Tests for Multiple Breaks in the Trend with Stationary or Integrated Shocks
| Date | 01 June 2016 |
| Published date | 01 June 2016 |
| DOI | http://doi.org/10.1111/obes.12116 |
394
©2015 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 78, 3 (2016) 0305–9049
doi: 10.1111/obes.12116
Tests for Multiple Breaks in theTrend with Stationary
or Integrated Shocks*
Nuno Sobreira† and Luis C. Nunes‡
†CEMAPRE and School of Economics and Management (ISEG), Universidade de Lisboa,
Lisbon, Portugal (e-mail: nsobreira@iseg.ulisboa.pt)
‡Nova School of Business and Economics, Universidade Nova de Lisboa, Lisbon, Portugal
(e-mail: lcnunes@novasbe.pt)
Abstract
In this paper, we propose new tests of the presence of multiple breaks in the trend of a
univariate time-series where the number and dates of the breaks are unknown and that
are valid in the presence of stationary or unit root shocks. These tests can also be used to
sequentially estimate the number of breaks. The behaviour of the proposed tests is studied
through Monte Carlo experiments.
I. Introduction
Many macroeconomic time series are characterized by a clear tendency to grow over
time, that is, as having a deterministic time trend component. There has been a large
debate in the literature regarding the appropriate methods to infer about the linearity and
stability of the trend function and the nature of the shocks affecting a time series. This is
a particularly important issue when it comes to make accurate economic forecasts or test
economic hypotheses. In fact, there are manyinteresting economic applications that involve
statistical inference on the parameters of the trend function, namely, in the continuous time
macroeconomic modelling (see Bergstrom, Nowman and Wymer, 1992, Nowman, 1998),
in international trade, for example, with the Prebish–Singer hypothesis testing (see Bunzel
and Vogelsang, 2005), in the empirical debate regarding regionalconvergence in per capita
income (see Sayginsoy andVogelsang, 2004), or in environmental economics on the future
consequences of global warming (see Vogelsang and Franses, 2005).
The stationarity properties of the shocks have important implications on the appropriate
methods to make inferences about the trend function. In particular, the correct approach
JEL Classification numbers: C12, C22.
*Weare grateful to three anonymous referees, Editor Anindya Banerjee, Iliyan Georgiev, Paulo M. M. Rodrigues,
Luis F.Mar tins,Vasco Gabriel and the participants in seminars at Universiteit Van Amsterdam and NovaSchool of
Business and Economics, in the QED Conference (Amsterdam, May 2009), in the Econometric Society European
meeting (joint congress with the European Economic Association; Barcelona, August 2009) and in the XXXV
Simposio de la Asociaci´on Espa˜nola de Econom´
ia (Madrid, December 2010) for helpful comments and suggestions
on earlier versions of this paper.We also thank DavidHarvey and Mohitosh Kejriwal for providing us with their Gauss
programs. The authors acknowledge support from FCT grants UID/ECO/00124/2013 and UID/Multi/00491/2013.
Tests for multiple breaks in the trend 395
to make inferences about the stability or the existence of breaks in the trend depends on
whether the shocks are I(0) or I(1). In the first case, one should use regressions on the
levels, while for the latter the correct approach is to model the first-differences of the
series. However, it is often not known a priori whether the shocks are stationary or contain
a unit-root. Moreover, stationarity or unit-root tests also sufferfrom similar problems since
their properties are in turn affected by the stability of the trend function.
Only recently some solutions to this dilemma have been proposed in the literature.
These resort to statistical tests of the null hypothesis of a constant linear trend against
the alternative of one break at some unknown date that do not require a priori knowledge
of whether the noise is I(0) or I(1). Sayginsoy and Vogelsang (2004) proposed a Mean
Wald and a Sup Wald statistic scaled by a factor based on unit root tests to smooth the
discontinuities in the asymptotic distributions of the test statistics as the errors go from
I(0) to I(1). The scaling factor approach is based on Vogelsang (1998) who proposed
test statistics for general linear hypothesis regarding the parameters of the trend function
which do not require knowledge as to whether the innovations are I(0) or I(1). Perron
and Yabu (2009) (hereafter PY) proposed a Feasible Quasi Generalized Least Squares
approach to estimate the slope of the trend function. By truncating the estimate of the sum
of the autoregressive coefficients of the disturbance term to take the value of one whenever
the estimate is in a neighbourhood of one, they have shownthat the limiting distribution of
the t-statistic becomes Normal regardless of the persistence of the error term. Kejriwal and
Perron (2010) proposed a sequential testing procedure based on PY. Harvey, Leybourne and
Taylor (2009) (hereafter HLT) employed a weighted average of the appropriate regression
t-statistics used to test the existence of a broken trend when the errors are I(0) and I(1).
However, as Lumsdaine and Papell (1997) point out with an example of Jones (1995),
allowing for only one break is not always the best characterization of a macroeconomic
variable, especially when analysing long historical time series.
This paper extends the results from HLT by providing tests of the null hypothesis of
no trend breaks against the alternative of one or more breaks in the trend slope which
do not require knowledge of the form of serial correlation in the data and are robust as to
whether the underlying shocks are stationary or havea unit-root. We build on the framework
proposed by HLT for the case of a single break, and construct test statistics that are weighted
averages of the appropriate F-statistics to test the existence of multiple trend breaks when
the disturbance term is I(0) and I(1). We adopt the weight function used in HLT and prove
that it has the same large sample properties regardless of the number of trend breaks being
tested.
We start by considering the case where the true break fractions are known and prove
that the proposed statistics converge in distribution to a chi-square distribution under the
null. Next, we consider the case where the trend break fractions are unknown and need to
be estimated. We transform our statistic in the same spirit as Andrews (1993) and Bai and
Perron (1998) and take the supremum of the F statistic over all possible break fractions
except those that are activelyrestricted by the trimming parameter. Here, the weightfunction
is evaluated at the estimated break fractions and we prove that its large sample behaviour is
similar regardless of the number of break fractions estimated and the number of structural
breaks in the trend function. However, the asymptotic null distributions of the appropriate
F-statistics for I(0) and I(1) environments are different and so, followingVogelsang (1998),
©2015 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd
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