Local Linear Impulse Responses for a Small Open Economy*

Date01 June 2012
DOIhttp://doi.org/10.1111/j.1468-0084.2011.00643.x
Published date01 June 2012
470
©Blackwell Publishing Ltd and the Department of Economics, University of Oxford, 2011. Published by Blackwell Publishing Ltd,
9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 74, 3 (2012) 0305-9049
doi: 10.1111/j.1468-0084.2011.00643.x
Local Linear Impulse Responses for a Small Open
Economy*
Alfred A. Haug† and Christie Smith
Department of Economics, University of Otago, Dunedin, New Zealand
(e-mail: ahaug@business.otago.ac.nz)
Economics Department, Reserve Bank of New Zealand, 2 The Terrace, PO Box 2498,
Wellington, New Zealand (e-mail: christie.smith@rbnz.govt.nz)
Abstract
Traditional vector autoregressions derive impulse responses using iterative techniques that
may compound specication errors. Local projection techniques are more robust to this
problem, and Monte Carlo evidence has suggested they provide reliable estimates of the
true impulse responses. We use local linear projections to investigate the dynamic proper-
ties of a model for a small open economy, New Zealand. We compare impulse responses
from projections to those from standard techniques, and consider the implications for mon-
etary policy. We pay careful attention to the dimensionality of the model, and focus on
effects of policy on gross domestic product, interest rates, prices and exchange rates.
I. Introduction
Impulse response functions (IRFs) are widely used in macroeconomics to assess the persis-
tence and relative effects of various macroeconomic shocks. These empirical observations
are also used in the development of theoretical models. Tocharacterize the effects of macro-
economic shocks, the standard approach is to estimate a vector autoregressive (VAR)
model.1To derive the impulse responses from such a VAR, it is transformed into a moving
average (MA) representation by appealing to Wold’s decomposition theorem.
In this article we take an alternative approach, following Jord`a (2005). We apply local
linear projections to obtain the impulse responses, as an alternative to the moving average
transformation. An impulse response can be regarded as a revision in the forecast of a
variable at a future horizon t+sto a one-time experimental shock at time t, assuming that
no other shocks hit the system. Based on this denition, Jord`a proposes using multi-step
ÅThe authors thank two anonymous referees of this journal, Rebecca Braeu, `
Oscar Jord`a, ¨
Ozer Karagedikli, Lutz
Kilian, Troy Matheson, Adrian Pagan, Ken West and participants at the New Zealand Econometrics Study Group
Meeting in Dunedin, the Econometric Society Australasian Meeting in Brisbane, the WEAI Pacic Rim Conference
in Beijing, the Annual Conference of the Economic Society of Australia in Hobart, and at seminars at the German
Bundesbank, Massey (Albany), Melbourne, and Otago universities for helpful comments. The usual caveat applies.
JEL Classication numbers: C51, E52, F41.
1To identify structural shocks from the estimated reduced-form shocks, either a recursive causal ordering or
structural relations are imposed on contemporaneous shocks, and/or some long run effects of shocks are restricted.
Local linear impulse responses 471
direct forecasts, which he refers to as local projections, to calculate impulse responses.
For every impulse horizon a new forecast is estimated instead of using iterated forecast
all based on the same coefcient estimates from one VAR estimation. Jord`a proves that
impulse responses derived from direct forecasts are consistent and asymptotically normal.
Standard impulse responses based on the MA representation face several potentially
serious problems because the iterative process used in the calculations magnies speci-
cation problems. First, the lag length required for estimating a VAR to produce reliable
impulse responses may be very large.2Second, the vector-MA (VMA) representation of a
VAR may not be unique and different invertibility assumptions can produce very differ-
ent impulses.3Third, the presence of unit roots and cointegration in the VAR leads to
inconsistent impulses at longer horizons.4
Local projections are more robust to misspecication than standard iterative forecasts.5
The potential for a misspecied VAR is manyfold, from variable selection to lag length
specication. Estimation procedures that are more robust to misspecication are there-
fore highly desirable. Jord`a’s Monte Carlo evidence shows that impulse responses can be
estimated more accurately by using local linear projections than by VMA-based methods.
Typical estimates of VARs are global approximations to the data-generating process, and
these approximations are optimally designed for one-period ahead forecasting. However,
given the ubiquity of model misspecication, local approximations may be preferable to
global approximations at longer horizons. Jord`a’s Monte Carlo results illustrate that the
loss of efciency from using local projection impulse responses (instead of a correctly
specied VAR) is very small, and that local projection impulses are much more accurate
at medium to longer horizons than VMA-based methods when the model is misspecied.
Our article contributes to the existing literature in several ways. Jord`a provides an
empirical example for his local projection-based impulses, using a recursive causal order-
ing to identify structural shocks for a large economy model. In our article we extend his
method to a non-recursive structural identication scheme and to a small open economy.
Our model is related to the open-economy structural VAR models of, among others, Mojon
and Peersman (2003), Kim and Roubini (2000) and Cushman and Zha (1997), though our
set of variables is tailored to New Zealand and we use local projections instead of traditional
VMA-based methods. We compare IRFs based on local projections to those from VMAs.
The dynamic responses to monetary policy shocks that we derive with Jord`a’s method are
consistent with standard macroeconomic theories, whereas traditional structural impulse
responses reveal several idiosyncracies or ‘puzzles’.
The rest of the article is organized as follows. Section II outlines the method used to
calculate IRFs. Section III discusses the specication of the models and the criteria used
2See Kapetanios, Pagan and Scott (2007) for an extreme scenario.
3See Hansen and Sargent (1980). Lippi and Reichlin (1993) provide an example with long-run identifying restric-
tions (which we do not use in our article).
4Consistent parameter estimates can be obtained by applying least squares to levels VARs,even when unit roots
and cointegration are ignored (Sims, Stock and Watson, 1990). However, standard impulse responses are highly
nonlinear functions of the VARparameters and the consistency results do not carry over to the impulse responses.
Phillips (1998) proves inconsistency for unit roots and roots near unity when the lead time of the IRFs is a xed
fraction of the sample size. See also Pesavento and Rossi (2006).
5See, among others, Lin and Tsay (1996), Ing (2003), Chevillon and Hendry (2005) and Marcellino, Stock and
Watson (2006).
©Blackwell Publishing Ltd and the Department of Economics, University of Oxford 2011

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