Minoas Koukouritakis
email: minoas@econ.soc.uoc.gr

Leo Michelis

E43 -Determination of Interest Rates|Term Structure of Interest Rates < , F42 -International Policy Coordination and Transmission < , F15 -Economic Integration

Keywords: Term Structure, Granger Causality, Common Trends, Cointegration, EU Enlargement JEL Classification: E43, F15, F42 Term Structure, EU Enlargement, Cointegration, Common Trends, Granger Causality

This paper uses cointegration and common trends techniques to investigate empirically the expectations hypothesis of the term structure of interest rates for the 10 new EU countries, along with Bulgaria and Romania.

The empirical results support the expectations theory of the term structure for all countries except Malta. By decomposing each term structure into its transitory and permanent components, we also analyze short run and long run interdependence among the term structures of interest rates in these countries. Our results indicate only weak linkages among the term structures of the 10 new EU countries, and strong linkages between Bulgaria and Romania that hope to join the EU in 2007.

Introduction

The enlargement process of an economic union is an important issue in the theory and practice of economic integration. In the framework of the European Union (EU), enlargement has been a concern since the foundation of the European Economic Community by the Treaty of Rome in 1957. The Treaty states explicitly that one of its main objectives is the continuous and balanced expansion. Indeed the current EU is the result of various expansions since 1957.

After growing in size from the original six members to twelve members and then to fifteen member states, the EU has recently experienced its biggest expansion ever in terms of scope and diversity. On May 1, 2004 ten countries joined the Union. These countries are Cyprus, the Czech Republic, Estonia, Hungary, Latvia, Lithuania, Malta, Poland, the Slovak Republic and Slovenia. In order to successfully join the EU these countries had to satisfy certain economic and political criteria, which include being stable democracies, respect human rights and the rule of law as well as having a functioning market economy. Currently, two more countries, Bulgaria and Romania, hope to join the EU in 2007.

The Maastricht Treaty has laid down explicit nominal convergence criteria that must be satisfied before a candidate country can join the European Monetary Union (EMU) successfully. One criterion concerns the convergence of long term interest rates to the average interest rate of the three EMU countries with the lowest inflation rates.

In the present paper we investigate interest rate linkages among the new and prospective EU countries using the expectations hypothesis of the term structure (EHTS) of interest rates. According to the EHTS, the interest rate on a long run government security is an average of the current short rate and the expected future rates on securities of shorter maturity. If future short rates are expected to be constant over time, then the yields curve will be a horizontal line at the level of the current short rate. If future short rates are expected to rise, then the yield curve will be upward slopping, while if future short rates are expected to fall, then the yield curve will be downward slopping. Thus, the EHTS provides a plausible link between short and long term interest rates over time.

Clearly the EHTS has important policy implications. The term structure is a channel through which government policies can affect the long term prospects of an economy. For example, if a government adopts policies that lower the expected future short rates, then the long term interest rate will be lower and consequently, investment and economic growth will be higher.

The literature on the term structure of interest rates is large and growing; see [START_REF] Shiller | The term structure of interest rates[END_REF] for an excellent survey of theory and empirical studies. The recent empirical literature has employed various econometric techniques such as cointegration, vector autoregressions and Kalman filtering and is, in general, supportive of the EHTS of interest rates. Among others, [START_REF] Hall | A cointegration analysis of treasury bill yields[END_REF] used monthly data from 1970:3 to 1988:12 for 12 yield series of US Treasury Bills and found evidence supportive of the EHTS. [START_REF] Hardouvelis | The term structure spread and future changes in long and short rates in the G7 countries[END_REF] used monthly data of different time spans for the G7 countries: Canada, France, Germany, Italy, Japan, the UK and the USA. He found that the expectations hypothesis holds for all countries except the USA. [START_REF] Gerlach | The term structure of Euro-rates: some evidence in support of the expectations hypothesis[END_REF] studied the term srtuctures in a sample 17 countries with time spans between 10 and 30 years, and monthly data for 1-month, 3-month, 6-month and 12-month euro rates. Using cross-sectional regression analysis, they concluded that for most of the countries the EHTS is compatible with the data. [START_REF] Wolters | Cointegration and German bond yields[END_REF] tested the EHTS for seven German interest rates, with monthly data from 1977:1 to 1993:12. He concluded that the strong form of the EHTS does not hold for the German bond market. [START_REF] Mcdermott | Testing the expectations model of the term structure in times of financial transition[END_REF] studied three English interest rates of the eighteenth century, using quarterly data from 1719:4 to 1797:3, while [START_REF] Balcombe | Bootstrap testing of the expectations hypothesis with the term structure of interest rates[END_REF] tested the EHTS for four UK interest rates with monthly data from 1975:1 to 2001:1. Each of these two studies found evidence in support of the EHTS. [START_REF] Dominguez | Can forward rates be used to improve interest rate forecasts?[END_REF] analyzed interest rates on euro-deposits from the London euro-currency market, using monthly data over the period 1979:1 to 1998:12, and found evidence consistent with the EHTS. [START_REF] Anatolyev | The term structure of Russian interest rates[END_REF] examined the Moscow interbank offered rates for a sample of daily observations from January 1, 2000 to May 1, 2003. They concluded that the yield curve is increasing and slightly concave, a result that is in favor of the EHTS. [START_REF] Chen | Inflation and real short-term interest rates -A Kalman filter analysis of the term structure[END_REF], [START_REF] Cassola | A two factor model of the German term structure of interest rates[END_REF] and [START_REF] Gravelle | A Kalman filter approach to characterizing the Canadian term structure of interest rates[END_REF] adopted the Kalman filter technique in order to test the EHTS. The first study used quarterly data for 12 US Treasury Bill rates over the period 1960:1-1991:2, while the second study used monthly data for German spot rates for two different samples (1986:1-1998:12 and 1972:9-1998:12). Both of these studies reported evidence in favour of the EHTS. The third study used Canadian end-of-the-month closing yields and forward rates for the period 1988:8-1998:6 and strongly rejected the expectations hypothesis.

Even though most of the studies to date have been concerned with testing the EHTS for a specific country or group of countries, the decomposition of the term structure into its transitory (i.e. the I(0) cointegration) and permanent (i.e. the I(1) common trend) components can be equally useful and insightful. The cointegration relation, which captures the spread between the long and short rates, contains information about the effects of short run monetary policies, while the common trend contains information about long run macroeconomic conditions and expectations about the course of future government policies. The interdependence among the transitory or the permanent components for a group of countries can thus reveal information about the degree of policy convergence among the countries. This is useful information for applied economists and policy makers. [START_REF] Hafer | Linkages in the EMS term structures: evidence from common trend and transitory components[END_REF] and [START_REF] Gonzalo | Estimation of common long-memory components in cointegrated systems[END_REF] to study linkages in the term structures of interest rates in 4 EU countries: Belgium, France, Germany and the Netherlands. Using a sample of monthly observations from 1979:3 to 1995:6, they found that the EHTS holds for these countries. Also, by decomposing each term structure into its transitory and permanent components, these authors found that the long term interest rate is the source of the common trend in each country, and that the common trends are cointegrated across countries and thus move together over time, but no single country dominates the common trends. [START_REF] Holmes | The term structure of interest rates and financial integration in the ERM[END_REF] reported similar results for 6 EU countries (Belgium, France, Germany, Italy, the Netherlands and the UK), using a sample of monthly observations from 1974:1 to 1996:3.

In the present paper we contribute to the existing literature in three ways. First, we use the most recent data available from the early 1990s to the present and the VECM approach of [START_REF] Johansen | Statistical analysis of cointegration vectors[END_REF][START_REF] Johansen | Estimation and hypothesis testing of cointegration in Gaussian vector autoregressive models[END_REF][START_REF] Johansen | The role of the constant and linear terms in cointegration analysis of nonstationary variables[END_REF][START_REF] Johansen | Likelihood-based Inference in Cointegrated Vector Autoregressive Models[END_REF] to test the EHTS of interest rates for the 10 new EU member states, as well as Bulgaria and Romania that hope to join the EU in 2007.

Second, we use the Gonzalo-Granger methodology to identify and estimate the common trend that drives the cointegrating relation between long and short rates in each country. Hypothesis testing in this framework provides information as to which interest rate contains the common trend. This is useful information for the design of monetary policies of the 10 new EU countries and the 2 prospective member states.

Third, using multivariate Granger causality and Johansen cointegration tests, we investigate the possibility of short run and long run interdependence among the term structures of the countries in our sample. This is also useful information, since interdependence among the term structures of interest rates means interdependence among the monetary policies of these countries.

The rest of the paper is organized as follows. In Section 2 we briefly describe the EHTS of interest rates and outline the models for cointegration and common trends that we use in the paper. In Section 3 we describe the data and analyze the empirical results. Briefly, the evidence suggests that the EHTS holds for all countries of our sample, except for Malta. Our results also indicate that the short run monetary policies are set independently for most of the new EU countries, while the long run monetary policies are only weakly interdependent. Further, our findings indicate strong short run and long run linkages between the monetary policies of Bulgaria and Romania.

In Section 4 we make some concluding remarks. The EHTS of interest rates states that the yield to maturity of an n-period bond R n,t will equal an average of the current and future rates on a set of m-period short yields r m,t , with m < n, plus the term premium. The relationship can be expressed in the following form

(1 + R n,t ) n = ϕ * n,t n-1 Y i=0 (1 + E t r m,t+i ) , ( 1 )
where ϕ * n,t is a possible non-zero but stationary n-period term premium and E t is the expectations operator conditional on information up to and including time t. The equality in equation ( 1) is established by the condition of no arbitrage opportunities to investors willing to hold both short term and long term bonds. Log-linearizing equation (1) we get

R n,t = ϕ n,t + (1/n) n-1 X i=0 E t r m,t+i . ( 2 )
where ϕ n,t = log(ϕ * n,t ). Equation (2) indicates that the yield of the n-period bond and the m-period short yields are functionally related. For the subsequent analysis it is convenient to re-express equation (2) as

R n,t -r m,t = ϕ n,t + (1/n) n X i=1 E t (r m,t+i-1 -r m,t ) . ( 3
)
The left hand side of equation (3) represents the spread between the n-period (long term) yield and the m-period (short term) yield. Assuming that the yields are I(1) and cointegrated the right hand side of equation ( 3) is stationary. It follows that the left hand side of equation ( 3) is stationary and that (1, -1) 0 is a cointegration vector linking the long term and short term interest rates. In what follows, we analyze the time series and cointegration properties of the long term and short term interest rates, given the insights of equation (3).

The Cointegration and Common Trends Models

This section outlines the basic maximum likelihood theory of cointegration and the models that employed in the subsequent empirical analysis. The maximum likelihood theory of cointegration assumes that the stochastic variables are integrated of order one, or I(1), and that the data generating process is a Gaussian 1 vector autoregressive 1 The Gaussian assumption is not necessary, but it is convenient for the derivation of asymptotic results. model of finite order k, or V AR(k) which may possibly include some deterministic components. Let Y t be a p-dimensional column vector of I(1) variables. Then the V AR(k) can be written in a vector error-correction model (VECM) form as

∆Y t = ΠY t-1 + k-1 X i=1 Γ i ∆Y t-i + µ 0 + µ 1 t + t , t= 1...T (4)
where Π and Γ i are p × p matrices of coefficients, µ 0 and µ 1 are p × 1 vectors of constant and trend coefficients, respectively and t is a p × 1 multivariate normal random error vector with mean vector zero and variance matrix Ω that is independent across time periods.

The hypothesis of cointegration can be stated in terms of the rank of the long run matrix Π in equation ( 4).

Under the hypothesis of cointegration, this matrix can be written as

Π = αβ 0 (5)
where α and β are p × r matrices of full rank. If r = 0, then Π = 0, which means that there is no linear combination of the elements of Y t that is stationary. The other extreme case is when the rank of the Π matrix equals p. In this case Y t is a stationary process. In the intermediate case, when 0 < r < p there are r stationary linear combinations of the elements of Y t and pr non stationary common trends.

Under the hypothesis Π = αβ 0 , the relation between α and the deterministic term µ t ≡ µ 0 + µ 1 t is crucial for the properties of the process Y t . To see this, first decompose µ 0 and µ 1 in the directions of α and α ⊥ , where α ⊥ is a p × (pr) matrix that is the orthogonal complement to α:

µ i = αβ i + α ⊥ γ i , i= 0, 1 (6)
where [START_REF] Johansen | The role of the constant and linear terms in cointegration analysis of nonstationary variables[END_REF], consider the following five submodels, which are ordered from the most to the least restrictive:

β i = (α 0 α) -1 α 0 µ i and γ i = (α 0 ⊥ α ⊥ ) -1 α 0 ⊥ µ i . Next, following
Model 0:

µ t = 0 Model 1*: µ t = αβ 0 Model 1: µ t = αβ 0 + α ⊥ γ 0 Model 2*: µ t = αβ 0 + α ⊥ γ 0 + αβ 1 t Model 2: µ t = αβ 0 + α ⊥ γ 0 + (αβ 1 + α ⊥ γ 1 )t
The interpretation of these models becomes clear in the context of the solution of Y t in equation ( 4).

Y t = C t X i=1 t + 1 2 τ 2 t 2 + τ 1 t + τ 0 + W t + A (7)
where W t is a stationary process, A is a vector such that

β 0 A = 0, C = β ⊥ (α 0 ⊥ Γβ ⊥ ) -1 α 0 ⊥ , Γ = I p - k-1 P i=1 Γ i , β ⊥ is a p × (p -r) matrix of full rank that is orthogonal to β and τ 2 = Cµ 1 .
Using equation ( 7), [START_REF] Johansen | The role of the constant and linear terms in cointegration analysis of nonstationary variables[END_REF] shows that the five submodels imply different behavior for the process Because of the normality assumption, one can easily test for the reduced rank of the Π matrix using the maximum likelihood approach. This procedure gives at once the maximum likelihood estimators (MLE) of α and β and the eigenvalues needed in order to construct the likelihood ratio test. The MLE of α and β are obtained by regressing ∆Y t and Y t-1 on ∆Y t-1 ...∆Y t-k and µ t (allowing for the restrictions imposed by each of the five models). These auxiliary regressions give residuals R 0t and R 1t respectively, and residual product matrices

Y
S ij = T -1 T X t=1 R it R 0 jt , i, j= 0, 1 (8)
Solving the eigenvalue problem ¯λS 11 -S 10 S -1 00 S 01 ¯= 0 (9) In testing the null hypothesis that rank(Π) ≤ r against the alternative hypothesis that rank(Π) = p, the likelihood ratio statistic, called also the Trace statistic by [START_REF] Johansen | Maximum likelihood estimation and inference on cointegration with applications to the demand for money[END_REF], is given by

for eigenvalues 1 > b λ 1 > ... > b λ p > 0 and eigenvectors b V = (b v 1 ...b v p ), normalized such that b V 0 S 11 b V = I,
T race = -T p X i=r+1 ln(1 -b λ i ) ( 10
)
The testing is performed sequentially for r = 0, ..., p -1 and it terminates when the null hypothesis is not rejected It is also possible to test the null hypothesis that rank(Π) = r against the alternative that rank(Π) = r + 1.

In this case, the likelihood ratio statistic, which is called the λ max statistic, is given by

λ max = -T ln(1 -b λ r+1 ). ( 11
)
Of course, the λ max statistic is equal to the Trace statistic when pr = 1.

MacKinnon, Haug and Michelis (1999) have computed highly accurate critical values for the Trace statistic in equation ( 10) and the λ max statistic in equation ( 11), using the response surface methodology. These critical values differ substantially from those in the existing literature, especially when the dimension of the VECM is large; e.g., see [START_REF] Osterwald-Lenum | A note with quantiles of the asymptotic distribution of the maximum likelihood cointegration rank test statistics[END_REF] or [START_REF] Johansen | Likelihood-based Inference in Cointegrated Vector Autoregressive Models[END_REF]. Since we deal with large dimensional systems in this study, we use these new critical values for testing hypotheses2 .

In respect to the common trends, it is clear from equation ( 7) that the common trends in Y t are contained in the first term of that expression. Given the definition of C, Johansen (1995, p. 41) defines the common trends by the cumulated disturbances α 0

⊥ t P i=1 t .
Assuming that the common trends are a linear combination of Y t , in the form [START_REF] Gonzalo | Estimation of common long-memory components in cointegrated systems[END_REF] Given this framework, it is easy to test whether or not certain linear combinations of Y t can be common trends.

f t = α 0 ⊥ Y t ,
Null hypotheses on α ⊥ have the following form

H 0 : α ⊥ = Gθ (13)
where G is a p × m known matrix of constants and θ is an m × (pr) matrix of unknown coefficients such that pr ≤ m ≤ p. To carry out the test, one solves the eigenvalue problem given by

¯λG 0 S 00 G -G 0 S 01 S -1 11 S 10 G ¯= 0 (14)
L = -T p X i=r+1 ln h (1 -b λ * i+(m-p) )/(1 -b λ i ) i . ( 15
)
In the next section, we use the L-statistic in (15) to test the statistical significance of the α ⊥ of the long term and the short term interest rate of the EU accession countries. A significant α ⊥ implies that the respective interest rate is weakly exogenous and dominates the common trend in the cointegrating system.

3 Data and Empirical Results

Data

We collected data for the 10 new EU countries and for the 2 prospective EU countries, Bulgaria and Romania. Due to lack of data availability we worked only on two interest rates for each country: either treasury bill yields (short term) and government bond yields (long term) or short term and long term commercial banks' lending rates. Our sample consists of monthly data of varying time spans for different countries determined by data availability. All interest rates are expressed in natural logarithms.

For Cyprus the time span is 1997:1 to 2004:12. Monthly average treasury bill rates were obtained from line 60c of the CD-ROM of the International Financial Statistics (IFS) of the IMF, 2004. Monthly average government bond yields were obtained from the Central Bank of Cyprus and refer to bonds with maturity greater than 5 years.

For the Czech Republic the time span is 1993:8 to 2004:12. Treasury bill rates were taken from line 60c of the IFS.

Long term government bond yields were obtained from line 61 of the IFS. This IFS data series begins at January 2000. For the period 1993:8-1999:12 we used long term government bond yields obtained from the Central Bank of the Czech Republic.

No data on treasury bill rates or government bond yields are available for Estonia, Latvia, Lithuania and Slovenia. For this reason, we used commercial banks' lending rates instead. For Estonia the time span is 1994:1 to 2004:12. Three-month lending rates were obtained from the Main Economic Indicators (MEI) of the OECD, 2004, and 10-year lending rates were taken from the Central Bank of Estonia. For Latvia short term and long term lending rates were obtained from the MEI and the time span is 1993:1 to 2004:12. For Lithuania 6 to12 months and over 5 years lending rates were taken from the Central Bank of Lithuania and the time span is 1997:1 to 2004:12. For Slovenia the time span is 1996:1 to 2004:12 and the short term and long term lending rates were obtained from the Central Bank of Slovenia.

For Hungary the time span is 1997:1 to 2004:12. The 3-month treasury bill rates and the 5-year government bond yields were obtained from the Central Bank of Hungary. For Malta the time span is 1993:1 to 2004:12. Monthly average treasury bill rates were obtained from line 60c of the IFS, while 5-year monthly average government bond yields were taken from the Central Bank of Malta.

For Poland the time span is 1994:2 to 2004:12. The 1-year treasury bill rates and the over 2 years government bond yields were obtained from the Polish Ministry of Finance. In the case of the Slovak Republic the time span is 1994:12 to 2004:12. The 1 to 6 months treasury bill rates were taken from the Central Bank of the Slovak Republic.

For government bond yields the IFS data series (line 61) for the Slovak Republic begins at September 2000 and refers to 10-year government bond yields. For the period 1994:12-2000:8 we collected data from the Central Bank of the Slovak Republic.

For Bulgaria the time span is 1993:7 to 2004:12. Monthly average treasury bill rates were obtained from the Central Bank of Bulgaria and monthly average government bond yields were obtained from line 61 of the IFS. For

Romania the time span is 1997:1 to 2004:12. Since data for treasury bill rates and government bond yields are not available, we used long term and short term lending rates that were taken from the MEI. The MEI data series ends at December 2003. For the period 2004:1-2004:12 we used long term and short term lending rates obtained from the Central Bank of Romania.

Testing for the EHTS

In this section we report and analyze the unit root and cointegration results between the short term and the long term interest rates for each country. Evidence of cointegration would validate empirically the EHTS of interest rates.

Before testing for cointegration, we tested each time series for unit roots using the Augmented Dickey-Fuller test at the 5 percent level of significance. The results are presented in Table 1. To select the appropriate lag length for the ADF test regression, we used the Akaike's information criterion. As shown in Table 1, we fail to reject the unit root hypothesis in the long and short rates in all countries except Poland. In all the cases where the unit root hypothesis was not rejected , we also tested for a second unit root. As shown in Table 1, this hypothesis was rejected in all cases. Based on these results we proceeded with cointegration analysis using the VECM in equation ( 4) above, where Y t = (R n,t , r m,t ) 0 . To select the appropriate lag length, k, in equation ( 4), we set up a separate VECM for each country and used the likelihood ratio test. Under the hypothesis Γ k = 0, the likelihood ratio test is asymptotically distributed as χ 2 with p 2 degrees of freedom (see Johansen 1995, p. 21). Further, to determine which submodel describes best each set of variables, we tested the submodels against each other using the likelihood ratio tests in Johansen (1995, Chapter 11, Corollary 11.2 and Theorem 11.3, pp. 161-162). These tests are also distributed as χ 2 with degrees of freedom determined by the pairs of models being tested as follows:

0 ⊂ r 1 * ⊂ p-r 1 ⊂ r 2 * ⊂ p-r 2 .
Table 2 reports the cointegration results between the long and short rates for each of the new EU countries (except Poland) plus Bulgaria and Romania. Based on the Trace and the λ max statistics at the 5 percent level of significance, we find evidence of one cointegrating vector between the short term and the long term interest rates in all the countries, except Malta. Clearly, this is indirect evidence in favour of the EHTS of interest rates in all countries in this sample, except for Malta.

Table 3 reports the parameter estimates of the cointegration vectors, normalized on the long rate, for the countries for which the EHTS holds (i.e. Cyprus, the Czech Republic, Estonia, Hungary, Latvia, Lithuania, the Slovak Republic, Slovenia, Bulgaria and Romania). Numbers in parentheses are likelihood ratio test statistics which are distributed as χ 2 1 asymptotically, under the null hypothesis that each component of the cointegration vector is insignificantly different from zero. As shown in Table 3, the parameters of the cointegrating vectors (β i 's) are statistically significant in all cases, which means that the short term and the long term interest rates enter significantly each cointegration vector. We also tested the hypothesis H 0 : β R + β r = 0, when (β R , β r ) = (1, -1).

Equivalently, we tested whether or not the spread between the long term and short term interest rates belongs in the cointegration space of the term structure, as suggested by the EHTS. Using the likelihood ratio test statistic at the 5 percent level of significance, this hypothesis is rejected for Cyprus, the Czech Republic, Hungary, Latvia, Slovenia, Bulgaria and Romania, but it cannot not be rejected for Estonia, Lithuania and the Slovak Republic.

Consequently, only the latter group of countries satisfy the EHTS exactly.

The parameter estimates of the adjustment coefficients α R and α r are also presented in Table 3. These are the coefficients of the error correction terms in the VECM and their subscripts denote the variable that adjusts to deviations from the long run equilibrium relation between the two rates. As shown in Table 3, α R is statistically significant and α r is statistically insignificant at the 5 percent level, for the three Baltic countries (i.e., Estonia, Latvia and Lithuania) and for Romania. This implies that, for this group of countries, the long term interest rate is an endogenous variable which adjusts to deviations from its long run equilibrium with the short rate. At the same time, the short rate is a weakly exogenous variable, changes of which have a permanent effect on both the long rate and the short rate. On the other hand, α R is statistically insignificant and α r is statistically significant for Cyprus, the Czech Republic, Hungary, the Slovak Republic, Slovenia and Bulgaria. Hence for the latter group of countries, the evidence suggests that the short term interest rate adjusts to deviations from the long run equilibrium, while the long rate is weakly exogenous, affected primarily by fundamental factors such as the future state of the economy and expectations about the future path of government policies.

These empirical findings are reinforced by direct tests on the α ⊥ 's, the components of the common trend in each country. In order to test the statistical significance of the the α ⊥ 's, we compute the L-statistic in equation ( 15) for specific choices of the G matrix. In particular, to test the null hypothesis that the long term interest rate has a permanent component in the common trend of a country, we set the G matrix to

G =    1 0    .
Alternatively, to test the hypothesis tat the short term interest rate has a permanent component in the common trend, we set the G matrix to

G =    0 1    .
Table 3 reports the computed L-statistics. For the countries for which α R is statistically significant (i.e., the three Baltic countries and Romania), the null hypothesis that the short term interest rate, r, has a permanent component in the common trend cannot be rejected at the 5 percent level of significance.

On the other hand, for the countries with a significant α r (i.e. Cyprus, the Czech Republic, Hungary, the Slovak Republic, Slovenia and Bulgaria), the null hypothesis that the long term interest rate, R, has a permanent component in the common trend cannot be rejected at the 5 percent level of significance. For these countries the short term interest rate adjusts to deviation from the long run equilibrium, while the long term interest rate is not affected by past disequilibria and thus "drives" the common trend.

Short Run Interdependence

In this section we analyze short run interdependence among the term structures of the new EU member states, using Granger causality tests. In order to include all the countries in our analysis, the time span is restricted to 1997:1-2004:12, in this and the next section. Poland is also included in the analysis of this section as its interest rates and their spread are stationary variables.

Fluctuations in the transitory component (or the spread) of the term structure are often thought to be caused by changes in monetary policy. If these fluctuations are related across the countries, then it is possible that policy actions are interdependent in the short run. Since the transitory components of the term structures are stationary, we test for such interdependence explicitly by conducting Granger-causality tests based on multivariate VAR models among the cointegrating relations. The appropriate lag length for each VAR model was selected using the likelihood ratio test.

The causality results are reported in Table 4. The tests results contain pairwise causality tests and joint causality tests that include all the countries in a given group of countries. The first panel of Table 4 includes results for all the new EU member states, except Malta. Malta was excluded from this analysis, since its interest rates do not cointegrate. It is clear from this panel that there is weak short run interdependence among the term structures of the countries in this group. Only 5 out of 72 possible pairwise test statistics (excluding the own effects) and 2 out of 9 joint test statistics (including only the cross effects) are significant at the 5 or 10 percent level of significance.

For example, only the Estonia's spread is influenced by changes in the spreads of the Czech Republic and Poland, both pairwise and jointly. Similar results hold for Slovenia with respect to the Slovak Republic.

The second panel of Table 4 reports the results for the group of the Eastern European countries: the Czech Republic, Hungary, Poland, the Slovak Republic and Slovenia. In this case, there is also evidence of loose short run linkages among the term structures. The Wald test statistics are significant in 5 out of 20 pairwise tests and 2 out of 5 joint tests. Changes in the spreads of the other Eastern European countries jointly influence only those of the Slovak Republic and Slovenia in the short run. These findings are expected, given relative independence of the Czech Republic, Hungary and Poland, which in recent times have followed policies tied more closely to Western Europe than to Eastern Europe.

The third panel of Table 4 includes each other, which reflects their independent monetary policies.

The last panel of Table 4 refers to Bulgaria and Romania, which hope to become EU members by 2007. In this case, the empirical results indicate strong short run interdependence, both pairwise and jointly, between the term structures of these two countries. The Wald test statistics are significant at the 5 percent level of significance for both countries. This evidence may reflect the historic, economic and political ties between the two countries, as well as their joint efforts, in recent times, to become ready to join the EU.

In summary, the above results indicate weak short run linkages among the term structures of the new EU countries including the sub-groups of the 5 Eastern European countries and the Baltic countries. The results also indicate stronger short run linkages between the term structures of Bulgaria and Romania.

Long Run Interdependence

In this section, we examine the long run interdependence among the term structures, by analyzing linkages among their common trends in a cointegration framework.

The common trends are often thought to capture information about future economic and monetary policies.

Following [START_REF] Hafer | A long run view of German dominance and the degree of policy convergence in the EMS[END_REF] we can claim that there are strong long run linkages among these policies in a group of p countries, if there exist r = p -1 cointegrating relations among the common trends. On the other hand, if 0 < r < p -1, then there is only partial interdependence among the future policies of the countries concerned.

In this sense, interdependence means that the countries' policies have converged enough, so that the permanent components of their term structures tend to move towards a long run equilibrium and do not drift too far apart over time.

The cointegration results for the common trends for different groups of countries are reported in Table 5 3 The last panel of Table 5 reports the results for Bulgaria and Romania. Both statistics indicate one cointegration vector between the two common trends, at the 5 percent level of significance. These results point to strong long run linkages between the term structures and monetary policies of these two countries. This evidence is encouraging as these two countries aspire to become EU members in 2007.

The results for Bulgaria and Romania seem plausible. Since the mid-1990s, both countries have implemented radical programs of structural reforms and macroeconomic stabilization. Their budgetary discipline and privatization programs have helped to stabilize their public deficits. They have also managed to reduce inflation gradually and stabilize their nominal exchange rates. Bulgaria adopted a currency board arrangement for its exchange rate, while Romania controlled tightly its monetary base. All these policy measures are reflected, in turn, in the evidence of interdependence between their term structures.

Concluding Remarks

In this paper we investigated empirically the term structure of interest rates among the 10 new and 2 prospective EU countries. Since the interest rates follow random walks, we evaluated the expectations hypothesis of the term structure using cointegration analysis and common trends techniques. Further we analyzed short run and long run interdependence among the term structures of these countries.

Our empirical findings indicate that the EHTS holds for all the new EU countries, except for Malta. For Cyprus, the Czech Republic, Hungary, the Slovak Republic, Slovenia and Bulgaria, the long term interest rate is weakly exogenous and drives the common trend in each term structure. On the other hand, for the three Baltic countries and Romania, the short term interest rate is weakly exogenous and has a permanent component in the common trend.

Our results also indicate weak short run and long run interdependence among the term structures of the new EU states and the subgroups of the Eastern European and the Baltic countries. This evidence, in turn, implies that these countries set their monetary policies independently, both in the short run and the long run.

Only the two prospective EU countries of Bulgaria and Romania appear to have interdependent term structures pointing to strong short run and long run interdependence between their monetary policies. These strong linkages will contribute positively in their efforts to join the EU easier in 2007.

Our analysis has been focused on term structure linkages among the new and the two prospective EU countries.

This is a useful exercise as it provides valuable knowledge about the degree of monetary convergence in these countries. More insights can be gained by analyzing the term structures of these countries in relation to some core 1* The value reported at the top of each column is for r = 0, so that pr = p, where p = 2 (i.e. the number of interest rates included). * denotes rejection of the null hypothesis of at most r cointegrating relations at the 5% level of significance. a k indicates the lag length. -6.90 L-statistic 6.79 0.04 R and r denote the long term and short term interest rate respectively. The β's are the parameters of the cointegrating vectors, normalized on the long term interest rates. The α's are the adjustment coefficients and α ⊥ s are their orthogonal complements. Numbers in parentheses are likelihood ratio statistics for H 0 : β i = 0 or H 0 : α i = 0. Numbers in the row of H 0 : β R + β r = 0 are likelihood ratio test statistics. L-statistics are for the null hypothesis that the respective interest rate (either R or r) determines the common trend. ** (*) denotes rejection of the null hypothesis at the 5% (10%) level of significance. The number in each cell is the Wald test statistic, which, under the null, is asymptotically distributed as χ 2 . b df stands for the degrees of freedom. ** (*) denotes rejection of the null hypothesis Γ k = 0 at the 5% (10%) level of significance. CY is for Cyprus, CZ for the Czech Republic, EE for Estonia, HU for Hungary, LV for Latvia, LT for Lithuania, PL for Poland, SK for the Slovak Republic, SV for Slovenia, BG for Bulgaria and RO for Romania. The value reported at the top of each column is for r = 0, so that pr = p, where p is the number of common trends included. * denotes rejection of the null hypothesis at most r cointegrating relations at the 5% level of significance. a k indicates the lag length. CY is for Cyprus, CZ for the Czech Republic, EE for Estonia, HU for Hungary, LV for Latvia, LT for Lithuania, SK for the Slovak Republic, SV for Slovenia, BG for Bulgaria and RO for Romania.

  one gets the MLE of α and β as b α = S 01 b β and b β = (b v 1 ...b v r ), where (b v 1 ...b v r ) are the eigenvectors associated with the r largest eigenvalues of equation (9).

  derived the MLE of α ⊥ as the eigenvectors corresponding to the (pr) smallest eigenvalues of the problem ¯λS 00 -S 01 S -1 11 S 10 ¯= 0. (12) Solving equation (12) for eigenvalues 1 > b λ 1 > ... > b λ p > 0 and eigenvectors c M = ( b m 1 ... b m p ), normalized such that c M 0 S 00 c M = I, one gets the MLE of α ⊥ as b α ⊥ = ( b m r+1 ... b m p ).

  .. > b λ * m > 0, and eigenvectors c M * = ( b m * 1 ... b m * m ), normalized such that c M * 0 (G 0 S 00 G) c M * = I. Choose b θ m×(p-r) = ( b m (m+1)-(p-r) ... b m m ) and b α ⊥ = G b θ. The likelihood ratio test statistic for testing H 0 is

  the three Baltic countries. As shown in the second column of this panel, change in the spread of Estonia affects the spread of Latvia at the 5 percent level of significance. The spread of Estonia is unaffected by changes in the spread of Latvia and Lithuania. The same result holds for Lithuania with respect to Estonia and Latvia. Clearly, the term structures of the three Baltic countries are minimally linked to

  . The Trace and λ max statistics refer to the same groups of countries as in the previous section 4 . As shown in the first panel of Table5, the Trace and the λ max statistics indicate four and three cointegration vectors respectively, at the 5 percent significance level, among the eight common trends of the new EU countries. For the four Eastern European countries, both statistics indicate one cointegration vector among the four common trends. Similarly for the three Baltic countries, both statistics indicate one cointegration vector among the three common trends. Overall, the above results indicate weak long run interdependence among the term structures in these groups of countries, and consequently, only partial long run convergence of their monetary policies.

  used the multivariate cointegration and common trends techniques of Gonzalo

  t and the cointegrating relations β 0 Y t . Briefly, in Model 0, Y t has no deterministic trend and all the stationary components have zero mean. In Model 1*, Y t has neither quadratic or linear trend. However, both Y

t and the cointegrating relations β 0 Y t are allowed a constant term. In Model 1, Y t has a linear trend, but the cointegrating relations β 0 Y t have no linear trend. In Model 2*, Y t has no quadratic trend but Y t has a linear trend that is present even in the cointegrating relations. In Model 2, Y t has a quadratic trend but the cointegrating relations β 0 Y t have only a linear trend.

Page 12 of 23 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

	Submitted Manuscript
	F o r
	P
	e
	e r
	R e v
	i e
	w

19 of 23 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

  EU countries such as France and Germany. We intend to undertake this research in the near future. The entry in each cell is the ADF test statistic. * denotes rejection of the unit root hypothesis at the 5% level of significance. For the countries of the table, the sample sizes are 96 for Cyprus, 137 for the Czech Republic, 132 for Estonia, 96 for Hungary, 144 for Latvia, 96 for Lithuania, 144 for Malta, 131 for Poland, 121 for the Slovak Republic, 108 for Slovenia, 138 for Bulgaria and 96 for Romania.

	F o r F o r F o r F o r Augmented Dickey -Fuller tests for a unit root a Table 1 Short term interest rate Long term interest rate Level First Level First Country difference difference Cyprus -1.42 -9.29 * -1.50 -4.86 * Czech Republic -1.60 -6.66 * -1.88 -9.37 * Estonia -2.46 -9.61 * -2.05 -8.44 * Hungary -1.64 -6.10 * -1.47 -7.40 * Latvia -1.87 -4.24 * -0.50 -5.99 * Lithuania -2.00 -13.40 * -1.85 -7.21 * Malta -0.78 -6.44 * -1.91 -4.58 * Poland -3.99 * -4.50 * Slovak Republic -1.56 -11.66 * -1.75 -10.42 * Slovenia -1.45 -10.36 * -0.55 -9.92 * Bulgaria -1.25 -4.59 * -1.03 -4.60 * Romania -2.02 -4.06 * -0.38 -6.46 * a Page Submitted Manuscript F o r	Page 16 of 23
	P P P P P
	e e e e e
	e r e r e r e r e r
	R e v R e v R e v R e v R e v
	i e i e i e i e i e
	w w w w w
	Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 4

 4 Testing for short run interdependence: Multivariate Granger Causality tests a

	Dependent				Explanatory variables (lag length = 3)
	variable	CY	CZ	EE	HU	LV	LT	PL	SK	SV	Joint
	CY	45.46 **	3.52	1.53	4.58	1.10	0.11	1.19	0.46	0.61	14.69
	CZ	5.59	76.01 **	3.07	2.18	1.44	2.89	1.32	4.26	3.75	21.52
	EE	5.65	13.33 **	17.13 **	1.07	1.07	2.82	11.91 **	3.72	3.19	33.70 *
	HU	2.43	4.02	4.95	67.80 **	2.08	2.19	1.51	4.10	1.66	19.05
	LV	2.19	2.13	1.49	0.43	21.34 **	1.26	6.45 *	1.80	0.09	19.57
	LT	0.45	1.29	1.81	1.65	2.12	80.62 **	5.89	0.35	1.48	18.53
	PL	1.41	0.13	3.46	1.41	1.16	0.66	339.16 **	2.36	1.29	15.24
	SK	0.93	2.97	10.38 **	0.20	4.02	2.26	1.43	9.71 **	2.12	27.82
	SV	1.13	1.60	5.40	2.96	1.03	3.44	1.17	13.80 ** 1004.19 ** 39.84 **
	df b	3	3	3	3	3	3	3	3	3	2 4
	Dependent				Explanatory variables (lag length = 2)
	variable	CZ	HU	PL	SK	SV					Joint
	CZ	193.59 **	3.31	0.12	2.81	0.27					6.34
	HU	1.21	78.27 **	0.05	2.34	2.91					7.03
	PL	0.67	0.64	332.06 **	2.03	1.73					6.87
	SK	6.62 **	0.53	4.74 *	8.94 **	6.63 **					19.08 **
	SV	5.87 *	2.29	3.31	12.30 ** 126.07 **					27.84 **
	d f	2	2	2	2	2					8
	Dependent				Explanatory variables (lag length = 4)
	variable	EE	LV	LT							Joint
	EE	79.54 **	3.41	3.91							7.14
	LV	11.06 **	64.82 **	3.18							13.74 *
	LT	3.38	0.76	100.21 **							4.34
	d f	4	4	4							8
	Dependent				Explanatory variables (lag length = 4)
	variable	BG	RO								Joint
	BG	22.41 **	25.81 **								25.81 **
	RO	21.18 **	200.21 **								21.18 **
	df	4	4								4
	a

Table 5

 5 Testing for long run interdependence: Trace and λ max statistics CY, CZ, EE, HU,

	5% critical values

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

The latest edition of EViews 5 has also adopted the[START_REF] Mackinnon | Numerical distribution functions of likelihood ratio tests for cointegration[END_REF] critical values.

The appropriate lag length k for the VECM and the sub-model that describes best each set of variables were chosen following the same procedures as in Section 3.2 above.

Poland was not included in this section, since its interest rates are stationary.


Arielle P Beyaert

Juan-Jose Perez-Castejon

MARKOV-SWITCHING MODELS, RATIONAL EXPECTATIONS AND THE TERM STRUCTURE OF INTEREST RATES

Keywords: Interest rates, Term structure, Rational expectations, Markov switching regimes, Non linearity JEL classification: C32, C12, E43 Interest rates, term structure, rational expectations, Markov switching regimes, non linearity Markov-Switching, rational expectations and term structure of interest rates

HAL is

In the past, [START_REF] Hamilton | Rational-expectations econometric analysis of changes in regime[END_REF] and [START_REF] Sola | Testing the term structure of interest rates using a stationary vector autoregression with regime switching[END_REF] applied this type of non-linear models to the U.S. term structure of interest rates with results that improved with respect to the case where linear models were used. Hamilton works with a one-equation model, whereas Sola and Driffill generalize bivariate VAR models of the type considered by [START_REF] Campbell | Cointegration and tests of present value models[END_REF], allowing for switching regimes in the variance matrix and in the intercept, but with constant autoregressive coefficients. In this particular case, testing rational expectations is very similar to the case where switching regimes are not considered. [START_REF] Kugler | The term structure of interest rates and regime shifts: some empirical results[END_REF] -with US and Swiss data-and [START_REF] Engsted | Regime shifts in the Danish term structure of interest rates[END_REF] -with Danish data-extended the Sola and Drifill approach 1 . They consider a VAR(1) model linking a one-period with a three-period interest rates, letting the autoregressive parameters depend on the current state of the economy; they obtain the expression for the Rational Expectations restrictions for this specific case only.

In the present paper, we also consider a two-state model where the mean vector, the variance matrix, as well as the autoregressive coefficients of the bivariate model are allowed to change and we obtain the restrictions on the latter coefficients that guarantee the fulfilment of the rational expectations hypothesis. However, our approach differs from Kluger and from Engsted and Nyholm in various aspects. On the one hand, there is a difference in how the structure of the VAR model alters with a change in the regime. On the other hand, we obtain very general expressions for the Rational Expectations restrictions that are valid for any order of the VAR and for any number of periods that separate the maturities of the short and the longer interest rate. We obtain the restrictions both for the case where there are no measurement errors and for the case where such errors affect the agents in their decisionmaking process and/or the econometrician in the model-building process. Moreover, these formulas can easily be adapted for testing other economic theories, which involve variables linked by a rational expectations relationship (such as the asset pricing model, the uncovered interest parity or even the rational expectations theory of intertemporal consumption). The paper is organized as follows. In section 2 we describe the model. In section 3 we obtain the general form of the testable restrictions implied by the rational expectations hypothesis.

In section 4 we apply these formulas to the Spanish inter-bank one-week, two-week and onemonth weekly interest rates. Section 5 concludes.

The model

Starting point: expectations theory and Campbell-Shiller methodology in a linear model

Let R t,n be the interest rate at time t of an asset with maturity in (t+n), and let r t be the interest rate of an asset maturing in (t+1). According to the Rational Expectations Theory of the term structure of interest rates, when R t,n and r t correspond to assets both with a short maturity, typically measured in terms of days, weeks or months, [START_REF] Shiller | Forward rates and future policy: interpreting the term structure of interest rates[END_REF] suggest the following formula:

[ ] k r E n 1 R 1 n 0 i i t t n , t + = ∑ - = + [1]
where t E represents the rational expectations of the agents conditional on the information available at moment t and k is a constant liquidity premium. An alternative expression is based on the spread between the longer-term and shorter-term rates:

2 [ ] [ ] k S E k r E n i 1 r R S * n , t t i t t 1 n 1 i t n , t n . t + = + ∆       - = - = + - = ∑ [2]
Campbell and [START_REF] Campbell | Cointegration and tests of present value models[END_REF] develop a procedure to test the present-value relation of the type expressed in [2] for the case when n is infinite. For any maturity, it is easy to show that if the interest rates are I(1) and the expectations theory is true there exists an Error Correction Model (ECM) that relates

      +       ∆       =       ∆ - - t 2 t 1 1 t n , 1 t t n , t u u r S ) L ( d ) L ( c ) L ( b ) L ( a r S [3]
where a(L), b(L), c(L) and d(L) are lag polynomials of order p . From [3], it is possible to deduce and test the restrictions implied by the expectations theory, whether n is infinite or finite and small. The difference takes place in the form of these restrictions: they are linear and very easy to test in the infinite horizon case, while they are highly non-linear in the finite horizon case 3 .

A two-state switching regime VAR model

In model [3], all the parameters are constant. The possibility of behavioural modifications of economic agents caused by political, institutional or economic changes are not considered.

However, as argued by [START_REF] Ang | Regime switches in interest rates[END_REF], changes in business cycle conditions and monetary policy may cause interest rates to behave differently in different time periods. We allow for such behavioural changes by introducing the possibility of stochastic changes of regime, generalizing the approach first considered by [START_REF] Hamilton | Rational-expectations econometric analysis of changes in regime[END_REF] and [START_REF] Sola | Testing the term structure of interest rates using a stationary vector autoregression with regime switching[END_REF].

We consider the same VAR(p) as in [3], with the added feature that the variance-covariance matrix of the errors, the mean of the multivariate process and the autoregressive coefficients are allowed to switch endogenously between two possible states or regimes; according to this view, the economy would switch stochastically between two regimes, associated with periods of different economic characteristics (such as expansion or recession, high or low risk, etc.). The state is a variable that the econometrician does not observe and has to be inferred from the data, along with the parameter estimates. The resulting Markov-Switching VAR model (MS-VAR) might be represented as follows: 4

F o r P e e r R e v i e w 6 ∑ ∑ ∑ ∑ = ∆ ∆ - = - ∆ = ∆ - = - + µ - ∆ + µ - = µ - ∆ + µ - ∆ + µ - = µ - - - - - - - - - p 1 i t , r x , r i t x , i p 1 i x , S i t x , i x , r t p 1 i t , S x , r i t x , i p 1 i x , S i t x , i x , S t u ) r ( d ) S ( c r u ) r ( b ) S ( a S i t i t i t i t t i t i t i t i t t [4]
where x t is an unobservable variable that takes the value 0 or 1, according to the state of the economy at date t. It is governed by a first-order Markov process, with transition probabilities P(x t = 0 / x t-1 = 0 ) = p 00 , and P(x t = 1 / x t-1 = 1 ) = p 11 . Note that p 01 = P(x t = 1 / x t-1 = 0 ) = 1p 00 , and p 10 = P(x t = 0 / x t-1 = 1 ) = 1 -p 11 . In this model, the distribution of the errors conditional on all past information

) , r , r ; , S , S ; , x , x ( 1 t t 1 t t 1 t t - - - ∆ ∆
is assumed to depend only on t

x and it has the following form:

        σ σ σ σ = Ω Ω ∆ ∆ ∆ 2 x , r x , r S x , r S 2 x , S x x t t t t t t t t with ) , 0 ( N x / u
This model extends the [START_REF] Hamilton | Rational-expectations econometric analysis of changes in regime[END_REF] approach in two directions: it allows the shorter rate r t to depend on the past values of the longer rate R t and it allows not only the means, the variances and covariances, but also the autoregressive coefficients to vary with the state 5 .

Only the first extension has been contemplated by [START_REF] Sola | Testing the term structure of interest rates using a stationary vector autoregression with regime switching[END_REF], who centre their study on the specific and simplest case in which the maturity of the longer rate R t is twice the maturity of the short rate r t (n=2). We also deal with other values of n.

In specification [4], the autoregressive coefficient corresponding to lag i (i=1,...,p) depends on the state the economy was in at time (t-i). A second possibility consists of having the coefficient at lag i depending on the state the economy is in at time t, instead of at time (t-i); in this case, model [4] would transform into: In model [4], the influence of past interest rates on present ones varies according to whether the market was in state 0 or 1 at that past date; on the other hand, in model [4'], the influence of past information depends on the state the market is in at the present date. Does it make more sense to consider that what happened in the past affects today's behaviour according to the state the economy today (Model 4'), or according to the state of the economy at the date at which that past information was generated (Model 4)? It is not easy to decide on a priori theoretical grounds which option is better 6 . It is wise to let the data tell which version is better. Beyaert and Perez-Castejón (2000) apply Schwartz information criterion to discriminate between [4] and [4'] on the same data (except that in this paper contains three additional years of information). They obtain a systematic and overwhelming dominance of version [4] over version [4'] for p=1,2,3 and 4. This strong dominance, together with the fact that estimating these models is not a straightforward task, justifies that we centre on model [4] in the rest of the paper.

∑ ∑ ∑ ∑ = ∆ ∆ - = - ∆ = ∆ - = - + µ - ∆ + µ - = µ - ∆ + µ - ∆ + µ - = µ - - - - - p 1 i t ,
Note that in model [4], the value of the implicit "intercept", which is a function of the means and of the autoregressive coefficients, will depend on

∆ ∆ ∆ ∆ σ σ σ σ µ µ µ µ = θ b , a , a , , a , a , , , , , , , , , , p , p ∆
This vector contains 12+8p parameters. They are estimated by Maximum Likelihood, applying numerical optimisation techniques. A 5-steps filter process is used, similar to the one described for instance in [START_REF] Hamilton | Rational-expectations econometric analysis of changes in regime[END_REF] and [START_REF] Sola | Testing the term structure of interest rates using a stationary vector autoregression with regime switching[END_REF]. This procedure slows down very fast as p increases. It is therefore essential to make a sensible selection of the starting values given to the parameters in the estimation algorithm. For that purpose, we use the procedure described in [START_REF] Beyaert | Switching-regime models in the Spanish inter-bank market[END_REF]. This allows to consider Sola and Drifill(1994), [START_REF] Kugler | The term structure of interest rates and regime shifts: some empirical results[END_REF] and [START_REF] Engsted | Regime shifts in the Danish term structure of interest rates[END_REF].

The restrictions implied by the expectations theory

In order to obtain the restrictions imposed on the coefficients of [4] by REH, we proceed in a similar way as in [START_REF] Blix | Rational expectations in a VAR with Markov switching[END_REF]. We obtain the expressions of the conditional forecasts of t r ∆ , from the companion form of the MS-VAR(p); these forecasts are therefore expressed in terms of the MS-VAR(p) coefficients. Plugging them in equation [2] provide testable restrictions on these coefficients that guarantee that the term structure fulfil the REH. The algebraic details are developed in Appendix A. It is shown there that the restrictions to be tested, for any finite value of p and n, are as follows:

∑ - = - = 1 n 1 i ) i ( X t C ) n i 1 , 0 ( J ) 0 , 1 ( [5] with ) p 2 2 ( 2 2 2 2 2 2 ) 0 , , 0 , I ( J × × × × = , t t X ) 1 ( X B J C ⋅ = , ( ) t t t p 1 X x , , x -+ = ) 2 n . ie ( 2 i if X / B E J p B J C i 1 m t X 1 0 x 2 i 0 m x x i 1 m X 1 0 x ) i ( X m i t 1 t 1 m t m t m i t 1 i t t > ≥       ⋅ =                   ⋅ = ∏ ∑ ∏ ∏ ∑ = = - = = = - + + + + + - + - + where p 2 x p 2 2 x 2 2 x 2 2 x 2 2 x 2 2 x 2 2 x 2 2 x 2 2 x 2 2 x 2 2 x 2 ) p ( x ) 2 ( x ) 1 ( x X 0 I 0 0 0 I 0 0 0 I B B B B 1 p t 1 t t t                 = + - - … … … … … … … … … … … … and F o r P e e r R e v i e w 9 p , , 1 j d c b a B t t t t t x , j x , j x , j x , j ) j ( x … = ∀         =
Expression [5] includes 2p one-dimensional restrictions. They have to be fulfilled for any possible value of the p-dimensional process t X , which includes 2 p alternatives. There are thus a total of p2 p+1 one-dimensional restrictions to be tested, which may include some redundant ones that have to be removed before testing. For small p and n, the restrictions simplify and the redundant ones are easily detected.

Expression [5] can be tested using a non-linear Wald test, asymptotically distributed as a chisquare variable with degrees of freedom equal to the number of non-redundant restrictions. This test is not invariant to the form in which the non-linear restrictions are expressed (see [START_REF] Gregory | Formulating Wald tests of nonlinear restrictions[END_REF]. An alternative consists of estimating the restricted model and applying an LR or LM test. However, as soon as p>1 or n>2, the restricted model is extremely difficult to estimate; therefore, the alternative based on an LR test is considered here only when p=1 and n=2.

Note also that the restrictions [5] can be extended to other contexts in which rational expectations theory linking variables has to be tested (such as the asset pricing model or the uncovered interest rate parity). The only thing to do is to adapt suitably the vectors j N .

Before testing [5], it might be wise to test weaker conditions deduced from

[2]. If [2] is true, then the spread t S should Granger-cause t r ∆ .
As is well known, this can be checked through a joint significance test of S t-i , i=1,...,p in The data correspond to the interest rates of the Spanish inter-bank money market between January 1986 and May 1995. There are several reasons why we have chosen this sample. On the one hand, Prats-Albentosa and [START_REF] Prats-Albentosa | The expectations theory in a market of short-term financial assets: an application to the Spanish Inter-bank market 1986-1995[END_REF] apply on these data a linear model of the type described by equation [3] which accepts the hypothesis for n=24, but rejects it for n=2,4 and 12; the rejection is very strong in the case of n=2 (two weeks) 7 ; it is our purpose to check whether these conclusions change for n=2 and n=4 when a non-linear model of the type of [4] is adjusted, using [5] to test the theory. On the other hand, Spain joined the European Economic Union in January 1986 and entered the European Monetary Union (EMU) in January 1999; the entrance in EMU was far from automatic and intensive efforts were made in order to fulfil compulsory economic conditions imposed by the Maastricht Treaty; one of these conditions referred to the level of the market interest rates, which by May 1998 could not surpass the average of the three lowest interest rates of the EU by more than 1.5 percentage points; this requirement provoked very steep downward trends in all the Spanish interest rates in the years preceding the entrance in the EMU. In order to make sure that these exceptional circumstances did not interfere in any direction with the results of the tests, we preferred to exclude from the sample the data corresponding to the last years before the entrance of Spain in the EMU.

From an initial sample of daily observations, the data of the days 1,8,15 and 22 of each month were selected, as representative of the first, second, third and fourth week respectively. In this way, there are 4 observations per month and 48 observations per year 8 . Three different terms have been considered: one week, two weeks and four weeks (relative to the one-month rate).

The short-term rate r t is in all cases the one-week rate, the other two interest rates play the role

4 , 2 n r R S t n , t n , t = - = .

The estimation results

Model [4] has been estimated for n=2 and n=4 by maximum likelihood methods in a 5-step algorithm, described for instance in [START_REF] Hamilton | Rational-expectations econometric analysis of changes in regime[END_REF] and [START_REF] Sola | Testing the term structure of interest rates using a stationary vector autoregression with regime switching[END_REF]. The convergence of this algorithm and its speed is very much affected by the number of parameters to be estimated, as well as by the starting values. For this reason, the maximum value of p we have considered is 5 and the starting values have been determined applying the procedure of [START_REF] Beyaert | Switching-regime models in the Spanish inter-bank market[END_REF].

Specification tests and model selection

To check the empirical validity of the models, we applied specification tests that constitute bivariate extensions of those developed by [START_REF] Hamilton | Specification testing in Markov-switching time-series models[END_REF]. They are based on the scores of the likelihood function with respect to the parameters at time t: if the model is correctly specified, these scores must be uncorrelated between t and (t-1). It can be shown that a test of correlation of the scores with respect to the elements of the mean vectors may be seen as a test of autocorrelation of the errors; by the same token, a test of correlation of the scores with respect to the variances can be seen as a test of generalized ARCH effect. 9 An illustration of the results of these tests is presented in Table 1, where we reproduce the results for the MS-VAR(1) case. The test statistics are joint tests corresponding in the first column to the scores with respect to all the means (16 pairs of parameters), and in the second column to the scores with respect to the variances (8 pairs of parameters). For that model, the absence of autocorrelation is accepted for both values of n: the test statistic χ critical value; with only one lag, the first order dynamics of the data is therefore adequately covered.

Things are different for the second-order dynamics: there are clear symptoms of non modelled ARCH effects. However, it must be noted that the heteroskedasticity still present in these models is far below the heteroskedasticity detected in linear VAR models with the same data 10 .

We used Schwartz criterion in order to select a model among those that do not present autocorrelation. With that procedure, a MS-VAR(1) model has been selected both for n=2 and n=4. It is interesting to note that when the linear model [3] had been adjusted to the same data, much higher values of p had to be considered. 11 On these MS-VAR(1) models, we applied the [START_REF] Hansen | The likelihood ratio test under non-standards conditions: testing the Markov switching model of GNP[END_REF][START_REF] Hansen | Erratum: the likelihood ratio test under non-standards conditions: testing the Markov switching model of GNP[END_REF] linearity test against the nonlinear switching-regime specification. This test tackles the problem of the existence of unidentified parameters under the null hypothesis, which rules out the application of standard Likelihood Ratio tests. It requires Monte-Carlo simulations in every application, in order to obtain the critical value. For multivariate model, this computer-intensive procedure is extremely long. Usually, it is not carried out, except for simple single-equation models. In Table 2, Various parameters of this table seem to accept some type of restriction that might be worth taking into account in order to simplify the estimation and increase the efficiency. The means of both models seem in most cases to be equal in both states; some of them even seem to F o r P e e r R e v i e w 13 be equal to zero. Note that this is compatible with the constant premium of equations [2], as well as with a stricter version of the rational expectations theory. Also, some of the autoregressive coefficients seem to be non-significant. We tested these simplifying restrictions, both jointly (all of them together) and separately (one at a time). When they were accepted, we proceeded to re-estimate the model under the corresponding restrictions. The results are presented in Table 4.a; in Table 4.b, we reproduce the corresponding Hamilton specification tests. As can be seen in that table, for n=2, the results do not differ qualitatively from those of the unrestricted model; for n=4, the null of no autocorrelation is now marginally rejected at 5% but accepted at 2.5% and 1%.

Besides the simplification restrictions, there are two more aspects of the estimation process that are worth mentioning. The probability 00 p of staying in state 0 is high and systematically above p 11 . This fact is reinforced by the relative size of variances: those of state 0 are small and far below those of state 1. The former may be qualified as a "low-variance high-persistence stable state" , whereas the latter would be more a "high-variance unstable state".

4.2..2 The smoothed probabilities

From these estimations, the so-called "smoothed probabilities" may be inferred: on the basis of the estimated vector of parameter and the full sample of T observations, an inference is drawn about the historical state the process was in at some date t. These probabilities are shown in Figure 1.a for the two-week simplified model, and in Figure 1.b for the one-month simplified model. The probability of being in the stable state (state 0) is represented on the y-axis, and the dates on the x-axis. An analysis of these graphs provides an additional proof of the usefulness of the types of model used in this paper. A striking feature of these graphs refers to the much higher stability of the market from the middle of 1989 onwards, with the exception of the period extending from October 1992 to the third term of 1993. These dates coincide with specific events Monetary System (EMS); the instability period of 92-93 coincides with the crisis of the EMS, marked by the "monetary turmoil" of 1992, and the depreciation of the Peseta in September and November of that same year and of May of 1993; the EMS crisis ended in August 1993 with the enlargement of the fluctuation bands, although the monetary authorities purposely let the rates increase to very high levels in the autumn of 1993. The short instability periods observable at the beginning of 1991 are also attributable to known events: the overvaluation of the Peseta with respect to the DM that called for increases in the interest rates, together with the Gulf war and the political and economic instability in the East European countries were events that all created tensions in the financial markets. As far as the first years of the graph are concerned, it is interesting to note the instability discernible in the figures during the first half of 1987: it corresponds to the serious liquidity crisis of the public system, with important frictions between the monetary and exchange policies. The end of 1988 and beginning of 1989 is also characterised by relative instability; it is well known that this was a period of expectations of high interest rates that alters the markets. To sum up, the overall impression transmitted by these graphs is that the model is able to perceive the characteristics of the market.

Testing the expectations theory

Exact Rational Expectations Hypothesis

The form and the complexity of the REH restrictions [5] vary substantially as a function of n and p. For the models estimated in Section 4.2, the parameters n and p take the following values: n= 2 and 4, p=1. The low value of p greatly simplifies the procedure. Note that when p=1 we have indeed:

        = = t t t t t t x , 1 x , 1 x , 1 x , 1 ) 1 ( x X
] 3 . 6 [ , ] 2 . 6 [ , ] 1 . 6 [ , 1 1 , 0 , , , 1 0 1 , 0 1), (for , , 2 0 1 , 0 2 = = ∀ = = > = ∀ = = ∀ = t x j t x j t x x p j d x p p j c x c t t t … … [6]
For further analysis it is useful to note that the restrictions [6.1] imply the same value of 2 for 1 c in both states. It is easy to see that the specific value of 2 stems from the value of the (1,1) element of the matrix that pre-multiplies C (i) in [5]; this element derives directly from the fact that the expectations theory as expressed in version [1] asserts that the long rate is a simple average of the present and future short rate when n=2:

[ ] k / r E 2 1 r 2 1 R t 1 t t n , t + Η + = + [7]
On the other hand, the fact that 1 c takes the same value in both states is a consequence of the fact that the weights in the average [7] are state-independent.

For n=4, that is for the one-month model, when p=1, [5] transforms into 4 restrictions that can be written as:

[ ] [ ] 1 , 0 ) 0 , 4 ( / ) 1 , 0 ( / ) 2 , 0 ( ) 3 , 0 ( 1 2 1 = ∀ = + + + + + + t x t x x x t x x x B x B B E B x B E B t t t t t t [8]
These restrictions are not as easily interpretable as in the case of n=2.

We tested [6] and [8] both in the unrestricted and in the simplified models; we tested also Granger causality from the spread t S to t r ∆ (the so-called "weak" restrictions). The results are reproduced in Table 5.a and 5.b. 5.a indicates for the two-week model that the "weak" implications of the expectations theory are strongly supported by the data, since the null hypothesis of no causality is strongly rejected. It is symptomatic that in the simplification process from the general unrestricted model to the simplified one, the only significant coefficients in the equation of t r ∆ are precisely those associated with the spread, i.e. with Granger causality. As far as the "strong" implications of the expectations theory (restrictions [6]) are concerned, the results in Table 5.b are quite different: these restrictions are strongly rejected by the data, both using a nonlinear Wald test and a LR test. Note, however, that the rejection is entirely attributable to the requirement that

2 c c 1 , 1 0 , 1 = = , since the requirement that 0 d d 1 , 1 0 , 1 = =
, checked in the simplification process of the model, is fulfilled. We therefore also tested

1 , 1 0 , 1 c c =
without imposing a common value of 2; this hypothesis is accepted at different probability levels depending on the version of the model and the test used (see Table 5.b).

For the one-month model (n=4), global Granger causality form t S to t r ∆ is confirmed (see Table 5.a), although this result has to be qualified, since we have seen in the simplification process that 1 t S -does not influence t r ∆ in state 0, that is, in the stable state (see Table4.a).

As far as the strong implications are concerned, these are rejected.

Rational Expectations Hypothesis with "measurement errors"

It is often the case that the strict version of REH, as considered in the Section 4.3.1, is rejected, but a weaker version that makes allowance for a random error term in the REH relation is accepted. [START_REF] Hamilton | Rational-expectations econometric analysis of changes in regime[END_REF] provides a detailed justification of the existence of such a random error, which is often called "measurement error". The existence of this measurement error is equivalent to assuming that the agents build their expectations for F o r P e e r R e v i e w 17 period t on the basis of the information relative to period (t-2), instead of using information from period (t-1) as in the stricter (or "exact") version of the theory.

The implication of this measurement error for the testing of the theory is that even with a MS-VAR of order 1 (p=1), all the REH restrictions on the parameters will now be non-linear. No linear restrictions subsist. Although it is relatively easy to derive the theoretical expression of these new restrictions form what has been done for the case where no measurement errors are assumed, their practical implementation has to be reduced to small values of p and n. In Appendix B, it is shown that the restrictions to be satisfied and tested are now as given in [9] instead of [5]:

( ) 0 C n j 1 , 0 C 0 , 1 n 2 j ) j ( X ) 1 ( X t t =       - + ⋅ - ∑ = [9]
Due to the difficulty of their practical implementation, we applied them only on our two-week model, in which case n=2 and p=1. When n=2, [9] simplifies to:

( ) ) 1 ( X ) j ( X t t C 0 , 1 C 2 1 , 0 ⋅ =      
With p=1, the right-hand side becomes:

( ) t t t t x ) 1 ( x X ) 1 ( X B ) 0 , 1 ( B ) 0 , 1 ( B ) 0 , 1 ( C 0 , 1 = = = ⋅
where the redundant superscript (1) is omitted. Similarly, the left-hand side simplifies to:

∑ = + + +       =       1 0 x x , x x x ) j ( X 1 t 1 t t t 1 t t p B B 2 1 , 0 C 2 1 , 0
Thus, there are four restrictions to be tested that may be represented as 6. The upper part of that table corresponds to the original unrestricted model, whereas the lower part corresponds to the simplified model. In the simplified model, the simplifying restrictions coincide with three of the four restrictions included in [10]. So, only one REH restriction is left to be tested. Thus, with this simplified model, REH is in fact tested step by step. Given the inherent complexity of the model, this is in fact an advantage.

( ) ( ) 2 , 1 x 0 B 0 , 1 p B B p B B 2 1 , 0 t x 2 , x x 1 1 , x x 0 t t t t t = ∀ = - +       [10]
According to the results of Table 6, three of the four REH restrictions are accepted at any conventional level of significance, while the fourth one is compatible with the data at least at a 2.5% level of probability. So, with the type of nonlinear Markov Switching model that we have used and assuming measurement errors (or that agents adjust their expectations with a lag of two periods), there are signs of reconciliation of the Rational Expectations Hypothesis with the data in the Spanish financial market.

Summary and Conclusions

In this paper, we obtain very general formulas to test the Rational Expectations Hypothesis in the term structure of interest rates in the framework of a Markov-switching VAR model. Our theoretical results cover not only the strict version of REH, but also the weaker one where socalled "measurement errors" affect the decision-making process of the economic agents.

Although our interest in this paper lies in testing REH for the term structure of interest rates, the formulas are extensible to other contexts whenever REH can be expressed as a presentvalue relationship. This is similar, in a Markov-Switching context, to what occurs with the [START_REF] Campbell | Cointegration and tests of present value models[END_REF] methodology in a linear framework.

We apply the theoretical results on Spanish weekly data of the inter-bank market, comparing the one-week interest rate with the two-week rate and with the one-month rate. The period extends from January 1986 to May 1995, mainly in order to avoid the exceptional evolution of the Spanish interest rates in the last years before the entrance of this country in the EMU. For these rates, the expectations theory had been overwhelmingly rejected in previous studies where linear VAR models were used. The estimation results and the specification tests indicate that the switching-regime models specified in this paper very strongly dominate the linear ones when applied to these data. This conclusion stems not only from the results of Hansen's linearity test, but also from the fact that the data are represented with a smaller number of lags and with far fewer heteroskedasticity symptoms than in a linear model. The analysis of the "smoothed probabilities", which provide information about the most probable state of the economy in each point in time, also indicates that the model correctly identifies the successive stability and instability periods through which the Spanish economy and the Spanish financial markets evolved between 1986 and 1995. As far as REH is concerned, it is rejected in its exact version although the weak conditions of Granger causality from the spread to the interest differential are supported by the data. When measurement errors are considered, that is, if we assume that the agents form their expectations on the basis of what happened in the economy two periods ago, there is some reconciliation of the REH theory with the data for the two-week model. For the one-month model, REH with measurement errors is much more complex to test and has not been considered so far.

In summary, according to our results, the hypothesis of linearity imposes severe misspecifications on the models and might be responsible for excessive rejection of the Rational Expectations hypothesis in the term structure of interest rates. A non-linear model fits much better the data and provides some signs of fulfilment of the rational expectations hypothesis in the term structure of the Spanish inter-bank interest rates. In order to obtain testable restrictions, let us operate as in Sola and Drifill(1994) and in [START_REF] Blix | Rational expectations in a VAR with Markov switching[END_REF].

In terms of our model, this theory is fulfilled if

[ ] ( ) 1 p t t t X t i t 1 n 1 i n . t x , , x X with H / r E n i 1 S t + - + - = = λ + ∆       - = ∑
or alternatively ( )

1 p t t t 1 n 0 j X t j t j x , , x X ) H / y ( E N t + - - = + = ∀ λ = ∑ [A.1] with { } 1 p t t
    - = ∀ - = = 1 n , , 1 j ) 1 n j , 0 ( 0 j ) 0 , 1 ( N j
In order to obtain the restrictions that equation [A.1] imposes on the parameters of model [4], it is useful to resort to the "companion form" of VAR model [4]. For that purpose, first define:

( ) ′ ′ ′ = + -1 p t t
… = ∀         = p 2 x p 2 2 x 2 2 x 2 2 x 2 2 x 2 2 x 2 2 x 2 2 x 2 2 x 2 2 x 2 2 x 2 ) p ( x ) 2 ( x ) 1 ( x X 0 I 0 0 0 I 0 0 0 I B B B B 1 p t 1 t t t                   = + - - … … … … … … … … … … … … ( ) ) p 2 x 2 ( 2 2 2 2 2 2 0 , , 0 , I J × × × = t t X ) 1 ( X B J C ⋅ = , 2 i if X / B E J p B J C i 1 m t X 1 0 x 2 i 0 m x x i 1 m X 1 0 x ) i ( X m i t 1 t 1 m t m t m i t 1 i t t ≥       ⋅ =                         ⋅ = ∏ ∑ ∏ ∏ ∑ = = - = = = - + + + + + - + - +
Writing model [4] in "companion form" with the use of the coefficient matrix t X B , we obtain:

j t c 1 j t X c j t u J Y B Y 1 j t + - + + ⋅ ′ + = - +
For j=1, multiplying by J , and transferring 1 t

x + µ to the right-hand side, we get:

1 t c t X x 1 t u Y B J y t 1 t + + + ⋅ + µ = + [A.3]
For j>1, we obtain by recursion:

- + - + + ′ + ′ ⋅         ⋅ + ⋅ = ′ + ′ + = - + - + - + - + j 1 i 1 j 1 h j t h j t h 1 i c t X c t X j t 1 j t c 2 j t X X c j t u J u J Y B Y B u J ) u J Y B ( B Y i j t i j t 2 j t 1 j t
Again, multiplying by J , and transferring j t

x + µ to the right-hand side, we get:

∏ ∑ ∏ = - = + - + = + > + ′ ⋅         ⋅ ⋅ + ⋅ ⋅ + µ = - + - + + j 1 i 1 j 1 h j t h j t h 1 i c t X c t X x j t 1 j for u u J Y B J Y B J y i j t i j t j t [A.4]
We need the expressions of [ ]

t
+ - - = Therefore: [ ] [ ] ) Y ( B J p Y B J p x , , x , I / y E H / y E t t t t t X t X 1 q 0 i i , x i c t X 1 q 0 i i , x i 1 p t t t 1 t t 1 t µ - ⋅ + µ = ⋅ + µ = = ∑ ∑ - = - = + - + + [A.5]
where q is the number of regimes (in our case, q=2).

For j>1, proceeding similarly, we obtain:

[ ] c t j 1 m X x 1 p t t j t t j t Y B J x , , x , , x , I / y E m j t j t         ⋅ + µ = ∏ = + - + + - + + Therefore: F o r P e e r R e v i e w 24 [ ] [ ] ) Y )( x / x , , x ( P B J ) x / i x ( P Y ) x / x , , x ( P B J ) x / i x ( P x , , x , x , I / y E H / y E t 1 j t 1 t m j t j t 0 1 t m j t x t t 1 t 1 j t 1 q 0 x 1 q 0 x j 1 m X 1 q 0 i t j t i c t t 1 t 1 j t 1 q 0 x 1 q x j 1 m X 1 q 0 i t j t i 1 p t 1 t t t j t t j t µ -                 ⋅ + = µ =                 ⋅ + = µ = = = + - + - = - = = - = + + - + - = - = - = + + - - + + ∑ ∑ ∏ ∑ ∑ ∑ ∏ ∑ - + + - + + = + - + [A.6]
Separating the sums on t X µ from those on t Y and using a more compact notation, we may write, both for j=1 and for j>1:

[ ] t ) j ( X X ) j ( X ) j ( x t j t Y C C A H / y E t t t t + µ ⋅ + = + [A.7] with i , x 1 q 0 i i ) 1 ( x t t p A ∑ - = µ = and ) x / i x ( P A t j t 1 q 0 i i ) j ( x t = µ = + - = ∑ for j>1. Entering [A.7] in [A.1] and taking into account that [ ] t t t t Y J y H / y E ⋅ = = , we obtain [ ] t t t t t X 1 n 1 j t ) j ( X X ) j ( X ) j ( x j t 0 Y C C A N Y J N λ = + µ ⋅ + + ⋅ ⋅ ∑ - =
Using the definition of j N , reordering the elements and multiplying by -1, this can be expressed as: )

0 Y C n j 1 , 0 J 0 , 1 C n j 1 , 0 A n j 1 , 0 t 1 n 1 j ) j ( X 1 n 1 j X X ) j ( X 1 n 1 j ) j ( x t t t t t =                       - + ⋅ - +       λ + µ       - +       - ∑ ∑ ∑ - = - = - = [A.8]
The sum of terms in the first bracket refers to the restrictions on the term premium X t λ .The second bracket involves the implications of most interest, which correspond to the way in which the short and long rates have to be related dynamically with each other in order for the theory to be true. These implications are often called the "cross-equations" restrictions and we centre the testing on them. 12 For these restrictions to be true for any value of t Y and any combination of past regimes t X , it must be that:

( ) 0 C n j 1 , 0 J 0 , 1 1 n 1 j ) j ( X t =       - + ⋅ - ∑ - = [A.9]
which are the restrictions [5] in the core of the paper. As is well known, a weaker version of the expectation theory consists of accepting that the agents are affected in their decisions by measurement errors of different types (see among others Campbell and[START_REF] Campbell | Cointegration and tests of present value models[END_REF][START_REF] Sola | Testing the term structure of interest rates using a stationary vector autoregression with regime switching[END_REF][START_REF] Sola | Testing the term structure of interest rates using a stationary vector autoregression with regime switching[END_REF]. As far as the testing is concerned, the existence of such errors amounts to assume that the agents use an information set that is lagged one period compared with what they would have used in absence of these errors. In other words, expression [A.1] has to be substituted by

[ ] ∑ - = - - + λ = 1 n 0 i 1 t X 1 t i t i H / E ) H / y ( E N t [B.1] which is equivalent to [ ] ∑ = + + λ = n 1 j t X t j t j H / E ) H / y ( E N ~1 t [B.

  above p=1 (we consider p=1,…,5), instead of limiting the estimation to the only case of a VAR(1) model as done by

  equation of[4], against the alternative that at least one of these coefficients differs from 0.

  on the term structure of interest rates of Spain 4.1. The data set and the motivation of its use

  term rate R t . The following spreads have thus been considered:

  we reproduce the value of this test for our MS-VAR(1) model. Based on 1000 simulations, the one percent critical value is below 40 for the two-week (n=2) model and below 35 for the one-month (n=4) model. As can be seen, the test statistics are far above these values, indicating the domination of the non-linear model [4] over the linear model [3].Table 3 reproduces the estimated value of the parameters of model [4] for n=2, n=4 and p=1.

  market: June 1989 is the date of the entrance of Spain into the European

  we may omit the redundant superscript (1).

  n=2, only one element intervenes in the sum of the right-hand side of[5], so that we have, for any value of p:

  testing [10] in the two-week model are reproduced in Table

  the coefficients of the model has already been obtained in Appendix A (see[A.6]) and that is all what we need to specify the restrictions on the dynamics of the short and the long rate that would substitute [5] (or equivalently[A.9]).Therefore, applying to [B.2] the same reasoning as to[A.1] in Appendix A, it results that the restrictions with measurement errors are:

Figure 1 :

 1 Figure 1: Smoothed probabilities for the Spanish data

Fig

  Fig 1a: Simplified two-week VAR(1) model

Table 2 :

 2 Hansen tests -VAR(1) :

		Value of the test	0.01 critical value
		statistic
	2 weeks, n=2	1088.08	<40
	1 month, n=4	983.818	<35

Table 3 :

 3 Estimated parameters of model [4]

			2 weeks -VAR(1)	1 month -VAR(1)
			n=2	n=4
			State 0	State 1	State 0	State 1
	a 1		.416(.060)	.100(.059)	.757(.038)	.441(.060)
	b1	.034(.030)	.026(.019)	.170(.041)	.075(.019)
	c 1		.239(.134)	.842(.208)	.142(.082)	.682(.136)
	d 1	-.126(.063)	.018(.044)	-.139(.059)	-.015(.030)
	σ σ σ σ 2	S,x	.004(.001)	.119(.020)	.0009(.001)	.170(.027)
	σ σ σ σ 2 ∆ ∆ ∆ ∆r,x	.020(.003)	1.567(.278)	.017(.003)	1.277(.196)
	σ σ σ σS ∆ ∆ ∆ ∆r,x	-.002(.001)	-.167(.050)	-.004(.001)	-.245(.055)
	µ µ µ µ S	.019(.006)	.018(.036)	.043(.019)	.131(.041)
	µ µ µ µ ∆ ∆ ∆ ∆r	-.017(.008)	.036(.122)	-.013(.007)	.029(.099)
	p00	.890(0.023)	.902(.021)
	p11	.595(0.064)	.709(.052)
		Standard errors in brackets

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

[START_REF] Bansal | Term structure of interest rates with regime shifts[END_REF] also make use of Markov-switching model but in a different approach: they introduce a two-state MS process in the Cox-Ingersoll-Ross Model for the term structure and

ACKNOWLEDGMENTS:

The authors acknowledge the support of the Spanish Ministerio de Educación y Ciencia, under CICYT Research Grant SEC971253. They also benefited from very helpful comments by Gabriel Perez-Quiros and by an anonymous referee.

Model

Null hypothesis p-value

One-month VAR(1) model n=4

[9] is true -"strong" restrictions 0.0% (Wald test)

Restricted onemonth VAR(1) model n=4

[9] is true -"strong" restrictions 0.0% (Wald test)

Table 6: Tests of the "strong" restrictions with measurement errors for the two-week model

Model

Null hypothesis p-value

Two-week VAR(1) model n=2

[10] is true-"strong" restrictions 0.0%

Simplified twoweek VAR(1) model n=2

Remaining restriction in [10] is true -"strong" restrictions 2.72%