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

Not surprisingly, the estimates of the persistence parameter ρ turn out to be highly significant and quite close to one for the measurement error in the level of the output gap. By contrast, the estimates of ρ for the measurement error in the change in the output gap as well as for real output growth are negative. On the other hand, the measurement errors in potential output growth are again quite persistent, but much smaller in size (with very low standard deviations).

Although the unconditional mean of the measurement error in the level of the output gap amounts to 3.10 for the sample period 1974Q1 to 1995Q1, the intercept term is not significant. This is not inconsistent but reflects the fact that a high positive serial correlation in the errors may create the appearance of a bias in the real-time data relative to the final series, even though the underlying process is in fact unbiased.

For comparison’s sake, we also report statistics on the measurement errors of the variables with respect to a second set of ex post series which is based on a much later vintage of GDP data (September 2005). Despite some differences in the distribution of the measurement errors over time (see Figure 4), the parameter estimates of the measurement error processes are very similar.

**3. Model specification**

3.1. Aggregate demand, aggregate supply and money demand The model that we use is a version of the canonical New Keynesian model which has been adapted by Rudebusch (2002) for empirical implementation with quarterly

**data. Specifically, the model contains a hybrid Phillips curve and a purely backwardlooking specification of aggregate demand:**

The corresponding statistics for the four-quarter-rates of change are available upon request.

where y is a measure of the output gap, i is the short-term nominal interest rate, π is the annualised percentage change in the price level, Et −1π t +3 is a measure of the rate of inflation expected to prevail over the subsequent four quarters (lagged one quarter), rt is the time-varying equilibrium real rate of interest, and ε tπ and ε ty are white noise shocks.

The generalized Phillips curve described by equation (1) captures the New Keynesian consensus on price dynamics. In the canonical New Keynesian model derived from first principles, inflation is purely forward-looking, that is γ equals zero.

This result can be derived, for instance, within a model of Calvo price setting (Calvo, 1983). However, a number of reasons have been advanced why inflation may depend on its own past values as well as on expected future inflation.7 The purely backwardlooking nature of the IS curve reflects the empirical problems associated with estimating hybrid IS curves (Stracca, 2007, p. 24).

The model features lags in the transmission of monetary policy (from interest rates to the output gap and, again, from the output gap to inflation) as well as an expectational lag in the Phillips curve. Rudebusch (2002, p. 405) argues that these lags are appropriate “given real-world recognition, processing and adjustment lags”. Stracca (2007) estimates the model on euro area data and finds coefficient values of γ=0.20, k=0.31, σπ2=0.94, α1=1.47, α2=-0.53, σ=0.17 and σy2=0.20 (sample period: 1987Q1Q2). The coefficient of particular interest is γ, or rather (1-γ), which measures the degree of explicitly forward-looking behaviour. With an estimated value of 0.80 for (1γ), Stracca finds the Phillips curve for the euro area to be quite forward-looking which is in line with other evidence on the low degree of intrinsic persistence in euro area inflation (see Galí et al., 2001, Smets and Wouters, 2003, ECB, 2005). By contrast, movements in the output gap are very persistent, implying that demand shocks have a more protracted effect on output and inflation than cost-push shocks.

For instance, following Gali and Gertler (1999), it is often assumed that a fraction of price setters adjust their prices in a backward-looking fashion (following simple rules of thumb).

Models of the type described by equations (1) and (2) are usually closed with an interest rate rule and/or a central bank objective function. However, as we want to analyse the role of money growth as a potential feedback variable in the interest rate rule, we have to add a money demand equation to the model. Following Rudebusch and Svensson (2002) and Coenen et al. (2005), we use a standard specification of the error

**correction type:**

Δmtr = −κm (mtr−1 − κq qt −1 + κiit −1 ) + κ1Δmtr−1 + κΔq Δqt + ε tm (3) where mtr = mt − pt is the real money stock, qt is the level of actual output and ε tm captures shocks to money demand. For the baseline version of the model, we use the parameter values κm=0.15, κq=1.20, κi=0.80, κΔm=0.40, κΔq=0.10 and σm2=0.20, which are in line with standard estimates for the euro area.

The fact that money demand depends on the level of actual output rather than on

**the output gap requires us to specify the relationship between these variables:**

as well as the process governing potential output, qt*. Here, we follow Ehrmann and Smets (2003) and assume that potential output follows a highly persistent AR(1)

**process:**

where ε tq* is a white noise shock.

3.2. Monetary policy rules As noted in the introduction, our analysis takes place in a simple rules framework and focuses on the relative performance of several variants of the basic Taylor rule, taking into account that policymakers observe only a noisy measure of the output gap.

These rules are simple because they model the interest rate as a function of a limited set of specified state variables while the fully optimal rule would involve all state variables of the model. Given the constraint on the number of feedback variables, the feedback coefficients are chosen so as to minimise policymakers’ expected loss (see Section 3.4).

A potential advantage of simple rules is that they are easier to understand and monitor for the public than the (complex) optimal commitment solution. Furthermore, simple rules may be more robust to model uncertainty.8

**The first simple rule that we consider is a Taylor rule with interest rate smoothing:**

ˆ where it is the deviation of the nominal interest rate from its steady state value and the subscript t|t indicates the information on the contemporaneous value of a specific variable available at time t.9 The second rule is a simple growth rate targeting or speed limit rule of the kind advocated by Orphanides (2003b) and Walsh (2003) which

**involves a response to the change rather than to the level of the output gap:**

ˆ ˆ it = φ1 ⋅ it −1 + φ2 (π t t − π t* ) + φ4 ⋅ ( yt t − yt −1 t ) (SPL) However, central banks need not be limited to a discrete choice among these two simple rules. Especially with output gap uncertainty, it may be advantageous to respond to the level as well as to the change in the output gap (see Rudebusch, 2002). Hence, we

**also consider a “hybrid” rule which nests both cases:**

Our motivation for including money growth among the right-hand-side variables of the policy rule is twofold. First, Söderström (2005) has shown that in models with forwardlooking expectations, stabilising money growth around a target can be a sensible strategy for a central bank acting under discretion because it introduces inertia and For further discussion, see Taylor (1999), Williams (2003), and Berg et al. (2006).

The steady state value of the nominal interest rate, it*, depends on the equilibrium real interest rate, rt*, and the inflation target, πt*. Both variables are assumed to be constant and normalised to zero. Hence, our analysis abstracts from uncertainty about the equilibrium real interest rate. However, Rudebusch (2001) has shown that in this kind of analysis, uncertainty about r* is of little importance in terms of altering the optimal rule coefficients or the expected loss.

history-dependence into monetary policy. Augmenting the Taylor rule by a response to the money growth gap allows us to test the relevance of this argument in a simple rules framework. Secondly, Coenen et al. (2005) have demonstrated that monetary aggregates may have substantial information content about the “true” level of aggregate output if the environment is characterised by (a) significant measurement errors in GDP data, (b) a strong contemporaneous linkage between money demand and real output and (c) a low variability of money demand shocks.

3.3. Measurement errors in the feedback variables Simple rules like the ones considered here typically model the interest rate in quarter t as a function of the contemporaneous values of key macro variables like the rate of inflation and the level of the output gap. However, as noted in Section 2, realtime data sets suggest that policymakers face substantial uncertainty about the “true” values of these variables, especially as regards the output gap. Here, we focus on errors in the measurement of the level and the change in the output gap, and ignore errors in the measurement of inflation and money growth, on the grounds that the latter have been shown to be relatively minor (see Section 2).

To capture the implications of real-time output gap uncertainty, we follow Rudebusch (2001, 2002), Orphanides (2003c) and others and assume that the estimates of the output gap available to policymakers at the time the decisions are made (t) differ

**from the true series by a measurement error ηy,t:**

** yt t = yt +η y,t (6a)**

According to this specification, the measurement error ηy,t is correlated with the initial estimates, but uncorrelated with the final estimates, implying that the initial estimates contain an element of inefficient noise relative to the final estimates.10 Alternatively, one could assume that the central bank uses optimal filtering to infer the true state of the economy. However, this presupposes that the central bank has the true model of the economy at its disposal (which, in practice, it does not have). Moreover, the “best” (model-consistent) estimate of unobservable variables like the output gap is a An alternative formulation would be y t = y t|t + η y, t, implying that the forecast errors are uncorrelated with the initial estimates, but correlated with the final estimates (the revisions are “news”). However, the correlations in the data favour a substantial noise element (results available on request).

complicated function of past observable variables which is at odds with the simple rules framework used here.11 This argument is reinforced by the fact that optimal filtering is even more intricate if the information set of the private sector differs from that of the central bank (see Svensson and Woodford, 2002), which is the case we consider here.12 To capture the potential persistence in the measurement error ηy,t, we follow Orphanides et al. (2000) and assume that it follows an AR(1) process:13

where ε tηy is the measurement error shock. The measurement error ηy,t subsumes errors in assessing the contemporaneous levels of actual and potential output, Qt t and Qt*t.

For the purpose of our analysis, it is not necessary to model each of the underlying error processes explicitly. However, we need to make an assumption about the measurement error (ME) in the change in the output gap, ηΔy,t,14 Here, we assume that this ME is approximately equal to the change in the ME of the level of gap, Δηy,t:15 yt t − yt −1 t = yt − yt −1 +ηΔy,t ≈ yt − yt −1 + Δη y,t (7a) It can be shown that under these assumptions, the variance of the ME in the change of the output gap is 2σε2/(1+ρηy), whereas the variance of the ME in the level of the output gap is σε2/(1-ρηy2). Thus, as long as ρηy 0.5, the error in the change is smaller than the error in the level. Estimates for the parameters of the ME process are obtained from the real-time data set presented in Section 2. As baseline values, we take For a discussion see Svensson and Woodford (2002, 2003), Orphanides (2003a), and Swanson (2004).

For an application of the method to a model of the euro area see Ehrmann and Smets (2003) and Coenen et al. (2005).