Inference in Structural VARs with External Instruments Jos Luis - PowerPoint PPT Presentation

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