The Sovereign Bond Yield Curves of Germany and The UK and their Interaction Effects

Master Thesis

MSc Economics and MSc Finance

The Sovereign Bond Yield Curves of Germany and The UK and

their Interaction Effects

Maaike Dijkstra1

Faculty of Economics and Business, University of Groningen

Supervisor: Dr J. J. (Jakob) Bosma

June 2019

Abstract: This study models the yield curves of Germany and the UK to examine their interaction effects from 1995 to 2015. The sovereign bond yields are modelled using the Nelson-Siegel latent factors, i.e. the level, slope and curvature factors, which provide a good fit. The findings show similar level factors but a higher slope and curvature factor for the German model. The estimates for the curvature and slope factor are used to examine the relationship between the medium- and short-term yield curve factors. Using the expectations hypothesis as a benchmark, the validity of this hypothesis is tested using a multivariate cointegration approach followed by a test for Granger causality. Finally, this study presents evidence that there exists a long run relationship between the medium- and short-term rate in the UK and for cross-country relation between the medium-term rate in Germany and short-term rate in the UK for the period between 1995 and 2015. The cointegration between the cross-country yield curves indicates signs of international financial integration.

Course codes: EBM877A20; EBM866B20

Keywords: Nelson-Siegel yield curve factors, time-series analysis, yield curve across-country spillover effects

JEL codes: C32, E43

1 _{Student number: S2542951}

1. Introduction

Over the past half century, economic and financial integration have been of paramount importance in the European financial system. The creation of the euro marks the culmination of a challenging European unification process. The degree to which the European sovereign bond markets are integrated is of particular interest of this study. Even though the differences in sovereign bond yield curves are not negligible, there are indications of sovereign bond market integration among European countries. For example, Ehrmann et al. (2011) show that the yield curves between European countries who have adopted the euro converge in terms of level and movement. This raises the question as to what extend the unification process leads to the integration of sovereign bond markets within Europe that do not share a common currency. Do the yield curves of these European countries show signs of integration? In this paper, the sovereign bond yield curves of Germany and the United Kingdom are studied to determine the coherence of the yield curve factors in order to construe the degree of interrelations between these sovereign bond markets.

Within the European Union, Germany and the UK have the strongest economy in terms of nominal gross domestic product. In the light of the major changes that have occurred in the political and financial environment recently, it is important to study the financial relations between these countries. Therefore, the yield curves of Germany and the UK are the focus of this research.

In these turbulent economic conditions, is it likely for the German and UK yield curves to show signs of convergence? Given the years of economic and financial integration within the EU, the yield curves are probably somehow related. In this paper, the hypothesis of interrelations is tested for two factors of the yield curve. The aim of this study is to understand in-depth the connectivity between the slope and curvature factors of the sovereign bond yield curve in Germany and the UK.

First, the yield curves of Germany and the UK are modelled with the Nelson-Siegel (NS) curve. As mentioned earlier, the yield curve is analyzed in terms of level, slope and curvature factors. These factors are estimated and compared for Germany and the UK to identify the similarities and discrepancies. The model is a good fit for the data. The results of the estimated factors show that the level factors are similar for the two countries, but there are some differences regarding the slope and curvature of the sovereign bond yield curves. Then, the estimated factors (i.e. latent factors) are compared with their empirical proxies and comparable macroeconomic variables. This comparison shows significant similarities between the latent factors and empirical proxies. For the macroeconomic variables, the linkages differ per factor.

Second, the estimated curvature and slope factors are used to investigate the mutual medium-term and short-medium-term relationships. These relations are founded by the expectation hypothesis (EH) theory and are tested with the vector error correction model (VECM) estimation technique and the Granger causality Wald test. The VECM shows that the EH holds within the UK borders and that there are signs of long run relations across country (i.e. between Germany and UK). The Wald test implies that the short-term rate in the UK Granger causes the medium-term rate in the UK.

2. Literature review

Nelson and Siegel (1981) were the first ones to introduce a simple model representing the movement of the yield curve for sovereign bonds. The NS curve consists of three coefficients indicated by the latent factors that describe the level, slope and curvature of the yield curve. They found a high correlation between the fitted values based on the exponential component framework and the actual values of the bond prices.

Thereafter, multiple authors have suggested to extend the model introduced by Nelson and Siegel (1981) to make it more applicable (Svensson 1995; Bliss 1997b; Soderlind and Svensson 1997; Björk and Christensen 1999; Filipovic 1999, 2000; Björk 2000; Björk and Landén 2000; Björk and Svensson 2001 and Diebold and Li 2006). The main criticism for the model relates to the arbitrage free condition. Björk and Christensen (1999) and Filipovic (1999) conclude that, theoretically speaking, the NS model does not ensure the absence of arbitrage opportunities. This means that the model does not ensure consistency between the dynamic evolution of yields over time, and the shape of the yield curve at a given point in time. Duffee (2002) and Brousseau (2002) refute the critique on the existence of arbitrage opportunities in the NS model by stating that the arbitrage free models are somewhat worse in portraying the dynamics of yield curve over time. Since this research does focus on the dynamics of the yield curve, it relies on the dynamic formulation of the model that was suggested by Diebold and Li (2006).

Svensson (1995) and Soderling and Svensson (1997) extend the NS model by adding a “second hump-term” to the model, to increase the flexibility and improve the fit. This addition imposes a horizontal asymptote on the curve to avoid the problem of unstable estimates on long-term rates. Christensen et al. (2008) also follow Diebold and Li (2006) to introduce a dynamic version of the Svensson extension. They discover that this version does not lead to an “arbitrage free approximation”, because this approximation requires for every curvature factor to be complemented with a slope factor. Therefore, for the approximation to hold, the model must be extended further.

The additions proposed by Diebold and Li (2006) are of particular interest due to their focus on the dynamic relations between the factors and the incorporation of macroeconomic variables. Furthermore, they are the first ones to interpret the three factors of the NS curve as level, slope and curvature. In their paper, a vector autoregression model is used to cover the dynamics of the model. In this paper, a VECM is estimated to discover possible long-term relations between particular latent factors, because of the existence of non-stationary series that are known to be cointegrated.

The dynamic factor model approach is fundamental for this research. The facture structure is applied, because of its attractive features. First, the factor structure offers a precise characterization of yield curve data. Second, factor models appear to be very desirable for statistical reasons. They compress an inapplicable high-dimensional modeling situation into an applicable low-dimensional situation. Third, financial economic theory invokes factor structure (Diebold et al., 2006b).

The loadings on the level, slope and curvature factors, which are further discussed in Section 3, determine the associated bond maturities (Diebold et al., 2006b). The level factor loading is constant at 1 and relates to the long-term factor, because it directly affects die long-term rates. The loading on the slope factor starts at 1 and decays to 0 and therefore the slope is connected to the short-term factor and directly influences the short-term rates. Lastly, the curvature factor linked to the medium-term factor which drives the medium-term rates, because the loading on the factor starts at 0, increases, and then decays to zero.

In this study, the EH theory lies on the basis of the analysis of potential convergence between latent factors. The expectation theory of the term structure of interest rates suggests that medium- or long-term rates are merely established by the current and future expected short-term rates. Therefore, the value that is created by investing in a medium- or long-short-term bond equals the cumulative value of investments in short-term bonds. The theory states that bonds of different terms are perfect substitutes. Short-term rates thus serve as the foundation for medium- and long-term bonds. In the context of monetary policy, the EH theory is an important indicator for the effectiveness of decision made by the central banks. If the short-term rates influence the medium- or long-term rates, the monetary policy resolutions effectively influence the economy in the long run in terms of changes in the long-term interest rates. As opposed to the EH theory, the market segmentation (MS) theory states that medium- or long-term rates and short-term rates are not related. Therefore, the MS theory suggests that the short-, medium- and long-term rates should be regarded separately. To which degree the EH holds true has thus strong implications for economic policy.

For the cross-country analysis, the degree of international financial integration is associated with the interdependencies between the German and UK term structure. As in Holmes et al. (2010), the existence of interdependencies is based on the stationarity of the difference between short- and longer-term factors of the yield curves. They suggest that if cross-country yield curves are stationary, this provides support for the uncovered interest parity but also for a considerable degree of financial integration between the observed countries. In this context, the uncovered interest parity states that the difference in bond yields between Germany and the UK in a specific period equals the relative change in currency foreign exchange rates over the same period.

3. Methodology

In this paper, the examination of the bond yield curves of Germany and the UK relies on two parts: (1) modelling the sovereign bond yield curves of both countries with the NS method, and (2) examining the EH theory and the degree of financial integration with cointegrating equations.

3.1 The Nelson and Siegel approach

First, the yield curve is modelled with the dynamic latent-factor approach used in Diebold and Li (2006). This approach is based on the model of Nelson and Siegel (1987). The Nelson and Siegel yield curve equation is formulated as

𝑓_"(𝜏) = 𝛽_("+ 𝛽_*"+(,-./0

12 3 − 𝛽5"𝑒

,12_{+ 𝜀}

The three-latent-factor model (Diebold and Li, 2006) gives the bond yield of a maturity at any point of time by 𝑦"(𝜏) = 𝐿"+ 𝑆"+ (,-./0 12 3 + 𝐶"+ (,-./0 12 − 𝑒,123 + 𝜖"(𝜏), (2)

in which 𝑦"(𝜏) is the bond yield of maturity 𝜏 at time t; 𝐿", 𝑆" and 𝐶" are the unobservable

time-varying level, slope and curvature factors of the yield curve. Therefore, the level, slope and curvature factors loadings are represented by, respectively, 1, (1 − 𝑒,1")/𝜆𝜏 and (1 −

𝑒,1"_{)/𝜆𝜏 − 𝑒},12_{. 𝜆 represents the exponential decay parameter, which determines the maturity}

for which the curvature factor assumes its maximum.

Equation (2) is expressed in matrix form by (3) and in vector notation by (4).

@ 𝑦"(𝜏() 𝑦_"(𝜏_*) ⋮ 𝑦_"(𝜏_B) C = ⎝ ⎜ ⎜ ⎜ ⎛ 1 (,-./0G 12G (,-./0G 12G − 𝑒 ,12G 1 ⋮ 1 (,-./0H 12H ⋮ (,-./0I 12I (,-./0H 12H − 𝑒 ,12H ⋮ (,-./0I 12I − 𝑒 ,12I ⎠ ⎟ ⎟ ⎟ ⎞ M 𝐿_" 𝑆_" 𝐶_"N + @ 𝜖"(𝜏() 𝜖_"(𝜏_*) ⋮ 𝜖_"(𝜏_B) C, (3) 𝑦_"= Λ𝑓_"+ 𝜖_", (4)

wherein 𝑦" is a column matrix of bond yield at time 𝑡, Λ is a matrix of factor loadings, and 𝑓"

is a column matrix of latent factors.

Parameter optimization is achieved by minimizing the function

min 1,UGV,UHV,UWV∑ Y𝛽("+ 𝛽*"Z (,-./0[ 12[ \ + 𝛽5"Z (,-./0[ 12[ − 𝑒 ,12[\ − 𝑦_"(𝜏_^)_ * ` ^ (5),

which represents the sum of squared residuals (SSR). 𝛽(", 𝛽*" and 𝛽5" present the unobservable

time-varying level, slope and curvature factors, respectively.

The SSR represented in Equation (5) can be simplified to

min

1,UV(𝑋1𝛽"− 𝑦" )b_(𝑋

1𝛽"− 𝑦"), (6)

optimization function fminsearch2_{. For every candidate for 𝜆, the optimization function solves} for the 𝛽^"’s. This process continuous until both the optimal 𝜆 and 𝛽^"vector is found. Hence, a

global estimate for 𝜆 is found for the whole sample.

3.2 The EH theory

Second, the interrelations between the latent factors are examined. The relation between the estimated slope and curvature factor that relate to the short- and medium-term rates, respectively, are explored based on the EH term structure. Campbell and Shiller (1991) describe the connection between short- and medium-term rates in terms of the EH as

𝑅_"(d)=(_e∑e,(^hi 𝐸"𝑅<sub>"g`^</sub>(`) + 𝑐, (7)

in which 𝑅_"(d) illustrates the n-period medium-term interest rate, 𝑅_"(`) defines the m-period short-term interest rate and 𝑛 > 𝑚. As mentioned in the previous section, the EH implies that the expected value from investing in an n-period rate is equal to the expected value that is created by investing in m-period rates up to 𝑛 − 𝑚 periods in the future plus a constant risk premium of 𝑐.

Subtracting 𝑅_"(`) from Equation (7) yields the spread between the n-period and m-period rates

𝑅_"(d)− 𝑅_"(`) =( e∑ ∑ 𝐸"nΔ(`)𝑅"gp` (`) _q ph^ ph( e,( ^hi + 𝑐, (8)

Interest rates are generally an I(1) processes. In this light, Campbell and Shillar (1998) state that Equation (8) gives some important stationarity restrictions to the process. Given the I(1) process of interest rates, the differenced value of the medium- and short-term rates, Δ𝑅_"(d) and

Δ𝑅_"(`), are I(0). Therefore, it is a stationary series, because 𝑐 is a constant. Muzindutsi and

Mposelwa (2016) state that the I(0) process of 𝑐 and Δ𝑅_"(`) imply stationarity for the right-hand side of Equation (8), thus 𝑅<sub>"</sub>(d)− 𝑅<sub>"</sub>(`) must be stationary as well. If this does not hold, the order of integration is inconsistent or the EH does not hold.

3.3 Cointegration and the VECM

Brooks (2014) advises not to use an OLS regression to estimate a relation like the one in Equation (8) if the dependent variables are nonstationary at levels, because the chance of

2 _{A non-linear minimization technique which uses the Nedler-Mead (1965) technique. This is a heuristic search}

spurious regression is significant. If variables are I(1) in levels and I(0) when differenced, the variables could be cointegrated. So, if two series are individually integrated and a linear combination between these series has a lower order of integration, then the two series are cointegrated. In this research, cointegration of the medium- and short-term factor would represent that the EH theory holds. The cointegration between factors is tested on a national (i.e. in Germany and in the UK) and international level (i.e. between Germany and the UK, and vice versa) with the Johansen cointegration test. Diebold and Li (2006), Shelile (2006), Koukouritakis (2010) and Muzindutsi and Mpolselwa (2016) also use this method to test for the EH. If the medium- and short-term factors are indeed cointegrated, a VECM is estimated.

The VECM is formulated as

Δ𝑦_" = 𝛼𝛽s_𝑦

",(+ ∑u,("h(Γ^Δ𝑦",^+ 𝜖", (9)

wherein 𝑦 is a vector of I(1) variables, 𝛼 and 𝛽 are (𝐾 × 𝑟) parameter matrices with rank 𝑟 < 𝐾, Γ^ is a (𝐾 × 𝐾) matrix of parameters, and 𝜖" is a (𝐾 × 1) vector of normally distributed

errors that are not serially correlated.

Before the VECM can be estimated, multiple steps have to be taken: 1) the optimal number of lags has to be specified based on AIC,3 _{2) the trend specifications have to be chosen, which is}

also done via AIC, 3) the number of cointegration equations is chosen based on Johansen’s maximum likelihood (ML) estimator of the parameters of a cointegrating VECM.The two types of Johansen tests are the test based on the trace statistic and the maximum-eigenvalue test (Johansen, 1995). Both tests search for the number of linear combinations (i.e. 𝐾), but they test different hypotheses. The trace statistic tests the null of 𝐾 = 𝐾_i against the alternative 𝐾 > 𝐾_i. The maximum-eigenvalue statistic tests the same null against a different alternative, namely 𝐾 = 𝐾_i+ 1.

This study examines the relations between the short- and medium-term factors on both the national and international level for Germany and the UK. Therefore, the multivariate Johansen co-integration approach tests 1) the relation between the short- and medium-term factors in Germany, 2) the relation between the short- and medium-term factors in the UK, 3) the relation between the short-term factor in the UK and medium-term factor in Germany, and 4) the relation between the short-term factor in Germany the medium-term factor in the UK. For the relationships that have more than zero cointegration equations, a VECM is estimated.

3 _{Akaike information criterion (AIC) measures the relative quality of statistical models for the medium- and}

3.4 Granger causality

Granger causality is a popular method for studying casual links between random variables (Granger, 1969). A variable X Granger-causes a variable Y if, given the past values of Y, the past values of X are useful for predicting Y. Granger causality between the medium- and short-term rate is tested by regressing the medium-short-term factor on its own lagged values and on lagged values of the short-term factor. It tests the null hypothesis that the estimated coefficients on the lagged values of the short-term factors are jointly zero. Failure to reject the null hypothesis is equivalent to failing to reject the hypothesis that the short-term factor does not Granger-cause the medium-term factor.

4. Data

Month-end zero-coupon yields with maturities of 12, 24, 36, 48, 60, 72, 84, 96, 108 and 120 months from January 1995 to December 2015 for Germany and the UK are obtained from Thomas Reuters. The date at the beginning of the sample period corresponds to roughly three years after the ratification of the Treaty of Maastricht, hence the start of the European Union. This marks the starting point of the European integration on political and economic front, which reaches its peak with the implementation of the euro from 1999 (the financial market introduction) to 2002 (the physical implementation of euro coins and banknotes). Hereafter follow approximately six years that can be characterized by an exorbitant rise in asset prices, an associated boom in economic demand, rising debt levels and trade imbalances until the subprime mortgage crisis in 2007 and later the Great Recession of 2008. The recession ended in the first and last quarter of 2009 for Germany and the UK, respectively. The period after the recession and before 2015 is defined as a period of slow recovery, higher growth and more economic stability. These macroeconomic and financial events are relevant for the data in this research and are observed accordingly.

The macroeconomic factors are the capacity utilization4 _{and the inflation rate}5_{. For Germany,}

this data is collected from the Directorate-General for Economic and Financial Affairs and Thomas Reuters, respectively. Analogous data for the UK is also retrieved from the Directorate-General for Economic and Financial Affairs and Thomas Reuters, respectively. Data on capacity utilization is only available on a quarterly basis. Therefore, monthly data for these variables is extracted via interpolation. The inflation rate is available on a monthly basis. In this way, the macroeconomic variables can all be observed per month.

4 _{Capacity utilization is a proxy of the output gap. The output gap is only available on a quarterly basis for}

Germany and the UK. Therefore, monthly data is calculated by linear interpolation.

The summary statistics of the data are shown in the Appendix. The descriptive statistic of the yields in Germany and the UK are displayed in Table A1 and A2, respectively. The yields of the UK (Table 12b) portray three specific yield curve characteristics. First, the yields increase with maturity. Risk aversion or liquidity preferences can account for the presence of term premia. Second, the yield variance or standard deviation decreases with maturity. Third, the autocorrelations indicate that the yields are highly persistence. The German yields (Table 12a) follow these same features, except for the second characteristic. The volatility of the German yields does not necessarily go down when the maturity increases. The yields even show an increase in the yields up until a maturity of 72 months. This can be explained because of the lower overall yield level in Germany compared to that in the UK. Since the yield level in Germany is lower, yield changes are more significant. After 84 months, the variance of the German yields lowers with maturity.

5. Results

The three-factor NS model has been fitted to the yield data for Germany and the UK. Figures 1a and 2a display the yields per maturity of Germany and the UK as observed from the data. The fitted NS yields per maturity for Germany and the UK are plotted in Figure 1b and 2b.

Figure 1a. Observed yield curve of Germany Figure 1b. Fitted yield curve of Germany

Notes: Figures 1a and 1b plot the observed and the NS fitted yield curve, respectively, per maturity for Germany in the period between 1995 and 2015. The yields are in percentages and the maturities are noted in months.

There is a minor difference between the observed and NS yield curve for Germany. This can be seen in the spacing between the curves. The NS yield curves are more cropped between 1998 and 2000 in comparison with the observed yield curves, but this difference is negligible.

Figure 2a. Observed yield curve of the UK Figure 2b. Fitted yield curve of the UK

Notes: Figures 2a and 2b plot the observed and the NS fitted yield curve, respectively, per maturity for the UK in the period between 1995 and 2015. The yields are in percentages and the maturities are noted in months.

seem to look identical. Figures 1a-2b would suggest that the NS model provides a good fit for the German and UK data. Next, the measurement errors of both models are examined to verify this statement.

Table 1 displays the resulting measurement errors. The second and third column of Table 1 show the measurement errors of the German model. For Germany, the mean error value shows peaks at 84, 96 and 120 months of maturity, but these are still very small values. The fourth and fifth column give the measurement errors of the UK model. For the UK, the measurement error is negligible at all maturities. Therefore, it can be concluded that the NS model is suitable for the yields of both countries. Additionally, the UK model fits the data somewhat better than the German model.

Table 1. Descriptive statistics of the measurement errors

German model UK model Maturity Mean St. Dev. Mean St. Dev. 12 0.352 2.713 -0.570 2.789 24 -0.406 3.381 0.816 4.761 36 -0.128 3.016 0.690 2.671 48 -0.168 1.911 -0.122 2.785 60 -0.714 2.068 -0.924 2.494 72 -0.154 3.417 -0.846 3.052 84 1.372 3.897 0.142 3.110 96 1.819 3.484 0.647 2.661 108 -0.220 1.305 0.173 1.552 120 -1.753 5.273 -0.005 3.515

Notes: This table contains the means and standard deviations of the measurement errors of the yields per maturity and per country for the period between 1995 and 2015. The second and third column show the corresponding values for the NS fit for the German yields. The last two columns display these values for the UK yields. The maturities are noted in months and all other values are expressed in basis points.

Table 2. Descriptive statistics of the estimated factors

Factor Mean St. Dev. Median Min Max 𝜌|(1) 𝜌|(12) 𝜌|(30) ADF German model 𝛽€_(" 4.999 1.696 4.942 0.658 8.904 0.974 0.662 0.288 -2.354 𝛽€_*" -2.756 1.525 -2.776 -6.202 0.008 0.971 0.451 -0.238 -2.543 𝛽€_5" -2.437 1.908 -2.445 -6.136 2.424 0.905 0.371 -0.221 -3.404 UK model 𝛽€(" 5.151 1.490 4.808 2.213 9.379 0.959 0.586 0.093 -2.000 𝛽€_*" -1.719 2.325 -1.518 -6.784 2.846 0.983 0.703 0.218 -2.321 𝛽€_5" -0.647 2.933 -0.188 -6.564 6.084 0.945 0.561 0.444 -3.129

Notes: This table shows the mean, standard deviations median, minimum and maximum values of the estimated factors per model, which are expressed in basis points. The three beta factors implicate the level, slope and curvature factor, respectively. The last four columns contain the sample autocorrelations at displacements of 1, 12, 30 months and the augmented Dickey-Fuller unit root test statistics, respectively. The number of lags that is used in the ADF test is selected using an AIC criterion.

For all estimated latent factors, an augmented Dickey-Fuller (ADF) test that a variable follows a unit-root process has been performed. The last column of Table 2 shows the ADF test statistic. The MacKinnon critical values for rejection of hypothesis of a unit root are -3.991 at the one percent level, -3.430 at the five percent level, and -3.130 at the ten percent level. Based on the five percent confidence level, the test statistics show that the level, slope and curvature factor for both countries may follow a unit root. The issue of non-stationary time series variables is addressed in Section 5.1.

Figure 3a. Estimated latent factors for Germany Figure 2b. Estimated latent factors for the UK

Notes: Figures 3a and 3b plot the level, slope and latent factor between 1995 and 2015 for Germany and the UK, respectively.

To assess the variations in the latent factor estimates, the latent factors are individually plotted against their equivalent observable macroeconomic factors and common empirical proxy. Diebold et al. (2006b) relate the macroeconomic factor inflation to the level factor and capacity utilization to the slope factor. They do not find a macroeconomic link for the curvature factor. Therefore, the curvature factor is only compared with the empirical proxy. According to Diebold et al. (2006b), 1) the average of the short, medium and long-term yields is an empirical proxy for the level factor, 2) the difference between the short and long-term yields is a proxy for the slope factor, and 3) two times the medium-term minus the long and short-term yields is a proxy for the curvature factor. In this study, the 12-month yield is regarded as the short-term yield, the 60-month yield equals the medium-term yield and the 120-month yield is the long-term yield.

Figure 3a. Estimated level factor for Germany Figure 4b. Estimated level factor for the UK

Notes: Figures 4a and 4b plot the level factor, inflation variable and related empirical proxy between 1995 and 2015 for Germany and the UK, respectively.

The estimated level factors are compared to the macroeconomic indicator inflation and the empirical proxy ((𝑦"(12) + 𝑦"(60) + 𝑦"(120))/3). For Germany, the inflation does not seem to

move parallel to the level factor, which is substantiated by the -0.37 correlation coefficient. By contrast, the movements of the proxy and the level factor show a close resemblance with a correlation of 0.84. The same is true for the UK, wherein the correlation between actual inflation and the level factor is -0.23. Additionally, the correlation of 0.64 between the proxy and the level factor shows more similarities.

Figures 5a and 5b illustrate a slope factor that fluctuates from year to year. For Germany, the slope factor stays negative throughout the sample, except for June 2008 when the slope factor is positive but very small (0.008 basis points). The slope factor for the UK shifts from positive to negative and vice versa and is positive for about a quarter of the sample.

Figure 5a. Estimated slope factor for Germany Figure 5b. Estimated slope factor for the UK

Notes: Figures 5a and 5b plot the slope factor, capacity utilization variable and the related empirical proxy between 1995 and 2015 for Germany and the UK, respectively.

Figures 6a and 6b reveal a constantly varying curvature factor. In 90 percent of the sample, the curvature factor of Germany is negative. For the UK case, this is true for 52 percent of the sample. Furthermore, the curvature factor of the UK exhibits higher positive values. In general, the German curvature seems to be varying more than the curvature factor of the UK. This is supported by the difference in standard deviation of the curvature factor, which is 1.91 for Germany and 2.93 for the UK (Table 2).

Figure 6a. Estimated curvature factor for Germany Figure 6b. Estimated curvature factor for the UK

Notes: Figures 6a and 6b plot the curvature factor and the related empirical proxy between 1995 and 2015 for Germany and the UK, respectively.

Since there is no evidence for a macroeconomic variable that is connected to the curvature factor, this factor is only explored by comparison with the empirical proxy (2 ∙ 𝑦"(60) −

Optimization of the German and the UK model by minimizing the SSR yield an exponential decay parameter value, 𝜆, of 0.4626 and 0.4468, respectively.

5.1 Interrelations between the yield curve factors

Using the EH as a benchmark, this study proceeds with an examination of the interrelations between dynamic latent factors. In particular, the relation between the medium-term and short-term yield curve factors is investigated via a VECM approach. Henceforth, the curvature and slope factor are referred to as the medium-term and short-term factor, respectively. To estimate a VECM, the three steps mentioned in Section 3 are followed.

Before the VECM steps are followed, there needs to be prove that the medium-term and short-term factors are I(1). As mentioned earlier, the medium-short-term and short-short-term are non-stationary at levels. The non-stationarity of the variables can be concluded based on the plots of the estimated latent factors in Figures 3a and 3b and the results of the ADF test in Table 2. Hence, the medium-term and short-term tend to not fluctuate around their average value.

Figure 7a. Differenced factors for Germany Figure 7b. Differenced factors for the UK

Notes: Figures 7a and 7b plot the differenced value of the medium- and short-term factors between 1995 and 2015 for Germany and the UK, respectively.

The method of first differences is applied. As displayed in Figures 7a and 7b, the medium- and short-term factors of both countries are stationary in first differences. This statement is supported by the ADF test conducted on the transformed variables in Table 3. In this case, the MacKinnon critical values for rejection of hypothesis of a unit root are -2.342 at the one percent level, -1.651 at the five percent level, and -1.285 at the ten percent level6_{. The stationarity of}

the differenced values substantiates the broad understanding of that interest rates are an I(1) process, as mentioned in Section 3.3.

Table 3. ADF test statistics for the differenced short- and medium-term factors

German model UK model Factor ADF Factor ADF

Δ𝛽€_*" -7.428 Δ𝛽€_*" -7.447 Δ𝛽€_5" -8.934 Δ𝛽€_5" -8.803

Notes: This table shows the Augmented Dickey-Fuller test statistics for the differenced value of the medium-term factor (i.e. Δ𝛽€5") and the short-term factor (i.e. Δ𝛽€*") for Germany and the UK.

The four different relations mentioned in Section 3.3 are examined according to the steps that are outlined. As a first step towards the VECM estimation, the optimal number of lags is determined. Table 4 gives the optimal lag specification for the four models. The optimal number of lags is set at one for the model estimating the factors in Germany and the model estimating the relationship between the short-term factor in the UK and the medium-term factor in Germany. This number equals four for the model estimating the UK factors and the model estimating the link between the short-term factor in Germany and medium-term factor in the UK.

Table 4. Lag specification per model

GermanyST → GermanyMT UKST → UKMT UKST → GermanyMT GermanyST → UKMT Number

of lags

1 4 1 4

Notes: This table indicates the optimal number of lags that has to be included in the VECM estimation to examine the relationship between the medium-term factors (i.e. MT) and short-term factors (i.e. ST). The first row defines the four different relations for which the number of lags is determined. The conclusion is based on the lowest AIC statistic.

Table 5. Trend specification per model

GermanyST → GermanyMT UKST → UKMT UKST → GermanyMT GermanyST → UKMT Trend Restricted trend None Restricted constant Restricted constant

Notes: This table indicates the trend specification that has to be included in the VECM estimation to examine the relationship between the medium-term factors (i.e. MT) and short-term factors (i.e. ST). The first row defines the four different relations for which the trend specification is determined. The conclusion is based on the lowest AIC statistic.

Third, following the model specifications indicated by Table 4 and 5, a Johansen test for cointegration is conducted for the four models. The test statistics of the model that relates the medium- to the short-term rate in Germany are based on one lag and a restricted trend. The statistics for this relationship in the UK based on four lags and no included trend nor constant. The test statistics of link between the medium-term factor in Germany and the short-term factor in the UK are based on one lag and a restricted constant. For the model that relates the medium-term factor in the UK to the short-medium-term factor in Germany, four lags and a restricted constant are incorporated.

Table 6. Johansen test statistics for the four models

GermanyST → GermanyMT UKST → UKMT

Rank Eigenvalue Trace statistic Critical value Eigenvalue Trace statistic Critical value

0 24.581* 25.32 14.695 12.53

1 0.079 3.888 12.25 0.052 1.543* 3.84

2 0.015 0.006

UKST → GermanyMT GermanyST → UKMT

Rank Eigenvalue Trace statistic Critical value Eigenvalue Trace statistic Critical value

0 26.284 19.96 15.008* 19.96

1 0.092 2.020* 9.42 0.043 4.031 9.42

2 0.008 0.016 5.269

Notes: This table gives the results of the Johansen tests for cointegration of the medium-term factors (i.e. MT) and short-term factors (i.e. ST) in Germany, the UK and across these countries. The maximum-eigenvalue and trace statistic are based on the tested number of cointegration equations mentioned in the first column. The critical value for the trace statistic is based on 5% confidence level. The * indicates the number of cointegrated equations specified by the trace statistic.

5.1.1 Long run relations between the medium- and short-term yield curve factors

Fourth, a VECM is estimated for the models that include the factors that are cointegrated. A VEC is a restricted VAR designed for use with non-stationary series that are known to be cointegrated.

Table 7. Cointegration equations for factors in the UK and between Germany and the UK.

UKST → UKMT UKST → GermanyMT Medium-term factor 1.000 1.000 Short-term factor -0.822*** (0.22) -0.549*** (0.13) Constant 0.016*** (0.00)

Notes: This table shows the cointegration equations of the two models that have cointegrating variables. A constantis only specified for the model that estimates the relationship between the medium-term (MT) factors in Germany and the short-term (ST) factors in the UK, as mentioned in the model specification criteria. The standard errors are stated in the parentheses. *** implies statistical significance at the 1% level, ** implies statistical significance at the 5% level, and * implies statistical significance at the 10% level.

Table 7 shows the results for the VECM estimation for the medium- and short-term factors in the UK and between Germany and the UK. For both models, the short-term factors are positive and significant at the one percent level. Furthermore, the constant in the second model is negative and highly significant. This result implies that the data for medium- and short-term factor supports the restriction that there is a long-run relationship between these factors in the UK and between Germany and the UK. The long run equilibrium relations are described in Equation (10) and (11) as follows

UK_…b= 0.822 UK_ˆb, (10)

Germany_…b = −0.016 + 0.549 UK_ˆb, (11)

wherein 𝑀𝑇 and 𝑆𝑇 denote the medium- and short-term factors, respectively. For the UK, the short-term factors positively influence the medium-term factors. Equation (10) indicates that in the long run, an increase of one percent in the short-term rate results in an increase of 82.2 percent in the medium-term rate (ceteris paribus). The short-term rate in the UK also positively influence the medium-term rate in Germany, as described by Equation (11). In the long run, a one percent increase in the short-term rate in the UK will lead to a 54.9 percent increase in the medium-term rate in Germany (ceteris paribus).

5.1.2 Short run relations between the medium- and short-term yield curve factors

Table 8a. Short run VECM estimation for the medium- and short-term factors in the UK. ΔUKMT(t) ΔUKST(t) CE 1 -0.075*** (0.02) 0.006 (0.01) ΔUKMT(t-1) 0.008 (0.07) -0.048 (0.032) ΔUKMT(t-2) -0.067 (0.07) -0.029 (0.03) ΔUKMT(t-3) 0.132** (0.07) 0.029 (0.03) ΔUKST(t-1) -0.082 (0.14) 0.055 (0.07) ΔUKST(t-2) -0.038*** (0.14) 0.086 (0.07) ΔUKST(t-3) 0.042 (0.14) 0.093 (0.07) Root MSE 0.007 0.004 R2 _{0.108 0.039} p-value 0.000 0.187 Log-likelihood 1852.89 Determinant residual covariance 1.11E-09

AIC -14.822

Notes: This table shows the short run dynamics for the relationship between the medium-term (MT) factors and the short-term (ST) factors in the UK estimated in the VECM. The differenced values are denoted by Δ. CE 1 gives the estimations for the first cointegrating equation. The standard errors are stated in the parentheses. *** implies statistical significance at the 1% level, ** implies statistical significance at the 5% level, and * implies statistical significance at the 10% level.

The adjustment coefficient of -0.075 in Table 8a is negative and highly significant, so it suggests that when the average medium-term factor in the UK is too high, it will decrease slowly toward the UK short-term factor level. In the UK, about 7.5 percent of the disequilibrium is corrected each month, thus short-term rate changes take approximately 13.33 months to have a full effect on the medium-term rate.

Table 8b. Short run VECM estimation for the medium- and short-term factors between Germany and the UK.

ΔGermanyMT(t) ΔUKST(t) CE 1 -0.153*** (0.03) -0.001 (0.02) Root MSE 0.007 0.004 R2 _{0.090 0.000} p-value 0.000 0.951 Log-likelihood 1898.26

Determinant residual covariance 9.25E-10

AIC -15.094

The adjustment parameter of -0.153 in Table 8b is negative and highly significant, so it suggests that when the average medium-term factor in the Germany is too high, it will decrease at a slow rate toward the UK short-term factor level. About 15.3 percent of the disequilibrium is corrected each month, thus short-term rate changes in the UK take approximately 6.54 months to have a full effect on the medium-term rate in Germany.

Therefore, both short- and medium-term rates for these models converge to a long run equilibrium. However, the adjustment speed differs. Surprisingly, the results indicate that a change in the short-term rate in the UK reaches the medium-term rate in Germany quicker than the medium-term rate in the UK.

Equation (9) specified for the UK relations yields

Y_ΔUKΔUK“”(•) –”(•)_ = + 0.055 −0.048 −0.082 0.008 3 Y ΔUK_{“”(•,()} ΔUK_{–”(•,()}_ + + 0.086−0.038 −0.067−0.0293 Y ΔUK_{“”(•,*)} ΔUK_{–”(•,*)}_ + +0.093 0.029 0.042 0.1323 Y ΔUK_{“”(•,5)} ΔUK_{–”(•,5)}_ + +0.0060 −0.0750 3 Y UK–”(•,()− (0) − (−0.822)UK“”(•,() UK–”(•,()− (0) − (0.822)UK“”(•,() _ + 𝜖_"

For the interrelations between Germany and the UK, this relation is expressed as

Y_ΔGermanyΔUK“”(•) –”(•)_ = +0.153 0 0 −0.0013 Y Germany_–”(•)− (0.016) − (−0.549)UK_{“”(•,()} Germany_–”(•)− (0.016) − (−0.549)UK_{“”(•,()}_ + 𝜖" 5.2 Postestimation VECM

To see if the interrelations between the medium- and short-term factors indicated by the VECM are substantiated, the residuals of both models are tested.

First, both differenced VECM residual terms, of UKMT(t) and GermanyMT(t), are regressed on the

lagged residual term of UKMT(t) and GermanyMT(t),respectively. The results of the regression are

Table 9. ADF test on VECM residual term

Dependent variable Independent variable Coefficient ADF Critical value p-value Δ res. UKMT(t) Lagged res. UKMT(t) -1.008

(0.06)

-15.798 -3.640 0.000 Δ res. GermanyMT(t) Lagged res. GermanyMT(t) -1.022

(0.06)

-16.132 -3.641 0.000

Notes: This table shows the results of the ADF test on the differenced residual (res.) term of the UKMT(t) and GermanyMT(t)

VECM. The differenced VECM residual term of UKMT(t) and GermanyMT(t) are regressed on the lagged residual term of UKMT(t)

and GermanyMT(t),respectively. The corresponding coefficients are stated in column three. The fourth column gives the

Augmented Dickey-Fuller statistics and the last two column states the MacKinnon critical value at the 1% confidence level and the corresponding p-value, respectively. The differenced values are denoted by Δ. The standard errors are stated in the parentheses.

The estimated coefficients are around the -1. Furthermore, the ADF test statistic shows to be much lower than the MacKinnon critical value at the one percent confidence level. Therefore, the null hypothesis that states that the residual terms contain a unit root can be rejected at the one percent confidence level. The residual terms of the VECM that estimates the long run relationship between the medium- and short-term factor in the UK andthe one that estimates the long run linkage between the medium-term factor in Germany and the short-term factor in the UK are stationary.

Second, a Lagrange multiplier (LM) test tests the degree of autocorrelation in the residuals of the VECMs. Table 10 gives the results of the LM test.

Table 10. Lagrange Multiplier test of serial correlation in the residuals

Lag Χ* df Critical value p-value UKMT(t) model 1 0.795 4 9.488 0.939 2 1.670 4 9.488 0.796 3 8.439 4 9.488 0.077 4 0.374 4 9.488 0.985 GermanyMT(t) model 1 4.683 4 9.488 0.321 2 8.100 4 9.488 0.088

Notes: This table shows the results of the Lagrange Multiplier test for autocorrelation in the residuals. This approach tests the null hypothesis of no autocorrelation against the alternative of serial correlation in the residuals. The first column shows the number of lags tested for autocorrelation. The second column implies the chi-square (Χ2_{) distribution test statistic. The critical}

value of the chi-square distribution is based on a five percent confidence level and the degrees of freedom (df) that is stated in the third column. The p-value that is indicated by the LM test is noted in the fifth column. The conclusion of the LM test is mentioned in the last column.

5.3 Granger causality

Since the UK model includes multiple lags, it is tested for Granger causality. A Granger causality Wald tests is performed for each equation in the VECM for the UK model (i.e. UKMT → UKST). The results of the tests are shown in Table 11.

Table 11. Granger causality test

Equation Excluded variable Χ* _{df Critical value p-value}

UKMT(t) UKST(t) 8.19 3 7.815 0.0422

UKST(t) UKMT(t) 4.08 3 7.815 0.2528

Notes: This table shows the results of the Granger causality test. This approach tests the null hypothesis of no evidence for Granger causality against the alternative that the data does imply evidence for Granger causality. The first column shows the equation that is tested. The second column shows the excluded variable. The third column states the chi-square (Χ2₎

distribution test statistic. The fourth column shows the degrees of freedom (df) that are included in the LM test. The critical value of the chi-square distribution is based on a five percent confidence level and the degrees of freedom. The p-value is noted in the last column.

Table 11 shows that the null of no Granger causality can be rejected for the first line. Therefore, the medium-term factor in the UK Granger causes the short-term factor in the UK. This evidence implies that the past values of UKMT(t) contain information that helps predict UKST(t)

more than the information contained in the past values of UKST(t) alone. However, the null

cannot be rejected for the second line. For this reason, the short-term factor in the UK does not Granger cause the medium-term factor in the UK.

This direction of Granger causality does not follow the EH theory mentioned in Section 1 and 2. The EH implies that the short-term rate in the UK should Granger cause the medium-term factor. According to the results in Table 11, only the reverse relationship holds true. Therefore, the Granger causality test does not provide proof for the EH theory in the way that the short-term factor causes the medium-short-term factor, but it does find proof for a reversed relation.

6. Discussion

The effectiveness of central bank’s policy is based on the EH theory to hold. Therefore, the EH is originally expected to be valid for Germany and the UK. According to Engle and Granger (1987), the EH holds if there exists a cointegrating relation between the medium- or long-term rates and the short-term rates. However, the results show that the EH is true for the UK, but not for Germany. Therefore, monetary policy in the UK influences the longer-term rates by operating at the short-end of the market. The rejection of the EH theory for Germany implies economically important deviations from the theory, which can be attributed to the MS theory as mentioned in Section 2 or to time-varying term premia (Jardet, 2008). The latter suggests that information in the spread includes information about the variation of time-varying term premia. Figure 1b shows evidence for the MS theory in Germany. From 2012, the short-term rate increases at a rate much faster than the medium- or long-term rates. This indicates that German bonds of different maturities are not perfect substitutes, because they might involve different risks.

Furthermore, the cross-country analysis points to interdependencies between the German and UK yield curve. Since this study suggests financial integration based on the cointegration between the short- and longer-term maturities as in Holmes et al. (2011), the results indicate signs of financial integration between yield curves of Germany and the UK. The stationarity between the cross-country yield curves can be supported by the uncovered interest parity.

7. Conclusion

In this research, the German and UK yield curves between 1995 and 2015 are modelled with the NS method. In particular, factor interpretation of the dynamic NS approach is applied. The factors that correspond with the level, slope and curvature of the yield curves are examined to capture the dynamics between the German and UK yield curves.

The factor estimations show similarities across the countries. However, the larger estimates for the German slope and curvature factor indicate greater shifts in the German yield curve in the sample period compared to the yield curve in the UK. The yield curve factors reveal great resemblances with their equivalent empirical proxies. However, the macroeconomic variables that correspond to the latent factors do not all run in parallel. Apart from the capacity utilization factor in the UK which is substantially correlated with the UK slope factor.

rate, respectively, and are both I(1) processes. The results of the Johansen cointegrations test show that there is no cointegrated relation between the medium- and short-term rate in Germany and between the medium-term rate in the UK and short-term rate in Germany. Therefore, the EH does not hold for these two models. Conversely, there are signs of cointegration for the medium- and short-term factor in the UK and between the medium-term factor in Germany and the short-term factor in the UK. This indicates that the EH does hold for in the UK and that there are signs of financial integration between Germany and the UK.

The cointegrating relations between these rates are modelled with a VECM. The VECM estimation method gives evidence for a long run relationship between the medium- and short-term rate in the UK and the medium-short-term rate in Germany and the short-short-term rate in the UK. Moreover, the short-term rate in the UK impacts the medium-term rate in the long run positively with 82.2 percent in the UK and with 54.9 percent in Germany (ceteris paribus). Furthermore, the adjustment towards the UK short-term rate happens at a much faster speed for the German medium-term rate compared to the UK medium-term rate.

This long run relationship is further investigated with a Granger causality test based on the EH theory. The EH theory suggest that medium-term rates are directly affected by short-term rates. In Granger causality terms, this statement can be rephrased as follows: the short-term yield Granger causes the medium-term yield. However, this study finds evidence for the reversed relationship. According to the Granger causality test, the medium-term rate in the UK Granger causes the short-term rate in the UK. Therefore, it can be concluded that this relationship does not follow the directions implied by the EH theory. Even though, it does find evidence for a long run relationship between the medium- and short-term rate on both a national (in the UK) and an international (between Germany and the UK) level in the years 1995 and 2015.

In summary, this research combines the factor formulation presented by Diebold and Li (2006) of the dynamic NS method and a cointegration and Granger causality tests to assess whether the EH holds in Germany and the UK within the specified time frame and if there are signs of financial integration between these two countries. A long run relationship based on the EH theory is found for the UK and this study finds evidence for interrelations between Germany and the UK.

Appendix

Table A1. Descriptive statistics German model yields

Maturity Mean St. Dev. Median Min Max 𝜌|(1) 𝜌|(12) 𝜌|(30) 12 2.3792 1.6598 2.7200 -0.4016 5.7019 0.980 0.706 0.409 24 2.5732 1.7017 2.8850 -0.4129 6.3717 0.976 0.720 0.460 36 2.7980 1.7365 3.1019 -0.3698 6.7584 0.975 0.734 0.491 48 3.0148 1.7561 3.3650 -0.2904 7.0197 0.975 0.738 0.501 60 3.2098 1.7645 3.5700 -0.1873 7.2210 0.975 0.737 0.499 72 3.3947 1.7692 3.6832 -0.0684 7.3951 0.976 0.737 0.496 84 3.5673 1.7692 3.8400 -0.0098 7.5397 0.976 0.738 0.494 96 3.7086 1.7508 3.9600 0.0528 7.5917 0.976 0.738 0.490 108 3.8066 1.7093 4.0243 0.1183 7.5405 0.976 0.730 0.476 120 3.8935 1.6671 4.0856 0.1846 7.4853 0.976 0.722 0.461

Notes: This table displays the descriptive statistics of the German yields used for this study. The mean, standard deviation, median, minimum and maximum value of the yields are expressed in basis points per bond maturity. The maturity is denominated in months. The last three columns contain the sample autocorrelations at displacements of 1, 12, 30 months.

Table A2. Descriptive statistics UK model yields

Maturity Mean St. Dev. Median Min Max 𝜌|(1) 𝜌|(12) 𝜌|(30) 12 3.6505 2.4429 4.3450 0.0760 7.4240 0.988 0.812 0.566 24 3.8591 2.4072 4.5100 0.0900 8.2100 0.985 0.813 0.585 36 4.0236 2.3044 4.5450 0.2110 8.6390 0.984 0.806 0.580 48 4.1554 2.1907 4.6050 0.3850 8.8160 0.982 0.795 0.566 60 4.2649 2.0867 4.6335 0.5870 8.8410 0.981 0.782 0.545 72 4.3643 1.9998 4.6650 0.8010 8.8720 0.979 0.769 0.523 84 4.4570 1.9301 4.6600 1.0160 8.9570 0.978 0.756 0.500 96 4.5318 1.8675 4.6550 1.2240 9.0030 0.977 0.744 0.474 108 4.5861 1.8079 4.6750 1.3190 8.9650 0.976 0.736 0.446 120 4.6345 1.7568 4.6720 1.4020 8.8950 0.976 0.729 0.419

Notes: This table displays the descriptive statistics of the UK yields used for this study. The mean, standard deviation, median, minimum and maximum value of the yields are expressed in basis points per bond maturity. The maturity is denominated in months. The last three columns contain the sample autocorrelations at displacements of 1, 12, 30 months.

Table A3. Descriptive statistics macroeconomic variables

Variable Mean St. Dev. Median Min Max German model Capacity utilization -0.7148 1.5224 -0.5883 -5.8900 2.4100 Inflation 1.4661 0.6933 1.4871 -0.5025 3.3229 UK model Capacity utilization 0.8886 2.5494 0.8200 -3.3900 5.1300 Inflation 2.0432 1.0507 1.9096 -0.1246 5.2116

References

Bae, B. Y., Kim, D. H., 2011. Global and regional yield curve dynamics and interactions: the case of some Asian countries. International Economics Journal 25, 717-738.

Björk, T., 2000. A geometric view of interest rate theory. Handbook of Mathematical Finance. Cambridge: Cambridge University Press.

Björk, T., Christensen, B., 1999. Interest rate dynamics and consistent forward rate curves. Mathematical Finance 9, 323-348.

Björk, T., Landén, C., 2000. On the construction of finite dimensional realizations for nonlinear forward rate models. Finance Stochast 6, 303-331.

Björk, T., Svensson, L., 2001. On the existence of finite dimensional realizations for nonlinear forward rate models. Mathematical Finance 11, 205-243.

Bliss, R., 1997b. Testing term structure estimation methods. Advances in Futures and Options Research 9, 97-231.

Brooks, C., 2014. Introductory econometrics for finance. 3rd. Cambridge: Cambridge University Press.

Brousseau, V., 2002. The functional form of yield curves. Working Paper 148, European Central Bank.

Campbell, J.Y., Shiller, R.J., 1991. Yield spreads and interest rate movements: a bird's eye view. The Review of Economic Studies 58, 495-514.

Christensen, J. H. E., Diebold, F. X., Rudebusch, G. D., 2008. An arbitrage-free generalized Nelson-Siegel term structure model. Econometrics Journal 12, 33-64.

Diebold, F. X., Li, C., 2006. Forecasting the term structure of government bond yields. Journal of Econometrics 130, 337-364.

Diebold, F. X., Rudebusch, G. D., Aruoba, B., 2006b. The macroeconomy and the yield curve: a dynamic latent factor approach. Journal of Econometrics 131, 309-338.

Duffee, G., 2002. Term premia and interest rate forecasts in affine models. Journal of Finance 57, 405-443.

Ehrmann, M., Fratzscher, M., Gürkaynak, R. S., Swanson, E. T., 2011. Convergence and anchoring of yield curves in the Euro Area. The Review of Economics and Statistics 93, 350-364.

Engle, R.F., Granger, C.W.J., 1987. Co-integration and error correction: representation, estimation, and testing. Journal of Econometric Society 55, 251-276.

Faini, R., 2006. Fiscal policy and interest rates in Europe. Economic Policy, 443-489.

Filipovic, D., 1999. A note on the Nelson–Siegel family. Mathematical Finance 9, 349–359.

Filipovic, D., 2000. Exponential-polynomial families and the term structure of interest rates. Bernoulli 6, 1-27.

Granger, C. W. J., 1969. Investigating causal relations by econometric models and cross-spectral methods. Econometrica 37, 424-438.

Holmes, M. J., Otero, J., Panagiotidis, T., 2011. The term structure of interest rates, the expectations hypothesis and international financial integration: evidence from Asian economies. International Review of Economics and Finance 20, 679-689.

Jaramillo, L., Weber, A., 2013. Global spillovers into domestic bond markets in emerging market economies. Working Paper, International Monetary Fund.

Jardet, C., 2008. Term structure anomalies: term premium of peso-problem? Journal of International Money and Finance 27, 592-608.

Johansen, S., 1995. Likelihood-based inference in cointegrated vector autoregressive models. Oxford: Oxford University Press.

Muzindutsi, P. F., Mpolselwa, S., 2016. Testing the expectation hypothesis of the term structure of interest rates in BRICS countries: a multivariate co-integration approach. Acta Universitatis Danubius: Oeconomica 12, 289-304.

Nelson, C. R., Siegel, A. F., 1987. Parsimonious modeling of yield curves. Journal of Business 60, 473-489.

Prasanna K., Subramaniam, S., 2017. Yield curve in India and its interactions with the US bond market. International Economics and Economic Policy 14, 353-375.

Sirichand, K., Coleman, S., 2015. International yield curve comovements: impact of the recent financial crisis. Applied Economics 43, 4561-4573.

Soderlind, P., Svensson, L.E.O., 1997. New techniques to extract market expectations from financial instruments. Journal of Monetary Economics 40, 383-430.

Steeley, J. M., 2014. Forecasting the term structure when short-term rates are near zero. Journal of Forecasting 33, 350-363.

Svensson, L.E.O., 1995. Estimating forward interest rates with the extended Nelson and Sicgel method. Quarterly Review. Sveriges Riksbank 3, 13-76.

Tam, C., Yu, I., 2008. Modelling sovereign bond yield curves of the US, Japan and Germany. International Journal of Finance and Economics 13, 82-91.