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

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Identification and Estimation of Dynamic Causal Effects in Macroeconomics

Jim Stock and Mark Watson IAAE 2017 Sapporo June 26, 2017

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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

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Dynamic causal effects and IRFs Yt = Θ0 εt + Θ1 εt −1 + … = Θ(L)εt nY nε εj,t are "structural shocks": i.i.d. (over j and t) with mean zero. Dynamic ACE: E(Yi,t+h | εj,t = 1) − E(Yi,t+h | εj,t = 0) = Θh,ij ( = ∂Yi,t+h / ∂ε j,t )

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Estimating dynamic causal effects in macroeconomics Standard Approach:

• Estimate VAR for Y
• Assume "invertibility" to relate εt 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.

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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

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A Running Empirical Example: Gertler-Karadi (2015)

• Yt = (Rt, 100×Δln(IP), 100×Δln(CPI), EBP)
• Monetary policy shock = ε1,t
• Causal Effects: E(Yi,t+h | ε1,t = 1) − E(Yi,t+h | ε1,t = 0) = Θh,i
• Kuttner (2001)-like instrument, Zt = change in Federal

Funds rate futures in short window around FOMC announcements.

• Zt correlated with ε1,t but uncorrelated with

ε•,t = (ε2,t,ε3,t,…,εnε ,t).

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ε1,t isDirect estimation of Θh,i1 Yt = Θ0 εt + Θ1 εt −1 + … = Θ(L)εt Yi,t+h = Θh,i1 ε1,t + ut (LP) ut = {εt+h, … , εt+1, ε•,t, εt −1, … } E(ε1,tut) = 0 But ε1,t is not observed

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Unit-Effect Normalization – 1 GK – IRF or Monetary Policy Shock

10 20 30 40 −0.2 0.2 0.4

One-year rate Percent

External instruments

10 20 30 40 −0.6 −0.4 −0.2 0.2

IP Percent

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Unit-Effect Normalization - 1 Parameters of interest: ∂IP

t+h / ∂ε1,t

∂Rt / ∂ε1,t

10 20 30 40 −0.2 0.2 0.4

One-year rate Percent

External instruments

10 20 30 40 −0.6 −0.4 −0.2 0.2

IP Percent

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Unit-effect normalization − 2 Θ0,11 = 1 GK: Y1,t = Rt and units are such that ∂Rt / ∂ε1,t = 1

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IV estimation of Θh,i2 Yi,t+h = Θh,i1ε1,t + {εt+h, … , εt+1, ε•,t, εt −1, … } Y1,t = Θ0,11ε1,t + { ε•,t, εt −1, … } = ε1,t + { ε•,t, εt −1, … } Yi,t+h = Θh,i1Y1,t + {εt+h, … , εt+1, ε•,t, εt −1, … } Condition LP-IV: (i) E(ε1,t Zt) = α ≠ 0 (ii) E( ε•,t Zt') = 0 (iii) E(εt+j Zt') = 0 for j ≠ 0

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Odds and ends

• HAR SEs
• Dyn. Causal Effects for levels vs. differences
• Weak-instrument robust inference
• "News" Shocks
• replace Θ0,11 = 1 normalization with Θk,11 = 1

normalization

• Smoothness constraints (B&B − today, Plagborg-

Møller,)

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Results for [R and 100 ×ln(IP)]

(1990m1 -2012:m6)

lag (h) (a) R 1.00 (0.00) 6

• 0.07 (1.34)

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• 1.05 (2.51)

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• 2.09 (5.66)

IP

• 0.59 (0.71)

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• 2.15 (3.42)

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• 3.60 (6.23)

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• 2.99 (10.21)

Controls none First-stage F 1.7

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Results for [R and 100 ×ln(IP)]

(1990m1 -2012:m6)

lag (h) (a) R 1.00 (0.00) 6

• 0.07 (1.34)

12

• 1.05 (2.51)

24

• 2.09 (5.66)

IP

• 0.59 (0.71)

6

• 2.15 (3.42)

12

• 3.60 (6.23)

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• 2.99 (10.21)

Controls none First-stage F 1.7

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Results for [R and 100 ×ln(IP)]

(1990m1 -2012:m6)

lag (h) (a) R 1.00 (0.00) 6

• 0.07 (1.34)

12

• 1.05 (2.51)

24

• 2.09 (5.66)

IP

• 0.59 (0.71)

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• 2.15 (3.42)

12

• 3.60 (6.23)

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• 2.99 (10.21)

Controls none First-stage F 1.7

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IV Estimation of Θh,i2 with additional controls -1 Yi,t+h = Θh,i1Y1,t + {εt+h, … , εt+1, ε•,t, εt −1, … } 2 Motivations: (1) add controls to eliminate part of error term

• controls should be uncorrelated with ε1,t.
• Examples: lags of Z, Y, other macro variables

(2) Zt may be correlated with error, but uncorrelated after adding controls (a) Example: GK Z = {ΔFFFt, ΔFFFt −1}. Add lags of FFFt.

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IV Estimation of Θh,i2 with additional controls - 2 Yi,t+h = Θh,i1Y1,t + γ'Wt + ut xt

⊥ = xt − Proj(xt | Wt)

Condition LP-IV⊥ (i) E ε1,t

⊥ Zt ⊥′

( ) =

′ α ≠ 0 (ii) E ε•,t

⊥ Zt ⊥′

( ) = 0

(iii) E εt+ j

⊥ Zt ⊥′

( ) = 0 for j ≠ 0.

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Results for [R and 100 ×ln(IP)] Yi,t+h = Θh,i1Y1,t + γ'Wt + {εt+h, … , εt+1, ε•,t, εt −1, … }

lag (h) (a) (b) (c) R 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)

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• 2.09 (5.66)
• 0.80 (1.53)
• 1.57 (1.48)

IP

• 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)

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• 3.60 (6.23)
• 6.70 (4.70)
• 6.86 (5.49)

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• 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

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Results for [R and 100 ×ln(IP)] Yi,t+h = Θh,i1Y1,t + γ'Wt + {εt+h, … , εt+1, ε•,t, εt −1, … }

lag (h) (a) (b) (c) R 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)

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• 2.09 (5.66)
• 0.80 (1.53)
• 1.57 (1.48)

IP

• 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)

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• 3.60 (6.23)
• 6.70 (4.70)
• 6.86 (5.49)

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• 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

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Results for [R and 100 ×ln(IP)] Yi,t+h = Θh,i1Y1,t + γ'Wt + {εt+h, … , εt+1, ε•,t, εt −1, … }

lag (h) (a) (b) (c) R 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)

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• 2.09 (5.66)
• 0.80 (1.53)
• 1.57 (1.48)

IP

• 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)

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• 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

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SVARs with External Instruments - 1 VAR: Yt = A1Yt-1 + A2Yt-2 + … + vt Structural MA: Yt = Θ0 εt + Θ1 εt −1 + … = Θ(L)εt Invertibility: εt = Proj(εt|Yt, Yt-1, … ) ⇒ vt = Θ0εt with Θ0 nonsingular (so ny = nε)

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SVARs with External Instruments - 2 A(L)Yt = vt = Θ0εt ⇒ Yt = C(L)Θ0εt with C(L)=A(L)-1 thus Θh,i1 = ChΘ0,i1 Unit-effect normalization yields: vi,t = Θ0,i1v1,t + { ε•,t} Condition SVAR-IV (i) E(ε1,tZt) = α ≠ 0 (ii) E( ε•,tZt') = 0

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SVAR with external instruments – estimation

• 1. Regress Yi,t onto Y1,t using instruments Zt and p lags of Yt

as controls. This yields ˆ Θ0,i1

SVAR−IV.

• 2. Estimate a VAR(p) and invert the VAR to obtain

ˆ C(L) = ˆ A(L)−1 .

• 3. Estimate the dynamic causal effects of shock 1 on the

vector of variables as ˆ Θh,1

SVAR−IV = ˆ

Ch ˆ Θ0,1

SVAR−IV

(odds and ends: (1) News shocks; (2) Dif. sample periods in (1) and (2))

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SVAR with external instruments – inference

• Strong instruments:

T ˆ A− A ˆ Θ0,1

SVAR−IV − Θ0,1

⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟

d

⎯ → ⎯ Normal + δ-method

• Weak instruments:
• T ( ˆ

A− A)

d

⎯ → ⎯ Normal.

• ˆ

Θ0,1

SVAR−IV − Θ0,1 d

⎯ → ⎯ NonNormal.

• Use weak-instrument robust methods.

(Montiel Olea, Stock and Watson (20xx)).

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SVAR with external instruments – invertibility 1 VAR forecast error: vt = Yt − Proj(Yt | Yt-1 ) Structural MA: Yt = Θ(L)εt ⇒ vt = Θ0εt + Θh εt−h − Proj(εt−h |Y t−1)

( )

h=1

∑

= Θ0εt + "omitted variables" Invertibility: εt = Proj(εt | Yt −1 ), so omitted variables vanish.

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SVAR with external instruments – invertibility 2 εt = Proj(εt | Yt −1 ) ⇒

• Proj(Yt| Yt-1, ε t−1) = Proj(Yt| Yt-1 )
• Proj(Yt| Yt-1, X t−1) = Proj(Yt| Yt-1 )

"Getting to invertibility"

• Many Ys
• But depends on timing dynamic causal effects (Θ(L))
• No guarantees

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Results for [R and 100 ×ln(IP)]

lag (h) LP-IV

1990m1-2012m6

SVAR-IV

IV: 1990m1-2012m6 VAR:1980m7-2012m6

R 1.00 (0.00) 1.00 (0.00) 6 1.12 (0.52) 0.89 (0.31) 12 0.78 (1.02) 0.78 (0.46) 24

• 0.80 (1.53)

0.40 (0.49) IP 0.21 (0.40) 0.16 (0.59) 6

• 3.80 (3.14)
• 0.81 (1.19)

12

• 6.70 (4.70)
• 1.87 (1.54)

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• 9.51 (7.70)
• 2.16 (1.65)

Controls 4 lags of (Z,Y) 12 lags of Y 4 lags of Z First-stage F 23.7 20.5

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Results for [R and 100 ×ln(IP)]

lag (h) LP-IV

1990m1-2012m6

SVAR-IV

IV: 1990m1-2012m6 VAR:1980m7-2012m6

R 1.00 (0.00) 1.00 (0.00) 6 1.12 (0.52) 0.89 (0.31) 12 0.78 (1.02) 0.78 (0.46) 24

• 0.80 (1.53)

0.40 (0.49) IP 0.21 (0.40) 0.16 (0.59) 6

• 3.80 (3.14)
• 0.81 (1.19)

12

• 6.70 (4.70)
• 1.87 (1.54)

24

• 9.51 (7.70)
• 2.16 (1.65)

Controls 4 lags of (Z,Y) 12 lags of Y 4 lags of Z First-stage F 23.7 20.5

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Comparing LP-IV and SVAR-IV - 1 Condition LP-IV⊥ (i) E ε1,t

⊥ Zt ⊥′

( ) =

′ α ≠ 0; (ii) E ε•,t

⊥ Zt ⊥′

( ) = 0

(iii) E εt+ j

⊥ Zt ⊥′

( ) = 0 for j ≠ 0.

Condition SVAR-IV (i) E ε1tZt′

( ) =

′ α ≠ 0; (ii) E ε•tZt′

( ) = 0

(iii) Invertibility

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Comparing LP-IV and SVAR-IV – 2 Suppose (i) E ε1tZt′

( ) =

′ α ≠ 0, (ii) E ε•tZt′

( ) = 0, and

E ε1,t+ jZt′

( ) = 0 for j > 0.

Then Wt = (Yt −1) are valid controls for LP-IV (ensuring E εt+ j

⊥ Zt ⊥′

( ) = 0) if and only if the SMA is invertible.

"No Free Lunch"

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Comparing LP-IV and SVAR-IV – 3 With strong instruments: Regardless of invertibility, ˆ Θh,i1

LP−IV is consistent and asy.

normal. With invertibility, ˆ Θh,i1

SVAR−IV is more efficient than ˆ

Θh,i1

LP−IV ,

but ˆ Θh,i1

SVAR−IV is inconsistent without invertibility ("omitted

variable bias"). Hausman-like test based on ˆ Θh,i1

SVAR−IV − ˆ

Θh,i1

LP−IV .

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Under strong instruments and "local-to-invertibility" T ˆ θ LP−IV − ˆ θSVAR−IV

( )

d

⎯ → ⎯ N(d,V ) ξ = T( ˆ θ LP−IV − ˆ θSVAR−IV )' ˆ V −1( ˆ θ LP−IV − ˆ θSVAR−IV )

• d is the local departure from invertibility
• V = (α2)Ω so instrument strength matters
• ξ is a Granger-Causality test (Z → Y), with power

focused on omitted variable bias in SVAR IRFs.

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Results for [R and 100 ×ln(IP)]

lag (h) LP-IV SVAR-IV Difference R 1.00 (0.00) 1.00 (0.00) 0.00 (0.00) 6 1.12 (0.52) 0.89 (0.31)

• 0.23 (1.19)

12 0.78 (1.02) 0.78 (0.46) 0.00 (1.79) 24

• 0.80 (1.53)

0.40 (0.49) 1.19 (2.57) IP 0.21 (0.40) 0.16 (0.59)

• 0.06 (0.35)

6

• 3.80 (3.14)
• 0.81 (1.19)

3.00 (2.32) 12

• 6.70 (4.70)
• 1.87 (1.54)

4.83 (4.00) 24

• 9.51 (7.70)
• 2.16 (1.65)

7.35 (6.40) Controls 4 lags of (Z,Y) 12 lags of Y 4 lags of Z First-stage F 23.7 20.5

(Overall tests ξ are also statistically insignificant)

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Left to do Theory: a little Applications: a lot