Macro-finance models of the term structure: a review

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1 Macro-finance models of he erm srucure: a review Fabio Filipozzi allinn Universiy of echnology Absrac: in his paper we presen a review of recen developmens in he erm srucure lieraure ha incorporae macroeconomic facors as explanaory variables of he behavior of he yield curve. he conribuions here analyzed have in common he resul ha macro economic variables improve he abiliy of erm srucure models o mach he yield curve dynamics. Some relevan differences exis amongs he papers here considered. Firsly, macroeconomic variables can be added wih differen degrees of complexiy of he macroeconomic srucure, from a simple aylor rule o a srucural macroeconomic model. Secondly, he way in which macro facors influence he shape and level of he yield curve (and vice versa) is an open quesion, and has received differen answers from differen researches. A shor review is given also o he problem of esimaion of hese models. Possible fuure developmens of his field of research are also highlighed. Inroducion he join macro-finance field of research is a quie recen developmen in finance. In his review we will ry o skech a brief hisory of he lieraure of erm srucure models, highlighing he las developmens, and also giving a review of he recen aemps o add macro-based models o he erm srucure models, in order boh o improve he explanaory power of erm srucure models and o use he inuiion ha real economy facors influence he erm srucure of ineres raes. Firs of all, i is worh spending some words for definiions. he lieraure on yield curve esimaion is very huge, and we have o race he limis of he research filed ha we wan o cover here, ha is only one par of he yield curve modeling lieraure. he saring poin is he goal of he yield curve models. Under any research field here is a problem (eiher pracical or inellecual) o be solved, and some early aemps o underake he ask. Yield curve modeling lieraure has a he origin he need of explaining ineres raes behavior. Ineres raes (boh shor and long erm ineres raes) have an imporan role in financial markes, for very differen reasons and purposes. hey are used for example as he price of borrowing/lending money, hey are needed o price bonds, o price derivaives on bonds and oher fixed income insrumens. he lis is very long, bu in essence he undersanding of behavior (dynamic) of ineres raes has been an imporan ask for economic and financial heory for cenuries 1. Ineres rae models have a shorer hisory, no longer han half a cenury, and are characerized by he aemp o give a quaniaive explanaion o he behavior of he yield curve. here are many differen reasons behind he aemp o explain in more deail he 1 For a shor review of he hisory of ineres raes behavior models, see for example James Webber

2 dynamic of ineres rae. he firs reason is moneary policy. he decisions of cenral banks are based on various facors, and one of hem is he level and expeced pah of long erm asses. hese also influences he behavior of consumers and firms, hence he performance of he enire economy. If we have a ool ha is able o help us forecasing boh shor and long erm raes, we can gain an imporan piece of knowledge abou moneary policy decisions. A second poin of ineres regards deb managemen policies. Sae reasuries, when financing public deb, mus decide he mauriy of he insrumen hey have o issue. he fuure pah of ineres raes, and, more generally, he behavior of he enire erm srucure, can help hem in deciding which poin of he yield curve o issue. he hird, and mos imporan, reason o model ineres raes srucure is he pricing of derivaive insrumens and hedging ineres rae exposure. In order o price ineres rae derivaives (opions, caps, floors ec.) hypoheses on he acual and fuure srucure of he yield curve mus be done. Also hedging risk exposure (necessary for banks, bu also for oher kind of insiuions) requires some forecas of he srucure of he yield curve 2. For he reasons above saed, modeling he behavior of he erm srucure of ineres raes can give relevan informaion o mos of he paricipans in he financial markes (banks, raders, cenral banks, sae reasuries ec.), and his fac explains why such a huge amoun of lieraure and research has been devoed o his field. In order o resric furher he area covered by his paper we sar defining, following Dai Singleon 2003a wha is a Dynamic erm Srucure Model (DSM). Wih erm srucure we will inend from here on (if no oherwise specified) he se of bond yields ranging from 0 o n years mauriy a a given poin in ime. Also, for racabiliy and compuaional purpose, he bonds are considered risk-free zero coupon bonds. hen here are differen ways in which coupon bonds, risk facors ec. can be inroduced in he base model o inroduce in he base model. he family of models here considered is called Dynamic in he sense ha he objecive is o formalize he evoluion (and comoven) in ime of shor and long erm bonds. In order o furher resric he area of his paper, we will deal only wih Affine erm srucure models (ASM) 3. he exac meaning of his resricion will be explained laer, when he model will be developed. For he momen we can say ha modeling he relaion beween bonds yields and sae variables wih an affine srucure makes he model more easy o rea boh for compuaion of soluions and for esimaion. his characerisic has made his class of model very popular. Affine erm srucure models: model skech In his secion he main characerisics of ASMs are oulined, leaving aside he more echnical deails. We will follow he mainly he framework se up by Piazzesi 2003, referring o oher conribuions when needed. 2 See Piazzesi 2003 for a more deailed descripion of he use of erm srucure models 3 For a shor overview of oher ype of models, see for example Dai Singleon 2003a and Dai Singleon 2003b 2

3 In order o undersand he basic srucure of he model, i is easier o sar from he definiion of he price of a zero coupon bond. Inuiively, we can say ha in absence of risk and of arbirage opporuniies, a zero coupon bond holding period reurn mus be equal o he average risk free shor erm rae during he same period. (1) P E exp r udu where P in he price of he zero coupon bond beween and, E is he expecaion a ime under he Q and r u is he risk free shor ime rae. his equaion is he resul of series of very imporan condiions and hypoheses ha mus be now considered, because hey shade a ligh on he framework on which erm srucure models are buil and give he echnical background of he analysis ha will follow. he model here is defined in coninuous ime. he reason of using coninuous ime is because he model is easier o deal wih; in paricular for some parameers specificaion i is possible o find closed soluions o he bond pricing equaion 4. A second elemen is he expecaion erm. I ells us ha we are operaing in a sochasic environmen; hence we mus find a way o model he behavior a leas of he shor erm rae in order o use he model for pricing zero coupon bonds (and consequenly model he erm srucure). he pricing of he bonds is made using Q, or he risk-neural probabiliy measure. he saisical deails of his concep go behind he scope of he paper 5, bu he basic concep is ha when expecaions are calculaed under his measure, some nice properies are guaraneed. he mos imporan is ha if his risk neural probabiliy measure exiss, hen he no-arbirage condiion is saisfied. his is a fundamenal elemen, which also assure ha he model is consisen. Following Piazzesi 2003, we can say ha he model for he yield curve (or, he model hrough which we define he prices of he zero coupon bonds ha characerize he yield curve), is defined hrough wo elemens: (I) he change of measure from Q o (II) he shor rae dynamics Q he former has been shorly explained above. Q can be considered he daa generaing measure. If all he agens in he marke would be risk neural, he wo pricing measures would be he same. Bu if we allowed uncerainy, we need, as explained above, a risk neural probabiliy measure Q. he second condiion allows us o model he behavior of he shor erm risk free rae, from 4 We will see laer ha when macro models and erm srucure models are considered ogeher and esimaed, i is necessary o use discree ime, because macroeconomic variables are observed in discree ime. Here he analysis is conduced in coninuous ime because mos of he models in lieraure have his represenaion. 5 a very good and deailed discussion on his subjec can be found in Duffie An informal discussion of risk neural probabiliy measure (or equivalen maringale measure) can be found in Sundaram

4 which any oher bond is priced. he way in which his dynamics is modeled can vary from differen conribuors, bu as a firs approximaion, we can describe he general affine approach as follow: (2) r R(x) wih n x R x is ime homogenous Markov process under Q. Again, we will here give an inuiive explanaion of he las definiion. x, as we will see below, will be defined as an affine diffusion process. Bu in general he definiion of x means ha his sochasic process is wihou memory (i depends only on presen values) and also he is behavior is ime independen. his model is made more racable imposing affine (linear) srucure on boh he relaion beween r and x and also in he diffusion srucure definiion of x. In paricular i is assumed ha: - R(x) is affine - x is an affine diffusion process under Q he firs condiion means ha he shor erm risk free rae is a linear funcion of he facors characerizing he model. he second condiion insead allows us o model he diffusion process of he facors of he model in a linear (affine) way. In paricular, he shor erm rae will be: (3) r x where 0 is a scalar and is a vecor in model. he affine diffusion process for every x is usually defined as follow: N R (4) dx ( x ) d ( x ) dz, where N is he number of facors of he For he momen we will no specify any funcional form for ( x ) and ( x ). Usually quie resricive limis are imposed o hese wo funcions in order o make he model racable. I is imporan here jus o skech he meaning of he wo. x ) is he drif of he process, and ells us how he process evolves in ime supposing no shock o i. ( x ) insead is he variance elemen of he process, and models exacly he perurbaions o he drif rajecory of his diffusion process. z is assumed o be a sandard Brownian moion (i means ha shocks are independenly normally disribued wih mean 0 and variance d. I is imporan here o noice also ha boh are funcion of he sae reached by he process a any ime (hey depends on x ). Before showing how we can find a soluion for he price of he bonds, i is here useful o look again a he difference beween Q and Q. he difference beween he wo can also be explained, again following Piazzesi 2003, adding a hird hypohesis o he wo above ( 4

5 menioned. If we suppose han he local expecaion hypohesis holds, hen equaion (1) holds also under he daa generaing process Q. he inuiive reason comes direcly from he definiion of expecaion hypohesis. If i holds in fac any bond a any mauriy has a yield ha i is simply he weighed average of he shor risk free rae over he life of he bonds, hence he excess reurn of every bond is zero 6. Now we wan o use all he seup above consruced in order o find he price of he bonds, his of course will depend from boh he evoluion of ime and he remaining mauriy of he bond. Affine erm srucure models are so widely used because hey allow us, hrough oher simplifying (and resricing) hypoheses, o find in some case a closed soluion o he problem. he firs hing we need o define is he funcional form of boh he drif and he variance of equaion (4). We will sar analyzing he case under risk neural measure. he usual form assumed is as follow: (5) ( x ) K ( ) x (6) ( x ) S were he former equaion ells us ha he drif is again an affine funcion of he sae variables, and he marix K is NxN. he laer is formed by he NxN marix and S, which is supposed o be diagonal, and each elemen on he diagonal is again affine in x ' (7) S i i x ii he meaning of he drif funcional form is ha when he process for x is differen from he mean, hen he process will end o reurn owards he mean of he diffusion (he erm in parenhesis will be posiive if x is below he mean and negaive in he opposie case). he K marix insead gives he speed of adjusmen of he deviaions from he mean. K can also no be diagonal. If i is indeed diagonal, hen he correcion of every sae variable will be driven only by iself, and will no depend on oher sae variables. A cross influence in he adjusmen process beween he facors will be presen if K is allowed o be non diagonal. Equaions (6) and (7) characerize he variance covariance srucure of he sae variables. In paricular again if is diagonal, he perurbance effec will be ransmied o he oher facors, bu if i is insead diagonal he perurbance on he pah of one sae variable will no affec he ohers. We will come back laer on he paricular hypoheses regarding he volailiy componen of he diffusion process dx. 6 As Piazzesi 2003 shows, here is a difference beween local expecaion hypohesis and expecaion hypohesis given by he Jensen inequaliy, bu we will no furher develop his concep here. 5

6 Under a series of regulariy condiions, he guess closed soluion for he price of he bond (or, he soluion o equaion (1) ) is of course affine in he sae variables, and is equal o: (8) P expa( ) B( ) x wih A( ) and B ( ) ha saisfy: da( ) ' ' ' (9) K B( ) B( ) i 0 db( ) ' ' (10) K B( ) B( ) i x hese wo las are ordinary differenial equaions, which can be solved aking as iniial condiion A ( ) 0 and B ( ) 0. A las consideraion mus be done regarding he acual daa generaing process. he closed form soluion of he price of he bond given in equaion (8) has been found using he risk neural price measure. When we wan o esimae he parameers of he model we use acual daa, hence for he implemenaion of he model we need o specify he missing link beween Q and Q. In paricular, we mus find wha is called he marke price of risk 7. Inuiively, he marke price of risk represens he deparure of he price of a bond from is risk neural level due o he fac ha a risk neural invesor requires a yield level higher han he average of fuure expeced shor erm raes, in order o compensae for he risk he is aking for holding a securiy wih longer mauriy. he sandard way in which i is inroduced ino he model is hrough he pricing kernel concep. Pricing kernel is a way o define he price of a zero coupon bond alernaive o (1). he fundamenal relaion is: 1 (11) P E m P 1 1 his is a recursive equaion ha assures no arbirage. I basically saes ha he price of a zero coupon bond wih mauriy in periods is he price of he same bond (wih mauriy in -1 periods) a ime +1, muliplied by he discoun facor m 1. he specificaion of m 1 is usually he following: 1 (12) m 1 exp r 1 2 is hen, in affine erm srucure models, modeled hrough a sandard affine process: N i1 N i1 2 i 2 i 7 For furher echnical deails on his subjec, see Duffie 2001 and Dai Singleon

7 (13) 0 1 x wo elemens can here be noed. Firsly, if 0 and 1 are boh equal o zero, hen here is no need for risk adjusmen and he invesors are no risk adverse. his is exacly he case when equaion (1) holds boh under he risk neural probabiliy measure and he daa generaing measure (see Ang Dong Piazzesi 2005). Secondly, as noed by Hördahl risani Vesin 2004, in a micro founded model he ime varying price of risk can be endogenously calculaed from he preferences of maximizing agens, bu in mos of he affine erm srucure models i is simply exogenously modeled hrough he affine represenaion exposed above. Wih also he price of risk specified, he erm srucure model has been compleely defined. A a heoreical level, we are now able o price zero coupon bonds and also model heir dynamics boh wih risk neural and risk adverse agens. Dai Singleon 2000 defines a general framework hrough which i is possible o classify all he affine erm srucure models. We will no here deal wih his conribuion, bu i is worh underlying ha he criical assumpion in facors models is no only he number of sae variables used, bu also how he srucure of (7) is defined, and in paricular how differen facors are allowed o ener he condiional variance of he process specified in (7). We will see ha his has imporan consequences in he esimaion of he model. Macro models Now we have defined all he main characerisics of an affine erm srucure model, we can see how recen conribuions have ried o join his framework wih a macro economic model. he firs naural approach has been influenced by he original approach o erm srucure models. One of he seminal conribuions o erm srucure analysis has been Vasicek he model proposed by Vasicek is a one facor model, and he single facor is he shor erm rae. he reason is of course ha he shor erm rae drives all he mauriies along he yield curve, due o equaion (1). If we wan o see from a differen perspecive he shor erm rae, in paricular from a macroeconomic poin of view, he aylor rule is he firs naural simple approach 8. he simple inuiion given by aylor is ha a cenral bank, when seing he repo rae, akes ino accoun wo main (ofen conflicing) goals. he firs one is he inflaion, because moneary aggregaes influence in he long run price dynamics, and because in mos of he counries cenral banks have as an explici goal he price sabiliy. he second one is growh, because growh dynamics can be heavily influenced in he shor erm by he ineres raes se by cenral banks. he ypical aylor rule hence ries o explain changes in he shor erm rae as a combinaion of oupu gap (difference beween he acual oupu level and an oupu level desired by he cenral bank) and inflaion gap (again difference beween he acual inflaion 8 See aylor 1993 for a descripion of he moneary policy rule esimaed by aylor 7

8 and he desired inflaion). his come firs of all by he observaion of he behavior ha cenral banks have in realiy, bu can also be modeled hrough a complee macroeconomic model, defining he whole srucure of he economy, making he usual hypoheses of agens maximizing heir uiliies. hese wo ways of modeling he aylor rule are refleced also in he way he macro economic variables are insered in he erm srucure model, or, beer, in he way in which macro srucure and erm srucure are combined. We will review more in deails in he las secion as some imporan conribuions differ in his approach. For he momen we skech here he idea behind he main conribuions. he firs way in which macro variables and he erm srucure can be combined is adding o he se of he (unobservable) sae variables some economic variables, and see how he laer influence he esimaion of he erm srucure and he relaion beween he wo groups of facors. In his case he saring poin is usually a radiional erm srucure model, and some economic variables are added in he se of he facors explaining he dynamic of he erm srucure. he work of Diebold, Rudebush and Aruoba 2005 akes his direcion. he model for he erm srucure is very simple: he parameers of Nelson-Siegel represenaion of he yield curve are esimaed, and heir inerpreaion is given 9. In paricular, he hree facors are inerpreed as he level of he curve, he slope of he curve and he curvaure. his is a quie sandard inerpreaion of he hree dimensions which characerize he yield curve. Wha Diebold, Rudebush and Aruoba find is ha he firs and he second dimension have a macro economic inerpreaion: he firs one (level) is highly correlaed wih he inflaion, and he second facor (slope of he curve) is more correlaed wih he level of he real aciviy. Ang Piazzesi 2003 defines he erm srucure in a differen way, very similar o he srucure we described in he firs secion, bu again he macro facors are added o he laen sae variables wihou any deailed srucural macro economic model behind hem. hey are chosen following on he sream of he aylor rule, herefore economic aciviy and inflaion. Ang Dong Piazzesi 2005 devoes more aenion o he way in which he aylor rule can be specified, aking also ino accoun he no/arbirage condiion. he second way in which erm srucure models and macro models are combined devoes more aenion o he way in which he macro srucure of he economy is defined and modeled. he saring poin is ofen a macroeconomic model specified in a srucural way, and no, as for he firs group described above, in a reduced form represenaion. Some examples of his approach are Hördahl risani Vesin 2004, Rudebush Wu 2003 and Bekaer Cho Moreno he firs conribuion sars, differenly from Ang Piazzesi 2003, from he macro model. hey model shor erm rae, economic aciviy and inflaion 9 In his approach no assumpion is made on he diffusion process of he laen facors. he curve is modeled wih a polynomial represenaion, and wih hisorical daa i is possible o find he value of he parameers of he polynomial funcion. For deails, see also James Webber For an applicaion o forecasing he yield curve wih he Nelson-Siegel approach, see Diebold Li

9 allowing a feedback from he second and hird variable o he firs. Assuming ha shor erm rae has a feedback influence on he macroeconomic variables implies he need of a defined endogenous macro model. Economic variables are no simply exogeously added, hey are influenced by he shor rae, even if no ye by he enire erm srucure. Bekaer Cho Moreno 2005 (described in more deails in he las secion) gives a more deailed macro model. In paricular hey define boh he Phillips curve, he IS equaion and he moneary policy rule. o he neo-keynesian model he erm srucure is added, and he unobservable facors are showed o have an economic srucural inerpreaion. Before giving some more deails on he wo kinds of models described in his secion, we will give a descripion of daa and esimaion mehods usually employed in his field. Esimaion and daa Mos of he lieraure in his field is dedicaed o he US marke. he obvious reason is ha he daa ime span is longer han for oher counries, and also he qualiy of he daa is higher. An imporan excepion is Hördahl risani Vesin 2004, which uses German daa for he esimaion. Imporan is also he fac ha zero coupon raes are no available on he marke. Usually nominal bonds yields are boosrapped in order o find zero coupon yields. When coupon bearing bonds are used, we remain wih he problem ha heir mauriy doesn mach exacly he canonical consan mauriies. I is herefore necessary o inerpolae hese yields, and his process can inroduce some measuremens errors. Bonds yields are also affeced by seasonal effecs. In paricular a he shor-end cenral banks open marke operaions and repo rae decisions can creae some echnical huge volailiy no linked o economic daa (see Piazzesi 2005). hese effecs are ofen ignored in esimaing models. I is also difficul o find reliable sources for daase of bond yields. In US he Federal Reserve publishes a consan mauriy US reasury bond yields daa se, bu in Europe does no exis a similar daa se. Even if no explicily saed during he descripion of he affine erm srucure models, i is always considered ha bonds are defaul risk free, bu i is no possible o consider even US reasury compleely free of defaul risk. his is even ruer for Europe. Also his effec is no considered in esimaion of erm srucure models. Some paper (for example Piazzesi 2005) recenly uses swap rae daa. hey have he advanage of being quoed a consan mauriy, hence here is no need for inerpolaion, bu hey can no be considered free of defaul risk. heir risk depends on he risk of he banks operaing in he swap markes, risk ha can vary hrough ime, as he Russian crisis in 1998 shows. Swap raes have also he problem ha heir ime span is relaively shor, being acively raded only from he beginning of he 90 s. Summarizing, he daa collecion problem and he qualiy of he daa are one of he mos imporan obsacle o he capaciy of erm srucure models esimaion o mach he real daa. 9

10 Abou esimaion mehods, here is no consolidaed way in which his ask is ackled. he main problem in esimaing models wih boh yields daa and macro daa is he number of parameers. If we look a equaions (4) o (6), i is necessary o esimae he elemens of ( x ) and ( x ), where he firs is a vecor and he second is a marix. his means ha he number of parameers o be esimaed grows exponenially wih he number of facors ha we consider. Also he prices of risk described by (12) and (13) mus be esimaed. I means ha wih five facors he number of parameers already exceeds 30, and his makes he esimaion very difficul. his difficuly has been handled in very differen ways by differen researchers. wo poins in common can be here highlighed. Firs of all resricion on parameers are imposed before performing he esimaion. his is necessary for compuaional reasons and he values of parameers derive eiher from previous researches resuls, or from he hypoheses of he model. For example hypoheses on he covariance amongs differen facors can reduce he number of parameers o be esimaed in he marix in equaion (6). Secondly, he esimaion implies ofen more hen one sep, and i is compuaionally heavy, due o he number of parameers. he resul is ha i is impossible here o give an accoun of he esimaion mehod, because each conribuion has a differen approach. wo kinds of esimaion mehods are usually employed, he likelihood-based mehods and he maching momens mehods (see for deails Piazzesi 2003). Deails are really differen in differen researches, and we will no devoe more ime o ha here 10. We will say somehing more specific for wo of hem in he nex secion. Inegraing erm srucure and macro models Now ha he wo classes of models (ASMs and macro) and he daa and esimaion approaches have been described, wo more aspecs mus be considered. Firs of all, how he wo kinds of models are brough ogeher, and wha are he main resuls of his research field a his sage of developmen. In order o see how erm srucure models and macro models are brough ogeher, we will ake here and analyze in some deails wo examples. he firs has a reduced form approach o he macro par, i.e. he macro variables are simply added o he laen sae variables se and he esimaion is run. he second one has a more srucural approach o he ineracion beween erm srucure and macro model. We use he conribuion of Ang Piazzesi 2003 (from here on AP) as an example of he firs approach and Bekaer Cho Moreno 2005 (from here on BCM) for he second approach. In his analysis we will discuss also he resuls of oher conribuors o his research field. 10 mehods acually used in he conribuions here analyzed are Maximum Likelihood mehod (as Ang Piazzesi 2003, described below, and Rudebush Wu 2004), General Mehod of Momens (as Bekaer Cho Moreno 20005, described below), Simulaed Mehod of Momens (Dai Singleon 2000), mehod of Simulaed Annealing (Hördahl risani Vesin 2004), or even Bayesian esimaion approach (Ang Dong Piazzesi 2005). 10

11 Ang piazzesi 2003 he facors used in AP are boh laen unobservable facors in modeling he behavior of he erm srucure and real observable macro facors, real aciviy and inflaion. he macro model is basically he aylor rule. Boh kinds of facors explain he behavior of he shor erm rae. he model obain in his way has a VAR represenaion. No-arbirage hypohesis implies ha he movemens of he all curve is coheren, or ha bonds a differen mauriies are no independen, as such o creae an arbirage opporuniies. Risk premia are in his seup ime varying. hey are defined hrough he pricing kernel concep exposed above by equaions (12) and (13). he model is specified as a discree ime erm srucure model. VAR approach is chosen because convenien o sudy he dynamic of he relaionship beween macro and laen (and yields) facors, via impulse response funcions and variance decomposiion. In paricular variance decomposiion allows he auhors o measure how much of he movemen in he yields is aribuable o laen facors or macroeconomic facors. here are wo elemens ha mus be highlighed here. Firsly, no srucural macro model sands behind he consrucion (macro variables are aken from he aylor rule), differenly from BCM. Secondly, in order o reduce he parameers o be esimaed, i is assumed ha here is no feedback from yields o macro variables. his is obained imposing in he esimaion of he model 0 value o he coefficiens which should explain he ineracion of macro variables wih laen facors. As saed by auhors, his means ha moneary policy (he shor erm rae) will no affec fuure inflaion and real aciviy 11. Daa used are US zero coupons (sripped from he acual curve), he period covered is quie ample, from 1952 o 2000, yield are for mauriies from 1 monh o 5 years, aken monhly. Macro variables are exraced principal componens from wo groups of daa, he firs being linked o inflaion (Consumer price index, Producer price index and Commodiy spo prices) and he second o real aciviy (help waned index, employmen, indusrial producion). he esimaion procedure implies he following seps. Firs of all, a parameerizaion of he model is done, and, as above saed, his is ypical in hese kinds of models because here is a huge number of parameers o be esimaed, and only imposing a priori some parameers values allows he esimaion of he ohers. he independence beween laen facors and macro variables has also he objecive of reducing he number of parameers o be esimaed. In order o esimae he model i is also necessary o infer he unobservable facors from he yields (AP, page 762). he mehod used is he maximum likelihood mehod. A wo 11 A model in which he feedback from moneary policy (or shor erm rae) o real variables is analyzed in Ang Piazzesi Wei hey use he shor erm rae o exrac informaion on fuure pah of real aciviy 11

12 sep procedure is used, in order o avoid explosive soluions. In he firs sep macro dynamics and coefficiens of macro facors in explaining he shor erm rae dynamics are esimaed, and kep fixed when oher parameers are esimaed in he second sep. Equaions from (20) o (23) in AP are he equaions esimaed. Basically hey define he behavior of macro facors (20), laen facors (21), shor erm rae (22) and marke price of risk (23). hree models are hen esimaed, a yields only model, a model wih macro variables wihou lag and a model wih macro variables and heir lags. his is done in order o see if he inroducion of macro variables effecively helps o increase he explanaory power of he model, and if aking ino accoun also lag values of macro variables can improve he capaciy of he model o fi he daa. he main resuls are he following: - hree unobservable facors can be inerpreed as level slope and curvaure, as in previous lieraure. - Esimaion of ime-varying risk premium depends heavily on model specificaion. - Impulse response and variance decomposiion sudy ells ha macro variables have an impac on yields. his influence is more marked a shorer mauriies. For longer mauriies he impac of laen facors (in paricular he level facor) is higher. - Auhors ry a forecas exercise for he hree differen models, and find ha adding boh no-arbirage resricions on parameers and macro variables helps o forecas yields ou of sample wih respec of unresriced VARs or yields only models. - Comparing facors, auhors find ha he laen firs (level) facor mainains is role in explaining he erm srucure when macro facors are added o he model. A he same ime macro facors (in paricular inflaion) can help explaining he level and slope of he yield curve. Bekaer Cho Moreno 2004 heir approach is differen from AP. he aim is firs of all improving he srucure and he capaciy of a Neo-Keynesian macro model o explain he daa (in paricular increasing he endogenous persisence of he macro model). his is done hrough he addiion of he erm srucure. he oher difference wih AP is ha all he macro variables are derived from he underlying model; herefore hey are he resuls of srucural macro relaionships. All he facors driving he erm srucure have a clear inerpreaion derived from he macro model. he macro model is based on hree equaions. An AS equaion ha specifies he Phillips curve, relaing inflaion o real aciviy, derived from he inflaion dynamics specificaion, wih an unobservable facor inerpreed as ime-varying markup process. An IS equaion, consruced from he demand equaion of opimizing raional agens, wih an unobservable facor inerpreed as risk aversion. Finally, he usual aylor rule specifies he moneary policy reacion funcion. he erm srucure par is, differenly from AP, derived direcly from he macro srucure. he pricing kernel defined in equaion (12) is micro-founded, in he sense ha is no 12

13 given exogenously, bu i is given by he ineremporal marginal rae of subsiuion of he macro model. Also he resricions on he prices of risk are derived by he model specificaion. he model is specified herefore by equaions (13) o (17) in BCM. here are hree observed facors (inflaion, real aciviy and shor erm ineres rae) and wo unobserved facors (here inerpreed by he model as markup process and risk aversion). he esimaion is conduced using again US daa for a long ime span, from 1961 o Granger causaliy es is before performed in order o check for he causaliy beween macro facors and he yield curve. Conemporaneous correlaion exiss, also he influence from macro variables o yield curve is clear, he opposie is no clearly rue. he model o be esimaed is a VAR, and he esimaion is no done wih maximum likelihood, bu General Mehod of Momens wih resricions is uilized, because probably he daa are no normally disribued and homoskedasiciy probably doesn hold, as maximum likelihood esimaion requires. We don analyze here he macro implicaions of he esimaion, bu only he erm srucure implicaions. From his poin of view, he main resuls are he following. - he model abiliy o fi hisorical yields daa is comparable wih previous yields only models. - Level, slope and curvaure are highly correlaed wih he ineres rae, bu also wih macro variables (inflaion and real aciviy). his again means ha macro daa and yield curve dimensions are linked. - Srucural shocks: he insananeous reacion of he yield curve o shocks on he five equaions and also he variance decomposiion are sudied. he main conclusions are: - Shock o IS provokes a parallel shif of he yield curve. - Shock o AS or moneary policy have a ransien effec o he slope of he shor end. I seems logical in a srucural model, bu i is counerfacual, usually moneary policy shocks have longer lasing effecs. - Unobservable facors have more effec on he slope and curvaure - he model is able o explain one of he regulariies in yield curve movemens, in paricular he excess volailiy a he longer end of he curve. Comparing he resuls of AP and BCM, firs of all boh finds ha macro variables are relevan in explaining he erm srucure. his is he main conclusion of basically all he recen lieraure in he field. he divergence of he wo conribuions is also a clear sign of he acual sae of ar in he field. AP find ha macro variables have more impac on he shor end of he curve, and less on he long end, hence explain more he slope and curvaure of he yield curve, where he level is sill explained by unobservable facor, as in yields only models. A he conrary, BCM find ha macro shocks, in paricular o IS curve, explain he level facor, and slope and curvaure are sill explained beer by unobservable facors. his shows as he resuls 13

14 are heavily influenced by he specificaion of he models, resricion imposed o he relaion beween macro facors and erm srucure, and esimaion mehods. Conclusions he analysis here conduced had he aim o give a review of he las developmens in erm srucure models ha employ also macro economic variables in he se sae variables ha drive he dynamics of he yield curve. Even if he ineracion of hese wo dimensions has been considered already a he early sages of erm srucure lieraure, only recenly i has been devoed more aenion in modelling and esimaing i. he rapid expansion of his field in a so shor period can explain he absence of commonly shared resuls, as he comparison of wo models conduced in he las secion shows. his will in fuure require a more deep analysis. In our opinion boh he conemporaneous ineracion and he feedback mechanisms beween macro variables and erm srucure are relevan par for fuure analysis and specificaion of hese kinds of models. ogeher wih he ineracion beween macro and erm srucure elemens, as highlighed by Diebold Piazzesi Rudebush 2005, oher componens of he lieraure above described are sill open quesions ha require furher analysis. For example he specificaion of erm premiums is one of hese, ogeher wih macro srucure specificaions. 14

15 References Ang, A.; Dong, S.; Piazzesi, M.; No-Arbirage aylor Rules, Working paper, Available a hp://gsbwww.uchicago.edu/fac/monika.piazzesi/research/adp2004.pdf Ang, A.; Piazzesi, M.; A No-Arbirage Vecor Auoregression of erm Srucure Dynamics wih Macroeconomic and Laen Variables, Journal of Moneary Economics Volume 50, Issue 4, May 2003, pp Ang, A.; Piazzesi, M.; Wei, M.; Wha does he yield curve ell us abou GDP growh?, Journal of Economerics 2005 (forhcoming). Bekaer, G.; Cho, S.; Moreno, A.; New-Keynesian macroeconomics and he erm srucure, Working paper, Columbia Universiy. Dai, Q.; Singleon, K.; Specificaion analysis of affine erm srucure models, he journal of finance, Volume 55, Issue 5, pp Dai, Q.; Singleon, K.; 2003a. erm srucure analysis in heory and realiy, Review of Financial Sudies, Volume 16, pp Dai, Q.; Singleon, K.; 2003b. Fixed Income Pricing, Handbook of Economics of Finance, Chaper 20, C. Consaninides, M. Harris, and R. Sulz, eds, Norh Holland, Diebold, F. X.; Piazzesi, M.; Rudebush, G. D.; Modeling Bond Yields in Finance and Macroeconomics, American Economic Review P&P, 2005(forhcoming). Diebold, F. X.; Li, C.; Forecasing he erm Srucure of Governmen Bond Yield, Journal of Economerics, forhcoming. Diebold, F. X.; Rudebusch, G. D.; Aruoba, B.; he Macroeconomy and he Yield Curve: A Dynamic Laen Facor Approach, Journal of Economerics, forhcoming. Duffie, D. ; 2001: Dynamic asse pricing heory, hird Ediion, Princeon Universiy Press Hördahl, P.; risani, O.;Vesin, D.; A join economeric model of macroeconomic and erm srucure dynamics, European Cenral Bank Working papers, No. 405, November James, J.; Webber, N.; Ineres rae modelling, John Wiley & Sons ld. Piazzesi, M.; Affine erm srucure models, prepared for he Handbook of financial economerics, available a he following link: hp://home.uchicago.edu/~lhansen/s.pdf 15

16 Piazzesi, M.; Bond yields and he Federal Reserve, Journal of Poliical Economy Volume 113, Issue 2, April 2005 pp Rudebush, G. D.; Wu,.; A macro-finance model of he erm srucure, moneary policy, and he economy, FRBSF Working Paper Series, 2003 Rudebush, G. D.; Wu,.; he recen shif in erm srucure behavior from a noarbirage macro-finance perspecive, FRBSF Working Paper Series, 2004 Sundaram, R. K; Equivalen maringale measures and risk/neural valuaion: an exposiory noe, Journal of Derivaives, Fall 1997, aylor, J. B.; 1993: Discreion versus policy rules in pracice, Carnegie-Rocheser conference series on public policy 39, Vasicek, O.; An equilibrium characerizaion of he erm srucure, Journal of Financial Economics 5,