Inference in Structural VARs with External Instruments

Transcription

1 Inference in Srucural VARs wih Exernal Insrumens José Luis Moniel Olea, Harvard Universiy (NYU) James H. Sock, Harvard Universiy Mark W. Wason, Princeon Universiy 3rd VALE-EPGE Global Economic Conference Business Cycles May 9-0, 203 Las revised 5/9/203

2 Srucural VAR Idenificaion Problem: Sims (980) Exernal Insrumen Soluion: Romer and Romer (989) Weak Insrumens: Saiger and Sock (997) Andrews-Moreira-Sock (2006) Las revised 5/9/203 2

3 Noaion Reduced form VAR: Y = A(L)Y + η ; A(L) = A L + + A p L p ; Y is r Srucural Shocks: η = H = H Srucural VAR: Y = A(L)Y + H H r, H is non-singular. r Srucural MA: Y = [I A(L)] H = C(L)H C(L)H is srucural impulse response funcion (dynamic causal effec) Las revised 5/9/203 3

4 SVAR esimands (focus on shock ) Pariioning noaion: η = H = H H = r r H H SMA for Y = CH k CH k k k k k where [I A(L)] = C 0 + C L + C 2 L 2 + Las revised 5/9/203 4

5 SVAR esimands: Wrie SMA for Y = CH k CH k k k k k Impulse Resp: IRF j,k = Y j k = C j,k H, where C j,k is he j h row of C k Hisorical Decomposiion: HD j,k = Cj, lh k l0 l Variance Decomposiion: VD j,k = var var k l0 k C l0 H j, l l C jl, l Las revised 5/9/203 5

6 wo approaches for srucural VAR idenificaion problem: = H. Inernal resricions: Shor run resricions (Sims (980)), long run resricions, idenificaion by heeroskedasiciy, bounds on IRFs) 2. Exernal informaion ( mehod of exernal insrumens ): Romer and Romer (989), Ramey and Shapiro (998), Seleced empirical papers Moneary shock: Cochrane and Piazzesi (2002), Faus, Swanson, and Wrigh ( ), Romer and Romer (2004), Bernanke and Kuner (2005), Gürkaynak, Sack, and Swanson (2005) Fiscal shock: Romer and Romer (200), Fisher and Peers (200), Ramey (20) Uncerainy shock: Bloom (2009), Baker, Bloom, and Davis (20), Bekaer, Hoerova, and Lo Duca (200), Bachman, Elsner, and Sims (200) Liquidiy shocks: Gilchris and Zakrajšek s (20), Basse, Chosak, Driscoll, and Zakrajšek s (20) Oil shock: Hamilon (996, 2003), Kilian (2008a), Ramey and Vine (200) Las revised 5/9/203 6

7 he mehod of exernal insrumens: Idenificaion Mehods/Lieraure Nearly all empirical papers use OLS & repor (only) firs sage However, hese shocks are bes hough of as insrumens (quasiexperimens) reamens of exernal shocks as insrumens: Hamilon (2003) Kilian (2008 JEL) Sock and Wason (2008, 202) Merens and Ravn (202a,b) same seup as here execued using srong insrumen asympoics Las revised 5/9/203 7

8 An Empirical Example: (Sock-Wason 202) Dynamic Facor Model Dynamic facor model: X = F + e (X conains 200 series, F = r = 6 facors, e = idiosyncraic disurbance) [I A(L)]F = (facors follow a VAR) = H (Inverible) U.S., quarerly daa, Q2 Las revised 5/9/203 8

9 -shocks and Insrumens. Oil Shocks a. Hamilon (2003) ne oil price increases b. Killian (2008) OPEC supply shorfalls c. Ramey-Vine (200) innovaions in adjused gasoline prices 2. Moneary Policy a. Romer and Romer (2004) policy b. Smes-Wouers (2007) moneary policy shock c. Sims-Zha (2007) MS-VAR-based shock d. Gürkaynak, Sack, and Swanson (2005), FF fuures marke 3. Produciviy a. Fernald (2009) adjused produciviy b. Gali (999) long-run shock o labor produciviy c. Smes-Wouers (2007) produciviy shock Las revised 5/9/203 9

10 -shocks and Insrumens, cd. 4. Uncerainy a. VIX/Bloom (2009) b. Baker, Bloom, and Davis (2009) Policy Uncerainy 5. Liquidiy/risk a. Spread: Gilchris-Zakrajšek (20) excess bond premium b. Bank loan supply: Basse, Chosak, Driscoll, Zakrajšek (20) c. ED Spread 6. Fiscal Policy a. Ramey (20) spending news b. Fisher-Peers (200) excess reurns gov. defense conracors c. Romer and Romer (200) all exogenous ax changes. Las revised 5/9/203 0

11 Idenificaion of SVAR esimands (IRF, HD, VD): Z is a k vecor of exernal insrumens = [ A(L)]Y and A(L) are idenified from reduced form o Y = C(L) C(L) is idenified from reduced form Express IRF, HD, VD as funcions of, ZZ, Z Las revised 5/9/203

12 Idenifying Assumpions: (i) E Z = 0 (relevance) (ii) E jz = 0, j = 2,, r (exogeneiy) (iii) E = 0 for j j Las revised 5/9/203 2

13 Idenificaion of IRF j,k = C j,k H Z = E( Z ) = E(H Z ) = = H H E( Z ) E( rz ) H = H H r r Normalizaion: he scale of H and is se by a normalizaion, he normalizaion used here: a uni posiive value of shock is defined o have a uni posiive effec on he innovaion o variable, which is u. his corresponds o: (iv) H = (uni shock normalizaion) where H is he firs elemen of H Las revised 5/9/203 3

14 Idenificaion of IRF j,k = C j,k H, cd Z = H so H = Z /( ) Impose normalizaion (iv): H Z = H H H so ꞌ = Z and H = Z Z /( Z Z ) If Z is a scalar (k = ): H = E Z E Z Las revised 5/9/203 4

15 h Idenificaion of HD = Ck, jh jrequires idenificaion of H k0 Proj(Z ) = Proj(Z ) = Proj(Z ) = b where b= 2 H = Proj( ) = Proj( b ) = Proj( Proj(Z )) = Proj( Z ) =, where = Z ( Z Z) + Z (Noe Z = H has rank, so pseudo inverse is used) Las revised 5/9/203 5

16 Idenificaion of VD = var var k l0 k C l0 H j, l l C jl, l Noe his requires idenificaion of var(h ), which from las slide is var( ) = ꞌ. Las revised 5/9/203 6

17 Overidenifying Resricions () Muliple Z s for one shock: Z = H has rank. Reduced rank regression of Z ono.) (2) Z idenifies, Z 2 idenifies 2, and and 2 are uncorrelaed. his implies ha Proj(Z ) is uncorrelaed wih Proj(Z 2 ) or = 0 Z Z 2 Las revised 5/9/203 7

18 Esimaion: GMM: Noe A,, and Z are exacly idenified, so concenrae hese ou of analysis. Focus on Z and SVAR esimands. Z = E( Z ), so vec( Z ) = E(Z ) or Z = H ꞌ so ha vec( Z ) = ( H ) High level assumpion (assume hroughou) [ Z ] vec( ) Z d N(0,) Las revised 5/9/203 8

19 GMM Esimaion: (Ignore esimaion of VAR coefficiens A and hese are sraighforward o incorporae). Efficien GMM objecive funcion: J( Z ) ( Z) vec( Z) ( Z) vec( Z) = where, Z = H ꞌ. (Similarly when more han one shock is idenified). k = (exac idenificaion): ˆ Z Z k > (Homo): ˆ Z can be compued from reduced rank regression esimaor of Z ono. Las revised 5/9/203 9

20 Esimaion of H (k = ) Z = H = H, so GMM esimaor solves, Z = ˆ ˆH ˆ Z GMM esimaor: Ĥ = Z IV inerpreaion: j = H j + u j, = j Z + v j Las revised 5/9/203 20

21 4. Srong insrumen asympoics d vec( ˆ ) N (0, V ) and asympoic disribuions of all saisics Z Z of ineres follow from usual dela- mehod calculaions. Overidenified case (k > ): o usual GMM formula o J-saisics, ec. are sandard exbook GMM Las revised 5/9/203 2

22 5. Weak insrumen asympoics: k = Z (a) Disribuion of Ĥ = Z Weak IV asympoic seup local drif (limi of experimens, ec.): = = a/, so Z = H a / = / becomes ( Z ) Z Z d N(0,) (*) d N(, ) (*-weakiv) Las revised 5/9/203 22

23 Las revised 5/9/ Z d N(, ) Weak insrumen asympoics for H, cd Ĥ = /2 /2 Z Z Sandardize (*): Z Z + z, (**) where = Z and z ~ N(0,/( 2 2 Z ) ), hus, in k = case, Ĥ = Z Z z z = * H

24 Weak insrumen asympoics for H, cd Z z Ĥ = Z z Commens = * H. In he no-hac case, convergence o srong insrumen normal is governed by a / Z 2 = = noncenraliy parameer of firs-sage F For he HAC case, see Moniel Olea and Pflueger (202) Las revised 5/9/203 24

25 Las revised 5/9/ Weak insrumen asympoics for H, cd Ĥ = Z Z z z = * H Commens 2. Consider unidenified case: a = 0 so = 0 so ˆ j H = j Z Z z j z ~ (, ) j j z N df z where j = plim of OLS esimaor in he regression, j = j + j o Ĥ is median-biased owards = E( )/ 2 = he firs column of he Cholesky decomposiion wih ordered firs

26 Weak insrumen asympoics for srucural IRFs Srucural IRF: C(L)H where C(L) = [I A (L)] = C 0 + C L + C 2 L 2 + Effec on variable j of shock afer h periods: C h,j H Weak insrumen asympoic disribuion of IRF ( Aˆ A) = O p () (asympoically normal) so ˆ ˆ * CLH ( ) C(L) H Esimaor of h-sep IRF on variable j: ˆ ˆ * Ch, jh Ch, jh his won be a good approximaion in pracice need o incorporae O p ( /2 ) erm Las revised 5/9/203 26

27 Numerical resuls for IRFs asympoic disribuions DGP calibraion: r = 2 Y = (lnpoil, lngdp ), US, 959Q-20Q2 Esimae (L),, and H, hen fix hroughou o A(L), : VAR(2) o H : esimaed using Z = Kilian (2008 RESa) OPEC supply shorfall (available 97Q-2004Q3) Las revised 5/9/203 27

28 Impulse:Oil; Response:Oil Cenraliy Parameer= Quarer Effec of oil on oil growh: 2 = 00 Las revised 5/9/203 28

29 Impulse:Oil; Response:Oil Cenraliy Parameer= Quarer Effec of oil on oil growh: 2 = Las revised 5/9/203 29

30 Impulse:Oil; Response:Oil Cenraliy Parameer= Quarer Effec of oil on oil growh: 2 = 0 Las revised 5/9/203 30

31 Impulse:Oil; Response:Oil Cenraliy Parameer= Quarer Effec of oil on oil growh: 2 = 20 Las revised 5/9/203 3

32 Impulse:Oil; Response:Oil Cenraliy Parameer= Quarer Effec of oil on oil growh: 2 = 50 Las revised 5/9/203 32

33 Impulse:Oil; Response:Oil Cenraliy Parameer= Quarer Effec of oil on oil growh: 2 = 00 Las revised 5/9/203 33

34 Impulse:Oil; Response:Oil Cenraliy Parameer= Quarer Effec of oil on oil growh: 2 = 000 Las revised 5/9/203 34

35 0.2 Impulse:Oil; Response:Oupu Cenraliy Parameer= Quarer Effec of oil on GDP growh: 2 = 00 Las revised 5/9/203 35

36 0.2 Impulse:Oil; Response:Oupu Cenraliy Parameer= Quarer Effec of oil on GDP growh: 2 = Las revised 5/9/203 36

37 0.2 Impulse:Oil; Response:Oupu Cenraliy Parameer= Quarer Effec of oil on GDP growh: 2 = 0 Las revised 5/9/203 37

38 0.2 Impulse:Oil; Response:Oupu Cenraliy Parameer= Quarer Effec of oil on GDP growh: 2 = 20 Las revised 5/9/203 38

39 0.2 Impulse:Oil; Response:Oupu Cenraliy Parameer= Quarer Effec of oil on GDP growh: 2 = 00 Las revised 5/9/203 39

40 0.2 Impulse:Oil; Response:Oupu Cenraliy Parameer= Quarer Effec of oil on GDP growh: 2 = 000 Las revised 5/9/203 40

41 Weak insrumen asympoics for HD and VD Le = Z ( Z Z ) - Z HD: H = VD: var(h ) = ˆ ˆ ˆ ˆ ˆ ( ' ) ' where ˆ = /2 d Z so ha vec( ˆ ) N( vec( ), ) ˆ d Funcion of noncenral Wishar r.v.s (Anderson & Girshick (944)) Las revised 5/9/203 4

42 Empirical Resuls: Example 2: Dynamic facor model: X = F + e, [I A (L)]F =, = H Firs sage : F : regression of Z on, F 2 : regression of on Z Srucural Shock F F 2. Oil Hamilon Killian..6 Ramey-Vine Moneary policy Romer and Romer Smes-Wouers Sims-Zha GSS Produciviy Fernald FP Smes-Wouers Srucural Shock F F 2 4. Uncerainy Fin Unc (VIX) Pol Unc (BBD) Liquidiy/risk GZ EBP Spread ED Spread BCDZ Bank Loan Fiscal policy Ramey Spending Fisher-Peers.3 0. Spending Romer-Romer axes Las revised 5/9/203 42

43 Correlaions among seleced srucural shocks O K M RR M SZ P F U B U BBD S GZ B BCDZ F R F RR O K.00 M RR M SZ P F U B U BBD L GZ L BCDZ F R F RR Oil Kilian oil Kilian (2009) M RR moneary policy Romer and Romer (2004) M SZ moneary policy Sims-Zha (2006) P F produciviy Fernald (2009) U B Uncerainy VIX/Bloom (2009) U BBD uncerainy (policy) Baker, Bloom, and Davis (202) L GZ liquidiy/risk Gilchris-Zakrajšek (20) excess bond premium liquidiy/risk BCDZ (20) SLOOS shock L BCDZ F R fiscal policy Ramey (20) federal spending F RR fiscal policy Romer-Romer (200) federal ax Las revised 5/7/203 43

44 Weak insrumen asympoics for cross-shock correlaion Correlaion beween wo idenified shocks: Le Z and Z 2 be scalar insrumens ha idenify and 2 : Z Z2 Cor( 2 ) = 2 = Z Z Z Z 2 2 ˆ /2 Z d 2 N, /2 Z ˆ ˆ ˆ ˆ ˆ ZZ 2 2 r 2 = ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Z Z Z Z under null, ꞌ 2 = 0 Las revised 5/7/203 44

45 ˆ d N 2, ˆ, (hom = ZZ ) r 2 ˆ ˆ 2 ˆ ˆ ˆ ˆ 2 2 ( )( ) ( )( ) ( )( ) = 2 /, 2 = /2 Z 2 / /2 2 Z 2 = 2 ~ N(0,I), = corr( Z, Z2) corr( Z, Z2) Las revised 5/7/203 45

46 Weak insrumen asympoics for cross-shock correlaion, cd. ( )( 2 2) r 2 ( )( ) ( )( ) Commens. Nonsandard disribuion funcion of noncenral Wishar rvs 2. Normal under null as and 2 2 p 2 3. Srong insrumens under alernaive: r Las revised 5/7/203 46

47 Weak insrumen asympoics for cross-shock correlaion, cd. Numerical resuls: Asympoic null disribuion is a funcion of =, 2 2 = 2 and corr(z, Z 2 ) Las revised 5/7/203 47

48 Weak insrumen asympoics for cross-shock correlaion, cd % Criical Value λ λ 2 Weak insrumen asympoic null disribuion of r 2 : Corr(Z,Z 2 ) = 0 Las revised 5/7/203 48

49 Weak insrumen asympoics for cross-shock correlaion, cd % Criical Value λ λ 2 Weak insrumen asympoic null disribuion of r 2 : Corr(Z,Z 2 ) = 0.4 Las revised 5/7/203 49

50 Weak insrumen asympoics for cross-shock correlaion, cd. 95% Criical Value λ λ 2 Weak insrumen asympoic null disribuion of r 2 : Corr(Z,Z 2 ) = 0.8 Las revised 5/7/203 50

51 Weak insrumen asympoics for cross-shock correlaion, cd. Sup criical values (wors case over and 2 2 ): corr(z, Z 2 ) 95 % criical value go back o empirical resuls Las revised 5/7/203 5

52 Weak insrumen asympoics for reduced rank resricion Le Z and Z 2 be scalar insrumens ha idenify : Z = H ꞌ has rank ˆ /2 Z d 2 N, /2 Z ˆ where 2 = b Las revised 5/7/203 52

53 Non-HAC case: var /2 vec Z' ZZ LR = k i where i are he eigenvalues of i2 Z Z ' ' ' /2 /2 /2 /2 ZZ ZZ Weak insrumen limi /2 /2 ' /2 ' /2 ' ' ZZ Z Z ZZ where vec() ~ N(0, I r k ) and = /2 /2 ZZ ' Las revised 5/7/203 53

54 /2 /2 ' /2 ' /2 ' ' ZZ Z Z ZZ Limiing disribuion of OID es depends on vec() vec(). vec() vec() large, OID d 2 n vec() vec().= 0, OID = sum of n- smalles eigenvalues of. n = 3 (vec() vec()) /2 95% CV (= 2 3 cv) Las revised 5/7/203 54

55 6. Weak-insrumen robus inference () All objecs of ineres are funcions of Z ( = / ) ˆ /2 Z' and vec ˆ Consruc Conf. Se for : d ( ) N( vec( ), ) CS() = ˆ ( vec( ) vec( ))' ( vec( ˆ) vec( )) cv Join CS for IRF(), VD(), HD(), ec. deermined by CS() (2) Some objecs have disribuions ha depend on, say vec() vec(). Bonferroni. Las revised 5/7/203 55

56 (3) Bes unbiased ess for a single IRF: IRF = C h,j H Consider null hypohesis IRF = C h,j H = 0 wih a single Z. hen H = /, so null hypohesis is C h,j 0 = 0. A single linear resricion on. d Wih ˆ N (, ), he bes unbiased es in limiing problem rejecs for large values of sa = C ˆ ˆ hk, 0 SE( C ˆ ˆ ) hk, 0 Which can be invered o find CS for IRF (). Las revised 5/7/203 56

57 Commens his is one degree of freedom es Conf. in. inversion can be done analyically (raio of quadraics) Srong-insrumen efficien (asy equivalen o sandard GMM es) Muliple Z: he esing problem of H 0 : = 0 can be rewrien as H 0 : = 0 in he sandard IV regression form, C( 0 )ꞌ = 0 η + u η = Z + v so for muliple Z he Moreira-CLR confidence inerval can be used. (Working on efficiency improvemens) Las revised 5/7/203 57

58 Examples: IRFs: srong-iv (dashed) and weak-iv robus (solid) poinwise bands Hamilon (996, 2003) oil shock (F 2 = 5.7) Las revised 5/7/203 58

59 Kilian (2008) oil shock (F 2 =.6) Las revised 5/7/203 59

60 Romer and Romer (2004) moneary policy shock (F 2 = 2.4) Las revised 5/7/203 60

61 Conclusions Work o do includes Inference on correlaions and on ess of overid resricions in general Efficien inference for k > (beyond Moreira-CLR confidence ses) exploi equivariance resricion o lef-roaions (respecify SVAR in erms of linear combinaion of Y s his should reduce he dimension of he sufficien saisics in he limi experimen) Inference in sysems imposing uncorrelaed shocks Formally aking ino accoun higher order (O p ( /2 )) sampling uncerainy of reduced-form VAR parameers HAC (non-kronecker) case: (a) robusify; (b) efficien inference? Las revised 5/7/203 6