRF k,ij CS P T ( θ T, d) Which depends on the degrees of freedom used in CS for θ. ) Brute Force: Calibrate the parameter d until C k (d) = 1 α. (theoretically, all we need to show is continuity of C k (d)) Equivalent to setting different MC exercises over a grid for θ. (making the minimum coverage probability over MCs 1 α) 34 / 58

87 Practically, projection can be calibrated over a grid of values of θ 35 / 58

88 inf θ Θ G Find d k (Θ G ) such that: ( ) inf P θ IRF k,ij CS P T ( θ T, d k (Θ G )) = 1 α. IRF k,ij I k,ij (θ) 36 / 58

89 The calibrated d k (Θ G ) is a lower bound for d k (Θ) (the grid can include only one point; namely θ T ) 37 / 58

90 calibrated degrees of freedom calibrated degrees of freedom Lower Bound on d k (Θ) 8 WAGE 8 EMP Months after shock Months after shock Grid of 1 point ( θ T ), only integer values d k allowed. (Time: 1.5 hours, using 25 parallel cores) 38 / 58

91 % change % change 95% Projection (blue) vs. 95% Bayesian Credible Set (gray) 5 WAGE 5 EMP Months after shock Months after shock Dotted: Calibrated Projection corresponding to d k ( θ T ) 39 / 58

92 % change % change 95% Projection (blue) vs. 95% Bayesian Credible Set (gray) 5 WAGE 5 EMP Months after shock Months after shock Dotted: Calibrated Projection corresponding to d k ( θ T ) 40 / 58

93 % change % change 95% Projection (blue) vs. 95% Bayesian Credible Set (gray) 5 WAGE 5 EMP Months after shock Months after shock Calibration over Θ would give lines between dashed and dotted. 41 / 58

94 Projection is conservative for IRF k,ij, yes / 58

95 but even if we were to eliminate projection bias / 58

96 % change % change 95% Projection (blue) vs. 95% Bayesian Credible Set (gray) 5 WAGE 5 EMP Months after shock Months after shock 95% CS gives different conclusions than a 95% Credible Set. 44 / 58

97 Is projection also conservative for the impulse-response function? 45 / 58

98 Projection can also be calibrated to cover (IRF 0,ij, IRF 1,ij,... IRF h,ij ) R h+1 46 / 58

99 % change % change 95% Projection (blue) vs. 95% Bayesian Credible Set (gray) 5 WAGE 5 EMP Months after shock Months after shock Dotted: Calibrated Projection for (IRF 0,ij,... IRF 20,ij ); d( θ T ) = 3 47 / 58

100 % change % change 95% Projection (blue) vs. 95% Bayesian Credible Set (gray) 5 WAGE 5 EMP Months after shock Months after shock Dotted: Calibrated Projection for IRF k,ij ; d k ( θ T ) 48 / 58

101 4. Empirical Application 49 / 58

102 An Unconventional Monetary Shock What is an unconventional monetary shock? (forward guidance, large-scale asset purchase program) 50 / 58

103 An Unconventional Monetary Shock What is an unconventional monetary shock? (forward guidance, large-scale asset purchase program) Hard to define exactly, but easy to state minimal properties 50 / 58

104 An Unconventional Monetary Shock What is an unconventional monetary shock? (forward guidance, large-scale asset purchase program) Hard to define exactly, but easy to state minimal properties At the very minimum: shock that long term rates... (sign restriction on the contemporaneous response) 50 / 58

105 An Unconventional Monetary Shock What is an unconventional monetary shock? (forward guidance, large-scale asset purchase program) Hard to define exactly, but easy to state minimal properties At the very minimum: shock that long term rates... (sign restriction on the contemporaneous response)... but perhaps no effect over the fed funds rate. (zero restriction on contemporaneous response) 50 / 58

106 An Unconventional Monetary Shock What is an unconventional monetary shock? (forward guidance, large-scale asset purchase program) Hard to define exactly, but easy to state minimal properties At the very minimum: shock that long term rates... (sign restriction on the contemporaneous response)... but perhaps no effect over the fed funds rate. (zero restriction on contemporaneous response) Extra: No contractionary effect over inflation and output 50 / 58

107 Monetary SVAR 4 variable SVAR(2) with monthly data ( : ): 1. Consumer Price Index, log 51 / 58

108 Monetary SVAR 4 variable SVAR(2) with monthly data ( : ): 1. Consumer Price Index, log 2. Industrial Production Index, log 51 / 58

109 Monetary SVAR 4 variable SVAR(2) with monthly data ( : ): 1. Consumer Price Index, log 2. Industrial Production Index, log 3. 2 year government bond rate, 51 / 58

110 Monetary SVAR 4 variable SVAR(2) with monthly data ( : ): 1. Consumer Price Index, log 2. Industrial Production Index, log 3. 2 year government bond rate, 4. Federal Funds rate, 51 / 58

111 % change % change % change % change 95% Projection (blue) vs. 95% Bayesian Credible Set (gray) IP CPI Months after shock GS Months after shock FFR Months after shock Months after shock 28 seconds vs. 283 seconds IntelCore i7) 52 / 58

112 Second Round of Quantititative Easing Use the pre-crisis bands to bound the post QE2 responses 53 / 58

113 Response of IP, CPI: 95% Uhlig s Credible Set Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May 99.5 Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May The Credible Set covers IP 7/9 (78%) and CPI 5/9 (56%) times 54 / 58

114 Response of IP, CPI: 95% Projection CS Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May 99.5 Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May The Projection CS covers IP 9/9 (100 %) and CPI 8/9 (89 %). 55 / 58

115 5. Conclusion 56 / 58

116 Main Messages from our Revised Project Projection CS for IRF k,ij R and (IRF 0,ij,... IRF h,ij ) R h+1 (as general as Uhlig s current Bayesian Approach) 57 / 58

117 Main Messages from our Revised Project Projection CS for IRF k,ij R and (IRF 0,ij,... IRF h,ij ) R h+1 (as general as Uhlig s current Bayesian Approach) Implementing projection requires evaluation of max θ CS v k,ij (θ) (value function of a nonlinear mathematical program) 57 / 58

118 Main Messages from our Revised Project Projection CS for IRF k,ij R and (IRF 0,ij,... IRF h,ij ) R h+1 (as general as Uhlig s current Bayesian Approach) Implementing projection requires evaluation of max θ CS v k,ij (θ) (value function of a nonlinear mathematical program) Projection is conservative, but in theory can be calibrated. (propose a lower bound on the calibrated degrees of freedom) 57 / 58

119 Main Messages from our Revised Project Projection CS for IRF k,ij R and (IRF 0,ij,... IRF h,ij ) R h+1 (as general as Uhlig s current Bayesian Approach) Implementing projection requires evaluation of max θ CS v k,ij (θ) (value function of a nonlinear mathematical program) Projection is conservative, but in theory can be calibrated. (propose a lower bound on the calibrated degrees of freedom) One structural shock: delta-method type inference is available (contemporaneous restrictions, we establish unif. validity) 57 / 58

120 Main Messages from our Revised Project Projection CS for IRF k,ij R and (IRF 0,ij,... IRF h,ij ) R h+1 (as general as Uhlig s current Bayesian Approach) Implementing projection requires evaluation of max θ CS v k,ij (θ) (value function of a nonlinear mathematical program) Projection is conservative, but in theory can be calibrated. (propose a lower bound on the calibrated degrees of freedom) One structural shock: delta-method type inference is available (contemporaneous restrictions, we establish unif. validity) max/min differentiable and strictly set identified model: Calibrated Projection Delta Method 57 / 58

121 Thanks! 58 / 58