Working Paper / Document de travail 97-10

Transcription

1 Working Paper / Documen de ravail The Srucure of Ineres Raes in Canada: Informaion Conen abou Medium-Term Inflaion by Jim Day and Ron Lange Bank of Canada Banque du Canada

2 ISSN ISBN Prined in Canada on recycled paper

3 Bank of Canada Working Paper May 1997 The Srucure of Ineres Raes in Canada: Informaion Conen abou Medium-Term Inflaion by Jim Day and Ron Lange Deparmen of Moneary and Financial Analysis Bank of Canada Oawa, Onario, Canada K1A 0G9 (613) The views expressed in his sudy are hose of he auhors and do no necessarily represen he views of he Bank of Canada. Correspondence concerning his paper should be addressed o Ron Lange.

4 Acknowledgmens We hank Bob Amano, Jean-Pierre Aubry, Kevin Clinon, Agahe Côé, Kevin Feig, Sco Hendry, Ken Kingsbury, Dave Longworh, Jack Selody, and Pierre S-Aman for commens and help in preparing his paper.

5 Absrac This paper examines he relaionship beween he erm srucure of ineres raes and fuure changes in inflaion for Canada using a newly consruced par-value yield series. The main conclusion of he empirical work is ha he slope of he nominal erm srucure from 1- o 5-year mauriies is a reasonably good predicor of fuure changes in inflaion over hese horizons. This resul is similar o ha obained for he Unied Saes and oher counries. Resuls for models ha also include compeing indicaors of inflaion sugges ha he medium-erm srucure of ineres raes conains unique informaion abou fuure inflaion. Alhough here is addiional informaion abou fuure changes in inflaion in M2+, commodiy prices, and he oupu gap, his does no affec he predicive conen of he medium-erm srucure. Résumé Les aueurs examinen la relaion enre la srucure des aux d inérê e l évoluion fuure de l inflaion au Canada en se servan d une nouvelle série de aux de rendemen consruie au moyen des valeurs nominales. La principale conclusion qu ils iren de ce examen empirique es que la pene de la courbe des rendemens (mesurés en ermes nominaux) pour les échéances de 1 à 5 ans es un indicaeur relaivemen fiable de l évoluion fuure de l inflaion à ces horizons. Différenes éudes réalisées au suje des Éas-Unis e d aures pays abouissen à la même conclusion. Selon les résulas obenus à l aide de modèles faisan inervenir d aures indicaeurs de l inflaion, la srucure des aux d inérê à moyen erme recèle des renseignemens disincs au suje de l évoluion fuure de l inflaion. Bien que l agréga M2+, les prix des produis de base e l écar de producion aiden aussi à prévoir l inflaion à venir, la valeur informaive de la srucure des aux d inérê à moyen erme es indépendane de celle de ces variables.

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7 CONTENTS 1 Inroducion The Basic Model of he Term Srucure and Inflaion The Expecaional Inerpreaion A Causal Inerpreaion Previous Research Daa Descripion and Analysis Theoreical Par-Value Yields Inflaion Daa Properies of he Daa Empirical Resuls for he Basic Term-Srucure Model Some Economeric Issues Esimaion Resuls Augmened Term-Srucure Models of Inflaion Inflaion and Forecas Errors Financial and Real Variables Forecasing Performance Concluding Remarks Summary of he Resuls Usefulness and Limiaions as an Indicaor...17 Chars Tables References...27

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9 1 Inroducion Recen empirical research for he Unied Saes and oher counries has found a close relaionship beween he slope of he erm srucure of ineres raes beyond one year and fuure changes in medium-erm inflaion. This is in srong conras o he shorer mauriies, where changes in shor-erm ineres raes end o reflec changes in real ineres raes and he sance of moneary policy more han hey reflec changes in expeced inflaion. In his paper, we examine wheher he srong relaionship beween he medium-erm srucure of ineres raes and fuure changes in inflaion holds for Canada using a newly consruced par-value yield series. Since moneary policy acions affec economic aciviy and inflaion wih a significan lag, he changes in inflaion beyond one year are he mos relevan for policy. The heoreical framework for his research is he well-known Fisher hypohesis, which saes ha nominal ineres raes adjus one-for-one wih expeced inflaion. The Fisher hypohesis, under raional expecaions and he assumpion of a consan slope of he real erm srucure over ime, implies ha he nominal erm srucure of ineres raes should be associaed wih changes in fuure inflaion. However, he slope of he real erm srucure is no consan bu saionary, so ha he relaionship beween he nominal erm srucure and changes in expeced inflaion will no necessarily hold in any paricular shor run; ha is, real shocks can have emporary effecs on he slope of he erm srucure of ineres raes. The main conclusion of our empirical work is ha he slope of he nominal erm srucure from 1- o 5-year mauriies conains considerable informaion abou fuure changes in inflaion over hese horizons. For example, oday s 5-year rae minus oday s 1-year rae is a reasonably good predicor of he average inflaion rae over he nex five years minus he average inflaion rae over he nex year. Furhermore, his informaion abou fuure changes in inflaion is unique o he medium-erm srucure of ineres raes. When he consrains of he raional expecaions heory of he yield curve are relaxed o allow for informaion from oher indicaors of inflaion (such as M2+ and he oupu gap), he slope of he medium-erm srucure of ineres raes reains is predicive power for fuure changes in inflaion.

10 2 The following secion oulines he basic heory underlying he esimaed equaions. Secion 3 briefly reviews he recen research and Secion 4 describes he consrucion of he daa se and he properies of he daa. Secion 5 discusses some economeric problems and presens he esimaion resuls for he basic erm-srucure model. Secion 6 presens he esimaion resuls for erm-srucure models ha also include compeing indicaors of inflaion, and Secion 7 analyses he forecasing performance of he models. The final secion summarizes he esimaion resuls and discusses how he erm-srucure models may be used o monior inflaion expecaions and o forecas fuure changes in inflaion. 2 The Basic Model of he Term Srucure and Inflaion This firs par of his secion oulines he heoreical relaionship beween he erm srucure and fuure changes in inflaion based on he Fisher equaion and he raional expecaions heory of he erm srucure of ineres raes. This is he mehodology used in he lieraure and he inerpreaion used in his paper. The second par briefly oulines an alernaive inerpreaion, which is based on a simple macroeconomic framework. 2.1 The Expecaional Inerpreaion The Fisher equaion relaes nominal ineres raes, real ineres raes and inflaion, so ha expeced inflaion over k years is equal o he k-year nominal ineres rae minus he k-year real ineres rae E ( π k, ) = i k, r k,, where E is expecaions a ime, π k, is he inflaion rae from ime o +k, i k, is he k-year nominal ineres rae a ime, and r k, is he k-year (ex ane) real ineres rae. Assuming ha expecaions are raional, he realized inflaion rae over he nex k years is π k, = E ( π k, ) + ε k,, where ε k is he forecas error over he k years wih a mean value of zero., These errors will be auocorrelaed wih monhly daa because of overlapping observaions. Subsiuing ino he Fisher equaion above yields π k, = i k, r k, + ε k,.

11 3 In order o examine he informaion in he erm srucure of ineres raes, he equaion for he n-year inflaion rae is subraced from he equaion for he k-year inflaion rae o yield an equaion for he fuure change in inflaion π k, π n, = i k, i n, r k, + r n, + ε k, ε n,. Following Mishkin and ohers, 1 we assume ha he slope of he real erm srucure is consan and esimae he following regression equaion for he change in inflaion π k, π n, = α k, n + β k, n ( i k, i n, ) + µ The β k, n coefficien in his regression reveals how much informaion here is in he slope of he nominal erm srucure of ineres raes abou fuure changes in inflaion. A value of β k, n saisically differen from zero provides evidence ha he erm srucure conains significan informaion abou fuure changes in inflaion and ha slopes of nominal and real erm srucures do no move one for one wih each oher. A value of β k, n saisically differen from one indicaes ha he slope of he real erm srucure of real ineres raes is no consan over ime and ha he nominal erm srucure conains informaion abou he real erm srucure. An inerpreaion of he size of he β-coefficiens follows from he char below. I plos he heoreical relaionship beween he β-coefficien on he erm srucure under he assumpion of raional expecaions, he raio of he sandard deviaions of he change in expeced inflaion and of he slope of he real erm srucure σ, and he correlaion beween he change in inflaion and he slope of he real erm srucure ρ. 3 As he variabiliy of inflaion increases relaive o ha of he real erm srucure, he coefficien on he erm srucure increases. Consequenly, since inflaion raes are less variable han real ineres raes in he shor run, See Gerlach (1995), Jorion and Mishkin (1991), and Mishkin (1990a; 1990b; 1991). 2. The error erm for his equaion conains a sochasic elemen if he slope of he real erm srucure is no consan over ime. 3. This simple relaionship falls ou of he sandard formula for he β-coefficien in a forecas equaion under he assumpion of raional expecaions, which ses he covariance of he inflaion forecas error wih he slope of he real erm srucure equal o zero. See Gerlach (1995), Hardouvelis (1988) and Mishkin (1990a; 1990b; 1991).

12 4 coefficiens on he erm srucure can be expeced o be smaller for shorererm mauriies han for longer mauriies. Furhermore, he coefficien is larger for higher correlaions beween he change in expeced inflaion and he slope of he real erm srucure. β = ( σ 2 + ρσ) ( 1 + σ 2 + 2ρσ). ρ = 0.95 β ρ = 0.8 ρ = 0.5 ρ = 0 ρ = 1 σ For example, he coefficien on he erm srucure would be expeced o be larger for he period of rising inflaion during he 1970s han for he period since he early 1980s. The coefficien would also be larger during he earlier period because of he large oil-price shocks, which probably increased he negaive correlaion beween he change in expeced inflaion and he slope of he real erm srucure. A value above 1 for he raio of he sandard deviaions and a value close o -1 for he correlaion coefficien would give slope coefficiens of 1.5 o A Causal Inerpreaion The heoreical relaionship beween he srucure of ineres raes and changes in medium-erm inflaion presened above is based on he assumpion ha he medium-erm srucure reflecs agens raional expecaions of fuure changes in inflaion. An alernaive inerpreaion is ha

13 5 changes in he slope of he erm srucure cause fuure changes in inflaion because he slope proxies he sance of moneary policy. Recen empirical research by Cozier and Tkacz (1994) for Canada, Esrella and Hardouvelis (1991) for he Unied Saes, and Harvey (1991) and Hu (1993) for he G-7 counries finds ha he slope of he erm srucure is a good predicor of fuure real economic aciviy and inflaion. The common explanaion for his srong link is ha he spread beween shor- and long-erm raes is a good measure of he real ineres rae relaive o is equilibrium. The erm spread is viewed as an indicaor of moneary sance, since moneary acions mainly affec he shor-erm rae. The link wih real economic aciviy exiss because aggregae demand depends on changes in he slope of he real erm srucure hrough an IS-curve relaionship wih a lag of several quarers. The changes in he level of real oupu relaive o he level of poenial oupu in urn affec inflaion hrough an augmened Phillips-curve relaionship, which allows for a shor-run rade-off beween oupu and inflaion. 3 Previous Research Much of he empirical lieraure on he relaionship beween he erm srucure of ineres raes and inflaion expecaions is associaed wih he work of Mishkin and Fama. Overall, his research finds ha he erm srucure beyond one year conains considerable informaion abou fuure inflaion, while he erm srucure less han one year conains very lile informaion. Mishkin (1990a) finds for he Unied Saes ha he difference beween yields on mauriies of Treasury bills from 6-12 monhs conains some informaion abou he pah of fuure inflaion, while he erm srucure for mauriies of 6 monhs or less conains almos no informaion. The fi of his equaions is modes, predicing less han 10 per cen of he change in inflaion. For Canada, Mishkin (1991) also finds ha he mauriies of less han 12 monhs provide virually no informaion abou fuure inflaion. Applying he same mehodology o he erm srucure of mauriies from 1-5 years, Mishkin (1990b) finds significan informaion conen abou fuure inflaion, wih he erm spread predicing as much as 45 per cen of

14 6 he change in inflaion over he 3- and 4-year horizons in he 1980s. The coefficiens on he erm spreads are no significanly differen from 1, suggesing ha he nominal erm srucure over his mauriy range conains very lile informaion abou he real erm srucure. Fama (1990) similarly finds ha he spread beween he 1- and 5-year yields predics as much as 30 per cen of he 2- and 3-year changes in inflaion. Jorion and Mishkin (1991) exend he research on he medium-erm srucure o Germany, he Unied Kingdom, and Swizerland, and similarly find srong evidence ha he medium-erm srucure has significan abiliy o forecas changes in inflaion. More recenly, Gerlach (1995) exends he work for Germany ou o he 10-year mauriy and finds considerable informaion abou fuure changes in inflaion, wih spreads agains 2-year raes being more informaive han spreads agains 1-year raes. The resuls of his research on he slope of he erm srucure and medium-erm inflaion are consisen wih he work by Fama and Bliss (1987) on he raional expecaions heory of he medium-erm srucure of ineres raes. They found ha spreads for mauriies longer han one or wo years have some predicive conen for movemens in fuure ineres raes. 4 Daa Descripion and Analysis 4.1 Theoreical Par-Value Yields Previous research for Canada on he erm srucure of ineres raes used he readily available average mauriy-class yield series of 1-3 years, 3-5 years, 5-10 years, and over 10 years. 4 Since he series is based on simple averages of he bond yields wihin a given mauriy class, hey poenially can suffer from large swings wihin-class average erm o mauriy. One of he conribuions of our empirical work is he esimaion of a heoreical par-value or consan-mauriy yield series for Canada going back o The yield series were esimaed on a daa se made up of endof-monh (las Wednesday) observaions for all domesic-pay Governmen of Canada bonds. The bond price daa were originally colleced from he 4. See Browne and Manasse (1989) and Hardouvelis (1994). 5. Currenly, he Bank of Canada mainains a heoreical par-value series for all mauriies ou o 30 years on a daily frequency going back o July 1985.

15 7 Bank of Canada Review (Table G7) by Rose and Schworm (1980) for he period , and were updaed by Boohe (1987) for he period and by he Bank of Canada for he period hereafer. Hypoheical par-value yield curves were esimaed for each monh wih a spline-funcion model ha was developed by Bell Canada (1969) and modified by he Bank of Canada o esimae a daily yield curve. 6 A sample of close-o-par bonds chosen by a number of filers is used for he esimaions. 4.2 Inflaion Daa The consumer price index (CPI) is he appropriae measure of inflaion since i is he one ha is mos closely wached by he bond marke. The erm-srucure model of inflaion was esimaed over he period January 1967 o February The CPI measure excludes food and energy and, for he period since 1984, also excludes he effecs of indirec axes. 7 The esimaions use he cumulaive inflaion rae, expressed a coninuously compounded annual raes, for he forecas horizons ou o 5 years π k, = ( 100 k) [ log( P + k P )], where k= 1, 2, 3, 4, or 5 years and P is he level of he consumer price index. Even hough monhly inflaion is only known for cerain when he daa is published in he following monh, he inflaion daa for he monh was aligned wih he ineres rae daa for las Wednesday of he monh. This alignmen reflecs he view ha he marke has developed a fairly good forecas of he curren level of inflaion by he end of he monh. 8 Thus, he expeced 1-year-ahead inflaion rae for January 1982 is he change in he CPI from January 1982 o January 1983; he 2-year-ahead inflaion rae is from January 1982 o January 1984; and so on. Since he las observaion is February 1995, he las observaion for he regression wih a 2-year horizon is February 1993, and for he 5-year horizon i is February A more parsimonious specificaion developed by Echols and Ellio (1979) was used for some monhs in when here were no many observaions. 7. Inflaion daa excluding he effecs of indirec axes are only available from Preliminary esimaions wih he end-of-he monh ineres rae daa aligned wih he previous monh s inflaion daa were very similar o hose presened in he paper.

16 8 4.3 Properies of he Daa The slope of he erm srucure of ineres raes and changes in 2- o 5-year-ahead inflaion relaive o 1-year-ahead inflaion are presened in Char 1. The chars show raher large movemens in he change in inflaion for he differen forecas horizons over he sample period. The seesaw-ype movemens reflec he effecs of he oil-price shocks in 1973 and 1979 and he subsequen moneary policy accommodaion. For example, in he char for he 3-year horizon, he change in inflaion rises as he 3-year-ahead inflaion rae is affeced by he shocks and hen drops abruply as he 1-yearahead rae is affeced. The inflaion series were also affeced by a smaller negaive oil-price shock in early The chars indicae ha he slope of he erm srucure missed he big swings in inflaion ha were associaed wih he oil-price shocks, which could no have been foreseen by he marke. The chars also sugges ha he marke may have underesimaed he longer-run effecs of he shocks or, possibly, he subsequen moneary accommodaion, since he erm srucure also appears o miss he periods of persisen high inflaion following he shocks. The relaionship appears o be much closer during he recen period of more sable inflaion. Table 1 presens he augmened Dickey-Fuller -ess for uni roos for he spreads agains he 1-year ineres rae ou o 5 years and changes in inflaion relaive o he 1-year rae ou o 5 years for he full sample from The auoregressive order for each es saisic was based on a recursive -saisic procedure wih a 5 per cen criical level for choosing he las lag. The ADF saisics for he full sample sugges ha he slope of he nominal erm srucure and changes in inflaion follow a mean-saionary process a he 5 per cen level. This also suggess ha he real erm srucure was mean-saionary over he sample period. 9. Mone Carlo evidence suggess ha ADF -ess perform quie well under he null even when he process generaing he error includes a large (negaive) moving-average componen (Davidson and Mackinnon 1993).

17 9 5 Empirical Resuls for he Basic Term-Srucure Model 5.1 Some Economeric Issues This secion discusses he correcion for he serial correlaion of he forecas errors menioned earlier and he problem of small-sample bias due o he large daa overlap. The problem of serial correlaion of he error erms in hese regression equaions arises because he monhly frequency of he daa is finer han he forecas horizon for inflaion and he forecas errors are realized only a he end of he forecas horizon. Alhough raional expecaions rule ou any correlaion beween he forecas error of inflaion over he k years ( ε k ) and, he variables enering he condiioning informaion se a ime, hey do no rule ou auocorrelaion in he forecas errors realized from +1 o +k since hey do no ener he condiioning se. Consequenly, hese error erms are likely o follow a moving-average process of order (12k-1) wih monhly daa. 10 This problem is correced in he esimaions wih he Newey and Wes (1987) adjusmen mehodology. The correced sandard errors will in general lead o correc inference asympoically. However, as he degree of overlapping increases wih he longer forecas horizons, he number of fully independen observaions diminishes. Consequenly, differences beween finie-sample disribuions and asympoic disribuions may be large when he daa have a large moving-average componen, 11 ranging from 23 periods wih monhly observaions for a 2-year forecas horizon o 59 periods for a 5-year horizon. Jorion and Mishkin (1991) idenified he possible magniude of he small-sample bias using Mone Carlo es simulaions. They found ha a sandard 5 per cen criical value from he asympoic disribuion wrongly rejecs he null hypohesis on β -coefficiens abou 20 per cen of he ime for he 2-year forecasing horizon and abou 50 per cen of he ime for he 5- year horizon. They found ha he criical values for -ess from Mone Carlo simulaions on he β -coefficien a he 5 per cen level range from The error erm, µ, could be serially correlaed for lags greaer han (12k-1) if he slope of he real erm srucure is no consan. 11. See Richardson and Sock (1989), and Huizinga and Mishkin (1984).

18 10 for k=2 years o 5.0 for k=5 years. These criical values are based on esimaes from four counries on he same model as above for a 15- year period. In he esimaions presened below, he -saisics on he esimaed slope coefficiens for he recen sample period are considerably larger han he criical values ha are calculaed by Jorion and Mishkin (1991). 5.2 Esimaion Resuls The esimaes of he equaions for he fuure change in inflaion for horizons ou o 5 years for he period 1967 o are presened in Table 2. The resuls indicae ha here is informaion in he medium-erm mauriy srucure abou fuure inflaion, wih he R 2 rising from 0.09 for he 2- year horizon o 0.33 for he 5-year horizon. This is in sharp conras o he lack of predicive power in money marke raes for Canada (Mishkin 1991). The coefficiens on he erm spreads are all posiive as expeced and increase wih he erm o mauriy, a resul similar o ha found for oher counries. The coefficiens rise from 0.87 for he 2-year horizon o 1.73 for he 5-year horizon. The -saisics based on Newey-Wes sandard errors sugges ha β-coefficiens are significanly greaer han zero for all horizons. Furhermore, he β-coefficiens are no significan differen from one for all horizons, suggesing ha he slope of he real erm srucure was consan for he sample period from The consan erm is insignifican for all horizons, suggesing ha real raes over he 1- o 5-year horizon were no saisically differen over his period; ha is, he real erm srucure was fla over his mauriy range. The rolling Chow ess on he forecas residuals presened in Char 2 indicae parameer insabiliy in he early 1970s and hrough mos of he 1980s for all horizons. The insabiliy appears o be associaed mainly wih oil-price shocks. These shocks were concenraed over a few quarers and riggered higher levels of inflaion for a number of years. The price shocks were no forecasable and he failure of he erm srucure o predic hem is expeced. In order o exclude he errors associaed wih he unforecasable oil-price shocks, slope dummy variables were inroduced ino he forecas equaions. The inroducion of oil-price dummy variables follows in spiri Perron (1991), who argues ha oil price shocks are exogenous and no a realizaion of he underlying daa-generaing mechanism of a series. The oil-

19 11 price dummy variables ake ou he bias due o hese he unforeseen shocks so ha he erm srucure can more closely reflec inflaion expecaions. Inspecion of he price of oil suggesed ha he firs shock covered he period from abou 1973M06 o 1974M12 and he second from 1979M05 o 1980M03. The dummy variables equal one when inflaion raes from 1- o 5-years ahead fall wihin hese periods. The resuls presened in he final panel in Table 2 indicae ha he β-coefficiens are now noiceably lower for all horizons and much closer o one for he 3- o 5-year horizons. 12 The forecas equaions were re-esimaed for a more recen period in order o ascerain he informaion conen in he erm srucure of ineres raes when inflaion and/or real ineres raes were no subjeced o large and unforeseen shocks. The resuls presened in Table 3 are for he oal CPI, excluding food, energy and he effecs of indirec axes, for he period 1984 o The coefficiens are highly significan and closer o one across all horizons. 14 The R 2 s peak a 0.59 a he 3- and 4-year horizons, much higher han he peak of 0.33 for he full sample. The negaive consan erms are now significan for he hree longer horizons, suggesing ha erm premiums increased wih he erm o mauriy over he recen period. 6 Augmened Term-Srucure Models of Inflaion 6.1 Inflaion and Forecas Errors Alhough we find evidence of a one-o-one relaionship beween he nominal erm srucure and changes in inflaion for Canada, he relaionship will no necessarily hold in any paricular shor run because he real erm srucure over his mauriy range is no consan, bu saionary; ha is, sochasic disurbances o real raes can have emporary effecs on he nominal erm srucure of ineres raes. In addiion, he one-o-one relaionship may no hold if he marke misses imporan informaion abou inflaion. An aemp o capure deviaions in he one-o-one relaionship beween he erm srucure and changes in inflaion is made by inroducing 12. The firs dummy variable was significan for all horizons and he second one for he wo longer horizons. 13. The resuls for he oal CPI and he CPI excluding food and energy are quie similar. 14. Alhough he slope of he erm srucure esed I(1) over his period, he slope and he change in inflaion were coinegraed for all horizons, wih slope coefficiens no saisically differen from one. These resuls are available upon reques.

20 12 addiional informaion and shor-run dynamics ino he basic equaions. As a preliminary es of he exen o which informaion oher han he nominal erm srucure may be imporan (perhaps because of he sochasic porion of real raes and/or because he marke does no incorporae all relevan informaion in predicing inflaion), he curren level of inflaion and lagged forecas errors are inroduced ino he erm-srucure equaions. The erm-srucure equaion wih he curren level of inflaion is π k, π 1, = α k + β k ( i k, i 1, ) + γ k π + ε k,, where π is he year-over-year inflaion rae a ime. The curren level of he inflaion rae also proxies lagged inflaion in his forecas equaion. The esimaes presened in Table 4 indicae ha he curren level of inflaion capures imporan informaion abou he change in medium-erm inflaion ha is missed by he erm srucure spreads. The R 2 s are much higher for all forecas equaions and he -saisics on he level of inflaion are highly significan. The size and significance of he β-coefficiens for hose horizons are no maerially affeced by he presence of he level of inflaion. The negaive sign suggess ha he level of inflaion has a endency oward mean reversion over hese horizons. The basic equaion wih he forecas errors realized a ime is π k, π 1, where e k, k is he inflaion forecas error for he k-year inflaion rae from ime -k ha is realized a ime minus he 1-year inflaion rae from ime -k. The resuls presened in Table 5 indicae ha he informaion conen in he forecas errors for he 3- and 4-year horizons increased he predicive conen of he erm-srucure equaions for hose mauriies by around 50 per cen. As wih he level of inflaion, he nominal erm srucure reains is predicive conen for fuure changes in inflaion. 6.2 Financial and Real Variables = α k + β k ( i k, i 1, ) + γ k e k k +, ε k, In his secion, compeing indicaors of inflaion are inroduced ino he basic equaions in an anoher aemp o idenify relevan missing,

21 13 informaion. 15 These augmened regressions also allow for an assessmen of he marginal predicive conen of he medium-erm srucure of ineres raes abou inflaion. The compeing financial variables include he M1 gap creaed from Hendry s (1995) monhly long-run demand-for-money funcion for M1 (m1gap), M2+ (m2+) as in Clinon (1995), he Torono Sock Exchange Index (se), as a measure of asse prices, and he Bank of Canada Commodiy Price Index (bcpi), as a measure of commodiy prices. The only real variable considered is he oupu gap (ygap), defined as he residual from a regression of monhly oupu agains a linear and a quadraic ime rend. GDP a facor cos was used since i is he only aggregae available on a monhly basis. The oupu gap is included in he basic model in order o assess he robusness of he expecaional inerpreaion aken in he paper. The causal inerpreaion of he relaionship beween he erm spread and he change in inflaion sresses ha he empirical link exiss because he erm spread affecs fuure inflaion via he shor-run rade-off agains he oupu gap. In his case, he presence of he oupu gap would be expeced o affec he predicive conen of he nominal erm spreads. The augmened equaions add oher variables as follows π k, π 1, = α k β k ( i k, i 1, ) γ k, j ( Z j, Z j, p ) + ε k,, + + where he Zs are financial and real variables. These variables correspond o he levels of he M1 gap and he oupu gap, and o he growh raes for M2+, asse and commodiy prices. Using he levels of he gap variables is consisen wih an acceleraionis view of inflaion. These compeing indicaor variables are expressed in difference form since he erm-srucure equaions are used o predic he k-year-ahead inflaion rae relaive o he 1- year-ahead rae. The values for p in are chosen on he basis of fi. The difference beween he 3- and 1-year growh of he indicaor variables provided he bes fi for he wo shorer horizons and he 2-year j Z j, p 15. Specificaions ha included shor-run dynamics, such as an error-correcion mechanism, did no noiceably improve he in-sample fi of he models.

22 14 difference generally did beer for he longer horizons. 16 The resuls for he curren level of he gaps are slighly beer han hose ha also included lagged gaps.the esimaion resuls are quie similar across forecas horizons. The resuls for he augmened model of he change in 3-year-ahead inflaion relaive o he 1-year-ahead inflaion rae are presened in Table 6. The erm-srucure spreads mainain heir predicive conen in he presence of he compeing indicaors of inflaion in virually all models for all forecas horizons. In addiion, he size and significance of he β -coefficiens are no generally affeced, wih virually all of hem remaining no saisically differen from one. A significan amoun of addiional informaion abou he fuure change in inflaion is conained in he difference in he growh of M2+ for all horizons and of commodiy prices for he 2- o 4-year horizons. The oupu gap conains addiional informaion for he 3- and 5- year horizons. The difference in he growh of asse prices and he M1 gap are insignifican across all horizons. In general, he improvemen in fi due o he inclusion of he compeing indicaors of inflaion is similar o ha resuling from adding he curren level of inflaion. The improvemen suggess ha he marke a imes misses informaion abou fuure inflaion ha is conained in hese indicaors. The improvemen may also reflec ha hese variables capure flucuaions in he real raes. 7 Forecasing Performance The esimaion resuls in he previous secions focussed on in-sample goodness-of-fi. In his secion, rolling regression ess are used o assess he abiliy of he erm srucure of ineres raes o forecas ou-of-sample. The firs se of regressions focusses on he sabiliy of he forecas errors of he medium-erm srucure. The second se assesses he incremenal forecasing abiliy of he erm srucure relaive o compeing inflaion indicaors. 16. The fi of he 2- and 3-year growh specificaions is noiceably beer han he difference beween he curren 1-year rae and he 3-year lag of he 1-year rae. However, he addiional informaion in he 2- and 3-year growh raes is relaively small.

23 15 The sabiliy of he forecas errors is assessed by calculaing rolling roo-mean-squared errors (RMSE). The period covered by each RMSE corresponds o he lengh of he forecas horizon. 17 This rolling es procedure reflecs he way ha bond marke paricipans wih specific horizons would likely evaluae heir forecas errors a each poin in ime. Char 3 presens he ou-of-sample RMSEs for forecass of 2- o 5- year-ahead inflaion relaive o he 1-year-ahead inflaion rae. The daes on he horizonal axis correspond o he dae of he las forecas error used in he calculaion, and for a 12k-monh horizon he RMSE rolls over k years. For example, he firs RMSE repored in he second panel for he 3-year horizon is for 36 consecuive 3-year-ahead forecass and is daed 1982M12. The rolling RMSEs indicae ha he forecasing performance of he erm srucure has been relaively more sable and accurae since he lae 1980s, especially for he 2- and 3-year horizons. The incremenal forecasing abiliy of he longer-erm srucure spreads is assessed by calculaing RMSEs from recursive regressions of he basic erm-srucure model and an augmened model ha also includes eiher he curren level of inflaion or he difference in he growh of M2+. Boh variables were saisically significan across all forecas horizons. The recursive mehod involves making a k-year-ahead forecas, where k = 2, 3, 4, and 5, and hen sequenially updaing he sample by one monh, re-esimaing and forecasing k years ahead. The ou-of-sample RMSEs for he recursive regressions are presened in he able below. The RMSEs indicae ha he errors for boh he basic and augmened indicaor models increase wih he forecas horizon, especially beyond he 3-year horizon. Despie he beer in-sample fi, he augmened model wih he curren level of inflaion has slighly poorer forecas power for he wo 17. The models for all forecas horizons were esimaed for a 10-year period from 1967 o The esimaed coefficiens from hese regressions were used o generae a k-yearahead forecas wih he k-mauriy erm spread saring in 1977M01. The sample was hen updaed by one monh, he model re-esimaed, and a second forecas was generaed from 1977M02. The process of updaing he esimaion and forecas was coninued unil here were 12k-consecuive, k-year-ahead forecass. A RMSE was hen calculaed for his se of 12k-consecuive forecass. Subsequen ses of k-year-ahead forecass are generaed and RMSEs calculaed by dropping he firs forecas in he previous se and adding a forecas for anoher monh.

24 16 shorer horizons. 18 The model wih he growh of M2+ has slighly beer ou-of-sample fi for he 3-year horizon. Roo Mean Squared Errors for 1989M01 o 1995M02 Indicaor Models i k, i 1, π 2, π 1, π 3, π 1, π 4, π 1, π 5, π 1, : β= i k, i 1, i k, i 1,, π , : β= i k, i 1, π i k, i 1,, m , m2+: β= i k, i 1, Variable m2+ is he difference beween he 3- and 1-year growh raes for he 2- and 3-year horizons and he difference beween he 2- and 1-year raes for he longer horizons. Neverheless, here is a marked improvemen in he forecasing power of he basic erm-srucure models beyond he 2-year horizon when he β-coefficiens on he erm spreads are resriced o equal one. The incremenal forecasing power of M2+ a he 3-year horizon disappears wih he resricion. Since he coefficiens are noiceably greaer han one (bu no saisically differen) for he full sample period, his resricion reduces he volailiy of he erm srucure forecass for he recen period of more sable inflaion Concluding Remarks 8.1 Summary of he Resuls The esimaion resuls for he period from 1967 o 1995M02 sugges ha a relaively close relaionship beween he slope of he erm srucure 18. The beer in-sample fi and poorer ou-of-sample performance may be a sympom of overfiing. 19. The slope coefficiens ha are esimaed on he sample period since 1984 are much smaller and quie close o one, especially for he 3- o 5-year horizons (see Table 3).

25 17 and fuure changes in inflaion exiss in he Canadian daa. The erm-srucure equaions predic up o one-hird of he variaion in he changes in inflaion ou o 5 years. The forecass based on esimaions for he period from 1984 predic as much as 59 per cen of he variaion in he fuure changes in inflaion. The coefficiens are highly significan and are no significanly differen from one for boh periods as prediced by he raional expecaions heory. Resuls for models ha also include compeing indicaors of inflaion sugges ha he medium-erm srucure of ineres raes conains unique informaion abou expeced inflaion. Alhough here is addiional informaion abou fuure changes in inflaion in M2+, commodiy prices and he oupu gap, his does no affec he predicive conen of he medium-erm srucure. The ou-of-sample performance of he erm-srucure models was assessed on a sample period ha includes he effecs of he large oil-price shocks in he 1970s. These shocks and he subsequen moneary policy accommodaion were no anicipaed by he marke. Consequenly, he forecasing performance is inferior o ha of models esimaed over a more recen period. Rolling ou-of-sample forecass indicae ha he addiional informaion in compeing indicaors does no noiceably improve he forecas performance. Rolling roo-mean-squared errors ha correspond o he lengh of each forecas horizon indicae ha he forecas values have been generally more sable and accurae since he lae 1980s. 8.2 Usefulness and Limiaions as an Indicaor The overall conclusion from he esimaion resuls is ha a persisen change in he slope of he medium-erm srucure srongly indicaes a change in expeced inflaion. Thus, if a seeper yield curve persiss, inflaion will likely rise several years in he fuure by he amoun of he seepening. A negaively sloped erm srucure ha persiss indicaes ha inflaion will decline. The char below plos he acual values of he change in inflaion from 1 o 3 years from 1988 o lae 1995 and he forecas values for boh unresriced and resriced versions of a erm-srucure model ou o The models are based only on he spread beween he 3- and 1-year yields. The

26 18 ou-of-sample values are generaed from recursive regressions beginning in 1985 ha are re-esimaed afer each monhly forecas. The inercep and slope coefficiens in he unresriced model are based on esimaions on he sample from In he resriced model, he slope coefficien is consrained o equal one and he (negaive) inercep erm is re-esimaed. Acual and Forecas Change in Inflaion (3-year minus 1-year inflaion rae) Acual Change in Inflaion Forecas from Unresriced Term Srucure Equaion Forecas from Resriced Term Srucure Equaion The acual and forecas values from he models sugges ha he erm srucure from 1 o 3 years racks he change in inflaion during periods when inflaion is relaively sable as well as i does during periods when i is changing rapidly. The erm srucure in 1995 was predicing relaively sable inflaion hrough 1998, while he erm srucure in 1996 was predicing inflaion o accelerae in he coming years. The resricion on he slope coefficien noiceably improves he ouof-sample performance of he erm srucure over he period, probably because of he absence of large supply-side shocks.

27 The medium-erm srucure, however, has some limiaions as a predicor of inflaion. Since he erm spreads from 1- o 5-year mauriies are no assumed o have a causal link wih inflaion, hey canno be expeced o be useful when he economy faces large unforeseen shocks. In addiion, when he erm srucure is responding o hisorically aypical informaion and o aypical expecaions abou changes in real reurns or aypical erm premiums, i migh be less useful for predicing variaions in inflaion. 19

28 Char 1 Change in Inflaion and Slope of he Term Srucure of Ineres Raes 2-year Horizon year Horizon year Horizon year Horizon Change in Inflaion Term Spread

29 Char 2 Rolling Chow Tess 2-year Horizon year Horizon year Horizon year Horizon

30 Char 3 Rolling RMSEs 2-year Horizon 3-year Horizon 4-year Horizon 5-year Horizon

31 23 Table 1 Uni Roo Tess for he Slope of he Term Srucure of Ineres Raes and he Change in Inflaion Series W/O Trend W/ Trend M02 i 2, - i 1, -3.75* -3.74* i 3, - i 1, -3.14* i 4, - i 1, -3.24* i 5, - i 1, -3.28* * -3.57* π 2, π 1, * -3.91* π 3, π 1, π 4, π 1, * -3.94* * -3.75* π 5, π 1, * Rejecion of he null hypohesis of non-saionariy a he 5 per cen level using he asympoic criical values of 2.86 and 3.41 for he uni-roo es wihou and wih rend, respecively.

32 24 Table 2: 1967 o M02 π k, π 1, = α k + β k ( i k, i 1, ) + ε k, Horizon (k) 2-year 3-year 4-year 5-year α (-sa) (0.54) (1.11) (1.38) (1.95) β (-sa) 0.87 (2.27) 1.41 (3.31) 1.66 (4.13) 1.73 (4.60) RBAR SEE β (p-value) 1.0 (0.73) 1.0 (0.34) 1.0 (0.10) 1.0 (0.05) : β + β i D β i i (-sa) (1.88) (2.85) (4.29) (3.14) β + β i D : β i i (p-value) 1.0 (0.10) 1.0 (0.64) 1.0 (0.21) 1.0 (0.51) The D i s are oil-price dummy variables ha equal one when inflaion raes ou o 5-years ahead fall wihin he periods 1973M M12 and 1979M M03. T-saisics are based on Newey-Wes sandard errors of he coefficiens. P-values for he resricion β=1 are disribued as χ 2 ( 1). Table 3: 1984 o M02 Horizon (k) 2-year 3-year 4-year 5-year α (-sa) β (-sa) RBAR 2 SEE (1.50) 0.66 (3.94) (3.71) 0.87 (10.66) (3.15) 0.90 (16.06) (3.55) 0.85 (7.44) β (p-value) 1.0 (0.05) 1.0 (0.13) 1.0 (0.09) 1.0 (0.18) Coefficien -saisics are based on Newey-Wes sandard errors of he coefficiens. P-values for he resricion =1 are disribued as ( ). β χ 2 1

33 25 Table 4: 1967 o M02 π k, π 1, = α k + β k ( i k, i 1, ) + γ k π + ε k, Horizon (k) 2 - year (0.54) 0.51 (1.88) 3 - year (1.11) 0.97 (2.55) 4 - year (1.38) 1.19 (2.47) 5 - year (1.95) 1.03 (1.66) α β γ (-sa) (-sa) (-sa) 0.87 (2.27) 0.70 (2.24) 1.41 (3.31) 1.12 (3.97) 1.66 (4.13) 1.35 (5.96) 1.73 (4.60) 1.40 (6.55) (2.47) (4.12) (3.88) (2.78) R Table 5: o M02 π k, π 1, = α k + β k ( i k, i 1, ) + γ k e k k +, ε k, 2 - year (0.54) (0.45) 3 - year (1.11) (1.69) 4 - year (1.38) (2.85) 5 - year (1.95) (2.43) 0.87 (2.27) 0.89 (2.21) 1.41 (3.31) 1.10 (3.19) 1.66 (4.13) 1.43 (4.77) 1.73 (4.60) 1.61 (3.79) 0.06 (0.41) (4.12) (3.83) (0.68)

34 26 Table 6 Indicaor Model Regressions Sample: M02, = α 3 β 3 ( i 3, i 1, ) γ 3j ( Z j, Z j, 3 ) + ε 3,, π 3, π where Z is he curren level of a gap or a 1-year growh rae and, for growh variables, Z 3 is he 3-year growh rae a ime. j Model m1- α β ygap m2+ se bcpi π gap τ R 2 1 (-sa) (1.22) 0.94 (3.28) (-sa) 0.35 (1.19) 0.74 (3.57) (1.64) (-sa) (1.33) 1.13 (3.59) 7.69 (2.47) (-sa) (0.97) 1.11 (4.28) 0.19 (3.27) (-sa) (1.31) 0.85 (3.82) 0.01 (0.66) (-sa) (1.02) 0.87 (3.82) 0.05 (2.97) (-sa) 0.88 (1.79) 0.80 (3.56) (2.98) (-sa) 0.65 (1.55) 0.78 (3.06) (0.54) (0.44) 0.07 (0.97) (0.10) 0.04 (2.33) 0.11 (1.55) 0.36

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37 Bank of Canada Working Papers Reconsidering Coinegraion in Inernaional Finance: Three Case Sudies of Size Disorion in Finie Samples M.-J. Godbou and S. van Norden 97-2 Fads or Bubbles? H. Schaller and S. van Norden 97-3 La courbe de Phillips au Canada: un examen de quelques hypohèses J.-F. Fillion and A. Léonard 97-4 The Liquidiy Trap: Evidence from Japan I. Weberpals 97-5 A Comparison of Alernaive Mehodologies for C. Dupasquier, A. Guay Esimaing Poenial Oupu and he Oupu Gap and P. S-Aman 97-6 Lagging Produciviy Growh in he Service Secor: Mismeasuremen, Mismanagemen or Misinformaion? D. Maclean 97-7 Moneary Shocks in he G-6 Counries: Is There a Puzzle? B. S.C. Fung and M. Kasumovich 97-8 Implemenaion of Moneary Policy in a Regime wih Zero Reserve Requiremens K. Clinon 97-9 Mesures du aux d inflaion endenciel T. Laflèche The Srucure of Ineres Raes in Canada: Informaion Conen abou Medium-Term Inflaion 1996 J. Day and R. Lange 96-9 Does Inflaion Uncerainy Vary wih he Level of Inflaion? A. Crawford and M. Kasumovich Uni-Roo Tess and Excess Reurns M.-J. Godbou and S. van Norden Avoiding he Pifalls: Can Regime-Swiching Tess Deec Bubbles? S. van Norden and R. Vigfusson The Commodiy-Price Cycle and Regional Economic Performance in Canada M. Lefebvre and S. Poloz Speculaive Behaviour, Regime-Swiching and Sock Marke Crashes S. van Norden and H. Schaller L endeemen du Canada e ses effes sur les aux d inérê réels de long erme J.-F. Fillion A Modified P*-Model of Inflaion Based on M1 J. Aa-Mensah Earlier papers no lised here are also available. Single copies of Bank of Canada papers may be obained from Publicaions Disribuion, Bank of Canada, 234 Wellingon Sree Oawa, Onario K1A 0G9 WWW: FTP: publicaions@bank-banque-canada.ca hp:// fp.bank-banque-canada.ca (login: anonymous, o subdirecory /pub/publicaions/working.papers/)