Term structure of interest rates and the expectation hypothesis: The euro area

European Journal of Operational Research 185 (2008) 1596–1606 www.elsevier.com/locate/ejor

Term structure of interest rates and the expectation hypothesis: The euro area Silvana Musti

a,*

, Rita Laura D’Ecclesia

b

a

b

Dipartimento di Scienze Economiche, Matematiche e Statistiche, Universita` degli Studi di Foggia, largo Papa Giovanni Paolo II, 71100 Foggia, Italy Dipartimento di Teoria Economica e Metodi Quantitativi per le Scelte Politiche, Universita` degli Studi di Roma ‘‘La Sapienza’’, p.le Aldo Moro 5, 00185 Roma, Italy Available online 27 November 2006

Abstract This paper investigates the informational content of the yield curve in the European market using data on the Italian term structures. According to the expectation hypothesis theory (EHT) the current forward rate equals the future short rate plus a constant risk premium that is time invariant but maturity dependent. This theory has been widely tested in the empirical literature providing various findings according to the country where it has been applied and to the segment of the yield curve examined or the period under study. The standard approach to test the EHT uses the regression techniques assuming data on spot rates and their first differences to be stationary. Recently an increasing number of studies evidenced the non stationarity of interest rates time series and some tests of the EHT are formulated using term spread and forward-spot spread which are stationary. A new strand of literature suggests to investigate the EHT using a restricted VAR framework. In this paper, following [Jondeau, E., Ricart, R., 1999. The expectations hypothesis of the term structure: tests on us, german, french and uk euro-rates. Journal of International Money and Finance 18, 725–750, Ghazali, N.A. Low, S.W., 2002. The expectations hypothesis in emerging financial markets: the case of malaysia. Applied Economics 34, 1147–1156 and Seo, B., 2003. Non linear mean reversion in the term structure of interest rates. Journal of Economic Dynamics and Control 27, 2243–2265], we test if the expectation hypothesis holds using cointegration and error correction analysis. For the period under study results suggest that the long and short term interest rates are cointegrated and therefore subject to a long equilibrium path, providing evidence that the EHT holds for the Italian and the European market.  2006 Elsevier B.V. All rights reserved. Keywords: Term structure of interest rates; Expectations hypothesis; Error correction model; Cointegration

1. Introduction

*

Corresponding author. Tel.: +39 881 753724; fax: +39 881 775616. E-mail addresses: [email protected] (S. Musti), rita.decclesia@ uniroma1.it (R.L. D’Ecclesia).

Over the last decade, many empirical studies have examined the information provided by the slope of the yield curve and how this information may be used to analyze future macroeconomic conditions, i.e. condition of recession, or to select securities that

0377-2217/$ - see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2006.08.034

S. Musti, R.L. D’Ecclesia / European Journal of Operational Research 185 (2008) 1596–1606

have to be included in a fixed income portfolio. Estrella and Mishkin (1998) documented that the spread between the ten-year Treasury bond rate and the three-month Treasury bill rate performs better than composite indices of leading indicators when tempting to forecast the economic recessions in the US, especially at a horizon beyond one quarter. The expectations theory formulated by Fisher (1930), Keynes (1930) and Hicks (1953) still provides a good explanation of the determination of long term interest rates and this theory states that long-term interest rates are determined by the expectations on the future short-term interest rate. According to the expectations hypothesis theory (EHT) of the term structure forward rates are useful predictors of future spot interest rates. Fama (1984) finds that forward rates can predict spot rates one month ahead. However, most of the empirical evidence suggests that the relation between forward rates and expected returns is stronger than the relation between forward rates and future spot rates. Davis (2000) provides an update of Fama’s (1984) results, showing that there is more information in the term structure about expected returns than there is about future interest rates. This means that forward interest rates cannot predict future interest rates but may provide information on expected returns of fixed income instruments. It is one of the most empirically tested explanation for the relationship between short-term and long-term interest rates. Empirical tests of the EHT for the term structure are too numerous to list here, tests have been applied to the US data and the results are mixed. An excellent survey of the empirical findings may be found in Cook and Hahn (1990) and Campbell and Shiller (1991). Most of the literature of the 90s provides evidence of the existing relationship between short term and long term interest rates for some countries of the European Union, and for the US. Campbell and Shiller (1991), Bekaert and Hodrick (2001) among others reject this hypothesis for the US market. One possible explanation for this rejection may be provided by the presence of single regime shifts in the observed data since they use data related to the post war period. Other studies as Longstaff (2000) and Mankiw and Miron (1986) find support of this theory when using data for the period before the founding of the Federal Reserve System. Hardouvelis (1994), Mills (2002), Dahlquist and Jonsson (1995), Gerlach and Smets (1996), Bekaert

1597

et al. (1997), Jondeau and Ricart (1999), Engsted (1996), Hejazi et al. (2000), Ghazali and Low (2002), Boero and Torricelli (2002) provide some evidence of the EHT for other countries. Most of the works examined show that EHT is easily supported in the European countries, while it is not supported in the US. Recent studies, Hurn et al. (1995) and Engsted (1996), provide evidence of EHT when short end of the yield curve is considered, in addition, using money market rates for the Danish and Swedish market, Dahlquist and Jonsson (1995), support this finding. An analysis centered exclusively on the Euro area is still unavailable, given the lack of a complete historical data set. However, since January 1999 money market rates have been showing a clear and stable relationship between all the countries of the euro area so, preliminarily, information of the Euro area may be detected using a set of domestic rates over this period. The EHT implies that the yield curve spread between the short rate and the long rate represents the market prediction on the future short-term rate evolution: the long-term interest rate is determined by averaging the current and expected future short rates of interest plus a constant risk premium, which is set to zero in the pure expectations hypothesis. The aim of this paper is to investigate the information provided by the yield curve of the italian market and verify if the EHT holds. Given that spot rates and forward rates are proven to be non-stationary, standard regression techniques do not provide an efficient tool to validate the EHT. Most of the formulations of the standard tests of the EHT are derived in terms of spreads and changes in market rates. In this paper we follow the approach adopted by Jondeau and Ricart (1999) and by Ghazali and Low (2002) and investigate the existence of the process of adjustment between two interest rates of different maturities using a cointegration analysis and an error correction representation (ECM). We find some long run equilibrium path between the short term and the long term rate providing evidence that the EHT holds in the Italian and in the Euro market. The article is organized as follows. Section 2 revises the expectation hypothesis theory and the methodology employed in the analysis. Section 3 introduces the data set and Section 4 describes the empirical analysis. Section 5 summarizes the results and some brief conclusions are presented.

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2. The expectation hypothesis of the term structure Theoretical studies (Vasicek (1977); Richard (1978); Cox et al. (1985); Hall et al. (1992)) of the modern term structure of interest rates suggest that there is an equilibrium (steady-state) relationship among the interest rates of different maturities. This implies that short-term rates cannot continually drift away from long-term rates. Many of the different theories advanced by economists to explain the shape of the yield curves include expectations, segmented markets and preferred habitat theories. The EHT states that longer-term interest rates are an average of the shorter-term interest rates expected to prevail during the life of the longer-term asset. This implicitly assumes that rational investors are risk-neutral, and thus would not pay a premium to lock in a longer-term interest rate. In addition, it assumes that the transaction costs are zero, so that there should be no expected difference in the returns from holding a long term bond or rolling over a sequence of short term bonds. It follows, from the traditional interpretation of the expectations view, that the long-term interest rate can be represented as an average of the future expected short term rates plus a term premium. Much of the recent research has focused on whether this prediction of the theory is supported by the data. A surprising finding is that parts of the yield curve have been useful in forecasting interest rates while other parts have not. Two concepts central to the tests of the expectations theory reviewed below are the ‘‘forward rate premium’’ and the ‘‘term premium’’. Suppose an investor can purchase a six-month Treasury bill now or purchase a three-month bill now and reinvest his funds three months from now in another three-month bill. The forward rate is the hypothetical rate on the three-month bill, three months in the future that equalizes the rate of return from the two options, given the current three- and six-month ð3;6Þ rates. The forward rate, F t , calculated from the ð6Þ current six-month rate Rt and the current threeð3Þ month rate Rt , is defined as ð3Þ ð3;6Þ Þ; ð1 þ Rð6Þ t Þ ¼ ð1 þ Rt Þð1 þ F t

ð1Þ

where the yields are simple unannualized yields. Using continuously compounded yields the forward rate can be expressed as an additive (rather than a multiplicative) function of the current six- and three-month rates. In terms of annualized yields the forward rate becomes: ftð3;6Þ ¼ 2rtð6Þ  rtð3Þ ;

ð2Þ

where annualized yields are denoted by lower case letters. The forward rate premium is defined as the difference between the forward rate and the current short-term spot rate: n ðmÞ ðmÞ ¼ ð2rðnÞ ¼ ðrtðnÞ  rðmÞ ftðm;nÞ  rðmÞ t t  rt Þ  rt t Þ: m ð3Þ When the maturity of the long-term rate is twice the maturity of the short-term rate the forward rate premium is simply twice the spread between the longand short-term rates ð3Þ ð3Þ ð6Þ ð3Þ ftð3;6Þ  rtð3Þ ¼ ð2rð6Þ t  r t Þ  rt ¼ 2ðr t  rt Þ:

ð4Þ 2.1. Methodology approach In general, for each maturity m and n, m < n, the term premium, kf, is defined as the difference between the forward rate and the corresponding expected spot rate: ðm;nÞ

kf

ðmÞ

¼ ftðm;nÞ  Et rtþnm ;

ð5Þ

where Et denotes the mathematical expectation conðm;nÞ ditional on information available at time t, ft is the yield on a forward investment at t in n  m periods on a security maturing at t + n and the expected ðmÞ yield, rtþnm , of an investment at time t + n  m having maturity m. Eq. (5) can be rewritten in terms of the forward rate premium by rearranging terms and ðmÞ subtracting rt from both sides: ðmÞ

ðm;nÞ

ðm;nÞ ½Et rtþnm  rðmÞ  rðmÞ t  ¼ ðft t Þ  kf

:

ð6Þ

This expression now decomposes the forward rate premium into the expected change in the spot rate and a term premium. An equivalent decomposition of the forward rate premium, used in some papers, employs the concept of holding period yield, which is the return earned on a security sold prior to maturity. The forward rate premium can be divided into: (1) the expected change in the m-month rate and (2) the difference between the expected holding period yield, earned by investing in a n-month bill and selling it when it is a m-month bill (n  m) months in the future, hðm;nÞ , and the t return from investing in a m-month bill:1 ðmÞ

ðm;nÞ ðmÞ ðftðm;nÞ  rðmÞ  rðmÞ t Þ ¼ ½Et r tþnm  rt  þ ½Et ht t ;

ð7Þ 1

Fama (1986) and Fama and Bliss (1987) derive this decomposition.

S. Musti, R.L. D’Ecclesia / European Journal of Operational Research 185 (2008) 1596–1606

where the holding term premium is defined as ðm;nÞ kh

¼

Et htðm;nÞ



rðmÞ t :

ð8Þ

This relationship, following Jondeau and Ricart (1999), may be rewritten expressing the relationship between the expected change in the yield of a long term bond and the term spread: h i m m ðnmÞ ðm;nÞ Et rtþm  rtðnÞ ¼ ðrðnÞ  rðmÞ k : t Þ nm t nm h ð9Þ The EHT is based on two assumptions about the behaviour of participants in the money market. The first is that the term premium that market participants demand for investing in one maturity rather than another is constant over time. The second is that the term spread- the spread between the long term and the shot term rate- is an unbiased predictor of future short run changes in long-term rates as well as of future cumulative changes in short term rates, Mankiw (1986), Mankiw and Miron (1986), Campbell (1995), Evans and Lewis (1994). Under this assumption Eq. (6) is the focus of most of the recent empirical work testing the expecðm;nÞ tations hypothesis and kf is assumed to be a conðmÞ stant term premium. In Eq. (6) the values of Et rtþnm is unknown. The procedure generally used to get these values is to assume that interest rate expectations are formed ‘‘rationally’’, so that: ðmÞ

ðmÞ

rtþnm ¼ Et rtþnm þ etþnm ;

ð10Þ

where et+nm is a forecast error that has an expected value of zero and is assumed to be uncorrelated with any information available at time t. The ideas behind the rational expectations assumption are that • there is a stable economic environment; • market participants understand this environment; • they should not systematically over- or underforecast future interest rates; • they should not ignore any readily available information that could improve their forecasts. This assumption specifically requires that forecast errors are not correlated with the forward rate premium at time t or its two components, the expected change in interest rates and the expected term premium. Substituting (10) into (5) yields the following regression equation: ðmÞ rtþnm



rðmÞ t

¼ af þ

bf ðftðm;nÞ



rðmÞ t Þ

þ uf ;tþnm : ð11Þ

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Under the rational expectations assumption the error term in equation (11) is uncorrelated with the right-hand side variable so that the coefficient bf can be estimated consistently. The theory predicts that bf should not differ significantly from one. A significantly different value would contradict either the assumption of a fixed term premium or the rational expectations assumption. An estimated coefficient of zero would be evidence that the forward rate premium has no forecasting power for the subsequent behaviour of the three-month rate. An alternative to Eq. (11), used by Fama (1984), replaces the change in the m-month spot rate with the holding period premium: m ðnmÞ rtþm  rðnÞ ½rðnÞ  rðmÞ t ¼ a h þ bh t  þ uh;tþm : nm t ð12Þ In Eqs. (11) and (12) the analysis of the coefficients ai and bi, for i = f, h, which represent the term premium, are used to verify the EHT assumptions. In its pure form the EHT implies that ai = 0, and bi = 1,i = f, h, but in empirical works the null premium hypothesis ai = 0 is often neglected to concentrate on parameter bi being equal to one. This is consistent with constant risk premia over time but maturity dependent. Formulations (11) and (12) to validate the EHT try to overcome the problem of running regression estimates using non-stationary variables (the interest rates levels). They use as dependent variable the term spread and try to work with rates spread which are in most cases stationary. In such a context we believe that a cointegration framework may be more adequate to analyze the relationship between interest rates of different maturities, so we suggest an alternative formulation of Eqs. (11) and (12). 2.2. An error correction model framework Recent studies have outlined how economic time series display significant stochastic trends over time. Evidence of non-stationarity in the interest rates series has been documented in various studies. This allows cointegration tests to be performed to validate the EHT of term structure employing alternatively the non-stationary level series as dependent and independent variables2. 2

The test proposed considers the possibility that a bivariate process may have only zero or one cointegrating vector. Tests for multiple cointegrating vectors are provided by Stock (1987) and Johansen (1991)

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S. Musti, R.L. D’Ecclesia / European Journal of Operational Research 185 (2008) 1596–1606

From the seminal work by Engle and Granger (1987) it was pointed out how, even though economic time series may be characterized as a random walk process, there may exist some linear combination of the variables that, over time, converges to an equilibrium. If separate economic series are stationary only after differencing and they have a stationary linear combination of their levels, then the series are said to be cointegrated. Cointegration tests are appealing in the context of the expectation hypothesis that calls for an equilibrium relationship between the interest rates of different maturities. If the short-term and long-term interest rates are cointegrated it means that, over the long run, they move in tandem with each other and this can be used as evidence of the EHT. In this context we perform a cointegration test on interest rates data and use an error correction model (ECM) framework to validate the EHT on the Italian market. We consider as dependent variable and indepenðmÞ dent variable alternatively the short term yields rt ðnÞ and the long term yields, rt : rðmÞ ¼ g0 þ g1 rtðnÞ þ t ; t

ð13Þ

rtðnÞ

ð14Þ

¼ u0 þ

u1 rðmÞ t

þ tt :

Eqs. (13) and (14) allow us to measure the pattern of adjustment between short and long term interest rates using their long run equilibrium path. We estimate the two error series, t and tt, to analyze the relative strength of the adjustment process long-toshort versus short-to-long hypothesis. The t represents deviation of actual short-term interest rates from the long-run equilibrium path driven by movement in long term interest rates, while the tt series is the reverse. In this context an ECM could be set up to represent a cointegration relationship between two time series. Two variables that are cointegrated can be seen as subjected to a common attractor. In the case of EHT it is interesting to identify which of the interest rates assume the role of attractor or if both are adjusting towards each other. The ECM is estimated using both of the error series separately in order to capture this process of adjustments to long run equilibrium. The two error correction equations are defined as follows: DrðmÞ ¼ det1 þ t

p X i¼1

ðmÞ

/r;i Drti þ

q X

ðnÞ

qrðnÞ ;i Drti þ xt ;

i¼1

ð15Þ

DrðnÞ t ¼ stt1 þ

p X i¼1

ðnÞ

krðnÞ ;i Drti þ

q X

ðmÞ

prðmÞ ;i Drti þ nt :

i¼1

ð16Þ Eq. (15) describes the movement of short-term interðmÞ est rates with maturity m, rt , m < n, in a given period t and it depends on how far it deviates from the long run equilibrium path at time t  1 that is ðnÞ dictated by long-term interest rates, rt . The error correction coefficient, d, measures the speed of adjustment toward the equilibrium path. If jdj is significantly different from zero, then, changes in short term rates are subjected to the ‘equilibrium error’ and the long-to-short version of the EHT is supported. Interpretation for (16) is the reverse with short-term interest rates determining the equilibrium path for long-term rates. Comparison between the relative size of absolute error correction coefficients, jdj, jsj leads us to identify the interest rates that act as the force of attractor. If jdj > jsj, then the long-term interest rate plays a stronger role as attractor that determines the longrun equilibrium path and vice versa. The significance of the coefficients, qrðnÞ ;i , for lagged differences of long-term rates in Eq. (15) provides evidence of causality from long- to short-term interest rates. Short run causality from short- to long-term interest rates is evidenced by the significance of the coefficient, prðmÞ ;i in Eq. (16). This simple approach is used to test if an equilibrium relationship between the interest rates of different maturities may be detected and therefore provide a first evidence of the EHT. 3. Data set The data set used in this analysis is composed of six interest rates measures of the securities issued by the Italian government, namely: three-, six- and twelve-month Buoni Ordinari del Tesoro (BOT3M, BOT6M and BOT12M) and three-, five- and tenyear Italian government bonds (BTP3Y, BTP5Y and BTP10Y). Data on Treasury bills rates and on Italian Government bonds are collected from the Monthly Statistical Bulletin of the Bank of Italy. Monthly observations of these rates are compiled for 1993:1 to 2003:6. In order to use the Italian market data as representative of the Euro area we analyze the relationship between the Italian term structure and the Euro term structure. We collect data on Euro rates. The Euro area has only few years of life, so it was possible to get interest rates

S. Musti, R.L. D’Ecclesia / European Journal of Operational Research 185 (2008) 1596–1606

for three different maturities, 3, 6 and 12 months, are reported for the entire Euro area and for the Italian market showing how the rates of the Italian market can be used to analyze the feature of the Euro market. The term structure for the 1993–1998 period in the Italian market is described in Fig. 4. One of the assumption of the EHT concerns the behaviour of participants in the money market. It states that the term premium the market participants demand for investing in one maturity rather than another is constant over time. Under this

for the Euro area from the ECB bulletin only starting from January 1999. The available data refer to 3 months, 1 year, 2 year, 5 years and 10 years maturity. There is an interesting piece of literature on creating historical Euro zone data (Gulde and Schulze-Gattas (1992); Winder (1997); Hong and Beilby-Orrin (1999)) but it is beyond the aim of this work. The analysis of the existing Euro rates for the period 1998–2003 provides some evidence of the strong relationship existing between the Euro rates and the Italian rates. In Figs. 1–3, the spot rates

Spot rates t=3 months

Euro Italy

8

%

6 4 2 0 Jan.98

Jan.99

Jan.200

Jan.01 1998-2003

Jan.02

Jan.03

Jan.02

Jan.03

ð3Þ

Fig. 1. qðreð3Þ ; rI Þ ¼ 0:823.

8

Spot rates t=6 months

%

6 4 2 0 Jan.98

Jan.99

Jan.200

Jan.01 1998-2003

Fig. 2.

ð6Þ qðreð6Þ ; rI Þ

¼ 0:883.

Spot rates t=12 months

6 5 %

4 3 2 1 0 Jan.98

Jan.99

Jan.200

1601

Jan.01 1998-2003 ð12Þ

Fig. 3. qðreð12Þ ; rI Þ ¼ 0:923.

Jan.02

Jan.03

1602

S. Musti, R.L. D’Ecclesia / European Journal of Operational Research 185 (2008) 1596–1606

7 6 5 4 % 3 2

Jan-98

1 3m 12m 5y 30y

0 Jun 2003

Fig. 4. Italian term structure over the period 1994–2003.

assumption, rearranging Eq. (6), using the expression for the forward rate obtained in (3) we obtain m ðm;nÞ n  m ðmÞ m ðmÞ rt þ Et rtþnm rtðnÞ ¼ kf þ ð17Þ n n n which shows that the long term rate is equal to and average of the current and expected short term rates plus a constant, in the case kf = c, which reflects the term premium. In Fig. 5 the term premium, defined by Eq. (5), and the forward-spot spread, for m = 3 and n = 6 over the period 1994–2003 are reported. It is interesting to notice how, starting from 1998, when the Euro area became effective, the term premium tends to stabilize and to provide some evidence for the EHT. 4. Empirical results We first test our data for stationarity using the Augmented Dickey Fuller (ADF) unit root tests Dickey and Fuller (1979, 1981). The test is based on the following ordinary least squares (OLS) estimation Dxt ¼ c þ bT þ axt1 þ

k X

ui Dxti þ lt ;

ð18Þ

i¼1

where D is the first-difference operator, xt is the series tested, T is a linear time trend and lt is a covariance stationary random error. The appropriate number of lagged difference k is determined by Akaike Information Criteria (AIC) due to Akaike (1970). Optimal choice of lag length removes autocorrelations in error term. Three versions of ADF tests are performed varying in the inclusions of con-

stant term c and time trend (T). The null hypothesis of unit root, jaj = 1, is tested against the alternative of stationarity, jaj < 1. Table 13 provides the ADF tests for the short and long-term interest rates in both levels and first differences. Table 2 provides the ADF test for the forward rates. At the 1% level of significance the null hypothesis of unit root cannot be rejected for all interest rates. Similar to many previous studies on interest rates dynamics, interest rates in Italy posses a unit root which preclude conclusions derived from OLS estimations using level measurement. However, the unit root hypothesis is uniformly rejected for the first difference series. Thus, Italian interest rates can be characterized as integrated of order one, I(1), fulfilling the precondition for applying the two-step EG cointegration test. The cointegration regression described by Eqs. (13) and (14) and the ADF tests are performed on both the error series, t and tt, separately in order to verify the cointegration relationship between the short and long-term interest rates. The ADF test results on t and tt are reported in Table 3. For all the different maturities the ADF test results significative showing that the couple of considered rates are cointegrated. In the first three columns of Table 3 the shorter-term interest rate is used as dependent variable in the cointegration regression, the results provide evidence of cointegration among all of the shorter and longer term rates. In the second part of Table 3 the cointegration relationship is considered using the longer-term rates as dependent variable. In this case the cointegration relationship is also confirmed for all the maturities considered. This evidence of cointegration supports the validity of expectation hypothesis in the context of the Italian financial markets. The short and the long-term interest rates show a long run equilibrium path. The ECM is also performed in order to identify the possible attractor in this long run equilibrium. We estimate Eqs. (15) and (16), and the estimates of the coefficients are reported in Table 4a and 4b. Results are reported in Table 4a. For 10 out of 15 estimations the error correction coefficient, d, is significantly different from zero supporting the cointegration results and suggesting that in the long run, movement of Italian short rates is influenced by

3

In the empirical analysis data refer to spot rates at time t = 0 so the time indicator is omitted and we simply refer to r(m).

S. Musti, R.L. D’Ecclesia / European Journal of Operational Research 185 (2008) 1596–1606

1603

6.000 4.000

term premium for-spot spread

%

2.000

Jan-03

Jan-02

Jan-01

Jan-00

Jan-99

Jan-98

Jan-97

Jan-96

Jan-95

-2.000

Jan-94

0.000

-4.000 1994-2003 Fig. 5. Term premium and forward- spot spread for m = 3 months, n = 6 months.

Table 1 Augmented Dickey–Fuller (ADF) tests for spot rates Rates

Level ADF

r(3) r(6) r(12) r(36) r(5y) r(10y) *

First difference ADF

ADF

ADF

ADF

ADF

No c, no T

With c

With c, T

No c, no T

With c

With c, T

1.6864 1.8829 2.9528* 1.3126 1.2600 1.2858

0.4522 0.6284 1.5883 0.2972 0.2964 0.3776

1.6510 1.5853 1.7261 1.7700 1.7733 1.6477

10.699* 12.382* 10.170* 4.2180* 4.0840* 4.1200*

10.956* 12.747* 10.552* 4.3312* 4.1851* 4.2092*

10.909* 12.692* 10.507* 4.3329* 4.1883* 4.2074*

Indicates significance at the 5% level.

Table 2 Augmented Dickey–Fuller (ADF) test for forward rates (3,6)

f f (3,12) f (3,36) f (6,12) f (6,36) f (12,36)

ADF on f(m,n)

ADF on Df(m,n)

1.5141 1.9462 0.4362 2.3898 0.5838 0.7844

14.4664 10.7897 4.40206 12.1758 9.60560 9.64602

the deviation from the long run equilibrium path. On the other hand, if we look at Table 4b we may notice that only 4 out of 15 estimates report an error coefficient, s, significatively different from zero. In this case we have to take into account the choice of the specification we are testing and may notice that the long term rate plays the role of attractor in the long run equilibrium path. The absolute size of the coefficients, jdj is greater than jsj, confirming the hypothesis that the long term rates plays the role of attractor in the long run equilibrium path, so movements in short term interest rates adjust more

significatively to its deviation from the equilibrium path driven by the long-term interest rates, while adjustments of long term interest rates are relatively slower where they occurs. If we look at the coefficient, qrðnÞ ;i , of the lagged variables to analyze the short run causality from long to short term interest rates, we notice that 10 out of 15 estimates support one way causation from long to short term interest rates. While in only three cases the coefficient, prðmÞ ;i , of the lagged short term rates in Eq. (16) results significative. It is interesting to notice that the short term spot rate r(3) shows significative relationship with the lagged long run rates. In addition, the error coefficient, d, results significatively different from zero in all the regressions performed, considering as independent variables the rates of longer maturities. This supports the evidence that longer term maturities are essential for the definition of short term rates. An analysis of the forward-term spread for m = 3 and n = 6 provides further support of the expectation hypothesis for the Italian market. This is in line

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S. Musti, R.L. D’Ecclesia / European Journal of Operational Research 185 (2008) 1596–1606

Table 3 Evidence of cointegration between interest rates Dependent variable

Independent variable

ADF test for error term

Dependent variable

Independent variable

ADF test for error term

r(3)

r(6) r(12) r(36) r(5ys) r(10ys)

5.753a 3.137a 2.258a 2.615a 2.586a

r(6) r(12) r(36) r(5ys) r(10ys)

r(3)

5.793a 3.515a 2.195a 2.559a 2.538a

r(6)

r(12) r(36) r(5ys) r(10ys)

4.057a 2.428a 2.398a 2.397a

r(12) r(36) r(5ys) r(10ys)

r(6)

4.307a 2.299a 2.266a 2.302a

r(12)

r(36) r(5ys) r(10ys)

2.921a 2.942a 2.903a

r(36) r(5ys) r(10ys)

r(12)

5.525a 2.328a 2.343a

r(3y)

r(5ys) r(10ys)

3.323a 2.012a

r(5ys) r(10ys)

r(36)

3.323a 2.040a

r(5ys)

r(10ys)

2.192a

r(10ys)

r(5ys)

2.196a

Dependent variable

Independent variable

Results reported employes the error terms of Eqs. (13) and (14). a Indicates significance at the 5% level.

Table 4a Estimates of error correction models (long to short hypothesis) Dependent variable

Independent variable

Lagged Dr(n)

ðmÞ Drt

r(3)

r(6)

r(6) {1,3} r(12) {8,10} r(36) {2,1} r(5ys) {3,1} r(10ys) {3,2} r(12) {9,3} r(36) {3,2} r(5ys) {6,2} r(10ys) {10,1}

d P P ðmÞ ðnÞ ¼ det1 þ pi¼1 /r;i Drti þ qi¼1 qrðnÞ ; iDrti þ xt

4.3467* [0.013] 0.1543 [0.792] 5.4146* [0.001] 3.9876* [0.005] 5.0963* [0.001] 0.0549 [0.0087] 4.8532* [0.002] 4.4685* [0.003] 0.0780 [0.767]

0.2770* [2.956] 0.0312 [0.839] 0.2686* [8.876] 0. 1410* [4.219] 0.1349* [3.931] 0.0299 [0.548] 0.1353* [3.969] 0.1272* [3.895] 0.1360 [3.900]

r(12)

r(36)

r(5ys)

r(36) {1,3} r(5ys) {3} r(10ys) {2,1} r(5ys) {6,3} r(10ys) {2,1} r(10ys) {1,3}

Lagged Dr(n)

d

4.8427* [0.001] 5.9843* [0.003] 4.9876* [0.000] 0.5555 [1.153]  0.0448 [0.133] 0. 4964 [0.957]

0.1023* [4.351] 0.1021* [4.480] 0.1152* [4.900] 0.2363 [0.777] 0.0806 [0.562] 0.0461 [0.217]

Single asterisk indicate significance at the 5% level. Figures in brackets is the lag length p and q. The reported figures in column 3 are F-statistics for the null hypothesis that all lagged terms are equal to zero. The p-values are in parenthesis. In the 4 column the t-statistics for the H0: d = 0.

with the standard test of the EHT that uses the three and six months rates to calculate a three month forward rate three months in the future and estimate

coefficients of Eq. (11). We run the regression described by (11) to test the Null H0 if b = 1 and a 5 0. Results are reported in Table 5.

S. Musti, R.L. D’Ecclesia / European Journal of Operational Research 185 (2008) 1596–1606

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Table 4b Estimates of error correction models (short to long hypothesis) Dependent variable

Independent variable

Lagged

Dependent variable

Independent variable

Dr(m) ðnÞ Drt

r(6)

r(12)

r(36)

r(3) {1,6} r(3) {1,3} r(6) {3,5} r(3) {1,4} r(6) {3,3} r(12) {5,3}

s P P ðnÞ ðmÞ ¼ set1 þ pi¼1 kr;i Drti þ qi¼1 prðmÞ ;i Drti þ nt

1.3467* [0.313] 4.5673 [0.002] 5.4146* [0.001] 4.1876* [0.001] 0.9603 [0.865] 1.965 [0.187]

0.0817 [1.079] 0.0157* [3.834] 0.2455* [4.756] 0.0991* [2.287] 0.0192 [0.501] 0.0138 [0.545]

r(5y)

r(10y)

r(3) {1,3} r(6) {3} r(12) {2,1} r(36) {6,3} r(3ys) {2,1} r(6) {1,5} r(12) {3,7} r(36) {2,3} r(5y) {2,6}

Lagged Dr(m)

s

4.8427* [0.001] 5.9843* [0.003] 4.9876* [0.000] 0.5555 [1.153] 0.0448 [0.133] 0.4964 [0.957] 0.9876 [0.765] 0.885 [0.897] 1.067 [0.745]

0.00313 [0.064] 0.0207 [0.587] 0.0194 [0.817]  0.0550 [0.185] 0.0006 [0.014] 0.0173 [0.509] 0.0180 [0.795] 0.0314 [0.252] 0.0585 [0.307]

Single asterisk indicate significance at the 5% level. Figures in brackets is the lag length p and q. The reported figures in column 3 are F-statistics for the null hypothesis that all lagged terms are equal to zero. The p-values are in parenthesis. In the 3 column the t-statistics for the H0 on s. Table 5 Estimates of the regression for the ‘‘forward term spread’’ Dependent variable ð3Þ

r3  rð3Þ ð6Þ r6  rð6Þ ð12Þ r12  rð12Þ

a

b

R2

0.15 (2.678)* 0.05 (1.945)* 0.04 (0.986)

0.20 (4.356)* 0.11 (2.987)* 0.09 (1.123)

0.35 0.25 0.10

The coefficients in this regressions are all positive and two of them are significant at the 5% level. This shows that the forward-term spread between the three and six months maturities has substantial forecasting power for the three months rate for the period 1994–2003, supporting the EHT on this section of the yield curve. This is opposite to what has been outlined by Hamburger and Platt (1975), Mankiw and Miron (1986) and Mankiw and Summers (1984) which report coefficients for the forward rate premium that are not significantly different from zero on the US data over the post-war period.

5. Conclusions This paper investigates the empirical validity of the expectations hypothesis in the Italian market

using a cointegration and correction model framework. Given the shown link existing among European term structures over the last 5 years, the set of Italian data can be used to provide evidence of the EHT for the entire Euro area. The results of cointegration and ECM analyses support the validity of the hypothesis. The shortand long-term interest rates are shown to be cointegrated and their relationship can be specified by an error correction representation. In addition to shedding some light on Italian financial markets, they also explicitly identify the process of adjustment towards a long-run equilibrium path. Two variables which can be represented with a cointegration relation are subjected to a common force of attractor. The results indicate that, in Italy, movement of short- and long-term interest rates adjust accordingly as they are out of the equilibrium path. However, relatively, long term interest rate is shown to be a stronger force of attraction that drives movement of short-term interest rates. The pattern of short-run causation, however, is the reversed with short-term rates generally causing changes in longterm interest rates. These results support the view that interest rates of different maturities are linked

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