Seemingly Unrelated Regression Estimation for VAR Models with Explosive Roots*

Published date01 August 2023
AuthorYe Chen,Jian Li,Qiyuan Li
Date01 August 2023
DOIhttp://doi.org/10.1111/obes.12551
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 85, 4 (2023) 0305-9049
doi: 10.1111/obes.12551
Seemingly Unrelated Regression Estimation
for VAR Models with Explosive Roots*
YECHEN,† JIAN LI and QIYUAN LI§
International School of Economics and Management, Capital University of Economics and
Business, Beijing, China (e-mail: zoeyechen_cueb@163.com)
College of Economics and Management, Huazhong Agricultural University, Wuhan, China
(e-mail: jli@mail.hzau.edu.cn)
§School of Economics, Singapore Management University, Singapore (e-mail:
qyli.2019@phdecons.smu.edu.sg)
Abstract
For VAR models with common explosive root, the OLS estimator of the autoregressive
coeff‌icient matrix is inconsistent (refer to Nielsen, 2009 and Phillips and Magdalinos,
2013). Although Phillips & Magdalinos (2013) proposed using the future observations
as the instrumental variable for removing the endogeneity from VAR models, type I
error occurs when testing for a common explosive root from the distinct explosive roots
before the implementation of IV estimation. Such error creates bias and variance in the
estimate and further causes incorrect inference in the structural analysis such as forecast
error decomposition (FEVD). Hence, we propose using of seemingly unrelated regression
(SUR) estimation for VAR models with explosive roots. Our SUR estimator is consistent
in the case of both distinct explosive roots and common explosive root. We also consider
models with drift in the system for generalization. Simulations show that the SUR estimate
performs better than OLS and IV estimate in the case of both a common explosive root
and distinct explosive roots case. In structural FEVD analysis, simulations show that SUR
yields a different result from OLS and IV. We demonstrate the use of SUR in FEVD for
agricultural commodity markets between 3 July 2010, and 29 January 2011.
I. Introduction
The explosive process is able to capture bubbles in asset prices (Diba and Grossman, 1988).
Hence, it has been used extensively in recent studies of asset price bubbles. The algorithms
provided by Phillips, Wu, and Yu (2011), Phillips, Shi, and Yu (2015a,2015b)areable
JEL Classif‌ication numbers: C12, C13, C58.
*We thank the Editor Anindya Banerjee and two anonymous referees for very helpful comments on earlier
versions of the paper, which improved the quality of the paper signif‌icantly. Chen acknowledges support from
National Natural Science Foundation of China (No. 71803138), the Project of Construction and Support for
high-level Innovative Teams of Beijing Municipal Institutions (BPHR20220119), and the Project of Cultivation
for Young Top-notch Talents of Beijing Municipal Institutions (BPHR202203171). Li acknowledges support
from National Natural Science Foundation of China (No.72173052 & No.71803058).
910
©2023 Oxford University and John Wiley & Sons Ltd.
Unrelated regression estimation for VAR models 911
to detect bubble behaviour and date-stamp its origination and collapse. These methods
are widely used in the empirical study of asset price bubbles in various markets,
including the stock market (see Basse et al.,2021;Horv
´
ath, Li, and Liu, 2021; Li, Wang,
and Zhao, 2021), housing market (See Phillips and Yu, 2013; Greenaway-McGrevy
and Phillips, 2016;Shiet al.,2016), cryptocurrency market (see Cheung, Roca, and
Su, 2015; Corbet, Lucey, and Yarovaya, 2018; Bouri, Shahzad, and Roubaud, 2019),
oil market (see Fantazzini, 2016; Caspi, Katzke, and Gupta, 2018; Gharib, Mefteh-
Wali, and Jabeur, 2021), precious metals market (see Figuerola-Ferretti, Gilbert, and
McCrorie, 2015;Pan,2018; and Ma and Xiong, 2021), exchange rate market and others
(see Etienne, Irwin, and Garcia, 2014;Kr
¨
aussl, Lehnert, and Martelin, 2016; Shi, Hurn,
and Phillips, 2020). These models are built in a non-stationary framework. In a stationary
framework, Gouri´
eroux and Zakoı¨an (2017) propose a non-causal autoregressive process
with heavy-tailed errors to capture the local explosive behaviour in the f‌inancial time
series.
In addition to the study of univariate explosive process, there is increased interest
in multivariate explosive processes. For example, Nielsen (2010) studied a vector
autoregressive model with one unit root process and one explosive process. From
Nielsen (2010), Engsted and Nielsen (2012) proposed a bubble detection mechanism for
asset prices in VAR regression. Phillips and Lee (2015) analyse a VAR system with
mixed explosive roots. This model allows for a local to unit root from the explosive side
and a mildly explosive root. Moreover, they study the Wald test and model selection
criterion for testing for common roots. Magdalinos and Phillips (2009) developed limit
theory for multivariate co-explosive processes. In particular, they consider the cases of
both the distinct explosive roots and common explosive root. Different from the distinct
explosive roots case, the common explosive root case yields the singular matrix for
the sample variance matrix, hence it requires coordinates rotation in developing the
asymptotics. When the regressors are endogenous, Phillips and Lee (2016) consider
self-generated instruments in a method called IVX in the co-explosive system. The IVX
procedure enables a robust Wald test for regressors with different levels of persistence.
The continuous-time counterpart of Magdalinos and Phillips (2009) in discrete time is
developed in Chen, Phillips, and Yu (2017). In a stationary framework, Gourieroux and
Jasiak (2017) consider a VAR(p) model with mixed causal and non-causal components.
They introduce a consistent semi-parametric estimator for model estimation. Cubadda,
Hecq, and Telg (2019) studies co-movement features in the non-causal time-series
models.
VAR models with explosive roots are of particular interest in this paper. VAR
models are the fundamental statistical tool for studying the relationship between multiple
time series over time. It provides a framework for structural analysis such as forecast
error variance decomposition, which are useful tools for analysing the effect of shocks
to the variables in the system. However, for VAR models with a common explosive
root, the OLS estimator of the autoregressive coeff‌icient matrix is inconsistent (see
Nielsen, 2009; Phillips and Magdalinos, 2013). Phillips and Magdalinos (2013) explained
the inconsistency problem in terms of endogeneity induced by co-explosive behaviour.
In particular, co-explosive behaviour results in the singularity of the sample variance
matrix in the limit. To address this asymptotic singularity, they rotate coordinates by
©2023 Oxford University and John Wiley & Sons Ltd.

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