CONSISTENT VARIANCE OF THE LAPLACE‐TYPE ESTIMATORS: APPLICATION TO DSGE MODELS

• Published date: 01 May 2016
• Author: Denis Nekipelov,Anna Kormilitsina
• Date: 01 May 2016
• DOI: http://doi.org/10.1111/iere.12169

INTERNATIONAL ECONOMIC REVIEW

Vol. 57, No. 2, May 2016

CONSISTENT VARIANCE OF THE LAPLACE-TYPE ESTIMATORS: APPLICATION

TO DSGE MODELS∗

BYANNA KORMILITSINA AND DENIS NEKIPELOV1

Southern Methodist University, U.S.A.; University of Virginia, U.S.A.

The Laplace-type estimator has become popular in applied macroeconomics, in particular for estimation of dynamic

stochastic general equilibrium (DSGE) models. It is often obtained as the mean and variance of a parameter’s quasi-

posterior distribution, which is defined using a classical estimation objective. We demonstrate that the objective must be

properly scaled; otherwise, arbitrarily small confidence intervals can be obtained if calculated directly from the quasi-

posterior distribution. We estimate a standard DSGE model and find that scaling up the objective may be useful in

estimation with problematic parameter identification. It this case, however, it is important to adjust the quasi-posterior

variance to obtain valid confidence intervals.

1. INTRODUCTION

In spite of the popularity of medium-scale dynamic stochastic general equilibrium (DSGE)

models in empirical macroeconomic research, their estimation is often associated with practical

difficulties. For an applied researcher, the problems with estimation range from the possibility

of multiple local solutions to poor identification of model parameters due to the flatness of the

objective function in the vicinity of the extremum. In estimations with classical objectives, it

has become popular to rely on Bayesian methods by using the Laplace-type estimator (LTE).

(See Christiano et al., 2010; Coibion and Gorodnichenko, 2011; Kormilitsina, 2011; Schmitt-

Groh´

e and Uribe, 2011, among others.) The LTE is a Bayesian alternative to the classical

extremum estimators. It consists in formulating the so-called “quasi-likelihood” function based

on a prespecified statistical criterion, which could be derived from the general method of

moments (GMMs) objective, the maximum likelihood, or another classical estimator. The

quasi-likelihood function implies the quasi-posterior distribution of model parameters, which

can be evaluated using Markov Chain Monte Carlo (MCMC) algorithms, and the estimate is

then obtained as the mean or a quantile of the quasi-posterior distribution.

The popularity of the LTE is largely due to the result in Chernozhukov and Hong (2003),

who demonstrate that the estimator is both theoretically and computationally attractive. From

the computational perspective, the LTE allows one to overcome the curse of dimensionality

problem related to the search of the extremum in classical estimation, because it relies on MCMC

methods instead of costly search procedures. From the theoretical point of view, Chernozhukov

and Hong (2003) establish that under mild assumptions, the LTE is asymptotically equivalent

to the corresponding frequentist extremum estimator. Moreover, if the generalized information

equality (GIE) holds, then the variance of the quasi-posterior distribution provides a consistent

estimate for the variance of the corresponding frequentist estimator. However, if the GIE

does not hold, then the variance of the parameter estimate cannot be approximated by the

∗Manuscript received February 2014; revised November 2014.

1This article has benefited from discussions with Han Hong, Atsushi Inoue, Frank Schorfheide, and James Stock.

We would also like to thank the editor, Jesus Fern´

andez-Villaverde, and two anonymous referees for their insightful

comments. Please address correspondence to: Anna Kormiltsina, Southern Methodist University, 3300 Dyer Street,

Suite 301, Umphrey Lee, Dallas, TX 75275. E-mail: annak@smu.edu.

603

C

(2016) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social

and Economic Research Association

604 KORMILITSINA AND NEKIPELOV

quasi-posterior distribution. Instead, one should transform the quasi-posterior variance using

the “sandwich formula” in Chernozhukov and Hong (2003).2

In this article, the focus is on situations where the GIE is not satisfied. More specifically,

we study the LTE derived using a GMM objective. These estimators are popular in empirical

macroeconomic research; however, it is often difficult to ensure the GIE in these problems,

because an efficient weighting matrix cannot be reliably computed given the sample size in

these applications.3Because relying on efficient weighting may significantly hinder the small

sample performance of the estimator, researchers often resort to diagonal or other inefficient

weighting matrices in formulating the GMM objective.

Within the class of GMM problems, our contribution is the following: First, we demonstrate

that even when the weighting matrix is efficient, the GIE may fail if the objective function is not

scaled correctly. We show that although in classical GMM estimation, the scaling of the objective

function is not essential for the calculation of variance, proper scaling is crucial in LTE, as it

modifies the quasi-posterior distribution. In particular, larger scaling implies smaller variance

of the quasi-posterior distribution. We therefore conclude that one can calculate confidence

intervals directly from quasi-posterior distributions only in efficient estimation problems with

proper scaling of the objective function.

Our second contribution is of practical nature. We find that in empirical applications, it may

be optimal to force deviation from the GIE by scaling up the objective function. In an empirical

exercise, we estimate a simple DSGE model using real and simulated data. We first document

that the variance of the quasi-posterior distribution is generally inversely proportional to the

scaling parameter. Moreover, the variance of the LTE calculated by properly transforming the

variance of the quasi-posterior distribution is robust to the choice of the scaling parameter.

However, we find that these conclusions fail when the scaling is absent (μ=1). In this case,

both the variance of the MCMC chains and the variance of the estimator are usually greater

than those at μ>1, contrary to the predictions of theory. This result is indicative of the

poor performance of the unscaled LTE, which we relate to the presence of poorly identified

parameters and small samples. We confirm this idea in a Monte Carlo experiment where we

repeatedly estimate the model using artificially generated data sets. We find that increasing the

scaling parameter of the objective function allows one to reduce both the bias and variance

of parameter estimates. We therefore conclude that in empirical applications, the scaling of

the objective can be used as an instrument to improve the outcome of estimation. It has to be

emphasized, however, that confidence intervals of the estimator in this case must be obtained

by appropriately transforming the variance of the quasi-posterior distribution.

Implementation of the LTE parallels that in the Bayesian estimation, which has also become

a popular approach in empirical macroeconomics (see, for example, An and Schorfheide, 2007;

Fern´

andez-Villaverde, 2010; Aruoba and Schorfheide, 2011; Fern´

andez-Villaverde et al., 2012;

and references therein). The uniqueness of the LTE, however, is that it relies on Bayesian

methods to address alternative, classical estimation problems.4The LTE based on the maxi-

mum likelihood estimator is most similar to the Bayesian estimation methods commonly used

to estimate DSGE models. Both the LTE and the Bayesian approach therefore face similar

difficulties in empirical applications, stemming from problematic parameter identification and

short data samples. However, although scaling of the quasi-likelihood function may help re-

solve these problems for LTE, it cannot be helpful for Bayesian estimation. The reason is that

the Bayesian approach assumes that the structural parameters are of a stochastic, instead of

a deterministic, nature. This means that a Bayesian economist is interested in evaluating the

2See theorems 2 and 4 in Chernozhukov and Hong (2003).

3This is usually the case in minimum-distance estimation problems that aim to match a large number of impulse

responses or moments of the model and data. See Christiano et al. (2010), Kormilitsina (2011), and DiCecio (2009).

4Although in this article, we focus on LTE based on GMM objective, our results can be easily extended to include

other classical estimation methods where the LTE is commonly applied, for example, extremum estimators that contain

nonparametric plug-in components. See Altonji and Segal (1996), Windmeijer (2005), and Newey and Windmeijer

(2009), among many others.