In recent years there has been a renewed interest in the study of optimal monetary policy under uncertainty. Typical formulations of optimal policy consider only additive sources of uncertainty, where in a linear-quadratic (LQ) 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 difficult outside the classical LQ framework.
One of the conclusions of the Onatski and Williams [31] 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 uncertainty that remains quite tractable. We use a so-called Markov jump-linear-quadratic (MJLQ) model, where model uncertainty takes the form of different “modes” (or regimes) 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 LQ framework that the analysis is trans-parent. We examine 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 models; more complex structural uncertainty about very different 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 [39])—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 algorithm 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 instruments. Our methods hence extend certainty equivalence and “mean forecast targeting,” where only the mean of future variables matter (Svensson [39]), to more gen-eral certainty non-equivalence and “distribution forecast targeting,” where the whole probability distribution of future variables matter (Svensson [38]).
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 for the optimal policy design problem 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 [15], and the references therein). More recently, Zampolli [45] 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 observable modes and discretion equilibria.
Our MJLQ approach is also closely related to the Markov regime-switching models which have been widely used in empirical work. These methods first gained prominence with Hamilton [21] which started a burgeoning line of research. Models of this type have been used to study a host empirical phenomena, with many developments and techniques. summarized in [25]. More recently, the implications of Markov switching in rational expectations models of monetary policy have been studied by Davig and Leeper [13] and Farmer, Waggoner, and Zha [16]. These papers focus on (and debate) the conditions for uniqueness or indeterminacy of equilibria in forward-looking models, taking as given a specified policy rule.
Contents
1 Introduction
2 The model
2.1 The baseline model
2.2 Informational assumptions
2.3 Reformulation according to the recursive saddlepoint method
- 2.3.1 Observable modes
2.3.2 Unobservable modes
3 Interpretation of model uncertainty in our framework
4 Examples
4.1 An estimated backward-looking model
4.2 An estimated forward-looking model
4.3 Uncertainty about whether the model is forward- or backward-looking
5 Arbitrary instrument rules and optimal restricted instrument rules
5.1 Instrument rules
5.2 Optimal Taylor-type instrument rules in a forward-looking model
6 Conclusions
Appendix
A Incorporating central-bank judgment
B An algorithm for the case of observable modes
C An algorithm for the case of unobservable modes
- C.1 Unobservable modes and forward-looking variables
C.2 An algorithm for unobservable modes with forward-looking variables
C.3 An algorithm for unobservable modes without forward-looking variables
D A unit discount factor
E Mean square stability
F Details of the estimation
G An algorithm for an arbitrary instrument rule
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Monetary Policy with Model Uncertainty: Distribution Forecast Targeting
