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May 31, 2018
Martin Bodenstein
James Hebden
Fabian Winkler
Learning and Misperception: Implications for Monetary Policy Strategies
Introduction
A key determinant of the transmission of monetary policy to the economy is the public’s
expectation about the central bank’s strategy for responding to economic developments. To
assess the effects of alternative monetary policy strategies, Board staff routinely use
macroeconomic models with specific assumptions about private-sector expectations formation.
In the Monetary Policy Strategies (MPS) section of Tealbook A, it is assumed that private sector
agents know the exact strategy pursued by the central bank and they believe that policymakers
are committed to responding according to this strategy in all future circumstances. These
assumptions may be reasonable when the monetary policy strategy is stable over time. However,
these assumptions are more questionable when policy changes occur and the public does not
immediately understand them.
In this memo, we expand the existing our conventional policy analysis to allow for the
examination of changes in monetary policy strategies when the public needs to learn about those
changes. We apply this learning framework to policy experiments like those routinely performed
in the MPS. Our framework builds on an important academic literature, including Tetlow and
von zur Muehlen (2001), and Cogley, Matthes and Sbordone (2015), which departs from rational
expectations and assumes that the public infers key aspects of a policymaker’s strategy using
observed data. 1 More specifically, we assume that the public does not directly observe the
parameters of the central bank’s reaction function, and instead must form beliefs about likely
current and future parameter values. To obtain estimates of these unknown parameters in the
reaction function, the private sector agents run regressions using data on policy interest rates and
their determinants (namely, measures of economic slack and price inflation). If agents learn the
1
The literature on learning is extensive and distinguishes amongst others between procedural, rational, and
educative learning. Our paper is closely related to Erceg and Levin (2003) since they also consider the situation in
which the public is learning about aspects of the policy rule. In their model, the public is learning only about the
inflation target in a rational manner. In contrast, we allow for the possibility that the public learns about other
parameters of the rule, and our least-square learning mechanism falls into the class of procedural learning.
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true nature of the policy change at a slow pace, their misperceptions about the course of
monetary policy can lead to economic outcomes that differ notably from the case in which agents
have instantaneous and complete information about the policy change.
Without loss of generality, we apply this framework to a specific policy change routinely
analyzed in the MPS. In this case, the policy rule changes from the inertial version of the Taylor
(1999) rule, which approximates the projected path of the nominal federal funds rate in the
Tealbook baseline, to its non-inertial counterpart, also known as the balanced-approach Taylor
(1999) rule. In recent years, the latter rule has prescribed a federal funds rate that remains
considerably above the Tealbook baseline for several years, but this different policy path does
not induce substantial differences in economic outcomes. This result is reasonable when the
public understands the change in the central bank’s policy reaction function (full information
case) and believes that policymakers are committed to no longer engage in interest rate
smoothing. In that case, the public correctly anticipates that the federal funds rate eventually
will fall persistently below the prescriptions of the inertial Taylor (1999) rule. This anticipatory
effect greatly mitigates the contractionary effects stemming from the large initial tightening
associated with the balanced-approach Taylor (1999) rule.
However, when agents must learn about the change in monetary policy strategy from observed
data (learning case), the initial unexpectedly fast tightening of monetary policy under the
balanced-approach Taylor (1999) rule can lead to large contractionary effects. The public does
not immediately attribute the observed higher level of the federal funds rate solely to the central
bank’s decision to no longer engage in interest rate smoothing. After all, the observed rise in the
federal funds rate could also reflect the beginning of an aggressive tightening of policy or a onetime discretionary adjustment in the policy stance. If the public views some of the rise in the
federal funds rate as signaling the beginning of a tightening cycle, then it can significantly push
up the public’s anticipated path of the federal funds rate and thus realized long-term interest
rates. As a result, resource utilization and in particular inflation can experience a more
pronounced fall than under the full information case routinely shown in the MPS.
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Finally, our approach is general enough to be used in a broad class of linearized economic
models. In fact, we confirm that the Small FRB/US (sfrb) model and the Estimated Dynamic
Optimization-based (EDO) model deliver similar responses to changes in monetary policy
strategy as the FRB/US model routinely used in the MPS. Moreover, the range of policy
changes to which we can apply our framework is considerably broader than the experiments we
consider in this memo. For instance, we can consider actual and perceived changes in the
inflation target or the equilibrium real interest rate as well as the addition or replacement of
economic variables in the policy reaction function (such as a switch to a rule targeting the price
level instead of inflation). At the end of this memo, we discuss possible extensions and
limitations to our framework.
Framework
We conduct our analysis in the Small FRB/US (sfrb) model and the Estimated Dynamic
Optimization-based (EDO) model. 2 The monetary policy strategy of the central bank is given by
an interest rate rule that prescribes a target value for the federal funds rate. Throughout our
analysis, we distinguish between the actual policy rule—the interest rate rule from which the
central bank derives its prescriptions—and the perceived policy rule—the interest rate rule that
the private sector believes the central bank is following.
When expressed in deviations from the steady state of the model, the actual policy rule is of the
form:
𝑖𝑖𝑡𝑡 = 𝜌𝜌𝑖𝑖,𝑡𝑡 𝑖𝑖𝑡𝑡−1 + �1 − 𝜌𝜌𝑖𝑖,𝑡𝑡 ��𝜌𝜌𝜋𝜋,𝑡𝑡 𝜋𝜋𝑡𝑡 + 𝜌𝜌𝑦𝑦,𝑡𝑡 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑡𝑡 � + 𝑒𝑒𝑡𝑡 .
The target value of the federal funds rate 𝑖𝑖𝑡𝑡 is determined by the value of inflation relative to its
long-run target value 𝜋𝜋𝑡𝑡 , the output gap 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑡𝑡 , and the lagged value of the federal funds rate
𝑖𝑖𝑡𝑡−1. Our notation makes explicit that the parameters 𝜌𝜌𝑡𝑡 = {𝜌𝜌𝑖𝑖,𝑡𝑡 , 𝜌𝜌𝜋𝜋,𝑡𝑡 , 𝜌𝜌𝑦𝑦,𝑡𝑡 } can in principle vary
over time to accommodate changes in the central bank’s systematic response to economic
outcomes.
2
Both models are linear and can be used to describe fluctuations of the economy around a baseline path. The sfrb
model is closely related to the much larger and nonlinear FRB/US model. Like the FRB/US model employed in the
MPS, the sfrb model features model-consistent expectation formation for inflation and asset prices under full
information, but not for other variables. In the EDO model, by contrast, all expectations are model-consistent under
full information.
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The component 𝑒𝑒𝑡𝑡 of the interest rate consists of a transitory and a persistent shock:
𝑒𝑒𝑡𝑡 = εt + vt
where εt is a white noise process, , and vt follows an AR(1) process
vt = 𝜌𝜌𝑣𝑣 vt−1 + 𝜂𝜂𝑡𝑡
where the innovations ηt are again a white noise process. We see the transitory shock as
representing a one-time discretionary adjustment to the federal funds rate, and the persistent
shock as representing a long-lasting time-varying intercept term in the policy rule. Shifts in the
public's beliefs about the intercept capture a variety of perceived persistent deviations from the
policy rule, including those associated with the equilibrium real federal funds rate or the inflation
target. 3
Private sector agents know the structure of the economy as captured by the underlying model
except for the details of the interest rate rule. They also observe all past and current realizations
of the endogenous variables, in particular the values of the federal funds rate, inflation, and the
output gap. While private sector agents know the general form of the interest rate rule, they do
not necessarily know the vector of parameters 𝜌𝜌𝑡𝑡 applied by the central bank. Neither do they
observe past or current values of the two shocks, εt and vt . Agents believe that shocks and
changes to the parameters 𝜌𝜌𝑡𝑡 are independent and normally distributed with variances
𝑉𝑉�ε𝑡𝑡 � = 𝜎𝜎 2 ,
2 2
𝑉𝑉�η𝑡𝑡 � = 𝜎𝜎�𝑣𝑣𝑣𝑣
𝜎𝜎 ,
𝑉𝑉(𝜌𝜌𝑡𝑡 − 𝜌𝜌𝑡𝑡−1 ) = Σ�𝑡𝑡 𝜎𝜎 2 .
The private sector agents use their observations of economic data and statistical inference to
inform their understanding of the central bank’s current and future monetary policy strategy,
summarized in the perceived policy rule. The perceived policy rule resembles in form the actual
policy rule with the parameters 𝜌𝜌𝑡𝑡 being replaced by their sample estimates 𝜌𝜌�𝑡𝑡 and similarly for
the transitory and persistent shocks. We denote the estimates of the two shocks by 𝜀𝜀̂𝑡𝑡 and 𝑣𝑣�𝑡𝑡 ,
respectively.
3
An increase in the value of the equilibrium real federal funds rate in the policy rule corresponds to an increase in
the intercept, while an increase in the inflation target corresponds to a decrease in the intercept.
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In a given period t, the public updates its beliefs regarding the properties of the rules as follows:
Let the private sector’s beliefs of the persistent shock and the rule parameters inherited
from the previous period be denoted by 𝑣𝑣�𝑡𝑡−1 and 𝜌𝜌�𝑡𝑡−1 .
Within the period, differences arise between the observed value of the federal funds rate
set by the policymaker under the actual interest rate rule and the value predicted by the
perceived rule. Agents ascribe this difference partly to the transitory and partly to the
persistent monetary policy shock. When computing the equilibrium prices and
allocations in period 𝑡𝑡, we assume that agents form expectations about the future under
anticipated utility (Kreps, 1998). 4
At the end of the period, agents update their beliefs about the parameters of the rule, the
persistent shock, and the transitory shock in light of the newly observed data. To do so,
they run a Bayesian regression. 5
The public’s revisions to its past estimates, 𝑣𝑣�𝑡𝑡 − 𝑣𝑣�𝑡𝑡−1 and 𝜌𝜌�𝑡𝑡 − 𝜌𝜌�𝑡𝑡−1 , depend on the
public’s beliefs about the volatility of changes in the persistent shock and the parameters
relative to the transitory shock. These beliefs, which are embodied in the parameters
2
𝜎𝜎�𝑣𝑣𝑣𝑣
and Σ�𝑡𝑡 , are an exogenous input to the learning process and can be time-varying. 6 If
the public believes that large changes of a parameter are relatively likely in a given
period, the Bayesian regression assigns a relatively large share of the discrepancy
between the actual and the predicted federal funds rate in that period to a change in this
specific parameter.
In addition, the magnitude of the revisions to past estimates are larger when the
discrepancy between the predicted and the actual value of the federal funds rate is larger.
A more detailed description of these steps can be found in a separate technical appendix to
this memo.
4
Anticipated utility refers to a widely used assumption in the learning literature that agents derive their decisions
and their expectations about future economic developments under the assumption that their current perception of the
policy rule parameters persists indefinitely. It is a simplifying assumption because at the same time, the public treats
the parameters in the policy rule as random variables when learning about them. See Cogley and Sargent (2008) on
interpreting anticipated utility as an approximation to Bayesian optimal learning.
5
Assuming that the private sector agents update the parameters in the policy rule only at the end of the period
simplifies the computational implementation, and is standard practice in the learning literature.
6
The matrix 𝑄𝑄𝑡𝑡 can also be understood as the collection of signal-to-noise ratios in the Bayesian regression problem.
Larger values of an element of 𝑄𝑄𝑡𝑡 indicate a higher signal-to-noise ratio for the corresponding parameter, and lead to
relatively larger revisions of its estimate.
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A question that arises naturally is whether the public’s beliefs about the rule will eventually
converge to the actual rule being followed by the central bank. For this to be the case, agents
have to entertain the possibility of changes in those rule parameters that the central bank actually
changes. But beyond that, it also has to be the case that the data agents observe are sufficient to
identify the parameters that agents are uncertain about. In practice, this implies that we have to
run stochastic simulations of the model, because deterministic simulations do not provide
sufficient variation in the observables to identify the parameters of the policy rule. 7
Application to the MPS
As noted above, we consider the economic effects of learning in a situation in which
policymakers switch from a rule with inertia to a rule without inertia, using the sfrb model and
the December 2017 Tealbook baseline. Up until 2017Q4, we assume that the central bank
follows the inertial version of the Taylor (1999) rule and that agents previously learned the
correct specification of the policy rule. In 2018Q1, the central bank switches from the inertial
version of the Taylor (1999) rule to the balanced-approach Taylor (1999) rule by lowering the
coefficient on the lagged value of the federal funds rate from 0.85 to zero. Expressed in terms of
the notation used earlier, it is 𝜌𝜌𝑖𝑖,𝑡𝑡 = 0.85 for 𝑡𝑡 ≤ 2017Q4 and 𝜌𝜌𝑖𝑖,𝑡𝑡 = 0 for 𝑡𝑡 ≥ 2018Q1, while
𝜌𝜌𝑥𝑥,𝑡𝑡 = 𝜌𝜌𝑥𝑥 = 1 and 𝜌𝜌𝜋𝜋,𝑡𝑡 = 𝜌𝜌𝜋𝜋 = 1.5 for all 𝑡𝑡. To simplify the analysis, we further assume that
agents know the true rule parameters with certainty up until 2017Q4, and also remain certain
about the parameters 𝜌𝜌𝜋𝜋 and 𝜌𝜌𝑥𝑥 afterwards. Beginning in 2018Q1, they become uncertain and
have to learn about the value of the inertia parameter 𝜌𝜌𝑖𝑖,𝑡𝑡 and the intercept term.
To work through the effects of learning in this particular experiment, we start with Figure 1,
which depicts how the private sector’s beliefs about the values of the smoothing coefficient 𝜌𝜌�𝑡𝑡𝑖𝑖
(top left panel) and the persistent shock 𝑣𝑣𝑡𝑡 (top right panel) evolve over time. At the beginning
of 2018Q1, agents still believe that monetary policy follows the inertial version of the Taylor
(1999) rule. However, by the end of that quarter, and after observing the realized values of the
federal funds rate, inflation, and the output gap, agents attribute some of the unanticipated hike in
7
Another condition for convergence is that the regression model that agents use has to produce unbiased estimates.
Unbiasedness holds for the experiments described in this memo, but it would need to be verified case by case in
different applications of our framework.
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the federal funds rate to a lower value of the smoothing coefficient 𝜌𝜌𝑖𝑖 . Initially, the updating
step is small as agents view large changes to the smoothing coefficient as relatively unlikely.
Given the uncertainty about the source of the abrupt federal funds rate increase, agents also raise
their perceived value of the persistent shock. This misperception turns out to be crucial for the
effects of learning because it translates into a persistently higher expected path of the federal
funds rate. The misperception about the future path of the federal funds rate is illustrated in the
bottom panel of Figure 1. The panel shows the paths of the federal funds rate as anticipated by
the public in a given quarter, as well as the realized path of the federal funds rate. In 2018Q1,
agents anticipate a path for the real federal funds rate that is considerably higher than the path
that is eventually realized. The reason for the divergence between the anticipated and realized
paths is precisely the misperception of the policy rule by the public, and in particular the public’s
estimate of the intercept in the rule.
Over time, agents adjust their beliefs about the smoothing parameter and the persistent shock
towards their true values—that is, the values governing the actual policy rule implemented by the
central bank. However, if the learning about the true value of the persistent shock is slow,
misperceptions about the anticipated path of the federal funds rate will persist.
Our mechanism influences economic outcomes through its effects on the anticipated path of the
federal funds rate. Figure 2 shows the evolution of the economy under three distinct assumptions
about monetary policy: (1) Monetary policy follows the inertial version of the Taylor (1999)
rule as in the Tealbook baseline (no switch, in blue). (2) Monetary policy switches to the
balanced-approach Taylor (1999) rule and agents know the rule followed by the central bank
(full information, in black). (3) Monetary policy switches to the balanced-approach Taylor
(1999) rule, but agents have to learn about the change in the rule (learning, in red). 8
The first two scenarios replicate the standard analysis shown in the MPS Section of the
Tealbook. Under the Tealbook baseline (the blue line) the federal funds rate slowly rises from
8
The lines shown in Figure 2 depict the average sample paths of the variables for stochastic simulations of the sfrb
model around the Tealbook baseline. Unexpected deviations from the forecasted path of the economy facilitate
private sector learning and guarantee that over time the agent’s beliefs held about the monetary policy rule converge
to the true rule.
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1 1/4 percent towards 4 percent over the next 4 years. The sizable projected overshooting of
output over its potential level occurs with inflation slowly returning to its long-run target of
2 percent. Whereas the balanced-approach Taylor (1999) rule calls for a significantly tighter
path of the federal funds rate—the prescribed value of the federal funds rate immediately jumps
above 3 percent—this monetary policy strategy induces a path for inflation that barely falls
below the Tealbook baseline. At the same time, the positive output gap is only somewhat below
its baseline path.
When the private sector agents do not understand the exact nature of the change in the monetary
policy strategy that caused the rise in the nominal federal funds rate, the switch from the inertial
version of the Taylor (1999) rule to the balanced-approach Taylor (1999) rule induces a larger
reduction in inflation and real activity than under full information. With agents attributing some
of the observed rise in the federal funds rate to a perceived increase in the persistent shock, the
upward shift in the anticipated path of the federal funds rate shown in the bottom of Figure 1
induces a larger run-up in the real 10-year Treasury yield that persists for several quarters longer
than under full information (shown in the bottom left panel of Figure 2). Inflation declines
sharply and further increases in the output gap are forestalled. Over time, as agents review their
current perception of the monetary policy strategy in the light of new data, they reduce their
estimates of the intercept term and the smoothing term towards their true values. As this process
occurs, the path for the 10 year yield converges to the path under full information and the paths
for the output gap and inflation converge toward their full information paths as well.
The decline in inflation under learning is large relative to the fall in the output gap, which might
seem surprising given that the Phillips curve in the sfrb (and FRB/US) model is relatively flat.
To understand this result, it is important to remember that inflation in sfrb under modelconsistent expectations is only to a small extent affected by the current output gap, and depends
mostly on expected future inflation, which in turn depends on the anticipated output gap path in
the future. 9 Figure 3 plots the anticipated paths of the real 10-year Treasury yield and the output
gap under learning. In the quarters after the policy switch, agents anticipate long term real yields
9
In the Risk and Uncertainty section of Tealbook, FRB/US is often simulated under VAR-based expectations for
which these considerations do not apply.
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to stay elevated, and therefore also expect the output gap to decline markedly for a prolonged
time. These anticipated paths result from misperception about the persistent monetary shock and
ultimately do not materialize. Nevertheless, they exert considerable downward pressure on the
inflation rate in the short term, much more than the realized path of the output gap path reveals.
Discussion
The public’s belief of the relative volatility of the parameters in the in policy rule and the policy
shocks over time is a key determinant of the degree of misperception and the speed of learning in
our model. The effects of a change in the monetary policy rule on the economy are stronger if
agents attribute a larger share of the initial discrepancy between the perceived value of the
federal funds rate and its actual value to the presence of a persistent shock. Figures 4 and 5
repeat our benchmark exercise when agents believe the (relative) volatility of the persistent
shock to be larger than assumed previously. In the top right panel of Figure 4, the estimated
persistent shock rises by three times as much as in Figure 1 and it remains at considerably higher
values throughout the horizon shown. The anticipated path of the real federal funds rate suggests
that agents expect even more tightening in the future and the real 10-year Treasury yield jumps
up to 3.5 percent. Although the central bank cuts the federal funds rate almost back to its
2017Q4 level under its newly adopted balanced-approach Taylor (1999) rule, inflation drops
down to 0.5 percent while the output gap narrows and the unemployment rate rises. As before,
the decline in inflation is mostly due to anticipated declines in the output gap that are much
larger than are ultimately realized.
Alternatively, if the public assigns zero probability to changes in the persistent shock, the change
in the actual policy rule induces outcomes that closely resemble those under full information (not
shown).
A key condition for our mechanism to matter quantitatively is that the change in the monetary
policy strategy induces a sizable change in the observed value of the federal funds rate. If the
inertial version of the Taylor (1999) rule and the balanced-approach Taylor (1999) rule
prescribed similar values for the current value of the federal funds rate (for example, suppose the
lagged value of the federal funds rate is close to the prescription of the balanced-approach Taylor
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(1999) rule), then a change in the monetary policy rule followed by the central bank would only
have a minor impact on the perceived intercept term and thus current and expected future longterm interest rates. Learning might still induce uncertainty about the true policy parameters, but
this uncertainty would leave expectations of future policy rates little changed in response to the
change in the policy strategy.
Our results are generally robust to the underlying economic model. In the EDO model, the
switch of monetary policy strategy underlying Figures 1 and 2 described earlier induces a large
drop in inflation under learning compared to the full information case. In comparison to the sfrb
model, the EDO model features larger narrowing of the output gap and increase in the
unemployment rate in response to the change in monetary policy strategy, as shown in Figure 6.
Extensions
Lowering the smoothing coefficient in the interest rate rule is only one possible way in which
monetary policy strategies can change over time. Our approach can in principle be applied to all
the simple rules currently shown in the MPS, including the price level targeting rule. Moreover,
we have analyzed cases of private sector misperception of monetary policy after changes in the
long-run inflation target (similar to Erceg and Levin, 2003) or in response a sequence of
unanticipated discretionary adjustments in the federal funds rate. To save space, we refrain from
including these experiments here.
One important caveat is that our current methodology only handles linear macroeconomic
models. This implies in particular that our analysis abstracts from the zero lower bound on
interest rates. Incorporating such nonlinearities would raise additional theoretical and numerical
challenges. If all policy rules under consideration prescribed keeping the nominal federal funds
rate at zero, agents would receive little or no signal from the policy rate about the change in
monetary policy strategy. Switching to a policy rule that keeps the federal funds rate low for
longer—is unlikely to provide as much easing of economic conditions under learning as under
full information. 10
10
See, for example, Bodenstein, Hebden, and Nunes (2012) or Gust, Herbst, and Lopez-Salido (2018) who develop
models of imperfect credibility at the effective lower bound.
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Similarly, we have not applied our framework to the case of monetary policy strategies derived
from optimal control settings. Learning in an optimal control environment introduces additional
theoretical challenges, but offers the possibility of having the policymakers learn about
unobserved changes to the structure of the economy.
References
Bodenstein, M., Hebden, J., and Nunes R., (2012). “Imperfect Credibility and the Zero Lower
Bound,” Journal of Monetary Economics, Elsevier, vol. 59, pages 135-149.
Cogley, T., Matthes, C., and Sbordone, A. M., (2015). “Optimized Taylor rules for disinflation
when agents are learning,” Journal of Monetary Economics, Elsevier, vol. 72(C), pages 131-147.
Cogley, T., and Sargent, T. J. (2008). “Anticipated Utility and Rational Expectations as
Approximations of Bayesian Decision Making,” International Economic Review, vol. 49(1),
pages 185-221, February.
Erceg, C. J. and Levin, A. T., (2003). “Imperfect credibility and inflation persistence,” Journal
of Monetary Economics, Elsevier, vol. 50(4), pages 915-944, May.
Gust, C., Herbst, E., and Lopez-Salido, D., (2018). “Forward Guidance with Bayesian Learning
and Estimation,” mimeo.
Kreps, D., (1998). “Anticipated Utility and Dynamic Choice,” 1997 Schwartz Lecture, in
Frontiers of Research in Economic Theory, Edited by D.P. Jacobs, E. Kalai, and M. Kamien,
Cambridge University Press, Cambridge, England.
Tetlow, R. J. and von zur Muehlen, P., (2001). “Robust monetary policy with misspecified
models. Does model uncertainty always call for attenuated policy?,” Journal of Economic
Dynamics and Control, Elsevier, vol. 25(6-7), pages 911-949.
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Figure 1: Policy expectations
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Figure 2: Outcomes
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Figure 3: Long-term yield and output gap expectations
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Figure 4: Policy expectations with increased misperception about the persistent shock
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Figure 5: Outcomes with increased misperception about the persistent shock
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Figure 6: EDO model simulations
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May 31, 2018
Martin Bodenstein
James Hebden
Fabian Winkler
Learning and Misperception:
Implications for Monetary Policy Strategies
– Technical Appendix –
1
Setup
We consider a standard linear rational expectations model with model variables yt and with a central
bank setting the nominal interest rate it according to an interest rate rule. The model equations are:
0 = F1 Et yt+1 + F0 yt + F−1 yt−1 + Fi it + Fu ut
(1)
it = φ (βt ) xt + vt + εt
(2)
vt = ρv vt−1 + ηt .
(3)
Equation 1 describes, through the matrices F , the set of all equilibrium conditions of the model except
for the interest rate rule followed by the central bank. This interest rate rule is described by equation (2).
It is linear in a subset xt of the endogenous model variables yt , and depends on potentially time-varying
parameters βt . It also contains a transitory policy shock εt and a persistent policy shock vt described in
equation 3 with innovation ηt . The other exogenous shocks in the economy are denoted ut . All variables
are expressed in deviations from a reference level, which is usually the steady state of the model.
Agents in the private sector only observe yt , ut and the nominal interest rate it , but they do not observe
the monetary policy shocks. They also do not know some or all of the time-varying coefficients βt and
instead need to form subjective beliefs about some or all of the parameters in the interest rate rule. The
operator Et stands for rational expectations given information available to agents in time t.
In the application of the memo, the policy rule is an inertial Taylor rule of the form
it = ρit it−1 + (1 − ρit ) (ρπ πt + ρy ygapt ) + vt + εt
where all variables are expressed in deviation from their long-run levels. We assume that all parameters are known to the agents except for the interest rate smoothing parameter ρi . Then βt = ρit , xt =
0
0
(it−1 , πt , ygapt ) , φ (βt ) = (βt , (1 − βt ) ρπ , (1 − βt ) ρy ) , and therefore
φ (βt ) xt = βt (it−1 − ρπ πt − ρy ygapt ) + ρπ πt + ρy ygapt .
.
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2
Belief Updating
At the end of each period, agents update their beliefs about the rule parameters φt and the policy shock vt ,
using their observations of the endogenous variables and the interest rate. They do this using a Bayesian
regression that can be cast as a Kalman filtering problem. The observation and state transition equations
of the filter are:
it = φ (βt ) xt + vt + εt
(4)
βt = βt−1 + βt
(5)
vt = ρv vt−1 + ηt .
(6)
We assume that agents believe that the transitory shock, innovations to the persistent shock, and changes
to the parameters at time t are normally distributed white noise with:
εt
1 0
2
ηt
∼ N 0, 0 σ̃vt
βt − βt−1
0 0
0
0 σ2 .
Σ̃t
(7)
We use the extended Kalman filter to describe the Bayesian filtering problem with the following updating
equations:
v̂t
β̂t
!
=
ρv
0
Pt =
ft =
ht =
!
ρv v̂t−1
β̂t−1
0
I
1 ht
+ Pt−1
1
h0t
!
!
Pt−1 − Pt−1
Pt−1
1
h0t
it − φ β̂t−1 xt − ρv v̂t−1 ft−1
1
h0t
ht
h0t ht
!
!
Pt−1 ft−1
ρv
0
0
I
(8)
!
+
2
σ̃vt
0
0
Σ̃t
!
(9)
!
(10)
+1
dφ
β̂t−1 xt .
dβ
(11)
In our application in the memo, ht = it−1 − φπ πt − φy ygapt .
2
The parameters σ̃vt
and Σ̃t act as signal-to-noise ratios in the Bayesian updating equations. A larger
2
value for σ̃vt indicates that forecast errors in the interest rate are more informative about v̂t , which will
therefore get updated by a larger amount for a given forecast error of the interest rate. The updating gains
also depend on the initial degree of parameter uncertainty as captured in the prior variance matrix P0 . In
2
the first exercise of the memo, we set σ̃vt
= 0.01, Σ̃t = 0.5 for all t ≥ 1, and P0 as the 2x2-diagonal matrix
with diagonal entries P0,11 = 0.1 and P0,22 = 0.01. These choices imply that agents believe that relative
to the value of the smoothing coefficient, the value of the intercept is more uncertain at the start of the
simulation but more stable over time. In the second exercise of the memo, the prior variance matrix is
changed to P0,11 = 0.5, implying that agents are even more uncertain about the value of the intercept,
leading to larger misperceptions of its value.
When ρv = 0, the persistent shock is in fact iid and not a latent state anymore. If additionally φ (βt ) yt =
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βt0 xt , then the updating equations simplify to:
0
β̂t = β̂t−1 + Pt−1 xt it − β̂t−1
xt ft−1
ft = x0t Pt−1 xt + 1
Pt = (Pt−1 − Pt−1 xt x0t Pt−1 ) + Σ̃t
which is the Bayesian equivalent of the discounted recursive least squares (DRLS) learning in Tetlow and
von zur Muehlen (2001). The variance matrix Σ̃t replaces the “forgetting factor” in the DRLS algorithm.
For | Σ̃t |> 0, the coefficients βt are preceived to be time-varying, and the estimator β̂t will overweigh the
information from recent forecast errors and asymptotically discard information from the distant past.
The prior variance matrix P0 and the variance Σ̃t determine the size of the Kalman gain, i.e. the speed of
belief updating. When Σ̃t = 0, then the coefficients are perceived to be unchanged between t − 1 and t.
Note that the regressors ht have to be uncorrelated with the innovations εt and ηt for the Kalman filter
to be a valid Bayesian estimator. Violation of this condition can lead to biased estimates. This condition
is not innocuous because monetary policy shocks can easily be correlated with the variables that enter
the policy rule. This is a well-known problem that arises when estimating reduced-form policy rules (cf.
Clarida, Gali and Gertler, 1999).
3
Computing the equilibrium
We have described how agents update their beliefs at the end of each period, after observing equilibrium
outcomes. We now describe how we compute these outcomes.
Each period, agents start with a belief about the rule coefficients φ̂t−1 = φ β̂t−1 and the past value of
the persistent policy shock v̂t−1 . They then form expectations about other variables using the anticipated
utility approximation (Kreps, 1998): Expectations about future variables in equation (1) are formed as if
the rule coefficients were known with certainty and fixed forever at φ̂t−1 . The equilibrium also depends on
agents’ beliefs about the time-t values of the two policy shocks. These will be a function of the discrepancy
between the observed interest rate and the prescription of the perceived policy rule. How much this
discrepancy gets ascribed to the persistent shock vt relative to the transitory shock εt depends on agents’
beliefs about the signal-to-noise ratio in their signal extraction problem, which we describe here in detail.
Formally, in period t we solve the linear model under the assumption that φt+s = φ̂t−1 ∀s ≥ 0. The
solution takes the form
yt = At yt−1 + Bt ut + cvt v̂ˆt + cεt ε̂ˆt .
(12)
where v̂ˆt and ε̂ˆt denote agents’ perceived values of the two monetary shocks before they update their
beliefs about φt . These perceived values are a function of agents’ forecast error on interest rates, which in
turn depends on the actual policy rate set by the central bank:
∆t = it − φ̂t−1 yt − ρv v̂t−1 = φ (βt ) − φ̂t−1 yt + vt + εt − ρv v̂t−1
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(13)
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The following updating equations describe how much of the forecast error gets ascribed to the persistent
shock vt relative to the transitory shock εt :
v̂ˆt = ρv v̂t−1 + kt ∆t , ε̂ˆt = (1 − kt ) ∆t , kt =
P11,t−1
.
P11,t−1 + 1
(14)
Here, the coefficient kt acts like a Kalman gain in the partial updating problem of agents beliefs about the
shocks, holding the parameter estimates φ̂t−1 constant.
To find the equilibrium, we have to impose that the observed interest rate equals the interest rate set
by the central bank according to equation (2). Substituting (14) into (12), and substituting the resulting
expression for yt into (13) yields:
φt − φ̂t (At yt−1 + Bt ut + cvt ρv v̂t−1 ) + vt − ρv v̂t−1 + εt
∆t =
.
1 + φ̂t − φt (kt cvt + (1 − kt ) cεt )
(15)
The solution for yt follows from equations (12) and (14).
References
Clarida, Richard, Jordi Gali, and Mark Gertler (1999). “The science of monetary policy: a new Keynesian
perspective,” Journal of Economic Literature, vol. 37(4), pp. 1661–1707.
Kreps, David M. (1998). “Anticipated utility and dynamic choice,” Econometric Society Monographs, vol.
29, pp. 242–274.
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