RDP 2012-08: Estimation and Solution of Models with Expectations and Structural Changes 4. Numerical Examples

4.1 A Credible Disinflation

In this example there is a credible disinflation in the context of the standard New Keynesian model described below in Equations (21)–(27).^[6]

In the equations above, x_t is the output gap defined as the deviation of output from a socially efficient level of output; π_t is the gross rate of inflation, that is ln(p_t/p_t−1); r_t is the log of the gross nominal interest rate; g_t is the growth rate of output; ŷ_t is the percentage deviation from steady state of the log of the stochastically detrended level of output. The log of total factor productivity follows a unit root with a drift, g. Finally, a_t is a demand shock, e_t is a cost-push shock and ε_z,t is the shock to total factor productivity. The ε's are identically and independently distributed shocks.

We construct a sample of 200 observations from this system with the following characteristics. First, the initial structure (model 1) shown in Table 1 governs the system up to period 159. Second, at the beginning of period 140, the monetary authority announces a disinflation program that involves a lower inflation target (π = 0.0125) and a more aggressive response to deviations of inflation from this target (ρ_r and ρ_π increase). The response to deviations of growth from trend also increase (ρ_g increases). This new policy will be implemented in period 160. Finally, there are no further structural changes until the end of the sample in period 200. Agents believe the announcement and revise their expectations accordingly. In terms of the sample parameters given earlier, T = 200, T_a = 140 and = 160. The parameters of the modified system are then shown in lower panel of Table 1 while data on the observable variables, r_t, π_t and g_t are shown in Figure 3.

In estimation, r_t, π_t, and g_t are taken to be observed without noise, that is V = 0. For our choice of observables, ω is unidentified. Moreover, in practice it is typically the case that β is not estimated. For these reasons we set these parameters prior to estimation. The task is then to estimate the values of the remaining 17 parameters, .

The results are given in Table 2. The point estimates obtained with the history of observables in Figure 3 correspond to the MLE column in Table 2. The standard error of the maximum likelihood estimators are computed using the theoretical bootstrap with 250 replications. That is, we generate, at the estimated values of the parameters, 250 histories for the observables and estimate the parameters each time.

There are three distinct sub-samples in the data. The first 139 observations are constructed using the initial structure (model 1), the last 41 observations are found using the final structure (model 2), and the observations during the transition period – 140 to 159 – involve using both model 1 and model 2 weights when forming expectations. The model parameters that change are those of the monetary policy rule, including the target rate of inflation. As one would expect, because there are more observations generated from the initial structure, the parameters of the initial policy rule are estimated more precisely than those of the final structure. In contrast, because the new policy rule penalises deviations from the new inflation target more strongly, the final inflation target is estimated more precisely. This example illustrates an important point – even though there are relatively fewer observations coming from the final structure, not all of its parameters are estimated less precisely.

These outcomes are illustrated in Figure 4 which shows distributions of the estimators of the inflation response for both structures, ρ_π and , and Figure 5 which shows distributions of the estimators of the inflation targets, π and π′.

4.2 A Slowdown in Trend Growth

For this example the monetary policy rule, Equation (23), is replaced with

This specification makes a distinction between the inflation target of the central bank and its estimate of trend growth, π_cb and g_cb, and those of the private sector, πand g. For the initial and final structures these are the same, that is π_cb = π and g_cb = g. At T_m = 32 there is a structural change: g falls to g′ and π_cb increases to There is another structural change at = 64, when the parameters revert back to their original values. Unlike the example above, in the period running from T_m = 32 to = 64, expectations are (incorrectly) based on the first model. The reduced-form therefore follows Equation (18). The parameters of this simulation are summarised in Table 3 along with the steady state real interest rate for both structures, rr.

While the temporary structure is in place trend growth falls. However, the central bank's view of trend growth does not. At the same time, the central bank runs looser monetary policy in an attempt to offset weaker growth outcomes. This is captured by an increase in the central bank's inflation target to Agents' beliefs are never updated and are based on the initial (and final) structure. So in this example, , etc and , etc. It is therefore unnecessary to specify T_b and . The reduced form therefore follows Equation (18).

The results are given in Table 4. The point estimates associated with the history of observables shown in Figure 6 correspond to the MLE column in Table 4. The standard error of the maximum likelihood estimator is, as before, computed using the theoretical bootstrap with 250 replications. Figure 6 shows also the non-stochastic path of the simulation which corresponds to the path the economy would have experienced in the presence of structural changes but in the absence of random shocks.

Footnote

See Ireland (2004) for more details. [6]