A Unified test for the Intercept of a Predictive Regression Model*
| Author | Yao Rao,Fucai Lu,Xiaohui Liu,Yuzi Liu |
| Published date | 01 April 2021 |
| DOI | http://doi.org/10.1111/obes.12408 |
| Date | 01 April 2021 |
571
©2020 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 83, 2 0305–9049
doi: 10.1111/obes.12408
A Unified test for the Intercept of a Predictive
Regression Model*
Xiaohui Liu,†Yuzi Liu,†Yao R ao ‡ and Fucai Lu§
†School of Statistics, Jiangxi University of Finance and Economics, Nanchang, Jiangxi
330013, China (e-mails: liuxiaohui@jxufe.edu.cn and yuzi liu960119@163.com)
‡Department of Economics, Management School, The University of Liverpool, Liverpool
L69 72H, UK (e-mail: Y.Rao@liverpool.ac.uk)
§School of Business Administration, Jiangxi University of Finance and Economics,
Nanchang, Jiangxi 330013, China (e-mail: lu-fucai@263.net)
Abstract
Testingthe predictability of the predictive regression model is of great interest in economics
and finance. Recently, (Zhu et al. (2014) Predictive regressions for macroeconomic data,
Vol. 8, pp. 577–594.) proposed a unified test to account for this issue. Their test has a
desirable property that its limit distribution is standard regardless of the regressor being
stationary, near unit root or unit root. However,this test depends on, a priori , whetherthere
is an intercept in the predictive regression while this is usually unknown in practice. In this
paper, using empirical likelihood inference, we develop a unified pretest for the intercept,
as a pretest to determine the choice of the predictability test. Simulations studies confirm
that the proposed pretest works well.Two real data examples are also provided to illustrate
the importance of such pretest. The first revisits the S&P 500 index data and the second
investigates stock return predictability and investor sentiment for six countries.
I. Introduction
As an important tool for modelling the relationship between a dependent variable and
the lagged value of a regressor, predictive regression models have been widely used in
economics and finance, especially when low frequency data are present. Seminal results
about these models include Campbell andYogo(2006), Cai and Wang(2014), and Kostakis,
Magdalinos and Stamatogiannis (2014), among others. See, for example the excellent
summary Phillips (2015) and references therein for details.
A typical predictive regression model has usually the following linear structure:
JEL Classification numbers: C12; C22; G1.
*Our thanks go to Professor Liang Peng for stimulating this research and ShanShan Ping for doing some simu-
lation studies. Wealso deeply grateful to referees for valuable suggestions and comments. Xiaohui Liu’s research is
supported by NSF of China (Grant No.11971208, 11601197), China Postdoctoral Science Foundationfunded project
(2016M600511, 2017T100475), the Postdoctoral Research Project of Jiangxi (No.2017KY10, xskt19393), NSF of
Jiangxi Province (No.2018ACB21002,20171ACB21030, 20192BAB201005). NSF project of Jiangxi provincial ed-
ucation department (No. GJJ190261). Yuzi Liu’s research is partly supported by the Postgraduate Innovation Project
of Jiangxi Province (No.YC2019-S216). The corresponding author is Fucai Lu’s research is supported by NSF of
China (No.71863015)
572 Bulletin
Yt=+Xt−1+Ut,
Xt=+Xt−1+"t,
B(L)"t=Vt,
(1)
where Ytdenotes the dependent variable such as the asset return at time t, and Xt−1denotes
a financial variable, for example the log dividend-priceratio at time t−1. The initial value
X0=op(√T) independent of {Vs}T
s=1,B(L)=1−(p
i=1biLi) with Li"t="t−i,B(1) =0, all
roots of B(L) are fixed and <1 in absolute value, and {(Ut,Vt)}T
t=1are independent and
identically distributed (i.i.d.) random vectors with zeros means.
In practice, an interesting issue on this model is to test the predictability, that is the
null hypothesis H0:=0, of Ytby some lagged regressor Xt−1. It is known that the tests
involving in the ordinary least squares estimator of have different limit distributions.
They depend on (i) whether ||<1 (stationary), or =1 (unit root), or =1+c/T (near
unit root) for some c=0, and (ii) whether is included or excluded in the AR process,
and (iii) whether there exists the so-called embedded endogeneity, that is Xtand Utmay
be correlated. They may depend on some non-estimable parameters, say c, when {Xt}is
non-stationary; See, for example Cai and Wang (2014); Campbell and Yogo (2006) and
references therein for details.
In view of this, it is important to develop some unified tests which are robust against
(i) – (iii). For theAR(1) process Xt’s, uniform inference procedures for havealready been
discussed by many authors; See So and Shin (1999), Mikusheva(2007), Chan, Li and Peng
(2012) and Hill, Li and Peng (2016). Recently, Zhu, Cai and Peng (2014) investigated the
unified predictability test for in (1) by empirical likelihood method.The most outstanding
property of this test is its robustness against (i)-(iii). However,this method depends in prior
on whether or not =0 in (1). A further data splitting technique has to be employed to get
rid of the impact of the intercept on the resulting testing statistic if =0. Although both
empirical likelihood methods have the same limit distribution, the data splitting technique
leads to a great loss of power (Zhu, Cai and Peng, 2014).They even may lead to different
conclusions over the same data set; See section IV in the sequel for an example.In practice,
whether or not =0 is unknown, and needs to be tested.
Unfortunately, it is nontrivial to construct a unified method for testing the existence of
intercept in a time series model regardless of whether the regressors are stationary or non-
stationary. For the AR(1) model, testing H*
0:=0 was studied by separately considering
the case =1 and the case ||<1 in Dickey and Fuller (1981) and Fuller, Hasza and
Goebel (1981), respectively. Recently, Dios-Palomares and Roldan (2006) proposed to
test H*
0:=0 by combining the tests in Dickey and Fuller (1981) and Fuller et al. (1981),
which depends on a prior test on testing H**
0:=0&=1 and does not work for the case
of =1+c/T with some c=0.
Motivated by the unified empirical likelihood inference in Chan et al. (2012) and Zhu
et al. (2014), in this paper we develop a similar unified empirical likelihood inference for
and studies its powerproperty. The proposed empirical likelihood function can be employed
to test H0:=0 and construct an interval for without knowing whether ||<1or=1
or =1+c/T for some c=0; We refer to Owen (2001) for an overview on empirical
likelihood method. It turns out that if the proposed test fails to reject H0:=0, we will
be able to use the first empirical likelihood method of Zhu et al. (2014), which explicitly
©2020 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd
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