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Working Paper Series Maximum Likelihood in the Frequency Domain: A Time To Build Example Lawrence J. Christano and Robert J. Vigfusson Working Papers Series Research Department (WP-99-4) Federal Reserve Bank of Chicago M aximum Likelihood inthe Frequency Domain: A T ime to B uild E xample Law rence J .Christano and R ob ert J .Vigf usson¤ Ab strac t A w ellknow nresult isthat the G aussianlog-likelihood c anb e expressed asthe sum over d i®erent f requency c om ponents.T hisim pliesthat the likelihood ratio statistic hasa sim ilar linear d ec om position.W e exploit these ob servationsto d evise d iagnostic m ethod sthat are usef ulf or interpreting m aximum likelihood param eter estim atesand likelihood ratio tests. W e apply the method sto the estim ationand testingoftw o realb usinessc yc le m od els.T he stand ard realb usinessc yc le mod elisrejec ted inf avor ofanalternative inw hich c apital investment requiresa planningperiod . J E L Cod es:E 2 ,E 2 2 ,E 1, C52 ,C32 ,C12 . ¤T he ¯rst author isgratef ulf or ¯nancialsupport f rom a NationalSc ience Found ation grant to the NationalB ureauofE c onom ic R esearc h.O pinionsexpressed inthispaper are those ofthe authorsand are not nec essarily those ofthe Fed eralR eserve B ank ofChic ago or ofthe Fed eralR eserve System . 1 In trod uc tion Fullinf orm ationec onom etric m ethod sinem piric alm ac roec onom ic s°ourished inthe early 19 80 s, stimulated inlarge part b y the w ork ofHansenand Sargent (1980 ). Sub sequently, lim ited inf orm ationm ethod sb ec am e m ore popular.A prom inent exam ple ofsuch m ethod s isthe c alib rationm ethod ology ad voc ated b y K yd land and P resc ott (19 82 ).1 T hisshif t in interest re°ec ted increased c oncernw ith the notionthat, since m od els are ab strac tions, any mod elisnec essarily misspec i¯ed onsom e d im ensions.A key perc eived shortc om ing of f ullinf orm ationmethod sisthat these spec i¯c ationerrorshave unpred ic tab le and hard to 2 d iagnose im plic ationsf or the parameter valuesand f or m od el¯t. M ore rec ently,it hasb een emphasized that lim ited inf orm ationm ethod shave their ow nprob lem s.For exam ple, their 3 sm allsample propertiesm ay b e poor c om pared w ith those off ullinf orm ationm ethod s. Consid erationssuc h asthese have helped to renew interest inf ullinf orm ationm ethod sin 4 empiric alm ac roec onom ic s. O ur ob jec tive isto d raw attentionto the potentialvalue ofthe f requency d om ainf or d iagnosing estim ationand testing results b ased onf ullinf orm ation, G aussianm aximum 5 likelihood m ethod s. Inthe proc essofillustrating these m ethod s, w e provid e evid ence in f avor ofa partic ular c lassofb usinessc yc le m od els. W e propose a set oftoolsf or evaluatingthe im pac t onparam eter estim ationand m od el ¯t ofd i®erent f requency c om ponentsofthe d ata.W e exploit the w ellknow nf ac t that the log, G aussiand ensity f unctionhasa linear d ec om positioninthe f requency d om ain. T his d ec ompositionhastw o implic ations.F irst, the likelihood ratio statistic f or testing a m od el c anb e represented asthe sum oflikelihood ratiosinthe f requency d om ain.Asa result,ifa m od elisrejec ted b ec ause ofa large likelihood ratio statistic ,thenit ispossib le to d eterm ine, 1 O ther, related methods include those based on H ansen and Singleton' s (1 982)generalized method of moments (G M M )framework. T hese include the exactly identi¯ed G M M methodology ofChristiano and Eichenbaum (1 992),and theoveridenti¯ed G M M known as simulated method ofmoments (D u±eand Singleton1 993).Inaddition,therearethediagnosticmethodsproposedbyW atson(1 993)andD iebold,O hanian and B erkowitz (1 998). 2 A recentexamplebyH ansenandSargent(1 993)illustratestheprinciple.T heyshowhowmisspeci¯cation ofthe seasonalcomponentofa modelcan, usingmaximum likelihood methods, lead todistortions in the estimated values ofallmodelparameters. 3 T he 1 996 issue ofthe JournalofB usiness and EconomicStatistics reports evidence on the smallsample properties oflimited information methods based on generalized method ofmomentestimators.Fora particularempiricalapplication, Fuhrer, M oore, and Schuh (1 995)made the case that the smallsample problems aresoseverethatmaximum likelihoodperforms betterthanlimitedinformationmethods,evenin thepresenceofplausibleforms ofspeci¯cation error.Cogley(1 998),however,displays an examplein which G M M performs betterthan maximum likelihood when thetechnologyshockis misspeci¯ed. 4 R ecentexamples includeA ltug(1 989);Christiano(1 988);Christiano,Eichenbaum,andM arshal(1 991 ); M cG ratten (1 994);H all (1 996);Ireland (1 997);Kim (1 998);L eeper and Sims (1 994);and M cG rattan, R ogerson,and W right(1 997). 5 A notherpaperwhich does this is A ltug(1 989).H ermethods complementours. 1 arithm etic ally,w hic h f requenciesofthe d ata are responsib le f or the poor m od el¯t.Sec ond , ifparam eter estim ateslook `strange',thenit ispossib le to d eterm ine w hich f requenciesare responsib le. W e illustrate the method b y estim ating and testing sim ple realb usinessc yc le m od els usingd ata onaggregate,quarterly,U S output grow th.W e start w ith a stand ard realb usiness c yc le m od el,inw hic h the tec hnology shoc kisa geom etric rand om w alk.W e ¯rst w orkw ith a versionofthe m od elinw hic h the only f ree param eter isthe variance ofthe technology shoc k. Allother param etersare ¯xed at the estim ated valuesreported inChristiano and E ic henb aum (199 2 ).T he likelihood ratio statistic ,testingthism od elagainst anunrestric ted alternative, rejec tsthe mod el.W henw e exam ine the likelihood ratio statistic inf requency d om ain, the reasonf or the rejec tionisc lear. T he m od el¯t isvery poor intw o f requency b and softhe d ata: those c orrespond ingto period sofosc illationinthe range of2 .5 - 8 years and those c orrespond ing to period sofosc illationinthe range of7- 7.5 m onths.W henw e f ree up som e ofthe other m od elparam eters, our G aussianestim ationc riteriond rivesthem into regionsthat c ause the m od elto c onf orm b etter to the d ata over allf requency b and s. How ever,the estim ated parameter valuesappear im plausib le onother ground s.O verall,our resultsare c onsistent w ith the ¯nd ingsreported inChristiano (1988, p. 2 74 ), Cogley and Nason(19 95),and W atson(19 93).T he poor ¯t inthe 2 .5- 8 year range re°ec tsthe d i± c ulty the stand ard realb usinessc yc le m od elhasingeneratingoutput persistence. W e next c onsid er a versionofthe realb usinessc yc le m od elw here c apitalinvestm ent requires f our period s to b uild . W e estim ate the f rac tionofoverallresourc es that m ust b e put into plac e ineac h ofthe ¯rst, sec ond , third , and f ourth period s ofc onstruc tion. T he parameter estim ates are plausib le f rom the perspec tive ofm ic roec onom ic stud ies of investment projec ts. T hey im ply that the am ount ofresourc esalloc ated inthe early part ofaninvestm ent projec t isrelatively sm all.For reasonsexplained inChristiano and T od d (19 96),incorporatingthisf eature ofinvestm ent projec tsinto the tim e to b uild m od elallow s that m od elto generate persistence inoutput grow th.T hisinturnhelpsthe m od elto m atch the 2 . 5 - 8 year c omponent ofthe d ata.Inad d ition, the estim ated m od elalso d oesw ellin m atc hingthe 7¡7:5 m onth c om ponent ofthe d ata.Asa result,our tim e to b uild m od elis not rejec ted b y the d ata. W e now c onsid er the relationship ofour paper to the existing literature.Severalother papersexploit the f ac t that the G aussiand ensity f unctionc anb e d ec om posed inthe f requencyd om ain.For example,Altug(1989) d em onstratesitsvalue f or estim atingm od elsw ith m easurement error.O ther papersem phasiz e itsvalue inthe estim ationoftim e-aggregated 6 m od els. Christiano and E ic henb aum (1987) and Hansenand Sargent (199 3) exploit the d e6 See, forexample, H ansen and Sargent(1 980 a), Christiano(1 985), Christianoand Eichenbaum (1 987) 2 c om positionto evaluate the c onsequencesf or m aximum likelihood estim atesofc ertaintypes 7 ofmod elspec i¯c ationerror. T he value ofc omparing mod eland d ata spec tra hasalso b eenem phasized inthe rec ent c ontrib utionsofW atson(19 93) and Dieb old , O hanianand B erkow itz (19 98).W atson (19 93)'sob jec tive isto provid e d esc riptive toolsonly, and so hisapproach isnot d esigned f or c ond uc tingstatistic alinf erence.O ursis, since our m ethod sare sim ply d esigned to help interpret the resultsofstand ard statistic alestim ationand testingproc ed ures. O ur approac h ismost c losely related to that ofDieb old ,O hanianand B erkow itz (19 98). T hey also d o estimationusingthe f requency d om aind ec om positionofthe G aussiand ensity f unction. T heir paper d i®ersf rom oursinthree respec ts. F irst, they use the f requency d om aind ec om positionasa c onvenient w ay to exc lud e f requency b and sf rom the analysis. W e incorporate allf requency b and sinto our analysis,and use the f requency d om aind ec om positionasa d evic e f or gaining insight into the resultsofanalysisb ased onallf requencies. Sec ond , their approac h to testing isd i®erent f rom ours. W e f oc usonthe likelihood ratio statistic and the value ofthe f requency d om ainf or d iagnosing itsm agnitud e. T hird , the applic ationw e use to illustrate the m ethod d i®ersf rom theirs. T he f ollow ingsec tionpresentsour ec onom etric f ram ew ork.Sec tion3presentsthe results. Sec tion4 c onclud es. 2 E c onom etric Fram ew ork T his sec tiond esc rib es the ec onometric f ram ew ork ofour analysis. F irst, w e d isplay the f requency d omaind ec om positionofthe G aussiand ensity f unction. Sec ond , w e d erive the log-likelihood f unctionofthe unrestric ted representationofthe d ata.T hird ,w e d isplay the likelihood ofthe representationrestric ted b y the variousrealb usinessc yc le m od elsthat w e c onsid er.F inally, w e d isplay the linear, f requency d om aind ec om positionofthe likelihood ratio statistic . 2. 1 Spec tralDec om positionofthe G aussianLikel ihood T he logarithm ofthe G aussiand ensity f unctionf or a T d im ensionalvec tor ofob servations, y1 ;:::;yT ; is: T 1 1 L(y) = ¡ log2 ¼ ¡ logjV j¡ y0V 2 2 2 ¡1 y and Christiano,Eichenbaum and M arshall(1 991 ). 7 T heseapproaches tospeci¯cationerroranalysis aresimilarinspirittotheearlyapproachtakeninSims (1 972). 3 w here V isthe T b y T c ovariance m atrixofy = [y1 ;:::;yT ]0.It isw ellknow n(Harvey,19 89, p.193) that f or T large,thisexpressionis,approxim ately, " # 1 TX¡1 I (! j) L(y) = ¡ 2 log2 ¼ + logf (! j) + 2 j= 0 f (! j) (1) w here I (!) isthe period ogram ofthe d ata: I (!) = 1 XT j ytexp(¡i!t)j; 2 ¼ T t= 1 (2 ) and 2 ¼j ; j= 0 ;1;:::;T ¡1: T F inally,f(!) isthe spec trald ensity ofy at f requency ! im plied b y V f or large T:8 W e ¯nd it c onvenient,f or later purposes,to expressthe likelihood f unctionasa w eighted likelihood ,asinDieb old ,O hanianand B erkow itz (1998): !j = " # 1 TX¡1 I (! j) L(y) = ¡ w j 2 log2 ¼ + logf (! j) + : 2 j= 0 f (! j) (3) Inour analysis,w e w illc onsid er w j 2 f0 ;1g: 2. 2 Likel ihood Fun c tionf or T he Struc turalM od el T hissub sec tiond erivesthe restric ted red uc ed f orm representationf or output grow th im plied b y tw o struc turalm od els,and their assoc iated loglikelihood f unctions. 2. 2. 1 R ealB usinessCyc l e M od el P T he representative agent inour mod elhaspref erences,E0 1t= 0 ¯ t[log(C t) + à log(1 ¡n t)]; w here C t d enotesc onsumptionand n t d enoteshoursw orked .T he tim e end ow m ent isnorm aliz ed to unity and the param eters ¯ and à satisf y 0 < ¯ < 1; à > 0 :T he resourc e c onstraint isC t+ I t·Y t; w here Y t= K tµ (z tn t)(1¡µ) ; 0 < µ < 1; w ith a tec hnology shoc k z 8 t L etV lj denotethejth elementofthelth rowofV :T hen, f(!)= V ll+ 2 1 X j=l+ 1 foranyl: 4 V lj cos(!(j ¡l)); log(z t) = log(z t¡1 ) + ´t w here ´t isi. i.d . Norm alw ith m ean¹ and variance ¾ 2 . Inthe realb usinessc yc le (R B C) versionofthism od el, K t+ 1 ¡(1 ¡±)K t = I t; 0 < ± < 1: W e d enote the unknow nparam eter valuesofthe R B C m od elb y the vec tor © :Inthe next sec tion's estim ationexerc ise, w e c onsid er tw o c ases. Inone, µ = 0 :34 4 ; à = 3:92 ; ± = 0 :0 2 1;¯ = 1:0 3¡0 :2 5 and © = ¾ ´ :Inthe other, à = 3:92 ; ¯ = 1:0 3¡0 :2 5 and © = (¾ ´ ;±;µ): T hese c hoic esare mad e to enhance the illustrative value ofthe applic ationstud ied inSec tion T hree. 2. 2. 2 T ime to B uil d M od el T he time to b uild mod eld i®ersf rom the R B C m od elonly inthe investm ent technology. P eriod tinvestment is: I t= Á 1 xt+ Á 2 xt¡1 + Á 3xt¡2 + Á 4 xt¡3; w here Á i ¸0 f or i= 1;2 ;3;4 ; and Á 1 + Á 2 + Á 3 + Á 4 ´1: T he investm ent tec hnology requiresthat ifxt unitsofnet investm ent are to oc c ur d uring period t+ 3,i.e., K t+ 4 ¡(1 ¡±)K t+ 3 = xt; then, resourc esinthe am ount Á 1 xt m ust b e applied inperiod t, Á 2 xt must b e applied in period t+ 1; Á 3xt must b e applied inperiod t+ 2 ,and ¯nally Á 4 xt m ust b e applied inperiod t+ 3:O nce initiated , aninvestm ent projec t'ssc ale c annot b e expand ed or c ontrac ted .As inthe R B C m od el, © d enotesthe vec tor ofparam etersto b e estim ated :Inour analysis, © = (¾ ´ ;Á 1 ;Á 2 ;Á 3). 2. 2. 3 R ed uc ed Form R epresen tationand Likel ihood Function W e used the und eterm ined c oe± c ient m ethod d esc rib ed inChristiano (19 98) to approxim ate the polic y rulesf or em ploym ent and c apitalthat solve the planningprob lem assoc iated w ith the ab ove tw o m od elec onom ies.W e m anipulated these approxim ate polic y rulesto ob tain a red uc ed f orm representationf or yt = log(Y t=Y t¡1 ): yt = ® (L;© )´t = ® 0 (© )´t+ ® 1(© )´t¡1 + ® 2 (© )´t¡2 + ::::: 5 (4 ) T hisrepresentationisa restric ted AR M A(4 ;8) m od el9 .T hat is, ® (L;© ) isthe ratio ofan 8th ord er polynomialinthe lagoperator,L; and a 4 th ord er polynom ialinL.W e restric t © so that 1 X ® i(© )2 < 1 ; i= 0 guaranteeing that the spec trald ensity ofyt exists.W e also restric t © so that ® (z ;© ) = 0 impliesjz j¸1: T he spec trald ensity ofyt at f requency ! is ¾ ´2 ® (e ¡i! ;© )® (e i! ;© ); 2¼ fr (!;© ) = w here the supersc ript, r; ind ic atesthe restric ted m od elf or yt:T he f requency d om ainapproxim ationto the restric ted likelihood f unctionis(1) w ith f(!) replac ed b y fr (!;© ): 2. 3 Un restric ted R ed uc ed Form Likel ihood Inord er to test our mod el,w e need to estim ate anunrestric ted versionof(4 ): yt= ® u(L)"t; (5) w here 2 ® u(L) = 1 + ® 1uL + ® u ::: 2L + : Also, 1 X i= 0 u 2 (® u i) < 1 ; and ® (z ) = 0 im pliesjz j¸1:T hese c orrespond to the analogousrestric tionsim posed onthe restric ted red uc ed f orm .T he polynom ialinL; ® u(L); isthe ratio ofan8th ord er polynom ial and a 4 th ord er polynom ial, w ith c onstant term snorm aliz ed to unity. T hisspec i¯c ation neststhe realb usinessc yc le m od eland the tim e to b uild m od el.It has13 f ree param eters: the 12 param etersof® u(L); and ¾ ":W e d enote these b y the 13 d im ensionalvec tor, °:Let fu(!;°) d enote the spec trald ensity ofyt: fu(!;°) = ® u(e ¡i! )® u(e i! ) 2 ¾ "; 2¼ T he f requencyd omainapproxim ationto the unrestric ted likelihood f unctionis(1) w ith f(!) replac ed b y fu(!;°): 9 T heappendixpresents thederivation ofthis A R M A representation. 6 2. 4 Cumul ative Likel ihood R atio T he likelihood ratio statistic is ¸ = 2 (L u¡L r ); w here L r and L u are the m axim ized valuesofthe restric ted and unrestric ted loglikelihood s, respec tively.U nd er the nullhypothesisthat the restric ted m od elistrue,thisstatistic hasa c hi-square d istrib utionw ith d egreesoff reed om equalto the d i®erence b etw eenthe numb er ofparam etersinthe restric ted and unrestric ted m od els(Harvey,1989,p.2 35).De¯ne " # fr (!;©b) 1 1 ¸ (!) = log u + I (!) r ¡ u ; (6) b b) b) f (!;° f (!;© ) f (!;° w here a hat over a variab le ind ic atesitsestim ated value.T hen,it iseasily c on¯rm ed that, ¸= TX ¡1 ¸(! j): j= 0 T hisexpressionc anb e sim pli¯ed b ec ause ofthe sym m etry propertiesof¸(!) : T ¸(! T ¡l) = ¸(! T + l); l = 1;2 ;:::; ¡1: 2 2 2 T hese imply that ¸ c anb e w ritten: T 2 ¸ = ¸(0 ) + 2 ¡1 X ¸(! j) + ¸(¼): (7) j= 1 T hisisour linear,f requency d om aind ec om positionofthe likelihood ratio statistic . If¸ islarge,thenw e should b e ab le to d eterm ine w hich ! j'sare responsib le f or this.T o assist inthis,w e d e¯ne the c umulative likelihood ratio: ¤ (!) = ¸(0 ) + 2 ¤ (0 ) = ¸(0 ); ¤ (¼ ) = ¸: X ! j·! ¸(! j); 0 < ! < ¼ (8) A sharp increase in¤ (!) insom e regionof!'ssignalsa f requency b and w here the m od el ¯tspoorly. 3 R esul ts T his sec tionpresents our results f or estim ating and testing the R B C and tim e to b uild m od els. T he period ogram ofthe d ata, (2 ), and the spec trald ensity ofthe unrestric ted red uc ed f orm are important ingred ientsinthe analysis,and so w e b eginb y presentingthese. T he f ollow ingtw o sub sec tionspresent the analysisofthe R B C and the tim e to b uild m od els, respec tively. 7 3. 1 P eriod ogram and Spec trum ofU nrestric ted R ed uc ed Form 10 F igure 1 presentsa sm oothed versionofI (!) f or ! 2 (0 ;2 ¼ ); b ased on(2 ). T he thic ksolid line inF igure 1 isthe spec trum ofour unrestric ted AR M A(4 ;8) representationofU S G DP grow th.Note how sim ilar these are.T hisisto b e expec ted ,since b oth represent c onsistent estim atesofthe spec trum ofthe d ata. Vertic alb arsd raw attentionto three f requency b and s, the low f requencies(those c orrespond ing to period 8 yearsto in¯nity), the b usinessc yc le f requencies(period 1 year to 8 years) and the high f requencies(period 2 quartersto 1 year).Note that the low and b usiness c yc le f requencieshave high pow er. Inad d ition, the spec trum haspronounced d ipsinthe 7¡7:5 months(near ! = 2 :5) range and inthe higher f requency c om ponent ofthe b usiness c yc le (near ! = 1:5): F igure 1: E stim ated Spec tralDensity Smoothed Periodogram log f(ω) Unrestricted ARMA(4,8) -5 10 Low Frequencies 0 3. 2 Business Cycle Frequencies 0.5 1 High Frequencies 1.5 2 ω 2.5 3 E stim ationand T estin gofR B C M od el W e b eginb y estim atingthe versionofthe R B C m od elinw hich only the innovationvariance ofthe tec hnology shoc k,¾ ´ ; isf ree.W e c allthisthe r̀estric ted R B C' m od el.W e thenturn to the version(the `unrestric ted R B C'm od el) inw hich ± and µ are also f ree. T he spec trum ofthe estimated restric ted R B C m od elisd isplayed inF igure 2 .For c onvenience,F igure 2 reprod uc esthe spec trum ofthe unrestric ted AR M A(4 ;8) representationof the d ata f rom F igure 1.Asem phasiz ed inW atson(199 3),the spec trum ofthe R B C m od el 10 T hedataareseasonallyadjusted,covertheperiod1 955Q 3 to1 997Q 1 ,andarefrom theCitibasedatabase. T hesamplemean ofyt is subtracted from thedata,sothatI(0 )is zero.W epresentthesmoothed version oftheperiodogram because,as is wellknown,theunsmoothedperiodo gram is quitevolatile.T hesmoothed P periodogram atfrequency!j is acentered,equallyweighted average, 3i=¡3 I(!j+ i)=7: 8 isessentially °at.T o a ¯rst approxim ation,the m od elim pliesthat aggregate output inherits the persistence propertiesofthe tec hnology shoc k,w hich isa rand om w alk b y assum ption. F igure 2 Spec tra R elevant to the Analysisofthe R B C M od el Unrestricted ARMA(4,8) Unrestricted RBC, Low log f(ω) Unrestricted RBC, High Restricted RBC -5 10 Unrestricted RBC, All Unrestricted RBC, Business Cycle 0 0.5 1 1.5 2 ω 2.5 3 For a f ormalevaluationofm od el¯t, c onsid er F igure 3 w hich d isplaysthe c umulative likelihood ratio, (8). Note that ¸ isjust und er 2 5 (see the c umulative likelihood ratio f or ! = ¼).U nd er the nullhypothesisthat the restric ted R B C m od elistrue,¸ isthe realiz ation ofa c hi-square statistic w ith 12 d egreesoff reed om .T he statistic hasa p-value of1.5perc ent and hence the m od elisrejec ted at the ¯ve perc ent signi¯c ance level.T o see w hy the m od el isrejec ted , note that the c umulative likelihood ratio d isplays sharp increases inthe low f requency c om ponent ofthe b usinessc yc le,and inthe f requenciesc orrespond ingto period s 7-7.5 m onths. F igure 3: Cumulative Likelihood R atio 30 Business Cycle Frequencies Low Frequencies High Frequencies 25 Λ(ω) 20 15 Time To Plan Model Restricted RBC Model 10 Estimated Time to Plan Model 5 0 0 0.5 1 1.5 2 ω 9 2.5 3 W e now turnto the unrestric ted R B C m od el.T he estim ated param eter valuesare µb= 0 :37and ±b= 0 :73:Although the estim ated value ofc apital'sshare isreasonab le,the estim ated value of± ismuc h larger thanseemsplausib le inlight ofd ata oninvestm ent and the stock ofc apital(Christiano and E ic henb aum ,19 92 ).T o see w hat f requency c om ponent ofthe d ata b± bseveraltim esusin d rivesthisresult,w e rec omputed µ; galternative w eightsinthe w eighted likelihood f unction,(3).T he estim ationresultsare d isplayed inT ab le 1 and F igure 2 . T ab l e 1: W eighted Likelihood E stim ationR esults Frequencies µ ± ¾´ ¸ ¸w O b servations U sed High 0. 2 5 0 .99 0 . 0 12 6 9.6 3. 7 50 % B usinessCyc le 0 . 51 0 .99 0 . 0 170 2 6. 1 2. 3 4 3% Low 0. 15 0 0. 0 10 0 37. 3 -0 . 2 7% All 0. 37 0 .73 0 . 0 14 4 8.5 8. 5 10 0 % ²Notes:T hesearethe results ofestimatingtheunrestricted R B C modelby weighted maximum likelihood (i. e. ,by maximizing(3)).L owfrequencies: w j equals 1 only forw j' s that belongtofrequenciescorrespondingtoperiods 8 years andup.B usiness cyclefrequencies:w j equals 1 onlyforw j' s thatbelongtofrequencies correspondingtoperiods 1 to8 years.H igh frequencies: w j equals 1 only forw j' s thatbelong tofrequencies corresponding to periods 2 quarters to 1 year;A llfrequencies: w j equals 1 forall j. P ercentofobservations used: fraction of j2 f0 ;1;:::;T ¡1g equalto unity in the weighted likelihood estimation. ¸ : likelihood ratiostatistic.¸ w :likelihood ratiostatisticbased only on thesubintervalforthe weighted likelihood function. T he b usinessc yc le and high f requency c om ponentsofthe d ata d rive ± to nearly unity. W ith ± near one,the m od elred uc esto the sc alar versionofthe m od elinLongand P losser (19 83),inw hic h output grow th isa ¯rst ord er autoregressionw ith autoregressive param eter µ:W ith the spec trum ofthisproc ess, proportionalto 1=(1 + µ2 ¡2 µ c os(!)); the m od elis ab le to matc h the shape ofthe d ata spec trum inthe b usinessc yc le and high f requencies(see Ù nrestric ted R B C, B usinessCyc le' and Ù nrestric ted R B C, High' inF igure 2 ).How ever, d i®erent valuesofµ w ork b etter inthe tw o f requency ranges. T o matc h the low f requencies,a very d i®erent param eteriz ationisneed ed ,w ith ± nearly 0 and µ sm all(see Ù nrestric ted R B C,Low 'inF igure 2 ).T he param eter estim atesb ased on allf requenciesare roughly anaverage ofthe resultsover the variousf requencies. 3. 3 T im e T o B uil d M od el R esultsf or estim ating the time to b uild m od elare d isplayed inF igure 4 .For c onvenience, F igure 4 d isplaysthe spec trum ofthe restric ted R B C m od el,and ofthe d ata.B oth ofthese are takend irec tly f rom F igure 2 .O ur estim atesofthe w eightsare: Á 1 = 0 :0 1, Á 2 = 0 :2 8, Á 3 = 0 :4 8; and Á 4 = 0 :2 3:Note that the ¯rst w eight isalm ost z ero. T hisim pliesthat in 10 the ¯rst period ofaninvestm ent projec t, essentially no resourc esare used .T hism otivates ref erringto this¯rst period asa planningperiod ,one inw hich plansare d raw nup,perm its 11 are sec ured ,etc . W e ref er to thisasthe estim ated tim e to planm od el. Note how w ellthe spec trum ofthe tim e to planm od elc onf orm sw ith the spec trum of the d ata.T he time to planm od elevenm atc hesthe d ip inthe spec trum inthe 7-7.5 m onth range.T hisisre°ec ted inthe good perf orm ance ofthe m od el'sc um ulative likelihood ratio (see F igure 3).T he c umulative likelihood ratio risesslow ly w ith f requency and achievesa m aximum value just und er 10 .U nd er the nullhypothesisthat the m od elistrue,thisisthe realiz ationofa c hi-square d istrib utionw ith 9 d egreesoff reed om .U nd er these c ond itions, the p-value is35 perc ent.Asa result,the m od elisnot rejec ted at c onventionallevels. F igure 4 E stimationR esultsf or the T im e to B uild M od el Unrestricted ARMA(4,8) log f(ω) Estimated Time to Plan Model Restricted RBC Model -5 10 0 0.5 1 1.5 2 ω 2.5 3 W e c om pare the estim ated tim e to b uild m od elw ith tw o others: the tim e to b uild m od el suggested inK yd land and P resc ott (1982 ),w here Á i = 0 :2 5; i= 1;2 ;3;4 ;and the versionof the tim e to planm od elanalyz ed inChristiano and T od d (199 6), w here Á 1 ¼ 0 ; Á i = 1=3, i= 2 ;3;4 :W e d o not d isplaythe spec trum im plied b yK yd land and P resc ott'sm od el,b ec ause that essentially c oincid esw ith the spec trum ofthe restric ted R B C m od el(K ing, 19 95).As a result, K yd land and P resc ott'sm od elisrejec ted like the restric ted R B C m od el. For a d etailed d isc ussionofthe similarity ofthese m od els, see Christiano and T od d (199 6) and R ouw enhorst (19 91).T he Christiano and T od d (199 6) param eteriz ationofthe tim e to plan m od elisanimprovement over the restric ted R B C m od elinthe b usinessc yc le c om ponents ofthe d ata (F igure 3).O ver allf requencies,the tw o m od els,how ever,have a c om parab le ¯t. 11 See Christianoand T odd (1 996), who argue thatthe notion ofa planning period conforms wellwith studies ofinvestmentprojects. 11 4 Concl usions W e have d esc rib ed som e ad vantages, f or d iagnosing m od elestim atesand ¯t, ofusing the f requency d om aind ec om positionofthe likelihood f unction. W e illustrate the approach w ith anem piric alanalysisofthe stand ard R B C m od eland a versionw ith a tim e to b uild tec hnology.W e rejec t the f orm er inf avor ofthe latter.T he tim e to b uild tec hnologythat ¯ts the d ata b est appearsto b e one inw hic h investm ent projec tsb eginw ith a planningperiod , d uringw hic h relatively f ew resourc esare expend ed .Christiano and T od d (1996) em phasize that thisspec i¯c ationc onf orm sw ellw ith m ic roec onom ic stud iesofinvestm ent projec ts,and d isc ussother ad vantagesto thism od elf or b usinessc yc le analysis. 5 Append ix: Show ingthat y isanAR M A(4 ,8) T he polic y rulesthat solve the tim e-to-b uild m od elare linear equationsinthe logofc apital lnK and ofhours-w orked lnn and the technology shoc ks(w here lnz t equalslnz t¡1 + ´t). lnK t = (1 ¡A(1))lnK + A(L)lnK t+ (1 ¡A(L))lnz t¡4 + + B (L)(´t¡4 ¡¹ ) lnn t = lnn ¡C (1)lnk+ C (L)lnK t+ 4 ¡C (L)lnz t+ D(L)(´t¡¹) T he term sA(L) and B (L) are polynom ialsofd egree f our and C (L) and D (L) are polynom ialsofd egree three inthe lagoperator.Capitalisa f unctionofthe past c apitaland the shoc ksto tec hnology f rom f our to eight period sago.Hoursw orked isa f unctionoff uture c apital(since youhave to w orkf or the investm ent that youhave alread y c om m itted to m aking) and the c urrent and lagged shocks. T he variab lesw ithout the tim e sub sc ript are the variab lesat stead y state. T akingthe ¯rst d i®erence ofthe ab ove tw o equationselim inatesthe stead y state values. 4 lnK t = = A(L) 4 lnK t+ (1 ¡A(L)) 4 lnz t¡4 + B (L) 4 (´t¡4 ¡¹) (1 ¡A(L) + B (L)(1 ¡L))L 4 ´t 1 ¡A(L) 4 lnn t = = C (L) 4 lnK t+ 4 ¡C (L) 4 lnz t+ D(L) 4 (´t¡¹) ! C (L)B (L)(1 ¡L) + D(L)(1 ¡L) ´t 1 ¡A(L) à T he next step isto d erive anequationf or y:O utput isprod uc ed usinga Cob b -Douglas prod uc tionf unction.Hence,output c anb e w rittenas 12 lnY t = µ lnK t+ (1 ¡µ)lnn t+ (1 ¡µ)lnz t T akingthe ¯rst d i®erence yt = 4 lnY t = µ 4 lnK t+ (1 ¡µ) 4 lnn t+ (1 ¡µ)´t Sub stitutinginthe valuesf or 4 lnK t and 4 lnn t w e have (1 ¡A(L))yt = à µ (1 ¡A(L) + B (L)(1 ¡L))L 4 + (1 ¡µ)(C (L)B (L)(1 ¡L) + (1 ¡A(L))D(L)(1 ¡L) + 1) ! ´t T he polynomialsA and C are f ourth ord er and the polynom ialsD and B are third ord er. Asthe ¯rst d i®erence operator isalso present, the m oving average c om ponent isaneighth ord er polynom ial.T he autoregressive term isthe sam e ord er asA.T he tim e-to-b uild m od el, theref ore, c anb e c harac teriz ed asa restric ted versionofanAR M A(4 ,8) m od el.T he R B C m od elnestsinsid e thisspec i¯c ation. 13 R ef erences [1] Altug,Sumru,1989 ,T̀ im e to B uild and Aggregate F luc tuationsSom e New E vid ence', InternationalE conomic R eview , 30 (4 ),Novem b er,889 -92 0 . [2 ] Christiano,Law rence J . ,1985,`A M ethod f or E stim atingthe T im ingIntervalina Linear E c onom etric M od el, W ith anApplic ationto T aylor'sM od elofStaggered Contrac ts. ' J ournalofE conomic Dynamic sand Control, 9,363-4 0 4 . [3] Christiano,Law rence J . , 1988, ` W hy DoesInventory Investm ent F luc tuate So M uc h?', J ournalofM onetary E conomic s,2 1(2 /3),M arch/M ay,2 4 7-80 . 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[19] Harvey,And rew ,1989,ForecastingStruc turalT im e SeriesM od elsand the K almanF ilter,Camb rid ge U niversity P ress. [2 0 ] Ireland , P eter N., 199 7, ` A Sm all, Struc turalQuarterly M od elf or M onetary P olic y E valuation', Carnegie R oc hester Conference SeriesonP ublic P olic y, 4 7(0 ),Dec emb er, 83-10 8. [2 1] K im,J inil,19 98,` Construc tingand E stim atinga R ealistic O ptim iz ingM od elofM onetary P olic y',unpub lished manusc ript. [2 2 ] K ing,R ob ert,19 95,` Quantitative T heory and E c onom etric s',Fed eralR eserve B ankof R ic hmond E conomic Quarterly,81(3),53-10 5. [2 3] K yd land , F innE .and E d w ard C.P resc ott, 19 82 , T̀ im e to B uild and Aggregate F luc tuations',E conometrica, 50 ,Novemb er,134 5-70 . 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