Inference in Structural VARs with External Instruments José Luis Montiel Olea, Harvard University (NYU) James H. Stock, Harvard University Mark W. Watson, Princeton University 3rd VALE-EPGE Global Economic Conference “Business Cycles” May 9-10, 2013 Last revised 5/9/2013 1

Structural VAR Identification Problem: Sims (1980) “External” Instrument Solution: Romer and Romer (1989) Weak Instruments: Staiger and Stock (1997) Andrews-Moreira-Stock (2006) Last revised 5/9/2013 2

Notation Reduced form VAR: Y t = A (L) Y t —1 + η t ; A (L) = A 1 L + + A p L p ; Y is r ×1 1 t Structural Shocks: η t = H t = H H , H is non-singular. 1 r rt Structural VAR: Y t = A (L) Y t —1 + H t Structural MA: Y t = [I − A (L)] − 1 H t = C (L) H t C (L) H is structural impulse response function (dynamic causal effect) Last revised 5/9/2013 3

SVAR estimands (focus on shock 1) Partitioning notation: 1 t η t = H t = = 1 t H H H H 1 r 1 t rt SMA for Y t = C H C H k 1 1 t k k t k k k where [I − A (L)] − 1 = C 0 + C 1 L + C 2 L 2 + … Last revised 5/9/2013 4

SVAR estimands: Write SMA for Y t = C H C H k 1 1 t k k t k k k Y jt Impulse Resp: IRF j,k = = C j,k H 1 , where C j,k is the j ’th row of C k 1 t k k l C H Historical Decomposition: HD j,k = j l , 1 1 t l 0 k var C H j l , 1 1 t l l 0 Variance Decomposition: VD j,k = k var C j l , t l l 0 Last revised 5/9/2013 5

Two approaches for structural VAR identification problem: = H 1. Internal restrictions: Short run restrictions (Sims (1980)), long run restrictions, identification by heteroskedasticity, bounds on IRFs) 2. External information (“method of external instruments”): Romer and Romer (1989), Ramey and Shapiro (1998), … Selected empirical papers Monetary shock : Cochrane and Piazzesi (2002), Faust, Swanson, and Wright (2003. 2004), Romer and Romer (2004), Bernanke and Kuttner (2005), Gürkaynak, Sack, and Swanson (2005) Fiscal shock : Romer and Romer (2010), Fisher and Peters (2010), Ramey (2011) Uncertainty shock : Bloom (2009), Baker, Bloom, and Davis (2011), Bekaert, Hoerova, and Lo Duca (2010), Bachman, Elstner, and Sims (2010) Liquidity shocks : Gilchrist and Zakrajšek’s (2011), Bassett, Chosak, Driscoll, and Zakrajšek’s (2011) Oil shock : Hamilton (1996, 2003), Kilian (2008a), Ramey and Vine (2010) Last revised 5/9/2013 6

The method of external instruments: Identification Methods/Literature Nearly all empirical papers use OLS & report (only) first stage However, these “shocks” are best thought of as instruments (quasi- experiments) Treatments of external shocks as instruments: Hamilton (2003) Kilian (2008 – JEL ) Stock and Watson (2008, 2012) Mertens and Ravn (2012a,b) – same setup as here executed using strong instrument asymptotics Last revised 5/9/2013 7

An Empirical Example: (Stock-Watson 2012) Dynamic Factor Model Dynamic factor model: X t = F t + e t ( X t contains 200 series, F t = r = 6 factors, e t = idiosyncratic disturbance) [I − A (L)] F t = t (factors follow a VAR) t = H t (Invertible) U.S., quarterly data, 1959-2011Q2 Last revised 5/9/2013 8

-shocks and Instruments 1. Oil Shocks a. Hamilton (2003) net oil price increases b. Killian (2008) OPEC supply shortfalls c. Ramey-Vine (2010) innovations in adjusted gasoline prices 2. Monetary Policy a. Romer and Romer (2004) policy b. Smets-Wouters (2007) monetary policy shock c. Sims-Zha (2007) MS-VAR-based shock d. Gürkaynak, Sack, and Swanson (2005), FF futures market 3. Productivity a. Fernald (2009) adjusted productivity b. Gali (1999) long-run shock to labor productivity c. Smets-Wouters (2007) productivity shock Last revised 5/9/2013 9

-shocks and Instruments, ctd. 4. Uncertainty a. VIX/Bloom (2009) b. Baker, Bloom, and Davis (2009) Policy Uncertainty 5. Liquidity/risk a. Spread: Gilchrist-Zakrajšek (2011) excess bond premium b. Bank loan supply: Bassett, Chosak, Driscoll, Zakrajšek (2011) c. TED Spread 6. Fiscal Policy a. Ramey (2011) spending news b. Fisher-Peters (2010) excess returns gov. defense contractors c. Romer and Romer (2010) “all exogenous” tax changes. Last revised 5/9/2013 10

Identification of SVAR estimands (IRF, HD, VD): Z t is a k ×1 vector of external instruments t = [1 − A (L)] Y t and A (L) are identified from reduced form o Y t = C (L) t … C (L) is identified from reduced form Express IRF, HD, VD as functions of , ZZ , Z Last revised 5/9/2013 11

Identifying Assumptions: (i) = 0 (relevance) E Z 1 t t (ii) = 0, j = 2,…, r (exogeneity) E Z jt t (iii) E = 0 for j ≠ 1 1 t jt Last revised 5/9/2013 12

Identification of IRF j,k = C j,k H 1 E ( Z ) 1 t t = Z = E ( t Z t ) = E ( H t Z t ) = H H 0 H H 1 r 1 r 0 E ( Z ) rt t = H 1 is set by a normalization, The 2 Normalization: The scale of H 1 and 1 normalization used here: a unit positive value of shock 1 is defined to have a unit positive effect on the innovation to variable 1, which is u 1 t . This corresponds to: (iv) H 11 = 1 (unit shock normalization) where H 11 is the first element of H 1 Last revised 5/9/2013 13

Identification of IRF j,k = C j,k H 1 , ctd Z = H 1 so H 1 = Z /( ’ ) Impose normalization (iv): H 1 Z = H 1 11 so ꞌ = 1 Z H H 1 1 and H 1 = Z ) /( Z Z 1 Z 1 1 E Z t t If Z t is a scalar ( k = 1): H 1 = E Z 1 t t Last revised 5/9/2013 14

h requires identification of H 1 1 t H C Identification of HD = k j , 1 1 t j k 0 Proj( Z t | t ) = Proj( Z t | t ) = Proj( Z t | 1 t ) = b 1 t where b = t 2 1 H 1 1 t = Proj( t | 1t ) = Proj( t | b 1 t ) t ) = Proj( t | Proj( Z t | t )) = Proj( t | Z 1 t , = 1 where = Z ( Z Z 1 Z ) + (Note Z = ’H 1 has rank 1, so pseudo inverse is used) Last revised 5/9/2013 15

k var C H j l , 1 1 t l l 0 Identification of VD = k var C j l , t l l 0 Note this requires identification of var( H 1 1 t ), which from last slide is var( t ) = ꞌ . 1 1 Last revised 5/9/2013 16

Overidentifying Restrictions (1) Multiple Z’s for one shock: Z = ’H 1 has rank 1. Reduced rank “regression” of Z onto .) (2) Z 1 identifies 1 , Z 2 identifies 2 , and 1 and 2 are uncorrelated. This implies that Proj( Z 1 | ) is uncorrelated with Proj( Z 2 | ) or 1 = 0 Z Z 1 2 Last revised 5/9/2013 17

Estimation: GMM: Note A , , and Z are exactly identified, so concentrate these out of analysis. Focus on Z and SVAR estimands. Z = E ( t Z t ), so vec( Z ) = E ( Z t t ) or Z = H 1 ꞌ so that vec( Z ) = ( H 1 ) High level assumption (assume throughout) 1 T N(0, ) d [ Z ] vec ( ) t t Z T t 1 Last revised 5/9/2013 18

GMM Estimation: (Ignore estimation of VAR coefficients A and − these are straightforward to incorporate). Efficient GMM objective function: J ( Z ) 1 T 1 T 1 = ( Z ) vec ( ) ( Z ) vec ( ) t t Z t t Z T T t 1 t 1 where, Z = H 1 ꞌ . (Similarly when more than one shock is identified). ˆ T k = 1 (exact identification): 1 T Z Z t t t 1 k > 1(Homo): ˆ can be computed from reduced rank regression Z estimator of Z onto . Last revised 5/9/2013 19

Estimation of H 1 ( k = 1) Z = H 1 = , H 1 ˆ T so GMM estimator solves, 1 = T Z ˆ t t ˆ H t 1 1 T 1 T Z ˆ t t t 1 GMM estimator: H = 1 T 1 T Z 1 t t t 1 + u jt , = H 1 j IV interpretation: jt 1 t = j Z t + v jt 1 t Last revised 5/9/2013 20

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