Monetary policy with model uncertainty: distribution forecast targeting

Monetary policy with model uncertainty:
distribution forecast targeting
Lars E.O. Svensson
(Princeton University)
Noah Williams
(Princeton University)
Discussion Paper
Series 1: Economic Studies
No 35/2005
Discussion Papers represent the authors’ personal opinions and do not necessarily reflect the views of the
Deutsche Bundesbank or its staff.

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Monetary Policy with Model Uncertainty:
Distribution Forecast Targeting
Abstract
We examine optimal and other monetary policies in a linear-quadratic setup with a relatively
general form of model uncertainty, so-called Markov jump-linear-quadratic systems extended to
include forward-looking variables. The form of model uncertainty our framework encompasses
includes: simple i.i.d. model deviations; serially correlated model deviations; estimable regime-
switching models; more complex structural uncertainty about very di erent models, for instance,
backward- and forward-looking models; time-varying central-bank judgment about the state of
model uncertainty; and so forth. We provide an algorithm for finding the optimal policy as well
as solutions for arbitrary policy functions. This allows us to compute and plot consistent distri-
bution forecasts–fan charts–of target variables and instruments. Our methods hence extend
certainty equivalence and “mean forecast targeting” to more general certainty non-equivalence
and “distribution forecast targeting.”
JEL Classification: E42, E52, E58
Keywords: Optimal policy, multiplicative uncertainty
We thank Pierpaolo Benigno, Marvin Goodfriend, Boris Ho man, Eric Leeper, Rujikorn Pavasuthipaisit, and
participants in the New York Area Workshop on Monetary Policy, New York, May 2005, the Conference on Macro-
economic Risk and Policy Responses, Berlin, May 2005, and the Conference on Computing in Economics and Finance,
Washington, DC, June 2005, for helpful comments. Expressed views and any remaining errors are our own responsi-
bility.

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Non-technical summary
Monetary Policy with Model Uncertainty: Distribution Forecast Targeting
Monetary policy is always conducted under considerable uncertainty about the
data and the state of the economy but also, in particular, about the transmission
mechanism of monetary policy, that is, the strength and the lags with which
interest changes affect inflation and output—so-called model uncertainty.
Determining the best design of monetary policy under model uncertainty is a
notoriously difficult problem and few general results are available.
In this paper we propose a flexible, powerful, and yet tractable framework to
investigate the appropriate design of monetary policy under model uncertainty.
We represent model uncertainty by the parameters of the transmission mechanism
shifting over time between different “modes” according to a Markov process with
an arbitrarily given probability distribution for the shifts between modes. This way
we can incorporate a number of very relevant different kinds of model uncertainty;
such as serially correlated random parameters and volatility; regime shifts; and
more complex but realistic uncertainty about structurally very different models,
such as backward-looking and forward-looking private-sector expectations
formation, different degrees of price and wage rigidity, and so forth. In particular,
we can incorporate shifting central-bank judgment about the nature and amount of
model uncertainty, such as temporary concerns with shifting degrees of exchange-
rate pass-through.
We develop an algorithm for determining the policy that is optimal under model
uncertainty given the central-bank objectives for monetary policy. We also show
how to determine the dynamics of the economy for arbitrary instrument rules and
instrument paths. These methods make it possible to illustrate policies and policy
choices for policymakers in terms of forecasts in the form of internally consistent
probability distributions (fan charts) for both target variables and policy
instruments. The framework allows the practical use of forecasts in the form of
probability distributions (distribution forecast targeting) rather than in the more
traditional form of probability averages (mean forecast targeting). This is a
significant improvement since the latter is not valid under model uncertainty.

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Nichttechnische Zusammenfassung
Geldpolitik mit Modellunsicherheit: Ziele für prognostizierte
Wahrscheinlichkeitsverteilungen
Geldpolitik muss immer unter beträchtlicher Unsicherheit betrieben werden: Das gilt
für die Daten, den Zustand der Wirtschaft und besonders hinsichtlich des
Transmissionsprozess der Geldpolitik, das heißt die Stärke und die Verzögerungen
mit denen Zinsänderungen die Inflation und den Output beeinflussen. Dies ist die so
genannte Modellunsicherheit. Das beste Design für die Geldpolitik unter
Modellunsicherheit zu bestimmen, ist ein schwieriges Problem und es gibt wenige
allgemein gültige Grundsätze.
In diesem Papier schlagen wir einen flexiblen, allgemeinen und doch handhabbaren
Rahmen vor, um das richtige Design für eine Geldpolitik unter Modellunsicherheit zu
untersuchen. Modellunsicherheit wird dadurch dargestellt, dass Parameter des
Transmissionsprozesses im Zeitverlauf entsprechend einem Markov- Prozess
zwischen verschiedenen Zuständen schwanken können. Die Verschiebungen
zwischen den Zuständen folgen einem beliebigen Wahrscheinlichkeitsprozess. Auf
diese Weise können wir eine Anzahl von sehr relevanten Arten von
Modellunsicherheit behandeln: Dazu gehören seriell korellierte Zufallsparameter und
Volatilität, Regimeshifts und komplexere, aber realistische Unsicherheiten über
strukturell sehr verschiedene Modelle, wie z. B. rückwärts- und vorwärtsschauende
Erwartungen des privaten Sektors, verschiedene Arten von Preis- und Lohnrigiditäten
und so weiter. Insbesondere können wir berücksichtigen, dass Zentralbanken im
Zeitverlauf ihre Meinung über die Natur und das Ausmaß von Modellunsicherheit
ändern. Dazu gehören etwa zeitweilige Besorgnisse, dass der Durchwirkungsprozess
bei den Wechselkursen sich geändert haben könnte.
Wir entwickeln einen Algorithmus, um die optimale Geldpolitik unter
Modellunsicherheit zu bestimmen, wenn die Ziele der Notenbank gegeben sind. Wir
zeigen auch für beliebige Instrumentenregeln und Instrumentenpfade, wie die
Dynamik der Volkswirtschaft bestimmt wird. Diese Methoden machen es möglich, die
Politikwahl in Form von konsistent Wahrscheinlichkeitsverteilungen (sog. fan charts )
für die Zielvariablen und die Politikinstrumente darzustellen. Der Modellrahmen
erlaubt weiterhin, die praktische Nutzung von Prognosen in der Form von
Wahrscheinlichkeitsverteilungen (distribution forecast targeting) statt der
traditionellen Prognose von Wahrscheinlichkeitsdurchschnitten (mean forecast
targeting). Das stellt eine wesentliche Verbesserung dar, da letztere unter
Modellunsicherheit nicht gültig ist.

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Contents
1
Introduction
1
2
The model
4
2.1
The baseline model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.2
Reformulation according to the recursive saddlepoint method . . . . . . . . . . . . .
6
2.3
Optimal policy and dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3
Interpretation of model uncertainty in our framework
9
4
Examples
11
4.1
An estimated backward-looking model . . . . . . . . . . . . . . . . . . . . . . . . . .
11
4.2
An estimated forward-looking model . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
5
Arbitrary time-varying instrument rules and instrument paths
19
5.1
Setup
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
5.2
Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
6
Arbitrary time-invariant instrument rules and optimal restricted instrument
rules
23
6.1
Setup
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
6.2
Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
6.3
Optimal Taylor-type instrument rules in a forward-looking model . . . . . . . . . . .
25
7
Unobservable modes
27
7.1
Optimal policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
7.2
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
8
Conclusions
32
Appendix
35
A Incorporating central-bank judgment
35
B An algorithm for the value function and optimal policy function
36
C A unit discount factor
39
D Mean square stability
40
E Alternative models with di erent predetermined and forward-looking variables 41
F Details of the estimation
43
G Details for arbitrary time-varying instrument rules
44
H Details for arbitrary time-invariant instrument rules
45

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I
Details with unobservable modes
48
I.1
Unobservable modes and forward-looking variables . . . . . . . . . . . . . . . . . . .
48
I.2
An algorithm for the model with forward-looking variables . . . . . . . . . . . . . . .
51
I.3
Unobservable modes without forward-looking variables . . . . . . . . . . . . . . . . .
53
I.4
An algorithm for the backward-looking model . . . . . . . . . . . . . . . . . . . . . .
54
J
Optimization under discretion
55

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1
Introduction
In recent years there has been a renewed interest in the study of optimal monetary policy under
uncertainty. Classical analyses of optimal policy consider only additive sources of uncertainty, where
in a linear-quadratic framework the well-known certainty-equivalence result applies and implies that
optimal policy is the same as if there were no uncertainty. Recognizing the uncertain environment
that policymakers face, recent research has considered broader forms of uncertainty for which
certainty equivalence no longer applies. While this may have important implications, in practice
the design of policy becomes much more di cult outside the classical linear-quadratic framework.
One of the conclusions of the Onatski and Williams [28] study of model uncertainty is that, for
progress to be made, the structure of the model uncertainty has to be explicitly modeled. In line
with this, in this paper we develop a very explicit but still relatively general form of model uncer-
tainty that remains quite tractable, using a so-called Markov jump-linear-quadratic (MJLQ) model,
where model uncertainty takes the form of di erent “modes” that follow a Markov process. Our
approach allows us to move beyond the classical linear-quadratic world with additive shocks, yet
remains close enough to the linear-quadratic framework that the analysis is transparent. We exam-
ine optimal and other monetary policies in an extended linear-quadratic setup, extended in a way
to capture model uncertainty. The forms of model uncertainty our framework encompasses include:
simple i.i.d. model deviations; serially correlated model deviations; estimable regime-switching mod-
els; more complex structural uncertainty about very di erent models, for instance, backward- and
forward-looking models; time-varying central-bank judgment–information, knowledge, and views
outside the scope of a particular model (Svensson [36])–about the state of model uncertainty; and
so forth. Moreover, while we focus on model uncertainty, our methods also apply to other linear
models with changes of regime which may capture boom/bust cycles, productivity slowdowns and
accelerations, switches in monetary and/or fiscal policy regimes, and so forth. We provide an algo-
rithm for finding the optimal policy as well as solutions for arbitrary policy functions. This allows
us to compute and plot consistent distribution forecasts–fan charts–of target variables and in-
struments. Our methods hence extend certainty equivalence and “mean forecast targeting,” where
only the mean of future variables matter (Svensson [36]), to more general certainty non-equivalence
and “distribution forecast targeting,” where the whole probability distribution of future variables
matter (Svensson [35]).1
1 The importance of the whole distribution of future target variables was recently emphasized by Greenspan [18]
at the 2005 Jackson Hole symposium, with reference to his [17] so-called risk-management approach:
1

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Certain aspects of our approach have been known in economics since the classic works of Aoki
[2] and Chow [8], who allowed for multiplicative uncertainty in a linear-quadratic framework. The
insight of those papers, when adapted to our setting, is that in MJLQ models the value function
remains quadratic in the state, but now with weights that depend on the mode. MJLQ models
have also been widely studied in the control-theory literature for the special case when there are no
forward-looking variables (see Costa and Fragoso [10], Costa, Fragoso, and Marques [11] (henceforth
CFM), do Val, Geromel, and Costa [14], and the references therein). More recently, Zampolli
[41] uses an MJLQ model to examine monetary policy under shifts between regimes with and
without an asset-market bubble, although still in a model without forward-looking variables. Blake
and Zampolli [4] provide an extension of the MJLQ model to include forward-looking variables,
although with less generality than in our paper and with the analysis and the algorithms restricted
to discretion equilibria.
Relative to this previous literature, our contribution is the development of a general approach
for handling MJLQ models that include forward-looking variables. This extension is key for policy
analysis under rational expectations, but the forward-looking variables make the model nonrecur-
sive. We show that the recursive saddlepoint method of Marcet and Marimon [26] can nevertheless
be applied to express the model in a convenient recursive way, and we derive an algorithm for
determining the optimal policy and value functions.
In addition to considering the optimal policy, we also consider the behavior of the model for
arbitrary time-varying or time-invariant instrument rules.
This allows us to construct model-
consistent probability distributions —fan charts–of the variables relevant to policy makers for any
arbitrary instrument-rate path. Moreover, much of the literature in monetary policy analysis has
focused on “simple” instrument rules which are restricted to respond to only a subset of all available
information, with Taylor rules and various generalizations being most prominent. We show how to
derive optimal restricted instrument rules in our setting. Importantly, our approach is not restricted
to instrument rules; any given or optimal restricted policy rule, including targeting rules, can be
considered.
For most of the paper, we focus on the case where agents can directly observe the mode. While
In this [risk management] approach, a central bank needs to consider not only the most likely [rather:
mean] future path for the economy but also the distribution of possible outcomes about that path. The
decisionmakers then need to reach a judgment about the probabilities, costs, and benefits of various
possible outcomes under alternative choices for policy.
We agree with Feldstein [15] that Greenspan’s risk-management approach is best interpreted as standard expected-
loss minimization and we consider the risk-management approach and the approach of this paper as completely
consistent. See Blinder and Reis [5] for further discussion of possible interpretations of the risk-management approach.
2

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