Mildly Explosive Autoregression with Anti‐persistent Errors*
| Date | 01 April 2021 |
| DOI | http://doi.org/10.1111/obes.12395 |
| Author | Yiu Lim Lui,Jun Yu,Weilin Xiao |
| Published date | 01 April 2021 |
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©2020 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 83, 2 (2021) 0305–9049
doi: 10.1111/obes.12395
Mildly Explosive Autoregression with Anti-persistent
Errors*
Yiu Lim Lui,†Weilin Xiao,‡Jun Yu§
†The Institute for Advanced Economic Research, Dongbei University of Finance and
Economics, Dalian, Liaoning China
‡School of Management, Zhejiang University, Hangzhou, Zhejiang China
§School of Economics and Lee Kong Chian School of Business, Singapore Management
University, Singapore, Singapore
Abstract
An asymptotic distribution is derived for the least squares (LS) estimate of a first-order
autoregression with a mildly explosive root and anti-persistent errors. While the sample
moments depend on the Hurst parameter asymptotically, the Cauchy limiting distribution
theory remains valid for the LS estimates in the model without intercept and a model with an
asymptotically negligibleintercept. Monte Carlo studies are designed to check the precision
of the Cauchy distribution in finite samples. An empirical study based on the monthly
NASDAQ index highlights the usefulness of the model and the new limiting distribution.
I. Introduction
The autoregressive (AR) model with an explosive root was first studied in White (1958)
and Anderson (1959) where the following process was considered:
yt=yt−1+ut,>1, t=1, 2,…,n.(1)
Under the assumptions of independent and identically distributed (iid) Gaussian errors
(i.e. ut
iid
∼N(0, 2)) and the zero initial condition (i.e. y0=0), White (1958) and Anderson
(1959) showed that the least squares (LS) estimate of (denoted by ˆ
) has the following
Cauchy limiting distribution:
JEL Classification numbers: C15, C22.
*We thank two referees for thoughtful comments and Jia Li for discussions. Yiu Lim Lui, The Institute
for Advanced Economic Research, Dongbei University of Finance and Economics, Dalian, Liaoning, 116025,
China, Email: allenlui@dufe.edu.cn. WeilinXiao, School of Management, Zhejiang University, Hangzhou, 310058,
China. Email: wlxiao@zju.edu.cn. Jun Yu, School of Economics and Lee Kong Chian School of Business,
Singapore Management University, 90 Stamford Rd, Singapore. Email for Jun Yu: yujun@smu.edu.sg. URL:
http://www.mysmu.edu/faculty/yujun/. Xiao gratefully acknowledges the financial support of the National Natu-
ral Science Foundation of China (No. 71871202), the Humanities and Social Sciences of Ministry of Education
Planning Fund of China (No. 17YJA630114) and the Fundamental Research Funds for the Central Universities.Yu
thanks the Singapore Ministry of Education for Academic Research Fund under grant number MOE2013-T3-1-009.
Author for correspondence is Weilin Xiao.
Explosive AR with Anti-persistent Errors 519
n
2−1(ˆ
−)as
→C,asn→∞,(2)
where as
→denotes the convergence almost surely and Cis a standard Cauchy variate.
It is noteworthy that the above limit theory is not obtained from an invariance princi-
ple because the distributional assumption ut
iid
∼N(0, 2) cannot be relaxed.1To relax the
assumption of Gaussian errors, and in the meantime, to allow for a non-zero initial condi-
tion, Phillips and Magdalinos (2007a) (PM hereafter) and Phillips, Magdalinos and Giraitis
(2010) (PMG hereafter) considered two variations which are analogous to Model (1). PM
designed a mildly explosive AR model by letting =n=1+c/n,c>0, ∈(0,1), while
PMG introduced a mildly explosive model by letting =m,n=1+cm/n,c>0. Under
some suitable assumptions but without the requirements of Gaussian errors and the zero
initial condition, applying different forms of the central limit theorem and the martingale
convergence theorem, PM and PMG obtained the asymptotic theory:
n
n
2
n−1(ˆ
−n)⇒C,asn→∞; (PM)
n
m,n
2
m,n−1ˆ
−m,n⇒C,asn→∞followed by m→∞.(PMG)
The pivotalness of the Cauchy distribution suggests that it is easy to test a hypothesis
about the AR coefficient. Not surprisingly, it has been used in the literature to test the
presence of rational bubbles in asset prices; see Phillips, Wu and Yu (2011). Moreover,
considerable efforts have been made in the literature to explore the explosive-type AR
models with dependent errors. The errors could be weakly dependent as in Phillips and
Magdalinos (2007b), or strongly dependent as in Magdalinos (2012), or could involvecon-
ditional heteroskedasticity as in Arvanitis and Magdalinos (2018). These generalizations
are important as the explosive-type model with dependent errors can potentially better
describe the movement of real data than the pure explosive AR(1) model. A number of
related studies in the literature allow for m-dependent errors (Pedersen and Sch¨utte, 2017),
errors with deterministic time-varying volatilities (Harvey, Leybourne and Zu, 2019a, b).
Tothe best of our knowledge, no limit theory has been developed to cover anyexplosive-
type AR model with anti-persistent errors. The goal of this paper is to fill the gaps in the
context of the explosive-type AR model of PMG. Why are the gaps important? To see the
empirical relevanceof an explosive model with anti-persistent errors, Figure 1 presents time
series plots of four logarithmic stock market indices (left axis) and the residuals obtained
from the fitted AR(1) model with and without intercept (right axis). In particular, we
consider four monthly indices over different sampling periods, namely FTSE 100 Index
from January 2003 to October 2007, Hang Seng Index from May 1989 to June 1997,
NASDAQ Composite Index from January 1990 to December 1999 and Nikkei 225 Index
from August 1982 to November 1989. The sampling periods are selected as these markets
1This is because what is used to derive equation (2) is the martingale convergencetheorem which gives the almost
sure convergence.
©2020 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd
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