Robust control and filtering of forward-looking models

Lars Peter Hansen

University of Chicago

Thomas J. Sargent

Stanford University and Hoover Institution

5 October 2000

ABSTRACT

This paper shows how to compute robust Ramsey (aka Stackelberg)plans for linear models with forward looking private agents. We formulatea Bellman equation for the robust plan. We describe robust filtering forwhen some of the forcing variables (like potential GDP or trend growth)are not observed, and how the decision problem interacts with the filteringproblem. We use a ‘new synthesis’ macro model of Woodford as an example.

1. Introduction

Milton Friedman expressed an enduring concern when he recommended that designersof macroeconomic policy rules acknowledge model uncertainty. His style of analysis re-vealed that he meant a kind of model uncertainty that could not be formalized in terms ofobjective or subjective probability distributions over models. For despite his willingness touse Savage’s (1954) axioms in other work (see Friedman and Savage (1948)), he declinedto use them in his writings on monetary policy. Instead Friedman indicated that he knewtoo little about the dynamic structure of the economy to allow him to specify the personalprobabilities needed to rationalize policy choices using Savage’s method. Friedman men-tioned unknown long and variable lags in the effects of policies to support his preferencefor policy rules that don’t depend on detailed knowledge about economic dynamics.

Recently Bennett McCallum and others have taken up Friedman’s theme by embracinga research program designed to evaluate the robustness of monetary policy rules across setsof alternative models. In his comment on Rotemberg and Woodford (1997), McCallumrepeated his preference for using multiple models when analyzing a proposed rule:1

∗ We thank Evan Anderson, Dirk Krueger, Hanno Lustig, and Michael Woodford forhelpful discussions. We thank Chao Wei and Stijn Van Nieuwerburgh for excellent helpwith the calculations and comments on earlier drafts.

1 Blinder (1998, pp. 12-13) advocates a procedure for assessing the robustness of proposed decisions.

2

I have favored a research strategy centering around a rule’s robustness, in thefollowing sense: Because there is a great deal of professional disagreement as tothe proper specification of a structural macroeconomic model, it seems likely to bemore fruitful to strive to design a policy rule that works reasonably well in a varietyof plausible quantitative models, rather than to derive a rule that is optimal in anyone particular model. (McCallum, 1987, p. 355).

McCallum also criticized analyses of monetary policy rules that unrealistically assume thatlatent variables like ‘potential GNP’ or ‘trend productivity growth’ are observed withouterror by the policy maker.2

This paper describes how robust control and filtering theory can be used to addressboth of McCallum’s concerns: robustness and hidden state variables. We extend earlierwork on robust control theory to solve Ramsey or Stackelberg problems – i.e., problemsof optimal government policy where the government can commit itself. Our use of robustcontrol theory in this context and our desire to put the private agents and the governmenton the same footing induces us to model robustness differently than do McCallum and hisco-workers. They take a small set of particular models and evaluate the performance of agiven rule across those models. Typically, several of those models are rational expectationsmodels in which the agents forecast using the model itself. In contrast, we explicitly specifyonly a single model, but treat it as an approximation. We surround the approximatingmodel with a continuum of unspecified alternative models that fit the data nearly as well.Both the government and the agents inside the model explicitly confront their concernabout model misspecification in making forecasts and designing policies. We attributeequal concerns about model misspecification to the private sector and the government. Theprivate sector and the government share the same approximating model of the stochasticvariables shaking the economy, and the same doubts about that model in the form of acommon penumbra of alternative models surrounding it. These alternative models canbe difficult to distinguish statistically from the approximating model based on finite datasets, so that the government’s and public’s concerns about model misspecification are wellfounded.

In Ramsey or Stackelberg problems, the government can commit itself to a policy fromtime 0 onward. To design its policy, the government takes into account that the privatesector’s current decisions respond to the government’s future actions. That strategic in-teraction requires us to develop new methods to compute robust Ramsey policies. Thebehavior of the private sector is described by a set of Euler equations, first-order condi-tions from the choice problems of private agents who take as given an entire sequence ofsettings of a government policy instrument. The government’s policy instruments enterthe private sector’s Euler equations as ‘forcing variables.’ Following Hansen, Epple, andRoberds (1985), Sargent (1987), Currie and Levine (1987), Pearlman, Currie, and Levine(1986), Pearlman (1992), Woodford (1998), King and Wolman (1999), and Marcet andMarimon (1999), we solve the Ramsey problem by forming a Lagrangian with sequences ofLagrange multipliers attaching to each of the private sector’s Euler equations. The privatesector’s Euler equations become ‘implementability constraints’ requiring the governmentto confirm the expectations about government policy settings that are reflected in privateagents’ previous decisions. These multipliers encode how the government’s policy choices

2 Several of the contributions in Taylor (1999) share McCallum’s concern for robustness.

3

depend on the history of the economy. The multipliers allow us to obtain a recursive repre-sentation of the Ramsey problem. We modify the government’s Lagrangian to incorporatea preference for robustness.

While robust decision and filtering theory can seem intimidating at first encounter,they are easy to use. A purpose of this paper is to convince the reader of this. In single-agent problems, robust decision rules are justified and can be computed as the NashMarkov equilibrium of a zero-sum two person game. The zero-sum feature means thatthere is only one value function. For a single-agent problem, the value function can becomputed by solving a Bellman equation that is as simple as ones we now routinely solvein macroeconomics. We show this in section 4. Then we show that computing robustdecision rules in forward looking models is also easy. There is another Bellman equationwhose solution yields a robust decision rule for the forward looking model. Because wework in a linear-quadratic context, associated with the Bellman equation is a standardmatrix Riccati equation.

We then show how to solve a joint robust filtering and control problem in a forwardlooking model. Here the government and the public both solve a filtering problem aswell as their control problems, taking into account their doubts about the laws of motionthat determine the usual optimal filtering formulas. We use earlier results of Basar andBernhard (1995) and Hansen and Sargent (2000) to adjust the ordinary Kalman filter toreflect a concern about robustness.

This paper is organized as follows. Section 2 summarizes basic ideas in single-agentrobust control theory. It can be skipped by readers familiar with these ideas. Section 3states the class of robust Ramsey problems that we want to solve. Section 4 describes aBellman equation and an associated Riccati equation for the robust single-agent problem.Section 5 shows how that same Riccati equation can be used to solve the robust Ramseyproblem. Section 6 describes the connection of our work to a Bellman equation that Marcetand Marimon (1999) have used to solve problems with implementability constraints likeours. Section 7 briefly describes a monetary policy example of Woodford. Section 8 showshow to adjust the computation of a robust Ramsey rule when some of the state variablesare latent, requiring that agents and the government filter. Section 9 computes an examplefor Woodford’s model, showing a sense in which the robust rule for setting the interestrate is less agressive than the rule formed without a concern for model misspecification.Section 10 concludes.

1.1. Related literature

Brunner and Meltzer (1969) and Von Zur Muehlen (1982) were early advocates ofzero-sum two person games (i.e., min−max) for representing model uncertainty and de-signing macroeconomic rules. Stock (1999), Sargent (1999), and Onatski and Stock (1999)have used versions of robust control theory to study robustness of pure backward lookingmacroeconomic models. They focussed on whether a concern for robustness would makepolicy rules more or less agressive in response to shocks. Vaughan (1970), Blanchard andKhan (1980), Whiteman (1983), and Anderson and Moore (1985) are early sources on solv-ing control problems with forward-looking private sectors. Anderson, Hansen, McGrattan,and Sargent (1996) survey efficient computational algorithms for such models. Without a

4

concern for robustness, Pearlman, Currie, and Levine (1986), Pearlman (1992) and Svens-son and Woodford (2000) study the control of forward looking models where part of thestate is unknown and must be filtered. DeJong, Ingram, and Whiteman (1996), Otrok(In press), and others study the Bayesian estimation of forward looking models. Theysummarize the econometrician’s doubts about parameter values with a prior distribution,meanwhile attributing no doubts about those parameter values to the private agents intheir models. Mark Giannoni (2000) assumes the same asymmetry when he studies robust-ness in a forward looking macro model. He models the policy maker (but not the agents)as knowing only bounds for two key parameter values, then computes the min−max pol-icy rules. Kasa (2000) also studies robust policy in a forward looking model. Christianoand Gust (1999) study robustness from the viewpoint of the determinacy and stability ofrules under nearby parameters, sharing a perspective of robust control theorists like Basarand Bernhard (1995), who are interested in finding rules that stabilize a system under thelargest set of departures from the model.

2. Primer on robust control theory

This section briefly summarizes ideas used in single-agent models of robust control.

2.1. The approximating model

A single approximating model defines transition probabilities. For example, below weextensively use the following linear-quadratic state space system:

yt+1 = Ayt + But + Cηt+1 (2.1)dt = Hyt + Jut (2.2)

where yt is a state vector, ut is a control vector, and dt is a target vector, all at datet. In addition, suppose that ηt+1 is a vector of independent and identically normallydistributed disturbances with mean zero and covariance matrix given by I. The targetvector is used to define preferences via:

−12

∞∑t=0

βtE|dt|2 (2.3)

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 of controllaw ut = −Fyt.

5

2.2. Distortions

A particular way of distorting transition probabilities defines a set of models that expressan agent’s model ambiguity. First we need a concise way to describe a class of alternativespecifications. For a given policy u = −Fy, the state equation (2.1) defines a Markovtransition kernel π(y′|y). For all y′, let w(y′|y) be positive for all y. We can use w(y′|y)to define a distorted model via πw(y′|y) = w(y′|y)π(y′|y)

Ew(y′|y) , where the division by Ew(y′|y)lets the quotient be a probability. Keeping w(y′|y) strictly positive means that the twomodels are mutually absolutely continuous, which makes the models difficult to distinguishin small data sets. Conditional relative entropy is a measure of the discrepancy betweenthe approximating model and the distorted model. It is defined as

ent =∫

logπw (y′|y)π (y′|y)

πw(dy′|y)

. (2.4)

Conditional entropy is thus the conditional expectation of the log likelihood ratio of thedistorted with respect to the approximating model, evaluated with respect to the distortingmodel.

In the models that occupy us in this paper, we represent perturbations to model (2.1)by distorting the conditional mean of the shock process away from zero:

yt+1 = Ayt + But + C (ηt+1 + wt+1) (2.5)wt+1 = ft (yt, . . . , yt−n) (2.6)

Here (2.6) is a set of distortions to the conditional means of the innovations of the stateequation. In (2.6), these are permitted to feed back on lagged values of the state andthereby represent misspecified dynamics. For the particular model of the discrepancy(2.6), by exploiting the conditional normality of xt+1 it can be established that

ent =12w′

t+1wt+1. (2.7)

2.3. Conservative valuation

We use a penalized minimization problem to define a conservative way of evaluatingcontinuation utility. Let V (yt+1) be a continuation value function. Fix a control lawut = −Fyt so that the transition law under a distorted model becomes

yt+1 = Aoyt + C (ηt+1 + wt+1) , (2.8)

where Ao = A − BF . We define a distorted expectations operator Rθ(V (yt+1)) for eval-uating continuation values as the indirect utility function for the following minimizationproblem:

Rθ (V (yt+1)) = minwt+1

θw′

t+1wt+1 + E (V (yt+1)), (2.9)

where the minimization is subject to (2.8). Here θ ≤ +∞ is a parameter that penalizesthe entropy between the distorted and approximating model and thereby describes the sizeof the set of alternative models over which the decision maker wants a robust rule. This

6

parameter is context-specific and depends on the confidence in his model that the historicaldata used to build it inspire. Anderson, Hansen, and Sargent (2000) show how detectionprobabilities for discriminating between models can be used to discipline the choice of θ.

Let wt+1 = Gθyt attain the minimum on the right side of (2.9). The minimization onthe right side of (2.9) produces the following useful robustness bound:

EV (Aoyt + C (ηt+1 + wt+1)) ≥ Rθ (V (yt+1)) − θw′t+1wt+1. (2.10)

The left side is the expected continuation value under a distorted model. The right sideis a lower bound on that continuation value. The first term is a conservative value of thecontinuation value under the approximating (wt+1 = 0) model. The second term on theright gives a bound on the rate at which performance deteriorates with the entropy of themisspecification. Decreasing the penalty parameter θ lowers RθV (yt+1), making RθV (yt+1)more conservative as an estimate under the approximating model and also decreasing therate at which the bound on performance deteriorates with entropy. Thus, the indirectutility function Rθ(V (yt+1)) induces a conservative way of evaluating continuation utilitythat can be regarded as a context-specific distortion of the usual conditional expectationoperator.

There is a class of operators indexed by the parameter θ that summarizes the size ofthe set of alternative models: θ = +∞ signifies no preference for robustness, because itgives the minimizing agent in (2.9) no scope to distort the transition law. Lowering θexpands the set of alternative models over which the minimizing agent can choose. Thereis typically a lowest value θ for which the min problem in (2.9) is well defined. This‘breakdown’ value θ indexes the largest set of models against which it is reasonable towant robustness.3

The operator R can be represented as

Rθ (V (yt+1)) = h−1Et (h (V (yt+1))) (2.11)= θw′

t+1wt+1 + Et (V (yt+1)) (2.12)

where h(V ) = exp(−Vθ ) and wt+1 is the minimizing choice of wt+1. Note that h is a convex

function. This is an operator used by Epstein and Zin (1989, 1991) and Weil (1989, 1990),but without a connection to robustness.

Hansen, Sargent, and Tallarini (1999) use the distorted expectation operator R usedto rework the theory of asset pricing. Also, the operator has been used by Hansen andSargent (1995) to define a discounted version of risk-sensitive preference specification ofJacobson (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.

3 See Whittle (1990) and Hansen and Sargent (2000) for discussions of the relationship of θ to H∞ controltheory.

7

2.4. Robust decision rule

A robust decision rule is produced by the Markov-perfect equilibrium of a two-personzero-sum game in which the maximizing agent chooses a policy and a minimizing agentchooses a model. To compute a robust control rule we use the Markov-perfect equilibriumof the two-person zero-sum dynamic game associated with the Bellman equation:4

V (y) = maxu

minw

−1

2d′d +

βθ

2w′w + βEtV (y∗)

(2.13)

subject toy∗ = Ay + Bu + C (η + w) .

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 pa-rameter θ takes the value +∞, we have ordinary control theory because it is too costlyfor the minimizing agent to set a nonzero distortion wt+1. Lower values of θ achieve somerobustness (again see the role of θ in the robustness bound (2.10)).

In section 4, we show how the Bellman equation (2.13) leads to a simple Riccatiequation that determines both the robust decision rule u = −Fy and the worst casedistortion w = Gy.

2.5. Uncertainty and certainty equivalence

The random disturbance ηt+1 in (2.1) plays an important role in motivating our concernfor robustness because it can make it difficult to detect nonzero wt+1’s from moderate–sized samples of observations of linear functions of yt. The penalty parameter θ affects thesize of possible model misspecifications. In applications, we can discipline the parameterθ by linking it to the statistical theory for discriminating between alternative models. SeeAnderson, Hansen, and Sargent (2000). By letting model misspecification be difficult todetect, the presence of randomness in (2.1) justifies an agent’s concern with robustness.

Nevertheless, the linear-quadratic structure produces a version of the certainty equiva-lence principle that allows us to derive the robust rule for the stochastic system by droppingthe random term ηt+1 from (2.1) and analyzing a nonstochastic system. This is what wedo in most of the remainder of this paper. See Sargent (1998) and Hansen and Sargent(2000) for more examples.

4 There is only one value function because it is a zero-sum game.

8

2.6. Frequency domain representation

We can show that concern about model misspecification manifests itself in a desire toflatten the frequency response of targets d to shocks w.5 Thus, for the complex scalar ζ,let G(ζ) be the transfer function from the shocks wt to the targets dt under a fixed rule F :G(ζ) = H0(I − (A − BF )ζ)−1C, where H0 = H − JF . Let ′ denote matrix transpositionand complex conjugation, Γ =

ζ : |ζ| =

√β

, and dλ(ζ) = 1

2πi√

βζdζ. Then the criterion

(2.3) can be represented as

H2 = −∫

Γtrace

[G (ζ)′ G (ζ)

]d λ (ζ) .

Where −δj(ζ) is the jth eigenvalue of G(ζ)′G(ζ), we have

H2 =∑

j

∫Γ−δj (ζ) d λ (ζ) . (2.14)

Hansen and Sargent (2000) show that a robust rule is induced by using the criterion

ent (θ) =∫

Γlog det

[θI −G (ζ)′ G (ζ)

]d λ (ζ)

orent (θ) =

∑j

∫Γ

log [θ − δj (λ)] d λ (ζ) (2.15)

Because log(θ − δ) is a concave function of −δ, (2.15) is obtained from (2.14) by puttinga concave transformation inside the integration. Aversion to model misspecification isthus represented as additional ‘risk aversion’ across frequencies instead of across states ofnature. Under (2.15), the decision maker prefers decision rules that render trace G(ζ)′G(ζ)flatter across frequencies than does (2.14).

5 See Whiteman (1985a, 1985b), Otrok (in press, b), Kasa (2000) and references therein for previous usesof frequency domain characterizations of objective functions.

9

3. The robust Ramsey problem

This section defines the ordinary Ramsey problem then states a robust version of it.Section 7 analyzes a particular example of this problem.

3.1. Ordinary Ramsey problem

Let zt be an nz×1 vector of natural state variables, xt be an nx×1 vector of endogenousvariables free to jump at t, and ut a vector of government instruments. The zt vector isinherited from the past. It is the job of the model to determine the ‘jump variables’ xt

at time t. Included in xt are prices and quantities that adjust to clear markets at time t.

Define yt =[zt

xt

]. Define the government ‘target’ vector

dt = Hyt + Jut. (3.1)

A government wants to maximize

−∞∑

t=0

βtd′tdt. (3.2)

Where Et(o) denotes a forecast, or conditional expectation, of variable o, the government’sapproximating model is

[I 0

G21 G22

] [Etzt+1

Etxt+1

]=

[A11 A12

A21 A22

] [zt

xt

]+ But. (3.3)

We assume that the matrix on the left is invertible, so that6

[Etzt+1

Etxt+1

]=

[A11 A12

A21 A22

] [zt

xt

]+ But. (3.4)

Here we appeal to certainty equivalence to allow us to work with non-stochastic models.We would attain the same decision rule if we add a shock process Cηt+1 to the right sideof (3.3) or Cηt+1 to the right side of (3.4), where ηt+1 is an i.i.d. random process withmean zero and identity covariance matrix. The private sector’s behavior is summarized inthe second block of equations of (3.3) or (3.4). These are typically Euler equations, thefirst-order conditions of private agent’s optimization problem. These equations summarizethe ‘forward looking’ aspect of private agents’ behavior. We describe an example in section7.

Let X t denote the history of any variable X from 0 to t. Hansen, Epple, and Roberds(1985), Currie and Levine (1987), Sargent (1987), Pearlman (1992) and others have studiedversions of the following problem: given that the private agents and the government share

6 For ease of presentation, we shall assume that the matrix on the left of (3.3) is invertible. However, byappropriately using the invariant subspace methods described in Anderson, Hansen, McGrattan, and Sargent(1986) and in section 4, it is straightforward to adapt the computational method when this assumption isviolated.

10

model (3.3) or (3.4), maximize (3.2) by finding a sequence of government rules that expressut as a function of the history yt. These authors show that the optimal rule is history-dependent. The history dependence comes from the forward-looking behavior of the privatesector that is embedded in the second block of equations in (3.3) or (3.4). Those equationsdescribe how private agents’ decisions at t are influenced by the government’s instrumentat t + j for all j ≥ 0. The government takes those influences into account in planning itsdecisions at t+ j. Doing so prevents the problem from being recursive in the natural statevariables z alone. The government’s optimal plan becomes time-inconsistent and history-dependent. The history dependence can be encoded in a set of multipliers on the last nx

equations of (3.3) or (3.4). These multipliers track past government promises about futuresettings of u. We shall say much more about these multipliers after we have introduced apreference for robustness into the Ramsey problem.

3.2. Misspecification and the robust Ramsey problem

Rather than regarding the model (3.4) as known, both the government and the privateagents believe it is an approximation to an unknown model that for some wt+1 processcan be described by

[Etzt+1

Etxt+1

]=

[A11 A12

A21 A22

] [zt

xt

]+ But + Cwt+1. (3.5)

Here wt+1 is a vector of unknown specification errors that are permitted to feedback,possibly nonlinearly, on (zt, xt). The wt+1’s represent misspecified dynamics. The analysisis meant to apply to a situation in which the random term Cηt+1 is also added to the rightside of (3.5), where ηt+1 is a sequence of i.i.d. random vectors with mean zero and identitycovariance matrix, uncorrelated with the wt+1 process. Certainty equivalence allows us todrop ηt+1 in computing robust and optimal rules. Government and private agents regardthe model as a good approximation to the unknown model in the sense that the unknownwt+1 sequence satisfies7

∞∑t=0

βtw′t+1wt+1 ≤ Ω0. (3.6)

The government wants a rule that will work well for any wt+1 sequence satisfying (3.6).Attach a positive multiplier θ to the single constraint (3.6). The government attains arobust rule by using the ut rule that solves the following zero-sum two-player game:

maxut∞t=0

minwt+1∞t=0

−∞∑

t=0

βtd′tdt − θw′

t+1wt+1

. (3.7)

subject to (3.5). The muliplier θ constrains the minimizing agent. The private agents solve‘pure forecasting’8 problem, using the objective (3.2) to find a worst-case shock to slanttheir beliefs for the purpose of generating robust predictions. Thus, we impute common

7 We would take the expectation of (3.6) when ηt+1 is present in (3.5).8 See Hansen and Sargent (2000) and Whittle (1996).

11

objective (3.2) and common approximating models and sets of alternative models to theprivate agents and the government.

We solve this game by exploiting analogies with another problem that is easier to solve,the optimal linear regulator.

4. A single-agent problem

Though we ultimately want to study macroeconomic control of forward-looking mod-els, we can introduce most of the important conceptual issues by first studying a simplerproblem, a single-agent robust decision problem. In addition to explaining the basic com-putational strategy in a familiar context, this also lets us introduce the approximatingmodel, a class of misspecified models, and a worst case model. We can also draw attentionto the distinction between the implied behavior of the system under the approximatingmodel and the worst case model. These issues will recur in our analysis of the robustRamsey problem, where their presence will be obscured by additional complications. Thissection shows how to compute a robust decision rule by solving an adjusted optimal linearregulator problem.

Currie and Levine (1996) and Pearlman (1992) recognized and exploited the connec-tions between the optimal linear regulator problem and the optimal control of ‘forwardlooking’ models under commitment. We adapt and extend their arguments to include apreference for robustness of decision rules to model misspecification.

Consider the discounted nonstochastic robust single-agent decision problem:9

maxut∞t=0

minwt+1∞t=0

−∞∑

t=0

βt[y′tQyt + u′

tRut − θw′t+1wt+1

], β ∈ (0, 1) (4.1)

where the maximization and minimization are subject to

yt+1 = Ayt + But + Cwt+1. (4.2)

Here yt is a state vector, ut is a vector of controls, and wt+1 is a vector of possible specifica-tion error shocks; R and Q are positive definite matrices and θ ∈ (θ, +∞) is a scalar multi-plier penalizing model misspecification shocks as measured by

∑∞t=0 βtw′

t+1wt+1. Equation(4.2) represents a class of models, of which the agent’s model, called the approximatingmodel , is a special case. Since we study a nonstochastic setting, the agent’s approximatingmodel is (4.2) with wt+1 = 0. In (4.2), the sequence wt+1, which is allowed to feed backon the history of the state y, represents model misspecification. The agent solves problem(4.1), (4.2) because he wants to design a decision rule that is robust to model misspecifi-cations that he models as an unknown nonzero sequence wt+1. See Hansen and Sargent(1995, 2000) for alternative equivalent formulations of problem (4.1), (4.2).

9 The problem assumes that there are no cross products between states and controls in the return function.A simple transformation converts problems whose return functions have cross products into an equivalentproblem that does not. See Anderson, Hansen, McGrattan, and Sargent (1996). For our Ramsey problem,there are cross products between states and controls unless H ′J = 0 in (3.1).

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4.1. Lagrangian

A standard way to solve the problem (4.1), (4.2) is to form the Bellman equation anditerate on it to convergence. The optimal value function has the form v(y) = y′Py, whereP satisfies a Riccati equation that is induced by iterating on the Bellman equation. Wedisplay this Riccati equation in (4.22) below. In this section, we briefly recall how we canalso solve the problem using Lagrangian methods. We describe how these methods providea way to compute P , and how the computation of P gives a recursive representationµt = Pyt for the shadow price µt of the state. Later, we will adapt these Lagrangianmethods and the optimal linear regulator to let us find a robust Ramsey plan.

We form the Lagrangian10

L = −∞∑

t=0

βt[y′tQyt + u′

tRut + 2βµ′t+1 (Ayt + But + Cwt+1 − yt+1) − θw′

t+1wt+1

]. (4.3)

We want to maximize (4.3) with respect to sequences for ut and yt+1 and minimize it withrespect to a sequence for wt+1. In (4.3), θ ∈ (θ, +∞) is a penalty parameter restrainingthe minimizing choice of the wt+1 sequence. The first-order conditions with respect tout, yt, wt+1, respectively, are:

0 = Rut + βB′µt+1 (4.4a)µt = Qyt + βA′µt+1 (4.4b)0 = θwt+1 − βC ′µt+1. (4.4c)

Solving (4.4a) and (4.4c) for ut and wt+1 and substituting into (4.2) gives

yt+1 = Ayt − β(BR−1B′ − θ−1CC ′)µt+1. (4.5)

Define

B = [B C ] (4.6a)

R =[R 00 −θI

]. (4.6b)

Then write (4.5) asyt+1 = Ayt − βBR−1B′µt+1. (4.7)

We seek a stable solution of the system formed from (4.7) and (4.4b), a system that wecan represent as [

I βBR−1B′0 βA′

] [yt+1

µt+1

]=

[A 0−Q I

] [yt

µt

](4.8)

or

L∗[yt+1

µt+1

]= N

[yt

µt

]. (4.9)

10 Note to Sargent: in the computations, the timing of the multiplier θ was actually βθ. We made thischange to make the joint controls BC matrix OK.

13

The matrix pencil ( λ√βL∗ − N) is symplectic,11 implying that the generalized eigen-

values of (L∗, N) occur in√

β-symmetric pairs: if λi is an eigenvalue, there is anothereigenvalue λ−i = 1

βλi. To find the optimum, we must solve stable roots backward and

unstable roots forward. In particular, we obtain the stable solution by finding an initial-ization µ0 = Py0 that replicates itself over time, in the sense that µt = Pyt, and thatimplies that

∑∞t=0 βty′tyt < ∞. In the optimal linear regulator problem, yt is a state vector

inherited from the past; the multiplier µt jumps at t to stablize the system. We now brieflydescribe how to find such a stabilizing P by using invariant subspace methods. Once P isfound, the optimal decision rules can be computed easily, as we show below.

4.2. Invariant subspace methods

We use a version of a standard trick for handling discounted problems: we transformvariables to make the eigenvalues of an altered system have reciprocal pairs of eigenvalues,then inverse transform after we have solved the transformed system. Let L = L∗β−.5 andtransform the system (4.9) to

L

[y∗t+1µ∗

t+1

]= N

[y∗tµ∗

t

], (4.10)

where y∗t = βt/2yt, µ∗t = µtβ

t/2. Now λL−N is a symplectic pencil, so that the generalizedeigenvalues of L,N occur in reciprocal pairs: if λi is an eigenvalue, then so is λ−1

i .We can use Evan Anderson’s Matlab program schurg.m to find a stabilizing solution of

system (4.10). The program computes the ordered real generalized Schur decompositionof the matrix pencil. Thus, schurg.m computes matrices L, N , V such that L is uppertrianglar, N is upper block triangular, and V is the matrix of right Schur vectors such thatfor some orthogonal matrix W the following hold:

WLV = L

WNV = N .(4.11)

Let the stable eigenvalues (those less than 1) appear first. Then the stabilizing solution is

µ∗t = Py∗t (4.12)

whereP = V21V

−111 ,

V21 is the lower left block of V , and V11 is the upper left block.If L is nonsingular, we can represent the solution of the system as12

[y∗t+1µ∗

t+1

]= L−1N

[IP

]y∗t . (4.13)

11 See Anderson, Hansen, McGrattan, and Sargent (1986) for a definition and properties of a symplecticpencil.12 The solution method in the text assumes that L is nonsingular and well conditioned. If it is not, the

following method proposed by Evan Anderson will work. We want to solve for a solution of the form

y∗t+1 = Ao∗y∗t .

14

The solution is to be initiated from (4.12). We can use the first half and then the secondhalf of the rows of this representation to deduce the following recursive solutions for y∗t+1and µ∗

t+1:y∗t+1 = Ao∗y∗tµ∗

t+1 = ψ∗y∗t(4.14)

Now express this solution in terms of the original variables:

yt+1 = Aoyt

µt+1 = ψyt,(4.15)

where Ao = Ao∗β−.5, ψ = ψ∗β−.5. We also have the representation

µt = Pyt. (4.16)

The matrix Ao = A− BF , where F is the matrix for the optimal decision rule in (4.24).We now describe an alternative way to compute the optimal P , one that is probably

more familiar to macroeconomists schooled in iterating on a Bellman equation.

4.3. The Riccati equation

The stabilizing P obeys a Riccati equation coming from the Bellman equation. Sub-stituting µt = Pyt into (4.7) and (4.4b) gives

(I + βBR−1BP

)yt+1 = Ayt (4.17a)

βA′Pyt+1 = −Qyt + Pyt. (4.17b)

A matrix inversion identity implies(I + βBR−1B′P

)−1= I − βB

(R + βB′PB

)−1B′P. (4.18)

Solving (4.17a) for yt+1 givesyt+1 =

(A− BF

)yt (4.19)

whereF = β

(R + βB′PB

)−1B′PA. (4.20)

Note that with (4.12),L [I;P ] y∗t+1 = N [I;P ] y∗t

The solution Ao∗ will then satisfyL [I;P ]Ao∗ = N [I;P ] .

Thus Ao∗ can be computed via the Matlab command

Ao∗ = (L ∗ [I;P ]) \ (N ∗ [I;P ]) .

15

Premultiplying (4.19) by βA′P gives

βA′Pyt+1 = β(A′PA− A′PBF

)yt. (4.21)

For the right side of (4.21) to agree with the right side of (4.17b) for any initial value of y0

requires that

P = Q + βA′PA− β2A′PB(R + βB′PB

)−1B′PA. (4.22)

Equation (4.22) is the algebraic matrix Riccati equation associated with the ordinary linearregulator for the system A, B, Q, R.

In summary, the solution of our problem is given by the feedback rule13

[ut

wt+1

]= −Fyt. (4.23)

We can find P either by solving a Riccati equation or by using an invariant subspacemethod that rearranges the generalized eigenvectors of L,N .14

Soon we show how to use these same methods to solve another system of the form(4.9) that arises in our robust Ramsey problem. But first, we interpret various aspects ofthe solution introduced by the concern for model misspecification.

4.4. Interpreting the solution

The solution of problem (4.1), (4.2) has a recursive representation in terms of a pairof feedback rules

ut = −F1yt (4.24a)wt+1 = −F2yt. (4.24b)

Here ut = −F1yt is the robust decision rule for the control ut. The rule wt+1 = −F2yt is arule for a worst case shock. This worst-case shock induces a distorted transition law

yt+1 = (A− CF2) yt + But. (4.25)

In effect, the agent devises a robust decision rule by choosing a sequence ut to maximize

−∞∑

t=0

βt[y′tQyt + u′

tRut

]

subject to (4.25). The distortion of (4.25) relative to (4.2) with wt+1 = 0 is the agent’stool for devising a robust rule. As noted above, the decision maker believes that the dataare generated by some model with an unknown process wt+1 = 0. By planning against theworst case process wt+1 = −F2yt, he designs a robust decision rule.

13 The preceding calculations show that we can solve a robust linear-quadratic control problem by forminganother artificial ordinary linear-quadratic control problem with additional controls being the worst-caseshocks wt+1. The ordinary Riccati equation and decision rule apply to this problem. We determine therobust decision rule by selecting the F1 part of the decision rule for the artificial problem, as in the text.14 See Hansen and Sargent (1995, 2000) for alternative ways of computing the robust rule.

16

4.5. Approximating and worst case models

Thus, the solution of the robust decision problem (4.1), (4.2) can be represented

yt+1 = Ayt −BF1yt − CF2yt. (4.26)

Equation (4.26) describes the behavior of the system under the robust decision rule ut =−F1yt and the worst case shock process wt+1 = −F2yt. However, the agent does notreally believe that the worst-case shock process will prevail. He designs his decision ruleby using wt+1 = −F2yt to slant the transition law in order to acquire a rule that will berobust against a range of departures from his approximating model. We want to knowthe behavior of the robust decision rule under other models. In particular, we sometimeswant to study the behavior of the system when the approximating model governs the data(so that the decision maker’s fears of model misspecification are unfounded). Under therobust decision rule but the approximating model, the law of motion is

yt+1 = (A−BF1) yt. (4.27)

We obtain (4.27) from (4.26) by replacing the worst case shock −F2yt by zero. Notice thatalthough we set F2 = 0 in (4.26) to get (4.27), F1 in (4.26) embodies a best response toF2, and thereby reflects the agent’s ‘pessimistic’ forecasts of future values of the state.

We call (4.27) the approximating model and (4.26) the worst-case or distorted model.These two models will have counterparts in our versions of robust Ramsey plans.

5. Ramsey with robustness

This section shows how the problem of devising a robust decision rule to control aforward-looking macro model under commitment can be formulated as an optimal linearregulator problem. We adapt arguments of Currie and Levine (1987) and Pearlman (1992)to include a preference for robustness.

Let zt be an nz ×1 vector of natural state variables and xt be an nx×1 vector of ‘jumpvariables’ to be determined at time t. We consider a class of forward-looking models thatcan be represented as [

zt+1

xt+1

]= A

[zt

xt

]+ [B C ]

[ut

wt+1

](5.1)

or

yt+1 = Ayt + B

[ut

wt+1

](5.2)

where

yt =[zt

xt

]and B = [B C ] .

Evidently, (5.2) assumes the form of (4.2). The last nx equations in (5.2) is a set of Eulerequations for private agents’ optimum problems. To construct a robust decision rule, thegovernment solves the problem (4.1), (4.2). Formally, this looks like an ordinary linear

17

regulator problem, but we have to keep in mind that the vector xt is not truly part of the‘state’ vector; its elements are all jump variables to be determined by the model at t. Thegovernment regards the last nx equations of (5.2) as a set of implementability constraintson its choice of sequences for zt, xt, ut, wt+1.

5.1. Key idea: make implementation multipliers state variables

The key idea is to make some of the ‘state’ variables in (5.1) (in particular, the xt’s)change places with some of the multipliers (namely, the µxt’s pertaining to the laws ofmotion for xt+1 in (5.1)), then to let the optimal linear regulator do the hard work.

Thus, we partition µt conformably with the partition of yt into zt, xt:

µt =[µzt

µxt

].

For the optimal linear regulator, the Lagrange multipliers µt are all jump variables whileyt is a state vector that is given at t. In the present problem, only the first nz componentsof yt are state variables: the last nx variables of yt, the xt vector, are all jump variables.However, the last nx Lagrange multipliers µxt now will play the role of state variables.Mechanically, the government’s problem is equivalent to that solved above, except forwhich variables are permitted to jump to guarantee stability, and which cannot. Thisalteration of the problem affects things only part way through the solution process, inparticular, after we have performed the key step of computing the P that solves theRiccati equation (4.22).

We set up the same Lagrangian and obtain the same first-order necessary conditionsas in section 4. As before, because we seek a stabilizing solution, we impose

µt = Pyt. (5.3)

It is only in how we impose (5.3) that the solution diverges from that for the linearregulator. Write the last nx equations of (5.3) as

µxt = P21zt + P22xt, (5.4)

where the partitioning of P is conformable with that of yt into zt, xt. In the present case,µxt is a part of the state at t, while xt is free to jump at t. Therefore, we solve (5.3) for xt

in terms of (zt, µxt):xt = −P−1

22 P21zt + P−122 µxt. (5.5)

Then we can write

yt =[

I 0−P−1

22 P21 P−122

] [zt

µxt

](5.6)

and from (5.4)µxt = [P21 P22 ] yt. (5.7)

With these modifications, the key formulas (4.20) and (4.22) from the optimal linearregulator for F and P , respectively, continue to apply. Using (5.6), the solutions for thecontrol and worst case shock are[

ut

wt+1

]=

[−F1

−F2

] [I 0

−P−122 P21 P−1

22

] [zt

µxt

]. (5.8)

18

Using the law of motion for yt+1, equation (4.19), together with (5.6) and (5.7) allows usto represent our solution recursively as[

zt+1

µx,t+1

]=

[I 0

P21 P22

](A−BF1 − CF2)

[I 0

−P−122 P21 P−1

22

] [zt

µxt

](5.9a)

xt = [−P−122 P22 P−1

22 ][

zt

µxt

](5.9b)

Equation (5.9a) is the worst-case law of motion for zt. To get the law of motion underthe approximating model and the robust Ramsey plan, we replace the first nz rows of (5.9)with

zt+1 = [ I 0 ] (A−BF1)[

I 0−P−1

22 P21 P−122

] [zt

µxt

]. (5.10)

Denote

A− BF =[A1 −B1F1 − C1F1

A2 −B2F1 − C2F2

]

where A1, A2 denote the first nz and last nx rows of A, respectively, with similar conventionsfor Bi, Ci. Then we have the following description of the approximating model under therobust Ramsey plan:[

zt+1

µx,t+1

]=

[I 0

P21 P22

] (A1 −B1F1

A2 −B2F1 − C2F2

) [I 0

−P−122 P21 P−1

22

] [zt

µxt

](5.11a)

xt = [−P−122 P21 P−1

22 ][

zt

µxt

](5.11b)

In summary, we solve the robust Ramsey problem by formulating a particular opti-mal linear regulator, solving the associated matrix Riccati equation (4.22) for P , thenpartitioning P to obtain representation (5.11).

6. A Bellman equation

We briefly indicate the connection of the preceding formulation to that of Kydland andPrescott (1980) and Marcet and Marimon (2000). For a class of problems with structuresclose to ours, they construct a Bellman equation in a state vector defined as (z, µx): theseare the ‘natural’ state variables and the vector of multipliers on the laws of motion for the‘jump’ variables xt. We show how to modify that Bellman equation to include a concernabout model misspecification.

Continuing to let (5.1) be the law of motion, let λt = µxt denote the subvector of mul-tipliers attaching to the implementability constraints that summarize the Euler equationsof the private sector. Then the Lagrangian for the optimum problem (4.3) can be written

L = −∞∑

t=0

βt

[zt

xt

]′Q

[zt

xt

]+ u′

tRut − θw′t+1wt+1

+ βλ′t+1 (A21zt + A22xt + B2ut + C2wt+1 − xt+1)

.

(6.1)

19

This Lagrangian is to be ‘extremized’ (i.e., maximized or minimized, as appropriate) withrespect to sequences zt, xt λ, wt+1 subject to λ0 = 0 and the transition law

zt+1 = A11zt + A12xt + B1ut + C1wt+1. (6.2)

Equation (6.1) can be rewritten

L = −∞∑t=0

βt

[zt

xt

]′Q

[zt

xt

]+ u′

tRut − θw′t+1wt+1

+(βλ′

t+1A22 − λ′t

)xt + βλ′

t+1 (A21zt + B2ut + C2wt+1),

(6.3)

which is to be extremized with respect to the same constraints (6.2). Define the one-

period return function −r(z, λ, x, λ∗, w) =[zx

]′Q

[zx

]+u′Ru− θw′w + (βλ∗′A22 −λ′)x+

βλ∗′(A21z +B2u+C2w), where ∗ superscripts denote one-period ahead values. Let v(z, λ)be the optimum value of the problem starting with augmented state (z, λ). Problem (6.3)is recursive and has the following Bellman equation:

v (z, λ) = maxu,x

minw,λ∗

r (z, λ, x, λ∗, w) + βv (z∗, λ∗) (6.4)

where the extremization is subject to

z∗ = A11z + A12x + B1u + C1w. (6.5)

The Bellman equation (6.4), (6.5) is a version of the recursive saddle problem describedby Kydland and Prescott (1980) and Marcet and Marimon (2000). We have added aconcern for robustness via the extra minimization with respect to the shock distortion w.In related contexts, Marcet and Marimon stress that while such problems are not recursivein the natural state variables z alone, they becomes recursive when the multipliers λ areincluded.

Although one could solve our problem by iterating to convergence on (6.4), (6.5), itis more convenient for us to use the method described in section 5 that solves the Riccatiequation (4.22) and its associated Bellman equation.

20

7. A ‘new synthesis’ model

To illustrate a robust Ramsey plan, we adopt the economic setting of Woodford (1998).Woodford’s model is one of a class of ‘new synthesis’ macroeconomic models (see Good-friend and King (1997), Rotemberg and Woodford (1997), Clarida, Gali, and Gertler(1999)) that add nominal price stickiness to a real business cycles model, thereby puttingnominal prices into the model and creating an avenue by which monetary policy can in-fluence output.

The heart of Woodford’s model is two log-linear approximations to Euler equationsthat describe decisions of households and firms. Households’ behavior is summarized by astandard Euler equation for consumption. Log linearizing this equation and using the pro-duction function delivers what Woodford calls the ‘IS curve’.15 Monopolistically competi-tive firms set nominal prices according to timing protocols that imply first order conditionsthat generate what Woodford calls a ‘Phillips curve.’ We begin our analysis from thesetwo log-linearized Euler equations. The expectational Phillips curve and the dynamic IScurve, respectively, are:

πt = κγt + βEtπt+1 + ε1t (7.1)

γt = Etγt+1 − σ−1[(rt − rn

t ) − Etπt+1

]+ ε2t (7.2)

where γt = ot − ont , the deviation of the log of output from the log of potential output, rt

is the nominal short term interest rate (the monetary authority’s instrument), and rnt is

the natural rate of interest. We have added the two shock processes ε1t, ε2t to Woodford’stwo equations. Woodford suppresssed these terms. He assumed an exact model. HereEt(·) is a (possibly distorted) conditional expectation operator. Woodford assumes rationalexpectations, so for him there is no distortion. We will find a class of distorted expectationsoperators indexed by θ.

Woodford assumes that the government seeks to maximize

−∞∑

t=0

βtLt (7.3)

where the loss function L is taken to be

Lt = π2t + λγ (γt − γ∗)2 + λr (rt − r∗)2 .

Following Woodford, we regard (7.3) as the log-linear- quadratic approximation of thewelfare of a representative household. Both firms and households use the criterion (7.3) toformulate their robust pure forecasting problems.16

Woodford shows that the natural rate of interest obeys

rnt = σEt

[(ont+1 − on

t

) − (gt+1 − gt)]. (7.4)

15 We use the label IS curve because Woodford and other authors in the literature do. This label offendssome friends from Minnesota. Every time we say IS curve, think ‘consumption Euler equation.’16 See Hansen and Sargent (2000) and Whittle (1996).

21

Rather than using (7.4) directly, Woodford adopted the specification that rnt is simply a

stationary univariate exogenous process, about which we shall soon say more.17 Then thearrival of information in the economy can be described by

St+1 = A11St + C1 (wt+1 + ηt+1) (7.5a)It = eSt (7.5b)

where It = [ rtn ε1t ε2t γ∗ r∗ ]′ and ηt+1 is an i.i.d. Gaussian sequence of shocks to

the state vector St and wt+1 is a vector of model specification errors. As above, we shallappeal to a certainty equivalence principle that is induced by the linear-quadratic structureto allow drop the ηt+1 term while computing the optimal rules. The approximating modelassumes that wt+1 = 0.

We want to add a preference for robustness to Woodford’s setup. To transform Wood-ford’s model into our setup, define

zt = St, ut = rt (7.6a)

xt =[πt

γt

](7.6b)

C =[C1

0

]. (7.6c)

Then the model (7.1), (7.2) (7.5) and the associated robust control problem fit within theframework of section 5.

As section 5 shows, the worst case law of motion for the state is[

St+1

µx,t+1

]=

[Ao

11 Ao12

Ao21 Ao

22

] [St

µxt

]+

[C1

0

]ηt+1 (7.7)

The law of motion according to the approximating model under the robust Ramsey planis [

St+1

µx,t+1

]=

[A11 0Ao

21 Ao22

] [St

µxt

]+

[C1

0

]ηt+1, (7.8)

which we obtain by substituting the law of motion for St+1 from the approximating model.

17 An important feature of this specification is that rnt be a stationary process to validate Woodford’s

procedure of using linear approximations around a nonstochastic steady state.

22

7.1. Specification of rnt

For the purposes of preparing the way for issues to be discussed in section 8, we saymore about the specification of the natural rate rn

t . Woodford’s specification that thenatural rate of interest rn

t is a first order autoregressive process is compatible with thefollowing law of motion for (on

t , gt):

ont+1 = 2on

t − ont−1 + εo,t+1 (7.9a)

gt+1 − ont+1 = ag + sg,t+1 (7.9b)

sg,t+1 = ρsg,t + εg,t+1, |ρg| < 1. (7.9c)

Remember that ont represents the log of potential output and gt the log of aggregate

demand. The system (7.9) says that the growth rate of ont takes a random walk. This

allows ‘trend breaks’ in potential output. Equations (7.9b), (7.9c) allow gt to deviate fromont by a constant plus a stable first-order autoregressive process. Equations (7.4) and (7.9)

imply

rnt = σ (q − ρg) sgt =

σ (1 − ρ)(1 − ρL

)εg,t, (7.10)

where L is the lag operator. Equation (7.10) is Woodford’s specification, which we adoptin the numerical calculation in section 9.

8. Filtering

This section analyzes the combined filtering and control of forward-looking models thatwas studied without a concern for model misspecification by Pearlman, Currie, and Levine(1986), Pearlman (1992), and Svensson and Woodford (1999). Drawing heavily on resultsfrom Hansen and Sargent (2000), we briefly describe how the calculations in section 5 canbe adapted when agents and the government have incentives to estimate a latent variable,like potential GDP. Designing a robust Ramsey rule now requires deriving a robust filterand adjusting the control problem to admit additional possible model distortions whenfiltering is done with a possibly misspecified model. As a laboratory, we use a version ofWoodford’s model, suitably adjusted to make potential GDP a latent variable.

23

8.1. Specification with observed state

For concreteness, we adopt the particular specification (7.9). To represent this speci-fication with matrix notation, let

sut = [ ont on

t−1 sgt ]′ (8.1)

Soon we shall assume that sut is not observed and that it must be estimated by both thepublic and the government. But for now let sut be observed at t. Let sot be a vector ofinformation variables whose histories up to t are observed at time t. Assume that[

su,t+1

so,t+1

]=

[Auu Auo

Aou Aoo

] [sut

sot

]+

[Cu

Co

]ηt+1. (8.2)

Let so,t = [ ε1t ε2t 1 ]′. Under the assumption that both sut, sot are observed at t, speci-fication (8.2) is consistent with system (7.5a), so that the Ramsey plan can be computedas in section 5.

8.2. Latent state variables

The solution procedure can be altered to handle the case when sut is not observed. Asan example, we now assume that potential output is observed by neither the governmentnor the private sector. Everyone treats potential output as a hidden variable that mustbe estimated as a function of the history of observed variables. Consistent with thisassumption, we adjust the model to be

πt = κγt + βEtπt+1 + ε1t (8.3)

γt = Etγt+1 − σ−1[(rt − rn

t ) − Etπt+1

]+ ε2t (8.4)

where γt = ot−ont and for any variable a, at+1 represents an estimate of a using observations

up to time t + 1. Now the natural rate of interest satisfies

rnt = σEt

[(ont+1 − on

t

) − (gt+1 − gt)]. (8.5)

Equation (8.5) assumes that gt+1 is observed but that ont+1 is not at t+ 1, so that a robust

estimator ont must be constructed.

The government and private agents now face a robust filtering problem on top of theirrobust decision problems. Results of Hansen and Sargent (2000) can be applied to showthat filtering and ‘control’ aspects of the problem separate so that the following two stepprocedure is appropriate.

Step 1. Solve an ordinary Kalman filtering problem for the sytem (8.2) to attain thecorresponding ‘innovations representation’18[

su,t+1

so,t+1

]=

[Auu Auo

Aou Aoo

] [sut

sot

]+

[Cu Cu

0 Co

] [ηt+1

ηt+1

](8.6)

18 See Anderson and Moore (1979) for a study of innovations representations.

24

where[ηη

]is an i.i.d. vector disturbance with mean zero and identity covariance ma-

trix, and Cu, Cu, Co are matrices determined by a Kalman filter, as described below.

Step 2. Using the following evolution for information[su,t+1

so,t+1

]=

[Auu Auo

Aou Aoo

] [sut

sot

]+

[Cu Cu

0 Co

] [wt+1

wt+1

], (8.7)

solve the problem using the method described in section 5. System (8.7) is obtainedfrom (8.6) by replacing the stochastic shocks η, η with distortions to the conditionalmeans , w, w.

We next briefly describe the filtering problem in enough detail to implement the procedure.

8.3. Filtering details

System (8.6) or (8.7) is obtained from (8.2) by an application of the standard Kalmanfilter.19 In particular, the steady-state covariance for the error in reconstructing the stateis given by Σ, the positive semi-definite matrix that solves the Riccati equation

Σ = AuuΣA′uu + CuC

′u

− [AuuΣA′

ou + CuC′o

] [AouΣA′

ou + CoC′o

]−1 [AouΣA′

uu + CoC′u

].

(8.8)

Define the innovation covariance matrix

Λ = E

[su,t+1 − su,t+1

so,t+1 − so,t+1

] [su,t+1 − su,t+1

so,t+1 − so,t+1

]′.

It can be computed from (8.2) that

Λ =[AouΣA′

ou + CoC′o AouΣA′

uu + CoC′u

AuuΣA′ou + CuC

′o AuuΣA′

uu + CuC′u

]. (8.9)

Obtain Cu, Co, Cu from the Cholesky factorization:

Λ =[Cu Cu

0 Co

] [Cu Cu

0 Co

]′. (8.10)

These are the objects we use to create the innovations representation (8.6).In (8.6), C0ηt+1 ≡ ao,t+1 is the innovation in so,t+1, the error in predicting so,t+1 linearly

from its own past. The term Cuηt+1 ≡ Kao,t+1 is the projection of the unobserved su,t+1

on the innovation ao,t+1, where K = CuC−1o is the stationary Kalman gain. The formula

for updating estimates sut of the hidden state is

su,t+1 = Aoosut + Auosot + Cuηt+1. (8.11)

The reconstruction error associated with su,t+1 is

su,t+1 − su,t+1 = Cuηt+1. (8.12)

Combining (8.11) and (8.12) leads us to use (8.6) as the representation for information attime t.

19 See Anderson and Moore (1979).

25

8.4. Interpretation

A concern for robustness leaves the basic form of the Kalman filter intact, but nev-ertheless contributes a new term to the decision problem: namely, the ηt+1. This termrepresents an additional avenue for model misspecification that originates in concern aboutrobustness of the filter. See Whittle (1996), Basar and Bernhard (1995, chapters 5 and7), Hansen and Sargent (2000), and Hansen, Sargent, and Wang (2000) for related discus-sions of the separation of combined filtering into a two-step process that is afforded by thelinear-quadratic structure.

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

From ηt

To

r t

Figure 8.1: Impulse responses of rt to ηt (the innovation to rnt ). Solid line

is θ = +∞, dotted line is θ = 22.75.

0 2 4 6 8 10 12 14 16 18 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

From etat

To

wt

Figure 8.2: Impulse responses of conditional mean of worst case shockswt to ηt. Solid line is θ = +∞, dotted line is θ = 22.75.

26

0 2 4 6 8 10 12 14 16 18 20−0.07

−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

From ηt

To

µ s1

0 2 4 6 8 10 12 14 16 18 20−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

To

µ s2

Figure 8.3: Responses of implementation multipliers µs1 (on IS curve)and µs2 (on Phillips curve) to ηt (innovation to rn

t ). Solid line is θ = +∞,dotted line is θ = 22.75.

9. Numerical example

For the version of Woodford’s model where the state is observed, we have computedordinary (θ = +∞) and robust (θ = 22.75) rules for Woodford’s model.20 Following Wood-ford, we assume the following parameter values: (β, σ, κ, λx, λr) = (.99, .157, .024, .047, .233).Woodford assumed a first-order a.r. process for rn

t with a.r. coefficient .35 and standarddeviation of rn

t of 3.72. Woodford had indirect evidence about the serial corrlation of rnt

and studied the consequences of varying it. In our example, we set ρ = .8 to magnify theeffects of robustness in particular directions. By trial and error, we discovered that thevalue θ = 22.75 is close to the ‘breakdown point’, so that it gives the maximum effectsof robustness for the particular parameter values we, following Woodford, have chosen forthe approximating model. Also, following Woodford, we assume that the IS and Phillipscurves are exact, so that ε1t = ε2t = 0 in (7.1), (7.2). Let µs1,t, µs2,t denote the multiplierson the implementability constraints (7.1), (7.2), respectively.

Here we report impulse responses under the approximating model and the robust de-cision rules. Figures 8.1, 8.2, and 8.3 report equilibrium impulse response functions forθ = +∞ (solid line) and θ = 22.75 (dotted line). In figure 8.1, the impulse response ofrt to innovations in rn

t shows that on impact, the government’s instrument rt respondssomewhat less aggressively under the robust (dotted) rule than under θ = +∞ rule. Notethat the response function under the robust rule eventually crosses that under the θ = +∞rule, indicating that the government draws out its response to the shock longer under therobust rule, even though it responds less immediately. Evidently, this response mirrorsthe altered response of the implementatibility multiplier on the IS curve, shown in thetop panel of Fig. 8.3. Qualitatively, such a result is potentially interesting in light of aliterature that often finds that empirical Taylor rules are less agressive than optimal rules

20 Note to Sargent: computed using managegv7.m, task24gv4.m in projects/robust/woodford.

27

computed from various structural models.21 However, quantitatively, the effect is not verylarge in our example. Before claiming that a concern for model misspecification can helpaccount for the apparently conservative nature of empirical Taylor rules, we would wantto carry out an an analysis with a richer model having more stochastic components andmore state variables.

Figures 8.2 and 8.3 display the impulse response functions of the worst case shocks andthe multipliers to an innovation ηt+1 in the natural rate.

10. Concluding remarks

This paper has generalized standard methods for solving Ramsey problems in linear-quadratic forward looking models to include a common concern for model misspecificationto both the government and private agents. The government and private agents sharean approximating model that describes the shocks and other exogenous variables hittingthe economy. We add one parameter to the standard rational expectations setup, thepenalty parameter θ that indexes the size of a set of models near the approximatingmodel with respect to which policy rules and private agents’ forecasts are to be robust.We show how the Ramsey rule can be computed in effect by forming an optimal linearregulator problem and carefully exchanging their roles in stabilizing the system of theforward looking model’s artificial state variables and the Lagrange multipliers on theirlaws of motion. Mechanically, robustness is achieved simply by adding another controlto the regulator problem, a distortion to the conditional mean of the disturbances thatis chosen by a fictitious evil agent. Anderson, Hansen, and Sargent (2000) and Hansen,Sargent, and Wang (2000) indicate how a Bayesian model detection problem can be usedto calibrate the parameter θ. We intend the preference for robustness as measured by θnot to be a primitive, but to be context-specific, as determined by model detection errorprobabilities for the approximating model and, say, the endogenous worst-case model (see,e.g., Hansen, Sargent, and Wang (2000)).

Using similar methods, it would be possible to generalize our results to relax ourassumption that private agents and the government share a common approximating modeland a preference for robustness (i.e., θ). In particular, one could construct a ‘two-θ’ model,reflecting different concerns about model uncertainty on the part of the private sector andthe government. Such a generalization would move part way toward the specification usedby the papers of Giannoni (1999), Otrok (in press) and others cited in section 1.

21 For analyses of this question in the context of backward-looking models, see comments by Sargent(1999) and Stock (1999) as well as the paper by Onatksi and Stock (2000). They show that with backwardlooking models, a concern for model misspecification typically makes the monetary authority respond moreaggressively than it does without concerns for robustness.

28

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