FACTOR AUGMENTED AUTOREGRESSIVE DISTRIBUTED LAG MODELS. Serena Ng Dalibor Stevanovic. November Preliminary, Comments Welcome

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1 FACTOR AUGMENTED AUTOREGRESSIVE DISTRIBUTED LAG MODELS Serena Ng Dalibor Sevanovic November 212 Preliminary, Commens Welcome Absrac This paper proposes a facor augmened auoregressive disribued lag (FADL) framework for analyzing he dynamic effecs of common and idiosyncraic shocks. We firs esimae he common shocks from a large panel of daa wih a srong facor srucure. Impulse responses are hen obained from an auoregression, augmened wih a disribued lag of he esimaed common shocks. The approach has hree disincive feaures. Firs, idenificaion resricions, especially hose based on recursive or block recursive ordering, are very easy o impose. Second, he dynamic response o he common shocks can be consruced for variables no necessarily in he panel. Third, he resricions imposed by he facor model can be esed. The relaion o oher idenificaion schemes used in he FAVAR lieraure is discussed. The mehodology is used o sudy he effecs of moneary policy and news shocks. JEL Classificaion: C32, E17 Keywords: Facor Models, Srucural VAR, Impulse Response Deparmen of Economics, Columbia Universiy, 42 W. 118 S. New York, NY 125. (serena.ng@columbia.edu) Déparemen des sciences économiques, Universié du Québec à Monréal. 315, Se-Caherine Es, Monréal, QC, H2X 3X2. (dsevanovic.econ@gmail.com) The firs auhor acknowledges financial suppor from he Naional Science Foundaion (SES )

2 1 Inroducion This paper proposes a new approach for analyzing he dynamic effecs of q common shocks such as due o moneary policy and echnology on q or more observables. We assume ha a large panel of daa X ALL = (X, X OT H ) is available and use he sub-panel X ha is likely o have a srong facor srucure o esimae he common shocks. Idenificaion is based on resricions on a q dimensional subse of X. The impulse response coefficiens are obained from an auoregression in each variable of ineres augmened wih curren and lagged values of he idenified common shocks. Observed facors can coexis wih laen facors. We refer o his approach as Facor Augmened Auoregressive Disribued Lag (FADL). An imporan feaure of he FADL is ha i esimaes he impulse responses using minimal resricions from he facor model. The approach has several advanages. Firs, while X is large in dimension, idenificaion is based on a subse of variables whose dimension is he number of common shocks. This reduces he impac of invalid resricions on variables ha are no of direc ineres. Second, he impulse responses are he coefficiens esimaed from a regression wih common shocks as predicors. Resricions are easy o impose, and for many problems he impulse responses can be esimaed on an equaion by equaion basis. Third, he analysis only requires a srong facor srucure o hold in X and is less likely o be affeced by he possibiliy of weak facors in X OT H. The proposed FADL mehodology les he daa speak whenever possible and is in he spiri of vecor-auoregressions (VAR) proposed by Sims (198). The FADL also shares some similariies wih he Facor Augmened Vecor Auoregressions (FAVAR) considered in Bernanke and Boivin (23). Their FAVAR expands he economerician s informaion se wihou significanly increasing he dimension of he sysem. Our FADL furher simplifies he analysis by imposing resricions only on he variables of ineres. Recursive and non-recursive resricions can be easily implemened. The FADL is derived from a srucural dynamic facor model which has a resriced FAVAR as is reduced form. A facor model imposes specific assumpions on he covariance srucure of he daa. Even hough many variables are available for analysis, a facor srucure may no be appropriae for every series. As noed in Boivin and Ng (26), more daa may no be beneficial for facor analysis if he addiional daa are noisy and/or do no saisfy he resricions of he facor model. We rea X like a raining sample. Using i o esimae he common shocks enables us o validae he facor srucure in X OT H, he series no in X. The FADL approach sands in conras o srucural FAVARs ha impose all resricions of a dynamic facor model in esimaion, as Forni, Giannone, Lippi, and Reichlin (29). The FADL esimaes will necessarily be less efficien if he resricions are correc, bu are more robus when he resricions do no hold universally. As in Sock and Wason (25), our FADL also permis 1

3 implicaions of he facor model o be esed. However, we go one-sep furher by leing he daa deermine he Wold represenaion insead of invering a large FAVAR. The paper proceeds as follows. Secion 2 firs ses up he problem of idenifying he effecs of common shocks from he perspecive of a dynamic facor model. I hen presens he FADL framework wihou observed facors. Esimaion and idenificaion of a FADL is discussed in Secion 3. Relaion of FADL o alernaive srucural dynamic facor analysis is discussed in Secion 4, and FADL is exended o allow for observed facors. Simulaions are presened in Secion 5. Secion 6 considers he idenificaion of moneary and news shocks. Boh examples highligh he wo main feaures of FADL:- he abiliy o perform impulse responses analysis and o es he validiy of he facor srucure of variables no used in esimaion or idenificaion of he common shocks. 2 Dynamic Facor Models and he FADL Framework Le N be he number of cross-secion unis and T be he number of ime series observaions where N and T are boh large. We observe daa X ALL = (X, X OT H ) which are saionary or have been ransformed o be covariance saionary. I is assumed ha X = (X 1,..., X N ) has a (srong) facor represenaion and can be decomposed ino a common and an idiosyncraic componen: X = λ(l)f + u X (1) where f = (f 1,..., f q ) is a vecor of q common facors and λ(l) = λ + λ L +... λ s L s is a polynomial marix of facor loadings in which he N q marix λ j = (λ j1..., λ jn ) quanifies he effec of he common facors a lag j on X. The series-specific errors u X = (u X1,..., u XN ) are muually uncorrelaed bu can be serially correlaed. We assume (I N D(L)L)u X = v X (2) where v X is a vecor whie noise process. The q laen dynamic facors are assumed o be a vecor auoregressive process of order h. Wihou loss of generaliy, we assume h = 1 and hus f = Γ 1 f 1 + Γ v f (3) where he characerisic roos of Γ 1 are sricly less han one. The q 1 vecor v f consiss of srucural common shocks (such as moneary policy or echnology). These srucural shocks can affec several dynamic facors simulaneously. Hence, he q q marix Γ need no be an ideniy. By assumpion, E(v Xi v Xj ) = and E(v Xi v fk ) = for all i j and for all i = 1,... N and k = 1,... q. 2

4 Assuming ha I D(L)L is inverible, he vecor-moving average represenaion of X in erms of he srucural common and idiosyncraic shocks is X = Ψ f (L)v f + Ψ X (L)v X. The srucural impulse response coefficiens Ψ X j and Ψ f j are defined from Ψ X (L) = Ψ X j L j = (I D(L)L) 1 Ψ f (L) = j= Ψ f j Lj = (I D(L)L) 1 λ(l)(i Γ 1 L) 1 Γ. j= For each j, Ψ f j is a N q marix summarizing he effec of a uni increase in v f afer j periods. We use Ψ f j,i 1 :i 2,k 1 :k2 o denoe he submarix in he i 1 o i 2 rows and k 1 o k 2 columns of Ψ f j. When i 1 = i 2 = i and k 1 = k 2 = k, we use ψ f j,i,k period + j. o denoe he effec of shock k in period on series i in The objecive of he exercise is o uncover he dynamic effecs (or he impulse response) of he srucural common shocks v f on variables of ineres. By using X 1,..., X N for facor analysis, he economerician s informaion se is of dimension N. Forni, Giannone, Lippi, and Reichlin (29) argue ha non-fundamenalness is generic of small scale models bu canno arise in a large dimensional dynamic facor model. The reason is ha Ψ f (z) is a recangular raher han a square marix and is rank is less han q for some z only if all q q sub-marices of Ψ f (z) are singular, which is highly unlikely. Assuming ha N is large ensures ha he common shocks are fundamenal for X. However, even if N is large, nohing disinguishes one common shock from anoher. In a VAR analysis wih q endogenous variables and q shocks, q(q 1)/2 resricions will be necessary. A popular approach is o impose conemporaneous exclusion resricions such ha a rank condiion is saisfied, see, eg. Deisler (1976), Rubio-Ramírez, Waggoner, and Zha (21). If he idenificaion resricions imply a recursive ordering, hen he parameers can be idenified sequenially and esimaion can proceed on an equaion by equaion basis. While Ψ X = I N in a dynamic facor model, he conemporaneous response of X o common shocks v f is given by λ,1,1 λ,1,2... λ,1,q. Γ,1,1... Γ,1,q Ψ f = Λ Γ =. λ,q,1 λ,q,2... λ,q,q.. Γ,q,1... Γ,q,q.. λ,n,1 λ,n,2... λ,n,q 3

5 The (i, k) enry of Λ is he conemporaneous effec of facor k on series i, and he (k, j) enry of Γ is he effec of he j-h common shock on facor k. In general, Ψ f will no be an ideniy marix. Two addiional problems make he idenificaion problem non-sandard. Firs, while having more oal shocks han endogenous variables should faciliae idenificaion, he common shocks also resric he co-movemens across series. Imposing consrains on an isolaed number of series is acually quie difficul wihin he facor framework. Zero resricions on he enries of Λ or Γ alone are no usually enough o ensure ha a paricular enry of Ψ f akes on he desired value (ofen zero). Second, he dynamic facors are hemselves laen. Thus, no only do we need o idenify he effecs of v f, we also need o idenify v f. Our analysis is based on he following assumpions. Assumpion 1: E(v f ) =, E(v f v f ) = I q. Assumpion 2 D(L) is a diagonal marix wih δ i (L) in he i-h diagonal, ie δ 1 (L)... D(L) =...,... δ N (L) Assumpion 3: For some j, a q q marix of Ψ f j is full rank. Assumpion 1 is a normalizaion resricion as we canno separae he size of he common shocks from heir impac effecs. Assumpion 2 is a form of exclusion resricion. We assume univariae auoregressive dynamics idiosyncraic errors: u Xi = δ i (L)u Xi 1 + v Xi. This implies ha dynamic correlaions beween any wo series are due enirely o he common facors, which is he defining feaure of a dynamic facor model. Diagonaliy of D(L) in urn allows X i o be characerized by an auoregressive disribued lag model X i = δ i (L)X i 1 + (1 δ i (L))λ i (L)f + v Xi (4) where λ i (L) = λ i + λ 1i L +... λ si L s is he i-h row of λ(l). A represenaion ha is more useful for impulse response analysis is an auoregressive disribued lag in he primiive shocks v f : X i = δ i (L)X i 1 + ψ f i (L)v f + v Xi (5) where ψ f i (L) = (1 δ i(l))λ i (L)(I Γ 1 L) 1 Γ. 4

6 We will henceforh refer o (5) as he FADL represenaion of X i. Noe ha ψ f i (L) = j= ψf j,i,1:q Lj is precisely he i-h row of Ψ f (L), wih ψ f,i,1:q = λ iγ = ( Γ,1,1... Γ,1,q ) λ,i,1 λ,i,2... λ,i,q. Γ,q,1... Γ,q,q The dynamic effecs of he common shocks v f on X i are defined by he coefficiens ψ f i (L). If v f were observed and N = q, equaion (5) defines a dynamic simulaneous equaions sysem in which idenificaion can be achieved by excluding some v f or is lags from cerain equaions. For example, conemporaneous resricions can be imposed so ha he q q marix Ψ f has rank q. As our sysem is all wih N q, Assumpion 3 is modified o require ha a q q submarix of Ψ f j is full rank. If all resricions are imposed on Ψ f, Assumpion 3 will hold if he op q q submarix of Ψ f has rank q. However, long run and sign resricions are also permied. Assumpions 1 o 3 are fairly sandard. Bu our facors are also laen and we can only idenify he space spanned by he facors and no he facors hemselves. To make he procedure operaional, we need o replace v f by esimaes v f which have he same properies as Assumpion 1. These idenificaion condiions will be furher developed below. 3 Esimaion and Idenificaion If here are q common shocks, we will need a leas q series for idenificaion.. Wihou loss of generaliy, le Y be he firs q series in X. Since each y Y admis a dynamic facor srucure, i holds ha y = α yy (L)y 1 + α yf (L)v f + v y. (6) Esimaion of (6) is no possible because we do no observe v f. Our impulse response analysis is based on leas squares esimaion of he FADL y = α yy (L)y 1 + α yf (L) v f + v y (7) where a prior resricions are be imposed on α yf (L) for idenificaion. We now explain how v f is esimaed and how resricions are imposed on he FADL. Le Λ be he N r marix of loadings, F be a r = q(s + 1) 1 vecor of saic facors, where Λ 1 f Γ 1 Γ 2... Γ s Λ 2 Λ =., F f 1 =., Φ I q.. F = I q. Λ i = ( ) λ i λ i1... λ is.... Λ N f max(h,s). I q. 5

7 The saring poin is he saic facor represenaion of he pre-whiened daa, x i = (1 δ i (L)L)X i : x i = Λ i F + v Xi (8) F = Φ F F 1 + ε F (9) ε F = Gε f. (1) The ε F are he reduced form errors of F and are hemselves linear combinaions of he srucural shocks v f and ε f = Γ v f is he vecor of reduced form common shocks, see (3),. The r q marix G maps he srucural dynamic shocks o he reduced form saic shocks. Since X is assumed o have a srong facor srucure, Λ Λ/N Λ > as N, and he N N marix 1 T T =1 x x has r eigenvalues ha diverge as N, T while he larges eigenvalue of he N N covariance marix of v X is bounded. From v Xi = x i Λ i F = x i Λ i (Φ F F 1 + ε F ), define ε Xi = x i Λ i Φ F F 1 = Λ i ε F + v Xi. (11) As noed in Sock and Wason (25), he rank of he r 1 vecor ε F is only q, since F is generaed by q common shocks. 1 In oher words, ε Xi iself has a facor srucure wih common facors ε f. Bu ε f are hemselves linear combinaions of v f. Le v f = Hε f. The q q marix H maps he reduced form dynamic shocks o he srucural dynamic shocks. The objecive is o idenify v f and o race ou is effecs on he variables of ineres. If here are q common shocks, q(q 1)/2 resricions are necessary o idenify v f via H. Esimaion proceeds in five seps. Sep E1: Esimae F from he full panel of daa X by ieraive principal componens (IPC). i Iniialize δ X i (L) using esimaes from a univariae AR(q) regression in X i. Le D(L) be a diagonal marix wih δ X i (L)L on he i-h diagonal. ii Ierae unil convergence min SSR = T ) ) ((I D(L)L)X ΛF ((I D(L)L)X ΛF. D(L),Λ,F =1 1 Bai and Ng (27) hus sugges using he number of diverging eigenvalues in he covariance of ε F o esimae q. 6

8 a Le F be he firs k principal componens of xx using he normalizaion ha F F/T = I k, where k is he assumed number of saic facors. b Esimae D(L) and Λ by regressing X i on F and lags of X i. The mehod of principal componens (PC) esimaes k facors as he eigenvecors corresponding o he k larges eigenvalues of XX /(NT ). Under he assumpion of srong facors, Bai and Ng (26) show ha he esimaes are consisen for he space spanned by he rue facors in he sense ha 1 T T =1 F 2 HF = Op (min(n, T )), where H is a k r marix of rank r. However, he idiosyncraic errors may no be whie noise. Sock and Wason (25) sugges using IPC o ieraively updae δi X(L), which is hen used o define x i. The saic facors form he common componen of x i. Sep E2: Esimae a VAR in F o obain Φ F and ε F and le ε Xi = x i Λ Φ i F F 1, where Λ and F 1, Φ F are obained from Sep (E1). Amengual and Wason (27) show ha he q principal componens of ε X can precisely esimae he space spanned by ε f. Sep E3: Idenificaion of v f : The common shocks ε f are unorhogonalized and, in general, are muually correlaed. We seek a marix H such ha v f = H ε f, (12) and v f is a vecor of muually uncorrelaed srucural common shocks. We consider wo approaches. The firs condiion (abbreviaed as RO) is lower riangulariy of a q q sub-marix so ha he shocks can be idenified recursively from q equaions. The second condiion (abbreviaed as BO) requires organizing he daa ino blocks using a priori informaion so ha he facors esimaed from each block can be given meaningful inerpreaion. Assumpion Recursive Ordering (RO) Mehod (a) is based on an assumed causal srucure. Jus like a VAR, his would require knowledge of which of he q variables o order firs. For j = 1 : q consider esimaing he regression: y j = α yy,j (L)y 1,j + q a yf,j,k (L) ε fk + v y,j where ε f are he q principal componens of he N residuals ê X. k=1 i Le Âf be he esimaed conemporaneous response o he q unorhogonalized shocks ε f : â yf,,1,1 â yf,,1,2... â yf,,1,q  f =.... â yf,,q,1 â yf,,q,2... â yf,,q,q 7

9 ii Define he q q marix H = [chol(âfâ f )] 1 Â f. Now le v f = H ε f α yf,j = α yf,j (L)H 1. By consrucion, v f is orhonormal. The mehod achieves exac idenificaion by using he causal ordering of he q variables seleced for analysis. Imposing a causal srucure hrough he ordering of variables is he mos common way o achieve idenificaion of FAVAR. Sock and Wason (25) also use Assumpion RO o idenify he primiive shocks. Their implemenaion differs from ours in ha we apply Choleski decomposiion o he FADL esimaes of α yf () and hence we do no impose all he resricions of he facor model. In conras, Sock and Wason (25) impose resricions implied by he FAVAR in X and F. The resuls are likely o be more sensiive o he choice of X. Assumpion Block Ordering (BO) Mehod (b) is useful when he daa can be organized ino blocks. Le X = (X 1, X 2,... X q ) be daa organized ino q blocks. To see how daa blocks faciliae idenificaion, observe ha he facor esimaes ε f are linear combinaions of ε X. Le ε f = ε X,:,:ζ be he T q marix of facor esimaes where for each, ζ11 ζ ζ 1N ε f = ζ21 ζ ζ2n..... ζq1 ζq ζqn Idenificaion requires a priori informaion on he ζ. ε X1 ε X2. ε XN. (13) i. For b = 1,... q, le ε b f be he marix of eigenvecor corresponding o he larges eigenvalues of he n b n b marix ε b X εb X. ii. Le H be he Choleski decomposiion of he q q sample covariance of ε f. Then v f = H ε f. The idenificaion sraegy can be undersood as follows. From (11), we see ha ε X = ( ε 1 X ε 2 X... ε q X) have ε f as common facors. Since he facors are pervasive by definiion, he facors are also common o all ε b X for arbirary b. Thus for each b = 1,... q, consider a facor model for ε b Xi = Λb i εb f + vb Xi. If εb Xi were observed, he facors for block b can be esimaed by principal componens which are linear combinaions of series in ε b X. We do no observe εb X, bu we have ε X = x Λ Φ F F 1 from Sep (E2). For example, if X 1 is a T N 1 panel of employmen 8

10 daa, he firs principal componen of ε 1 X ε1 X is a labor marke facor ε f1, and if X 2 is a panel of price daa, ε f2 is a price facor. Collecing he facors esimaing from all blocks ino ε f, we have ζ1,1:n ζ1,1:n ε f =.... ε 1 X ε 2 X. (14)... ζ q 1,1:N q ε q X Obviously, he facors are defined by assuming a srucured covariance relaion in he observables. The appeal is ha we can now associae he q facors wih he block of variables from which hey are esimaed. However, hese facors can sill be correlaed across blocks. To orhogonalize hem, sep (ii) performs q regressions beginning wih v f1 = ε 1 f. For m = 2,... q, v fb = M b ε b f are he residuals from projecing ε b f ono he space orhogonal o v f1,..., v f,b 1, and M b is he corresponding projecion marix. Bernanke, Boivin, and Eliasz (25) rea he ineres rae as an observed facor, organize he macro variables ino a fas and a slow block, and esimae he one facor from he slow variables. Their idenificaion is based on a Choleski decomposiion of he residuals in he slow variables and he observed facor. Their implemenaion is specific o he quesion under invesigaion while our mehodology is general. Our idenificaion algorihm is generic, provided blocks of variables wih meaningful inerpreaion can be defined. 2 In convenional VAR models, he srucural impulse responses are obained by roaing he reduced form impulse response marix by a marix, say, H. The primiive shocks are hen obained by roaing he reduced form errors wih he inverse of he same marix. In our seup, idenificaion of srucural common shocks precedes esimaion of he impulse responses. This allows us o impose economic resricions on he impulse response funcions wihou simulaneously affecing he srucural shocks. As presened, H is a lower riangular marix. However, sign, long run and oher srucural resricions can be imposed. Sep E4: Consruc Impulse Response Funcion: wih resricions on α Y f (L): Esimae a q dimensional FADL by OLS Y = α Y Y (L)Y 1 + α Y f (L) v f + v y (15) where α Y Y (L) is a diagonal polynomial in he L of order p y, and α Y f is of order p f. Given inerpreaion of v f idenified from Sep E3, shor and long-run economic resricions on he impulse 2 Moench and Ng (211) consruc regional facors from daa organized geographically. Ludvigson and Ng (29) sudy he relaive imporance of he facor loadings and find ha facor one loads heavily on real aciviy series, facor wo on money and credi variables, while facor hree loads on price variables. 9

11 responses can be direcly imposed on α yf. The esimaed responses of y o a uni increase in he common shocks v f and idiosyncraic shocks v y are defined by ψ f y (L) = α yf (L) 1 α yy (L)L ψ y y(l) = 1 1 α yy (L)L. Since α yy (L) is a scalar raional polynomial, he impulse responses are easy o compue using he filer command in malab. Noe ha by Assumpion 1, he sandard deviaion of all common shocks are normalized o uniy. The response o a uni shock is hus he same as he response o a sandard deviaion shock. Sep E5: Model Validaion Our mainained assumpions are ha F are pervasive amongs X raher han (X, X OT H ) and by assumpion, X have a srong facor srucure. We refer o X as a raining sample. This is useful because once he esimaed common shocks v f are available, hey can be reaed as regressors in a FADL model for z (scalar) no necessarily in X. This is because if (X, X OT H ) have a facor srucure, he shocks v f common o X are also common o variables in X OT H. If he common facors are imporan for z X OT H, hen FADL coefficiens on v f and is lags should be saisically significan. 4 Relaion o he Oher Mehods and Allowing for Observed Facors An imporan difference beween our approach and exising srucural FAVAR analysis is ha we esimae he impulse responses direcly raher han invering a VAR. Chang and Sakaa (27). esimaes he shocks as residuals from long vecor auoregressions in observed variables. auhors show ha heir esimaed impulse responses are asympoically equivalen o he local projecions mehod proposed by Jorda (25). Our analysis has he addiional complicaion ha he facors are laen. Thus, we firs esimae he space spanned by common facors, hen esimae he space spanned by he common shocks, before finally esimaing he impulse response funcions. I is useful o relae our esimae of Ψ f (L) wih he convenional FAVAR approach which sars wih he represenaion ( F X from which i follows ha ) ( ) ( Φ F 1 = ΛΦ D(L) X 1 ) ( + ε F Λε F + v X ) The x = ΛΦL(I ΦL) 1 ε F + Λε F + v X ( ) = ΛΦL(I ΦL) 1 + Λ ε F + v X. 1

12 The dynamic effecs of shocks ε F o he saic facors on (prewhiened) daa are deermined by ( ) Λ ΦL(I ΦL) 1 + I = Λ Φ i+1 L i+1. (16) j= A lag j, he N N response marix ΛΦ j = ( Λ i Λ 2. Λ N) Φ j. Inuiively, he oal effec of ε F depends on X hrough F and hence depends on he dynamics of F and he imporance of he facor loadings on X. Assuming ha he reduced form shocks are relaed o he srucural shocks via ε F = A v f, he response o he srucural shocks esimaed by a FAVAR is Ψ f = Λ Φ j A 1 which is a produc of hree erms: wo ha are he same for all i, and one ( Λ) ha is specific o uni i s. Since Λ i is only available for any x i X, Ψ f can be consruced only for N series. This is a consequence of he fac ha he FAVAR esimaes he impulses wihou direcly esimaing v f. Since we esimae v f, we can consruc impulse responses for series no in X. In conras, our esimaor of Ψ f is Λ i Φ j A, which may no equal Λ i Φ j A, because we do no fully impose resricions of he dynamic facor model on he saic facor represenaion. Insead of a large FAVAR sysem, we esimae he FADL one variable a a ime. Cross parameer resricions beween α yf (L) and α yy (L) are also no imposed. As is usually he case, sysem esimaion is more efficien if he resricions are rue. However, misspecificaion in one equaion can adversely affec he esimaes of all equaions. This possibiliy increases wih N. The single equaion FADL esimaes are more robus o misspecificaion han hose ha rely on a large number of overidenifying resricions which are ofen imposed on variables ha are no of primary ineres, or whose facor srucure may no be srong. Finally, resricions on Γ and Λ alone may no be enough for idenificaion. Consequenly, i is no always easy o direcly define A. FAVARs ypically require several auxiliary regressions o deermine A. In addiion o incurring sampling variaions a each sep, he idenificaion procedure requires ricks ha are problem specific. In a FADL seing, he resricions are direcly imposed when he FADL is esimaed. I is more sraighforward, as will be illusraed in Secions 6 and Exension o m Observed Facors Some economic analysis involves idenificaion of shocks o observed variables in he presence of laen shocks. For example, Bernanke, Boivin, and Eliasz (25), Sock and Wason (25) and Forni and Gambei (21) consider idenificaion of moneary policy shocks in he presence of oher 11

13 shocks, using he informaion ha some variables have insananeous, while ohers have delayed response o shocks o he observed facor, being he Fed Funds Rae. These sudies, summarized in Appendix A, impose resricions of he facor models on all series. Our proposed FADL approach imposes significanly fewer resricions on he facor model. To exend he dynamic facor model o allow for m observed common facors W, le X = λ f (L)f + λ w (L)w + u X u X = D(L)u X 1 + v X ) ( ) ( ) ( ) ( ) Γ1,ff Γ = 1,fw f 1 Γ,ff Γ +,fw vf ( f w Γ 1,wf Γ 1,ww w 1 Γ,wf Γ,ww wih Γ 1,fw and Γ,wf. Wihou hese assumpions, w is weakly exogenous and can be excluded from he analysis. Le W be a vecor consising of w and is lags. Assume ha is dynamics can be represened by a VAR(1): v w W = Φ W W 1 + ε W. (17) The reduced form model is X = (I D(L)L) 1 ( λ f (L) λ w (L) ) ( ) (I Γ 1 ) 1 vf Γ v w = Ψ f (L)v f + Ψ w (L)v w + Ψ X (L)v X. + (I D(L)L) 1 )v X The saic facors are esimaed from prewhiened daa ha also nes ou he effecs of he observed facors, and he consrucion of he srucural shocks v f mus ake ino accoun ha he reduced form innovaions o he saic facors can be correlaed wih he innovaions o he reduced form represenaion of he observed facors. Le x i = X i δ i X i 1 and define x i = Λ if F + λ W iw + ε i where W = (w w 1,... w p). The seps can be summarized as follows. Sep W1: Esimae F condiional on W by ieraing unil convergence min = T ) ) ((I D(L)L)X Λ F F Λ F W ((I D(L)L)X Λ F F Λ W W. D(L),Λ,F =1 i Le F be he k principal componens of xx using he normalizaion ha F F/T = I k. ii Esimae D(L), Λ F and Λ W by regressing X i on F and W. Sep W2: Esimae Φ F and Φ W from a VAR in F and W, respecively. Also le ε W be he residuals from esimaion of (17), he VAR in W. 12

14 Sep W3: Esimae v f : i. Le ε Xi = x i Λ Φ if F F 1 Λ Φ iw W W 1, where F are he ieraive principal componens of he full panel. ii. Le X = (X 1, X 2,... X q ) be daa organized ino q blocks. For b = 1,... q, le ε fb be he eigenvecor corresponding o larges eigenvalue of he n b n b marix ε ε X b X b. iii. Orhogonalize ε = ( ε f ε w) using he causal or block ordering of he variables. Sep W4: Consruc he impulse response: and α yw (L): Then ψ f (L) = 5 Simulaions Esimae by OLS wih resricions on α yf (L) y = α yy y 1 + α yf (L) v f + α yw (L)v w + v y. (18) α yf (L) (1 α yy(l)) gives he response of y o v f holding W fixed. We use simulaions o evaluae he finie sample properies of he idenified impulse responses. Daa are simulaed from equaions (1)-(3) wih λ(l) being a polynomial of degree s = 1. The persisence parameer δ i is uniformly disribued over (.2,.5). The errors v Xi, v f and he non-zero facor loadings are normally disribued wih variances σx 2, 1, σ2 λ respecively. We se T = 2 and N = 12 o mimic he macroeconomic panels used in empirical work. The srucural moving-average represenaion is X i = 1 δ i L) 1 ( λ i λ 1i L ) ( ) 1 ( ) vf1 I Γ 1 L Γ + v Xi. This implies ha he impac response of X i o he shocks is summarized by ( ) ( ) γ,11 γ X i = (λ,i,1 λ,i,2 ),12 vf1 + v γ,21 γ Xi. (19),22 DGP 1: q = 2 facors, Γ 1 = ( ).75, σ.7 λ,1k = 1. ( ) 1 case a: Γ = I, case b: Γ =, σ.5 1 λ,2k =.8. The N variables are ordered such ha he firs N/2 variables respond conemporaneously o boh shocks and are labeled fas. The las N/2 do no respond conemporaneously o shock 2 and are labeled slow. By design, X 1 is a fas variable and X N is a slow variable. This srucure is achieved by specifying (λ,i,1 λ,i,2 ), i = 1,..., N/2 and (λ,i,1 ) i = N/2 + 1,..., N. v f2 v f2 13

15 Le Y = (X 1, X N, ) be he wo variables whose impulse responses are of ineres. Since here are no observed facors, esimaion begins wih E1 and E2. We consider boh idenificaion sraegies and esimae wo FADL regressions, one for each variable in Y. As a benchmark, we also esimae he (infeasible) FADL regressions using he rue common shocks, v f. The resuls are summarized in Table 1. The op panel of Table 1 shows ha for DGP 1a, he correlaion beween v fj and v fj are well above.9 for boh idenificaion sraegies. For DGP 1b, Mehod (a) is more precise han (b) bu he laer is sill quie precise. The correlaion beween v fj and v fk are saisically differen from zero, bu are numerically small. Panel B of Table 1 repors he RMSE of he esimaed impulse responses when he shocks are observed. Given ha here are wo shocks, here are wo impulse responses o consider for each of he wo variables. We use v fj X k o denoe he response of X k o shock j, where k = 1 is he fas variable, and k = N is he slow variable. Panel C repors resuls when he common shocks have o be esimaed. The ψ are pracically idenical o he analyical ones given by (19). Furhermore, he impac response of slow variable o second shock is no saisically differen from zero. When he FADL models are esimaed on v f insead of v f, we observe ha (i) corr(v f, v f ) I; In case 2, off-diagonal elemens (ii) ψ(l) are very close o rue impulse response coefficiens (iii) he non-zero coefficiens have saisically significan esimaes. DGP 2: q = 2 laen and m = 1 observed facors.75 σ λ,3k =.7 and Γ 1 = Le σ λ,jk = 1, σ λ,1k = 1, σ λ,2k =.8, and 1 case a: Γ = I, case b: Γ = The ordering of srucural shocks is v f = (vf slow, v mp f, vfas f ). The goal is (parial) idenificaion of he effecs of v mp f. Again, he N variables are divided beween fas and slow: slow variables do no respond on impac o second and hird shocks, and a leas one variable does no respond immediaely only o he hird shock, such ha he causal ordering holds. Afer he common shocks are esimaed and idenified according o Mehods RO and BO, wo FADL regressions are esimaed for he wo componens in Y : one fas and one slow variable. As we are ineresed in parial idenificaion of he second shock, we only repor resuls on he approximaion of v mp f, and impulse responses of wo variables o his shock. As in he previous exercise, FADL regressions wih rue shocks produce impulse response coefficiens pracically idenical o he analyical ones. The esimaed second shock is very close o he rue one and ψ(l) very close o rue impulse response coefficiens. 14

16 6 Two Examples In his secion, we use FADL o analyze wo problems:- measuring he effecs of moneary policy in he presence of oher common shocks, and news shocks. 6.1 Example 1: Effecs of a Moneary Policy Shock As in Bernanke, Boivin, and Eliasz (25), he moneary auhoriy observes N slow variables (such as measures of real aciviy and prices) colleced ino X slow when seing he ineres rae R bu does no observe N fas variables (such as financial daa) colleced ino X fas. In his exercise, R is an observed facor. Le v f = (v slow f, v mp v mp f is he moneary policy shock, v fas f of q 2 shocks, specific o X slow f, vfas f ) be he vecor of q common shocks, where is a vecor is a vecor q 1 shocks, specific o X fas, and vf slow respecively, wih q = q 1 + q The issue of ineres is (parial) idenificaion of he effecs of moneary policy shock, meaning ha he effecs due o vf slow are no of ineres. v fas f Bernanke, Boivin, and Eliasz (25) idenify he moneary policy shock by assuming ha Ψ f is a block lower riangular srucure. This involves resricions o on N slow > q 2 variables. In a daa rich environmen, some of hese resricions could well be invalid. We consider wo alernaive idenificaion sraegies, boh using fewer resricions. The firs is based on Assumpion RO which can be achieved by choosing he firs q variables o compose of q 1 (slow) indicaors of real aciviy and prices, followed by he moneary policy insrumen. 3 The second is based on Assumpion BO which idenifies he shocks a he block level. The daa are ordered as Y = (X slow, R, X fas ). Afer esimaing v slow f from X slow and and v fas f and from X fas, he moneary policy shocks are he residuals from a regression of R on curren and lag values of v f slow. By consrucion, he esimaed srucural shocks are muually uncorrelaed under boh RO and RO assumpions. A FADL in all he shocks is hen esimaed for each variable of ineres. In erms of marix Ψ f, Bernanke, Boivin, and Eliasz (25) assumes: ψ,1,1 Ψ f = N slow q N 1 slow 1 ψ,2,1 ψ,2,2 1 q ψ,3,1 N fas q 1 ψ,3,2 N fas 1 N slow q 2 1 q 2 ψ,3,3. N fas q 2 3 One may also add q 2 financial indicaors a he end of he recursion, bu Bernanke, Boivin, and Eliasz (25) found ha here is lile informaional conen in he fas moving facors ha is no already accouned for by he federal funds rae. 15

17 Assumpions RO and BO boh assume ha he op q q block of Ψ f is lower riangular: ψ,1:q,1 q 1 q 1 q 1 1 q 1 q 2 Ψ f,1:q,1:q = ψ,2:q,1 ψ,2:q,2 1 q q 2 ψ,3:q,1 q 2 q 1 ψ,3:q,2 q 2 1 ψ,3:q,3 q 2 q 2. However, he Ψ f marix and v f idenified by RO will be differen from hose idenified by BO. Under Assumpion RO, all N series are used o esimae he q vecor ε f. Thus any q series in he raining sample can be used o idenify primiive shocks v. Under Assumpion BO, ε j f is esimaed from block j of X. Thus, he j shock in v f is idenified from one of he N j series in block j of X. Assumpion BO also allows a priori economic resricions o be imposed on some or all variables wihin he blocks. For example, we can resric all N slow series no o reac on impac o a moneary policy shock, while he response of fas moving variables is unresriced. Since hese resricions are imposed on equaion by equaion basis, hey do no affec he esimaion nor he idenificaion of srucural shocks Daa and Resuls The raining sample used o esimae he facors consiss of 17 quarerly aggregae macroeconomic and financial indicaors over he exended sample 1959Q1-29 Q1. This daa se consiss of fas and slow moving variables. The Federal funds rae (FFR) is reaed as an observed facor. All daa are assumed saionary or ransformed o be covariance saionary. The complee lis of variables is given in he Appendix. Our esimaion differs from Bernanke, Boivin, and Eliasz (25) in wo ways. Firs, we use quarerly daa. Second, we esimae he facors by IPC o ake care of auocorrelaion in residuals. According o informaion crieria in Amengual and Wason (27) and Bai and Ng (27), here are q = 3 laen dynamic facors in he raining sample. Idenificaion is achieved by imposing a causal ordering. We order commodiy price inflaion firs, followed by GDP deflaor inflaion, unemploymen rae, and hen FFR. Hence moneary policy is he las variable in his causal ordering, which implies zero conemporaneous response o moneary policy by he slow moving variables. We only impose resricions on q series (one from each block) while Bernanke, Boivin, and Eliasz (25) impose resricions on all series belonging o he slow moving block. Compared o Sock and Wason (25), we impose he same minimal number of resricions o idenify he srucural shocks, bu our approach differs in esimaing he impulse response funcions. 4 The resricions can vary across series in he block. For example, one series could be resriced o respond only 2 periods afer he shock, he sign of anoher variables could be fixed, he shape of he impulse response funcion could be consrained for a hird variables, and so on. 16

18 Insead of consrucing impulse response coefficiens of X as (I D(L)) Λ(I Γ 1 (L)) 1 Γ, we raher esimae he produc, ψ f i (L), equaion by equaion for any elemen of X and X OT H. The 12 period impulse responses are presened in Figure 1. As in Bernanke, Boivin, and Eliasz (25), conrolling for he presence of common shocks resolves anomalies found in he lieraure. Afer a moneary policy shock, he fas moving variables such as Treasury bills increase immediaely, while sock prices, housing sars, and consumer expecaions fall. Furhermore, many measures of he slow variables including real aciviy and prices decline as a resul of he shock wihou evidence of a price puzzle. The exchange rae appreciaes fully on impac, wih no evidence of overshooing. The resuls for he variables of ineres are in line wih Chrisiano, Eichenbaum, and Evans (2) who use recursive and non-recursive idenificaion schemes o sudy he effecs of moneary policy, using small VARs. However, once he common shocks are esimaed, he effecs of moneary policy can be sudied for many variables, no jus he q variables used in idenificaion. The scope of he analysis is much larger han a small VAR. X OT H To check he validiy of he facor srucure in series no in he raining sample, we consider consising of 17 disaggregaed series. Amongs hese are (i) 3 secoral CPI, 55 PCE, and 3 PPI measures of inflaion, (ii) 1 disaggregaed employmen series, (iii) 18 invesmen measures, and (iv) 18 consumpion series. For each of hese addiional variables, he Wald es is used o es he null hypohesis ha all coefficiens in α yf (L) are joinly zero. The null hypohesis canno be rejeced a he five percen level for many series including one secoral CPI, 15 PCE, one employmen, one invesmen and wo consumpion series. For hese series, he daa does no suppor he presence of a facor srucure. We hen proceed o analyze he effecs of moneary policy on variables in X OT H. Ineresingly, he impulse responses of variables no affeced by v f display price-puzzle like feaures. As seen in he op panel of Figure 2 for some of hese variables, an increase in he Fed Funds rae increases raher han lowers prices. The boom panel displays resuls for four series wih significan α yf (L). For hese laer se of variables, he impulse responses are similar o hose repored for he variables in he raining sample, namely, ha an increase in he Fed funds rae lowers prices. The impulse responses of all secoral variables are presened in Figure 3. The responses of many disaggregaed series are in line wih heory: a decline of real aciviy and price indicaors across several secors afer an adverse moneary policy shock. In case of employmen variables, only mining and governmen secor series diverge from ohers during he firs year afer he shock, while he price indicaors of some nondurable goods secors presen he price puzzle behavior. 6.2 Example 2: Effecs of a News Shock Beaudry and Porier (26) consider echnology shock and news shocks, v f = (v T F P v NS ), iner- 17

19 preed as an announcemen of fuure change in produciviy. They are ineresed in he effecs of hese wo shocks on produciviy X 1. Consider idenificaion by he shor run resricions. Suppose ha he firs N 1 variables X 1 o v T F P X do no respond immediaely o v NS, bu heir response is unresriced. Then Ψ f is lower block riangular, viz: ψ f,1,1.. Ψ f Ψ f = ψ f,n 1 +1,1 ψ f,1:n 1,1 1:N1,1,N 1 +1, Ψ f,n 1 +1:N,1 Ψ f,n 1 +1:N,2. ψ f,n,1 ψ f,n,2 Λ N 2 This srucure can be achieved if Λ and Γ are boh lower block riangular, ie. λ,1,1.. = λ,n1 +1,1 λ,n1 +1,2 = Λ,1:N 1,1. N1 1 and Γ =.. Λ,N1 +1:N,1. Λ,N1 +1:N,2 2 2 λ,n,1 λ,n,2 The zero resricion should hold for all series in he firs block. ( ) Γ,11 Γ,21 Γ,22. Bu since here are only wo shocks, any wo series permi exac idenificaion provided one is from X 1, one from X 2, and one resricion is imposed on Ψ f. Beaudry and Porier (26) only uses wo variables (X 1, X N ) for analysis where X 1 is a measure of TFP and X N is sock price. We allow for N > 2 variables. Bu unlike sandard VARs which require resricions of order N 2 o idenify N shocks, we use q series o exacly idenify q = 2 shocks. As discussed earlier, insead of puing resricions on Γ or Λ separaely, our resricions are imposed on he relevan row(s) of Ψ f = Γ Λ. The bivariae sysem has he propery ha ( X1 X N ) = ( ψ f,11 ψ f,21 ψ f,22 ) (v ) T F P v NS + ( ψ f j,11 ψ f j,12 ψ f j,21 ψ f j,22 j=1 ) (v ) T F P j v j NS. The number of idenifying resricions used in he FADL is of order q 2 irrespecive of N. This also conrass wih sandard FAVARs which impose many overidenifying resricions. In our seup, a large N is desirable for FADL because i improves esimaion of v f. Long run resricions can similarly be imposed so ha Ψ f (1) is block lower riangular. A FADL leads o exac idenificaion using he salien feaures of he facor model Daa and Resuls Our daa consiss of X = (X T F P FRB San Francisco, X SP, X SP, X OT H ), where X T F P conains six TFP measures from is a vecor of eigh S&P and Dow Jones aggregae sock price indicaors, 18

20 and X OT H is a vecor of 14 macroeconomic ime series used in he previous example bu wih he sock prices removed 5. Beaudry and Porier (26) only use one series of he six series in X T F P and one series in X SP a he ime. Forni, Gambei, and Sala (211) use he same TFP series and some of our sock price measures. Two idenificaion sraegies are considered: i (Causal Ordering) esimae wo common shocks from X = (X T F P from X T F P and one from X SP ha idenifies he echnology and he news shock., X SP ). Two series, one are seleced. By ordering he TFP series ordered firs, he H ii (Block Ordering) ε T F P is esimaed exclusively from X T F P The idenificaion is based on he srucure ( ) ( ) ( ) ε T F P a11 v T F P ε SP = a 21 a 22 v NS. and ε SP is esimaed from X SP. Effecively, v T F P = ε T F P and v NS are he residuals from a projecion of ε SP ono v T F P. Noe ha under boh idenificaion sraegies he esimaed shocks are muually uncorrelaed. Once v T F P and v NS are available, variable by variable FADL equaions are esimaed for all series in X. The zero impac resricions are imposed for all TFP measures, while all oher FADL regressions are lef unresriced. echnology and news shocks (v T F P resuls for differenced daa. The resuls for he wo idenificaion sraegies and for boh and v NS respecively) are given in Figures 4-7. We repor The Table 3 conains p-values for Wald es for he null hypohesis of no facor srucure in X T F P, X SP and X OT H variables. The abbreviaions RO, BO sand for Assumpion RO and BO respecively. The null hypohesis is srongly rejeced for many series. Turning o he impulse responses, he effecs of echnology shocks are in line wih Chrisiano, Eichenbaum, and Vigfusson (23) who sugges ha echnology improvemens are pro-cyclical for real aciviy and hours measure, bu conrary o Basu, Fernald, and Kimball (26) and Gali (1999). Of special ineres here are he responses o a posiive news shock. The forward looking variables such as sock prices, housing sars, new orders and consumer expecaions increase on impac. Consumpion reacs posiively. The wealh effec does no seem imporan enough such ha he worked hours also increase on impac. Our resuls are in line wih Beaudry and Porier (26) for he pro-cyclical response of worked hours. However, Barsky and Sims (211) also esimae posiive response of consumpion and find an immediae decrease of hours. Forni, Gambei, and Sala (211) find ha boh consumpion and 5 The complee lis of addiional variables used in news shock applicaion is available in Appendix 19

21 hours respond negaively on impac. These differences can be due o he choice of variables used o idenify he shocks and o he variables seleced for analysis. In paricular, hese sudies used a small se of worked hours measures. We check he sensiiviy of our resuls o a much broader se of available indicaors. To his end, we assess he sensiiviy of our resuls (under he assumpion of a block srucure) o addiional variables as follows: a Esimae ε OT H from he macro daa X OT H. Idenificaion is now based on ε T F P ε SP ε OT H a 11 = a 21 a 22 a 31 a 32 a 33 v T F P v NS v OT H. b change he ordering o ε T F P, ε OT H naure of sock prices. wih ε SP ordered las in view of he forward looking These resuls are denoed Block 2 and Block 3 respecively. In a VAR seup, here would be 14 VARs o consider when here are 14 macro variables ha migh no be economerically exogenous o TFP and sock prices. In he facor seup, we only need o esimae one se of macro shocks from 14 macro series. As shown in Figure 5, he effecs of news shocks are smaller when he macro shocks are presen. In oher words, omied variables from he VAR could have biased he esimaed effecs of news shocks. However, for an assumed q, he idenified impulse responses are robus o he ordering of he variables. As is well known, VARs involving hours worked are sensiive o wheher he hours series is in level or in difference, see for example, Féve and Guay (29). We use he specificaion labeled Block 3 o furher undersand he dynamic response he level (Figure 8) and growh (Figure 9) of average weekly hours (AWH) level o news shock. The dynamic responses of AWH and oal hours indices are ploed in Figure 9. Regardless of he daa ransformaion, he hours variables are pro-cyclical afer he news echnology shock. This exercise illusraes he FADL can be used o check he robusness of he resuls o many oher measures wihou affecing he idenificaion of srucural shocks. 7 Conclusion In his paper, we have proposed a new approach o analyze he dynamic effec of common shocks in a daa-rich environmen. Afer esimaing he common shocks from a large panel of daa and imposing a minimal se of idenificaion resricions, he impulse responses are obained from an auoregression in each variable of ineres, augmened wih a disribued lag of srucural shocks. 2

22 The FADL framework presens several advanages. The mehod is more robus o a fully srucural facor model when he idenifying facor resricions do no hold universally. Since he impulse responses are obained from a se of regressions, he resricions are easy o impose, and implicaions of he facor model can be esed. The esimaion of common shocks is less likely o be affeced by he presence of weak facors. The FADL mehodology is used o measure he effecs of moneary policy shocks, and o news and echnology shocks. The approach allows us o go beyond exising srucural FAVAR, and o validae resricions of he facor model. 21

23 Appendix: Relaion o Oher Mehods wih Observable Facors The esimaed common shocks are reaed as regressors of a FADL. As such, a priori resricions on he impulse response funcions can be direcly imposed in esimaion of he FADL by leas squares. The approach is simpler and more ransparen han exising implemenaions of srucural FAVARs. Consider he idenificaion of moneary policy shocks in he presence of oher shocks as in Bernanke, Boivin, and Eliasz (25). Their poin of deparure is a saic facor model wih laen and observed facors: [ F R X = Λ F F + Λ R R + u (2) ] [ ] F 1 = Φ + η (21) where F is vecor of r laen facors and R is he observed facor (usually Federal Funds Rae or 3-monh Treasury Bill). The auhors organize he N = 12 daa vecor X ino a block of slowmoving variables ha are largely predeermined, and anoher consising of fas moving variables ha are sensiive o conemporaneous news. The idiosyncraic errors are assumed o be serially uncorrelaed. R 1 BBE Idenificaion 1 Esimae F. i Le Ĉ(F, R ) be he K principal componens of X. ii Le X S be N S slow moving variables ha do no respond immediaely o a moneary policy shock. Le he K principal componens of X S be C (F ). iii Define F = Ĉ(F, R ) b R R where b R is obained by leas squares esimaion of he regression Ĉ(F, R ) = b C C (F ) + b R R + e. 2 Esimae he loadings by regressing X on F and R : ΛF and Λ R. 3 Esimae he FAVAR given by (21) and le η be he residuals. From he riangular decomposiion of he covariance of η, le A be a lower riangular marix wih ones on he main diagonal Then η =  ε are he moneary policy shocks. 4 Obain IRFs for F and R by invering (21) and using  5 Muliplying he IRFs in (3) by Λ F and Λ R o obain he IRFs for X. 22

24 The novely of he BBE analysis is ha Sep (1) accommodaes he observed facor R when F is being esimaed. By consrucion, Ĉ(F, R ) spans he space spanned by F and R while C (F ) spans he space of common variaions in variables ha do no respond conemporaneously o moneary policy. Since R is observed, he regression hen consrucs he componen of Ĉ ha is orhogonal o R. Once F is available, Sep (2) is sraighforward. Under he BBE scheme, he common shocks are idenified in Sep (3) when a FAVAR in ( F, R ) is esimaed. Because F may be correlaed conemporaneously wih R, he moneary policy shocks are idenified by ordering R afer F in (21). 6 The lower riangular of A is no enough o idenify he srucural shocks as he response depends on he produc ( Λ F Λ R) A. 7 Thus, BBE impose addiional resricions. In paricular, he K slow moving variables are ordered firs in X. Furhermore, he K K block of Λ F is ideniy, and he firs elemen in Λ R is zero. As a resul, he firs K + 1 K + 1 par of he produc ( Λ F Λ R) A is lower riangular. For K = 2, The srucural model is jus-idenified a 21 1 λ 31 λ 32 λ 33 a 31 a 32 1 Sock and Wason (25) The SW approach reas moneary policy as a dynamic facor. The idenificaion assumpions are ha (i) he moneary policy shock does no affec he slow-moving variables conemporaneously; and (ii) he slow-moving shock and moneary policy affecs he Fed Funds rae conemporaneously. Thus, as in Bernanke, Boivin, and Eliasz (25), he slow-moving variables firs, followed by he Fed funds rae, and hen he fas-moving variables. The poin of deparure is ha ε X = Λε f + v X is assumed o have a facor srucure and ε f = Gη = GHv f. Leing C = GH, he errors are relaed by ε X = ΛCv f + v X where v f is of dimension q. The seps are as follows: 6 Boivin, Giannoni, and Sevanović (29) suggess an alernaive way o esimae F ha does no rely on organizing he variables ino fas and slow. 1 Iniialize F o be he K firs principal componens of X. 2 (i) Regress X on F and R, o obain firs K principal componens of X Λ F,j and Λ R,j. (ii) Compue X j = X R, Λ R (iii) Updae F as he By consrucion, F is conemporaneously uncorrelaed wih R This is possible because he sep ha esimaes he laen facors conrolling for he presence of he observed facors is separaed from idenificaion of srucural shock. In BBE, η depends on he choice of variables used in he firs sage o esimae F. 7 In BBE applicaion, Sep 1 esimaes he loadings of slow moving variables o R close o zero. 23