Is Stock Price Correlated with Oil Price? Spurious Regressions with Moderately Explosive Processes
| Date | 01 October 2019 |
| Published date | 01 October 2019 |
| Author | Yundong Tu,Ye Chen |
| DOI | http://doi.org/10.1111/obes.12302 |
1012
©2019 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 81, 5 (2019) 0305–9049
doi: 10.1111/obes.12302
Is Stock Price Correlated with Oil Price? Spurious
Regressions with Moderately Explosive Processes*
Ye Chen† and Yundong Tu‡
†Capital University of Economics and Business, Beijing, China (e-mail:
zoeyechen cueb@163.com)
‡Guanghua School of Management and Center for Statistical Science, Peking University,
Beijing, 100871, China (e-mail: yundong.tu@gsm.pku.edu.cn)
Abstract
This study explores the spurious effects in linear regressions with moderately explosive
processes. Asymptotic results are developed for the least square estimator, the typical
t-statistic, the Durbin–Watson statistic, and the coefficient of determination. The typical
t-statistic is unable to detect the presence of a spurious relationship, due to the presence
of nuisance parameters that characterize deviations from unity. Moreover, the t-statistic
for common explosive processes has different asymptotics compared to that for distinct
explosive processes. Such differences further complicate the use of the t-statistic. We
demonstrate that two popular methods available in the literature are incapable for this
purpose due to similar difficulties. To overcome these limitations, we propose a t-test
based upon balanced regressions that induces asymptotic inference based on the standard
normal distribution, which is therefore robust to deviations from unity. These results are
further generalized to spurious regressions with multivariate mildly explosive processes.
Simulation results confirm that our test is effectivein finite samples, while other alternatives
are not. An empirical example that demonstrates the phenomenon of spurious correlation
between the NASDAQ stock index and crude oil price in the US is provided to show the
practical merit of our proposed method.
I. Introduction
The discovery of the spurious relationship phenomenon dates back to the mid-1920s, when
Yule(1926) first explored why nonsense correlations between time series sometimes occur.
Granger and Newbold (1974) and Phillips (1986) extended the study of this phenomenon
to econometrics for independent unit root processes. The simulation studies of Granger and
JEL Classification numbers: C12, C13, C58.
*Chen acknowledgesthe financial suppor t from National Natural Science Foundationof China (NSFC 71803138).
Tu (corresponding author) thanks support from NSFC (71532001, 71671002), China’s National Key Research Spe-
cial Program (2016YFC0207705), the Center for Statistical Science at Peking University and Key Laboratory of
Mathematical Economics and Quantitative Finance (Peking University),Ministry of Education.
Spurious Regressions with Moderately Explosive Processes 1013
Newbold (1974) reveal some of the likelyempirical consequences of nonsense regressions
involving(two independent) unit root processes. Phillips (1986) provides a theoretical anal-
ysis of the spurious regressions considered by Granger and Newbold(1974). Phillips (1998)
further uses the orthonormal representation theorem to analyse the spurious regression of
stochastic trends on time polynomials and regression between two unit root processes.
The phenomenon of the spurious relationship can also occur in other scenarios. For
instance, see Abeysinghe (1994) for spurious regressions in deterministic seasonal mod-
els; Cappuccio and Lubian (1997) for spurious regressions with I(1) processes with long
memory errors; Choi, Hu and Ogaki (2008) for robust estimation of structural spurious
regressions; Entorf (1997) for spurious regressions involving random walks with drift
terms; Granger, Hyung and Jeon (2001) for stationary processes; Hassler (1996) and
Stewart (2006) for spurious regressions when stationary regressors are included; Kao
(1999) for spurious regressions with panel data; Kim, Lee and Newbold (2004) for sta-
tionary processes around linear trends; Marmol (1995, 1996a,b, 1998) for higher order
integrated processes; Newbold and Davies (1978) for spurious regressions with error
misspecification; Sun (2006) and Tsay and Chung (2000) for long memory fractionally
integrated processes.
The empirical examination of the spurious relationship is common in economic and
financial studies. For example, Hendry (1980) explores the spurious relationship between
cumulative rainfall and inflation rate in the UK; Jacobsen and Marquering (2008) iden-
tify a spurious correlation between stock return and weather; Ferson, Sarkissian and Simin
(2003, 2008) note the spurious effect of financial returns in predictive regression, for which
Deng (2014) further provides an analytical explanation; Bauer and Hamilton (2018) find
a spurious effect in the context of bond risk premium; and Shintani, Yabu and Nagakura
(2012) investigate the spurious effect in forecasting asset returns when signals from tech-
nical trading rules are used as predictors. In addition, simulation evidence provided by
Mart´
inez-Rivera and Ventosa-Santaul`aria (2012) demonstrate that the spurious regression
problem cannot always be fixed by using the standard autocorrelation correction proce-
dures suggested by McCallum (2010); therefore it remains a not-so spurious problem. Solis
(2011), Zhang (2013) and Tu (2017) also provide related simulation studies.
This paper is concerned with spurious regressions that involve moderately explosive
processes, an important issue that has been overlooked by the aforementioned literature.
The potential explosive behaviour of the financial time series has attracted significant
attention in the past decade in the wake of the recent global financial crisis, and has been
used for the detection of bubbles in financial markets. Phillips and Magdalinos (2007)
develop the asymptotic theory for time series with mildly explosive behaviour. Phillips,
Wu and Yu (2011) and Phillips, Shi and Yu (2015a,b) use mildly explosive processes to
capture market exuberance and develop the bubble dating algorithms with the regression
analysis. Phillips and Yu (2011) consider a recursive regression based on the approach to
date the timeline of financial bubbles. Magdalinos and Phillips (2009) propose a general
cointegrated system to model the co-movement of multiple mildly explosive processes.
Recently Chen, Phillips and Yu (2017) have connected Magdalinos and Phillips’ (2009)
results with their counterparts in the continuous time.
Unlike Phillips (1986), our paper examines spurious regressions with time series which
have autoregressive roots of the form n=1+c/kn. Here, (kn)n∈Nis a deterministic
©2019 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd
1014 Bulletin
sequence increasing to infinity at a rate slower than nand c>0. Such parameterizations,
following Phillips and Magdalinos (2007), represent moderate deviations from unity in
the sense that the regressive roots belong to larger neighbourhoods of one than conven-
tional local to unity roots (Chan and Wei, 1987; Phillips, 1987). It includes a wide family
of moderate deviations by considering kn=n, where ∈(0,1), accommodating both the
conventional moderately explosive process when →1 and the explosive autoregression
of >1 when →0. A similar framework has later been used in Magdalinos and Phillips
(2009) to study the limit theory for cointegrated systems with moderately integrated and
moderately explosive regressors.
We establish the asymptotic theory for spurious regressions with moderately explosive
processes. However, we identify two issues that hinder its practical implementation. First,
similar to the findings of Phillips (1986), the associated statistics are found to have nonstan-
dard limiting distributions. As a result, simulations are needed to obtain critical values for
the subsequent inferences. Second, the limiting distributions all involve a nuisance para-
meter c, which cannot be consistently estimated in the discrete time context, and therefore
are not robust to different levels of persistence in the observed data. Similar problems with
inference are discussed by Phillips and Magdalinos (2007), and Magdalinos and Phillips
(2009), among others. These two issues pose a great challenge to practical applications of
the derived results.
This paper provides an easy-to-use solution as a remedy. To achieve robust inference,
we suggest augmenting the spurious regression model by the lagged-dependent variable
and the lagged regressors. This type of augmentation helps to balance the persistence level
on both sides of the regression. As a result, the unknown parameters of interest can then be
formulated as transformations of stationary variables and a central limit theorem can apply.
The resulting limiting distributions of related statistics in the balanced regression therefore
become standard and are free of the nuisance parameter c. This augmentation technique
has been applied to I(1) systems (Park and Phillips, 1989), fractionally cointegrated sys-
tems (Dolado and Marmol, 2004), unit root testing (Choi, 1993), Granger causality testing
(Bauer and Maynard, 2012), and predictive regressions (Ren, Tu and Yi, 2018), among
others, to achieve standard inference. To emphasize, this paper demonstrates the relevance
of balanced regressions induced standard inference in mildly explosive spurious regres-
sions; in particular its role in removing the nuisance parameters in the derived asymptotic
distributions. We further extend our results to spurious regressions with multivariate mildly
explosive regressors, and find that the Wald statistic for detecting the spurious relationship
follows the usual Chi-squared distribution.
Our theory has empirical relevance for economists and financial analysts, especially
those concerned with time series involving bubbles. The real data problem we are par-
ticularly interested in is the relationship between the oil market and the stock market,
which serves as an empirical illustration of a spurious regression with mildly explosive
processes. There is a large body of literature examining the relationship between the oil
market and the stock market, as evidenced by the recent studies of Broadstock and Filis
(2014), Kang, Ratti and Yoon (2015), Kilian and Park (2009), Nandha and Faff (2008),
Park and Ratti (2008), and references therein. Vector autoregressions of these series are
examined byHammoudeha, Dibooglub and Aleisac (2004), Henriques and Sadorsky (2008),
and Miller and Ratti (2009), among others. In particular, Miller and Ratti (2009) analyse
©2019 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd
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