NBER WORKING PAPER SERIES FORECASTING THE TERM STRUCTURE OF GOVERNMENT BOND YIELDS. Francis X. Diebold Canlin Li

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1 NBER WORKING PAPER SERIES FORECASTING THE TERM STRUCTURE OF GOVERNMENT BOND YIELDS Francis X. Diebold Canlin Li Working Paper hp:// NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachuses Avenue Cambridge, MA Ocober 2003 The Naional Science Foundaion, he Wharon Financial Insiuions Cener, and he Guggenheim Foundaion provided research suppor. For helpful commens we are graeful o he Edior (Arnold Zellner), he Associae Edior, and hree referees, as well as Dave Backus, Rob Bliss, Michael Brand, Todd Clark, Qiang Dai, Ron Gallan, Mike Gibbons, David Marshall, Monika Piazzesi, Eric Renaul, Glenn Rudebusch, Til Schuermann, and San Zin, and seminar paricipans a Geneva, Georgeown, Wharon, he European Cenral Bank, and he Naional Bureau of Economic Research. We, however, bear full responsibiliy for all remaining flaws. The views expressed herein are hose of he auhors and no necessarily hose of he Naional Bureau of Economic Research by Francis X. Diebold and Canlin Li. All righs reserved. Shor secions of ex, no o exceed wo paragraphs, may be quoed wihou explici permission provided ha full credi, including noice, is given o he source.

2 Forecasing he Term Srucure of Governmen Bond Yields Francis X. Diebold and Canlin Li NBER Working Paper No Ocober 2003 JEL No. G1, E4, C5 ABSTRACT Despie powerful advances in yield curve modeling in he las weny years, comparaively lile aenion has been paid o he key pracical problem of forecasing he yield curve. In his paper we do so. We use neiher he no-arbirage approach, which focuses on accuraely fiing he cross secion of ineres raes a any given ime bu neglecs ime-series dynamics, nor he equilibrium approach, which focuses on ime-series dynamics (primarily hose of he insananeous rae) bu pays comparaively lile aenion o fiing he enire cross secion a any given ime and has been shown o forecas poorly. Insead, we use variaions on he Nelson-Siegel exponenial componens framework o model he enire yield curve, period-by-period, as a hree-dimensional parameer evolving dynamically. We show ha he hree ime-varying parameers may be inerpreed as facors corresponding o level, slope and curvaure, and ha hey may be esimaed wih high efficiency. We propose and esimae auoregressive models for he facors, and we show ha our models are consisen wih a variey of sylized facs regarding he yield curve. We use our models o produce erm-srucure forecass a boh shor and long horizons, wih encouraging resuls. In paricular, our forecass appear much more accurae a long horizons han various sandard benchmark forecass. Francis X. Diebold Deparmen of Economics Universiy of Pennsylvania 3718 Locus Walk Phiadelphia, PA and NBER fdiebold@sas.upenn.edu Canlin Li Universiy of California, Riverside canlin.li@ucr.edu

3 1. Inroducion The las weny-five years have produced major advances in heoreical models of he erm srucure as well as heir economeric esimaion. Two popular approaches o erm srucure modeling are no-arbirage models and equilibrium models. The no-arbirage radiion focuses on perfecly fiing he erm srucure a a poin in ime o ensure ha no arbirage possibiliies exis, which is imporan for pricing derivaives. The equilibrium radiion focuses on modeling he dynamics of he insananeous rae, ypically using affine models, afer which yields a oher mauriies can be derived under various assumpions abou he risk premium. 1 Prominen conribuions in he no-arbirage vein include Hull and Whie (1990) and Heah, Jarrow and Moron (1992), and prominen conribuions in he affine equilibrium radiion include Vasicek (1977), Cox, Ingersoll and Ross (1985), and Duffie and Kan (1996). Ineres rae poin forecasing is crucial for bond porfolio managemen, and ineres rae densiy forecasing is imporan for boh derivaives pricing and risk managemen. 2 Hence one wonders wha he modern models have o say abou ineres rae forecasing. I urns ou ha, despie he impressive heoreical advances in he financial economics of he yield curve, surprisingly lile aenion has been paid o he key pracical problem of yield curve forecasing. The arbirage-free erm srucure lieraure has lile o say abou dynamics or forecasing, as i is concerned primarily wih fiing he erm srucure a a poin in ime. The affine equilibrium erm srucure lieraure is concerned wih dynamics driven by he shor rae, and so is poenially linked o forecasing, bu mos papers in ha radiion, such as de Jong (2000) and Dai and Singleon (2000), focus only on in-sample fi as opposed o ou-of-sample forecasing. Moreover, hose ha do focus on ou-of-sample forecasing, noably Duffee (2002), conclude ha he models forecas poorly. In his paper we ake an explicily ou-of-sample forecasing perspecive, and we use neiher he no-arbirage approach nor he equilibrium approach. Insead, we use he Nelson-Siegel (1987) exponenial componens framework o disill he enire yield curve, period-by-period, ino a hreedimensional parameer ha evolves dynamically. We show ha he hree ime-varying parameers may 1 The empirical lieraure ha models yields as a coinegraed sysem, ypically wih one underlying sochasic rend (he shor rae) and saionary spreads relaive o he shor rae, is similar in spiri. See Diebold and Sharpe (1990), Hall, Anderson, and Granger (1992), Shea (1992), Swanson and Whie (1995), and Pagan, Hall and Marin (1996). 2 For comparaive discussion of poin and densiy forecasing, see Diebold, Gunher and Tay (1998) and Diebold, Hahn and Tay (1999). 1

4 be inerpreed as facors. Unlike facor analysis, however, in which one esimaes boh he unobserved facors and he facor loadings, he Nelson-Siegel framework imposes srucure on he facor loadings. 3 Doing so no only faciliaes highly precise esimaion of he facors, bu, as we show, i also les us inerpre he facors as level, slope and curvaure. We propose and esimae auoregressive models for he facors, and hen we forecas he yield curve by forecasing he facors. Our resuls are encouraging; in paricular, our models produce one-year-ahead forecass ha are noiceably more accurae han sandard benchmarks. Relaed work includes he facor models of Lizenberger, Squassi and Weir (1995), Bliss (1997a, 1997b), Dai and Singleon (2000), de Jong and Sana-Clara (1999), de Jong (2000), Brand and Yaron (2001) and Duffee (2002). Paricularly relevan are he hree-facor models of Balduzzi, Das, Foresi and Sundaram (1996), Chen (1996), and especially he Andersen-Lund (1997) model wih sochasic mean and volailiy, whose hree facors are inerpreed in erms of level, slope and curvaure. We will subsequenly discuss relaed work in greaer deail; for now, suffice i o say ha lile of i considers forecasing direcly, and ha our approach, alhough relaed, is indeed very differen. We proceed as follows. In secion 2 we provide a deailed descripion of our modeling framework, which inerpres and exends earlier work in ways linked o recen developmens in mulifacor erm srucure modeling, and we also show how i can replicae a variey of sylized facs abou he yield curve. In secion 3 we proceed o an empirical analysis, describing he daa, esimaing he models, and examining ou-of-sample forecasing performance. In secion 4 we offer inerpreive concluding remarks. 2. Modeling and Forecasing he Term Srucure I: Mehods Here we inroduce he framework ha we use for fiing and forecasing he yield curve. We argue ha he well-known Nelson-Siegel (1987) curve is well-suied o our ulimae forecasing purposes, and we inroduce a novel wis of inerpreaion, showing ha he hree coefficiens in he Nelson-Siegel curve may be inerpreed as laen level, slope and curvaure facors. We also argue ha he naure of he facors and facor loadings implici in he Nelson-Siegel model faciliae consisency wih various empirical properies of he yield curve ha have been caaloged over he years. Finally, moivaed by our inerpreaion of he Nelson-Siegel model as a hree-facor model of level, slope and curvaure, we conras i o various muli-facor models ha have appeared in he lieraure. 3 Classic unresriced facor analyses include Lierman and Scheinkman (1991) and Knez, Lierman and Scheinkman (1994). 2

5 Consrucing Raw Yields Le us firs fix ideas and esablish noaion by inroducing hree key heoreical consrucs and he relaionships among hem: he discoun curve, he forward curve, and he yield curve. Le P ( ) denoe he price of a -period discoun bond, i.e., he presen value a ime of $1 receivable periods ahead, and le y ( ) denoe is coninuously-compounded zero-coupon nominal yield o mauriy. From he yield curve we obain he discoun curve, P ( ) e y ( ), and from he discoun curve we obain he insananeous (nominal) forward rae curve, f ( ) P ( )/P ( ). The relaionship beween he yield o mauriy and he forward rae is herefore y ( ) 1 0 f (u)du, which implies ha he zero-coupon yield is an equally-weighed average of forward raes. Given he yield curve or forward curve, we can price any coupon bond as he sum of he presen values of fuure coupon and principal paymens. In pracice, yield curves, discoun curves and forward curves are no observed. Insead, hey mus be esimaed from observed bond prices. Two popular approaches o consrucing yields proceed by esimaing a smooh discoun curve and hen convering o yields a he relevan mauriies via he above formulae. The firs discoun-curve approach o yield consrucion is due o McCulloch (1975) and McCulloch and Kwon (1993), who model he discoun curve wih a cubic spline. The fied discoun curve, however, diverges a long mauriies insead of converging o zero. Hence such curves provide a poor fi o yield curves ha are fla or have a fla long end, which requires an exponenially decreasing discoun funcion. A second discoun-curve approach o yield consrucion is due o Vasicek and Fong (1982), who fi exponenial splines o he discoun curve, using a negaive ransformaion of mauriy insead of mauriy iself, which ensures ha he forward raes and zero-coupon yields converge o a fixed limi as mauriy increases. Hence he Vasicek-Fong model is more successful a fiing yield curves wih fla 3

6 long ends. I has problems of is own, however, because is esimaion requires ieraive nonlinear opimizaion, and i can be hard o resric he implied forward raes o be posiive. A hird and very popular approach o yield consrucion is due o Fama and Bliss (1987), who consruc yields no via an esimaed discoun curve, bu raher via esimaed forward raes a he observed mauriies. Their mehod sequenially consrucs he forward raes necessary o price successively longer-mauriy bonds, ofen called an unsmoohed Fama-Bliss forward raes, and hen consrucs unsmoohed Fama-Bliss yields by averaging he appropriae unsmoohed Fama-Bliss forward raes. The unsmoohed Fama-Bliss yields exacly price he included bonds. Throughou his paper, we model and forecas he unsmoohed Fama-Bliss yields. Modeling Yields: The Nelson-Siegel Yield Curve and is Inerpreaion A any given ime, we have a large se of (Fama-Bliss unsmoohed) yields, o which we fi a parameric curve for purposes of modeling and forecasing. Throughou his paper, we use he Nelson- Siegel (1987) funcional form, which is a convenien and parsimonious hree-componen exponenial approximaion. In paricular, Nelson and Siegel (1987), as exended by Siegel and Nelson (1988), work wih he forward rae curve, f ( ) 1 e 2 3 e. The Nelson-Siegel forward rae curve can be viewed as a consan plus a Laguerre funcion, which is a polynomial imes an exponenial decay erm and is a popular mahemaical approximaing funcion. 4 The corresponding yield curve is y ( ) e 3 1 e e. The Nelson-Siegel yield curve also corresponds o a discoun curve ha begins a one a zero mauriy and approaches zero a infinie mauriy, as appropriae. Le us now inerpre he parameers in he Nelson-Siegel model. The parameer exponenial decay rae; small values of mauriies, while large values of governs he produce slow decay and can beer fi he curve a long produce fas decay and can beer fi he curve a shor mauriies. 4 See, for example, Couran and Hilber (1953). 4

7 also governs where he loading on achieves is maximum We inerpre, and as hree laen dynamic facors. The loading on is 1, a consan ha does no decay o zero in he limi; hence i may be viewed as a long-erm facor. The loading on 2 is (1 e ) /, a funcion ha sars a 1 bu decays monoonically and quickly o 0; hence i may be 3 ((1 e ) / ) e viewed as a shor-erm facor. The loading on is, which sars a 0 (and is hus no shor-erm), increases, and hen decays o zero (and hus is no long-erm); hence i may be viewed as a medium-erm facor. We plo he hree facor loadings in Figure 1. They are similar o hose obained by Bliss (1997a), who esimaed loadings via a saisical facor analysis. 6 An imporan insigh is ha he hree facors, which following he lieraure we have hus far called long-erm, shor-erm and medium-erm, may also be inerpreed in erms of level, slope and curvaure. The long-erm facor 1, for example, governs he yield curve level. In paricular, one can easily verify ha y ( ) 1. Alernaively, noe ha an increase in increases all yields equally, as he loading is idenical a all mauriies, hereby changing he level of he yield curve. The shor-erm facor year yield minus he hree-monh yield. In paricular, 2 1 is closely relaed o he yield curve slope, which we define as he en- as Frankel and Lown (1994), moreover, define he yield curve slope as 2. Some auhors such y (120) y (3) y ( ) y (0), which is exacly equal o. Alernaively, noe ha an increase in increases shor yields more han long yields, because he shor raes load on 2 2 more heavily, hereby changing he slope of he yield curve. 1 2 We have seen ha governs he level of he yield curve and governs is slope. I is ineresing o noe, moreover, ha he insananeous yield depends on boh he level and slope facors, because. Several oher models have he same implicaion. In paricular, Dai and y (0) 1 2 Singleon (2000) show ha he hree-facor models of Balduzzi, Das, Foresi and Sundaram (1996) and Chen (1996) impose he resricions ha he insananeous yield is an affine funcion of only wo of he hree sae variables, a propery shared by he Andersen-Lund (1997) hree-facor non-affine model. Finally, he medium-erm facor 3 is closely relaed o he yield curve curvaure, which we define as wice he wo-year yield minus he sum of he en-year and hree-monh yields. In paricular, 5 Throughou his paper, and for reasons ha will be discussed subsequenly in deail, we se = for all. 6 Facors are ypically no uniquely idenified in facor analysis. Bliss (1997a) roaes he firs facor so ha is loading is a vecor of ones. In our approach, he uni loading on he firs facor is imposed from he beginning, which poenially enables us o esimae he oher facors more efficienly. 5

8 2y (24) y (3) y (120) Alernaively, noe ha an increase in 3 will have lile effec on very shor or very long yields, which load minimally on i, bu will increase medium-erm yields, which load more heavily on i, hereby increasing yield curve curvaure. Now ha we have inerpreed Nelson-Siegel as a hree-facor of level, slope and curvaure, i is appropriae o conras i o Lizenberger, Squassi and Weir (1995), which is highly relaed ye disinc. Firs, alhough Lizenberger e al. model he discoun curve P ( ) using exponenial componens and we model he yield curve y ( ) using exponenial componens, he yield curve is a log ransformaion of he discoun curve because y ( ) log P ( )/, so he wo approaches are equivalen in he one-facor case. In he muli-facor case, however, a sum of facors in he yield curve will no be a sum in he discoun curve, so here is generally no simple mapping beween he approaches. Second, boh we and Lizenberger e al. provide novel inerpreaions of he parameers of fied curves. Lizenberger e al., however, do no inerpre parameers direcly as facors. In closing his sub-secion, i is worh noing ha wha we have called he Nelson-Siegel curve is acually a differen facorizaion han he one originally advocaed by Nelson and Siegel (1987), who used y ( ) b 1 b 2 1 e 6 b 3 e Obviously he Nelson-Siegel facorizaion maches ours wih b 1 1, b 2 2 3, and b 3 3. Ours is preferable, however, for reasons ha we are now in a posiion o appreciae. Firs, (1 e ) / and e have similar monoonically decreasing shape, so if we were o inerpre b 2 and b 3 as facors, hen heir loadings would be forced o be very similar, which creaes a leas wo problems. Firs, concepually, i would be hard o provide inuiive inerpreaions of he facors in he original Nelson- Siegel framework. Second, operaionally, i would be difficul o esimae he facors precisely, because he high coherence in he facors produces mulicolineariy. Sylized Facs of he Yield Curve and he Model s Poenial Abiliy o Replicae Them A good model of yield curve dynamics should be able o reproduce he hisorical sylized facs concerning he average shape of he yield curve, he variey of shapes assumed a differen imes, he srong persisence of yields and weak persisence of spreads, and so on. I is no easy for a parsimonious model o accord wih all such facs. Duffee (2002), for example, shows ha muli-facor affine models are inconsisen wih many of he facs, perhaps because erm premia may no be adequaely capured by affine models..

9 Le us consider some of he mos imporan sylized facs and he abiliy of our model o replicae hem, in principle: (1) The average yield curve is increasing and concave. In our framework, he average yield curve is he yield curve corresponding o he average values of, and. I is cerainly possible in principle ha i may be increasing and concave. (2) The yield curve assumes a variey of shapes hrough ime, including upward sloping, downward sloping, humped, and invered humped. The yield curve in our framework can assume all of hose shapes. Wheher and how ofen i does depends upon he variaion in, and. (3) Yield dynamics are persisen, and spread dynamics are much less persisen. Persisen yield dynamics would correspond o srong persisence of dynamics would correspond o weaker persisence of., and less persisen spread (4) The shor end of he yield curve is more volaile han he long end. In our framework, his is refleced in facor loadings: he shor end depends posiively on boh and, whereas he long end depends only on. raes. Overall, i seems clear ha our framework is consisen, a leas in principle, wih many of he key sylized facs of yield curve behavior. Wheher principle accords wih pracice is an empirical maer, o which we now urn. 3. Modeling and Forecasing he Term Srucure II: Empirics In his secion, we esimae and assess he fi of he hree-facor model in a ime series of cross secions, afer which we model and forecas he exraced level, slope and curvaure componens. We begin by inroducing he daa. The Daa We use end-of-monh price quoes (bid-ask average) for U.S. Treasuries, from January 1985 hrough December 2000, aken from he CRSP governmen bonds files. CRSP filers he daa, eliminaing bonds wih opion feaures (callable and flower bonds), and bonds wih special liquidiy problems (noes and bonds wih less han one year o mauriy, and bills wih less han one monh o mauriy), and hen convers he filered bond prices o unsmoohed Fama-Bliss (1987) forward raes. Then, using programs and CRSP daa kndly supplied by Rob Bliss, we conver he unsmoohed (5) Long raes are more persisen han shor raes. In our framework, long raes depend only on 1 1. If is he mos persisen facor, hen long raes will be more persisen han shor

10 Fama-Bliss forward raes ino unsmoohed Fama-Bliss zero yields. Alhough mos of our analysis does no require he use of fixed mauriies, doing so grealy simplifies our subsequen forecasing exercises. Hence we pool he daa ino fixed mauriies. Because no every monh has he same mauriies available, we linearly inerpolae nearby mauriies o pool ino fixed mauriies of 3, 6, 9, 12, 15, 18, 21, 24, 30, 36, 48, 60, 72, 84, 96, 108, and 120 monhs, where a monh is defined as days. Alhough here is no bond wih exacly days o mauriy, each monh here are many bonds wih eiher 30, 31, 32, 33, or 34 days o mauriy. Similarly we obain daa for mauriies of 3 monhs, 6 monhs, ec. 7 The various yields, as well as he yield curve level, slope and curvaure defined above, will play a prominen role in he sequel. Hence we focus on hem now in some deail. In Figure 2 we provide a hree-dimensional plo of our yield curve daa. The large amoun of emporal variaion in he level is visually apparen. The variaion in slope and curvaure is less srong, bu neverheless apparen. In Table 1, we presen descripive saisics for he yields. I is clear ha he ypical yield curve is upward sloping, ha he long raes are less volaile and more persisen han shor raes, ha he level (120-monh yield) is highly persisen bu varies only moderaely relaive o is mean, ha he slope is less persisen han any individual yield bu quie highly variable relaive o is mean, and he curvaure is he leas persisen of all facors and he mos highly variable relaive o is mean. 8 I is also worh noing, because i will be relevan for our fuure modeling choices, ha level, slope and curvaure are no highly correlaed wih each oher; all pairwise correlaions are less han In Figure 3 we display he median yield curve ogeher wih poinwise inerquarile ranges. The earlier-menioned upward sloping paern, wih long raes less volaile han shor raes, is apparen. One can also see ha he disribuions of yields around heir medians end o be asymmeric, wih a long righ ail. Fiing Yield Curves As discussed above, we fi he yield curve using he hree-facor model, y ( ) e 3 1 e e. 7 We checked he derived daase and verified ha he difference beween i and he original daase is only one or wo basis poins. 8 Tha is why affine models don fi he daa well; hey can generae such high variabiliy and quick mean reversion in curvaure. 8

11 We could esimae he parameers by nonlinear leas squares, for each monh. { 1, 2, 3, } Following sandard pracice racing o Nelson and Siegel (1987), however, we insead fix prespecified value, which les us compue he values of he wo regressors (facor loadings) and use ordinary leas squares o esimae he beas (facors), for each monh. Doing so enhances no only simpliciy and convenience, bu also numerical rusworhiness, by enabling us o replace hundreds of poenially challenging numerical opimizaions wih rivial leas-squares regressions. The quesion arises, of course, as o an appropriae value for. Recall ha deermines he mauriy a which he loading on he medium-erm, or curvaure, facor achieves i maximum. Two- or hree-year mauriies are commonly used in ha regard, so we simply picked he average, 30 monhs. The maximizes he loading on he medium-erm facor a exacly 30 monhs is = a a value ha Applying ordinary leas squares o he yield daa for each monh gives us a ime series of esimaes of {ˆ, 1 ˆ, 2 ˆ } 3 and a corresponding panel of residuals, or pricing errors. Noe ha, because he mauriies are no equally spaced, we implicily weigh he mos acive region of he yield curve mos heavily when fiing he model. 9 There are many aspecs o a full assessmen of he fi of our model. In Figure 4 we plo he implied average fied yield curve agains he average acual yield curve. The wo agree quie closely. In Figure 5 we dig deeper by ploing he raw yield curve and he hreefacor fied yield curve for some seleced daes. Clearly he hree-facor model is capable of replicaing a variey of yield curve shapes: upward sloping, downward sloping, humped, and invered humped. I does, however, have difficulies a some daes, especially when yields are dispersed, wih muliple inerior minima and maxima. Overall, however, he residual plo in Figure 6 indicaes a good fi. In Table 2 we presen saisics ha describe he in-sample fi. The residual sample auocorrelaions indicae ha pricing errors are persisen. As noed in Bliss (1997b), regardless of he erm srucure esimaion mehod used, here is a persisen discrepancy beween acual bond prices and prices esimaed from erm srucure models. Presumably hese discrepancies arise from persisen ax and/or liquidiy effecs. 10 However, because hey persis, hey should vanish from fied yield changes. In Figure 7 we plo {ˆ 1, ˆ 2, ˆ 3 } along wih he empirical level, slope and curvaure defined earlier. The figure confirms our asserion ha he hree facors in our model correspond o level, slope 9 Oher weighings and loss funcions have been explored by Bliss (1997b), Soderlind and Svensson (1997), and Baes (1999). 10 Alhough, as discussed earlier, we aemped o remove illiquid bonds, complee eliminaion is no possible. 9

12 and curvaure. The correlaions beween he esimaed facors and he empirical level, slope, and curvaure are (ˆ = 0.97,,l ) 1 (ˆ = -0.99, and,s ) 2 (ˆ,c ) = 0.99, where (l, s, c ) 3 are he empirical level, slope and curvaure of he yield curve. In Table 3 and Figure 8 (lef column) we presen descripive saisics for he esimaed facors. From he auocorrelaions of he hree facors, we can see ha he firs facor is he mos persisen, and ha he second facor is more persisen han he hird. Augmened Dickey-Fuller ess sugges ha ˆ and ˆ may have a uni roos, and ha ˆ does no. 11 Finally, he pairwise correlaions beween he esimaed facors are no large. Modeling and Forecasing Yield Curve Level, Slope and Curvaure 1 ŷ ( h/ ) ˆ ˆ 1, h/ 2, h/ 2 We model and forecas he Nelson-Siegel facors as univariae AR(1) processes. The AR(1) models can be viewed as naural benchmarks deermined a priori: he simples grea workhorse auoregressive models. The yield forecass based on underlying univariae AR(1) facor specificaions are: 1 e ˆ 3, h/ 1 e e, 3 where ˆ i, h/ ĉ i ˆ ˆ i i, i 1, 2, 3, and and ˆ are obained by regressing ˆ on an inercep and ˆ. 12 ĉ i i i For comparison, we also produce yield forecass based on an underlying mulivariae VAR(1) specificaion, as: ŷ ( h/ ) ˆ ˆ 1, h/ 2, h/ 1 e i, h ˆ 3, h/ 1 e e, where 11 We use SIC o choose he lags in he augmened Dickey-Fuller uni-roo es. The MacKinnon criical values for rejecion of hypohesis of a uni roo are a he one percen level, a he five percen level, and a he en percen level. 12 Noe ha we direcly regress facors a +h on facors a, which is a sandard mehod of coaxing leas squares ino opimizing he relevan loss funcion, h-monh-ahead RMSE, as opposed o he usual 1-monh-ahead RMSE. We esimae all compeior models in he same way, as described below. 10

13 ˆ h/ ˆ ĉ ˆ. We include he VAR forecass for compleeness, alhough one migh expec hem o be inferior o he AR forecass for a leas wo reasons. Firs, as is well-known from he macroeconomics lieraure, unresriced VARs end o produce poor forecass of economic variables even when here is imporan cross-variable ineracion, due o he large number of included parameers and he resuling poenial for in-sample overfiing. 13 Second, our facors indeed display lile cross-facor ineracion and are no highly correlaed, so ha an appropriae mulivariae model is likely close o a sacked se of univariae models. In Figure 8 (righ column) we provide some evidence on he goodness of fi of he AR(1) models fi o he esimaed level, slope and curvaure facors, showing residual auocorrelaion funcions. The auocorrelaions are very small, indicaing ha he models accuraely describe he condiional means of level, slope and curvaure. Ou-of-Sample Forecasing Performance of he Three-Facor Model A good approximaion o yield-curve dynamics should no only fi well in-sample, bu also forecas well ou-of-sample. Because he yield curve depends only on {ˆ, 1 ˆ, 2 ˆ } 3, forecasing he yield curve is equivalen o forecasing {ˆ, 1 ˆ, 2 ˆ } 3. In his secion we underake jus such a forecasing exercise. We esimae and forecas recursively, using daa from 1985:1 o he ime ha he forecas is made, beginning in 1994:1 and exending hrough 2000:12. In Tables 4-6 we compare h-monh-ahead ou-of sample forecasing resuls from Nelson-Siegel models o hose of several naural compeiors, for mauriies of 3, 12, 36, 60 and 120 monhs, and forecas horizons of h = 1, 6 and 12 monhs. Le us now describe he compeiors in erms of how heir forecass are generaed. (1) Random walk: ŷ h/ ( ) y ( ). The forecas is always no change. (2) Slope regression: 13 Tha, of course, is he reason for he ubiquious use of Bayesian analysis, feauring srong priors on he VAR coefficiens, for VAR forecasing, as pioneered by Doan, Lierman and Sims (1984). 11

14 ŷ ( h/ ) y ( ) ĉ( ) ˆ( )(y ( ) y (3)). The forecased yield change is obained from a regression of hisorical yield changes on yield curve slopes. (3) Fama-Bliss forward rae regression: ŷ ( h/ ) y ( ) ĉ( ) ˆ( )(f h ( ) y ( )), where f h ( ) is he forward rae conraced a ime for loans from ime h o ime h. Hence he forecased yield change is obained from a regression of hisorical yield changes on forward spreads. Noe ha, because he forward rae is proporional o he derivaive of he discoun funcion, he informaion used o forecas fuure yields in forward rae regressions is very similar o ha in slope regressions. (4) Cochrane-Piazzesi (2002) forward curve regression: ŷ ( h/ ) y ( ) ĉ( ) ˆ 0 ( ) y (12) 9 k 1 ŷ ( h/ ) ĉ( ) ˆ y ( ). ˆ ( k )f 12k (12). Noe ha he Fama-Bliss forward regression is a special case of he Cochrane-Piazzesi forward regression. 14 (5) AR(1) on yield levels: (6) VAR(1) on yield levels: ŷ h/ ĉ ˆ y. where y [y (3), y (12), y (36), y (60), y (120)]. (7) VAR(1) on yield changes: ẑ h/ ĉ ˆ z, 14 Noe ha his is an unresriced version of he model esimaed by Cochrane and Piazzesi. Imposiion of he Cochrane-Piazzesi resricions produced qualiaively idenical resuls. 12

15 where z [y (3) y (3), y (12) y 1 (12), y (36) y 1 (36), y (60) y 1 (60), y (120) y 1 (120)] 1. (8) ECM(1) wih one common rend: ẑ h/ ẑ h/ ˆ ĉ where z [y (3) y (3), y (12) y (3), y (36) y (3), y (60) y (3), y (120) y (3)] 1. (9) ECM(1) wih wo common rends: ĉ ˆ z, where z [y (3) y 1 (3), y (12) y 1 (12), y (36) y (3), y (60) y (3), y (120) y (3)]. (10) Direc regression on hree AR(1) principal componens z, We firs perform a principal componens analysis on he full se of seveneen yields y, effecively decomposing he yield covariance marix as ˆx i, h/ ĉ i ˆ x i i, i 1, 2, 3, Q Q T ŷ h ( ) q 1 ( ) ˆx 1, h q 2 ( ) ˆx 2, h q 3 ( ) ˆx 3, h,, where he diagonal elemens of are he eigenvalues and he columns of Q are he associaed eigenvecors. Denoe he larges hree eigenvalues by eigenvecors by 1, 2, and 3, and denoe he associaed q 1, q 2, and q 3. The firs hree principal componens x [x 1, x 2, x 3 ] are hen defined by i y, i 1, 2, 3. We hen use a univariae AR(1) model o x i q produce h-sep-ahead forecass of he principal componens: and we produce forecass for yields y [y (3), y (12), y (36), y (60), y (120)] as where q i ( ) is he elemen in he eigenvecor q i ha corresponds o mauriy. We define forecas errors a +h as forecas ( y h ( ) y h ( ) ŷ h/ ( ). Noe well ha, in each case, he objec being ) is a fuure yield, no a fuure Nelson-Siegel fied yield. We will examine a number of descripive saisics for he forecas errors, including mean, sandard deviaion, roo mean squared error (RMSE), and auocorrelaions a various displacemens. Our model s 1-monh-ahead forecasing resuls, repored in Table 4, are in cerain respecs humbling. In absolue erms, he forecass appear subopimal: he forecas errors appear serially correlaed. In relaive erms, RMSE comparison a various mauriies reveals ha our forecass, alhough 13

16 slighly beer han he random walk and slope regression forecass, are indeed only very slighly beer. Finally, he Diebold-Mariano (1995) saisics repored in Table 7 indicae universal insignificance of he RMSE differences beween our 1-monh-ahead forecass and hose from random walks or Fama-Bliss regressions. The 1-monh-ahead forecas defecs likely come from a variey of sources, some of which could be eliminaed. Firs, for example, pricing errors due o illiquidiy may be highly persisen and could be reduced by including variables ha may explain mispricing. I is worh noing, moreover, ha relaed papers such as Bliss (1997b) and de Jong (2000) also find serially correlaed forecas errors, ofen wih persisence much sronger han ours. Maers improve radically, however, as he forecas horizon lenghens. Our model s 6-monhahead forecasing resuls, repored in Table 5, are noiceably improved, and our model s 12-monh-ahead forecasing resuls, repored in Table 6, are much improved. In paricular, our model s 12-monh ahead forecass ouperform hose of all compeiors a all mauriies, ofen by a wide margin in boh relaive and absolue erms. Seven of he en Diebold-Mariano saisics in Table 7 indicae significan 12-monhahead RMSE superioriy of our forecass a he five percen level. The srong yield curve forecasabiliy a he 12-monh-ahead horizon is, for example, very aracive from he vanage poin of acive bond rading and he vanage poin of credi porfolio risk managemen. 15 Moreover, our 12-monh-ahead forecass, like heir 1- and 6-monh-ahead counerpars, could be improved upon, because he forecas errors remain serially correlaed. 16 I is worh noing ha Duffee (2002) finds ha even he simples random walk forecass dominae hose from he Dai-Singleon (2000) affine model, which herefore appears largely useless for forecasing. Hence Duffee proposes a less-resricive essenially-affine model and shows ha i forecass beer han he random walk in mos cases, which is appropriaely viewed as a vicory. A 15 Noe ha Nelson-Siegel loadings imply a very smooh yield curve, which in urn suggess ha our model, alhough no arbirage-free, would no likely generae exreme porfolio posiions. Compeiors such as regression on principal componens, in conras, have no smooh cross-secional resricions and may well generae exreme porfolio posiions in pracice. This is one imporan way in which our approach is superior o direcs regression on principal componens, despie he fac ha our esimaed facors are close o he firs hree principal componens. (Four more are given below.) 16 We repor 12-monh-ahead forecas error serial correlaion coefficiens a displacemens of 12 and 24 monhs, in conras o hose a displacemens of 1 and 12 monhs repored for he 1-monh-ahead forecas errors, because he 12-monh-ahead errors would naurally have moving-average srucure even if he forecass were fully opimal, due o he overlap. 14

17 comparison of our resuls and Duffee s, however, reveals ha our hree-facor model produces larger percenage reducions in ou-of-sample RMSE relaive o he random walk han does Duffee s bes essenially-affine model. Our forecasing success is paricularly noable in ligh of he fac ha Duffee forecass only he smoohed yield curve, whereas we forecas he acual yield curve. 17 Finally, we noe ha alhough our approach is closely relaed o direc principal componens regression, neiher our approach nor our resuls are idenical. Ineresingly, here is reason o prefer our approach on boh empirical and heoreical grounds. Empirically, our resuls indicae ha our approach has superior forecasing performance on our sample of yields. Theoreically, oher mehods, including regression on principal componens and regression on ad hoc empirical level, slope and curvaure, ofen have unappealing feaures, including: (1) hey can no be used o produce yields a mauriies oher han hose observed in he daa, (2) hey do no guaranee a smooh yield curve and forward curve, (3) hey do no guaranee posiive forward raes a all horizons, and (4) hey do no guaranee ha he discoun funcion sars a 1 and approaches 0 as mauriy approaches infiniy. 4. Concluding Remarks We have re-inerpreed he Nelson-Siegel yield curve as a modern hree-facor dynamic model of level, slope and curvaure, and we have explored he model s performance in ou-of-sample yield curve forecasing. Alhough he 1-monh-ahead forecasing resuls are no beer han hose of random walk and oher leading compeiors, he 1-year-ahead resuls are much superior. A number of auhors have proposed exensions o Nelson-Siegel o enhance flexibiliy, including Bliss (1997b), Soderlind and Svensson (1997), Björk and Chrisensen (1999), Filipovic (1999, 2000), Björk (2000), Björk and Landén (2000) and Björk and Svensson (2001). From he perspecive of ineres rae forecasing accuracy, however, he desirabiliy of he above generalizaions of Nelson-Siegel is no obvious, which is why we did no pursue hem here. For example, alhough he Bliss and Soderlind- Svensson exensions can have in-sample fi no worse han ha of Nelson-Siegel, because hey include Nelson-Siegel as a special case, here is no guaranee of beer ou-of-sample forecasing performance. Indeed, accumulaed experience sugges ha parsimonious models are ofen more successful for ou-of- 17 We noe, however, ha our enhusiasm mus be empered by he fac ha our in-sample and ou-of-sample periods are no idenical o Duffee s, so definiive comparisons can no be made. 15

18 sample forecasing. 18 Some of he exensions alluded o above are designed o make Nelson-Siegel consisen wih noarbirage pricing. I is no obvious o us, however, ha use of arbirage-free models is necessary or desirable for producing good forecass. 19 Indeed we have shown ha our model (which is no arbiragefree) produces good forecass, whereas Duffee (2002) and ohers have recenly shown ha he popular affine no-arbirage models produce very poor forecass. Moreover, alhough our model is no heoreically arbirage-free, we expec i o be empirically nearly arbirage-free. The U.S. Treasury bond marke is very liquid, which should make Treasury bond yields nearly arbirage-free, so ha given he very good fi of our model, i should also be nearly arbirage-free. In closing, we would like o elaborae on he likely reason for he forecasing success of our approach, which relies heavily on a broad inerpreaion of he shrinkage principle. The essence of our approach is inenionally o impose subsanial a priori srucure, moivaed by simpliciy, parsimony, and heory, in an explici aemp o avoid daa mining and hence enhance ou-of-sample forecasing abiliy. This includes our use of a ighly-parameric model ha places sric srucure on facor loadings in accordance wih simple heoreical desideraa for he discoun funcion, our decision o fix, our emphasis on simple univariae modeling of he facors based upon our heoreically-derived inerpreaion of he model as one of approximaely orhogonal level, slope and curvaure facors, and our emphasis on he simples possible AR(1) facor dynamics. All of his is in keeping wih a broad inerpreaion of he shrinkage principle, which has a firm foundaion in Bayes-Sein heory, in empirical inuiion, and in an accumulaed rack record of good performance (e.g., Garcia-Ferrer e al., 1987; Zellner and Hong, 1989; Zellner and Min, 1993). Here we inerpre he shrinkage principle as he insigh ha imposiion of resricions, which will of course degrade in-sample fi, may neverheless be helpful for ou-of-sample forecasing, even if he resricions are false. The fac ha he shrinkage principle works in he yieldcurve conex, as i does in so many oher conexs, is precisely wha heory and empirical experience would lead one o expec. This is no o say, of course, ha our specificaion is in any sense uniquely bes, and we make no claims o ha effec. Raher, he broad lesson of he paper is o show in he yieldcurve conex ha he shrinkage perspecive, which ends o produce seemingly-naive bu ruly sophisicaedly-simple models (of which ours is one example), may be very appealing when he goal is 18 See Diebold (2004). 19 See Dai and Singleon (2002) for an ineresing analysis ha explores cerain aspecs of he radeoff beween freedom from arbirage and forecasing performance. 16

19 forecasing. Pu differenly, he paper emphasizes in he yield curve conex Zellner s (1992) KISS principle of forecasing Keep I Sophisicaedly Simple. 17

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24 Table 1 Descripive Saisics, Yield Curves Mauriy (Monhs) Mean Sandard Deviaion Minimum Maximum ˆ(1) ˆ(12) ˆ(30) (level) slope curvaure

25 Table 2 Descripive Saisics, Yield Curve Residuals Mauriy (Monhs) Mean Sandard Deviaion Min. Max. MAE RMSE ˆ(1) ˆ(12) ˆ(30)

26 Table 3 Descripive Saisics, Esimaed Facors Facor Mean Sd. Dev. Minimum Maximum ADF

27 Table 4 Ou-of-Sample 1-Monh-Ahead Forecasing Resuls Nelson-Siegel wih AR(1) Facor Dynamics Mauriy ( ) Mean Sd. Dev. RMSE 3 monhs year years years years Random Walk Mauriy ( ) Mean Sd. Dev. RMSE 3 monhs year years years years Slope Regression Mauriy ( ) Mean Sd. Dev. RMSE 3 monhs NA NA NA NA NA 1 year years years years Fama-Bliss Forward Rae Regression Mauriy ( ) Mean Sd. Dev. RMSE 3 monhs year years years years

28 Table 4 (Coninued) Ou-of-Sample 1-Monh-Ahead Forecasing Resuls Cochrane-Piazzesi Forward Curve Regression Mauriy ( ) Mean Sd. Dev. RMSE 3 monhs NA NA NA NA NA 1 year years years years Univariae AR(1)s on Yield Levels Mauriy ( ) Mean Sd. Dev. RMSE 3 monhs year years years years VAR(1) on Yield Levels Mauriy ( ) Mean Sd. Dev. RMSE 3 monhs year years years years VAR(1) on Yield Changes Mauriy ( ) Mean Sd. Dev. RMSE 3 monhs year years years years

29 Table 5 Ou-of-Sample 6-monh-Ahead Forecasing Resuls Nelson-Siegel wih AR(1) Facor Dynamics Mauriy ( ) Mean Sd. Dev. RMSE 3 monhs year years years years Random Walk Mauriy ( ) Mean Sd. Dev. RMSE 3 monhs year years years years Slope Regression Mauriy ( ) Mean Sd. Dev. RMSE 3 monhs NA NA NA NA NA 1 year years years years Fama-Bliss Forward Rae Regression Mauriy ( ) Mean Sd. Dev. RMSE 3 monhs year years years years

30 Table 5 (Coninued) Ou-of-Sample 6-monh-Ahead Forecasing Resuls Cochrane-Piazzesi Forward Curve Regression Mauriy ( ) Mean Sd. Dev. RMSE 3 monhs NA NA NA NA NA 1 year years years years Univariae AR(1)s on Yield Levels Mauriy ( ) Mean Sd. Dev. RMSE 3 monhs year years years years VAR(1) on Yield Levels Mauriy ( ) Mean Sd. Dev. RMSE 3 monhs year years years years VAR(1) on Yield Changes Mauriy ( ) Mean Sd. Dev. RMSE 3 monhs year years years years

31 Table 6 Ou-of-Sample 12-monh-Ahead Forecasing Resuls Nelson-Siegel wih AR(1) Facor Dynamics Mauriy ( ) Mean Sd. Dev. RMSE 3 monhs year years years years Nelson-Siegel wih VAR(1) Facor Dynamics Mauriy ( ) Mean Sd. Dev. RMSE 3 monhs year years years years Random Walk Mauriy ( ) Mean Sd. Dev. RMSE 3 monhs year years years years Slope Regression Mauriy ( ) Mean Sd. Dev. RMSE 3 monhs NA NA NA NA NA 1 year years years years

32 Table 6 (Coninued) Ou-of-Sample 12-monh-Ahead Forecasing Resuls Fama-Bliss Forward Rae Regression Mauriy ( ) Mean Sd. Dev. RMSE 3 monhs year years years years Cochrane-Piazzesi Forward Curve Regression Mauriy ( ) Mean Sd. Dev. RMSE 3 monhs NA NA NA NA NA 1 year years years years Univariae AR(1)s on Yield Levels Mauriy ( ) Mean Sd. Dev. RMSE 3 monhs year years years years VAR(1) on Yield Levels Mauriy ( ) Mean Sd. Dev. RMSE 3 monhs year years years years

33 Table 6 (Coninued) Ou-of-Sample 12-monh-Ahead Forecasing Resuls VAR(1) on Yield Changes Mauriy ( ) Mean Sd. Dev. RMSE 3 monhs year years years years ECM(1) wih one Common Trend Mauriy ( ) Mean Sd. Dev. RMSE 3 monhs year years years years ECM(1) wih Two Common Trends Mauriy ( ) Mean Sd. Dev. RMSE 3 monhs year years years years Direc Regression on Three AR(1) Principal Componens Mauriy ( ) Mean Sd. Dev. RMSE 3 monhs year years years years

34 Table 7 Ou-of-Sample Forecas Accuracy Comparisons Mauriy ( ) 1-Monh Horizon agains RW agains FB 12-Monh Horizon agains RW agains FB 3 monhs * -2.43* 1 year * -2.31* 3 years * -2.18* 5 years * 10 years

35 y ( ) 1 2 Noes o Tables Noes o Table 1: We presen descripive saisics for monhly yields a differen mauriies, and for he yield curve level, slope and curvaure, where we define he level as he 10-year yield, he slope as he difference beween he 10-year and 3-monh yields, and he curvaure as he wice he 2-year yield minus he sum of he 3- monh and 10-year yields. The las hree columns conain sample auocorrelaions a displacemens of 1, 12, and 30 monhs. The sample period is 1985: :12. Noes o Table 2: We fi he hree-facor model, 1 e 3 1 e e, using monhly yield daa 1985: :12, wih fixed a , and we presen descripive saisics for he corresponding residuals a various mauriies. The las hree columns conain residual sample auocorrelaions a displacemens of 1, 12, and 30 monhs. Noes o Table 3: We fi he hree-facor Nelson-Siegel model using monhly yield daa 1985: :12, wih fixed a , and we presen descripive saisics for he hree esimaed facors ˆ 1, ˆ 2, and ˆ 3. The las column conains augmened Dickey-Fuller (ADF) uni roo es saisics, and he hree columns o is lef conain sample auocorrelaions a displacemens of 1, 12, and 30 monhs. Noes o Table 4: We presen he resuls of ou-of-sample 1-monh-ahead forecasing using eigh models, as described in deail in he ex. We esimae all models recursively from 1985:1 o he ime ha he forecas is made, beginning in 1994:1 and exending hrough 2000:12. We define forecas errors a +1 as y 1 ( ) ŷ 1/ ( ), and we repor he mean, sandard deviaion and roo mean squared errors of he forecas errors, as well as heir firs and welfh sample auocorrelaion coefficiens. Noes o Table 5: We presen he resuls of ou-of-sample 6-monh-ahead forecasing using eigh models, as described in deail in he ex. We esimae all models recursively from 1985:1 o he ime ha he forecas is made, beginning in 1994:1 and exending hrough 2000:12. We define forecas errors a +6 as y 6 ( ) ŷ 6/ ( ), and we repor he mean, sandard deviaion and roo mean squared errors of he forecas errors, as well as heir sixh and eigheenh sample auocorrelaion coefficiens. Noes o Table 6: We presen he resuls of ou-of-sample 12-monh-ahead forecasing using welve models, as described in deail in he ex. We esimae all models recursively from 1985:1 o he ime ha he forecas is made, beginning in 1994:1 and exending hrough 2000:12. We define forecas errors a +12 as y 12 ( ) ŷ 12/ ( ), and we repor he mean, sandard deviaion and roo mean squared errors of he forecas errors, as well as heir welfh and weny-fourh sample auocorrelaion coefficiens. Noes o Table 7: We presen Diebold-Mariano forecas accuracy comparison ess of our hree-facor model forecass (using univariae AR(1) facor dynamics) agains hose of he Random Walk model (RW) and he Fama- Bliss forward rae regression model (FB). The null hypohesis is ha he wo forecass have he same mean squared error. Negaive values indicae superioriy of our hree-facor model forecass, and aserisks denoe significance relaive o he asympoic null disribuion a he en percen level.

36 Figure 1 Facor Loadings 1 β 1 Loadings 0.8 Loadings 0.6 β 2 Loadings β 3 Loadings τ (Mauriy, in Monhs) Noes o Figure 1: We plo he facor loadings in he hree-facor model, y ( ) e 3 1 e where he hree facors are,, and, he associaed loadings are 1,, and, and denoes mauriy. We fix = e 1 e 1 e e

37 Figure 2 Yield Curves, Noes o Figure 2: The sample consiss of monhly yield daa from January 1985 o December 2000 a mauriies of 3, 6, 9, 12, 15, 18, 21, 24, 30, 36, 48, 60, 72, 84, 96, 108, and 120 monhs.