A THEORY OF PRUNING

• Author: Giovanni Lombardo,Harald Uhlig
• Published date: 01 November 2018
• Date: 01 November 2018
• DOI: http://doi.org/10.1111/iere.12321

INTERNATIONAL ECONOMIC REVIEW

Vol. 59, No. 4, November 2018 DOI: 10.1111/iere.12321

A THEORY OF PRUNING∗

BYGIOVANNI LOMBARDO AND HARALD UHLIG1

Bank for International Settlements, Switzerland; University of Chicago, U.S.A.

Often, numerical simulations for dynamic, stochastic models in economics are needed. Kim et al. (Journal of

Economic Dynamics and Control 32 2008, 3397–414) proposed “pruning” to deal with the challenge of generating

explosive paths when employing second-order approximations. In this article, we provide a theory of pruning

and formulas for pruning of any order. Our approach builds on Judd’s Numerical Methods in Economics (1998),

chapter 13. We provide a comparison to existing methods.

1. INTRODUCTION

Solutions to dynamic stochastic general equilibrium (DSGE) models can often be thought of

to take a recursive, state-space form, say

xt=h(xt−1;σ)+σηt

(1)

for the vector of state endogenous variables xt∈Rnxand

yt=g(xt;σ)(2)

for the vector of endogenous control variables yt∈Rny, where t∈Rnis a disturbance with

some given distribution, σis a parameter linking the latter with the former, and ηis an nx×n

scaling matrix. The law of motion h(·) and the mapping g(·) are not known: Instead, they need

to be derived from the underlying equations of the DSGE model. In the rest of the article, we

will focus on h(·), as our results can easily be extended to g(·).

It is typically not possible to provide an exact or closed-form expression for h(·). Instead,

numerical approximation methods need to be used; see Judd (1998). First-order approximations

are well understood and popular; see, for example, Uhlig (1999). Following Jin and Judd (2002),

researchers have made increasing use of higher order polynomial approximations to h, obtained

from a perturbation approach or Taylor series around some steady state. These higher order

approximations contain powers of xt−1. As Kim et al. (2008), henceforth KKSS, have pointed

out, these higher order terms (quadratic in their case) can be problematic for simulations. KKSS

suggest “pruning” to deal with this problem by “rewriting” higher order terms as products of

lower order terms and thus delivering stationarity. Their approach has been expanded in the

recent literature. Andreasen et al. (2018) extend the “pruning” technique to higher orders of

approximation. Lan and Meyer-Gohde (2013a, 2013b) use a Volterra expansion approach to

derive a “pruned” approximation. We discuss these two approaches in some detail in Section 5.

Den Haan and de Wind (2012) emphasize the potential pitfalls of particular implementations of

the “pruning” technique to higher orders of approximation and propose an alternative approach.

∗Manuscript received November 2017; revised November 2017.

1Harald Uhlig has an ongoing consulting relationship with a Federal Reserve Bank, the Bundesbank, and the ECB.

Please address correspondence to: Giovanni Lombardo, Bank for International Settlements, Centralbahnplatz 2, 4051

Basel, Switzerland. Phone: +41 76 3509283. E-mail: giannilmbd@gmail.com.

1825

C

(2018) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social

and Economic Research Association