Functional Coefficient Cointegration Models Subject to Time–Varying Volatility with an Application to the Purchasing Power Parity
| DOI | http://doi.org/10.1111/obes.12309 |
| Author | Ying Wang,Yundong Tu |
| Date | 01 December 2019 |
| Published date | 01 December 2019 |
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©2019 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 81, 6 (2019) 0305–9049
doi: 10.1111/obes.12309
Functional Coefficient Cointegration Models Subject
to Time–Varying Volatility with an Application to the
Purchasing Power Parity*
Yundong Tu† and Ying Wang‡
†Guanghua School of Management and Center for Statistical Science, Peking University,
Beijing, 100871, China (e-mail: yundong.tu@gsm.pku.edu.cn)
‡Department of Economics, Business School, The University of Auckland, Auckland 1010,
New Zealand (e-mail: wangyingstat@gmail.com)
Abstract
This paper analyses functional coefficient cointegration models with both stationary and
non-stationary covariates, allowing time-var ying (unconditional) volatility of a general
form. The conventional kernel weighted least squares (KLS) estimator is subject to po-
tential efficiency loss, and can be improved by an adaptive kernel weighted least squares
(AKLS) estimator that adapts to heteroscedasticity of unknown form. TheAKLS estimator
is shown to be as efficient as the oracle generalized kernel weightedleast squares estimator
asymptotically,and can achieve significant efficiency gain relative to the KLS estimator in
finite samples. An illustrative example is provided by investigating the Purchasing Power
Parity hypothesis.
I. Introduction
Cointegration has been a fundamental tool to analyse long-term equilibrium relationships
since the seminal work of Granger (1981) and Engle and Granger (1987). While cointe-
gration models have attracted a huge amount of attention in economic studies, concerns on
structural instabilities lead to development on testing parameter constancy in cointegrated
regressions (Hansen, 2002; Quintos and Phillips, 1993; Qu and Perron, 2007), on general-
izing the models to incorporate stochastic cointegrations (Harris, McCabe and Leybourne,
2002; Cai, Li and Park, 2009; Xiao, 2009) and deterministic time-varying coefficients
(Park and Hahn, 1999), and so on. In particular, Xiao (2009) considered the following
functional coefficient cointegration (FCC) model
JEL Classification numbers: C14; C22.
*The authors thank the Editor, Prof. Anindya Banerjee, an Associate Editor and two anonymous referees for
helpful suggestions. The paper has benefited from discussions with Jiti Gao, Peter C. B. Phillips, QiyingWang and
Zhijie Xiao. Tuthanks support from the National Natural Science Foundation of China (Grant 71472007, 71532001,
71671002), China’sNational Key Research Special Program (2016YFC0207705), the Center for Statistical Science
at Peking University,and Key Laboratory of Mathematical Economics and Quantitative Finance (Peking University),
Ministry of Education. Wang (corresponding author) acknowledges the support from Marsden Grant 16-UOA-239.
1402 Bulletin
yt=(zt)xt+ut,t=1,…,T,(1)
where yt,ztand utare scalars, xt=(xt1,…, xtk)isak-dimensional vector of integ rated
covariates of order one (I(1)), (·)isak×1 column vector function. The advantage of
such an FCC specification, compared with a fully non-parametric cointegration, is not only
that it attenuates the ‘curse of dimensionality’ problem, but also it provides a framework
under which structural non-parametric cointegrations (Wangand Phillips, 2009) of multiple
integrated variables can be analysed.
Related to model (1) is a large literature on functional coefficientregressions. Functional
coefficient models wereintroduced by Cleveland, Mallows and McRae (1993) and extended
to various cases by Hastie and Tibshirani (1993). Chen and Tsay (1993), Cai, Fan andYao
(2000), Li et al. (2002), among others, applied the model to stationary (I(0)) time-series
data. When zt=t, model (1) has been tackled by Robinson (1989, 1991), Cai (2007) and
Chen and Hong (2012) for stationary xt, by Park and Hahn (1999) and Chang and Martinez-
Chombo (2003) for non-stationary xt, and by Cai and Wang (2009) for nearly integrated xt.
When ztis I(1), model (1) has been considered by Wang and Phillips (2009) and Karlsen
et al. (2007) for xt=1, by Cai et al. (2009) for stationary xt, and by Sun, Cai and Li (2013)
for xtbeing I(1). When ztis I(0) and xtis I(1), model (1) becomes that considered by Xiao
(2009), Cai et al. (2009), Li, Lin and Hsiao (2015), Sun, Cai and Li (2016) and Wang,Tuand
Chen (2016). Recently, Gao and Phillips (2013) considered the case when ztis multivariate
and contains both I(1) and I(0) covariates while xtis non-stationary, Cai, Juhl and Yang
(2015) considered the variable selection with indexedfunctional coefficients for stationary
covariates, and Hirukawa and Sakudo (2018) considered the case where non-stationary xt
contains both stochastic and deterministic trends when ztis I(0).
However, little work has been done when utin model (1) possesses non-stationary
volatility, despite the fact that time-varying behaviour is revealed as a quite common
phenomenon in the unconditional volatility of the disturbances driving macroeconomic
time series. For the recent applied studies on heteroscedasticity in time series of stock
returns, interest rates, GDP and other macroeconomic variables, see, for example, Pagan
and Schwert (1990), Loretan and Phillips (1994), Watson (1999), Kim and Nelson (1999),
McConnell and Perez-Quiros (2000), van Dijk, Osborn and Sensier (2002), Sensier and
van Dijk (2004), St˘aric˘a, Herzel and Nord (2005), St ˘aric˘a and Granger (2005), Tu and
Yi (2017) and references therein. Ignoring such heteroscedasticity may lead to inefficient
estimation and unreliable inference on the FCC models, and result in erroneous conclusions
on functional cointegrations.
The literature has witnessed a number of influential studies that model the time-varying
volatility using a smooth deterministic non-parametric framework, under the assumption
that time-changing is the main feature to be captured in the volatility dynamics. These
include research papers on unit root tests (Hamori and Tokihisa, 1997; Kim, Leybourne
and Newbold, 2002; Cavaliere, 2005; Cavaliere and Taylor, 2005, 2007), on stationarity
tests (Busetti and Taylor, 2003; Cavaliere, 2004; Cavaliere and Taylor, 2008), on estima-
tion in stable autoregressions (Phillips and Xu, 2006; Xu, 2008; Xu and Phillips, 2008),
and on cointegration testing (Cavaliere, Rahbek and Taylor, 2010, rank selection (Cheng
and Phillips, 2012), and model averaging (Tu and Yi, 2017), among others. Compared
to stochastic heteroscedasticity modelling such as GARCH-type models, this framework
©2019 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd
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