Identification and Inference in Structural VARs: Recent Developments

Transcription

1 Weak Insrumens and Weak Idenificaion wih Applicaions o Time Series James H. Sock, Harvard Universiy RES Easer School 202, April 5 7, 20 Universiy of Birmingham Lecure 6 Idenificaion and Inference in Srucural VARs: Recen Developmens 6-

2 Ouline ) VARs, SVARs, and he Idenificaion Problem 2) Long-run resricions: idenificaion 3) Long-run resricions: inference 4) Exernal insrumens: idenificaion 5) Exernal insrumens: inference 6-2

3 6. VARs, SVARs, and he Idenificaion Problem Wha is he effec of a policy inervenion (ineres rae increase, fiscal simulus) on macroeconomic aggregaes of ineres oupu, inflaion, ec? r Le Y be a vecor of macro ime series, and le denoe an unanicipaed moneary policy inervenion. We wan o esimae he dynamic causal effec of on Y : r Y h r, h =, 2, 3,. where he parial derivaive holds all oher inervenions consan. This dynamic causal effec is also called he impulse response funcion (IRF) of Y o he shock (unexpeced inervenion). r 6-3

4 The SVAR idenificaion problem Consider a 2-variable sysem of linear simulaneous equaions: Le and 2 be uncorrelaed srucural shocks, where E( Y, Y 2, ) = 0: Y = B 0,2 Y 2 + B,2 Y B p,2 Y 2 p + B, Y + + B p, Y p + Y 2 = B 0,2 Y + B,2 Y + + B p,2 Y p + B,22 Y B p,22 Y 2 p + 2 Given he B s, we could compue srucural impulse responses from his sysem (formulas below). Bu he coefficiens of his sysem are no idenified. To idenify hem, we eiher need an insrumen Z, or a resricion on he parameers. 6-4

5 VAR background and noaion Y = B 0,2 Y 2 + B,2 Y B p,2 Y 2 p + B, Y + + B p, Y p + Y 2 = B 0,2 Y + B,2 Y + + B p,2 Y p + B,22 Y B p,22 Y 2 p + 2 This simulaneous equaions sysem can be wrien, B(L)Y =, where B(L) = B 0 B L B 2 L 2 B p L p and in general B 0 is no diagonal. are he srucural shocks. The sysem B(L)Y = is called a srucural VAR (SVAR). The SVAR has a reduced form (Sims (980)), which is idenified: Reduced form VAR(p): Y = A Y + + A p Y p + or A(L)Y =, where A(L) = I A L A 2 L 2 A p L p innovaions: = Y Proj(Y Y -,, Y p ) E = 6-5

6 The SVAR idenificaion problem, cd. Suppose: (i) A(L) is finie order p (known or knowable) (ii) spans he space of srucural shocks, ha is, = H, H is square and inverible (iii) A(L),, and H are ime-invarian Then IRFs can be obained from he SVAR, A(L)Y = = H : SVAR: B(L)Y =, where B(L) = H A(L) Reduced form VAR: A(L)Y = MA represenaion: Y = D(L), D(L) = B(L) = A(L) H Impulse response: Y h = D h 6-6

7 The SVAR idenificaion problem, cd. Summary of VAR and SVAR noaion Reduced form VAR A(L)Y = Srucural VAR B(L)Y = Y = A(L) = C(L) A(L) = I A L A 2 L 2 A p L p E = (unresriced) E = = Y = B(L) = D(L) B(L) = B 0 B L B 2 L 2 B p L p k = H B(L) = H A(L) (B 0 = H ) D(L) = C(L)H D(L) is he srucural IRF of Y w.r... srucural forecas error variance decomposiions are compued from D(L) and 6-7

8 The SVAR idenificaion problem, cd. We focus on idenificaion of he IRF wih respec o he firs shock, Two akes on idenificaion:. Idenificaion of H (or column of H). = H = H H r r Thus idenifying H idenifies he IRF of The idenificaion ask hus is o idenify H. 6-8

9 The SVAR idenificaion problem, cd. 2.Idenificaion of shocks If you knew, you could esimae is srucural IRF of Y: u = H = H H r = H + H H r r r Because is uncorrelaed wih 2,, r, H can be recovered by he (populaion) regression of u ono. Because u = H, = H = H H r where H i is he i h row of H. Thus idenifying H idenifies he IRF of. I can be shown using he pariioned marix inverse formula ha idenificaion of H plus he assumpion ha E is diagonal idenifies H. 6-9

10 Digression on inveribiliy Recall he SVAR assumpion: (ii) spans he space of srucural shocks, ha is, = H, where R is square This is ofen called he assumpion of inveribiliy: he VAR can be invered o span he space of srucural shocks. If here are more srucural shocks han u s, hen condiion (ii) will no hold. One response is o add more variables so ha spans - see Forni, Giannone, Lippi, and Reichlin (2009) (inveribiliy in FAVAR). If agens see fuure shocks, inveribiliy fails. Or, does he definiion of shock jus become more suble (an expecaions shock)? See Lippi and Reichlin (993, 994), Sims and Zha (2006b), Fernandez-Villaverde, Rubio-Ramirez, Sargen, and Wason (2007), and Hansen and Sargen (2007). 6-0

11 6.2 Exernal Insrumens: Idenificaion The approach of exernal insrumens enails finding some exernal informaion (ouside he model) ha is relevan (correlaed wih he shock of ineres) and exogenous (uncorrelaed wih he oher shocks) Seleced references. Applicaions: Romer and Romer (989, 2004, 2008), Ramey and Shapiro (998), Ramey (2009) Mehods: Sock and Wason (2008, 202), Merens and Ravn (20). Terminological digression. We rea he following as having essenially he same meaning: Exernal insrumens Quasi-experimens Naural experimens 6-

12 Idenificaion using exernal insrumens u = H = H H r r Suppose you have an insrumenal variable Z (no in Y ) such ha E Z = 0 (relevance) (i) (ii) E jz (iii) = 0, j = 2,, r (exogeneiy) E = = I r (or diagonal) Under (i) and (ii), you can idenify H up o scale E(u Z ) = E(H Z ) = H H r E( Z ) E( rz ) = H 6-2

13 Idenificaion using exernal insrumens, cd.. Idenificaion of H E( Z ) = E(H Z ) = H H r E( Z ) E( rz ) = H The scale of H can be se by a normalizaion e.g. a uni posiive value of shock is defined o have a uni posiive effec on he innovaion o variable, which is u. This corresponds o (iv) H = (H is he firs elemen of H ) (uni shock normalizaion) 6-3

14 Idenificaion using exernal insrumens, cd. 2. Idenificaion of The shock is idenified using (iii) (recall he pariioned marix inversion argumen). I is also idenified by leing be he coefficien marix of he populaion regression of Z ono u : = EZ ( ) = HH = H ( ) = H H H H because H H = ( 0 0) Thus is idenified up o scale by u = H 6-4

15 Idenificaion using exernal insrumens, cd. Commens.Nearly all papers ha use his approach don acually do IV, hey repor reduced-form regressions of variables of ineres ono Z. In general he reduced form regressions don give you he srucural coefficiens of ineres. 2.Muliple insrumens per shock resul in overidenificaion 3.For r = 2, a single insrumen idenifies boh srucural shocks (why?). Thus if each shock has is own insrumen, he sysem is overidenified. (This generalizes o r > 2.) 6-5

16 Exernal Insrumens: Inference GMM esimaion of H Impose he normalizaion condiion H = and le H = (H H )The momen condiion is, E(u Z ) = H or E( Z = H This can be esimaed by GMM using he sample momens, ˆ T T Z Specialize o case of a single insrumen (exac idenificaion): E Z = H 2 so he GMM esimaor is, 2 Ĥ = ˆ ˆ T T T Z T Z where u = (u u ).

17 GMM esimaion of H The momen condiion is, EZ ( ) = H or EZ ( ) = This can be esimaed by GMM using he sample momens, T ˆ ˆ T Z ˆ Exac idenificaion: H is esimaed (up o scale) by he regression of Z on ˆ Overidenificaion/no-HAC. If hese momens have a Kronecker srucure (no serial correlaion/no heeroskedasiciy), he GMM esimaor simplifies o reduced rank regression: Z = ˆ +, where E( ) = 0 Overidenifying resricions can be esed by esing he reduced rank regression resricions. 6-7

18 6.4 Long Run Resricions: Idenificaion Reduced form VAR: A(L)Y = Srucural VAR: B(L)Y =, = H, B(L) = H A(L) This approach idenifies H by imposing resricions on he long run effec of one or more s on one or more Y s. Long run variance marix from VAR: = A() A() Long run variance marix from SVAR: = B() B() Digression: B() = D() is he long-run effec on Y of ; his can be seen using he Beveridge-Nelson decomposiion, Ys = D() s s + D*(L), where s * D i = j i Noaion: hink of Y as being growh raes, e.g. if Y is employmen growh, lnn, hen Ys is log employmen, lnn s D j 6-8

19 Long run resricions, cd. From VAR: From SVAR: = A() A() = B() B() = HA() A() H Sysem idenificaion by long run resricions. The SVAR is idenified if RA() A() R = () can be solved for he unknown elemens of R and.. There are k(k+)/2 disinc equaions in (), so he order condiion says ha you can esimae (a mos) k(k+)/2 parameers. If we se = I (jus a normalizaion), i is clear ha we need k 2 k(k+)/2 = k(k )/2 resricions on R. If k = 2, hen k(k )/2 =, which is delivered by imposing a single exclusion resricion (ha is, R is lower or upper riangular). This ignores rank condiions, which maer This is a momen maching approach; an IV inerpreaion comes laer 6-9

20 Long run resricions, cd. The long run neuraliy resricion. The main way long resricions are implemened in pracice is by seing = I and imposing zero resricions on D(). Imposing D ij () = 0 says ha he effec he long-run effec on he i h elemen of Y, of he j h elemen of is zero If = I, he momen equaion () can be rewrien, = D()D() (2) where D() = B(). Because RA() = B(), R is obained from D() as R = A() B(), and B(L) = RA(L) as above. Commens: If he zero resricions on D() make D() lower riangular, hen D() is he Cholesky facorizaion of. 6-20

21 Long run resricions, cd. Blanchard-Quah (989) had 2 variables (unemploymen and oupu), wih he resricion ha he demand shock has no long-run effec on he unemploymen rae. This imposed a single zero resricion, which is all ha is needed for sysem idenificaion when k = 2. King, Plosser, Sock, and Wason (99) work hrough sysem and parial idenificaion (idenifying he effec of only some shocks), hings are analogous o he parial idenificaion using shor-run iming. This approach has been a he cener of a spiried debae abou wheher echnology shocks lead o a shor-run decline in hours, based on long-run resricions (Gali (999), Chrisiano, Eichenbaum, and Vigfusson (2004, 2006), Erceg, Guerrieri, and Gus (2005), Chari, Kehoe, and McGraan (2007), Francis and Ramey (2005), Kehoe (2006), and Fernald (2007)) 6-2

22 Long run resricions, cd. In his lieraure, is esimaed using he so-called VAR-HAC esimaor, VAR-HAC esimaor of : ˆ = A() A() D() and R are esimaed as: D ˆ () = Chol( ˆ), ˆR = ˆ () ˆ D A() Commens: A recurring heme is he sensiiviy of he resuls o apparenly minor specificaion changes, in Chari, Kehoe, and McGraan s (2007) example resuls are sensiive o he lag lengh. I is unlikely ha ˆ is sensiive o specificaion changes, bu A ˆ() is much more difficul o esimae. These observaions are closely linked o he criiques by Faus and Leeper (997), Pagan and Roberson (998), Sare (997), Cooley and Dwyer (998), Wason (2006), and Gospodinov (200); we reurn o his below. An alernaive is o use medium-run resricions, see Uhlig (2004) ˆ ˆ ˆ ˆ 6-22

23 6.5 Long-run resricions: Inference Recall he esimaor of R under he long-run neuraliy condiion wih lower riangular resricions on D(): ˆR = Chol ˆ Aˆ () = ˆ ˆ Chol Aˆ () Aˆ () Aˆ () Convenional inference requires ha ˆR be consisen wih a sampling disribuion ha is well-approximaed by a normal. However, inference abou is difficul. There are wo ways o hink abou hese inference issues: as a HAC esimaor (Lecure 5) and as an IV esimaor (now). 6-23

24 IV inerpreaion of LR resricions Shapiro and Wason (988); Pagan and Roberson (998), Sare (997), Cooley and Dwyer (998); Wason (2006), Gospodinov (200) Preliminaries (i) Resricions on D() mean resricions on B(): SVAR: B(L)Y = D() = B() - Consider 2-variable VAR: D D () D () () D () = B B () B () () B () = B B () B () () B () de( B()) so D 2 () = 0 is equivalen o B 2 () = 0. Esimaion of D() wih D 2 () = 0 is equivalen o esimaion of B() wih B 2 () =

25 IV inerpreaion of LR resricions, cd. (ii) Lag manipulaion. Recall he Beveridge-Nelson decomposiion for a lag polynomial of degree p: c(l) = c() + c*(l), where c * j = p ij c i This is no unique; you can load c() on any lag, in paricular, lag p: c(l) = c()l p + c + (L), where c j = j i c i Call his he reverse BN decomposiion. 6-25

26 IV inerpreaion of LR resricions for a 2-variable SVAR Le B(L) = b( L) b2( L) b2( L) b22( L) so B(L)Y = becomes, Y = b 2 (L)Y 2 + b (L)Y + Y 2 = b 2 (L)Y + b 22 (L)Y Apply he reverse BN decomposiion : Y = b 2 ()Y 2 p + b 2 ( L) Y 2 + b (L)Y + Y 2 = b 2 ()Y p + b 2 ( L) Y + b 22 (L)Y Impose he long-run neuraliy resricion D 2 () = 0, i.e. b 2 () = 0: Y = b2( L) Y 2 + b (L)Y + Y 2 = b 2 ()Y p + b 2 ( L) Y + b 22 (L)Y

27 IV inerpreaion of LR resricions for a 2-variable SVAR, cd Y = b 2 ()Y 2 p + b 2 ( L) Y 2 + b (L)Y + (3) Y 2 = b 2 ()Y p + b 2 ( L) Y + b 22 (L)Y (4) The long-run resricion b 2 ()=0 implies an exclusion resricion: Y 2 p doesn appear in (3), bu i does appear in (4). Thus: he coefficien b 2,0 on Y 2 in (3) can be esimaed by IV, using Y 2 p as an insrumen for Y 2. Because Y,, Y p+ appear as regressors in (3), his is equivalen o: he coefficien b 2,0 on Y 2 in (3) can be esimaed by IV, using Y 2 as an insrumen for Y

28 IV inerpreaion of LR resricions for a 2-variable SVAR, cd Weak insrumen inerpreaion Is Y 2 a weak or srong insrumen? Firs-sage regression: regress Y 2 on Y 2, Y 2, Y 2 2,, Y, Y 2, Back of he envelope calculaion: approximae Y 2 as he AR(), Approximaion: Y 2 = Y 2 + or Y 2 = ( )Y 2 + (5) In IV noaion: Y = Z + v Concenraion parameer: 2 = ZZ/ Translaed o he firs-sage regression (5): E 2 = ( ) 2 2 var(y 2 )T/ = 2 2 v 2 T = T 6-28

29 IV inerpreaion of LR resricions for a 2-variable SVAR, cd Values of 2 = T for T = 00: These are probably bes-case numbers, in higher order ARs and in VARs he marginal conribuion of Y 2 given addiional lags would be less In he local o uni case (very persisen), 2 = O p () random variable (Gospodinov (200)) (bu noe, Y is supposed o be saionary) Some simulaions from Pagan and Roberson (998) of esimaed long-run effecs: 6-29

30 MC resuls of long run effecs esimaed by imposing LR neuraliy resricions, from Pagan and Roberson (998) 6-30

31 IV inerpreaion of LR resricions cd Commens The IV inerpreaion and he esimaion inerpreaion boh sugges ha (in some applicaions) here can be considerable sensiiviy o sample period and especially lag lengh which we would expec if idenificaion is weak. Wheher his is an issue depends on he amoun of persisence. If persisence is small, Y 2 will be a sronger insrumen (and will be easier o esimae) Some pracical advice: perform a MC simulaion and don rus boosrap SEs wihou checking in a MC Francis, Owyang, and Roush (2005) change he infinie-run resricion o a finie long-run resricion using Faus s algorihm a sensible approach worh following up. More work is sill needed (especially ools for handling weak IVs) 6-3