E
t
R
t1 1
5 p
t
E
t
R
t1 1
up 5 p
L
1 ~1 2 p
t
E
t
R
t1 1
up 5 p
H
, 17
where p
t
is given by equation 12. As p
t
is a function of past inflation, the rational expectations version of the model will now exhibit some of the backward-looking
characteristics of traditional adaptive expectations.
33
Our specification of credibility in the form of beliefs about two possible and known values of p
is obviously a simplification of the complicated learning problem faced by real-world agents following a policy regime change. To achieve a more realistic set-up,
one could possibly assume that agents employ a Kalman filter or a least squares regression algorithm to continually update their estimate of p
and the matrices A and c in equation
9 as the economy evolves over time.
34
Our simple specification will serve to illustrate some basic points which we believe are likely to carry over to more elaborate learning
schemes.
III. Estimation and Calibration
For the purpose of estimating parameters, we adopted a baseline model specification that incorporates rational expectations and full credibility. The resulting parameter set was
then used for all model specifications to maintain comparability in the simulations. The data used in the estimation procedure are summarized in Table 1.
The model’s reduced-form parameters are assumed to be structural, in the sense that they are invariant to changes in the monetary policy reaction function equation 3. We
attempted to gauge the reasonableness of this assumption by examining the sensitivity of the parameter estimates to different sample periods. Following Fuhrer 1996, we did not
estimate the duration parameter but instead calibrated it to the value D 5 28. This coincides with the sample average duration in quarters of a 10-year constant-maturity
Treasury bond. Equations 1–4 form a simultaneous system that we estimated using full-information maximum likelihood.
35
The estimation results are summarized in Table 2.
With the exception of a
r
and g, the parameter estimates from the full sample 1965:1–1996:4 were all statistically significant. These results are very much in line with
33
A similar effect obtained in the models of Fisher 1986, Ireland 1995, Blake and Westaway 1996, King 1996, Bomfim et al. 1997, and Bomfim and Rudebusch 1997. In these models, credibility was determined
by a backward-looking, linear updating rule. In contrast, Ball 1995 modeled credibility using a purely time-dependent probability measure.
34
This type of approach to learning has been taken by Friedman 1979, Cripps 1991, Fuhrer and Hooker 1993, Sargent 1999, Marcet and Nicolini 1997, and Tetlow and von zur Muehlen 1999, among others.
35
We used the Matlab programs developed by Fuhrer and Moore 1995b, as modified to reflect the differences in our model specification and data.
Table 1.
Quarterly Data, 1965:1 to 1996:4
Variable Definition
y˜
t
Deviation of log per capita real GDP fixed-weight from its linear trend. p
t
Log-difference of GDP price deflator fixed-weight. r
t
Yield on 3-month Treasury bill. R
t
Yield on 10-year constant-maturity Treasury bond. u
t
Nonfarm civilian unemployment rate.
Expectations, Credibility and Disinflation
63
those obtained by Fuhrer and Moore 1995b, Table 4, despite some differences in our model specification and data. Estimates from the first subsample 1965:1–1979:4 were
highly imprecise, most likely due to the strong upward trends in U.S. inflation and nominal interest rates over this period. Estimates from the second subsample 1980:1–
1996:4 were much closer to the full-sample results. A comparison of the subsample estimates of a
p
and a
y
suggests that the Fed placed more emphasis on targeting inflation and less emphasis on stabilizing output in the period after 1980. Evidence of subsample
instability seems to be concentrated mostly in the I–S curve parameters, a
1
, a
2
, and a
r
. Notice, however, that all subsample point estimates lie within one standard error of each
other. We interpret these results to be reasonably supportive of the hypothesis that the reduced-form parameters, a
1
, a
2
, a
r
, r, and g, do not vary across monetary policy regimes. For the simulations, we required values for p
H
and p
L
. Given the imprecise nature of the first subsample estimate of p
, we chose p
H
5 0.06 to coincide with the sample mean from 1965:1 to 1979:4. Thus, we assumed that the U.S. inflation rate prior to October
1979 could be characterized by a stationary distribution centered at 6. Although this assumption is undoubtedly false, it serves to illustrate the effects of partial credibility on
the disinflation episode. As p
L
is intended to represent the new steady state after the disinflation has been completed, we chose p
L
5 0.03 to coincide with the sample mean from 1985:1 to 1996:4. In computing this average, we omitted the period of rapidly falling
inflation from 1980:1 to 1984:4, because this can be interpreted as the transition to the new steady state.
36
For the other model parameters, we adopted the full-sample estimates in Table 2.
Our disinflation simulations abstract from stochastic shocks to clearly illustrate the differences in the dynamic propagation mechanisms of the various model specifications.
37
We assumed, however, that agents make decisions as if uncertainty were present. This assumption is necessary for a meaningful analysis of credibility because without uncer-
tainty, agents can always learn the true value of p within two periods. To compute the
integrals in equations 13 and 14, we simply calibrated the standard deviations of the two long-run inflation distributions centered at p
H
and p
L
. For both distributions, we
36
The values, p
H
5 0.06 and p
L
5 0.03, are very close to those used by Fuhrer 1996, figure IIb to help reconcile the pure expectations theory of the term structure with U.S. nominal interest rate data.
37
Because equation 12 is nonlinear, the addition of stochastic shocks would affect the mean length and speed of the disinflation under partial credibility. See Orphanides et al. 1997 and Bomfim and Rudebusch
1997 for studies which investigate disinflation dynamics in stochastic models with a nonlinear monetary policy rule.
Table 2.
Maximum Likelihood Parameter Estimates
Parameter 1965:1 to 1996:4
1965:1 to 1979:4 1980:1 to 1996:4
Estimate Std. Error
Estimate Std. Error
Estimate Std. Error
a
1
1.23 0.09
0.94 4.97
1.24 0.10
a
2
20.26 0.08
0.10 4.62
20.31 0.09
a
r
20.20 0.12
20.57 2.17
20.05 0.05
r 0.02
0.01 0.02
0.36 0.00
0.04 g
0.01 0.01
0.04 0.47
0.01 0.01
a
p
0.06 0.03
0.07 1.04
0.10 0.05
a
y
0.08 0.03
0.07 1.05
0.05 0.06
p 0.05
0.01 0.04
0.45 0.05
0.01
64
C. G. Huh and K. J. Lansing
chose s
p
5 0.023 to coincide with the sample standard deviation from 1965:1 to 1979:4. This reflects our interpretation that the Fed’s announcement on October 6, 1979 concerned
only a change in the target level of inflation, not a change in the target variability of inflation.
38
We used the same value of s
p
for all model specifications. For the steady-state unemployment rate, we chose u 5 0.06 to coincide with the
average over the full sample. Given u, we estimated the parameters of Okun’s law equation 8, using ordinary least squares to obtain b
1
5 0.96, b
2
5 20.30, b
3
5 0.10, and b
4
5 0.18, which are all statistically significant.
The matrix A in equation 9 was estimated by an unrestricted first-order VAR on U.S.
data from 1965:1 to 1996:4. The result is:
A 5
3
0.953 0.002 20.197 0.096
0.111 0.628 0.290 20.211
0.093 0.073 0.839
0.055 0.012 0.053
0.039 0.907
4
. 18
Given A, we defined two versions of the matrix c so that equation 9 would be consistent with the two steady states associated with p
H
and p
L
, respectively. This procedure yields:
c 5
5
3
0.008 0.016
0.004 0.001
4
when p 5 p
H
5 0.06,
3
0.005 0.007
0.003 0.001
4
when p 5 p
L
5 0.03.
19
Our solution procedure can be briefly summarized as follows. Given a set of parameters and an assumption regarding the way that expectations are formed rational or adaptive,
we solved the full-information version of the model for each of the two cases: p 5 p
H
and p
5 p
L
. In each case, the solution consists of a set of time-invariant linear decision rules for p
t
, r
t
, and R
t
, defined in terms of the state vector, s
t
5 { y˜
t2 1
, y˜
t2 2
, p
t2 1
, r
t2 1
, r
t2 1
}. The decision rules for y˜
t
and r
t
are simply given by equations 1 and 3, respectively. For each value of p
[ {p
H
, p
L
}, we constructed linear expressions for the conditional expectations, E
t
[p
t1 1
up], E
t
[r
t1 1
up], and E
t
[R
t1 1
up]. Under rational expectations, these expressions were constructed using the decision rules, whereas under adaptive expecta-
tions, the expressions were constructed using equations 9 and 10. Next, we formed the unconditional expectations, E
t
p
t1 1
, E
t
r
t1 1
, and E
t
R
t1 1
, using the current value of p
t
which does not depend on p
t
and equations 15–17. Finally, the unconditional expectations were substituted into equations 2, 4, and 6 which, together with
equations 1 and 3, form a system of linear equations in the variables, y˜
t
, p
t
, r
t
, r
t
, and R
t
.
38
We relaxed this assumption in Huh and Lansing 1998 by allowing the reaction function parameters, a
p
and a
y
, to shift in conjunction with the Fed’s announcement.
Expectations, Credibility and Disinflation
65
Under full credibility, it is straightforward to show that the model possesses a unique, stable equilibrium for the parameter values we employed.
39
Under partial credibility, agents use observations of an endogenous variable inflation to form expectations which
are crucial for determining the period-by-period values of that same variable. The presence of this dynamic feedback effect between the trajectory of inflation and the inputs
to the learning process can create an environment where learning can go astray. In particular, there is no way to guarantee that the model will converge to a new steady state
with p
5 p
L
.
40
We found that convergence is always achieved in the numerical simula- tions, however.
IV. Quantitative Results
