A Guide to Autoregressive Distributed Lag Models for Impulse Response Estimations*
| Published date | 01 October 2022 |
| Author | ChaeWon Baek,Byoungchan Lee |
| Date | 01 October 2022 |
| DOI | http://doi.org/10.1111/obes.12494 |
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 84, 5 (2022) 0305-9049
doi: 10.1111/obes.12494
A Guide to Autoregressive Distributed Lag Models
for Impulse Response Estimations*
CHAEWON BAEK† and BYOUNGCHAN LEE‡
†Department of Economics, Tufts University, 02155, Massachusetts, Medford, 177 College Ave
USA (e-mail: Chaewon.Baek@tufts.edu)
‡Department of Economics, Hong Kong University of Science and Technology, 6068 Lee Shau
Kee Business Building, Clear Water Bay Kowloon, Hong Kong (e-mail: bclee@ust.hk)
Abstract
We provide a guide to using autoregressive distributed lag models for impulse response
estimations with an identified structural shock or an external instrument for the shock. We
illustrate how specifications widely used in practice can lead to inconsistent and inefficient
estimators. We further review empirical results from previous papers and show that some
results appear to be statistical artefacts. We propose a simple method to avoid such a
false conclusion from biases or large standard errors and obtain consistent and precise
estimates.
I. Introduction
Understanding the dynamic responses of aggregate variables to structural shocks has
been a central question in macroeconomics. While many researchers employ traditional
vector autoregressive (VAR) models to estimate responses, single equation methods are
becoming popular alternatives in applied macroeconomic research. These methods are
simple, flexible, and easy to understand. Moreover, these methods allow researchers to
concentrate on a single shock and a single endogenous variable because single equation
methods do not require full specifications of other shocks.
This paper studies a specific single equation method that estimates structural impulse
responses via an autoregressive model with distributed lags of an identified shock or an
external instrument for the shock. We use autoregressive distributed lag (ADL) models
as a tool to estimate the impulse response coefficients rather than describe the underlying
data generating process (DGP) in every respect. Therefore, our focus differs from that
of the classic literature concerning distributed lag models mainly in the search for the
JEL Classification numbers: C13, C29, C52, G01.
*We are greatly indebted to Yuriy Gorodnichenko, Christina Romer, and David Romer for their continual support
and advice. We also received helpful comments from Marc Dordal-i-Carreras, Olivier Coibion, Pierre-Olivier
Gourinchas, Woojin Kim, Peter McCrory, Todd Messer, Emi Nakamura, Piyush Panigrahi, Nick Sander, J´
on
Steinsson, and the seminar participants at the University of California at Berkeley. We also thank the editor
and the referee for helpful comments and suggestion. Support from the Hong Kong University of Science and
Technology Grant No. R9213 is gratefully acknowledged. All errors are our own.
1101
©2022 Oxford University and John Wiley & Sons Ltd.
1102 Bulletin
identification of ‘true’ models (see Sims, 1973; Hendry, Pagan, and Denis Sargan, 1984,
and references therein). Instead, we investigate the properties of the estimated structural
impulse response function (IRF) based on ADL models when an identified shock or an
external instrument is available.
ADL models, often referred to as single equation regressions following (Romer and
Romer, 2004,2010), have a long history and have been actively used until recently
(e.g. Romer and Romer, 2004; Cerra and Saxena, 2008; Romer and Romer, 2010;
Coibion, 2012; Alesina, Favero, and Giavazzi, 2015; Arezki, Ramey, and Sheng, 2017).
However, knowledge concerning their properties in terms of the derived impulse responses
is limited, and guidance for practitioners is lacking. We investigate the properties of ADL
models and provide a simple recipe to tailor these models according to various research
questions. Our findings could help researchers better understand their estimators and
estimate impulse responses consistently.
Specifically, we focus on the following three parameters: the length of the distributed
lags of the identified shocks or the external instruments (denoted by A), the order of
autoregression (B), and the longest horizon of the impulse responses relevant to the
research questions (C). We examine how A,B,andCaffect the estimators of IRFs. We
further provide a practitioner’s guide to selecting Aand Bin a manner that guarantees
consistency of the estimator up to horizon C. Our result holds in a generic setup regardless
of the true ADL representation of the data generating process (DGP). When researchers
do not know the true lag orders, which is usually the case because macroeconomic
variables are driven by multiple structural shocks, our guide can provide a reliable method
to consistently estimate impulse responses using ADL models. Furthermore, when a
structural shock is not observable, our procedure can point-identify impulse responses up
to normalization using an external instrument. Our findings echo similar results reported
in Stock and Watson (2018) regarding the method of external instruments for structural
VARs and local projections (LP).
We show that we can consistently estimate impulse response coefficients up to horizon
Cby making Agreater than or equal to C. This result holds for a generic setup, including
cases in which the true lag orders are infinite. However, it is not unusual for Ato be much
smaller than Cin applied research. We explain why, in this case, it is easy to falsely
conclude that a shock has long-run effects on a dependent variable with no relation to
the true impulse responses. Given that consistency is achieved by long distributed lags
(A≥C), Bcan further affect the variability of the estimates; the estimator can become
more efficient by properly choosing B. Therefore, given C, which naturally follows from
the research question, we suggest setting Aat Cand relying on standard order selection
criteria for autoregressive models to choose B. Depending on DGPs, our consistency
results may be accompanied by an increase in variance of the impulse response estimator,
especially at long horizons. Thus, regarding the bias-variance trade-off, setting A≥C
prioritizes eliminating (asymptotic) biases in a generic setup.
Our suggestion differs from the approaches employed in previous papers. Without an
established procedure, researchers have relied on statistical tests (Cerra and Saxena, 2008),
used an information criterion to jointly choose Aand B(Coibion, 2012), or selected
a number that might be justified by reasonable results (Romer and Romer, 2004;
Kilian, 2008; Romer and Romer, 2010; Hsiang and Jina, 2014; Arezki et al.,2017). All
©2022 Oxford University and John Wiley & Sons Ltd.
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