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#  Predictability Tests Robust against Parameter Instability

Christis Katsouris
Department of Economics, University of Southampton
blueJob Market Paper I blue(unedited)  PhD Candidate, Department of Economics,
University of Southampton - Highfield Campus, UK.
<EMAIL_ADDRESS>I am grateful to my academic advisors Jean-
Yves Pitarakis, Jose Olmo and Tassos Magalinos for their guidance, support and
continuous encouragement. Moreover, I would like to thank session participants
of the IX Workshop in Time Series Econometrics at the University of Zaragoza,
the 2019 Conference of the International Association of Applied Econometrics
(IAEE) at the University of Cyprus, as well as the participants of the 2018
and 2019 Econometric Workshops at the Department of Economics, for helpful
conversations. Financial support from the University of Southampton Vice
Chancellors PhD Scholarship is gratefully acknowledged. The author also
acknowledge the use of the IRIDIS High Performance Computing Facility, and
associated support services at the University of Southampton, in the
completion of this work. All the usual disclaimers apply.

(January 25, 2021)

Abstract

We consider Wald type statistics designed for joint predictability and
structural break testing based on the instrumentation method of Phillips and
Magdalinos, (2009). We show that under the assumption of nonstationary
predictors: (i) the tests based on the OLS estimators converge to a
nonstandard limiting distribution which depends on the nuisance coefficient of
persistence; and (ii) the tests based on the IVX estimators can filter out the
persistence under certain parameter restrictions due to the supremum
functional. These results contribute to the literature of joint predictability
and parameter instability testing by providing analytical tractable asymptotic
theory when taking into account nonstationary regressors. We compare the
finite-sample size and power performance of the Wald tests under both
estimators via extensive Monte Carlo experiments. Critical values are computed
using standard bootstrap inference methodologies. We illustrate the usefulness
of the proposed framework to test for predictability under the presence of
parameter instability by examining the stock market predictability puzzle for
the US equity premium.

JEL classification: C12, C22, C26, C32, C53, C58.

Keywords: Stock Return Predictability, Parameter Instability, Predictive
Regression, IVX filtration, Local-to-unity asymptotic theory, Ornstein-
Uhlenbeck process, bootstrap.

##  1 Introduction

The stock predictability puzzle is an important research theme appeared in the
financial econometrics literature which has seen growing attention in recent
years111In empirical finance the predictive regression model provides a
suitable testing framework for aspects such as conditional asset pricing and
investment strategy performance evaluation. An extensive study about the
latter is presented by Pesaran and Timmermann, (1995). The author examines the
predictability of stock returns from the perspective of optimal portfolio
decision. Moreover, the author introduces the concept of episodic
predictability, arguing that predictability can be explained in accordance to
certain macroeconomic events. Moreover, related applied macroeconometric
applications include the forecasting of macroeconomic fundamentals as well as
the examination of short-horizon vis-a-vis long-horizon predictability..
Various studies have demonstrated that forecast performance via predictive
regressions with macroeconomic variables as predictors, is countercyclical,
that is, there is a co-movement with the business cycle. Further studies
focused on identifying periods of episodic predictability; a terminology
adapted to describe the phenomenon of predictors "switching on" during certain
periods, while appear to have no predictive ability during other periods
(e.g., see Gonzalo and Pitarakis, (2012, 2017), Chinco et al., (2019),
Demetrescu et al., (2020)). These periods of unstable forecasting ability
appear in time series models in the form of parameter instability (see, Rossi
and Inoue, (2012), Inoue et al., (2017), Pitarakis, (2017) and Georgiev et
al., (2018)). Considering this vibrant discussion, we aim to study how the
degree of persistence affects the asymptotic theory of related to the above
aspects. Our goal is to tackle the following econometric question: "How does
the time series properties of predictors affect predictability testing under
the presence of parameter instability?"

The predictive regression model operates under the strong assumption of
parameter stability, which could be violated in certain regions of the
sample222A related study to parameter instability in prediction models is
presented by Paye and Timmermann, (2006). In this paper, the authors propose a
framework for identifying and estimating multiple breaks.. Moreover, the
majority of the structural break literature propose testing methodologies
under the assumption of stationary time series. Therefore, the development of
a robust joint test against both predictability and structural break is an
important aspect to tackle. Related issues from the predictability literature,
include the high persistence of predictors as well as the econometric
implications of testing for structural break given the existence of
nonstationary regressors as captured via the local-to-unit root specification
of the predictive regression. In this paper we propose an econometric
framework for jointly testing against both predictability and structural
break. The tests are constructed in a similar manner as the Wald type
statistics proposed by Gonzalo and Pitarakis, (2012) in a threshold predictive
regression model333The particular framework and in the subsequent paper of
Gonzalo and Pitarakis, (2017) the authors propose tests which capture the
effects of linearity and the presence of a threshold effect in predictive
regressions with persistent predictors.. Our contributions are threefold: (i)
We propose a test statistic which jointly tests against the alternative
hypothesis of both predictability and structural break and show that the test
is robust to the persistence properties of predictors in the model;

(ii) We show that the test statistic has a nuisance parameter free limiting
distribution under the assumption of stationary and mildly stationary
regressors, while it converges to a nonpivotal asymptotic distribution under
the assumption of local-to-unit root (LUR) or integrated regressors; (iii) We
provide an extensive Monte Carlo simulation study where we compare the
empirical size and power of the sup IVX-Wald test. We also present simulations
to examine the finite-sample properties of the corresponding sup OLS-Wald
statistic. Lastly, we employ the proposed testing framework to investigate the
predictability puzzle based on the equity premium of US stock returns.

##### Related Literature

The ideas in this paper are related to research done in two different fields.
From finance, it is related to the stock return predictability literature and
from the econometrics and statistics perspective it is related to the
literature of parameter instability and structural break testing
methodologies. We aim to bridge the gap in the literature by proposing a
testing framework which incorporates both aspects.

Firstly, we provide a general overview of the predictability literature,
starting with the implementation of simple t-tests to detect statistical
significance in stable relations of predictants such as equity index returns,
on the lagged time series of predictors. From extensive empirical
applications, this practise has been proved to cause distorted inference due
to the presence of high persistent predictors, since nonstandard terms appear
in the limiting distribution of the t-test. In particular, Stambaugh, (1999)
observed this finite sample bias444Notice that the Stambaugh bias correction
is based on the studies of Marriott and Pope, (1954) and Kendall, (1954) who
proposed suitable bias correction in autocorrelations. which occurs when the
classical least squares estimator is employed for statistical inference.
Furthermore, Amihud and Hurvich, (2004) consider a second-order bias-
correction and propose a reduced-bias OLS based estimator. Both aforementioned
methods are considered to provide a post-estimation bias correction in finite
samples.

Secondly, the predictability literature was extended to nonstandard inference
to account for the presence of nonstationary predictors; some of the most
notable contributions include the Bonferroni-type approach as in Cavanagh et
al., (1995) and Campbell and Yogo, (2006), the conditional likelihood method
based on sufficient statistics555The idea of sufficient statistics considers
optimal tests invariant under transformation based on the curved exponential
family. This framework allows to use the conditional restrictions testing
problem in the presence of nuisance parameters and is particularly appealing
in the case of near integrated regressors. proposed by Jansson and Moreira,
(2006) and the control function method proposed by Elliott, (2011). The
drawback of these testing methodologies is that the asymptotic theory has some
undesirable properties such as the uncorrectable bias due to the presence of
nuisance parameter in the limit distribution (see, Phillips and Lee, (2013)).
Moreover, the particular approaches are computational complex especially in
multivariate predictive regression settings. For instance, Kasparis et al.,
(2015) propose a set of nonparametric predictive tests which allows for
nonlinearities and offers robustness to integration order.

Thirdly, a novel approach that recently has been attracting much attention is
the instrumental variable-based test proposed by Kostakis et al., (2015) (KMS,
hereafter), which is build upon the theoretical framework developed by
Phillips and Magdalinos, (2009). This methodology, referred to as IVX-Wald
test, provides a robust framework for predictive regression models which is
valid for predictors with general persistence properties. Specifically, the
asymptotic theory shows that the IVX estimator which converges to a mixed
Gaussian distribution, successfully removes the long-run endogeneity, that
appears due to the innovation structure of the model, and provides a pivotal
statistic robust under different degrees of persistence or even regressors of
mixed integration (e.g., see Phillips and Lee, (2016)). Hence, a self-
normalized Wald statistic can be constructed that converges to a nuisance
parameter free $\chi^{2}$ limiting distribution. More importantly, the IVX
filtration implies a direct inference procedure via the various moment
approximations (e.g., long-run covariance matrices) and can be easily extended
to the multivariate predictive regression model under certain regulatory
conditions.

The aforementioned literature operates under the assumption of parameter
constancy666Note that, the literature of structural change goes back to 1940s
and 1950s with the pioneering work of Wald on sequential hypothesis testing as
well as the seminal work of Page, (1954) which proposes methodologies for
detecting anomalies in control charting, an idea further developed by Chu et
al., (1995) and Chu et al., (1996) who consider testing for structural break
in the sense of contaminated and non-contaminated periods in linear regression
models. Further seminal studies for structural break tests include Chow,
(1960), Hawkins, (1987) and Ploberger and Krämer, (1990) among others. which
implies a stable predictive relationship over the sample period. However, due
to the nature of economic conditions, shifting between periods of market
tranquillity and periods of market exuberance, the phenomenon of episodic
predictability777Note that, formal econometric methods to test for episodic
predictability have been recently proposed by Demetrescu et al., (2020), who
consider related subsampling techniques. has been proposed to capture these
"pockets" of predictability across business cycles (see, Farmer et al.,
(2019)). This implies the existence of time-varying predictability which can
be examined within an econometric framework which accommodates time-varying
parameters. In this paper, we approach this aspect by proposing to test for
the existence of joint predictability and structural break, in the form of a
single parameter shift at an unknown break-point. Thus, our approach is closer
to the framework of Andrews, (1993) who proposed Wald-type statistics for
testing for a single structural break at an unknown break-point.

The presence of structural break in predictive regression models implies
variation in predictability across time. As a result, standard testing
methodologies for break detection, such as Andrew’s Wald statistics for linear
regression models as well as the family of tests proposed by Bai and Perron,
(1998) can no longer be valid for nonstationary888The frameworks proposed by
Phillips and Magdalinos, (2009) and Magdalinos and Phillips, (2009) explicitly
examine the various forms of nonstationarity attributed due to the structure
of the predictive and cointegrated system, as captured by specific
characteristics of the system. This is a major distinction in the literature
which previously thought that the presence of nonstationarity in the
regressors as a charateristic occurring due to the autoregressive
specification of the model predictors solely. predictors

as captured by the autoregressive specification of the predictive regression
model. This invalidity is demonstrated by the simulation studies of Paye and
Timmermann, (2006). More specifically, the authors show that in the case of
highly correlated innovations and (unfiltered) persistent predictors, an
implementation of a sup-F test and UDMax test can cause severe size
distortions when testing for structural breaks in time series. Moreover, the
optimal test of Elliot and Muller has good empirical size performance, but
distortions appear in the asymptotic properties of the power function of the
test.

The literature of structural break tests has recently focused in the
construction of suitable testing frameworks for predictive regression models.
Cai et al., (2015) propose modelling smooth structural breaks using an
$L_{2}-$ type statistic, in predictive regressions with nonstationary
predictors. A different approach include the study of Pitarakis, (2017) who
consider a CUSUM-type statistic. Additionally, due to the occurrence of
nonstandard limiting distributions in standard structural break test
implementations, Georgiev et al., (2018) and Georgiev et al., (2019) propose
the use of a fixed regressor wild bootstrap procedure999Notice that the
particular fixed regressor wild bootstrap procedure is an extension to the
fixed regressor bootstrap approach employed by Hansen, 2000b  to deal with the
appearance of a nonstandard limiting distribution due to nonstationary
regressors. to approximate critical values. Another related study to the
bootstrapping approach of structural break tests is presented by Boldea et
al., (2019). However, the proposed framework is restricted to the case of
exogenous regressors.

Despite the recent developments of the structural break literature for
predictive regressions, these methodologies consider the detection of
parameter instability in predictive regression models without simultaneously
testing whether the tests are robust to the presence of predictability around
the break-point location. The only existing study that jointly tests for the
existence of both effects is the paper of Demetrescu et al., (2020), who
propose to combine testing for predictability using appropriate subsampling
techniques101010The subsampling technique is similarly employed by Davidson
and Monticini, (2010) in the context of detecting structural break due to
break in cointegration of the relation under investigation.(see also similar
techniques employed by Hansen, 2000a ). The authors propose to use bootstrap-
based inference due to the existence of a non-pivotal limiting distribution,
which is a robust approach to provide statistical validity of the tests.

Further aspects related to the IV based approach of KMS that we follow in this
paper, can be also examined within the above parameter instability testing
procedures. Recent applications include Magdalinos, (2020) who consider
predictability tests with GARCH-type effects (see also, Gungor and Luger,
(2020) ) as well as Yang et al., (2020) who consider a modification of the KMS
test that accounts for serial correlation in the error term of the linear
predictive regression. Moreover, Pang et al., (2020) propose testing
methodologies for multiple structural breaks under the presence of
nonstationary predictors with an application to financial bubble detection.
All these features demonstrate additional refinements one can consider as
future research related to our proposed framework.

##### Outline

The paper is organized as follows. Section 2, presents the predictive
regression model along with the background assumptions. This Section also
includes a review of the IVX instrumentation procedure of KMS, which is
employed for the construction of the proposed test statistic. Section 3,
presents the asymptotic theory of the test statistic under different degrees
of regressors persistence. Section 4 presents an extensive Monte Carlo
simulation study. Section 5 illustrates an application to the US stock returns
which provides evidence of the empirical relevance of the proposed tests for
jointly testing predictability and structural break. Section 6 concludes. All
the mathematical proofs and related results are included in the Appendix of
the paper.

##  2 Econometric Model and main assumptions

In this section we present the theoretical background within which we operate.
We denote with $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ a suitable
probability space on which all of the random elements are defined111111Notice
that this is an important assumption, since our framework can be extended to
Hilbert spaces within which we can denote the expectation operator and higher
moments as inner products. In that scenario, structural break has a slightly
different interpretation. We leave this as a future research endeavour. .
Moreover, throughout the paper, all limits are taken as $T\to\infty$, where
$T$ is the sample size. The symbol $"\Rightarrow"$ is used to denote the weak
convergence of the associated probability measures as $T\to\infty$. The symbol
$\overset{d}{\to}$ denotes convergence in distribution and
$\overset{\text{plim}}{\to}$ denotes convergence in probability, within the
probability space. Let $\left\\{Y_{t},X_{t}\right\\}_{t=1}^{T}$ denote the
corresponding random variables of the underline distributions.

###  2.1 Predictive Regression Model

Consider the predictive regression model with a possible single structural
break

$\displaystyle y_{t}$
$\displaystyle=\alpha_{t}+\beta_{t}^{\prime}x_{t-1}+u_{t}\ \ \ \ \ \ \ 0\leq
t\leq T,$ (2.1) $\displaystyle x_{t}$
$\displaystyle=\mathbf{R}_{T}x_{t-1}+v_{t},$ (2.2)

where $y_{t}\in\mathbb{R}$ is an one$-$dimensional vector and
$x_{t}\in\mathbb{R}^{p\times 1}$ is a $p-$dimensional vector of predictors,
with an initial condition $x_{0}=\mathcal{O}_{p}(1)$ which we assume that does
not affect the limit theory. Moreover, the autocorrelation matrix is expressed
as below

$\displaystyle\mathbf{R}_{T}$
$\displaystyle=\left(\mathbf{I}_{p}-\frac{\mathbf{C}}{T^{\gamma_{x}}}\right)$
(2.3)

where $\mathbf{C}=diag\\{c_{1},...,c_{p}\\}$ is a $p\times p$ matrix. The
degree of persistence in the regressors is determined by the unknown
persistence coefficients $c_{i}$’s which are assumed to be positive constants
and the exponent rate $\gamma_{x}\in\mathbb{R}$, that is, $\gamma_{x}<1$,
$\gamma_{x}\in(0,1)$ or $\gamma_{x}>1$.

The predictive regression model given by (2.1)-(2.2) accommodates the
existence of a single unknown structural break at location $k$. Therefore, the
model parameters take the following form which indicate a time-varying
parameter vector (i.e., parameter instability)

$\displaystyle\alpha_{t}=\begin{cases}\alpha_{1},&0\leq t\leq k\\\
\alpha_{1},&k\leq t\leq T\end{cases}\ \ \ \ \text{and}\ \ \ \
\beta_{t}=\begin{cases}\beta_{1},&0\leq t\leq k\\\ \beta_{2},&k\leq t\leq
T\end{cases}$ (2.4)

where $\beta_{1}\in\mathbb{R}^{p\times 1}$ and
$\beta_{2}\in\mathbb{R}^{p\times 1}$ while for the univariate model
$\beta_{1}\in\mathbb{R}$ and $\beta_{2}\in\mathbb{R}$.

To develop asymptotic theory suitable for robust inference in the predictive
regression model under the presence of parameter instability and
nonstationarity we impose the following assumptions and regulatory conditions
as described below.

######  Assumption 1.

Let $e_{t}=\left(u_{t},v_{t}^{\prime}\right)^{\prime}$ be a
$(p+1)-$dimensional vector. The innovation sequence $e_{t}$ is a conditionally
homoscedastic martingale difference sequence ( m.d.s ) such that the following
two moment conditions hold:

  1. A1.

$\mathbb{E}\left[e_{t}|\mathcal{F}_{t-1}\right]=0$, where
$\mathcal{F}_{t}=\sigma\left(u_{t},u_{t-1},...\right)$ is the natural
filtration.

  2. A2.

$\mathbb{E}\left[e_{t}e_{t}^{\prime}|\mathcal{F}_{t-1}\right]=\mathbf{\Sigma}_{ee}$,
where $\mathbf{\Sigma}_{ee}\in\mathbb{R}^{(p+1)\times(p+1)}$ is a positive-
definite covariance matrix, which has the following form:

$\displaystyle\mathbf{\Sigma}_{ee}=\begin{bmatrix}\sigma_{u}^{2}&\sigma^{\prime}_{uv}\\\
\sigma_{vu}&\mathbf{\Sigma}_{vv}\end{bmatrix}>0.$

with $\sigma_{u}^{2}\in\mathbb{R}$, $\sigma_{uv}\in\mathbb{R}^{p\times 1}$ and
$\mathbf{\Sigma}_{vv}\in\mathbb{R}^{p\times p}$, where $p$ is the number of
predictors.

Assumption A1 indicates that the innovation vector is a martingale difference
sequence while A2 indicates that the innovation sequence is conditionally
homoscedastic. Under conditions A1 and A2, the following Functional Central
Limit Theorem (FCLT) applies

$\displaystyle\frac{1}{\sqrt{T}}\sum_{t=1}^{\lfloor
Tr\rfloor}e_{t}=\frac{1}{\sqrt{T}}\sum_{t=1}^{\lfloor
Tr\rfloor}\begin{bmatrix}u_{t}\\\ v_{t}\end{bmatrix}\Rightarrow\
\text{BM}\left(\mathbf{\Sigma}_{ee}\right)=\begin{bmatrix}B_{u}(r)\\\
B_{v}(r)\end{bmatrix}=\mathbf{\Sigma}_{ee}^{1/2}W(r),$

where $\mathbf{\Sigma}_{ee}^{1/2}W(r)$ is a $(p+1)-$dimensional Brownian
motion with covariance matrix $\mathbf{\Sigma}_{ee}$.

The above error structure provides a realistic interpretation of macroeconomic
shocks. In other words, the shocks to $y_{t}$ and $x_{t-1}$, that is, $u_{t}$
and $v_{t}$ respectively, appear to be contemporaneously correlated, a
commonly used assumption in predictive regression models. Related definitions
can be found in Phillips and Durlauf, (1986), Phillips, (1987) and Phillips,
(1988). Moreover, the implementation of a FCLT121212A standard FCLT for linear
regression models is introduced by Theorem 7.17 in White, (2001). allows to
derive the limiting distribution of Wald type statistics for jointly testing
predictability and structural break within the framework of this paper.

######  Assumption 2.

The innovation to $x_{t}$ is a linear process with the following
representation

$\displaystyle
v_{t}:=\Phi(L)\epsilon_{t}\equiv\displaystyle\sum_{j=0}^{\infty}\Phi_{j}\epsilon_{t-j},\
\ \ \epsilon_{t}\sim^{i.i.d}\left(0,\mathbf{\Sigma}_{ee}\right)$

where $\left\\{\Phi\right\\}_{j=0}^{\infty}$ is a sequence of absolute
summable constant matrices such that $\sum_{j=0}^{\infty}\Phi_{j}$ has full
rank and $\Phi_{0}=I_{p}$ with $\Phi(1)\neq 0$, allowing for the presence of
serial correlation.

To allow for the detection of possible structural breaks via the predictive
regression model and avoid the presence of nuisance parameters at the boundary
of the parameter space we also impose the following assumption.

######  Assumption 3.

Let $k=\pi T$, where $0<\pi<1$. Then, the fraction of structural break defined
as $\pi_{0}=k_{0}/T$ is within the interior of $(0,1)$ for some fixed
$\pi_{0}$ parameter131313Notice that this mathematical statement does not
imply that we consider only known break-points..

######  Assumption 4.

The estimator $\widehat{\pi}$, is $\mathcal{F}_{t}-$measurable and
$[0,1]-$valued. Then, consistent estimation of the break-point implies that
$\exists$ some $m_{0}$ such that
$\widehat{\pi}-\pi=\mathcal{O}_{p}\left(T^{-m_{0}}\right)$.

######  Remark 1.

Assumption 2 gives the linear process representation of innovations proposed
by Phillips and Solo, (1992) and can accommodate the case of serial
correlation or even conditional heteroscedasticity with suitable
specifications. Assumption 3 excludes structural breaks at the boundaries of
the sample. Assumption 4 ensures the existence of a consistent estimator of
the unknown break-point which imply that
$\underset{T\to\infty}{\text{lim}}\mathbb{E}\big{[}|\hat{\pi}-\pi|\big{]}=0$.
For instance, assuming that we are testing for an unknown break-point, to
avoid unidentified parameters under the null hypothesis the supremum
functional is employed for the construction of the Wald type statistics we
propose in the next section.

When the time-varying parameters $\alpha_{t}$ and $\beta_{t}$ are both
constant over the full sample, that is, $\alpha_{1}=\alpha_{2}=\alpha$ and
$\beta_{1}=\beta_{2}=\beta$, then the regression specification given by
(2.1)-(2.2) reduces to the standard predictive regression model. In that case,
the hypothesis testing of interest is finding statistical evidence against the
null hypothesis of no predictability141414For instance, the nonparametric
predictive test of Kasparis et al., (2015) is constructed by comparing the
nonparametric functional form estimator with the parametric counterpart
estimator. In our setup, the estimation procedure is simpler to that aspect,
however it still requires to carefully examine the stochastic properties of
the Wald type statistics under nonstationarity and the presence of a single
structural break.. Furthermore, it has been shown in the literature when the
ordinary least squares estimation is employed, then statistical inference is
nonstandard due to the fact that the limiting distribution of the tests
depends on the unknown persistence parameter, $c_{i}$, and results to
uncorrectable asymptotic bias. Since in our setup we are interested in
detecting a single structural break an appropriate bias correction which
occurs due to endogeneity should take into consideration the presence of these
two regimes151515Notice that the underline regimes of our framework are
motivated by the hypothesis of parameter instability in the predictive
regression model rather than the presence of an independent threshold variable
which induces the presence of the two regimes, as in Gonzalo and Pitarakis,
(2012, 2017)..

###  2.2 Robust Inference with the IVX filtration

To deal with the aforementioned challenges for inference in predictive
regression models, a robust procedure has been introduced by Phillips and
Magdalinos, (2009) and extended by Kostakis et al., (2015) to detect the
presence of predictability via the IVX-Wald test. The particular methodology,
called IVX filtation, allows for the endogenous formulation of instruments
based on the information contained in the regressors of the predictive
regression. As a result, the degree of persistence of the instrumental
variable has degree of persistence explicitly controlled so that the process
is mildly integrated.

Specifically, the IVX filtration considers the first order difference of the
corresponding autocorrelation regression such as it can be expressed as below

$\displaystyle\Delta x_{t}=\frac{\mathbf{C}}{T^{\gamma_{x}}}x_{t-1}+v_{t},\ \
\ \gamma_{x}\in(0,1).$ (2.5)

The particular first difference is not an innovation process unless the
regressor belongs to the persistence class of integrated processes. However,
it behaves asymptotically as an innovation after linear filtering by a matrix
consisting of near-stationary roots161616Note this assumption is a key idea in
the development of the asymptotic theory in cointegrated systems for
regressors with various types of nonstationarity (see, Phillips and
Magdalinos, (2009))..

Therefore, the procedure requires to choose an artificial coefficient matrix
of the form

$\displaystyle\mathbf{R}_{Tz}=\left(\mathbf{I}_{p}-\frac{\mathbf{C}_{z}}{T^{\delta_{z}}}\right),\
\delta_{z}\in(0,1),c_{z}>0.$ (2.6)

where $\mathbf{C}_{z}=diag\\{c_{z,1},...,c_{z,p}\\},c_{z}>0$ for all
$i\in\\{1,...,p\\}$. Then, the instrumental regressor matrix
$\widetilde{z}_{t}\in\mathbb{R}^{T\times p}$ can be constructed as below

$\displaystyle\widetilde{z}_{t}=\sum_{j=1}^{t}\mathbf{R}_{z}^{t-j}\Delta
x_{j},\ \ \
\mathbf{R}_{z}=\left(\mathbf{I}_{p}-\frac{\mathbf{C}_{z}}{T^{\delta_{z}}}\right),\delta_{z}\in(0,1),\mathbf{C}_{z}>0.$
(2.7)

To see this, notice that using expression (2.5) we obtain the following
decomposition

$\displaystyle\widetilde{z}_{t}$
$\displaystyle=\sum_{j=1}^{t}\mathbf{R}_{z}^{t-j}\left(\frac{\mathbf{C}}{T^{\gamma_{x}}}x_{j-1}+u_{j}\right)=\sum_{j=1}^{t}\mathbf{R}_{z}^{t-j}u_{j}+\frac{\mathbf{C}}{T^{\gamma_{x}}}\sum_{j=1}^{t}\mathbf{R}_{z}^{t-j}x_{j-1}.$
(2.8)

which can be written via the following expression

$\displaystyle\widetilde{z}_{t}=z_{t}+\frac{\mathbf{C}}{T}\psi_{t},\ \ \ \
z_{t}=\sum_{j=1}^{t}\mathbf{R}_{z}^{t-j}v_{j}\ \ \text{and}\ \
\psi_{t}=\sum_{j=1}^{t}\mathbf{R}_{z}^{t-j}x_{j-1}.$ (2.9)

Therefore, the IVX filtration proposes to use the constructed
$\widetilde{z}_{t}$ instruments for the regressors $x_{t}$ which are
considered to behave asymptotically as mildly-integrated processes.

More explicitly, by replacing $x_{t}$ with the instrument $z_{t}$ which has a
controllable degree of persistence, result to a robust inference procedure
which accounts for the effects of nonstationarity. Then, the tuning
parameters, which are the exponent rate $\delta_{z}$ and the diagonal matrix
$C_{z}$ are selected to ensure that $z_{t}$ is mildly integrated; less
persistence than a unit root or a regressor assumed to be generated via a
local-unit-root process.

Thus, the IV based approach, implies that the IVX estimator is expressed as
below

$\displaystyle\widetilde{\beta}^{IVX}=\left[\sum_{t=1}^{T}\left(x_{t}-\bar{x}_{T-1}\right)\widetilde{z}_{t-1}^{\prime}\right]^{-1}\sum_{t=1}^{T}\big{(}y_{t}-\bar{y}_{T}\big{)}\widetilde{z}_{t-1}^{\prime},$
(2.10)

where $\bar{x}_{T-1}=\frac{1}{T}\sum_{t=1}^{T}x_{t-1}$ and
$\bar{y}_{T}=\frac{1}{T}\sum_{t=1}^{T}y_{t}$, are the corresponding sample
means.

As shown by Theorem A in the Appendix of KMS, the IVX estimator converges to a
mixed Gaussian171717The mixed Gaussianity property of the IVX estimator is
also extensively examined in the papers of Phillips and Lee, (2013) and
Phillips and Lee, (2016) under various integration orders. limiting
distribution, which holds regardless of the degree of persistence of the
regressors in the model. In turn, this property allows to construct a self-
normalized Wald-type statistic which is shown to converge to a standard
$\chi^{2}-$distribution.

The classical testing hypothesis implies that,
$\mathbb{H}_{0}:\mathcal{R}\beta=0$ vs $\mathbb{H}_{1}:\mathcal{R}\beta\neq
0$, where $\mathcal{R}$ the full rank $q\times p$ restriction matrix with rank
$q$. Then, the IVX-Wald statistic can be used to test the null hypothesis of
no predictability in predictive regression models.

The IVX-Wald statistic is expressed as below

$\displaystyle\mathcal{W}^{IVX}_{T}=\widetilde{\beta}^{IVX\prime}\mathbf{Q}_{\mathcal{R}}^{-1}\widetilde{\beta}^{IVX}$
(2.11)

where $\mathbf{Q}_{\mathcal{R}}$ is a consistent estimator of the asymptotic
variance-covariance matrix of $\widetilde{\beta}^{IVX}$ that accommodates both
long-run endogeneity caused by the correlation between the error terms of the
system, that is, $u_{t}$ and $v_{t}$, and the finite-sample distortion which
results from removing the model intercept. The covariance matrix
$\mathbf{Q}_{\mathcal{R}}$, is derived via the following fully modified (FM)
estimation of the system covariance terms

$\displaystyle\mathcal{\mathbf{Q}}_{\mathcal{R}}$
$\displaystyle=\left(\sum_{t=1}^{T}\widetilde{z}_{t-1}x_{t-1}^{\prime}\right)^{-1}\mathbf{M}\left(\sum_{t=1}^{T}x_{t-1}^{\prime}\widetilde{z}_{t-1}\right)^{-1}$
(2.12) $\displaystyle\mathbf{M}$
$\displaystyle=\widehat{\sigma}^{2}_{u}\left(\sum_{t=1}^{T}\widetilde{z}_{t-1}\widetilde{z}_{t-1}^{\prime}\right)-T\bar{z}_{T-1}\bar{z}_{T-1}^{\prime}\widehat{\mathbf{\Omega}}_{FM}$
(2.13) $\displaystyle\widehat{\mathbf{\Omega}}_{FM}$
$\displaystyle=\widehat{\mathbf{\Sigma}}_{uu}-\widehat{\mathbf{\Omega}}_{uv}\widehat{\mathbf{\Omega}}_{uu}\widehat{\mathbf{\Omega}}_{uv}^{\prime}$
(2.14)

where $\widetilde{z}_{T-1}=\frac{1}{T}\sum_{t=1}^{T}\tilde{z}_{t-1}$. Notice
that for the univariate predictive regression then
$\widehat{\mathbf{\Omega}}_{FM}\equiv\sigma^{2}_{FM}$ and
$\widehat{\mathbf{\Sigma}}_{uu}\equiv\widehat{\sigma}^{2}_{u}$, where
$\widehat{\sigma}^{2}_{u}$ is a consistent estimator of $\sigma^{2}_{u}$. In
that case, we can use the bias correction of KMS for the univariate model
which is given by

$\displaystyle
M=\left[\sum_{t=1}^{T}\widetilde{z}^{2}_{t-1}-T\bar{z}_{t-1}^{2}\left(1-\widehat{\rho}^{2}_{uv}\right)\right]\widehat{\sigma}^{2}_{u},\
\ \ \ \ \text{with}\ \ \
\widehat{\rho}^{2}_{uv}=\displaystyle\frac{\widehat{\Omega}_{uv}}{\widehat{\Sigma}_{uu}\widehat{\Omega}_{vv}}.$
(2.15)

A key aspect for the computation of the IVX-Wald statistic is the estimation
procedure for the matrices $\widehat{\mathbf{\Omega}}_{uv}$ and
$\widehat{\mathbf{\Omega}}_{uu}$, which represent the estimated long-run
covariance between $u_{t}$ and $v_{t}$ and the long-run variance of $u_{t}$
respectively. More specifically, these covariance matrices can be constructed
using nonparametric kernels with preselected bandwidth parameters, such as the
Newey-West type estimators (see, KMS for detailed description and related
studies such as Newey and West, (1987) and Andrews, (1991)).

Furthermore, it can be proved that under the null hypothesis which imposes a
set of linear restrictions in the parameter vector $\beta$, we obtain that
$\mathcal{W}^{IVX}_{T}\implies\chi^{2}(q)$ as $T\to\infty$ (see, Theorem 1 in
KMS). This important limit theory result provides a unified framework for
robust inference and testing regardless the persistence properties of
regressors. A simple example is the application of the IVX-Wald test for
inferring the individual statistical significance of predictors under abstract
degree of persistence. For additional scenarios such as predictors of mixed
integration order see Phillips and Lee, (2013) and Phillips and Lee, (2016).
In these cases the powerful property of the IVX-Wald statistic still holds.

###  2.3 Comparing OLS against IVX based estimators

In this section, we present some preliminary comparisons between the OLS-Wald
and the IVX-Wald statistics, under the null hypothesis of no structural break,
to motivate further our research. These comparisons aim to shed some light on
the ability of the OLS estimator to detect parameter instability under the
assumption of nonstationary predictors. Furthermore, we examine the
performance of the IVX estimator for robustifying inference in predictive
regression models with persistent181818Notice that the assumption of
persistent or integrated regressors is commonly used in the time series
econometrics literature due to the nature of economic and financial variables.
predictors under the presence of parameter instability which has been recently
a stylized fact in various economic data.

We consider that the pair $\left\\{\left(y_{t},x_{t}\right):1\leq t\leq
T\right\\}$ is generated by the following bivariate predictive regression
system drawn from an i.i.d normally distributed error sequence

$\displaystyle y_{t}$ $\displaystyle=\beta_{t}^{\prime}x_{t-1}+u_{t},$ (2.16)
$\displaystyle x_{t}$ $\displaystyle=\left(1-\frac{c}{T}\right)x_{t-1}+v_{t},\
\ x_{0}=0.$ (2.17)

where
$\left(u_{t},v_{t}\right)\overset{i.i.d}{\sim}\mathcal{N}\begin{pmatrix}\sigma_{u}^{2}&\rho\sigma_{u}\sigma_{v}\\\
\rho\sigma_{u}\sigma_{v}&\sigma_{v}^{2}\end{pmatrix}$ with $c_{i}>0$ or
$c_{i}<0$ where $y_{t}\in\mathbb{R}$ and $x_{t}\in\mathbb{R}$.

We are particularly interested to design Wald statistics robust to
nonstationarity which allows to test jointly for the presence of
predictability and parameter instability in the predictive regression model
given by (2.16) and (2.17). To demonstrate that the estimator of the model
parameter can affect the limiting distribution of the test and thus
statistical inference we consider via a Monte Carlo experiment testing for a
structural break. Thus, we can apply the sup-Wald statistics using each
estimator separately by partitioning the sample, estimating the test statistic
for each partition and then taking the supremum over the generated sequence of
test statistics191919Formal assumptions and asymptotic theory regarding the
implementation of the sup-Wald test can be found in Andrews, (1993). In this
section, we demonstrate simple comparisons of the finite sample performance of
the two Wald-type statistics based on the OLS and IVX estimators
respectively..

Initially, we consider that the covariance matrix $\Sigma$ which captures the
dependence between the predictive regression and the persistent regressor has
a unit variance. Notice that in this paper we consider a single structural
change as in the seminal paper of Andrews, (1993). When the break-point
location, denoted with $\pi$ is known, then to construct the Wald statistic we
estimate model parameters $\beta_{1}(\pi)$ for $t=\left\\{1,...,T\pi\right\\}$
and $\beta_{2}(\pi)$ for $t=\left\\{T\pi+1,...,T\right\\}$ based on the
observations of the corresponding sub-samples. When the break-point location
is unknown then we consider the closed interval $\pi\in[\pi_{1},\pi_{2}]$.

Under the null hypothesis of no structural break,
$\mathbb{H}_{0}:\beta_{1}=\beta_{2}$, which implies parameter constancy across
the full sample. We report rejection frequencies from 5,000 Monte Carlo
replications, using the predictive regression model with a fixed parameter
$\beta=0.25$. For the purpose of comparability we use the critical values that
correspond to the NNB asymptotic critical values given by Table 1 of Andrews,
(1993). The empirical size results for the sup OLS-Wald statistic are
presented on Table 1 and 3, while additional results with 10,000 replications
can be found on Table LABEL:tableC1 and LABEL:tableC2.

As we can see from these rejection frequencies, under the assumption of
persistent predictors (low values of $c$) when we use the critical values that
correspond to the standard NBB limiting distribution we obtain size
distortions which considerably deteriorate for larger correlation values
between the error sequences $u_{t}$ and $v_{t}$. Thus, using the OLS estimator
especially when modelling nonstationarity can produce biased inference. The
econometric intuition is clear, constructing structural break statistics based
on the OLS estimator when the predictor follows a local-to-unit root process
causes not correctable size distortions of the tests due to the fact that the
degree of persistence coefficient $c_{i}$ is unknown a prior and cannot be
consistently estimated (see, Proposition 1). Furthermore, when we implement
the sup IVX-Wald statistic as explained in the previous section then we will
need to carefully examine the stochastic properties of the proposed tests.

Firstly, we briefly examine in this setting the asymptotic theory of the two
estimators to understand which components of the limiting distribution result
into nonstandard sampling results especially due to the introduction of
parameter instability.

Notice that the appearance of nonstandard limiting distribution occurs in the
case we estimate the sup OLS-Wald statistic, since when we estimate the model
parameter $\beta$ by OLS its limiting distribution depends on the following
two components

$\displaystyle\sum_{t=1}^{\lfloor\pi
T\rfloor}\left(x_{t-1}-\bar{x}_{T-1}\right)\left(x_{t-1}-\bar{x}_{T-1}\right)^{\prime}\
\ \ \text{and}\ \ \ \sum_{t=1}^{\lfloor\pi
T\rfloor}\left(x_{t-1}-\bar{x}_{T-1}\right)\left(v_{t-1}-\bar{v}_{T}\right)^{\prime}.$
(2.18)

The first component converges to the integral of a squared demeaned OU
process, while the second component weakly converges to a stochastic integral
of that demeaned OU process. Therefore, under the assumption of nonstationary
regressors and due to the appearance of the corresponding stochastic integral
approximations, the second moment of the asymptotic distribution of the model
estimator have a nonconstant variance and a nonstandard limiting distribution.
For instance, Georgiev et al., (2018) derive the limiting distribution of the
sup-Wald test statistic under the local-to-unit-root generating process, which
results to a non-pivotal asymptotic theory. In order to deal with this
problem, the authors propose to use the fixed regressor bootstrap which is
asymptotically valid even under the nonstationarity assumption.

Secondly, in the case we estimate the model parameter using the IVX
instrumentation, then the following matrix moments appear

$\displaystyle\sum_{t=1}^{\lfloor\pi
T\rfloor}\widetilde{z}_{t-1}\left(x_{t-1}-\bar{x}_{T-1}\right)^{\prime}\ \ \
\text{and}\ \ \ \sum_{t=1}^{\lfloor\pi
T\rfloor}\widetilde{z}_{t-1}\left(v_{t-1}-\bar{v}_{T}\right)^{\prime}.$ (2.19)

The asymptotic theory of the above moment matrices is examined extensively by
KMS. A summary of the main results is presented by Lemma 1 in Appendix A of
this paper.

Both OLS and IVX estimators weakly converge to a mixed Gaussian distribution,
thus to better understand the differences between the OLS and the IVX
estimator202020The asymptotic distribution of the IVX estimator in the
univariate predictive regression model is proved to be mixed Gaussian. A proof
of this asymptotic result can be found in Phillips and Lee, (2016). and how
these manifest in the asymptotic theory of the tests, we examine the
stochastic quantity

$\displaystyle\mathcal{P}^{IVX}_{c}\left(\pi\right):=\frac{\displaystyle\pi+\int_{0}^{\pi}\mathbf{J}_{C}dJ_{C}}{\displaystyle
1+\int_{0}^{1}\mathbf{J}_{C}dJ_{C}}$ (2.20)

The $\mathcal{P}_{c}\left(\pi\right)$ quantity which appears in the limiting
distribution of the Wald statistics when an IVX estimator is employed, is a
random variable. Thus, considering the analytical form of this random quantity
will require to derive an analytical approximation with stochastic terms of
higher order, which is a challenging task.

Similarly, the corresponding expression for the OLS estimator is is a function
of the stochastic quantity

$\displaystyle\mathcal{P}^{OLS}_{c}\left(\pi\right):=\frac{\displaystyle\int_{0}^{\pi}\mathbf{J}^{2}_{C}dJ_{C}}{\displaystyle\int_{0}^{1}\mathbf{J}^{2}_{C}dJ_{C}}.$
(2.21)

To obtain some insights regarding the asymptotic behaviour of the stochastic
quantities $\mathcal{P}^{IVX}_{c}\left(\pi\right)$ and
$\mathcal{P}^{OLS}_{c}\left(\pi\right)$ we use integral approximation
simulation techniques with 10,000 replications for a fixed
$\pi\in[0.15,0.85]$. Specifically, to approximate the underline OU process
which drives these stochastic quantities we use a $1000-$step random walk.

Moreover, to make these heuristic arguments more apparent, we plot the mean
and $95\%$ confidence intervals of the particular empirical fluctuation
processes for values of the persistence parameter
$c\in\left\\{1,5,100\right\\}$. As we can see from Figure 1, for low
persistence levels the asymptotic variance of these processes exhibit a linear
growth rate for both the OLS and the IVX estimators since
$\mathcal{P}_{c}\left(\pi\right)$ has approximate linear increments with
$\pi$. In other words, since the predictive regression model does not assume
stationarity for the persistence properties of regressors then by allowing for
breaks if we use the OLS based estimator then the accumulation of information
as seen by the the second moments has a faster convergence rate especially in
the case of high persistence. On the other hand, when we use the IVX estimator
this accumulation of information is smoother for different values of $\pi$ due
to the properties of the particular estimator in filtering out abstract
degrees of persistence, even under time-varying settings.

FIGURE 1. black

black

##  3 Joint Predictability and Structural Break Testing

We focus on the linear predictive regression model and in particular the
univariate case. Our proposed testing framework and related asymptotic theory
can be extended to the multivariate predictive regression with suitable
notation modification. Testing for a single structural break at an unknown
break point location within the full sample requires that the multiple
predictive regression is expressed as below

$\displaystyle y_{t}$
$\displaystyle=\left(\alpha_{1}+\beta_{1}^{\prime}x_{t-1}\right)\mathbf{1}\\{t\leq
k\\}+\left(\alpha_{2}+\beta_{2}^{\prime}x_{t-1}\right)\mathbf{1}\\{t>k\\}+u_{t}$
(3.1)

where $k=\lfloor T\pi\rfloor$ for some $\pi\in(0,1)$, with $\lfloor.\rfloor$
denoting the integer part operator.

To derive the asymptotic theory of the tests we employ the local-unit-root
specification proposed by Phillips, (1987). Specifically, since the set of
regressors are assumed to be generated via the LUR process
$x_{t}=\left(I_{p}-\frac{C}{T}\right)$ $x_{t-1}+v_{t}$, we consider the
following $p-$dimensional Gaussian process

$\displaystyle\displaystyle K_{c}(r)=\int_{0}^{r}e^{(r-s)C}dB_{v}(s),\ \ \ \
r\in(0,1).$ (3.2)

which satisfies the Black-Scholes differential equation $dK_{c}(r)\equiv
cK_{c}(r)+dB_{v}(r)$, with $K_{c}(r)=0$, implying also that
$K_{c}(r)\equiv\sigma_{v}J_{c}(r)$, where $\displaystyle
J_{c}(r)=\int_{0}^{r}e^{(r-s)C}dW_{v}(s)$ and $K_{c}(r)$ the Ornstein-
Uhlenbeck, (OU) process,212121The OU is a stationary Gaussian process with an
autocorrelation function that decays exponentially over time. Moreover, the
continuous time OU diffusion process has a unique solution. which encompasses
the unit root case such that $J_{c}(r)\equiv B_{v}(r)$, for $C=0$.

###  3.1 Classical Least Squares Estimation

In this Section, we derive the asymptotic theory result that corresponds to a
standard sup-Wald test, based on the OLS estimator, when testing for a single
structural break in predictive regressions with multiple predictors.

We denote with $\mathbf{X}_{1}:=\big{[}\mathbf{1}\\{t\leq k\\}\ \
x_{t}\mathbf{1}\\{t\leq k\\}\big{]}$,
$\mathbf{X}_{2}:=\big{[}\mathbf{1}\\{t>k\\}\ \
x_{t}\mathbf{1}\\{t>k\\}\big{]}$ and define $\mathbf{X}=[\mathbf{X}_{1}\
\mathbf{X}_{2}]\in\mathbb{R}^{T\times 2(p+1)}$ the corresponding partitioned
matrix and $\mathbf{\mathcal{R}}=\left[\mathbf{I}_{p+1}\
-\mathbf{I}_{p+1}\right]$ with $\mathbf{I}_{p+1}$ denoting an identity matrix.
Moreover, we denote with $\beta:=(\beta_{1},\beta_{2})^{\prime}$, the
parameter vector, then the predictive regression is $y=X\beta+u$. Thus, the
OLS Wald statistic for testing the null hypothesis of no structural break,
that is, $\mathbb{H}_{0}:\theta_{1}=\theta_{2}$ against
$\mathbb{H}_{1}:\theta_{1}\neq\theta_{2}$, where
$\theta_{j}=(\alpha_{j},\beta_{j})$ for $j=1,2$, is given by the following
expression

$\displaystyle\mathcal{W}_{T}^{OLS}(\pi)$
$\displaystyle=\frac{1}{\hat{\sigma}_{u}^{2}}\left(\widehat{\theta}_{1}(\pi)-\widehat{\theta}_{2}(\pi)\right)^{\prime}\left[\mathcal{R}\left(X^{\prime}X\right)^{-1}\mathcal{R}^{\prime}\right]^{-1}\left(\widehat{\theta}_{1}(\pi)-\widehat{\theta}_{2}(\pi)\right)$
(3.3)

Statistical inference under the null hypothesis of no structural break in the
predictive regression are conducted using the supremum functional. The sup
OLS-Wald statistic is

$\displaystyle\widetilde{\mathcal{W}}^{OLS}(t):=\underset{\pi\in[\pi_{1},\pi_{2}]}{\text{
sup }}\mathcal{W}_{T}^{OLS}(\pi),\ \ \ \text{for}\ \ t_{1}\leq t\leq t_{2}.$
(3.4)

where $0<\pi_{1}<1$ and $0<\pi_{2}<1$ with $\pi_{2}=1-\pi_{1}$.

Under the null hypothesis the breakpoint $k$ is unidentified and so the
supremum functional selects the maximum Wald statistic corresponding to a
sequence of Wald statistics evaluated at values within the interval
$[\pi_{1},\pi_{2}]$. Specifically, in order to construct the corresponding
supremum Wald tests we split the full sample
$\left\\{y_{j},x_{j-1}\right\\}_{j=1}^{T}$ into two sub-samples; the first
sub-sample corresponds to the time period before time $t$,
$\left\\{y_{j},x_{j-1}\right\\}_{j=1}^{t}$ and the second sub-sample
corresponds to the time period after time $t$,
$\left\\{y_{j},x_{j-1}\right\\}_{j=t+1}^{T}$. Due to the fact that we operate
within the Skorokhod topology the sample moments that correspond to these sub-
samples weakly converge to the corresponding asymptotic result which are based
on the $\mathcal{D}[0,1]$ topology. For instance, when deriving the sample
moments we can use the notation $t\in\left[t_{1},t_{2}\right]$, where $t_{1}$
and $t_{2}$ are the lower and upper bounds for the possible break-point
location.

######  Theorem 1.

If Assumptions 1 and 3 hold and $\pi$ denotes the unknown break-point, then
the sup OLS-Wald statistic under the null hypothesis
$\mathbb{H}_{0}:\theta_{1}=\theta_{2}$ against
$\mathbb{H}_{1}:\theta_{1}\neq\theta_{2}$, where
$\theta_{j}=(\alpha_{j},\beta_{j})^{\prime}$ with $j=1,2$ based on the
predictive regression model (2.1)-(2.2) with $\gamma_{x}=1$222222Note that
this parameter restriction for the exponent rate implies that $x_{t}$ follows
a LUR process. weakly converges to the following limiting distribution

$\displaystyle\widetilde{\mathcal{W}}^{OLS}(\pi)\equiv\underset{\pi\in[\pi_{1},\pi_{2}]}{\text{
sup }}\
\mathcal{W}_{T}^{OLS}(\pi)\Rightarrow\displaystyle\underset{\pi\in[\pi_{1},\pi_{2}]}{\text{
sup }}\
\bigg{\\{}\widetilde{\mathbf{N}}^{\prime}_{c}(\pi)\widetilde{\mathbf{M}}_{c}(\pi)^{-1}\widetilde{\mathbf{N}}_{c}(\pi)\bigg{\\}}$
(3.5)

where

$\displaystyle\widetilde{\mathbf{M}}_{c}(\pi)^{-1}=\widetilde{\mathbf{G}}_{c}(\pi)-\widetilde{\mathbf{G}}_{c}(\pi)\widetilde{\mathbf{G}}_{c}(1)^{-1}\widetilde{\mathbf{G}}_{c}(\pi)$
(3.6)

and

$\displaystyle\widetilde{\mathbf{N}}_{c}(\pi)=\left\\{\widetilde{\mathbf{G}}_{c}(\pi)^{-1}\widetilde{H}_{c}(\pi)-\bigg{[}\widetilde{\mathbf{G}}_{c}(1)-\widetilde{\mathbf{G}}_{c}(\pi)\bigg{]}^{-1}\bigg{[}\widetilde{H}_{c}(1)-\widetilde{H}_{c}(\pi)\bigg{]}\right\\}^{\prime}$
(3.7)

such that

$\displaystyle\widetilde{\mathbf{G}}_{c}(\pi):=\int_{0}^{\pi}\widetilde{K}_{c}(r)\widetilde{K}^{\prime}_{c}(r)dr\
\ \text{and}\ \
\widetilde{H}_{c}(\pi):=\int_{0}^{\pi}\widetilde{K}_{c}(r)dB_{u}(r)$ (3.8)

with $\widetilde{\mathbf{G}}_{c}(\pi)\in\mathbb{R}^{(p+1)\times(p+1)}$ a
positive-definite stochastic matrix and
$\widetilde{H}_{c}(\pi)\in\mathbb{R}^{(p+1)\times 1}$.

Theorem 1 provides a novel result in the literature and our first theoretical
contribution, demonstrating that the sup OLS-Wald statistic as a structural
break test on an unknown location based on the predictive regression model
with persistent predictors, does not converge to the NBB result as proposed by
Andrews, (1993) in linear regressions. Thus, we show that the asymptotic
theory of the test depends on the nuisance parameter of persistence $c_{i}$.
However, when the break-point is known a prior, such as $\pi\equiv\pi_{0}$, it
can be easily proved that the limit theory of the OLS-Wald test converges to a
NBB even in the case of persistent predictors. The proof of Theorem 1 is shown
in Appendix C.

###  3.2 IVX based Estimation

In this section, we examine the IVX-Wald based tests for jointly testing for
the presence of predictability and parameter instability based on the
predictive regression model. The null hypothesis remains the same as in the
IVX-Wald test, but the alternative hypothesis is constructed so that it
captures both effects. We examine separately the case of stable model
intercept and the case of unstable model intercept. The model estimates of the
two sub-samples are used to construct the proposed test statistics. We denote
with $\widetilde{\beta}_{1}^{IVX}\left(t\right)$ and
$\widetilde{\beta}_{2}^{IVX}\left(t\right)$ the IVX estimates of the two sub-
samples; and with $\widetilde{Q}_{1}\left(t\right)$ and
$\widetilde{Q}_{2}\left(t\right)$ the corresponding covariance matrices. The
sample estimates are given by

$\displaystyle\widetilde{\beta}_{1}^{IVX}(t)$
$\displaystyle=\left(\frac{1}{t}\sum_{j=1}^{t}\widetilde{z}_{1,j-1}x_{1,j-1}^{\prime}\right)^{-1}\left(\frac{1}{t}\sum_{j=1}^{t}\widetilde{z}_{1,j-1}y_{j}\right)$
(3.9) $\displaystyle\widetilde{\beta}_{2}^{IVX}(t)$
$\displaystyle=\left(\frac{1}{T-t}\sum_{j=t+1}^{T}\widetilde{z}_{2,j-1}x_{2,j-1}^{\prime}\right)^{-1}\left(\frac{1}{T-t}\sum_{j=t+1}^{T}\widetilde{z}_{2,j-1}y_{j}\right).$
(3.10)

####  3.2.1 Joint Wald Tests under stable model intercept

We begin our asymptotic theory analysis by considering the case in which the
model intercept $\alpha$ is assumed to be stable, that is, under the null
hypothesis there is no structural break in the model intercept. Then, the
testing hypothesis of interest is given by

$\displaystyle\mathbb{H}_{0}:\alpha_{1}=\alpha_{2}\ \ \text{and}\ \
\beta_{1}=\beta_{2}$ (3.11)

For evaluating our hypothesis we use the IVX-Wald test which has the following
form232323Note that the covariance matrices are computed based on the long-
covariance matrices and corresponding FM corrections given by KMS, see also
Kasparis et al., (2015). These definitions are employed since we do not rule
out the strong assumption of a weakly covariance dependence in the model.

$\displaystyle\mathcal{W}_{\beta}^{IVX}(t)=\left(\widetilde{\beta}^{IVX}_{1}(t)-\widetilde{\beta}^{IVX}_{2}(t)\right)^{\prime}\bigg{[}\widetilde{\mathbf{Q}}_{1}(t)+\widetilde{\mathbf{Q}}_{2}(t)\bigg{]}^{-1}\left(\widetilde{\beta}^{IVX}_{1}(t)-\widetilde{\beta}^{IVX}_{2}(t)\right).$
(3.12)

We denote with
$\widetilde{\mathcal{W}}^{IVX}_{\beta}(t)\equiv\underset{t\in[t_{1},t_{2}]}{\text{sup}}\mathcal{W}_{\beta}^{IVX}(t)$,
the corresponding sup IVX-Wald statistic, since we consider the case of the
unknown break-point. The two IVX estimates of $\beta$ and their corresponding
asymptotic variances are computed, using the data from each sub-sample
separately. Thus, the $\mathcal{W}_{\beta}^{IVX}(t)$ statistic can be thought
as a Chow-type statistic for detecting break at time $t$, such that $t_{1}\leq
t\leq t_{2}$. Then, due to the unknown nature of the break-point the sup Wald
test is simply a sequence of Chow-type statistics within the same probability
space. Theorem 2 presents the limiting distribution of
$\widetilde{\mathcal{W}}^{IVX}_{\beta}(t)$, under the null hypothesis of no
parameter instability in the predictive regression model.

######  Theorem 2.

If Assumptions 1 and 3 hold and $\pi$ denotes the unknown break-point, then
the sup IVX-Wald statistic under the null hypothesis (with $\alpha$ known to
be stable a priori) $\mathbb{H}_{0}:\alpha_{1}=\alpha_{2}$ and
$\beta_{1}=\beta_{2}$, based on the predictive regression model (2.1)-(2.2)
and no restriction on the exponent rate $\gamma_{x}$, has the following
asymptotic behaviour

$\displaystyle\widetilde{\mathcal{W}}^{IVX}_{\beta}(t)\Rightarrow\underset{\pi\in[\pi_{1},\pi_{2}]}{\text{sup}}\bigg{\\{}\mathbf{N}(\pi)^{\prime}\mathbf{M}(\pi)^{-1}\mathbf{N}(\pi)\bigg{\\}}$
(3.13)

with $t\in[t_{1},t_{2}]$242424Note that, since we operate within the Skorokhod
topology $\mathcal{D}\left(0,1\right)$ we can apply standard weakly
convergence arguments. and $\pi\in[\pi_{1},\pi_{2}]$, where

$\displaystyle\mathbf{N}(\pi)$
$\displaystyle=B_{p}(\pi)-\mathbf{R}(\pi)B_{p}(1)$ (3.14)
$\displaystyle\mathbf{M}(\pi)$
$\displaystyle=\pi\big{(}\mathbf{I}_{p}-\mathbf{R}(\pi)\big{)}\big{(}\mathbf{I}_{p}-\mathbf{R}(\pi)\big{)}^{\prime}+(1-\pi)\mathbf{R}(\pi)\mathbf{R}(\pi)^{\prime}$
(3.15)

such that

$\mathbf{R}(\pi)=\begin{cases}\displaystyle\left(\pi\mathbf{\Omega}_{xx}+\int_{0}^{\pi}\mathbf{\underline{B}}dB^{\prime}\right)\left(\mathbf{\Omega}_{xx}+\int_{0}^{1}\mathbf{\underline{B}}dB^{\prime}\right)^{-1}&,\text{if}\
\gamma_{x}>1\\\ \\\
\displaystyle\left(\pi\mathbf{\Omega}_{xx}+\int_{0}^{\pi}\mathbf{\underline{J}}_{C}dJ_{C}^{\prime}\right)\left(\mathbf{\Omega}_{xx}+\int_{0}^{1}\mathbf{\underline{J}}_{C}dJ_{C}^{\prime}\right)^{-1}&,\text{if}\
\gamma_{x}=1\\\ \\\ \displaystyle\pi\mathbf{I}_{p}&,\text{if}\
\gamma_{x}<1\end{cases}$ (3.16)

where $B(.)$ is a $p-$dimensional standard Brownian motion,
$J_{C}(\pi)=\int_{0}^{\pi}e^{C(\pi-s)}dB(r)$ is an Ornstein-Uhkenbeck (OU)
process and we denote with
$\underline{J}_{C}(\pi)=J_{C}(\pi)-\int_{0}^{1}J_{C}(s)ds$ and
$\underline{B}(\pi)=B(\pi)-\int_{0}^{1}B(s)ds$ the demeaned processes of
$J(\pi)$ and $B(\pi)$ respectively.

Theorem 2 demonstrates that the supremum functional of the IVX-Wald test
weakly convergence to a stochastic quadratic functional and does not have the
same asymptotic behaviour as the corresponding IVX-Wald statistic which is
proved to follow a $\chi^{2}_{p}$ distribution (see, KMS, Phillips and Lee,
(2013) and Phillips and Lee, (2016)).

An important implication of Theorem 2, which consists our second contribution
to the limit theory of structural break tests in predictive regression models
with nonstationary predictors, is that we show that the dependence of the
limiting distribution to the unknown parameter of persistence takes different
forms by restricting further the parameter space of the exponent rate.
Specifically, Theorem 2 demonstrates that when testing for a structural break
in predictive regression models, under the assumption that the predictors are
integrated or nearly integrated, then the limiting distribution of the test,
weakly converge to a nonstandard process, which verifies the corresponding
result already mentioned by Hansen, 2000b ). On the other hand, when the
predictors are stationary or mildly stationary, then the test statistic weakly
converges to the familiar squared tied-down Bessel process as proved by
Andrews, (1993) in the case of linear regression models. The latter implies
that when we consider separately the special case for which $\gamma_{x}<1$,
which covers both the cases of mildly integrated regressors, i.e.,
$\gamma_{x}\in(0,1)$, and stationary regressors, i.e., $\gamma_{x}=0$, then it
can be proved that the limiting distribution of the sup IVX-Wald test
converges to the standard NBB result252525Note that in the case of a known
break-point we can easily prove a convergence to a $\chi^{2}_{p}$ distribution
which is free of nuisance parameters and so conventional inference methods
apply. Corollary 1 below is a direct implication of Theorem 2 and summarizes
this finding.

######  Corollary 1.

Under the assumptions and definitions given by Theorem 2, when $\gamma_{x}<1$
then the following asymptotic distribution holds

$\displaystyle\widetilde{\mathcal{W}}^{IVX}_{\beta}(t)\Rightarrow\underset{\pi\in[\pi_{1},\pi_{2}]}{\text{sup}}\frac{\mathcal{BB}_{p}(\pi)^{\prime}\mathcal{BB}_{p}(\pi)}{\pi(1-\pi)},$
(3.17)

where $\mathcal{BB}_{p}(.)$ is a $p-$dimensional standard Brownian bridge.

######  Remark 2.

The weakly convergence of the sup IVX-Wald test into a normalized squared
Brownian bridge, when the exponent rate is restricted such that
$\gamma_{x}<1$, implies standard statistical inference due to the known
distribution. One rejects $\mathbb{H}_{0}$ for large values of the sup IVX-
Wald test based on a significance level $\alpha$ such that $0\leq\alpha\leq 1$
and thus the limit distribution can be used to derive associated critical
values, denoted with $c_{\alpha}$ such that
$\mathbb{P}\big{(}\widetilde{\mathcal{W}}_{\beta}^{IVX}(\pi)>c_{\alpha}\big{)}>0$
with
$\underset{T\to\infty}{\text{lim}}\mathbb{P}\big{(}\widetilde{\mathcal{W}}_{\beta}^{IVX}(\pi)>c_{\alpha}\big{)}=1$.

Our findings presented by Theorem 2 verify the conjuncture of Hansen, 2000b
who argues that when testing for a structural break based on a sup-functional
induces an asymptotic non-pivotal distribution under the assumption of
nonstationarity. However, this seminal study has not examined in details
certain forms of nonstationarity which can occur and how these are manifested
in the limit theory of the tests. In particular, within our framework we
demonstrate using local-to-unit root asymptotic arguments that the nonstandard
limiting distribution occurs when $x_{t}$ is properly modelled as a
nonstationary stochastic process, and specifically the NBB result no longer
holds when $x_{t}$ is either a nearly integrated or an integrated process.

The proofs of both Theorem 2 and Corollary 1 can be found in Appendix A. Next,
we focus on designing test statistics for jointly testing against
predictability and structural break in predictive regression models. We define
a joint Wald test based on the IVX estimator under the assumption of an
unknown break-point as below

$\displaystyle\widetilde{\mathcal{W}}_{\beta}^{joint}:=\underset{t\in[t_{1},t_{2}]}{\text{sup}}\bigg{\\{}\mathcal{W}_{T}^{IVX}+\mathcal{W}_{\beta}^{IVX}(t)\bigg{\\}},\
\ \ \text{for}\ \ t_{1}\leq t\leq t_{2}.$ (3.18)

where the supremum functional applies only on the second component of the test
above. Then, the Corollary 2 below gives the related limit theory result.

######  Corollary 2.

(i) If the conditions of Theorem 2 hold, then under the null hypothesis,
$\mathbb{H}_{0}:\alpha_{1}=\alpha_{2}$ and $\beta_{1}=\beta_{2}=0$ and no
restriction on the exponent rate $\gamma_{x}$, the large sample theory of the
test statistic specified by (3.18) has the following form

$\displaystyle\widetilde{\mathcal{W}}_{\beta}^{joint}\Rightarrow
B(1)^{\prime}B(1)+\underset{\pi\in[\pi_{1},\pi_{2}]}{\text{sup}}\bigg{\\{}\mathbf{N}(\pi)^{\prime}\mathbf{M}(\pi)^{-1}\mathbf{N}(\pi)\bigg{\\}},$
(3.19)

where $B(.)$, $\mathbf{N}(\pi)$ and $\mathbf{N}(\pi)$ are defined in Theorem
2.
(ii) As a special case, when $\gamma_{x}<1$, it follows that

$\displaystyle\widetilde{\mathcal{W}}_{\beta}^{joint}\Rightarrow\chi^{2}_{p}+\underset{\pi\in[\pi_{1},\pi_{2}]}{\text{sup}}\frac{\mathcal{BB}_{p}(\pi)^{\prime}\mathcal{BB}_{p}(\pi)}{\pi(1-\pi)},$
(3.20)

where $\mathcal{BB}_{p}(.)$ is a $p-$dimensional standard Brownian bridge and
$\chi^{2}_{p}$ denotes the $\chi^{2}$ random variable with $p$ degrees of
freedom. Furthermore, the two stochastic quantities of the limiting
distribution are assumed to be independent.

######  Remark 3.

Notice that Corollary 2 demonstrates that when $\gamma_{x}<1$, the large
sample theory of the test statistic $\mathcal{W}_{\beta}$ is pivotal.
Therefore, in this special case by having a limiting distribution being free
of nuisance parameters, asymptotic critical values for testing the null
hypothesis can be easily obtained. The particular test statistic provides a
methodology for testing for both predictability and structural break which is
robust to the persistence properties of regressors after replacing the OLS
with an IVX estimator.

The main arguments we use for the proof of Corollary 2 is to consider the
joint testing hypothesis as a composite hypothesis based on the mutually
exclusive parameter space, and thus we construct the limit theory of this test
based on the joint formulation of the two separate testing hypotheses. By
employing the asymptotic matrix moments based on the IVX instrumentation we
prove that the limiting distribution of the joint test can be decomposed into
two components. These two components are considered to be independent random
variables and therefore practically we could use the critical values of the
limiting distributions that corresponds to each of these stochastic
quantities.

####  3.2.2 Joint Wald Tests under unstable model intercept

Notice that when a model intercept is included then under the null hypothesis,
when $\alpha_{1}=\alpha_{2}$, for example if we are testing whether
$H_{0}:\beta_{1}=\beta_{2}$, then the particular hypothesis would be
equivalent to testing the null hypothesis of no structural break. However, in
the case of an unstable model intercept, that is, $\alpha_{1}\neq\alpha_{2}$,
then testing the null hypothesis $H_{0}:\beta_{1}=\beta_{2}$ can result to a
different asymptotic theory due to the break in the model intercept. In this
section, we consider the case that a potential structural break occurs in both
the intercept and the slope coefficient $\beta$ of the predictive regression
model. The two proposed tests aim to test: (i) the null hypothesis of no break
on model intercepts while the slope coefficients remain at a fixed level; (ii)
the null hypothesis of no break on model intercepts while the slope
coefficients are both equal to zero.

The proposed test statistic is expressed as below

$\displaystyle\mathcal{W}^{IVX}_{\alpha}(t)=\bigg{(}\tilde{\alpha}_{1}(t)-\tilde{\alpha}_{2}(t)\bigg{)}^{\prime}\bigg{[}\widetilde{\mathbf{\Omega}}_{1}(t)+\widetilde{\mathbf{\Omega}}_{2}(t)\bigg{]}^{-1}\bigg{(}\tilde{\alpha}_{1}(t)-\tilde{\alpha}_{2}(t)\bigg{)}$
(3.21)

where the covariance estimators are computed as below

$\displaystyle\widetilde{\mathbf{\Omega}}_{1}(t)$
$\displaystyle=\frac{1}{t^{2}}\widehat{u}_{1}(t)^{\prime}\widehat{u}_{1}(t)+\bar{z}_{t}^{\prime}\widetilde{\mathbf{Q}}_{1}(t)\bar{z}_{t}$
(3.22) $\displaystyle\widetilde{\mathbf{\Omega}}_{2}(t)$
$\displaystyle=\frac{1}{(T-t)^{2}}\widehat{u}_{2}(t)^{\prime}\widehat{u}_{2}(t)+\bar{z}_{T-t}^{\prime}\widetilde{\mathbf{Q}}_{2}(t)\bar{z}_{T-t}$
(3.23)

Then, the joint Wald test is expressed using the following form

$\displaystyle\mathcal{W}_{\alpha\beta}^{joint}:=\bigg{\\{}\mathcal{W}_{T}^{IVX}+\mathcal{W}^{IVX}_{\alpha}(t)+\mathcal{W}^{IVX}_{\beta}(t)\bigg{\\}},\
\ \ \text{for}\ \ t_{1}\leq t\leq t_{2}.$ (3.24)

To distinguish between the standard Wald statistics in the case of known
break-point and the Wald type statistics that correspond to the unknown break-
point we denote with
$\widetilde{\mathcal{W}}^{IVX}_{\alpha}(t)=\underset{t\in[t_{1},t_{2}]}{\text{sup}}\mathcal{W}^{IVX}_{\alpha}(t)$
and
$\widetilde{\mathcal{W}}^{joint}_{\alpha\beta}(t)=\underset{t\in[t_{1},t_{2}]}{\text{sup}}\mathcal{W}^{joint}_{\alpha\beta}(t)$.
Both of these test statistics require to split the sample into subsamples of
window size $[t_{1},t_{2}]\subset(1,T)$ and estimate the joint Wald test based
on the observations from each subsample. For instance, the last two components
of the joint test requires to estimate the Wald IVX for the full sample plus
the supremum statistics for the model intercept and the model slope
separately. Notice that the model intercept has different asymptotic
properties, in particular, convergence rate to the true population parameter
when the IVX estimator is used for the slope parameter. Therefore, we use the
covariance estimators given by (3.22) and (3.23) which is constructed based on
the residuals from the fitted predictive regression of each subsample while we
add a bias correction based on the covariance estimator of the slope
coefficients since the estimator of the intercept is conditional on the IVX
estimator in the case we switch from the OLS to the IVX estimation procedure.

######  Proposition 1.

Consider the predictive regression model given by expressions (2.1)-(2.2). If
Assumption 1-4 hold and $\alpha$ is known to be unstable a priori, then under
the null hypothesis $\mathbb{H}_{0}:\alpha_{1}=\alpha_{2}\ \ \text{and}\ \
\beta_{1}=\beta_{2}=\beta$, as T $\to\infty$ the following limit result holds

$\displaystyle\widetilde{\mathcal{W}}^{IVX}_{\alpha}(t)\Rightarrow\underset{\pi\in[\pi_{1},\pi_{2}]}{\text{sup}}\frac{\mathcal{BB}_{1}(\pi)^{\prime}\mathcal{BB}_{1}(\pi)}{\pi(1-\pi)}$
(3.25)

where $\mathcal{BB}_{1}(.)$ is a one-dimensional standard Brownian bridge.

######  Remark 4.

Notice that Proposition 1 provides an asymptotic result for a composite
hypothesis since we consider jointly testing for a structural break in the
model intercept and the slope coefficients while we test, that under the null
hypothesis the slope coefficient has a fixed parameter value $\beta$.
Additionally, we can investigate the limiting distribution of the joint Wald
test when we assume that under the null hypothesis there is no predictability.

######  Proposition 2.

Consider the predictive regression model given by expressions (2.1)-(2.2). If
Assumption 1-4 hold and $\alpha$ is known to be unstable a priori, then under
the null hypothesis $\mathbb{H}_{0}:\alpha_{1}=\alpha_{2}$ and
$\beta_{1}=\beta_{2}=0$, as $T\to\infty$ the following limit results hold: (i)

$\displaystyle\widetilde{\mathcal{W}}^{joint}_{\alpha\beta}(t)\Rightarrow
B(1)^{\prime}B(1)+\underset{\pi\in[\pi_{1},\pi_{2}]}{\text{sup}}\bigg{\\{}\widetilde{\mathbf{N}}(\pi)^{\prime}\widetilde{\mathbf{M}}(\pi)^{-1}\widetilde{\mathbf{N}}(\pi)\bigg{\\}}$
(3.26)

where
$\widetilde{\mathbf{N}}(\pi)=\big{(}\mathcal{BB}_{1}(\pi),\mathbf{N}(\pi)\big{)}^{\prime}$
and $\widetilde{\mathbf{M}}(\pi)=\begin{pmatrix}\pi(1-\pi)&0\\\
0&\mathbf{M}(\pi)\end{pmatrix}$. The terms, $\mathbf{N}(\pi)$ and
$\mathbf{M}(\pi)$ are defined in Theorem 2.
(ii) As a special case, when $\gamma_{x}\in(0,1)$, it holds that

$\displaystyle\widetilde{\mathcal{W}}^{joint}_{\alpha\beta}(t)\Rightarrow\chi^{2}_{p}+\underset{\pi\in[\pi_{1},\pi_{2}]}{\text{sup}}\frac{\mathcal{BB}_{p+1}(\pi)^{\prime}\mathcal{BB}_{p+1}(\pi)}{\pi(1-\pi)}$
(3.27)

where $\mathcal{BB}_{p+1}(.)$ is a $(p+1)-$dimensional standard Brownian
bridge, and $\chi^{2}_{p}$ is a random variable following a $\chi^{2}$
distribution with $p$ degrees of freedom.

######  Remark 5.

Notice that, Proposition 2 shows that the limiting distribution of the joint
test for both predictability and structural break, when we consider
simultaneously testing whether there is a structural break to the model
intercept and no predictability using the set of regressors of the model, has
an asymptotic distribution which takes a different form when we consider
different values of the parameter space of the exponent rate.

As we have seen by our extensive asymptotic theory analysis provided in this
Section, the asymptotic distribution of the Joint IVX-Wald tests can be
affected by various scenarios, such as the inclusion of model intercept in the
predictive regression as well as the parameter space of the exponent rate.

##  4 Monte-Carlo Simulation Study

In this section, we present a Monte Carlo simulation study in order to examine
the finite size properties of the proposed Wald-type statistics in terms of
their empirical size and power performance, under the null hypothesis of no
joint parameter instability and predictability. In practise, the degree of
persistence in the time series of the regressors is unknown. That is, both the
coefficient of persistence $c_{i}$ as well as the exponent rate $\gamma_{x}$
are both unknown parameters to the researcher. Moreover, we have proved that
the limiting distribution of the Wald-type statistics for detecting structural
break in predictive regression models depend on these unknown properties of
the regressors for certain parameter value restrictions on the coefficient of
persistence. In particular when the exponent rate of persistence has an
exponential rate , then for both cases i.e., testing for structural break or
jointly testing for predictability and structural break we have a nonstandard
limiting distribution which depends on the corresponding stochastic integrals
which are functions of the unknown break-point. Now, in the case we known a
prior that the exponent rate equals to one, which means we are within the
realm of near integrated or LUR regressors then the tests converge to a
standard NBB distribution which is easy to tabulate critical values.
Nevertheless, in any case in empirical applications critical values can be
obtained by either simulations or more advanced bootstrap methodologies.

The MC simulation study aims to shed light on the main theoretical results of
the paper. Therefore, to demonstrate the above theoretical result, under the
null hypothesis of no structural break, we can generate a DGP with no breaks
in the coefficients of the predictive regression. Then, constructing the sup-
Wald test and using the Andrews’ critical values we can observe whether size
distortions indeed occur in this scenario262626Recall that under the
assumption of persistent predictors, we prove via Proposition 1 of the paper
that when testing for parameter instability in the predictive regression the
limiting distribution of the sup-Wald statistic no longer follows a normalized
squared brownian bridge, (NBB) for an unknown structural break. In particular,
when contacting inference or assessing the statistical validity of the sup-
Wald statistic via a MC experiment, using the corresponding critical values of
the sup-Wald test proposed by Andrews, (1993) we can observe that leads to
size distortions due to the non-NBB limiting distribution.. Secondly, via
Proposition 2 of the paper we propose an alternative approach to overcome this
problem. In particular, using an IV based sup-Wald test, which is constructed
using the IVX instrumentation, we prove that the limiting distribution of the
statistic indeed weakly converges to a NBB, which allow us to use the Andrews’
critical values, avoiding this way to simulate critical values which can be
computational complex. Below, we present the DGP, the test statistics as well
as the size and power comparisons for the proposed tests of our econometric
framework.

###  4.1 Experiment Design

We use the following data generating process (DGP) where $y_{t}$ is a scalar
and $x_{t}$ is a vector of LUR predictors (with the property of being highly
persistent).

$\displaystyle y_{t}$
$\displaystyle=\beta_{0}+\beta_{1}x_{t-1}+\beta_{2}x_{t-1}+u_{t}$ (4.1)
$\displaystyle\underline{x}_{t}$
$\displaystyle=\begin{pmatrix}1-\displaystyle\frac{c_{1}}{T}&0\\\
0&1-\displaystyle\frac{c_{2}}{T}\end{pmatrix}\underline{x}_{t-1}+\underline{v}_{t},\
\underline{x}_{0}=\underline{0}.$ (4.2)

with $t\in\\{1,...,T\\}$ and $T=\\{100,250,500,1000\\}$ for $B=5,000$
replications.

Furthermore, we consider the effect of different localizing coefficients of
persistence across the predictors. We use $c_{i}\in\\{1,5,10,20\\}$ for
$i=1,2$, which cover various cases of LUR regressors, with smaller values
implying that we impose the assumption of higher persistence and lower values
implying that existence of mild persistence in the predictor.

The covariance matrix of the innovations
$\underline{e}_{t}=\left(u_{t},\underline{v}_{t}\right)^{\prime}\sim\mathcal{N}\left(\underline{0}_{3\times
1},\Sigma_{ee}\right)$ we assume that is parametrised with the following
structure

$\displaystyle\Sigma_{ee}=\begin{bmatrix}\sigma^{2}_{u}&\sigma_{uv_{1}}&\sigma_{uv_{2}}\\\
\sigma_{uv_{1}}&\sigma^{2}_{v_{1}}&\sigma_{v_{1}v_{2}}\\\
\sigma_{uv_{2}}&\sigma_{v_{1}v_{2}}&\sigma^{2}_{v_{2}}\end{bmatrix}$ (4.3)

Clearly, we can see that the covariance matrix given by expression (4.3)
allows to consider various scenarios regarding the contemporaneous correlation
of the regressors and the dependent variables which has related economic
interpretation. We use the following predetermined covariance matrices

$\displaystyle\Sigma_{ee}=\begin{bmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\\\
\end{bmatrix},\ \ \ \text{or}\ \ \ \Sigma_{ee}=\begin{bmatrix}1&0.10&-0.29\\\
0.10&1&-0.03\\\ -0.29&-0.03&1\\\ \end{bmatrix}$

Under both the null and the alternative hypothesis, the predictive regression
is generated using the following econometric specification (using (B.2) and
(4.3))

$\displaystyle y_{t}$
$\displaystyle=\left(\beta_{01}+\underline{\beta}_{11}^{\prime}x_{t-1}\right)\mathbf{1}\\{t\leq
k\\}+\left(\beta_{02}+\underline{\beta}_{12}^{\prime}x_{t-1}\right)\mathbf{1}\\{t>k\\}+u_{t}$
(4.4)

where $k=\lfloor T\pi\rfloor$ for some $\pi\in(0,1)$.

Additionally under the null hypothesis, of no parameter instability, we
simulate the DGP using the same vector of regression parameters before and
after the breakpoint, such as $\underline{\beta}_{1}=\underline{\beta}_{2}$.
Furthermore, under the alternative hypothesis of parameter instability, we can
generate a sequence of local alternatives by using a different vector of
regression parameters before and after the breakpoint. Note that, since our
framework consider a single unknown structural break, the DGP is re-
constructed for each window $[\pi_{1},\pi_{2}]$ in order to apply the supremum
functional.

###  4.2 Test statistics

We assess the statistical validity in finite and large samples of the Wald
based statistics within our proposed framework, that is, the sup Wald-OLS test
and the sup-Wald IVX test. In particular, we are interested to verify any size
distortions under the null hypothesis of no structural change, when using the
sup Wald-OLS test and we expect to observe improvements in the empirical size
of the sup Wald-IVX test. Moreover, we examine the size and power performance
of the two statistics across different degree of persistence as well as the
rate at which we allow the IVX instrumentation procedure to create a more
mildly integrated regressor (as given by the parameters $c_{z}$ and
$\delta\equiv\delta_{0}$).

For the large sample properties of the test statistics a standard convergence
result apply, that is, $\mathcal{W}_{T}(\pi;\delta)\to\mathcal{W}(\pi;\delta)$
as $T\to\infty$. Furthermore, the testing hypothesis of interest is a two-
sided type hypothesis which is expressed as below

$\displaystyle\mathbb{H}_{0}:\underline{\beta}_{1}=\underline{\beta}_{2}\ \ \
\text{vesus}\ \ \
\mathbb{H}_{1}:\underline{\beta}_{1}\neq\underline{\beta}_{2}$ (4.5)

The test statistics are computed via expressions (4.6) and (4.7), while
standard regularity and invariance principles holds (e.g, uniform convergence
for the compact space $\underline{\theta}\in\Theta$ such that
$(\underline{\beta}_{1},\underline{\beta}_{2})^{\prime}\subset\underline{\theta}$
and $\Theta\in\mathbb{R}^{q}$).

####  4.2.1 Wald-OLS statistic

$\displaystyle\mathcal{W}^{\text{OLS}}_{T}(\pi)$
$\displaystyle=\frac{1}{\hat{\sigma}_{u}^{2}}\left(\hat{\underline{\beta}}_{1}-\hat{\underline{\beta}}_{2}\right)^{\prime}\left[\left(X_{1}^{\prime}X_{1}\right)^{-1}+\left(X_{2}^{\prime}X_{2}\right)^{-1}\right]^{-1}\left(\hat{\underline{\beta}}_{1}-\hat{\underline{\beta}}_{2}\right)$
(4.6)

####  4.2.2 Wald-IVX statistic

$\displaystyle\mathcal{W}^{\text{IVX}}_{T}(\pi)$
$\displaystyle=\frac{1}{\hat{\sigma}_{u}^{2}}\left(\hat{\underline{\beta}}^{\text{IVX}}_{1}-\hat{\underline{\beta}}^{\text{IVX}}_{2}\right)^{\prime}\mathcal{Q}_{\mathcal{R}}^{-1}\left(\hat{\underline{\beta}}^{\text{IVX}}_{1}-\hat{\underline{\beta}}^{\text{IVX}}_{2}\right)$
(4.7)

where $\mathcal{Q}_{\mathcal{R}}$ is defined as below

$\displaystyle\mathcal{Q}_{\mathcal{R}}=\left\\{\left(Z_{1}^{\prime}X_{1}\right)^{-1}\left(Z_{1}^{\prime}Z_{1}\right)\left(X_{1}^{\prime}Z_{1}\right)^{-1}+\left(Z_{2}^{\prime}X_{2}\right)^{-1}\left(Z_{2}^{\prime}Z_{2}\right)\left(X_{2}^{\prime}Z_{2}\right)^{-1}\right\\}$
(4.8)

In the case of a known break-point we can use the critical values proposed by
Andrews, (1993) with an appopriate significance size such as $\alpha=5\%$ for
both test statistics. Since we consider an unknown break point, then the sup
functional gives the statistic with the maximum value after estimating a
sequence of statistics over the interval $\pi\in[\pi_{1},1-\pi_{2}]$, and thus
critical values need to be used due to the fact that the limiting distribution
in this case is not a standard $\chi^{2}-$distribution. Furthermore, the data
that we generate from the DGP given by (B.1) and (B.2) only differ over the
values of $c_{1}$ and $c_{2}$ and are the same for different values of the
sample size $T$. Thus, as the sample size increases the degree of persistence
of the endogenous regressors remain the same one (for both predictors).
Moreover, the increase in the sample size aims to reflect the properties of
finite versus large sample asymptotics for the Wald type statistics in testing
for a single unknown break-point in predictive regression models with
persistent regressors. Furthermore, in order to control for the existence of
empirical size distortions when testing for structural break via the Wald-IVX
statistic, we use the bias corrected IVX estimators as proposed by Phillips
and Magdalinos, (2009). The particular bias correction, implies that we
account for an overestimation effect when projecting the instrumental variable
towards the direction of the dependent variable (due to endogeneity).

The bias corrected IVX estimator has the following form

$\displaystyle\tilde{\underline{\beta}}^{\text{IVX}}=\left(Y^{\prime}\widetilde{Z}-T\hat{\Delta}_{T}(h,m)\right)\left(X^{\prime}\widetilde{Z}\right)^{-1}$
(4.9)

where the estimator of $\Delta$ is given by

$\displaystyle\hat{\Delta}_{T}(h,m)=\frac{1}{T}\sum_{h=0}^{m}\left(1-\frac{h}{m+1}\right)\sum_{t=h+1}^{T}\hat{u}_{t}\hat{u}^{\prime}_{t-h}$
(4.10)

The above non-parametric estimator is known as Newey-West type estimator (see,
Newey and West, (1987)) and allows to estimate the bias correction without
imposing additional parametric assumptions. Optimal bandwidth choices can be
of the form $m=\eta T^{1/5}$, where $\eta$, is a positive constant. The bias
correction is applied to both $\tilde{\underline{\beta}_{1}}^{\text{IVX}}$ and
$\tilde{\underline{\beta}_{2}}^{\text{IVX}}$.

###  4.3 Size Comparison

To conduct a size comparison of the test statistics, we examine the empirical
rejection rates of the sup-Wald OLS and sup-Wald IVX tests for detecting
single structural change with persistent predictors, under the null hypothesis
of no structural change, that is,
$\mathbb{H}_{0}:\underline{\beta}_{1}=\underline{\beta}_{2}$ using a
significance level $\alpha=5\%$ and compare using different critical value
approximations. In particular, we repeat the empirical-size experiment using
the asymptotic critical values from (i) Table 4 of Gonzalo and Pitarakis,
(2012), (ii) Table 1 of Andrews, (1993)), (iii) the bootstrap generated
critical values within the MC step. More specifically, to do this, we generate
5,000 datasets from DGP (B.1) and (B.2) for various values of
$\underline{\beta}^{0}$ and compute the frequency of rejecting the null
hypothesis.

$\displaystyle\textbf{Design 1}\ \ \ \ \ \ \ \ \ \underline{\beta}^{0}$
$\displaystyle:=\big{(}\beta_{0}=0.25,\beta_{1}=0.50\big{)}$
$\displaystyle\textbf{Design 2}\ \ \ \ \ \ \ \ \ \underline{\beta}^{0}$
$\displaystyle:=\big{(}\beta_{0}=1,\beta_{1}=0.8,\beta_{2}=1.2\big{)}$

Table 1 (for sup Wald-OLS) and Table 3 and LABEL:Table3 (for sup Wald-IVX),
present the probabilities of rejection of the two Wald-type statistics at the
$5\%$ nominal rate, under the null hypothesis (for Design 1). In particular,
we consider different values for the exponent rate of the IVX parameter, such
as $\delta\in\\{0.75,0.95\\}$ in order to investigate the varying effect of
the degree of persistent of the instrumental variable for detecting structural
change in predictive regressions with persistent regressors, as well as
different localizing coefficient of persistence, $c_{i}\in\\{1,5,10,20\\}$.
Comparing the empirical size results for Table 1 versus Table 3 and
LABEL:Table3, we can see that the sup Wald IVX produces values for the
empirical size quite close to the nominal size for critical value
$c_{\alpha}=13.42$, $\delta=0.95$ and $-0.5\leq\rho\leq 0.5$. The particular
critical value is a closer representation to the $\alpha-$quantile from the
corresponding limiting distribution.

In Table 1 and Table 3 we present the empirical size under the null hypothesis
for the model with a single predictor and no-intercept. We can observe that
size distortions appear for larger values of correlation between the $u_{t}$
and $v_{t}$ and this is more severe for high persistence regressors (i.e., low
values of the coefficient of persistence). Moreover, we also consider the case
of explosive regressors, that is, $c_{i}<0$. Clearly, the empirical size in
this scenario has higher values since we use the critical value of Andrews
(8.85) which is not the corresponding critical value of the asymptotic
distribution of the test statistic for detecting structural break in the
predictive regression model under the assumption of explosive regressors.
Notice also that in the case we have $p>1$ and we use the sup OLS-Wald
statistic for testing for joint predictability and structural break then the
size distortions are much higher than in the case of $p=1$ and this is due to
the nonstandard limiting distribution of the test statistic under the
assumption of nonstationary regressors.

All main conclusions are in line with similar findings in the literature of
predictive regression models, such as larger size distortions appear as the
correlation between u(t) and v(t) increases, and this effect is more apparent
in the case of persistent predictors (i.e., lower values of the persistence
coefficient c). Moreover, we see that the Sup OLS-Wald statistic with the
standard asymptotics (Andrews) is clearly immune to persistence when there is
no correlation between u(t) and v(t). Furthermore, size distortions appear
under the existence of nonzero correlation between the error sequences $u_{t}$
and $v_{t}$.

black

In other words, when Cov $\left(u_{t},v_{t}\right)\neq 0$, then under the
assumption of nonstationary predictors which is captured via the Local to unit
root specification, then the limiting distribution no longer follows the
standard NBB result of Andrews. In fact in this case it depends on the degree
of persistence $c_{i}$.

We observe that even though the simulated asymptotic values of the sup IVX-
Wald test for detecting a single structural break in the predictive regression
model with no model intercept is not quite close to the 8.85 cv that
corresponds to the NBB result of Andrews, we can verify that the asymptotic
result does not depend on the nuisance parameter $c$.

Moreover, from the empirical simulations across different values of $\rho$ we
can see that the simulated asymptotic critical values are quite close when
observing at a specific value of $c$. This is not surprising since the fully
modified covariance estimator incorporated in the construction of the
covariance matrix for the IVX takes into account the dependence structure of
the regressors by applying a common long-run covariance structure. This holds
across different values of $c_{i}$. black

###  4.4 Illustrative Examples

We consider for instance the case of bivariate predictor persistence and
examine the finite-sample performance of the proposed tests. In this case, we
use the $c_{1}=1$ and $c_{2}=5$ considering this way two predictors with
different degree of persistence, however which belongs to the same persistence
class as defined in KMS. In particular, the quantile results we obtain since
are based on finite-sample approximations then they have slightly lower value
of the true asymptotic quantile that corresponds to the asymptotic
distribution of the test.

black

###  4.5 Power Comparison

To conduct a power comparison we compare the rejection rates under the
alternative hypothesis of structural change,
$\mathbb{H}_{1}:\underline{\beta}_{1}\neq\underline{\beta}_{2}$. Specifically,
under the alternative hypothesis, we generate data using the predictive
regression given by expression (3.1) and then find statistical evidence
against the null hypothesis by computing the two test statistics. In
particular, comparing the same value of the localizing coefficient across
different sample sizes, the power function is indeed monotonically increasing.

We consider a sequence of local alternatives of the form
$\widetilde{\underline{\beta}}=\left(\underline{\beta}^{0}+b/T\right)$ and
implement the power function for breaks in both the model intercept and the
coefficients of the predictors. We introduce parameter instability via the
following:

$\displaystyle\textbf{Design 1}\ \ \ \ \ \ \ \ \ \underline{\beta}^{0}$
$\displaystyle:=\big{(}\beta_{0}=1,\beta_{1}=1.6,\beta_{2}=0.6\big{)}$
$\displaystyle\textbf{Design 2}\ \ \ \ \ \ \ \ \ \underline{\beta}^{0}$
$\displaystyle:=\big{(}\beta_{0}=2,\beta_{1}=0.8,\beta_{2}=0.4\big{)}$

Table 4 and 5 presents the probabilities of rejection of the two Wald-type
statistics at the $5\%$ nominal rate, under the alternative hypothesis. The
proposed methodology is recommended in cases in which the practitioner has
some prior information regarding the persistent properties of predictors
included in the predictive regression. Moreover, for a given value of $c_{z}$
(e.g., $c_{z}\equiv 1$), then for a larger exponent rate $(\delta=\delta_{0})$
in the IVX instrument can lead to higher power in exprense of lower values for
the empirical size, while a smaller $\delta_{0}$ can achieve better size
corrections. The number of predictors can be another source of distorted
inferences and thus in high dimensional settings further corrections might be
needed to correct the size and power of the proposed tests.

The Monte Carlo experiments verify that our proposed IV based Wald test
provides a solution to the problem of possible size and power distortions when
testing for parameter instability in predictive regressions with persistent
predictors. Implementation is therefore straightforward via the use of
standard statistical tables. In particular, once the magnitude of the sup-Wald
IVX statistic has been computed in the case of a known break-point it then
suffices to obtain the relevant critical values from Table 1 in Andrews,
(1993) (see e.g., Pitarakis, (2008)).

######  Remark 6.

Note that simulating critical values as the results given by Table 1 of
Andrews, (1993), can be done with other numerical approximation methods.
Further studies among others include the papers of Estrella, (2003) and
Anatolyev and Kosenok, (2012). Moreover, Hansen, (1997) proposes a methodology
to obtain simulated p-values for stability tests in linear regression models,
which can be used for the empirical size of the Wald-OLS statistic in our
framework.

Table 1: Empirical size with nominal level $\alpha=5\%$. sup-Wald OLS
---
$\mathbf{c=1}$
T | -0.9 | -0.7 | -0.5 | -0.3 | 0 | 0.3 | 0.5 | 0.7 | 0.9
100 | 0.0804 | 0.0736 | 0.0704 | 0.0672 | 0.0556 | 0.0540 | 0.0512 | 0.0664 | 0.0700
250 | 0.0968 | 0.0812 | 0.0700 | 0.0628 | 0.0564 | 0.0608 | 0.0696 | 0.0764 | 0.0904
500 | 0.0924 | 0.0812 | 0.0720 | 0.0644 | 0.0608 | 0.0668 | 0.0744 | 0.0916 | 0.1044
1000 | 0.1060 | 0.0824 | 0.0788 | 0.0716 | 0.0684 | 0.0668 | 0.0760 | 0.0860 | 0.1032
$\mathbf{c=5}$
T | -0.9 | -0.7 | -0.5 | -0.3 | 0 | 0.3 | 0.5 | 0.7 | 0.9
100 | 0.0588 | 0.0564 | 0.0592 | 0.0548 | 0.0492 | 0.0480 | 0.0500 | 0.0552 | 0.0560
250 | 0.0740 | 0.0608 | 0.0596 | 0.0548 | 0.0516 | 0.0504 | 0.0568 | 0.0692 | 0.0744
500 | 0.0860 | 0.0784 | 0.0696 | 0.0648 | 0.0564 | 0.0572 | 0.0632 | 0.0680 | 0.0848
1000 | 0.0908 | 0.0816 | 0.0732 | 0.0708 | 0.0636 | 0.0596 | 0.0620 | 0.0724 | 0.0864
$\mathbf{c=10}$
T | -0.9 | -0.7 | -0.5 | -0.3 | 0 | 0.3 | 0.5 | 0.7 | 0.9
100 | 0.0448 | 0.0524 | 0.0508 | 0.0488 | 0.0444 | 0.0448 | 0.0480 | 0.0516 | 0.0492
250 | 0.0572 | 0.0540 | 0.0528 | 0.0488 | 0.0472 | 0.0472 | 0.0548 | 0.0608 | 0.0604
500 | 0.0788 | 0.0768 | 0.0676 | 0.0604 | 0.0544 | 0.0592 | 0.0592 | 0.0648 | 0.0736
1000 | 0.0796 | 0.0768 | 0.0704 | 0.0620 | 0.0568 | 0.0528 | 0.0552 | 0.0652 | 0.0704
$\mathbf{c=20}$
T | -0.9 | -0.7 | -0.5 | -0.3 | 0 | 0.3 | 0.5 | 0.7 | 0.9
100 | 0.0380 | 0.0416 | 0.0440 | 0.0432 | 0.0476 | 0.0416 | 0.0440 | 0.0440 | 0.0388
250 | 0.0560 | 0.0476 | 0.0480 | 0.0408 | 0.0436 | 0.0460 | 0.0496 | 0.0480 | 0.0484
500 | 0.0620 | 0.0592 | 0.0600 | 0.0560 | 0.0512 | 0.0504 | 0.0564 | 0.0596 | 0.0648
1000 | 0.0676 | 0.0656 | 0.0608 | 0.0548 | 0.0496 | 0.0520 | 0.0548 | 0.0576 | 0.0572

Table 1 presents finite-sample empirical sizes for the sup Wald-OLS test, with
nominal level $\alpha=5\%$ for $B=5,000$ replications. The predictive
regression model under the null hypothesis, $H_{0}:\beta_{1}=\beta_{2}$, is
given by, $y_{t}=0.25x_{t-1}+u_{t}$, $x_{t}=(1-\frac{c}{T})x_{t-1}+v_{t}$,
with $\Sigma_{ee}=\begin{bmatrix}1&\rho\\\ \rho&1\end{bmatrix}$.

Table 2: Empirical size with nominal level $\alpha=5\%$. sup-Wald OLS
---
$\mathbf{c=-1}$
T | -0.9 | -0.7 | -0.5 | -0.3 | 0 | 0.3 | 0.5 | 0.7 | 0.9
100 | 0.0692 | 0.0636 | 0.0656 | 0.0600 | 0.0584 | 0.0604 | 0.0584 | 0.0616 | 0.0616
250 | 0.0724 | 0.0616 | 0.0596 | 0.0660 | 0.0584 | 0.0636 | 0.0640 | 0.0668 | 0.0748
500 | 0.0804 | 0.0680 | 0.0664 | 0.0604 | 0.0612 | 0.0628 | 0.0720 | 0.0740 | 0.0868
1000 | 0.0728 | 0.0636 | 0.0700 | 0.0680 | 0.0636 | 0.0644 | 0.0680 | 0.0764 | 0.0768
$\mathbf{c=-5}$
T | -0.9 | -0.7 | -0.5 | -0.3 | 0 | 0.3 | 0.5 | 0.7 | 0.9
100 | 0.0660 | 0.0672 | 0.0732 | 0.0740 | 0.0664 | 0.0732 | 0.0732 | 0.0696 | 0.0656
250 | 0.0700 | 0.0756 | 0.0804 | 0.0832 | 0.0836 | 0.0820 | 0.0812 | 0.0820 | 0.0816
500 | 0.0692 | 0.0660 | 0.0752 | 0.0784 | 0.0792 | 0.0856 | 0.0836 | 0.0772 | 0.0764
1000 | 0.0804 | 0.0880 | 0.0916 | 0.0896 | 0.0864 | 0.0828 | 0.0864 | 0.0748 | 0.0708
$\mathbf{c=-10}$
T | -0.9 | -0.7 | -0.5 | -0.3 | 0 | 0.3 | 0.5 | 0.7 | 0.9
100 | 0.1124 | 0.1084 | 0.1060 | 0.1072 | 0.1020 | 0.1056 | 0.1104 | 0.1176 | 0.1160
250 | 0.1264 | 0.1396 | 0.1384 | 0.1372 | 0.1244 | 0.1244 | 0.1276 | 0.1304 | 0.1368
500 | 0.1220 | 0.1224 | 0.1260 | 0.1260 | 0.1240 | 0.1280 | 0.1288 | 0.1376 | 0.1432
1000 | 0.1528 | 0.1468 | 0.1424 | 0.1340 | 0.1308 | 0.1272 | 0.1316 | 0.1384 | 0.1400
$\mathbf{c=-20}$
T | -0.9 | -0.7 | -0.5 | -0.3 | 0 | 0.3 | 0.5 | 0.7 | 0.9
100 | 0.1576 | 0.1500 | 0.1464 | 0.1532 | 0.1564 | 0.1532 | 0.1584 | 0.1632 | 0.1704
250 | 0.2104 | 0.2036 | 0.1988 | 0.2004 | 0.1908 | 0.1976 | 0.1956 | 0.2016 | 0.1928
500 | 0.2144 | 0.2144 | 0.2040 | 0.2024 | 0.2040 | 0.2176 | 0.2200 | 0.2224 | 0.2460
1000 | 0.2520 | 0.2552 | 0.2368 | 0.2376 | 0.2284 | 0.2320 | 0.2336 | 0.2360 | 0.2528

Table 2 presents finite-sample empirical sizes for the sup Wald-OLS test, with
nominal level $\alpha=5\%$ for $B=5,000$ replications. The predictive
regression model under the null hypothesis, $H_{0}:\beta_{1}=\beta_{2}$, is
given by, $y_{t}=0.25x_{t-1}+u_{t}$, $x_{t}=(1-\frac{c}{T})x_{t-1}+v_{t}$,
with $\Sigma_{ee}=\begin{bmatrix}1&\rho\\\ \rho&1\end{bmatrix}$.

red OLD TABLE (to update with the ones after bootstrapping) black

Table 3: Empirical size with nominal level $\alpha=5\%$. sup Wald-IVX
statistic
---
c = 1 ($c_{z}=1$, $\delta=0.75$, $c_{\alpha}=13.42$)
T | -0.9 | -0.7 | -0.5 | -0.3 | 0 | 0.3 | 0.5 | 0.7 | 0.9
100 | 0.1672 | 0.1070 | 0.0718 | 0.0480 | 0.0368 | 0.0496 | 0.0710 | 0.1106 | 0.1726
250 | 0.1336 | 0.0878 | 0.0486 | 0.0324 | 0.0232 | 0.0322 | 0.0506 | 0.0838 | 0.1354
500 | 0.1316 | 0.0808 | 0.0504 | 0.0306 | 0.0206 | 0.0282 | 0.0462 | 0.0772 | 0.1226
1000 | 0.1076 | 0.0644 | 0.0412 | 0.0264 | 0.0200 | 0.0260 | 0.0398 | 0.0668 | 0.1044
c = 5 ($c_{z}=1$, $\delta=0.75$, $c_{\alpha}=13.42$)
T | -0.9 | -0.7 | -0.5 | -0.3 | 0 | 0.3 | 0.5 | 0.7 | 0.9
100 | 0.1134 | 0.0804 | 0.0580 | 0.0426 | 0.0350 | 0.0446 | 0.0596 | 0.0850 | 0.1210
250 | 0.0998 | 0.0662 | 0.0456 | 0.0348 | 0.0248 | 0.0326 | 0.0452 | 0.0666 | 0.0986
500 | 0.0944 | 0.0592 | 0.0410 | 0.0250 | 0.0210 | 0.0282 | 0.0448 | 0.0628 | 0.0916
1000 | 0.0752 | 0.0488 | 0.0318 | 0.0236 | 0.0190 | 0.0250 | 0.0364 | 0.0522 | 0.0768
|  |  |  |  |  |  |  |  |
c = 1 ($c_{z}=1$, $\delta=0.75$, $c_{\alpha}=12$)
T | -0.9 | -0.7 | -0.5 | -0.3 | 0 | 0.3 | 0.5 | 0.7 | 0.9
100 | 0.2386 | 0.1610 | 0.1108 | 0.0768 | 0.0632 | 0.0814 | 0.1138 | 0.1612 | 0.2482
250 | 0.2004 | 0.1336 | 0.0816 | 0.0538 | 0.0390 | 0.0558 | 0.0822 | 0.1338 | 0.2042
500 | 0.1938 | 0.1280 | 0.0828 | 0.0538 | 0.0360 | 0.0536 | 0.0792 | 0.1232 | 0.1900
1000 | 0.1644 | 0.1048 | 0.0672 | 0.0448 | 0.0356 | 0.0454 | 0.0686 | 0.1048 | 0.1664
c = 5 ($c_{z}=1$, $\delta=0.75$, $c_{\alpha}=12$)
T | -0.9 | -0.7 | -0.5 | -0.3 | 0 | 0.3 | 0.5 | 0.7 | 0.9
100 | 0.1722 | 0.1234 | 0.0936 | 0.0704 | 0.0568 | 0.0752 | 0.0966 | 0.1332 | 0.1818
250 | 0.1588 | 0.1094 | 0.0714 | 0.0548 | 0.0458 | 0.0544 | 0.0766 | 0.1080 | 0.1548
500 | 0.1476 | 0.0984 | 0.0716 | 0.0506 | 0.0366 | 0.0494 | 0.0708 | 0.1008 | 0.1422
1000 | 0.1190 | 0.0828 | 0.0578 | 0.0428 | 0.0358 | 0.0456 | 0.0630 | 0.0898 | 0.1276

Table 3 presents finite-sample sizes for the sup Wald-IVX test, with nominal
size $\alpha=5\%$. The predictive regression model under the null hypothesis
is given by $y_{t}=0.25+0.5x_{t-1}+u_{t}$,
$x_{t}=(1-\frac{c_{1}}{T})x_{t-1}+v_{t}$, with
$\Sigma_{ee}=\begin{bmatrix}0.25&\sigma_{uv}\\\
\sigma_{uv}&0.75\end{bmatrix}$,
$\rho=\displaystyle\frac{\sigma_{uv}}{\sigma_{u}\sigma_{v}}$, as given above.
Furthermore, for the IVX estimation step, we use an IVX persistence parameter
$\delta\in\left\\{0.75,0.95\right\\}$ and the localizing coefficient is set to
$c_{z}=1$. The number of replications is $B=5,000$.

##  5 Empirical Application

In this section we present an empirical application aiming to shed light on
the literature of stock return predictability. Identifying periods of
predictability has important implications in various aspects of finance.
Related modern reviews of these aspects are presented by Kostakis et al.,
(2018) and Chinco et al., (2019), among others. However, despite the extensive
research of the field, the findings are still rather mixed (Kasparis et al.,,
2015) (see, Welch and Goyal, (2008) for a full discussion). For instance,
aspects such as the chosen sample period or the selected predictors can give
different conclusions. Additionally, parameter instability due to certain
economic events can also affect the reliability of predictability tests.
Therefore, it is of paramount importance to develop robust testing
methodologies for inferring predictability under conditions such as parameter
instability or the presence of nonstationary regressors. Using the predictive
regression model studied in this paper272727Alternative model specifications
can be considered; for instance a model which considers expected returns in
relation to macroeconomic conditions and forecasting uncertainty. A first move
towards this direction is presented by Atanasov et al., (2020) who examine
consumption fluctuations and expected returns with respect to the
predictability literature. More specifically, the authors indeed find
statistical evidence of predictability at the one-quarter horizon using the
IVX testing approach of KMS. our primary focus is to examine the robustness of
the proposed tests.

###  5.1 Data Description

We focus on examining the presence of predictability for monthly US stock
market excess returns over the period 1990-2019 using the set of variables
considered in Welch and Goyal, (2008)282828The dataset can be retrieved from
Amit Goyal’s website at http://www.hec.unil.ch/agoyal/. Detailed descriptions
of variables can be found in the Online Appendix of Welch and Goyal, (2008).
which capture economic and financial conditions for the US economy.

##### Predictant

The dependent variable is the monthly equity premium (excess return) of the US
stock market based on the S$\&$P500 index. We construct the excess return as
in Kasparis et al., (2015), that is, the difference between the total rate of
return and the risk-free rate for the same sample period. As a proxy of the US
stock market return, we use the value-weighted S$\&$P500 total stock market
return including dividends. The risk-free rate is the 3-month T-bill rate
obtained from the database of FRED292929Time series of macroeconomic
variables, such as the US inflation rate and the T-bill rate can be found at
https://fred.stlouisfed.org/. Notice also that the proxy of equity premium and
the other financial variables we consider in this paper, are commonly used in
the predictability literature, see Gonzalo and Pitarakis, (2012), Kasparis et
al., (2015),Kostakis et al., (2015) and Kostakis et al., (2018)..

##### Predictors

The predictor variables we consider include: dividend-payout ratio (d/e),
long-term yield (lty), dividend yield spread (dy), dividend-price-ratio (d/p),
T-bill rate (tbl), earnings-price-ratio (e/p), book-to-market ratio (b/m),
default yield spread (dfs), net equity expansion (ntis), term spread (tms) and
inflation rate (inf).

###  5.2 Predictability Tests

To begin with, we apply the simple predictability test on the full sample for
the period 1946-2019 using as predictant the S$\&$P500 Equity Premium.
However, in this paper we take a slightly different approach than the
literature. In particular, we focus on the subsample spanning the period
1990Q1 to 2019Q4, and consider monthly sampling frequency. Our first goal is
to examine the stock return predictability of this subsample, that is, to
identify the financial variables which are individually statistical
significant as well as to identify for evidence of joint statistical
significance. Furthermore, our second goal is to repeat the same exercise
using the proposed joint test of predictability and parameter instability and
compare the results we obtain. The chosen subsample includes the period of the
2008 financial crisis, so it is natural to assume that certain predictors
might exhibit structural break around that economic event. Therefore, this is
a suitable sample to assess the statistical validity of the proposed
methodology for the case of a single structural break. Certain limitations of
our approach are on sight, however these do not invalidate our findings. In
particular, we do not consider the existence of multiple structural breaks
neither we consider sample splitting techniques which can affect the power of
the tests especially when using an out-of-sample forecasting scheme.

Firstly, summarizing our findings is Table C1 which presents predictability
tests based on both the classical least squares estimator and the IVX
estimator. Notice that for these set of tests we consider the regressors on
their original form in order to preserve the degree of persistence and have
comparability between the two estimators under examination. Table C2 presents
structural break tests for the regressors. In this case the traditional
structural break tests are implemented under the assumption of stationary time
series, by taking the first difference of the regressors before fitting the
AR(1) models.

Secondly, we examine the short-horizon predictability via the proposed joint
predictability and structural break Wald statistics.

##  6 Conclusion

In this paper, we have extensively examined the asymptotic theory of tests for
Joint predictability and parameter instability under the assumption of
nonstationary regressors. We compare our results with previous seminal work in
the fields of both structural break testing in linear regressions as well as
predictability testing in predictive regression models. We find some
interesting results not previously presented in the literature. Firstly, using
the OLS estimator for the parameters of the predictive regression model we
show that the limiting distribution of the Wald statistic for testing for a
single structural break has a nonstandard limiting distribution which depends
on the unknown coefficient of persistence. Secondly, by employing the IVX
estimator proposed in the literature as a robust estimator which filters out
the abstract degree of persistence in regressors, we have proved that the
limiting distribution of the Joint tests takes different forms which weakly
converges to a functional of a Brownian bridge in some instances while
converges to a nonstandard limiting distribution which depends on the
coefficient of persistence when regressors are assume to be highly persistent.

Conducting inference, such as structural break testing, on the regression
coefficient of the predictive regression model with multiple highly persistent
regressors can lead to a nonstandard limiting distribution. The proposed
testing methodology ensures that the limiting distribution of the structural
break tests is free of any nuisance parameters, such as the unknown localizing
coefficient of persistence. Moreover, we consider "pure" structural change as
it is defined by Andrews, (1993), in the sense that the entire parameter
vector is subject to structural change under the alternative hypothesis. The
asymptotic distribution of the IV based Wald test is found to be given by the
supremum of the normalized squared Brownian Bridge (NBB). Thus, the exact
limiting distribution of the IV based Wald test allows to determine critical
values similar to the case of the linear regression model, without further
simulations and additional computational cost. This holds in the case of
mildly integrated or integrated regressors, while in the case of nonstationary
regressors further investigation is needed to determine exact critical values
since the limiting distribution has a dependence on the nuisance coefficient
of persistence.

The developed asymptotic theory for the sup-Wald IVX tests under various
degrees of persistence we consider in this paper, indicate that the asymptotic
behaviour of the tests when the supremum functional is included, is different
from the corresponding limit theory in the case of only linear restrictions to
the parameters of the predictive regression. Nevertheless, the robustness of
the IVX instrumentation provides a way to determine an analytic form of the
asymptotic distribution for different levels of persistence, a feature often
seen in time series data. This feature appears in various empirical finance
applications in which the available information set for current or future
economic conditions with many times persistent properties and existence of
parameter instability.

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##  Appendix A Asymptotic Theory

In this Appendix we present the main mathematical derivations and proofs
related to the results reported in the main body. We begin by summarizing via
Lemma 1 below the limit theory results which can be found in Phillips and
Magdalinos, (2009) and Kostakis et al., (2015). We introduce the shorthand
notation $\alpha\wedge\beta\equiv\text{min}(\alpha,\beta)$ to denote the
minimum operator, employed for the stochastic dominance of the convergence
rates.

######  Lemma 1.

Let
$\mathbb{V}_{xz}:=\displaystyle\int_{0}^{\infty}e^{rC}\mathbf{V}_{xx}e^{rC_{z}}dr$,
where
$\mathbf{V}_{xx}:=\displaystyle\int_{0}^{\infty}e^{sC}\mathbf{\Omega}_{xx}e^{sC}ds$,
and $\mathbf{\Omega}_{xx}$ is the long-run covariance of $u_{t}$. Then, under
the null hypothesis of no structural break in the predictive regression model,
the following asymptotic results hold:

  1. (i)

the sample covariance satisfies that

$\displaystyle\frac{1}{T^{\frac{1+\gamma_{x}\wedge\delta_{z}}{2}}}\sum_{t=1}^{\lfloor
T\pi\rfloor}\widetilde{z}_{t-1}\big{(}u_{t}-\bar{u}_{T}\big{)}\Rightarrow
U\left(\pi\right)$ (A.1)

where $U\left(.\right)$ is a Brownian motion with variance
$\sigma^{2}_{u}\widetilde{\mathbf{V}}$, where $\widetilde{\mathbf{V}}$ is
defined as

$\widetilde{\mathbf{V}}=\begin{cases}\displaystyle\int_{0}^{\infty}e^{rC_{z}}\mathbf{\Omega}_{xx}e^{rC_{z}}dr&,\text{if}\
\gamma_{x}>\delta_{z}\\\ \\\
\displaystyle\int_{0}^{\infty}e^{rC_{z}}\left(\mathbf{C}\mathbb{V}_{xz}+\mathbf{C}_{z}\mathbb{V}_{xz}^{\prime}\mathbf{C}\right)e^{rC_{z}}dr&,\text{if}\
\gamma_{x}=\delta_{z}\\\ \\\
\displaystyle\int_{0}^{\infty}e^{rC}\mathbf{\Omega}_{xx}e^{rC}dr&,\text{if}\
0<\gamma_{x}<\delta_{z}\\\ \\\
\mathbb{E}\left(x_{0,1}x_{0,1}^{\prime}\right)&,\text{if}\
\gamma_{x}=0.\end{cases}$ (A.2)

where
$x_{0,t}=\sum_{j=0}^{\infty}\left(\mathbf{I}_{p}+\mathbf{C}\right)^{j}u_{t-j}$
is the corresponding stationary sequence of the regressor vector $x_{t}$ when
$\gamma_{x}=0$.

the sample second moment satisfies that

$\displaystyle\frac{1}{T^{\big{(}1+\gamma_{x}\wedge\delta_{z}\big{)}}}\sum_{t=1}^{\lfloor
T\pi\rfloor}\widetilde{z}_{t-1}\big{(}x_{t-1}-\bar{x}_{T-1}\big{)}^{\prime}\Rightarrow\mathbf{\Psi}\left(\pi\right)$
(A.3)

where $\mathbf{\Psi}\left(\pi\right)$ has a different asymptotic convergence
result as below, depending on the exponent rates $\gamma_{x}$ and $\delta_{z}$
of the original regressor and instrumental regressor respectively.

$\mathbf{\Psi}\left(\pi\right)=\begin{cases}\displaystyle-\mathbf{C}_{z}^{-1}\left(\pi\mathbf{\Omega}_{xx}+\int_{0}^{\pi}\underline{\mathbf{B}}dB^{\prime}\right)&,\text{if}\
\gamma_{x}>1\\\ \\\
\displaystyle-\mathbf{C}_{z}^{-1}\left(\pi\mathbf{\Omega}_{xx}+\int_{0}^{\pi}\underline{\mathbf{J}}_{C}dJ_{C}^{\prime}\right)&,\text{if}\
\gamma_{x}=1\\\ \\\
\displaystyle-\pi\mathbf{C}_{z}^{-1}\bigg{(}\mathbf{\Omega}_{xx}+\int_{0}^{\infty}e^{rC}\mathbf{\Omega}_{xx}e^{rC}dr\mathbf{C}\bigg{)}&,\text{if}\
\delta_{x}<\gamma_{x}<1\\\ \\\
\displaystyle-\pi\mathbf{C}\mathbb{V}_{xz}&,\text{if}\
\gamma_{x}=\delta_{x}\\\ \\\
\displaystyle\pi\int_{0}^{\infty}e^{rC}\mathbf{\Omega}_{xx}e^{rC}dr&,\text{if}\
0<\gamma_{x}<\delta_{x}\\\ \\\
\displaystyle\pi\mathbb{E}\left(x_{0,1}x_{0,1}^{\prime}\right)&,\text{if}\
\gamma_{x}=0.\end{cases}$ (A.4)

where $B(.)$ is a $p-$dimensional standard Brownian motion,
$J_{C}(\pi)=\int_{0}^{\pi}e^{C(\pi-s)}dB(\pi)$ is an Ornstein-Uhkenbeck (OU)
process and we denote with
$\underline{J}_{C}(\pi)=J_{C}(\pi)-\int_{0}^{1}J_{C}(s)ds$ and
$\underline{B}(\pi)=B(\pi)-\int_{0}^{1}B(s)ds$ the demeaned processes of
$J(\pi)$ and $B(\pi)$ respectively.

The weakly joint convergence result applies and the asymptotic terms given by
expressions in (i) and (ii) are stochastically independent.

Notice that for summarizing the above results we used that

$\displaystyle\frac{1}{T^{1+\delta_{z}}}\sum_{t=1}^{T}z_{t-1}z_{t-1}^{\prime}\overset{\text{plim}}{\to}\mathbf{V}_{zz}:=\int_{0}^{\infty}e^{rC_{z}}\mathbf{\Omega}_{xx}e^{rC_{z}}dr$
(A.5)

Moreover, we have the weakly convergence result from Phillips and Magdalinos,
(2009):

$\displaystyle\frac{1}{T^{\frac{1+\delta_{z}}{2}}}\sum_{t=1}^{T}\left(z_{t-1}\otimes
u_{t}\right)\Rightarrow\mathcal{N}\big{(}0,\mathbf{V}_{zz}\otimes\mathbf{\Sigma}_{uu}\big{)}$
(A.6)

Expression (A.6) proves a mixed Gaussian limiting distribution. This, shows
that the limit distribution of
$T^{-(1+\delta_{z})/2}\sum_{t=1}^{T}\left(z_{t-1}\otimes u_{t}\right)$ is
Gaussian with mean zero and covariance matrix equal to the probability limit
of $T^{-(1+\delta)/2}\sum_{t=1}^{T}\left(z_{t-1}\otimes u_{t}\right)$, which
is equal to $\mathbf{V}_{zz}\otimes\mathbf{\Sigma}_{uu}$, where
$\mathbf{V}_{zz}:=\int_{0}^{\infty}e^{rC_{z}}\mathbf{\Omega}_{xx}e^{rC_{z}}dr$.
Specifically, the above Mixed Gaussianity convergence, is a powerful result
within the IVX framework and ensures the robustness of the methodology and the
estimation procedure. The dependence of the covariance matrix on the degree of
persistence of the IVX instrumentation methodology, induces exactly the Mixed
Gaussianity. Similarly, the limit distribution of
$T^{-(1+\delta_{z})/2}\sum_{t=1}^{T}\big{(}x_{t-1}\otimes
u_{t}\big{)}\Rightarrow\mathcal{N}\big{(}0,\mathbf{V}_{xx}\otimes\mathbf{\Sigma}_{uu}\big{)}$,
where $\mathbf{V}_{xx}:=\int_{0}^{\infty}e^{rC}\mathbf{\Omega}_{xx}e^{rC}dr$,
is proved in Lemma 3.3 of PM.

Proof of Theorem 1.

###### Proof.

We denote with $\widetilde{x}_{t}=\left(1,x_{t}^{\prime}\right)^{\prime}$ and
with $\theta_{j}=(\alpha_{j},\beta_{j})^{\prime}$ for $j=1,2$ the parameter
vector which is obtained via the OLS estimator. Then, under the null
hypothesis of no structural break, $\mathbb{H}_{0}:\theta_{1}=\theta_{2}$,
against $\mathbb{H}_{1}:\theta_{1}\neq\theta_{2}$, we obtain the following
expressions

$\displaystyle\left(\widehat{\theta}_{1}-\theta^{0}\right)$
$\displaystyle=\left(\sum_{t=1}^{T}\widetilde{x}_{t-1}\widetilde{x}_{t-1}^{\prime}I_{1t}\right)^{-1}\left(\sum_{t=1}^{T}\widetilde{x}_{t-1}u_{t}I_{1t}\right)$
$\displaystyle\left(\widehat{\theta}_{2}-\theta^{0}\right)$
$\displaystyle=\left(\sum_{t=1}^{T}\widetilde{x}_{t-1}\widetilde{x}_{t-1}^{\prime}I_{2t}\right)^{-1}\left(\sum_{t=1}^{T}\widetilde{x}_{t-1}u_{t}I_{2t}\right)$

with $I_{1t}$ and $I_{2t}$ the dummy time variables and
$\theta^{0}=(\alpha_{0},\beta_{0})^{\prime}$, the population value of the
parameter vector $\theta$. Therefore, we have that

$\displaystyle T\left(\widehat{\theta}_{1}-\theta^{0}\right)$
$\displaystyle=\left(\frac{1}{T^{2}}\sum_{t=1}^{\lfloor
T\pi\rfloor}\widetilde{x}_{t-1}\widetilde{x}_{t-1}^{\prime}\right)^{-1}\left(\frac{1}{T}\sum_{t=1}^{\lfloor
T\pi\rfloor}\widetilde{x}^{\prime}_{t-1}u_{t}\right)$ $\displaystyle
T\left(\widehat{\theta}_{2}-\theta^{0}\right)$
$\displaystyle=\left(\frac{1}{T^{2}}\sum_{t=\lfloor
T\pi\rfloor+1}^{T}\widetilde{x}_{t-1}\widetilde{x}_{t-1}^{\prime}\right)^{-1}\left(\frac{1}{T}\sum_{t=1}^{\lfloor
T\pi\rfloor}\widetilde{x}^{\prime}_{t-1}u_{t}\right)$

Then, the weakly convergence result for the estimator of $\beta_{1}$ follows

$\displaystyle T\left(\widehat{\theta}_{1}-\theta^{0}\right)$
$\displaystyle\Rightarrow\left(\int_{0}^{\pi}\widetilde{K}_{c}(r)\widetilde{K}^{\prime}_{c}(r)dr\right)^{-1}\left(\int_{0}^{\pi}\widetilde{K}_{c}(r)dB_{u}\right)$
(A.7)

Similarly, for the estimator of $\beta_{2}$ we have the following weakly
convergence result

$\displaystyle T\left(\widehat{\theta}_{2}-\theta^{0}\right)$
$\displaystyle\Rightarrow\left(\int_{\pi}^{1}\widetilde{K}_{c}(r)\widetilde{K}^{\prime}_{c}(r)dr\right)^{-1}\left(\int_{\pi}^{1}\widetilde{K}_{c}(r)dB_{u}\right)$
(A.8)

In order to simplify the expression of the Wald OLS statistic we denote with

$\displaystyle\widetilde{G}_{c}(\pi):=\int_{0}^{\pi}\widetilde{K}_{c}(r)\widetilde{K}^{\prime}_{c}(r)dr\
\ \text{and}\ \ H_{c}(\pi):=\int_{0}^{\pi}\widetilde{K}_{c}(r)dB_{u}(r)$ (A.9)

which implies that due to the argument $\pi$ in the expressions for
$\widetilde{G}_{c}(\pi)$ and $\widetilde{H}_{c}(\pi)$

$\displaystyle\widetilde{G}_{c}(1):=\int_{0}^{1}\widetilde{K}_{c}(r)\widetilde{K}^{\prime}_{c}(r)dr\
\ \text{and}\ \
\widetilde{H}_{c}(1):=\int_{0}^{1}\widetilde{K}_{c}(r)dB_{u}(r)$ (A.10)

Notice that, for example we can deduce that

$\displaystyle\left(\int_{\pi}^{1}\widetilde{K}_{c}(r)\widetilde{K}^{\prime}_{c}(r)dr\right)$
$\displaystyle=\left(\int_{0}^{1}\widetilde{K}_{c}(r)\widetilde{K}^{\prime}_{c}(r)dr\right)-\left(\int_{0}^{\pi}\widetilde{K}_{c}(r)\widetilde{K}^{\prime}_{c}(r)dr\right):=\widetilde{\textbf{G}}_{c}(1)-\widetilde{\textbf{G}}_{c}(\pi)$

Thus, the statistical distance component of the sup Wald-OLS statistic is
given by

$\displaystyle
T\left(\widehat{\theta}_{1}-\widehat{\theta}_{2}\right)=\left\\{\widetilde{\textbf{G}}_{c}(\pi)^{-1}\widetilde{H}_{c}(\pi)-\big{[}\widetilde{\textbf{G}}_{c}(1)-\widetilde{\textbf{G}}_{c}(\pi)\big{]}^{-1}\big{[}\widetilde{H}_{c}(1)-\widetilde{H}_{c}(\pi)\big{]}\right\\}$
(A.11)

Denote with $X=[x_{t}I_{1t}\ \ x_{t}I_{2t}]\equiv[X_{1}\ X_{2}]$ then the
convergence of the covariance matrix

$\displaystyle\widetilde{M}_{c}(\pi):=\left[\mathcal{R}\left(X^{\prime}X\right)^{-1}\mathcal{R}^{\prime}\right]$
$\displaystyle=\left[\left(\frac{X_{1}^{\prime}X_{1}}{T^{2}}\right)^{-1}+\left(\frac{X_{2}^{\prime}X_{2}}{T^{2}}\right)^{-1}\right]$
$\displaystyle\Rightarrow\left\\{\left(\int_{0}^{\pi}\
\widetilde{K}_{c}(r)\widetilde{K}_{c}^{\prime}(r)dr\right)^{-1}+\left(\int_{\pi}^{1}\widetilde{K}_{c}(r)\widetilde{K}_{c}^{\prime}(r)dr\right)^{-1}\right\\}$
$\displaystyle\equiv\left\\{\widetilde{\textbf{G}}_{c}(\pi)^{-1}+\left[\widetilde{\textbf{G}}_{c}(1)-\widetilde{\textbf{G}}_{c}(\pi)\right]^{-1}\right\\}$

Recall that the expression for the Wald statistic is as below

$\displaystyle\mathcal{W}_{T}^{OLS}(\pi)$
$\displaystyle=\frac{1}{\widehat{\sigma}_{u}^{2}}\left(\widehat{\theta}_{1}-\widehat{\theta}_{2}\right)^{\prime}\left[\mathcal{R}\left(X^{\prime}X\right)^{-1}\mathcal{R}^{\prime}\right]^{-1}\left(\widehat{\theta}_{1}-\widehat{\theta}_{2}\right)$
(A.12)

Therefore, we can now derive the limiting distribution of the sup OLS-Wald
statistic in the case of the multiple predictive regression with persistent
predictors.

$\displaystyle\widetilde{\mathcal{W}}^{OLS}(\pi)$
$\displaystyle\Rightarrow\underset{\pi\in[\pi_{1},\pi_{2}]}{\text{ sup
}}\left\\{\widetilde{\textbf{G}}_{c}(\pi)^{-1}H_{c}(\pi)-\big{[}\widetilde{\textbf{G}}_{c}(1)-\widetilde{\textbf{G}}_{c}(\pi)\big{]}^{-1}\big{[}\widetilde{H}_{c}(1)-\widetilde{H}_{c}(\pi)\big{]}\right\\}^{\prime}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \
\times\left\\{\widetilde{\textbf{G}}_{c}(\pi)^{-1}+\bigg{[}\widetilde{\textbf{G}}_{c}(1)-\widetilde{\textbf{G}}_{c}(\pi)\bigg{]}^{-1}\right\\}^{-1}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \
\times\left\\{\widetilde{\textbf{G}}_{c}(\pi)^{-1}\widetilde{H}_{c}(\pi)-\big{[}\widetilde{\textbf{G}}_{c}(1)-\widetilde{\textbf{G}}_{c}(\pi)\big{]}^{-1}\big{[}\widetilde{H}_{c}(1)-\widetilde{H}_{c}(\pi)\big{]}\right\\}$

By applying the related inverse matrix formula to
$\widetilde{\textbf{M}}_{c}(\pi)^{-1}$ we obtain that

$\displaystyle\widetilde{\textbf{S}}_{c}(\pi)^{-1}$
$\displaystyle\equiv\left\\{\widetilde{\textbf{G}}_{c}(\pi)^{-1}+\bigg{[}\widetilde{\textbf{G}}_{c}(1)-\widetilde{\textbf{G}}_{c}(\pi)\bigg{]}^{-1}\right\\}^{-1}$