MONETARY AND ECONOMIC STUDIES (SPECIAL EDITION)/FEBRUARY 2001

Acknowledging Misspecificationin Macroeconomic Theory
Lars Peter Hansen and Thomas J. Sargent

We explore methods for confronting model misspecification in
macroeconomics. We construct dynamic equilibria in which private
agents and policy makers recognize that models are approximations.
We explore two generalizations of rational expectations equilibria.
In one of these equilibria, decision makers use dynamic evolution
equations that are imperfect statistical approximations, and in theother misspecification is impossible to detect even from infinite
samples of time-series data. In the first of these equilibria, decisionrules are tailored to be robust to the allowable statistical discrepancies. Using frequency domain methods, we show that robust
decision makers treat model misspecification like time-series
econometricians.
Key words: Robustness; Model misspecification; Monetary policy;
Rational expectations

Lars Peter Hansen: University of Chicago (E-mail: l-hansen@uchicago.edu)
Thomas J. Sargent: Stanford University (E-mail: sargent@leland.stanford.edu)
This paper was prepared for the Society of Economic Dynamics Conference in Costa
Rica in June 2000 and for the Ninth International Conference sponsored by the Institutefor Monetary and Economic Studies, Bank of Japan, in Tokyo on July 3–4, 2000. We
thank Lars Svensson and Grace Tsiang for helpful comments.

213

I. Rational Expectations versus Misspecification
Subgame perfect and rational expectations equilibrium models do not permit aself-contained analysis of model misspecification. But sometimes model builders
suspect misspecification, and so might the agents in their model.1 To study that, wemust modify rational expectations. But in doing so, we want to respect and extend
the inspiration underlying rational expectations, which was to deny that a modelbuilder knows more about the data generating mechanism than do the agents inside
his model.
This paper describes possible reactions of model builders and agents to two
different types of model misspecification. The first type is difficult to detect in timeseries samples of the moderate sizes typically at our disposable. A second type ofmodel misspecification is impossible to detect even in infinite samples drawn froman equilibrium.
A. Rational Expectations Models
A model is a probability distribution over a sequence. A rational expectations
equilibrium is a fixed point of a mapping from agents’ personal models of an
economy to the actual model. Much of the empirical power of rational expectations
models comes from identifying agents’ models with the data generating mechanism.
Leading examples are the cross-equation restrictions coming from agents’ using
conditional expectations to forecast and the moment conditions emanating from
Euler equations. A persuasive argument for imposing rational expectations is thatagents have incentives to revise their personal models to remove readily detectablegaps between them and the empirical distributions. The rational expectations
equilibrium concept is often defended as the limit point of some more or lessexplicitly specified learning process in which all personal probabilities eventuallymerge with a model’s population probability.
B. Recognitions of Misspecification
A rational expectations equilibrium (indexed by a vector of parameters) is alikelihood function. Many authors of rational expectations models express or reveal
concerns about model misspecification by declining to use the model (i.e., thelikelihood function) for empirical work. One example is the widespread practice ofusing seasonally adjusted and/or low-frequency adjusted data. Those adjustments
have been justified formally by stressing the model’s inadequacy at particular frequencies and by appealing to some frequency domain version of an approximation
criterion like that of Sims (1972), which is minimized by least squares estimates of a
misspecified model. Sims (1993) and Hansen and Sargent (1993) describe explicit
justifications that distinguish the model from the unknown true data generatingprocess. They posit that the true generating process has behavior at the seasonalfrequencies that cannot be explained by the model except at parameter values that
1. By studying how agents who fear misspecification can promote cautious behavior and boost market prices of risk,
we do not intend to deny that economists have made tremendous progress by using equilibrium concepts that
ignore model misspecification.

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Acknowledging Misspecification in Macroeconomic Theory

cause bad fits at the non-seasonal frequencies. Maximum likelihood estimates makethe model best fit the frequencies contributing the most variance to the data set.
When the model is most poorly specified at the seasonal frequencies, then using
seasonally adjusted data can trick the maximum likelihood method to emphasize
frequencies where the model is better specified. That can give better estimates ofparameters describing tastes and technologies.2
Less formal reasons for divorcing the data analysis from the model also appeal tomodel misspecification. For example, calibrators say that their models are approximations aimed at explaining only “stylized” facts or particular features of the time series.
Notice that in both the formal defenses of data filtering and the informal practiceof calibration, the economist’s model typically remains a rational expectations model
inhabited by agents who do not doubt the model. Thus, such analyses do not let theagents inside the economist’s model share his doubts about model specification.

II. Agents Who Share Economists’ Doubts
But the intent of rational expectations is to put the economist and the agents inside hismodel on the same footing. Letting the agents contemplate model misspecification
reopens fundamental issues that divided Knight, Fellner, and Ellsberg from Savage,
and that were set aside when, by adopting rational expectations, macroeconomists
erased all model ambiguity from their agents’ minds.
A. Savage versus Knight
Knight (1921) distinguished risky events, which could be described by a probability
distribution, from a worse type of ignorance that he called uncertainty and that could
not be described by a probability distribution. He thought that profits compensatedentrepreneurs for bearing uncertainty. Especially in some urn examples that prefigured
Ellsberg (1961), we see Knight thinking about decision making in the face of possiblemodel misspecifications.3 Savage contradicted Knight. Savage (1954) proposed a setof axioms about behavior that undermined Knight’s distinction between risk anduncertainty. A person behaving according to Savage’s axioms has a well-defined
personal probability distribution that rationalizes his behavior as an expected utility
maximizer. Savage’s system undermined Knight by removing the agent’s possiblemodel misspecification as a concern of the model builder.
B. Muth versus Savage
For Savage, it was not an aspect of rationality that personal probabilities be “correct.”
But for followers of Muth (1961), it was. By equating personal probabilities with
objective ones, rational expectations assumes away possible model misspecifications
2. The remarkable feature of these results is that better estimates of taste and technology parameters are acquiredby imposing false cross-equation restrictions and by accepting worse estimates of the parameters governing
information and agents’ forecasts. Two-sided seasonal adjustment distorts the temporal and information properties
processes that agents are trying to forecast.
3. As Ellsberg (1961) points out, Knight’s introspection about urns did not produce the paradox that Ellsberg isfamous for.

215

and disposes of diversity of personal probabilities. Rational expectations substantiallyweakens the appeal of Savage’s “solution” of the model specification problems thatconcerned Knight because it so severely restricts personal probabilities.
C. Ellsberg versus Savage
On the basis of experimental evidence, Ellsberg (1961) and Fellner (1961) challenged
Savage’s theory. Fellner (1965) proposed a semiprobabilistic framework in which
agents used context-specific “slanted probabilities” to make decisions in ways that
violate the Savage axioms. The Ellsberg paradox motivated Gilboa and Schmeidler(1989) to formulate a new set of axioms that accommodate model ambiguity. Gilboa
and Schmeidler’s axioms give agents not a unique personal probability distribution
but a set of distributions. They posit that agents make decisions as the min-max
outcomes of a two-person game in which the agent chooses a utility maximizing
decision and a malevolent nature chooses a minimizing probability distribution
from within the set chosen by the agents. They show that such behavior can explain
the Ellsberg paradox.
Convinced by the Ellsberg paradox and inspired by Gilboa and Schmeidler’s
formulation, Epstein and Wang (1994), Epstein and Melino (1995), and Chen and
Epstein (1998) have constructed dynamic models in which agents are adverse
to model ambiguity. Some of this work represents model ambiguity by a class ofprobability distributions generated by the epsilon-contaminations used in the robust
statistics literature.
D. Brunner and Meltzer
Brunner and Meltzer (1967) discussed the role model misspecification in the designof monetary policy. They challenge the existence of a “fully identified, highly
confirmed theory of macroeconomic processes.” They write:
we acknowledge that many of the questions raised here can be answered more
fully if (and only if ) more useful knowledge about the structure of the economy
is assumed or obtained. Put otherwise, the theorist may choose to ignore thisproblem by assuming the possession of reliable information currently outside
the scope of quantitative economics. The policy maker is not as fortunate.4
Brunner and Meltzer went on to suggest a min-max strategy for selecting among
endogenous indicators of monetary policy. They are led to this rule by acknowledging and hence confronting the ambiguity that policy makers have about models of
the macroeconomic economy.
We share the concern of Brunner and Meltzer. We will operationalize this concern
by introducing formally perturbations or potential model misspecifications into abenchmark dynamic model. These perturbations could be viewed as indexing large
families of dynamic models as in dynamic extensions of the Gilboa-Schmeidler
multiple prior formulation. We prefer to think about these as errors in a convenient,
4. See Brunner and Meltzer (1967), p. 188.

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Acknowledging Misspecification in Macroeconomic Theory

but misspecified dynamic macroeconomic model. Our choice of perturbations comesfrom a source that served macroeconomists well before, especially at the dawn ofrational expectations.

III. Control Theory
The mathematical literatures on control and estimation theory were the main sourcesof tools for building rational expectations models. That was natural, because before
1975 or so, control theory was about how to design optimal policies under theassumption that a decision maker’s model, typically a controlled Markov process, iscorrectly specified. This is exactly the problem that rational expectations modelers
want the agents inside their models to solve.
But just when macroeconomists were importing the ordinary correct-modelversion of control theory to refine rational expectations models, control theorists
were finding that policies they had designed under the assumption that a model iscorrect sometimes performed much worse than they should. Such practical problems
caused control theorists to soften their assumption about knowing the correct modeland to think about ways to design policies for models that were more or less good
approximations. Starting in the mid-1970s, they created tools for designing policies
that would work well for a set of possible models that were in a sense close to a basic
approximating model. In the process, they devised manageable ways of formulating a
set of models surrounding an approximating model. To design policies in light ofthat set of models, they used a min-max approach like that of Gilboa and Schmeidler.

IV. Robust Control Theory
We want to import, adapt, and extend some of the robust control methods to build
models of economic agents who experience model ambiguity. We briefly sketch
major components of our analysis.
A. The Approximating Model
There is a single approximating model defining transition probabilities. For example,
consider the following linear-quadratic state space system:
xt +1 = Axt + But + Cwt +1(1)

zt = Hxt + Jut(2)

where xt is a state vector, u t is a control vector, and z t is a target vector, all at date t .
In addition, suppose that {wt +1} is a vector of independent and identically normally
distributed shocks with mean zero and covariance matrix given by I. The target vector
is used to define preferences via

217

1 ∞ tE |z | 2
– ––
∑β t
2 t =0(3)

where 0 < β < 1 is a discount factor and E is the mathematical expectation operator.
The aim of the decision maker is to maximize this objective function by choice ofcontrol law ut = –Fxt .
B. Distortions
A particular way of distorting those probabilities defines a set of models that express
an agent’s model ambiguity. First, we need a concise way to describe a class of alternative specifications. For a given policy u = –Fx, the state equation (1) defines a Markov
transition kernel π (x' |x). For all x, let v (x' |x ) be positive for all x . We can usev (x' |x) to define a distorted model via π v (x' |x ) = v (x' |x )π (x' |x )/Ev(x' |x ), where
the division by Ev(x' |x) lets the quotient be a probability. Keeping v (x' |x ) strictly
positive means that the two models are mutually absolutely continuous, which makes
the models difficult to distinguish in small data sets. Conditional relative entropy isa measure of the discrepancy between the approximating model and the distorted
model. It is defined as

π v (x' |—x) π v (d x' |x).
ent = ∫ log ———π (x' |x)(4)

Conditional entropy is thus the conditional expectation of the log likelihood ratio ofthe distorted with respect to the approximating model, evaluated with respect to the
distorting model.
The distortion just described preserves the first-order Markov nature of themodel. This occurs because v (x' |x) because only x shows up in the conditioning.
More general distortions allow lags of x to show up in the conditioning information
set. To take a specific example, we represent perturbations to model (1) by distortingthe conditional mean of the shock process away from zero:
xt +1 = Axt + But + C (wt +1 + vt )(5)

vt = ft (xt , . . . , xt –n).

(6)Here equation (6) is a set of distortions to the conditional means of the innovationsof the state equation. In equation (6), these are permitted to feed back on lagged
values of the state and thereby represent misspecified dynamics. For the particularmodel of the discrepancy (6), it can be established that1 t vt .
ent = —v'
2
This distortion substitutes a higher-order nonlinear Markov model for the first-orderlinear approximating model. More generally, the finite-order Markov structure canbe relaxed by supposing that vt depends on the infinite past of x t . The essential

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Acknowledging Misspecification in Macroeconomic Theory

requirement is the mutual absolute continuity between the approximating model andits perturbed counterpart.
C. Conservative Valuation
We use a min-max operation to define a conservative way of evaluating continuation
utility. Let V (x t +1) be a continuation value function. Fix a control law ut = –Fxt sothat the transition law under a distorted model becomes
x t +1 = A 0x t + C (wt +1 + vt ),(7)

where A 0 = A – BF. We define an operator R θ (V(x t +1)) for distorting the continuation
values as the indirect utility function for the following minimization problem:R θ (V(x t +1)) = min {θv't vt + E (V(x t +1))},vt(8)where the minimization is subject to constraint (7). Here θ ≤ +∞ is a parameter thatpenalizes the entropy between the distorted and approximating model and thereby
describes the size of the set of alternative models for which the decision maker wantsa robust rule. This parameter is context-specific and depends on the confidence in
his model that the historical data used to build it inspire. Below, we illustrate how
detection probabilities for discriminating between models can be used to discipline
the choice of θ.
Let vˆt = Gθ x t attain the minimum on the right side. The following useful robustness bound follows from the minimization on the right side of equation (8):EV (A 0 + C (wt +1 + vt )) ≥ R θ (V(x t +1)) – θv't vt .

(9)

The left side is the expected continuation value under a distorted model. The rightside is a lower bound on that continuation value. The first term is a conservativevalue of the continuation value under the approximating (vt = 0) model. The second
term on the right gives a bound on the rate at which performance deteriorates
with the entropy of the misspecification. Decreasing the penalty parameter θ lowers
the R θV (x t +1), thereby increasing the conservative nature of R θV (x t +1) as an estimate
under the approximating model and also decreasing the rate at which the bound on
performance deteriorates with entropy. Thus, the indirect utility function induces aconservative way of evaluating continuation utility that can be regarded as a contextspecific distortion of the usual conditional expectation operator. There is a classof operators indexed by a single parameter that summarizes the size of the set ofalternative models.
It can be shown that the operator R θ can be represented as
R θ (V(x t +1)) ≈ h –1Et (h (V(x t +1))
≈ θv̂'t v̂t + Et (V(x t +1))

219

( )

–V
where h (V ) = exp —— and vˆt is the minimizing choice of vt .5 Note that h is a convex
θfunction. This is an operator used by Epstein and Zin (1989), Weil (1993), andothers, but with no connection to robustness.
The operator R θ can be used to rework the theory of asset pricing or to designrobust decision rules. The operator also has been used by Hansen and Sargent (1995)
to define a discounted version of risk-sensitive preference specification of Jacobson(1973) and Whittle (1990). They parameterize risk-sensitivity by a parameterσ ≡ –θ –1, where the σ < 0 imposes an additional adjustment for risk that acts like apreference for robustness.
D. Robust Decision Rule
A robust decision rule is produced by the Markov-perfect equilibrium of a two-person
zero-sum game in which maximizing agent chooses a policy and a minimizing
agent chooses a model. To compute a robust control rule, we use the Markov-perfectequilibrium of the two-agent zero-sum dynamic game:6βθ
1V (x) = max min –—z'z+——v'v + βEtV (x*)
u
v
2
2

{

}

(10)

subject tox* = Ax + Bu + C (w + v).
Here, θ > 0 is a penalty parameter that constrains the minimizing agent; θ governs thedegree of robustness achieved by the associated decision rule. When the robustness
parameter θ takes the value + ∞, we have ordinary control theory because it is too
costly for the minimizing agent to set a nonzero distortion vt . Lower values of θ
achieve some robustness (again see the role of θ in the robustness bound (9)).
To illustrate robustness, we present some figures from Hansen and Sargent(2000c) based on a monetary policy model of Ball (1999). Ball’s model is particularly
simple because the private sector is “backward-looking.” This reduces the monetary
policy problem to a simple control problem. While this simplifies our depiction ofrobustness, it is of more substantive interest to investigate models in which private
sector decision-makers are “forward-looking.” Hansen and Sargent (2000b) study
models in which both the Federal Reserve and private agents are forward-looking andconcerned about model misspecification.
Under various worst-case models indexed by the value of σ = –θ –1 on the horizontal axis, Figure 1 shows the values of –Ez't z t for a monetary policy model of Ball
(1999) for three rules that we have labeled with values of σ = 0, –.04, –.085. Later,
we shall describe how these different settings of σ = –θ –1 correspond to different
sizes of the set of alternative models for which the decision maker seeks robustness.
5. We use the notation ≈ because there is a difference in the constant term that becomes small when we take acontinuous-time diffusion limit.
6. There is only one value function because it is a zero-sum game.

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Acknowledging Misspecification in Macroeconomic Theory

Figure 1 Value of –Ez't z t for Three Decision Rules
Value
–2.4
–2.6
–2.8
–3.0
–3.2

σ = –.085
–3.4
–3.6

σ = –.04
–3.8

σ=0

–4.0
–0.09

–0.08

–0.07

–0.06

–0.05

–0.04

–0.03

–0.02

-0.01

0.00

σ
Note: When the data are generated by the worst-case model associated withthe value of σ on the horizontal axis: σ = 0 rule (solid line), σ = –.04 rule(dashed-dotted line), σ = –.085 (dashed) line. The robustness parameter isgiven by θ = –1/σ.

For Ball, –Ez't z t is the sum of variances of inflation and a variable measuring
unemployment. The three fixed rules solve equation (10) for the indicated value of
σ. The value of –Ez't z t is plotted for each of the fixed rules evaluated for the law of~ )x ) where G (σ~ )x denotes the minimizing
motion x t +1 = (A – BF )xt + C (wt +1 + G (σtt~rule for vt associated with the value σ = σ on the horizontal axis. The way the curves
cross indicates how the σ = –.04 and σ = –.085 rules are not optimal if the model is
specified correctly (if σ = 0 on the horizontal axis), but do better than the optimal
rule against the model misspecifications associated with the distortions associatedwith movements along the horizontal axis. Notice how the σ used to design therule affects the slope of the payoff line. We now briefly turn to describe how σ might
be chosen.
E. Detection Probabilities
The Bellman equation (10) specifies and penalizes model distortions in terms of theconditional relative entropy of a distorted model with respect to an approximating
model. Conditional relative entropy is a measure of the discrepancy between twomodels that appears in the statistical theory of discriminating between two models.
We can link detection error probabilities to conditional relative entropy, as inAnderson, Hansen, and Sargent (1999) and Hansen, Sargent, and Wang (2000).
This allows us to discipline our choice of the single free parameter θ that ourapproach brings relative to rational expectations.

221

For a sample of 147 observations, Figure 2 displays a set of Bayesian detection error
probabilities for comparing Ball’s model with the worst-case model from equation (10)that is associated with the value of σ = –θ –1 on the axis. The detection error probability
is .5 for σ = 0 (Ball’s model and the σ = 0 worst-case model are identical, and therefore
detection errors occur half the time). As we lower σ, the worst-case model fromequation (10) diverges more and more from Ball’s model (because v't vt rises) and thedetection error probability falls. For σ = –.04, the detection error probability is still .25:
this high fraction of wrong judgments from a model comparison test tells us that it isdifficult to distinguish Ball’s model from the worst-case σ = –.04 model with 147 observations. Therefore, we think it is reasonable for the monetary authority in Ball’s model
to want to be robust against misspecifications parameterized by such values of σ. In thisway, we propose to use a table of detection error probabilities like that encoded in
Figure 2 to discipline our selection of σ. See Anderson, Hansen, and Sargent (1999)
and Hansen, Sargent, and Wang (2000) for applications to asset pricing.
Figure 2 Detection Error Probability
p (σ)
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
–0.12

–0.10

–0.08

–0.06
–0.04
–0.02
σ
Note: (Coordinate axis) as a function of σ = –1/θ for Ball’s model.

–0.00

F. Precaution
A preference for robustness induces context-specific precaution. In asset pricing
models, this boosts market prices of risk and pushes a model’s predictions in the
direction of the data with respect to the equity premium puzzle. See Hansen,
Sargent, and Tallarini (1999) and Hansen, Sargent, and Wang (2000). In permanent
income models, it induces precautionary savings. In sticky-price models of monetary
policy, it can induce a policy authority to be more aggressive in response to shocks
than one who knows the model. Such precaution has an interpretation in terms of a

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Acknowledging Misspecification in Macroeconomic Theory

frequency domain representation of the criterion function under various model
perturbations.
For the complex scalar ζ, let G (ζ ) be the transfer function from the shocks wt to| |the—targets z t . Let' denote matrix—— transposition and complex conjugation, Γ = {ζ : ζ= √ β }, and d λ (ζ ) = 1/[2πi √βζ ]d ζ. Then the criterion (3) can be represented as
H2 = – ∫ trace[G(ζ )' G (ζ )]d λ (ζ ).
ΓWhere –δj (ζ ) is the j -th eigenvalue of G(ζ )' G(ζ ), we have
H2 = ∑ ∫ –δj (ζ )d λ (ζ ).
j

Γ

(11)

Hansen and Sargent (2000a) show that a robust rule is induced by using the criterionent (θ ) = ∫ log det[θI – G(ζ )' G(ζ )]d λ (ζ )Γorent (θ ) = ∑ ∫ log[θ – δj (λ )]d λ (ζ ).
j

Γ

(12)

Because log(θ – δ ) is a concave function of –δ, criterion (12) is obtained fromcriterion (11) by putting a concave transformation inside the integration. Aversion
to model misspecification is thus represented as additional “risk aversion” acrossfrequencies instead of across states of nature. Under criterion (12), the decision makerprefers decision rules that render trace G(ζ )' G(ζ ) flat across frequencies.
For example, the curve for σ = 0 in Figure 3 depicts a frequency domain decomposition of Ez't z t for the optimal rule under Ball’s model. Notice that it is biggest
at low frequencies. This prompts the minimizing agent in problem (10) to makewhat Ball’s model are supposed to be i.i.d. shocks instead be serially correlated(by making vt feed back appropriately on xt ). The maximizing agent in problem(10) responds by changing the decision rule so that it is less vulnerable to lowfrequency misspecifications. In Ball’s model, this can be done by having the monetary
authority adjust interest rates more aggressively in its “Taylor rule.” Notice how thefrequency decompositions under the σ = –.04 and σ = –.085 rules are flatter.7 They
thereby achieve robustness by rendering themselves less vulnerable to low-frequency
misspecifications.
A permanent income model is also most vulnerable to misspecifications of theincome process at the lowest frequencies, since it is designed to do a good job atsmoothing high-frequency movements. For a permanent income model, a preference
for robustness with respect to the specification of the income process then induces
precautionary saving of a type that does not depend on the third derivative of thevalue function.

7. These are frequency decompositions of the Ball’s criterion function operating under the robust rules when theapproximating model governs the data.

223

Figure 3 Frequency Decompositions of Ez't z t for Objective Function of Ball’s Model
under Three Decision Rules

σ =0
9

σ = –.04

8

σ = –.085

7
6
5
4
3
2
1
0
0.0

0.5

1.0

1.5

2.0

2.5

3.0

Note: (σ = 0, –.04, –.085). The robustness parameter satisfies: θ = –1/σ.

G. Multi-Agent Settings
We have discussed only single-agent decision theory, despite the presence of the minimizing second agent. The minimizing agent in the Bellman equation (10) is fictitious,
a computational device for the maximizing agent to attain a robust decision rule.
Macroeconomists’ routines use the idea of a representative agent to study aggregate
phenomena using single-agent decision theory, for example by studying planningproblems and their decentralizations. We can use a version of equation (10) in conjunction with such a representative agent device. A representative agent formulation
would attribute a common approximating model and a common set of admissiblemodel perturbations to the representative agent and the government. Robust Ramsey
problems can be based on versions of problem (10) augmented with implementability
constraints. See Hansen and Sargent (2000a, 2000b) for some examples.

V. Self-Confirming Equilibria
We have focused on misspecifications that are difficult to detect with moderate-sizeddata sets, but that can be distinguished with infinite ones. We now turn to a more
subtle kind of misspecification, one that is beyond the capacity of detection error probabilities to unearth even in infinite samples. It underlies the concept of self-confirming
equilibrium, a type of rational expectations that seems natural for macroeconomics.
A self-confirming equilibrium attributes possibly distinct personal probabilities(models) to each agent in the model. Those personal probabilities (1) are permitted to

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Acknowledging Misspecification in Macroeconomic Theory

differ on events that occur with zero probability in equilibrium, but (2) must agree on
events that occur with positive probability in equilibrium. Requirement (2) means thatthe differences among personal probabilities cannot be detected even from infinite
samples from the equilibrium. Fudenberg and Kreps (1995a, 1995b), Fudenberg
and Levine (1993), and Sargent (1999) advocate the concept of self-confirming
equilibrium partly because it is the natural limit point of a set of adaptive learning
schemes (see Fudenberg and Levine [1998]). The argument that agents will eventuallyadapt to eliminate discrepancies between their model and empirical probabilities
strengthens the appeal of a self-confirming equilibrium but does nothing to promotesubgame perfection.8 A self-confirming equilibrium is a type of rational expectations
equilibrium. However, feature (1) permits agents to have misspecified models that fit
the equilibrium data as well as any other model but that miss the “causal structure.”
The beliefs of a large player about what will occur off the equilibrium path can
influence his choices and therefore outcomes along the equilibrium path.
That self-confirming equilibria permit large players—in particular, governments—
to have wrong views provides ways to resolve what we at Minnesota used to call“Wallace’s conundrum” in the mid-1970s.9 Wallace had in mind a subgame perfectequilibrium. He noted that there is no room for policy advice in a rational expectations equilibrium where private agents know the conditional probabilities of future
choices of the government. In such a (subgame perfect) equilibrium, there are no freevariables for government agents to choose: their behavior rules are already known andresponded to by private agents. For example, if a researcher believes that the historical
data obey a Ramsey equilibrium for some dynamic optimal tax problem, he has noadvice to give other than maybe to change the environment.
Self-confirming equilibria contain some room for advice based on improved
specifications of the government’s model.10 But such advice is likely to be resisted
because the government’s model fits the historical data as well as the critic’s
model. Therefore, those who criticize the government’s model must do so on purely
theoretical grounds, or else wait for unprecedented events to expose an inferior fitof the government’s model.

8. Bray and Kreps (1987) present a Bayesian analysis of equilibrium learning without misspecification. They express
regret that they precluded an analysis of learning about a model by having set things up so that their agents can be
Bayesians (which means that they know the model from the beginning and learn only within the model). Agent’s
model misspecification, which disappears only eventually, is a key part of how Fudenberg and Levine (1998) andothers analyze learning about an equilibrium.
9. See Sargent and Wallace (1976).
10. Much of macroeconomists’ advice has been of that form: think of the arguments about the natural unemployment rate theory, which were about using new cross-equation restrictions to reinterpret existing econometric
relations.

225

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Chen, Z., and L. G. Epstein, “Ambiguity, Risk, and Asset Returns in Continuous Time,” mimeo, 1998.
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pp. 643–669.
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———, and T. Wang, “Intertemporal Asset Pricing under Knightian Uncertainty,” Econometrica, 62(3), 1994, pp. 283–322.
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———, and ———, The Theory of Learning in Games, MIT Press, 1998.
Gilboa, I., and D. Schmeidler, “Maxmin Expected Utility with Non-Unique Prior,” Journal of
Mathematical Economics, 18 (2), 1989, pp. 141–153.
Hansen, L. P., and T. Sargent, “Seasonality and Approximation Errors in Rational Expectations
Models,” Journal of Econometrics, 55, 1993, pp. 21–55.
———, and ———, “Discounted Linear Exponential Quadratic Gaussian Control,” IEEE
Transactions on Automatic Control, 40, 1995, pp. 968–971.
———, and ———, “Elements of Robust Filtering and Control for Macroeconomics,” mimeo, 2000a.
———, and ———, “Robust Control and Filtering in Forward-Looking Models,” unpublished manuscript, 2000b.
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Sargent, T. J., Conquest of American Inflation, Princeton University Press, 1999.
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Comment

Sims, C. A., “Approximate Prior Restrictions in Distributed Lag Estimation,” Journal of the American
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———, “Rational Expectations Modeling with Seasonally Adjusted Data,” Journal of Econometrics, 55,
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Weil, P., “Precautionary Savings and the Permanent Income Hypothesis,” Review of Economic Studies,60, 1993, pp. 367–383.
Whittle, P., Risk-Sensitive Optimal Control, John Wiley and Sons, 1990.

Comment
FUMIO HAYASHI
University of Tokyo
Serving as a discussant for a Lars Hansen paper is not only an honor but also achallenge. Throughout my career as an academic, I have read his writings and learned
a lot from them. I have found his style to be optimal for presenting ideas in theinherently difficult field that he covers. However, this particular paper I agreed to
discuss was hard to read. I therefore devote more than half of my discussion tosummarize the paper. This is followed by a list of my comments on the paper.
To describe the robust control problem considered in the paper, let us start fromthe relatively familiar ground, which is the standard linear control theory withquadratic objectives described at the beginning of the paper’s Chapter IV. There is astate vector xt and a control vector ut , which follow a system of linear difference
equations given in equations (1) and (2) of the paper. I will refer to this system as themodel. Here, {wt } is a sequence of shocks with unit variance. A confusing feature of
the paper is that in some parts of the paper this shock sequence is a random sequencewith mean zero, while in other parts it is either a deterministic sequence or (if I
am not mistaken) a random sequence with nonzero mean (as in the part of the paperthat discusses Ball’s macro model). To encompass all these cases, I will allow themean of wt , denoted µt , to differ from zero. So µt is a sequence of time-dependentintercepts of the model.
The objective is to maximize the negative of the expected discounted value ofa loss function given in equation (3) of the paper, that is, to minimize the expecteddiscounted loss:∞∑ β E z  .
t

2

t

(1)

t =0

The controller’s objective is to choose a feedback rule ut = Fxt that minimizes thiscost. In the context of monetary policy, the model is a macro model involving suchvariables as the inflation rate and output as the state variables and the interest rate as
the control variable.
The model’s parameter vector α consists of the coefficients of the differential
equations (A, B, C, H, J ). Write the expected loss shown above as
R (F, α, { µt }).

(2)

227

Here, F is the feedback rule to be chosen by the controller and { µt } is the sequence ofthe mean of the shocks.
With this notation, the standard control problem can be written as
min R (F, α, {0}),(3)Fwhere {0} is a sequence of zeros. As the notation hopefully makes clear, the controller
here is assumed to know the true model parameter α and the shock sequence is
known to have mean zero (i.e., µt = 0 for all t ).
In the emerging economic literature on robust control (where Lars Hansen andThomas Sargent are two of the main contributors), the controller is not assumed toknow the model parameter; the coefficients α can differ from the true parametervalue. From what I know about the literature, there are two distinct formulations.
Both formulations employ the “minimax” criterion. Let ∆ be the unknown nonrandom deviation of the true parameter value from α , and let D be the set of
possible deviations. The minimax problem considered by the first formulation can be
written as
Robust Control I: min max R (F, α + ∆, { 0}).

(4)

∆ ∈D

F

Therefore, it is assumed, as above, that the shock sequence is known to have zeromean. This formulation has been applied to a macro model similar in spirit to Ball’s
model by Stock (1999) for a particular specification of the set D.
The second formulation, which is the one taken by Hansen and Sargent (andalso, I gather, in the engineering literature on robust control), differs from the first inthat the controller knows the true coefficient parameters to be α but the true model
has nonzero shock means. Let M be the set of possible mean sequences. The second
formulation can be written as
Robust Control IIa: min max R (F, α, { µt }).
F(5){ µ t }∈M

A natural idea, which arises in any constrained optimization problem, is torepresent the constraint as a penalty in the objective function. A variant of thissecond formulation can be written as

[

∞

]

Robust Control IIb: min max R (F, α, { µt }) – θ ∑ β µ't µt .
F

{µt}

t

t =0

(6)

Now, there is no constraint on { µt } (or, the set M is replaced by a very large set), but∞ tthere is a penalty term θ ∑ t = 0 β µ't µt in the objective function. If θ = ∞, then the { µt }that maximizes the objective function is such that µt = 0 for all t . Thus, the case with
θ = ∞ is the standard control problem discussed above. This formulation of indexing
the problem by θ is attractive for several reasons. First, it provides a convenient way
of measuring the degree of ignorance on the part of the controller. Second, as shownin Section IV.E of the paper, some intelligent discussion is possible about how θ
might be chosen. Third, as extensively discussed in Hansen and Sargent (2000),

228 MONETARY AND ECONOMIC STUDIES (SPECIAL EDITION)/FEBRUARY 2001

Comment

the whole apparatus of linear-quadratic control can be employed to compute thesolution. Sargent (1999) applied this formulation IIb to compute the robust feedback
rule for Ball’s macro model for different choices of θ . Some of the results of thiscomputation are also reported in the paper.
Having described what I think is the crux of the paper, I now present several
comments in ascending order of substantiveness.
• Stock (1999), which adopted Robust Control I described above, finds that therobust rule entails policy aggressiveness , at least for the particular macro modeland for a particular range of deviations D. That is, in his words, the robust
monetary policy “exhibits very strong reaction to inflation and the output gap
to guard against the possibility that the true response of the economy to monetary policy is weak.” Sargent (1999), who applied Robust Control IIb to Ball’s
macro model, also reports policy aggressiveness. Is policy aggressiveness ageneric feature of robust control? If not, what are the features of the modelthat entails policy aggressiveness?
• I would have liked to see a discussion of how the three formulations—RobustControl I, IIa, and IIb—are related. In particular, is there some sort of equivalence theorem stating that for some choice of D and M the feedback rule given
by Robust Control I is the same as that of Robust Control IIa?
• My understanding is that Gilboa and Schmeidler (1989) provided a set of
axioms under which the minimax rule is a rational behavior when the decision
maker does not know the model. It seems to me that Robust Control I and IIa
can be justified on those grounds. I would like to know if there is any similar
theoretical basis for Robust Control IIb.
• There is a question of which formulation of robust control is more useful to thecentral bank. Although there seems to be a broad agreement as to what the loss
function should be, economists and policy makers disagree about what the true
macro model is. It might be plausibly argued that the model uncertainty is
about the slope coefficients, rather than about the intercept. If so, RobustControl I should be preferred.
• I am not sure exactly how this paper relates to the theme of the conference,
which is about monetary policy in the age of near-zero interest rate. It is probably true that the effect of monetary policy is more uncertain when the interest
rate is zero than when positive, but this increased uncertainty is due to a nonlinearity brought about by the non-negativity constraint on the nominal interest
rate. It seems to me that the first-order business for researchers is to examine theimplication of this nonlinearity.
• The Bank of Japan’s monetary policy is determined by committee. My impression from reading the minutes of the Policy Board meeting is that the board
members subscribe to different models and that each member thinks his or her
own model is the true one. Here, robust control theory may be useful as apositive theory of monetary policy. A rational way of making policy would be totake the range of models held by the board members as the D in RobustControl I and adopt the solution to the minimax problem.

229

References

Gilboa, I., and D. Schmeidler, “Maxmin Expected Utility with Non-Unique Prior,” Journal of
Mathematical Economics, 18 (2), 1989, pp. 141–153.
Hansen, L., and T. Sargent, “Elements of Robust Control and Filtering in Macroeconomics,”
manuscript, 2000.
Sargent, T., “Comment on ‘Policy Rules for Open Economies,’” in J. B. Taylor, ed. Monetary Policy
Rules, University of Chicago Press, 1999, pp. 144–154.
Stock, J., “Comment on ‘Policy Rules for Open Economies,’” in J. B. Taylor, ed. Monetary Policy Rules,
University of Chicago Press, 1999, pp. 253–259.

Comment
TIFF MACKLEM11
Bank of Canada
A great deal has changed in a decade. The average rate of inflation in the Organisation
for Economic Co-operation and Development (OECD) countries in 1990 was
6.2 percent. By the end of the decade, it had fallen to 2.5 percent. With this success inunwinding inflation, the focus of monetary policy has shifted away from reducing
inflation to how best to maintain low and stable inflation so as to reap the benefits ofprice stability in terms of improved economic performance. With this shift in the focus
of policy has come a welcome rebirth of research on monetary control. This latest
contribution by Lars Hansen and Thomas Sargent is a leading example of how
economic theory and control theory can be combined to confront very real issues facedby central bankers who must make decisions in an uncertain world. Central bankers
like low inflation. We are also very pleased that the decision problems we face are being
addressed by some of the most influential thinkers in the profession.
My comments proceed in three parts. First, by making reference to some of the
inputs into monetary policy decisions at the Bank of Canada, I want to convince youthat the issues Hansen and Sargent address really are of concern to central banks.
Second, I compare the outcomes of robust control to observed policy, and speculate on
the relevance of robust control to practical central banking. Finally, I contrast and compare traditional optimal control in models with multiplicative parameter uncertainty
to robust control and suggest that maybe they are not as different as they appear.

I. Uncertainty and Robustness in Policy Making
Hansen and Sargent’s starting point can be summarized by two assertions: (1) economic
models are approximations; and (2) economic policy should take this into account. It
is hard to imagine anyone at a central bank disagreeing with either of these assertions.
11. These comments benefited from discussions with David Longworth, John Murray and James Yetman. The viewsin this comment are my own. No responsibility for them should be attributed to the Bank of Canada.

230 MONETARY AND ECONOMIC STUDIES (SPECIAL EDITION)/FEBRUARY 2001

Comment

Anyone who has worked at a central bank has seen economic thinking and the models
change much more quickly than the economy could possibly be changing. Hopefully,
this evolution in economic thinking reflects progress toward the “true” model.
However, even if—with enough time—we could build a nearly exact model of theeconomy, the economy is not a constant. Thus, the reality is that the best we can hopefor is better approximations. This raises the question: What are central banks doing totake into account the fact that their models are approximations?
The short answer is they diversify. Policy makers do not rely on a single set ofassumptions with a single model. Rather, they consider a variety of assumptions
and a variety of models, as well as a wide range of evidence that does not fit neatly
into their models (precisely because these models are approximations). To make thismore concrete, I will briefly describe the inputs into policy decisions at the Bank of
Canada—matters I know something about. Most central banks do somethingroughly similar. Then I want to compare what central banks do to Hansen andSargent’s recommendation—robust monetary control.
Policy decisions at the Bank of Canada are based on a portfolio of analysis, somemodel-based, some more judgmental. The starting point is the staff economic
projection. The projection is based on a structural model of the Canadian economy(the Quarterly Projection Model). The projection is essentially a deterministic simulation of the model using the base case monetary rule (basically a forward-looking
Taylor rule of the type estimated by Clarida, Galí, and Gertler [1997] for several
countries). Around this base case, we also routinely simulate a variety of alternative
scenarios. A first set of alternatives considers the implications of using different
monetary rules. These include simulations with the nominal interest rate held
constant for six to eight quarters (then reverting to the base case rule), simulations
using the markets expectations for interest rates and the exchange rate over the next
year or so (again reverting to the base case rule beyond this horizon), and simulations
with a more standard Taylor rule that uses the contemporaneous deviation between
inflation and its target (rather than the forecast deviation). A second set of alternatives considers the implications of varying key assumptions regarding variables thatare exogenous to the model (such as global demand, world commodity prices, anddomestic productivity growth), or the dynamics of key endogenous relationships in
the model (such as expectations). These alternatives are chosen on the basis of ajudgmental assessment of the major risks facing the Canadian economy. Some ofthese are “bad” outcomes, such as a hard-landing scenario for the U.S. economy oran exogenous upward shift in inflation expectations. Others are more attractive
outcomes, such as an acceleration in productivity growth, which nonethelessrequire the appropriate monetary policy response. So, in the context of the model,
substantial effort is expended to consider the robustness of the monetary policy
advice coming out of the model.
Moreover, the projection using the structural macro model is only one input into
policy. Formal forecasts are also conducted using money-based vector-error-correction
models that seek to exploit information in the monetary aggregates. As with thestructural-model forecast, several scenarios are considered for these money-based
forecasts. In addition, a short-run national forecast is built up from judgmental

231

regional outlooks. Finally, augmenting these forecasts are information from financial
markets, from industry liaison activities, and from our ongoing assessment of abroad range of economic indicators. All this is to say that a lot of time and energy
is devoted to looking at the issues from a variety of perspectives. This diversity is
motivated by uncertainty about the current state of the economy, uncertainty about
how the economy will respond to shocks, and uncertainty about the appropriate
policy response.
Is this robust monetary control? At a formal level, clearly it is not. But, justas households maximize utility without any knowledge of utility functions or the
Kuhn-Tucker theorem, perhaps central bank decision makers use robust rules
without formal knowledge of robust control theory or the axiomatic treatment of
Gilboa and Schmeidler. To begin to address this issue, it is useful to consider theoutcomes implied by robust control theory.

II. Outcomes of Robust Control and Observed Policy Compared
At the heart of robust control is the injection of pessimism into monetary decisions.
Formally, pessimism is injected by an inner minimization in which the policy makerperforms a worst-case analysis. Against this worst-case scenario, the policy makerthen maximizes by choosing the monetary response that leads to the best outcome inthis worst-case scenario. The implication of the injection of pessimism is that policy
is adjusted (relative to standard control theory) in such a way as to provide insurance
against the worst-case outcome.
Hansen and Sargent’s latest paper does not discuss in any detail what this implies
for the behavior of inflation, output, and interest rates, relative to the standardcontrol case. And I will confess up front that I do not know what the implications arein general. But at least in the context of simple, linear-quadratic, backward-looking
models, previous work by Sargent (1999) has shown that robust rules are moreaggressive than the rules produced by “standard” or “modern” control theory. The
intuition for this result is straightforward. Taking out insurance against bad outcomesmeans moving aggressively so as to maximize the chance of heading off these badoutcomes or at least making them less bad.
Does this type of aggressive action aimed at taking out insurance against very bad
outcomes look like observed central bank behavior? The short answer is no and yes.
A. No
In simple backward-looking models in which the central bank minimizes the squared
deviations of inflation from target and output from potential output, Svensson(1997) and Ball (1997) have shown that standard control theory yields an optimal
rule with the Taylor form. However, empirical versions of the Taylor rule that areeither estimated or calibrated to fit actual central bank behavior generally yield coefficients on inflation and output that are well below those suggested as optimal in the
Ball-Svensson models. This result has spawned a growing literature on why central
banks appear to smooth interest rates relative to what is implied by standard control

232 MONETARY AND ECONOMIC STUDIES (SPECIAL EDITION)/FEBRUARY 2001

Comment

theory.12 Robust rules, at least in the context of these models, are even more aggressive
than standard control rules. While several plausible explanations have been offered
for why observed central bank behavior is not as aggressive as standard control rules
suggest (e.g., Sack [1998], Woodford [1998]), robust rules raise the bar further.
Perhaps even more ingenious arguments can be used to square theory with reality.
Nevertheless, robust rules would appear to be moving theory and observed behavior
further apart, not closer together. So by this standard, observed policy outcomes do
not look much like those implied by robust control.
B. YesThe discussion so far has focused on the role of the central bank in maintaining pricestability. Formally, this is represented by a quadratic objective function in inflation andoutput. But central banks have another role—financial stability. And here, I think,observed policy looks much more like robust control. In particular, there are numerous
examples of situations in which central banks changed interest rates aggressively inwhat looks to be an attempt to take out insurance against financial instability and thepossibility that financial instability may endanger price stability. A few examples:• The stock market crash of 1987. Following the crash, the Federal Reserve
dropped the Federal Funds rate by 125 basis points against a background ofrobust growth and potential inflationary pressures.
• In September 1998, following the difficulties at Long-Term Capital Management and concern about financial fragility, the Federal Reserve lowered theFederal Funds rate by 75 basis points, again against a background of robust
economic growth with the potential for inflationary pressures.
• Following the 1997 financial crisis in Asia, most countries responded by tightening monetary and fiscal policies despite the likely negative impact of the financial
turmoil on economic activity and the absence of obvious macro problems—e.g.,
output growth had been strong, inflation was in hand, and fiscal situations weregenerally healthy.
• In August 1998, the Bank of Canada raised the bank rate 100 basis points torestore confidence in the external value of the Canadian dollar, despite the factthat the domestic macro situation was being adversely affected by the Asian crisis.
These examples all describe situations in which the central bank took actions
that appear puzzling relative to a “narrow macro” view that focuses on inflation andoutput, but can be rationalized as taking out insurance against a very bad outcome.
In this sense, they look very much like a maxi-min strategy in which the central bankaction is designed to maximize welfare in the worst-case scenario.
Why does central bank behavior look like robust control in some situations but
not in others? The occasions in which behavior looks like robust control share two
common features. First, they are typically situations that depend on things that wehave only very rudimentary models of—things like the psychology of perceptions
and financial market disintermediation. Second, they are situations in which things
can change very quickly. These two ingredients appear to fit neatly into Hansen and
12. See Goodhart (1999) for an eloquent exploration of the issue.

233

Sargent’s framework. The absence of good models means there is a considerable
degree of Knightian uncertainty in the sense that policy makers do not know the
distribution of outcomes. And the reality that the situation can deteriorate very
rapidly means the worst-case scenario could develop before the policy makers haveanother opportunity to reconsider the stance of policy.
In contrast, in the more narrowly defined macro view typically studied in theliterature on monetary rules, we do have reasonable models. While crude, thesemodels are based on estimated relationships that do hold more or less over time.
Policy makers are therefore better equipped to analyze risks and make probabilistic
statements about outcomes. Second inflation and output, unlike perceptions, do notchange overnight. Policy makers can therefore afford to take a more gradualist
approach, trading off some preemptiveness for the benefits of the more complete
information that comes with waiting.
To summarize, robust monetary control appears to have a lot to do with howmonetary policy reacts when facing a crisis or a potential crisis, but less to do with
monetary policy in more “normal” situations. Looking at the inputs into central bankdecisions (at least at the Bank of Canada) the diversified approach suggests that in“normal” times policy makers do not put special emphasis on the worst-case outcomes. Rather they look at all the outcomes and try to chart a “middle” course basedon an assessment of the relative probabilities of these outcomes. This produces
behavior that is much more gradual than suggested by robust control. In moving
forward, I encourage Hansen and Sargent (and others) to apply robust control to thegrowing literatures on financial stability and dealing with crises.

III. Robust Control and Standard Control Compared
Following Brainard’s seminal 1967 paper, the conventional wisdom has been thatparameter (or multiplicative) uncertainty results in more muted monetary responses
relative to the certainty equivalence case. This conventional wisdom has been reinforced by empirical applications, such as Sack (1998), that show that incorporating
multiplicative uncertainty goes at least partway to explaining why observed policy
actions are less aggressive than those suggested by standard control theory in models
with only additive uncertainty. As shown by Hansen and Sargent, robust control
theory, like multiplicative uncertainty, results in a breakdown of certainty equivalence. But while the conventional wisdom is that multiplicative uncertainty results inmore muted policy responses, robust control calls for more aggressive actions.
Apparently, Brainard and robust control are at odds.
In closing, I want to challenge this interpretation and suggest that the outcomesof robust control and standard control with multiplicative uncertainty may havemore in common than has been suggested to date in the literature. To do this, I draw
on a recent study by one of my colleagues—Gabriel Srour.
Srour (1999) considers the simple backward-looking Ball-Svensson model inwhich the policy maker seeks to minimize the expected weighted sum of deviationsof inflation from the target and deviations of output from potential:

234 MONETARY AND ECONOMIC STUDIES (SPECIAL EDITION)/FEBRUARY 2001

Comment

αE (y – y *)2 + (1 – α)E (π – π *)2,

(1)

where π is the inflation rate, π * is the inflation target, and y and y * are actual andpotential output (in logs). The economy is described by a Phillips curve and an
aggregate demand equation:

πt +1 – π * = a t (πt – π *) + dt (yt – y *) + εt +1
yt +1 – y * = bt (yt – y*) – ct (rt – r*) + ηt +1,(2)

where rt is the short-term real interest rate (the instrument of monetary policy), r*
is the equilibrium real interest rate, and ε t +1 and ηt +1 are white-noise shocks. The
twist that Srour adds relative to Ball and Svensson is that a, b, c, and d are mutually––uncorrelated and i.i.d. random variables with means 1, b , c–, and d , respectively.
Srour shows that the optimal rule with multiplicative uncertainty continues tohave the Taylor form (as in the Ball-Svensson case with only additive uncertainty):
rt – r* = A (yt – y *) + B(πt – π *).

(3)

The impact of uncertainty is to alter the optimal coefficients A and B relative to what
they would be in the certainty equivalence case with no parameter uncertainty (call––these values A and B ). Srour shows that, in general, the relationship between A and––
A and between B and B is ambiguous, and depends on the relative uncertainties of
all the parameters in the model. In other words, multiplicative uncertainty does notnecessarily lead to more muted monetary responses.
How does this arise? To answer this question, Srour looks at the implications of
uncertainty in the parameters a, b, c, and d one at a time. What he shows is that
uncertainty in c and d —the coefficients that translate interest rate changes to changes–
in output and inflation—leads to less aggressive monetary responses (i.e., A < A and–B < B ). Thus, for example, uncertainty about the interest elasticity of demand leads
the policy maker to move interest rates less in response to shocks. This is simply theclassic result obtained by Brainard (1967). In contrast, uncertainty about a—how
surprises in the Phillips curve translate into future inflation—results in a more––aggressive monetary response (i.e., A > A and B > B ). The intuition for this last result
is that uncertainty about a raises the expected variance of future inflation. To mitigate
this, the monetary authority has the incentive to bring inflation back to target more
quickly so as to lower the volatility of inflation in future periods.
What I take away from this is that standard control theory with multiplicative
uncertainty can produce outcomes that look like the more aggressive outcomes predicted by robust control theory. Perhaps this example is just a special case of robust
control theory. Ostensibly, however, this is not the case in the sense that there is no
inner minimization in Srour’s example. In other words, the aggressive response does
not arise because of the injection of pessimism. It arises simply because a moreaggressive response will minimize the expected variance.

235

My final point is a question for Hansen and Sargent: Does robust control always(or even almost always) lead to more aggressive responses than in the certainty equivalence case? I do not pretend to know the answer to this question. But one feature of
the examples considered to date with backward-looking, linear-quadratic models
raises a question. In the example considered in Sargent (1999), the Phillips curve is of
the accelerationist variety—i.e., a = 1. This is the case in which inflation surprises
lead to permanently higher inflation (other things equal), and I wonder if it is thisassumption that is interacting with the pessimistic element to produce the moreaggressive monetary responses. To put it another way, suppose a = 0.5, would robust
control still produce a more aggressive response than in the certainty equivalence
case? Would robust control rules look more like observed behavior?
I raise this point because with credible inflation control, Sargent (1971) taught us
almost three decades ago that a should be less than one. And, indeed consistent with
Sargent’s classic result, there is now evidence for a number of countries that a has
fallen in the 1990s as inflation control has improved and the credibility of pricestability has increased. Does this mean that robust control might not predict suchaggressive responses after all?

References

Ball, Laurence, “Efficient Rules for Monetary Policy,” NBER Working Paper No. 5952, National
Bureau of Economic Research, 1997.
Brainard, William, “Uncertainty and the Effectiveness of Policy,” American Economic Review, 57, Papers
and Proceedings, 1967, pp. 411–425.
Clarida, Richard, Jordi Galí, and Mark Gertler, “Monetary Policy Rules in Practice: Some International
Evidence,” NBER Working Paper No. 6254, National Bureau of Economic Research, 1997.
Goodhart, Charles, “Central Bankers and Uncertainty,” Annual Keynes Lecture at the British Academy,
published in Bank of England Quarterly Bulletin, February 1999, pp. 102–114.
Sack, Brian, “Does the Fed Act Gradually? A VAR Analysis,” Finance and Economics Discussion Paper
Series No. 1998-17, Board of Governors of the Federal Reserve System, 1998.
Sargent, Thomas, “A Note on the Accelerationist Controversy,” Journal of Money, Credit and Banking,8, 1971.
———, “Comment on ‘Policy Rules for Open Economies,’” in John B. Taylor, ed. Monetary Policy
Rules, University of Chicago Press, 1999, pp. 144–154.
Srour, Gabriel, “Inflation Targeting under Uncertainty,” Bank of Canada, Technical Report No. 85,
1999.
Svensson, Lars E. O., “Inflation Forecast Targeting: Implementing and Monitoring Inflation Targets,”
European Economic Review, 41, 1997, pp. 1111–1146.
Woodford, Michael, “Optimal Monetary Policy Inertia,” mimeo, Princeton University, 1998.

General Discussion
In reply to Tiff Macklem’s comment on how monetary policy should respond tocrisis situations, Lars P. Hansen explained that while it is not explicitly stated, his
paper did envision two types of exogenous shocks: a “jump diffusion process” with

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General Discussion

rapid changes, and a “standard Brownian process.” He said that it would be possible
to describe the former as “crisis conditions” and the latter as “ordinary conditions.”
Marvin Goodfriend noted that the model used in Hansen’s paper was an“engineering model” with a high degree of universality, but to make it applicableto the practical design of economic policy, the idiosyncrasy of economic conditions
should be carefully examined. Vítor Gaspar pointed out differences in the frameworks
appropriate to analyze individual decision-making and collective decision-making.
In this context, he emphasized the importance of analyzing the decision-making
process in policy committees. Spencer Dale commented on the difficulty of comparative evaluations among various models, which the central bank must confront whenseeking robustness for its economic model.
Hiroshi Fujiki agreed with Hansen on the usefulness of the application of asset
prices for the central bank, but asked two questions: (1) what kind of asset pricing
model was assumed to underlie the argument that changes in the subjective discountfactor and robustness could be identified; and (2) whether robustness or thepermanent income hypothesis could better explain the weakness of consumption
experienced by Japan in recent years. On the latter question, Hansen responded thatthe central bank should actively utilize the information obtained from asset prices,
particularly by estimating equity price models to clarify the role played by robustness.
Shin-ichi Fukuda hinted to the possibility of the existence of multiple equilibria
for asset prices in Hansen’s model. Lars E. O. Svensson pointed out that Brunner andMeltzer had discussed robust optimal control more than 30 years ago, and felt that
this ought to be referred to in Hansen’s paper.
As a practical implication of the exercise on the monetary policy, Guy M.
Meredith suggested that the success of the flexible U.S. monetary policy during thelast several years could have many indications for current Japanese monetary policy.

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