Testing for a Moderately Explosive Process with Structural Change in Drift*
| Published date | 01 April 2022 |
| Author | Jingjie Xiang,Gangzheng Guo,Qing Zhao |
| Date | 01 April 2022 |
| DOI | http://doi.org/10.1111/obes.12469 |
Testing for a Moderately Explosive Process with
Structural Change in Drift*
JINGJIE XIANG,†GANGZHENG GUO‡,§and QING ZHAO¶
†School of Economics and Business Administration, Central China Normal University, Wuhan,
China
‡International Institute, China Construction Bank, Beijing, China
§School of Economics and Management, Tsinghua University, Beijing, China
¶School of Finance, Dongbei University of Finance & Economics, Dalian, China (e-mail:
qingzhao@dufe.edu.cn)
Abstract
This paper studies large sample properties of a moderately explosive autoregression with a
structural change in the unobservable drift term, and develops asymptotic tests for the null
of moderate explosiveness under different dependence structures. When the innovation
sequence is independently and identically distributed (i.i.d.), we show that the tstatistic is
asymptotically standard normal. When the innovations are weakly dependent in the form
of homoskedasticity or conditional heteroskedasticity, we invoke the fixed-smoothing
asymptotics to construct the heteroskedasticity and autocorrelation robust standard error,
under which the tstatistic follows Student’stdistribution in large samples. Monte Carlo
simulations show that our tests have small size distortion and high power in finite samples.
As we impose no restrictions on the occurrence time and magnitude of the drift, our
proposed asymptotic tests enjoy strong robustness and applicability.
I. Introduction
The past several years have witnessed growing studies on the moderately explosive
process first proposed by Phillips and Magdalinos (2007a). These moderately explosive
models share the very interesting property that the limiting distribution of the
coefficient-based statistic is Cauchy, which is invariant to the distribution of the
innovation term, as well as the dependence structures in the innovation sequence. For
example, Phillips and Magdalinos (2007b) imposed a linear process structure on the
error term; Arvanitis and Magdalinos (2018) and Lee (2018) encompassed general
classes of conditional heteroskedasticity. Interestingly, the limiting distribution of the
JEL Classification numbers:C12. C22.
*Guo equally shares the first authorship with Xiang. The work of Zhao is supported by MOE (Ministry of
Education in China) Project of Humanities and Social Sciences (No.20YJA790084). The authors are grateful for
very helpful comments from Anindya Banerjee (the editor) and two anonymous referees.
300
©2021 The Department of Economics, University of Oxford and John Wiley & Sons Ltd
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 84, 2 (2022) 0305-9049
doi: 10.1111/obes.12469
coefficient-based statistic is not affected by the process of the innovation sequence, and
continues to be asymptotically Cauchy distributed with the same rate of convergence.
Guo, Sun and Wang (2019) developed the theory of Phillips and Magdalinos
(2007a, b) by allowing for an intercept in the moderately explosive process, so that the
data process has two dominant effects: the moderately explosive stochastic effect and
the nonlinear deterministic drift effect. In the work of Guo et al. (2019), they discussed
the impact of different sizes of drift on the least-squares limit theory. Their results
show that under the null of moderate explosiveness, the limiting distribution of the
OLS tstatistic is invariant to the drift size. This invariance property extends the results
obtained in Wang and Yu (2015), Fei (2018) and Liu and Peng (2019) who also
accommodated a drift, but assumed it to be a fixed constant.
In practice, however, the fixed deterministic trend component of an autoregressive
process is insufficient to capture some dramatic shifts which suddenly occur at a
certain time point. The dramatic shift may be the result of a major event, such as
industrial revolution, financial crisis or the outbreak of coronavirus. The profound
influence of these events can usually be characterized as a structural break. Currently,
studies that consider structural changes in the drift term of a moderately explosive
model are rather limited. Whether the above invariance property remains valid under
the presence of a structural change in the nonlinear deterministic trend and how the
structural change specification affects the asymptotic behaviour of a moderately
explosive model are yet to be seen.
Therefore, this paper proposes a moderately explosive model with a structural
change in drift, and develops an asymptotically valid test for moderate explosiveness.
We impose no restrictions on the occurrence time and magnitude of the structural
change, so that the dominant trend component can be either a stochastic trend
component or a nonlinear deterministic trend component. We allow the structural
change to occur at any time point within the sample period.
Our limit theories are derived under different scenarios of innovation sequence.
When the innovation sequence is independently and identically distributed (i.i.d.), the
asymptotic distribution of the OLS tstatistic is standard normal, regardless of when the
structural change occurs and whether the pre and post drifts are large or small. When
the innovation sequence is assumed to be a stationary linear process generated from a
homoskedastic process or a square-integrable conditionally heteroskedastic process, we
construct the heteroskedasticity and autocorrelation robust (HAR) standard error of the
OLS estimator of ρTbased on Guo et al. (2019), and show that the tstatistic based on
the LRV estimator follows Student’stdistribution in large samples.
Simulation results show that the asymptotic ttest has satisfactory finite sample
performance with rather small size distortion and high power under various settings.
The performance of the HAR ttest is robust to the occurrence time and magnitude of
structural change. If it is not clear whether the errors are i.i.d., which is common in
practice, we recommend using the HAR ttest with a data-driven smoothing parameter.
Finally, we apply the HAR ttest to investigate the moderate explosiveness in the
Bitcoin markets. The market capitalization of Bitcoins experienced a rapid increase
over the fourth quarter of 2017, which complies with a moderately explosive process,
while a sudden sharp drop on November 10th can be viewed as a structural change.
©2021 The Department of Economics, University of Oxford and John Wiley & Sons Ltd
Moderately explosive process with break in drift301
The results of our proposed HAR ttest further confirm that the variation of market
capitalization of Bitcoins could be explained by a moderately explosive process with a
structural change in drift.
The rest of the paper is organized as follows. Section II establishes the model of a
moderately explosive process with a structural break in the drift term. Section III
derives the asymptotic properties of the tstatistic under various scenarios. Section IV
delivers simulation results. Section V provides the empirical application. The last
section concludes.
1
II. Model
We consider a moderately explosive process with a structural break in drift:
yt¼μ1;Tþðμ2;Tμ1;TÞDtþρTyt1þut,t¼1, 2, ...,T(1)
where Dtdenotes a dummy variable for the structural change defined as:
Dt¼0,if1 ≤t≤TB,
1,if TB<t≤T:
(2)
Here, the drift changes from μ1;Tto μ2;Tat the time point TB.Wedefine a constant δ
as the structural change fraction such that TB¼bδTc, where the symbol ⌊⌋signifies
the integer part.
We maintain the following assumptions.
Assumption 1.
(a) ρT¼1þc=kT, where c>0, kT!∞and kT=T!0asT→∞.
(b) μ1;Tffiffiffiffiffi
kT
p!τ1∈½0, ∞and μ2;Tffiffiffiffiffi
kT
p!τ2∈½0, ∞as T→∞.
(c) y0is independent of fut,t¼1, ...,Tgand y0¼opðffiffiffiffiffi
kT
pÞ.
(d) δ∈(0, 1).
Assumption 1(a) states that the autoregressive root approaches unity from the right
side at a certain rate. Similar assumptions are made in Phillips and Magdalinos (2007a, b)
and Magdalinos (2012). The parametrization of ρT¼1þc=kThas clear implications.
First, the autoregressive root is larger than 1 and falls on the explosive side area.
Compared with the data process generated by a traditional stationary root, the data
process generated by an explosive root exhibits an exponential divergence growth,
thereby forming a spike effect. Second, by imposing bounds on the rate of divergence
of kT, that is, kT¼oðTÞ!∞, we impose restrictions on the rate of divergence of the
data process. Specifically, an interesting family of ρToccurs when we consider
kT¼Tα, where the exponent parameter αlieson(0, 1).Insuchcases,therateof
divergence of yTdecreases with α, from Oðð1þcÞTÞas α→0toOðffiffiffiffi
T
pÞas α→1.
The specification of ρT¼1þc=kTleads to more divergent behaviour than the usual
1
Appendix A presents some technical lemmas and their proofs. Appendix B proves the key results in the main
text.
©2021 The Department of Economics, University of Oxford and John Wiley & Sons Ltd
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