Projection Estimators for Structural Impulse Responses*
| Published date | 01 December 2023 |
| Author | Jörg Breitung,Ralf Brüggemann |
| Date | 01 December 2023 |
| DOI | http://doi.org/10.1111/obes.12562 |
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 85, 6 (2023) 0305-9049
doi: 10.1111/obes.12562
Projection Estimators for Structural
Impulse Responses*
J¨
ORG BREITUNG† and RALF BR¨
UGGEMANN‡
†Institute of Econometrics, University of Cologne, K¨
oln, 50923, Germany (e-mail:
breitung@statistik.uni-koeln.de)
‡Department of Economics, University of Konstanz, Konstanz, 78457, Germany
Abstract
In this paper we provide a general two-step framework for linear projection estimators
of impulse responses in structural vector autoregressions (SVARs). This framework is
particularly useful for situations when structural shocks are identified from information
outside the VAR (e.g. narrative shocks). We provide asymptotic results for statistical
inference and discuss situations when standard inference is valid without adjustment for
generated regressors, autocorrelated errors or non-stationary variables. We illustrate how
various popular SVAR models fit into our framework. Furthermore, we provide a local
projection framework for invertible SVAR models that are estimated by instrumental
variables (IV). This class of models results in a set of quadratic moment conditions
used to obtain the asymptotic distribution of the estimator. Moreover, we analyse
generalized least squares (GLS) versions of the projections to improve the efficiency
of the projection estimators. We also compare the finite sample properties of various
estimators in simulations. Two highlights of the Monte Carlo results are (i) for invertible
VARs our two-step IV projection estimator is more efficient compared to existing
projection estimators and (ii) using the GLS projection variant with residual augmentation
leads to substantial efficiency gains relative to standard OLS/IV projection estimators.
I. Introduction
The analysis of dynamic effects in vector autoregressive (VAR) models by means of
impulse responses has become a standard tool in empirical macroeconomics (cf. Kilian
and L¨
utkepohl, 2017). Following Sims (1980) the dynamic effects of shocks are typically
measured by the moving average (MA) coefficients derived from the finite-order VAR
representation of the time series. In recent years it has become popular to estimate the
effects of structural shocks by ‘local projections’ (e.g. Jord`
a, 2005;Jord
`
a, Schularick,
and Taylor, 2015; Ramey and Zubairy, 2018;Jord
`
a, Schularick, and Taylor, 2020). This
method is based on a direct representation of the time series vector shifted hperiods
JEL Classification numbers: C32, C51.
*Financial support of the German Science Foundation (DFG grant number: BR 2941/3-1/2) is gratefully
acknowledged. Part of the research was carried out while the second author was visiting Monash University,
Melbourne. We thank the editor and two anonymous referees for many useful comments. Open Access funding
enabled and organized by Projekt DEAL.
1320
©2023 The Authors. Oxford Bulletin of Economics and Statistics published by Oxford University and John Wiley & Sons Ltd.
This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and
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Projection estimators for structural impulse responses 1321
ahead, whereas the traditional method traces out the impulse responses iteratively from
thefirstuptothehth period.
The use of (local) projection estimators in structural VAR (SVAR) applications has
been motivated by different arguments. First, local projection estimators are more robust
against model misspecification. Second, in some applications there may not exist a suitable
VAR framework (e.g. if shocks are determined outside the VAR models). Third, the VAR
model may not be invertible with respect to the structural shocks (see e.g. Stock and
Watson, 2018). Furthermore, asymptotic inference on structural impulse responses based
on iterated VARs may be inaccurate as the quantity of interest is a highly nonlinear
transformation of the VAR parameters. Inference for projection estimators, which are
linear in the parameters, may be easier to implement and statistical inference is typically
more accurate in small samples. Consequently, despite being potentially less efficient than
iterated response estimators, local projections are nowadays a popular tool in empirical
economics (for a review of the earlier literature see Ramey, 2016).
Our paper considers a general modelling framework for linear projection estimators
of impulse responses in SVAR models and makes the following contributions: First,
we adapt a two-step framework for shock identification and response estimation. This
generalizes the VAR approach and can incorporate structural shocks that stem from a
different information set (e.g. narrative or high-frequency shocks). Second, we provide an
asymptotic framework for statistical inference on the projection estimator, which takes into
account that the structural shocks are estimated in the first step. Third, we propose a new
local projection estimator (2S-IV) for invertible SVARs identified by instrumental or proxy
variables and compare it to existing methods. Although not robust against non-invertibility,
our simulations indicate that in invertible settings this estimator provides substantial
efficiency gains over existing projection estimators. Based on our approach, we derive a
test for non-fundamental shocks that is similar to the test proposed by Plagborg-Møller and
Wolf (2022). Fourth, we characterize applications where standard regression inference
applies no matter of serial correlation, generated regressors or non-stationary time series.
Fifth, we provide guidance on how to adapt projection estimators to various popular
identification schemes. Sixth, we argue that the iterative VAR approach is asymptotically
equivalent to a particular GLS version of the projection estimator. Finally, we compare
the small sample properties of various estimators and of tests for non-fundamental shocks.
We stress that our paper does not offer any new reasons for preferring local projections
over iterated VARs in empirical practice. Rather we start from the fact that many empirical
economists apply this methodology in order to assess the effects of macroeconomic shocks.
Therefore, we analyse popular strategies for identifying structural shocks such as recursive
(Cholesky) schemes, proxy-VARs, narrative shocks, systems of simultaneous equations
(the AB-model) and shocks identified by long-run restrictions. We show how all these
popular variants of SVARs can be cast into a general framework consisting of two separate
steps, the identification of shocks and the estimation of impulse responses. More formally
the analysis can be characterized by two different steps (e.g. Ramey, 2016, Sec. 2.4):
εj,t=f(xj,t,βj),(1)
yi,t+h=θh
ijεj,t+z
j,tπh
ij +eh
ij,t,(2)
©2023 The Authors. Oxford Bulletin of Economics and Statistics published by Oxford University and John Wiley & Sons Ltd.
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