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Federal Reserve Bank of Chicago

Robustness and Macroeconomic Policy
Gadi Barlevy

WP 2010-04

Robustness and Macroeconomic Policy∗
Gadi Barlevy
Economic Research
Federal Reserve Bank of Chicago
230 South LaSalle
Chicago, IL 60604
e-mail: gbarlevy@frbchi.org
June 29, 2010

Abstract
This paper considers the design of macroeconomic policies in the face of uncertainty. In recent years,
several economists have advocated that when policymakers are uncertain about the environment they
face and ﬁnd it diﬃcult to assign precise probabilities to the alternative scenarios that may characterize
this environment, they should design policies to be robust in the sense that they minimize the worstcase loss these policies could ever impose. I review and evaluate the objections cited by critics of this
approach. I further argue that, contrary to what some have inferred, concern about worst-case scenarios
does not always lead to policies that respond more aggressively to incoming news than the optimal
policy would respond absent any uncertainty.
Key Words: Robust Control, Uncertainty, Ambiguity, Attenuation Principle

∗ Posted with permission from the Annual Review of Economics, Volume 3 (c) 2011 by Annual Reviews,
http://www.annualreviews.org. I would like to thank Charles Evans for encouraging me to work in this area and Marco
Bassetto, Lars Hansen, Spencer Krane, Charles Manski, Kiminori Matsuyama, Thomas Sargent, and Noah Williams for helpful discussions on these issues.

Introduction

A recurring theme in the literature on macroeconomic policy concerns the role of uncertainty. In practice,
policymakers are often called to make decisions with only limited knowledge of the environment in which
they operate, in part because they lack all of the relevant data when they are called to act and in part
because they cannot be sure that the models they use to guide their policies are good descriptions of the
environment they face. These limitations raise a question that economists have long grappled with: How
should policymakers proceed when they are uncertain about the environment they operate in?
One seminal contribution in this area comes from Brainard (1967), who considered the case where a
policymaker does not know some of the parameters that are relevant for choosing an appropriate policy
but still knows the probability distribution according to which they are determined. Since the policymaker
knows how the relevant parameters are distributed, he can compute the expected losses that result from
diﬀerent policies. Brainard solved for the policy that yields the lowest expected loss, and showed that when
the policymaker is uncertain about the eﬀect of his actions, the appropriate response to uncertainty is to
moderate or attenuate the extent to which he should react to the news he does receive. This result has
garnered considerable attention from both researchers and policymakers.1
However, over time economists have grown increasingly critical of the approach inherent in Brainard’s
model to dealing with uncertainty. In particular, they have criticized the notion that policymakers know the
distribution of variables about which they are uncertain. For example, if policymakers do not understand
the environment they face well enough to quite know how to model it, it seems implausible that they can
assign exact probabilities as to what is the right model. Critics who have pushed this line of argument have
instead suggested modelling policymakers as entertaining a class of models that can potentially describe
the data, which is usually constructed as a set of perturbations around some benchmark model, and then
letting policymakers choose policies that are “robust” in the sense that they perform well against all models
in this class. That is, a robust policy is one whose worst performance across all models that policymakers
contemplate exceeds the worst performance of all other policies.
The recommendation that macroeconomic policies be designed to be robust to a class of models remains
controversial. In this paper, I review the debate over the appropriateness of robustness as a criterion for
choosing policy, relying on simple illustrative examples that avoid some of the complicated technicalities
that abound in this literature. I also address a question that preoccupied early work on robustness and
macroeconomic policy, namely, whether robust policies contradict the attenuation result in the original
Brainard model. Although early applications of robustness tended to ﬁnd that policymakers should respond
1 See,

for example, the discussion of Brainard’s result in Blinder (1998), pp. 9-13.

more aggressively to the news they do receive than they would otherwise, I argue that aggressiveness is not
an inherent feature of robustness but is speciﬁc to the models these papers explored. In fact, robustness can
in some environments lead to the same attenuation that Brainard obtained, and for quite similar reasons.
This oﬀers a useful reminder: Results concerning robustness that arise in particular environment will not
always themselves be robust to changes in the underlying environment.
Given space limitations, I restrict my attention to these issues and ignore other applications of robustness
in macroeconomics. For example, there is an extensive literature that studies what happens when private
agents — as opposed to policymakers — choose strategies designed to be robust against a host of models,
and examines whether such behavior can help resolve macroeconomic problems such as the equity premium
puzzle and other puzzles that revolve around attitude toward risk. Readers interested in such questions
should consult Hansen and Sargent (2008) and the references therein. Another line of research considers what
policymakers should do when they know private agents employ robust decisionmaking, but policymakers
are themselves conﬁdent about the underlying model of the economy. Examples of such analyses include
Caballero and Krishnamurthy (2008) and Karantounias, Hansen, and Sargent (2009). I also abstract from
work on robust policy in non-macroeconomic applications, for example treatment rules for a heterogeneous
population where the eﬀect of treatments is uncertain. The latter is surveyed in Manski (2010).
This article draws heavily on Barlevy (2009) and is similarly organized. As in that paper, I ﬁrst review
Brainard’s (1967) original result. I then introduce the notion of robustness and discuss some of the critiques
that have been raised against this approach. Next, I apply the robust control approach to a variant of
Brainard’s model and show that it too recommends that the policymaker attenuate his response to incoming
news, just as in the original Brainard model. Finally, using simple models that contain some of the features
from the early work on robust monetary policy, I oﬀer some intuition for why concern for robustness can
sometimes lead to aggressive policy.

The Brainard Model

Brainard (1967) studied the problem of a policymaker who wants to target some variable so that it will
equal some prespeciﬁed level. For example, consider a monetary authority that wants to maintain inﬂation
at some target rate or to steer short-run output growth toward its natural rate. These objectives may force
the authority to intervene to oﬀset shocks that would otherwise cause these variables to deviate from the
desired targets. Brainard was concerned with how this intervention should be conducted when the monetary
authority is uncertain about the economic environment it faces but can assign probabilities to all possible
scenarios it could encounter. Note that this setup assumes the policymaker cares about meeting only a
single target. By contrast, most papers on the design of macroeconomic policy assume policymakers try
2

to meet multiple targets, potentially giving rise to tradeoﬀs between meeting conﬂicting targets that this
setup necessarily ignores. I will discuss an example with multiple tradeoﬀs further below, but for the most
part I focus on meeting a single target.
Formally, denote the variable the policymaker wants to target by . Without loss of generality, we can
assume the policymaker wants to target  to equal zero, and can aﬀect it using a policy variable he can set
and which I denote by . In addition,  is determined by some variable  that the policymaker can observe
prior to setting . For example,  could reﬂect inﬂation,  could reﬂect the short-term nominal rate, and
 could reﬂect shocks to productivity or the velocity of money. For simplicity, let  depend on these two
variables linearly; that is,
 =  − 

(1)

where  measures the eﬀect of changes in  on  and is assumed to be positive. Absent uncertainty, targeting
 in the face of shocks to  is simple; the policymaker would simply have to set  to equal  to restore 
to its target level of 0.
To incorporate uncertainty into the policymaker’s problem, suppose  is also aﬀected by random variables
whose values the policymaker does not know, but whose distributions are known to him in advance. Thus,
let us replace equation (1) with
 =  − ( +  ) +  

(2)

where  and  are independent random variables with means 0 and variances and  2 and  2 , respectively.


This formulation lets the policymaker be uncertain both about the eﬀect of his policy, as captured by the
term  that multiplies his choice of , and about factors that inﬂuence , as captured by the additive term
 . The optimal policy depends on how much loss the policymaker incurs from missing his target. Brainard
assumed the loss is quadratic in the deviation between the actual value of  and its target — that is, the loss
is equal to  2 — and that the policymaker chooses  so as to minimize his expected loss, that is, to solve
h
i
£ ¤
2
min   2 = min  ( − ( +  )  +  ) 




(3)

Solving this problem is straightforward, and yields the solution
=



 +  2 


(4)

Uncertainty about the eﬀect of the policy instrument  will thus lead the policymaker to attenuate his
response to  relative to the case where he knows the eﬀect of  on  with certainty. In particular, when
 2 = 0, the policymaker will set  to undo the eﬀect of  by setting  = . But when  2  0, the policy


will not fully oﬀset . This is Brainard’s celebrated result: A policymaker who is unsure about how the
policy instrument he controls inﬂuences the variable he wishes to target should react less to news about

missing the target than he would if he were fully informed. By contrast, uncertainty about  has no
implications for policy, as evident from the fact that the optimal rule for  in (4) is independent of  2 .

To understand this result, note that the expected loss in (3) is essentially the variance of . Hence, a
policy that leads  to be more volatile will be considered undesirable. From (2), the variance of  is equal
to 2  2 +  2 , which is increasing in the absolute value of . An aggressive policy that uses  to oﬀset


nonzero values of  thus implies a more volatile outcome for , while an attenuated policy that sets  closer
to 0 implies a less volatile outcome for . This asymmetry introduces a bias toward less aggressive policies.
Even though a less aggressive response to  would cause the policymaker to miss the target on average, he
is willing to do so in order to make  less volatile. Absent this asymmetry, there would be no reason to
attenuate policy. Indeed, this is precisely why uncertainty in  has no eﬀect on policy, since this source of
uncertainty does not generate any asymmetry between aggressive and attenuated policy.
The fact that uncertainty about  does not lead the policymaker to attenuate his response to news
serves as a useful reminder that the attenuation principle is not a general result, but depends on what the
underlying uncertainty is about. This point has been reiterated in subsequent work such as Chow (1973),
Craine (1979), Soderstrom (2002), who show that uncertainty can sometimes imply that policymakers
should respond to information about missing a target more aggressively than they would in a world of
perfect certainty. The same caveat about drawing general conclusions from speciﬁc examples will turn out
to apply equally to alternative approaches of dealing with uncertainty, such as robustness.

Robustness

As noted above, one often cited critique of the approach underlying Brainard’s (1967) analysis is that it
may be unreasonable to expect policymakers to know the distribution of the variables about which they are
uncertain. For example, there may not be enough historical data to infer the likelihood of certain scenarios,
especially those that have yet to be observed but remain theoretically possible. Likewise, if policymakers do
not quite understand the environment they face, they will have a hard time assigning precise probabilities
as to which economic model accurately captures the key features of this environment. Without knowing
these probabilities, it will be impossible to compute an expected loss for diﬀerent policy choices and thus
to choose the policy that generates the smallest expected loss.
These concerns have led some economists to propose an alternative criterion for designing policy that
does not require assigning probabilities to diﬀerent models or scenarios. This alternative argues for picking
the policy that minimizes the damage a policy could possibly inﬂict under any scenario the policymaker
is willing to contemplate. That is, policymakers should choose the policy under which the largest possible
4

loss across all potential scenarios is smaller than the maximal loss under any alternative policy. Such a
policy ensures the policymaker will never have to incur a bigger loss than the bare minimum that cannot be
avoided. This rule is often associated with Wald (1950, p. 18), who argued that this approach, known as the
minimax or maximin rule, is “a reasonable solution of the decision problem when an a priori distribution
in [the state space] Ω does not exist or is unknown.” In macroeconomic applications, this principle has been
applied to dynamic decision problems using techniques borrowed from the engineering literature on optimal
control. Hence, applications of the minimax rule in macroeconomics are often referred to as robust control,
and the policy that minimizes the worst possible loss is referred to as a robust policy.2
By way of introduction to the notion of robustness, it will help to begin with an example outside of
economics where the minimax rule appears to have some intuitive appeal and which avoids some of the
complications that arise in economic applications. The example is known as the “lost in a forest” problem,
which was ﬁrst posed by Bellman (1956) and which evolved into a larger literature that is surveyed in Finch
and Wetzel (2004).3 The lost in a forest problem can be described as follows. A hiker treks into a dense
forest. He starts his trip from a road that cuts through the forest, and he travels one mile into the forest
along a straight path that is perpendicular to the road. He then lies down to take a nap, but when he wakes
up he realizes he forgot which direction he originally came from. He wishes to return back to the road — not
necessarily the point where he started, but anywhere on the road where he can ﬂag down a car and head
back to town. Moreover, he would like to reach the road using the shortest possible route. If he knew which
direction he came by originally, this task would be easy: He could reach the road in exactly one mile by
simply retracing his original route in reverse. But the assumption is that he cannot recall from whence he
came, and because the forest is dense with trees, he cannot see the road from afar. Thus, he must physically
reach the road to realize he found it. The problem is more precisely described geometrically, as illustrated
in panel (a) of ﬁgure 1. First, draw a circle of radius one mile around the hiker’s initial location. The road
the hiker is searching for is assumed to be an inﬁnite straight line that lies tangent to this circle. The point
of tangency lies somewhere along the circle, and the objective is to reach any point on the tangent line — not
necessarily the point at which the line is tangent to the circle but any point on the line — using the shortest
possible path starting at the center of the circle. Panel (a) in ﬁgure 1 illustrates three of the continuum of
possible locations where the road might lie.
What strategy should the hiker follow in searching for the road? Solving this problem requires a criterion
2 The term “robust control” is used by some interchangeably with the term “optimal decisionmaking under ambiguity.”
However, the latter terminology is more closely associated with the literature following Gilboa and Schmeidler (1989) that asks
whether it is logically consistent for individuals to be averse to ambiguously posed choice problems in which the probability of
various events remain unspeciﬁed. By contrast, robust control is motivated by the normative question of what a decisionmaker
ought to do when faced with such ambiguity. I comment on the connection between these two literatures further below.
3 The problem is sometimes referred to as the “lost at sea” problem. Richard Bellman, who originally posed the problem,
is well known among economists for his work on dynamic programming for analyzing sequential decision problems.

to determine what constitutes the “best” possible strategy. In principle, if the hiker knew his propensity to
wake up in a particular orientation relative to the direction he travelled from, he could assign a probability
that his starting point lies in any given direction. In that case, he could pick the strategy that minimizes
the expected distance he would need to reach the main road. But what if the hiker has no good notion
of his propensity to lie down in a particular orientation or the odds that he didn’t turn in his sleep? One
possibility, to which I shall return below, is to assume all locations are equally likely and choose the path that
minimizes the expected distance to reach any point along the road. While this restriction seems natural
in that it does not favor any one direction over another, it still amounts to imposing beliefs about the
likelihood of events the hiker cannot fathom. After all, it is not clear why assuming that the road is equally
likely to lie in any direction should be a good model of the physical problem that describes how likely the
hiker is to rotate a given amount in his sleep.
Bellman (1956) proposed an alternative criterion for choosing the strategy that does not require assigning
any distribution to the location of the road. He suggested choosing the strategy that minimizes the amount
of walking required to ensure reaching the road regardless of where it is located. That is, for any strategy, we
can compute the longest distance one would have to walk to make sure he reaches the main road regardless
of which direction the road lies. It will certainly be possible to search for the road in a way that ensures
reaching the road after a ﬁnite amount of hiking: For example, the hiker could walk one mile in any
particular direction, and then, if he didn’t reach the road, turn to walk along the circle of radius one mile
around his original location. This strategy ensures he will eventually ﬁnd the road. Under this strategy,
the most the hiker would ever have to walk to ﬁnd the road is 1 + 2 ≈ 728 miles. This is the amount he

would have to walk if the road was a small distance from where he ended up after walking one mile out from
his original location, but the hiker unfortunately chose to turn in the opposite direction when he started
to traverse the circle. Bellman suggested choosing the strategy for which the maximal distance required to
guarantee reaching the road is shortest. The appeal of this rule is that it requires no more walking than
is absolutely necessary to reach the road. While other criteria have been proposed for the lost in a forest
problem, many have found the criterion of walking no more than is absolutely necessary to be intuitively
appealing. But this is precisely the robust control approach. The worst-case scenario for any search strategy
involves exhaustively searching through every wrong location before reaching the true location. Bellman’s
suggestion thus amounts to using the strategy whose worst-case scenario requires less walking than the
worst-case scenario of any other strategy. In other words, the “best” strategy is the one that minimizes the
amount of distance the hiker would need to run through the gamut of all possible locations for the road.
Although Bellman (1956) ﬁrst proposed this rule as a solution to the lost in a forest problem, it was Isbell
(1957) who derived the strategy that meets this criterion. Readers who are interested in the derivation of
the optimal solution are referred to Isbell’s original piece. It turns out that it is possible to search for the

road in a way that requires walking no more than approximately 640 miles, along the path illustrated by
the heavy line in panel (b) of ﬁgure 1. The idea is that by deviating from the circle and possibly reaching the
road at a diﬀerent spot from where the hiker originally left, one can improve upon the strategy of tracking
the circle until reaching the road just at the point in which it lies tangent to the circle of radius one mile
around the hiker’s starting point.
An important feature of the lost in a forest problem is that the set of possibilities the hiker must consider
in order to compute the worst-case scenario for each policy is an objective feature of the environment: The
main road must lie somewhere along a circle of radius one mile around where the hiker fell asleep, we just
don’t know where. By contrast, in most macroeconomic applications, there is no objective set of models
an agent should use in computing the worst-case scenario. Rather, the set of models agents can consider
is typically derived by ﬁrst positing some benchmark model and then considering all the models that are
“close” in the sense that they imply a distribution for the endogenous variables of the model that is not
too diﬀerent from the distribution of these same variables under the benchmark model — often taken to be
the entropy distance between the two distributions. Alternatively, agents can contemplate any conceivable
model, but relying on a model that is further away from the benchmark model incurs a penalty in proportion
to the entropy distance between that model and the benchmark one.4 Thus, in contrast with the lost in
a forest problem, economic applications often involve some arbitrariness in that the set of scenarios over
which the worst-case scenario is calculated involves terms that are not tied down by theory — a benchmark
model and a maximal distance or a penalty parameter that governs how far the models agents contemplate
can be from this benchmark. One way of making these choices less arbitrary is to choose the maximal
distance or penalty parameter in such a way that policymakers will not consider models that given the
relevant historical time-series data can be easily rejected statistically in favor of the benchmark model.
The robust strategy is viewed by many mathematicians as a satisfactory solution for the lost in a forest
problem. However, just as economists have critiqued the use of robustness in economics, one can ﬁnd fault
with this search strategy. The remainder of this section reviews the debate over whether robustness is an
appropriate criterion for coping with uncertainty, using the lost in a forest problem for illustration.
One critique of robustness holds that the robust policy is narrowly tailored to do well in particular
scenarios rather than in most scenarios. This critique is sometimes described as “perfection being the
enemy of the good”: The robust strategy is chosen because it does well in the worst-case scenario, even if
that scenario is unlikely and even if the same strategy performs much worse than alternative strategies in
most, if not all, remaining scenarios. Svennson (2007) conveys this critique as follows: “If a Bayesian prior
probability measure were to be assigned to the feasible set of models, one might ﬁnd that the probability
4 Hansen

and Sargent (2008) refer to these two approaches as constraint preferences and multiplier preferences, respectively.

assigned to the models on the boundary are exceedingly small. Thus, highly unlikely models can come to
dominate the outcome of robust control.”5 This critique can be illustrated using the lost in a forest problem.
In that problem, the worst-case scenario for any search strategy involves guessing each and every one of
the wrong locations ﬁrst before ﬁnding the road. Arguably, guessing wrong at each possible turn is rather
unlikely, yet the robust policy is tailored to this scenario, and the hiker does not take advantage of shortcuts
that allow him to search through many locations without having to walk a great distance.
The main problem with this critique is that the original motivation for using the robustness criterion to
design policy is that the policymaker cannot assign a probability distribution to the scenarios he considers.
Absent such a distribution, one cannot argue that the worst-case scenario is unlikely. That is, there is no
way to tell whether the Bayesian prior probability measure Svennson alludes to is reasonable. One possible
response is that even absent an exact probability distribution, we might be able to infer from common
experience that the worst-case scenario is not very likely. For example, returning to the example of the lost
in a forest problem, we do not often run through all possibilities before we ﬁnd what we are searching for,
so the worst-case scenario should be viewed as remote even without attaching an exact probability to this
event. But such intuitive arguments are tricky. Consider the popularity of the adage known as Murphy’s
Law, which states that whatever can go wrong will go wrong. The fact that people view things going wrong
at every turn as a suﬃciently common experience to be humorously elevated to the level of a scientiﬁc law
suggests they might not view looking through all of the wrong locations ﬁrst as such a remote possibility.
Even if similar events have proven to be rare in the past, there is no objective way to assure a person who
is worried that this time might be diﬀerent that he should continue to view such events as unlikely.
In addition, in both the lost in a forest problem and in many economic applications, it is not the case that
the robust strategy performs poorly in all scenarios aside from the worst-case one. In general, the continuity
present in many of these models implies that the strategy that is optimal in the worst-case scenario must
be approximately optimal in nearby situations. Although robust policies may be overly perfectionist in
some contexts, in practice they are often optimal in various scenarios rather than just the worst-case one.
Insisting that a policy do well in many scenarios may not even be feasible, since there may be no single
policy that does well in a large number of scenarios. More generally, absent a probability distribution, there
is no clear reason to prefer a strategy that does well in certain states or in a large number of states, since
it is always possible that those states are themselves unlikely.
Setting aside the issue of whether robust control policies are dominated by highly unlikely scenarios,
economists have faulted other features of robustness and argued against using it as a criterion for designing
5 A critique along the same spirit is oﬀered in Section 13.4 in Savage (1954), using an example where the minimax policy is
preferable to alternative rules only under a small set of states that collapses to a single state in the limit.

macroeconomic policy. For example, Sims (2001) argues that decisionmakers should avoid rules that violate
the sure-thing principle, which holds that if one action is preferred to another action regardless of which
event is known to occur, it should remain preferred if the event were unknown. As is well known, individuals
whose preferences fail to satisfy the sure-thing principle can in principle be talked into entering gambles
known as “Dutch books” they will almost surely lose.6 But the robust control approach violates the surething principle. Sims instead advocates that policymakers proceed as Bayesians; that is, they should assign
subjective beliefs to the various scenarios they contemplate and then choose the strategy that minimizes
the expected loss according to their subjective beliefs. Proceeding this way ensures that they will satisfy
the sure-thing principle.7 Al-Najjar and Weinstein (2009) describe other anomalies that arise from minimax
behavior, which they ﬁnd suﬃciently compelling that they argue for rejecting preferences that naturally give
rise to such behavior. The Bayesian prescription that Sims advocates as an alternative to robustness seems
particularly natural for the lost in a forest problem, in which the hiker is always free to assume the road is
equally likely to lie in any direction and then choose the search program that minimizes expected walking
distance given this distribution. Indeed, when Bellman (1956) originally posed his problem, he suggested
both minimizing the longest path (minmax) and minimizing the expected path assuming a uniform prior
(min-mean) as possible solutions.
Do the anomalies that Sims and others point out provide suﬃcient grounds to reject robustness as
a suitable criterion for choosing policy? As Siniscalchi (2009) notes in his discussion of Al-Najjar and
Weinstein (2009), “there is no fundamental canon of rationality according to which every decisionmaker
should feel similarly uncomfortable” with the implications of minimax strategies. Ultimately, whether one
feels uncomfortable with these implications and prefers to act like a Bayesian is a matter of taste.8 Indeed,
there is some evidence that distaste for the Bayesian prescription advocated by Sims is quite common in
6 See,

for example, Yaari (1998).

7 Sims (2001) goes on to argue that while robust control is an inappropriate policy recommendation, it might still aid
policymakers who rely on simple procedural rules to guide their decisions rather than explicit optimization. The idea is to
back out what beliefs would justify using the robust strategy, and then reﬂect on whether these beliefs seem reasonable. As
Sims (2001, p. 52) notes, deriving the robust strategy “may alert decision-makers to forms of prior that, on reﬂection, do
not seem far from what they actually might believe, yet imply decisions very diﬀerent from that arrived at by other simple
procedures.” Interestingly, Sims’ prescription cannot be applied to the lost in a forest problem: There is no distribution over
the location of the road for which the minimax path minimizes expected distance. However, the lost in a forest problem oﬀers
an analogous interpretation consistent with Sims’ view. Suppose that the cost of hiking can increase nonlinearly with eﬀort.
In that case, the hiker can ask what cost functions would support the minimax algorithm as a solution to the problem of
minimizing expected eﬀort, assuming the location of the road is distributed uniformly. This can alert him to paths that are
diﬀerent from mechanical search algorithms he already contemplated that were not derived from his true cost of eﬀort, and to
make him reﬂect on whether the cost functions that favor the minimax path seem reasonable.
8 On this point, note that Yaari (1998) shows that expected utility preferences can also be subject to certain types of Dutch
book manipulation. Rejecting minimax rules on the grounds that they allow for Dutch books while advocating alternative
rules that allow for other types of Dutch books must therefore reﬂect a subjective distaste for certain types of Dutch books.
But more generally, even if only minimax preferences allowed for Dutch books, the point is that if a policymaker who was
made aware of this vulnerability preferred to stick to his original decision, we could not fault him for these preferences.

practice. Take the “Ellsberg paradox” laid out in Ellsberg (1961). This paradox is based on a thought
experiment in which people are asked to choose between a lottery with a known probability of winning
and another lottery featuring identical prizes but with an unknown probability of winning. Ellsberg argued
that most people would not behave as if they assigned a ﬁxed subjective probability to the lottery whose
probability of winnings they did not know, and would instead be averse to participating in a lottery that was
ambiguously speciﬁed. In support, he cites several prominent economists and decision theorists to whom
he presented his thought experiment.9 Subsequent researchers who conducted experiments oﬀering these
choices to real-life test subjects, starting with Becker and Brownson (1964), conﬁrmed that people often
fail to behave in accordance with Sims’ prescription of adopting a Bayesian prior. Another example of the
limited appeal of Bayesian decision rules is the aforementioned popularity of Murphy’s Law. A believer in
Murphy’s Law would approach the lost in a forest problem expecting that they are cursed, so that they
will always start their search in the worst possible location. But this implies the location of the road would
depend on where they choose to search, which is inconsistent with assigning a subjective distribution to
the location of the road. Thus, anyone who ﬁnds Murphy’s Law appealing would be uncomfortable with
imposing subjective beliefs over the location of the road. Since any recommendation on policy must respect
the policymaker’s tastes, we cannot objectively fault decisionmakers for not behaving like Bayesians.
Note that not all of those who have criticized the robustness criterion have advocated for Bayesian rules.
For example, Savage (1951) argued that the minimax rule is unduly pessimistic, and oﬀered an alternative
approach that has come to be known as “minimax regret.” This view argues for choosing not the policy
that maximizes the lowest possible payoﬀ that could ever be achieved under any conceivable scenario, but
the policy that minimizes the largest possible regret, deﬁned as the diﬀerence between the payoﬀ under a
given policy in a given state of the world and the payoﬀ that would have been optimal to pursue in that
state. In other words, a policy should seek to minimize losses relative to what could have been achieved.10
This alternative approach has not been used in macroeconomic applications, but has been used in other
economic applications, for example, in Manski (2004). Such applications are surveyed in Manski (2010),
who considers decision problems where an uncertain decisionmaker must choose an appropriate treatment
rule for a population that may respond diﬀerently to diﬀerent treatments. Manski notes several features
of the minmax-regret approach that diﬀer from the recommendations of either the minmax or Bayesian
approach. Unfortunately, little is known as to whether minimax regret exhibits similarly appealing features
9 Ellsberg (1961, p. 655-6) cites these reactions as reﬂecting the spirit of Paul Samuelson’s reply when confronted with the
fact that his preferences were at odds with certain conventional axioms on preferences: “I’ll satisfy my preferences. Let the
axioms satisfy themselves.” Interestingly, Ellsberg cites Samuelson as one who reported that he would not violate the Savage
axioms when asked about the thought experiment concerning the two lotteries.
1 0 For the lost in a forest problem, the optimal policy given a speciﬁc location for the road entails walking exactly one mile
back to the hiker’s starting location. In this case, the policy that minimizes regret turns out to be identical to the one that
minimizes the loss from the worst-case outcome. But the two criterion often lead to diﬀerent policy recommendations.

in macroeconomic applications that involve meeting targets as opposed to assigning treatments. Again,
though, any preference for minimax regret is a matter of subjective tastes, just as with the preference for
Bayesian decision rules.
The theme that runs through the discussion thus far is that if decisionmakers cannot assign probabilities
to scenarios they are uncertain about, there is no inherently correct criterion on how to choose a policy. As
Manski (2000, p. 421) put it, “there is no compelling reason why the decision maker should or should not
use the maximin rule when [the probability distribution] is a ﬁxed but unknown objective function. In this
setting, the appeal of the maximin rule is a personal rather than normative matter. Some decision makers
may deem it essential to protect against worst-case scenarios, while others may not.” Thus, one can point to
unappealing elements about robust control, but these do not deﬁnitively rule out this approach. What, then,
is the appeal of robustness that defenders point to? Aside from the intuitive appeal some ﬁnd in minimizing
risk exposure, proponents of robustness often cite the work on ambiguity aversion, starting with the work
of Gilboa and Schmeidler (1989), as motivation for robustness.11 The latter literature is concerned not with
the normative question of what a decisionmaker should do if he does not know the probability associated
with various relevant scenarios, but with why people often seem averse to situations in which they do not
know the probabilities of various outcomes. More precisely, this literature asks whether there exist coherent
preferences over “acts” in the sense of Savage (1954) that are consistent with preferring clearly posed choice
problems to ambiguously posed ones. Savage deﬁned an “act” as a detailed description of the outcomes
that would occur in every possible state of the world without any reference to the probability these states
will be realized. Gliboa and Schmeidler asked if there are preferences that would tend to favor scenarios in
which the probabilities of ﬁnal outcomes are clearly stipulated over scenarios in which the probabilities of
ﬁnal outcomes are not clearly stipulated. For example, could individuals systematically prefer an act where
in each state of the world the individual is oﬀered the same lottery with known odds of winning each prize
to an act where the prizes and odds of winning vary across states, so the odds of winning a particular prize
depends on one’s subjective perceptions of a given state occurring? Some of the preference speciﬁcations
that can generate ambiguity aversion imply the agent behaves as if he contemplated a set of subjective
probability distributions over the diﬀerent states, and then chose the act that guaranteed the highest worst
case expected utility over these distributions, that is, the strategy that was robust to the set of subjective
beliefs he entertains. These preferences can rationalize why people would appear to choose strategies that
are robust to the models they are subjectively willing to contemplate.
Of course, just as the appeal of Sims’ directive to accept the sure-thing principle amounts to a matter
of taste rather than rationality, there is no compelling justiﬁcation for why a decisionmaker ought to
1 1 Gilboa and Schmeidler (1989) consider only static decision problems. Various authors have since worked on extending
their analysis to dynamic decision problems. See, for example, Epstein and Schneider (2003), Maccheroni, Marinacci, and
Rustichini (2006), and Strzalecki (2009).

ﬁnd compelling the particular axiomatic restrictions on preferences that give rise to minimax behavior.
Ultimately, the aforementioned literature only shows that there exist coherent preferences that justify such
decisions, implying that minimax behavior does not rest on logical contradictions. But this literature
does not argue that these preferences are somehow natural, nor do they oﬀer empirical evidence that
these preferences are common. Instead, these papers largely appeal to the Ellsberg paradox, which can
be reconciled by appealing to preferences that imply subjective multiple priors. But as Al-Najjar and
Weinstein (2009) point out, other explanations for the Ellsberg paradox exist, and the paradox does not by
itself conﬁrm all of the restrictions on preferences needed to give rise to minimax behavior.
Equally important, the axiomatic approach provides a theory of robustness to subjective uncertainty.
That is, it can tell under what conditions decision makers will behave as if they entertain a set of beliefs
regarding the likelihood of various states, and then choose the policies that are robust against this set. But
robust control is typically presented as a theory of what policymakers should do in the face of objective
uncertainty, when they cannot fathom the environment they face. To appreciate this distinction, consider
the lost in a forest problem. The axiomatic approach tells us that if individuals have certain preferences
over proposals for diﬀerent search algorithms, their decisions can be described as if they had a set of beliefs
over the location of the road, and they chose their strategy to be robust to these beliefs. But there is
no restriction that their preferences must carry over to other decision problems that involve a completely
diﬀerent set of Savage acts than the original lost in a forest problem. Thus, there is no guarantee that
individuals who follow a minimax rule in the original lost in a forest problem will also follow a minimax rule
if they had to search for a road located two miles away or if they also forgot the distance they originally
hiked before falling asleep. By contrast, Bellman advocated the minimax search strategy as a general rule
for any search problem in which the location of the object is unknown.
In sum, whether policymakers ﬁnd robustness appealing is subjective. Given the sizable engineering
literature on robust control, the notion of designing systems that minimize the worst-case scenario appears
to have had some appeal among engineers. Interestingly, Murphy’s Law is also an export from the ﬁeld
of engineering.12 The two observations may be related: If worst-case outcomes are treated as everyday
occurrences, robustness would seem like an appealing criterion. Policymakers who are equally nervous
about worst-case outcomes would presumably like the notion of keeping their risk exposure to a bare
minimum. But less nervous policymakers may ﬁnd robustness unappealing upon reﬂection, either because
of its implications in particular models or because it leads to the anomalous results pointed out by detractors.
1 2 According to Spark (2006), the law is named after aerospace engineer Edward Murphy, who complained after a technician
attached a pair of sensors in a precisely incorrect conﬁguration during a crash test Murphy was observing. Engineers on the
team Murphy was working with began referring to the notion that things will inevitably go wrong as Murphy’s Law, and the
expression gained public notoriety after one of the engineers used it in a press conference.

Robustness and the Brainard Model

Now that I have described what it means for a policy to be robust, I can return to the implications of robust
control for macroeconomic policy. Various authors have solved for the robust policy in particular macro
models, and have carried out comparative static exercises on how the robust policy changes with underlying
features of the model, with the set of models the policymaker entertains, or with the penalty parameter that
governs the disutility from using a model that is diﬀerent from the benchmark. Not surprisingly, the results
tend to be model-speciﬁc, and there are few general results. This is also true for the literature on optimal
policy under uncertainty in the Brainard tradition, which assumes policymakers are uncertain about some
parameters but still know the distribution. For example, Chow (1973) ﬁnds few general results when he
considers a broader model that encompasses Brainard’s framework.
However, early applications of robustness did have one result in common: In all of these applications,
the robust policy tended to contradict the attenuation result in Brainard’s model, and instead implied
that uncertain policymakers should react more aggressively to news than they would in the absence of
uncertainty. Examples of this result include Sargent (1999), Stock (1999), Tetlow and von zur Muehlen
(2001), Giannoni (2002), and Onatski and Stock (2002). These ﬁndings were sometimes interpreted to
mean that concern for robustness leads to more aggressive policy.13 In this section, my aim is to dispel this
interpretation. In particular, I argue that the lack of attenuation is not driven by concern for robustness per
se, but by the environments these early papers considered. When we consider the implications of robustness
in the same environment that Brainard considered, we obtain similar results to his.14
Consider the Brainard model above, but suppose now that the policymaker no longer knows the distribution of all of the parameters that he is uncertain about. More precisely, suppose the variable to be targeted,
, is still determined by (2):
 =  − ( +  ) +  

(5)

Since the attenuation result only arises when the policymaker is uncertain about  , I will continue to
assume the policymaker knows the distribution of  for simplicity. By contrast, I assume the policymaker
only knows that the support of  is restricted to some interval [ ] that includes 0, that is,   0  , and
does not know its distribution. Thus, the eﬀect of  on  can be less than, equal to, or higher than , but
the policymaker cannot ascribe probabilities to these events.15
1 3 For example, Bernanke (2007) notes: “The concern about worst-case scenarios emphasized by the robust-control approach
may likewise lead to ampliﬁcation rather than attenuation in the response of the optimal policy to shocks.”
14 A

similar point was made by Onatski (1999), who considers a closely related example to the one I discuss in this section.

1 5 The

case where a decisionmaker is uncertain about the value of one or more parameters in a model is known as structured

The robust control approach in this environment can be viewed as a two-step process. First, for each
value of , we compute its worst-case scenario over all values  ∈ [ ], or the largest expected loss the
policymaker could incur. Deﬁne this expected loss as  (); that is,

n
o
£ ¤
2
 () ≡ max   2 = max ( − ( +  )) +  2 

 ∈[]

 ∈[]

(6)

Second, we choose the policy  that implies the smallest value for  (). The robust strategy is deﬁned as
the value of  that solves min  (); that is,
min max


 ∈[]

n
o
( − ( +  ))2 +  2 


(7)

Below, I describe the solution to this problem and provide some intuition for it. For a rigorous derivation,
see Barlevy (2009). It turns out that the robust policy hinges on the lowest value that  can assume. If
 ≤ −, so the coeﬃcient  +  can be either positive or negative, the solution to (7) is given by
 = 0
If instead   −, so the policymaker is certain that the coeﬃcient  +  is positive but is unsure of its

exact value, the solution to (7) is given by

=



 + ( + ) 2

(8)

Thus, if the policymaker knows the sign of the eﬀect of  on , he should respond to changes in  in a way
that depends on the extreme values  can assume, that is, the endpoints of the interval [ ]. But if the
policymaker is unsure about the sign of the eﬀect of policy on , he should not respond to  at all.16
To better understand why concerns for robustness imply this rule, consider ﬁrst the result that if the
policymaker is uncertain about the sign of  +  , he should altogether abstain from responding to . This
is related to Brainard’s (1967) attenuation result: There is an inherent asymmetry in that a passive policy
where  = 0 leaves the policymaker unexposed to risk from  , while a policy that sets  6= 0 leaves him
exposed to such risk. When the policymaker is suﬃciently concerned about this risk, which turns out to
hinge on whether he knows the sign of the coeﬃcient on , he is better oﬀ resorting to a passive policy that
uncertainty. Unstructured uncertainty refers to the case where the decisionmaker is unsure about the probability distribution
for the endogenous variables in the model, and considers any distribution that is some entropy distance from a benchmark
model. The original ﬁnding that robust policy tends to be aggressive was demonstrated for both types of uncertainty.
1 6 Interestingly, Kocherlakota and Phelan (2009) also provide an example where doing nothing is the minmax solution to
a policy problem. They show that if policymakers rely on an interventionist mechanism, there is an environment in which
intervention plays no useful role, since it does not allow the policymaker to achieve a superior outcome, but it admits an
additional equilibrium that yields an inferior outcome. Thus, an interventionist mechanism is more vulnerable to bad outcomes
than a mechanism that does nothing. The example here yields a similar outcome, but does not involve multiple equilibria.

protects him from this risk than trying to oﬀset nonzero values of .17 However, the attenuation here is
both more extreme and more abrupt than what Brainard found. In Brainard’s formulation, the policymaker
will always act to oﬀset , at least in part, but he will moderate his response to  continuously with  2 .

By contrast, robustness considerations imply a threshold level for the lower support of  , which, if crossed,
leads the policymaker to radically shift from actively oﬀsetting  to not responding to it at all.
The abrupt shift in policy in response to small changes in  demonstrates one of the criticisms of robust
control cited earlier — namely, that this approach formulates policy based on how it performs in speciﬁc
states of the world rather than how it performs in general. When  is close to −, it turns out that the
policymaker is almost indiﬀerent among a large set of policies that achieve roughly the same worst-case loss.

When  is just below −, setting  = 0 performs slightly better under the worst-case scenario than setting
 according to (8). When  is just above −, setting  according to (8) performs slightly better under the

worst-case scenario than setting  = 0. When  is exactly equal to −, both strategies perform equally

well in the worst-case scenario, as do all values of  that fall between these two extremes. However, the

two strategies lead to quite distinct payoﬀs for values of  that are between  and . Hence, concerns for
robustness might encourage dramatic changes in policy to eke out small gains under the worst-case scenario,
even if these changes result in substantially larger losses in most other scenarios. A dire pessimist would
feel perfectly comfortable guarding against the worst-case scenario in this way. But in situations such as
this, where the policymaker chooses his policy based on minor diﬀerences in how the policies perform in
one particular case even though the policies result in large diﬀerences in other cases, the robust control
approach has a certain tail-wagging-the-dog aspect to it that some may ﬁnd unappealing.
Next, consider what concern for robustness dictates when the policymaker knows the sign of  +  but
not its precise magnitude. To see why  depends on the endpoints of the interval [ ], consider ﬁgure 2.
2

This ﬁgure depicts the expected loss ( − ( +  ) ) +  2 for a ﬁxed  against diﬀerent values of  . The


loss function is quadratic and convex, which implies the largest loss will occur at one of the two extreme
values for  . Panel A of ﬁgure 2 illustrates a case in which the expected losses at  =  and  =  are
unequal: The expected loss is larger for  = . But if the losses are unequal under some rule , that value
of  fails to minimize the worst-case scenario. This is because, as illustrated in panel B of ﬁgure 2, changing
 will shift the loss function to the left or the right (it might also change the shape of the loss function, but
for our purposes this can eﬀectively be ignored). The policymaker should thus be able to reduce the largest
possible loss over all values of  in [ ]. Although shifting  would lead to a greater loss if  happened to
1 7 An alternative intuition is as follows. Suppose the policymaker could achieve a lower loss function by using . If the
support of the coeﬃcient includes , an evil agent disposed to lower the policymaker’s utility could always shut oﬀ the eﬀect
of  by setting the coeﬃcient on  to zero. Thus, once the support of  includes 0, policymakers should not be able to use
intervention to make things better oﬀ. What the results show is that in fact, the evil agent can make the policymaker worse
oﬀ for choosing  6= 0 when the coeﬃcient on  can assume either positive or negative values.

equal , since the goal of a robust policy is to minimize the largest possible loss, shifting  in this direction
is desirable. Robustness concerns would therefore lead the policymaker to adjust  until the losses at the
two extreme values were balanced, so that the loss associated with the policy being maximally eﬀective is
exactly equal to the loss associated with the policy being minimally eﬀective.
When there is no uncertainty, that is, when  =  = 0, the policymaker would set  = , since this
would set  exactly equal to its target. When there is uncertainty, whether the robust policy will respond
to  more or less aggressively than this benchmark depends on how the lower and upper bounds are located
relative to 0. If the region of uncertainty is symmetric around 0 so that  = −, uncertainty has no eﬀect
2

on policy. To see this, note that if we were to set  = , the expected loss would reduce to () 2 +  2 ,


which is symmetric in  . Hence, setting  to oﬀset  would naturally balance the loss at the two extremes.
But if the region of uncertainty is asymmetric around 0, setting  =  would fail to balance the expected
losses at the two extremes, and  would have to be adjusted so that it either responds more or less to 
than in the case of complete certainty. In particular, the response to  will be attenuated if   −, that
is, if the potential for an overly powerful stimulus is greater than the potential for stimulus that is far too
weak, and this response will be ampliﬁed in the opposite scenario.
This result begs the question of when the support for  will be symmetric or asymmetric in a particular
direction. If the region of uncertainty is constructed using past data on , , and , any asymmetry would
have to be driven by diﬀerences in detection probabilities across diﬀerent scenarios — for example, if it is
more diﬃcult to detect  when its value is large than when it is small. This may occur if the distribution of
 were skewed in a particular direction. But if the distribution of  were symmetric around 0, policymakers
who rely on past data should ﬁnd it equally diﬃcult to detect deviations in either direction, and the robust
policy would likely react to shocks in the same way as if  were known with certainty.
The above example shows that the robustness criterion does not inherently imply that policy should be
more aggressive in the face of uncertainty. Quite to the contrary, the robust policy exhibits a more extreme
form of the same attenuation principle that Brainard demonstrated, for essentially the same reason: The
asymmetry between how passive and active policies leave the policymaker exposed to risk tends to favor
passive policies. More generally, whether uncertainty about the economic environment leads to attenuation
or more aggressive policy depends on asymmetries in the underlying environment. If the policymaker
entertains the possibility that policy can be far too eﬀective but not that it will be very ineﬀective, he will
naturally tend to attenuate his policy. But if his beliefs are reversed, he will tend to magnify his response
to news about potential deviations from the target level.18
1 8 Rustem, Wieland, and Zakovic (2007) also consider robust control in asymmetric models, although they do not discuss
the implications of these asymmetries for attenuation.

Robustness and Aggressive Rules

Given that robustness considerations do not necessarily imply more aggressive policies, which features of
the macro models that were used in early applications of robust control account for the ﬁnding that robust
policies tend to be more aggressive? The papers cited earlier consider diﬀerent models, and their results
are not driven by one common feature. To give some ﬂavor of why some factors may lead robust policies
to be aggressive, I now consider two simpliﬁed examples inspired by these papers. The ﬁrst assumes the
policymaker is uncertain about the persistence of shocks, following Sargent (1999). The second assumes
the policymaker is uncertain about the tradeoﬀ between competing objectives, following Giannoni (2002).
I show that both features can tilt the robust policy toward reacting more aggressively to incoming news.

5.1

Uncertain persistence

One of the ﬁrst to argue that concerns for robustness could dictate more aggressive policy was Sargent
(1999). Sargent asked how optimal policy would be aﬀected in a particular model when we account for the
possibility that the model is misspeciﬁed — and especially the possibility that speciﬁcation errors are serially
correlated. To gain some insight on the implications of serially correlated shocks, consider the following
extension of the Brainard model in which the policymaker attempts to target only one variable.19
Let  denote the value at date  of the variable that the policymaker wishes to target to 0. As in (1), I
assume  is linear in the policy variable  and in an exogenous shock term  :
 =  −  
I now no longer assume the policymaker is uncertain about the eﬀect of his policy on  . As such, it will
be convenient to normalize  to 1. However, I now assume the policymaker is uncertain about the way 
is correlated over time. That is, suppose  follows the AR(1) process
 = −1 +  
where  are independent and identically distributed over time with mean 0 and variance  2 , and  captures

the autocorrelation of  . At each date , the policymaker can observe −1 and condition his policy on its
realization. However, he must set  before observing  . His uncertainty about  is due to two diﬀerent
factors. First, I assume he does not know  at the time he chooses his policy. Second, to capture uncertainty
1 9 Sargent assumed the policymaker cares about both inﬂation and unemployment, following the model he was discussing.
Here, I continue to assume the policymaker cares about only one target. But my objective is not to replicate Sargent’s results.
Rather, it is to illustrate why a concern that shocks are persistent can lead to robust policies that tend to be aggressive.

about the persistence of variables over which the policymaker is uncertain, I assume the policymaker does
not know the autocorrelation coeﬃcient  that governs how −1 aﬀects  .
Suppose the policymaker discounts future losses at rate   1 so that his expected loss is given by
#
"∞
#
"∞
X 
X 
2
2
  = 
 ( −  ) 

=0

=0

If the policymaker knew  with certainty, his optimal strategy would be to set  =  [ |−1 ] = −1 .

But in line with the notion that the policymaker remains uncertain about the degree of persistence, suppose
£ ¤
he only knows that  falls in some interval   . Let ∗ denote the midpoint of this interval:
¡
¢
∗ =  +  2

To emphasize asymmetries inherent to the loss function as opposed to the region of uncertainty, suppose
the interval of uncertainty is symmetric around the certainty benchmark. An important and empirically
plausible assumption in what follows is that ∗  0; that is, the beliefs of the monetary authority are
centered around the possibility that shocks are positively correlated.
Once again, we can derive the robust strategy in two steps. First, for each rule  , deﬁne  ( ) as the
biggest loss possible among the diﬀerent values of ; that is,
"∞
#
X 
2
  
 ( ) = max 
∈[]
=0
We then choose the policy rule  that minimizes  ( ); that is, we solve
"∞
#
X 
2
  
min max 
 ∈[]
=0

(9)

Following Sargent (1999), I assume the policymaker is restricted in the type of policies  he can carry out:
He must choose a rule of the form  = −1 , where  is a constant that cannot vary over time. This
restriction is meant to capture the notion that the policymaker cannot learn about the parameters over
which he is uncertain and then change the way policy reacts to information as he observes  over time
and potentially infer . I further assume the expectation in (9) is the unconditional expectation of future
losses; that is, the policymaker calculates his expected loss from the perspective of date 0. To simplify the
calculations, I assume 0 is drawn from the stationary distribution for  .
The solution to (9), subject to the constraint that  = −1 , is derived in Barlevy (2009). The key
result is that as long as ∗  0, the robust policy would set  to a value in the interval that is strictly greater
than the midpoint ∗ . In other words, starting with the case in which the policymaker knows  = ∗ , if
we introduce a little bit of uncertainty in a symmetric fashion so that the degree of persistence can deviate
18

equally in either direction, the robust policy would react more to a change in −1 in the face of uncertainty
than it would react to such a change if the degree of persistence were known with certainty.
To understand this result, suppose the policymaker set  = ∗ . As in the previous section, the loss
function is convex in , so the worst-case scenario will occur when  assumes one of its two extreme values,
that is, when either  =  or  = . It turns out that when  = ∗ , setting  =  imposes a bigger cost on the
policymaker than setting  = . Intuitively, for any given , setting  = ∗ will imply  = ( − ∗ )−1 +  .

The expected deviation of  from its target given −1 will have the same expected magnitude in both
£ ¤
cases; that is, |( − ∗ )−1 | will be the same when  =  and  = , given   is symmetric around ∗ .
However, the process  will be more persistent when  is higher, and so deviations from the target will be
more persistent when  =  than when  = . More persistent deviations imply more volatile  and hence
a larger expected loss. Since the robust policy tries to balance the losses at the two extreme values of ,
the policymaker should choose a higher value for  to reduce the loss when  = .
The basic insight is that, while the loss function for the policymaker is symmetric in  around  = 0, if we
focus on an interval that is centered around positive values of , the loss function will be asymmetric. This
asymmetry will tend to favor policies that react more to past shocks. In fact, this feature is not unique to
policies guided by robustness. If we assumed the policymaker assigned a symmetric distribution to values of
£ ¤
 in the interval   and acted to minimize his expected losses, the asymmetry in the loss function would
hX∞
i
2
  (−1 − −1 +  ) will exceed ∗ .
continue to imply that the value of  that solves min 


=0

Hence, the feature responsible for the aggressiveness of the robust policy is not disproportionate attention

to worst-case scenarios, since even if the policymaker assigned small probabilities to extreme values of , he
might still respond aggressively to past shocks given such beliefs. Rather, because more persistent shocks
create bigger losses than less persistent shocks, policymakers will want to minimize the damage in the event
that shocks are persistent. As in the Brainard model, what tilts policy away from its certainty benchmark
is an underlying asymmetry that favors certain types of policies over others.

5.2

Uncertain tradeoﬀ parameters

Next, I turn to the Giannoni (2002) model. Giannoni focused on the case where policymakers are uncertain
about the parameters in the underlying model the policymaker believes, speciﬁcally the parameters that
dictate the eﬀect of shocks on the endogenous variables in the model. If policymakers wish to target
more than one of these variables, this can make them uncertain about the tradeoﬀ between meeting their
competing objectives, which in Giannoni’s framework turns out to favor more aggressive policies. Up to
now, I have conveniently assumed policymakers were only interested in targeting a single variable. However,
tradeoﬀs in trying to target multiple variables can be important in designing robust policy, and so it is worth

exploring a simpliﬁed example to illustrate some of these eﬀects.
Consider the following simple static framework. Suppose the monetary authority cares about two variables, denoted  and , where  and  represent the output gap and inﬂation, respectively. The monetary
authority has a quadratic loss function in both terms:
 2 +  2 

(10)

The variables  and  are assumed to be related linearly, speciﬁcally
 =  + 

(11)

where  is an observable shock.20 This relationship implies a tradeoﬀ between targeting  and . In
particular, if we set  = 0 as desired, then  would vary with  and deviate from 0. But if we set  = 0,
then  would vary with  and deviate from 0. Substituting in (11) into the loss function allows us to express
the policymaker’s problem as choosing  to minimize the loss
2



( − )
+ 2
2

Taking the ﬁrst-order condition with respect to  gives us the optimal choices for  and  as
=




2 = −
+
 + 2

(12)

¤
£
Suppose the policymaker were uncertain about , knowing only that it lies in some interval   . However,
the policymaker still has to commit to a rule that determines  and  as functions of . Given a value for
, the worst-case scenario over this range of  is given by
max 
∈[]

( − )2
+ 2 
2

For any choice of  except  = , the worst case corresponds to  = . If  = , the value of  has no
eﬀect on the loss function. Hence, the robust strategy would be to choose a rule that sets  and  to their
values in (12) as if  = . As long as the certainty benchmark  lies in the interior of the uncertainty
¤
£
interval   , concerns for robustness will lead the policymaker to have  respond less to  and  respond
more to . Intuitively, when a shock  causes  to deviate from its target, a lower value of  implies that

pushing  back to its target rate of 0 would require that  deviate by a larger extent from its desired target.
The worst-case scenario would thus amount to the policymaker fearing that meeting one target comes at a
much higher cost in terms of missing the other target. Hence, the worst-case scenario would suggest that
2 0 This relationship is meant to capture the Phillips curve relationship in Giannoni (2002), which implies inﬂation at date 
depends on the output gap at date , expected inﬂation at date  + 1, and a shock term, so that   =  +  +1 +  .
Using a static model ignores an important contribution of Giannoni’s work in explicitly incorporating a forward-looking term
  +1 that is a staple of New Keynesian models, but this simpliﬁcation makes the analysis considerably more transparent.

the policymaker should not try to stabilize  too much for fear of missing his target for . Put another way,
the robust policy implies more aggressively stabilizing  and less aggressively stabilizing .
Figure 3 illustrates this result graphically. The problem facing the policymaker is to choose a point from
the line given by  =  + . Ideally, it would like to move toward the origin, where  =  = 0. Changing
 will rotate the line from which the policymaker must choose, as depicted in the ﬁgure. A lower value of 
corresponds to a ﬂatter curve. Given the policymaker prefers to be close to the origin, a ﬂatter curve leaves
the policymaker with distinctly worse options that are farther from the origin, since one can show that
the policymaker would only choose points in the upper left quadrant of the ﬁgure. This explains why the
worst-case scenario corresponds to the ﬂattest curve possible. If we assume the policymaker must choose his
relative position on the line before knowing the slope of the line (that is, before knowing ), then the ﬂatter
the line could be, the greater his incentive will be to locate close to the -axis rather than risk deviating
from his target on both variables, as indicated by the path with the arrow. This explains why he will act
more aggressively to isolate the eﬀects of  on .
Robustness concerns thus encourage the policymaker to proceed as if he knew  was equal to its lowest
possible value. Note the diﬀerence from the two earlier models, in which the robust policy recommended
balancing losses associated with two polar opposite scenarios. Here, the policy equates marginal losses for
a particular risk (the loss from letting  deviate a little more from its target and the loss from letting 
deviate a little more), namely the risk that  will be low so that stabilizing inﬂation comes at a large cost
of destabilizing output.

Conclusion

My objective in this paper was to review the robustness criterion that some macroeconomists have advocated
for designing policy when policymakers are at a loss to assign probabilities to the scenarios they contemplate.
The key theme I have stressed is that in the absence of objective probabilities, there is no right criterion
by which to choose policy. Critics of robust control are really just expressing their own subjective views
about some of the implications of this approach, just as advocates of robust control are expressing their
own subjective views that this approach has some appealing features. While there is no one right answer for
whether robustness is an appropriate criterion for designing policy, even critics of robustness are willing to
acknowledge that deriving the robust policy and understanding its properties can be useful, not necessarily
because policymakers should adopt this policy but because solving for this rule can alert policymakers to
where their policies might go amiss and whether acting to minimize the potential for losses is costly. Since
the exact features of robust policies will depend on the underlying environment, typically one would have to
solve for the minimax policy in each application rather than rely on general results on the nature of robust
21

policy rules. Indeed, one of the points of this survey was that concern for robustness does not inherently
contradict the attenuation principle derived by Brainard (1967), contrary to what some have inferred based
on the results that emerged from the particular models used in early applications of robust control.
My survey focused exclusively on the normative question of how macroeconomic policymakers should
behave when they are uncertain about the environment they face. However, equally important questions
left out from this survey hinge on whether policymakers and agents do in fact exhibit a preference for
robustness and choose policies this way. For example, Ellison and Sargent (2009) argue that concern for
robustness can help to understand monetary policy and why the Federal Open Market Committee seems to
ignore projections made by their staﬀ even though they know these are good forecasts of future economic
conditions. In addition, various authors have investigated the nature of optimal policy when private agents
follow robust decision rules, both when policymakers themselves face no uncertainty and when policymakers
are themselves concerned about robustness. Even if there are compelling arguments for why robustness may
not always be desirable, evidence that agents have a taste for robustness may still force policymakers to
take robustness concerns into account when designing macroeconomic policy.

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24

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(a) Graphical representation of the problem
210⁰

2
√3

30⁰
1
√3

(b) Min‐max path, derived by Isbell (1957)

Figure 1: Graphical Representation of the “Lost in a Forest” Problem

Expected loss
(x‐(k+εk)r)2 + σ2

εk

(a) Expected loss function with unequal expected losses at extremes

Expected loss
(x‐(k+εk)r)2 + σ2

εk

(b) Adjusting r to balance expected losses at the two extremes

Figure 2: Loss Functions for the Robust Brainard Model

x
lower λ

higher λ

Figure 3: Graphical Representation of Uncertain Tradeoffs Model

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