Identification and Estimation of Dynamic Causal Effects in - PowerPoint PPT Presentation

Identification and Estimation of Dynamic Causal Effects in Macroeconomics Jim Stock and Mark Watson IAAE 2017 Sapporo June 26, 2017 1

Causal effects and IV regression Avg. Causal Effect: E( Y | X = 1) – E( Y | X = 0) Linear Regression: Y = γ + β X + u E( u | X ) = 0 With Controls: Y = β X + γ ’W + u X ⊥ = X – Proj( X | W ) E( u ⊥ X ⊥ ) = 0 With Instruments, Z : Relevance: E( X ⊥ Z ⊥ ) ≠ 0 Exogeneity: E( u ⊥ Z ⊥ ) = 0 2

Dynamic causal effects and IRFs Y t = Θ 0 ε t + Θ 1 ε t − 1 + … = Θ (L) ε t n Y n ε ε j,t are "structural shocks": i.i.d. (over j and t ) with mean zero. Dynamic ACE: E( Y i,t + h | ε j,t = 1) − E( Y i,t + h | ε j,t = 0) = Θ h , ij ∂ Y i , t + h / ∂ ε j , t ) ( = 3

Estimating dynamic causal effects in macroeconomics Standard Approach: • Estimate VAR for Y • Assume "invertibility" to relate ε t to VAR forecast errors. • Impose some restrictions on Θ (L) for identification Alternative Approach: • Find an "external" instrument Z that captures exogenous variation in one of the structural shocks. • Use instrument (with or without VAR step) to estimate dynamic causal effects. 4

Some references on external instruments VARs: Stock (2008), Stock and Watson (2012), Mertens and Ravn (2013, 2014), Gertler and Karadi (2015), Caldera and Kamps (2017), Montiel Olea, Stock and Watson (2012), Lumsford (2015), Jentsch and Lunsford (2016), Carriero, Momtaz, Theodoridis and Theophilopoulou (2015), … Local-projections: Jordà, Schularick, and Taylor (2015), Ramey and Zubairy (2017), Ramey (2016), Mertens (2015), Fieldhouse, Mertens, Ravn (2017) … This talk: Survey-ish 5

A Running Empirical Example: Gertler-Karadi (2015) • Y t = ( R t , 100 × Δ ln( IP ), 100 × Δ ln( CPI ), EBP ) • Monetary policy shock = ε 1, t • Causal Effects: E( Y i,t + h | ε 1 ,t = 1) − E( Y i,t + h | ε 1 ,t = 0) = Θ h , i • Kuttner (2001)-like instrument, Z t = change in Federal Funds rate futures in short window around FOMC announcements. o Z t correlated with ε 1, t but uncorrelated with ε • , t = ( ε 2, t , ε 3, t , … , ε n ε , t ) . 6

ε 1, t is Direct estimation of Θ h,i 1 Y t = Θ 0 ε t + Θ 1 ε t − 1 + … = Θ (L) ε t Y i,t + h = Θ h,i 1 ε 1, t + u t (LP) u t = { ε t + h , … , ε t +1 , ε • , t , ε t − 1 , … } E( ε 1, t u t ) = 0 But ε 1, t is not observed 7

Unit-Effect Normalization – 1 GK – IRF or Monetary Policy Shock One-year rate 0.4 External instruments Percent 0.2 0 − 0.2 10 20 30 40 IP 0.2 Percent 0 − 0.2 − 0.4 − 0.6 10 20 30 40 8

Unit-Effect Normalization - 1 One-year rate 0.4 External instruments Percent 0.2 0 − 0.2 10 20 30 40 IP 0.2 Percent 0 − 0.2 − 0.4 − 0.6 10 20 30 40 ∂ IP t + h / ∂ ε 1 , t Parameters of interest: ∂ R t / ∂ ε 1 , t 9

Unit-effect normalization − 2 Θ 0,11 = 1 ∂ R t / ∂ ε 1, t = 1 GK: Y 1, t = R t and units are such that 10

IV estimation of Θ h,i 2 Y i,t + h = Θ h,i 1 ε 1, t + { ε t + h , … , ε t +1 , ε • , t , ε t − 1 , … } Y 1, t = Θ 0,11 ε 1, t + { ε • , t , ε t − 1 , … } = ε 1, t + { ε • , t , ε t − 1 , … } Y i,t + h = Θ h,i 1 Y 1, t + { ε t + h , … , ε t +1 , ε • , t , ε t − 1 , … } Condition LP-IV: (i) E( ε 1, t Z t ) = α ≠ 0 ε • , t Z t ') = 0 (ii) E( (iii) E( ε t + j Z t ') = 0 for j ≠ 0 11

Odds and ends • HAR SEs • Dyn. Causal Effects for levels vs. differences • Weak-instrument robust inference • "News" Shocks o replace Θ 0,11 = 1 normalization with Θ k ,11 = 1 normalization • Smoothness constraints (B&B − today, Plagborg- Møller,) 12

Results for [ R and 100 × ln( IP )] (1990m1 -2012:m6) lag ( h ) (a) R 0 1.00 (0.00) 6 -0.07 (1.34) 12 -1.05 (2.51) 24 -2.09 (5.66) IP 0 -0.59 (0.71) 6 -2.15 (3.42) 12 -3.60 (6.23) 24 -2.99 (10.21) Controls none First-stage F 1.7 13

Results for [ R and 100 × ln( IP )] (1990m1 -2012:m6) lag ( h ) (a) R 0 1.00 (0.00) 6 -0.07 (1.34) 12 -1.05 (2.51) 24 -2.09 (5.66) IP 0 -0.59 (0.71) 6 -2.15 (3.42) 12 -3.60 (6.23) 24 -2.99 (10.21) Controls none First-stage F 1.7 14

Results for [ R and 100 × ln( IP )] (1990m1 -2012:m6) lag ( h ) (a) R 0 1.00 (0.00) 6 -0.07 (1.34) 12 -1.05 (2.51) 24 -2.09 (5.66) IP 0 -0.59 (0.71) 6 -2.15 (3.42) 12 -3.60 (6.23) 24 -2.99 (10.21) Controls none First-stage F 1.7 15

IV Estimation of Θ h,i 2 with additional controls -1 Y i,t + h = Θ h,i 1 Y 1, t + { ε t + h , … , ε t +1 , ε • , t , ε t − 1 , … } 2 Motivations: (1) add controls to eliminate part of error term • controls should be uncorrelated with ε 1, t . o Examples: lags of Z , Y , other macro variables (2) Z t may be correlated with error, but uncorrelated after adding controls (a) Example: GK Z = { Δ FFF t , Δ FFF t − 1 }. Add lags of FFF t . 16

IV Estimation of Θ h,i 2 with additional controls - 2 Y i,t + h = Θ h,i 1 Y 1, t + γ 'W t + u t ⊥ = x t − Proj( x t | W t ) x t Condition LP-IV ⊥ ( ) = ⊥ Z t ⊥ ′ E ε 1, t α ≠ 0 ′ (i) ( ) = 0 ⊥ Z t ⊥ ′ E ε • , t (ii) ( ) = 0 for j ≠ 0. ⊥ Z t ⊥ ′ E ε t + j (iii) 17

Results for [ R and 100 × ln( IP )] Y i,t + h = Θ h,i 1 Y 1, t + γ 'W t + { ε t + h , … , ε t +1 , ε • , t , ε t − 1 , … } lag ( h ) (a) (b) (c) R 0 1.00 (0.00) 1.00 (0.00) 1.00 (0.00) 6 -0.07 (1.34) 1.12 (0.52) 0.67 (0.57) 12 -1.05 (2.51) 0.78 (1.02) -0.12 (1.07) 24 -2.09 (5.66) -0.80 (1.53) -1.57 (1.48) IP 0 -0.59 (0.71) 0.21 (0.40) 0.03 (0.55) 6 -2.15 (3.42) -3.80 (3.14) -4.05 (3.65) 12 -3.60 (6.23) -6.70 (4.70) -6.86 (5.49) 24 -2.99 (10.21) -9.51 (7.70) -8.13 (7.62) Controls none 4 lags of ( z,y ) 4 lags of ( z,y,f ) First-stage F 1.7 23.7 18.6 18

Results for [ R and 100 × ln( IP )] Y i,t + h = Θ h,i 1 Y 1, t + γ 'W t + { ε t + h , … , ε t +1 , ε • , t , ε t − 1 , … } lag ( h ) (a) (b) (c) R 0 1.00 (0.00) 1.00 (0.00) 1.00 (0.00) 6 -0.07 (1.34) 1.12 (0.52) 0.67 (0.57) 12 -1.05 (2.51) 0.78 (1.02) -0.12 (1.07) 24 -2.09 (5.66) -0.80 (1.53) -1.57 (1.48) IP 0 -0.59 (0.71) 0.21 (0.40) 0.03 (0.55) 6 -2.15 (3.42) -3.80 (3.14) -4.05 (3.65) 12 -3.60 (6.23) -6.70 (4.70) -6.86 (5.49) 24 -2.99 (10.21) -9.51 (7.70) -8.13 (7.62) Controls none 4 lags of ( z,y ) 4 lags of ( z,y,f ) First-stage F 1.7 23.7 18.6 19

Results for [ R and 100 × ln( IP )] Y i,t + h = Θ h,i 1 Y 1, t + γ 'W t + { ε t + h , … , ε t +1 , ε • , t , ε t − 1 , … } lag ( h ) (a) (b) (c) R 0 1.00 (0.00) 1.00 (0.00) 1.00 (0.00) 6 -0.07 (1.34) 1.12 (0.52) 0.67 (0.57) 12 -1.05 (2.51) 0.78 (1.02) -0.12 (1.07) 24 -2.09 (5.66) -0.80 (1.53) -1.57 (1.48) IP 0 -0.59 (0.71) 0.21 (0.40) 0.03 (0.55) 6 -2.15 (3.42) -3.80 (3.14) -4.05 (3.65) 12 -3.60 (6.23) -6.70 (4.70) -6.86 (5.49) 24 -2.99 (10.21) -9.51 (7.70) -8.13 (7.62) Controls none 4 lags of ( z,y ) 4 lags of ( z,y,f ) First-stage F 1.7 23.7 18.6 20

SVARs with External Instruments - 1 VAR: Y t = A 1 Y t -1 + A 2 Y t -2 + … + v t Structural MA: Y t = Θ 0 ε t + Θ 1 ε t − 1 + … = Θ (L) ε t Invertibility: ε t = Proj( ε t | Y t , Y t -1 , … ) ⇒ v t = Θ 0 ε t with Θ 0 nonsingular (so n y = n ε ) 21

SVARs with External Instruments - 2 A(L) Y t = v t = Θ 0 ε t ⇒ Y t = C(L) Θ 0 ε t with C(L)=A(L) -1 thus Θ h , i 1 = C h Θ 0, i 1 Unit-effect normalization yields: v i,t = Θ 0, i 1 v 1, t + { ε • , t } Condition SVAR-IV (i) E( ε 1, t Z t ) = α ≠ 0 ε • , t Z t ') = 0 (ii) E( 22

SVAR with external instruments – estimation 1. Regress Y i,t onto Y 1, t using instruments Z t and p lags of Y t as controls. This yields ˆ SVAR − IV . Θ 0, i 1 2. Estimate a VAR( p ) and invert the VAR to obtain C (L) = ˆ ˆ A (L) − 1 . 3. Estimate the dynamic causal effects of shock 1 on the vector of variables as SVAR − IV = ˆ ˆ C h ˆ SVAR − IV Θ h ,1 Θ 0,1 (odds and ends: (1) News shocks; (2) Dif. sample periods in (1) and (2)) 23

SVAR with external instruments – inference • Strong instruments: ⎛ ⎞ ˆ A − A ⎯ → ⎯ Normal + δ -method d ⎜ ⎟ T SVAR − IV − Θ 0,1 ˆ Θ 0,1 ⎜ ⎟ ⎝ ⎠ • Weak instruments: T ( ˆ A − A ) ⎯ → ⎯ Normal. d o SVAR − IV − Θ 0,1 ˆ Θ 0,1 ⎯ → d ⎯ NonNormal. o o Use weak-instrument robust methods. (Montiel Olea, Stock and Watson (20xx)). 24

SVAR with external instruments – invertibility 1 VAR forecast error: v t = Y t − Proj( Y t | Y t -1 ) Structural MA: Y t = Θ (L) ε t ⇒ ( ) ∑ Θ h ε t − h − Proj( ε t − h | Y t − 1 ) v t = Θ 0 ε t + h = 1 = Θ 0 ε t + "omitted variables" Invertibility: ε t = Proj( ε t | Y t − 1 ), so omitted variables vanish. 25

SVAR with external instruments – invertibility 2 ε t = Proj( ε t | Y t − 1 ) ⇒ • Proj( Y t | Y t -1 , ε t − 1 ) = Proj( Y t | Y t -1 ) • Proj( Y t | Y t -1 , X t − 1 ) = Proj( Y t | Y t -1 ) "Getting to invertibility" • Many Y s • But depends on timing dynamic causal effects ( Θ (L)) • No guarantees 26