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Government bond yield spreads and the euro Lorenzo Pozzi∗ September 2009 Abstract This paper estimates a latent factor decomposition of the weekly 10 year government bond yield spreads of Belgium, France, Italy, and the Netherlands versus Germany over the pe- riod 1991-2006. Each spread is decomposed into a country-speciﬁc and a common factor. The country-speciﬁc factors and the country-speciﬁc factor loadings on the common factor are allowed to converge to a situation of full government bond market integration. The re- sults suggest that, in the period after the introduction of the euro, government bond market integration has increased JEL Classiﬁcations: E43, G12 Keywords: government bond yield spreads, euro, state space methods ∗ Tinbergen Institute and Department of Economics, Erasmus University Rotterdam, Burgemeester Oudlaan 50, PO Box 1738, 3000DR Rotterdam, the Netherlands. Tel: +31 (0)10 408 12 56. Fax: +31 (0)10 408 91 61. Email: pozzi@few.eur.nl. Website: http://people.few.eur.nl/pozzi. 1 1 Introduction This paper analyzes the integration of euro area government bond markets by investigating the importance of the common international risk factor in euro area government bond yield spreads. In the literature on (euro area) government bond yield spreads two approaches have been followed to take this common factor into account. First, a number of studies investigating euro area countries include a proxy for this common factor in their analysis (e.g., Codogno et al., 2003, and Favero et al., 2007). Second, a number of studies ﬁlter the common factor out of the government bond spreads through the use of factor analysis and state space methods (e.g., Dungey et al., 2000). This paper follows the second approach. Using a state space approach I decompose the 10 year government bond spreads of Belgium, France, Italy, and the Netherlands versus Germany into a common factor and an idiosyncratic country-speciﬁc factor. Data used are weekly and cover the period 1991-2006. The paper adds to the literature by investigating whether the country-speciﬁc factors and the country-speciﬁc factor loadings on the common factor in the spreads have converged towards a situation of full government bond market integration, i.e. a state that I deﬁne as characterized by zero country-speciﬁc factors and equalized country-speciﬁc factor loadings on the common factor. The results suggest that the country-speciﬁc factors and factor loadings in the bond spreads are signiﬁcantly reduced for all four countries in the period after the introduction of the euro implying a signiﬁcant increase in euro area government bond market integration. I report, ﬁrst, a reduction of country-speciﬁc premiums (e.g. liquidity premiums) in bond spreads since I ﬁnd a decrease towards zero of the country-speciﬁc factors in the latent factor decomposition of the spreads. Second, I report a decrease in the country-speciﬁc exposure to international risk since I ﬁnd a decrease towards a common value (of zero) of the country-speciﬁc factor loadings on the common factor in the latent factor decomposition of the spreads. The outline of the paper is as follows. Section 2 presents the empirical speciﬁcation and the estimation method. Data issues are also discussed. Results from the estimations are reported in section 3. The ﬁnal section concludes. 2 2 Empirical speciﬁcation, data, and estimation method 2.1 Empirical speciﬁcation I estimate the following system (where denotes time and denotes country with = 1 ), +1 = +1 + +1 +1 (1) +1 = +1 + + +1 +1 (2) +1 = + +1 (3) +1 = + + +1 (4) Eq.(1) presents the decomposition of the government bond spreads +1 into an idiosyncratic factor +1 and a common factor +1 where the latter is multiplied by country-speciﬁc and time-varying factor loadings +1 . For +1 I assume that lim→+∞ +1 = 0. I therefore model +1 as an (1) process, i.e. ∗ ∗ +1 = + + +1 with 0 1, multiplied by a deterministic convergence operator +1 where lim→+∞ +1 = 0. Thus, +1 = ∗ +1 . To obtain eq.(2) write +1 ∗ +1 = (1 − ) + +1 (1 − ) where is the lag operator. Multiplication by +1 then gives +1 = +1 (1 − ) + +1 +1 (1 − ). Multiplication of both sides of this expression by (1 − ) gives eq.(2). For I use the following logistic speciﬁcation1 , = exp [ ( − )] (1 + exp [ ( − )]) (5) where 0 is the rate of convergence. Since 0 I ﬁnd = 0 for → +∞ and = 1 for → −∞. In a sample of size the fact that 0 implies that ≈ 0 for and that ≈ 1 for . The parameter with 1 determines the mid-point of the change. For +1 I assume that lim→+∞ +1 = where is common across countries. Hence in eq.(3) I model +1 as a constant plus a country-speciﬁc constant that is multiplied by the convergence operator deﬁned above. 1 See e.g. Pozzi and Wolswijk (2008) who model the idiosyncratic components in government bond risk premia instead of government bond spreads. 3 As can be seen in eq.(4) I assume that the common component +1 follows a standard (1) process with 0 1. The error terms in the system +1 ( = with = 1 ) follow (1 1) processes (e.g., Dungey et al., 2000), 12 +1 = +1 +1 (6) where +1 ∼ (0 1) and where +1 = [+1 ] = + 2 + (7) with 0, 0 1, 0 1, and 0 + 1. The unconditional variance of +1 is given by (1 − − ). From the model the start of the convergence process is characterized by +1 = 1. This implies +1 = ∗ +1 and +1 = + . After complete integration +1 = 0 so that +1 = 0 and +1 = , i.e. the country-speciﬁc factors have disappeared and the country-speciﬁc impacts of the common component are identical across countries. 2.2 A look at the data All data are taken from Datastream/Thomson Financial. The available data cover the period 6/28/1991 to 8/4/2006 (789 weekly observations). I average daily data to weekly data to avoid day-of-the-week eﬀects (e.g., Dungey et al., 2000). For the spreads +1 I use the yield to maturity of 10 year government bonds issued by country (i.e. Belgium, France, Italy, and the Netherlands so that = 4) minus the yield to maturity of the benchmark country (i.e. Germany). To remove exchange rate changes from the spreads before the introduction of the euro on 1/1/1999 I subtract from these spreads the diﬀerence between the 10 year ﬁxed interest rates on swaps denominated in currency of country and those denominated in DEM (e.g., Codogno et al., 2003). In Figure 1 the joint movement of the spreads suggests that they are driven by a common component. Moreover, from table 1, the correlation between the spreads is higher after the intro- duction of the euro indicating that the idiosyncratic components in the spreads have become less important. 4 2.3 Estimation 2.3.1 Method I obtain estimates for the unobserved states +1 and +1 , for the conditional variance series +1 and +1 , for the convergence operator series +1 , and for the parameters in the model by putting the model described in section 2.1 in state space form.2 Estimates of the state vector are obtained with the Kalman ﬁlter and smoother while parameter estimates are obtained by maximum likelihood. The time-varying conditional variances complicate the otherwise standard Gaussian linear state space framework. To deal with this I follow the approach by Harvey et al. (1992) and augment the state vector with the shocks +1 and +1 . The Kalman ﬁlter then provides estimates of the conditional variance of the shocks, i.e. estimates for +1 and +1 . 2.3.2 Identiﬁcation First, note that eq.(1) can be written as +1 = +1 ∗ ++1 + +1 +1 . The ﬁrst term +1 in this expression +1 ∗ can be multiplied and divided by a constant , i.e. (+1 )(∗ ) +1 +1 so that a new term is obtained which is equally plausible. Hence the system is unidentiﬁed. To avoid this I set 1 = 1 (∀). The same problem holds for the second and third terms. To identify these terms I impose an unconditional variance of unity on +1 , i.e 2 = 1. This amounts to setting = 1 − − (e.g., Dungey et al., 2000). Second, the sign of the factor loadings +1 cannot be identiﬁed because of the sign invariance of the factor variance decompositions of the spreads. Therefore, I impose +1 0 by setting 0 and 0 (∀). Third, since only constants can be estimated but the model contains + 1 constants I identify the constants by setting = 0. 3 Results In table 2 I present the results of estimating the system given by eqs.(1) to (7). In ﬁgure 2 the smoothed estimates for +1 are presented for all countries. In ﬁgure 3 the smoothed estimates for +1 are reported. In ﬁgure 4 the estimates for +1 are presented.3 2 The state space representation of the model is available from the author by request. 3 Figures for the estimated GARCH series +1 and +1 are available from the author by request. 5 From table 2 I note that there are signiﬁcant convergence eﬀects since is signiﬁcantly lower than zero for all countries. The estimates for and imply an estimated series +1 for each country (see ﬁgure 4). It is obvious from this ﬁgure that the convergence of the government bond spreads towards full government bond market integration has occurred after the introduction of the euro for all countries. However, full convergence to zero had not yet occurred by the end of the sample period. While the magnitude of +1 and +1 is much lower after the introduction of the euro the non-zero values found for +1 at the end of the sample period for all countries suggest that the idiosyncratic components still had a non-negligible impact. To investigate the adequacy of the presented model I report the Akaike Information Criterion for model comparison. I calculate this statistic for the model given by eqs.(1) to (7) () and for three alternative models. denotes this statistic for a model with no convergence eﬀects, i.e. when +1 = 1 ∀ . denotes this statistic for a model with logistic convergence in +1 only and a time-invariant but country-speciﬁc +1 , i.e. +1 = . denotes the statistic for the same model as for but with an euro dummy in +1 (which takes on value 1 before 1/1/1999). According to the comparison of these statistics, the model presented in this paper - with (logistic) convergence eﬀects on both +1 and +1 - is preferred. 4 Conclusions I use weekly data over the period 1991-2006 to decompose the 10 year government bond spreads of Belgium, France, Italy, and the Netherlands versus Germany into a common component and an idiosyncratic component. Convergence operators are used to investigate government bond market integration. The results suggest that, after the introduction of the euro, both the country-speciﬁc factors and the country-speciﬁc factor loadings on the common factor in the bond spreads have converged towards zero for all four countries. Full convergence to zero of these components had not yet occurred by the end of the sample period however. 6 References Codogno, L., C. Favero, and A. Missale (2003): “Yield spreads on EMU government bonds,” Economic Policy, 18, 503—532. Dungey, M., V. Martin, and A. Pagan (2000): “A multivariate latent factor decomposition of international bond yield spreads,” Journal of Applied Econometrics, 15, 697—715. Favero, C., M. Pagano, and E. von Thadden (2007): “How does liquidity aﬀect government bond yields ?,” CSEF Working Paper 181. Harvey, A., E. Ruiz, and E. Sentana (1992): “Unobserved component time series models with ARCH disturbances,” Journal of Econometrics, 52, 129—157. Pozzi, L., and G. Wolswijk (2008): “Have euro area government bond risk premia converged to their common state ?,” mimeo. Web References http://ideas.repec.org/a/bla/ecpoli/v18y2003i37p503-532.html http://ideas.repec.org/a/jae/japmet/v15y2000i6p697-715.html http://ideas.repec.org/p/sef/csefwp/181.html http://ideas.repec.org/a/eee/econom/v52y1992i1-2p129-157.html http://ideas.repec.org/p/dgr/uvatin/20080042.html Tables and Figures 7 Table 1: Correlation matrix of corrected 10 year government bond spreads versus Germany (weekly data). Full sample Sample after introduction euro 6/28/1991 to 8/4/2006 1/1/1999 to 8/4/2006 BE FR NL IT BE FR NL IT Belgium 1 - - - 1 - - - France 0.2884 1 - - 0.8780 1 - - Netherlands 0.6106 0.1987 1 - 0.8891 0.8193 1 - Italy 0.3095 -.4663 -.0735 1 0.8608 0.7978 0.7264 1 8 Table 2: Maximum likelihood estimation of the common factor model with GARCH errors and con- vergence eﬀects (eqs. 1-7). Country-speciﬁc parameters Common parameters Belgium France Netherlands Italy 0.9716 0.9811 0.7003 0.9954 0.9667 (0.0077) (0.0062) (0.0644) (0.0035) (0.0076) 0.0063 0.0012 0.0262 0.0008 - (0.0020) (0.0011) (0.0075) (0.0011) - -0.0068 -0.0115 -0.0224 -0.0340 - (0.0003) (0.0021) (0.0054) (0.0051) - 737.82 681.38 725.65 713.31 - (0.0053) (34.821) (14.735) (8.596) - 2.5E-5 0.0001 2.4E-5 3.3E-6 0.0055 (8.8E-6) (3.5E-5) (6.4E-6) (1.4E-6) (0.0036) 0.1459 0.3699 0.4291 0.1253 0.0994 (0.0455) (0.0651) (0.0714) (0.0138) (0.0204) 0.7906 0.5397 0.5551 0.8739 0.8951 (0.0515) (0.0677) (0.0703) (0.0139) (0.0211) 0.0252 0.0222 0.0223 0.0223 - (0.0049) (0.0045) (0.0043) (0.0044) - - - - - 1.7E-8 - - - - (1.4E-6) Goodness of ﬁt -18.2915 -18.2687 -18.2510 -18.2003 Note: Hessian based standard errors between brackets. For the common state the point estimate and standard error of are obtained from the restriction = 1 − − . denotes the Akaike Information Criterion for the full model with convergence eﬀects on and , is the statistic if converges but is country-speciﬁc and contains an euro dummy, is the statistic if converges but is country-speciﬁc but constant over time, and is the statistic if there is no convergence, i.e. if = 1 ∀ . A model with a smaller is preferred. 9 Figure 1: Corrected 10 year government bond spreads versus Germany 2 .5 2 1. 5 1 0 .5 0 - 0 .5 J un-91 J un-94 J un-97 Jun-00 Jun-03 J un-06 B elg ium Fr an c e N e t he r lan d s It a l y Figure 2: Idiosyncratic state for all countries 2.5 2 1.5 1 0.5 0 -0. 5 J un-91 J un-94 J un-97 J un-00 J un-03 J un-06 B el gi um Fr a n c e N et her l ands Ital y 10 Figure 3: Common state 9 4 -1 -6 - 11 - 16 Jun- 91 Jun- 94 Jun- 97 Jun- 00 Jun- 03 Jun- 06 Figure 4: Convergence dynamics of the idiosyncratic components and the idiosyncratic factor loadings (convergence operators ) 1.2 1 0 .8 0 .6 0 .4 0 .2 0 J un- 91 J un-94 Jun-97 J un-00 J un-03 J un- 06 B elgium Fr a n c e N et her lands It a l y 11