2.6 Estimation Results
2.6.3 Estimation of the Slope Factor
value is -3.27.
This chapter focuses on the impact of macroeconomic volatility on the e¤ects of the macroeconomy on the long term interest rate, which is the First Principal Component of the term structure of interest rates. The tests for cointegration between the t-values of the macroeconomic variables explaining the long term interest rate and realized macro-economic volatility show that the e¤ects of the macroeconomy on the long term interest rate are signi…cantly cointegrated with macroeconomic volatility. To research on the e¤ect of macroeconomic volatility on the long term interest rate, this chapter focuses on the analysis of the t-values and does not provide coe¢ cient estimates. The latter would enable a comparison between interest rates implied by the model and actual interest rates.
expected values of in‡ation. The deviation of GDP growth from its potential growth rate (output gap) is approximated by the deviation of the Ifo-Index from its 60-month moving average. The sentiment indicator for the German economy includes an expectations component. Consequently, the expectations of real GDP growth are taken into account by the forward-looking Taylor rule.
To calculate the slope of the yield curve (empirical counterpart of the Second Prin-cipal Component), the short term interest rate is explained by a forward-looking Taylor rule and combined with the long term interest rate. The estimation results of the Taylor rule in equation 2.16 (it = c+ t+ y (yt y ) + t) are given in equation 2.26, which is estimated with OLS based on 328 monthly observations between July 1978 and October 2005. The short term interest rate it is represented by the three-month money market rate3Mt, because it is strongly in‡uenced by the target rate of the central bank. In‡ation expectations tare approximated by theCP It. The output gap variable yt y is calculated as the Ifo-Index If ot relative to its average over the whole sample mean(If o) (t-values in parenthesis):
3Mt = 2:11
(11:33)+ 1:46
(21:16) CP It+ 4:33
(2:90) (If ot mean(If o));
R2 = 0:61, DW= 0:13, obs.= 328: (2.26)
The t-values show that all estimated parameters are signi…cant. The coe¢ cients of the output gap and in‡ation are positive and therefore in line with economic theory, i.e. an increase in in‡ation or a GDP growth higher than the potential GDP growth increases the short term interest rate. Based on equation 2.15 (it = (r ) + (1 + ) t+ y(yt y ) + t) and on the estimation results of equation 2.26, it is possible to estimate the short term real interest rate r for Germany during the period August 1978 to October 2005. Combining c=r and the estimation results forc(2.11), for1 + (1.46) and an in‡ation target of 2% indicates a real interest rater around 3%.
The residuals of the regression for the short term interest rate (equation 2.26) show a signi…cant positive autocorrelation (the Durbin-Watson statistic is 0.13), because the short term interest rate usually tends to be integrated of order one. As the estimation
equation does not include an AR(1) term, the residuals are positively autocorrelated.
The reason for the non-stationarity of the short term interest rate might be that both the Deutsche Bundesbank did and the ECB still does interest rate smoothing, i.e. they avoid fast changes between an expansionary and a restrictive monetary policy stimulus.
Consequently, interest rates change only slowly.
Sack and Wieland (2000) o¤er three explanations for interest rate smoothing, which are empirically validated. First, …nancial market participants are forward-looking. So, forward-looking monetary policy rules which avoid large interest rate movements are more appropriate to in‡uence output and in‡ation. Second, interest rate smoothing avoids unnecessary movements in output and in‡ation caused by a central bank which reacts too aggressively to the …rst announcement of macroeconomic data. As the initial announcement might include an error, it is likely to be revised (especially potential output and the natural rate of unemployment). Third, a slow adjustment of the target rate causes low disruptions in in‡ation and unemployment, because the e¤ects of discrete monetary policy through transmission channels on the real economy are uncertain.30
To integrate interest rate smoothing in the estimation, the following equation of the short term interest rate is used (Sauer and Sturm (2004)):
it = (1 ) it + it 1; (2.27)
where the current nominal short term interest rateitat timetgradually converges to the optimal target ratei . The adjustment speed is given by (smoothing parameter). The optimal target rate i is determined by a Taylor rule (equation 2.16). The combination of equations 2.27 and 2.16 results in equation 2.28 for the short term interest rate based on a Taylor rule and interest rate smoothing, which can be transformed to equation 2.29,
it = (1 )(r ) + (1 )[(1 + ) t+ y(yt y )] + it 1+ t; (2.28)
30As central banks are deciding as a committee on interest rate decisions, the process of switching to another target rate is slow. This might be another reason for interest rate smooth-ing, which is not empirically validated (Sack and Wieland (2000)).
it= (1 ) c+ (1 ) [ t+ y (yt y )] + it 1+ t: (2.29) An OLS estimation of equation 2.29 (based on 328 monthly observations) yields the following results (t-values in parenthesis):
3Mt = (1 0:99) ( 2:01
( 0:36)
) + (1 0:99) 2:52
(1:49) CP It+ 1:62
(0:92)(If ot mean(If o)) + 0:99
(88:16) 3Mt 1;
R2 = 0:99, DW= 1:42, obs.= 328: (2.30)
The inclusion of the autoregressive term3Mt 1 in the estimation of the short term inter-est rate 3Mt reduces the autocorrelation in the residuals signi…cantly (Durbin-Watson statistic is 1.42). The results show that almost all of the variation in the current short term interest rate is explained by the lagged short term interest rate, if interest rate smoothing is taken into account: only the coe¢ cient of the autoregressive term is signif-icant and has a parameter estimate of 0.99. Consequently, the part of variation in the short term interest rate explained by macroeconomic information becomes negligible and the short term interest rate is only explained according to Time Series Analysis based on its past value. As this chapter researches on an empirical Macro-Finance model of the term structure of interest rates, the short end of the yield curve is modelled by a Taylor rule without interest rate smoothing.
Figure 2.15 shows the t-values of the e¤ects of the business cycle (Ifo-Index), in‡ation (CPI) and a constant on the short term interest rate estimated by OLS over a rolling window of 60 months of equation 2.26. Even though the t-values of the explanatory variables estimated in the moving window regressions are time-varying, a Cusum of squares test of regression 2.26 over the whole sample signals that the coe¢ cients are stable during the sample (appendix A.4).
Similar to the research on the e¤ect of realized macroeconomic volatility on the im-pact of macroeconomic variables on the long term interest rate in section 2.6.2, the e¤ect of macroeconomic volatility on the time varying impact of macroeconomic variables on the short term interest rate is analysed according to the approach to test for cointegra-tion by Banerjee et al. (1993) and Banerjee, Dolado and Mestre (1998). The t-values
-10 -5 0 5 10 15 20
1983 1986 1989 1992 1995 1998 2001 2004
0 10 20 30 40 50 60
t-value (Ifo) t-value (CPI) t-value (constant) [RS]
Figure 2.15: Time series of t-values of macroeconomic variables and a constant explaining the short term interest rate (regression 2.26) from July 1983 to October 2005.
of the two explanatory variables in regression 2.26 are explained by realized macroeco-nomic volatilities, whereas only the variables of macroecomacroeco-nomic volatility are included in the regression which are signi…cant at the …ve percent level. A constant is included even if it is insigni…cant in order to make use of the critical values provided by Banerjee et al. to test for cointegration.
Equation 2.31 quanti…es the e¤ect of vola(3M)t, vola(CP I)t and vola(If o)t on the impact of the Ifo-Index on the short term interest rate, i.e. the t-values estimated by a rolling window OLS estimation of equation 2.26. The estimated coe¢ cients of realized volatility of the three-month money market rate, CPI and the Ifo-Index are all signi…cantly di¤erent from zero (t-values in parenthesis). Analogous, equation 2.32 states signi…cant e¤ects of vola(CP I)t and vola(If o)t on the impact of the CPI on the short term interest rate.
tvalue(If o; short)t = 0:44
(0:89)
0:04
( 10:10)
vola(3M)t+ 0:02
(2:72)
vola(CP I)t+ 0:01
(4:95)
vola(If o)t R2 = 0:29, DW= 0:06, obs.= 268: (2.31)
-8 -4 0 4 8 12 16
1983 1985 1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 Actual Value Fitted Value Residual
Figure 2.16: Time series of actual values, …tted values and residuals of regression 2.31 which explains the impact (t-value) of the Ifo-Index on the short term interest rate by realized macroeconomic volatilities.
tvalue(CP I; short)t = 0:21
( 0:22)
+ 0:15
(11:60) vola(CP I)t 0:02
( 6:55) vola(If o)t;
R2 = 0:33, DW= 0:03, obs.= 268: (2.32)
Figures 2.16 and 2.17 plot the actual and …tted values as well as the residuals of regressions 2.31 and 2.32, respectively. In both …gures, the time series of the residuals crosses the zero line several times. This might be an indication of cointegration between the impact of the Ifo-Index on the short term interest rate and realized macroeconomic volatility and of cointegration between the impact of the CPI on the short term interest rate and realized macroeconomic volatility, respectively.
Similar to the analysis of the long term interest rate, the t-values of regression 2.31 and their dependency on realized macroeconomic volatility are analysed in a single equa-tion error correcequa-tion framework. In equaequa-tion 2.33, the negative sign of the coe¢ cient of the error correction term (-0.04) signals that there is an error correction mechanism.
The corresponding critical value at the one percent signi…cance level reported by Baner-jee et al. (1998) is -4.22. Hence, the cointegration relationship is signi…cant due to a t-value of the error correction term of -4.39:
-8 -4 0 4 8 12 16
1983 1985 1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 Actual Value Fitted Value Residual
Figure 2.17: Time series of actual values, …tted values and residuals of regression 2.32 which explains the impact (t-value) of the CPI on the short term interest rate by realized macroeco-nomic volatilities.
d(tvalue(If o; short))t = 0:01
(0:14)
0:04
( 4:39)
[(tvalue(If o; short)t 1 0:04
( 2:90)
vola(3M)t 1 +0:01
(1:96)
vola(If o)t 1] + 0:73
(16:97)
d(tvalue(If o; short))t 1; R2 = 0:52, DW= 2:11, obs.= 267: (2.33)
Equation 2.34 tests for cointegration between the e¤ect of the CPI on the short term interest rate and realized macroeconomic volatility. As the t-value of the error correction term is -3.38 and the critical value at the …ve percent signi…cance level is -3.27, this cointegration relationship is signi…cant, too:
d(tvalue(CP I; short))t = 0:17
( 1:37) 0:03
( 3:38)[(tvalue(CP I; short)t 1 +0:15
(2:42) vola(CP I)t 1] + 0:64
(13:52) d(tvalue(CP I; short))t 1; R2 = 0:42, DW= 2:23, obs.= 267: (2.34)