Robust and Powerful Tests for Nonlinear Deterministic Components

Date01 December 2015
Published date01 December 2015
©2014 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
doi: 10.1111/obes.12079
Robust and Powerful Tests for Nonlinear
Deterministic Components*
Sam Astill, David I. Harvey, Stephen J. Leybourne‡ and A. M.
Robert Taylor§
Department of Economics, University of Warwick, CV4 7AL, Coventry (email:
Granger Centre for Time Series Econometrics and School of Economics, University of
Nottingham, NG7 2RD, Nottingham (email:,
§Essex Business School, University of Essex, Colchester, CO4 3SQ,(email:
We develop a test for the presence of nonlinear deterministic components in a univariate
time series, approximated using a Fourier series expansion, designed to be asymptotically
robust to the order of integration of the process and to any weak dependence present. We
show that our proposed test has uniformly greater local asymptotic powerthan the existing
tests of Harvey, Leybourne and Xiao (2010) when the shocks are I(1), identical local
asymptotic power when the shocks are I(0), and also improved finite sample properties.
Wealso consider the issue of determining the number of Fourier frequencies used to specify
any nonlinear deterministic components.
I. Introduction
The ability to detect the presence and magnitude of deterministic components in a financial
or economic time series is of key importance when conducting empirical analysis, partic-
ularly for the purposes of forecasting and testing for a unit root. For example, in the latter
case, failure to correctly specify a relevant deterministic component present in the data is
known to result in non-similar and (usually) inconsistent tests. Moreover, the power of unit
root tests to reject the null under the I(0) alternative when deterministic components are
unnecessarily included in a model specification is also reduced.
Traditionally,attention in the literature has focused on a linear deterministic component,
most often the case of a constant and/or linear trend. The possibility of breaks in such linear
deterministics has also received considerable attention, due in particular to the effect these
breaks have on standard unit root and stationarity tests; see, inter alia, Perron (1998).
*We are grateful to the Co-Editor, Debopam Bhattacharya and two anonymous referees for their helpful and
constructive comments on an earlier draft.
JEL Classification numbers: C22, C15.
Robust and powerful tests 781
However, as noted by Harvey et al. (2010) [HLX hereafter], these may not necessarily take
the form of an instantaneous break in the trend function. Theynote that changes in economic
aggregates are affected by the response of a large number of potentially heterogeneous
individuals who are unlikely to respond instantaneouslyto shocks. As a consequence, they
suggest that a smoothly evolvingnonlinear deter ministic component might providea better
approximation to the underlying deterministic component of these aggregates.
One possible method to capture such nonlinear behaviour is to approximate the de-
terministic component using a Fourier series expansion. This approach is explored by
Becker, Enders and Lee (2006) and is found to provide a good approximation for a variety
of functions. Enders and Lee (2012) show that modelling the deterministic components
of an economic time series via a Fourier function can approximate changes of various
forms, such as a number of sharp breaks or deterministic smooth transitions, e.g. exponen-
tial or logistic smooth transtions (ESTR or LSTR). Becker et al. (2006) derive a test for
the presence of nonlinear deterministic components using a Fourier expansion under the
assumption that the shocks are I(0). Enders and Lee (2012) propose a corresponding test
under the assumption that the data are I(1). The practical implementation of these tests is
clearly problematic since both assume the order of integration of the data to be known.
This results in a circular testing problem as we wouldneed to know the order of integration
of our data before performing the tests of either Becker et al. (2006) and Enders and Lee
(2012); however, in order to perform a unit root or stationarity test we would need to specify
the form of the deterministic component.
Motivated by this problem, and drawing on the robust linear trend tests of Vogelsang
(1998), HLX suggest a test that is robust to the order of integration of the data, thereby
eliminating this circular testing problem. This is achieved by using a composite statistic
based around a Wald statistic (that has a well defined limit distribution for both I(0) and
I(1) shocks) multiplied by a function of an auxiliary unit root test statistic. This function
is specified such that when the shocks are I(0) it converges in probability to one, leaving
the asymptotic distribution of the Wald statistic unaffected, but when the shocks are I(1),
it converges to a well-defined limit distribution. Judicious choice of the precise function
to be used then allows the asymptotic critical values of the composite test statistic to be
lined up in the I(0) and I(1) environments, for a given significance level. This approach,
therefore, yields tests that display correct asymptotic size for both I(0) and I(1) shocks.
Here we propose an alternative to the methodologyof HLX. In our approach, a function
of an auxiliary unit root test statistic is used to select between the asymptotic I(0) and I(1)
critical values for the Wald test, rather than creating a composite test via multiplication
with the Wald statistic as in HLX. The motivation underlying this is that the presence
of the multiplicative function of a unit root test statistic in the HLX procedure impacts
negatively on the local asymptotic power when the shocks are I(1), relative to a test that
compares the unmodified Wald statistic with its asymptotic I(1) critical value directly.
In contrast, the approach considered in this paper always uses the Wald statistic without
modification, and ensures that the correct asymptotic critical value is used, appropriate
to the order of integration. We show that this new procedure achieves the same local
asymptotic power as the HLX test in the I(0) setting, while delivering (often substantial)
local asymptotic power gains for I(1) shocks. The new approach also provides improved
finite sample behaviour.
©2014 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd

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