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October 13, 2015
Monetary Policy at the Lower Bound with Imperfect Information about 𝒓∗
David López-Salido, Christopher Gust, Benjamin K. Johannsen, and Robert Tetlow1
1. Introduction
Most central banks implement monetary policy by setting a target (or a range) for a nominal
overnight interbank interest rate. But the level of the short-term policy rate does not, by itself,
convey the stance of monetary policy, partly because spending decisions depend on real, rather
than nominal, interest rates and partly because the level of the real rate needs to be compared to a
benchmark level to determine if the stance of monetary policy is tight or loose. That benchmark,
or 𝑟 ∗ , is not observable and must therefore be estimated, and those estimates are subject to
uncertainty stemming from statistical errors as well as model misspecification errors. As we
discuss below, acknowledgement of this uncertainty is crucial for operationalizing the 𝑟 ∗ concept
in policy analysis, particularly when the policy rate is near its effective lower bound.
Many economists have argued that the recent financial crisis and Great Recession resulted in a
decline in 𝑟 ∗ , one that might be expected to persist for a long time. As discussed in the memo
titled “Real Interest Rates over the Long-Run,” shifts in low-frequency factors—such as long-run
productivity growth, trend population and labor force participation, and in the trend capital-labor
ratio—may affect longer-run 𝑟 ∗ , with implications for monetary policy. In addition, a myriad of
temporary but persistent “headwinds”—the effects of the financial crisis on credit conditions, the
slow recovery in the housing sector, restrictive fiscal policies, and global financial strains—may
themselves cause transitory movements in 𝑟 ∗ away from its longer-run value, as discussed in the
memo titled “Estimates of Short-Run 𝑟 ∗ from DSGE Models.” At the practical level, it may be
difficult to disentangle factors inducing temporary fluctuations in 𝑟 ∗ from those with longer-run
effects. It follows that policymakers may misperceive the nature and persistence of changes in
𝑟 ∗.
This memo focuses on two questions. First, given that measurements of 𝑟 ∗ are imperfect, how
should policymakers use those measures when conducting monetary policy? And second, what
1
In preparing this memo we have benefitted from the comments of Bill English, Thomas Laubach, Steve
Meyer, Marc Giannoni, and Marco del Negro. Kathryn Holston has provided outstanding research
assistance.
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are the implications of uncertainty about 𝑟 ∗ for using simple policy rules as guides for
conducting monetary policy? Our analysis uses a simple New Keynesian (NK) model.2 Our
choice of model means that in terms of the taxonomy laid out in the memo on “𝑟 ∗ : Concepts,
Measures and Uses,” we are equating 𝑟 ∗ with the level of the short-term natural real rate of
interest.
We begin by studying how optimal discretionary policy (ODP) responds to a change in the
natural rate of interest when policymakers have imperfect information about the magnitude of
the change. Under discretion, policymakers cannot credibly commit to carrying out a plan that
requires them to make future choices that would be suboptimal at future times. The discretion
concept limits policymakers’ ability to influence private-sector expectations regarding the federal
funds rate and other variables. Instead, the private sector knows that future Committees will
always re-optimize without regard for policymakers’ past promises. We pay particular attention
to the case where the effective lower bound (ELB) on nominal interest rates is initially binding,
and we characterize the optimal path for the policy rate around the time of departure from the
ELB. We next compare the macroeconomic outcomes under the ODP to outcomes when policy
is governed by simple rules (Taylor (1999) and first-difference rules). For some of these
comparisons, the coefficients of our simple rules are optimized, given our model economy, a step
that ensures that the comparison with the ODP is evenhanded. We also discuss the performance
of simple rules that have been modified to incorporate a time-varying intercept term as a proxy
for changes in the natural real rate, as well as the communications challenges associated with
committing to policy actions that follow a simple rule with a time-varying intercept term.
Broadly, the results highlight the interplay of the effective lower bound and incomplete
information about the natural rate in shaping optimal discretion as a basis for pursuing a riskmanagement approach to policymaking. Under the risk management approach, policymakers
make their policy rate decisions while considering the entire distribution of future outcomes,
including the fact that certain choices in the future might not be feasible, because of the ELB,
and thus choose policy settings today that reduce the cost of such outcomes. This approach to
decision making can be thought of as “taking out insurance” against adverse outcomes.
2
The main benefits of the NK model we employ are as follows: (1) it accounts for the nonlinearities
associated with the effective lower bound; (2) it formalizes the relationship between several concepts of
short-run 𝑟 ∗ ; (3) it can outline the monetary policy implications of imperfect knowledge about the natural
rate of interest; and (4) it can account for implications of the stochastic shocks borne by the economy in
the presence of uncertainty about economic conditions. The choice of model can be seen as a
compromise between tractability and relevance in generating qualitative outcomes that may help guide
important elements of the normalization of US monetary policy. Evans et al. (2015) use a similar model
to study the effects of uncertainty about the natural rate on optimal discretionary policy near the effective
lower bound. Our analysis is complementary, and extends their work by analyzing the information
acquisition problem facing the central bank and also considering simple policy rules. The details of the
model can be found in a companion technical appendix.
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We summarize our principal findings as follows:
1. If policymakers have imperfect information about 𝑟 ∗ , then there is an attenuation in ODP
responses to perceived shifts in 𝑟 ∗ . That is, when policymakers receive potentially
erroneous information about changes in 𝑟 ∗ , they optimally respond by less than would be
the case given full information.
2. When the economy is close to, or already at, the ELB, policymakers optimally respond to
incoming information about 𝑟 ∗ by even less than they would during normal times. The
reason for the relatively muted responses to signals about 𝑟 ∗ is to “take out insurance”
against situations in which their misperceptions about 𝑟 ∗ might cause policy to be “too
tight,” the ELB makes the policymaker unable to offset those misperceptions in the next
period, and discretionary policy makes the policymaker unable to offset those
misperceptions by committing to easier policy in future periods.
3. The performance of the ODP at the ELB can be reasonably approximated by a Taylortype rule with a constant intercept term if the feedback coefficient on inflation is set to a
high value. Similarly, the performance of the ODP at the ELB can be approximated by a
Taylor-type rule where the natural rate of interest is used as the intercept term and the
feedback coefficients on output and inflation are kept at benchmark values. The use of a
first-difference rule can also come close to performing as well as the ODP if the inflation
coefficient is set to a high value.
4. Including unobservable variables, like a time-varying 𝑟 ∗ , in simple rules introduces
communication challenges for policymakers as they need to justify their policy actions by
referring to information that cannot be verified by private agents.
2.
Optimal Discretionary Policy with Imperfect Information about 𝒓∗
As was discussed in the memo on concepts of 𝑟 ∗ , the NK literature has studied the monetary
policy implications of the fundamental factors affecting the economy by translating those factors
into a measure of the natural real rate and assessing how optimal monetary policy can be
identified with the natural real rate; see, e.g., Woodford (2003) and Galí (2008). For the simplest
linear NK model—a two-equation model with monopolistic competition, sticky prices, and
shocks from a narrow range of sources—the optimal, natural, and efficient real rates are
identical, in the absence of the ELB.3 More complicated models in this class, such as models
with sticky wages in addition to sticky prices and with shocks to the mark-up of prices over
See the Gust et al. memo on “𝑟 ∗ : Concepts, Measures and Uses” for a discussion of the definitions of
optimal, natural, and efficient real interest rates and the conditions under which they would all be the
same.
3
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costs, introduce trade-offs in monetary policy setting and therefore distinctions between these
different concepts of 𝑟 ∗ .4
The effective lower bound on short-term nominal interest rates introduces important
nonlinearities that sharpen the distinction between the optimal and the natural real rates.5 Near
the ELB any shock—supply or demand—creates a tradeoff between inflation and output for the
policymaker because of the restrictions on monetary policy actions implied by the ELB. Most of
the extant literature considers the case in which policymakers have complete information about
the nature of fluctuations in the natural real rate. But because the natural rate is unobservable,
we consider uncertainty about the level of the natural real rate. Formally, we assume that
policymakers observe only a noisy signal about the natural rate, so that in every period they have
to filter incoming information about fundamental factors and at the same time set policy
optimally. To simplify the analysis, we assume that policymakers learn the true value of 𝑟 ∗ with
a one-period lag. In all instances, policymakers are assumed to understand the structure of the
model economy, including their own uncertainty.
We study optimal monetary policy under discretion, by which we mean the optimal plan for the
short-term nominal interest rate that does not require commitment to future policy actions.6 We
could have alternatively studied optimal monetary policy under commitment, which allows
policymakers to achieve better outcomes for current inflation and output by influencing future
expectations. However, this implies strong assumptions, including that current policymakers can
impose constraints on future policymakers.
Uncertainty about the level of the natural real rate interacts with the ELB in ways that have
important policy implications. To demonstrate this fact, Figure 1 compares the ODP under
incomplete information with that under complete information. The policymaker optimally sets
the nominal policy rate (on the vertical axes) in response to a signal indicating that the natural
rate has increased (the horizontal axes show the magnitude of that signal, measured in percentage
points, relative to a baseline measure that varies). Where the two panels differ is in their
assumptions regarding the initial value for the natural rate. In the upper panel, the economy is in
“normal times,” meaning that the (initial) level of the policy rate is well away from the ELB. In
4
We will not explicitly refer to the neutral real rate because in our model the stance of monetary policy
can be defined in terms of the deviations of the actual real rate from the natural real rate.
5
See the original work by Eggertsson and Woodford (2003) and Levin et al. (2010) for further references.
6
In the model, optimal policy under commitment will outperform optimal discretion since the former
would imply that, through commitments on expected future short-term interest rates, the policy plan will
affect long-term interest rates that are the key channel through which monetary policy strategies affect
current inflation and output. Of course, the role of expectations implies that the benefits of commitment
strategies rely on credible communication that allows the private sector to understand such commitments.
In the absence of such understanding, it is likely that some of the benefits are weaker or could even be
overturned: For example, commitment strategies yield no benefits, and would even be costly, if the
commitments had no influence on long-term interest rates and aggregate demand; the benefits of such
strategies would also be smaller if expectations adjusted only very slowly to the announced strategy.
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the lower panel of the same figure (and several that follow) the policy rate is modestly
constrained by the ELB; that is, even small changes in 𝑟 ∗ will move the ODP away from the
ELB. In both panels, we compare the central bank’s optimal response to the signal of an upward
shift in 𝑟 ∗ when the policymaker has full information—so that the signal perfectly reveals the
value of 𝑟 ∗ —with the case of incomplete information, in which the signal is distorted by random
noise. Regardless of initial conditions, the policymaker operating under incomplete information
optimally attenuates the policy response under incomplete information (the red dot-dashed line),
relative to the complete information case (the blue dashed line).7,8
The bottom panel considers the case where the natural rate is low enough to convince
policymakers that the ELB is modestly binding (this scenario is labeled in the figure as the
economy is “at the cusp of the ELB”). For a positive 𝑟 ∗ signal of sufficient magnitude, the
policy rate now departs from the ELB, although it can take a perceived innovation to 𝑟 ∗ of some
magnitude to bring about that result when policymakers make decisions based on noisy signals.
The logic of this finding is straightforward: When the ELB is in play, in the sense that there is
non-trivial likelihood of its becoming a binding constraint and thereby impairing the operation of
conventional monetary policy, the optimal policy is to implement a strategy that limits the
likelihood of returning to the ELB at a subsequent post-departure date, thus reducing the
macroeconomic cost of a return, should it occur. Deferral of departure from the ELB is one
viable step toward achieving these objectives.
In Figure 2 we isolate the implications of the ELB by comparing experiments where information
is incomplete in all cases, but the ELB may or may not bind, depending on initial conditions.9
Whereas Figure 1 compared complete and incomplete information, Figure 2 maintains the
hypothesis that information is incomplete and examines how the effects of the ELB constraint
interact with the level of rates. The top panel shows results for normal times, meaning
conditions where the ELB is far enough away to be (probabilistically) ignored, and the output
gap and inflation are close to their target values. In this case, given the calibration of the model,
policymakers translate perceived or actual changes in 𝑟 ∗ into the same change in the policy
interest rate. The bottom panel repeats the exercise assuming that the inherited natural rate is
low enough that the effective lower bound is just binding, initially. When policymakers account
for the ELB (the red dot-dashed line) they react much less in response to positive signals about
the natural rate than would have been the case in the absence of the ELB (the blue dashed line).
7
Without making too much of the model calibration, under full information, the first-period response of
the policy rate to a positive signal in 𝑟 ∗ is one-for-one. The response under incomplete information has
been calibrated to be one-fourth of the full-information response. These magnitudes are sensitive to
model calibration, but the direction of the effect is not.
8
The reaction function displayed in the figure shows different responses to the signal that under perfect
information fully reveals the “true” shock to 𝑟 ∗ . Without perfect information, only policymaker
perceptions matter for determining the optimal policy, assuming that policymakers make use of all
available information efficiently.
9
It follows that the red dot-dashed line in Figure 2 is the same as the one in Figure 1.
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In particular, the policymaker keeps the nominal rate at the lower bound unless the signal about
the natural rate is large enough (about 0.7 percent in our exercise), whereas in the absence of the
ELB, the policy rate would rise even for a signal of zero.10 Of note is the fact that as the signal
grows larger, the reaction of the policymaker who optimally takes into account the ELB
approaches, nonlinearly, the unconstrained optimal policy. Taken together, Figures 1 and 2
show that incomplete information is sufficient to induce attenuation of policy, relative to the fullinformation case, and that the addition of concern about the ELB accentuates that tendency. In
effect, uncertainty about 𝑟 ∗ leads policymakers to keep the policy rate lower than would be the
case under complete information. The reason for the relatively muted responses to signals about
𝑟 ∗ is to “take out insurance” against situations in which their misperceptions about 𝑟 ∗ might
cause policy to be “too tight” and the ELB constrains the policymaker from offsetting those
misperceptions in the next period.11
Figure 3 shows the policy implications of increased uncertainty about the natural rate by adding
to Figure 2 an instance where uncertainty about 𝑟 ∗ is greater.12 The two panels contrast these
new results with those reported in Figure 2 with the optimal policy response in the more
uncertain economy (the solid black lines). As one might expect, the added uncertainty has only a
marginal effect on policy responses to signals about 𝑟 ∗ when the policy rate is far from the ELB,
as shown in the upper panel. But the effect gets large as the ELB gets closer, as in the lower
panel. The value of “insurance” rises with uncertainty about 𝑟 ∗ . Staying at the ELB for longer
when there is more uncertainty is akin to the risk-management approach described by Evans et
al. (2015).
A complementary way of demonstrating the effects of uncertain natural rates near the ELB is to
look at the distribution of economic outcomes. Figure 4 shows the distribution of the natural
rate, the nominal policy rate, the output gap, and the inflation rate during normal times (the black
solid lines) versus occasions when the ELB is just binding (the red dot-dashed lines).13 As
shown in the upper-left panel, in normal times the mean of this distribution is higher than when
the economy is in the vicinity of the ELB. Not surprisingly then, during normal times the
distribution of the nominal policy rate (the upper-right panel) is toward the right-hand side of the
panel and itself looks normal. However, when economic conditions are such that the policy rate
is initially at or near the ELB, there is a substantial probability that the nominal interest rate will
10
The reason for this result is that ignoring the ELB shifts the expected value of inflation and output up
because it removes the asymmetric downside risks associated with the ELB.
11
This observation is in accord with the conclusions of Evans et al. (2015).
12
More precisely, we increase the volatility of the underlying shocks (supply and demand) by 20 percent
while keeping the same signal-to-noise ratio as in the previous exercise. Because uncertainty matters for
optimal policy, this illustrates that such a policy is not certainty equivalent.
13
For this figure, the density for the natural rate is constructed according to its exogenous (stochastic)
distribution. The other densities are for outcomes at time 𝑡 given 𝑟 ∗ at time 𝑡 − 1.
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remain at the ELB, as shown by the red vertical bar at zero in the upper-right panel.14 The ELB
constraint, and the skewed distribution for the policy rate it induces, imparts skewness on the
output gap and inflation, as shown in the bottom panels. It is the adverse outcomes identified by
the long negative tails of the latter two distributions that lead to the risk management approach to
policy noted above; in the presence of incomplete information, this approach provides
precautionary stimulus to offset the downward bias associated with the effective lower bound.
3. Simple Policy Rules with Incomplete Information about 𝒓∗
3.1. Why Simple Rules?
The analysis to this point has focused on ODP, meaning a strategy in which the policymaker sets
the nominal rate optimally in response to incoming data on a period-by-period basis but does not
commit to future actions in order to achieve better economic outcomes today by influencing
private sector expectations of future monetary policy. Optimal policy under discretion in the
neighborhood of the ELB prescribes a complex, nonlinear reaction function for the policy rate
that may, in practice, be difficult to communicate and implement. An alternative approach might
be to commit to a policy rule in which the policy rate responds to the state of the economy in a
systematic manner. In this section, we turn to the potential benefits of committing to a rulebased approach in setting monetary policy within the same economic framework. That is, our
analysis in this section builds around the simple NK model employed in Section 2 and, as before,
takes into account the proximity of the ELB on nominal interest rates and the imperfect
knowledge that the central bank faces regarding 𝑟 ∗ .
A shift from ODP to policy based on a simple rule involves several considerations. A simple
rule cannot, in general, be optimal because the rule will adjust policy in response to only a subset
of the state variables that would be considered by the ODP. But a commitment to monetary
policy governed by a simple rule can, in principle, affect private sector expectations in a helpful
way that is amenable to policymakers’ achieving their objectives. Whether the benefits of
commitment outweigh the costs of suboptimal feedback is something that will vary from case to
case. Even so, as Taylor and Williams (2011) emphasized, because simple rules adjust policy in
response to a small list of variables that are central to entire classes of models, rather than
optimizing with respect to the idiosyncratic features of one particular model, such rules may
avoid some of the problems of model misspecification. Some simple rules restrict the variables
to which policy responds only to those that are directly observable, trading off the sub-optimality
of this restriction when measurement is reliable against the benefits of avoiding errors when
14
The thick vertical bar in the upper-right panel represents the probability of the short-term nominal
interest rate being at the effective lower bound, while the lines that plot the density are conditional on it
being above the lower bound.
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measurement turns out to be unreliable. In the context of this memo, this means that advocates
of rules must take a stand on how to incorporate imperfect information about 𝑟 ∗ .
In the next subsection, we build upon our results by first examining two versions of the
Taylor (1999) rule. The assumption regarding 𝑟 ∗ is the only difference between the two
specifications. In one polar case, the central bank assumes that the constant intercept of the rule
is the steady-state real rate; in the other polar case, the central bank acknowledges the short-term
fluctuations in the natural rate and adjusts the intercept term in the Taylor rule, but takes into
account its imperfect information about this unobserved variable by attenuating the adjustment as
suggested by the earlier discussion of optimal policy in the face of uncertainty about 𝑟 ∗ . Then in
Subsection 3.3, we consider a Taylor rule with optimized coefficients.
Because of uncertainty surrounding 𝑟 ∗ , some researchers and policymakers have advocated
alternative rules that do not depend on measurement of 𝑟 ∗ , such as the first-difference rule. In
light of this, we devote Subsection 3.4 to an examination of a version of the first-difference rule
studied by, among others, Orphanides and Williams (2002). In this case, to determine the
change in the policy rate the policymaker relies on a now-cast of current inflation and output
growth, about which the central bank has only imperfect information.
3.2. The Taylor rule
As we noted above, advocates of simple rules generally must decide how to incorporate
information about 𝑟 ∗ . Many rules, including Taylor (1993, 1999), simply assume that it is
constant.
To be general, we specify the Taylor (1999) rule in the following way:
𝑖𝑡 = max[0, 𝐸𝑡 {𝑟𝑡∗ + 𝛼(𝑟 ∗ − 𝑟𝑡∗ ) + 𝜋𝑡 + 0.5(𝜋𝑡 − 2) + (𝑦𝑡 − 𝑦𝑡∗ )}]
(1)
The notation 𝐸𝑡 indicates that expectations are conditional on the information available to the
policymaker at time 𝑡.15 Two comments are in order. First, nested within our specification is the
standard Taylor rule, which does not include a time-varying 𝑟𝑡∗ and instead includes an intercept
term equal to the long-run real rate (which was set in the original rule to 2 percent).16 This
corresponds to the case where 𝛼 = 1. Second, because the policymaker has incomplete
information, expectations must be formed about the current-dated variables in the rule:
inflation𝜋𝑡 , the output gap (𝑦𝑡 − 𝑦𝑡∗ ), and the natural rate.
At time 𝑡, the policymaker knows previous values of output, inflation and 𝑟 ∗ , but does not see inflation
and output in the current period, and only receives a noisy signal about 𝑟𝑡∗ .
16
Gust et al. memo on “𝑟 ∗ : Concepts, Measures and Uses” and the memo by Kei-Mu Yi and Jing Zhang,
titled “Real Interest Rates over the Long-Run” offer discussions both on the definition and the current
estimates of the long-run real rate.
15
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The left-hand column of Figure 5 compares the Taylor (1999) rule with 𝛼 = 1, the black solid
line, to the ODP when the economy is in the vicinity of the effective lower bound. As before, the
upper panel displays the policymaker’s reaction function to signals about 𝑟𝑡∗ . Even though 𝑟𝑡∗
does not appear directly in the specification of the rule—because the intercept, in this instance,
corresponds to the steady-state real rate and not the natural rate of interest—policymakers react
to signals about the underlying shocks affecting 𝑟𝑡∗ through associated changes in the expected
rate of inflation and the expected output gap. The prescribed response of the Taylor (1999) rule
is familiar to readers of the recent versions of the MPS section of Tealbook B: the rule counsels
notably higher nominal interest rates than the ODP even when the signal is low; indeed, the
policy rate departs from the ELB for any signal of 𝑟𝑡∗ shown. Accordingly, near the ELB, the
systematically high short-term nominal interest rate prescribed by the Taylor (1999) rule with a
constant intercept results in wide dispersions of the outcomes for the output gap and inflation
(shown in the lower two panels of the left-hand column) and average outcomes that are well
below target.17
It is arguably suboptimal for a policymaker to not react to changes in the natural real rate,
especially if changes in the natural rate can be seen as driving movements in the output gap and
inflation. To address that concern, the blue dashed lines in the same column of the figure show
the outcomes when 𝛼 = 0, meaning that the intercept of the rule is allowed to change, one-forone, with perceived movements in the natural rate. The upper panel shows that policymakers’
reactions to signals about 𝑟𝑡∗ are similar to the optimal discretionary policy (the red dot-dashed
line). And because policymakers respond directly to signals about 𝑟𝑡∗ , economic outcomes are
notably closer to the ODP results. As such, the distributions of outcomes for the output gap and
inflation resemble the distributions under discretion and are centered near their target levels
(shown in the lower panels).
As comforting as this result might appear, it bears noting that introducing a time-varying
intercept into the Taylor (1999) rule—particularly an intercept that entertains as much variability
as does the natural real rate—is subject to the understandable criticism that the natural rate is
unobservable and difficult to measure. It follows that even if policymakers commit to such a
rule, its behavior could appear discretionary to the public, a fact that would substantially
complicate communications and might impair public accountability.
17
One might argue that one should not take as given the particular level of the long-run real rate of 2
percent from Taylor (1999), particularly in the context of a simple model like the one we use here. We
have also tried the staff estimate for the long-run real rate of 1.5 percent; the results barely differed from
those shown in Figure 5. Interestingly, a lower and constant 𝑟 ∗ would also have the effects of making the
economy more prone to return to the ELB, which would result in undesirable inflation and output
outcomes.
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3.3 An Optimized Taylor rule
An alternative to the regular Taylor (1999) rule, with its essentially ad hoc specification of the
feedback coefficients on inflation and the output gap, is to optimize the rule coefficients,
conditional on the model, the form of the rule including the presence of the ELB, and the
shocks.18 Optimizing the coefficients in this way ensures that whatever performance is turned in
by the rule is not an artifact of an arbitrary calibration, which provides a solid foundation for
comparing the performance of the simple rule with the ODP. As it happens, in this model, the
optimized coefficient on inflation turns out to be quite large, while the coefficient on the output
gap is little different from that in the regular Taylor (1999) rule. The green solid line in the righthand column of Figure 5 shows the results for the optimized Taylor rule. These results can be
compared with other outcomes, including the ODP, the red dot-dashed line. As can be seen, the
optimized Taylor rule advises that the policy rate stay at the ELB for a wider range of signals
about 𝑟 ∗ than does the ODP, but with a steeper response when departure does occur. The
performance rendered by the optimized Taylor rule, while inferior to the ODP, is much better
than that of the regular Taylor rule with a constant intercept (the black solid line in the left-hand
column). More generally, while not as efficacious as the ODP, the optimized Taylor rule
performs fairly well; its performance is remarkably similar to that of the regular Taylor rule with
a time-varying intercept (the blue dashed lines in the left-hand column). Evidently, optimization
of Taylor rule coefficients and allowing for a time-varying intercept term are close substitutes in
their ability to provide accommodation under proper circumstances. Given that both policies
respond to the same set of model shocks, just in somewhat different ways, the similarity in their
performances is perhaps not so surprising.
3.4 A first-difference rule
Still another alternative to the simple Taylor (1999) rule that does not depend on 𝑟𝑡∗ , equation
(1), is a first-difference (FD) rule, our version of which is specified with changes in the policy
rate on the left-hand side, and current output growth on the right-hand side (as well as inflation):
𝑖𝑡 = max[0, 𝑖𝑡−1 + 𝐸𝑡 {𝛾𝜋 (𝜋𝑡 − 2) + 𝛾𝑦 (𝑦𝑡 − 𝑦𝑡−1 )}]
18
(2)
More precisely, the optimized coefficients are those that minimize a (discounted) quadratic loss
function written in terms of the output gap and the deviation of inflation from target, given the form of the
rule and the shocks. Because of the ELB, the density forecast for the model needs to be computed to
carry out the maximization; this optimization is doable for a small model such as the one we use here, but
quickly becomes infeasible as the size of the model grows. That this optimization is for unconditional
welfare means that the specifics of initial conditions, such as whether or not the economy is already at, or
near, the ELB, is not taken into account in carrying out the optimization, which in turn means that the rule
is not generic to those specific initial conditions.
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where the feedback parameters, 𝛾𝜋 and 𝛾𝑦 , are optimized.19 By specifying the policy rate in first
differences, a rule like equation (2) eschews the dependence of policy on the natural rate of
interest, which Orphanides and Williams (2002) have emphasized can, in some circumstances,
outperform a simple rule like the Taylor rule that depends on unobservable levels of the natural
rate. This advantage notwithstanding, it is still the case that the policymaker does not have
complete information about current inflation and the current output gap, and thus must rely on
potentially inaccurate now-casts of inflation and output. And because this specification carries
forward any changes in the nominal interest rate, any misperceptions about inflation or output
growth will persist for a long time.
The magenta dotted lines in the right-hand column of Figure 5 show outcomes for policy
governed by the optimized FD rule, which, like the optimized Taylor rule, carries a fairly
substantial feedback coefficient on inflation. Here, as before, the simulations are conducted for
initial values of 𝑟 ∗ where the ELB is initially (just) binding. The upper panel shows that the
policy rate responds to noisy signals about the natural rate of interest in a fashion quite similar to
the ODP. In essence, the shocks that would be encompassed by fluctuations in 𝑟 ∗ , which are
responded to under the ODP, are instead captured by changes in current inflation and output
growth that appear on the right-hand side of equation (2). The similarity in (first-period)
responses of the FD rule and ODP notwithstanding, there are noteworthy differences in
performance across the two policies, shown in the lower panels in this column. There is more
dispersion in economic outcomes under the FD rule—and, in particular, a higher incidence of
positive output gaps and inflation above target—than under the ODP. This is because while the
ODP tailors the policy response to the specific initial conditions of the day—the fact that the
ELB is just binding in this instance—the FD rule is optimized for economic conditions and
shocks, on average. Thus, in a greater set of circumstances under the FD rule than under ODP,
policymakers will find themselves having not tightened enough, or as quickly, as they would
have preferred had they benefitted from complete information. It is because the initial
misperceptions are propagated for as long as they are that the FD rule turns in a somewhat less
attractive performance than the ODP.
4. Conclusions
We began this memo with the premise that the use of some benchmark real rate might be useful
for the conduct of monetary policy, at least under some conditions. Our point of departure was
to consider carefully the implications of the uncertainty in the measurement of such a benchmark
real rate for policy, and to do so in a concrete model-based fashion. To this end, we employed a
19
The policymaker chooses the values of the parameters to minimize the discounted sum of weighted
squared deviations of inflation relative to a 2 percent inflation objective and output relative to potential
output.
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simple New Keynesian model, and narrowed down the concept of 𝑟 ∗ under study to be
equivalent to the natural rate of interest, to study the uncertainty question both as it relates to
optimal discretionary policy and to policy as governed by (modified) simple rules.
We documented that the unobservable nature of the natural rate of interest introduces
inescapable uncertainties, and that these uncertainties become particularly relevant near the
effective lower bound. Further, this uncertainty justifies attenuation in the policy response
relative to the case when policymakers have full information. We noted that this attenuation,
which can be thought of as a manifestation of a policy of risk management that “takes out
insurance” against possible adverse outcomes, is heightened by the proximity of the effective
lower bound. Finally, we examined how policy might adapt to 𝑟 ∗ uncertainty, finding that, in
our framework, a policy that increased the response coefficient on inflation in the context of
simple rules, or one that allowed for time variation in the intercept term of a Taylor-type rule,
could come reasonably close to replicating the performance of optimal discretionary policy.
Such findings always come with caveats, of course. In the current instance, besides the usual
disclaimers that come with any model-based analysis, there is the issue of the implementability
of each of the policies we studied. The elegance of simple rules is usually said to be their
simplicity, and associated with that, their time invariance and robustness. Using a simple timeinvariant rule—and sticking, more-or-less, to it—gives private agents the opportunity to learn
how the rules works and formulate their expectations with that sort of systematic behavior in
mind. Discretionary policy, even though it is optimal on a period-by-period basis, is a timevarying policy that does not share this feature. Taylor rules with time-varying intercept terms
are, in many ways, an attempt to slip in through the back door something close to the optimality
of the optimal discretionary policy without giving up on the simplicity and commitment of a
simple rule. Even so, this approach substitutes the relative clarity of simple rules for the opaque
process of estimating the natural real rate, an unobservable variable that may not be verifiable by
the private sector. Whether a simple rule with a time-varying intercept term is any easier to
communicate and adhere to than the optimal discretionary policy remains an open question.
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References
Gust, C., Johannsen, B. López-Salido, and R. Tetlow (2015), “𝑟 ∗ : Concepts, Measures, and
Uses,” memorandum to the FOMC, October 13, 2015.
Eggertsson, Gauti B., and Michael Woodford (2003), “The Zero Interest-Rate Bound and
Optimal Monetary Policy,” Brookings Papers on Economic Activity, 1, 139-211.
Evans, Charles, Fisher, J. Gourio F. and S. Krane (2015), “Risk Management for Monetary
Policy Near the Zero Lower Bound,” Brookings Papers on Economic Activity, March 19, 2015.
Galí, J. (2008), Monetary Policy, Inflation, and the Business Cycle: An Introduction to the New
Keynesian Framework and Its Applications, Princeton University Press.
Levin, A., David Lopez-Salido, E. Nelson, and T. Yun (2010), “Limitations on the Effectiveness
of Forward Guidance at the Zero Lower Bound,” International Journal of Central Banking, 6
(1), 143-189.
Orphanides, Athanasios and John C. Williams (2002), “Robust Monetary Policy Rules with
Unknown Natural Rates,” Brookings Papers on Economic Activity, 2:2002, 63-118.
Taylor, John B. (1993), “Discretion versus Policy Rules in Practice,” Carnegie Rochester
Conference Series on Public Policy, 39, 195-214.
Taylor, John B. (1999), Monetary Policy Rules, (editor), University of Chicago Press.
Taylor, J. and John C. Williams (2011), “Simple and Robust Rules for Monetary Policy,” in
Benjamin Friedman and Michael Woodford(ed.), Handbook of Monetary Economics, Volume
3B, North-Holland, 2011, 829-860.
Woodford, Michael (2003), Interest and Prices. Foundations of a Theory of Monetary Policy,
Princeton University Press.
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