Relative loss calculations
So far we have summarised the counterfactual results through the effects on the mean and variability of a number of key variables. In the following we will investigate how the interest rate rules in Table 10.1 perform when we select different weights X, ф in the monetary authorities’ loss function. We write the loss function as a linear combination of the unconditional variances of output growth A4yt and underlying inflation, A4put, with a possible extension in terms of the variance of interest rate changes ARSt.
£(X, ф) = V[A4put] + XV[Д4yt] + фУ[ARSt]. (10.2)
In Table 10.3 we report the square root of the loss according to equation (10.2) for different values of central bank preference parameters (i. e. the weights X and ф). The loss calculations reported in the upper part of Table 10.3 are calculated on the basis of the pure measures of volatility (sdev’s). The lower part of Table 10.3 reports similar loss calculations based on the RMSTEs. The first column in the table shows the results from the flexible rule FLX relative to the baseline scenario where we assume that interest rates are kept at their historical values.
Let us first consider losses based on sdev’s for the FLX rule. When we assume that the central bank pays no regard to interest rate variability (i. e. when ф = 0) we find that the loss is reduced by 1117% under the FLX rule depending on X. This is because both underlying inflation and output growth show less variability under the FLX rule (cf. columns two and three in Table 10.2) compared to under the baseline alternative. The loss reduction grows larger with increased weight X on output. If X is set to 0 the loss reductions is 11%, whereas X = 2, leads to a loss reduction. This is because the FLX rule gives rise to a larger relative reduction in variability for output than for underlying inflation. As we increase the weight ф on interest rate variability from 0 to 1, we find that relative losses increase from 0.89 to 1.62 when X = 0, since the variability in interest rate changes is 86% higher under the FLX rule compared with the baseline. As more weight is put on the variability of output, the partial effect from interest rate variability counts less and we find that when X = 2, relative losses only increase from 0.83 to 0.91 as we increase ф from 0 to 1. We find qualitatively similar results when we apply RMSTEs but since the bias for underlying inflation is relatively larger compared with that for output growth, we find the largest differences between losses based on RMSTE compared with those based on sdev for small values of X.
The strict rule ST puts zero weight on output growth and gives rise to considerably less variation in interest rate changes compared with the FLX rule and also compared with the baseline scenario. This puts the ST rule at an advantage as we increase the weight on interest rate variability ф from 0 to 1. When we span the relative loss measures in 3dimensional plots in Figures 10.5(a) and (b) we note that the ST rule in both cases gives rise to a relatively flat surface
Table 10.3
Counterfactual simulations 1995(1)2000(4). Loss function evaluation based
on relative sdev (upper half) and relative RMSTE (lower half)—relative to
the baseline scenario (actual observations of interest rates)
Central Bank preferences А ф 
FLX 
ST 
SM 
RX 
UR 
WF 
CR 

Loss based on relative sdev 

0 
0 
0.894 
0.895 
0.876 
0.915 
0.752 
0.851 
0.834 
0 
0.1 
1.125 
0.873 
1.003 
1.218 
0.785 
1.167 
1.104 
0 
0.5 
1.475 
0.829 
1.212 
1.654 
0.846 
1.612 
1.495 
0 
1.0 
1.617 
0.807 
1.301 
1.828 
0.873 
1.788 
1.652 
0.5 
0 
0.836 
0.908 
0.836 
0.856 
0.885 
0.891 
0.835 
0.5 
0.1 
0.867 
0.906 
0.852 
0.898 
0.885 
0.928 
0.868 
0.5 
0.5 
0.972 
0.896 
0.908 
1.034 
0.888 
1.054 
0.981 
0.5 
1.0 
1.073 
0.886 
0.965 
1.162 
0.891 
1.173 
1.088 
1 
0 
0.833 
0.909 
0.834 
0.853 
0.891 
0.893 
0.835 
1 
0.1 
0.850 
0.908 
0.842 
0.876 
0.891 
0.913 
0.853 
1 
0.5 
0.910 
0.903 
0.874 
0.955 
0.892 
0.985 
0.918 
1 
1.0 
0.975 
0.897 
0.910 
1.039 
0.894 
1.062 
0.987 
2 
0 
0.832 
0.909 
0.832 
0.852 
0.894 
0.894 
0.835 
2 
0.1 
0.840 
0.909 
0.837 
0.863 
0.894 
0.904 
0.844 
2 
0.5 
0.873 
0.906 
0.854 
0.907 
0.894 
0.943 
0.879 
2 
1.0 
0.911 
0.903 
0.874 
0.956 
0.895 
0.988 
0.919 
Loss based on relative RMSTE 

0 
0 
0.946 
0.926 
0.807 
0.972 
1.024 
0.979 
0.961 
0 
0.1 
1.057 
0.913 
0.883 
1.120 
1.016 
1.120 
1.078 
0 
0.5 
1.312 
0.877 
1.061 
1.447 
0.993 
1.433 
1.346 
0 
1.0 
1.465 
0.850 
1.171 
1.638 
0.977 
1.617 
1.505 
0.5 
0 
0.856 
0.915 
0.835 
0.876 
0.920 
0.912 
0.860 
0.5 
0.1 
0.883 
0.913 
0.849 
0.912 
0.920 
0.944 
0.889 
0.5 
0.5 
0.975 
0.904 
0.901 
1.033 
0.920 
1.055 
0.987 
0.5 
1.0 
1.066 
0.894 
0.953 
1.150 
0.921 
1.163 
1.084 
1 
0 
0.846 
0.914 
0.837 
0.866 
0.909 
0.905 
0.849 
1 
0.1 
0.861 
0.913 
0.845 
0.886 
0.909 
0.923 
0.866 
1 
0.5 
0.917 
0.908 
0.875 
0.960 
0.910 
0.990 
0.925 
1 
1.0 
0.978 
0.902 
0.909 
1.038 
0.910 
1.062 
0.990 
2 
0 
0.840 
0.913 
0.839 
0.860 
0.902 
0.901 
0.843 
2 
0.1 
0.849 
0.913 
0.843 
0.870 
0.902 
0.910 
0.852 
2 
0.5 
0.880 
0.910 
0.859 
0.912 
0.903 
0.948 
0.885 
2 
1.0 
0.916 
0.907 
0.879 
0.959 
0.903 
0.990 
0.924 
£(А, в) = m[A4put] + m[Ayyt + ^m[ARSt]
for А Є (0, 0.5,1, 2), ф Є (0, 0.1,0.5,1), m = (sdev, RMSTE).





compared with the other rules, which means that relative losses by adopting this rule are constant across values assigned to the central bank preferences parameters.
The smoothing rule SM gives rise to a more expansive monetary policy, with higher output growth and inflation. This entails an increase in the bias of output growth and a decrease in the bias for inflation, which we would expect to give different results depending on whether we calculate losses based on sdev’s or RMSTEs. As we increase ф the relative loss increases less sharply than the FLX rule due to the smaller volatility in interest rate changes under smoothing. Figures 10.5(a) and (b) show that SM does well compared to many of the other rules although the surface is far from being as flat as, for example, the strict ST rule.
The real exchange rate based RX rule gives increased interest rate volatility, and as we increase ф this translates into the largest relative loss compared with the other rules. The RX rule gives a slightly more contractive monetary policy compared with FLX, and the relative loss increases for all values of A (no matter which measure we base the loss calculations on). The RX stands out in Figures 10.5(a) and (b) showing the largest relative loss as we increase ф from 0 to 1. For large values of A this ‘open economy’ rule performs as well as or even better than many of the other rules.
Finally, we compare the results for the ‘realtime’ rules where output growth is replaced by either unemployment, wage growth, or credit growth. The unemployment based rule, UR, shows a remarkably flat surface in Figures 10.5(a) and (b). This is due to the fact that interest rate volatility is almost as low as in the case with the strict rule ST. The UR rule gives rise to the most contractionary monetary policy, and this is why that rule has a markedly different impact depending on whether the loss function is based on sdev’s or RMSTEs. This is mainly due to the increase in the inflation bias under a more contractionary monetary policy.
When we use wage growth or credit growth as basis for the ‘realtime’ rule we find higher interest rate volatility and this translates into a rising surface in Figures 10.5(a) and (b) as we increase ф from 0 to 1. Again these rules are (on average) more contractionary than the FLX rule and the increase in the inflation bias makes these rules score less well with the RMSTE based losses. For large values of A or for small values of ф the WF and CR rules stand out as superior to the other rules.
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