Consistent Estimation Of The Number Of Dynamic Factors In A Large N And T Panel

Transcription

1 Conssen Esmaon Of he Number Of Dynamc Facors In A Large N And Panel July 005 (hs Draf: May 9, 006) Dane Amengual Dearmen of Economcs Prnceon Unversy Prnceon, NJ amengual@rnceon.edu and Mark W. Wason* Woodrow Wlson Prnceon Unversy Prnceon, NJ mwason@rnceon.edu

2 Dane Amengual s graduae suden, Dearmen of Economcs, Prnceon Unversy, Prnceon, NJ (amengual@rnceon.edu); and Mark W. Wason s Professor, Dearmen of Economcs and Woodrow Wlson School, Prnceon Unversy, Prnceon, NJ (mwason@rnceon.edu). hs research s an ougrowh of jon work wh Jm Sock, who we hank for commens and suggesons. hanks also o Jushan Ba, Serena Ng, wo referees and he Assocae Edor for useful commens. he work was funded n ar by NSF gran SBR-043.

3 Absrac Ba and Ng (00) roose a conssen esmaor for he number of sac facors n a large N and aroxmae facor model. hs aer shows how he Ba-Ng esmaor can be modfed o conssenly esmae he number of dynamc facors. he modfcaon s sraghforward: he sandard Ba-Ng esmaor s aled o resduals obaned by rojecng he observed daa ono lagged values of rncal comonen esmaes of he sac facors. Key Words: aroxmae facor model, dynamc facor model, Ba-Ng esmaor

4 . Inroducon Panel daases wh large me seres dmenson () and cross secon dmenson (N) are beng ncreasngly used n macroeconomcs for boh forecasng and srucural analyss. Ofen, hese daa are analyzed n he conex of an assumed laen facor srucure of he form X = ΛF + e, (.) for =,,, where X denoes an N vecor of observed varables, F s an r vecor of laen facors, Λ s a marx of coeffcens, and e s a vecor of errors. When he elemens of e have weak cross seconal and seral correlaon, he facors F summarze he moran cross covarance roeres of he varables. A queson of fundamenal neres s he number of laen facors, r, ha are requred n (.). Sgnfcan rogress on addressng hs roblem was made n Ba and Ng (00) who roosed conssen esmaors of r based on a enalzed leas squares objecve funcon assocaed wh he classc rncal comonens esmaor. However, n dynamc models s moran o dfferenae beween he number of sac facors (necessary o f he covarance marx of X) and he number of dynamc facors (necessary o f he secral densy marx of X). Whle he Ba-Ng esmaor was develoed o esmae he number of sac facors, hs aer shows ha can be easly modfed o conssenly esmae he number of dynamc facors.

5 Dynamcs can be ncororaed n he model by assumng ha F evolves as a VAR: F = ΦF + ε (.) = wh nnovaons ε ha can be reresened as ε = Gη where G s r q wh full column rank and η s sequence of shocks wh mean zero and covarance marx Σ ηη = I q ; η s he vecor of dynamc facor shocks. Several aers show how (.) and (.) can be derved from a resrced verson of a general dynamc facor model drven by q dynamc facors; n hs case F conans lnear combnaons of curren and lagged values of he dynamc facors and (.)-(.) s analogous o he comanon form reresenaon of he dynamc facor model. (See Ba and Ng (005), Forn, Halln, L, and Rechln (005), Gannone, Rechln, and Sala (004), and Sock and Wason (004, 005).) o see how he Ba-Ng esmaor mgh be used o esmae he number of dynamc facors, q, subsue (.) no (.) o oban Y = Γη + e (.3) where Y = X ΛΦ F = and Γ = ΛG. hus Y can be reresened as a facor model wh q facors ha corresond o he common shocks η. Were Y observed daa, q could be conssenly esmaed by alyng he Ba-Ng esmaor o Y. hs s nfeasble because Y deends on unknown arameers and lags of he unobserved facors.

6 Y = X ˆ hs aer sudes he conssency roeres of he Ba-Ng esmaor aled o Π ˆ F ˆ = where Π ˆ s an esmaor of ΛΦ and F ˆ s an esmaor of F. he analyss roceeds n wo ses. In he frs se, he Ba-Ng esmaor s shown o reman conssen f he esmaon error Y ˆ Y s suffcenly small (secfcally n = = ( Yˆ Y ) = O [max(n,)]). he second se shows ha he rncal comonens esmaor of F and feasble esmaors of Π yeld esmaors Y ˆ ha acheve hs degree of accuracy. ogeher hese resuls yeld a feasble conssen esmaor of he number of dynamc facors. he esmaor suded n hs aer was roosed n Sock and Wason (005) and aled o he roblem of esmang he number of dynamc facors n a large anel of U.S. macroeconomc me seres. Sock and Wason (005) dd no sudy he conssency roeres of he esmaor, and ha s he urose of he resen aer. Oher esmaors have also been roosed and used n aled work. Noably, Forn, Halln, L and Rechln (000) sugges nformal mehods based on he relave sze of egenvalues from he esmaed secral densy marx for X, relaed mehods have been roosed and aled n he emrcal analyss of Forn, L and Rechln (003), Gannone, Rechln and Sala (004) and elsewhere, and Halln and Lška (005) show how a conssen esmaor of q can be consruced from he esmaed secrum. Ba and Ng (005a) roose an esmaor for q based on he resdual covarance marx of he VAR n (.3) esmaed usng he rncal comonens esmaor of F, and show ha he esmaor s conssen. Secon 3 sudes he relave erformance of varous conssen esmaors usng a smulaon sudy. 3

7 More generally, he lan of hs aer s as follows. Secon brefly summarzes he Ba-Ng esmaor, shows he esmaor remans conssen when aled o daa conamnaed wh a small amoun of measuremen error, and uses hs resul o show ha he Ba-Ng esmaor aled o ˆ Y s a conssen esmaor of he number of dynamc facors. A Mone Carlo sudy s resened n Secon 3 o gauge he erformance of he esmaor, Secon 4 conans some concludng remarks, and he Aendx ncludes he roofs o he resuls gven n Secon.. Assumons and Asymoc Resuls. Revew of Exsng Work wh a Small Exenson We begn by revewng resuls for he model (.) under a sandard se of assumons. ransosng (.) and sackng he equaons yelds X = FΛ + e (.) where X s N, F s r, Λ s N r, and e s N. he h rows of X, F and e are X, F, and e ; he h row of Λ s λ ; he h elemen of X s X and smlarly for e, so ha X = λ F + e. Asymoc roeres of varous sascs generaed by hs model have been suded n Sock and Wason (00), Ba and Ng (00), Ba (003), and Ba and Ng (005a, 005b) under a smlar se assumons. he focus s on daases n whch boh N and are large, so ha he asymocs assume ha, jonly (equvalenly ha N = N( ) wh lm ( ) = ). he mnmum value of N and lays an moran 4

8 role n he analyss and hs value s denoed by s = mn(n,). he remanng assumons concern momens and deendence roeres of he varables; for he uroses of hs aer, he followng assumons suffce: (A.) EFF ( ) = I. r (A.) E(λ λ ) = Σ ΛΛ, where Σ ΛΛ s a dagonal marx wh elemens σ >σ jj >0 for <j. (When Λ s deermnsc, Σ ΛΛ s nerreed as he lmng emrcal average.) (A.3) (A.4) FF Ir = N N = λλ Σ. ΛΛ. N e. = = (A.5) () e σ > 0 (A.6) For some neger m and for all negers j m, Erace[(ee ) j ] = O( max[n,] j ). (A.7) N λ s = ( ). = s= = E Fe O (A.8) N λλ = ( ). = = E e O 5

9 (A.9) N. = = E Fe = O( ) Assumons (A.)-(A.5) rule ou exlosve or rendng behavor n boh he me seres and cross secon dmensons; he arcular values of E(F F ) and E(λ λ ) lsed n (A.) and (A.) are normalzaons (because ΛF = ΛHH F for arbrary H), and assumon (A.5) rules ou degenerae cases n whch he facors exlan all of he varance of he X s. Assumon (A.6) lms he varably and deendence n he errors e. For j =, mles ha N = = E e ( ) ( ) τ ( τ ) N N N = j= = j = = = = O(); for j =, mles ha ee = ee = O ( max[, ]), and so forh for larger values of j. Assumons (A7)-(A9) lm he deendence across elemens of Λ, F and e. Imoranly, all of hese assumons hold for sequences of..d. random varables wh he arorae number of momens, and assumons (A6)-(A9) can be nerreed as relaxng he..d. assumon o allow weak deendence. he Ba-Ng esmaors of r are based on enalzed leas squares objecve funcons. he enaly funcon deends on a deermnsc funcon g(n,) ha sasfes g(n,) 0 and s δ g(n,) for δ = (m )/m, where m s gven n assumon (A6). he leas squares objecve funcon s convenenly wren n erms of he egenvalues of he XX momen marx. Le ω denoe he h larges egenvalue of () XX, and consder he leas squares roblem: mn{ k }{ k ( ) N ( X k k ) λ } λ F k λ and F = =, where k F are arbrary k vecors. he usual rncal comonens calculaons mly 6

10 ha he average redced sum of squares assocaed wh he leas squares soluon s gven by R(k,X) = k = ω. Leng ˆX σ = ( ) denoe he average oal N X = = sum of squares, he enalzed average sum of squared resduals s PC(k,X) = + kg(n,), and he Ba-Ng PC esmaor s ˆX σ R(k,X) PC BN ( X) = argmn PC( k, X). (.) max 0 k r Leng ICP(k,X) = ln[ σ R(k,X)] + kg(n,), he Ba-Ng ICP esmaor s ˆX ICP BN ( X) = argmn ICP( k, X), (.3) max 0 k r where r max s a fne consan ha sasfes r r max. Conssency of he Ba-Ng esmaor s gven n he followng lemma: Lemma (Ba-Ng): Under assumons (A)-(A9), PC BN ( X ) r and ICP BN ( X ) r. As dscussed on he las secon, we wll sudy conssency of he Ba-Ng esmaors aled o varables measured wh error ( Y ˆ n he noaon of he las secon). he followng resul shows ha he Ba-Ng esmaors reman conssen n he resence of suffcenly small measuremen error. 7

11 Lemma : Suose (A)-(A9) are sasfed and X = X + b where, hen PC ( ) N = = = N b O ( s ) BN X r and ICP BN ( X ) r. (hese lemmas are based on a enaly facor ha sasfes s δ g(n,) for δ = (m )/m, where m s gven n assumon (A6). Ba and Ng (00) roose verson of PC BN and IPC BN usng δ =. he Ba and Ng (00) roof o he conssency heorem wh δ =, redcaed on assumons lke hose n (A.)-(A.9), conans an error. Ba and Ng (006) roves conssency wh δ = under he assumon e = Rξ H where R and H are N N and marces wh bounded egenvalues and ξ s a N marx of ndeendenly dsrbued random varables wh mean zero and bounded 7 h momens.). Conssen Esmaon of he Number of Dynamc Facors he resuls from Lemma sugges ha he esmaors PC BN ( Y ˆ ) and ICP BN ( Y ˆ ) wll be conssen for he number of dynamc facors f he error Ŷ Y s small. We consder wo versons of ˆ Y ha are suffcenly accurae for hs urose. Boh rely on a frs-sage esmae of F. hus, le ˆF and ˆΛ denoe he rncal comonens esmaors of F and Λ consruced from (.) usng a conssen esmaor of r. Le ( Φˆ ˆ ˆ, Φ,..., Φ ) denoe he OLS esmaors from he regresson of F ˆ ono ( Fˆ ˆ,..., F ). he frs verson of Ŷ s A ˆ = ΛΦ = Yˆ X ˆ Fˆ. (.4) 8

12 he second verson of ˆ Y uses drec esmaes of he regresson of X ono lags of F. Le ( Πˆ, Πˆ,..., Πˆ ) denoe he OLS esmaors from he regresson of X ono OLS OLS OLS Fˆ,..., Fˆ ( ). he second verson of Ŷ s B ˆ OLS ˆ = Π = Yˆ X F (.5) whch does no mose he cross-equaon consran Π = ΛΦ. Conssency of he Ba-Ng esmaor for q s hen readly shown f () he facor model (.3) for Y sasfes he analogues of condons (A.)-(A.9) above, and () he esmaors ˆF, ˆΛ, ˆΦ and Π ˆ OLS are suffcenly accurae. hus o begn, assume ha (A.)-(A.9) hold wh η relacng F and Γ relacng Λ. (Noe ha he normalzaon n (A.)-(A.) can be acheved by arorae choce of G.) Sock and Wason (00) and Ba (003) dscuss he accuracy of he esmaors ˆF and ˆΛ under assumons lke hose lsed as (A.)-(A.9) above. As n Ba and Ng (005), ˆΦ wll be / -conssen under a sandard se of assumons for he VAR for F : (A0) Le F = ( F,..., F ), hen () he sochasc rocess {F } s saonary and ergodc; () E(F F ) s non-sngular; () vec(f η ) s a marngale dfference sequence wh fne second momens. 9

13 Fnally, accuracy of Π ˆ OLS requres he addonal assumon: (A.) N F. = = E e = O( ) We hen have: heorem: Consder he model (.)-(.3). Suose ha (.) sasfes (A.)-(A.9), ha he analogous assumons are sasfed for (.3) and ha (A.0) s sasfed. hen (a) PC ( ˆ A BN Y ) q and ICP ( ˆ A BN Y ) q. (b) In addon, suose ha (A.) s sasfed. hen PC ( ˆ B BN Y ) q and ICP ( ˆ B BN Y ) q. he remanng ngreden n he esng roblem s, he number of lags n he VAR. I s sraghforward o show ha, under he usual VAR assumons, can be esmaed conssenly by BIC. In some models, nnovaons n a subse of he X varables may deend on only a subse of he dynamc shocks η. For examle n Bernanke, Bovn and Elasz (005) and Sock and Wason (005), X s aroned no a se of slow movng varables and oher varables, X = ( Slow X Oher X ) where nnovaons n Slow X deend on only a subse of he η. I s sraghforward o show ha he sze of hs subse can be conssenly esmaed (N Slow, ) usng IPC BN aled o he relevan subse of elemens of ˆ Y. 0

14 3. Comarng he Esmaors Usng Smulaed Daa 3. Exermenal Desgn he exermenal desgn s aken from Ba and Ng (005a) where four daa generang rocesses (DGPs) are consdered. DGPs. In he frs desgn (DGP), X = λ F + e and F = ΦF + Gη, where F s 5 and η s 3, so ha r = 5 and q = 3; {λ }, {e }, and {η } are muually ndeenden, wh {λ }and {η }..d. sandard normal random varables/vecors; Φ s a dagonal marx wh elemens (0., 0.375, 0.55, 0.75, 0.90), and he columns of G are randomly chosen from he un shere and are ndeenden of he oher random varables. o allow cross seconal deendence n he dosyncrac errors, e s N(0, Ω), where Ω j = ρ j. Resuls are resened for ρ = 0 and ρ = 0.5. DGP s he same as DGP, bu wh r = 3 and Φ = 0.5 I 3. In he fnal wo desgns, X s a movng average of facors f ha follow an AR (DGP3) or MA (DGP4) rocess. In DGP3, X = (λ 0 + λ L) f + e and f = φf + η, where f s, so ha r = 4 and q =. hs model can be wren as (.) and (.) wh F = (f f ), λ = (λ 0 λ ), Φ = φ I 0 0, and G = [ I 0 ]. he facor loadngs and errors are generaed as n DGP, and φ = 0.5 I. In DGP4, X = (λ 0 + λ L + λ L ) f + e and f = (I + ΘL)η, where f s, so ha r = 6 and q =. In hs desgn F = (f f f ), bu now F follows an MA rocess, so ha he VAR n (.) serves as an aroxmaon. he MA coeffcen marx s dagonal wh elemens 0. and 0.9.

15 Esmaors. he ICP BN esmaors are mlemened usng he enaly facor g(n,) = ln(s )/A, where A = /(N+). (hs s he ICP enaly facor n Ba and Ng (00).) r s esmaed usng ICP BN ( X ), where X s he sandardzed verson of he daa generaed by DGP-DGP4, and where r max = 0. q s esmaed usng ICP ( ˆ A BN Y ) and ICP ( ˆ B BN Y ) consruced usng an esmaed VAR() for F ˆ. wo alernave esmaors, ˆq 3 and ˆq 4 from Ba and Ng(005a), were also consruced. hese esmaors use he egenvalues of he resdual covarance marx of he VAR for F ˆ o esmae q. Secfcally, le ˆ ε ˆ ˆ = F ΦF where F ˆ s he ˆr vecor of facors esmaed by rncal comonens usng he normalzaon N ΛΛ= ˆ ˆ I r and FF ˆˆ = dag(σ ), ˆˆ εε = + denoe he ordered egenvalues of = Σ ˆ = ˆˆ ε ε denoe he esmaed covarance marx, and Σ ˆ ˆˆ εε by c c c ˆr. Le D,k = rˆ / k+ c = [ c / ] rˆ rˆ / = k+ = ; he esmaors are and D,k = [ c / c ] /5 qˆ 3 mn k[ k: D, k m/ s ] = <, and /5 qˆ 4 mn k[ k: D, k m/ s ] = <, where m s a osve consan. Followng Ba and Ng(005a) we mlemen hese esmaors usng m = Resuls Resuls are shown n able for each DGP and varous values of N and =00. Panel A shows resuls wh ρ = 0 (so ha he e errors are muually uncorrelaed) and anel B shows resuls wh ρ = 0.5 (so ha e are correlaed n he cross secon).

16 Lookng frs a anel A, fve resuls sand ou. Frs, he esmaors are que accurae for N as small as 50, a leas for he smle desgns consdered. All of he esmaors roduce he correc answer n more ha 98% of he smulaons when N = 50, and erform nearly as well when N = 40. Second, he consran Π = ΛΦ used by ICP ( ˆ A BN Y ) bu gnored by ICP ( ˆ B BN Y ) s useful: ICP ( ˆ A BN Y ) has a smaller roo mean squared error han ICP ( ˆ B BN Y ) n all of he cases consdered n he able. hrd, for DGP and DGP, ICP ( ˆ A BN Y ) acheves a hgher rooron of correc values of q han he oher esmaors; for DGP3 and DGP4, ˆq 3 acheves he hghes rooron of correc values. Fourh, n DGP, whle Σ FF has rank 5, of s egenvalues are small and IPC BN ( X ) ends o underesmae he number of sac facors when N s large. In se of hs, IPC ( ˆ a BN Y ) and IPC ( ˆ b BN Y ) accuraely esmae he number of dynamc facors. Fnally, comarng he resuls from DGP3 and DGP4, he AR aroxmaon for DGP4 does no aear o lead o a serous deeroraon of erformance n any of he esmaors. Panel B shows ha he erformance of he IPC BN deeroraes when here s cross seconal correlaon n he errors: IPC BN ( X ) ends o overesmae r, he number of sac facors, and, whle no as severe, hs uward bas s also evden n IPC ( ˆ a BN Y ) and IPC ( ˆ b BN Y ). ˆq 3 and ˆq 4 suffer only a small deeroraon n accuracy. Boh BN and ˆq rovde accurae esmaes of he number of dynamc facors when N = 00. IPC ( Y ˆ ) 3

17 4. Summary and Concludng Remarks hs aer has roosed a modfcaon of he Ba-Ng (00) esmaor and shown ha he modfcaon rovdes a conssen esmaor for he number of dynamc facors n an aroxmae dynamc facor model. he modfcaon uses a resul (Lemma ) ha shows ha he Ba-Ng esmaor remans conssen even when he daa are conamnaed wh a suably small amoun of error. hs resul may rove useful n oher sengs, for examle n models n whch he equaon for X has he form X = λ F + β Z + e, where Z are observed regressors and β mus be esmaed. We leave hese calculaons for fuure work. 4

18 Aendx hs aendx summarzes key deals of roofs o he resuls gven n he ex. A comlee se of roofs are avalable n he dealed aendx (D-Aendx hereafer) avalable a h:// Proof of Lemma : hs s a verson of heorem and Corollary n Ba and Ng (00) under slghly dfferen assumons. See D-Aendx for a dealed roof usng he assumons lsed above. Proof of Lemma : Le ω k denoe he k h ordered egenvalue of () XX. As shown n D-Aendx, Lemma s mled by () ω k ω k = o () for k r and () ω k ω k = O ( s δ ) for k > r. o verfy () and (), le µ denoe he larges egenvalue of () bb, hen ω + µ ( ω µ ) ω ω + µ + ( ω µ ) (.6) / / k k k k k follows from Horn and Johnson (99). By he assumon of he lemma, race(bb ) = O( s ), so ha µ = O ( s ). For k r ω σ (D-Aendx R), so k kk ha ω k ω k = o () for k =,, r follows from (.6), and hs shows (). For k > r, ω O ( s δ ( + δ )/ = ) (D-Aendx R8), hus (.6) mles ω ω = O ( s ) + O ( s ), k and hs shows (). 5 k k Proof of heorem : Le Φ = (Φ, Φ,, Φ ) and Π = ΛΦ, so ha F = ΦF + Gη and Y = X ΠF ; le π denoe he h row of Π and γ denoe he h row of Γ, hen X =

19 η γ + F π + e. he followng resuls are versons of heorem n Ba and Ng (00) (see D-Aendx): = = Fˆ J F O ( s ), where J J a non-sngular marx, (.7) = Fˆ J F = O ( s ), where J J a non-sngular marx, (.8) N N ˆ λ J λ = = O ( s ). (.9) he followng lemma s useful: Lemma 3: Le ˆ π denoe an esmaor of π and b = Fˆ ˆ π F π. If N ˆ = N π J π = O ( s ), hen N = = = O ( N b s ). Proof: Wre F ˆ = J F + ( F ˆ J F ) and ˆ π = J π + ( ˆ π J π ), so ha b = FJ ˆ J + ( π π ) ( ˆ F J F) J π + ( Fˆ J F) ( ˆ π J π ). hus N N ˆ π F J J π = = = = N b N N ˆ + F JF J N π = = + ˆ = F J F N N = ˆ π - J π 6

20 and he resul follows from F = O () (A.0), = J J <, N ˆ = N π J π = O ( s ) (assumon of he lemma), and = Fˆ J F = O ( s ) (from (.8)). Par (a) of heorem : he feasble OLS esmaor of Φ s ˆ ˆˆ ˆˆ Φ= FF FF. Usng F = ΦF + Gη, F ˆ can be wren as = + = + ˆ ˆ ˆ ˆ = Φ + η + Φ F J J F J G ( F J F) J J ( F J F ). hus ˆ - ˆ ˆ ˆ ( ) ( ˆ ) ˆ ΦJΦ J JG η F J F J = F + F ΦJ F JF F = + = + = + ˆˆ FF = + Sraghforward calculaons (see D-Aendx) show ha each of he erms η ˆ F, = + ( Fˆ ) ˆ JF = + F, and = + ( Fˆ J F) F ˆ are O ( s / ), and ha = + FF ˆˆ J E(F F )J whch s nonsngular. hus Φ ˆ J Φ J = O ( s ). / o comlee he roof, le π ˆ ˆ =Φ λ, wre ˆ J ( ˆ λ = λ + λ J λ ) and ˆ ˆ Φ= J Φ J +( Φ ˆ J ΦJ ), so ha π ˆ J π = J ΦJ ( ˆ λ J λ ) 7

21 + ( ˆ ΦJ J ) J λ + ( Φ ˆ J ΦJ )( ˆ λ J λ ). hus Φ N ˆ = N π J π = O ( s ) follows from Φ ˆ J Φ / J = O ( s ) and N ˆ λ J λ = O ( N = s ). Par (a) hen follows from Lemma 3. Par (b) of heorem : he feasble OLS esmaor of π s ˆ π OLS = ˆˆ ˆ = + = + FF F X. Usng X = F π + η γ + e = FJ ˆ π ( ˆ F J F) J π +η γ + e, ˆ π OLS ˆˆ J π FF = + = ˆ ˆ ˆ ( ) ˆ F F J π + η γ + F J F Fe. = + = + = + Sraghforward calculaons (see D-Aendx) show ha each of he erms N ˆ ˆ ( ) = = + N F F J F J π, N N F ˆ η γ, and = = N N F ˆ η γ are O ( = = s ), and ha = + FF ˆˆ J E(F F )J whch s nonsngular. hus N OLS ˆ = N π J π = O ( s ), and he resul follows from Lemma 3. 8

22 References Ba, J. (003), Inferenal heory for facor models of large dmensons, Economerca 7:35-7. Ba, J., and S. Ng (00), Deermnng he number of facors n aroxmae facor models, Economerca 70:9-. Ba, J. and S. Ng (005a), Deermnng he number of rmve shocks n facor models, Journal of Busness and Economc Sascs, forhcomng. Ba, J. and S. Ng (005b), Confdence nervals for dffuson ndex forecass and nference for facor-augmened regressons, manuscr, Unversy of Mchgan. Ba, J. and S. Ng (006), Deermnng he number of facors n aroxmae facor models, Erraa, manuscr, Unversy of Mchgan. Bernanke, B.S., J. Bovn and P. Elasz (005), Measurng he effecs of moneary olcy: a facor-augmened vecor auoregressve (FAVAR) aroach, Quarerly Journal of Economcs 0: Forn, M., M. Halln, M. L and L. Rechln (000), he generalzed facor model: denfcaon and esmaon, he Revew of Economcs and Sascs 8: Forn, M., M. Halln, M. L and L. Rechln (005), he generalzed dynamc facor model: One-sded esmaon and forecasng, Journal of he Amercan Sascal Assocaon 00: Gannone, D., L. Rechln, and L. Sala (003), Oenng he Black Box: Srucural Facor Models versus Srucural VARs, CEPR Dscusson Paer No

23 Gannone, D., L. Rechln and L. Sala (004), Moneary Polcy n Real me, NBER Macroeconomcs Annual, 004. Halln, M. and R. Lška (005), he generalzed dynamc facor model: deermnng he number of facor, mmeo, E.C.A.R.E.S. Unversé Lbre d Bruxelles. Horn, R.A. and C.R. Johnson (99), ocs n Marx Analyss, Cambrdge Unversy Press, Cambrdge. Sock, J.H., and M.W. Wason (00), Forecasng usng rncal comonens from a large number of redcors, Journal of he Amercan Sascal Assocaon 97:

24 N able : Smulaon Resuls j Cov(e e j ) = ρ A. ρ = 0 ICP ICP ( ˆ A BN Y ) ( ˆ B ˆq 3 4 BN Y ) ˆq ICP BN ( X ) < q = q > q rmse < q = q > q rmse < q = q > q rmse < q = q > q rmse < r = r > r A. DGP X = λ F + e ; F = ΦF + Gη ; r = 5, q = B. DGP X = λ F + e ; F = ΦF + Gη ; r = 3, q = C. DGP3 X = (λ 0 + λ L) f + e ; f = Φf + η ; r = 4, q = D. DGP4 X = (λ 0 + λ L + λ ) f + e ; f = (I + ΘL )η ; r = 6, q =

25 N able (Connued) B. ρ = 0.5 ICP ICP ( ˆ A BN Y ) ( ˆ B ˆq 3 4 BN Y ) ˆq ICP BN ( X ) < q = q > q rmse < q = q > q rmse < q = q > q rmse < q = q > q rmse < r = r > r A. DGP X = λ F + e ; F = ΦF + Gη ; r = 5, q = B. DGP X = λ F + e ; F = ΦF + Gη ; r = 3, q = C. DGP3 X = (λ 0 + λ L) f + e ; f = Φf + η ; r = 4, q = D. DGP4 X = (λ 0 + λ L + λ ) f + e ; f = (I + ΘL )η ; r = 6, q = Noes: he frs wo columns show he values of N and used n he smulaons. he nex four columns summarze he resuls for he esmaor ICP ( ˆ A BN Y ); he columns labeled < q, = q and > q shows he fracon of esmaes ha were less han, equal o, and greaer han q; he column labeled rmse s he roo mean square error of he esmaes. he same enres are rovded for he oher esmaors of q. he fnal hree columns summarze he resuls for he esmaes of r. Resuls are based on 5,000 smulaons.