The data set of interest rates is obtained by the Deutsche Bundesbank.16 As it is common in empirical studies on the yield curve, the short end is represented by money market interest rates with maturities between one month and six months.17 The rest of the yield curve is represented by default free zero-coupon government bonds with maturities between one and ten years. The data of the default free zero-coupon government bonds is calculated by the Deutsche Bundesbank according to the parametric Svensson approach with market data for listed Federal coupon-bearing securities without default risk. Due to the parametric approach, the time series of the interest rates have a constant maturity so that the data can be used in an empirical Macro-Finance model of the yield curve.
The data consists of 398 monthly observations between September 1972 and October 2005 of money market and government bond interest rates. The money market interest rates with maturities of one month (1M), three months (3M) and six months (6M) are monthly averages and reported by Frankfurt banks.18 Most of the term structure of interest rates consists of interest rates of government bonds with maturities between
15The estimation results of equation 2.16 are presented in section 2.6.3.
16The data can be obtained on the webpage www.bundesbank.de. A description of the construction of the data can be found in Deutsche Bundesbank (2005).
17For example, Deutsche Bundesbank (2006) uses money market interest rates with a ma-turity of six months for the short end of the yield curve.
18In the period between September 1972 and March 1981, data for the six-month money market rate is not available. In order to …ll the gap for the corresponding period, the missing values are calculated as the mean between the three-month money market rate and the one-year government bond interest rate.
0 2 4 6 8
1M 3M 6M 9M 1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y
Time to Maturity
Interest Rate [%]
Money Market
(Default Risk) Bond Market (without Default Risk)
Figure 2.4: Average yield curve in Germany between September 1972 and October 2005, whereas the nine-month interest rate is the average between the six-month money market rate and the one-year government bond rate. Source: Deutsche Bundesbank.
one year (1Y) and ten years (10Y). The interest rates of government bonds are end of month data.
As the data for one, three and six months interest rates is reported by banks in Frankfurt for money transactions, it is not default free, because banks are exposed to default risk. That is the reason why money market interest rates include a default risk premium in contrast to default free zero-coupon yields. This can be seen in …gure 2.4, where the mean yield curve of short term maturities has a hump. Consequently, the mean interest rate of a one-year default free zero-coupon bond is lower than the mean money market interest rate for three and six months. The average spread between the interest rate of ten-year government bonds and the three-month money market rate, i.e.
the average slope of the yield curve, is 123 basis points. The reason for the positive slope might be that investors expect on average an increase of the short term interest rate and that the risk premium is positive.
In addition to the plot of the historical German yield curve (…gure 1.4), table 2.2 summarizes descriptive statistics of the German term structure for both interest rates in levels and in …rst di¤erences. Interest rates with short term maturities are more volatile than interest rates with long term maturities, which is a typical characteristic of interest rate data in levels. For example, the standard deviation of the level of the
three-Yields in Levels Yields in First Di¤erences
3M 2Y 5Y 10Y 3M 2Y 5Y 10Y
Mean 5.77 5.97 6.57 7.00 -0.01 -0.01 -0.01 -0.01
Median 4.82 5.64 6.45 6.98 0.00 -0.03 -0.04 -0.03
Maximum 14.57 12.33 11.49 11.30 2.93 1.52 1.25 1.20
Minimum 2.01 2.04 2.56 3.21 -1.42 -1.21 -0.91 -0.84
Standard Deviation 2.86 2.28 2.00 1.75 0.39 0.34 0.27 0.25
Skewness 0.92 0.36 0.07 -0.02 1.57 0.57 0.51 0.64
Kurtosis 3.17 2.33 2.21 2.42 14.23 5.94 4.52 5.04
Autocorrelation 0.989 0.987 0.987 0.984 0.442 0.215 0.214 0.107 Table 2.2: Descriptive statistics of German yields in levels and …rst di¤erences.
month interest rate of 2.86 is larger than of the ten-year interest rate of 1.75, because the central bank reacts to the business cycle by changing the short term interest rate, which only has a smaller e¤ect on long term interest rates. Skewness and kurtosis of the level of German interest rates indicate that the various interest rates are not normally distributed, as the Gaussian distribution has a skewness equal to zero and a kurtosis equal to three.
The …rst di¤erence of the interest rates shows the typical non-normal distribution of changes in …nancial market data: the skewness is positive and indicates that a decrease in the interest rate is more likely than an increase, which is in line with the decline in the level of interest rates in Germany during the sample (…gure 1.4). The kurtosis is larger than three and indicates that the probability distribution of the …rst di¤erences has thicker tails than the normal distribution. The autocorrelation of the time series of the level of the interest rates is very high. Therefore, next period’s value of a certain interest rate heavily depends on the value of the interest rate in the period before.
The correlation between interest rates in levels and the correlation between interest rates in …rst di¤erences is also high. In …gure 2.5, the correlation of yields in levels is close to one for yields with similar time to maturities. The larger the di¤erence in the time to maturity of two yields, the lower the correlation between them. In …gure 2.6, the correlation between the …rst di¤erence of two yields with di¤erent maturities n and m (Corr(int int 1; imt imt 1)) is plotted. At the short end of the yield curve, the correlation between yields in …rst di¤erences is not as large as between yields in levels, because the correlation between yields in …rst di¤erences declines faster as the di¤erence in the time
Figure 2.5: Correlation of German interest rates in levels.
to maturity of two yields widens. Interest rates with medium and long term maturities have similar correlations in …rst di¤erences as in levels.19
The macroeconomic data set consists of time series for real economic activity and for in‡ation. Even though real GDP growth is the best information variable for the real economy, the business sentiment indicator Ifo-Index is used as an approximation for GDP in this analysis. The Ifo-Index is published monthly and therefore has a larger number of observations than GDP, which is only published quarterly. Furthermore, the Ifo-Index is a leading indicator for the real economy and therefore it is possible to include expectations of the future path of the real economy into this empirical macroeconomic model of the yield curve. The time series of the Ifo-Index is seasonally adjusted and the source is the Ifo institute. The development of prices in the economy is represented by the CPI. The data considers wholesale and retail sales and does not take into account energy prices, which is useful as central banks and long term investors do not react to short term ‡uctuations in prices due to volatile energy prices. The data is also seasonally adjusted and the source is the Deutsche Bundesbank.
19The correlation of yields in levels and in …rst di¤erences is also shown in appendix A.2.
Figure 2.6: Correlation of German interest rates in …rst di¤erences.