Testing for Flexible Nonlinear Trends with an Integrated or Stationary Noise Component

Date01 October 2017
Published date01 October 2017
©2017 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
doi: 10.1111/obes.12169
Testing for Flexible Nonlinear Trends with an
Integrated or Stationary Noise Component*
Pierre Perron†, Mototsugu Shintani‡,§,and Tomoyoshi Yabu
Department of Economics, Boston University, 270 Bay State Rd. Boston, MA, 02215, USA
(e-mail: perron@bu.edu)
RCAST, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8904, Japan
(e-mail: moto.shintani@gmail.com)
§Department of Economics, Vanderbilt University, 2301 Vanderbilt Place Nashville, TN
37235-1819, USA
Department of Business and Commerce, Keio University, 2-15-45 Mita, Minato-ku, Tokyo,
108-8345, Japan (e-mail: tyabu@fbc.keio.ac.jp)
This paper proposes a new test for the presence of a nonlinear deterministic trend ap-
proximated by a Fourier expansion in a univariate time series for which there is no prior
knowledge as to whether the noise component is stationary or contains an autoregressive
unit root. Our approach builds on the work of Perron and Yabu (2009a) and is based on
a Feasible Generalized Least Squares procedure that uses a super-efficient estimator of
the sum of the autoregressive coefficients when =1. The resulting Wald test statistic
asymptotically follows a chi-square distribution in both the I(0) and I(1) cases.To improve
the finite sample properties of the test, we use a bias-corrected version of the OLS estimator
of proposed by Roy and Fuller (2001). We show that our procedure is substantially more
powerful than currently available alternatives. We illustrate the usefulness of our method
via an application to modelling the trend of global and hemispheric temperatures.
I. Introduction
It is well known that economic time series often exhibit trends and serial correlation. As
the functional form of the deterministic trend is typically unknown, there is a need to
determine statistically whether a simple linear trend or a more general nonlinear one is
appropriate. The main issue is that the limiting distributions of statistics to test for the
presence of nonlinearities in the trend usually depend on the order of integration which is
also unknown. On the other hand, testing whether the noise component is stationary, I(0),
JEL Classification numbers: C22.
*Wethank the associate editor, two anonymous referees and Vadim Marmer for helpful comments and Francsico
Estrada for providingthe data used in the empirical applications. Wealso thank the seminar and conference participants
for helpful comments at Boston University, Osaka University, University ofAlabama, 23th Annual Symposium of
the Society for Nonlinear Dynamics and Econometrics, 11th International Symposium on Econometric Theory and
Applications (SETA),26th Annual Meeting of the Midwest Econometrics Group, and 32nd Meeting of the Canadian
Econometric Study Group.
Flexible nonlinear trend tests 823
or has an autoregressive unit root, I(1), depends on the exact nature of the deterministic
trend (e.g., Perron, 1989, 1990, for the cases of abrupt structural changes in slope or level).
In particular, if the trend is misspecified, unit root tests will lose power and can be outright
inconsistent (e.g., Perron, 1988; Campbell and Perron, 1991).This loss in power can also be
present if the components of the trend function are over-specified,for example, by including
an unnecessary trend; Perron (1988). Hence, we are faced with a circular problem and what
is needed is a procedure to test for nonlinearity that is robust to the possibilities of an I(1)
or I(0) noise component.
We propose a Feasible Generalized Least Squares (FGLS) method to test for the pres-
ence of a smooth nonlinear deterministic trend function that is robust to the presence of
I(0) or I(1) errors. A similar issue was tackled by Perron and Yabu (2009a) in the context
of testing for the slope parameter in a linear deterministic trend model when the integration
order of the noise component is unknown. The key idea is to make the estimate of the sum
of the autoregressive (AR) coefficients from the regression residual ‘super-efficient’when
the error is I(1). This is achieved by replacing the least squares estimate of the sum of
the AR coefficients by unity whenever it reaches an appropriately chosen threshold. The
limiting distribution of the test statistic is then standard normal regardless of the order of
integration of the noise.
As a class of smooth nonlinear trend functions, we consider a Fourierexpansion with an
arbitrary number of frequencies, as in Gallant (1981) and Gallant and Souza (1991) among
others. Its advantage is that it can capture the main characteristics of a very general class
of nonlinear functions. This specification of the nonlinear trend function has been used
in recent studies. For example, Becker, Enders and Hurn (2004) use a Fourier expansion
to approximate the time-varying coefficients in a regression model and propose a test for
parameter constancy when the frequency is unknown. Becker, Enders and Lee (2006)
recommend pretesting for the presence of a Fourier-type nonlinear deterministic trend
under the assumption of I(0) errors before employing their test for stationarity allowing
a nonlinear trend. Similarly, Enders and Lee (2012) propose a Lagrange Multiplier (LM)
type unit root test allowing for a flexible nonlinear trend using a Fourier approximation
and use it along with a nonlinearity test under the assumption of I(1) errors. Rodrigues
and Taylor (2012) also consider the same nonlinear trend in their local GLS detrended test
for a unit root.
Our analysis is not the first to propose a nonlinear trend test using a flexible Fourier
approximation while maintaining robustness to both I(0) and I(1) noise. At least two
previous studies share the same motivation. Harvey, Leybourne and Xiao (2010, hereafter
HLX) extend the robust linear trend test of Vogelsang (1998) to the case of a flexible
Fourier-typetrend function. Vogelsang’s (1998) approach requires the choice of an auxiliary
statistic so that the multiplicative adjustment term on the Wald statistic approaches one
under I(0) errors and has a non-degenerate distribution under I(1) errors in the limit
under the null hypothesis. By controlling the coefficient on the auxiliary statistic, the
Modified Wald (MW) test can have a critical value common to both I(0) and I(1) cases.
HLX suggest employing a unit root test to be used as the required auxiliary statistic.
Astill et al. (2015, hereafter AHLT) suggest instead an adjustment to the critical values
using a similar auxiliary statistic. AHLT show that their procedure is also robust to I(0)
and I(1) errors, yet dominates the HLX method in terms of local asymptotic and finite
©2017 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd

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