# «Christina Gerberding (Deutsche Bundesbank) Franz Seitz (University of Applied Sciences Amberg-Weiden) Discussion Paper Series 1: Economic Studies No ...»

For a justification of the assumption of asymmetric information, see Aoki (2006). In Aoki’s model, the economy behaves as if it is a representative-agent economy in which the representative agent has perfect information while the central bank has partial information, although each agent observes only a subset of the data (that is, the factors influencing her/his own consumption decisions).

We do not explore the implications of a significantly positive intercept term here (see Nelson and Nikolov, 2001).

It may be argued that the measurement error in potential output growth should be modelled explicitly since the money growth target depends on the central bank’s real-time estimate of potential output growth. However, as shown in Section 2, the historical measurement errors in the Bundesbank’s estimates of potential output growth were quite small, so that modelling them would not change the results.

Strictly speaking, this is only true if the measurement error in the level of the output gap is so persistent that the second estimate of the output gap, y t −l|t, does not differ noticeably from the initial estimate, y t −l|t −l, which is a feature of the historical measurement errors described in Section 2.

the estimates for the shorter sample period 1980-1995 which excludes the large measurement errors of the 1970s. In addition, we consider a high-uncertainty scenario which is based on the estimates for the full sample period (1974-1995), and a lowuncertainty scenario which is characterised by the baseline degree of persistence, but a smaller variance of the shocks. As shown in Table 2, the parameter values underlying our analysis are in fact very close to the estimates reported by Orphanides et al. (2000) for the US.

3.4. Central bank preferences Deriving the optimal feedback coefficients requires an objective function, and we use a fairly standard one in which the central bank is assumed to minimize the variation in inflation around its target (which is normalized to zero), in the output gap, and in the change in the interest rate:16

where the parameters ωπ, ωy and ωΔi are the relative weights on the three elements of the loss function. If the discount factor ß approaches unity from below, this loss function can be rewritten as the weighted sum of the unconditional variances of the

**three target variables (see Rudebusch and Svensson, 1999):**

E ( Lt ) = ωπ Var (π t ) + ω yVar ( yt ) + ω ΔiVar (Δit ) (4a) This specification has been widely used in the literature on monetary policy rules (see Ehrmann and Smets, 2003, or Coenen et al., 2005). In the initial exercise, we follow Coenen et al. and set ωπ =1, ωy =0.5 and ωΔi = 0.1. This may be viewed as a reasonable representation of a policymaker whose primary objective is to stabilise inflation around target, while also seeking to stabilize output and to avoid large interest rate volatility.17 Alternatively, it is sometimes assumed that policymakers care about the deviation of the interest rate from its steady-state level (rather than about its change against the previous The target for output is assumed to be equal to the natural rate, so the target for the output gap is also zero.

Within a welfare-optimising framework, Calvo-pricing with reasonable parameters typically suggests that the central bank should care relatively more about inflation variability.

period).18 Below, we will perform some sensitivity analysis regarding the robustness of our results to the details of the loss function (such as the exact specification of the interest rate variable and the weights on the elements of the loss function).

4. Performance of the rules

4.1 Results of model simulations with optimised feedback coefficients As a first step, we use the model described in Section 3.1. and summarized in Table 3 to compare the relative performance of the five rules defined above under different degrees of output gap uncertainty (no uncertainty, low uncertainty, baseline uncertainty, high uncertainty). We assume that the central bank minimises Equation (4a) subject to the rule in question and the model, while taking into account that its estimate of the output gap is imperfect. Furthermore, we assume that the policy rule is perfectly credible, so agents know the rule and assume (correctly) that it will be followed.19 Table 4 reports the values of the optimised coefficients, the standard deviations of the variables which enter the loss function, and the values of the period loss function. In order to gain a better understanding of the role of output gap uncertainty, we first consider the hypothetical case of perfectly observable output gaps. Here, our results regarding the Taylor rule (TR) and the speed limit rule (SPL) closely resemble the ones presented by Stracca (2007) despite the fact that we use a slightly different objective function. In particular, the optimal Taylor rule is found to have a very low degree of inertia, while the optimal speed limit rule is found to be very persistent (in fact, it is identical to a first difference rule). Stracca argues that the difference between the values of Φ1 is likely to reflect the fact that the Taylor rule feeds back strongly from the highly persistent level of the output gap, while the SPL rule reacts (again strongly) to the less persistent change in the output gap. Another interesting result is that the reaction to the output variable is much stronger than the response to current inflation, especially as regards the SPL rule. Again, this makes sense, since in an environment characterised by transmission lags and a low degree of inflation inertia, demand shocks which affect See, for instance, Stracca (2007). As shown by Woodford (2003), concern about the level of the nominal interest rate (relative to some target value) can be motivated by the presence of non-negligible transactions frictions and/or by the desire to keep away from the zero bound on nominal interest rates.

All calculations are done using DYNARE for Matlab. The optimization is based on the OSR routine.

current output are much more relevant for future inflation than cost-push shocks which matter only for current inflation. Allowing for an additional response to money growth somewhat changes the optimal coefficients of the Taylor rule, but the associated reduction in the overall loss is fairly limited. Augmenting the speed limit rule by a response to the output gap (TRSPL) or to money growth (SPLM) has even less impact on the optimal coefficients and on the overall losses.

Allowing for measurement error in the output gap changes these results in several directions. First of all, output gap uncertainty attenuates the optimal response to the output gap and to the output growth gap across all policy rules. The intuition for this result is straightforward: as the reliability of an indicator is reduced, one should place less emphasis on the information it conveys. Secondly, the optimal reaction to inflation increases with the degree of output gap uncertainty. While this result is in line with the literature on the consequences of output gap uncertainty in an optimal targeting rules framework (see Swanson, 2004), Rudebusch (2001) and Smets (2002) find that higher output gap uncertainty moderates the reaction to the inflation rate in the optimal simple rules they consider. As pointed out by Leitemo and Lonning (2006), this apparent contradiction can be explained by the presence of two countervailing effects. On the one hand, in the case of a demand shock, a stronger policy reaction to the inflation rate can substitute for a reaction to an imprecisely measured output gap. Ceteris paribus, this effect will increase the optimal coefficient on inflation. On the other hand, in the presence of cost-push shocks, a stronger reaction to inflation will destabilize the output gap even further. Hence, with increasing output gap uncertainty, it will be optimal for the central bank to reduce its response to both the output gap and inflation. Apparently, in the model considered here, the first effect dominates.

A third important result is that output gap uncertainty generates a non-trivial role for money growth as a feedback variable. Allowing for output gap uncertainty significantly increases the optimal coefficient on money growth, Φ5, in both the moneyaugmented Taylor rule and the money-augmented speed limit rule.20 At baseline (high) levels of uncertainty, Φ5 reaches a value of 1.42 (1.56) in the TRM rule and of 1.08 The parameterization of the measurement error process for the baseline and the high-uncertainty case is based on Table 2. For the low-uncertainty case, the standard deviation of the innovation is lowered to 0.60.

(1.20) in the SPLM rule. More importantly, even at a low degree of uncertainty, the additional response to money growth reduces the loss by 4.6% relative to the standard Taylor rule and by 3.9% relative to the standard speed limit rule (without money).

Under baseline (worst case) assumptions about output gap uncertainty, the welfare gain increases to 6.4% (8.0%) for the Taylor rule and to 6.2% (6.4%) for the speed limit rule.

One explanation for the welfare gain compared to the standard rules is that responding to money growth allows the central bank to reduce its response to inflation in both the TRM and the SPLM rule, thus enabling it to avoid inefficient reactions to cost push shocks. By contrast, augmenting the speed limit rule with a response to the output gap (TRSPL) reduces the loss relative to the standard SPL rule only marginally.

Figure 5 plots the optimised coefficients of the standard Taylor rule, the standard speed limit rule and its money-augmented variants for different levels of persistence (left) and of shock variability (right) in the measurement error process. It shows that the main insights to be gained from tables 4 and 5, such as the negative impact of increasing output gap uncertainty on the optimal response to the output gap (and the change in the output gap) and the corresponding rise in the coefficient on the money growth gap, are independent of whether the increased uncertainty comes in the form of higher persistence or higher shock variability. The vertical dashed lines mark the baseline assumptions about the measurement error process.

Figure 6 plots the rule-specific losses as a function of the degree of persistence in the measurement error (left) and of the variability of the measurement error shock (right). Again, the main insight is that, for realistic degrees of output gap uncertainty, the speed limit rule outperforms the classic Taylor rule, especially if it is augmented with an additional response to the money growth gap.

4.2 Some sensitivity analysis In this section, we carry out some robustness checks regarding the key results of the paper. In particular, we try to find out whether the superior performance of the money-augmented speed limit rule is robust to changes in the parameters of the central bank loss function and to variations in key coefficients of the underlying model.

Figure 7 shows the efficiency frontiers of the Taylor rule, the speed limit rule and the money-augmented speed limit rule for the baseline level of output gap uncertainty.

The frontiers trace out the minimum standard deviation of the goal variables as the relative weight on the output gap, ωy, in the period loss function is increased from 0.1 to 0.9.21 According to Figure 7, the efficiency frontier of the money-augmented speed limit rule is always below the frontiers of the other two rules, implying that it delivers a lower variability in both the output gap and inflation for any choice of the relative weight.

Hence, the ranking of the policy rules is robust to the choice of the relative weight on output gap versus inflation stabilisation.