WORKING PAPER SERIES WORKING PAPER NO 11, 2004 ESI TESTING THE EXPECTATIONS HYPOTHESIS WITH SURVEY DATA. Ulf Luthman.

WORKING PAPER SERIES WORKING PAPER NO 11, 2004 ESI TESTING THE EXPECTATIONS HYPOTHESIS WITH SURVEY DATA wih an inroducion of an analysis of surveyed ineres raes by Ulf Luhman hp://www.oru.se/esi/wps SE-701 82 Örebro SWEDEN ISSN 1403-0586

CONTENTS I II III IV V VI INTRODUCTION PRESENTING AND EXAMINING THE SURVEY DATA 1 The news agency Dire s measuremen of he expeced ineres raes 2 The predicive power of he mean surveyed ineres rae 2.1 Presening he survey daa and he mean surveyed forecas errors 2.2 Do he survey forecas erros behave raional? 2.3 The diversiy in he surveyed answers THEORY - THE EXPECTATIONS HYPOTHESIS TESTS OF THE EXPECTATIONS HYPOTHESIS WITHOUT A DIRECT MEASURE OF EXPECTATIONS 1 Foreign sudies of he EH 2 Swedish sudies of he EH TESTING THE EXPECTATIONS HYPOTHESIS WITH SURVEY DATA 1 Decomposiion of he forward premium 2 Decomposiion of he β-coefficien 3 Comparison wih Froo s resuls and ess made by oher auhors SUMMARY AND CONCLUSIONS FIGURES AND TABLES APPENDIX REFERENCES 2

I INTRODUCTION This paper is an empirical sudy of he properies of he erm srucure of ineres raes. I ess saisically o wha exen he forward ineres raes ha are implici in he erm srucure can be used as a forecas of he fuure ineres raes, i.e. i ess wha is nown as he expecaions hypohesis of he erm srucure of ineres raes (EH 1. I is esed in a grea number of aricles of Modigliani and Shiller (1973, Shiller (1979, Shiller, Campbell and Schoenholz (1983, Friedman (1979, Fama (1984, Mariw (1986 and Campbell and Shiller (1987. Gerlach and Smes (1995 esed he EH for 17 counries a he shor end of he mauriy srucure. In abou half of cases (including Sweden hey could no rejec he EH. USA and Ausria are wo counries where he EH does no hold. On UK daa, MacDonald and Macmillan (1994 do no find suppor for he EH. In daa from he USA i is ofen found ha forward raes are worse predicors of fuure ineres raes han he naive maringale mehod - ha he fuure ineres rae is he same as he ineres oday 2. The null hypohesis in mos ess of he expecaions heory is a join hypohesis - ha he expecaions are raional, and ha he ineres rae differenials beween differen mauriies depend on expeced ineres rae changes. The explanaion of he rejecions of he EH varies from auhor o auhor. Mos auhors argue ha he rejecions are a consequence of ime-varying ris premia. Among hose we find Sarz (1982, Maniw and Miron (1986. Auhors who rejec he EH because of his explanaion have ofen sudied ineres raes wih shor mauriies. Ohers sugges ha he rejecion is due o he mare s over- or underreacion in relaion o wha is raional. Among hose Shiller (1981, Campbell and Shiller (1984, Maniw and Summers (1984 can be menioned. Their ess suppor he idea ha he rejecion of he EH depends on underreacion of long ineres raes o shor ineres raes' changes. In Sweden, he EH has been esed by Hörngren (1986, Edahl and Warne (1990, Dahlquis and Jonnson (1994 and Hördahl (1994. The common feaure of hese sudies is ha, wih excepion of Hörngren s sudy, hey do no rejec he EH. Edahl and Warne use a VAR-model and Hördahl an ARCH-M model. As his sudy uses he same model as Hörngren and Dahlquis and Jonsson heir resuls will be more carefully compared o he resuls in his sudy. In esing he EH Froo (1989 goes one sep furher han all he oher auhors. He performs he sandard es of wheher he differenial beween he implici forward rae and oday s spo rae is an unbiased predicor of fuure ineres rae changes 3. By decomposing his spread s bias ino one componen aribuable o a ris premium and one componen aribuable o a sysemaic expecaion error he maes i possible o measure o wha exen a ime-varying ris premium and an expecaion error give he greaes conribuion o he rejecion of he EH. To be able o do he 1 In he lieraure, he EH is someimes called he efficiency-es. However, he EH may or may no hold in an efficien mare. 2 However, Hsu and Kugler (1997 show ha he EH canno be rejeced on U.S. daa for he las of he four subperiods sudied, 1987-1995. 3 Apar from he sandard es, in he lieraure here are several varians of his model o es he expecaions hypohesis. Ofen eiher he change in he long-erm ineres rae or he realized excess holding reurn is used as he dependen variable, and he spread of he forward premium above he long-erm rae is used as he regressor. For a more complee se of univariae regression ess of he expecaions hypohesis, see Froo (1987. 3

decomposiion of he spread s bias ino he wo componens a ime series of he expeced ineres rae is required. In his respec Froo uses a survey daa se of expeced ineres raes. A he end of each quarer from mid-1969 o he end of 1986 a survey insiue 4 has ased financial mare paricipans abou heir expecaions on ineres raes on hree-monh T-bills, hree-monh Eurodollar deposis and bonds wih longer mauriies. Each responden was ased o give his or her expecaion of he level of each of he raes in hree and six monhs' ime. This sudy uses he sandard es of he EH, which maes i is possible, direcly, o compare he resuls in his sudy wih he resuls of ess made by oher auhors - in he firs place Froo, Hörngren and Dahlquis and Jonsson. This sudy is no moivaed by a concern for ineres rae forecasing per se, bu i shows how reliable he EH is in forecasing ineres raes. If his sudy shows ha he EH canno be rejeced, of course, i gives he acors in he money mare a good mehod for forecasing fuure ineres raes wih differen mauriies. The sudy follows Froo s mehod of decomposing he spread s bias ino he wo componens - one aribuable o he ris premium and one aribuable o he sysemaic expecaion error. In his respec his sudy deepens previous ess of he EH on Swedish daa - paricularly Hörngren and Dahlquis and Jonsson s ess. The decomposiion is he exra conribuion his paper gives he Swedish lieraure of ess of he EH. The resuls of his decomposiion migh indicae a sriing difference in he imporance of he compeing explanaions of he U.S.- and Swedish money mares 5. If our es of he EH on Swedish daa shows ha he EH canno be rejeced, he decomposiion is sill ineresing o do. I is possible ha he decomposiion of he forward premia ino is wo componens shows ha boh he ris premium and he sysemaic expecaion error componens are boh quie large - bu show differen signs - so ha he EH is no rejeced. In spie of he grea imporance which has been aribued o ineres rae expecaions here have been very few aemps, in Sweden, o direcly invesigae and measure he expecaions of fuure ineres raes ha differen caegories are expeced o have. The firs survey in Sweden was conduced by he news agency Dire. Since Sepember 1992 i regulary surveyed mare acors abou heir expecaions of he fuure ineres rae. Hence, Sweden has for a long ime been wihou an empirical basis o judge many of he argumens ha have been carried forward o value moneary policy and invesmen decisions made by porfolio managers. However, here are mehods o indirecly measure he mare s ineres rae expecaions. The basis for hese mehods is raional expecaions - ha expecaions are no sysemaically wrong and use all available informaion. One example of an indirec measure of he expeced ineres rae is he yield curve - where he implici forward rae is regarded as a measure of he expeced ineres rae. The news agency Dire s survey of he expeced ineres rae was he firs of his ind in Sweden. Therefore a presenaion and analysis of he ime series iself is moivaed. The purpose of he firs par of he paper is o presen and analyse he surveyed ineres raes. We presen he mean surveyed answers and analyse heir power o predic he fuure ineres rae. The quaniaive analysis uses he level of he answers - he mean - o measure he capaciy o forecas fuure ineres raes. In addiion o he mean he maximum and minimum surveyed answers are announced. This gives us he opporuniy o map he degree of diversiy in he survey answers. In heoreical and empirical sudies of he erm srucure of ineres raes, as a rule, i is assumed ha he sudied group has idenical ineres rae expecaions. This quesion is discussed in relaion o he resuls from he survey. 4 Froo uses survey daa from he Goldsmih-Nagan Bond and Money Mare Leer publicaion. 5 According o Froo (1989 he rejecion of he EH on U.S. daa is due o a posiive ris premium. 4

Consequenly, he purpose of he paper is o hrow ligh on several quesions. Some of hem have no been analysed earlier on Swedish daa. The paper is organized as follows. In secion II we presen and analyse, empirically, he answers o he survey by he news agency Dire. Secion II:1 describes he agency s procedure in surveying financial-mare paricipans. Secion II:2 analyses he mean surveyed ineres raes and ess he predicive power of he mean surveyed daa se. In secion II:2.1 we presen he survey daa and he mean forecas errors and in 2.2 we answer he quesion if he surveyed forecas errors behave raional. We measure o wha exen he mean survey ineres raes predics he acual fuure ineres raes. Secion II:2.3 analyses he uncerainy in he survey answers where he uncerainy is measured by he difference beween he maximum and minimum surveyed answers. Secion III is devoed o disinguish beween differen forms of he EH. In secion IV we presen how o es he EH wihou a direc measure of expecaions - he sandard es of he EH. We also presen he resuls of his es. Secion IV:1 conains a comparison wih oher ess of he EH on foreign daa and in secion IV:2 we do Swedish comparisons of ess of he EH. Secion V is devoed o he EH wih survey daa. In secion V:1 we show how he so-called forward premium is decomposed and secion V:2 performs Froo s (1989 decomposiion of he regression coefficien of he difference beween he forward rae and oday s spo rae ino one componen aribuable o a ris premium and one componen aribuable o a sysemaic expecaion error. This secion is also devoed o he empirical resuls of he decomposiion of he β-coefficien ino is wo componens. In secion V:3 we mae comparisons wih Froo s and oher auhors who have made he decomposiion of he β-coefficien ino a ime-varying ris premium and an expecaion error. The paper ends wih conclusions and a summary in secion VI. II PRESENTING AND EXAMINING THE SURVEY DATA 1 The news agency Dire s measuremen of he expeced ineres raes Before we presen and analyse he Swedish survey daa we wan o presen survey daa sudies from oher counries. Froo (1989 is already menioned o be he pioneer o perform he sandard es of he EH and mae he decomposiion of he differenial beween he implici forward rae and oday s spo rae. MacDonald/Macmillan (1994 and Bachelor (1990 perform exacly he same ess as Froo wih UK and U.S. daa, respecively. To es he unbiasedness of survey daa Kim (1996 and Friedman (1980 use Ausralian and U.S. survey daa, respecively. Ferderer/Shadbegian (1993 and Prell (1973 sudied he survey forecas errors measured as roo mean square error and mean average error. The resuls from hese sudies will be compared o he resuls in his sudy. Tradiionally, economiss have been criical o resuls from surveys 6. One argumen is ha he resuls from survey do no describe decisions and acions ha he respondens acually choose o realize. In his respec he respondens rue comprehension is no revealed and hence he expecaions canno be measured in a correc way. So, he disadvanage is ha he respondens never can be pu in a compleely realisic decision maing conex where a correc answer is rewarded and a wrong guess is punished. Anoher argumen is ha differen survey respondens have differen beliefs. If here is one single mare expecaion he medium survey measures i wih error. Froo discusses ha he 6 See Froo (1989 p 285 and Jonung (1990 p 15. 5

mares expecaion should be a complicaed weighed average of individual respondens expecaions. A possible measuremen error can arise when survey respondens and he corresponding ineres raes are no recorded a he same ime. From his poin of view a mehod where he respondens mare behaviour, in some way or oher, is exposed o he average ineres rae expecaion is o be preferred. However, such experimenal evidence is no available. Lahiri and Zaporowsi (1988 sudied inflaion forecass from he Livingsone survey. Several sudies have found ha hese daa ypically underesimae he rue 7 mare expecaions. The auhors find ha surveyed answers had less predicive accuracy han raional measures. The roo mean square predicion error on he Livingsone survey is nearly hree imes as grea as ha of he raionally expecaion inflaion forecass. 8 These findings mae hem doubful on he reliabiliy of he Livingsone survey daa. Survey measuremen of ineres rae expecaions is of wo inds, qualiaive and quaniaive. The firs group is based on surveys where respondens sae wheher hey expec ineres raes o rise, say consan or fall. The second ype of measuremen of ineres rae expecaions is based on surveys which as for direc quaniaive measuremen of he expeced level of he ineres raes. This paper describes a survey of expeced ineres raes in Sweden ha belongs o he second caegory of sudies described. A he end of each monh he news agency Dire surveyed financial-mare paricipans abou heir expeced level of he six-monh Treasury bill rae ( sassuldväxel hree- and six monhs ahead and he expeced level of he five- year bond rae six monhs in he fuure. The survey sared in Sepember 1992. However, he daa we presen in his sudy begins in he end of December 1992. There are wo reasons why we drop he observaions up o December 1992. Firs, in his sudy we wan o cover he ime when Sweden applied a flexible exchange rae regime. Sweden lef he fix rae exchange rae regime on November 19, 1992. I is no unrealisic o hin ha i aes some ime boh for he mare o sele down and for mare paricipans o learn how he new sysem wors well enough o avoid sysemaic errors. Second, he money mare was exremely urbulen in auumn 1992, before he cenral ban had o give up he defence of he fixed exchange rae. The ineres raes were very difficul o forecas, so he forecas errors during his ime period migh no be represenaive compared o a more normal developmen of he money mare. The news agency Dire erminaed he survey in he end of January 1996. From March 1996 Six Mares Esimaes have coninued he survey. To be able o compare he survey resuls wih similar sudies in oher counries hey have changed he surveyed ineres rae - from he six-monh rae o he hree-monh rae. This means ha our daa series of surveyed six-monh rae hree-and sixmonhs ahead end in January 1996. The survey respondens were analyss and raders in bans and invesmen companies 9. In oher words he respondens do no cover a broad represenaive sample bu are resriced o a group ha 7 True is in he conex of heir raional expecaions hypohesis (REH model o esimae inflaion expecaions. 8 Lahiri and Zaporowsi (1988 also show ha he Livingsone series significanly underesimaes acual inflaion, while inflaion expecaions esimaed by he raional expecaions model exhibis no such underesimaion. 9 In he end of Ocober 1994, e.g., he following respondens were ased abou he expeced fuure ineres raes: Aragon, Consensus, Föreningsbanen, Handelsbanen, Midland Ban, Nordbanen, S-E-Banen, Sparbanen, Transferaor, Unied Securiies and Öhman. I was no exacly he same respondens every monh. Cerain monhs a few can have been subsiued by ohers. 6

professionally follow he ineres rae developmen. The news agency Direc sen he survey forms a he end of each monh. The respondens answered on average wo days afer he send ou. The send ou was made by fax and he answers were also made by fax or by elephone, no ordinary mail. The send ou daes vary from monh o monh. The forms were sen ou beween he 20h and he 28h each monh (during he summer monhs beween he 8h and he 11h. We dae he replys wo days afer ha day, i.e. if he send ou was made Sepember 28, 1993, he reply dae is Sepember 30, 1993. The acual ineres raes, spo and forward, are of course aen from he same day - Sepember 30, 1993, and hree- and six monhs ahead. As he dae of he send ou varies from monh o monh he ime series of he acual six-monh rae hree monhs ahead and he acual six-monh rae six monhs ahead (lagged hree imes are no exacly idenical. Suppose he send ou in December is made December 20, 1993 and he answer is received December 22, 1993. The acual hree monhs ahead rae is on March 22, 1994, which is differen from he acual six monhs ahead rae March 30, 1994 (six monhs afer he answer received Sepember 30, 1993. This is he explanaion why he acual hree- and six monhs ahead raes are no idenical in Figures 2a and 2b. The mean expeced ineres raes were hen announced as he mare s expeced ineres raes. In addiion o he mean he maximum and minimum answers were announced. Unforunaely we do no have access o each responden s answer so i is no possible o analyse he saisical disribuion of he survey answers. Some observaions in he survey daa se are simply unavailable for reasons unnown o he auhor. The missing values have no been replaced. Griliches (1986 calls his he ignorable case 10, in ha for purposes of esimaion, if we are no concerned wih efficiency, we may simply ignore he problem. The missing observaions are displayed in Table 1. The daase of he acual ineres raes consiss of monhly observaions for Swedish T-bills ( sassuldväxlar wih 3, 6, 9 and 12 monhs o mauriy. We need he las wo mauriies o be able o calculae he implici forward raes. The sample period covers he ime from December 1992 o January 1996 providing 38 observaions in oal. We have had access o daily observaions. The acual-and expeced ineres raes are quoed as annualized simple ineres raes. From hese we compue coninuously compounded ineres raes and forward raes 11. The erm srucure of ineres raes observed in our daa is ploed agains ime o mauriy in Figure 1. The yield curve has a negaive slope unil mid 1994 when i urns o a posiive slope. In mid 1995 i is horizonal bu 10 Griliches (1986 disinguishes beween hree ypes of missing observaions: undercoverage, uni non-response, and iem non-response. The firs ype relaes o he possibiliy ha a cerain fracion of he populaion is excluded from he sample by acciden or design. The uni non-response relaes o he refusal of a responden o answer a quesionnaire or he inabiliy of he inerviewer o find i. Iem non-response is used when responses are missing for some fracion of he sample, i.e., quesions no answered, iems no filled in, in a larger daa collecion effor. The discussion deals wih randomly missing observaions or he noion of he ignorable case in anoher erminology. Given he assumpion of a consan β irrespecive of he level of he explanaory variable, x, he observaions can be missing non-randomly as long as he condiional expecaion of y for a given x does no depend on which x s are missing. For example, here is nohing wrong if all high x s are missing, provided he error erm and x are uncorrelaed over he whole range of he daa. 11 In he Appendix we illusrae how coninuously compounded ineres raes and implici forward raes are calculaed. 7

receives a negaive slope a he end of he year. Figure 1 also shows ha he ineres raes have decreased from December 1992 o he end of 1993. The ineres raes have increased from he beginning of 1994 o mid 1995 and hen decreased again o January 1996. 2 The predicive power of he mean surveyed ineres rae 2.1 Presening he survey daa and he mean surveyed forecas errors Figures 2a,b display he acual- and mean surveyed ineres raes of he six-monh rae hree- and six monhs ahead, respecively. The daing is such ha he numbers for monh refer o he expecaion held a wih respec o +3 (or +6 and he acual rae a +3 (or +6, respecively. When he acual ineres rae decreases (increases he mean ineres raes expecaions end o be above (under he acual ineres raes. How good are he respondens a esimaing he fuure ineres raes? Wha accuracy is here in he ineres rae expecaions? This is illusraed in Figures 3a,b where we pu correc expecaions along he 45 -line, where he acual ineres raes coincide wih he expeced. Observaions above he 45 line represen overesimaions of he ineres raes in he surveys, poins under i represen underesimaions of he acual developmen. Figure 3a shows ha he respondens boh have overand underesimaed he fuure six-monh rae hree monhs ahead. Of he 32 mean surveyed observaions during he ime from March 1993 o April 1996 here are 14 overesimaions and 17 underesimaions and one mean surveyed ineres rae which is exacly equal o he acual fuure ineres rae. Figure 3b displays he corresponding resuls for he six-monh rae six monhs ahead. In his far disan horizon he number of over- and underesimaions are he same - or 17 (17, ou of 34, overesimaions (underesimaions. Thus, he wo cases are very idenical. Table 2 shows ha he mean surveyed ineres raes, on average, have very slighly underesimaed he acual ineres raes a boh horizons 12. The sandard deviaions of he mean surveyed ineres raes are however greaer han he sandard deviaions of he acual ineres raes. The correlaions beween he acual and expeced six-monh raes hree- and six monhs ahead are 0.51 and 0.39, respecively. No surprisingly, he correlaion is greaer for he shorer horizon. Table 3 shows he forecas errors of he mean surveyed daa se. We see ha boh he roo mean square error (RMSE and he mean absolue error (MAE show abou wice as grea values for he six monhs ahead rae compared o he hree monhs ahead rae. The mean absolue forecas error hree (six monhs ahead is abou 0.5 (1.0 percen. In addiion o he resuls displayed in Table 3 we found ha he mean surveyed forecas errors are no significanly differen from zero boh for ineres raes hree- and six monhs ahead. One way o assess he accuracy is o compare wih forecass made wih oher mehods. We compare he survey forecas errors wih forecas errors of he implici forward rae and he maringale mehod. This laer mehod predics ha oday s spo rae will remain unchanged. The 12 Noice ha he surveyed ineres raes comprise 32 (34 observaions for he six-monh rae hree (six monhs ahead while he acual ineres rae comprises 38 observaions. Therefore, we show he acual six-monh rae hree-and six monhs ahead which exacly correspond o he survey ineres raes wih 32 and 34 observaions, respecively. For missing observaions in he survey daa se see Table 1. 8

hree meods forecas errors are displayed in Table 3. Noice ha he surveyed ineres rae comprises 32 (34 observaions for he hree (six monhs ahead raes. Therefore, we show he forward raes- and he maringale forecas errors which exacly correspond o he survey ineres raes. Ou of he hree mehods he survey- and forward raes are abou idenical for forecasing fuure ineres raes for boh horizons measured boh as RMSE and MAE. The maringale mehod is slighly worse for forecasing fuure ineres raes. The resuls should be inerpreed wih cauion because of he small number of observaions. Now we are going o compare our resuls wih he corresponding daa from USA. Since Sepember 1969, he Goldsmih-Nagan Leer has conduced a quarerly survey of he ineres expecaions of a seleced panel of approximaely fify of is subscribers who are nown o he publisher o be mare professionals. The panel members ypically represen a variey of differen inds of financial insiuions. In one issue per quarer he Goldsmih-Nagan Leer repors o is subscribers he resuls of he survey in he form of he median of he individual responses. Ferderer and Shadbegian (1993 used he consensus (median ineres rae forecas from he Goldsmih-Nagan Leer o forecas he hree- and six monhs ahead U.S. hree-monh Treasury bill rae during 1969:3 o 1990:2. The survey forecas errors are compared o he forward rae forecas errors in Table 4. All means are insignificanly differen from zero. Boh average- and mean absolue errors indicae ha he forward raes forecass are more accurae for he hree monhs forecas horizon while he survey forecass are slighly more accurae for he six monhs forecas horizon. Tables 3 and 4 indicae ha he forecas errors in Ferderer and Shadbegian s sudy 13 are greaer, boh for survey- and forward raes, compared o he forecas errors in his sudy. The wo sudies boh show mean survey forecas errors ha are no significanly differen from zero. Prell (1973 sudied he Goldsmih-Nagan median surveyed ineres raes for he hree-monh Eurodollar rae and he hree-monh Treasury bill rae hree- and six monhs ahead during Sepember 1969-December 1972. The wo mauriies and horizons underesimaed he acual ineres rae. The underesimaion was larger for he six monh forecas horizon. He finds ha he surveyed forecas errors for he wo ineres raes and horizons, measured as average absolue error, are smaller han he average absolue errors in Ferderer and Shadbegian s sudy bu larger han in his sudy. Prell s resuls show ha he mean survey forecas errors for he wo ineres raes and horizons are smaller han he forecas mehod ha predics ha oday s spo rae will remain unchanged. This is in line wih he resuls in his sudy as indicaed in Table 3 14. There was no significan endency oward under- or overesimaion of levels. 2.2 Do he survey forecas errors behave raional? Figures 4a,b show he forecas errors a boh horizons; i.e. he differenial beween acual and forecas in Figures 2a,b. I is no surprising ha he six-monh forecass show greaer errors han he 13 Noe ha he mauriy of he ineres raes compared differs in he wo sudies. This sudy uses he six-monh rae while Ferderer and Shadbegian use he hree-monh rae. 14 Noe ha even in his case he wo compared mauriies differ in he wo sudies. Prell (1973 uses he hreemonh rae while his sudy uses he six-monh rae. 9

hree-monh forecass. The survey forecass are raional if hey use all available informaion. This implies ha here is no auocorrelaion in he forecas errors. A quic glance a Figures 4a,b shows ha he forecas errors reveal clear signs of auocorrelaion. This means ha here is informaion which is no made use of. The auocorrelaion coefficiens are displayed in Table 5. Since he ime series of he surveyed ineres raes conain missing observaions we spli he series ino subseries. For he hree (six-monh horizon we loo a wo (hree subseries, December 1992-June 1994 and Augus 1994-June 1995 (December 1992-June 1994, Augus 1994-March 1995, and July 1995- January 1996. Firs, we es wheher a paricular value of he auocorrelaion is equal o zero 15. For he hree (six-monh horizon, lags one and wo (one, wo and hree show auocorrelaion during he period December 1992-June 1994. In his case we rejec ha he rue auocorrelaion coefficiens are zero. We also es he join hypohesis ha all auocorrelaions are zero by using he Q-saisic inroduced by Box and Pierce 16. We rejec ha he rue auocorrelaion forecas errors are all zero boh for he hree- and six-monh horizons during December 1992-June 1994. To summarise, he survey forecass behave raionally neiher for he hree-monh horizon nor he six-monh horizon. Le us now as o wha exen he mean survey ineres rae predics he acual fuure ineres rae. This could be done using he following regression model. (1 r + = α 1 + β 1 E ( r + + e + where E ( r + is he mean surveyed expeced ineres rae a ime periods ahead, and e + = r + E ( under he null hypohesis α 1 =0 and β 1 =1, i.e. he residual erm reflecs purely random - r + news. Noe ha equaion (1 concerns he relaion beween he levels of he expeced rae and he acual fuure spo rae. This version of he regression gives consisen esimaion if he ime series are saionary. The ineres raes used in his sudy appear o be non-saionary according o Table 6. To give regression (1 saionariy we rewrie i as (2 r + - r = α 2 + β 2 [ E ( r + - r ] + e + The subsequen change in he spo rae is regressed on he expeced change over he previous period, E ( r + - r. This is a es of he unbiasedness of he survey daa. The resuls from fiing equaion (2 o our daa are displayed in Table 7. The β-coefficiens are boh less han one bu no significanly, confirming he impression given by Figures 3a,b. The consan erms are boh small and no significanly differen from zero. The chi-square ess show ha we canno rejec he join hypohesis ha α 2 =0 and β 2 =1, for he six-monh rae neiher hree- nor six monhs ahead. 15 To es wheher a paricular value of he auocorrelaion is equal o zero we use Barle s es. 16 They show ha he saisic Q=N $p K =1 (K is he number of lags and $p he auocorrelaion coefficien 2 is (approximaely disribued as chi square wih K degrees of freedom 10

Swedish resuls sand in conras o wo foreign sudies, namely Kim (1996 and Friedman (1980. Kim (1996 examined he properies of survey based expecaions 17 of he 90-day ban acceped bill rae and he 10-year governmen bond rae, respecively, in he Ausralian financial mares. The sudied period covers he ime from Augus 1985 o January 1993. One aim of he paper was o es he unbiasedness of he surveyed ineres raes. Using equaion (2 18 he finds ha he unbiasedness of he surveyed forecass are rejeced boh for he 90-day bill rae and he 10-year governmen bond rae wo- and four wees ahead 19. The β:s are 0.426 (-0.025 and 0.507 (0.089 for he 90-day bill (10-year governmen bond rae wo- and four wees ahead, respecively. Friedman (1980 used Goldsmih-Nagan survey daa o es, among oher hings, he unbiasedness of surveyed answers - H 0 : (α, β = (0, 1 20. The analyses focused on six ineres raes, included in he survey, which are all yields on asses acively raded in he financial mares. The six ineres raes are: The federal funds, hree-monh U.S. Treasury bills, six-monh Eurodollar cerificaes of deposi, welve-monh U.S. Treasury bills, new issues of high-grade municipal bonds, long-erm uniy bonds, and seasonal issues of high-grade long-erm. The sample period consiss of hiry quarerly observaions, beginning wih predicions made in Sepember 1969. The horizons are hree and six monhs. The ess show ha he survey respondens did no mae unbiased predicions 21. The general endency is ha α>0 and β<1. 2.3 The diversiy in he surveyed answers In heoreical and empirical sudies of he erm srucure of ineres raes, as a rule, i is assumed ha he sudied group has idenical ineres rae expecaions. In our survey daa se we see ha he respondens do no have idenical ineres rae expecaions. In Figures 5a,b 22 we see he acual, maximum and minimum surveyed ineres raes of he six-monh rae hree-and six monhs ahead, respecively, and in Figures 6a,b 23 he difference beween he maximum and minimum values are 17 The surveyed answers were colleced from various issues of he Ausralian Financial Review. The mare expecaions of fuure values of hese ineres raes were surveyed by Money Mare Services Ausralia (MMS. They carry ou weely elephone surveys on one- and four wees ahead poin forecass of he 90-day bill rae and he 10-year governmen bond rae. They survey he forecas of 20 o 25 financial mare economiss and mare paricipans and repor he median of he survey. 18 Kim (1996 disinguishes beween he unbiased expecaions hypohesis, UEA, - esed according o equaion (2 - and he wea raional expecaions hypohesis, WREH, in he sense forecasers use all available informaion when forming expecaions. He also ess he WREH and finds he surveyed forecass o be wealy raional. 19 Kim (1996 also examined he unbiasedness of surveyed USD/$A exchange rae one- and four wees ahead. Even in he exchange rae mare he finds he surveys o be no unbiased predicors of fuure exchange raes. The same resuls are shown in Chinn and Franel (1991 daa on monhly survey expecaions of fuure USD exchange raes agains 25 currencies for he period February 1988 o February 1991. They conclude ha he expecaions appear o be biased. MacDonald (1992 finds he same resuls of Briish survey-based monhly forecass conduced on companies of G7 counries for he hree monhs ahead USD exchange raes agains he Briish Pound, he Yen and he Deusche Mar for he period Ocober 1989 o March 1991. 20 Friedman (1980 used equaion (1. He refers o Muh (1961 and says ha a ey propery of raional ineres rae expecaions is ha hey are unbiased (p 456. 21 Friedman sresses ha he evidence of serial correlaion invalidaes he F ess. He presens resuls from esing he null hypohsis of unbiasedness by applying Zeller s (1962 seemingly unrelaed regression procedure. The es saisic λ warrans rejecing he join hypohesis of unbiasedness across all he seemingly unrelaed regressions a 90 precen confidence inerval for he hree monhs ahead predicions and a 99 percen confidence inerval for he six monhs ahead predicions. 22 These figures should be inerpreed in he same way as Figures 2a,b. 23 These figures should also be inerpreed in he same way as Figures 2a,b. 11

displayed. The mean difference beween he maximum and he minimum value for he whole sample period is 0.0070 and 0.0121 for hree- and six-monh horizons, respecively. The spread is greaer for he six-monh horizon compared o he hree-monh horizon. This is naural since he uncerainy is greaer for he longer horizon. The maximum and minimum values comprises 30 and 32 observaions for he hree- and six monh horizons, respecively. The acual ineres raes fall wihin (ouside he range given by he survey maximum and minimum values, 15 (15 imes a he hree-monh horizon and he six-monh horizon 13 (19 imes. Despie he fac ha he spread is nearly wice as large for he longer horizon, here is a somewha lesser share of acual observaions ha fall wihin he range given by he survey maximum and minimum values. We may regard he difference beween he maximum and minimum value for each survey day as a measure of he uncerainy referring o he ineres rae developmen. The degree of uncerainy should be refleced as ris premia, rp. The ris premium is defined by he differenial beween he implici forward rae, f, and he expeced spo rae, E ( r +, or formally (3 rp = f - E ( r + We es his hypohesis by running he following regression (4 rp = α 3 + β 3 [max E ( r + - min E ( r + ] + ' where max (min E ( r + is he maximum (minimum surveyed ineres rae. The regression resuls are displayed in Table 8. The R 2 - values are exremely low, -0.0142 and 0.0034 for he hree- and six-monh horizons, respecively. β 3 has a posiive sign as hypohesized, bu neiher he consan erm- nor he β 3 -coefficiens are significanly differen from zero on he 95 percen level. III THEORY - THE EXPECTATIONS HYPOTHESIS In he nex secion we will es he EH wihou a direc measure of expecaions. Before doing ha we need o disinguish beween differen forms of he EH. A sandard formulaion of he EH uses he implici forward rae 24. This can be decomposed ino wo pars, a forecas of he fuure ineres rae and a erm ha may be labelled a ris premium, or (5 f = E ( r + + rp where f is he implici forward rae a ime for a six monhs invesmen ha sars a ime, E ( r + is he expeced spo rae and rp he ris premium which is some form of compensaion for bearing ris. 24 How he forward rae is calculaed is shown in he Appendix. 12

The mos popular simple heory of he erm srucure is nown as he pure expecaions hypohesis (PEH. This requires ha he expeced ris premium is zero for all mauriies, rp =0, or formally (6 E ( r + = f The PEH should be disinguished from he expecaions hypohesis, EH, which says ha he expeced ris premium, rp, is consan over ime and he same for all mauriies. We al abou he liquidiy preference hypohesis (LPH when he ris premium depends on he erm o mauriy of he m bond and rp, 6, 6m 1 > rp 25... ec. Noe ha rp has a subscrip because i may vary over ime. In his case we al abou a ime-varying ris premium. When he heory is esed wihou a direc measure of expecaions we need an assumpion abou expecaions. A popular assumion is raional expecaions, RE, or (7 E ( r + - r + = v + where v + is a serially uncorrelaed forecas error. Hence, a es of he PEH+RE implies ha he expeced excess reurn should have a zero mean, he forecass be made uilizing all available and relevan informaion a he ime of maing he forecas, and should be serially uncorrelaed. The EH+RE yields similar predicions as he PEH+RE. The excess yield is now equal o he consan ris premium under EH+RE and equal o an increasing ris premium for each bond o mauriy, under he LPH+RE. Apar from he consan ris premium, he main esable implicaions using regression analysis of he PEH, EH and LPH are idenical. Equaion (7 suggess a sraighforward regression-based es on: (8 r + = α 4 + β 4 f + ε + where ε + = r + - E ( r + under he null hypohesis α 4 = 0 and β 4 =1, i.e. he residual erm reflecs purely random news. For he PEH we have a zero consan erm and for he EH α 4 0. LPH suggess increasing α when increasing he ime o mauriy. Under PEH, EH and LPH we expec β 4 =1. Since E ( is (under RE independen of informaion a ime, he OLS yields unbiased r + esimaes of he parameer. Noe, however, ha if he ris premium is ime-varying and correlaed wih he expeced fuure ineres rae he parameer esimaes in equaion (8 will be biased. To adop he erminology in his sudy wih he erminology in he erm srucure lieraure we simply al abou he EH no maer we mean he PEH or he EH. The mos common used es, in he lieraure, of he EH is a join hypohesis which implies ha α=0, and β=1 and he assumpion of raional expecaions, EH+RE. Wih direc measured expecaions i is, however, possible o isolae ess of he EH wihou assuming raional expecaions. This is wha we are going o do in secion V. 25 6m is he lengh of he invesmen - six monhs. 13

In his secion we have disingushed beween differen forms of he EH. In he nex secion we perform he es of he EH+RE wihou a direc measure of expecaions. IV TESTS OF THE EXPECTATIONS HYPOTHESIS WITHOUT A DIRECT MEASURE OF EXPECTATIONS When we use daa wihou a direc measure of expecaions a join es of he EH and raional expecaions (EH and RE is used. To show his he saring poin is equaion (8. However, his concerns he relaion beween he levels of he forward rae and he fuure spo rae. This is a version of he es of he EH+RE ha is no commonly used in he lieraure (see Shiller 1990. The reason is ha he ineres raes migh show signs of mean non-saionariy. To achieve saionariy we rewrie i, following Shiller (1990, Fama (1984 and ohers, as (9 r + - r = α 5 + β 5 ( f - r + ε + ; The change in he spo rae is regressed on he forward premium in he previous period, defined by fp, (10 fp = f - r The ineres rae differenials in equaion (9 are saionary. I is herefore legiimae o esimae his equaion using OLS. In he lieraure i is he mos frequenly used model (see Shiller 1990. I is used by Froo (1989, Hörngren (1986 and Dahlquis and Jonsson (1994. Wih his model i is possible o mae comparisons wih oher sudies. The null hypohesis of he expecaions heory posulaes ha he inercep erm is equal o zero (α 5 =0 and he coefficien of he forward premium is equal o one (β 5 =1 in equaion (9. Noe ha his is a join es. I ess ha here is no ris premium implying ha he forward premium is equal o he expeced fuure ineres rae change, condiional on he mainained assumpion ha he mare s expecaions are made raionally, meaning ha he expecaion is an efficien forecas of he fuure ineres rae change and he residual, ε +, is purely random. If he wo assumpions hold, he forward premium is an unbiased predicor of he fuure ineres rae change. If he coefficien of he forward premium is significanly differen from one i depends eiher on an expecaion error or a premium ha is correlaed wih he level of he ineres rae. If he consan erm is differen from zero i depends, in he same way, eiher on a sysemaic error or a consan ris premium or boh. In addiion o he sandard regression here are oher approches o es he EH. Edahl and Warne (1990 used an vecor auoregressive (VAR model and Hördahl (1994 used an auoregressive condiional heeroscedasiciy in mean (ARCH-M model o es he EH on Swedish daa. We presen he resuls from fiing equaion (9 o daa from he Swedish T-bill mare in Table 9. The coefficiens of he forward premia are boh less han one bu no significan. The coefficien of he forward premium decreases wih he horizon. I is easier o forecas ineres raes in he near fuure han ino he more disan fuure. The consan erms are boh less han zero bu no significan. 14

Furher, we canno rejec he join hypohesis ha α 5 =0 and β 5 =1. This means ha he EH+RE canno be rejeced for eiher of he wo horizons. 1 Foreign sudies of he EH The numbers of foreign sudies of he EH are enormous. Only a small number of hese ess, which sar wih Macauley (1938 have found suppor for he EH. A sill smaller number saisically rejec he alernaive hypohesis ha he spreads beween long- and shor ineres raes have no forecas power a all for fuure ineres raes' changes. The null hypohesis in hese ess is he join hypohesis ha all variances in he spread beween shor and long raes depend on fuure ineres rae changes and ha expecaions are raional. Wihou furher informaion i is apparenly no possible o draw any conclusions as o which caegory he rejecion of he EH+RE should be classified. Gerlach and Smes (1995 26 used hree-, six- and welve monhs Euro-raes o es he EH+RE for a sample of 17 counries. They found ha all β-coefficians were posiive and significanly differen from zero (Sweden showed he greaes β-values 27. In 35 ou of 51 cases hey canno rejec β=1. Furhermore, in 26 ou of 51 cases hey canno rejec he join hypohesis ha α=0 and β=1. In Table10 a compilaion of some foreign esimaions of he EH+RE where only he shor ineres raes are displayed. The models used are in general regression (9, or some varian of i. The slope coefficiens are far below one. Cerain sudies on U.S. daa even show a slope coefficien no significanly differen from zero. The resuls in Table 10 loo specially bad for he EH+RE. However, Hsu and Kugler (1997 28 deec ha he predicive power of he spread for shor ineres rae changes has improved when hey find suppor for he EH+RE for he mos recen of four subperiods sudied, 1987-85. For he subperiods 1973-79, 1979-82 and 1982-87 and he whole sample period, 1973-95, hey do no find suppor for he EH+RE. Table 11 displays some resuls from ess of he EH+RE using long ineres raes. Beaer, Hodric and Marshall (1997 rejec he EH+RE on U.S. daa boh for hree- and five years ineres raes. Froo (1989 ess he one (hiry year (s rae six monhs ahead and finds no suppor for he EH+RE. In line wih Beaer, Hodric and Marshall (1997 one of Froo s regressions gives negaive slope coefficien. The resuls loo bad for he EH+RE, on U.S. daa, which correspond well wih Shiller (1990 when he forecas horizon is small and he mauriy of he forward rae is long (no shown in he able. Tesing he EH+RE for long ineres raes on European daa pain a differen picure. Engsed (1996 finds suppor for he EH+RE on Danish daa. Hardouvelis canno rejec he EH+RE for he en-year rae neiher for France nor Ialy. 29 The predominan approach ha he variance in he forward premia reflecs ris has already been shown for oher mares. Hodrich and Srivasava (1984,1986 have shown ha forward- and fuure 26 They used he following model o es he EH+RE: ( r + j 1 + r+ j 2 +... + r /j - r = α + β( where j R is he j-period invesmen rae a ime and R r j + v r is he 1-period ineres rae a ime. The heory implies ha he realised fuure reurn on rolling over an 1-monh invesmen over j periods should equal he curren spread beween a j-period- and an 1-period ineres rae. 27 The β-coefficiens, no significanly differen from one, are 1.24, 1.19 and 1.13 for he hree-, six- and welve monh raes, respecively. 28 They used McCallum s model. 29 However, esed on U.S. daa he EH+RE is rejeced. 15

raes in he exchange rae mares are an unbiased predicor of fuure spo rae exchange raes. Their inerpreaion is ha he variance in he exchange ris premium is higher han he variance in expeced depreciaion. The resul is confirmed by Nessén (1994 in a sudy of Sweden s, Norway s, Denmar s and Finland s exchange raes mares ha he erm-premium in he preceding period is an unbiased predicor of he fuure exchange rae. Her sudy also shows ha he variance of he ris premium is higher han he variance of he expeced exchange rae changes for all counries. 2 Swedish sudies of he EH In his sudy, equaion (9 is chosen o es he EH+RE. The reason is ha Froo (1989, Hörngren (1986 and Dahlquis and Jonsson (1994 have used his expression o esimae he spo rae differenial. Hörngren s sudy is similar o Fama s (1984 pioneer paper bu conrary o he laer, Hörngren sudies he capabiliy of forward raes o forecas fuure ineres raes wih differen mauriies while Fama sudies he forecas power several periods in he fuure. Hörngren s sudy comprises he period 1980-1985 i.e. when Swedish bans had he righ o borrow a sum of 25 per cen of heir equiy a he discoun rae and a he penaly rae unlimied borrowing was permied. Bans could also deposi excess reserves i.e. have negaive borrowing, and on hose deposis hey earned a fixed ineres rae. Needless o say, his was lower han he discoun rae and he penaly rae. The resuls from Hörngren s sudy are shown in Table 12. He uses he forward premia one-monh rae one monh ahead (1m1m, one-monh rae wo monhs ahead (1m2m and hree-monh rae hree monhs ahead (3m3m. Hörngren finds ha he slope coefficiens for he forward premia onemonh rae one monh ahead (1m1m and one-monh rae wo monhs ahead (1m2m are less han one. These are significanly differen from zero on he 16- and 10 percen levels, respecively, which sugges some evidence ha he forward-spo rae differenials conain informaion abou fuure premia. The hree-monh rae hree monhs ahead (3m3m displays he invered resul. The slope coefficien is greaer han one bu no significanly. In his case he forward-spo rae differenial conains un unbiased predicor of he fuure change in he hree monh rae. Hörngren s resuls are in conras o he resuls in his sudy which canno rejec he EH+RE. The differen resuls may depend on he fac ha Hörngren used ineres raes on ban cerificaes of deposis wih possible defaul premiums. Furher, he capial mare in he early eighies was undeveloped and highly regulaed. Dahlquis and Jonsson (1994 use T-bills and heir sudy also covers he ime when Sweden applied a fixed exchange rae regime or specifically, January 1984 o July 1992. They also use equaion (9 o esimae he spo rae differenial. Their sudy conains he forward one-monh rae one monh ahead (1m1m, he one-monh rae wo monhs ahead (1m2m, he hree-monh rae hree monhs ahead (3m3m, and he six-monh rae six monhs ahead (6m6m. The resuls are displayed in Table 12. The slope coefficiens are all no significanly differen from one, which is good for he EH. The conclusion is ha he EH+RE canno be rejeced. Dalquis and Jonsson s resuls correspond well o he resuls in his sudy. The EH+RE canno be rejeced in he wo sudies. In he Swedish money mare he EH+RE seems o be valid boh under fixed- and flexible exchange rae regimes. 16

Edahl and Warne (1990 use a VAR-model 30 o sudy he Swedish erm srucure of ineres raes during 1984-1989. They used daa for he one monh ineres rae of T-bills and five year governmen bonds. The resuls show ha he forecas power - in general - decreases wih he horizon for he shores ineres raes bu increases wih he horizon for forecass several years ino he fuure. As in his sudy he EH+RE canno be rejeced. Hördahl (1994 uses an ARCH-M ( auoregressive condiional heeroscedasiciy in mean model 31 o sudy he erm srucure of ineres raes for four ineres raes series - 2 vs 1 monh, 4 vs 2 monhs, 6 vs 3 monhs and 10 years vs 1 monh - of T-bills respecively bonds. The resuls show ha here is no suppor for a variance-dependen ris premium. However, he auhor found a wea suppor for a general ime-varying ris premium. The conclusion is ha he EH canno be rejeced wih he excepion of 4 vs 2 monhs' T-bills. Hence, in general, he resuls coincide wih he resuls of his and Dahlquis and Jonsson s sudy. In he nex secion we follow he discussion how o derive he decomposiion of he forward premium. V TESTING THE EXPECTATIONS HYPOTHESIS WITH SURVEY DATA 1 Decomposiion of he forward premium 30 A vecor auoregressive model, or VAR for shor, is a ime-series model used o forecas values of wo or more economic variables. Suppose y 1... y n are economic variables whose values we wish o forecas. A VAR model for hese variables is given by he following n-equaion model, wih he number of lags equal o periods. Z = i= 1 A i Z i + ε. All variables are reaed as endogenous. Noice ha no curren values for any variables appear on he righ hand side of any equaion. Suppose we wish o use he VAR model for forecasing, say y 1 + 1. We use he firs equaion in he model. y ˆ 1 + 1 = â 11 y 1 +...+ aˆ 1 n y n + ˆb 11 y 1 1 +...+ bˆ 1n y n 1 + ˆ 11 y 1 ( 1 +...+ ˆ 1n y n ( 1 yˆ + where 1 1 is he forecas and he â :s, bˆ :s and ˆ :s are esimaed coefficiens. 31 The ARCH-M model used is: y =β x +δh + u = h 2 and u, V[ ] A 1 2 h =γ+α i= q w i 1 u. 2 i y is he ex-pos excess holding yield over a long erm T-bill wih L days o mauriy over a shor erm T-bill wih S days o mauriy, where L=2S. I is defined as y = (1 + r (1 + r L 2 S + 1 - (1+ r s. x is a vecor of explanaory variables and 1 2 all available informaion a ime -1. h is he condiional variance as a funcion of pas squared errors, u. The 2 parameers of he ARCH-M model are esimaed wih a number of differen lags in he weighed average of squared residuals. The mos common mehod uilized is he maximum lielihood mehod. The log lielihood funcion may be expressed as ln ( φ T L = = 1 ln L ( φ where ln L ( φ = - ln( u 2 h -. The expression is 2 maximized wih respec o φ. If one or more of he independen variables in he firs equaion are significan he excess holding yield is no consan. Such a ime-varying ris premium maes he EH o be rejeced. 2h i A is 17

In line wih oher Swedish ess of he EH+RE our resuls repored in Table 9 show ha we canno rejec i. One inerpreaion is ha forward raes reflec expeced fuure ineres raes and ha expecaions are raional. However, in ligh of he rejecion of he EH+RE in a large number of U.S. sudies an alernaive inerpreaion is ha neiher of hese condiion holds. For Sweden we have already given an indicaion of ha neiher RE nor EH holds. This is wha we are going o es in his secion. Using he survey daa i is possible o decompose he forward premium ino an expecaional componen and a ris premium. By definiion, he ris premium, rp, is (11 or (12 f - r = E ( fp = E (, r r + - + rp r + rp where fp and E (, r are boh observable. The variance of fp is Var( fp = Var( E (, r + Var( rp + 2Cov( E (, r, rp The ime series for he ris premia, rp, are shown in Figures 7a,b ogeher wih he expeced ineres rae differenials, E (, r and he forward premia, fp. The variance of he E (, r is smaller (abou he same han (as he variance of he variances in he fp :s are abou wice as grea as he variances in he are boh greaer han zero. The variances of he fp for he hree (six monhs ahead rae. The rp :s. The mean of he rp :s rp for he hree monhs ahead rae is smaller han he variance of he E (, r. They show negaive correlaion. This is confirmed in Table 13 and 14 which conain he variances and correlaions of he ime series. For he six monhs ahead rae he variance of he rp is abou half he size of he variance in he E (, r and hey show posiive correlaion. A grea variance in rp indicaes ha he EH does no hold. Equaion (12 reveals ha if fp he covariance beween rp fp canno be high. This is he variance in he rp is smaller (greaer han he variance in he and E (, r mus be posiive (negaive (oherwise he variance in confirmed in Tables 13 and 14. The variances in he ineres rae differenials, By definiion he expecaion error,, r. This means ha he u, is fp :s are smaller han he variances in he acual fp :s do no eep up wih he, r :s. (13 u = - E (, r r, The variances in he u :s are nearly wice as grea as he variances in he fp :s and here is posiive correlaion beween he variables. This indicaes ha RE does no hold. Figures 7a,b show imevarying ris premia while Figures 8a,b display expecaion errors. To summarize, he above analysis of he variances of rp and u suggess ha neiher RE nor EH hold. However, according o Table 9 he EH+RE canno be rejeced. To invesigae his we urn o decompose he β-coefficien ino wo componens one measuring he forecas error and one he ris premium. 18