Testing for Asymmetric Comovements*
| Published date | 01 October 2022 |
| Author | O‐Chia Chuang,Xiaojun Song,Abderrahim Taamouti |
| Date | 01 October 2022 |
| DOI | http://doi.org/10.1111/obes.12485 |
Testing for Asymmetric Comovements*
O-CHIA CHUANG,†XIAOJUN SONG‡and ABDERRAHIM TAAMOUTI§
†Department of Mathematical Economics and Mathematical Finance, Economics and
Management School, Wuhan University, Luojia Hill, Wuhan 430072, China
‡Department of Business Statistics and Econometrics, Guanghua School of Management and
Center for Statistical Science, Peking University, Beijing 100871, China (e-mail:
sxj@gsm.pku.edu.cn)
§Department of Economics, Liverpool University Management School, University of Liverpool,
Chatham St, Liverpool L69 7ZH, UK
Abstract
This paper aims to provide non-parametric tests for asymmetric comovements between
random variables. We consider the popular Cram´
er-von Mises and Kolmogorov–
Smirnov test statistics based on the distance between positive and negative joint
conditional exceedance distribution functions. These tests can capture both linear and
nonlinear dependence in the data and do not require selecting kernel functions and
bandwidths. We derive the asymptotic distributions of the tests and establish the
validity of a block multiplier-type bootstrap that one can use in finite-sample settings.
We also show that these tests are consistent for any fixed alternative and have non-
trivial power for detecting local alternatives converging to the null at the parametric
rate. Monte Carlo simulations and a real financial data analysis illustrate satisfactory
performance of the proposed tests.
I. Introduction
The knowledge of the symmetric or asymmetric nature of the comovements between
asset returns is essential for optimal portfolio allocation and risk management. In a
recent paper, Dahlquist, Farago and T´
edongap (2017) show that, when coupled with
generalized disappointment aversion preferences, asymmetric comovements between
asset returns yield qualitatively different optimal portfolios from those of the standard
model, in which the asymmetry is ignored; see also Ang and Chen (2002); Patton
(2004). Furthermore, Tsafack (2009) shows that in the presence of asymmetric
comovements, an optimal portfolio which is based on a multivariate symmetric
JEL Classification numbers: C12, C14, C15.
*Chuang would like to acknowledge the financial support from Youth Foundation of Social Science and
Humanity (Grant 18YJC790245), China Ministry of Education. Song would like to acknowledge the financial
support from the National Natural Science Foundation of China (Grant No. 71973005) and the Key Laboratory
of Mathematical Economics and Quantitative Finance (Peking University), Ministry of Education, China. The
authors thank very much two anonymous referees and the Editor Heino Bohn Nielsen for their constructive
comments and suggestions, which have led to a much improved version of the article.
1153
©2022 The Authors. Oxford Bulletin of Economics and Statistics published by Oxford University and John Wiley & Sons Ltd.
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in
any medium, provided the original work is properly cited.
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 84, 5 (2022) 0305-9049
doi: 10.1111/obes.12485
comovements assumption will lead to an underestimation of portfolio Value-at-Risk
and Expected Shortfall. Several statistical procedures have been developed to test
symmetric comovements between returns. However, most of the proposed tests were
built based on the coefficient of exceedance correlation, thus they are expected to have
low power for detecting full and nonlinear dependence, since by construction this
coefficient only measures linear dependence and cannot capture the full dependence
structure that can be due to the dependence in higher order moments. In this paper, we
develop model-free tests for symmetric comovements which account for nonlinear and
full dependence in the data. Our ultimate goal is to derive tests that are fully non-
parametric, but converge at the parametric rate and, in particular, do not require
selecting bandwidths and kernel functions.
Early studies have tested the symmetric comovements between asset returns in the
context of parametric models; see Longin and Solnik (2001); Ang and Bekaert (2002);
Ang and Chen (2002); Patton (2004) and references therein. For a pair of variables
Xt,Yt
ðÞ[hereafter both Xtand Ytare standardized to have zero mean and unit
variance], these parametric tests are based on the classical Pearson-type exceedance
correlations: ρþðcÞ¼corrðXt,YtjXt>c,Yt>cÞand ρðcÞ¼corrðXt,YtjXt<c,Yt<cÞ,
for a given exceedance level c≥0.
1
They test the null hypothesis
HL
0:ρþðcÞ¼ρðcÞ
against the alternative hypothesis
HL
1:ρþðcÞ≠ρðcÞ:
The above tests, however, focus on only linear types of asymmetric comovements and
cannot reflect vast asymmetric structures beyond the second moments; see Li (2014).
Modelling asymmetric exchange rate dependence, Patton (2006) shows that using
linear correlation is not sufficient to describe the dependence between exchange rates,
and consequently this might have a negative effect (or low degree of protection) on a
hedging strategy constructed using linear correlation.
Moreover, few recent papers have proposed alternative tests for symmetric
comovements; see Hong, Tu and Zhou (2007); Jiang, Wu and Zhou (2018a); Jiang
et al.(2018b). Hong et al.(2007) derive a test for symmetry in correlation that does
not require the specification of a statistical model for the data. However, their test only
captures linear dependence and is unable to detect asymmetric comovements beyond
the second moment. This has motivated Jiang et al.(2018a) to introduce an entropy
measure that quantifies asymmetry in the joint distribution of an individual stock return
and market return. Their measure is used to build a model-free test to determine if
asymmetric comovements exist in the stock markets. Their test statistic is a bivariate
normalized version of the entropy measure originally proposed by Granger, Maasoumi
and Racine (2004). The calculation of their test statistic is based on a non-parametric
kernel smoothing method that depends on the selection of kernel function and
1
Observe that the standardization of the data (Xtand Ytin the current paper), which is quite common in the
literature of symmetry testing for financial data, might induce some estimation effect; see, for example, Chen
(2016) among others. However, the examination of this effect is left for future research.
©2022 The Authors. Oxford Bulletin of Economics and Statistics published by Oxford University and John Wiley & Sons Ltd.
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