Mind the gap! Stylized facts and structural models. Fabio Canova, - - PowerPoint PPT Presentation

Mind the gap! Stylized facts and structural models. Fabio Canova, Norwegian Business School and CEPR Filippo Ferroni, Chicago Fed June 2019

Introduction Common in macroeconomics to compare dynamics induced by distur- bances using SVAR and DSGE; see e.g. Gali (1999); Christiano et al. (2005); Iacoviello (2005), Basu and Bundik (2017), etc. Non-invertibility/truncation problems: Ravenna (2007), Fernandez et al. (2007), Giacomini (2013), Plagborg-Moller (2018), Pagan and Robinson (2018), Chahrour and Jurado (2018). Typically, if VAR has q shocks, use a theory with q or less disturbances. Does the DGP only has q disturbances? Which ones? Here DGP has q disturbances; empirical model q1 < q variables. Iden- tied shocks and identied dynamics become mongrels with little eco- nomic interpretation.

Cross sectional deformation:

• Identied shocks need not combine "types" of structural disturbances.
• Appropriate theoretical restrictions may be insucient.

Dicult to match e.g, identied technology shocks to TFP disturbances. Time deformation:

• Identied shocks are, in general, linear combinations of current and past

structural disturbances. Perceived internal transmission stronger than in the DGP.

Punchlines VARs can not be too small: dicult to make sense of identied shocks. If VARs can not be suciently large, compare data VARs with the theory reduced to the same VAR observables. Some structural disturbances may not be obtained from a given VAR. (Corollary) VARs used to derive dynamic facts might change depending

• n the DGP and the disturbances of interest. To identify monetary policy

disturbances may need VARs with dierent variables if the DGP has nancial disturbances or not.

Deformation vs. invertibility.

• Problems distinct.
• Long lags do not help to reduce cross sectional deformation.

Early literature: Lutkepohl (1984), Hansen and Sargent (1991), Marcet (1991), Braun and Mittnik (1991), Faust and Leeper (1998), Forni and Lippi (1999). Related literature: Canova and Sahneh (2018), Wolf (2018).

Intuition Growth model with log preferences, full depreciation, iid shocks to TFP (Zt), investment (Vt), preferences (Bt). Solution: Kt+1 = VtZtK

t

(1) Ct = (1 )BtZtK

t

(2) Yt = ZtK

t

(3) System invertible if 0 < 1. Recursive system. All three shocks identiable if VAR has three variables.

System 1: (log Kt+1; log Yt) log Kt+1 = log() + log kt + u1t (4) log Yt = log kt + u2t (5) u1t = log Vt + log Zt, u2t = log Zt. Cannot recover Bt ! cross sec- tional deformation. System maintain recursivity: identication works for log Vt; log Zt. System 2: (log Kt+1; log Ct) log Kt+1 = log() + log Kt + u1t (6) log Ct = log(1 ) + log Kt + u2t (7) u1t = log Vt + log Zt, u2t = log Bt + log Zt. ut mix demand and supply

• disturbances. Cross sectional deformation. Recursivity lost; identication

does not work.

System 3: (log Ct; log Yt) log Ct = log() + log Ct1 + u1t (8) log Yt = log() + log Yt1 + u2t (9)

• u1t = log Bt log Bt1 + log Zt + log Vt1.
• u2t = log Zt + log Vt1.
• Time and cross sectional deformation.
• Impossible to go from ujt; j = 1; 2 to demand and supply disturbances.
• Dynamics to identied ujt shocks more persistent than dynamics to

log Bt; log Vt; log Zt.

Relationship structural disturbances/empirical innovations (Log-) linear DGP: xt = A()xt1 + B()et (10) yt = C()xt1 + D()et (11) xt is k 1 vector of endogenous and exogenous states, et (0; ), diagonal, is q 1 vector of disturbances, yt is m 1 vector of endogenous

• controls. A() is k k, B() is k q, C() is m k, D() is m q,

structural parameters. Observables zit = Si[xt; yt]0, Si is qi q matrix.

Case 1: Empirical system eliminates some controls S1 = [I; S12] Innovations u1t generated by u1t = z1t E[z1tj1t1] z1t ~ F1z1t1 (12) Proposition 1 i) u1t = 1()et, where 1() is qi q. ii) A sucient condition for the identication of ej it is that the k-th row

• f G1()

B() S12D()

!

has at most one non-zero element in the j-th position. Related to Faust and Leeper (1998).

Cases 2-3: The empirical system eliminates/repackages states S2 = [S21; S22]; S3 = [S31; 0]. Innovations uit; i = 2; 3 generated by uit = zit E[zitjit1] zit ~ Fizit1 (13) Proposition 2 i) uit = i(; L)et, i is qi q, each L, i=2,3. ii) uit = i(; L)u1t; i = 2; 3.

Dynamics Proposition 3 i) If a shock can be identied from u1t and if ~ F1 = A() S12C()

!

, structural dynamics in the empirical system proportional to those of the DGP. ii) With uit; i = 2; 3 responses to identied shocks distorted at all horizons. Braun and Mittnik (1991): expression for response biases in VARs.

An example t = t+1 1 1 h gt+1 + h 1 h gt + rt t+1 (14) t = t+1 + kp

• h

1 h gt + (1 + n) nt

• + kp (t t)

(15)

• t

= t + (1 ) nt (16) rt = rrt1 + (1 r)

• y gt + p t
• + "t

(17) gt = at + ot ot1 (18) t = z t1 + "zt (19) at = a at1 + "at (20) t = t1 + "t (21) t = t1 + "t (22) t = "mpt (23)

Minimal state vector xt1 = [ot1; rt1; t1; at1; t1; t1]0 (6 1) Control vector yt = [gt; ot; t; nt; rt]0 (5 1). Shock vector et = ["zt; "at; "t; "t; "mpt]0 (5 1) Set: = 0:33; = 0:99; n = 1:5; h = 0:9; kp = 0:05; y = 0:1; p = 1:5; r = 0:8; z = 0:5; a = 0:2; = 0:5; = 0:0. How would the shocks/dynamics of an empirical system with q1 4 compare with the shocks/dynamics of the original model?

System with zt = (ot; t; nt; rt).

t = t+1 1 1 h (at+1 + ot+1 ot) + h 1 h (at + ot ot1) + rt t+1 (24) t = t+1 + kp

• h

1 h (at + ot ot1) + (1 + n) nt

• + kp (t t)

(25)

• t

= t + (1 ) nt (26) rt = r rt1 + (1 r)

• y (at + ot ot1) + pt
• + "mpt

(27)

State vector: xt1 = [ot1; rt1; t1; at1; t1; t1]0. Law of motion of the states (A, B matrices) unaltered. Cross sectional deformation, no time deformation distortions.

System with zt = (ot; t; nt).

((1 + r) rL)t = t+1 1 1 h (at+1 + ot+1 ot) + (h + r 1 h + (1 r)y) (at + ot ot1)

• ( hr

1 h) (at1 + ot1 ot2) + (r + (1 r)p) t + "mpt t+1 (28) t = t+1 + kp

• h

1 h (at + ot ot1) + (1 + n) nt

• + kp (t t) (29)
• t

= t + (1 )nt (30)

State vector: ^ xt1 = [ot1; ot2; t1; at1; t1; t1]0. Law of motion of the states (A, B matrices) altered. Cross-sectional and time deformation distortions.

System with zt = (t; nt; rt).

t = t+1 1 1 h (at+1 + t+1 t + (1 ) (nt+1 nt)) + h 1 h (at + t t1 + (1 ) (nt nt1)) + rt t+1 (31) t = t+1 + kp

• h

1 h (at + t t1 + (1 ) (nt nt1)) + (1 + n) nt

• +

kp (t t) (32) rt = rt1 + (1 )

• y (at + t t1 + (1 ) (nt nt1)) + p t
• + "mpt

(33)

State vector: ^ xt1 = [nt1; rt1; t1; at1; t1; t1]0. Law of motion of the states unchanged (given production function nt1 proxies for ot1). Cross-sectional deformation, limited time deformation distortions.

Cross correlation function: zt = (ot; t; nt; rt)

Cross correlation function: zt = (ot; t; nt)

Cross correlation function: zt = (t; nt; rt)

Structural disturbances at t t t t (y; t; nt; rt) u1t 0.018

• 0.722

0.087

• 0.005
• 0.303

innovations u2t

• 0.158
• 0.306

0.042 0.042

• 0.716

u3t

• 1.464
• 1.078

0.131

• 0.007
• 0.452

u4t

• 0.047
• 0.086

0.014 0.012 0.778 (yt; t; nt) u1t

• 0.05

0.71 0.11 0.03

• 0.29

innovations u2t

• 0.19
• 0.30

0.05 0.05

• 0.70

u3t

• 1.57
• 1.06
• 0.17

0.05

• 0.43

1 u1t

• 0.07
• 0.92

0.12 0.04

• 0.41

u2t

• 0.01
• 0.28

0.03 0.01

• 0.52

u3t

• 0.25
• 1.37

0.18 0.06

• 0.61

2 u1t

• 0.05
• 0.90

0.11 0.04

• 0.46

u2t

• 0.01

0.28 0.03

• 0.01
• 0.52

u3t

• 0.09
• 1.35

0.16

• 0.07
• 0.69

Wolf (2018): masquerading eects.

Impulse responses Monetary shocks, zt = (ot; t; nt; rt)

Monetary shocks, zt = (t; nt; rt).

Cost push shocks, zt = (ot; t; nt; rt)

Preference shocks, zt = (ot; t; nt; rt).

Permanent TFP shocks: (gt; nt). Assume = (r + (1 r)p)1. (1 + r)t = rt1 + t+1 + 1 1 hgt+1 + (r + h 1 h + (1 r)y)gt

• hr

1 hgt1 + mpt + p( h 1 hgt + (1 + n)nt) + p(t t) (34) gt = at + t + (1 )nt t1 (1 )nt1 (35) ^ xt1 = [gt1; nt1; t1; at1; t1; t1]0. Identify permanent TFP shocks via long run restrictions.

Hours fall; but impact and propagation is wrong if DGP has 5 shocks.

Transmission of house price shocks: Iacoviello (2005) Theory has preferences, technology, cost push, monetary policy distur-

• bances. Only 4 disturbances in DGP? Add LTV constraint and impatient

consumers wealth shocks (Rabanal,2018, Linde',2018).

Innovations Disturbances eRt ejt eut eat ehast ei1t ei2t Rt 1.0 t

• 0.53
• 0.003

1.43

• 0.11
• 0.13

0.18 0.24 qt

• 1.83

0.05

• 0.80

0.13 0.33

• 1.27
• 0.51

yt

• 3.92

0.03

• 1.14
• 0.02
• 0.09

2.46 0.92 Cross correlation coecients:

• qt innovations - preference shocks ejt: 0.67 (0.60,0.72).
• qt innovations - preference shocks ei1t:-0.63 (-0.67,-0.59).
• qt innovations - preference shocks ejt: 0.92 (0.88, 0.96). (DGP with 4

disturbances)

Transmission of uncertainty shocks: Basu and Bundick (2017) Model with TFP, preference and preference volatility disturbances. Solved with third order perturbation. Pruned solution linear but state and shock vector huge (432 states, 1112 shocks). Use a linear VAR with 8 variables for the data. Potentially huge defor- mation distortions. Monetary policy shock is missing. Nominal rate included in the VAR. What happens to the theory-data match if it is included?

Cross correlation coecients:

• V IX innovations - uncertainty disturbances: 0.63 (0.50, 0.74).
• V IX innovations - monetary disturbances: -0.46 (-0.50, -0.41).
• V IX innovations - uncertainty disturbances: 0.77 (0.68, 0.86). (DGP

with 3 disturbances)

Conclusions Deformation may make identied shocks and dynamics mongrels with little economic interpretation. Magnitude and sign distortions. Recovered shocks do not necessarily aggregate only structural distur- bances of the same type (cross sectional deformation). Sound theory re- strictions insucient. Recovered shocks are linear combinations of current and past structural disturbances (time deformation). Empirical model used to derive dynamic facts might change depending

• n the theoretical model and the disturbances of interest.