Measuring the Effects of Monetary Policy: A Factor-Augmented Vector Autoregressive (FAVAR) Approach * Ben S. Bernanke, Federal Reserve Board

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1 Measuring he Effecs of Moneary Policy: A acor-augmened Vecor Auoregressive (AVAR) Approach * Ben S. Bernanke, ederal Reserve Board Jean Boivin, Columbia Universiy and NBER Pior Eliasz, Princeon Universiy irs draf: April 22 Revised: December 23 Absrac Srucural vecor auoregressions (VARs) are widely used o race ou he effec of moneary policy innovaions on he economy. However, he sparse informaion ses ypically used in hese empirical models lead o a leas wo poenial problems wih he resuls. irs, o he exen ha cenral banks and he privae secor have informaion no refleced in he VAR, he measuremen of policy innovaions is likely o be conaminaed. A second problem is ha impulse responses can be observed only for he included variables, which generally consiue only a small subse of he variables ha he researcher and policymaker care abou. In his paper we invesigae one poenial soluion o his limied informaion problem, which combines he sandard srucural VAR analysis wih recen developmens in facor analysis for large daa ses. We find ha he informaion ha our facor-augmened VAR (AVAR) mehodology explois is indeed imporan o properly idenify he moneary ransmission mechanism. Overall, our resuls provide a comprehensive and coheren picure of he effec of moneary policy on he economy. * hanks o Chrisopher Sims, Mark Wason, ao Zha and paricipans a he 23 NBER Summer Insiue for useful commens. Boivin would like o hank Naional Science oundaion for financial suppor (SES- 244).

2 . Inroducion Since Bernanke and Blinder (992) and Sims (992), a considerable lieraure has developed ha employs vecor auoregression (VAR) mehods o aemp o idenify and measure he effecs of moneary policy innovaions on macroeconomic variables (see Chrisiano, Eichenbaum, and Evans, 2, for a survey). he key insigh of his approach is ha idenificaion of he effecs of moneary policy shocks requires only a plausible idenificaion of hose shocks (for example, as he unforecased innovaion of he federal funds rae in Bernanke and Blinder, 992) and does no require idenificaion of he remainder of he macroeconomic model. hese mehods generally deliver empirically plausible assessmens of he dynamic responses of key macroeconomic variables o moneary policy innovaions, and hey have been widely used boh in assessing he empirical fi of srucural models (see, for example, Boivin and Giannoni, 23; Chrisiano, Eichenbaum, and Evans, 2) and in policy applicaions. he VAR approach o measuring he effecs of moneary policy shocks appears o deliver a grea deal of useful srucural informaion, especially for such a simple mehod. Naurally, he approach does no lack for criicism. or example, researchers have disagreed abou he appropriae sraegy for idenifying policy shocks (Chrisiano, Eichenbaum, and Evans, 2, survey some of he alernaives; see also Bernanke and Mihov, 998). Alernaive idenificaions of moneary policy innovaions can, of course, lead o differen inferences abou he shape and iming of he responses of economic variables. Anoher issue is ha he sandard VAR approach addresses only he effecs of unanicipaed changes in moneary policy, no he arguably more imporan effecs of he

3 sysemaic porion of moneary policy or he choice of moneary policy rule (Sims and Zha, 998; Cochrane, 996; Bernanke, Gerler, and Wason, 997). Several criicisms of he VAR approach o moneary policy idenificaion cener around he relaively small amoun of informaion used by low-dimensional VARs. o conserve degrees of freedom, sandard VARs rarely employ more han six o eigh variables. his small number of variables is unlikely o span he informaion ses used by acual cenral banks, who are known o follow lierally hundreds of daa series, or by he financial marke paricipans and oher observers. he sparse informaion ses used in ypical analyses lead o a leas wo poenial ses of problems wih he resuls. irs, o he exen ha cenral banks and he privae secor have informaion no refleced in he VAR analysis, he measuremen of policy innovaions is likely o be conaminaed. A sandard illusraion of his poenial problem, which we explore in his paper, is he Sims (992) inerpreaion of he so-called price puzzle, he convenional finding in he VAR lieraure ha a conracionary moneary policy shock is followed by a sligh increase in he price level, raher han a decrease as sandard economic heory would predic. Sims s explanaion for he price puzzle is ha i is he resul of imperfecly conrolling for informaion ha he cenral bank may have abou fuure inflaion. If he ed sysemaically ighens policy in anicipaion of fuure inflaion, and if hese signals of fuure inflaion are no adequaely capured by he daa series in he VAR, hen wha appears o he VAR o be a policy shock may in fac be a response of he cenral bank o new informaion abou inflaion. Since he policy response is likely only o parially offse he inflaionary pressure, he finding ha a policy ighening is followed by rising Leeper, Sims, and Zha (996) increase he number of variables included by applying Bayesian priors, bu heir VAR sysems sill ypically conain less han 2 variables. 2

4 prices is explained. Of course, if Sims explanaion of he price puzzle is correc, hen all he esimaed responses of economic variables o he moneary policy innovaion are incorrec, no jus he price response. A second problem arising from he use of sparse informaion ses in VAR analyses of moneary policy is ha impulse responses can be observed only for he included variables, which generally consiue only a small subse of he variables ha he researcher and policymakers care abou. or example, boh for policy analysis and model validaion purposes, we may be ineresed in he effecs of moneary policy shocks on variables such as oal facor produciviy, real wages, profis, invesmen, and many ohers. Anoher reason o be ineresed in he responses of many variables is ha no single ime series may correspond precisely o a paricular heoreical consruc. he concep of economic aciviy, for example, may no be perfecly represened by indusrial producion or real GDP. o assess he effecs of a policy change on economic aciviy, herefore, one migh wish o observe he responses of muliple indicaors including, say, employmen and sales, o he policy change. 2 Unforunaely, as we have already noed, inclusion of addiional variables in sandard VARs is severely limied by degrees-of-freedom problems. Is i possible o condiion VAR analyses of moneary policy on richer informaion ses, wihou giving up he saisical advanages of resricing he analysis o a small number of series? In his paper we consider one approach o his problem, which combines he sandard VAR analysis wih facor analysis. 3 Recen research in dynamic 2 An alernaive is o rea economic aciviy as an unobserved facor wih muliple observable indicaors. ha is essenially he approach we ake in his paper. 3 Lippi and Reichlin (998) consider a relaed laen facor approach ha also explois he informaion from a large daa se. heir approach differs in ha hey idenify he common facors as he srucural shocks, 3

5 facor models suggess ha he informaion from a large number of ime series can be usefully summarized by a relaively small number of esimaed indexes, or facors. or example, Sock and Wason (22) develop an approximae dynamic facor model o summarize he informaion in large daa ses for forecasing purposes. 4 hey show ha forecass based on hese facors ouperform univariae auoregressions, small vecor auoregressions, and leading indicaor models in simulaed forecasing exercises. Bernanke and Boivin (23) show ha he use of esimaed facors can improve he esimaion of he ed s policy reacion funcion. If a small number of esimaed facors effecively summarize large amouns of informaion abou he economy, hen a naural soluion o he degrees-of-freedom problem in VAR analyses is o augmen sandard VARs wih esimaed facors. In his paper we consider he esimaion and properies of facor-augmened vecor auoregressive models (AVARs), hen apply hese models o he moneary policy issues raised above. he res of he paper is organized as follows. Secion 2 lays ou he heory and esimaion of AVARs. We consider boh a wo-sep esimaion mehod, in which he facors are esimaed by principal componens prior o he esimaion of he facoraugmened VAR; and a one-sep mehod, which makes use of Bayesian likelihood mehods and Gibbs sampling o esimae he facors and he AVAR simulaneously. Secion 3 applies he AVAR mehodology and revisis he evidence on he effec of using long-run resricions. In our approach, he laen facors correspond insead o conceps such as economic aciviy. While complemenary o heirs, our approach allows ) a direc mapping wih exising VAR resuls, 2) measuremen of he marginal conribuion of he laen facors and 3) a srucural inerpreaion o some equaions, such as he policy reacion funcion. 4 In his paper we follow he Sock and Wason approach o he esimaion of facors (which hey call diffusion indexes ). We also employ a likelihood-based approach no used by Sock and Wason. Sargen 4

6 moneary policy on wide range of key macroeconomic indicaors. In brief, we find ha he informaion ha he AVAR mehodology exracs is indeed imporan and leads o broadly plausible esimaes for he responses of a wide variey of macroeconomic variables o moneary policy shocks. We also find ha he advanages of using he compuaionally more burdensome Gibbs sampling procedure insead of he wo-sep mehod appear o be modes in his applicaion. Secion 4 concludes. An appendix provides more deail concerning he applicaion of he Gibbs sampling procedure o AVAR esimaion. 2. Economeric framework and esimaion Le Y be an M vecor of observable economic variables assumed o have pervasive effecs hroughou he economy. or now, we do no need o specify wheher our ulimae ineres is in forecasing he Y or in uncovering srucural relaionships among hese variables. ollowing he sandard approach, we migh proceed by esimaing a VAR, a srucural VAR (SVAR), or oher mulivariae ime series model using daa for he Y alone. However, in many applicaions, addiional economic informaion, no fully capured by he Y, may be relevan o modeling he dynamics of hese series. Le us suppose ha his addiional informaion can be summarized by an K vecor of unobserved facors,, where K is small. We migh hink of he unobserved facors as diffuse conceps such as economic aciviy or credi condiions and Sims (977) firs provided a dynamic generalizaion of classical facor analysis. orni and Reichlin (996, 998) and orni, Hallin, Lippi, and Reichlin (2) develop a relaed approach. 5

7 ha canno easily be represened by one or wo series bu raher are refleced in a wide range of economic variables. Assume ha he join dynamics of (, Y ) are given by: (2.) ( L) ν Y =Φ + Y where Φ (L) is a conformable lag polynomial of finie order d, which may conain a priori resricions as in he srucural VAR lieraure. he error erm ν is mean zero wih covariance marix Q. Equaion (2.) is a VAR in (, Y ). his sysem reduces o a sandard VAR in Y if he erms of Φ (L) ha relae Y o are all zero; oherwise, we will refer o equaion (2.) as a facor-augmened vecor auoregression, or AVAR. here is hus a direc mapping ino he exising VAR resuls, and (2.) provides a way of assessing he marginal conribuion of he addiional informaion conained in. Besides, if he rue sysem is a AVAR, noe ha esimaion of (2.) as a sandard VAR sysem in Y ha is, wih he facors omied will in general lead o biased esimaes of he VAR coefficiens and relaed quaniies of ineres, such as impulse response coefficiens. Equaion (2.) canno be esimaed direcly because he facors are unobservable. However, if we inerpre he facors as represening forces ha poenially affec many economic variables, we may hope o infer somehing abou he facors from observaions on a variey of economic ime series. or concreeness, suppose ha we have available a number of background, or informaional ime series, collecively 6

8 denoed by he N vecor X. he number of informaional ime series N is large (in paricular, N may be greaer han, he number of ime periods) and will be assumed o be much greaer han he number of facors ( K + M << N ). We assume ha he informaional ime series observable facors Y by: X are relaed o he unobservable facors and he f y (2.2) X ' =Λ ' +Λ Y ' + e ' where f Λ is an N K marix of facor loadings, y Λ is N M, and he N vecor of error erms e are mean zero and will be assumed eiher weakly correlaed or uncorrelaed, depending on wheher esimaion is by principal componens or likelihood mehods (see below). Equaion (2.2) capures he idea ha boh Y and, which in general can be correlaed, represen pervasive forces ha drive he common dynamics of X. Condiional on he Y, he X are hus noisy measures of he underlying unobserved facors. he implicaion of equaion (2.2) ha X depends only on he curren and no lagged values of he facors is no resricive in pracice, as can be inerpreed as including arbirary lags of he fundamenal facors; hus, Sock and Wason (998) refer o equaion (2.2) wihou observable facors as a dynamic facor model. In his paper we consider wo approaches o esimaing (2.)-(2.2). he firs one is a wo-sep principal componens approach, which provides a non-parameric way of uncovering he space spanned by he common componens, C = ( ', Y ')', in (2.2). he 7

9 second is a single-sep Bayesian likelihood approach. hese approaches differ in various dimensions and i is no clear a priori ha one should be favored over he oher. he wo-sep procedure is analogous o ha used in he forecasing exercises of Sock and Wason. In he firs sep, he common componens, firs K+M principal componens of C, are esimaed using he X. 5 Noice ha he esimaion of he firs sep does no exploi he fac ha Y is observed. However, as shown in Sock and Wason (22), when N is large and he number of principal componens used is a leas as large as he rue number of facors, he principal componens consisenly recover he space spanned by boh and Y. ˆ is obained as he par of he space covered by Ĉ ha is no covered by Y. 6 In he second sep, he AVAR, equaion (2.), is esimaed by sandard mehods, wih replaced by ˆ. his procedure has he advanages of being compuaionally simple and easy o implemen. As discussed by Sock and Wason, i also imposes few disribuional assumpions and allows for some degree of crosscorrelaion in he idiosyncraic error erm e. However, he wo-sep approach implies he presence of generaed regressors in he second sep. o obain accurae confidence inervals on he impulse response funcions repored below, we implemen a boosrap procedure, based on Kilian (998), ha accouns for he uncerainy in he facor esimaion. 7 5 A useful feaure of his framework, as implemened by an EM algorihm, is ha i permis one o deal sysemaically wih daa irregulariies. In heir applicaion, Bernanke and Boivin (23) esimae facors in cases in which X includes boh monhly and quarerly series, series ha are inroduced mid-sample or are disconinued, and series wih missing values. 6 How his is accomplished depends on he specific idenifying assumpion used in he second sep. We describe below our procedure for he recursive assumpion used in he empirical applicaion. 7 Noe ha in heory, when N is large relaive o, he uncerainy in he uncerainy in he facor esimaes can be ignored; see Bai (22). 8

10 In principle, an alernaive is o esimae (2.) and (2.2) joinly by maximum likelihood. However, for very large dimensional models of he sor considered here, he irregular naure of he likelihood funcion makes MLE esimaion infeasible in pracice. In his paper we hus consider he join esimaion by likelihood-based Gibbs sampling echniques, developed by Geman and Geman (984), Gelman and Rubin (992), Carer and Kohn (994) and surveyed in Kim and Nelson (999). heir applicaion o large dynamic facor models is discussed in Eliasz (22). Kose, Orok and Whieman (2, 23) use similar mehodology o sudy inernaional business cycles. he Gibbs sampling approach provides empirical approximaion of he marginal poserior densiies of he facors and parameers via an ieraive sampling procedure. As discussed in Appendix A, we implemen a muli-move version of he Gibbs sampler in which facors are sampled condiional on he mos recen draws of he model parameers, and hen he parameers are sampled condiional on he mos recen draws of he facors. As he saisical lieraure has shown, his Bayesian approach, by approximaing marginal likelihoods by empirical densiies, helps o circumven he high-dimensionaliy problem of he model. Moreover, he Gibbs-sampling algorihm is guaraneed o race he shape of he join likelihood, even if he likelihood is irregular and complicaed. Idenificaion Before proceeding, we need o discuss idenificaion of he model (2.) (2.2), specifically he resricions necessary o idenify uniquely he facors and he associaed loadings. In wo-sep esimaion by principal componens, he facors are obained enirely from he observaion equaion (2.2), and idenificaion of he facors is sandard. 9

11 f f In his case we can choose eiher o resric loadings by Λ ' Λ / N = I or resric he facors by f / = I. Eiher approach delivers he same common componen Λ ' and he same facor space. Here we impose he facor resricion, obaining ˆ = Z ˆ, where he Ẑ are he eigenvecors corresponding o he K larges eigenvalues of X X, sored in descending order. his approach idenifies he facors agains any roaions. In he one-sep (join esimaion) likelihood mehod, implemened by Gibbs sampling, he facors are effecively idenified by boh he observaion equaion (2.2) and he ransiion equaion (2.). In his case, ensuring idenificaion also requires ha we idenify he facors agains roaions of he form * = A BY, where A is K K and nonsingular, and B is K M. We prefer no o resric he VAR dynamics described by equaion (2.), and so we need o impose resricions in he observaion equaion, (2.2). Subsiuing for in (2.2) we obain f * y f (2.3) X = Λ A + ( Λ + Λ A B) Y + e Hence unique idenificaion of he facors and heir loadings requires Λ f A = Λ f and Λ y + Λ f A B = Λ y. Sufficien condiions are o se he upper K K block of f Λ o an ideniy marix and he upper K M block of y Λ o zero. he key o idenificaion here is o make an assumpion ha resrics he channels by which he Y s conemporaneously affec he X s. In principle, since facors are only esimaed up o a roaion, he choice of he block o se equal o an ideniy marix should no affec he space spanned by he esimaed facors. he specific choice made resrics, however, he conemporaneous

12 impac of Y on hose K variables and herefore such variables should be chosen for ha block ha do no respond conemporaneously o innovaions in Y. A separae idenificaion issue concerns he idenificaion of innovaions in he VAR par of he model, such as idenifying moneary policy innovaions which is he subjec of he nex secion. Imporanly, AVAR approach affords flexibiliy in idenifying innovaions - once facors are esimaed sandard procedures (e.g., srucural VAR procedures as in Bernanke and Mihov, 998) can be applied. One cavea is ha use of he Gibbs sampling mehodology may impose significan compuaional coss when complex idenificaion schemes are employed. or example, if we impose resricions ha overidenify he ransiion equaion, we need o perform numerical opimizaion a each sep of he Gibbs sampling procedure. his may easily become excessively ime consuming. In par for compuaional simpliciy we use a simple recursive ordering in our empirical applicaion below. he wo mehods differ on many dimensions. A clear advanage of he wo-sep approach is compuaional simpliciy. However, his approach does no exploi he srucure of he ransiion equaion in he esimaion of he facors. Wheher or no his is a disadvanage depends on how well specified he model is, and from a comparison of he resuls from he wo mehods we may be able o assess wheher he advanages of joinly esimaing he model are worh he compuaional coss.

13 3. Applicaion: he dynamic effecs of moneary policy As discussed in he Inroducion, an exensive lieraure has employed VARs o sudy he dynamic effecs of innovaions o moneary policy on a variey of economic variables. A variey of idenificaion schemes have been employed, including simple recursive frameworks, conemporaneous resricions (on he marix relaing srucural shocks o VAR disurbances), long-run resricions (on he shape of impulse responses a long horizons), and mixures of conemporaneous and long-run resricions. 8 Alernaive esimaion procedures have been employed as well, including Bayesian approaches (Leeper, Sims, and Zha, 996). However, he basic idea in virually all cases is o idenify shocks o moneary policy wih he esimaed innovaions o a variable or linear combinaion of variables in he VAR. Once his idenificaion is made, esimaing dynamic responses o moneary policy innovaions (as measured by impulse response funcions) is sraighforward. he fac ha his simple mehod ypically gives plausible and useful resuls wih minimal idenifying assumpions accouns for is exensive applicaion, boh by academic researchers and by praciioners in cenral banks. Neverheless, a number of criiques of he approach have been made (see, for example, Rudebusch, 998). Here we focus on wo issues, boh relaed o he fac ha degrees-of-freedom problems necessarily limi he number of ime series ha can be included in an esimaed VAR. We hen evaluae he 8 Recursive frameworks are employed, iner alia, in Bernanke and Blinder (992), Sims (992), Srongin (995), and Chrisiano, Eichenbaum, and Evans (2). Examples of papers wih conemporaneous, nonrecursive resricions are Gordon and Leeper (994), Leeper, Sims, and Zha (996), and Bernanke and Mihov (998a). Long-run resricions are employed by Lasrapes and Selgin (995) and Gerlach and Smes (995). Gali (992) and Bernanke and Mihov (998b) use a mixure of conemporaneous and long-run resricions. aus and Leeper (997) and Pagan and Roberson (998) poin ou some dangers of relying oo heavily on long-run resricions for idenificaion in VARs. 2

14 abiliy of AVARs which, poenially, can include much more informaion han sandard VARs o ameliorae hese problems. irs, as emphasized by Bernanke and Boivin (23), cenral banks rouinely monior a large number of economic variables. One raionale for his pracice is ha many variables may conain informaion ha is useful in predicing oupu, inflaion, and oher variables which ener ino he cenral bank s objecive funcion (Sock and Wason, 22; Kozicki, 2). Sandard VARs of necessiy include only a relaively small number of ime series, implying ha he informaion se employed by he economerician differs from (is a subse of) ha of he moneary policy-makers. o he exen ha policy-makers reac o variables no included in he VAR, moneary policy shocks and he implied dynamic responses of he economy will be mismeasured by he economerician. 9 A possible example of he effecs of shock mismeasuremen is he price puzzle discussed in he Inroducion. We will check below wheher including broader informaion se amelioraes he price puzzle. Even if moneary policy shocks are properly idenified, sandard VAR analyses have he shorcoming ha he dynamic responses of only hose few variables included in he esimaed VAR can be observed. As discussed in he Inroducion, his limiaion may be a problem for a leas wo reasons. irs, for purposes boh of policy analysis and model validaion, i is ofen useful o know he effecs of moneary policy on a lenghy 9 Anoher source of mismeasuremen arises from he fac ha mos VAR sudies ypically use revised, as opposed o real-ime daa. Croushore and Evans (999) do no find his issue o be imporan for he idenificaion of moneary policy shocks, a view consisen wih evidence presened in a forecasing conex by Bernanke and Boivin (23). However, Orphanides (2) argues ha assessmen of ed policy depends sensiively on wheher revised or real-ime daa are used. 3

15 lis of variables. Second, he choice of a specific daa series o represen a general economic concep (e.g., indusrial producion for economic aciviy, he consumer price index for he price level ) is ofen arbirary o some degree, and esimaion resuls may depend on idiosyncraic feaures of he paricular variable chosen. o assess he effecs of moneary policy on a concep like economic aciviy, i is of ineres o observe he responses of a variey of indicaors of aciviy, no only one or wo. he AVAR framework is well-suied for addressing boh issues. irs, he esimaed sysem (2.)-(2.2) can be used o draw ou he dynamic responses of no only he main variables Y bu of any series conained in X. Hence he reasonableness of a paricular idenificaion can be checked agains he behavior of many variables, no jus hree or four. Second, one migh also consider consrucing he impulse response funcions of facors (or linear combinaions of he facors) ha can be shown o sand in for a broad concep like economic aciviy. Empirical Implemenaion We applied boh he wo-sep and one-sep (join esimaion) mehodologies o he esimaion of moneary AVARs. In our applicaions, X consiss of a balanced panel of 2 monhly macroeconomic ime series (updaes of series used in Sock and Wason, 998 and 999). hese series are iniially ransformed o induce saionariy. he descripion of he series in he daa se and heir ransformaion is described in Appendix B. he daa span he period from January 959 hrough Augus 2. One approach o his problem is o assume no feedback from variables ouside he basic VAR, ha is, a block-recursive srucure wih he base VAR ordered firs (see Bernanke and Gerler, 995). However, he no-feedback assumpion is dubious in many cases. 4

16 or he baseline analysis, we assume ha he federal funds rae is he only observable facor, i.e. he only variable included in Y. In doing so, we rea he federal funds rae as a facor and inerpre i as he moneary policy insrumen. his is based on he presumpion ha moneary policy has pervasive effec on he economy, Moreover, he federal funds rae should no suffer from measuremen error issues, which would oherwise imply he presence of an idiosyncraic componen in he federal funds rae. he laen facors are hen undersood as capuring real aciviy and general price movemens. A key advanage of his specificaion is ha we do no have o ake a sand on he appropriae measure of he real aciviy or inflaion. X. We order he federal funds rae las and rea is innovaions as moneary policy shocks, in he sandard way. his ordering imposes he idenifying assumpion ha laen facors do no respond o moneary policy innovaions wihin he monh. o implemen his idenificaion scheme, i is useful o define wo caegories of variables: slow-moving and fas-moving. A slow-moving variable is one ha is largely predeermined as of he curren period, while a fas-moving variable hink of an asse-price is highly sensiive o conemporaneous economic news or shocks. he classificaion of variables beween each caegory is provided in he daa Appendix. As discussed above, he join likelihood esimaion only requires ha he firs K variables in he daa se are seleced from he se of slow-moving variables and ha he recursive srucure is imposed in he ransiion equaion. or he wo-sep esimaion his idenificaion requires firs conrolling for he par of Ĉ ha corresponds o he federal funds rae. his is achieved in he following way. irs, slow-moving facors, s, are 5

17 esimaed as he principal componens of he slow-moving variables. Second, he following regression, (3.) C ˆ s = b ˆ s + by Y + e, is esimaed and ˆ consruced from Cˆ bˆ Y. Noe ha in so far as s ˆ and Y are correlaed, so are ˆ and Y. inally, he VAR in ˆ and Y, is esimaed and idenified recursively using his ordering. Y he recursive assumpion may be subjec o criicism if componens of he esimaed facors, no accouned for by he federal funds rae, neverheless respond conemporaneously o ineres rae shocks. One way o address his poenial problem would be o exrac slow-moving and fas-moving facors from he respecive blocks of daa and order he fas-moving facors afer he federal funds rae in he VAR ordering. However he fas-moving facors obained in his way follow ineres rae movemens very closely and consequenly inroduce collineariy in he sysem. We inerpre he resuls of his exercise as suggesing ha here is lile informaional conen in he fas-moving facors ha is no already accouned for by he federal funds rae. We herefore adhere o our original formulaion. Empirical Resuls Our main resuls are shown in igures -4 below. Each igure shows impulse responses wih 9% confidence inervals of a selecion of key macroeconomic variables o a moneary policy shock. igures and 2 show he resuls for he AVAR model wih 3 laen facors, esimaed by principal componens and likelihood mehods, respecively. 6

18 We used 3 lags bu employing 7 lags led o very similar resuls as found wih he greaer number of lags. Likelihood-based esimaes employed, ieraions of he Gibbs sampling procedure (of which he firs 2, were discarded o minimize he effecs of iniial condiions). o assure convergence of he algorihm, we imposed proper bu diffuse priors on parameers of he observaion and he VAR equaions. Prior specificaions are discussed in he Appendix. here seemed o be no problems achieving convergence, and alernaive saring values or he use of 2, ieraions gave essenially he same resuls. We sandardize he moneary shock o correspond o a 25- basis-poin innovaion in he federal funds rae. 2 An imporan pracical quesion is how many facors are needed o capure he informaion necessary o properly model he effec of moneary policy. Bai and Ng (22) provide a crierion o deermine he number of facors presen in he daa se, X. However, his does no address he quesion of how many facors should be included in he VAR and due o compuaional consrain canno be readily implemened in he likelihood-based esimaion. o explore he effec of increasing he number of facors, we hus consider an alernaive specificaion wih 5 laen facors. he resuls are repored in igures 3 and 4. Increasing he number of facors beyond his did no change qualiaive naure of our resuls. As we have discussed, an advanage of he AVAR approach is ha impulse response funcions can be consruced for any variable in he informaional daa se, ha is, for any elemen of X. his gives boh more informaion and provides a more We have also experimened wih fla priors which yielded he same qualiaive resuls. 2 Noe ha he figures repor impulse responses, in sandard deviaion unis, o 25 basis poins shock in he federal funds rae. 7

19 comprehensive check on he empirical plausibiliy of he specificaion. In ha respec, he mos successful specificaion, in erms of plausibiliy, appears o be he wo-sep principal componen approach wih 5 facors, repored in igure 3. In his case, he responses are generally of he expeced sign and magniude: following a conracionary moneary policy shock, real aciviy measures decline, prices evenually go down and money aggregaes decline. he dividend yields iniially jump above he seady sae and evenually go down. Overall hese resuls seem o provide a consisen and sensible measure of he effec of moneary policy. Noe ha we display only 2 responses of all 2 ha in principle could be invesigaed. he AVAR model appears successful in capuring relevan informaion. irs, he price puzzle is no presen in our AVAR model esimaed by wo-sep approach, even when only hree facors are included. Given ha our recursive idenificaion of he policy shocks is consisen wih exising srucural VARs ha display he price puzzle, our resul migh sugges ha a few facors are sufficien o properly capure he informaion ha Sims argued migh be missing from hese VARs. Second, increasing he number of facors generally ends o produce resuls more consisen wih convenional wisdom. his is paricularly obvious when comparing he response of money aggregaes for he 2-sep approach in igure and 3: he apparen liquidiy puzzle in igures disappears when more facors are included. he amoun of informaion included in he empirical analysis is hus crucial o yield a plausible picure of he effecs of moneary policy, and he AVAR approach shows some success a exploiing his informaion. Bu, as is obvious from he likelihood-based resuls repored in igures 2 and 4, informaion is no all he sory. In his case, responses of prices and money aggregaes are 8

20 very imprecisely esimaed and display boh a price and liquidiy puzzle. Increasing he number of facors does no appear o improve he resuls. his migh sugges in fac ha he policy shock has no been properly idenified. his is a possibiliy ha would be worh considering in fuure research. I is imporan o sress, however, ha alhough we considered a recursive idenificaion of he policy shock, here is nohing in our proposed approach oher han he compuaional consrains menioned above ha prevens using alernaive, non-recursive, idenificaion schemes. However, he fac ha he wosep approach is relaively successful wih he same idenificaion scheme migh sugges ha he likelihood-based esimaion suffers from he addiional srucure i imposes, which migh no be enirely suppored empirically. o assess if differences beween resuls of he wo esimaion mehods are due o heir alernaive idenificaion or he esimaion mehod iself, we also generaed facors under he same idenificaion. I was accomplished by seing loadings on Y o zero in he observaion equaion for he likelihood-based esimaion and by omiing a cleaning regression (3.) in case of he principal componens mehod. hese are he alernaive ways of parialling ou he effecs of he federal funds rae from he esimaed facors. As i urns ou, he wo ses of facors generaed in his way are significanly differen. he facors esimaed by principal componen fully explain he variance of likelihoodesimaed facors bu he opposie is no rue. Moreover, he principal componen facors have greaer shor run variaion. We inerpre hese findings as evidence ha he differences in idenificaion are no he sole source of he differences in resuls. Since i is he likelihood mehod ha imposes addiional srucure on he model, we may expec PC facors o carry more informaion. 9

21 While he wo mehods yield somewha differen responses for money aggregaes and he consumer price index, overall he poin esimaes of he responses are quie similar. We find i remarkable ha he wo raher differen mehods, producing disinc facor esimaes as discussed above, give qualiaively similar resuls. On he oher hand, he degree of uncerainy abou he esimaes implied by he wo mehods is quie differen. In fac, for some series such as he consumer price index and indusrial producion, he likelihood based approach yields much wider confidence inervals. his migh sugges ha he likelihood-based facors do no successfully capure informaion abou hese variables. he nex subsecion invesigaes his possibiliy by including in he se of observable facors, Y, he consumer price index and indusrial producion. VAR AVAR Comparison he benchmark specificaion considered hus far has he advanage of imposing minimal assumpions abou he common componens. In paricular, we did no impose specific observable conceps for real aciviy or prices. Our mehodology does no preven, however, assuming ha facors, oher han he federal funds rae, are also observed. or insance, we can expand Y o also include indusrial producion, as a measure of real aciviy and he consumer price index as a measure of prices. he resuling AVAR sysem hus ness a sandard VAR in he variables ha are direcly suggesed by sandard moneary models: a moneary policy indicaor, a real-aciviy measure and a price index. By comparing he resuls wih and wihou he facors, i is hen possible o deermine he marginal conribuions of he informaion conained in he facors. 2

22 he impulse response funcions from his alernaive AVAR specificaion are presened in igure 5, for no facor, one facor and hree facors. he igure also reproduces he response obained from he benchmark specificaion, wih he federal funds rae assumed o be he only observable facor. he op panel shows he resuls from he wo-sep esimaion and he boom panel from he likelihood-based esimaion. When here is no facor, i.e. he sandard VAR specificaion, here is a srong price puzzle and he response of indusrial producion is very persisen, inconsisen wih long-run money neuraliy. or he wo-sep esimaion, adding one facor o sandard VAR changes he responses dramaically. he price puzzle is considerably reduced and he response of indusrial producion evenually reurns oward zero. In his case, adding one facor appears o be all ha is needed. or he likelihood-based esimaion, adding hree facors ends o produce qualiaively he same responses as for he wo-sep esimaion, alhough somewha more pronounced. he esimaed facors from boh mehods hus seem o conain useful informaion, beyond ha already conained in he sandard VAR. An ineresing aspec of hese resuls is ha he responses from he wo-sep esimaion of he benchmark AVAR are essenially he same as he one obained from expanding he sandard VAR by hree facors from eiher esimaion mehods. his suggess ha he wo-sep esimaion of he benchmark AVAR properly capures informaion abou real-aciviy and prices, even hough no such measure is imposed as observable facor. his is no he case for he likelihood-based facors and his seems o explain, a leas in par, he appeared less successful for he benchmark AVAR. 2

23 his comparison suggess ha he AVAR approach is successful a exracing perinen informaion from a large daa se of macroeconomic indicaors. ha does no mean, however, ha he AVAR approach is he only way o obain reasonable resuls. here exis, of course, oher VAR specificaions ha could lead o reasonable resuls over some periods. or example, some auhors have improved heir resuls by adding variables such as an index of commodiy prices o he VAR. 3 Bu unless hese variables are par of he heoreical model he researcher has in mind, i is no clear on wha grounds hey are seleced, oher han he fac ha hey work. he advanage of our approach is o pu discipline on he process, by explicily recognizing in he economeric model he scope for addiional informaion. As a resul, he fac ha adding he commodiy price index or any oher variables fixes or no he price puzzle is no direcly relevan o his comparison. Variance Decomposiion Oher han impulse response funcions, anoher exercise ypically performed in he sandard VAR conex is variance decomposiion. his consiss of deermining he fracion of he forecasing error of a variable, a a given horizon, ha is aribuable o a paricular shock. Variance decomposiion resuls follow immediaely from he coefficiens of he MA represenaion of he VAR sysem and he variance of he srucural shocks. or insance he fracion variance of ( Y Yˆ ) due o he moneary + k + k policy shock could be expressed as: 3 or insance, Sims (992), Bernanke and Mihov (998) and Chrisiano, Eichenbaum and Evans (999). 22

24 var( Y ˆ + k Y + k ε var( Y Yˆ ) A sandard resul of he VAR lieraure is ha he moneary policy shock explains a relaively small fracion of he forecas error of real aciviy measures or inflaion. + k + k Bu, as emphasized by equaion (2.2), par of he variance of he macroeconomic variables comes from heir idiosyncraic componen, which migh reflec in par measuremen error and upon which business cycle deerminans should have no influence. As a resul, i is no clear ha he sandard VAR variance decomposiion provides an accurae measure of he relaive imporance of he srucural shocks. In his conex, he AVAR framework suggess a poenially more appealing version of his decomposiion, where he relaive imporance of a srucural shock is assessed relaive only o he porion of he variable explained by he common facors. More precisely, his variance decomposiion for X i can be expressed as: i + k MP Λ ˆ i var( C k C+ k Λ var( C Cˆ ) MP + ε + k ) Λ ) Λ i ' i ' where Λ denoes he i h f y line of Λ = [ Λ, Λ ] and i var( C Cˆ ε ) / var( C Cˆ ) is he sandard VAR variance decomposiion MP + k + k + k + k based on (2.). able repors he resuls for he same weny macroeconomic indicaors analyzed in he previous igures. hese are based on he wo-sep esimaion of he benchmark specificaion. he firs column repors he conribuion of he moneary policy shock o he variance of he forecas of he common componen, a he sixy-monh horizon. he second column conains he R 2 of he common componen for each of hese 23

25 variables. 4 he produc of he wo columns is he equivalen of he sandard VAR variance decomposiion. Apar from he ineres raes and he exchange rae, he conribuion of he policy shock is beween 3.2% and 3.2%. his suggess a relaively small bu sill non-rivial effec of he moneary policy shock. In paricular, he policy shock explains 3.2%, 2.9% and 2.6% of capaciy uilizaion, new orders and unemploymen respecively, and 7.6% of indusrial producion. Looking a he R 2 of he common componen, hree observaions sand ou. irs, he facors explain a sizeable fracion of hese variables, in paricular for he mos ofen used macroeconomic indicaors: indusrial producion (7.7%), employmen (72.3%), unemploymen (8.6%) and he consumer price index (86.9%). his confirms ha he AVAR framework, esimaed by he wo-sep principal componen approach, does capure imporan dimensions of he business cycle movemens. Second, given he R 2 of he common componens, he discrepancies beween he sandard VAR decomposiion and he one inroduced here are considerable: for insance, he sandard VAR decomposiion of indusrial producion would imply a conribuion of he policy shock equal o 5.3% insead of 7.6%, and for new orders, 8.% insead of 2.9%. inally, he R 2 of he common componens is paricularly low for he money aggregaes, being.3% for he moneary base and 5.2% for M2. his implies ha we should have less confidence on he impulse response esimaes for hese variables. Ineresingly, hese are variables for which he impulse response funcions from he wo esimaion mehods differ he mos. 4 Noe ha since R is assumed o be an observed facor, he corresponding R 2 is one by consrucion. 24

26 4. Conclusion his paper has inroduced a mehod for incorporaing a broad range of condiioning informaion, summarized by a small number of facors, in oherwise sandard VAR analyses. We have shown how o idenify and esimae a facoraugmened vecor auoregression, or AVAR, by boh a wo-sep mehod based on esimaion of principal componens and a more compuaionally demanding, Bayesian mehod based on Gibbs sampling. Anoher key advanage of he AVAR approach is ha i permis us o obain he responses of a large se of variables o moneary policy innovaions, which provides boh a more comprehensive picure of he effecs of policy innovaions as well as a more complee check of he empirical plausibiliy of he underlying specificaion. In our moneary applicaion of AVAR mehods, we find ha overall he wo mehods produce qualiaively similar resuls, alhough he wo-sep approach ends o produce more plausible responses, wihou having o impose explici measures of realaciviy or prices. Moreover, he resuls provide some suppor for he view ha he price puzzle resuls from he exclusion of condiioning informaion. he condiioning informaion also leads o reasonable responses of money aggregaes. hese resuls hus sugges ha here is a scope o exploi more informaion in empirical macroeconomic modeling. uure work should invesigae more fully he properies of AVARs, alernaive esimaion mehods and alernaive idenificaion schemes. In paricular, furher comparison of he esimaion mehods based on principal componens and on Gibbs sampling is likely o be worhwhile. Anoher ineresing direcion is o ry o inerpre he 25

27 esimaed facors more explicily. or example, according o he original Sims (992) hypohesis, if he addiion of facors miigaes he price puzzle, hen he facors should conain informaion abou fuure inflaion no oherwise capured in he VAR. he marginal conribuion of he esimaed facors for forecasing inflaion can be checked direcly. 5 5 Sock and Wason (999) and Bernanke and Boivin (23) have shown ha, generally, facor mehods are useful for forecasing inflaion. 26

28 Appendix A: Esimaion by Likelihood-Based Gibbs Sampling his appendix discusses he esimaion of AVARs by likelihood-based Gibbs sampling. or furher deails see Eliasz (22). o esimae equaions (2.) and (2.2) joinly via likelihood mehods, we ransform he model ino he following sae-space form: (A.) X Y f Λ = y Λ e + I Y (A.2) L Y = Φ( ) Y +ν where Y is an M vecor of observable economic variables in whose dynamic properies we are ineresed, is an K vecor of unobserved facors, and X is an N vecor of ime series ha incorporae informaion abou he unobserved facors, all as described in he ex. ime is indexed =,2,...,. he coefficien marices f Λ and y Λ are N K and M finie order d. he loadings N, respecively, and Φ(L) is a conformable lag polynomial of f Λ and y Λ are resriced as discussed in he ex. he error vecors e and ν are N and ( K + M ), respecively, and are assumed o be disribued according o e ~ N(, R) and ν ~ N(, Q) and R diagonal., wih e and ν independen (A.) is he measuremen or observaion equaion, and (A.2), which is idenical o (2.), is he ransiion equaion. Inclusion of Y in he measuremen equaion (A.) as 27

29 well as in he ransiion equaion (A.2) does no change he model bu allows for boh noaional and compuaional simplificaion. We ake a Bayesian perspecive, reaing he model s parameers f y θ = ( Λ, Λ, R, vec( Φ), Q) as random variables; and where we define vec (Φ) as a column vecor of he elemens of he sacked marix Φ of he parameers of he lag operaor Φ (L). Likelihood esimaion by muli-move Gibbs sampling (Carer and Kohn, 994), proceeds by alernaely sampling he parameers θ and he unobserved facors. o be more specific, define X = ( X, Y ), e = (,), and = (, Y ) and e rewrie he measuremen and ransiion equaions (A.) and (A.2) as (A.3) (A.4) X = Λ + e = Φ( L) + ν where Λ is he loading marix from (A.) and R cov( e e ) is he covariance marix R = augmened by zeros in he obvious way. or his exposiion we assume ha he order d of Φ (L) equals one, oherwise we would rewrie (A.4) in a sandard way o express i as ~ a firs-order Markov process (see Eliasz, 22). urher, le X = ( X, X,..., X ) be he 2 ~ hisory of X from period hrough period, and likewise define = (,,..., ). 2 Our problem is o characerize he marginal poserior densiies of ~ and θ, respecively ~ ~ p( ) = p(, θ dθ and ) ~ ~ ~ p( θ ) = p(, θ ) d, where p(, θ ) is he join poserior densiy and he inegrals are aken wih respec o he suppors of θ and 28

30 ~, respecively. Given hese marginal poserior densiies, esimaes of ~ obained as he medians or means of hese densiies. and θ can be o obain empirical approximaions o hese densiies, we follow Kim and Nelson (999, chaper 8) and apply muli-move Gibbs sampling o he sae-space model (A.3)- (A.4). he Gibbs sampling mehodology proceeds as follows: irs, choose a se of saring values for he parameers θ, say θ. Second, condiional on θ and he daa X ~, draw a se of values for ~, say ~ ~ from he condiional densiy p( X, θ ). ~ hird, condiional on he sampled values of ~ and he daa, draw a se of values of he ~ ~ parameers θ, say θ, from he condiional disribuion p( θ X, ). he final wo seps consiue one ieraion, and are repeaed unil he empirical disribuions of s ~ and s θ converge, where s indexes he ieraion. I has been shown (Geman and Geman, 994), ha as he number of ieraions s, he marginal and join disribuions of he sampled values of s ~ s and θ converge o he rue corresponding disribuions a an exponenial rae. In pracice, hough, convergence can be slow and should be carefully checked, for example by using alernaive saring values. More deails on each sep are given below.. Choice of θ In general, i is good pracice o ry a variey of saring parameer values o see if hey generae similar empirical disribuions. As Gelman and Rubin (992) argue, a single sequence from he Gibbs sampler, even if i has apparenly converged, may give a 29

31 false sense of securiy. A he same ime, in a problem as large as he one a hand, for which compuaional capaciy consrains he number of feasible runs, a meaningful choice of θ may be advisable. An obvious choice was o use parameer esimaes obained from principal componens esimaion of (A.) and he vecor auoregression (A.2). We consrained hese parameer esimaes o saisfy he normalizaion, discussed in he ex, ha he upper K ( K + M ) block of loadings Λ is resriced o be [ I, K M K ]. We used hese parameer esimaes as saring values for θ in mos runs, bu we have confirmed he robusness of he key resuls for alernaive saring values. or example, we also ried saring values such ha () vec (Φ) =, (2) Q = I, (3) Λ f =, (4) Λ y = OLS esimaes from he regression of X on Y, and (5) R = residual covariance marix from he regression of X on Y, and obained similar resuls o hose repored in he ex. ~ ~ 2. Drawing from he condiional disribuion p( X, θ ) As in Nelson and Kim (p. 9), he condiional disribuion of he whole hisory ~ ~ of facors p( X, θ ) can be expressed as he produc of condiional disribuions of facors a each dae as follows: ~ ~ ~ ~ (A.5) p X, θ ) = p( X, θ ) p(, X, θ ) ( + = 3

32 3 where ),...,, ( ~ 2 X X X X =. (A.5) relies on he Markov propery of, which implies ha ),, ( ),,,...,, ( 2 θ θ p p X X =. Because he sae-space model (A.3)-(A.4) is linear and Gaussian, we have (A.6),..., ), ( ~, ~, ), ( ~, ~,, = N N P X P X θ θ where (A.7) ),, ( ), ~, ( ),, ( ), ~, ( ), ~ ( ), ~ (,, θ θ θ θ θ θ Cov Cov E E Cov E X P X X P X = = = = = = where he noaion refers o he expecaion of condiional on informaion daed or earlier. o obain hese, we firs calculae and P,,...,2 =, by Kalman filer, condiional on θ and he daa hrough period, X ~, wih saring values of zeros for he facors and he ideniy marix for he covariance marix (Hamilon, 994). he las ieraion of he filer yields and P, which ogeher wih he firs line of (A.6) allows us o draw a value for. reaing his drawn value as exra informaion, we can move backwards in ime hrough he sample, using he Kalman filer o obain updaed

33 values of, and P, ; drawing a value of using he second line of (A.6); and coninuing in similar manner o draw values for, = 2, 3,...,. If he order d of Φ (L) exceeds one, as i does in our applicaions, hen lags of he facors appear in he sae vecor and Q is singular, as is P for any <., + (he singulariy of hese wo covariance marices follows from he fac ha, in his case, and + have common componens.) In his case we canno condiion on he full vecor + when drawing, bu only on he firs d elemens of +. Kim and Nelson (999, p. 94-6) show how o modify he Kalman filer algorihm in his case. ~ ~ 3. Drawing from he condiional disribuion p θ X, ). ( Condiional on he observed daa and he esimaed facors from he previous ieraion, a new ieraion is begun by drawing a new value of he parameers θ. Wih known facors, (A.3) and (A.4) amoun o sandard regression equaions, wih (A.3) specifying he disribuion of Λ and R, and (A.4) he disribuion of vec (Φ) and Q. Consider (A.3) firs. Because he errors are uncorrelaed, we can apply OLS o (A.3) equaion by equaion o obain Λˆ and ê. We se R ij =, i j and assume a proper (conjugae) bu diffuse Inverse-Gamma (3,.) prior for R ii. Sandard Bayesian resuls (see Bauwens, Lubrano and Richard, 999, p. 58) deliver poserior of he form: R X%, % ~ ig( R, +.) ii ii ( ) ( ) where ˆ i i R = 3 + eˆ ' eˆ +Λ '[ M + ( % ' % ) ] Λˆ. Here ii i i i i M denoes variance parameer in he prior on he coefficiens of he i -h equaion, Λ i, which, condiional on he drawn 32

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