The comovement of US and German bond markets


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1 The comovement of US and German bond markets Tom Engsted and Carsten Tanggaard The Aarhus School of Business, Fuglesangs alle 4, DK8210 Aarhus V. s: (Engsted); (Tanggaard). February Abstract We use a vectorautoregression, with parameter estimates corrected for smallsample bias, to decompose US and German unexpected bond returns into three news components: news about future inflation, news about future real interest rates, and news about future excess bond returns (term premia). We then crosscountry correlate these news components to see which component is responsible for the high degree of comovement of US and German bond markets. For the period we find that inflation news is the main driving force behind this comovement. When news is coming to the US market that future US inflation will increase, there is a tendency that German inflation will also increase. This is regarded bad news for the bond market in both countries whereby bond prices are bid down leading to immediate negative return innovations and changing expectations of future excess bond returns. Thus, comovement in expected future inflation is the main reason for bond market comovement. Keywords: International bond markets, VARmodel, return variance decomposition, smallsample bias, bootstrap simulation. JEL codes: C32, E43, E44, F21, F36, G12, G15. 1 Introduction It is wellestablished that bond markets in di erent countries tend to move together, i.e. bond yields and returns are positively correlated across countries. A number of earlier studies have used cointegration analysis to document this fact, e.g. Hafer et al. (1997). Longterm Comments from seminar participants at the Aarhus School of Business, and participants at the International Bond and Debt Market Integration Conference in Dublin, are gratefully acknowledged. 1
2 interest rates seem to be cointegrated across countries, indicating that international bond markets are linked together. These cointegration analyses do not, however, examine comovement in the underlying factors determining bond yields. Ilmanen (1995) suggests a number of factors determining international bond returns, and he finds that a small set of global (world) factors accounts for the predictable variation in bond returns and their crosscountry correlation. In particular, wealthdependent riskaversion of bond investors appears to be an important source of the international comovement. Barr and Priestley (2004) also find that bond returns in di erent countries are predictable over time, and based on an international CAPM they find that 70 percent of the variation in expected returns is due to world risk factors while the remaining 30 percent is due to local countryspecific risk factors. They interpret this result as indicating that national bond markets are only partially integrated into world markets. Driessen et al. (2003) use a linear factor model and principal components analysis to analyze international bond returns, and they find that the positive correlation of bond returns is driven by the positive correlation between the levels of the term structures across countries. 1 In the present paper we approach the international comovement of bond markets from a di erent angle. We do not search for plausible underlying factors that can explain the comovement, or test specific international asset pricing models. Instead, we make use of the return variance decompositions developed by Campbell and Ammer (1993) and Engsted and Tanggaard (2001): we decompose excess bond return innovations in each country into three news components: news about future longterm inflation, news about future real interest rates, and news about future excess bond returns (term premia). We then use a vectorautoregressive (VAR) model to compute these news and innovation components. The VAR model contains variables from each country, and we measure international bond market linkages by crosscountry correlating the VARgenerated news and innovation components. The appealing feature of our approach is that, apart from a linearization error, the return variance decomposition holds as a dynamic identity, i.e. from the way bond returns are defined unexpected excess bond returns can always be stated in terms of changes in expectations ( news ) of future inflation, real interest rates, and term premia. Thus, these three news components jointly comprise all possible underlying economic factors governing the variation in bond returns. 1 An obvious economic candidate for this level factor is the rate of inflation. As will become clear, our analysis explicitly investigates the importance of expected inflation in explaining international bond market comovement. 2
3 Our analysis is close in spirit to the analysis in Sutton (2000). He suggests a method for examining whether comovement in longterm bond yields can be explained in terms of comovement in shortterm interest rates (in accordance with the Expectations Theory), or is rather a consequence of comovement in term premia across countries. He finds that term premia at the long end of the term structure are both timevarying and positively correlated across countries. Sutton s analysis has some restrictive features that distinguishes it from our analysis. First, Sutton assumes that term premia and interest rates are independent. Second, his approach does not make it possible to identify the separate e ects on bond yield comovement from inflation and real interest rates. Finally, his analysis is based on computation of perfect foresight longterm interest rates, which reduces the e ective samplesize dramatically. Our approach, by contrast, makes no assumptions on the relation between term premia and interest rates, identifies the separate e ects from inflation and real interest rates, and uses a loworder VAR model to compute expected values which implies no large reduction in the e ective sample size. Recently it has been documented that in VAR models the traditional OLS parameter estimates may be severely biased in small samples, and that this bias may seriously distort statistics generated from the VAR, see e.g. Bekaert et al. (1997). For this reason we use the analytical bias formula from Pope (1990) to correct the VAR parameter estimates for smallsample bias. In addition, we use standard bootstrap techniques to compute smallsample standard errors and confidence intervals of the VAR generated statistics. We apply our suggested procedure on monthly data from the US and German bond markets over the period Our most important empirical results can be summarized as follows. First, there is significant predictable variation in onemonth excess bond returns in both countries. Second, there are important spillover e ects from the US market to the German market, but only limited spillover e ects the opposite way, and onemonth excess bond returns in the US and Germany show a contemporanous positive correlation of 0.54, indicating that bond markets in the two countries are closely linked together. Third, inflation news is the main driving force behind this comovement. When news is coming to the US market that future US inflation will increase, there is a tendency that German inflation will also increase. This is regarded bad news for thebondmarketinbothcountrieswherebybondpricesarebiddown leading to immediate negative return innovations and changing expectations of future excess bond returns. Thus, in contrast to the results reported by Sutton (2000) where comovement in term premia is found 3
4 to be the main reason for bond market comovement, our results suggest that expected future inflation is the main reason. Finally, our results are mostly robust to changes in the laglength of the VAR and to changes in the sample periods. The rest of the paper is organized as follows. In section 2 we present the basic bond return variance decomposion based on a VARmodel, and we briefly describe the procedures for biasadjustment of the VAR parameter estimates and the bootstrap simulations. Section 3 reports the empirical results, and o ers some thoughts on the possible causes for inflation news comovement. Finally, in section 4 we relate our findings to some other recent findings in the literature. 2 A VARbased variance decomposition of bond returns In section 2.1 we present the basic excess bond return innovation formula that will form the basis for the subsequent empirical work. In section 2.2 we decribe how we construct the return variance decomposition based on a vectorautoregression. Finally, in section 2.3 we explain the procedure used to biascorrect the VAR parameter estimates, and the bootstrap technique used to compute smallsample standard errors and confidence intervals of the VAR generated statistics. 2.1 The excess bond return innovation formula Campbell and Ammer (1993) derive an expression for the unexpected excess return on an period zerocoupon bond. According to this expression the oneperiod return on an period zerocoupon bond in excess of the shortterm interest rate can be decomposed into changes in expectations of three variables 1 periods into the future: inflation, real interest rates, and excess bond returns. However, often zerocoupon yields on long bonds do not exist many years back, so one will have to work with yields on coupon bonds. In Engsted and Tanggaard (2001) we use the loglinearized formula for the return on a longterm coupon bond in Shiller and Beltratti (1992) to derive an excess return innovation expression similar to the one in Campbell and Ammer (1993), but for a coupon bond instead of a zerocoupon bond. The expression looks as follows: =( +1 ) X ( ) (1) =1 where +1 is the nominal oneperiod log gross bond return from to +1 in excess of the continuously compounded nominal oneperiod interest 4
5 rate, +1 is the inflation rate from to +1, +1 is the oneperiod log real interest rate from to +1, is the conditional expectations operator based on information at time, and is constant slightly less than one. 2 Expression (1) shows that unexpected excess bond returns must be due to news (i.e. changes in expectations) coming to the market about either future inflation, future real interest rates, or future excess returns (or a combination of the three). News that either inflation, real interest rates, or excess returns will be higher (lower) in the future, will lead to an immediate drop (increase) in excess bond returns. The expression is a dynamic accounting identity with no economic theory imposed; it is derived from the definition of the linearized oneperiod return on a longterm coupon bond, so apart from the linearization error, it holds by construction. Excess bond return innovations can always be decomposed into changes in expectations of future inflation, real rates, and excess returns. Thus, empirical analysis of the volatility of the bond market based on (1) is not conditional on a particular economic model for the bond market. However, it is illustrative and informative to ask what traditional economic theories for bond yields imply for equation (1). For example, the Fisher Hypothesis states that nominal bond yields move oneforone with expected inflation so that ex ante real rates are constant. This implies that the news about future real rates component in (1), ( +1 ) P =1 +1+, will be zero. Similarly, the Expectations Theory of the term structure says that the longterm bond yield is given as expected future short rates plus a timeinvarying term premium. This implies that expected excess bond returns are constant so that the news about future excess returns component, ( +1 ) P =1 +1+, is zero. Thus, if both the Expectations Theory and the Fisher Hypothesis hold, inflation news is the only source of variation in excess bond return innovations. However, these economic theories will not hold exactly (and according to many empirical studies notevenapproximately),soingeneralvariationinbondreturnswillbe caused by all three news components in (1). Equation (1) holds for each country, so we will use a VAR model, to be described in more detail in the next subsection, containing variables from the US and German bond markets to estimate the components in (1) for each country, and we will then crosscountry correlate the components to see whether the high positive correlation of US and German bond returns is caused mainly by high positive correlation of inflation 2 is defined as =exp( ), where is the mean nominal yield to maturity on the long coupon bond, see Shiller and Beltratti (1992) and Engsted and Tanggaard (2001). 5
6 news, real interest rate news, or term premia news. In order to simplify the notation, we write (1) as e +1 = e +1 e +1 e +1, = (2) where denotes either US (us) orgermany(ge), and e e +1 ( +1 ) P =1 +1+ e +1 ( +1 ) P =1 +1+ and e +1 ( +1 ) P = The VARbased return variance decomposition In order to compute each of the components in (2) based on available data samples, we need an econometric timeseries model that can be used to estimate the conditional expectations of returns, inflation, and interest rates. The standard choice is a linear VAR containing excess bond returns and real interest rates, and possibly other variables with predictive power for returns and real rates. The VAR needs to include inordertobeabletocompute +1, +1 and +1, and +1. Simi +1 and larly, and are needed in the VAR in order to compute +1. Notethatonlythreeofthefourcomponentsin(2)arecomputed directly from the VAR; the last component is computed as a residual to make sure that (2) holds exactly. Thus, e +1, e +1, ande +1 are computed directly, and the remaining component e +1 is computed as e +1 = e +1 e +1 e Our benchmark VAR model contains the following variables, collected in the vector : =(,,,,, ) 0. is the spread between the longterm bond yield and the shortterm interest rate in country. 4 Weincludethisvariablebecausemanyprevious studies have found that the longshort spread has significant predictive 3 In Campbell and Ammer (1993) the VAR model does not include excess bond returns directly. Instead, the firstdi erence of the nominal shortterm interest rate and the spread between the longterm and shortterm interest rates are included, and the excess return innovation is then computed from these two variables. Campbell and Ammer include as a state variable in the VAR, which makes it possible to compute the real interest rate news component directly. In addition, by combining the real interest rate with the firstdi erence of the nominal interest rate, the inflation news component can be computed. Finally, the excess return news component is computed as a residual from the other components using the dynamic identity. In Engsted and Tanggaard (2001) we show that Campbell and Ammer s procedure only works for zerocoupon bonds. For coupon bonds it will be necessary to include excess bond returns directly in the VAR, and then compute the inflation news component as a residual. 4 Specifically, is defined as =(1 ) 1, where is the log of the nominal longterm bond yield in country and is the continuously compunded nominal shortterm interest rate in country (see Shiller and Beltratti, 1992). 6
7 ability for bond returns. The firstorder VAR model for is written as (higherorder models are easily handled using the companion form) +1 = + +1 (3) where is the VAR parameter matrix and is the vector of error terms. 5 From (3) the VAR estimate of is +1. Similarly, the VAR estimate of ( +1 ) +1+ is +1. Thus, by defining row selection vectors 1, 1, 2,and 2 that pick out, respectively, the first, second, third, and fourth element of, VAR estimates of each of the components in (2) are given as e +1 = 1 +1 = (4) e +1 = 1 ( ) 1 +1 = (5) e +1 = 2 ( ) 1 +1 = (6) e +1 = e +1 e +1 e +1 = (7) where the VAR residuals and parameter estimates are inserted in +1 and, respectively. Note that the inflation news component, e +1, is computed as a residual from the dynamic identity. This implies that we do not have to include inflation directly as a state variable in the VAR, which is quite appealing because monthly inflation in both the US and Germany shows strong seasonal variation. 6 From (2) it follows that (e )= (e )+ (e )+ (e )+ 2 (e e )+2 (e e )+2 (e e ) (8) We calculate the relative magnitude of each of the news components by computing the variances and covariances in (8) and then normalize by (e ). The covariances, however, make these measures di cult to interpret. Thus, we also measure the relative magnitudes by orthogonalizing (using a Cholesky decomposition) the components and then compute the 2 values in regressions of e +1 onto each of the orthogonalized components. The sum of the 2 values then add up to one, 5 All variables in the VAR are measured in deviation from their unconditional means. 6 The seasonal pattern is slightly changing over time in both countries, so we could not account for it by simply including seasonal dummies in the VAR. 7
8 so each individual 2 number can be interpreted as the fraction of the return innovation variance explained by that particular news component. 2.3 Smallsample biascorrection and the bootstrap technique OLS estimates of VAR parameters are biased in small samples and hence the news and innovation components (4) to (7) are also biased. One way to correct for this bias is to use Monte Carlo procedures, as in e.g. Bekaert et al. (1997) and Engsted and Tanggaard (2001), by either assuming normally distributed VAR errors or by using bootstrap techniques to resample the original VAR residuals, i.e. use the empirical smallsample distribution of the errors. An alternative is to use the analytical bias formula derived by Pope (1990). Based on a higherorder Taylor series expansion, Pope derives the following formula for the bias,, in VAR models like (3): where is = + ( 3 2 ) = [( ) 1 + ( 2 ) 1 + X ( ) 1 ] (0) 1 and ( ) = +, 0 are the eigenvalues of, is the conditional variance of,and isthesamplesize. As seen, the approximation error in the bias formula vanishes at the rate 3 2 which is at least as fast as in standard Monte Carlo or bootstrap biasadjustment. In our empirical analysis we will use Pope s biasadjustment procedure. We use bootstrap simulation to compute smallsample standard errors and equaltailed confidence intervals of the VAR generated statistics, i.e. variance and covariance ratios, correlation coe cients and 2 values from the regressions with the orthogonalized components. 7 Basically, Bose s (1988) and Kilian s (1998) resampling technique is used with the special feature that residuals are resampled from the biasadjusted VAR system. 7 We report equaltailed confidence intervals in addition to standard errors since the statistics may have nonsymmetric finitesample distributions. Thus, confidence intervals summarize better than standard errors the distribution of the point estimates. See Davison and Hinkley (1997) for definitions of bootstrap confidence intervals. 8
9 3 Empirical results The data to be used in the empirical analysis are from the International Monetary Fund, International Financial Statistics. Monthly US and German data for longterm government bond yields were collected for the period July 1975 to February From these yields log onemonth holding period returns, +1, were computed using the formula +1 = +1 +, where is the log of the nominal yield to maturity on the long bond ( ), =exp( ) is the mean nominal yield over the sample, and is a linearization constant (see Shiller and Beltratti (1992) and Engsted and Tanggaard (2001)). 8 Excess log bond returns, +1, were then computed by subtracting from +1 the continuously compounded short term interest rate, taken to be the 1month interest rate on government Treasury bills. Real interest rates were computed by subtracting the monthly inflation rate from the short term nominal interest rate. Finally, the yield spread,, was obtained as described in footnote 4. The correlation between US and German monthly excess bond returns, and, over the period 1975 to 2003, is which indicates a quite high degree of comovement of US and German bond markets. In order to investigate in more detail the nature of this comovement we estimate a comprehensive VAR model containing excess bond returns, real interest rates, and yield spreads for each country, cf. section 2.2. Table 1 reports the results from a onelag VAR model. First, we see that excess bond returns are highly positively autocorrelated in both countries: the coe cient to in the +1 equations is positive and strongly significant. Second, lagged domestic real interest rates significantly predict excess bond returns in both countries. Third, none of the lagged German variables significantly predict US excess bond returns, whereas lagged US excess returns strongly predict German excess bond returns. Similarly, US variables serve as good predictors for German real interest rates and yield spreads, while German variables generally have no predictive power for US real rates or yield spreads, except for in the +1 equation. These results indicate significant predictable timevariation in USandGermanbondreturns,andthatthereareimportantspillover e ects from the US market to the German market, but not vice versa. Based on the VAR parameter estimates in Table 1, we next compute the innovation and news components as in equations (4) to (7). Table 2 reports the variance decomposition of these components. As seen, in both countries the news about future inflation component, e, is clearly the dominating factor behind movements in excess bond return 8 The values of are: =0 9932; =
10 innovations. Var(e ) is much higher than Var(e )andvar(e ), and the dominant covariance term also involves e. By looking at the orthogonalized components we see that inflation news accounts for 85 percent of the variation in unexpected excess bond returns in the US and 69 percent in Germany (both statistically significant), while news about future real interest rates is completely unimportant, both statistically and economically. News about future excess returns (term premia) is also statistically insignificant, but accounts for an economically important part of the variation (14 percent in the US and 29 percent in Germany), although still much less than the inflation news component. These results are in fact quite similar to the results documented by Campbell and Ammer (1993) for the US. They also find that inflation news is the most important factor driving US bond returns. Since US and German bond returns are highly positively correlated, and since news about future longterm inflation seems to be the most important force behind movements in bond returns in both the US and Germany, we expect that and are also highly positively correlated. This is confirmed in Table 3, which reports pairwise correlations of all computed innovation and news components. and have a positive correlation of Excess return news, and to a lesser extent real interest rate news, are also highly crosscountry correlated, but since these components are much less important in explaining bond return variability in the two countrries, the overall conclusion must be that the main reason for the high degree of comovement of US and German bond markets is a high degree of comovement of inflation news. Another noteworthy result to be seen in Table 3 is that in both countries the inflation news component is highly negatively correlated with both the return innovation component, e, and with the return news component, e.this means that news that inflation is going to increase in the future is bad news for the bond markets in both the short and long term. The results reported in Tables 1 to 3 are based on a VAR model with one lag. Diagnostic tests (not reported) do not show signs of misspecification, i.g. there are no strong indications of autocorrelation in the VAR residuals. However, in order to investigate the robustness of the results, we also estimate a twolag VAR model, and we split the sample period into two subsamples ( and ) and carry out the whole analysis with these shorter data samples. Table 4 summarizes the main results. 9 9 We only report the variance decompositions based on the orthogonalized components, and we only report the crosscountry correlations of the three news components. Furthermore, we do not report the bootstrapped standard errors and confidence intervals. The full set of results are available from the authors upon request. 10
11 First, we see that increasing the laglength to two does not in any substantive way change the results. Thus the results are robust to the laglength of the VAR. Second, the finding that inflation news is the most important part of bond return volatility remains true when the sample is split into subperiods, although the relative magnitude of e is smaller in the subperiods compared to the whole sample period. However, there seems to be an interesting change in the relative importance of e and e from the early period to the more recent period: in the period real interest rate news is completely unimportant in both countries, whereas excess return news is nonnegligible, especially in Germany. But in the period the relative importance of these two news components change; now real interest rate news becomes nonnegligible while the excess return news component becomes quite small. Third, both e and e remain highly positively crosscountry correlated in the subperiods and, in fact, the correlations increase over time. By contrast, the correlation between and changes from positive to negative when we go from the early period to the later period. However, since US and German bond returns are still highly positively correlated in the period, and since excess return news is quite unimportant in this subperiod, the implication of the above findings must be that comovement in real interest news cannot be responsible for the positive bond return correlation over the period ; instead comovement in inflation news must be the explanation, just as we concluded for the whole period The main conclusion from our analysis is that inflation news is the maindrivingforcebehindmovementsinboththeusandgermanbond market, and that this inflation news component shows a high degree of crosscountry correlation which generates a high degree of comovement of the two markets. When news is coming to the US market that future US inflation will increase, there is a tendency that German inflation will also increase. This is regarded bad news for the bond market in both countries whereby bond prices are bid down leading to immediate negative return innovations and changing expectations of future excess bond returns. Theresultsreportedaboveindicatethatinflationhasastrongroleto play in explaining bond return movements within countries and across countries. Termpremia(excessreturns)alsohavesomeroletoplay (especially if we focus on the period before 1990), but not nearly to the same extent as inflation. Thus, our results to some extent conflict with those reported by Sutton (2000). He measures the extent to which bond market comovement can be explained in terms of comovement of short term nominal interest rates and term premia, respectively, but he does 11
12 not attempt to isolate the e ects from inflation and real interest rates. He finds that term premia comovement is the most important factor behind the comovement of international longterm interest rates. Our results, by contrast, imply only a minor role for term premia comovement and instead refer most of the comovement to comovement in inflation news. We leave to future research the question of what then generate comovement in inflation news across countries. But naturally the increasing globalization and international trade, and the gradual elimination of barriers to trade between the US and Europe during the last 30 years, have contributed to a common development in inflation rates across countries. Hence, inflation in di erent countries to a large extent respond to a set of global factors in addition to countryspecific factors, and this implies that investors in bond markets in di erent countries revise their inflation expectations in response to the same kind of news. The identification of those global factors that generate common movement in inflation rates across countries marks an interesting topic for future research. 4 Concluding remarks InthispaperwehaveanalyzedthecomovementofUSandGerman bond returns over the period using VAR based return variance decompositions similar to the ones in Campbell and Ammer (1993) and Engsted and Tanggaard (2001). While the results reported in this paper contrast with some of the results reported in the earlier literature (e.g. Sutton, 2000), our findings complement the findings in other recent studies. Our finding of predictable variation in US and German bond returns and that common movement in inflation news is the main factor behind the comovement of the bond markets in the two countries, complement Barr and Priestley s (2004) finding that most of the variation in international bond returns is due to world (global) risk factors rather than to countryspecific factors. Furthermore, our analysis suggests a natural explanation for Dreissen et al. s (2003) finding that positive correlation between the levels of the term structures across countries accounts for the positive correlation of international bond returns: the level of the term structure is most naturally determined by the rate of inflation, so crosscountry correlation of inflation news will cause crosscountry correlation of term structure levels. Of course, further analysis will be needed in order to explain the precise reasons for common movements in inflation news over time, and that the correlation seems to have increased recently (c.f. Table 4). We 12
13 leave that for future research. 5 References Barr, D.G., and R. Priestley (2004): Expected returns, risk and the integration of international bond markets. Journal of International Money and Finance 23, Bekaert, G., R.J. Hodrick, and D.A. Marshall (1997): On biases in tests of the expectations hypothesis of the term structure of interest rates. Journal of Financial Economics 44, Bose, A. (1988): Edgeworth correction by bootstrap in autoregression. Annals of Statistics 16, Campbell, J.Y., and J. Ammer (1993): What moves the stock and bond markets? A variance decomposition for long term asset returns. Journal of Finance 48, Davison, A.C., and D.V. Hinkley (1997): Bootstrap Methods and Their Applications. Cambridge University Press. Driessen, J., B. Melenberg, and T. Nijman (2003): Common factors in international bond returns. Journal of International Money and Finance 22, Engsted, T., and C. Tanggaard: The Danish stock and bond markets: comovement, return predictability and variance decomposition. Journal of Empirical Finance 8, Hafer, R.W., A.M. Kutan, and S. Zhou (1997): Linkage in EMS term structures: evidence from common trends and transitory components. Journal of International Money and Finance 16, Ilmanen, A. (1995): Timevarying expected returns in international bond markets. Journal of Finance 50, Kilian, L. (1998): Smallsample confidence intervals for impulse response functions. Review of Economics and Statistics 80, Pope, A.L. (1990): Biases of estimators in multivariate nongaussian autoregressions. Journal of Time Series Analysis 11, Shiller, R.J., and A. Beltratti (1992): Stock prices and bond yields: can their comovement be explained in terms of present value models? Journal of Monetary Economics 30, Sutton, G.D., (2000): Is there excess comovement of bond yields between countries? Journal of International Money and Finance 19,
14 6 Tables (.062) (.004) (.001) (.051) (.005) (.000) [.248,.491] [.006,.010] [.001,.003] [.068,.268] [.004,.023] [.002,.000] (.755) (.047) (.009) (.629) (.061) (.004) [.145, 3.12] [.474,.660] [.009,.025] [.391, 2.09] [.062,.176] [.004,.013] (1.859) (.115) (.021) (1.553) (.148) (.010) [1.15, 6.24] [.317,.142] [.914,.996] [.973, 5.30] [.000,.590] [.020,.060] (.072) (.005) (.001) (.061) (.006) (.000) [.272,.009] [.012,.006] [.002,.001] [.133,.368] [.022,.001] [.000,.002] (.714) (.045) (.008) (.585) (.056) (.004) [1.26, 1.56] [.046,.220] [.009,.024] [.357, 2.67] [.096,.317] [.013,.003] (2.544) (.159) (.028) (2.058) (.200) (.014) [4.75, 5.40] [.348,.283] [.079,.032] [3.74, 4.32] [.997, .207] [.929,.985] (.000) (.000) (.000) (.000) (.000) (.000) Notes: The table reports biasadjusted VAR parameter estimates (in bold), with bootstrap standard errors in parentheses and bootstrap 95% equaltailed confidence intervals in brackets. The number of bootstrap simulation runs is 10,000. The number in parenthesis under the 2 value is the pvalue for joint significance tests of the explanatory variables. Table 1: VAR parameter estimates, July February
15 US bonds German bonds Shares of Var(e ) (1.046) [0.005, 3.966] (1.377) [0.155, 5.154] Var(e ) (0.007) [0.010, 0.016] (0.011) [0.020, 0.022] Var(e ) (0.827) [1.838, 1.189] (1.143) [2.016, 2.300] 2Cov(e e ) (0.117) [0.291, 0.184] (0.182) [0.420, 0.302] 2Cov(e e ) (1.826) [4.038, 2.803] (2.470) [6.319, 3.168] 2Cov(e e ) (0.099) [0.184, 0.221] (0.158) [0.284, 0.352] 2 (e ) (0.136) [0.755, 1.282] (0.135) [0.530, 1.056] 2 (e ) (0.059) [0.194, 0.015] (0.097) [0.310, 0.039] 2 (e ) (0.136) [0.242, 0.266] (0.152) [0.041, 0.530] Notes: The table presents the results of the variance decomposition (8), where each variance and covariance component has been normalized by Var(e ) The numbers in bold are point estimates based on the biasadjusted VAR parameter estimates, with bootstrap standard errors in parentheses; the numbers in brackets are bootstrap 95% equaltailed confidence intervals. The number of bootstrap simulation runs is 10,000. Table 2: Variance decomposition of bond returns, July February
16 (0.191) [0.472, 1.194] (0.156) (0.408) [0.091, 0.509] [0.382, 1.098] (0.118) (0.049) (0.294) [1.325, ] [1.013, ] [0.961, 0.174] (0.038) (0.159) (0.138) (0.104) [0.430, 0.579] [0,177, 0.440] [0.221, 0.321] [0.618, ] (0.156) (0.184) (0.379) (0.114) [0.583, 1.185] [0.779, 1.480] [0.342, 1.067] [1.205, ] (0.150) (0.489) (0.335) (0.354) [0.199, 0.769] [0.588, 1.089] [0.093, 1.346] [1.200, 0.116] (0.099) (0.205) (0.299) (0.111) [1.040, ] [1.117, ] [0.844, 0.301] [0.568, 1.000] Table 3: Correlations of innovations and news components, July February 2003 Table continues on next page. 16
17 Table 3 continued: (0.152) [0.300, 0.894] (0.128) (0.446) [0.060, 0.445] [0.247, 1.331] (0.119) (0.039) (0.340) [1.214, ] [0.972, ] [1.157, 0.126] Notes: The numbers in bold are correlation coe cients based on the biasadjusted parameter estimates. The numbers in parentheses and brackets are bootstrap standard errors and 95% equaltailed confidence intervals, respectively, based on 10,000 simulation runs. Table 3: Correlations of innovations and news components, July February
18 US GE US GE US GE VAR(1): 2 (e ) (e ) (e ) ( ) ( ) ( ) VAR(2): 2 (e ) (e ) (e ) ( ) ( ) ( ) Notes: "VAR( )" denotes a VAR model with lags. The precise sample periods are: July February 2003; July March 1989; April February Table 4: Robustness check: di erent laglengths and subsample analyses 18
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