Nested Models and Model Uncertainty

• Author: Alexander Kriwoluzky,Christian A. Stoltenberg
• Published date: 01 April 2016
• DOI: http://doi.org/10.1111/sjoe.12134
• Date: 01 April 2016

Scand. J. of Economics 118(2), 324–353, 2016

DOI: 10.1111/sjoe.12134

Nested Models and Model Uncertainty∗

Alexander Kriwoluzky

Martin Luther University of Halle-Wittenberg, DE-06099 Halle (Saale), Germany

alexander.kriwoluzky@wiwi.uni-halle.de

Christian A. Stoltenberg

University of Amsterdam, NL-1018 WB Amsterdam, The Netherlands

c.a.stoltenberg@uva.nl

Abstract

Uncertainty about the appropriate choice among nested models is a concern for optimal

policy when policy prescriptions from those models differ. The standard procedure is to

specify a prior over the parameter space, ignoring the special status of submodels (e.g., those

resulting from zero restrictions). Following Sims (2008, Journal of Economic Dynamics and

Control 32, 2460–2475), we treat nested submodels as probability models, and we formalize

a procedure that ensures that submodels are not discarded too easily and do matter for

optimal policy. For the United States, we find that optimal policy based on our procedure

leads to substantial welfare gains compared to the standard procedure.

Keywords: Bayesian model estimation; model uncertainty; optimal monetary policy

JEL classification:C51; E32; E52

I. Introduction

The empirical evaluation of monetary dynamic stochastic general equilib-

rium (DSGE) models employing Bayesian methods (Smets and Wouters,

2003, 2007; Castelnuovo, 2012) has made substantial progress. Policymak-

ers nowadays correspondingly employ estimated DSGE models, including

various features and frictions, in their policy analysis. However, a central

concern for policymakers is the uncertainty about the correct model, which

has led to the practice of using several models at the same time. The lat-

ter practice has encouraged research to find policy recommendations that

perform well over a set of distinct models.

Correspondingly, model uncertainty in non-nested models has been ex-

tensively studied in the literature (e.g., Levin and Williams, 2003; Levin

et al., 2003; Kuester and Wieland, 2010). For the US, Levin and Williams

∗We are especially grateful to Fabio Canova, Wouter Den Haan, Martin Ellison, Bartosz

Ma´

ckowiak, Morten Ravn, Harald Uhlig, and seminar participants at the Sveriges Riksbank.

Previous versions circulated under the title “Optimal Policy under Model Uncertainty: A

Structural-Bayesian Estimation Approach”.

CThe editors of The Scandinavian Journal of Economics 2015.

A. Kriwoluzky and C. A. Stoltenberg 325

(2003) and Levin et al. (2003) have studied optimal policies in the case

of competing reference models. For the Euro area, Kuester and Wieland

(2010) have compared the performance of worst-case or minimax policy

versus Bayesian policies over a range of different models. It is clearly un-

derstood that optimal policies between different models can be conflicting.

Little attention, however, has been given to the uncertainty nested within a

given model that results from the inclusion of various features and frictions.

In this paper, we seek to make two contributions. We document quan-

titatively that uncertainty about nested models is an important issue for

policymakers in practice. Furthermore, we propose a procedure to guard

against uncertainty about the appropriate choice of nested models. The

procedure follows the spirit of Sims (2008) who argues that accounting

for uncertainty in nested models requires a policymaker who treats nested

submodels as probability models. Probability models are models that char-

acterize the uncertainty incorporated in them; that is, the probability that

the models indeed generated a given set of time series (model uncertainty)

and the corresponding probability distribution of the structural parameters

(parameter uncertainty). A policymaker should take into account both types

of uncertainty when choosing an optimal course of action. This is exactly

the approach we formalize in this paper.

Starting with a baseline model, we subsequently estimate a set of com-

peting and nested models, including one model that comprises all features

and frictions. This information puts us into a position to separately eval-

uate the gain in explanatory power of each extension. We compare two

approaches to compute optimal simple rules under model uncertainty. The

first approach computes optimal policy in the model that nests all features

and frictions and ignores the set of nested models. As a methodological

contribution, we propose a second approach that takes into account the

whole set of nested models. The approach computes optimal policies over

the set of nested models by weighting each model by its posterior proba-

bility. By weighting over the set of models, the policymaker can construct

reasonable extensions of the baseline model, and thereby avoid the pitfalls

of only employing one potentially misspecified model.

In our application, we ask whether there is a quantitatively important

welfare difference between the two approaches to model uncertainty for

monetary policy in the US as one example. To answer this question, we

employ US data and choose as a baseline model one of the most popular

models employed in monetary policy: a standard cashless New Keynesian

economy with staggered price-setting (Woodford, 2003a). As examples of

uncertainty linked to the choice between nested models, we subsequently al-

low for more lags in endogenous variables (indexation and habit formation).

The baseline model and these extensions resemble a situation in which

the main focus of the policymaker is on the trade-off between stabilizing

CThe editors of The Scandinavian Journal of Economics 2015.