Recent Developments in Structural VAR Modeling

Transcription

1 NBER Summe Insiue Wha s New in Economeics: Time Seies Lecue 7 July 15, 2008 Recen Developmens in Sucual VAR Modeling Revised July 23,

2 Ouline 1) VARs, SVARs, and he Idenificaion Poblem 2) Idenificaion by Sho Run Resicions 3) Idenificaion by Long Run Resicions 4) Idenificaion by Sign Resicions 5) Idenificaion fom Heeoskedasiciy 6) DSGE Pios 7) Idenificaion fom Regional/Mulicouny Resicions 8) Infeence: Challenges and Recenly Developed Tools Revised July 23,

3 1) VARs, SVARs, and he Idenificaion Poblem A classic quesion in empiical macoeconomics: wha is he effec of a policy inevenion (inees ae incease, fiscal simulus) on macoeconomic aggegaes of inees oupu, inflaion, ec? Le Y be a veco of maco ime seies, and le ε denoe an unanicipaed moneay policy inevenion. We wan o know he dynamic causal effec of ε on Y : Y+ h, h = 1, 2, 3,. ε whee he paial deivaive holds all ohe inevenions consan. In maco, his dynamic causal effec is called he impulse esponse funcion (IRF) of Y o he shock (unexpeced inevenion) ε. The challenge is o esimae Y ε + h fom obsevaional maco daa. Revised July 23,

4 Two concepual appoaches o esimaing dynamic causal effecs (IRF) 1) Sucual model (Cowles Commission) a) ighly paameeized (many esicions): FMP,, DSGE b) Sucual veco auoegessions (SVARs) 2) Naual expeimens The idenificaion poblem Conside a 2-vaiable sysem of linea simulaneous equaions: Le ε 1 and ε 2 be uncoelaed sucual shocks, whee E(ε Y 1, Y 2, ) = 0: Y 1 = B 0,12 Y 2 + B 1,12 Y B p,12 Y 2 p + B 1,11 Y B p,11 Y 1 p + ε 1 Y 2 = B 0,21 Y 1 + B 1,21 Y B p,21 Y 1 p + B 1,22 Y B p,22 Y 2 p + ε 2 Given he B s, we could compue sucual impulse esponses fom his sysem (fomulas below). Bu he coefficiens of his sysem ae no idenified. To idenify hem, we eihe need an insumen Z, o a esicion on he paamees. Revised July 23,

5 VAR backgound and noaion: Y 1 = B 0,12 Y 2 + B 1,12 Y B p,12 Y 2 p + B 1,11 Y B p,11 Y 1 p + ε 1 Y 2 = B 0,21 Y 1 + B 1,21 Y B p,21 Y 1 p + B 1,22 Y B p,22 Y 2 p + ε 2 This simulaneous equaions sysem can be wien, B(L)Y = ε, whee B(L) = B 0 B 1 L B 2 L 2 B p L p and in geneal B 0 is no diagonal. ε ae he sucual shocks. The sysem B(L)Y = ε is called a sucual VAR (SVAR). This SVAR has a educed fom (Sims (1980)), which is idenified: Reduced fom VAR(p): Y = A 1 Y A p Y p + u o A(L)Y = u, whee A(L) = I A 1 L A 2 L 2 A p L p innovaions: u = Y Poj(Y Y -1,, Y p ) Eu u = Σ u Revised July 23,

6 Reduced fom o sucue: Suppose: (i) A(L) is finie ode p (known o knowable) (ii) u spans he space of sucual shocks ε, ha is, ε = Ru, whee R is squae (implicily his is assuming ha Y is linea in he sucual shocks) (iii) A(L), Σ u, and R ae ime-invaian, e.g. A(L) is invaian o policy changes ove he elevan peiod Then IRFs can be obained fom he SVAR, RA(L)Y = Ru o B(L)Y = ε : SVAR: B(L)Y = ε, whee B(L) = RA(L) and Ru = ε Reduced fom VAR: A(L)Y = u MA epesenaion: Y = D(L)ε, D(L) = B(L) 1 = A(L) 1 R 1 Y + h Impulse esponse: ε = D h Revised July 23,

7 Summay of VAR and SVAR noaion Reduced fom VAR A(L)Y = u Sucual VAR B(L)Y = ε Y = A(L) 1 u = C(L)u Y = B(L) 1 ε = D(L)ε A(L) = I A 1 L A 2 L 2 A p L p B(L) = B 0 B 1 L B 2 L 2 B p L p Eu u = Σ u (unesiced) Eε ε = Σ ε = Ru = ε B(L) = RA(L) (B 0 = R) D(L) = C(L)R 1 σ σ k Noe he assumpion ha he sucual shocks ae uncoelaed D(L) is he sucual IRF of Y w... ε. vaiance decomposiions w... ε ae compued fom D(L) and Σ ε Revised July 23,

8 Idenificaion of R and idenificaion of shocks Two akes on idenificaion: 1. Idenificaion of R. In populaion, we can obseve A(L). If we can idenify R, we can obain he SVAR coefficiens, B(L) = RA(L). 2. Idenificaion of shocks. If you knew (o could esimae) one of he shocks, you could esimae he sucual IRF of Y w... ha shock. Paiion Y ino a policy vaiable and all ohe vaiables: Y = ( k 1 1) X (1 1), u = u u The IRF/MA fom is Y = D(L)ε, o X, ε = X ε Y = ( DYX ( L) DY ( L) ) ε = D Y (L) ε + v, whee v = D YX (L) ε X. Because E ε v = 0, he IRF of Y w... ε, D Y (L) is idenified by he populaion OLS egession of Y ono ε ε X, ε. Revised July 23,

9 The naual expeimen appoach Rome and Rome (1989, 2004, 2008); Ramey and Shapio (1998); Ramey (2008). Suppose you have an insumenal vaiable Z (no in Y ) such ha (i) EZ u 0 (elevance) X (ii) EZ ε = 0 (exogeneiy) Then you can idenify (esimae) ε. To show his, again, paiion Y : Y = ( k 1 1) X (1 1), u = u u so Ru = ε becomes: R XX u = R X + o X R u = R X X u u + X u = R 1 R X + X XX u u = R 1 R X u + X, ε = X ε ε X ε ε, and R = 1 R X XX ε 1 R R R XX X R R X ε whee R is a scala Revised July 23,

10 The naual expeimen appoach, cd. u = R 1 R X + X XX u u = R 1 R X u + X 1 R XX 1 R ε X (1) ε (2) Suppose Z (no in Y ) is such ha (i) EZ u 0 (elevance) (ii) EZ ε = 0 (exogeneiy) X Then Z can be used as an insumen fo u in (1): (i) Esimae R 1 XX R X by IV esimaion of (1) (ii) Esimae X ε = 1 R XX ε as ˆ X X ε = (iii) Use ε ˆ X as an insumen fo (iv) Esimae ε = 1 R u ε as + X u + R 1 R X XX X u u in (2) o esimae R 1 R X R R 1 X X u : his delives ε up o scale. (v) Impulse esponses can be compued by egessing Y on ε, 1 ε, Revised July 23,

11 The naual expeimen appoach, cd. Commens 1.The IV appoach oulined above needs one IV o idenify one shock. Moe han one Z pe shock yields ove idenificaion. 2.The papes ha use his appoach don acually do IV, hey epo educed-fom egessions of vaiables of inees ono Z (Rome and Rome (1989, 2004, 2008); Ramey and Shapio (1998); Ramey (2008)). In geneal he educed fom egessions don give you he sucual coefficiens of inees. On he ohe hand one eason fo choosing educed fom ove IV is ha hee migh be heeogeneiy in eamen effecs of diffeen ypes of shocks. In his case he IV esimao using Z would have he addiional complicaion ha he Revised July 23,

12 The naual expeimen appoach, cd. local aveage eamen effec (LATE) fo he Z used in he sudy would diffe fom he aveage eamen effec, o fom he LATE using a diffeen insumen. This se of issues has eceived a lo of aenion in he pogam evaluaion lieaue bu almos no aenion in empiical maco. 3.This naual expeimen appoach is no efeed o by his em in he maco SVAR lieaue, i is ypically called he naaive appoach because i is based on uning ex-based infomaion ino quaniaive infomaion Implemenaion. The logic howeve is based on he logic in he naual expeimen appoach in micoeconomeics. Revised July 23,

13 The SVAR appoach Benanke (1986), Blanchad and Wason (1986), Sims (1986) Sysem idenificaion. In geneal, he SVAR is fully idenified if RΣ u R = Σ ε (3) can be solved fo he unknown elemens of R and Σ ε.. Thee ae k(k+1)/2 disinc equaions in (3), so he ode condiion says ha you can esimae (a mos) k(k+1)/2 paamees. If we se Σ ε = I (jus a nomalizaion), i is clea ha we need k 2 k(k+1)/2 = k(k 1)/2 esicions on R. If k = 2, hen k(k 1)/2 = 1, which is deliveed by imposing a single esicion (commonly, ha R is lowe o uppe iangula). This ignoes ank condiions, which mae! This descipion of idenificaion is via mehod of momens (equaion (3)), howeve idenificaion can equally be descibed via IV, e.g. see Blanchad and Wason (1986). Revised July 23,

14 The SVAR appoach, cd. Paial idenificaion. Many applicaions now ake a limied infomaion appoach, in which only a ow of R is idenified. Paiion ε = Ru, and paiion Y so ha: ε ε X = R R XX X R R X u u X (4) If R X and R ae idenified, hen (in populaion) ε can be compued using jus he final ow of (4), and D Y (L) can be compued by he egession of Y on ε, ε 1, as discussed above. Revised July 23,

15 A wod on inveibiliy: Blaschke Recall he SVAR assumpion: (ii) u spans he space of sucual shocks ε, ha is, ε = Ru, whee R is squae This is ofen called he assumpion of inveibiliy: he VAR can be inveed o span he space of sucual shocks. If hee ae moe sucual shocks han u s, hen condiion (ii) will no hold. One esponse is o add moe vaiables so ha u spans ε. This esponse is an impoan moivaion of he FAVAR appoach, which will be discussed in Lecue 12. See Lippi and Reichlin (1993, 1994), Sims and Zha (2006b), Fenandez-Villavede, Rubio-Ramiez, Sagen, and Wason (2007), and Hansen and Sagen (2007). Revised July 23,

16 This alk Ealy pomise of SVARs and he ciiques of he 1990s Suvey of new ideas abou how o ackle he idenificaion poblem Issues of infeence, new and old, including some ools Commens and efeences This is a maue lieaue I will suvey he developmens in he pas 10 yeas o so on idenificaion and infeence, focusing on he economeic issues Some geneal backgound efeences: Chisiano, Eichenbaum, and Evans (1999) Lükepohl (2005) Sock and Wason (2001) Wason (1994) Revised July 23,

17 Ouline 1) VARs, SVARs, and he Idenificaion Poblem 2) Idenificaion by Sho Run Resicions 3) Idenificaion by Long Run Resicions 4) Idenificaion by Sign Resicions 5) Idenificaion fom Heeoskedasiciy 6) DSGE Pios 7) Idenificaion fom Regional/Mulicouny Resicions 8) Infeence: Challenges and Recenly Developed Tools Revised July 23,

18 2) Idenificaion by Sho Run Resicions This exposiion follows CEE (1999). Paiion Y as, X S Y = X f The benchmak iming idenificaion assumpion is S S ε RSS 0 0 u ε = RS R 0 u f ε R fs Rf R f ff u o S slow-moving vaiables: u = policy insumen: u = fas-moving vaiable: f 1 R SS ε 1 S R S R S + u 1 R ε u = R 1 S R fs + R R f + ff u 1 ff u 1 R f ff ε Revised July 23,

19 Idenificaion by Sho Run Resicions, cd. S slow-moving vaiables: u = policy insumen: u = fas-moving vaiable: The space spanned by S egessing on. u u f 1 R SS ε 1 S R S R S + u 1 R ε u = R 1 S R fs + R R f + ff u 1 ff u 1 R f ff ε S ε is spanned by (is idenified as) he esidual fom Seleced ciicisms of iming esicions (Rudebusch (1998), ohes) he implici policy eacion funcion doesn accod wih heoy o pacical expeience Implemenaions ofen ignoe changes in policy eacion funcions quesionable cedibiliy of lack of in-peiod esponse of X s o VAR infomaion is ypically fa less han sandad infomaion ses Esimaed moneay policy shocks don mach fuues make daa Revised July 23,

20 Using high fequency daa o esimae he moneay shock Recall, Y = ( D ( L) D ( L) ) YX whee v = D YX (L) ε X X ε Y ε = D Y (L) ε + v,, so if you obseved ε you could esimae D Y (L). A vaian on sho-un iming idenificaion is o esimae ε diecly fom daily daa on moneay announcemens o policy-induced FF ae changes Cochane and Piazessi (2002) aggegaes daily ε (Euodolla ae changes afe FOMC announcemens) o a monhly ε seies Faus, Swanson, and Wigh ( ) esimaes IRF of w ε fom fuues make, hen maches his o a monhly VAR IRF (esuls in se idenificaion discuss lae) Benanke and Kune (2005) Revised July 23,

21 Ouline 1) VARs, SVARs, and he Idenificaion Poblem 2) Idenificaion by Sho Run Resicions 3) Idenificaion by Long Run Resicions 4) Idenificaion by Sign Resicions 5) Idenificaion fom Heeoskedasiciy 6) DSGE Pios 7) Idenificaion fom Regional/Mulicouny Resicions 8) Infeence: Challenges and Recenly Developed Tools Revised July 23,

22 3) Idenificaion by Long Run Resicions Reduced fom VAR: A(L)Y = u Sucual VAR: B(L)Y = ε, Ru = ε, B(L) = RA(L) This appoach idenifies R by imposing esicions on he long un effec of one o moe ε s on one o moe Y s. Long un vaiance maix fom VAR: Ω = A(1) 1 Σ u A(1) 1 Long un vaiance maix fom SVAR: Ω = B(1) 1 Σ ε B(1) 1 Digession: B(1) 1 = D(1) is he long-un effec on Y of ε ; his can be seen using he Beveidge-Nelson decomposiion, Ys = D(1) ε s + D*(L)ε s= 1 s= 1 Noaion: hink of Y as being gowh aes, e.g. ΔlnRGDP ; e.g. if Y is employmen gowh, ΔlnN, hen Ys is log employmen, lnn s= 1 Revised July 23,

23 Long un esicions, cd. Fom VAR: Ω = A(1) 1 Σ u A(1) 1 Fom SVAR: Ω = B(1) 1 Σ ε B(1) 1 = RA(1) 1 Σ ε A(1) 1 R Sysem idenificaion by long un esicions. The SVAR is idenified if RA(1) 1 Σ ε A(1) 1 R = Ω (5) can be solved fo he unknown elemens of R and Σ ε.. Thee ae k(k+1)/2 disinc equaions in (5), so he ode condiion says ha you can esimae (a mos) k(k+1)/2 paamees. If we se Σ ε = I (jus a nomalizaion), i is clea ha we need k 2 k(k+1)/2 = k(k 1)/2 esicions on R. If k = 2, hen k(k 1)/2 = 1, which is deliveed by imposing a single exclusion esicion (ha is, R is lowe o uppe iangula). This ignoes ank condiions, which mae This is a momen maching appoach; an IV inepeaion comes lae Revised July 23,

24 Long un esicions, cd. The long un neualiy esicion. The main way long esicions ae implemened in pacice is by seing Σ ε = I and imposing zeo esicions on D(1). Imposing D ij (1) = 0 says ha he effec he long-un effec on he i h elemen of Y, of he j h elemen of ε is zeo If Σ ε = I, he momen equaion (5) can be ewien, Ω = D(1)D(1) (6) whee D(1) = B(1) 1. Because RA(1) = B(1), R is obained fom D(1) as R = A(1) 1 B(1), and B(L) = RA(L) as above. Commens: If he zeo esicions on D(1) make D(1) lowe iangula, hen D(1) is he Cholesky facoizaion of Ω. Revised July 23,

25 Long un esicions, cd. Blanchad-Quah (1989) had 2 vaiables (unemploymen and oupu), wih he esicion ha he demand shock has no long-un effec on he unemploymen ae. This imposed a single zeo esicion, which is all ha is needed fo sysem idenificaion when k = 2. King, Plosse, Sock, and Wason (1991) wok hough sysem and paial idenificaion (idenifying he effec of only some shocks), hings ae analogous o he paial idenificaion using sho-un iming. This appoach has been a he cene of a spiied debae abou whehe echnology shocks lead o a sho-un decline in hous, based on long-un esicions (Gali (1999), Chisiano, Eichenbaum, and Vigfusson (2004, 2006), Eceg, Gueiei, and Gus (2005), Chai, Kehoe, and McGaan (2007), Fancis and Ramey (2005), Kehoe (2006), and Fenald (2007)) Revised July 23,

26 Long un esicions, cd. In his lieaue, Ω is esimaed using he so-called VAR-HAC esimao, VAR-HAC esimao of Ω: ˆΩ = 1 1 A(1) Σ A(1) ˆ ˆ ˆ u 1 D(1) and R ae esimaed as: D ˆ (1) = Chol( ˆΩ), ˆR = ˆ (1) ˆ D A(1) Commens: A ecuing heme is he sensiiviy of he esuls o appaenly mino specificaion changes, in Chai, Kehoe, and McGaan s (2007) example esuls ae sensiive o he lag lengh. I is unlikely ha Σ ˆ u is sensiive o specificaion changes, bu A ˆ(1) is much moe difficul o esimae. These obsevaions ae closely linked o he ciiques by Faus and Leepe (1997), Pagan and Robeson (1998), Sae (1997), Cooley and Dwye (1998), Wason (2006), and Gospodinov (2008); we eun o his below. An alenaive is o use medium-un esicions, see Uhlig (2004) Revised July 23,

27 Ouline 1) VARs, SVARs, and he Idenificaion Poblem 2) Idenificaion by Sho Run Resicions 3) Idenificaion by Long Run Resicions 4) Idenificaion by Sign Resicions 5) Idenificaion fom Heeoskedasiciy 6) DSGE Pios 7) Idenificaion fom Regional/Mulicouny Resicions 8) Infeence: Challenges and Recenly Developed Tools Revised July 23,

28 4) Idenificaion by Sign Resicions Conside esicions of he fom: a moneay policy shock does no decease he FF ae fo monhs 1,,6 does no incease inflaion fo monhs 6,..,12 These ae esicions on he sign of elemens of D(L). Sign esicions can be used o se-idenify D(L). Le D denoe he se of D(L) s ha saisfy he esicion. Thee ae cuenly hee ways o handle sign esicions: 1.Faus s (1998) quadaic pogamming mehod 2.Uhlig s (2005) Bayesian mehod 3.Uhlig s (2005) penaly funcion mehod I will descibe #2 (he fis seps ae he same as #3) Revised July 23,

29 Sign esicions, cd. SVAR idenificaion: RΣ u R = Σ ε Nomalize Σ ε = I; hen Σ u = R 1 R 1 One saemen of he SVAR idenificaion poblem is ha, wihou addiional esicions, R is idenified only up o a oaion. Le Chol(Σ u ) so 1 R 1 c c R = Σ u. Then i is also ue ha R = 1 c Σ u = R 1 HH R 1 c c fo any ohonomal maix H. Le R 1 = R H, o 1 c R = H 1 R c. Then R is also a soluion o RΣ u R = Σ ε. Revised July 23,

30 Sign esicions, cd. Uhlig s algoihm (sligh modificaion): (i) Daw H andomly fom he space of ohonomal maices (ii) Compue R = H 1 (iii) Compue he IRF (iv) If DL ( DL ( R c R 1 DL ( ) = C(L) ) D, discad his ial ) D, eain R hen go o (i) = A(L) 1 1 R H c R and go o (i). Ohewise, if (v) Compue he poseio (using a pio on A(L) and Σ u, plus he eained R s) and conduc Bayesian infeence, e.g. compue poseio mean (inegae ove A(L), Σ u, and he eained R s), compue cedible ses (Bayesian confidence ses), ec. This algoihm implemens Bayes infeence using a pio popoional o π(a(l), Σ u ) 1( DL ( ) D)μ(H) whee μ(h) is he disibuion fom which H is dawn. Revised July 23,

31 Sign esicions, cd. Sign esicion pio: π(a(l), Σ u ) 1( DL ( ) D)μ(H) Commens: This pocedue esuls in se idenificaion. This aises difficul issues fo infeence and is an acive aea of eseach in economeic heoy. Fom a fequenis pespecive, he idenified se is esimaed by he collecion of nonejeced impulse esponses, { DL ˆ ( ) = ˆ 1 CL ( ) R H : DL ˆ ( ) D} This se is eadily compued by saving he nonejeced DL ˆ ( ) s. c H s ha esul in The nonejeced DL s ˆ ( ) ae no daws ha can be used o compue fequenis confidence bands hey all fall in he se of poin esimaes! Fom a Bayes pespecive, he choice of μ maes (is μ infomaive?) Revised July 23,

32 Sign esicions, cd. Fequenis infeence abou esimaion of he se is difficul i equies imagining wha he se would have been, had diffeen ( A ˆ( L ), Σˆ u ) s been ealized. A confidence se fo he idenified se is a se-valued funcion of he daa ha conains he ue idenified se in 95% of all ealizaions. Mehodologically elaed example se idenificaion. Faus, Swanson, and Wigh (2004) have a elaed idenificaion scheme no sign esicions ha also gives se idenificaion (hey equie seleced SVAR IRFs o mach IRFs based on high fequency asse daa). Thei appoach also delives se idenificaion. They ake a sab a pefoming infeence abou he idenified se iself. As an illusaion look fis a hei esimaes of he idenified se (figue 3), hen hei confidence inevals fo he idenified se: Revised July 23,

33 Esimaes of idenified ses (Faus, Swanson, Wigh (2003), Fig 3) Revised July 23,

34 Confidence ses fo idenified ses (Faus, Swanson, Wigh (2003), Fig 4) Revised July 23,

35 Ouline 1) VARs, SVARs, and he Idenificaion Poblem 2) Idenificaion by Sho Run Resicions 3) Idenificaion by Long Run Resicions 4) Idenificaion by Sign Resicions 5) Idenificaion fom Heeoskedasiciy 6) DSGE Pios 7) Idenificaion fom Regional/Mulicouny Resicions 8) Infeence: Challenges and Recenly Developed Tools Revised July 23,

36 5) Idenificaion fom Heeoskedasiciy Suppose: (a) The sucual shock vaiance beaks a dae s: Σ ε,1 befoe, Σ ε,2 afe (b) R doesn change beween vaiance egimes (c) nomalize R o have 1 s on he diagonal, bu no ohe esicions; hus he unknowns ae: R (k 2 k); Σ ε,1 (k), and Σ ε,2 (k). Fis peiod: RΣ u,1 R = Σ ε,1 k(k+1)/2 equaions, k 2 unknowns Second peiod: RΣ u,2 R = Σ ε,2 k(k+1)/2 equaions, k moe unknowns Numbe of equaions = k(k+1)/2 + k(k+1)/2 = k(k+1) Numbe of unknowns = k 2 + k = k(k+1) Rigobon (2003), Rigobon and Sack (2003, 2004) ARCH vesion by Senana and Fioenini (2001) Revised July 23,

37 Revised July 23,

38 Idenificaion fom Heeoskedasiciy,cd. Commens: 1. Thee is a ank condiion hee oo fo example, idenificaion will no be achieved if Σ ε,1 and Σ ε,2 ae popoional. 2. The beak dae need no be known as long as i can be esimaed consisenly 3. Diffeen inuiion: suppose only one sucual shock is homoskedasic. Then find he linea combinaion wihou any heeoskedasiciy! 4. This idea also can be implemened exploiing condiional heeoskedasiciy (Senana and Fioenini (2001)) 5. Bu, some cauionay noes: a. R mus emain consan despie change in Σ ε (hink abou i ) b.song idenificaion will come fom lage diffeences in vaiances Revised July 23,

39 Ouline 1) VARs, SVARs, and he Idenificaion Poblem 2) Idenificaion by Sho Run Resicions 3) Idenificaion by Long Run Resicions 4) Idenificaion by Sign Resicions 5) Idenificaion fom Heeoskedasiciy 6) DSGE Pios 7) Idenificaion fom Regional/Mulicouny Resicions 8) Infeence: Challenges and Recenly Developed Tools Revised July 23,

40 6) DSGE Pios Use pios based on a DSGE owads which o shink he VAR paamees. Fo example compue he appoximae VAR(p) implied by he DSGE, use hese as poin esimaes fo B(L), and cene conjugae pio a hose. Moe on his in Lecue 8. Seleced efeences Ingam and Whieman (1994) Del Nego and Schofheide (2004) Del Nego, Schofheide, Smes, and Woues (2004) Revised July 23,

41 Ouline 1) VARs, SVARs, and he Idenificaion Poblem 2) Idenificaion by Sho Run Resicions 3) Idenificaion by Long Run Resicions 4) Idenificaion by Sign Resicions 5) Idenificaion fom Heeoskedasiciy 6) DSGE Pios 7) Idenificaion fom Regional/Mulicouny Resicions 8) Infeence: Challenges and Recenly Developed Tools Revised July 23,

42 7) Idenificaion fom Regional/Mulicouny Resicions The idea is o impose esicions aising fom couny boundaies: (a) disinguish beween couny-specific and common shocks, e.g. using a faco sucue; (b) impose addiional esicions aising fom ansmission via ade shaes Seleced efeences Canova and Ciccaelli (2008) Dees, di Mauo, Pesaan, and Smih (2007) (many ohe Pesaan/Smih) Ellio and Faás (1996) Nobin and Schlagenhauf (1996) Sock and Wason (2005) Revised July 23,

43 Ouline 1) VARs, SVARs, and he Idenificaion Poblem 2) Idenificaion by Sho Run Resicions 3) Idenificaion by Long Run Resicions 4) Idenificaion by Sign Resicions 5) Idenificaion fom Heeoskedasiciy 6) DSGE Pios 7) Idenificaion fom Regional/Mulicouny Resicions 8) Infeence: Challenges and Recenly Developed Tools Revised July 23,

44 8) Infeence: Challenges and Recenly Developed Tools Two opics: (a) Infeence using long-un esicions (b) Infeence abou IRFs Infeence using long-un esicions Recall he esimao of R unde he long-un neualiy condiion wih lowe iangula esicions on D(1): ˆR = Chol ( Ω) ˆ Aˆ (1) 1 = ( ) 1ˆ 1 Σ u Chol Aˆ(1) Aˆ(1) Aˆ(1) Convenional infeence equies ha ˆR be consisen wih a sampling disibuion ha is well-appoximaed by a nomal. Howeve, infeence abou Ω is difficul. Thee ae wo ways o hink abou hese infeence issues: as a HAC esimao (Lecue 9) and as an IV esimao (now). 1 Revised July 23,

45 IV inepeaion of LR esicions Shapio and Wason (1988); Pagan and Robeson (1998), Sae (1997), Cooley and Dwye (1998); Wason (2006), Gospodinov (2008) Peliminaies (i) Resicions on D(1) mean esicions on B(1): SVAR: B(L)Y = ε D(1) = B(1) -1 Conside 2-vaiable VAR: D D (1) D (1) (1) D (1) = B B (1) B (1) (1) B (1) = B B (1) B (1) (1) B (1) de( B(1)) so D 12 (1) = 0 is equivalen o B 12 (1) = 0. Esimaion of D(1) wih D 12 (1) = 0 is equivalen o esimaion of B(1) wih B 12 (1) = 0. Revised July 23,

46 IV inepeaion of LR esicions, cd. (ii) Lag manipulaion. Recall he Beveidge-Nelson decomposiion fo a lag polynomial of degee p: c(l) = c(1) + c*(l)δ, whee c * j = p i= j+ 1 c i This is no unique; you can load c(1) on any lag, in paicula, lag p: c(l) = c(1)l p + c + (L)Δ, whee c + j = j i= 1 c i Call his he evese BN decomposiion. Revised July 23,

47 IV inepeaion of LR esicions fo a 2-vaiable SVAR Le B(L) = 1 b11( L) b12( L) b21( L) 1 b22( L) so B(L)Y = ε becomes, Y 1 = b 12 (L)Y 2 + b 11 (L)Y ε 1 Y 2 = b 21 (L)Y 1 + b 22 (L)Y ε 2 Apply he evese BN decomposiion : + Y 1 = b 12 (1)Y 2 p + b L) ΔY 2 + b 11 (L)Y ε 1 12( + Y 2 = b 21 (1)Y 1 p + b L) ΔY 1 + b 22 (L)Y ε 2 21( Impose he long-un neualiy esicion D 12 (1) = 0, i.e. b 12 (1) = 0: + Y 1 = b L) ΔY 2 + b 11 (L)Y ε 1 12( + Y 2 = b 21 (1)Y 1 p + b L) ΔY 1 + b 22 (L)Y ε 2 21( Revised July 23,

48 IV inepeaion of LR esicions fo a 2-vaiable SVAR, cd + Y 1 = b 12 (1)Y 2 p + b L) ΔY 2 + b 11 (L)Y ε 1 (7) 12( + Y 2 = b 21 (1)Y 1 p + b L) ΔY 1 + b 22 (L)Y ε 2 (8) 21( The long-un esicion b 12 (1)=0 implies an exclusion esicion: Y 2 p doesn appea in (7), bu i does appea in (8). Thus: he coefficien b + 12,0 on ΔY 2 in (7) can be esimaed by IV, using Y 2 p as an insumen fo ΔY 2. Because ΔY 1,, ΔY p+1 appea as egessos in (7), his is equivalen o: he coefficien b + 12,0 on ΔY 2 in (7) can be esimaed by IV, using Y 2 1 as an insumen fo ΔY 2. Revised July 23,

49 IV inepeaion of LR esicions fo a 2-vaiable SVAR, cd Weak insumen inepeaion Is Y 2 1 a weak o song insumen? Fis-sage egession: egess ΔY 2 on Y 2 1, ΔY 2 1, ΔY 2 2,, Y 1 1, Y 1 2, Back of he envelope calculaion: appoximae Y 2 as he AR(1), Appoximaion: Y 2 = αy ε 1 o ΔY 2 = (α 1)Y ε 1 (9) In IV noaion: Y = ZΠ + v Concenaion paamee: μ 2 = Π Z ZΠ/ σ Tanslaed o he appoximae egession (9): Eμ 2 = (α 1) 2 2 va(y 2 1 ) T/ σ ε = ( α 1 )2 T α Revised July 23, v

50 IV inepeaion of LR esicions fo a 2-vaiable SVAR, cd α 1 Values of μ 2 = ( )2 2 1 α α μ T fo T = 100: These ae pobably bes-case numbes, in highe ode ARs and in VARs he maginal conibuion of Y 2 1 given addiional lags would be less In he local o uni case (vey pesisen), μ 2 = O p (1) andom vaiable (Gospodinov (2008)) Some simulaions fom Pagan and Robeson (1998) of esimaed long-un effecs: Revised July 23,

51 MC esuls of long un effecs esimaed by imposing LR neualiy esicions, fom Pagan and Robeson (1998) Revised July 23,

52 IV inepeaion of LR esicions cd Commens The IV inepeaion and he Ω esimaion inepeaion boh sugges ha (in some applicaions) hee can be consideable sensiiviy o sample peiod and especially lag lengh which we would expec if idenificaion is weak. Whehe his is an issue depends on he amoun of pesisence. If pesisence is small, Y 2 will be a songe insumen (and Ω will be easie o esimae) Some pacical advice: pefom a MC simulaion and don us boosap SEs wihou checking in a MC Fancis, Owyang, and Roush (2005) change he infinie-un esicion o a finie long-un esicion using Faus s algoihm a sensible appoach woh following up. Moe wok is sill needed (especially ools fo handling weak IVs) Revised July 23,

53 (b) Confidence Inevals fo IRFs The goal is o povide confidence inevals fo IRFs ha povide he saed coveage ae and ae as igh as possible. The naual saing poin is fis ode asympoic heoy. Remembe he dela mehod? If T ( ˆ d θ θ 0 ) N(0, Σ ) and if g( ) has coninuous deivaives, hen ˆ θ T [g( ˆ g θ ) g(θ 0 )] T ( ˆ d g g θ θ 0 ) N 0, Σ ˆ θ θ θ θ 0 θ θ 0 θ 0 Fo his o povide a good appoximaion, ˆ θ should be nealy nomal o sa wih and g mus be essenially linea ove mos of he mass of he sampling disibuion of ˆ θ. Fo SVAR IRFs, ˆ θ = ( A ˆ( L ), ˆR), and g( ˆ θ ) = DL ˆ ( ) = ˆ( ) A L 1 ˆR 1. Revised July 23,

54 Confidence Inevals fo IRFs, cd. ˆ ˆ( θ = ( A L ), ˆR), and g(θ ) = DL) = ˆ( ) ˆ ˆ ( A L 1 ˆR 1 In SVAR applicaions hee ae wo main poblems wih he dela mehod: 1.The funcion g is vey nonlinea so ha even if Aˆ ( L ) wee exacly nomally disibued, he impulse esponse funcions migh no be. Le 4 8 ˆα ~ N(.25,.25), wha is he disibuion of ˆα? ˆα? 2.Moeove, A ˆ( L ) is no well appoximaed by a nomal. I is well known ha if he oos of A(L) ae lage, hen A ˆ( L ) will exhibi subsanial bias owads zeo. In fac, in he limi ha he oos ae local o uniy, A ˆ ( L ) will no have a nomal asympoic disibuion. The poblem of pesisen oos complicaing infeence on D(L) is paiculaly impoan fo medium and longe hoizons Revised July 23,

55 Confidence Inevals fo IRFs, cd. An example: he disibuion of esimaed IRF of an AR(1) a long hoizons (Sock (1996, 1997), Phillips (1998)). Suppose Y = αy 1 + ε and model α as local o uniy: α = 1+c/T, whee c is a consan. The esimaed IRF a hoizon h is, ˆα h = (1+ ĉ/t) h Suppose ha h/t κ (he hoizon is a facion of he sample size). Then h c ˆ T e a diec applicaion of local-o-uniy asympoic heoy yields, ˆα h Jc() s dws κc e exp κ 2 Jc () s ds whee J c is an Onsein-Uhlenbeck pocess. The ue IRF is e κc ; he esimaed IRF is ha, imes a nonnomal andom faco. Revised July 23,

56 Confidence Inevals fo IRFs, cd. The poblem posed by lage oos wosens as he hoizon inceases. The facion of he sample a which poblems fo he nomal appoximaion aise is supisingly sho, say 10%. Pesaveno and Rossi (2005, 2006) povide one mehod fo handling his (and povide an applicaion o he LR echnology shock debae). Thei pocedue is heoeically jusified conols coveage aes bu is cumbesome and can poduce wide inevals. The ohe mehods in he lieaue (he mehods we now un o) do no handle lage oos in heoy, bu some seem o wok OK in Mone Calo sudies (and mos ceainly some ae bee han ohes). Revised July 23,

57 Confidence Inevals fo IRFs, cd. Sandad mehods fo consucing confidence inevals fo D(L): 1.Dela mehod (see Lükepohl (2005)) 2.Boosap mehods (Runkle (1987), Kilian (1998a, 1998b, 2001)) 3.Bayesian mehods (Sims and Zha (1999)) Kilian and Chang (2000) esuls MC simulaion evidence compaing he dela mehod, he Sims- Zha (1999) Bayesian mehod, he Runkle (1987) unadjused boosap, Kilian (1998) adjused boosap. Resuls pesened hee ae fo he Benanke-Gele (1995) VAR(12), 4 vaiables, 348 monhly obsevaions Revised July 23,

58 Revised July 23,

59 Confidence Inevals fo IRFs, cd. Commens: The coveage aes of IRFs depend on he VAR (he MC design). The mehod ha seems o wok bes (even hough i echnically isn valid when oos ae local o uniy) is Kilian s bias-adjused boosap. Hee is Kilian s algoihm: (i) Compue VAR esimaes A ˆ( L) (ii) Compue bias-adjused VAR esimaes (hee ae wo ways o do his using Pope s (1990) bias fomulas fo VARs o by boosap simulaion; see Kilian (1998b, 2001 appendix) fo deails) (iii) Boosap he foegoing (ha is, he bias-adjused esimaes) (iv) Use peceniles of he boosap IRF daws, hoizon by hoizon, o compue confidence bands (i.e. use he pecenile, no pecenile-, boosap mehod). Revised July 23,

60 Concluding commens Idenificaion emains a he coe of successful SVAR modeling Thee have been some ceaive and pomising developmens (he DSGE, high fequency, IRF sign esicions, and heeoskedasiciy echniques all have plausibly valid applicaions) Thee ae also some suble issues of infeence which ae no fully esolved you need o be awae of hese (a leas): o Se idenificaion issues and inepeaion of confidence bands compued using sign esicion mehods o Infeence wih long un esicions: can be viewed eihe as a (possibly) weak insumens poblem o a difficul HAC esimaion poblem he poblem is mos seious when he daa have subsanial low fequency componens. o Deeioaion of IRF confidence bands a long hoizons (and someimes a sho hoizons); limiaions of sandad IRF boosap. Revised July 23,