Testing Affine Term Structure Models in Case of Transaction Costs Driessen, Joost; Melenberg, Bertrand; Nijman, Theo

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1 Tilburg Universiy Tesing Affine Term Srucure Models in Case of Transacion Coss Driessen, Joos; Melenberg, Berrand; Nijman, Theo Publicaion dae: 1999 Link o publicaion Ciaion for published version (APA): Driessen, J. J. A. G., Melenberg, B., & Nijman, T. E. (1999). Tesing Affine Term Srucure Models in Case of Transacion Coss. (CenER Discussion Paper; Vol ). Tilburg: Finance. General righs Copyrigh and moral righs for he publicaions made accessible in he public poral are reained by he auhors and/or oher copyrigh owners and i is a condiion of accessing publicaions ha users recognise and abide by he legal requiremens associaed wih hese righs. - Users may download and prin one copy of any publicaion from he public poral for he purpose of privae sudy or research - You may no furher disribue he maerial or use i for any profi-making aciviy or commercial gain - You may freely disribue he URL idenifying he publicaion in he public poral Take down policy If you believe ha his documen breaches copyrigh, please conac us providing deails, and we will remove access o he work immediaely and invesigae your claim. Download dae: 26. apr. 2018

2 Tesing Affine Term Srucure Models in case of Transacion Coss Joos Driessen Berrand Melenberg Theo Nijman Firs Version: November 26, 1998 This Version: Sepember 15, 1999 We hank Lars Hansen, Pieer-Jelle van der Sluis, Bas Werker and paricipans of he Workshop Financial Modeling and Economeric Analysis, Lille 1999, and of he EFA conference, Helsinki 1999, for heir helpful commens. All hree auhors are from he Deparmen of Economerics, Tilburg Universiy. Corresponding auhor: Joos Driessen, Deparmen of Economerics, Tilburg Universiy, PO Box 90153, 5000 LE Tilburg, The Neherlands. Tel: The mos recen version of his paper is available a hp://cener.kub.nl/phd_sud/jdries/index.hm.

3 Tesing Affine Term Srucure Models in case of Transacion Coss Absrac In his paper we empirically analyze he impac of ransacion coss on he performance of affine ineres rae models. We es he implied (no arbirage) Euler resricions, and we calculae he specificaion error bound of Hansen and Jagannahan o measure he exen o which a model is misspecified. Using daa on T-bill and bond reurns we find, under he assumpion of fricionless markes, srong evidence of misspecificaion of one- and wo-facor affine ineres rae models. This is in line wih earlier research. However, we show ha he pricing errors of hese models are reduced considerably, if relaively small ransacion coss are aken ino accoun. The average ransacion coss for T-bills, due o he bid-ask spread, are around 1.5 basis poins. A his size of ransacion coss and for monhly holding periods, he misspecificaion of one- and wo-facor affine ineres rae models becomes saisically insignifican and economically very small. For quarerly holding periods, higher ransacion coss of around 3 basis poins are required o avoid misspecificaion. JEL Codes: G12, E43. Keywords: Ineres Rae Models; Marke Fricions; Transacion Coss; Model Misspecificaion.

4 1 Inroducion Nowadays erm srucure models are used exensively for many purposes, including risk managemen of porfolios conaining bonds and he valuaion of ineres-rae derivaives. No surprisingly, ess of he empirical validiy of he commonly used erm srucure models have araced considerable aenion in he lieraure. In line wih a large par of he empirical asse pricing lieraure, he ess are based on he assumpion of rading in fricionless markes. For example, Sambaugh (1988), Gibbons and Ramaswamy (1993), and Pearson and Sun (1994) es affine ineres rae models using daa on Treasury bills and bonds under he assumpion of rading in fricionless markes. However, marke fricions such as ransacion coss or shor selling consrains are an imporan fac of life for invesors. The implici assumpion when ignoring ransacion coss is ha hese coss are sufficienly small, so ha hey do no seriously affec he empirical resuls. In his paper we will explicily ake ransacion coss ino accoun in he empirical esing of affine erm srucure models, and show ha including ransacion coss considerably affecs ess of affine ineres rae models. We shall analyze he ineres rae models by esing wheher he sochasic discoun facor of each of hese ineres rae models saisfies he Euler resricions. These Euler resricions are implied by he noarbirage assumpion, and can be derived in boh fricionless markes and markes wih fricions. Based on hese Euler resricions, we will use wo approaches o analyze and es he models. Firs, we use Wald-ype ess o es he implied Euler resricions. For he fricionless case, he analysis of Euler resricions using Wald-ess is exensively discussed by Cochrane (1996). A disadvanage of his approach is ha, if one rejecs a model, here is no clear indicaion of he direcion of misspecificaion, for example which individual asses are possibly mispriced by he model and which are no. Also, if one applies his approach o wo non-nesed models and boh are rejeced, no indicaion is obained wheher one model is more misspecified han he oher. To overcome hese problems we also consider he specificaion error bound (SEB) developed by Hansen, Heaon and Lumer (1995) and Hansen and Jagannahan (1997). This bound measures he exen o which a model misprices a given se of asses. Hansen and Jagannahan (1997) show ha his bound can be inerpreed as he maximum pricing error for all porfolios ha can be consruced from he asses under consideraion. Also, his specificaion error bound allows for direc comparison across (non-nesed) models and he mehod indicaes which (porfolios of) asses conribue mos o he misspecificaion. Hansen and Jagannahan (1997) only consider fricionless economies; Hansen, Heaon and Lumer (1995) exend he seup o allow for marke fricions. We apply heir approach o affine erm srucure models and compare he resuls wih sandard ess using he Euler resricions. Our work is relaed o Lumer (1996) and He and Modes (1995), who boh analyze he influence of ransacion coss and oher marke fricions on he size of he volailiy bounds of Hansen and Jagannahan (1991), ha give a lower bound on he variabiliy of valid sochasic discoun facors. Empirically, Lumer (1996) finds ha small ransacion coss grealy influence he size of he volailiy bounds; especially, he -1-

5 volailiy bounds based on T-bill reurns are very sensiive o he size of ransacion coss. The resuls of Lumer (1996) imply ha he conclusion of rejecion of several asse pricing models in Hansen and Jagannahan (1991), based on he volailiy bounds, changes if ransacion coss are aken ino accoun. Our work exends he work of Lumer (1996), because he volailiy bound is a special case of he specificaion error bound. Also, Lumer (1996) focuses on consumpion-based asse pricing models, whereas we analyze bond pricing models and bond reurns. The bond pricing models ha we consider are discree-ime versions of he affine-yield models of Duffie and Kan (1996). This class includes he well-known Vasicek (1977) and Cox, Ingersoll and Ross (CIR, 1985) models. There is by now a large lieraure ha empirically invesigaes hese affine-yield models in fricionless markes (for example, Babbs and Nowman (1999), Backus and Zin (1994), Brown and Schaefer (1994), Chen and Sco (1993), Gibbons and Ramaswamy (1993), De Jong (1999), and Pearson and Sun (1994)). Our resuls indicae ha, wihou marke fricions, here is considerable evidence ha boh one- and wo-facor Vasicek, CIR and general affine Duffie-Kan (1996) models significanly misprice he reurns on bond porfolios, and, especially, he reurns on porfolios ha conain boh exreme long and shor posiions in shor-mauriy T-bills. This resul is in line wih mos of he empirical work menioned above, alhough in his lieraure he resuls for wo-facor models are somewha mixed. We also find ha he wo-facor Vasicek and CIR models have much higher specificaion errors han he general affine wofacor model. Afer invesigaing he fricionless case, we analyze how he empirical resuls are affeced if ransacion coss are aken ino accoun. The average bid-ask spread for T-bill prices in our daa is approximaely 3 basis poins, which is of a similar size as he spread repored by Lumer (1996). We find ha, for monhly holding periods, he bond pricing implicaions of he one- and wo-facor affine models are consisen wih he daa for ransacion coss of only 1.5 basis poins: he misspecificaion of he simple one- and wofacor models is hen boh saisically insignifican and economically very small. Because of he ransacion coss, he porfolios wih boh long and shor posiions in T-bills (and bonds) are no longer mispriced. For quarerly holding periods, on he conrary, he one- and wo-facor affine models are rejeced a he same size of ransacion coss. Transacion coss around 3 basis poins are required in his case o avoid any evidence of model misspecificaion. The remainder of his paper is organized as follows. In secion 2, we briefly review he lieraure on affine ineres rae models and asse pricing in markes wih fricions. In secion 3, we describe a Wald-es of he Euler resricions in a marke wih fricions. Secion 4 discusses he specificaion error bound. In secion 5, we describe our daa as well as he parameer esimaes for he affine ineres rae models. In secion 6 we presen he empirical resuls for he Wald-ess and specificaion error bound boh for he oneand wo-facor models. In secion 7 we summarize and conclude. -2-

6 2 Pricing Implicaions of Affine Ineres Rae Models in case of Transacion Coss Le he n-dimensional vecor R +1 conain he monhly gross reurns from ime o ime +1 of n asses (in our case bonds of n differen mauriies). Wihou marke fricions, i is well known (see, for example, Campbell, Lo and MacKinlay (1997), Chaper 11) ha absence of arbirage opporuniies in erms of he n asses requires he exisence of a sricly posiive sochasic discoun facor (SDF) Y +1 saisfying E [R i,%1 Y %1 ] ' 1, i'1,..,n. (1) Here he expecaion is condiional on he informaion se a ime. A model formulaed in erms of a paricular SDF can be esed by verifying some implied uncondiional version of equaion (1). If here are shor-selling consrains on he asses, absence of arbirage opporuniies requires he exisence of a sricly posiive SDF saisfying E [R i,%1 Y %1 ] # 1, i'1,...,n, (2) see, for example, Jouini and Kallal (1995) or Lumer (1996). For any given SDF Y +1, hese Euler inequaliies can be esed by applying, for example, he mehodology of Kodde and Palm (1986). When considering ransacion coss, we resric ourselves o he case of a proporional spread s ha is equal a he ask and bid side, and he same for all asses under consideraion. Le of asse i a ime. Then he gross reurn on aking a long posiion is equal o P i, denoe he midprice (1&s/2) P i,%1 (1%s/2) P i, / J l P i,%1 P i, / J l R i,%1, (3) and for shor posiions he gross reurn is equal o (1%s/2) P i,%1 (1&s/2) P i, / J s P i,%1 P i, / J s R i,%1. (4) -3-

7 In esing, ransacion coss can be aken ino accoun by rewriing he problem as one wih resricions on shor and long posiions (see Lumer (1996)) and inroducing separae asses for a long posiion in asse i wih reurn J l R i,%1 and for a shor posiion wih reurn J s R i,%1. As a consequence, he absence of arbirage opporuniies in he presence of ransacion coss requires he exisence of a sricly posiive SDF Y +1 such ha 1 J s # E [Y %1 R i,%1 ] # 1 J l, i'1,...,n. (5) We will analyze affine ineres rae models using he Euler resricions in (5), by esing wheher he SDF ha is implied by an affine model saisfies hese resricions for bond reurns, where we shall consider differen values of J l and J s. Duffie and Kan (DK, 1996) describe he class of coninuous-ime muli-facor ineres rae models, ha imply an affine relaionship beween ineres raes and a vecor of sae variables. Our seup is in discree ime. Therefore, we will use discree-ime versions of hese models, as described by, for example Backus e al. (1997) and Campbell, Lo and MacKinley (1997). These auhors show ha a discree-ime erm srucure model is affine, if he one-period ahead condiional join disribuion of he log-sdf, y %1 ' log(y %1 ), and an N-dimensional vecor of sae variables x %1 is mulivariae normal, and he condiional expecaion and covariance marix are boh affine funcions of he sae-variables x. Therefore, he N facor discree-ime DK model wih condiionally normal innovaions can be wrien as 1 &y %1 ' 4 ) N x %() > %1 x %1 ' µ%7x %E> %1 > %1 I ~ N(0,V) (6) V ' diag($ 1 %" ) 1 x,...,$ N %") N x ). Here > %1 represens an N-dimensional condiionally normally disribued random vecor wih zero condiional mean and condiional variance marix V, 7 and G are N N-marices conaining unknown parameers, " 1,...," N, $ ' ($ 1,...,$ N ) ), µ, and ( are N-dimensional unknown parameer vecors, 4 N is an 1 In Campbell, Lo and MacKinley(1997), he specificaion of he log-sdf also conains a normally disribued variable ha is independen from >. This variable only influences he mean of he yield curve in a way ha is very similar o he way he mean of he sae-variable influences he mean yield curve. In our analysis we do no include his variable (in line wih Backus and Zin (1994), Backus e al. (1997) and Bansal (1998)), allowing us in a sraighforward way o calculae he SDF in erms of observables. -4-

8 N-dimensional vecor conaining ones, and I represens he informaion se of ime. The componens of he vecor ( can be inerpreed as he marke prices of risk, as hey measure he sensiiviy of he SDF (and hus bond reurns) for he underlying facors. The N-facor Vasicek model is obained by seing ", whereas he N-facor CIR model is obained if. 2 j ' 0, j'1,..n $ j ' 0, " j '(0,..,0," jj, 0,...,0) ), j'1,..n In he empirical analysis, we will presen resuls boh for one- and wo-facor Vasicek and CIR models and for general affine one- and wo-facor models. Using equaions (1) and (6), one can derive ha bond prices are exponenial-affine funcions of he sae variable x, &log P n ' nr n ' A n %B ) n x, (7) where P n, is he price of an n-period zero-coupon bond a ime, and r n is he corresponding ineres rae. The facor loadings A n and B n are funcions of he underlying parameers, and do no depend on ime. By rewriing (6) he SDF can be expressed in erms of observables. Firs, by subsiuing > +1 in equaion (6), we ge > %1 ' E &1 (x %1 &µ&7x ) (8) and, hus, he SDF is given by &y %1 ' &( ) E &1 µ%(4 ) N &() E &1 7)x %( ) E &1 x %1 (9) Because all ineres raes are affine funcions of he sae-variables, as shown in equaion (7), his relaionship can be used o obain a log-sdf ha is affine in ineres raes of N differen mauriies and heir lagged values. If we define becomes r x as he N-dimensional vecor conaining hese N ineres raes, he SDF &y %1 ' 2 1 %2 ) 2 r x %2 ) 3 r x %1 (10) 2 Furhermore, no all parameers in hese affine models are idenified. Following Dai and Singleon (1997), we normalize he Vasicek and CIR models by seing G equal o he ideniy marix, and by imposing ha he marix 7 is a diagonal marix. Moreover, in he Vasicek model we normalize by seing all elemens of he vecor µ equal o zero, excep for he firs elemen µ

9 where he parameer 2 1, and he N-dimensional vecors 2 2 and 2 3 are implicily defined. Hence, hese reduced form parameers 2 ' (2 1,2 ) 2,2 ) 3 ) ) are funcions of he srucural parameers in equaion (6). Equaion (10) gives he funcional form of he SDF ha is implied by he discree-ime DK model (6) wih condiionally normally disribued innovaions. As noed by Kan and Zhou (1999), even if he SDF in (10) saisfies he Euler resricions exacly, here are oher SDFs ha also saisfy he Euler resricions. Tesing wheher a sochasic discoun facor saisfies he Euler resricions does no require a fully specified equilibrium model for asse reurns. The advanage of esing Euler resricions is, herefore, ha one can es a large class of models wihou making very specific assumpions. In our case, we can es all affine erm srucure models wih a given number of facors, by using he Euler resricions boh o esimae he reduced-form parameers 2 and o es he specificaion of he SDF in (10). Khan and Zhou (1999), demonsrae ha, if esimaion and esing are based on Euler resricions only, he es procedure migh have low power in deecing misspecified models. Therefore, we will also analyze specific equilibrium models, namely he Vasicek and CIR models, by esimaing he srucural parameers of hese models using oher momen resricions. These momen resricions are derived from he specificaion of he Vasicek and CIR models in equaion (6). Given esimaes of he srucural parameers of he Vasicek or CIR model, we can direcly consruc esimaes for he reduced-form parameers 2 and consruc he SDF of he Vasicek or CIR model using hese parameer esimaes. Subsequenly, we es wheher his Vasicek (or CIR) SDF saisfies he Euler resricions. In he nex secions, we will use he pricing implicaions of affine erm srucure models as described above o es he empirical validiy of hese models, using Wald ess as well as he specificaion error bound of Hansen and Jagannahan (1997). 3 Tesing he Euler Resricions using a Wald-Tes For every affine erm srucure model, Wald-ype ess of he Euler resricions for boh he fricionless case (equaion (1)) and he case wih ransacion coss (equaion (5)) are relaively sraighforward o implemen. For he case of ransacion coss, he inequaliy consrains can be esed along he lines of Kodde and Palm (1986). In his secion, we briefly review his es. The es for fricionless markes is a special case. We analyze resuls for differen holding periods, and, herefore, we firs of all inroduce producs of monhly SDFs and monhly reurns Y k ' k k j'1 Y %j, R k i, ' k k j'1 R i,%j. (11) -6-

10 In he empirical analysis, we will analyze boh monhly and quarerly holding periods, i.e., values for k of, respecively, 1 and 3. Equaion (5) implies ha hese lower frequency SDFs and reurns saisfy 1 J s # E [Y k R k i, ] # 1 J l, i'1,...,n. (12) Following Hansen and Jagannahan (1991) and many ohers, we incorporae condiional informaion by consrucing reurns on managed porfolios wih payoffs x k ' z ¼R k and corresponding prices q ' z ¼4, where z is a m-dimensional vecor wih variables ha are in he informaion se a ime. The implied uncondiional Euler resricions are 1 J s k x k i, ] # E[Y E[q,i ] # 1, i'1,...,m n. (13) l J In general, his muliplicaive approach is no an opimal way of incorporaing condiional informaion. For he volailiy bounds of Hansen and Jagannahan (1991), Ferson and Siegel (1998) discuss how o use condiional informaion opimally. Because similar resuls do no seem o be available ye for he specificaion error bound, and because of he simpliciy of he muliplicaive approach, we prefer o use his approach. In he sequel we refer o in (13) by n, insead of m n, o avoid oo cumbersome noaion. x k as he vecor of reurns, and we denoe he number of reurns The null hypohesis is ha he SDF saisfies he Euler resricions (13). Given T ime-series observaions on he n-dimensional vecor of reurns sample analogue x k and a SDF, we esimae he raio of expecaions in (13) by is ˆv i / 1 T jt '1 Y k x k i,, i'1,...,n. (14) 1 T jt '1 q i, Then he es-saisic > w as proposed by Kodde and Palm (1986) is given by -7-

11 > w ' min v0ú n T ( ˆv&v) ) Ŵ &1 ( ˆv&v) s.. 1 # v J s i # 1 J, i'1,...,n, (15) l where Ŵ is an esimaor of he asympoic covariance marix of ˆv ' ( ˆv 1,..., ˆv n ) ). For quarerly holding periods, we use he Newey-Wes (1987) mehod wih 2 lags in order o correc for he overlapping naure of he quarerly pricing errors 3. In he absence of ransacion coss, he es-saisic reduces o he J-saisic of Hansen (1982), and follows asympoically a chi-square disribuion wih n degrees of freedom. In case of ransacion coss, Kodde and Palm (1986) show ha, under he null hypohesis, his es-saisic is asympoically disribued as a mixure of chi-square disribuions 4. In his case simulaion can be used o obain p-values for a given value of he es-saisic. A disadvanage of his esing mehodology is ha, if a model is rejeced, here is lile indicaion of he direcion of he misspecificaion. Also, if one rejecs wo non-nesed models, no indicaion is obained wheher one model is more misspecified han he oher. In he nex secion, we will argue ha he use of he specificaion error bound overcomes hese problems. As described in he previous secion, we will boh es fully specified condiionally normal Vasicek and CIR models, as well as he general class of affine erm srucure models. To es all affine models wih a given number of facors, we esimae he parameers 2 in equaion (10) by minimizing equaion (15), wih J s 'J l '1, over 2 as well 5. Subsequenly, we use he Wald-es saisic (15) o es he (overidenifying) Euler resricions, boh for he fricionless markes case and he case of ransacion coss. The esimaion sraegy for 2 lowers he number of degrees of freedom of he limi disribuion, so ha he resuls of Kodde and Palm (1986) are no direcly applicable anymore. In he appendix we herefore derive he limi disribuion of he Wald es-saisic in (15), given he esimaion sraegy for 2. The Wald-es described above is useful o decide upon he correc number of facors in he SDF. > w 3 We have also esimaed he marix W using Newey-Wes (1987) wih 10 lags. This hardly changes he empirical resuls. 4 As suggesed by Wolak (1989, 1991), we inerpre his es as a local es of he inequaliy consrains. A global inerpreaion of our es procedure would imply ha we overesimae he size of ransacion coss ha is needed o avoid saisical rejecion of he model, or equivalenly, ha we would underesimae he influence of ransacion coss on model misspecificaion. 5 We do no re-esimae he parameer vecor 2 when we es he Euler resricions in markes wih ransacion coss, because his complicaes he calculaion of he limi disribuion of he es-saisic in (15). This is consisen wih esimaion sraegy for he parameers of he Vasicek and CIR models, where he (srucural) parameers are also no re-esimaed in case of ransacion coss. -8-

12 However, given he number of facors, his es does no indicae which erm srucure model has he bes empirical performance. Therefore, as described in he previous secion, we also es he Vasicek and CIR models, which are special cases of he DK-class of condiionally normal affine models in equaion (6). To analyze hese models, we firs derive momen resricions ha reflec he cross-secional and dynamic implicaions of he models, and esimae he srucural parameers of he models using GMM for hese momen resricions. These srucural parameer esimaes are hen used o obain esimaes of he reduced form parameers in he SDF. Subsequenly, we es wheher he SDF of he Vasicek or CIR model saisfies he Euler equaions using he es-saisic in (15). Because he momen resricions ha are used in he firs sep o esimae he parameers of he SDF are differen from he Euler resricions used in he second sep o es he model, he number of degrees of freedom of he limi disribuion of he es-saisic in (15) does no change. In his case, we only have o correc for he esimaion of 2 in he calculaion of he asympoic covariance marix in (15). 4 Analyzing he Euler Resricions using he Specificaion Error Bound As sressed by Hansen and Jagannahan (1997), an asse pricing model is an approximaion of realiy and, herefore, i will ypically no exacly saisfy he Euler resricions in an empirical analysis. They propose o measure he size of misspecificaion of a given proxy model, wih SDF Y, by measuring in some way he pricing errors of his proxy model. In his secion, we briefly describe he par of heir approach ha is relevan for our applicaion. In our case, he proxy model is given by one of he models ha we described in secion 2. For a given k holding period k, we sar by inroducing he se M of admissible SDFs consising of random variables m (which are in he informaion se a ime +k) ha saisfy he Euler resricions E[q,i ] J s # E[m k x k i, ] # E[q,i ] J l, i'1,...,n. (16) A SDF is hus admissible if i prices all (linear combinaions of) he asses under consideraion correcly. k The SDF Y ha is associaed wih he proposed model can be used o calculae model prices of he payoffs, ha, in general, may no saisfy he resricions in (16). Hansen and Jagannahan (1997) propose o measure he size of his misspecificaion by -9-

13 * 2 k ' min k m E[(Y 0M &m k ) 2 ]. (17) The square roo of (17) is called he Specificaion Error Bound (SEB), and can be inerpreed as a k (minimum) disance beween he proxy SDF Y and he se of admissible SDFs. 6 For he case wihou marke fricions, Hansen and Jagannahan (1997) show ha he SEB following from equaion (17) has an inerpreaion as he maximal pricing error of all porfolios in he n asses * ' max ë0ú n E[Y k (8 ) x k )&8 ) q ] s.. E[(8 ) x k ) 2 ] ' 1 (18) I is easy o show ha his inerpreaion of he SEB sill holds in he case of ransacion coss. More precisely, given he se M defined by (16), one can show ha * saisfies * ' max ë 0 ú n min v0ú n E[Y k (8 ) x k )&8 ) v] s.. E[(8 ) x k ) 2 ] ' 1 E[q,i ] J s # v i # E[q,i ] J l, i'1,...,n. (19) This shows ha * gives a bound on pricing errors of porfolio payoffs ha are normalized in a paricular way. Noe ha his normalizaion does no imply ha he componens or weighs in 8, which are equal o he Kuhn-Tucker mulipliers of he binding Euler resricions (see Hansen and Jagannahan (1997)), sum up o one. A sligh modificaion of a fricionless resul in Hansen and Jagannahan (1997) reveals ha he SEB can also be calculaed as follows 6 Hansen and Jagannahan (1997) also inroduce a bound where he se of admissible SDFs only conains SDFs wih he same uncondiional mean as he proxy SDF, and show ha his condiion is auomaically saisfied if one analyzes models wih a sochasic discoun facor ha conains an addiive, unknown consan erm, ha is chosen such as o minimize he SEB. We do no analyze sochasic discoun facors wih his propery, and we also do no impose his resricion on he mean of he SDF, because his would imply ha any model ha we analyze prices he reurn of a one-period bond wihou error. -10-

14 * 2 ' min v0ú n E[Y k x k &v] ) E[x k x k ) ] &1 E[Y k x k &v] s.. E[q,i ] J s # v i # E[q,i ] J l, i'1,...,n (20) Comparing his wih equaion (15) shows ha he SEB is closely relaed o he populaion analogue of he Wald es-saisic. The only difference is he weighing marix. Below we reurn o his difference and o he relaive advanages of he wo approaches. By replacing populaion momens wih heir sample analogues in equaion (20), an esimae ˆ* for he SEB can be obained. Hansen, Heaon and Lumer (1995) show, under he assumpion ha he rue * is sricly posiive, ha his esimaor has asympoically a normal limiing disribuion; hey also provide a consisen esimae for he asympoic variance 7. The assumpion ha he rue bound is sricly posiive is crucially differen from he seup of he Wald-es, where he null hypohesis is ha he model is correcly specified. For he Wald es, we handled he presence of unknown parameers in he SDF in wo ways, (i) by esimaing he reduced form parameers from he fricionless Euler resricions, which resuled in a es of he validiy of all affine models, and (ii) by esimaing he srucural form parameers of a paricular model, in our case he Vasicek or CIR model, in a firs sep using GMM. For he SEB, we will also use hese wo approaches. Hence, o analyze general N-facor affine models, we will presen resuls for he all affine models SEB, which is defined by he square roo of * 2 min ' min è 0 È max m k 0 M E[(Y k (2)&m k ) 2 ], (21) This bound gives a lower bound on he specificaion error of all affine ineres rae models wih a cerain number of facors. In order o obain esimaes of he SEB for he Vasicek and CIR models, he srucural parameers of hese models are esimaed using GMM wih some basic ineres rae momens, and he SDFs of he Vasicek and CIR models are consruced using hese parameer esimaes. In he second sep, he SEB is esimaed, along wih is asympoic variance 8. In our applicaion, we shall hus compare he esimaes of he all affine models SEB wih he esimaes of he Vasicek SEB and CIR SEB. A large difference beween he all affine models SEB and each of hese laer wo bounds is also an indicaion of 7 If he rue * is equal o zero, Hansen, Heaon and Lumer (1995) argue ha he limi disribuion is mixed chi-square if here are no ransacion coss. Then, his es-saisic is less efficien han he Wald-es discussed in he previous secion. 8 Of course, he asympoic variance as repored in Hansen, Heaon, and Lumer (1995) has o be adjused if he parameers of he SDF are esimaed in he firs sep. -11-

15 misspecificaion of he Vasicek or CIR ineres rae model ha is examined. Alhough he mahemaical difference beween he Wald es-saisic and he SEB is only he form of he weighing marix, he Wald-es and he SEB are wo complemenary approaches. The Wald-es allows for efficien saisical esing based on he Euler resricions of a given model, bu i does no provide informaion on he direcion of misspecificaion. If he model is misspecified, he properies of he ess are no easy o derive. For he SEB, i is a priori acceped ha he model is misspecified; herefore, he size of misspecificaion is measured, along wih he conribuions of individual asses o his misspecificaion size by means of he Kuhn-Tucker Mulipliers. 5 Daa and Esimaed Affine Term Srucure Models The daase ha we use conains monhly daa on ineres raes and bond holding reurns. The ineres rae daa are drawn from he CRSP Fama Files, and consis of ineres raes of mauriies ranging from 1 monh o 5 years. The shor-mauriy ineres raes are derived from T-bill prices, and he long-mauriy ineres raes are calculaed from bond prices. We use a subsample from , consising of 312 observaions. In able 1 some basic sample saisics of daa are presened. The monhly holding reurns daa ha we use also come from he CRSP Fama Files. For mauriies up o one year, we use he nominal holding reurns ha are calculaed from T-bill prices. For longer mauriies, we use he so-called mauriy porfolio reurns ha are consruced by averaging he nominal holding reurns of all bonds ha have a mauriy ha lies in a cerain inerval. The inervals we use are: 2 o 3 years, 4 o 5 years, 5 o 10 years, and larger han 10 years. Again we use he subsample from In able 2 we repor some sample properies of hese daa. From his able, i is clear ha he average holding reurns differ considerably for he various shor mauriies, whereas he differences in average holding reurns for he long-mauriy asses are quie small, relaive o he sandard deviaions and he difference in mauriy. In able 3 we repor informaion on he bid-ask spreads on T-bill prices, which are derived from he CRSP daa. I follows ha he size of he ransacion coss due o he bid-ask spread is around 1.5 basis poins, averaged over ime and over all T-bills. For bonds we do no have daa on he bidask spread, bu since long-mauriy bond prices are more volaile han shor-mauriy T-bill prices, and because he bid-ask spread of T-bills is increasing wih ime-o-mauriy, he bid-ask spreads on longmauriy bond prices are probably higher han he bid-ask spreads of T-bills. Table 3 also shows ha he bid-ask spreads have decreased considerably during he las 25 years. The following ses of asses reurns will be used in he empirical analysis: 9 Before 1972 here are missing observaions for some variables in he daa. -12-

16 1. Shor-Mauriies Asse Se: Four T-bills wih mauriies of 1, 3, 6, and 9 monhs wih condiional informaion consising of a consan and he raio of he 1-year and he 3-monh ineres rae. 2. Long-Mauriies Asse Se: Four porfolio holding reurns wih mauriy inervals equal o 2 o 3 years, 4 o 5 years, 5 o 10 years, and larger han 10 years, wih condiional informaion consising of a consan and he raio of he 5-year ineres rae and he 1-year ineres rae. 3. All-Mauriies Asse Se: Se 1 and se 2. We hus consider hree subses of asses, one ha conains only shor-mauriy T-bills, anoher one ha conains only long-mauriy bonds and a hird one ha conains bonds of boh shor and long mauriies. The mauriies of he T-bills are he same as in Lumer (1996). Following ha paper, he condiioning variables z are consruced in such a way ha hey are always posiive, so ha shor-selling consrains or ransacion coss are sraighforward o impose on he condiional asses as well. We choose o use he yield spread as condiioning variable because i migh proxy for a possible missing second facor. In he nex secion we presen resuls for one- and wo-facor Vasicek and CIR models, as well as for he general class of affine models wih one or wo facors. We use GMM o esimae he parameers of he one- and he wo-facor Vasicek and CIR models. For his GMM-esimaion, we choose our momen condiions such ha basic properies of boh a shor and a long ineres rae and he mean reurns on boh shor and long bonds are mached. Hence, as momens we include he mean, variance and auocovariance of boh he 3-monh and 5-year ineres rae, and he covariance beween boh he levels and he firs differences of hese wo raes. We also include in our momen se he mean of four bond reurns, wih mauriies of 3 monhs, 9 monhs, 2-3 years and 5-10 years. The parameers of he models are subsequenly esimaed by GMM using a consisen esimae for he opimal weighing marix (see, for example, Campbell, Lo and MacKinley (1997)). The resuls of he GMM esimaion are presened in able 4. For boh he one- and wo-facor Vasicek and CIR models, i can easily be verified ha he parameer esimaes imply an upward sloping mean yield curve and hus mean holding reurns ha are increasing wih mauriy. For he one-facor models, i urns ou ha he mean yield curve is relaively fla, whereas he wo-facor models are capable of fiing he upward sloping shor end of he yield curve. From he value of he J-saisic repored in his able i follows ha he overidenifying resricions lead o a srong rejecion of he one-facor Vasicek and CIR model. The fi of he wo-facor models is comparable o he fi of wo-facor affine models in, for example, Backus e al. (1997) and De Jong (1999), who boh conclude ha a wo-facor affine ineres rae model can quie reasonably fi he basic properies of ineres raes wih mauriies from a few monhs up o 10 years. However, for he wo-facor Vasicek model, no all parameers are esimaed accuraely. Given hese firs sep esimaion resuls, we can consruc observable SDFs of he one- and wo-facor -13-

17 Vasicek and CIR models, relevan for he SEB and he Wald es. We use equaions (9) and (10) and he 3-monh ineres rae o consruc an observable SDF for he one-facor models, by insering he GMM parameer esimaes. Similarly, we consruc a SDF for he wo-facor models using he 3-monh and 5-year ineres rae, and insering he GMM parameer esimaes of he Vasicek or CIR model. 6 Empirical Resuls for Term Srucure Models 6.1 One-Facor Models In his subsecion we presen empirical resuls for he specificaion ess of one-facor affine erm srucure models, firs of all for a seup wihou ransacion coss and hen wih ransacion coss, and boh for monhly and quarerly holding periods. We sar wih he case wihou ransacion coss, and monhly holding periods. In able 5, we presen in he upper panel he corresponding resuls for he one-facor models. Jus like he J-ess on he momen resricions ha were used in he previous secion, he Wald-es on he fricionless Euler resricions rejecs boh he one-facor Vasicek and CIR models for all asse ses. The bound esimaes are also large and far from zero, and he difference beween he SEBs of he Vasicek and CIR model is small. This is confirmed by he correlaions beween he pricing errors of he differen models, given in able 6, which shows ha he average correlaion beween pricing errors of he one-facor Vasicek and CIR models is The bounds based on he T-bills are much larger han he bounds based on long-mauriy bonds. As Lumer (1996) noices, an explanaion for he high T-bill bounds is ha, because he holding reurns on he differen T-bills are highly correlaed, differences in average holding reurns on hese T-bills can lead o somehing close o an arbirage opporuniy. Thus, he admissible se of SDFs is relaively small. For he long-mauriy bonds he differences in average holding reurns are no very large, especially relaive o he volailiy of he holding reurns, and hus he admissible se of SDFs is larger in his case. The Wald-es for he general class of affine one-facor models rejecs his class of models, and he differen asse ses yield SEBs which are large and far from zero. As expeced, we can conclude ha he one-facor affine models are no capable of pricing bond reurns of several mauriies correcly, given he fricionless markes assumpion. Also, he all affine models SEB is no very differen from he Vasicek and CIR SEBs, and able 6 shows ha he pricing errors of he general affine model are srongly correlaed wih he Vasicek and CIR pricing errors. This implies ha he one-facor Vasicek and CIR models are no much more misspecified han he general affine one-facor model. Given he assumpion of fricionless markes, he economic significance of he esimaed bounds is large. For example, based on he resuls for he All-Mauriies Asse Se and he Vasicek SEB, we can conclude ha for he one-facor Vasicek model here exiss a porfolio, normalized as in equaion (19), wih a pricing -14-

18 error of abou This porfolio has an observed (mid)price of 0.704, whereas he Vasicek model assigns a price of o his porfolio. In figure 1, we plo he -raios of he SEB-mulipliers for his one-facor Vasicek model in a fricionless marke. As shown in equaion (19), hese mulipliers are equal o he weighs of he maximum pricing error porfolio. This figure shows ha he mos severely mispriced porfolios, which drive he model rejecions irrespecive of he esing mehodology, are characerized by exreme shor and long posiions in adjacen mauriies. This implies ha he model is rejeced in his fricionless seing because he observed behaviour of bond-reurns of differen mauriies is less smooh han implied by he model. In his sudy of Euler equaions for equiy reurns, Cochrane (1996) also finds ha porfolios wih long and shor equiy posiions are largely mispriced. To obain furher insigh in hese resuls, we calculae he pricing errors for wo ypes of porfolios: porfolios in only one T-bill or bond, and wo-asse porfolios ha have a long posiion in one T-bill (bond) and an equally large shor posiion in anoher T-bill (bond). To faciliae he comparison beween hese porfolio pricing errors and he SEBs in able 5, we normalize hese porfolio weighs in he same way as he SEB-weighs 8 in equaion (19) are normalized. Table 7 presens he monhly pricing errors, in case of he one-facor Vasicek model 10. I follows ha individual T-bill and bond reurns have low pricing errors; he normalized pricing errors are around 0.01 for all asses. The normalized pricing errors for he porfolios in wo asses are much larger han he pricing errors for he individual asses, especially for he T-bills. Hence, he difference beween he small pricing errors of wo highly correlaed T-bill reurns implies a large pricing error for he porfolio ha has a long posiion in one T-bill and a shor posiion in he oher T-bill. Alhough he individual pricing errors of he shor-mauriy asses are comparable o hose of he longmauriy asses, he higher correlaion and lower variance of he shor-mauriy asse reurns gives higher pricing errors for he shor-mauriy wo-asse porfolios. Thus, under he assumpion ha here are no marke fricions, we find clear evidence of he misspecificaion of one-facor affine erm srucure models, which is line wih oher empirical work on onefacor models. To assess wheher his misspecificaion is sill presen if we correc for he presence of ransacion coss, we include ransacion coss of J (= s/2 in erms of equaions (3) and (4)) basis poins per holding period, where we le J vary beween 0 and 3 basis poins. We assume for simpliciy ha he ransacion coss are he same for all ransacions. In able 8, we summarize in he upper panel he Waldes resuls for a marke wih such ransacion coss and monhly holding reurns. I follows ha, for relaively small amouns of ransacion coss of around 1 basis poin, he one-facor Vasicek, CIR and affine one-facor models are no rejeced anymore, given our sample size and sample period. In figure 2A, we include a plo of he sensiiviy of he SEBs o he size of ransacion coss. For comparison, we also graph he SEB of he risk-neural pricing model, ha is obained if in a one-facor affine model he marke price of risk is equal o zero. Hence, he monhly SDF of his model is simply equal 10 The resuls for he CIR model are similar. -15-

19 o exp(&r 1 ). The graph shows ha he SEB for his simple model is always larger han he SEBs of all oher models, as could be expeced. Sill, he difference beween he SEB of he risk-neural pricing model and he SEBs of he oher models is no very large. The graph also shows ha, for he Vasicek, CIR and affine models, he size of he SEB is around 0.01 a ransacion coss of 1.8 basis poin, which is economically raher small. In he fricionless case, exreme shor and long posiions in T-bills and bonds blow up he differences beween pricing errors of T-bills and bonds so ha sandard es procedures rejec he affine models. We, however, show ha, if small ransacion coss are aken ino accoun, hese differences in pricing errors are no large enough o cause rejecion of he models. In figure 1, he -raios of he Vasicek SEB-mulipliers indicae ha, for ransacion coss of 1 basis poin, only he pricing errors of some individual asses conribue o he misspecificaion size of he onefacor models. In he absence of ransacion coss, boh he uncondiional and he condiional 6-monh and 9-monh T-bill reurns conribue mos significanly o he size of he wo-sep SEB. A ransacion coss of 1 basis poin, none of he individual asses conribues significanly o he size of he SEB. In fac, for several bonds, he mulipliers are exacly equal o zero; herefore, hese asses do no conribue a all o he size of he SEB. So far, all resuls discussed are based on he assumpion ha he holding period is equal o one monh. To analyze he influence of his assumpion, we now urn o he resuls of a quarerly holding period. In he lower panel of able 5 we give he p-values of he Wald-es and he SEBs of he one-facor models for a quarerly holding period, wihou ransacion coss. Again, in all cases, he models are rejeced saisically. If pricing errors and reurns were independenly and idenically disribued over ime, he SEB and is asympoic sandard error should increase approximaely by a facor 3. From able 5 we see ha his rule of humb is no exacly rue: we find an increase in he SEB ha is a lile smaller han 3. For he case of ransacion coss, he resuls for he quarerly holding period are given in he lower panel of able 8. Compared o he monhly holding period, larger ransacion coss of 3.1 basis poins are required o accep he models saisically and o obain a small SEB. Because he monhly pricing errors are only very weakly correlaed over ime, he quarerly pricing errors are larger han he monhly pricing errors and, herefore, also larger ransacion coss are required o accep he model saisically and o obain a small SEB. The difference beween he resuls for he monhly and quarerly SEBs is sriking. I implies ha he choice of holding period is a non-rivial one. Figure 2B shows ha here is sill a srong influence of small ransacion coss on he SEBs, alhough i is clearly less srong han for monhly holding periods. 6.2 Two-Facor Affine Models Nex we presen resuls for wo-facor Vasicek, CIR and general affine models, again for he cases wihou and wih ransacion coss and monhly and quarerly holding periods. In able 9 we presen he resuls for he Wald es and he SEB for he wo-facor models, for boh monhly and quarerly holding reurns and -16-

20 no ransacion coss. The wo-facor models are all rejeced by he Wald-es, under he assumpion of fricionless markes. The resuls also show ha he SEBs of he Vasicek and CIR models do no decrease significanly compared o he one-facor case when we add he second facor, which is consisen wih he high correlaion beween pricing errors of one-facor and wo-facor Vasicek (and CIR) models repored in able 6. However, he SEB of he general wo-facor affine model is much lower han he SEB of he onefacor affine model and he wo-facor Vasicek and CIR models. Because here are many oher wo-facor affine models han he wo-facor Vasicek and CIR models, his resul indicaes ha here is a wo-facor affine model, differen from he wo-facor Vasicek and CIR models, ha has a much lower specificaion error. Sill, under he assumpion of fricionless markes, he general class of wo-facor affine models is rejeced by he Wald es. These resuls are comparable o, for example, he resuls of Pearson and Sun (1994), who esimae a wo-facor CIR model using ML based on wo asse reurns and who, subsequenly, analyze he implicaions for all oher T-bill and bond prices. They conclude ha he wo-facor CIR model canno price all hese asses correcly and rejec his model saisically. In he previous secion we have indicaed how he presence of ransacion coss can provide a possible explanaion for he misspecificaion of he onefacor models in fricionless markes, whereas i hus seems ha adding a second facor wihou ransacion coss does no saisfacorily solve his misspecificaion. On he basis of able 7, we find ha he pricing errors of he one- and wo-asse porfolios are roughly he same for he one- and wo-facor Vasicek models, which is again consisen wih he high correlaion beween he pricing errors of he one- and wo-facor Vasicek models. This is also confirmed by figures 1 and 3 in which we find quie similar SEB-mulipliers for boh he one- and wo-facor Vasicek models. For he case of ransacion coss, he resuls are given in able 10 and in figures 4A-B. Qualiaively, he resuls are quie similar o he resuls for he one-facor models, alhough he ransacion coss ha are needed o accep he model, he so-called criical ransacion coss, are a lile smaller for he wo-facor Vasicek and CIR models, namely 2.2 basis poins for he quarerly holding period and he All Mauriies Asse Se. For he general class of affine wo-facor models and quarerly holding periods, ransacion coss of around 2.8 basis poins are needed o avoid rejecion of he model, given our sample size and sample period. Figures 4A-B demonsrae again he srong sensiiviy of he SEB o he size of ransacion coss. These figures also show ha he difference beween he SEB of he general affine wo-facor model and he SEBs of he Vasicek and CIR wo-facor models does no disappear when ransacion coss are included. -17-

21 7 Summary and Conclusions In his paper we analyze he bond pricing implicaions of affine ineres rae models, allowing for he presence of ransacion coss. The goal of he paper is o assess he imporance of incorporaing marke fricions for ess of asse pricing models. We es he models formally for differen sizes of ransacion coss, using a Wald-es, and we measure he size of misspecificaion of one- and wo-facor affine ineres rae models and analyze how sensiive he misspecificaion size is o he size of he ransacion coss. Our analysis can be seen as an exension of Lumer (1996), because we use he sronger specificaion error bound es, as opposed o he volailiy bound ha is used by Lumer (1996), which is a special case of he specificaion error bound. Also, Lumer (1996) focuses on consumpion-based asse pricing models, whereas we analyze models for he erm srucure of ineres raes. We find ha, under he assumpion of fricionless markes, one-facor affine ineres rae models misprice T-bill and bond reurns in a significan way. Small differences in he pricing errors of highly correlaed T-bill reurns lead o a srong rejecion of he one-facor models. Adding a second facor o he models does no explain he misspecificaion of one-facor models: wo-facor models are also srongly rejeced. We find, however, if we ake ransacion coss of abou 1.5 basis poins ino accoun, ha he misspecificaion of he one- and wo-facor models disappears, in case of a monhly holding period. For quarerly holding periods, hese models are no rejeced a ransacion coss of around 3 basis poins. -18-

22 Appendix: The Limi Disribuion of he All Affine Models Wald Tes To obain Wald-ess of general affine models, we esimae he parameers 2 in he SDF (10) by minimizing he es-saisic in (15), wih J s 'J l '1, over he parameers 2 as well. Subsequenly, we subsiue hese parameer esimaes in he SDF in (6), and es he (overidenifying) Euler resricions, boh in case of fricionless markes and in case of ransacion coss, using he es-saisic in (15). The esimaion sraegy for 2 influences he limi disribuion of he Wald-es in (15). In his appendix we derive his limi disribuion. We will show ha he limi disribuion of his minimized es-saisic is sill a mixure of chisquare disribuions, bu wih less degrees of freedom han he es-saisic ha is no minimized over he parameers 2. The Euler resricions used for esimaion are of he form v(2) / E[Y k (2) x k i, ] E[q i, ] ' 1, i'1,..,n, (A.1) where 2 is a q-dimensional parameer vecor, wih n>q. To simplify noaion, we define he esimaor of he lef-hand side of (A.1) for a given value of 2 as follows ˆv(2) i / 1 T jt '1 Y k (2)x k i, 1 T jt '1 q i,, i'1,...,n. (A.2) We hen esimae he parameer 2 by applying GMM o he momen resricions (A.1), wih a consisen esimae for he opimal weighing marix. This is equivalen o minimizing (15) over 2 wih ransacion coss se o zero. Denoe he unique probabiliy limi of he GMM-esimae ˆ2 wih 2 0. The asympoic covariance marix of ˆv( ˆ2) is given by T( ˆv(ˆ2)&4) 6 N(0,(I&M 0 )W(2 0 )(I&M ) 0 )), (A.3) where W(2 0 ) is he asympoic covariance marix of ˆv(2 0 ), and where M 0 is given by (see Gourieroux and Monfor (1995)) -19-

23 Mv(2 0 ) M2 ) [ Mv(2 0 )) M2 W(2 0 )&1Mv(2 0 ) M2 ) ] &1 Mv(2 0 )) M2 W(2 0 )&1. (A.4) The covariance marix in (A.3) has rank n-q. As noed by Gourieroux and Monfor (1995), a generalized inverse of he marix (I&M 0 )W(2 0 )(I&M ) 0 ) is given by W(2 0 )&1, so ha we can consisenly esimae his generalized inverse by he asympoic covariance marix. Ŵ( ˆ2) &1, which consiss of replacing populaion momens wih sample momens in In a second sep, he es-saisic is obained from he following minimizaion > w ' min v0ú n T[ ˆv( ˆ2)&v] ) Ŵ( ˆ2) &1 [ ˆv( ˆ2)&v] s.. 1 # v J s i # 1 J, i'1,...,n. (A.5) l Then, condiional on he even ha all resricions are esimaed o be binding in (A.5), he saisic in (A.5) is asympoically -disribued under he null hypohesis. Similarly, condiional on he even ha P 2 n&q exacly p resricions in (A.5) are esimaed o be binding, he disribuion of he es-saisic under he null hypohesis is P 2 max(p&q,0), where P 2 0 is he uni mass a he origin. The probabiliy weighs for each of hese evens can be found using he degeneraed mulivariae normal limi disribuion of ˆv( ˆ2) under he null hypohesis given in equaion (A.3). Hence, he limi disribuion of he es-saisic (A.5) under he null hypohesis is a mixure of chi-square disribuions wih degrees of freedom beween 0 and n-q. -20-