Tests of the Expectations Hypothesis and Policy Reaction to the Term Spread: some comparative evidence. Gianna Boero. Costanza Torricelli

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1 Tess of he Expecaions Hypohesis and Policy Reacion o he Term Spread: some comparaive evidence Gianna Boero Universiy of Warwick and Universiy of Cagliari and Cosanza Torricelli Universiy of Modena July 998 Absrac The aim of his paper is o evaluae he impac of moneary policy in ess of he Expecaions Hypohesis of he erm srucure of ineres raes. We apply he model developed by McCallum (994b), in which he Expecaions Hypohesis ineracs wih a policy reacion funcion and wih a ime-varying erm premium, o eigh counries wih differen moneary policy sances, wihin he period 985 o 995. The resuls sugges he imporance of he reamen of moneary policy in explaining he empirical performance of he Expecaions Hypohesis. Amongs oher resuls, we also find ha he model performs beer for some counries han ohers depending upon he moneary policy sance adoped. Keywords: Expecaions Hypohesis, ineres raes, moneary policy, erm srucure. JEL classificaion: C22, E43, E52 The auhors wish o hank Luisa Malagui for helpful commens on a previous version of he presen paper and acknowledge suppor from CNR CT0 and CNR97.02.CT0.

2 . INTRODUCTION There is an ongoing debae concerning he validiy of he Expecaions Hypohesis (EH) of he erm srucure of ineres raes (TS). The hypohesis has been widely esed, for differen counries and differen ime periods, wih mixed resuls. Typically, he hypohesis has been acceped wih daa for European counries (see, iner alia, Boero and Torricelli 997, Engsed and Tanggaard, 995, Gerlach and Smes, 997) and rejeced for he US (see Rudebusch, 995, for a summary of differen US sudies). However, he mos recen findings for he US provide new evidence in favour of he EH (Hsu and Kugler, 997). In order o inerpre his disparae evidence on he EH of he TS, hree main explanaions have been proposed in he lieraure. The firs one ress on a deparure from he assumpion of Raional Expecaions (RE), which is normally esed joinly wih he EH. An example of his is he overreacion explanaion pu forward by Hardouvelis (994), according o which agens do no reac raionally, bu insead overreac o expeced changes in he shor rae signalled by he TS spread. A second possible explanaion aribues he empirical failures of he EH o he exisence of ime-varying erm premia. Due o he unobservable naure of he erm premium, is influence on ess of he EH can be assessed only indirecly by including a proxy for he erm premium in sandard regression ess. Aemps along hese lines can be found in Simon (989), Boero, Madjlessi and Torricelli (996) and Tzavalis and Wickens (997). A number of papers (e.g. Kugler, 990, Gerlach and Smes, 997) have anyway shown ha ime-varying erm premia are no as imporan as he scarce variabiliy of shor raes in diminishing he predicive power of he EH. The hird explanaion involves policy behaviour, and assers ha he limied variabiliy of shor raes is due o paricular moneary policy sances. The basic idea sems from he argumen suggesed by Mankiw and Miron (986) ha he abiliy of he spread o predic fuure ineres rae movemens is enhanced in he presence of a money supply arge policy and is diminished 2

3 under ineres rae sabilisaion. In order o invesigae he issue furher, wo possible lines of invesigaion have been proposed in he lieraure: he firs empirical, he second heoreical. The former has been pu forward by Dosey and Orok (995) and Rudebusch (995), who have empirically formalised he argumen of Mankiw and Miron, by generaing synheic ineres rae daa from a Federal Reserve ineres rae argeing model, and hen using hese o es he EH. This ype of empirical analysis canno be replicaed for counries where moneary policy is officially moneary argeing and public ineres rae arges are no available. An alernaive line of research is represened by he heoreical model proposed by McCallum (994b). The auhor develops a model of he ineracion beween he EH of he TS, a ime-varying auoregressive erm premium, and an ineres rae smoohing moneary policy combined wih a reacion o changes in he spread. The model shows ha resuls in suppor of he EH can be explained by a srong policy response o changes in he spread and a highly posiively auocorrelaed erm premium. In a recen paper, Kugler (997) exends he McCallum model, deriving an exac soluion for he N-period long rae case. Applicaions of he model o US daa (Hsu and Kugler, 997) and o a number of differen counries (Kugler, 997) indicae he model is able o explain he resuls of sandard regression ess of he EH. The aim of he presen paper is o explore furher he influence of moneary policy on he oucome of ess of he expecaions hypohesis by applying he McCallum model o a wider range of counries wih differen moneary policy sances, and o differen sample periods. The counries considered are: USA, Japan, Germany, UK, France, Ialy, Canada and Swizerland. We find ha he McCallum model is beer able o explain resuls for counries where he moneary policy involves ineres rae smoohing and/or clearly responds o This model is closely relaed o he policy reacion model developed by McCallum (994a) which explains failures of Uncovered Ineres Pariy as a consequence of sysemaic moneary policy behaviour. 3

4 changes in he long-shor spread, han for counries operaing wih a more complex moneary policy. The empirical analysis is conduced wih weekly Euro-raes, for differen subperiods beween 985 and 995, and ress on wo sandard regression ess involving erm spreads: one predicing fuure shor raes, he oher fuure long raes. We sar he empirical analysis wih he implemenaion of sandard ess of he EH. The resuls from hese ess are confroned wih boh asympoic and small sample disribuions. The laer are hose newly derived by Bekaer, Hodrick and Marshall (BHM) (997), where he main source of bias is represened by he high persisence in shor ineres raes. When we conduc inference wih he BHM empirical disribuions we find ha small sample biases can seriously affec he inerpreaion of sandard ess of he EH, paricularly hose based on he long raes. Then, in he second sep of he analysis, we compare sandard regression ess wih he resuls obained wih he esimaion of he McCallum model and evaluae he abiliy of his model o explain deviaions from he EH. The plan of he paper is as follows. Secion 2 oulines he main feaures of he McCallum model and he exac soluion derived by Kugler (997), and Secion 3 presens he mehodology followed for he implemenaion of he model. Secions 4 and 5 repor evidence on ess of he EH for eigh differen counries: firs we perfom sandard regression ess ignoring moneary policy consideraions and conduc inference using boh asympoic and small sample disribuions (Secion 4); hen we esimae he McCallum model and evaluae is abiliy o explain conflicing evidence from sandard ess of he EH (Secion 5). Secion 6 closes he paper wih conclusions and furher remarks. 2. THE McCALLUM MODEL In his secion we describe he heoreical model esed in he presen paper. The model was originally se up by McCallum (994b) and laer developed by Kugler (997). In he 4

5 following, we presen he McCallum model and he exac soluion provided by Kugler o his model. McCallum (994b) develops an N-period model, characerised by an equaion for he erm srucure and an equaion for he moneary policy rule. The former is represened by he expecaions hypohesis modified by he exisence of an auoregressive ime-varying erm premium, which implies ha he reurn on an N-period bond is given by: R N N = r + Er + i N i= + ξ () ξ = ρξ + u (2) where N R is he reurn on an N-period long-erm bond, N is ime o mauriy of he longerm bond, r is he reurn on a one-period bond, ξ is he erm premium on he long bond wih ρ <, and u is a whie noise error. 2 For N large, i is reasonable o assume he approximaion: E R E R (3) N = N + + Hence () can be approximaed as follows: R N ( E R+ R ) = r + ξ (4) where from now on we drop superscrips and R sands for he reurn on a N period bond. For empirical ess, (4) can be more usefully rewrien as: N ( R R ) = ( R + r ) + ξ Nε (5) where ε = R E R is he expecaional error, which under RE is uncorrelaed wih R + + and r. The moneary policy rule is supposed o be aimed a ineres rae smoohing combined wih a reacion o he erm spread, i.e.: 2 ξ is no exacly he erm-premium on an N-period bond, bu insead a linear combinaion of erm premia. A deeper discussion of his poin and of oher assumpions underlying McCallum s model can be found in Malagui-Torricelli (997). 5

6 ( R r ) ζ r = r r = λ + (6) where λ 0 is he policy parameer, and ζ represens oher componens of policy behaviour and, for simpliciy, is assumed o be whie noise 3. The rule is based on he observaion ha acual moneary policy in many counries involves manipulaion of a shor-erm ineres rae insrumen (McCallum, 994b). Obviously, (6) represens a srong sylisaion of acual moneary policy rules since Cenral Banks generally use a wider range of policy indicaors oher han he spread. Neverheless, given he correlaion beween he spread and oher indicaors (e.g. real economic growh and inflaion expecaions), his simple rule can be hough of as also capuring policy sances of hose Cenral Banks which officially do no use he spread as an indicaor (e.g. he Bundesbank). Combining (4) and (6) gives: [ ] ( + N) R = NE R+ + ( + λ) r + λr + ζ + ξ (7) which has o be solved for R. The RE soluion procedure is based on he minimum-sae-variable (MSV) crierion discussed by McCallum (983), whereby he soluion is assumed o have he following form: R and is given by: = φ r + φ 2ξ + φ 3 ζ (8) R + λ = r N Nρ( + λ) ξ ζ (9) The relevan regressions, accordingly, become: r Nλ r = λρ ( R r ) + u + ζ (0) + N ( ρ( + λ)) and ( + λ) R R = ( λρ + ρ )( R r ) + u + N( ρ( + λ)) + ζ () 3 The analysis would no change if ζ was allowed o be auocorrelaed (McCallum, 994b, p. 5). 6

7 McCallum underlines ha, excep for very large values of ρ and/or λ, he coefficien of he spread in () will be negaive, hus maching some empirical resuls for he US (e.g. Evans and Lewis, 994, Campbell and Shiller, 99) which canno be reconciled wih he consan erm premium version of he EH. Kugler (997) offers an exac soluion o he N-period case in McCallum s model. Specifically, he soluion o he model hinges on he approximaion assumed in equaion (3), which allows one o eliminae he expeced values for he shor rae as far as period N. In order o avoid he use of he approximaion in (3), Kugler calculaes he (N-) RE values of he shor rae up o dae N. The RE soluions are sill aained according o he MSV crierion and by means of he mehod of undeermined coefficiens. Kugler s regression equaions for he spread and for he shor rae are respecively he following: ( R N u (2) j ( N j) ρ r ) = ρ( R r ) + N N λ j= and Nλ ( r r ) = λρ ( R r ) + u + ζ (3) j ( N j) ρ N N λ j= The laer has he implicaion ha he informaion conen of he spread vanishes whenever λ or ρ ends o zero. 4 Kugler inerpres his resul by noing ha he predicive conen of he spread is based on predicable moneary policy reacion o he spread. If λ and/or ρ are zero, here is no predicable exogenous movemen of he spread which deermines he predicabiliy of policy reacion. Kugler does no presen he regression equaion for he long raes, which we have derived from his soluions as follows: 4 This resul exends McCallum s equivalen resul for he 2-period case. 7

8 ( + λ) R R = ( λρ + ρ )( R r ) + j λ ( ) ρ N + N j i= u + ζ (4) By comparaive inspecion of () and (4), i is clear ha he exacness of he soluion calculaed by Kugler is relevan only for he coefficien of he whie noise erm u. Since he coefficien of he spread is in all cases he same, he implicaions for he ess of he EH based on he value of he spread coefficien are as in McCallum. 3. MODELLING STRATEGY In order o es wheher he model presened in he previous secion is able o explain empirical deviaions from he expecaions hypohesis, we follow he approach adoped by Kugler (997) which consiss in comparing he esimaed value of he spread coefficien in sandard regressions for ess of he EH wih he value implied by he McCallum model. The ype of regressions considered by Kugler use he spread o predic fuure shor raes. In our empirical analysis we presen evidence also for regressions which relae he spread o changes in he long rae. Firs of all recall ha he EH formulaed in equaion () implies ha he slope coefficien β of he following regression equaions should be equal o : N N ( / N ) ( N j) r + j = + β ( R r ) + ε + N j= α (5) R ( N ) + N N R = α + β R r + + ( N ) ( ) ε (6) where R N and r are he N- and -period ineres raes respecively, and r +j in (5) is equal o r +j - r +j-. Equaion (5) uses he spread o predic (a weighed average of) changes in he shor rae over an N-period horizon; in (6) he spread should predic he change in he N- period rae over he -period horizon. Taking expeced values of (2) and (3) and rearranging, i can be shown ha Kugler s 8

9 model implies: e j r + j = λρ ( R r ) Hence he counerpar of (5) in Kugler s model becomes: (/ N ) N j= ( N j) r e + j N j N = (/ N ) λ ( N j) ρ ( R r ) (7) j= where he implied β is: N j= j β = (/ N) λ ( N j) ρ (8) MC The counerpar of (6) is (4) where he implied β is: β MC = (N-)(λρ+ρ-) (9) The suggesion from Kugler is o esimae β MC by means of Indirec Leas Squares in wo sages: firs, esimae he wo reduced form equaions (2) and (3) wih OLS o obain values for ρ and λρ, hen divide he esimaed value of λρ by ρ o obain an esimae of λ, and use (8) and (9) o obain values of β MC. These values are hen compared wih he esimaes of β obained from regressions (5) and (6). Kugler (997) applies quie successfully his mehodology o regressions for he shor rae, using he one- and hree- monh ineres raes for he US, Japan, Germany and Swizerland in he period He found ha for he case of Japan, he good predicive power of he spread can be explained by boh a high reacion of he moneary policy o he spread and a srong auocorrelaion of he erm premium. On he oher hand, he low predicive power of he spread is explained by eiher low correlaion of he erm premium which is he case for Germany and Swizerland or low reacion of he moneary policy o he spread as is he case for he US during ha paricular sample period. The laer case is furher invesigaed in Hsu and Kugler (997) where resuls in favour of he EH were found for he US for he more recen sample period

10 In order o assess furher he empirical validiy of he model proposed by McCallum, in he following secions we presen new evidence on ess of he expecaions hypohesis for a wider range of counries wih differen moneary policy rules, and for differen sample periods. We sar wih sandard ess ignoring policy behaviour and subsequenly esimae he McCallum policy reacion model. 4. RESULTS FROM STANDARD REGRESSION TESTS Differen ess of he EH have been proposed in he lieraure. In his secion we presen evidence based on he implicaions of equaion () ha he erm spread should predic fuure changes in he shor rae and in he long rae. The wo regressions used o es hese implicaions of he EH are equaions (5) and (6). In hese equaions, ε +N- and ε + are forecas errors which under raional expecaions are orhogonal o informaion a ime, and herefore uncorrelaed wih he regressor R N -r, so OLS will give consisen esimaes. However, he errors in (5) will be serially correlaed following a MA(N-2) process, while he errors in (6) will follow a MA(m-) process when he shor rae has mauriy m>. So, sandard errors are usually calculaed wih he Newey- Wes or Hansen and Hodrick correcions. Tess of he predicive conen of he spread imply esing for he significance of β (β=0), while ess of he EH wih RE and consan erm premium imply esing for β=. The daa used in his sudy are weekly Euro-raes for he period o for USA, Japan, Germany, UK, France, Ialy, Canada and Swizerland. As we ulimaely wan o invesigae he effecs of moneary policy on ess of he EH, we only use he -monh and 3-monh raes, as hese are more direcly linked o moneary 0

11 policy. 5 In sub-secion 4. we consider regressions for he shor rae and in 4.2 regressions for he long rae. 4. SHORT RATE REGRESSIONS The equaion esimaed for he shor rae is equaion (5), where N=3-monhs. Wih monhly daa he equaion would be he following: ( j) r = α + β( R r ) + ε j= + j + 3 However, as we are using weekly daa, we approximae he, 2 and 3-monhs horizons as 4, 9 and 3 weeks respecively, so he regression becomes: = α + β + ε + (20) 3 ( r 4 r ) ( r 9 r 4 ) ( R r ) The errors in (20) follow an MA(h-) process, where h is he forecas horizon (9 weeks in our regressions) and hence we compued Newey-Wes correced sandard errors wih runcaion lag equal o 8. The resuls are summarised in Table. The esimaion period seleced for each counry varies wihin he inerval 6//985 and //995, and, in each case, reflecs he longes period for which he esimaed β was found o be sable. We firs discuss evidence based on sandard asympoic disribuions, and subsequenly consider he effecs of small sample bias. Inference based on asympoic disribuions Table shows ha all esimaes for β are significanly differen from zero, wih he excepion for Swizerland, hus confirming an overall informaion conen of he spread for 5 We hank Peer Kugler for kindly providing us wih hese daa.

12 fuure shor raes. However, ess of he EH (β=), repored in he las column of Table, indicae ha he EH is rejeced for all counries a convenional significance levels, excep for France, Ialy and Canada. The evidence on France and Ialy is in line wih resuls in previous sudies which manain ha he EH beer describes weak-currency counries han srongcurrency counries (see Gerlach and Smes, 997). A possible raionale behind his explanaion is ha Cenral Banks of weak-currency counries ofen have o manage shor raes in order o achieve mid-erm inermediae exchange rae objecives. As hese are usually well-known, shor raes become more predicable, and his jusifies higher values for β in hose counries 6. The effecs of small sample bias A common criicism of hese regression-based ess of he EH is ha hey can be seriously biased in small samples. This is paricularly rue for he long rae regressions, as we discuss below, bu in a recen paper Bekaer, Hodrick and Marshall (BHM) (997) found ha regressions for he shor rae can also be affeced by subsanial posiive bias. According o ha sudy, he posiive bias arises because under he assumpion ha he shor rae is generaed by an AR() process, he slope coefficien in hese regressions can be shown o be a negaive ransformaion of serial correlaion coefficiens. This ransformaion, combined wih he negaive bias in OLS esimaes of auocorrelaion coefficiens for highly persisen daa, generaes a posiive bias in he slope coefficiens (see Bekaer e al., 997, eqs. 7, 8 and 0). In order o evaluae he effecs of his kind of small sample bias in he ess presened so far, in wha follows we conduc inference by using he 5% quaniles of he empirical disribuions of he slope coefficien derived by BHM under wo alernaive daa generaing 6 An empirical analysis of he imporance of shor rae predicabiliy in ess of he EH can be found in a previous version of his paper (see Boero, Di Lorenzo and Torricelli, 998). 2

13 processes (d.g.p.): an AR() for he shor rae (see BHM, Panel C, Table 3) and a VAR- GARCH model for boh he shor rae and he spreads (see BHM, Panel C, Table 6). These empirical disribuions are characerised by subsanial posiive bias (which would srenghen rejecion of he EH) bu also increased dispersion (which would weaken rejecion). According o hese disribuions, o have a 5% rejecion of he hypohesis β = in a oneailed es, he slope coefficien should be smaller han 0.64 wih he AR() d.g.p., and smaller han 0.62 wih he VAR-GARCH d.g.p.. As eviden from Table, by conducing inference wih he small sample disribuions, our resuls remain virually unaffeced, he only excepions are USA and UK for which rejecion would be weakened. These resuls are quie ineresing, alhough hey should be inerpreed wih some cauion, as he criical values derived by BHM are no exacly applicable o our regressions for a leas wo reasons. Firs, we are using a very shor erm spread (3- monh), while he closes spread considered in he BHM Mone Carlo experimen is 2- monh. Second, BHM repor criical values for only one sample size (524 observaions) which is very close o he number of observaions used in some of our regressions, bu larger han ha used in ohers. 4.2 LONG RATE REGRESSIONS The equaion esimaed for he long rae is equaion (6), wih mauriies N=3-monhs for he long rae, and -monh for he shor rae. Wih monhly daa he equaion would be he following: R + R = + ( 3 ) 3 R r 3 α β ( ) + ε ( 2) + However, in our empirical ess his regression is modified for weekly daa o obain 3 3 R+ R = + R r α β ( ) ε 4 (2) 2 In equaion (2) we have used approximaion (3), discussed in Secion 2, E R N N + = E R +, 3

14 which is commonly adoped in regressions of his ype. However, while his approximaion is irrelevan for large N, as we see shorly, i may have significan bias effecs on he esimaed value of β for small N. Regressions for he long rae have been he focus of aenion of many sudies aemping o explain failures of he EH. In fac, while he EH implies ha he slope coefficien should be equal o one, mos of he empirical lieraure has repored very low values for he R 2, and esimaed coefficiens below uniy, becoming negaive as yields of longer-erm bonds are used o form he dependen variable and he erm spread. Negaive values indicae ha long raes move in he opposie direcion o ha implied by he heory. Our esimaes, repored in Table 2, are apparenly a odds wih previous findings, as hey seem o suppor he EH in mos cases. In wha follows we firs discuss he resuls using sandard asympoic disribuions and hen proceed o reinerpre he resuls using he BHM small sample disribuions. Inference based on asympoic disribuions Table 2 repors our esimaes of regression equaion (2) and ess of he EH (β = ) for he whole sample period (58 observaions), and for he sub-period (206 observaions). The las wo columns repor he R 2 values. Conrary o previous findings in he lieraure, mos poin esimaes are posiive, wih he excepion of Swizerland, and some are close o one. However, some esimaes are no significanly differen from zero, and he very low R 2 values conform wih previous resuls indicaing ha he spread beween he long and shor erm ineres raes has poor predicive conen for changes in he longer rae. The finding of esimaed slope coefficiens close o may depend on: (i) he paricular naure of our regressions which only look a he very shor end of he erm srucure, whereas regressions for he long raes are ypically esimaed in he longer end of he erm 4

15 srucure, and (ii) he approximaion used o consruc he dependen variable. This poin will be discussed furher below. The effecs of small sample bias We now reurn o he resuls in Table 2 and consider how hey can be affeced by small sample biases. In he sudy menioned before by BHM, he small sample bias which affecs all regression-based ess of he EH is shown o be paricularly srong for he long rae regressions. BHM show ha approximaion (3) which we used in he esimaion of equaion (2) inroduces a furher error in he regression which exacerbaes he small sample bias. In paricular, hey found ha for mauriy of he long rae N=2 and sample size T=524 he average of he OLS esimaes of β is abou 2, wih similar value for he sandard deviaion. So, as already seen for he shor rae regressions, he small sample disribuion is biased upward and has an increased dispersion. However, he bias for he long rae regressions is much higher. Moreover, in conras o he resuls obained from regressions for he shor raes, inference based on he small sample disribuions for he long raes is no uniformly conclusive abou rejecion of he EH, as hese disribuions seem o be very sensiive o he daa generaing process used in he BHM Mone Carlo simulaions. Specifically, according o he BHM empirical quaniles, he EH should be rejeced a he 5% for values of β<.203 when he d.g.p. for he shor rae is an AR() model (BHM, Panel B, able 3), and for values of β<0.3 if inference is conduced under he assumpion of a VAR-GARCH model for he shor rae and he spreads (BHM, Panel B, Table 6). So, inference based on hese criical values would indicae evidence agains he EH for mos counries (excep France and marginally Canada) under he AR() d.g.p., while under he alernaive d.g.p. he evidence would be generally in favour of he EH, wih he sole excepion for Swizerland. To summarise, in his secion we have conduced inference on regression-based ess of 5

16 he EH wih boh asympoic and small sample disribuions. The empirical criical values ha we used are hose derived in Bekaer e al. (997). Alhough hese small sample disribuions are affeced by subsanial posiive bias (which depends on he persisence of he shor rae), hey are also characerised by increased dispersion which, in he case of regressions for he shor rae, leads o resuls ha are in general invarian o hose based on asympoic disribuions. On he oher hand, inference based on he empirical criical values for regressions for he long rae is inconclusive, due o he high sensiiviy of he BHM small sample disribuions o he daa generaing process. 7 In he nex secion we esimae he McCallum model following he mehodology inroduced in Secion 3, and evaluae is abiliy o explain deviaions from he EH. We do his by comparing he values of he coefficiens β implied by his model wih hose obained from sandard regression ess. 5. AN EVALUATION OF THE McCALLUM MODEL In he presen secion we apply he McCallum model o he 8 counries considered in his sudy, and compare he implied values of β (β MC ) wih hose esimaed from sandard regression ess discussed in he previous secions. As described in Secion 3, in he McCallum model he EH ineracs wih a policy reacion funcion, in he presence of a imevarying erm premium, so he implied β in ess of he EH is a composie parameer reflecing policy behaviour (he λ coefficien in (6)) and he auoregressive componen of he erm premium (he ρ coefficien in (2)). The values of β MC are obained from equaions (8) and (9) modified according o he weekly frequency of he daa as follows: 4 8 j j shor rae regressions: β MC = ( / 3) λ( 2 ρ + ρ ) (22) j= j= 5 7 In a recen paper, Schoman (997) assumes ha ineres raes (shor and long) are generaed by ARIMA (,,) models and finds very large bias for he spread coefficien β. The bias is posiive or negaive depending on wheher he sum of he auoregressive and moving average coefficiens is negaive or posiive. 6

17 long rae regressions: β MC = 2(λρ+ρ-)(+ρ+ρ 2 +ρ 3 ) (23) Esimaes of λ and ρ are obained from he wo Reduced Form (RF) equaions (2) and (3). In paricular, ρ is obained by applying OLS o RF (2), while λ is obained by Indirec Leas Squares applied o RF (3) or, equivalenly, by IVE applied o he policy reacion equaion (6) wih insrumen (R - -r - ). We firs consider he abiliy of he McCallum model o explain resuls from sandard regressions for he shor rae. Then we urn o regressions for he long rae. 5. SHORT RATE REGRESSIONS In his secion we see wheher he McCallum model is able o explain he resuls discussed in Secion 4. and summarised in Table. For convenience, hose resuls are repored again in Table 3 (columns 4 and 6) wih esimaes of λ and ρ, he implied β MC compued as in equaion (22), and a es for β = β MC. An imporan resul is ha he implied slope coefficiens β MC are in general consisen wih he β esimaes obained from sandard EH regressions: he Wald es never rejecs he hypohesis β = β MC. This resuls indicaes ha he McCallum model can raionalise differen values for β, including low values as in he case of Swizerland, and suggess ha explici consideraion of a moneary policy reacion funcion is imporan in providing an explanaion of boh failures and successes of he EH. Table 3 also shows ha differences in he values of β MC are due more o differen esimaes of he moneary policy coefficien han o a differen paern of ime variaion in he erm premium. In fac, while esimaes of λ range from a minimum of 0.94 for Germany o a maximum of 0.79 for France, hose for ρ display a much lower variaion (beween for Ialy and 0.87 for Germany). 7

18 In paricular, resuls poin o a srong policy reacion o he spread for France (λ=0.79) and Ialy (λ=0.75), a moderae reacion for he UK (λ=0.348), he US (λ=0.480) and Canada (λ=0.497), and a low reacion for Germany (λ=0.94), Swizerland (λ=0.272) and Japan (λ=0.30). Table 4 cass furher ligh on he abiliy of he McCallum model o explain differen resuls from ess of he EH, showing ha he ranking beween he counries is similar according o he hree parameers λ, esimaed β and β MC. However, a his poin, a noe of cauion is necessary, paricularly wih respec o he esimaes of λ. In fac, hese are based on an exremely simplified policy reacion funcion, where policy responds only o he spread (reflecing he Cenral Bank reacion o changes in expeced fuure inflaion), and may herefore suffer from omied variable bias. Moreover, he esimaes of λ are based on he assumpion ha he error erm ζ is no auocorrelaed. To improve on he empirical esimaion of λ, ideally one would include oher poenially imporan policy indicaors (recen inflaion, exchange rae, oupu), bu his roue would require specificaion of an expanded macroeconomeric model which endogenously explains he added variables. Insead, following Kugler, we have considered he possibiliy ha Cenral Banks reac o lagged shor rae changes, as well as he spread, and reesimaed he policy reacion funcion wih insrumenal variables plus an AR() error, using as insrumens he lagged change in he shor rae (r - -r -2 ) and he lagged spread (R - -r - ). Wih his esimaion procedure we found, in general, values of λ very close o hose repored in Table 3, which we have chosen no o repor here for reasons of space. 8 However, here were wo excepions, France and Ialy, for which he esimaed λ did no seem o be robus. The value for France, for example, changed from 0.79 o 0.44 when he AR() procedure was employed, implying significan changes in he value of β MC, and suggesing ha he highly 8 Resuls are available from he auhors on reques. 8

19 sylised policy reacion funcion canno adequaely describe moneary policy in ha counry. Similar findings obained for Ialy. Boh hese counries operae wih inermediae exchange rae arges, so he simple policy reacion funcion which combines ineres rae smoohing wih a reacion o he spread may need o be expanded wih he inclusion of a more complex se of policy indicaors, o obain beer esimaes for λ. On he oher hand, he sylised reacion funcion of he McCallum model seems paricularly adequae o describe he moneary policy in counries where he spread is clearly used as an imporan indicaor. For example, wih regard o he US, he moderaely high value of λ is in line wih recen findings (Hsu and Kugler, 997) and reflecs he increased reliance of he Federal Reserve on he spread as a policy indicaor raher han on moneary aggregaes. This esimae of λ combined wih a high value for ρ (0.8) implies a value for he slope coefficien β MC close o (0.97). Anoher ineresing resul is ha for Germany, where he low value of λ corresponds o official saemens of he Bundesbank ha he spread is no used as a policy indicaor. Moreover, while he Bundesbank is officially moneary argeing, he esimaion of he McCallum policy reacion model suggess ha here may be elemens of ineres rae smoohing in he German moneary policy. The low esimae of λ for Germany combined wih a high persisence in he spread (ρ=0.87) imply a value for β MC of 0.49 which is no oo far away from he esimaed β (0.60). 5.2 LONG RATE REGRESSIONS Finally, in Table 5 we compare esimaes of he β coefficiens obained from regressions for he long rae (repeaed from Table 2) wih he value for β implied by he policy reacion model (equaion (23)). As we can see, his model can again raionalise differen resuls for β, including negaive values, and, in mos cases, he Wald es for he 9

20 equaliy of he β MC and he β esimaed from regressions for he long rae canno rejec his hypohesis. Alhough hese resuls imply ha he McCallum model can raionalise differen findings from ess of he EH, i is imporan o emphasise ha resuls from hese regressions for he long raes are o be inerpreed wih cauion no only because of he small sample bias discussed in Secion 4.2, bu also because of he very low R 2 values. 6. CONCLUSIONS In his paper we have esed he Expecaions Hypohesis of he erm srucure across eigh counries wih differen moneary policy rules and have examined he abiliy of he McCallum model (994b) o explain conflicing evidence from sandard ess. The empirical analysis was conduced in wo seps. Firs, we performed sandard regression ess of he EH using he erm spread o predic fuure changes in he shor raes and in he long raes. Second, we esimaed he McCallum model, in which he expecaions hypohesis ineracs wih a policy reacion funcion and wih an auoregressive ime-varying erm premium, and compared he esimaes of he erm srucure coefficiens implied by his model wih hose obained in sandard regressions. Wih respec o he firs sep, we found ha in he case of regressions for he shor raes he coefficien of he erm spread was significanly differen from zero in mos cases, hus confirming an overall informaion conen of he spread for fuure shor raes. However, ess of he Expecaions Hypohesis (β=) indicaed rejecion for five counries ou of eigh. In he case of regressions for he long raes, conrary o previous findings in he lieraure, we found ha mos poin esimaes were posiive, and some were close o one. However, for hese regressions we also found very low R 2 values, which conforms wih previous resuls, 20

21 indicaing ha he spread beween he long and shor erm ineres raes has poor predicive conen for changes in he longer rae. To complee he firs sep of he analysis, we also confroned he resuls of ess of he EH wih he empirical criical values recenly derived in Bekaer e al. (997). These ake ino accoun a paricular kind of bias due o persisence in he shor raes. We found ha his kind of bias did no affec subsanially he resuls of ess from he shor rae regressions, whereas inference based on he small sample disribuions for he long rae regressions produced conflicing resuls, depending on he assumpions underlying he daa generaing process. Wih respec o he second sep, we found ha he McCallum model was in general able o raionalise differen values for β. This resul suggess ha values of he slope coefficiens saisically differen from one are consisen wih he EH and deviaions from he EH reflec he way in which moneary policy responds o changes in he erm spread. Amongs oher resuls we also found ha he model performs beer for some counries han ohers depending upon he moneary policy sance adoped. Specifically, he model is beer able o explain resuls for counries where he moneary policy involves ineres rae smoohing and/or clearly responds o changes in he long-shor spread, han for counries operaing wih a more complex moneary policy. REFERENCES Bekaer, Hodrick and Marshall, 997, On biases in ess of he expecaions hypohesis of he erm srucure of ineres raes, Journal of Financial Economics, 44, Boero G., F. Madjlessi F. and C. Torricelli, 996, The informaion in he erm srucure of German ineres raes, Proceedings of he VI Inernaional AFIR Colloquium, Nürnberg, Ocober 996, VVW Karlsruhe. Boero G. and C.Torricelli, 997, The Expecaions Hypohesis of he erm srucure of ineres raes: evidence for Germany, Conribui di ricerca CRENOS, n. 4, Universià di Cagliari, seembre. Campbell J.Y. and R.J. Shiller, 99, Yield spread and ineres rae movemens: a 2

22 bird s eyes view, Review of Economic Sudies, 58, Dosey M. and C. Orok, 995, The raional expecaion hypohesis of he erm srucure, moneary policy and ime-varying erm premia, Economic Quarerly, Federal Reserve Bank of Richmond, 8, Engsed, T. and C. Tanggaard, 995, The predicive power of yield spreads for fuure ineres raes: Evidence from he Danish erm srucure, Scandinavian Journal of Economics, 97, Evans M.D.D. and K.K. Lewis, 994, Do saionary risk premia explain i all? - Evidence from he erm srucure, Journal of Moneary Economics, 33, Gerlach, S. and F. Smes, 997, The Term Srucure of Euro-raes: some evidence in suppor of he expecaions hypohesis, Journal of Inernaional Money and Finance, Vol. 6, no. 2, Hsu C. and P.Kugler, 997, The revival of he Expecaions Hypohesis of he US erm srucure of ineres raes, Economics Leers, 55, Kugler, P., 990, The erm srucure of ineres raes and raional expecaions, Journal of Inernaional Money and Finance, Vol. 9, pp Kugler P., 997, Cenral Bank policy reacion and he Expecaion Hypohesis of erm srucure, Inernaional Journal of Finance and Economics,2, Malagui L. and C. Torricelli, 997, The ineracion beween moneary policy and he Expecaion Hypohesis of he erm srucure of ineres raes in a N-period Raional Expecaion model, Maeriali di Discussione, n.8, Diparimeno di Economia Poliica, Universià di Modena, luglio. Mankiw N.G. and J.A..Miron, 986, The changing behaviour of he erm srucure of ineres raes, Quarerly Journal of Economics, 0, McCallum B.T., 983, On non-uniqueness in raional expecaions models: an aemp a perspecive, Journal of Moneary Economics,, McCallum B.T., 994a, A reconsideraion of he Uncovered Ineres Rae Pariy Relaionship, Journal of Moneary Economics, 33, McCallum B.T., 994b, Moneary policy and he erm srucure of ineres raes, NBER Working Paper Series, N Rudebusch G.D., 995, Federal Reserve ineres rae argeing, raional expecaions, and he erm srucure, Journal of Moneary Economics, 35, Schoman P.C., 997, Small sample properies of he regression es of he expecaions model of he erm srucure, Economics Leers, 54,

23 Simon, D.P., 989, Expecaions and risk in he Treasury bill marke: An insrumenal variables approach, Journal of Financial and Quaniaive Analysis, 24, Tzavalis E. and M.R. Wickens, 997, Explaining he failures of he Term Spread Models of he Raional Expecaions Hypohesis of he Term Srucure, Journal of Money, Credi and Banking, Vol. 29, No. 3,

24 TABLE - Esimaes of β in regressions for he shor rae (equaion 20) Counry and sample period (no. of obs.) USA 6//9- //95 (20) Japan 6//90- //95 (253) Germany 6//85- //95 (53) UK 6//90- //95 (253) France 6//90- //95 (253) Ialy 6//85- //95 (53) Canada 6//90- //95 (253) Swizerland 6//90- //95 (253) Esimaes of β (i). (correced SEs) (ii) 0.77 (0.05) 0.48 (0.098) 0.60 (0.60) 0.64 (0.4).2 (0.223) 0.80 (0.02) 0.95 (0.259) 0.27 ** (0.93) Wald es (iii) Chi-Sq for H 0 : β = 4.36 * ** 6.5 * 9.67 ** ** Noes: (i)** indicaes ha he coefficien is no saisically differen from zero a he 5% significance level; (ii) he number in parenhesis are heeroscedasiciy and auocorrelaion correced sandard errors wih runcaion lag 8; (iii) * indicaes rejecion of H 0 : β = a he 5%; ** indicaes rejecion a he %. 24

25 TABLE 2 - Esimaes of β in regressions for he long rae (equaion 2) Sample periods: 6//85-//95 (no. obs 58); 6//9-//95 (no. obs. 206) Counry Esimaes of β (i). (correced SEs) (ii) USA 0.54** 0.92 (0.33) (0.33) Wald es Chi-Sq for H 0 : β = (prob. of rejecion) (iii) (.7) (.8) R Japan 0.92 (0.25) 0.50 (0.22) 0.09 (.77).78 (.8) Germany 0.54** (0.39) 0.64 (0.30).42 (.23).50 (.22) UK.07 (0.3) 0.84 (0.38) 0.05 (.82) 0.8 (.67) France.30 (0.6).86 (0.5) 0.24 (.62) 2.8 (.09) Ialy 0.58 (0.20) 0.5 (0.4) 4.4* (.04) 2.2** (.0) Canada.23 (0.38) 0.89** (0.82) 0.38 (.54) 0.02 (.89) Swizerland 0.8** (0.24) -0.8 ** (0.38).9** (.00) 9.4** (.002) Noes: (i):** indicaes ha he coefficien is no saisically differen from zero a he 5% level; (ii) he number in parenhesis are heeroscedasiciy and auocorrelaion correced sandard errors, wih runcaion lag 3; (iii) * indicaes rejecion of H 0 : β = a 5%, ** indicaes rejecion a %. 25

26 TABLE 3- Esimaion of he McCallum model: comparison wih resuls from regressions for he shor rae Counry and sample period Esimaes of λ and ρ β MC (i) Esimaes of β Wald es (iii) H 0 :β =β MC Wald es (iii) H 0 : β = () USA /9-/95 (2) λ = (.079) ρ = 0.80 (.040) (3) (4) (ii) (0.05) (5) (6) (ii) * Japan /90-/95 λ = 0.30 (.074) ρ = 0.80 (.035) (.098) ** Germany /85-/95 λ = 0.94 (.042) ρ = 0.87 (.02) (.60) * UK /90-/95 λ = (.059) ρ = (.034) (.4) ** France /90-/95 λ = 0.79 (.207) ρ = (.046) (.223) Ialy /85-/95 λ = 0.75 (.24) ρ = (.035) (.02) Canada /90-/95 λ = (.0) ρ = 0.77 (.039) (.259) Swizerland /90-/95 λ = (.2) ρ = (.038) ** (.93) ** 4 8 j j Noe: (i) compued as in equaion (22) : β MC = ( / 3) λ( 2 ρ + ρ ) ; (ii) The values in columns 4 and j= j= 5 6 are aken from Table ; ** in column 4 indicaes ha he coefficien is no saisically differen from zero a he 5%; * and ** in column 6 indicaes rejecion a he 5% and % respecively; (iii) Wald es (chi-square 26

27 values) compued wih Newey-Wes correced sandard errors, consisen in he presence of MA(8) errors and heeroscedasic. TABLE 4- Counries ranked according o values of λ, esimaed β and β MC. λ esimaed β β MC France 0.79 France.2 USA 0.97 Ialy 0.75 Canada 0.95 France 0.93 Canada Ialy 0.80 Canada 0.86 USA USA 0.77 UK 0.74 UK UK 0.64 Ialy 0.69 Japan 0.30 Germany 0.60 Japan 0.58 Swizerland Japan 0.48 Swizerland 0.49 Germany 0.94 Swizerland 0.27 Germany

28 TABLE 5 Esimaion of he McCallum model: comparison wih resuls from regressions for he long raes. Sample period: 6//99-//95 (no. obs 206). Counry Esimaes of λ and ρ (i) β MC (ii) Esimaes of β Wald es (iv) H 0 : β = β MC (prob. of rejecion) Wald es (iv) H 0 : β= (prob. of rejecion) () (2) (3) (4) (iii) (5) (6) (iii) USA λ = (.079) ρ = 0.80 (.040) Japan λ = (.090) ρ = (.04) Germany λ = 0.26 (.059) ρ = 0.94 (.026) UK λ = (.080) ρ = (.043) France λ = (.263) ρ = 0.63 (.053) Ialy λ = (.50) ρ = (.053) Canada λ = (.2) ρ = (.045) Swizerland λ = 0.92* (.0) ρ = (.038) (.42) (.28) (.4) (.82) (.08) ** (.00) ** 0.07 (.80) ** 0.07 (.78) 0.05 (.8).78 (.8).50 (.22) 0.8 (.67) 2.8 (.09) 2.2** (.0) 0.02 (.89) 9.4** (.002) Noe. (i): λ and ρ are esimaed as explained in he ex; * indicaes ha λ is no significanly differen from zero a he % level; (ii) compued as in equaion (23): β MC =2(λρ+ρ-)(+ρ+ρ 2 +ρ 3 ); (iii) he values in columns 4 and 6 are aken from Table 2; ** in column 4 indicaes ha he coefficien is no saisically differen from zero a he 5%; ** in column 6 indicaes rejecion a he %; (iv): Wald es (chi-square values) compued wih Newey-Wes correced sandard errors, consisen in he presence of MA(3) errors, and heeroscedasic. 28