Confidence Sets for the Break Date in Cointegrating Regressions
| Date | 01 June 2018 |
| DOI | http://doi.org/10.1111/obes.12223 |
| Published date | 01 June 2018 |
514
©2017 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 80, 3 (2018) 0305–9049
doi: 10.1111/obes.12223
Confidence Sets for the Break Date in Cointegrating
Regressions*
Eiji Kurozumi† and Anton Skrobotov‡,§
†Department of Economics, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo, Japan
(e-mail: kurozumi@stat.hit-u.ac.jp)
‡Institute of Applied Economic Studies, Russian Presidential Academy of National Economy
and Public Administration, Moscow, Russia
§Innopolis University, Kazan, Russia (e-mail: antonskrobotov@gmail.com)
Abstract
In this paper, we propose constructing confidence sets for a break date in cointegrating
regressions by inverting a test for the break location, which is obtained by maximizing the
weighted average of power. It is found that the limiting distribution of the test depends on
the number of I(1) regressors whose coefficients sustain structural change and the number
of I(1) regressors whose coefficients are fixed throughout the sample. By Monte Carlo
simulations, we then show that compared with a confidence interval developed by using
the existing method based on the limiting distribution of the break point estimator under
the assumption of the shrinking shift, the confidence set proposed in the present paper has
a more accurate coverage rate, while the length of the confidence set is comparable. By
using the method developed in this paper, we then investigate the cointegrating regressions
of Russian macroeconomic variables with oil prices with a break.
I. Introduction
This paper proposes constructing a confidence set for the change point in cointegrating
regressions. Cointegration has long been an important concept for investigating the long-
run relationships among macroeconomic variables.1To capture the long-run relationship,
data overrelatively long time-frames are often used in such an investigation. In this case, we
should take into account that the economic structure may change during the sample period.
For example, Campos, Ericsson and Hendry (1996) investigate the effect of structural
change on cointegration tests and Gregory and Hansen (1996a,b) propose tests for the
JEL Classification numbers: C12, C21.
*Weare grateful to Editor, Anindya Banerjee, three anonymous referees and Andrey Polbinfor helpful comments
and valuable suggestions. All errors are our responsibility. Financial support through grants from the Russian Sci-
ence Foundation (A. Skrobotov, Project No. 16-18-10432) and the JSPS KAKENHI (E. Kurozumi, Grant Number
16K03594) for various and non-overlapping parts of this research is gratefully acknowledged.
1Recently we can find a lot of work on large-dimensionalVAR and/or Bayesian VAR methods but we focus on
classical cointegration models, using frequentist methods in this paper.
Confidence sets for the break date 515
null hypothesis of no cointegration that are robust to the existence of structural change,
while tests for the null hypothesis of cointegration with a structural break are proposed by
Carrion-i-Silvestre and Sans´o (2006) and Arai and Kurozumi(2007). On the contrar y, tests
for structural change in the framework of cointegrating regressions have been proposed
by Bai, Lumsdaine and Stock (1998, BLS hereafter) and Kejriwal and Perron (2010). By
using the tests presented in the literature in addition to the careful inspection of original
data and economic events, we may find cointegrating relations with structural change. In
this case, a statistical inference about the change point can be made by using the method
proposed by BLS in the case of a single break, while multiple breaks were investigated by
Kejriwal and Perron (2008a).
In the case of regressions using stationary variables, the break point is estimated by
minimizing the sum of the squared residuals or by using the quasi-maximum likelihood
method, while the confidence interval is constructed by using the limiting distribution of
the break point estimator, as suggested by Bai (1997) and Bai and Perron (1998).2In this
case, the crucial assumption made for the construction of the confidence interval is that the
magnitude of the structural break shrinks to 0 at a rate slower than 1/√T, as also assumed
in BLS and Kejriwal and Perron (2008a). However, as demonstrated by Elliott and M¨uller
(2007) and Chang and Perron (2015), a confidence interval based on the limiting distribu-
tion of the break point estimator tends to be too liberal when the magnitude of the break is
not so large. Instead of using the limiting distribution of the change point estimator, Elliott
and M¨uller (2007) propose constructing a confidence interval by inverting the locally best
invariant test for the break location, which helps control the coverage rate.3However, the
drawback of their method, as pointed out byChang and Perron (2015), is that the confidence
interval tends to be too wide. Indeed, it covers most of the sample period in some cases,
thereby offering no useful information in practice. To overcome this drawback,Yamamoto
(2016) pays attention to the estimation of the long-run variance for the construction of the
test for the break location and proposes estimating it by taking the estimated break point
into account, while Kurozumi and Yamamoto (2015) consider a similar method to Elliott
and M¨uller (2007) but propose inverting the sup-type, average-type, and exponential-type
tests for the break location, which can be obtained by maximizing the average power of
a test. By Monte Carlo simulations, it is shown that these methods can better control the
coverage rate and that the length of the confidence set becomes close to or smaller than that
based on Bai (1997). On the contrary, Eo and Morley (2015) investigate a confidence set
based on the likelihood ratio, while Harvey and Leybourne (2015) propose constructing a
confidence set for the date of a break in level and trend without stochastic regressors using
the LBI test. Their method is valid for both I(0) and I(1) processes. Further, Kurozumi
(2016) extends the method of Kurozumi andYamamoto (2015) to linear regression models
with non-homogeneous regressors, particularly with a linear trend.
As in the above case of stationary regressions, controlling the coverage rate of the
confidence interval of the break date in the case of cointegrating regressions may be difficult
based on the methods of BLS and Kejriwal and Perron (2008a). Indeed, the simulation
2Change point estimators have been investigated in the statistical and econometric literature. See, for example,
Cs¨org´o´ and Horv´ath (1997), Perron (2006) and Aue and Horv´ath (2013) among others.
3The duality between confidence regions and tests is a well-knownproperty in the statistical literature.
©2017 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd
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