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

## Inference in Structural VARs with External Instruments Jos 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

1. 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

2.  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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9.  -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

10.  -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

11. 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

12. 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

13. 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

14. 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

15. 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

16.    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

17. 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

18. 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

19. 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

20. 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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