Does the euro area forward rate provide accurate forecasts of the short rate?

Transcription

1 Does he euro area forward rae provide accurae forecass of he shor rae? Ana Beariz Galvao Queen Mary Universiy of London Sonia Cosa Bank of Porugal June 2012 Absrac The forward rae delivers accurae forecass of euro area shor-erm ineres raes depending on he ime period. During periods of macroeconomic uncerainy, forecass obained wih a model of yield and macro facors are more accurae han forward-based forecass. We provide evidence ha a ime-varying forward premium explains he variaion in forecasing performance. We develop a mehod o compue forward premium confidence inervals o ex-ane idenify periods during which forward-based forecass are inaccurae. Key words: ineres raes, forecasing, forward premium, euro area. JEL codes: C32, E43. We graefully acknowledge commens and suggesions of he edior (E. Ruiz) and wo anonymous referees of his journal. 1

2 1 Inroducion A popular way of exracing marke expecaions abou fuure ineres raes is he compuaion of implici forward raes from he yield curve (Söderlind and Svensson, 1997). However, he forward rae may differ from he marke expecaions of he shor rae by a forward premium. This paper shows ha here is indeed a ime-varying forward premium in he euro area forward raes, even when he forward raes are employed o exrac expecaions of nex quarer s shor rae. The forward premium is significan in periods of macroeconomic uncerainy, which are associaed wih he growh slowdown and he recession. 1 Our resuls are in conras wih Gerlach and Smes (1997) ha conclude ha euro-marke raes are adequae o obain privae secor s expecaions of fuure shor-rae changes in shor horizons using daa up o Jongen, Verschoor and Wolff (2011) found evidence agains he expecaions hypohesis and in suppor of a ime-varying risk premium using German daa for a period similar o ours ( ). They also show ha he expecaions hypohesis is no rejeced when survey forecass are employed as marke expecaions. As a resul, Jongen e al. (2011) use he difference beween he forward rae and he survey forecas as a measure of he forward premium. In his paper we compare he ou-of-sample forecasing performance of he forward rae and of a forecasing model ha incorporaes he informaion on yield and macroeconomic facors. Based on he comparison of heir relaive forecasing accuracy in wo periods ( ; ), we show ha he forward rae may deliver forecass ha are unbiased and accurae depending on he ime period. In he recen period, forward-based forecass are biased, and a forecasing model ha includes informaion on macroeconomic facors improves forecas accuracy by reducing he roo mean squared error in 20%. Then we invesigae wheher hese changes in forecasing performance are explained by a ime-varying forward premium by measuring he forward premium as he difference beween forward-based and model-based forecass. The saisical significance of hese differences are assessed using confidence inervals ha ake ino accoun he forecasing and parameer uncerainies which are embodied in he model-based forecass. We show ha he forward premium is significan in periods of macroeconomic urbulence. The forecasing model is based on he Diebold and Li (2006) approach, modified by he inclusion of macroeconomic facors, o obain forecass of yields of differen mauriies, compaible wih a 1 The CEPR Business Cycle Daing Commiee has classified he period 2008:Q1-2009:Q2 as a euro area recession. See: hp:// 2

3 smooh curve, using all informaion available a. Alhough he forecasing model used in his paper does no impose no-arbirage, Diebold, Piazzesi and Rudesbush (2005) show ha an approach similar o ours is consisen wih a wo-facor affi ne no-arbirage erm-srucure model, under some condiions. In addiion, Coroneo, Nyholm and Vidova-Koleva (2008) non-parameric ess sugges ha he facor loadings obained wih he Diebold and Li (2006) approach using US daa are no saisically differen from he ones from an affi ne dynamic erm-srucure model. We compue forecass wih German zero-coupon ineres raes, and also wih a shorer sample available afer he inroducion of he euro, which consiss of euro money marke raes and swap raes. Alhough he German daa have been exploied before (Hordähl, Trisani and Vesin, 2006; Cappiello e al., 2006, and Jongen e al. 2011), we are no able o find oher papers using only euro daa (an excepion when only looking a he money marke is Durré, Eujen and Pilegaard (2003)). 2 Comparing Forward-based and Model-based Forecass In his secion we evaluae he ou-of-sample forecas performance of he forward rae as predicor of he fuure shor rae in comparison wih a forecasing model. We sar by describing how we compue forward raes based on observed yields and fied yield curve. Then we explain how a VAR model of yield and macroeconomic facors is employed o compue forecass of he shor rae. Finally, we presen he resuls of our ou-of-sample forecasing exercise. 2.1 Using he Forward Rae as Forecas of he shor rae The τ monh forward rae is he difference of he ineres rae paid on wo zero-coupon bonds observed a ime : one wih mauriy a + n + τ, and he oher wih mauriy a + n, ha is: f (n,n+τ) = 1 τ [ ] τy (n+τ) + n(y (n+τ) y (n) ) = 1 [ ] (n + τ)y (n+τ) ny (n) ), (1) τ where y (n) is he yield o mauriy n, observed a, and n is measured in monhs. The forward rae is a n-sep ahead forecas for he τ-monh yield. If he expecaions hypohesis of he erm srucure of ineres raes holds, hen he forward rae is a good measure of he marke expecaions on he fuure shor-rae because f (n,n+τ) The forward raes f (n,n+τ) = E (y (τ) +n ) are compued using he yields y (n+τ) and y (n). However, he mauriies of observable yields may no mach all required forecas horizons. A soluion is o fi a curve 3

4 for he observable yields, such ha fied yields are employed o compue forward raes for any desired horizon. We use he Nelson and Siegel (1987) parameric approach for fiing he yield curve. The yield curve is fied for a group of observed zero-coupon bonds of mauriies τ (τ [3, 120]; mauriies measured in monhs) a ime. The Nelson and Siegel equaion for he spo (yield) rae is: y (τ) = β 1 + β 2 ( 1 e θ 1τ θ 1 τ ) + β 3 ( 1 e θ 1 τ θ 1 τ ) e θ 1τ, (2) where β 1, β 1 + β 2, and θ 1 mus all be posiive. The parameers β 1, β 2 and β 3 are called yield facors, and are inerpreed as he level (L ), (minus) he slope ( S ), and he curvaure (C ) of he yield curve. θ 1 is he parameer ha measures he rae of he exponenial decay of he loading of he second and he hird facors. Smaller values of θ 1 imply slower decay. This parameer also defines he mauriy a which β 3 has larger weigh. Diebold and Li (2006) keep θ 1 fixed, such ha he hree-year yield has he larges loading for β 3. A fixed θ 1 value has advanages: he facors can be esimaed by he usual leas squares formula, and he esimaes of he yield facors are more sable over ime. Svensson (1994) suggesed he inclusion of an addiional erm (oal of four facors) o increase he flexibiliy and o improve he fi of he yield curve in order o capure an addiional hump- or u- shape in he yield curve. Because generally parsimonious models are more accurae forecasers, i is no clear ha he addiional erm will improve he implied forward rae forecass. Boh approaches by Svensson (1994) and Nelson and Siegel (1987) are popular in cenral banks for he modelling of he yield curve (see BIS, 2005). Preliminary resuls wih euro area daa indicae ha, alhough he Svensson approach delivers a beer fi of he yield curve, he loss from using only hree facors wih a fixed θ 1 = is small: average roo squared error of 4 o 5 basis poins for mauriies τ = 3, 12 and smaller values elsewhere. As consequence, we use eq. (2) o fi he yield curve wih θ 1 = in he remainder of he paper. Based on his fied yield curve, we compue 3-monh forward raes as described in eq. (1). When predicing y (3) (3,6) +3, for example, we use f ; and if he arge is y (3) +12, he adequae forward rae is f (12,15). 2.2 Using a Forecasing Model We compue forecass of ineres raes based on a modificaion of Diebold and Li (2006), which is based on he Nelson and Siegel (1987) parameric yield curve fiing. The firs sep of he procedure 4

5 applies he Nelson and Siegel (1987) regression o compue yield facors. These facors are hen modelled joinly wih macroeconomic variables in a vecor auoregression (VAR). The esimaes of he VAR are used o compue forecass of he yields facors up o + h using informaion a. Using facor forecass, yield forecass are obained based on he Nelson and Siegel regression. Diebold and Li (2006) show, wih US daa, ha heir approach generaes forecass of shor raes ha are more accurae han he sandard benchmark forecass, including he ones obained wih he Cochrane and Piazzesi (2005) forward curve regression. In addiion, in comparison o accurae forecass of simple saisical models, he advanage of he Diebold and Li (2006) approach is ha he forecass are consisen for raes of each mauriy, so ha he implied yield curve is smooh and he forward raes are posiive. Using he facors names and a fixed θ 1, he Nelson and Siegel equaion for forecasing a yield of mauriy τ a h-sep ahead condiional on informaion a is: ( ) 1 e θ 1 τ ŷ (τ) +h = ˆL +h h Ŝ+h θ1 τ + Ĉ+h ( 1 e θ 1 τ θ1 τ e θ 1 τ ). (3) For example, if we are ineresed in a 3-monh shor-rae ha will be observed in hree monhs, he forecas arge is y (3) +3 and he required forecas from he above equaion is ŷ(3) +3. Diebold and Li (2006) sugges he esimaion of an AR(1) for each facor o be able o compue ˆL +h, Ŝ+h and Ĉ+h. However, here is some imporan dynamic correlaion beween he slope, he level and he curvaure. Hence we consider a VAR(p) o be more adequae, in which he number of lags is empirically chosen. Using he esimaed facors for each = 1,..., T, we define a VAR(p) for modelling he vecor x = ( ˆL Ŝ Ĉ ) as: x = c + A 1 x A p x p + ε, (4) where ε is a vecor of disurbances wih full variance-covariance marix. Condiional on he parameer esimaes (ĉ, Â1,..., Âp), he VAR is used o generae h-sep ahead forecass by ieraion, such ha ˆx +h is compued for each. This approach will be refereed as "DL" in he remaining of his secion. An exension of he "DL" approach is o include observed facors ha could help o forecas he yield curve facors in he VAR. In his paper, he DL + macro approach uses a VAR(p) for he following 5 1 vecor: x = ( ˆL Ŝ Ĉ g π ), (5) where g is a measure of economic aciviy growh and π is measure of inflaion. 5

6 There is an exensive lieraure on modelling he relaion beween facors of he yield curve and some imporan macroeconomic variables (for example, Ang and Piazzesi, 2003; Diebold, Rudebusch and Aruoba, 2006; Rudebusch and Wu, 2004; Hordähl e al., 2006). Boh economic aciviy growh and inflaion were chosen as observed facors, because he lieraure indicaes a srong dynamic relaion beween hese observed facors and yield facors. For he euro area or for some euro area counries, predicive regressions show ha he slope helps o predic real aciviy growh and inflaion (see, for example, Esrella e al., 2003; Monea, 2003; and Duare e al., 2005). Hordähl e al. (2006) esimae an affi ne erm-srucure model, including srucural equaions for inflaion, oupu gap and he shor-erm ineres rae, and hey conclude ha macroeconomic variables help in predicing he yield curve, even hough he yield curve does no provide useful addiional informaion in forecasing macroeconomic variables. In paricular, heir resuls sugges ha inflaion and he oupu gap affec he curvaure of he yield curve. Cappiello e al. (2006) examine he model of Hordähl e al. (2006) for he pre- and pos-euro periods, and hey conclude ha he relevance of macro facors in explaining he behaviour of he risk premium does no change wih he inroducion of he euro. An alernaive o he approach of esimaing he facors in a firs sep, and he dynamic relaion of he facors in he second sep, is he one proposed by Diebold e al. (2006). The auhors show how o joinly esimae he yield facors (eq. 2) and he coeffi ciens of a VAR of yield facors and macroeconomic variables (eq. 4). A disadvanage of he join esimaion of he parameers and he unobserved facors, which are non-linearly relaed wih observable yields, is he possibiliy of convergence problems in he numerical procedure required o maximise he likelihood funcion of he sae-space represenaion. The wo-sep esimaion also allows for more flexibiliy on he number of observed facors included, and on possible choices of he auoregressive order of he VAR. Anoher issue is ha he hree facors explain mos of he variaion of he yields, implying ha he inclusion of he macro variables does no affec he esimaion of he facors. However, when using he VAR o forecas yield facors, i is imporan o consider he dynamic relaion beween he macro variables and he yield facors because here is a srong relaion beween hem. Wih he suppor of previous saemens, his paper s wo-sep esimaion may generae yield forecass similar o he ones obained wih he single sep esimaion. In addiion, he use of a mehod ha is less demanding in compuaion reduces he problem of using a sample as shor as he one available afer he euro inroducion. Similar wo-sep approaches have been employed by Carriero, Favero and Kaminska (2006), and Favero and Kaminska (2006). 6

7 2.3 Daa Descripion and Model Specificaion We use wo differen ses of daa of coninuously compound zero-coupon ineres raes sampled a he end of each monh. These ses of daa are described in deail in Table 1. The firs daase uses Euribor raes for mauriies up o one year, and swap raes for mauriies from wo up o 10 years. The chosen sources for differen mauriies are similar o he ones ha he European Cenral Bank used o employ o compue marke expecaions of he shor-erm ineres rae using forward raes implied by an esimaed yield curve (ECB, 2007, p. 66, and ECB, 2008, p. 80). I would be preferable o use zero-coupon yields compued from governmen bonds for all mauriies. However, euro-area yields using cenral governmen bonds are only available from he European Cenral Bank daase from 2004, and long-erm ineres raes would be srongly affeced by he recen increase in sovereign risk. The impac of he financial crisis (from Augus 2007) on our euro area daase is ha yields of shor mauriies compued using he Euribor raes are more affeced han long mauriies yields based on he swap raes because of he changes in liquidiy and perceived credi risk in he euro area money marke. The second daa se consiss of zero-coupon yields from German governmen bonds. The advanage of he second daase is ha daa are available for a longer period, alhough he inroducion of he euro currency may affec he sabiliy of he esimaes. Boh daases include a measure of economic aciviy (annual growh of indusrial producion) and of inflaion (annual growh in consumer prices). There are several argumens in favour of using German daa as a proxy for euro ineres raes in he period before he inroducion of he euro. As poined ou in Cappiello e al. (2006), he German bond marke o some exen has been a benchmark for he European bond marke as a whole. In addiion, Germany did no observe currency crisis in he Exchange Rae Mechanism period; as a consequence, he daa are less affeced by inra-area currency effecs. The VAR (eq. 4), which is used o generae h-sep ahead forecass of he hree-monh ineres rae, is compued using only he informaion of yield facors, and also adding informaion of real aciviy growh and inflaion. Using he Schwarz informaion crierion, we choose an auoregressive order of 2 o model he dynamics of he erm-srucure and macro facors for boh daases. We also check for parameer insabiliy in he VAR esimaes. Wih he full sample of German daa, a es for a srucural break of an unknown dae deecs a break poin on he response of he slope o lagged level and curvaure in 1994:1. 2 When esimaing he VAR, wih daa afer 1994:M1, 2 The es is applied for each equaion of he VAR separaely. The null hypohesis is of no break on he coeffi ciens 7

8 no addiional breaks are deeced. This suppors he choice of using only daa for he 1994:M1-2009:M5 period when compuing forecass using he German daa. When using euro area daa, no evidence of a srucural break in he VAR parameers was found, so he sample period indicaed in Table 1 was employed. 2.4 Ou-of-sample Forecasing Exercise In his secion, we compare he forecas accuracy of he forward rae wih a VAR model of yield and macro facors. An imporan issue when assessing he accuracy of forward-base forecass is ha he forecasing performance may change over ime. As a consequence, we spli our ou-of-sample period ino wo subperiods of equal size. Because he euro area daase span a 10-year period, we use a 6-year period for he ou-of-sample evaluaion spli ino wo subperiods of 3 years (36 monhly observaions). Table 2 presens measures of forecas accuracy for boh subperiods (2003:M3-2006:M2 and 2006:M3-2009:M2) and daases (euro area and German daa). We evaluae forecass for nex quarer (3-monhs ahead), semeser (6-monhs ahead) and year (12-monhs ahead). The periods daes indicae forecas origins, so for h = 6, 12, he number of forecas errors of he second subperiod is respecively 33 and 27 since he las observaion available refers o 2009:M5. Table 2 includes he mean forecas bias and he roo mean squared forecas errors (RMSFE) of forecass using he forward, he "DL" approach, and he "DL + macro" approach. 3 The esimaion of he forecasing models are execued wih increasing sample sizes during he ou-of-sample period (recursive esimaion) in he lef panel, and wih fixed windows of daa in he righ panel (rolling; windows of 51 observaions wih euro area daa and of 101 observaions wih German daa). The advanage of forecass based on rolling esimaes is ha hey are more robus o breaks in he model parameers and he disadvanage is ha he shorer sample size exacerbaes small sample esimaion biases. Is he forward rae an accurae forecas of euro shor raes? I depends on he ime period. In he earlier period, he average bias when forecasing wih he forward is 3 basis poins in boh of lagged endogenous variables considered one a a ime. This implies ha he es was applied for 5 5 null hypoheses depending on which parameers he resricion of no break were imposed. The ess were performed using he "Quand-Andrews unknown breakpoin es" available in Eviews 6 as an implemenaion of Andrews (1993) wih asympoic p-values. 3 Forecass from he "DL+macro" model are compued as max(0, ŷ +h ) in order o impose a zero lower bound consrain (see, for example, Swanson and Williamson, 2012). Secion 3 provides addiional discussion on his issue. 8

9 daases, while forecass wih boh DL approaches are worse. In he laer period, however, he average bias increases up o 70 basis poins wih German Daa a h = 12, and he "DL + macro" delivers more accurae forecass. When comparing he accuracy beween he "DL" and he DL + macro forecass is also clear ha macroeconomic variables only improve forecass in he laer period. The RMSFE reducions from using he "DL+macro" approach insead of he forward rae in he period depend on he horizon, daase and mehod of esimaion, bu hey are up o 20-25%. In conras, in he period, he RMSFE increases are beween 20% and 80%. An explanaion for hese resuls is ha he forward premium in he period 2003:M3-2006:M2 is small for all horizons, bu i is large in he 2006:M3-2009:M2 period ha includes he financial crisis period (from Augus 2007) and he downurn (from Ocober 2008). In he nex secion, he imporance of he ime-varying forward premium as a candidae explanaion is exploied. The European Cenral Bank is currenly compuing marke expecaions for he euro area shor raes from Euribor fuure conracs insead of he forward raes implied by a fied yield curve (ECB, 2008, p. 80). I is beyond he scope of his paper o assess wheher he forecass of he shor rae obained using Euribor fuures include a risk premium. However, we colleced daa on Euribor fuures o evaluae he forecas performance of prediions compued using he fuures conracs in comparison wih he forward and he "DL+ macro" approach. Table 2 also presens he forecas performance of Euribor-fuures forecass for horizons +3, +6, and + 12, compued for he 2007:M1-2009:M2 period. 4 In erms of RMSFEs, he Euribor-fuures forecass are more accurae han he "DL+macro" forecass for predicing he shor rae of he nex quarer, bu are worse a long horizons. The remaining average bias of using Euribor-fuures forecass is larger han "DL+macro" forecass for all horizons; for + 12, he bias,.3, is hree imes as large as he "DL+macro" bias. Therefore, Euribor-fuures forecass are a beer measure of marke expecaions han he forward rae because hey reduce forecas bias and variance error a leas in he laer period. However, he forecasing model wih macroeconomic variables delivers he smales forecasing bias a all horizons. 4 The forecass of he hree-monh shor-rae were compued using Euribor fuures conracs for March, Sepember, June and December, observed a he end of monh. Because he forecass are compued a he end of monh from quarerly fuures conracs wih mauriy approximaely a middle of he monh, linear inerpolaion of he observed implied fuures raes is used. Finally, he implied hree-monh raes were ransformed from acual/360 o acual/acual and from simple raes ino annualised raes. 9

10 3 The Time-Varying Forward Premium In he previous secion we presen resuls ha suppor he claim ha he forward rae may provide adequae forecass of he fuure shor rae in specific ime periods. In his secion, we invesigae wheher his variaion in forecas performance is linked o a ime-varying risk premium. We sar by describing our approach o compue he forward premium. Subsequenly we show how o compue confidence inervals for our premium esimae. Finally, we evaluae in-sample and ou-of-sample esimaes of he premium measure and associaed confidence inervals. 3.1 The Forward Premium We compue he forward premium as he difference beween he forward rae and he forecas obained wih he "DL+macro" approach. Forecass wih he "DL+macro" use informaion on he yield curve and macro variables o compue expecaions abou fuure shor rae. By using a measuremen of he uncerainy on "DL+macro" forecass, we can assess he saisical significance of he forward premium a based on he assumpion ha he "DL+macro" is a measure of ineres rae expecaions since i uses publicly available informaion a. Our approach has some advanages in comparison o he use of measures of marke expecaions based on he median of professional forecasers forecass (such as he survey measures employed by Jongen e al. (2011)). The comparison beween forward-based and model-based forecass can be easily implemened and updaed in real ime, and he uncerainies around he forward premium values can be measured. A disadvanage is ha our model-based measure of shor-rae expecaions may differ from oher measures of he marke expecaions obained using eiher nonarbirage models or survey daa. However, his disadvanage is also characerisic of all possible empirical measures of an unobserved componen. A dae, he esimae of he τ-period forward premium for horizon h is he difference beween he implici forward rae of mauriy τ conraced in o sar in + h and he expeced value a of he yield of mauriy τ in + h. The τ period forward premium is: frp (h,h+τ) = f (h,h+τ) E (y (τ) +h ). (6) The expeced value of he fuure 3-monh rae is subsiued by "DL+macro" forecass in order o compue he 3-monh forward premium as: frp (h,h+3) = (h,h+3) ˆf max(0, ŷ (3) +h ), (7) 10

11 where he forward rae ˆf (h,h+3) requires he esimaion of he yield curve as described in secion 2.1, and he zero lower bound (ZLB) consrain of nominal ineres raes is imposed. The ZLB consrain is binding for he DL model wih macroeconomic facors during few monhs in he period because i includes variables ha deermine he policy rae. Swanson and Williams (2012) have also imposed he ZLB on compuing he shor-rae implied by a Taylor rule. 3.2 Confidence Inervals for he Forward Premium The compuaion of he forward premium (7) for a sample of size T is summarized as follows: 1. Use he eq. (2) o compue he erm-srucure facors ˆL, Ŝ, and Ĉ for = 1,..., T where T is he available number of monhly observaions on he yields. 2. Use he erm-srucure facors plus observed macroeconomic facors such as economic aciviy g and inflaion π o esimae a VAR(p) for = p + 1,..., T. 3. Use he VAR(p) esimaes o compue h-sep ahead forecass for he erm srucure facors ˆL +h, Ŝ+h and Ĉ+h, and he eq. (3) o compue h-sep ahead forecass of he shor rae ŷ (3) +h for = p + 1,..., T. 4. Compue he forward raes f (n,n+3) using he fied yield curve for n = 3, 6,...N and = p + 1,..., T, and use he forecass ŷ (3) (h,h+3) +h o compue he forward premium frp as described in eq. (7). Based on his summarized descripion, hree sources of uncerainy affec frp (h,h+3) : (i) he measuremen error from fiing he yield curve in sep (1); (ii) he parameer uncerainy from he VAR esimaion in sep (2); (iii) he forecasing uncerainy affecing he forecass in sep (3). Before describing he boosrap procedure o compue in-sample and ou-of-sample confidence inervals, i is useful o wrie he VAR(p), equaion (4), wih x defined as (5), in he companion (VAR(1)) form: X = ΦX 1 + ɛ, (8) where X and ɛ are boh 5p 1 vecors such ha X = [x µ,..., x p+1 µ] and ɛ = [ε, 0,.., 0]. Φ is a 5p 5p companion marix as described in Hamilon (1994, p. 259), and µ is a 5 1 vecor of uncondiional means of x ha depends on vecor of inerceps c and he marix Φ. 11

12 3.2.1 In-Sample Confidence Inervals When compuing confidence inervals o assess he expecaion hypohesis resricions in a VAR similar o ours, Carriero e al. (2006) chose o consider he uncerainy described in (ii), since he compuaion of expeced values is of ineres, so he pas hisory of unexpeced shocks (iii) could be disregarded. The auhors assume ha he sample size T is large enough, so i is adequae o use an asympoic normal disribuion for ˆΦ (eq. 8). The problem of applying a similar approach here is ha sample sizes available for he euro ineres raes are much shorer han he ones available for US ineres raes, so he assumpion of normaliy may be oo srong. Therefore, we propose o compue in-sample confidence inervals of he esimaed forward premium by using a boosrap approach o assess he same kind of uncerainy described by Carriero e al. (2006). The proposed boosrapping procedure is based on a bias-correcing boosrap o compue confidence inervals for impulse responses in vecor auoregressions proposed by Kilian (1998). The boosrap procedure akes ino accoun ha esimaes of ˆΦ are generally biased in small samples, and does no require he assumpion of normaliy of he disurbances. Kilian s original procedure needs o be adaped such ha he confidence inervals are cenered a each. As a consequence, he boosrapped forward premium is condiioned on he same informaion as he acual forward premium. A similar problem is found when using his ype of boosrap o compue densiy forecass from auoregressive models (Clemens and Taylor, 2001). The in-sample boosrap procedure has wo main pars. The firs par is employed for he bias correcion of ˆΦ. Using he residuals ˆε = X ˆΦX 1 for = p+1,..., T, ε is drawn wih reposiion T p imes from he marix of residuals. For each boosrapped series of residuals, one compues boosrapped sequences x for = p + 1,...T using he original esimaes ĉ and ˆΦ, and observed iniial values x 1,..., x p. The empirical disribuion for he VAR parameers can be compued by esimaing a VAR(p) for each boosrapped sequence of x, while he boosrapped procedure is repeaed B imes. If all he roos of ˆΦ are inside he uni circle, he bias compued using he previous boosrapping procedure is employed o bias-correc he VAR esimaes, delivering c and Φ. The second par of he boosrap uses c and Φ o generae B sequences of size T p of x, using x 1,..., x p as iniial values. For each one of he sequences, he compuaions described above in seps (2) up o (4) are underaken. However, when compuing forecass for each = p + 1,.., T, he boosrapped esimaes of he parameers are used, bu he forecass are condiioned on observed 12

13 daa, ha is, ˆx +h is obained condiional on ĉ and ˆΦ, and x,..x p 1. This sep is required so ha confidence inervals are cenred on he bias-correced esimaes of he forward premium. Noe also ha he forward raes required in sep (4) are also he observed ones. The second par of he boosrapping procedure delivers an empirical disribuion for frp (h,h+τ) for each = p + 1,..., T, aking ino accoun he uncerainy of sep (2) on he compuaion of he forward premium. The procedure described is a boosrapped version of Carriero e al. (2006) ha does no consider uncerainies (i) and (iii). The effec of he measuremen error (i) is small in comparison wih he parameer uncerainy of he VAR esimaion, so we are confiden ha i has very limied effec on confidence inerval measuremen Ou-of-Sample Confidence Inervals The compuaion of he forward premium depends inrinsically on he uncerainy on he shor rae forecass ŷ (3) +h. The previous procedure is applied o he full sample esimaes of he VAR of he erm-srucure and macro facors; herefore, i is reasonable o only consider he effec of parameer uncerainy on ŷ (3) +h. Clemens and Taylor (2001) show how o use boosrap o ake ino accoun boh parameer and forecas uncerainy when compuing forecas densiies from auoregressive models. When evaluaing ou-of-sample esimaes of he forward premium from = T +1,..., T +R, where R is he number of observaions in he ou-of-sample period, we examine boh parameer (ii) and forecasing uncerainies (iii). Suppose ha a each forecas origin in he ou-of-sample period, he VAR (eq. 4) is re-esimaed by eiher adding a new observaion (recursive) or moving he daa window (rolling), such ha ĉ and ˆΦ are available for = T + 1,..., T + R. When only considering forecasing uncerainy, he boosrap procedure requires he use of he forecas densiy of ŷ (3) +h o compue he confidence inerval for he forward premium a = T +1,..., T +R. For each forecas origin, ĉ, ˆΦ, and h draws wih reposiion from he residuals are combined o compue a sequence of forecass ŷ (3) +1,..., ŷ (3) +h. By repeaing his procedure B imes, he empirical disribuion of he forward premium for each ( = T +1,..., T +R) is obained by using he empirical disribuion of (h,h+τ) f max(0, ŷ (τ) +h ). Noe ha he forward premium is compued jus once a he end-of-sample, and ha he compuaion is repeaed for each change of forecas origin. The compuaion of he forward premium confidence inervals aking ino accoun he parameer uncerainy of ˆΦ is also carried ou by using he in-sample boosrapping procedure for he las observaion. 13

14 Finally, boh parameer and forecasing uncerainies may be combined o compue ou-ofsample confidence inervals. This requires a change in he second par of he in-sample boosrap procedure. When compuing facor forecass ˆx +h, he in-sample procedure uses ĉ and ˆΦ condiional on x,..., x p+1 o generae a single sequence of ˆx +h. A his sage, he ou-ofsample boosrapped procedure can be nesed o generae B sequences of ˆx +h. Insead of saring he ou-of-sample boosrap procedure wih observed daa and ˆΦ, he procedure is applied o boosrapped daa x and esimaes ˆΦ. Empirically, we can use he hree measures of ou-of-sample confidence inervals o esablish wheher he forward-based and he model-based forecass are expeced o differ a each forecas origin. The inference generaed from hese differen measures does no need o be conradicory, and hey are informaive abou differen kinds of uncerainies ha one should ake ino accoun when assessing he uncerainy of model-based forecass. 3.3 In-Sample Variaion of he Forward Premium Figure 1 presens he in-sample esimaes of he forward premium for mauriies (horizons) of 3 and 12 monhs, wih confidence inervals compued as described in Secion The esimaes are obained using he full sample (afer 1994 wih German daa) and Figure 1 shows 68% and 90% confidence inervals compued based on 2,000 boosrap replicaions. Remember ha hese inervals consider only he impac of parameer uncerainy on he compuaion of he forward premium. The inervals are no cenred since hey are compued using a bias correcion which was no incorporaed in he esimaes shown. Using he confidence inervals, he forward premium is differen from zero in periods of large macroeconomic uncerainy, such as in he slowdown and he recession. In he case of German daa, he period also shows significan premium values, while he Conference Board 5 has daed a recession for he 1995:M5-1996:M2 period. The ime changes of he euro area daa forward premium confidence inervals are raher similar o he values obained wih German daa, even hough he + 12 inervals are wider. Looking a one-quarer-ahead forward premium ( + 3) plos of euro area and German daa, he associaion wih he financial marke crisis of he increasing premium values afer Augus of 2007 is clear. In conras, he premium in he preceding period (from 2002:M6 hrough 2007:M7) is equal o zero. The ZLB consrain is biding in he las few observaions in Figure 1, creaing an upper bound in he foward premium 5 Turning poins for Germany available a hp:// 14

15 parameer uncerainy such ha he 68% and he 90% inerval limis are very similar. 3.4 Ou-of-Sample Variaion of he Forward Premium The forward premium is compued for horizons of 3, 6 and 12 monhs for each forecas origin from 2003:M3-2009:M2 wih boh daases. The resuls in Figure 2 are based on recursive esimaes wih euro area daa and on rolling esimaes wih German daa, since we chose he mehod ha delivers more accurae forecass in he second period (see Table 2). Figure 2 presens he forward premia and 68% confidence inervals ha include (a) parameer uncerainy, (b) forecasing uncerainy, and (c) boh uncerainies, compued wih 2,000 replicaions. Figure 3 presens he 68% confidence inervals of he forward premium compued wih boh uncerainies for easier comparison of he esimaes wih euro area and German daa. The confidence inervals are very narrow owards he end of he ou-of-sample period because he ZLB consrain is binding: "DL+macro" forecass considering all possible se of parameers and fuure shocks are such ha he forward premium his he upper bound defined by he compued forward rae. The plos in Figures 2 and 3 confirm he evidence in Tables 2 and Figure 1 ha in he four-year period ha precedes he 2007 financial crisis, he forward premium is saisically equal o zero. However, he forward premium increases fas from Augus 2007 and i is saisically differen from zero up o 2009:M2 for all horizons wih euro area daa (see paricularly Figure 3). Similar ime variaion in he forward premium is found when he German daase is employed, alhough he increase in he forward premium is in general smaller. Wrigh (2011) uses an affi ne erm-srucure model o compue he forward premium in German bonds and also idenifies an increase in he premium in similar period. The confidence inervals in Figure 3 are no very differen from he in-sample confidence inervals presened in Figure 1 for he same period, and he period of significan forward premium values in Figure 3 refers approximaely o he period of he forecasing evaluaion in Table 2. These wo resuls sugges ha we can effecively use he informaion on he forward premium and is confidence inerval compued wih he "DL + macro" forecasing model o ex-ane evaluae wheher he forward rae is an adequae forecas of he shor-rae a each forecas origin. These resuls also confirm ha he variaion in he forecasing accuracy of forward-based forecass in Table 2 using boh daases is driven by a ime-varying risk premium. Even when considering all he uncerainies surrounding in-sample and ou-of-sample model-based forecass, we are sill able o find ha he difference beween forward-based and model-based forecass is saisically 15

16 significan in specific periods of ime, generally associaed wih low growh and recession periods. 4 Concluding Remarks This paper presens evidence of a ime-varying forward premium in euro area ineres raes. This implies ha euro area forward raes may be an inadequae measure of marke expecaions. We propose a mehod o evaluae he impac of risk premium on he ex-ane forecas accuracy of forward rae forecass. The mehod is based on forecass compued wih a forecasing model of yield and macro facors. The inclusion of macroeconomic facors improves he forecass of shor raes in he period. Because he forecass employed o compue he forward premium are based on a saisical model, we are able o provide a mehod o compue confidence inervals of he ex-ane forward premium a each forecas origin. The proposed mehod correcly idenifies he period in which he forward raes deliver inferior forecass due o he forward premium, anicipaing he resuls of he ex-pos ou-of-sample forecasing evaluaion. The forecasing model employed o assess he relevance of he forward premium includes informaion from boh economic aciviy and inflaion. Hence he source of significan large forward premium may be eiher business cycle uncerainy, characerising a counercyclical erm premium, or inflaion uncerainy as argued by Wrigh (2011). References [1] Andrews. D. W. K. (1993). Tes for parameer insabiliy and srucural change wih unknown change poin. Economerica, 61: [2] Ang, A. and Piazzesi, M. (2003). A no-arbirage vecor auoregression of erm srucure dynamics wih macroeconomic and laen variables. Journal of Moneary Economics, 50: [3] BIS (2005). Zero-coupon yield curve: echnical documenaion. Moneary and Economic Deparmen. BIS Papers n. 25. [4] Bundesbank (2007). Esimaing he erm srucure of ineres raes. Deusche Bundesbank Monhly Repor. Ocober. [5] Cappiello L., Hördahl P., Kadareja A. and Maganelli S. (2006). The impac of he euro on financial markes. European Cenral Bank. Working Paper n

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