PARAMETER IDENTIFICATION IN DYNAMIC ECONOMIC MODELS*

Aicles Auumn PARAMETER IDENTIFICATION IN DYNAMIC ECONOMIC MODELS Nikolay Iskev. INTRODUCTION Paamee idenifi caion is a concep which evey suden of economics leans in hei inoducoy economeics class. The usual exbook eamen of idenifi caion leads one o hink of idenifi caion as a echnical issue elevan only o empiical wok, and o egad idenifi caion poblems as caused by eihe defi ciencies of he available daa, o of he saisical mehodology used o esimae he models. In his noe I will ague ha he analysis of idenifi caion has an impoan economic modeling aspec, and ha i may be vey useful o eseaches who ae no ineesed in esimaion. I will focus he discussion on he class of dynamic sochasic geneal equilibium (DSGE) models which have become one of he main analyical ools of moden macoeconomics. The essence of my agumen is ha when he economic model supplies a complee chaaceiaion of he daa geneaing pocess, paamee idenifi caion may be eaed as a popey of he undelying heoeical model. Paamees will be unidenifi able o weakly idenifi ed if he economic feaues hey epesen have no empiical elevance a all, o vey lile of i. This may occu eihe because hose feaues ae unimpoan on hei own, o because hey ae edundan given he ohe feaues epesened in he model. These issues ae paiculaly elevan o DSGE models, which ae someimes ciicied of being oo ich in feaues, and possibly ovepaameeied (Chai, Kehoe, and McGaan, 9). A second eason why i is impoan o sudy idenifi caion is is economeic implicaions. The eliable esimaion of a model is impossible unless is paamees ae well idenifi ed. Again, his is cucial fo DSGE models as hei use fo quaniaive policy analysis ofen hinges upon having accuae paamee esimaes. Teaing paamee idenifi caion as a popey of he model means ha we can sudy i wihou a efeence o a paicula daa se. Such an a pioi appoach o idenifi caion is no always possible in economeics since ypically he elaionship beween he economic model and he obseved daa is known only paially. Fo insance, he degee of coelaion beween insumens and endogenous vaiables in he simple linea insumenal vaiables model depends on nuisance paamees which, in he absence of a fully-aiculaed economic model, have no sucual inepeaion. In conas, when we ae in a geneal equilibium seing, as in he case of DSGE models, all educed-fom paamees become funcions of sucual paamees. In his seing we can sudy how he insumens sengh is deemined by he popeies of he undelying model. The auho hanks he commens of João Sousa. The opinions expessed in he aicle ae hose of he auho and do no necessaily coincide wih hose of Banco de Pougal o he Euosysem. Any eos and omissions ae he sole esponsibiliy of he auho. Banco de Pougal, Economics and Reseach Depamen. Economic Bullein Banco de Pougal 69

Auumn Aicles In wha follows I will use hee examples, one puely saisical and wo simple DSGE models, o illusae he a pioi analysis of idenifi caion and he kind of quesions we can answe wih is help. The pesenaion hee is based on seveal papes: in Iskev (a) i is explained how o deemine if he paamees of a DSGE model ae idenifi ed; Iskev (a) shows how o evaluae he sengh of idenifi caion of idenifi ed paamees; Iskev (b) discusses he ole of obsevables in he esimaion of DSGE models.. A SIMPLE EXAMPLE In his Secion I use a simple model o discuss he poblem of idenifi caion and o explain he main idea behind he a pioi appoach o idenifi caion analysis. Conside he following auoegessive moving aveage (ARMA(,)) pocess: x = x ε ε, <, <, ε (, ) + N (.) Panel (a) of Cha shows obsevaions geneaed by (.) wih = =.4, =. Panel (b) shows he ealiaions of ε, =,..., T used o geneae he obsevaions fo x. The wo seies x and ε ae idenical. Cha OBSERVATIONAL EQUIVALENCE WITH AN ARMA(,) PROCESS (a) x =.4x + ɛ.4ɛ 3 3 (b) x = ɛ 3 3 4 4 6 8 4 4 6 8 Souce: Auho s calculaions. This example illusaes wha in economeics is called obsevaional equivalence: hee ae wo values of he veco of paamees θ =[,, ] ', θ = [.4,.4,] ' and θ = [,,] ', which can poduce he same obsevaions fo x. In fac, in he ARMA(,) model hee ae infi niely many such values; as long as is kep he same, and is equal o, he ealiaions of x would be indisinguishable fom hose of ε. The eason fo his obsevaional equivalence is easy o undesand if we conside he auocovai- 7 Banco de Pougal Economic Bullein

Aicles Auumn ance funcion (ACF), which fo an ARMA(,) pocess is given by: ( + ) =( x )= ( )( ) =( xx )= (.) xx h h h h =( )=, Fom h e defi niion i is clea ha = is equivalen o =, =, k k. Theefoe, when he auoegessive and moving aveage coeffi ciens ae equal, he ACF of he ARMA(,) pocess x is idenical o ha of he whie noise pocess ε. This implies ha we canno disinguish daa geneaed fom ARMA(,) pocess wih abiay = fom daa geneaed fom ARMA(,) pocess wih = =. Now conside Cha, which shows wo seies of obsevaions geneaed by (.) wih θ = [,,] ' (solid line) and θ = [.7,.8,] ' (dashed line), using he same ealiaions of ε. Clealy, he wo seies ae vey simila, hough no idenical. In his case we have an example of nea obsevaional equivalence: daa geneaed fom ARMA(,) model wih is diffi cul o disinguish fom daa geneaed by he model wih abiay = and he same value of. How can we deec obsevaional equivalence (lack of idenifi caion) and nea obsevaional equivalence (weak idenifi caion)? A poweful esul, due o Rohenbeg (97), povides a geneal necessay and suffi cien condiion fo idenifi caion, namely, ha he infomaion maix is non-singula. Cha NEAR OBSERVATIONAL EQU IVALENCE WITH AN ARMA(,) PROCESS 3 x = x + ɛ ɛ = = =.7, =.8 3 3 4 5 6 7 8 9 Souce: Auho s calculaions. Economic Bullein Banco de Pougal 7

Auumn Aicles As Rohenbeg (97) poins ou, he infomaion maix is a measue of he amoun of infomaion abou he unknown paamees available in he sample. A paamee is unidenifi ed when hee is no infomaion abou i in he sample, o if he exising infomaion is insuffi cien o disinguish ha paamee fom ohe paamees in he model. Boh cases esul in a singula infomaion maix. In he case of he ARMA (,) model, he infomaion maix is given by: I (, )= (.3) Fom (.4) we can compue he deeminan of I (, ) ( ) de ( I (, )) = (.4) ( ) ( )( ) Since non-singulaiy is equivalen o he deeminan of he maix being diffeen fom eo, fom (.4) i is immediae ha is necessay and suffi cien fo idenifi caion in he ARMA(,) model. The infomaion maix is also useful fo deecing weak idenifi caion poblems. A paamee is idenifi ed bu pooly when he infomaion in he sample is vey lile, o if i is baely possible o disinguish ha paamee fom he ohe paamees. In his case he infomaion maix has full ank, bu is vey close o being singula. The sengh of idenifi caion may be measued using he esul ha he asympoic covaiance maix of an effi cien esimao is equal o he invese of he infomaion maix divided by he sample sie. Thus, he asympoic vaiances of he esimaos of he ARMA paamees and ae: ˆ ( ) ( ) ( ) ( ) va( ) =, va ( ) = ˆ T( ) T( ) (.5) The fomulas in (.5) eveal ha he asympoic vaiances ae lage when. This suggess ha he esimaes of he auoegessive and moving aveage paamees will be vey impecise when hei ue values ae simila. Theefoe, and ae weakly idenifi ed. Noe ha boh vaiances in (.5) depend on he values of and. Thus, fo a given sample sie T, he sengh of idenifi caion of eihe paamee is deemined by he ue values of boh paamees. This can be seen vey clealy in Cha 3 which shows how he asympoic vaiances vay acoss diffeen egions in he paamee space. To gain some inuiion abou he elaionship beween he paamee values and he sengh of idenifi caion, conside he following decomposiion of he infomaion maix (.4) 7 Banco de Pougal Economic Bullein

Aicles Auumn Cha 3 ASYMPTOTIC VARIANCES OF THE PARAMETERS OF AN ARMA(,) PROCESS Souce: Auho s calculaions. ( )( ) I(, )= ( )( ) (.6) Noe ha he fi s and he las ems on he igh hand side ae he same diagonal maix wih elemens equal o he squae oos of he diagonal elemens of I (, ). This maix ells us how much infomaion hee is in he sample abou each paamee if he ohe paamee was known. Fo insance, ( ) /T is he asympoic vaiance of an effi cien esimao of if was known. Theefoe, he close is o, he moe infomaion hee is abou, fo a given value of. Similaly, he close is o, he moe infomaion hee is abou, fo a given value of. Nex, conside he maix in he middle. I is a coelaion maix which ells us how simila is he effec on he disibuion of x of a small change in one paamee, say, o ha of a small change in he ohe paamee. Noe ha I (, ) is singula only when he coelaion maix in (.6) is singula, which occus if and only if he off-diagonal elemen, is equal o -. In his case a ( )( ) small change, say incease in, is exacly he same as a small decease in he ohe paamee. When he coelaion is close o, bu diffeen fom in absolue value, he effec of changing one paamee is almos he same as, hough diffeen fom, ha of changing he ohe one. Theefoe, he middle em in (.6) accouns fo he loss of infomaion abou eihe paamee due o he unceainy egading he ue value of he ohe paamee. The infomaion maix appoach o idenifi caion is possible only when he disibuion of he daa is known. Wha if we can no o do no wan o assume ha ε in (.) nomally disibued? A easonable appoach in his case is o base he idenifi caion analysis on he ACF of x. As we aleady saw, i is saighfowad o esablish he non-idenifi abiliy of he auoegessive and moving aveage pa- Economic Bullein Banco de Pougal 73

Auumn Aicles amees a = using he heoeical ACF of he ARMA(,) pocess. Moe fomally, we may poceed as follows: le = [,,..., ] ' k be he veco of he fi s k -auocovaiances of x. Then θ is idenifi ed a θ if he ( k 3) -dimensional maix / θ has ank equal o 3 when evaluaed a θ. The inuiion behind his condiion is vey simple: he maix has full column ank (equal o he dimension of θ) if and only if he vecos /, /, / ae linealy independen. Fo his o hold i mus be impossible o mach he effec on he momens of changing one paamee by changing he ohe wo paamees. Tha is, each paamee plays a disinc ole in deemining he popeies of he model, which is wha idenifi caion equies. Weak idenifi caion, on he ohe hand, means ha he effec of changing one paamee on he momens of x can be appoximaed vey closely by ha of changing ohe paamees. This esuls in deivaives which ae almos linealy dependen; fo insance, having collineaiy beween / and / of nealy one (in absolue value) means ha he effec of changing on is vey simila o ha of changing. Table illusaes he momens-based appoach o idenifi caion in he ARMA(,) model. Columns o 4 show he values of he deivaives of he fi s auocovaiances when he ue values of he paamees ae = =, =. As we can see, he deivaives wih espec o and ae pefecly negaively coelaed. Thus he ank of / θ is only and he and ae no idenifi ed. Columns 5 o 7 similaly show he deivaives of evaluaed a =.7, =.8, =. The degee of collineaiy beween / and / is.98, which is high bu less han -. Thus, and ae sill idenifi ed hough weakly. Table DERIVATIVE OF THE ACF OF A NARMA PROCESS () i =, =, = =.7, =.8, = / / / / / /. -. 4. -.35.57 4.8 4. -4.. 3.3 -.9 -.35.6 -.6..85 -.3 -.4 3.64 -.64..5 -.4 -.7 4.6 -.6..57 -. -. 5. -...8 -.7 -.8 6.4 -.4.. -.49 -.6 7. -... -.34 -.4 8. -.. -.3 -.4 -.3 9. -.. -.5 -.7 -.. -.. -.5 -. -. Souce: Auho s calculaions. 74 Banco de Pougal Economic Bullein

Aicles Auumn 3. DSGE MODELS In his secion I discuss paamee idenifi caion in DSGE models. I will sa wih a bief ouline of he geneal seup and hen un o analysis of wo pooypical DSGE models. 3.. Genealiies A DSGE model is summaied by a sysem of non-linea equaions. Cuenly, mos sudies involving eihe simulaion o esimaion of DSGE models use linea appoximaions of he oiginal models. Tha is, he model is fi s expessed in ems of saionay vaiables, and hen lineaied aound he seady-sae values of hese vaiables. Once lineaied, mos DSGE models can be wien in he following fom: Γ () θ = Γ () θ E +Γ () θ +Γ () θ u (3.) + 3 whee is a m dimensional veco of endogenous and exogenous sae vaiables, and he s ucual shocks u ae independen and idenically disibued n -dimensional andom vecos wih Eu =, Eu u ' = I n. The elemens of he maices Γ, Γ, Γ and Γ ae funcions of a k dimensional veco of deep paamees θ, whee Θ R is a poin in. The paamee space Θ is 3 k k defi ned as he se of all heoeically admissible values of θ. Thee ae seveal algoihms fo solving linea aional expecaions models (see fo insance Blanchad and Kahn (98), Andeson and Mooe (985), Klein (), Chisiano (), Sims ()). Depending on he value of θ, hee may exis eo, one, o many sable soluions. Assuming ha a unique soluion exiss, i can be cas in he following fom = A( θ) + B( θ) u (3.) whee he ( m m) maix A and he ( m n) maix B ae unique fo each value of θ. The model in (3.) canno be aken o he daa diecly since some of he vaiables in ae no obseved. Insead, he soluion of he model is expessed in a sae space fom, wih a ansiion equaion given by (7), and a measuemen equaion x = s ( θ) + C ( θ) (3.3) whee x is a l -dimensional veco of obseved sae vaiables, s is a l -dimensional veco, and C is a l m maix. The log-likelihood funcion of he daa X = [ x,..., x ] T may be compued using he Kalman fi le if he sucual shocks u ae (assumed o be) joinly nomally disibued. In his case he expeced infomaion maix may be deived analyically as discussed in Iskev (8). Economic Bullein Banco de Pougal 75

Auumn Aicles 3.. Idenificaion in he RBC model The fi s model I conside is a vesion of he one-seco sochasic gowh model of Hansen (985) wih invesmen-specifi c echnology shock. Below I ouline he main feaues of he model. 3.. The model The epesenaive household pefeences ae chaaceied by he lifeime uiliy funcion: E c n = ( ln( ) ) (3.4) whee c is consumpion in peiod and n is he oal labo supplied by he household. Aggegae oupu is poduced using capial k and labo using he following poducion funcion: y =exp( ) k n α (3.5) whee is oal faco poduciviy and follows an AR() pocess: = ε, ε (, ) + (3.6) The law of moion fo aggegae capial is: k =( δ) k + exp( u ) i (3.7) + whee u is invesmen-specifi c echnology and follows an AR() pocess: u u u = u ε, ε (, ) u u + (3.8) The esouce consain of he economy is: c + i = y (3.9) 3.. Idenificaion analysis The model is log-lineaied aound he deeminisic seady sae of he vaiables, and he sysem is expessed as in (3.). Thee ae fou poenially obsevable vaiables: oupu, consumpion, hous woked and invesmen. Since hee ae only wo sucual shocks, we can use a mos wo vaiables o esimae he model wih maximum likelihood; hose may be any wo of he fo vaiables, o some linea combinaions of hem. The model has 8 deep paamees, which ae colleced in he veco θ = [ αδ,,,,,,, ]. u u Le us fi s conside he case of using only one vaiable. This is an useful execise as i ells us which vaiable is mos infomaive fo which of he (idenifi able) paamees In his case in he measuemen equaion (3.3) x and s ae scalas, and C is a ow veco wih in he posiion of he obseved 76 Banco de Pougal Economic Bullein

Aicles Auumn vaiable, and eos elsewhee. The idenifi abiliy of θ may be esablished using eihe he infomaion maix o he momen-based appoach. Boh show ha of he 8 paamees ae no idenifi ed; hese ae and δ, which, when hee is only one obsevable vaiable, and iespecively which one i is, canno be idenifi ed sepaaely. This is easy o see fom he fac ha he deivaives of he momens wih espec o and δ ae collinea. Howeve, if eihe o δ is known, he emaining 7 paamees ae idenifi ed. Table shows he elaive asympoic sandad deviaions, defi ned as sd( θˆ ) i θ i, wih each obsevable assuming ha eihe o δ ae known. Noe ha hee ae subsanial diffeences in he pecision wih which he paamees may be esimaed depending on which vaiable is used and also on whehe o δ is known. Fo insance, oupu (y) is mos infomaive fo α if is known and δ is esimaed, bu hous woked (n) is mos infomaive when is esimaed and δ is known. The eason why he elaive sandad deviaions ae epoed is ha hey povide a measue of he idenifi caion sengh which is independen of he value of he paamee. This pemis us o deemine which paamees ae elaively bee and which ae elaively wose idenifi ed. The esuls in Table sugges ha alhough i is possible o esimae mos paamees wih only one obsevable, he esimaes ae likely o be vey impecise. Wih wo obseved vaiables hee is much moe infomaion abou he paamees, and hus he esimaion unceainy, capued by he asympoic sandad deviaion, is gealy educed. This can be seen in Table 3, which epos he elaive asympoic sandad deviaions wih each pai of obsevables. Fom he able we can see ha all paamees ae idenifi ed; geneally, he bes idenifi ed paamees ae, and u, while he wos idenifi ed ae, and u. To deemine he causes fo why some paamees ae bee and ohe wose idenifi ed, we can use a decomposiion of he infomaion maix analogous o ha in equaion (.6). Using i, we can expess he elaive sandad deviaion fo a given paamee as a poduc of wo ems: a sensiiviy componen, which is lage fo paamees which do no play an impoan ole in he model, and a collineaiy componen, which is lage fo paamees whose ole in he model is easy o appoximae wih ohe paamees. This decomposiion is shown in Table 4. We can see ha he eason why Table IDENTIFCATION STRENGTH IN THE RBC MODEL WITH ONE OBSERVABLE Pa. ue c y i n c y i n α.67 87 35 35 6 97 97 797 3.5 98 33 656 68 fi xed fi xed fi xed fi xed δ.98 fi xed fi xed fi xed fi xed 6 98 667 6.. 87 65 98 8.4 89 46 58 47.95 5.6 37 9 5.6 37 9 u.97 8.6 7 3 8.6 7 3. 4 3 545 95 5 57 7497 97. 36 843 4 5 89 97 99 u Souce: Auho s calculaions. Noe: Each column of he able shows he elaive asympoic sandad deviaions of θ when hee is only one obseved vaiable (shown in he fi s ow)and eihe o δ is assumed known. The esuls ae obained using he expeced infomaion maix and T=. Economic Bullein Banco de Pougal 77

Auumn Aicles Table 3 IDENTIFCATION STRENGTH IN THE RBC MODEL WITH TWO OBSERVABLES Pa. ue (c,y) (c,i) (c,n) (y,i) (y,n) (i,n) α.67.35.74.3.85.37.88 δ.5.7.454.5.454.9.453 u u.98.4.5.5.8.5.. 3.5.958.78.4.64.76.95.33.59.33.5.33.43.97.5.5.5.5.5.5..99.38.95.64.83.346..64.66.59.679.58.77 Souce: Auho s calculaions. Noe: Each column of he able shows he elaive asympoic sandad deviaions of θ when hee ae only wo obseved vaiables (shown in he fi s ow) The esuls ae obained using he expeced infomaion maix and T=. is so well idenifi ed is ha is sensiiviy componen is vey low; his implies ha is a vey impoan deeminan of he empiical popeies of he model vaiables. On he ohe exeme is, which has vey lage sensiiviy componen, and because of ha is he wos idenifi ed paamee. Song collineaiy explains he diffeen sengh of idenifi caion of and u which have he same sensiiviy componens. Ohe paamees wih song collineaiy ae α, δ and u. As was aleady discussed in Secion, song collineaiy implies ha wo o moe paamees play simila ole in he model. I is ineesing o know wha hese paamees ae. A simple way o fi nd ou is o compue coeffi ciens of paiwise collineaiy, which measue how simila he effecs of wo paamees ae. This is done in Table 5 and we can see ha hee is a song negaive collineaiy beween u and u on one hand and beween and δ, on he ohe. Thus, having highe volailiy of he invesmen specifi c shock is simila o having lowe pesisence of he same shock, and having moe paien consumes is simila o having lowe depeciaion ae. Fuhemoe, we can also see ha when he included obsevables Table 4 SENSITIVITY AND COLLINEARITY IN THE RBC MODEL WITH TWO OBSERVABLES Pa. (c,y) (c,i) (c,n) (y,i) (y,n) (i,n) sens. col. sens. col. sens. col. sens. col. sens. col. sens. col. α.56 5.8.8 9.7.57 5.7.4 9.8.57 5.4.3 93. δ.9 5.4.87 5..9 5.9.45..9 5.8.5 3. u u.5 4.8.5 5..5 5.. 3.6.5 5.. 73.4.374.3.799..7..95....647..4.4.4 4..4.4.4 3.6.4.4.4 3..8 6.5.8 6.5.8 6.4.8 6.5.8 6.4.8 6.5.7 4..7 5.4.7 4..7 3.7.7 4..7 4.9.7 8.5.7 8.5.7 8.3.7 9.6.7 8..7. Souce: Auho s calculaions. Noe: Each column of he able shows he sensiiviy and collineaiy componens of he elaive asympoic sandad deviaions of θ when hee ae wo obseved vaiables (shown in he fi s ow).the esuls ae obained using he expeced infomaion maix and T=. 78 Banco de Pougal Economic Bullein

Aicles Auumn Table 5 STRONGEST PAIRWISE COLLINEARITY IN THE RBC MODEL WITH TWO OBSERVABLES Pa. (c,y) (c,i) (c,n) (y,i) (y,n) (i,n) pcol pa. pcol pa. pcol pa. pcol pa. pcol pa. pcol pa. α..7 δ -.97 -.95 -.74 α -.98.7 -.98 δ.7 -.98 α -.98 -.999 -.997 α -.97 δ -.95 α -.98 δ -.96 α -.98 δ -.999 α. α.4 α -.4 α.4 α -.6 α -.4 α -.56 -.65 α -.56 -.49 α -.56 -. u u.7 α.7 α.7 α.45 α.7 α -.7 δ Souce: Auho s calculaions. Noe: The able shows which paamees ae mos songly elaed o each deep paamee as well as he value of he paiwise collineaiy (pcol) coeffi ciens. The esuls ae obained using he expeced infomaion maix and T=. ae oupu and invesmen, α is songly collinea wih boh and δ. This means ha he effec of hese paamees on he momens and coss momens of oupu and invesmen ae diffi cul o disinguish. 3.3. Idenificaion in he New Keynesian model In his secion I conside a small-scale New Keynesian model sudied in An and Schofheide (7). A bief descipion of he model follows. 3.3. The model The epesenaive household maximies lifeime uiliy funcion ( C / A ) E [ ( )], τ s + s + s N + s s= τ (3.) subjec o a budge consain: PC + B + T = PW N + R B + PD + PSC, (3.) whee C + s is consumpion, N () j + s is hous woked, P is he pice of he fi nal good, W is he eal wage, R is he inees on he govenme n bonds B, D is he esidual eal pofi, T is lump-sum axes and SC is ne cash fl ow fom ading sae-coningen secuiies. A is sock of habi given by he level of echnology in he inemediae good seco, and evolves accoding o Δ + + N ln A = ln ln, ln = ln ε, ε (, ) Thee is a pefecly compeiive seco poducing a single fi nal good fom inemediae inpus Y () j using he echnology Economic Bullein Banco de Pougal 79

Auumn Aicles ν ν Y =( ( ) ) Y j dj (3.) The fi nal goods fi m maximie pofi s given by PY P () i Y () i di, (3.3) whee P() i is he pice of inemediae good Y () i. Inemediae goods ae poduced in a monopolisically compeiive seco. Each vaie y i is poduced by a single fi m using he following poducion echnology: Y ()= i AN () i (3.4) The inemediae goods fi m j maximies he pesen value of is fuue pofi s P () j E [ Q ( Y () j W N () j AC ())], j s + s + s + s + s + s + s s= P+ s (3.5) whee s Q + is he ime value o he consumes of a uni of he fi nal good in peiod s P() j AC j π Y j P ( j) ()= ( ) () is he cos of a djusing pices and π is he seady sae ae of infl aion. The cenal bank ses he nominal inees ae accoding o he following ule + ; R R π Y = exp( )( ) [( ) ( ) ], ψ ψ ε π π π Y (3.6) whee is he seady sae eal inees ae, ae, and ε N is a moneay policy shock. (, ) π is he goss infl aion ae, π is he infl aion age The govenmen collecs lump-sum axes in ode o fi n ance is consumpion so as o espec he following budge consain PG + B R = T + B, (3.7) whee G = ζ Y is govenmen consumpion in ems of fi nal good, and ζ = / g whee g is andom vaiable evolving accoding o g g ln g = ( )ln g + ln g + ε, ε N(, ) g g g 3.3. Idenificaion analysis Again, he model is log-lineaied aound he deemini sic seady sae of he vaiables, and he sysem may be expessed as in (3.). Thee ae fou poenially obsevable vaiables: oupu, consumpion, infl aion and he nominal inees ae. Since hee ae only hee sucual shocks, we can use a mos hee vaiables o esimae he model wih maximum likelihood. The model has 4 deep 8 Banco de Pougal Economic Bullein

Aicles Auumn paamees, which ae colleced in he veco θ = [ τ, ν,, ψ, ψ,,,,, π,,,, ]. g g Le us fi s conside idenifi caion wih only wo obseved vaiables. Two of he 4 paamees, and ν, ae no idenifi ed wih any pai of obsevables. Examining he deivaives of he momens shows ha his is due o he pefec collineaiy of he deivaives wih espec o hese wo paamees. Theefoe, if eihe one of he wo paamees is fi xed, he ohe one would be idenifi ed along wih he ohe paamees. An excepion o his conclusion is he case when only oupu and consumpion ae obseved. Then we have o fi x hee moe paamees, in addiion o ν o. Fo example, if we fi x ν, ψ, π and, we could idenify he emaining paamees. The eason why he (oupu,consumpion) pai is less infomaive is ha he behavio of he wo vaiables in he model is vey simila. Theefoe, consumpion adds vey lile infomaion o ha aleady conained in oupu. This can be seen in Table 6, which shows he asympoic sandad deviaions fo each pai of obsevables assuming ha some of he elemens of θ ae known. The esimaion unceainy of mos paamees is much lage, compaed o he ohe pais of obsevables, even hough moe paamees ae assumed known. Noe ha, as in he RBC model, hee is a subsanial diffeence in he infomaion conen of diffeen vaiables. Also, which pai of vaiables is bes o use fo esimaion depends on he paamees one is mos ineesed in. Fo insance, he policy esponse o infl aion paamee ψ is bes idenifi ed wih ( π, ) while he policy esponse o oupu gowh ψ is bes idenifi ed wih ( y, ). Nex, conside using hee ou of he fou obsevables o esimae θ. Table 7 epos he asympoic Table 6 PARAMETER IDENTIFCATION IN THE NKM MODEL WITH TWO OBSERVABLES Pa. ue (, y π ) (, y ) (,) yc ( π, ) ( π, c) (, c ) (, y π ) (, y ) (,) yc ( π, ) ( π, c) (, c) τ. 3.9 3.3 554 689.3 3.9 3.3 554 689.3 ν. fi xed fi xed fi xed fi xed fi xed fi xed 7.5 786 64 3.7 36 7.5 8.3 873 68 36 3. 35 fi xed fi xed fi xed fi xed fi xed fi xed ψ ψ g.5 79 8.9 6 9 79 8.9 6 9. 8 4 fi xed 3 93 7.7 8.4 4.5 fi xed 3 93 7.96.7.5 8 7.5 4.5.7.5 8 7.5 4.5.95.95.7. fi xed fi xed..7. fi xed.7..65.8.9 34.3.3.8.9 34.3.3.4 397 443 938 3845 56 785 397 443 938 3845 56 785 π 4..3 443 fi xed.3.3 78.3 443 fi xed.3.3 78.5.3.3.3 769.3.3.3.3.3 769.3.3 g. 6.9 fi xed 493 8 6.9 fi xed 493 8.8.7.. 596 76 8.7.. 596 76 8.45.5.7 7 39.5.5.5.7 7 39.5.5 Souce: Auho s calculaions. Noe: Each column of he able shows he elaive asympoic sandad deviaions of θ when hee ae wo obseved vaiables (shown in he fi s ow)and some deep paamees ae assumed known. The esuls ae obained using he expeced infomaion maix and T=. Economic Bullein Banco de Pougal 8

Auumn Aicles Table 7 PARAMETER IDENTIFCATION IN THE NKM MODEL WITH THREE OBSERVABLES Pa. ue (, y π, ) (, y π,) c ( yc,, ) ( π, c, ) (, y π, ) (, y π,) c ( yc,, ) ( π, c, ) τ..36.56.6.3.36.56.6.3 ν. fi xed fi xed fi xed fi xed.7.36 88.38.7 7.5.79.5 9.3.8 fi xed fi xed fi xed fi xed ψ ψ g.5.4.37.6 3.58.4.37.6 3.58..4.36.8 3.8.4.36.8 3.8.96.3..3.4.3..3.4.95.3.4.4.3.3.4.4.3.65.7.9.7..7.9.7..4 3.4 33. 6.3 3.39 3.4 33. 6.3 3.39 π 4..3.3 6..3.3.3 6..3.5.5.5.5.5.5.5.5.5 g..39.3.8.35.39.3.8.35.8.5.7.7..5.7.7..45.3.35.33.7.3.35.33.7 Souce: Auho s calculaions. Noe: Each column of he able shows he elaive asympoic sandad deviaions of θ when hee ae wo obseved vaiables(shown in he fi s ow) and and eihe ν o is assumed known. The esuls ae obained using he expeced infomaion maix and T=. sandad deviaions fo each iple of obsevables assuming ha eihe ν o is known. As in Table 6, which one of he wo paamees is fi xed has no effec on he sandad deviaion of he ohe paamees. Wos idenifi ed wih all combinaions of obsevables ae he esponse coeffi ciens of he Taylo ule ( ψ and ψ ), he pice sickiness and invese elasiciy of demand paamees ( and ν ), and he seady sae inees ae ( ); bes idenifi ed ae he inees ae smoohing paamee ( ) and he govenmen consumpion shock paamee g. Table 8 shows he decomposiions of he elaive sandad deviaions ino sensiiviy and collineaiy componens. Noe ha mos of he wos idenifi ed paamees ae also he ones wih he lages collineaiy componens. Thus, hese paamees ae pooly idenifi ed because hei effecs on he empiical popeies of he obsevables ae easy o mimic wih ohe paamees. An excepion is, which is pooly idenifi ed because of he vey lage sensiiviy componen. This implies ha he value of is of lile consequence empiically. Noe ha boh and π have huge collineaiy compo- nens when π is no among he obsevables. Fo example, he value fo π anslaes ino a muliple collineaiy coeffi cien of.999999875. This means ha π is almos impossible oo disinguish fom ohe model paamees unless is effec on infl aion is accouned fo. Compuing he paiwise collineaiy coeffi ciens, epoed in Table 9, shows ha when infl aion is no among he obsevables, he collineaiy beween π and is.966. Thee we also see ha he policy esponse o infl aion ψ is highly collinea wih eihe he pice sickiness paamee o he inees ae smoohing paamee () The muliple collineaiy coeffi cien measues he degee of collineaiy beween a given paamee and all ohe model paamees. 8 Banco de Pougal Economic Bullein

Aicles Auumn Table 8 SENSITIVITY AND COLLINEARITY IN THE NKM MODEL WITH THREE OBSERVABLES Pa. (, y π, ) (, y π,) c ( yc,, ) ( π, c, ) sens. col. sens. col. sens. col. sens. col. τ.45 8..43 3..7 9.7.3.7 ψ ψ g. 36.8.67.6.4 5.5.38.3. 6.4.73 8.9.9 36.5.6.6. 66.7.95 4.6.58.4.35 93.6. 44.5.3 34.3. 5.7. 5.7..9.4..6.5.3.4.9.4..9.78 3.5.88.4.43 7.9.597 5..55 93.4.43 7.9 π. 9.7.33 9.3.53 3.3.7 8..8 3..47..7.4.8 3. g.7 5.5.7 33..7 4..7 5..7..7..7..7 3.4.7 4.3.7 4.9.7 4.6.7.4 Souce: Auho s calculaions. Noe: Each column of he able shows he sensiiviy and collineaiy componens of he elaive asympoic sandad deviaions of when hee ae hee obseved vaiables (shown in he fi s ow).the esuls ae obained assuming ν =. is known, and using he expeced infomaion maix wih T=. Table 9 STRONGEST PAIRWISE COLLINEARITY IN THE NKM MODEL WITH THREE OBSERVABLES Pa. (, y π, ) (, y π,) c ( yc,, ) ( π, c, ) pcol. pa. pcol. pa. pcol. pa. pcol. pa. τ -.76 ψ ψ g.96.9 -.9 π.99 ψ -.9 -.96.94 π.96 g g -.7 ψ -.9.94. g -.7 ψ.4 -.97 -.87.99 g -.4.87.98.99.3 -.95 π.76 -.9 ψ -.89.99 g -.4.87 -.68.9 π -.89 ψ -.89 -.95 τ ψ -.89 ψ -..95.97.97 π.9.76 π.94.9.94 g -..95 g -.9.8.35 g -.9.8 Souce: Auho s calculaions. Noe: The able shows which paamees ae mos songly elaed o each deep paamee as well as he value of he paiwise collineaiy coeffi ciens (pcol). The esuls ae obained assuming ν =. is known, and using he expeced infomaion maix wih T=. Economic Bullein Banco de Pougal 83

Auumn Aicles, while he esponse o oupu ψ is highly collinea wih eihe ψ o. 4. CONCLUDING REMARKS In he ecen yeas DSGE models ae inceasingly becoming an impoan ool fo quaniaive policy analysis. This has lead o a consideable eseach effo aimed o inceasing he models complexiy and ealism. As he numbe of numbe of feaues epesened in he models inceases, i becomes vey diffi cul o undesand by easoning alone hei sepaae conibuion o he model pefomance vis-a-vis he ealiy hey ae supposed o explain. In his noe I have ied o show ha sudying paamee idenifi caion may povide useful insighs egading he model paamees and he sucual feaues hey epesen. The sengh of paamee idenifi caion efl ecs hei impoance in deemining he empiical implicaions of he model. Weak idenifi caion aises when some model feaues have lile empiical elevance; his may occu eihe because hey ae unimpoan on hei own, o because hey ae edundan given he ohe feaues epesened in he model. Since DSGE models povide a complee chaaceiaion of he dynamics of he model vaiables, paamee idenifi caion may be eaed as a popey of he undelying model and sudied wihou a efeence o a paicula daa se. I have illusaed his appoach o paamee idenifi caion using wo canonical macoeconomic model - a eal business cycle model and a new Keynesian model. One limiaion of his analysis is ha only a single paamee value was consideed. To obain a complee picue of idenifi caion as a popey of he model, one has o sudy i acoss diffeen heoeically plausible paamee values. Fo a moe deailed discussion of his and ohe impoan aspecs of he a pioi analysis of idenifi caion, he eade may consul he papes cied in he inoducion. 84 Banco de Pougal Economic Bullein

Aicles Auumn REFERENCES An, S., e F. Schofheide (7): Bayesian Analysis of DSGE Models, Economeic Reviews, 6(- 4), 3-7. Andeson, G., e G. Mooe (985): A linea algebaic pocedue fo solving linea pefec foesigh models, Economics Lees, 7(3), 47-5,hp://ideas.epec.og/a/eee/ecole/ v7y985i3p47-5.hml. Blanchad, O. J., e C. M. Kahn (98): The Soluion of Linea Diffeence Models unde Raional Expecaions, Economeica, 48(5), 35-, hp://ideas.epec.og/a/ecm/emep/ v48y98i5p35-.hml. Chai, V. V., P. J. Kehoe, e E. R. McGaan (9): New Keynesian Models: No Ye Useful fo Policy Analysis, Ameican Economic Jounal: Macoeconomics, (), 4-66. Chisiano, L. J. (): Solving dynamic equilibium models by a mehod of undeemined coeffi ciens, Compuaional Economics, (-). Hansen, G. D. (985): Indivisible labo and he business cycle, Jounal of Moneay Economics, 6(3), 39--37, hp://ideas.epec.og/a/eee/moneco/v6y985i3p39-37.hml. Iskev, N. (8): Evaluaing he infomaion maix in lineaied DSGE models, Economics Lees, 99(3), 67-6. (a): Local idenifi caion in DSGE models, Jounal of Moneay Economics, 57(), 89-. (b): On he choice of obsevables in DSGE models, mimeo. Klein, P. (): Using he genealied Schu fom o solve a mulivaiae linea aional expecaions model, Jounal of Economic Dynamics and Conol, 4(), 45-43, hp://ideas.epec. og/a/eee/dyncon/v4yip45-43.hml. Rohenbeg, T. J. (97): Idenifi caion in Paameic Models, Economeica, 39(3), 577-9, hp:// ideas.epec.og/a/ecm/emep/v39y97i3p577-9.hml. Sims, C. A. (): Solving Linea Raional Expecaions Models, Compuaional Economics, (- ), -, hp://ideas.epec.og/a/kap/compec/vyi-p-.hml. Economic Bullein Banco de Pougal 85