THE INFORMATION IN THE TERM STRUCTURE OF GERMAN INTEREST RATES


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1 THE INFORMATION IN THE TERM STRUCTURE OF GERMAN INTEREST RATES Gianna Boero CRENoS, University of Cagliari, and University of Warwick Eail: and Costanza Torricelli University of Modena and Reggio Eilia Eail: Final version Deceber 2000 ABSTRACT This paper tests the Expectations Hypothesis (EH) of the ter structure of interest rates using new data for Gerany. The Geran ter structure appears to forecast future shortter interest rates surprisingly well, copared with previous studies with US data, while it has lower predictive power for longter interest rates. However, the direction suggested by the coefficient estiates is consistent with that iplied by the EH, that is when the ter spread widens, long rates increase. The use of instruental variables to deal with possible easureent errors in the data significantly iproves regressions for the long rates. Moreover, reestiation with proxy variables to account for the possibility of tievarying ter preia confirs that the evolution of both short and long rates corresponds to the predictions of the EH and that ost of the inforation is in the ter spread. These results are iportant as they suggest that onetary policy in Gerany could be guided by the slope of the ter structure. Keywords: expectations hypothesis, interest rate, ter structure, ter preia, forward rates, easureent errors, volatility. JEL classification: C22, E43, G14 Aknowledgeents: The authors are grateful to Foruhar Madjlessi for valuable research assistance, and to Herann Göppl for aking the KKMDB data available. We also thank two anonyous referees for particularly helpful coents. Gianna Boero acknowledges financial support fro CRENoS and CNR. Costanza Torricelli acknowledges financial support fro CNR and MURST. Correspondence address: Gianna Boero, Departent of Econoics, University of Warwick, Coventry, CV4 7AL, UK. 1
2 1. INTRODUCTION The inforation in the ter structure of interest rates has been the subject of extensive research using data for the US. The ai of this literature is to exaine whether the relationship between interest rates at different aturities, that is, the ter structure, helps to predict future oveents in interest rates. Many epirical studies have focused on tests of the Expectations Hypothesis (EH) relating longter interest rates to expected future shortter rates. The validity of the EH is of interest since it has iportant iplications for econoic policy: if the EH is supported by the data, it suggests that onetary policy could be guided by the yield curve. However, the epirical evidence on the EH for the US is by no eans conclusive. Typically, the hypothesis has been rejected with data related to the postwar period (see, inter alia, Capbell and Shiller, 1991, Hardouvelis, 1994, Evans and Lewis, 1994, and Rudebusch, 1995) and accepted with data for the period before the founding of the Federal Reserve Syste (see Mankiw and Miron, 1986). According to Mankiw and Miron, the lack of predictive power of the ter spread after the founding of the Federal Reserve has to be attributed to its short rate stabilisation procedure, under which the short rate evolves as a rando walk, and interest rate changes becoe unpredictable. Finally, a recent study by Hsu and Kugler (1997) suggests a revival of the EH with evidence for the period Hsu and Kugler attributed this finding to changes in the conduct of onetary policy in the US during the ost recent period, particularly the introduction in 1988 of the use of the ter spread as a policy indicator by the Federal Reserve. Studies of the EH also differ depending on the ethodology and the type of data used. With regard to the ethodology, the ost direct test of the EH uses standard regressions of changes in interest rates on different easures of the slope of the ter structure. In this literature, one can distinguish between regressions which use the ter spread as a regressor, that is the difference between a long rate and a short rate (see Capbell and Shiller, 1991), and regressions which adopt the forwardspot spread (see Faa, 1984, and Faa and Bliss, 1987). Moreover, within the ter spread regressions, there are two different specifications: regressions which predict changes in the short rate, and regressions which predict changes in the long rate. Tests based on regressions for the short rate are usually in favour of the EH, while tests based on regressions for the long rate indicate that actual long rates ove in the opposite direction fro that predicted by the EH. Another strand of the literature uses the Vector Autoregression (VAR) approach (Shiller, 1979, and Capbell and Shiller, 1987). In these studies, VAR odels are specified for 2
3 changes in the short rate and the ter spread, and the estiated coefficients are used to assess the deviation in the behaviour of the actual ter structure spread fro the theoretical spread under the EH. This approach is based on the result that the EH of the ter structure can be expressed in the for of the present value odel, and it is therefore strictly valid only when the long rate is a perpetuity or when bonds have very long life (for exaple, twenty years). With regard to differences in the type of data used, soe studies use estiated ter structures based on interpolation ethods such as McCulloch, 1971, and Chabers, Carleton and Waldan, 1984; other studies use interest rates on bonds of aturities that are actively traded, i.e. treasury bill yields, governent bond rates, oney arket interest rates and Eurorates. Estiated ter structures have soe potential advantages over other types of data in that they enable one to exaine a wider range of aturities and provide ore coprehensive results than those obtained with other data. On the other hand, the use of estiated ter structure data introduces easureent errors, but this proble can be easily overcoe with instruental variable estiation. Studies which have used estiated ter structures for the US are, aongst others, Faa (1984, 1990), Jorion and Mishkin (1991), and Evans and Lewis (1994). In contrast to the large nuber of studies for the US, only liited evidence exists for other countries (see, inter alia, Mills, 1991, Hardouvelis, 1994, Engsted, 1996, Gerlach and Sets, 1997 and Jondeau and Ricart, 1999). Although these studies are in general ore favourable to the EH, results vary, to soe extent, according to the regression specification adopted and the type of data used. Moreover, there is a clear suggestion in support of the arguent by Mankiw and Miron (1986) that the predictive power of the slope of the ter structure is stronger under onetary targeting (or during periods with relative volatile interest rates) than under interest rate targeting (or periods with relative sooth interest rates). For exaple, the study by Engsted (1996) provides results for the Danish arket that are uch ore supportive of the EH in a period when the Danish onetary authorities shifted fro a policy of interest rate stabilisation, and interest rates becae ore volatile. Thus, it is iportant to ascertain whether assessents in favour or against the EH can be replicated over independent data sets and different econoies. This paper contributes to the existing evidence on the EH in two ways. First, we present evidence for Gerany based on new data derived fro estiated ter structures. Only few studies have analysed the EH using data for Gerany, and none has used data obtained fro estiated ter structures (for exaple, the studies by Gerlach and Sets, 1997 and Jondeau and Ricart, 1999 are based on Eurorates). The use of estiated ter structures, which is 3
4 ore coon in applications for the US, perits us to present results for a finer aturity grid copared to studies based on different data sets, and to ascertain whether previous findings can be replicated over independent data sets. As a second contribution, we report results for both the long and shortter interest rates, while previous studies for Gerany have concentrated ainly on the predictability of the short rates. So our study fills a gap in the epirical literature concerning the longrate predictions in interestrate spreads for Gerany. The data used in the epirical analysis cover the period , and have a onthly frequency. The period is approxiately the sae as that used in previous studies for Gerany, and this enables a ore direct coparison with the results in those studies. The procedure adopted to estiate the ter structure is based on the interpolation ethod suggested by Chabers, Carleton and Waldan (1984). We have applied this approach to Geran Governent bond data provided by the Karlsruher Kapitalarkt Datenbank (KKMDB) 1. During the whole period under investigation the Bundesbank has pursued a onetary targeting (rather than interest rate soothing) policy. Hence, according to the Mankiw and Miron (1986) arguent, we would expect to find support for the EH in the Geran data. The EH is tested in this paper by eploying standard regressions, and using both the ter spread and forwardspot spread approaches. The VAR ethodology is not appropriate in our context because the data span a relatively short tie period, and the long rates are not of infinite life. Our results reinforce the finding docuented in Gerlach and Sets (1997) regarding the predictability of the shortter interest rates. Specifically, our data offer further support for the inforation content of the ter structure in regressions which predict shorterter interest rates, and the results are broadly consistent with the EH in ters of the coefficient value of the spread. On the other hand, the ter spread does not have uch predictive power for longter interest rate oveents, although the sign of the coefficient estiates on the ter spread is in ost cases consistent with the EH; that is, when the ter spread increases, long rates increase. This latter result contrasts with previous evidence for the US suggesting that the spread does not even predict the right direction for the long rate oveents (see, for exaple, Evans and Lewis, 1994, and Hardouvelis, 1994). Results for the long rate regressions iprove when we use instruental variables to correct for possible easureent errors in the data. We also attept to evaluate the effects of a tie varying ter preiu in tests of the EH. This analysis is conducted by estiating extended regressions which include alternative proxies based on different easures of interest rate volatility. We find that, in general, the volatility ters do not have predictive 4
5 power, confiring that ost of the inforation in the ter structure for oveents in interest rates is contained in the ter spread, and that both long rates and short rates ove in the direction consistent with the predictions of the EH. The rest of the paper is organised as follows. In Section 2, we explain the theory of the EH of the ter structure and describe different types of odels used in tests of the EH. In Section 3, we perfor tests of these odels and consider the possible effects of easureent errors, ignoring the ter preiu. In Section 4, we describe the characteristics of tievarying ter preia within stochastic odels of the ter structure and reestiate the standard regressions by including different proxies for the riskless rate volatility (the ter preiu) in the inforation set. Section 5 closes the paper with conclusions and further rearks. 2. THE THEORETICAL FRAMEWORK A great deal of epirical research on the ter structure of interest rates has focused upon the EH, relating longter interest rates to actual and expected future short rates. According to the ost coon version of the EH, a longerter interest rate is a siple average of the current and expected future shortter interest rate plus a constant ter preiu. In the case of an nperiod rate, R n, and a sequence of oneperiod short rates, r, in linearised for the EH can be stated as follows: n 1 n R t = ( 1 / n) rt + E trt+ i n i + π ( ) (1) = 1 where π(n) is a constant ter preiu on the longer rate, and E t is the rational expectations operator conditional on inforation at tie t. The EH can also be expressed in ters of the relationship between R n and any period rate R, for which k=n/ is an integer: n t k 1 R = ( 1 / k) E R + π ( n, ) i= 0 t t+ i (1a) where π(n,) is the ter preiu between the n and period rates. This relationship iplies that an upwardsloping ter structure curve predicts an increase in short rates and subsequently in long rates, and viceversa. The literature has investigated the inforation in the ter structure of interest rates using different spreads as easures of the ter structure and using substantially different data. In the present paper, we use two approaches: the spread between a long rate and a short 5
6 rate as a predictor of future changes in the short rate and in the long interest rate, and the forwardspot differential to predict future changes in the short rate. 2.1 The longshort yield spread approach After soe rearrangeents, equation (1a) can be shown to have testable iplications regarding the predictive power of the spread (R n tr t) for future interest rates: the spread should predict (a weighted average of) changes in the short rate over the nperiod horizon, and the change in the nperiod rate over the period horizon. These iplications can be tested by using the following regressions : ( n/ ) 1 ( 1 ( i / n)) R = α + β( R R ) + ε (2) i= 1 n t+ i t t t+ n with n/ an integer, and R = R R, and R ( n ) t+ t+ i t+ i t+ i n R t n R n t R = α + β t + t+ ( ) ( ) ε (3) In these equations ε t+n and ε t+ are forecast errors which under Rational Expectations are orthogonal to inforation at tie t, and therefore uncorrelated with the spread. Hence, assuing no easureent errors, OLS will give consistent estiates. The errors in (2) will follow a oving average process of order n1 if the difference between n and is larger than the data frequency (onthly in our case). The errors in (3) will follow a oving average process of order 1 if is larger than the data frequency. So, standard errors are usually calculated with the NeweyWest (1987) or Hansen and Hodrick (1980) corrections. Since these corrections becoe less reliable when the degree of overlap is large, ost of the results in this paper are presented for regressions with n up to 18 onths. Tests of the predictive content of the spread iply testing for the significance of β (that is β=0), while tests of the EH with Rational Expectations and constant ter preiu iply testing for β= The forwardspot yield spread approach Siilar tests can be conducted using the forwardspot yield spread approach which is based on the version of the EH stating that forward rates are (unbiased ) predictors of future 6
7 short rates. The corresponding regression odel for these tests takes the following for: ( 1 ) r r = α + β F ( n, n) r + ε (4) t+ n 1 t t t t+ n 1 where r t is the oneperiod rate and F t (n1,n) denotes the oneperiod forward rate, i.e. the rate at trade date t for a loan between periods t+n1 and t+n. ε t+n1 is the expectational error orthogonal to the inforation available at tie t. Equation (4) states that the forwardspot yield spread F t (n1,n)r t is an unbiased predictor of the expected change in interest rates over the n1 horizon. Hence, assuing a tieinvariant ter preiu, the EH iplies β = Previous evidence Most previous evidence relates to US data. Evidence based on equation (2) is well suarised in Dotsey and Otrok (1995) and Rudebusch (1995): while short or ediu ter spreads have very little or no predictive inforation for future changes in the short or ediu rates, long ter spreads show soe predictive content for oveents in future short interest rates. Evans and Lewis (1994) coent on evidence for the US based on equation (3) and conclude that the spread does not even predict the right direction of the long rates oveents, obtaining negative β coefficients. Finally, evidence on tests using forward rates, based on equation (4), is consistent with the results for the US suarised above for equation (2), naely that the EH perfors poorly at the short end of the aturity spectru, but iproves at longer aturities (see, inter alia, Faa and Bliss, 1987, and Mishkin, 1988). Only few studies have analysed the EH using data for Gerany, and none has used data obtained fro estiated ter structures. Gerlach and Sets (1997) present soe results in favour of the EH for Gerany, using 1, 3, 6 and 12onth Eurorates, for the period , focusing only on regressions based on equation (2). Jondeau and Ricart (1999) find less epirical support for the EH than Gerlach and Sets, using Eurorates on a different saple ( ). Hardouvelis (1994) presents evidence based on equations (2) and (3) for the countries which belong to the Group of Seven (G7), using data on a 3onth and a 10 year governent yield. Estiates of equation (2) for Gerany, for the period , show that the evolution of future short rates corresponds closely to the predictions of the EH. By contrast, OLS estiation of equation (3) shows oveents of the long rate in the opposite direction to that iplied by the EH, however, the negative regression sign is reversed with 7
8 the use of instruental variables. The discrepancy between the behaviour of long and short rates is anifested priarily in the US (Hardouvelis, p. 258). In the next section, we start the epirical analysis with the ipleentation of standard tests of the EH. As the data are derived fro the estiation of the ter structure, approxiation errors can have an effect on the OLS estiates. We explore this possibility by using an instruental variables approach in the estiation of the regressions used for the tests and copare the results with the OLS estiates. Next we extend the standard fraework of equations (2)(4) by including different proxies for tie varying ter preia in the inforation set. 3. EMPIRICAL REGULARITIES AND ECONOMETRIC RESULTS 3.1 Epirical regularities in the data The epirical results of this study are based on tie series drawn fro estiated ter structure curves using the Chabers, Carleton and Waldan (1984) approach, as explained in Boero, Madjlessi and Torricelli (1995). The data are onthly annualised rates, refer to the 15th of each onth, and cover the period fro Deceber 1983 to Deceber Figure 1 reports the interest rates both against tie and against tie to aturity. Panel (a) shows the tie period fro Noveber 1983 to June 1989, and Panel (b) covers the tie period fro July 1989 to Deceber Fro the Figure, it is evident that there have been parallel shifts over tie in the ter structure of interest rates, but we can also see that the ter structure has been at ties upward sloping, and at ties downward sloping. [FIGURE 1] In Figure 2 we show the evolution of the annualised rates with aturity 1, 3, 9 and 18onths, for the period to 94.12; in Table 1 we report their suary statistics. The interest rates ove ostly in the sae direction, as do those at interediate aturities, not shown in the figure. All interest rates have approxiately the sae iniu, around 1988, ranging fro 3.37% for the 1onth rate to 3.67% for the 18onths rate, while the axiu, exhibited in response to the inflationary pressures created by the Geran unification of 1990, decreases with aturity (9.98% for the 1onth rate, 8.83% for the 18onths rate). The ter structure 8
9 is upward sloping for ost of the period up to 1990, and is inverted for ost of the period fro 1990 onwards. [FIGURE 2] [TABLE 1] An iportant issue to be addressed is that of the tie series properties of the variables which appear in equations (2)(4). Estiation of these equations requires that the variables are stationary. Unit root tests to deterine the order of integration of the variables were perfored. The test statistics, not reported for reasons of space, suggest that all interest rate changes and all spreads used in the regressions below are stationary, which iplies that the inference presented below, using the t and Fdistributions, is valid. Furtherore, it is iportant to note that during the whole period under investigation, the onetary policy followed by the Bundesbank has been officially onetary targeting, so our regression estiates are free fro structural breaks caused by regie shifts. Before turning to the regressions results, it is useful to inspect Figure 3, where we report a selection of scatter plots of the dependent variables in equations (2)(4) against the relevant spreads. Panel (a) shows the cases of n,=6,1 and n,=9,3 for regressions based on equation (2); Panel (b) considers the cases of n,=3,1 and n,=6,3 for regressions based on equation (3); finally, Panel (c) plots the change in the oneonth interest rate against the forwardspot spread for the cases of n=3 and n=6, as forulated in equation (4). The EH with a constant ter preiu iplies that the observations should scatter around a line with unit slope. Inspection of Figure 3 suggests that while it is realistic to expect a slope close to unity for ost regressions relating the change in the shortter interest rate to the ter spread (equation 2) and to the forwardspot spread (equation 4), there is less visual evidence a fitted line would have a unit slope for regressions relating the change in the longter rate to the ter spread (equation 3). For these regressions, in fact, the scatter plots suggest a very low. However, s fro these regressions are typically low, so ost work on the EH concentrates on statistical testing rather than inforal evaluation of the fit of the odels. [FIGURE 3] 3.2 Standard regression tests: OLS results 9
10 The results fro OLS estiation are reported in Panels A of Tables 2, 3 and 4. The saple period used for each regression is the longest possible using data fro 1983:12 through 1994:12. Regressions based on equation (2) are shown in Table 2 for different pairs of aturities n and. Due to overlapping data, the equations are estiated with OLS with corrections based on NeweyWest (1987) for a oving average of order n1, and for conditional heteroscedasticity. The corrected standard errors are reported in parentheses below the coefficient estiates of β, in square brackets are the Wald tests for the expectations hypothesis β=1. Our results copleents previous findings for Gerany, and give further evidence of the ability of the Geran ter structure to predict future short rates. Unlike siilar regressions for the US, the results given in Table 2 indicate that the coefficient of the spread is significantly different fro zero at the 1% level in all cases 2. Moreover, the values are uch higher than those reported in earlier studies for the US, indicating that the slope of the ter structure has higher inforation content for predicting future short rates in Gerany than in the US. Furtherore, the estiates of β are close to the theoretical value of 1 for alost any two pairs of aturities n and, according to the predictions of the EH, although there are cases where the EH is rejected in a strict statistical sense. For exaple, in regressions with spread 61 and 63 the hypothesis is rejected at the 1% level, and in the regression with spread 91 at the 10% level. Finally, the regressions show higher inforation content (higher ) and increasing support for the EH when longer horizons are considered. This result is in line with the evidence for the US in Faa (1984, 1990) and Faa and Bliss (1987) and suggests that it is easier to predict changes in short rates over longer horizons. [TABLE 2] Regressions for the long rate have been the focus of attention of any studies attepting to explain failures of the EH. In fact, while the EH iplies that the slope coefficient should be equal to one, ost of the epirical literature, using data for the US, has reported very low values for the, and estiated coefficients below unity, becoing negative as yields of longerter bonds are used to for the dependent variable and the ter spread. This socalled sign puzzle or predictability sirk has been docuented for the US by Capbell and Shiller (1991) and ore recently by Roberds and Whitean (1999). Negative values 10
11 indicate that long rates ove in the opposite direction to that iplied by the theory. There is only liited evidence for Gerany regarding the relation between the spread and the future evolution of long rates. For exaple, Hardouvelis (1994) in his study for the G7 countries, focusing on the behaviour of a 10year governent yield and a 3onth yield during the period , finds that the long rate ove in the opposite direction to that iplied by the EH. However, this oveent is apparently due to a white noise error that does not affect the inforation in the ter structure and the use of instruental variables reverses the negative regression sign (Hardouvelis, p. 258). Table 3 reports our estiates of the slope coefficients based on equation (3). The results show that for nearly all pairs of aturities the coefficient estiates of the spread are consistently positive, although not always significantly so. Thus, to the extent that the ter spread predicts changes in the long rates, it does so in the direction iplied by the EH. Moreover, it is interesting to note that while for large n and sall the estiated coefficient is significantly saller than one (first row of the table), its value approaches one as increases. This result again suggests that it is easier to predict changes in interest rates over longer horizons. 3 In general, values for these regressions are all very low, as already suggested by visual inspection of the scatter plots in Figure 3, and Wald tests for the EH that the spread coefficient is equal to one show rejections in a strict statistical sense. These results are quite siilar to previous evidence in the ter structure literature, indicating that the spread between the long and short ter interest rates has poor predictive content for changes in the longer rate. On the other hand, an interesting finding is the positive sign for the coefficient of the spread obtained in ost regressions, suggesting that longter rates ove in the direction predicted by the EH. In a recent study Roberds and Whitean (1999) show that for the US the puzzle with the longrate predictions becoes particularly striking for aturities beyond 24 onths. The results in the first row of Table 3 see to suggest that the oneonth forecasts of interest rates with aturity n1 becoe increasingly puzzling with rising n. In order to see whether soe sort of predictability sirk for the long rate also applies to Gerany, we have extended the analysis to consider regressions with aturities =1, and n=24 and 36 onths. The estiated coefficients were not statistically significant, confiring the overall picture already outlined in Table 3. However, the coefficient values were increasingly positive (0.29 and 0.75, for n=24 and n=36, respectively) therefore suggesting that a 'sile' rather than a 'sirk' sees ore appropriate to describe the predictability of the longter rates in Gerany. 11
12 [TABLE 3] Finally, in Table 4 we report estiates of equation (4) in which we use the spread between the oneonth forward rate and the oneonth spot rate to predict changes in the spot rate over n1 periods, with n = 3, 6, 9, 12 and 15 onths. Results fro specification (4) show close siilarities to those based on equation (2). In fact, also for these regressions the slope coefficient is always significantly different fro zero at the 1% level, so there is significant predictive power of the forwardspot spread. The high inforation content of the forwardspot spread also eerges fro the relatively high values of the. Moreover, tests on the restriction β=1 are in general in favour of the EH, with only few exceptions: the hypothesis is rejected at the 5% level for the regression with spread f(2,3)r, and at the 10% level for the regression with spread f(14,15)r. [TABLE 4] Overall, the epirical analysis presented in this section suggests that both the longshort rate and the forwardspot rate spreads are very powerful predictors of future short interest rate changes, in accordance with the EH. On the other hand, the spread between the long and shortter interest rates has poor predictive content for changes in the longer rate, although our estiates suggest that long rates ove in the direction consistent with the EH 4. However, because our data are derived fro estiates of the ter structure, they ay introduce an approxiation error in our regressions, in which case the slope of the coefficient estiates obtained with OLS ay be biased. To explore the possible effect of easureent errors in tests of the EH, in the next section we use an instruental variable (IV) approach. 3.3 Standard regression tests: IV results In the presence of easureent error, the OLS estiators of the slope coefficients in equations (2)(4) will not converge to unity (see Mankiw, 1986, and Hardouvelis, 1994). To avoid the possible bias that a easureent error on short rates and long rates would generate, we reestiate the equations using instruental variables. The instruents are lags of the spread and lags of interest rates changes, and were selected on the basis of their ability to predict the ter spread 5. The IV coefficients should equal one, as iplied by the EH. The 12
13 results are displayed in Panel B of Tables 2, 3 and 4. Tables 2 and 4 show only inor evidence of a white noise error in the short rates. The IV estiates are in fact very siilar to the OLS estiates in Panel A, although they are now closer to the value of one, supporting the EH in all cases. On the other hand, the use of instruental variables significantly iproves the regressions for the long rates. As shown in Table 3 the coefficient estiates are uch closer to the value of one, according to the EH, for all pairs of aturities; we also notice that the negative signs in Panel A for aturities n=12,18 and =1 are now reversed. This result is robust with previous findings by Hardouvelis (1994) using different data for Gerany, and contrasts with previous results for the US. Overall, the results presented in this section suggest that the slope of the ter structure between alost any two pairs of aturities n and predicts changes in the short and long rates according to the EH. This result has iportant policy iplications for the conduct of onetary policy in Gerany. For exaple, a currently high spread reflects expectations in the arket of higher future short rates. Therefore, the interest rate spread provides onetary policy akers with useful inforation on how the arket expects future onetary policy to be conducted. 4. TIME VARYING TERM PREMIA AND EXTENDED REGRESSIONS 4.1 Proxies for tievarying ter preia Several studies have investigated the effect of tievarying ter preia in tests of the EH. The basic assuption is that the spread cobines inforation about the variation of expected future rates and ter preia. Therefore, if ter preia are tievarying and correlated with the ter spread, then estiates of the spread coefficient β in equations (2)(4) will be biased, due to oitted variables probles. Tievarying ter preia cannot be easily reconciled with the EH, and the need for a theory able to endogenise the has been underlined by any authors (see for exaple Mankiw and Suers, 1984, and Mankiw, 1986). General equilibriu odels, such as Cox, 13
14 Ingersoll and Ross (1985) and Longstaff and Schwartz (1992) offer soe answers to this proble 6. The functional for of the ter preiu resulting fro the Longstaff and Schwartz odel is particularly interesting for our epirical analysis. In fact, the ter preiu is, for any fixed aturity, a linear function of both the interest rate level and its volatility, where the sign of the relationship with the latter is indeterinate. This suggests that the short rate level and/or its volatility can be added as further regressors in equations (2)(4). Previous studies (e.g. Faa, 1990) have shown that the level of the short rate does not add uch inforation to that already contained in the spread. Therefore, in this paper we use volatility as a further source of inforation and include it as a second regressor in the extended regressions: ( n/ ) 1 ( 1 ( i / n)) R = α + β( R R ) + γtp + ε (2a) i= 1 n t+ i t t t t+ n R ( n ) t+ n R n R n t t R = α + β t + TP t + t+ ( ) ( ) γ ε (3a) ( 1 ) r r = α + β F ( n, n) r + γtp + ε (4a) t + n 1 t t t t t+ n 1 TP t is the proxy for the ter preiu, and γ is its coefficient, whose sign, according to the Longstaff and Schwartz odel, is indeterinate and depends essentially on the aturity length 7. Theoretically, in Longstaff and Schwartz (1992) the volatility is defined as the instantaneous variance of changes in the riskless rate and for its estiation the GARCH fraework is suggested. However, as the choice of the volatility proxy is ainly an epirical atter, to see if results are sensitive to the particular proxy chosen, we use three alternative easures and estiate the extended regressions (2a)(4a) for each of the. The proxies used are: (i) (ii) a oving average of absolute changes in the short rate, coputed over the previous 6 periods: 5 MA, t t i t i 1 / i= 0 TP = R R the expected square of excess holding period returns: 6 14
15 where H 2 n TP (exreturns), t = E t[ H t+ R t ] n R n n n = t R t is the the period holding period return n n t+ + on an nperiod zero coupon bond between t and t+; 2, (iii) estiates of conditional variances fro GARCH odels: TP GARCH,t = h t, where we have chosen the lag structure of GARCH(1,1) : h t = α 0 + α 1 ε 2 t1 + β 1 h t1 although various extended lag specifications were attepted. The first easure has been used in previous work by Faa (1976), Jones and Roley (1983), and Sion (1989), the second has been used by Sion (1989) and ore recently by Harris (1998). The third easure was initially proposed by Engle, Lilien and Robins (1987), and extensively used in subsequent studies. Previous studies that have allowed for the possibility of tie varying ter preia have provided contrasting results, depending on the choice of the proxy considered (see, for exaple, Sion, 1989, and Tzavalis and Wickens, 1997, for evidence with US data). By using different proxy variables we are able to check on the robustness of our results. 4.2 Extended results: the inforation content of volatility Estiates of equations (2a) and (3a) are reported in Tables 5 and 6 for each of the three proxies. Panel A shows results for regressions with short rate aturity =1, and Panel B for =3. The aturity of the longer rate selected for this exercise is n=3,6,9,12,18 8. The Tables also report the OLS and IVE results fro Tables 2 and 3 to facilitate the coparison. As the expected squared excess holding period return is replaced in the equations by its realisation, the actual squared excess holding period return contains an expectation error which is also present in the regression error, so the equations with this proxy are estiated with instruental variables. Following Sion (1989), we use as instruents lagged values of the squared excess returns. 15
16 Tables 5 and 6 show that although the volatility ter is not totally uninforative, these extended regressions represent only inor changes with respect to siple regressions which use only knowledge at tie t of the slope of the ter structure. These results indicate that ost of the inforation in the ter structure for oveents in the interest rates is contained in the spread, and confir that the evolution of both short rates and long rates is consistent with the predictions of the EH. Previous studies that have allowed for the possibility of tie varying ter preia with US data have provided contradictory results, depending on the choice of the proxy considered. A possible interpretation of such a difference ight be that Geran ter preia are relatively sall and constant, copared with US, so that ost of the inforation about future rates is actually given by the ter spread. This would explain why volatility does not add uch inforation. This result is in line with the finding by Gerlach and Sets (1997) based on Eurorates, showing that the higher inforation content concerning future shortter rates for soe countries is due to ore predictable oveents in shortter interest rates downplaying the possible ipact of tie varying risk preia. [TABLE 5] [TABLE 6] Finally, in Table 4, Panel C, we report the results for regressions with the forwardspot spread (equation 4a), but only for two of the ter preiu proxies: TP MA and TP GARCH. These results are qualitatively very siilar to those for regressions in Table 5. Specifically, the volatility ter is significant in only 2 cases out of 10, and there is only a arginal iproveent in ters of. The coefficient estiates of β reain highly significant, confiring that the forward preiu is a powerful predictor of expected changes in the short rate. 5. CONCLUSIONS AND FURTHER REMARKS In this paper we have exained the inforation content of the ter structure and tested the Expectations Hypothesis for the case of Gerany, using a new data set constructed fro the estiation of the ter structure. The EH has been tested by eploying two approaches: one 16
17 has used the spread between the long rate and the short rate to predict future oveents in both longer and shorterter interest rates; the other the spread between the forward rate and the spot rate to predict changes in the spot rate. Standard regression tests of the EH have been conducted by using both OLS and IV estiation. The latter has been adopted to account for possible easureent errors introduced by the use of data derived fro estiated ter structures. Moreover, inspired by the ost recent general equilibriu stochastic odels of the ter structure, we have extended the standard fraework by including alternative easures of interest rate volatility as proxies for the ter preiu. The data used in this paper have enabled us to produce ore coprehensive results than those obtained in previous studies for Gerany, and several interesting findings have eerged fro the epirical analysis. First, our results suggest that both the ter spread and the forwardspot spread are very powerful predictors of future short interest rate changes, in accordance with the EH. This is in strong contrast with previous evidence for the US, where, unlike in Gerany, interest rate targeting has been the priary target of onetary policy. In this respect, our results support the arguent put forward by Mankiw and Miron (1986), and later confired by epirical evidence for other countries (see aong others, Kugler, 1988, and Engsted, 1996), that the predictive power of the spread is stronger under onetary targeting than under interest rate targeting. Second, although the slope of the ter structure alone has less predictive power for longer ter interest rates, an interesting result fro our analysis is that both the sign and the value of the coefficient estiates are coherent with the predictions of the EH. A high spread is followed by an increase in the long rate, and once easureent errors are taken into account with instruental variables, the coefficient estiates are close to one. So, the Geran yield curve confors to the EH in its predictions of changes in the long rates as well as in the short rates. This results contrast with previous findings for the US, and suggest that in Gerany the spread can be used as an iportant indicator for the conduct of onetary policy. Notes 1 For further details on the data see Boero, Madjlessi and Torricelli (1995). The data are available fro the authors on request. 2 All regressions presented in this section include a constant ter which is never significantly different fro zero at the 1% level, and is therefore not tabulated. 3 One of the referees suggested to us that we should not interpret this result as a sign for no puzzle regarding the predictions of longter interest rates. In fact, with approaching n we have the longter forecast of a shortter interest rate. 17
18 4 These results are robust to different saple periods. Specifically, estiation over different subsaples yielded a predoinance of coefficient estiates close to one in regressions for the short rates, and a predoinance of positive signs in regressions for the long rates. 5 For errors which follow an MA(q) process, we use as instruents variables lagged tq1 periods or earlier. 6 See Boero and Torricelli (1996) for a coparative discussion of these and other stochastic odels of the ter structure. 7 See Longstaff and Schwartz (1992), page Results for other pairs of aturities were qualitatively siilar, and are therefore not reported here. 18
19 REFERENCES Boero G. and C. Torricelli (1996) A coparative evaluation of alternative odels of the ter structure of interest rates, European Journal of Operational Research, 93, Boero G. and C. Torricelli (1997) The Expectations Hypothesis of the Ter Structure: Evidence for Gerany, Contributi di Ricerca CRENoS, 97/4. Boero G., F. Madjlessi F. and C. Torricelli (1995) The ter structure of Geran interest rates: tests of the inforation content, University of Karlsruhe, Diskussionspapier N.187, pp Capbell, J.Y. and R.J. Shiller (1987) Cointegration and tests of present value odels, Journal of Political Econoy, 95, Capbell, J.Y. and R.J. Shiller (1991) Yield spreads and interest rate oveents: a bird s eye view, Review of Econoic Studies, 58, Chabers D., W. Carleton, D. Waldan (1984) A new approach to estiation of the ter structure of interest rates, Journal of Financial and Quantitative Analysis, 19, Cox J., Ingersoll J.E. and S.A. Ross (1985) A theory of the ter structure of interest rates, Econoetrica, 53, Dotsey M. and C. Otrok (1995) The rational expectation hypothesis of the ter structure, onetary policy, and tie.varying ter preia, Econoic Quarterly, Federal Reserve Bank of Richond, 81, Engle, R.F., D.M. Lilien and R.P. Robins (1987) Estiating TieVarying Risk Preia in the Ter Structure: the ARCHM Model, Econoetrica, 55, Engsted, T. (1996) The predictive power of the oney arket ter structure, International Journal of Forecasting, 12, Estrella A. and G.A. Hardouvelis (1991) The ter structure as a predictor of real econoic activity, Journal of Finance, Evans M.D.D. and K.K. Lewis (1994) Do stationary risk preia explain it all?  Evidence fro the ter structure, Journal of Monetary Econoics, 33, Faa E.J. (1976) Forward rates as predictors of future spot rates, Journal of Financial Econoics, 3, Faa E.J. (1984) The inforation in the ter structure, Journal of Financial Econoics, 13, Faa E.J. (1990) Terstructure forecasts of interest rates, inflation, and real returns, Journal of Monetary Econoics, 25,
20 Faa E.J. and R.R. Bliss (1987) The inforation in longaturity forward rates, Aerican Econoic Review, 77, Gerlach S. (1995) The inforation content of the ter structure: evidence for Gerany, Epirical Econoics, 22, Gerlach, S. and F. Sets (1997) The Ter Structure of Eurorates: soe evidence in support of the expectations hypothesis, Journal of International Money and Finance, 16, Hansen, L.P. and R.J. Hodrick (1980) Forward Rates as Optial Predictors of Future Spot Rates, Journal of Political Econoy, 88, Hardouvelis G.A. (1994) The ter structure spread and future changes in long and short rates in the G7 countries  Is there a puzzle?, Journal of Monetary Econoics, 33, Harris, R.D. (1998) A Test of the Expectations Hypothesis of the Ter Structure Using CrossSection Data, Paper presented at the Royal Econoic Society Conference, Warwick, Hsu C. and P. Kugler (1997) The revival of the Expectations Hypothesis of the US ter structure of interest rates, Econoics Letters, 55, Jondeau E. and R. Ricart (1999) The expectations hypothesis of the ter structure: tests on US, Geran, French, and UK Eurorates, Journal of International Money and Finance (18)5, Jones, D. S. and V.V. Roley (1983) Rational Expectations and the Expectations Model of the Ter Structure: a Test Using Weakly Data, Journal of Monetary Econoics, 12, Jorion P. and F. Mishkin (1991) A ulticountry coparison of terstructure forecasts at long horizons, Journal of Financial Econoics, 29, Longstasff F.A. and E.S.Schwartz (1992) Interest rate volatility and the ter structure: a twofactor odel, Journal of Finance, 47, Mankiw N.G. (1986) The ter structure of interest rates revisited, Brookings Papers on Econoic Activity, 1, Mankiw N.G. and L.H. Suers (1984) Do longter interest rates overreact to shortter interest rates?, Brookings Papers on Econoic Activity, 1, Mankiw N.G. and J. Miron (1986) The changing behaviour of the ter structure of interest rates, Quarterly Journal of Econoics, CI, McCulloch, J.H. (1971) Measuring the ter structure of interest rates, Journal of Business, 44,
21 McCallu B.T. (1994) Monetary policy and the ter structure of interest rates, NBER Working Paper Series, N Mills, T.C. (1991) The ter structure of UK interest rates: tests of the expectations hypothesis, Applied Econoics, 23, Mishkin, F.S. (1988) The Inforation in the Ter Structure: Soe Further Results, Journal of Applied Econoetrics, 3, 4, Newey, W.K. and K.D.West (1987) A Siple, Positive Definite, Heteroscedasticity and Autocorrelation Consistent Covariance Matrix, Econoetrica, 55, Roberds, W. and C. H. Whitean (1999) Endogenous ter preia and anoalies in the ter structure of interest rates: Explaining the predictability sile, Journal of Monetary Econoics, 44, Rudebusch G.D. (1995) Federal Reserve interest rate targeting, rational expectations, and the ter structure, Journal of Monetary Econoics, 35, Shiller, R.J. (1979) The volatility of long ter interest rates and expectations odels of the ter structure, Journal of Political Econoy, 87, Sion, D.P. (1989) Expectations and risk in the Treasury bill arket: An instruental variables approach, Journal of Financial and Quantitative Analysis, 24, Tzavalis, E. and M. R. Wickens (1997) Explaining the Failures of the Ter Spread Models of the Rational Expectations Hypothesis of the Ter Structure, Journal of Money, Credit, and Banking, 29, 3,
22 FIGURE 1 Ter structure of Geran interest rates (a) Ter structure evolution (b) Ter structure evolution
23 FIGURE 2 Interest rates R1M R3M R18M R9M 23
24 FIGURE 3 n=6, =1 n=9, = V V SP61 (a) : Shortrates regressions SP93 n=3, =1 n=6, = L L SP31 (b) : Longrates regressions SP D3R D6R FR31 (c) : Forwardspot spread regressions The figure shows scatterplots of the dependent variables of equations (2)(4) against the relevant spreads. In Panel (a), Vn denotes the change in the short rate with aturity, over the n period horizon; SPn is the spread between the long rate R n and the short rate R. In Panel (b), Ln denotes the change in the long rate R n over the period horizon; SPn is the spread (R n  R ) ultiplied by (/(n)). In Panel (c), DnR1 denotes the change in the oneonth rate over the n1 horizon; FRn1 is the spread between the forward rate F(n1, n) and the oneonth spot rate. FR61 24
25 TABLE 1 Suary statistics for the Geran interest rates R 1 R 3 R 9 R 18 Mean Maxiu Miniu Std. Dev
26 ( n/ ) 1 TABLE 2 Short rate regressions : OLS and IV estiates ( 1 ( i / n)) R = α + β( R R ) + ε (2) i= 1 n t+ i t t t+ n Panel A: OLS estiates n = β OLS =3 β OLS =6 β OLS =9 β OLS 0.74 (0.15) [2.97] c (0.10) [7.19] a (0.13) [7.69] a (0.10) [4.10] b (0.13) [3.20] c (0.10) [1.44] (0.14) [0.76] (0.22) [0.10] (0.15) [1.74] (0.26) [0.27] (0.28) [0.03] (0.36) [0.04] continued Panel A reports OLS estiates of β in regressions for the short rates, NeweyWest standard errors in parentheses below estiated coefficients, Ftests associated with the hypothesis that β=1 in square brackets, and the value. a, b, and c denote statistical significance at the 1%, 5% and 10% respectively. The estiation period used for each regression is the longest possible using data fro 83:12 through 94:12 26
27 TABLE 2 (continued) Panel B: IV estiates n = β IVE (0.21) [1.83] (0.35) [0.04] (0.39) [0.35] (0.16) [0.003] =3 β IVE (0.30) (0.39) (0.17) [0.002] [0.01] [0.28] =6 β IVE 1.29 (0.50) [0.34] =9 β IVE 1.24 (0.16) [2.29] 1.12 (0.26) [0.22] 1.20 (0.67) [0.09] 1.07 (0.58) [0.02] Panel B reports results obtained with IVE to account for sall saple bias due to easureent errors. The instruents used are lags of the ter spread and lags of interest rates changes, and were selected on the basis of their ability to predict the ter spread. are not reported in the case of IV regressions as their use as easures of goodness of fit is not valid. 27
28 R ( n ) t+ TABLE 3 Long rate regressions: OLS and IV estiates n R t n R n t R = α + β t + t+ ( ) ( ) ε (3) Panel A: OLS estiates n = β OLS =3 β OLS =6 β OLS =9 β OLS 0.70 (0.36) b [0.73] 0.03 See notes to Table (0.33) [4.53] b (0.27) [7.70] a (0.38) [5.72] b (0.16) [29.15] a (0.44) [1.70] (0.46) [5.28] b (0.14) [44.5] a (0.44) b [0.10] (0.71) [0.57] continued (0.66) [2.89] c (0.13) [49.5] a (0.55) c [0.003] (0.72) [0.04]
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