*

#  Structural Break Detection in Quantile Predictive Regression Models with
Persistent Covariates††thanks: I would like to thank Jose Olmo, Tassos
Magdalinos and Jean-Yves Pitarakis for guidance, support and continuous
encouragement throughout the PhD programme. Furthermore, this study has been
inspired by the work of Peter C. B. Phillips in time series econometrics. The
author acknowledge the use of the Iridis 5 HPC Facility and associated support
services at the University of Southampton in the completion of this work.
Financial support from the VC PhD scholarship of the University of Southampton
is also gratefully acknowledged. All the usual disclaimers apply. Parts of
this paper are derived from my Ph.D. thesis at the University of Southampton
with title: "Aspects of Estimation and Inference for Predictive Regression
Models".

Christis Katsouris111Ph.D Candidate, Department of Economics, University of
Southampton, Southampton, SO17 1BJ, UK. E-mail Address:
<EMAIL_ADDRESS>Website: sites.google.com/view/christiskatsouris
University of Southampton

JOB MARKET PAPER

First version: November 13, 2021
This version: November 13, 2021

###### Abstract

We propose an econometric environment for structural break detection in
nonstationary quantile predictive regressions. We establish the limit
distributions for a class of Wald and fluctuation type statistics based on
both the ordinary least squares estimator and the endogenous instrumental
regression estimator proposed by Phillips and Magdalinos,  (2009). Although
the asymptotic distribution of these test statistics appears to depend on the
chosen estimator, the IVX based tests are shown to be asymptotically nuisance
parameter-free regardless of the degree of persistence and consistent under
local alternatives. The finite-sample performance of both tests is evaluated
via simulation experiments. An empirical application to house pricing index
returns demonstrates the practicality of the proposed break tests for
regression quantiles of nonstationary time series data.

JEL classification: C12, C22

Keywords: Quantile predictive regression model, persistence, local to unity,
IVX instrument, structural break tests, weak convergence, brownian bridge.

###### Contents

  1. 1 Introduction
    1. 1.1 Literature Review
  2. 2 Econometric Environment and Testing Problem
    1. 2.1 Econometric Model and Assumptions
      1. 2.1.1 Quantile Predictive Regression Model
    2. 2.2 Estimation Methodology
      1. 2.2.1 OLS based estimation
      2. 2.2.2 IVX based estimation
    3. 2.3 Testing Hypotheses
  3. 3 Testing for structural break for a fixed quantile level
    1. 3.1 Preliminary Setting
      1. 3.1.1 OLS based functionals
      2. 3.1.2 IVX based functionals
    2. 3.2 Invariance principles for partial sum processes
    3. 3.3 Test Statistics
      1. 3.3.1 Fluctuation type tests
      2. 3.3.2 Wald type tests
    4. 3.4 Asymptotic Theory
      1. 3.4.1 OLS-Wald test statistic
      2. 3.4.2 IVX-Wald test statistic
  4. 4 Monte Carlo Experiments
    1. 4.1 Experimental Design
    2. 4.2 Implementation
    3. 4.3 Simulation Results
  5. 5 Empirical Application
    1. 5.1 Data Description
    2. 5.2 Data Analysis
  6. 6 Conclusion
  7. 7 APPENDIX
    1. 7.1 Proofs of asymptotic theory results
      1. 7.1.1 Proof of Corollary 2
      2. 7.1.2 Proof of Corollary 3
      3. 7.1.3 Proof of Corollary 4
      4. 7.1.4 Alternative IVZ-QR estimator
      5. 7.1.5 Proof of Lemma 1
      6. 7.1.6 Proof of Proposition 1
      7. 7.1.7 Proof of Proposition 2
      8. 7.1.8 Proof of Theorem 1
      9. 7.1.9 Proof of Corollary 5
    2. 7.2 Asymptotic results on stochastic integrals

Notation and Abbreviations

$x\vee y$ |  $\text{max}\left\\{x,y\right\\}$
---|---
$x\wedge y$ |  $\text{min}\left\\{x,y\right\\}$
$:=$ |  equality by definition
$\equiv$ |  equivalent statement
$\lfloor\ .\ \rfloor$ |  integer part of the argument
$|\ .\ |$ |  absolute value of the argument
$\mathds{1}\big{\\{}.\big{\\}}$ |  indicator function
$\mathbb{E}\left[\ .\ \right]$ |  expectation operator
$\mathbb{P}\left(\ .\ \right)$ |  probability operator
$\text{Var}\left(\ .\ \right)$ |  variance operator
$\mathcal{O}_{\mathbb{P}}\left(\ .\ \right)$ |  Order of convergence
$\overset{\mathcal{D}}{\to}$ |  Convergence in distribution
$\overset{\mathbb{P}}{\to}$ |  Convergence in probability
$\Rightarrow$ |  Weakly convergence argument
$\mathcal{D}\left([0,1]\right)$ |  Skorokhod topology
$\mathcal{SQ}^{ols}_{n}(\lambda,\uptau)$ |  sup-$\mathcal{Q}$ test based on OLS estimator
$\mathcal{SQ}^{ivx}_{n}(\lambda,\uptau)$ |  sup-$\mathcal{Q}$ test based on IVX estimator
$\mathcal{SW}^{ols}_{n}(\lambda,\uptau)$ |  sup-Wald test based on OLS estimator
$\mathcal{SW}^{ivx}_{n}(\lambda,\uptau)$ |  sup-Wald test based on IVX estimator
$\mathcal{W}^{ols}_{n}(\lambda,\uptau)$ |  (self-normalized) sup-Wald test based on OLS estimator
$\mathcal{W}^{ivx}_{n}(\lambda,\uptau)$ |  (self-normalized) sup-Wald test based on IVX estimator
$\mathcal{DQ}^{ols}_{n}(\lambda,\uptau)$ |  double sup-$\mathcal{Q}$ test based on OLS estimator
$\mathcal{DQ}^{ivx}_{n}(\lambda,\uptau)$ |  double sup-$\mathcal{Q}$ test based on IVX estimator
$\mathcal{DW}^{ols}_{n}(\lambda,\uptau)$ |  double sup-Wald test based on OLS estimator
$\mathcal{DW}^{ivx}_{n}(\lambda,\uptau)$ |  double sup-Wald test based on IVX estimator

  * •

Summary: We propose a set of structural break tests for testing the null
hypothesis of no parameter instability in nonstationary quantile predictive
regressions models. More specifically, we derive the asymptotic distribution
of Wald type and fluctuation type statistics based on both the OLS and IVX
estimators. We show that the limiting distribution of both the fluctuation and
Wald test statistics under high persistence weakly convergent into a
nonstandard and nonpivotal limiting distributions. In those cases, the
underline stochastic processes depend on the innovation distribution and
cannot be identified with conventional processes commonly employed in the
structural break literature such as a Brownian motion or Brownian bridge in
the topological space.

  * •

Related Literature: The relevant literature to the econometric environment
studied in the paper is the framework of predictive regression models proposed
by Phillips and Magdalinos,  (2009), Kostakis et al.,  (2015) and Lee,  (2016)
who extend the asymptotic theory to the case of quantile predictive regression
models. Further relevant studies include the framework proposed by Xiao,
(2009) for the quantile cointegrating regression model as well as the paper of
Qu,  (2008) who propose a framework for break testing in regression quantiles.

  * •

Research Objective: Our main research objective is to investigate the
implementation of structural break tests in quantile predictive regressions
under the assumption of nonstationarity, that is, regressors being generated
as near unit root processes. To do this, we study the asymptotic distribution
of the testing procedures and evaluate their statistical performance via Monte
Carlo simulations.

  * •

Research Contributions: To the best of our knowledge the proposed framework in
this paper that unifies the persistence properties of regressors in quantile
predictive regression models and structural break testing is a novel aspect
not previously examined in the literature. Our contributions in the literature
is a statistical framework for testing for structural breaks in the relation
between the regressand and the predictors based on a conditional quantile
functional form.

  * •

Research Findings: The main research findings of our study are summarized. We
derive the limit distributions of structural break tests suitable for quantile
predictive regression models with regressors being either high persistent or
mildly integrated. We focus on fluctuation type statistics as well as Wald
type statistics which are constructed based on either the OLS or IVX
estimators. Our tests can be used as diagnostic tools for break detection when
estimating predictive regressions under nonstationarity based on a conditional
quantile functional form.

##  Introduction

Time series predictability is a research question that sparked the development
of various methodologies for estimation and inference in predictive regression
models. Related studies include Jansson and Moreira,  (2006), Mikusheva,
(2007), Phillips,  (2014), Cai and Wang,  (2014), Kostakis et al.,  (2015),
Kasparis et al.,  (2015), Gonzalo and Pitarakis,  (2012, 2017), Ren et al.,
(2019), Demetrescu et al.,  (2020), Yang et al.,  (2020), Georgiev et al.,
(2021), Andersen and Varneskov,  (2021), Harvey et al.,  (2021) as well as Dou
and Müller,  (2021) among others222The problem of distorted statistical
inference in predictive regression models under nearly integrated predictors
has been also reported in previous studies such as Elliott and Stock,  (1994),
Elliott,  (2011), Campbell and Yogo,  (2006) and references therein.. The
aforementioned frameworks operate under the general assumption of a stable
relation between the predictant and the predictors of the model. However, the
possible presence of parameter instability under these conditions require a
different treatment (e.g., see Pitarakis,  (2017) and Georgiev et al.,
(2018)). Thus, to derive limits for structural break tests suitable for
predictive regressions, necessitate to employ certain regularity conditions
and invariance principles of partial sum processes as in Phillips and
Magdalinos,  (2007, 2009), to obtain stochastic integral approximations.

Most of the current literature focuses on detecting structural break in the
conditional mean of the distribution of $f(y_{t}|\boldsymbol{x}_{t})$ or
$f(y_{t}|\boldsymbol{x}_{t-1})$ where $\boldsymbol{x}_{t}$ is assumed to
follow a near unit root process such as in Cai et al.,  (2015), Georgiev et
al.,  (2018) and Katsouris,  (2021), leaving the conditional quantile
distribution vastly unexplored. Consequently, this paper addresses these
issues by developing a framework suitable for structural break testing for
quantile regressions under nonstationarity which has not seen much attention.
Specifically, we build on related studies such as Lee,  (2016) and Fan and
Lee,  (2019) who propose a framework for estimation and inference in quantile
predictive regressions as well as the study of Qu,  (2008) who focus on break
testing procedures for regression quantiles. More precisely, the latter
approach studies a linear quantile model with stationary covariates, while we
consider the quantile predictive regression model with possibly nonstationary
predictors as in Lee,  (2016). Another related study is presented by Aue et
al.,  (2017) in the context of piecewise quantile autoregressions. Therefore,
proposing tests which bridge the gap between these two approaches is a further
development of the current toolkit while the local-to-unity theory proposed by
the seminal work of Phillips, 1987a ; Phillips, 1987b ; Phillips and Perron,
(1988) can facilitate the asymptotic theory.

Consider the $\uptau-$th conditional quantile of $y_{t}$ which is defined as
following

$\displaystyle\mathsf{Q}_{y}\left(\uptau|x\right)=F^{-1}_{y|x}(\uptau|x):=\mathsf{inf}\big{\\{}\mathsf{s}:F_{y|x}(\mathsf{s}|x)\geq\uptau\big{\\}}$
(1.1)

Specifically, using the conditional quantile function (1.1) (see, Kiefer,
(1967)) to be the underline functional form for the predictive regression
model then the induced econometric environment permits to investigate the
presence of quantile predictability.

As a result, the parameter vector of the predictive regression model is
quantile dependent for some fixed quantile level within a compact set. In
practise, the parameter vector of a quantile regression can be estimated as
the unique solution of the following unconstrained optimization problem (see,
Koenker and Portnoy,  (1987) and Portnoy,  (1991))

$\displaystyle\underset{\boldsymbol{b}\in\mathbb{R}^{p}}{\mathsf{arg\ min}}\
\sum_{t=1}^{n}\uprho_{\uptau}\big{(}y_{t}-\boldsymbol{x}_{t-1}^{\prime}\boldsymbol{b}\big{)}$
(1.2)

where
$\uprho_{\uptau}(\mathsf{u})\equiv\mathsf{u}\left(\uptau-\mathds{1}\left\\{\mathsf{u}<0\right\\}\right)$,
is the check function as defined by Koenker and Bassett,  (1978),
$\boldsymbol{b}$ is some parameter vector and $\boldsymbol{x}_{t-1}$ is the
lagged regressor of the model. Our research interest is on the asymptotic
behaviour of the parameter estimators of $\boldsymbol{b}$ under both the null
hypothesis of no parameter instability as well as under the alternative
hypothesis of structural break at an unknown break-point location within the
full sample.

The structural break literature has various applications for different
modelling conditions, however there are currently limited studies related to
break testing in quantile time series models even under the assumption of
stationarity and ergodicity such as in the seminal paper of Andrews,  (1993)
for linear models. Furthermore, when considering predictive regression models
inference regarding structural break involves deriving nonstandard asymptotic
theory due to the presence of the nuisance coefficient of persistence. More
precisely, testing based on conventional estimation methods (such as OLS-based
tests) has been proved to have size distortions for persistent regressors
(e.g., see Georgiev et al.,  (2018) and Katsouris,  (2021)). Our study is
considered as a unified framework for structural break detection in quantile
predictive regressions (see, Lee,  (2016) and Fan and Lee,  (2019)) which
encompasses regressor properties such as high persistent, mildly integrated or
stationary under certain parameter space restrictions on the persistence
parameters. In particular, the proposed tests can statistically evaluate for
breaks in the model coefficients at both fixed and multiple quantiles of the
underline conditional quantile distribution. We investigate the asymptotic
theory and implementation of both Wald type statistics as in Andrews,  (1993)
as well as fluctuation type statistics as in Qu,  (2008) but within the
setting of nonstationary quantile predictive regressions.

Our contributions are threefold. Firstly, we study the estimation problem of
quantile predictive regressions with multiple predictors assumed to be
generated as either near unit root or mildly integrated processes. We
construct quantile predictability Wald statistics under the null of no
predictability and derive the asymptotic distributions (as in Lee,  (2016)).
Secondly, we propose a structural break testing procedure for quantile
predictive regressions which permits testing for the presence of breaks at the
tails. Thirdly, we examine the statistical performance of these tests for
detecting parameter instability in the coefficients of quantile predictive
regressions with extensive simulation experiments of empirical size.

Under the null hypothesis of no structural break the specific weakly
convergence argument permits to establish convergence to stochastic integral
approximations defined with respect to a two-parameter process; limit results
employed to derive the asymptotic distributions of fluctuation type
statistics. Similarly, for the Wald type statistics we employ the asymptotic
theory developed by Phillips and Magdalinos,  (2009) and Lee,  (2016). Thus,
we introduce invariance principles for partial sum processes of matrix moments
based on these functionals to obtain asymptotic results for the proposed
econometric environment.

Throughout the paper, we assume that all random elements are defined within a
probability space denoted with the triple
$\left(\Omega,\mathcal{F},\mathbb{P}\right)$. All limits are taken as
$n\to\infty$, where $n$ is the sample size. Denote with
$\mathcal{D}\left([0,1]\right)$ to be the set of functions on $[0,1]$ that are
right continuous and have left limits, equipped with the Skorokhod metric.
Then, the symbol $"\Rightarrow"$ is used to denote the weak convergence of the
associated probability measures as $n\to\infty$. The symbol
$\overset{\mathcal{D}}{\to}$ and $\overset{\mathbb{P}}{\to}$ are employed to
denote convergence in distribution and convergence in probability
respectively. Moreover, we denote with $\mathcal{O}_{\mathbb{P}}(.)$ and
$o_{\mathbb{P}}(.)$ the stochastic order of convergence in probability (see,
Billingsley,  (1968)). In terms of vector and matrix notation, for any random
matrix $X$, $\left\lVert X\right\rVert_{p}$ denotes the $L_{p}-$norm, that is,
$\left\lVert X\right\rVert_{p}=\left(\mathbb{E}\left\lVert
X\right\rVert^{p}\right)^{1/p}$ where $p$ is some positive constant. The
proposed test statistics are constructed based on the Euclidean norm.
Moreover, we employ the term stationary regressor meaning a regressor of the
predictive regression model which is generated from the local-to-unity
specification that satisfies $\upgamma_{x}=0$ and $|1+c_{i}|<1\ \forall i$.

The paper is organized as follows. Section 2 discusses the econometric
environment of the quantile predictive regression model along with main
assumptions, the estimation methodology as well as the testing hypotheses of
interest. Section 3 presents the testing procedure and asymptotic theory of
structural break testing for a fixed quantile level. Section 4 investigates
the finite sample performance of the proposed tests via Monte Carlo
experiments. Section 5 illustrates the implementation of of the proposed
testing procedures with an empirical application. Section 6 concludes. Proofs
of main limit results can be found in the Appendix (see Section 7).

###  Literature Review

Estimation and inference with quantile models has been proposed to the
literature with the seminal work of Koenker and Bassett,  (1978) and Koenker
and Bassett,  (1982). In particular, the uniform Bahadur-type representation
established by Koenker and Portnoy,  (1987) as well as the representation of
quantiles proposed by Knight,  (1989) (see also Koul and Saleh,  (1995)) are
commonly used to derive limit distributions for estimators and test
statistics. Several papers in the literature follow these methodologies that
examine related aspects to the proposed estimation environment and testing
procedures.

Firstly, related literature to the problem of structural change for regression
quantiles include the studies of Su and Xiao,  (2008) and Qu,  (2008). Both
frameworks develop diagnostic tools for break detection in quantile regression
models under the assumption of stationary covariates. In the former case the
alternative hypothesis of a single break is formulated with respect to the
magnitude of the break-point. In the latter case, the author also propose a
multiple break point testing procedure. Additionally, Furno,  (2014)
implements these quantile regression based statistics using the methodology
proposed by Chow,  (1960) which implies testing for break at a known location
in the sample.

A different perspective is presented by Hoga,  (2018) who consider detecting
for breaks using tail dependence measures (see also Hoga,  (2017)333The
framework proposed by Hoga,  (2017) considers a set of change point tests for
the tail index of random variables and contributes to the literature of
nonparametric modelling methods.) based on an empirical estimator of extremal
dependence. An extension of the method to an alternative hypothesis with
multiple breaks is also examined. All aforementioned procedures correspond to
structural break tests suitable for quantile models444Notice that the
literature of break tests presented here differs from the literature of slope
heterogeneity in quantile models. Related aspects to inference are Koenker and
Hallock,  (2001), Chernozhukov,  (2005), Portnoy,  (2012) and Escanciano and
Goh,  (2018) which are of independent interest. while in our study we focus on
implementing testing procedures specifically for quantile predictive
regression models in which the time series properties of predictors are
modelled with the nuisance parameter of persistence. In particular, a separate
autoregressive model with a local unit root coefficient matrix is used to
model the unknown degree of persistence which implies that conventional
approaches for deriving large sample theory are no longer valid. However, the
fluctuation based tests implemented by Qu,  (2008) for break detection in
quantile models with stationary covariates provides a suitable econometric
environment for investigating the effect of nonstationarity to the asymptotic
distribution of the tests.

Secondly, asymptotic theory for quantile time series regressions has been
examined by Koenker and Xiao,  (2002, 2004, 2006) in the context of
autoregressive model specifications and unit root testing555A unified
framework for econometric inference with nearly integrated regressors and unit
roots is proposed by the seminal work of Phillips,  (1988); Phillips, 1987b ;
Phillips, 1987a . Further relevant literature includes the study of Stock,
(1994) who discuss aspects related to testing with unit roots and structural
breaks in time series models as well as Chapter 14 in Davidson,  (2000) that
has additional derivations and examples.. Further applications in the time
series literature include testing procedures for threshold effects under the
assumption of a conditional quantile function as in the studies of Galvao et
al.,  (2011, 2014) and the case of nonstationary (nonlinear) quantile
regressions by Xiao,  (2009), Cho et al.,  (2015), Li et al.,  (2016) and
Uematsu,  (2019). Moreover, Kato,  (2009) derives related limit results which
are useful when considering the asymptotic behaviour of quantile estimators
for a wide class of modelling approaches and econometric conditions. All these
studies cover both stationary and nonstationary models however they operate
under the null hypothesis of no parameter instability in model parameters.
Therefore, our proposed testing procedure, is considered to be a novel
contribution to the time series econometrics literature.

Thirdly, the proposed structural break testing methodology is closely related
to the problem of unidentified parameters under the null hypothesis which is
well-known in the econometrics and statistics literature as the Davies problem
(see, Davies,  (1977)). The particular aspect which is relevant to parameter
admissibility has been investigated in problems of estimation and testing such
as in the studies of Hansen,  (1996), Andrews and Ploberger,  (1994),
Pitarakis,  (2004) and Elliott et al.,  (2015). A more recent approach to the
unidentified parameter problem under the null is presented by McCloskey,
(2017)666Specifically, McCloskey,  (2017) proposes a framework for a set of
flexible size-corrected critical values construction methods that lead to
tests with correct asymptotic size and desirable power properties in testing
problems with nuisance parameter under the null hypothesis.. Therefore, to
overcome this challenge, we employ the supremum operator when constructing
Wald type statistics. Furthermore, a nonparametric estimation approach for
quantile regressions such in the study of Qu and Yoon,  (2015), may require
different moment conditions for estimation and inference which is beyond the
scope of this paper.

Lastly, in terms of the model structure of the proposed modelling approach of
the paper, we impose standard econometric assumptions and conditions in the
literature of predictive regressions. More precisely, by imposing a standard
martingale difference condition on the equation innovations $u_{t}$, this
implies an orthogonality condition between the innovations and the model
regressors, such that
$\mathsf{Cov}\big{(}u_{t},\boldsymbol{x}_{t}\big{)}=\mathbb{E}\big{[}\boldsymbol{x}_{t}\mathbb{E}\left(u_{t}|\mathcal{F}_{t-1}\right)\big{]}=0$.
Furthermore, the model structure we follow does not allow for endogeneity even
though the IVX instrumentation method implies the construction of endogenous
instruments. Specifically, as explained by Phillips and Magdalinos,  (2009)
and Kostakis et al.,  (2015), the IVX filtration implies the construction of
instrumental covariates based on information obtained only from the regressors
of the model. According to (Wang and Phillips, , 2012, p. 731) the model
structure would permit for the presence of endogeneity when the equation error
could be serially dependent and cross-correlated under certain moment
restrictions (see also Yang et al.,  (2020)777Specifically, Yang et al.,
(2020) propose a unified IVX-AR Wald statistic that accounts for serial
correlation in the error terms of the linear predictive regression model. The
IVX-AR estimator corrects the size distortions arising from serially
correlated error under the presence of high persistence.). However, the
particular aspect can complicate the derivation of the limit distributions as
additional considerations will be needed, such as to incorporate conditional
heteroscedasticity, which is beyond the scope of our study and we leave as
future research related to the proposed framework.

Based on the aforementioned challenges in the literature, we are motivated to
develop an econometric framework for structural break testing in quantile
predictive regressions with persistent covariates. Overall our research
objectives are concentrated around two main pillars: (i) to introduce a new
set of weak convergence results for certain partial sums of functionals which
are based on either the OLS or the IVX estimators and involve nonstationary
time series. Specifically, these functionals are instrumental for deriving the
asymptotic behaviour of the proposed test statistics; and (ii) to demonstrate
the practicality of a new set of structural break tests for detecting
parameter instability.

##  Econometric Environment and Testing Problem

###  Econometric Model and Assumptions

Consider the (linear) predictive regression model

$\displaystyle y_{t}$
$\displaystyle=\alpha+\boldsymbol{\beta}^{\prime}\boldsymbol{x}_{t-1}+u_{t},\
\ \ \ 1\leq t\leq n,$ (2.1) $\displaystyle\boldsymbol{x}_{t}$
$\displaystyle=\boldsymbol{\Phi}_{n}\boldsymbol{x}_{t-1}+\boldsymbol{v}_{t}$
(2.2)

where $y_{t}\in\mathbb{R}^{n\times 1}$ is a scalar dependent variable and
$\boldsymbol{x}_{t-1}\in\mathbb{R}^{n\times p}$ is a $p-$dimensional vector of
predictors such that $\boldsymbol{x}_{t}$ is generated as a near unit root
process (or local unit root process) with an autocorrelation coefficient
matrix as defined by the studies of Phillips and Magdalinos,  (2009) and
Kostakis et al.,  (2015) expressed as below

$\displaystyle\boldsymbol{\Phi}_{n}=\left(\boldsymbol{I}_{p}+\frac{\boldsymbol{C}_{p}}{n^{\upgamma_{x}}}\right),\
\ \text{for some}\ \upgamma_{x}>0.$ (2.3)

where $n$ is the sample size, and
$\boldsymbol{C}_{p}=\mathsf{diag}\left\\{c_{1},...,c_{p}\right\\}$ is a
$p\times p$ diagonal matrix with the coefficients of persistence $c_{i}$ for
$i=1,...,p$. We consider that the predictors of the model are allowed to
belong only to one of the two degree of persistence as specified below

  * •

Local Unit Root (LUR): $\upgamma_{x}=1$ and $c_{i}\in(-\infty,0),\ \forall\
i=1,...,p$.

  * •

Mildly Integrated (MI): $\upgamma_{x}\in(0,1)$ and $c_{i}\in(-\infty,0),\
\forall\ i=1,...,p$.

Let $\mathcal{F}_{t}$ denote the natural filtration, then for the error term
of the predictive regression we assume that
$\mathbb{E}\left(u_{t}|\mathcal{F}_{t-1}\right)=0$ and
$\mathbb{E}\left(u^{2}_{t}|\mathcal{F}_{t-1}\right)=\sigma^{2}_{uu}$.
Specifically, the innovation structure of the predictive regression model
allows to impose a linear process dependence for $\boldsymbol{v}_{t}$, with a
conditionally homoscedastic martingale difference sequence condition such that

$\displaystyle\boldsymbol{v}_{t}=\sum_{j=1}^{\infty}\boldsymbol{\varphi}_{j}\boldsymbol{\varepsilon}_{t-j},\
\
\boldsymbol{\varepsilon}_{t}\sim\textit{mds}\left(\boldsymbol{0},\boldsymbol{\Sigma}\right),\
$

with necessary conditions for the linear process representation to hold given
as below

$\displaystyle\boldsymbol{\Sigma}>0,\ \ \ \ \
\sum_{j=0}^{\infty}j\left\lVert\boldsymbol{\varphi}_{j}\right\rVert<\infty\ \
\text{such that}\ \
\boldsymbol{\varphi}_{o}(\mathsf{z})=\sum_{j=0}^{\infty}\mathsf{z}^{j}\boldsymbol{\varphi}_{j}.$
(2.4)

Denote with
$\boldsymbol{e}_{t}=\left(u_{t},\boldsymbol{v}_{t}^{\prime}\right)^{\prime}$,
then under regularity conditions, the following invariance principle (FCLT)
holds (see Phillips and Solo,  (1992))

$\displaystyle\frac{1}{\sqrt{n}}\sum_{t=1}^{\lfloor
nr\rfloor}\boldsymbol{e}_{t}:=\frac{1}{\sqrt{n}}\sum_{j=1}^{\lfloor
nr\rfloor}\begin{bmatrix}u_{t}\\\
\boldsymbol{v}_{t}\end{bmatrix}\equiv\begin{bmatrix}B_{un}(r)\\\
\boldsymbol{B}_{vn}(r)\end{bmatrix}\Rightarrow\begin{bmatrix}B_{u}(s)\\\
\boldsymbol{B}_{v}(s)\end{bmatrix}:=\mathcal{BM}\begin{bmatrix}\sigma^{2}_{uu}&\boldsymbol{\sigma}^{\prime}_{uv}\\\
\boldsymbol{\sigma}_{vu}&\boldsymbol{\Sigma}_{vv}\end{bmatrix}_{{\color[rgb]{1,0,0}(p+1)\times(p+1)}}$
(2.5)

where $\boldsymbol{\Sigma}_{vv}\in\mathbb{R}^{p\times p}$ is a positive
definite covariance matrix and $0<r<1$.

Specifically, the individual components of the vector sequence
$\boldsymbol{e}_{t}=\left(u_{t},\boldsymbol{v}_{t}^{\prime}\right)^{\prime}$
have partial sums processes that weakly converge into their Brownian motion
counterparts as below

$\displaystyle B_{un}(r):=\frac{1}{\sqrt{n}}\sum_{t=1}^{\lfloor
nr\rfloor}u_{t}$ $\displaystyle\Rightarrow
B_{u}(r):=\mathcal{N}\big{(}0,r\sigma_{uu}^{2}\big{)}$ (2.6)
$\displaystyle\boldsymbol{B}_{vn}(r):=\frac{1}{\sqrt{n}}\sum_{t=1}^{\lfloor
nr\rfloor}\boldsymbol{v}_{t}$
$\displaystyle\Rightarrow\boldsymbol{B}_{v}(r):=\mathcal{N}\big{(}\boldsymbol{0},r\boldsymbol{\Sigma}_{vv}\big{)}$
(2.7)

where
$\boldsymbol{B}(r)=\big{(}B_{u}(r),\boldsymbol{B}_{v}(r)^{\prime}\big{)}^{\prime}$
being a $(p\times 1)$ Brownian motion with long-run covariance matrix
$\boldsymbol{\Sigma}_{ee}$, that is, a Gaussian vector process with almost
surely continuous sample paths. More precisely, since $\boldsymbol{x}_{t}$ is
an adapted process to $\mathcal{F}_{t}$ then in practise there exists a
correlated vector Brownian motion
$\boldsymbol{B}_{n}(r)=\big{(}B_{un}(r),\boldsymbol{B}_{vn}(r)^{\prime}\big{)}^{\prime}$
such that

$\displaystyle\left(\frac{1}{\sqrt{n}}\sigma_{uu}^{-1}\sum_{t=1}^{\lfloor
nr\rfloor}u_{t},\frac{1}{\sqrt{n}}\boldsymbol{\Sigma}_{vv}^{-1/2}\sum_{t=1}^{\lfloor
nr\rfloor}\boldsymbol{v}_{t}\right)^{\prime}\Rightarrow\boldsymbol{B}(r)=\big{(}B_{u}(r),\boldsymbol{B}_{v}(r)^{\prime}\big{)}^{\prime},\
\ 0<r<1$ (2.8)

on $\mathcal{D}\left([0,1]\right)^{2}$ as $n\to\infty$, with covariance matrix
as in (2.5) implying joint convergence. Assuming the above conditions hold,
then the following local to unity principle applies (as proposed by Phillips,
1987b ; Phillips, 1987a )

$\displaystyle\frac{\boldsymbol{x}_{[nr]}}{\sqrt{n}}\Rightarrow\boldsymbol{J}_{c}(r),\
\ \ \text{where}\ \
\boldsymbol{J}_{c}(r)=\int_{0}^{r}e^{(r-s)\boldsymbol{C}_{p}}d\boldsymbol{B}_{v}(s).$
(2.9)

The functional $\boldsymbol{J}_{c}(r)$ represents the Ornstein-Uhlenbeck
process888The OU process 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 and this property
allows to approximate asymptotic terms for estimators and corresponding test
statistics as a function of the nuisance parameter of persistence (see,
Perron,  (1991)). which is employed to derive stochastic integral
approximations for the nonstationary predictive regression model (2.1)-(2.2).
Denote with
$\boldsymbol{K}_{c}(r):=\boldsymbol{\Sigma}_{vv}\boldsymbol{J}_{c}(r)$, where
$\boldsymbol{K}_{c}(r)$ is a $p-$dimensional Gaussian process defined as
$\boldsymbol{K}_{c}(r)=\displaystyle\int_{0}^{r}e^{(r-s)\boldsymbol{C}_{p}}d\boldsymbol{B}_{v}(s)$
is the solution of Black-Scholes differential equation
$d\boldsymbol{K}_{c}(r)\equiv c\boldsymbol{K}_{c}(r)+d\boldsymbol{B}_{v}(r)$,
with $\boldsymbol{K}_{c}(r)=0$ as the initial condition.

The specific autoregression matrix specification (2.3) allows to examine other
persistence properties such as unit root processes, when $c_{i}=0$ for all
$i\in\left\\{1,...,p\right\\}$, or explosive processes when $c_{i}>0$. In this
paper, we consider two types of nonstationarity, that is, the near unit or
high persistent regressors and the mildly integrated regressors. Therefore, in
both cases the coefficient of persistent, $c_{i}$, lies below the unit root
boundary. Thus, the main difference between these two persistence classes is
that the mildly integrated regressors have an exponent rate below the unit
boundary, which implies that these regressors are less persistent than
regressors generated from near unit root processes. Furthermore, one can
consider extending our estimation and testing framework to the case of
explosive regressors; we leave this aspect for future research.

####  Quantile Predictive Regression Model

Our main goal is to investigate the asymptotic theory and empirical
implementation of structural break tests suitable for quantile predictive
regression models. Therefore, we consider suitable modifications of the
innovation structure that corresponds to the linear predictive regression
model, following standard conditions and assumptions employed for quantile
time series models, currently presented in literature. More precisely, the
conditional quantile function of $y_{t}$ denoted with
$\mathsf{Q}_{y_{t}}\left(\uptau|\mathcal{F}_{t-1}\right)$, replaces the
conditional mean function of the predictive regression which implies the
following model specification

$\displaystyle\mathsf{Q}_{y_{t}}\left(\uptau|\mathcal{F}_{t-1}\right):=F^{-1}_{y_{t}|\boldsymbol{x}_{t-1}}(\uptau)\equiv\alpha(\uptau)+\boldsymbol{\beta}(\uptau)^{\prime}\boldsymbol{x}_{t-1}.$
(2.10)

such that
$F_{y_{t}|\boldsymbol{x}_{t-1}}(\uptau):=\mathbb{P}\big{(}y_{t}\leq\mathsf{Q}_{y_{t}}\left(\uptau|\mathcal{F}_{t-1}\right)\big{|}\mathcal{F}_{t-1}\big{)}\equiv\uptau$,
where $\uptau\in(0,1)$ is some quantile level in the compact set $(0,1)$.
Therefore, in order to define the innovation structure that corresponds to the
quantile predictive regression, we employ the piecewise derivative of the loss
function such that
$\psi_{\uptau}(\mathsf{u})=\big{[}\uptau-\mathds{1}\left\\{\mathsf{u}<0\right\\}\big{]}$.
Consequently, this implies that $u_{t}(\uptau):=u_{t}-F^{-1}_{u}(\uptau)$
where $F^{-1}_{u}(\tau)$ denotes the unconditional $\uptau-$quantile of the
error term $u_{t}$. Then, the corresponding invariance principle for the
nonstationary quantile predictive regression model is formulated as below

$\displaystyle\frac{1}{\sqrt{n}}\sum_{t=1}^{\lfloor
nr\rfloor}\begin{bmatrix}\psi_{\uptau}\big{(}u_{t}(\uptau)\big{)}\\\
\boldsymbol{v}_{t}\end{bmatrix}\Rightarrow\begin{pmatrix}B_{\psi_{\uptau}}(r)_{{\color[rgb]{1,0,0}(1\times
n)}}\\\ \boldsymbol{B}_{v}(r)_{{\color[rgb]{1,0,0}(p\times
n)}}\end{pmatrix}\equiv\mathcal{BM}\begin{bmatrix}\uptau(1-\uptau)&\boldsymbol{\sigma}_{\psi_{\uptau}v}^{\prime}\\\
\boldsymbol{\sigma}_{v\psi_{\uptau}}&\boldsymbol{\Omega}_{vv}\end{bmatrix}$
(2.11)

######  Assumption 1.

The following conditions for the innovation sequence hold:

  * (i)

The sequence of stationary conditional probability distribution functions
(pdf) denoted with $\big{\\{}f_{u_{t}(\uptau),t-1}(.)\big{\\}}$ evaluated at
zero with a non-degenerate mean function such that
$f_{u_{t}(\uptau)}(0):=\mathbb{E}\left[f_{u_{t}(\uptau),t-1}(0)\right]>0$
satisfies a FCLT given as below

$\displaystyle\frac{1}{\sqrt{n}}\sum_{t=1}^{\lfloor
nr\rfloor}\big{(}f_{u_{t}(\uptau),t-1}(0)-\mathbb{E}\left[f_{u_{t}(\uptau),t-1}(0)\right]\big{)}\Rightarrow
B_{f_{u_{t}(\uptau)}}(r).$ (2.12)

  * (ii)

For each $t$ and $\uptau\in(0,1)$, $f_{u_{t}(\uptau),t-1}(.)$ is uniformly
bounded away from zero with a corresponding conditional distribution function
$F_{t}(.)$ which is absolutely continuous with respect to Lebesgue measure on
$\mathbb{R}$ (see, Neocleous and Portnoy,  (2008), Goh and Knight,  (2009) and
Lee,  (2016)).

Assumption 1 (i) provides a standard weak convergence argument to a Brownian
motion process that corresponds to the underline distribution generating the
innovation sequence of the quantile predictive regression model (see, Lee,
(2016) and Fan and Lee,  (2019)). Furthermore, Assumption 1 (ii) provides the
weak convergence argument for the sparsity function of the model to its
Brownian motion counterpart for some $0<r<1$.

###  Estimation Methodology

We investigate the statistical properties of the estimation methodology in
relation to the handling of the nuisance parameter of persistence. We derive
the asymptotic distributions of the associated test statistics based on two
optimization methods which are known to have different convergence rates in
the time series predictability literature.

####  OLS based estimation

We consider the OLS based estimation using the check function
$\uprho_{\uptau}(.)$, which is common practise for optimization problems of
quantile series. Specifically, the asymptotic behaviour of the quantity
$\boldsymbol{\mathcal{E}}_{n}(\uptau)\equiv\sqrt{n}\big{(}\boldsymbol{\beta}_{n}(\uptau)-\boldsymbol{\beta}_{0}(\uptau)\big{)}$
is of interest. The traditional approach to asymptotics for
$\hat{\boldsymbol{\beta}}(\uptau)$ is to employ a Bahadur representation which
allows to decompose the expression into a Brownian bridge component and an
error term (see, Portnoy,  (2012) and Kato,  (2009)). Furthermore, various
studies are concerned with the determination of sharp error bounds for the
specific error term. However in our setting, $\sqrt{n}-$consistent asymptotics
do not always apply due to the presence of nonstationarity. Additionally, the
chosen estimator affects the stochastic rates of convergence.

Similar to the linear predictive regression (a model with a conditional mean
functional form), under the assumption of persistent regressors, the OLS
estimator has been proved to be biased due to the presence of nuisance
parameters (e.g., see Campbell and Yogo,  (2006)), resulting to distorted
statistical inference999Relevant studies in the literature which examine the
asymptotic behaviour of standard $t-$tests under these conditions include
Phillips and Lee,  (2013, 2016), Lee,  (2016), Fan and Lee,  (2019), Kostakis
et al.,  (2015) and Kasparis et al.,  (2015) among others.. Nevertheless,
focusing on the two persistence properties we introduced previously, we derive
its limit distribution which is useful when considering structural break tests
under nonstationarity.

Denote with
$\boldsymbol{\theta}(\uptau)=\big{[}\alpha(\uptau),\boldsymbol{\beta}(\uptau)^{\prime}\big{]}^{\prime}\in\mathbb{R}^{(p+1)\times
1}$ and
$\boldsymbol{X}_{t-1}=\left(\boldsymbol{1},\boldsymbol{x}_{t-1}^{\prime}\right)^{\prime}\in\mathbb{R}^{(p+1)\times
n}$, then the OLS based estimator is obtained by solving the following
optimization problem

$\displaystyle\widehat{\boldsymbol{\theta}}_{n}^{qr}\left(\uptau\right):=\underset{\boldsymbol{\theta}\in\mathbb{R}^{p+1}}{\mathsf{arg\
min}}\
\sum_{t=1}^{n}\uprho_{\tau}\big{(}y_{t}-\boldsymbol{X}_{t-1}^{\prime}\boldsymbol{\theta}\big{)}$
(2.13)

where
$\uprho_{\uptau}(\mathsf{u})=\mathsf{u}\big{(}\uptau-\mathds{1}\left\\{\mathsf{u}<0\right\\}\big{)}$
with $\uptau\in(0,1)$, represents the asymmetric quantile regression function.
Following Lee,  (2016), we use the normalization matrices below which are
different according to the persistence properties of predictors such that

$\displaystyle\boldsymbol{D}_{n}:=\left\\{\begin{array}[]{ll}\mathsf{diag}\big{(}\sqrt{n},n\boldsymbol{I}_{p}\big{)}&\text{for}\
\textit{LUR},\\\
\mathsf{diag}\left(\sqrt{n},n^{\frac{1+\upgamma_{x}}{2}}\boldsymbol{I}_{p}\right)&\text{for}\
\textit{MI}.\end{array}\right.$ (2.16)

Then, Corollary 1 summarizes the asymptotic distribution of the OLS-QR
estimator (see also Theorem 2.1 Lee,  (2016)) for mildly integrated and high
persistent regressors.

######  Corollary 1.

Under Assumption 1 and FCLT (2.5)-(2.5) it follows that:
$\boldsymbol{D}_{n}\left(\widehat{\boldsymbol{\theta}}_{n}^{qr}\left(\uptau\right)-\boldsymbol{\theta}\left(\uptau\right)\right)$

$\displaystyle\Rightarrow\left\\{\begin{array}[]{ll}f_{u_{t}(\uptau)}(0)^{-1}\begin{bmatrix}1&\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)^{\prime}dr\\\
\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)dr&\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)\boldsymbol{J}_{c}(r)^{\prime}dr\end{bmatrix}^{-1}_{{\color[rgb]{1,0,0}(p+1)\times(p+1)}}\begin{bmatrix}B_{\psi_{\uptau}}(1)_{{\color[rgb]{1,0,0}(1\times
n)}}\\\
\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)dB_{\psi_{\uptau}}dr_{{\color[rgb]{1,0,0}(p\times
n)}}\end{bmatrix}&\textit{LUR},\\\
\mathcal{N}\displaystyle\left(0,\frac{\uptau(1-\uptau)}{f_{u_{t}(\uptau)}(0)^{2}}\begin{bmatrix}1&\boldsymbol{0}^{\prime}\\\
\boldsymbol{0}&\boldsymbol{V}_{xx}^{-1}\end{bmatrix}_{{\color[rgb]{1,0,0}(p+1)\times(p+1)}}\right)&\textit{MI}.\end{array}\right.$

where the stochastic matrix $\boldsymbol{V}_{xx}$ is defined by the following
expression

$\displaystyle\boldsymbol{V}_{xx}:=\int_{0}^{\infty}e^{r\boldsymbol{C}_{p}}\boldsymbol{\Omega}_{xx}e^{r\boldsymbol{C}_{p}}dr,\
\text{where}\
\boldsymbol{\Omega}_{xx}:=\sum_{m=-\infty}^{\infty}\mathbb{E}\left(\boldsymbol{v}_{t}\boldsymbol{v}_{t-m}^{\prime}\right)=\boldsymbol{\varphi}_{o}(1)\boldsymbol{\Sigma}\boldsymbol{\varphi}_{o}(1)^{\prime}.$

Consequently, the limiting joint distribution of the model intercept and
slopes for the nonstationary quantile predictive regression model under the
assumption of mildly integrated regressors, is a mixed normal of the form
$\mathcal{MN}\big{(}0,\boldsymbol{\Sigma}^{\star}\big{)}$, where

$\displaystyle\boldsymbol{\Sigma}^{\star}:=\frac{\uptau(1-\uptau)}{f_{u_{t}(\uptau_{0})}(0)^{2}}\begin{bmatrix}1&\boldsymbol{0}^{\prime}\\\
\boldsymbol{0}&\boldsymbol{V}_{xx}^{-1}\end{bmatrix},\ \ \text{for some}\
\uptau\in(0,1).$ (2.17)

while under high persistence the asymptotic behaviour of
$\boldsymbol{D}_{n}\left(\widehat{\boldsymbol{\theta}}_{n}^{qr}\left(\uptau\right)-\boldsymbol{\theta}\left(\uptau\right)\right)$
depends on functionals of OU processes which are more challenging to
approximate, especially if one is interested to obtain sharp error
bounds101010In particular, the study of Portnoy,  (2012) obtains a near
$\sqrt{n}-$consistent error bound by employing the ”Hungarian construction”
which requires to approximate the quantity
$\boldsymbol{\mathcal{E}}_{n}(\uptau)$ using a Brownian bridge limit which
converges to this non-zero Gaussian process with an appropriate rate of
convergence.  as in Portnoy,  (2012).

Furthermore, we assume that the sparsity coefficient can be consistently
estimated with a fixed unbiased estimator for all $t$. This a strong
assumption which can be regarded as a trade-off relation between the
complicated objective function (i.e., non-differentiable and nonstationary)
and the tractable error term of the model (Uematsu, , 2019). On the other
hand, it permits to consider the limiting distributions when testing for a set
of parameter restrictions. Thus, to overcome the problem of nonstandard
statistical inference due to the presence of the nuisance coefficient of
persistence, we employ the instrumental variable regression approach proposed
by Phillips and Magdalinos,  (2009).

The estimator of Phillips and Magdalinos,  (2009) performs reasonably well in
finite samples and even performs better that the OLS counterpart (see Georgiev
et al.,  (2021)), demonstrating the robustness of the method in filtering out
abstract degree of peristence when testing for linear restrictions in
predictive regressions. In addition, the suggested instrumental variable
approach is by definition neither spurious, since it is always correlated with
the corresponding regressor, not a poor instrument, because the correlation of
unit root processes tends to one asymptotically.

####  IVX based estimation

The endogenous instrumentation (IVX) procedure for predictive regression
models111111Notice that the related asymptotic theory which is robust to
abstract degree of persistence and results to nuisance-parameter free
inference was pioneered by Magdalinos and Phillips,  (2009) in the context of
cointegration models. proposed by Phillips and Magdalinos,  (2009) implies the
use of a mildly integrated instrumental variable. The instrumented variable is
constructed as below

$\displaystyle\widetilde{\boldsymbol{z}}_{t}=\sum_{j=0}^{t-1}\left(\boldsymbol{I}_{p}+\frac{\boldsymbol{C}_{z}}{n^{\upgamma_{z}}}\right)\big{(}\boldsymbol{x}_{t-j}-\boldsymbol{x}_{t-j-1}\big{)},$
(2.18)

where $\boldsymbol{C}_{z}=\mathsf{diag}\\{c_{z1},...,c_{zp}\\}$ is a $p\times
p$ diagonal matrix such that $c_{zj}<0\ \forall\ j\in\left\\{1,...,p\right\\}$
with $0<\upgamma_{z}<1$, where $\upgamma_{z}$ is the exponent rate of the
persistence coefficient of the instrumental variable, such that
$\upgamma_{z}\neq\upgamma_{x}$. The IVX filtering methodology transforms a
possibly nonstationary autoregressive process that generates the set of
predictors, $\boldsymbol{x}_{t}$, which encompasses both stable or unstable
processes based on the behaviour of the local unit root coefficient, into a
mildly integrated process which is less persistent than the endogenous
variables. Another statistical property is the choice of the exponent rate for
the coefficient of persistence that corresponds to the instrumental variable
$c_{zj}$. Specifically, the econometric literature has documented a choice of
$\upgamma_{z}$ close to 0.95 as a reasonable value with desirable finite-
sample properties when constructing predictability tests (see, Lee,  (2016),
Phillips and Lee,  (2016) and Kostakis et al.,  (2015)). Thus, to account for
the different convergence rates due to nonstationarity and obtain the
asymptotic distribution of the IVX-QR estimator we employ the following
normalization matrices

$\displaystyle\tilde{\boldsymbol{Z}}_{t-1,n}:=\tilde{\boldsymbol{D}}_{n}^{-1}\tilde{\boldsymbol{z}}_{t-1}\
\ \text{and}\ \
\tilde{\boldsymbol{X}}_{t-1,n}:=\tilde{\boldsymbol{D}}_{n}^{-1}\tilde{\boldsymbol{x}}_{t-1}$
(2.19)

where
$\tilde{\boldsymbol{D}}_{n}=n^{\frac{1+\upgamma_{x}\wedge\upgamma_{z}}{2}}\boldsymbol{I}_{p}$,
such that
$\upgamma_{x}\wedge\upgamma_{z}\equiv\mathsf{min}\left(\upgamma_{x},\upgamma_{z}\right)$
which is identical for both the case of local unit root and mildly integrated
regressors. Furthermore, we denote with
$y_{t}(\uptau):=y_{t}-\alpha(\uptau)+\mathcal{O}_{\mathbb{P}}(n^{-1/2})$ to be
the zero-intercept QR dependent variable. The particular dequantiling
procedure permits to reformulate the quantile model as
$y_{t}(\uptau)=\boldsymbol{x}_{t-1}^{\prime}\beta(\uptau)+u_{t}(\uptau)$ that
simplifies the derivations for the asymptotics of the model estimator which is
known to have different convergence rates when an intercept is included (e.g.,
see Gonzalo and Pitarakis,  (2012, 2017)). Then, the IVX-QR estimator for the
quantile regression, is defined by the following unconstrained optimization
problem

$\displaystyle\widehat{\boldsymbol{\beta}}_{n}^{ivx-
qr}(\uptau):=\underset{\boldsymbol{\beta}\in\mathbb{R}^{p}}{\mathsf{arg\
inf}}\ \frac{1}{2}\
\left\\{\left(\sum_{t=1}^{n}h_{t}\left(\boldsymbol{\beta}\right)\right)^{\prime}\left(\sum_{t=1}^{n}h_{t}\left(\boldsymbol{\beta}\right)\right)\right\\},$
(2.20)

where $h_{t}(.)$ is defined below such that
$\psi_{\uptau}(\mathsf{u}):=\big{[}\mathds{1}\left\\{\mathsf{u}\leq
0\right\\}-\uptau\big{]}$ and $\psi_{\uptau}(\mathsf{u})\overset{\mathsf{maps\
to}}{\mapsto}\uprho_{\uptau}^{-1}(\mathsf{u})$

$\displaystyle
h_{t}\left(\boldsymbol{\beta}\right):=\tilde{\boldsymbol{z}}_{t-1}\times\psi_{\uptau}\big{(}u_{t}(\uptau)\big{)}\equiv\tilde{\boldsymbol{z}}_{t-1}\times\big{[}\uptau-\mathds{1}\big{\\{}y_{t}(\uptau)<\boldsymbol{x}_{t-1}^{\prime}\boldsymbol{\beta}\big{\\}}\big{]}$
(2.21)

The minimization of expression (2.20) leads to the following first order
condition:

$\displaystyle\sum_{t=1}^{n}\tilde{\boldsymbol{Z}}_{t-1,n}\times\big{[}\uptau-\mathds{1}\big{\\{}y_{t}(\uptau)<\boldsymbol{x}_{t-1}^{\prime}\boldsymbol{\beta}_{n}^{ivx-
qr}(\uptau)\big{\\}}\big{]}=o_{\mathbb{P}}(1).$ (2.22)

Furthermore, it can be proved that for both the cases of high persistent and
mildly integrated predictors (LUR or MI) the asymptotic distribution of the
IVX-QR estimator is identical as presented by Corollary 2. Notice that for the
interested reader the limit distribution for other classes of persistence
(e.g., such as predictors which exhibit near stationary or mildy explosive
persistence) can be found in Theorem 3.1 of Lee,  (2016).

######  Corollary 2.

(IVX-QR Limit Theory) Under Assumption 1 it follows that

$\displaystyle\tilde{\boldsymbol{D}}_{n}\left(\widehat{\boldsymbol{\beta}}_{n}^{ivx-
qr}\left(\uptau\right)-\boldsymbol{\beta}\left(\uptau\right)\right)\Rightarrow\mathcal{N}\displaystyle\left(0,\frac{\uptau(1-\uptau)}{f_{u_{t}(\uptau)}(0)^{2}}\big{(}\boldsymbol{\Gamma}_{cxz}\boldsymbol{V}_{cxz}^{-1}\boldsymbol{\Gamma}_{cxz}^{\prime}\big{)}^{-1}\right)$
(2.23)

which is a mixed Gaussian distribution due to the stochastic covariance
matrix.

Analytic definitions of the covariance matrices $\boldsymbol{\Gamma}_{cxz}$
and $\boldsymbol{V}_{cxz}$ can be found in expressions (3.4) and (3.5) in Lee,
(2016). Lastly, Lemma 1 presents the asymptotic behaviour of the self-
normalized Wald statistic based on the IVX-QR estimator (see, Proposition 3.1
in Lee,  (2016)) for both the cases of LUR and MI regressors.

######  Lemma 1.

(Self-normalized IVX-QR) Under Assumption 1 it holds that,

$\displaystyle\frac{\widehat{f_{u_{t}(\uptau)}}(0)^{2}}{\uptau(1-\uptau)}\left(\widehat{\boldsymbol{\beta}}^{ivx-
qr}_{n}(\uptau)-\boldsymbol{\beta}(\uptau)\right)^{\prime}\big{(}\boldsymbol{X}^{\prime}\boldsymbol{P}_{\tilde{\boldsymbol{Z}}}\boldsymbol{X}\big{)}\left(\widehat{\boldsymbol{\beta}}^{ivx-
qr}_{n}(\uptau)-\boldsymbol{\beta}(\uptau)\right)\Rightarrow\chi^{2}_{p}$
(2.24)

where

$\displaystyle\big{(}\boldsymbol{X}^{\prime}\boldsymbol{P}_{\tilde{\boldsymbol{Z}}}\boldsymbol{X}\big{)}:=\left(\boldsymbol{X}^{\prime}\tilde{\boldsymbol{Z}}\right)\left(\tilde{\boldsymbol{Z}}^{\prime}\tilde{\boldsymbol{Z}}\right)^{-1}\left(\tilde{\boldsymbol{Z}}^{\prime}\boldsymbol{X}\right)\equiv\left(\sum_{t=1}^{n}\boldsymbol{x}_{t-1}\tilde{\boldsymbol{z}}_{t-1}^{\prime}\right)\left(\sum_{t=1}^{n}\tilde{\boldsymbol{z}}_{t-1}\tilde{\boldsymbol{z}}^{\prime}_{t-1}\right)^{-1}\left(\sum_{t=1}^{n}\tilde{\boldsymbol{z}}_{t-1}\boldsymbol{x}_{t-1}^{\prime}\right)$

such that $\widehat{f_{u_{t}(\uptau)}}(0)^{2}$ is a consistent estimator of
$f_{u_{t}(\uptau)}(0)^{2}$.

Furthermore, the above result can be generalized when testing for a set of
linear restrictions under the null hypothesis,
$\mathcal{H}_{0}:\boldsymbol{R}\boldsymbol{\beta}(\uptau)=\boldsymbol{q}(\uptau)$
where $\boldsymbol{R}$ is a $r\times p$ known matrix and
$\boldsymbol{q}(\uptau)$ is a prespecified vector. The corresponding
asymptotic distribution for the IVX-Wald statistic for the quantile predictive
regression is given by the following expression

$\displaystyle\frac{\widehat{f_{u_{t}(\uptau)}}(0)^{2}}{\uptau(1-\uptau)}\left(\boldsymbol{R}\widehat{\boldsymbol{\beta}}^{ivx-
qr}_{n}(\uptau)-\boldsymbol{q}(\uptau)\right)^{\prime}\bigg{[}\boldsymbol{R}\big{(}\boldsymbol{X}^{\prime}\boldsymbol{P}_{\tilde{\boldsymbol{Z}}}\boldsymbol{X}\big{)}^{-1}\boldsymbol{R}^{\prime}\bigg{]}^{-1}\left(\boldsymbol{R}\widehat{\boldsymbol{\beta}}^{ivx-
qr}_{n}(\uptau)-\boldsymbol{q}(\uptau)\right)\Rightarrow\chi^{2}_{r}$

where $\chi^{2}_{r}$ denotes the chi-square random variate with $r$ degrees of
freedom such that
$\mathbb{P}\left(\chi^{2}\geq\chi^{2}_{r;\upalpha}\right)=\upalpha$, where
$0<\upalpha<1$ denotes the fixed significance level.

###  Testing Hypotheses

In this section we present the testing problem of interest in this paper. More
precisely, we consider two type of testing hypotheses, that is: (i) testing
for structural break for a fixed quantile level $\uptau\in(0,1)$ and (ii)
testing for a structural break multiple quantile levels. For both testing
hypotheses we operate under the assumption of a single structural break at an
unknown location. Notice, that similar formulations can be found in the study
of Qu,  (2008), however the focus of our study in the quantile predictive
regression model with possibly nonstationary predictors in the context of
return predictability literature.

##### Testing Hypothesis A.

The first testing hypothesis of interest is concerned with testing for
structural break in a pre-specified quantile with the null and alternative
hypothesis given as below

$\displaystyle\mathcal{H}^{(A)}_{0}$
$\displaystyle:\boldsymbol{\beta}_{t}(\uptau)=\boldsymbol{\beta}_{0}(\uptau)\
\ \ \text{for all}\ \ 1\leq t\leq n,\ {\color[rgb]{0,0,1}\text{for a fixed}\
\uptau\in(0,1)},$ $\displaystyle\mathcal{H}^{(A)}_{1}$
$\displaystyle:\boldsymbol{\beta}_{t}(\uptau)=\begin{cases}\boldsymbol{\beta}_{1}(\uptau)&\text{where}\
1\leq t\leq\kappa\\\ \boldsymbol{\beta}_{2}(\uptau)&\text{where}\ \kappa+1\leq
t\leq n\end{cases}$

for a fixed $\uptau\in(0,1)$, where $\kappa=\lfloor\lambda n\rfloor$ the
unknown break-point with $\lambda\in(0,1)$.

##### Testing Hypothesis B.

The second testing hypothesis of interest is concerned with testing for
structural break across multiple quantiles, that is, quantiles contained in a
set $\mathcal{T}_{\iota}$, with the null and alternative hypothesis given as
below

$\displaystyle\mathcal{H}^{(B)}_{0}$
$\displaystyle:\boldsymbol{\beta}_{t}(\uptau)=\boldsymbol{\beta}_{0}(\uptau)\
\ \ \text{for all}\ \ 1\leq t\leq n,\ {\color[rgb]{0,0,1}\text{and for all}\ \
\uptau\in\mathcal{T}_{\iota}},$ $\displaystyle\mathcal{H}^{(B)}_{1}$
$\displaystyle:\boldsymbol{\beta}_{t}(\uptau)=\begin{cases}\boldsymbol{\beta}_{1}(\uptau)&\text{where}\
1\leq t\leq\kappa\\\ \boldsymbol{\beta}_{2}(\uptau)&\text{where}\ \kappa+1\leq
t\leq n\end{cases}$

for some $\uptau\in\mathcal{T}_{\iota}$, where $\kappa=\lfloor\lambda
n\rfloor$ the unknown break-point with $\lambda\in(0,1)$.

where $\boldsymbol{\beta}_{0}(\uptau)$ is the value of the true population
parameter under the null hypothesis of no parameter instability.

######  Remark 1.

Notice that the statistical problem given by Testing Hypothesis A allow us to
focus on a particular quantile of interest, e.g., any fixed quantile level
$\uptau\equiv\uptau_{0}\in(0,1)$. On the other hand, the inference problem
given by Testing Hypothesis B permits testing for structural break in the
coefficients of the quantile predictive regression model by investigating the
presence of breaks in the conditional distribution, that is, at any possible
quantile level within the compact set $\mathcal{T}_{\iota}:=[\iota,1-\iota]$
where $0<\iota<1/2$.

Both Testing Hypothesis A and B summarize the modelling environment under the
null as well as under the alternative hypothesis. More precisely, under the
null hypothesis the parameter vector is taken to be constant throughout the
sample such that
$\boldsymbol{\beta}_{t}(\uptau)\equiv\boldsymbol{\beta}_{0}(\uptau)$ for
$t=1,...,n$ where $\boldsymbol{\beta}_{0}(\uptau)$ is the unknown quantile
dependent regression parameter. Therefore, in practise we are interested in
testing the null hypothesis that $\boldsymbol{\beta}_{t}(\uptau)$ remains
constant, that is,
$\boldsymbol{\beta}_{t}(\uptau)=\boldsymbol{\beta}_{0}(\uptau)$ for all $t$
against the alternative that the quantile dependent parameter vector
$\boldsymbol{\beta}_{t}(\uptau)$ has a single structural break at an unknown
location within the full sample, resulting to two regimes121212Notice that the
two regimes we refer to here, are not equivalent to testing methodologies
proposed in studies such as Gonzalo and Pitarakis,  (2012, 2017) and Galvao et
al.,  (2014) in which emphasis is given to testing the null hypothesis of
linearity based on the presence of no threshold effect.. Under the alternative
hypothesis:

$\displaystyle\mathsf{Q}_{y_{t}}\left(\uptau|\mathcal{F}_{t-1}\right)=\boldsymbol{\beta}_{1}(\uptau)\boldsymbol{x}_{t-1}\mathds{1}\big{\\{}t\leq\kappa\big{\\}}+\boldsymbol{\beta}_{2}(\uptau)\boldsymbol{x}_{t-1}\mathds{1}\big{\\{}t>\kappa\big{\\}}+u_{t}$
(2.27)

where $\mathcal{F}_{t}$ denotes the $\sigma-$field generated by
$\left\\{\boldsymbol{x}_{t-1},\boldsymbol{x}_{t-2},...,\right\\}$. Therefore,
it is convenient to write the hypotheses with a different formulation. Denote
with $\boldsymbol{\upbeta}_{(1)}(\uptau)=\boldsymbol{\beta}_{1}(\uptau)$ and
$\boldsymbol{\upbeta}_{(2)}(\uptau)=\boldsymbol{\beta}_{2}(\uptau)-\boldsymbol{\beta}_{1}(\uptau)$.
Furthermore, to construct the model so that it can capture the magnitude of
the structural break we denote with
$\boldsymbol{\mathcal{X}}_{t-1}=\big{(}\boldsymbol{x}_{t-1}^{\prime},\boldsymbol{x}_{t-1}^{\prime}\mathds{1}\left\\{t>\kappa\right\\}\big{)}^{\prime}$
and
$\boldsymbol{\vartheta}(\uptau)=\big{(}\boldsymbol{\upbeta}_{(1)}(\uptau)^{\prime},\boldsymbol{\upbeta}_{(2)}(\uptau)^{\prime}\big{)}^{\prime}$.
Thus, we express the null hypothesis as following

$\displaystyle\mathcal{H}_{0}^{(A)}:\mathsf{Q}_{y_{t}}\left(\uptau|\mathcal{F}_{t-1}\right)=\boldsymbol{\mathcal{X}}_{t-1}^{\prime}\boldsymbol{\vartheta}(\uptau),\
\text{with}\ \boldsymbol{\upbeta}_{(2)}(\uptau)=\boldsymbol{0}\ \text{for some
fixed}\ \uptau_{0}\in\mathcal{T}_{\iota}$ (2.28)

Then, the alternative hypothesis can be formulated as below

$\displaystyle\mathcal{H}_{1}^{(A)}:\mathsf{Q}_{y_{t}}\left(\uptau|\mathcal{F}_{t-1}\right)=\boldsymbol{\mathcal{X}}_{t-1}^{\prime}\boldsymbol{\vartheta}(\uptau),\
\text{with}\ \boldsymbol{\upbeta}_{(2)}(\uptau)\neq\boldsymbol{0}\ \text{for
some fixed}\ \uptau\in\mathcal{T}_{\iota}$ (2.29)

Furthermore, since we consider the nonstationary quantile predictive
regression model without an intercept, following the formulations given by
expressions (2.28) and (2.29) for the null and alternative hypotheses
respectively, then we define the quantile dependent estimator as the
optimization problem below

$\displaystyle\widehat{\boldsymbol{\vartheta}}_{n}\big{(}\lambda,\uptau\big{)}:=\underset{\boldsymbol{\vartheta}\in\mathbb{R}^{2p}}{\mathsf{arg\
min}}\
\sum_{t=1}^{n}\uprho_{\tau}\big{(}y_{t}-\boldsymbol{\mathcal{X}}_{t-1}^{\prime}\boldsymbol{b}\big{)},$
(2.30)

Therefore, with the above formulation of the estimator
$\widehat{\boldsymbol{\vartheta}}_{n}(\uptau,\lambda)$ is the quantile
dependent regression estimator when we employ $\boldsymbol{\mathcal{X}}_{t-1}$
to be the model predictor variables. Specifically, when the
$\mathcal{H}_{0}^{(A)}$ is true, under suitable regularity conditions,
$\widehat{\boldsymbol{\vartheta}}_{2}(\lambda,\uptau)$ converges in
probability to $\boldsymbol{0}$ for each
$(\lambda,\uptau)\in\Lambda_{\eta}\times\mathcal{T}_{\iota}$. On the other
hand, when $\mathcal{H}_{1}^{(A)}$ is true,
$\widehat{\boldsymbol{\vartheta}}_{2}(\lambda;\uptau_{0})$ converges in
probability to
$\boldsymbol{\upbeta}_{(2)}(\uptau_{0})=\big{(}\boldsymbol{\beta}_{2}(\uptau_{0})-\boldsymbol{\beta}_{1}(\uptau_{0})\big{)}\neq\boldsymbol{0}$.
In summary, since the quantile level $\uptau_{0}$ especially for Testing
Hypothesis B is unknown a prior, then it is reasonable to reject
$\mathcal{H}_{0}$ when the magnitude of
$\widehat{\boldsymbol{\vartheta}}_{2}(\lambda,\uptau)$ is suitable large for
some $(\lambda,\uptau)\in\Lambda_{\eta}\times\mathcal{T}_{\iota}$. Thus, an
example of a suitable test statistic to test whether $\mathcal{H}_{0}$ against
the alternative hypothesis $\mathcal{H}_{1}$ is to employ the supremum of the
Wald process.

In general, a Wald type statistic has the following form

$\displaystyle\mathcal{SW}_{n}\big{(}\uptau,\lambda\big{)}:=\underset{(\lambda,\uptau)\in\Lambda_{\eta}\times\mathcal{T}_{\iota}}{\mathsf{sup}}n\hat{\boldsymbol{\upbeta}}_{(2)}(\uptau_{0})\bigg{[}\boldsymbol{V}_{n}(\lambda;\uptau_{0})\bigg{]}\hat{\boldsymbol{\upbeta}}_{(2)}(\uptau_{0})$
(2.31)

where $\boldsymbol{V}_{n}(\lambda;\uptau_{0})$ is the asymptotic covariance
matrix of the stochastic process
$\sqrt{n}\hat{\boldsymbol{\upbeta}}_{(2)}(\uptau_{0})$, under the null
hypothesis. However, since the covariance matrix that corresponds to the
population regression parameters is in practise unknown, is replaced by a
suitable consistent estimate that holds under the null hypothesis of no
structural break in the quantile predictive regression model. Obviously within
the aforementioned structural break setting which is our main research focus
in the paper, the break-point location is not identified under the null
hypothesis as we explained in the introduction.

Moreover, additionally to Remark 1, the Testing Hypotheses of interest A and B
correspond to two different test functions such that Testing Hypothesis A
requires to formulate test statistics by employing the supremum funcitonal
while Testing Hypothesis B requires to construct test statistics with the use
of the double supremum functional. Intuitively, we are interested to examine
both hypotheses since it might be the case that A is not rejected, that is,
there are no statistical evidence of the presence of a structural break for a
given quantile level $\uptau_{0}\in(0,1)$, while when employing Testing
Hypothesis B, it could be the case that the test statistic provides
statistical evidence of rejecting the null hypothesis, implying that a
structural break still exist at some other quantile level not the one which is
kept fixed, within the set (0,1).

##  Testing for structural break for a fixed quantile level

Next, we focus on the structural break testing procedures for the quantile
regression model with regressors generated as near unit root processes based
on two test statistics.

###  Preliminary Setting

We consider structural break tests for a fixed quantile level, say
$\uptau_{0}\in(0,1)$. Consider the subgradient131313The required convexity
arguments for obtaining estimators based on the conditional quantile function
are based on the convexity lemma result presented by Pollard,  (1991).
Furthermore, related results are presented by Koenker and Portnoy,  (1987).
$S_{n}\big{(}\lambda,\uptau_{0},\boldsymbol{b}\big{)}$, based on the subsample
$1\leq t\leq\kappa$

$\displaystyle
S_{n}\big{(}\lambda,\uptau_{0},\boldsymbol{b}\big{)}=n^{-1/2}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{x}_{t-1}\psi_{\uptau}\big{(}y_{t}-\boldsymbol{x}_{t-1}^{\prime}\boldsymbol{b}\big{)},$
(3.1)

where $\boldsymbol{b}$ corresponds to an estimator of the parameter vector
$\boldsymbol{\beta}(\uptau_{0})$ which encompasses both the OLS and IVX
estimators under suitable parametrizations.

The continuous function $\psi_{\uptau}(.)$ is defined as
$\psi_{\uptau}(\mathsf{u})=\big{[}\uptau_{0}-\mathds{1}\left\\{\mathsf{u}\leq
0\right\\}\big{]}$ and $\kappa=\lfloor\lambda n\rfloor$ denotes the unknown
break-point location implying a break fraction
$\lambda\equiv\underset{n\to\infty}{\mathsf{lim}}\kappa/n$ such that
$\lambda\in\Lambda_{\eta}:=[\eta,1-\eta]$ is a compact set. Thus, under the
null hypothesis of no structural break with stationary and ergodic regressors,
the quantity
$\psi_{\uptau}\big{(}y_{t}-\boldsymbol{x}_{t-1}^{\prime}\boldsymbol{\beta}(\uptau_{0})\big{)}$
is a pivotal statistic. In particular, since
$\boldsymbol{x}_{t-1}^{\prime}\boldsymbol{\beta}(\uptau_{0})$ is equal to the
conditional $\uptau-$quantile of $y_{t}$ given $\boldsymbol{x}_{t-1}$, then
the random variables
$\mathds{1}\left\\{y_{1}\leq\boldsymbol{x}_{t-1}^{\prime}\boldsymbol{\beta}(\uptau_{0})\right\\},...,\mathds{1}\left\\{y_{n}\leq\boldsymbol{x}_{t-1}^{\prime}\boldsymbol{\beta}(\uptau_{0})\right\\}$
are independent Bernoulli trials with success probability $\uptau$ (see,
Galvao et al.,  (2014)), which implies a sequence of random variables with
mean zero and variance $\uptau_{0}(1-\uptau_{0})$. A similar result should
hold in our setting of possibly nonstationary regressors. Furthermore, denote
with $\boldsymbol{X}=\left(x_{1}^{\prime},...,x_{n}^{\prime}\right)^{\prime}$
and define the following auxiliary quantity

$\displaystyle\mathcal{J}_{n}\big{(}\lambda,\uptau_{0},\boldsymbol{\beta}_{0}(\uptau)\big{)}:=\left(n^{-1}\boldsymbol{X}^{\prime}\boldsymbol{X}\right)^{-1/2}S_{n}\big{(}\lambda,\uptau_{0},\boldsymbol{\beta}_{0}(\uptau)\big{)}.$
(3.2)

Furthermore, replacing the unknown parameter vector
$\boldsymbol{\beta}_{0}(\uptau_{0})$ with the quantile dependent regression
estimator based on the full sample, under the null hypothesis of no structural
break in the model, the quantity given by (3.2) can be formulated as below

$\displaystyle\hat{\mathcal{J}}_{n}\big{(}\lambda,\uptau_{0},\widehat{\boldsymbol{\beta}}_{n}(\uptau_{0})\big{)}=\big{(}\boldsymbol{X}^{\prime}\boldsymbol{X}\big{)}^{-1/2}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{x}_{t-1}\psi_{\uptau}\left(y_{t}-\boldsymbol{x}_{t-1}^{\prime}\widehat{\boldsymbol{\beta}}_{n}(\uptau_{0})\right).$
(3.3)

for some $0<\lambda<1$ and $\uptau_{0}\in(0,1)$.

The use of fluctuation type statistics provide a way for statistical inference
regarding the presence of structural breaks in model coefficients (Leisch et
al.,  (2000)). Intuitively for these class of tests we consider the asymptotic
behaviour of the corresponding empirical processes to decide whether to accept
or reject the null. In particular, the fluctuation type test converges to a
nondegenerate limiting distribution under the null hypothesis, since the
random quantity given by expression (3.3) is essentially governed by the
invariance principle under the null. In practise, when the quantile dependent
parameters exhibit no structural break in the sample, then
$\widehat{\boldsymbol{\beta}}_{n}(\uptau_{0})$ is a consistent estimator and
as a result,
$\hat{\mathcal{J}}_{n}\big{(}\lambda,\uptau_{0},\widehat{\boldsymbol{\beta}}_{n}(\uptau_{0})\big{)}$
has the same stochastic order as its population counterpart. On the other
hand, when the null hypothesis is false, the underline stochastic process
exhibit excessive fluctuations. Specifically, under the alternative
hypothesis, model parameters have a break at some unknown location in the
sample, which implies that $\widehat{\boldsymbol{\beta}}_{n}(\uptau_{0})$ will
differ significantly from the true value for some sub-sample and the estimated
residuals will have high fluctuations (beyond the usual increments of a Wiener
process) resulting to falsely rejecting the null due to a large value of the
statistic (Qu, , 2008, p. 172).

The focus of the proposed econometric environment in this paper is the
structural break detection in the model parameters of quantile predictive
regression with possibly nonstationary regressors, under the assumption that
these are generated as near unit root processes. Intuitively, the two
persistence classes (mildly integrated and near unit root) we consider
encompasses moderate deviations from the unit boundary similar to the case of
near integrated (see Phillips,  (1988)). Therefore, both the value (and sign)
of the coefficient of persistence as well as its exponent
rate141414Practically, these are nuisance parameters however via Monte Carlo
simulations we can choose suitable values for $c_{i}$ and $\upgamma_{x}$ in
order to simulate these experimental conditions and thus evaluate the finite-
sample performance of the test statistics with high persistence or mildly
integrated regressors. (a tuning parameter), determine the asymptotic
behaviour of functionals based on these near unit root process. As a result,
the chosen estimator can affect the asymptotic theory of the proposed test
statistics as well as the corresponding functionals which we examine
separately below.

####  OLS based functionals

Within the proposed econometric environment which corresponds to the modelling
of nonstationary quantile time series models, regressors are assumed to follow
a local unit root process. Thus, we expect that OLS based functionals will
dependent on the nuisance coefficient of persistence151515Notice that this is
the standard inference problem in the predictability literature. Further
details regarding the bias (nonstandard distortion) occurred in predictability
tests (i.e., $t-$tests) in quantile predictive regression models can be found
in the study of Lee,  (2016). In our study, we aim to compare both the OLS as
well as the instrumental variable approach of Phillips and Magdalinos,
(2009).. Furthermore, due to the presence of both a model intercept and the
set of nonstationary regressors, we also need to modify the functionals given
by expressions (3.1)-(3.2) in order to account for the different convergence
rates.

Therefore, to obtain equivalent representations to the quantity
$S_{n}\big{(}\lambda,\uptau_{0},\boldsymbol{b}\big{)}$, we consider the
corresponding partial sum process of the functional
$\boldsymbol{K}_{nx}\big{(}\uptau_{0},\boldsymbol{\theta}^{ols}_{n}(\uptau_{0})\big{)}$
as given by Definition 1. We obtain the limit result for these functionals
based on the full sample and then focus on deriving invariance principles for
the corresponding partial sum processes for the two estimators under
examination (see, Section 3.2). More precisely, the functionals given by
Definition 1 correspond to a quantile regression ordinary least squares
estimator and are employed when the asymptotic behaviour of the OLS based test
statistics is concerned (see also Lemma A1 in Lee,  (2016)).

######  Definition 1.

$\displaystyle\boldsymbol{K}_{nx}\big{(}\uptau_{0},\boldsymbol{\theta}^{ols}_{n}(\uptau_{0})\big{)}$
$\displaystyle:=\boldsymbol{D}_{n}^{-1}\sum_{t=1}^{n}\boldsymbol{X}_{t-1}\psi_{\uptau}\big{(}u_{t}(\uptau_{0})\big{)}$
(3.4)
$\displaystyle\boldsymbol{L}_{nx}\big{(}\uptau_{0},\boldsymbol{\theta}^{ols}_{n}(\uptau_{0})\big{)}$
$\displaystyle:=\boldsymbol{D}_{n}^{-1}\left[\sum_{t=1}^{n}f_{u_{t}(\uptau),t-1}(0)\boldsymbol{X}_{t-1}\boldsymbol{X}_{t-1}^{\prime}\right]\boldsymbol{D}_{n}^{-1}$
(3.5)

for some $\uptau_{0}\in(0,1)$ where
$\psi_{\uptau}\big{(}u_{t}(\uptau_{0})\big{)}=\big{[}\uptau_{0}-\mathds{1}\big{\\{}y_{t}-\boldsymbol{X}_{t-1}^{\prime}\boldsymbol{\theta}^{ols}_{n}(\uptau_{0})\leq
0\big{\\}}\big{]}$.

A key observation is that under the null hypothesis of no parameter
instability these functional converge to a nondegenerate limit distribution.
Corollary 3 demonstrates the asymptotic distributions of the functionals given
by Definition 1.

######  Corollary 3.

Under the assumption that the pair
$\left\\{y_{t},\boldsymbol{x}_{t-1}\right\\}_{t=1}^{n}$ is generated by the
model (2.1)-(2.2) then for both LUR and MI regressors it holds that

  * (i)

$\boldsymbol{K}_{nx}\big{(}\uptau_{0},\boldsymbol{\theta}^{ols}_{n}(\uptau_{0})\big{)}\Rightarrow\boldsymbol{K}_{x}\big{(}\uptau_{0},\boldsymbol{\theta}_{0}(\uptau_{0})\big{)}$,
for some $\uptau_{0}\in(0,1)$ as $n\to\infty$,

  * (ii)

$\boldsymbol{L}_{nx}\big{(}\uptau_{0},\boldsymbol{\theta}^{ols}_{n}(\uptau_{0})\big{)}\Rightarrow\boldsymbol{L}_{x}\big{(}\uptau_{0},\boldsymbol{\theta}_{0}(\uptau_{0})\big{)}$,
for some $\uptau_{0}\in(0,1)$ as $n\to\infty$,

where

$\displaystyle\boldsymbol{K}_{x}\big{(}\uptau_{0},\boldsymbol{\theta}_{0}(\uptau_{0})\big{)}$
$\displaystyle\equiv\begin{cases}\begin{bmatrix}B_{\psi_{\uptau}}(1)_{{\color[rgb]{1,0,0}(1\times
n)}}\\\
\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)dB_{\psi_{\uptau}}\end{bmatrix}_{{\color[rgb]{1,0,0}(p+1)\times
n}}&\ \ \ \ \ \ \ \ \ \ \ \ \textit{LUR},\\\ \\\
\mathcal{N}\displaystyle\left(\boldsymbol{0},\uptau_{0}(1-\uptau_{0})\times\begin{bmatrix}1&\boldsymbol{0}^{\prime}\\\
\boldsymbol{0}&\boldsymbol{V}_{xx}\end{bmatrix}_{{\color[rgb]{1,0,0}(p+1)\times(p+1)}}\right)_{{\color[rgb]{1,0,0}(p+1)\times
n}}&\ \ \ \ \ \ \ \ \ \ \ \ \textit{MI}.\end{cases}$ (3.6)
$\displaystyle\boldsymbol{L}_{x}\big{(}\uptau_{0},\boldsymbol{\theta}_{0}(\uptau_{0})\big{)}$
$\displaystyle\equiv\begin{cases}f_{u_{t}(\uptau)}(0)\times\begin{bmatrix}1&\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)^{\prime}\\\
\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)&\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)\boldsymbol{J}_{c}(r)^{\prime}\end{bmatrix}_{{\color[rgb]{1,0,0}(p+1)\times(p+1)}}&\
\ \ \textit{LUR},\\\ \\\
f_{u_{t}(\uptau)}(0)\times\begin{bmatrix}1&\boldsymbol{0}^{\prime}\\\
\boldsymbol{0}&\boldsymbol{V}_{xx}\end{bmatrix}_{{\color[rgb]{1,0,0}(p+1)\times(p+1)}}&\
\ \ \textit{MI}.\end{cases}$ (3.7)

where the stochastic matrix $\boldsymbol{V}_{xx}$ is defined by Phillips and
Magdalinos,  (2009) as

$\displaystyle\boldsymbol{V}_{xx}:=\int_{0}^{\infty}e^{r\boldsymbol{C}_{p}}\boldsymbol{\Omega}_{xx}e^{r\boldsymbol{C}_{p}}dr,\
\text{where}\
\boldsymbol{\Omega}_{xx}:=\sum_{m=-\infty}^{\infty}\mathbb{E}\left(\boldsymbol{v}_{t}\boldsymbol{v}_{t-m}^{\prime}\right)=\boldsymbol{\varphi}_{o}(1)\boldsymbol{\Sigma}\boldsymbol{\varphi}_{o}(1)^{\prime}.$

######  Remark 2.

Notice that an important aspect for robust inference in quantile
regressions161616In some studies presented in the literature the use of the
check function is defined to be the difference of the indicator function from
the quantile level, as in Zhou and Portnoy,  (1998); however both expressions
are equivalent due to the monotonicity property of the check function. is the
consistent estimation of the sparsity coefficient (see, discussion presented
in Koenker and Machado,  (1999)) and also conditions proposed by Koltchinskii,
(1997), especially in finite samples. In our setting the self-normalized
property of Wald type tests ensures that the sparsity coefficient does not
affect the estimation accuracy.

####  IVX based functionals

In this Section, we derive the asymptotic distribution of the IVX based
functionals which are useful to obtain the asymptotic behaviour of the
proposed structural break tests under the assumption of nonstationary
regressors in the model. We employ the embedded normalization version of the
instruments such that
$\tilde{\boldsymbol{Z}}_{t-1,n}:=\tilde{\boldsymbol{D}}_{n}^{-1}\tilde{\boldsymbol{z}}_{t-1}$.

######  Definition 2.

$\displaystyle\boldsymbol{K}_{nz}\big{(}\uptau_{0},\boldsymbol{\beta}_{n}^{ivx}(\uptau_{0})\big{)}$
$\displaystyle:=\sum_{t=1}^{n}\tilde{\boldsymbol{Z}}_{t-1,n}\psi_{\uptau}\big{(}u_{t}(\uptau_{0})\big{)}$
(3.8)
$\displaystyle\boldsymbol{L}_{nz}\big{(}\uptau_{0},\boldsymbol{\beta}_{n}^{ivz}(\uptau_{0})\big{)}$
$\displaystyle:=\left[\sum_{t=1}^{n}f_{u_{t}(\uptau),t-1}(0)\tilde{\boldsymbol{Z}}_{t-1,n}\tilde{\boldsymbol{Z}}_{t-1,n}^{\prime}\right]$
(3.9)
$\displaystyle\boldsymbol{M}_{nz}\big{(}\uptau_{0},\boldsymbol{\beta}^{ivx}_{n}(\uptau_{0})\big{)}$
$\displaystyle:=\left[\sum_{t=1}^{n}f_{u_{t}\left(\uptau\right),t-1}(0)\tilde{\boldsymbol{Z}}_{t-1,n}\boldsymbol{X}^{\prime}_{t-1,n}\right]$
(3.10)

for some $\uptau_{0}\in(0,1)$ where
$\psi_{\uptau}\big{(}u_{t}(\uptau_{0})\big{)}=\big{[}\uptau_{0}-\mathds{1}\big{\\{}y_{t}-\boldsymbol{\beta}^{ivx}_{n}(\uptau_{0})^{\prime}\boldsymbol{x}_{t-1}\leq
0\big{\\}}\big{]}$.

######  Corollary 4.

Under the assumption that the pair
$\left\\{y_{t},\boldsymbol{x}_{t-1}\right\\}_{t=1}^{n}$ is generated by the
model (2.1)-(2.2) then for both LUR and MI regressors it holds that

  * (i)

$\boldsymbol{K}_{nz}\big{(}\uptau_{0},\boldsymbol{\beta}_{n}^{ivx}(\uptau_{0})\big{)}\Rightarrow\boldsymbol{K}_{z}\big{(}\uptau_{0},\boldsymbol{\beta}_{0}(\uptau_{0})\big{)}\equiv\mathcal{N}\big{(}\boldsymbol{0},\uptau_{0}(1-\uptau_{0})\boldsymbol{V}_{cxz}\big{)}$,

  * (ii)

$\boldsymbol{L}_{nz}\big{(}\uptau_{0},\boldsymbol{\beta}_{n}^{ivz}(\uptau_{0})\big{)}\Rightarrow\boldsymbol{L}_{z}\big{(}\uptau_{0},\boldsymbol{\beta}_{0}(\uptau_{0})\big{)}\equiv
f_{u_{t}(\uptau)}(0)\times\boldsymbol{V}_{cxz}$,

  * (iii)

$\boldsymbol{M}_{nz}\big{(}\uptau_{0},\boldsymbol{\beta}_{n}^{ivx}(\uptau_{0})\big{)}\Rightarrow\boldsymbol{M}_{z}\big{(}\uptau_{0},\boldsymbol{\beta}_{0}(\uptau_{0})\big{)}\equiv
f_{u_{t}\left(\uptau\right)}(0)\times\boldsymbol{\Gamma}_{cxz}$,

where the definition of the asymptotic matrix $\boldsymbol{V}_{cxz}$ depends
on the stochastic dominance of the two exponent rates (see, Phillips and
Magdalinos,  (2009) and Lee,  (2016)) such as

$\displaystyle\boldsymbol{V}_{cxz}\equiv\begin{cases}\boldsymbol{V}_{zz}=\displaystyle\int_{0}^{\infty}e^{r\boldsymbol{C}_{z}}\boldsymbol{\Omega}_{xx}e^{r\boldsymbol{C}_{z}}dr,&\
\text{when}\ 0<\upgamma_{z}<\upgamma_{x}<1,\\\ \\\
\boldsymbol{V}_{xx}=\displaystyle\int_{0}^{\infty}e^{r\boldsymbol{C}_{p}}\boldsymbol{\Omega}_{xx}e^{r\boldsymbol{C}_{p}}dr,&\
\text{when}\ 0<\upgamma_{x}<\upgamma_{z}<1.\end{cases}$ (3.11)

Moreover, the definition of the moment matrix $\boldsymbol{\Gamma}_{cxz}$ is
presented by Lee,  (2016) via expression (3.4) which is the corresponding
asymptotic limit given by expression (20) in Phillips and Magdalinos,  (2009)
as given below

$\displaystyle\boldsymbol{\Gamma}_{cxz}:=\begin{cases}\displaystyle-\boldsymbol{C}_{z}^{-1}\left(\boldsymbol{\Omega}_{xx}+\int_{0}^{1}\boldsymbol{J}_{c}(r)d\boldsymbol{J}_{c}^{\prime}\right),&\
\text{when}\ \upgamma_{x}=1,\\\
-\boldsymbol{C}_{z}^{-1}\bigg{(}\boldsymbol{\Omega}_{xx}+\boldsymbol{C}_{p}\boldsymbol{V}_{xx}\bigg{)},&\
\text{when}\ 0<\upgamma_{z}<\upgamma_{x}<1,\\\ \boldsymbol{V}_{xx},&\
\text{when}\ 0<\upgamma_{x}<\upgamma_{z}<1.\end{cases}$ (3.12)

The proofs of Corollary 3 and 4 can be found in the Appendix of the paper.
Note that the stochastic convergence of these functional holds for large
sample size, $n\to\infty$, and the existence of well-defined moment matrices
with negligible higher-order terms. Therefore, to facilitate the development
of the asymptotic theory we define the following empirical process for some
parameter vector $\boldsymbol{b}\in\mathbb{R}^{p}$ such that

$\displaystyle\boldsymbol{G}_{n}\left(\uptau,\boldsymbol{b}\right):=n^{-(1+\upgamma_{x})/2}\sum_{t=1}^{n}\boldsymbol{z}_{t-1}\times\left\\{\psi_{\uptau}\big{(}u_{t}\left(\uptau\right)-\boldsymbol{x}_{t-1}^{\prime}\boldsymbol{b}\big{)}-\mathbb{E}_{\mathcal{F}_{t-1}}\left[\psi_{\uptau}\big{(}u_{t}\left(\uptau\right)-\boldsymbol{x}_{t-1}^{\prime}\boldsymbol{b}\big{)}\right]\right\\}$

where $\uptau\in(0,1)$ and $0<\upgamma_{x}<1$. In particular, the empirical
process $\boldsymbol{G}_{n}\left(\uptau,\boldsymbol{b}\right)$ is consider
stochastically $\varrho-$equicontinuous over $\mathcal{T}_{\iota}\times B$,
such that for any $\epsilon>0$,

$\displaystyle\underset{\delta\to 0}{\mathsf{lim}}\
\underset{n\to\infty}{\mathsf{lim\ sup}}\
\mathbb{P}\left(\underset{[\delta]}{\mathsf{sup}}\
\big{|}G_{n}\big{(}\uptau_{1},\boldsymbol{b}_{1}\big{)}-G_{n}\big{(}\uptau_{2},\boldsymbol{b}_{2}\big{)}\big{|}>\epsilon\right)=0,$
(3.13)

where
$[\mathcal{\delta}]:=\big{\\{}(\uptau_{1},\boldsymbol{b}_{1}),(\uptau_{2},\boldsymbol{b}_{2})\in\left(\mathcal{T}\times
B\right)^{2}:\
\varrho\big{(}(\uptau_{1},\boldsymbol{b}_{1}),(\uptau_{2},\boldsymbol{b}_{2})\big{)}<\delta\big{\\}}$.

######  Remark 3.

The above expression is often employed to derive asymptotics for quantile
regression models (with stationary regressors). Specifically, one can consider
the validity of the stochastic equicontinuity proof of Bickel,  (1975) under
nonstationarity. Practically, since the regressors employed when estimating
the inverse of the quantile function $\psi_{\uptau}(.)$, that is,
$\tilde{\boldsymbol{z}}_{t-1}$ is mildly integrated, inducing a nearly
stationary process, then the conditions given by Bickel,  (1975) are valid and
the proof follows with modifications to accommodate the nonstationary quantile
predictive regression (Lee,  (2016)).

An additional condition for convergence in probability for the empirical
process is imposed by Lemma 2, which can be employed to derive the convergence
rate of the IVX estimator for the nonstationary quantile predictive regression
model.

######  Lemma 2.

For a generic constant $\mathcal{C}_{1}>0$

$\displaystyle\mathsf{sup}\big{\\{}\big{\|}\boldsymbol{G}_{n}(\uptau,\boldsymbol{b})-\boldsymbol{G}_{n}(\uptau,\boldsymbol{0})\big{\|}:\left\lVert\boldsymbol{b}\right\rVert\leq
n^{(1+\delta)/2}\mathcal{C}_{1}\big{\\}}=o_{\mathbb{P}}(1).$ (3.14)

where $\boldsymbol{b}$ is some estimator of the model parameter vector.

More precisely, Lemma 2 provides a simplified way to derive the convergence
limit for the IVX-QR estimator (see, also Lee,  (2016)) that ensures
consistent estimation of the model parameters for the quantile predictive
regression model. A related study to our setting with detailed derivations for
nonstandard inference problems, (Wald type statistics), for nonstationary
quantile regressions is presented in the study of Goh and Knight,  (2009).
Overall, the asymptotic theory of this paper aims to combine unit root
asymptotics with empirical process methods. Specifically, we employ a two-
parameter empirical process that converges weakly to a two-parameter Brownian
motion. Therefore, our asymptotic distributions involve stochastic integrals
with respect to this two-parameter process.

###  Invariance principles for partial sum processes

To obtain the asymptotic distributions of the test statistics, we consider the
asymptotic behaviour of the partial sum processes of the functionals defined
in the previous section. We focus in the case of nonstationary regressors
which are either high persistent or mildly integrated (see, Section 2 for
definitions and Kostakis et al.,  (2015)). Moreover, since we derive and
compare the limit distributions of structural break tests based on the chose
estimation methodology, we derive invariance principles that correspond to
each of these two estimators. Therefore, we define with

$\displaystyle
S_{nx}^{ols}\big{(}\lambda,\uptau_{0},\boldsymbol{\theta}^{ols}_{n}(\uptau_{0})\big{)}:=\boldsymbol{D}_{n}^{-1}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{X}_{t-1}\psi_{\uptau}\big{(}u_{t}(\uptau_{0})\big{)},\ \
\text{for some}\ \ 0<\lambda<1,$ (3.15)

where
$u_{t}(\uptau_{0})=\big{(}y_{t}-\boldsymbol{X}_{t-1}^{\prime}\boldsymbol{\theta}^{ols}_{n}(\uptau_{0})\big{)}$
for $\uptau_{0}\in(0,1)$, which can be determined uniquely, making the mapping
$\psi_{\uptau}(\mathsf{u})\mapsto\uprho_{\uptau}^{-1}(\mathsf{u})$ one-to-one
and well-defined. Moreover, we denote with
$\boldsymbol{X}_{t-1}=\big{(}\boldsymbol{1},\boldsymbol{x}^{\prime}_{t-1}\big{)}^{\prime}$
the regressors and
$\boldsymbol{\theta}(\uptau_{0})=\big{(}\alpha(\uptau_{0}),\boldsymbol{\beta}^{\prime}(\uptau_{0})\big{)}^{\prime}$
the parameters.

Recall that for mildly integrated regressors it holds that (see, Corollary 3)

$\displaystyle\boldsymbol{K}_{nx}\big{(}\uptau_{0},\boldsymbol{\theta}^{ols}_{n}(\uptau_{0})\big{)}$
$\displaystyle:=\boldsymbol{D}_{n}^{-1}\sum_{t=1}^{n}\boldsymbol{X}_{t-1}\psi_{\uptau}\big{(}u_{t}(\uptau_{0})\big{)}\Rightarrow\mathcal{N}\displaystyle\left(\boldsymbol{0},\uptau_{0}(1-\uptau_{0})\begin{bmatrix}1&\boldsymbol{0}^{\prime}\\\
\boldsymbol{0}&\boldsymbol{V}_{xx}\end{bmatrix}\right)$ (3.16)

Similarly, we can show that
$S_{nx}^{ols}\big{(}\lambda,\uptau,\boldsymbol{\theta}^{ols}_{n}(\uptau_{0})\big{)}\Rightarrow
S_{x}\big{(}\lambda,\uptau_{0},\boldsymbol{\theta}_{0}(\uptau_{0})\big{)}$ as
$n\to\infty$, where

$\displaystyle
S_{x}\big{(}\lambda,\uptau_{0},\boldsymbol{\theta_{0}}(\uptau_{0})\big{)}\equiv\mathcal{N}\displaystyle\left(\boldsymbol{0},\uptau_{0}(1-\uptau_{0})\lambda\begin{bmatrix}1&\boldsymbol{0}^{\prime}\\\
\boldsymbol{0}&\boldsymbol{V}_{xx}\end{bmatrix}\right)$ (3.17)

for some $0<\lambda<1$. Then, for the corresponding IVX based functional it
holds that

$\displaystyle
S_{nz}^{ivx}\big{(}\lambda,\uptau_{0},\boldsymbol{\beta}^{ivx}_{n}(\uptau_{0})\big{)}:=\sum_{t=1}^{\lfloor\lambda
n\rfloor}\tilde{\boldsymbol{Z}}_{t-1,n}\psi_{\uptau}\big{(}u_{t}(\uptau_{0})\big{)}\Rightarrow\mathcal{N}\big{(}\boldsymbol{0},\uptau_{0}(1-\uptau_{0})\lambda\boldsymbol{V}_{cxz}\big{)}$
(3.18)

where
$\tilde{\boldsymbol{Z}}_{t-1,n}:=\tilde{\boldsymbol{D}}_{n}^{-1}\tilde{\boldsymbol{z}}_{t-1}$
since we employ the corresponding dequantiled model.

######  Definition 3.

$\displaystyle\hat{\mathcal{J}}^{ols}_{nx}\big{(}\lambda,\uptau_{0},\widehat{\boldsymbol{\theta}}^{ols}_{n}(\uptau_{0})\big{)}$
$\displaystyle:=\big{(}\boldsymbol{X}^{\prime}\boldsymbol{X}\big{)}^{-1/2}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{X}_{t-1}\psi_{\uptau}\left(y_{t}-\boldsymbol{X}_{t-1}^{\prime}\widehat{\boldsymbol{\theta}}^{ols}_{n}(\uptau_{0})\right),$
(3.19)
$\displaystyle\hat{\mathcal{J}}^{ivx}_{nx}\big{(}\lambda,\uptau_{0},\widehat{\boldsymbol{\beta}}^{ivx}_{n}(\uptau_{0})\big{)}$
$\displaystyle:=\big{(}\boldsymbol{X}^{\prime}\tilde{\boldsymbol{Z}}\big{)}^{-1/2}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\tilde{\boldsymbol{Z}}_{t-1,n}\psi_{\uptau}\left(y_{t}-\boldsymbol{X}_{t-1,n}^{\prime}\widehat{\boldsymbol{\beta}}^{ivx}_{n}(\uptau_{0})\right),$
(3.20)
$\displaystyle\hat{\mathcal{J}}^{ivz}_{nx}\big{(}\lambda,\uptau_{0},\widehat{\boldsymbol{\beta}}^{ivz}_{n}(\uptau_{0})\big{)}$
$\displaystyle:=\big{(}\tilde{\boldsymbol{Z}}^{\prime}\tilde{\boldsymbol{Z}}\big{)}^{-1/2}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\tilde{\boldsymbol{Z}}_{t-1,n}\psi_{\uptau}\left(y_{t}-\tilde{\boldsymbol{Z}}_{t-1,n}^{\prime}\widehat{\boldsymbol{\beta}}^{ivz}_{n}(\uptau_{0})\right).$
(3.21)

for some $0<\lambda<1$ and $\uptau_{0}\in(0,1)$.

Consider the functionals given by Definition 3, then when we employ the OLS
estimator for a model with mildly integrated regressors,
$\upgamma_{x}\in(0,1)$, the following limit result holds

$\displaystyle\hat{\mathcal{J}}^{ols}_{nx}\big{(}\lambda,\uptau_{0},\widehat{\boldsymbol{\theta}}^{ols}_{n}(\uptau_{0})\big{)}$
$\displaystyle=\big{(}\boldsymbol{X}^{\prime}\boldsymbol{X}\big{)}^{-1/2}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{X}_{t-1}\psi_{\uptau}\left(y_{t}-\boldsymbol{X}_{t-1}^{\prime}\widehat{\boldsymbol{\theta}}^{ols}_{n}(\uptau_{0})\right)$
$\displaystyle\equiv\left(\boldsymbol{D}_{n}^{-1}\left[\sum_{t=1}^{n}\boldsymbol{X}^{\prime}_{t-1}\boldsymbol{X}_{t-1}\right]\boldsymbol{D}_{n}^{-1}\right)^{-1/2}\left\\{\boldsymbol{D}_{n}^{-1}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{X}_{t-1}\psi_{\uptau}\big{(}u_{t}(\uptau_{0})\big{)}\right\\}$
$\displaystyle\Rightarrow\left\\{\begin{bmatrix}1&\boldsymbol{0}^{\prime}\\\
\boldsymbol{0}&\boldsymbol{V}_{xx}\end{bmatrix}\right\\}^{-1/2}\times\mathcal{N}\displaystyle\left(\boldsymbol{0},\uptau_{0}(1-\uptau_{0})\lambda\begin{bmatrix}1&\boldsymbol{0}^{\prime}\\\
\boldsymbol{0}&\boldsymbol{V}_{xx}\end{bmatrix}\right)$
$\displaystyle=\sqrt{\uptau_{0}(1-\uptau_{0})}\times\mathcal{N}\big{(}\boldsymbol{0},\lambda\boldsymbol{I}_{p}\big{)}.$
(3.22)

since the term $\frac{1}{n^{1+\upgamma_{x}}}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{x}_{t-1}\boldsymbol{x}_{t-1}^{\prime}\overset{\mathbb{P}}{\to}\lambda\boldsymbol{V}_{xx}$
converges in probability. A similar limit result holds for the IVZ based
functional such that
$\hat{\mathcal{J}}^{ivz}_{nx}\big{(}\lambda,\uptau_{0},\widehat{\boldsymbol{\theta}}^{ivz}_{n}(\uptau_{0})\big{)}\Rightarrow\sqrt{\uptau_{0}(1-\uptau_{0})}\times\mathcal{N}\big{(}\boldsymbol{0},\lambda\boldsymbol{I}_{p}\big{)}$,
regardless of whether the regressors exhibit high persistence. On the other
hand, the OLS based functional has a nonstandard limit distribution with
regressors of high persistence. In the case of the IVX functional one needs to
consider the limit result for these two classes of persistence separately.
These conjectures are summarized and proved by Proposition 1 in the next
section where we formalize the test statistics.

###  Test Statistics

We consider as detectors two types of test statistics commonly employed in the
literature related to structural break testing methodologies. The first type
of test corresponds to the fluctuation type statistic studied by Qu,  (2008)
specifically for a quantile regression model, while the second type of test
corresponds to the Wald statistic proposed by the seminal paper of Andrews,
(1993) for the linear regression model. Both test statistics utilize the
supremum functional since the underline assumptions allow for a structural
break for the coefficients of the nonstationary quantile predictive regression
model at an unknown break-point location within the full sample.

Therefore, the null hypothesis of interest (e.g., see (2.3)) is formulated as
below

$\displaystyle\mathcal{H}^{(A)}_{0}:\boldsymbol{\theta}^{(1)}_{n}\big{(}\lambda;\uptau_{0}\big{)}=\boldsymbol{\theta}^{(2)}_{n}\big{(}\lambda;\uptau_{0}\big{)}\
\ \ \text{versus}\ \ \
\mathcal{H}^{(A)}_{1}:\boldsymbol{\theta}^{(1)}_{n}\big{(}\lambda;\uptau_{0}\big{)}\neq\boldsymbol{\theta}^{(2)}_{n}\big{(}\lambda;\uptau_{0}\big{)}$
(3.23)

where
$\boldsymbol{\theta}^{(j)}_{n}(\lambda;\uptau_{0})=\big{(}\alpha^{(j)}_{n}(\lambda;\uptau_{0}),\boldsymbol{\beta}^{(j)}_{n}(\lambda;\uptau_{0})^{\prime}\big{)}^{\prime}$,
for $j\in\left\\{1,2\right\\}$ and the location of the break-point is denoted
with $\kappa=\lfloor\lambda n\rfloor$ for some $0<\lambda<1$. Specifically,
the implementation of structural break tests for the purpose of detecting
parameter instability in nonstationary quantile predictive regressions is a
novel aspect in the literature. To facilitate for the development of large
sample theory, Assumption 2 presents necessary conditions relating the matrix
moments to the quantile structure of the model.

######  Assumption 2.

The regressors of the nonstationary quantile predictive regression model which
follow a near unit process, are assumed to satisfy the following conditions:

  * (a)

$\displaystyle\underset{n\to\infty}{\mathsf{plim}}\
\frac{1}{n^{1+\upgamma_{x}}}\sum_{t=1}^{\lfloor\lambda
n\rfloor}f_{u_{t}(\uptau),t-1}(0)\boldsymbol{x}_{t-1}\boldsymbol{x}_{t-1}^{\prime}=\lambda
f_{u_{t}(\uptau)}(0)\boldsymbol{V}_{xx}$, uniformly for $0<\lambda<1$,

  * (b)

$\displaystyle\underset{n\to\infty}{\mathsf{plim}}\
\frac{1}{n^{1+\upgamma_{x}}}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{x}_{t-1}\boldsymbol{x}_{t-1}^{\prime}=\lambda\boldsymbol{V}_{xx}$,
uniformly for some $0<\lambda<1$, where $\boldsymbol{V}_{xx}$ is a $p\times p$
non-random positive definite matrix and $\upgamma_{x}\in(0,1)$,

  * (c)

$\mathbb{E}\left(\boldsymbol{x}_{t-1}\boldsymbol{x}_{t-1}^{\prime}\right)^{2+s}<L$
with $s>0$ and $L<\infty$ for all $1\leq t\leq n$,

  * (d)

there exists a $\delta>0$ and an $M<\infty$, such that
$n^{-1}\sum_{t=1}^{n}\mathbb{E}\left\lVert\boldsymbol{x}_{t-1}\right\rVert^{3(1+\delta)}<M$
and
$\mathbb{E}\left(n^{-1}\sum_{t=1}^{n}\left\lVert\boldsymbol{x}_{t-1}\right\rVert^{3}\right)^{(1+\delta)}<M$
hold for any $n$.

Assumption 2 (a) and (b) are standard convergence in probability limits for
the nonstationary quantile predictive regression model. Assumption 2 (c) is
employed for the convergence of the weighted empirical process
$n^{-1/2}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{x}_{t-1}\big{(}\uptau_{0}-\mathds{1}\left\\{F_{y|x}(y_{t})\leq\uptau_{0}\right\\}\big{)}$.
Furthermore, Assumption 2 (d) ensures stochastic equicontinuity (see, Chapter
2 in Van Der Vaart and Wellner,  (1996)) of the sequential empirical process
based on estimated quantile regression residuals, which is needed to establish
weak convergence of the tests (see Bai,  (1996)). Moreover, in the case of
possibly nonstationary regressors, since standard quantile regression
estimators follow a locally uniform weak convergence (see De Haan and
Ferreira,  (2006)) then invariance principles hold uniformly for
$\lambda\in(0,1)$.

####  Fluctuation type tests

Specifically, since we assume that the true break point is unknown, we need to
search over all possible candidate subsets within the full sample.
Furthermore, according to Qu,  (2008) recentering
$\hat{\mathcal{J}}_{n}\big{(}\lambda,\uptau_{0},\widehat{\boldsymbol{\beta}}_{n}(\uptau_{0})\big{)}$
by the quantity
$\lambda\hat{\mathcal{J}}_{n}\big{(}1,\uptau_{0},\widehat{\boldsymbol{\beta}}_{n}(\uptau_{0})\big{)}$
often yields better finite sample performance. Such considerations lead to the
following test statistic:

$\displaystyle\mathcal{SQ}_{n}(\lambda;\uptau_{0})=\underset{\lambda\in[0,1]}{\mathsf{sup}}\bigg{\|}\frac{1}{\sqrt{\uptau_{0}(1-\uptau_{0})}}\bigg{[}\hat{\mathcal{J}}_{n}\big{(}\lambda,\uptau_{0},\widehat{\boldsymbol{\beta}}_{n}(\uptau_{0})\big{)}-\lambda\hat{\mathcal{J}}_{n}\big{(}1,\uptau_{0},\widehat{\boldsymbol{\beta}}_{n}(\uptau_{0})\big{)}\bigg{]}\bigg{\|}_{\infty}$
(3.24)

where $\left\lVert.\right\rVert_{\infty}$ is the $\mathsf{sup}-$norm such that
for a generic vector
$\mathsf{z}=\left(\mathsf{z}_{1},...,\mathsf{z}_{p}\right)$ implies that
$\left\lVert.\right\rVert_{\infty}:=\mathsf{max}\left(|\mathsf{z}_{1}|,...,|\mathsf{z}_{p}|\right)$
(see, Koenker and Xiao,  (2002)).

We focus on the implementation of two different estimation methodologies.
Therefore, to investigate the practical use of the proposed fluctuation type
test for structural break detection in the nonstationary quantile predictive
regression model, we consider the asymptotic distribution of the test
statistics according to the estimator employed to construct the test function.
Thus, Proposition 1 summarizes the formulations of the test according to the
estimation methodology employed for a fixed quantile level
$\uptau_{0}\in(0,1)$.

######  Proposition 1.

Under the null hypothesis $\mathcal{H}_{0}^{(A)}$ and given that Assumptions
1-2 hold, then the fluctuation type statistics weakly converge to the limit
distributions below

$\displaystyle\textbf{({i})}\ \mathcal{SQ}^{ols}_{n}(\lambda;\uptau_{0})$
$\displaystyle:=\underset{\lambda\in[0,1]}{\mathsf{sup}}\
\bigg{\|}\frac{1}{\sqrt{\uptau_{0}(1-\uptau_{0})}}\bigg{[}\hat{\mathcal{J}}_{n}\left(\lambda,\uptau_{0},\widehat{\boldsymbol{\theta}}^{ols}_{n}(\uptau_{0})\right)-\lambda\hat{\mathcal{J}}_{n}\left(1,\uptau_{0},\widehat{\boldsymbol{\theta}}^{ols}_{n}(\uptau_{0})\right)\bigg{]}\bigg{\|}_{\infty}$
$\displaystyle\Rightarrow\begin{cases}\underset{\lambda\in[0,1]}{\mathsf{sup}}\
\big{\|}\boldsymbol{\mathcal{BB}}_{p+1}(\lambda)\big{\|}_{\infty},&\ \
\text{when}\ \upgamma_{x}\in(0,1)\\\ \underset{\lambda\in[0,1]}{\mathsf{sup}}\
\mathbb{S}_{xx}^{-1/2}\times\left\\{\begin{bmatrix}\mathbf{\mathcal{BB}}_{\psi_{\uptau}}(\lambda)_{{\color[rgb]{1,0,0}(1\times
n)}}\\\
\displaystyle\mathbf{\mathcal{JB}}_{\psi_{\uptau}}(\lambda)_{{\color[rgb]{1,0,0}(p\times
n)}}\end{bmatrix}_{{\color[rgb]{1,0,0}(p+1)\times n}}\right\\},&\ \
\text{when}\ \upgamma_{x}=1\end{cases}$ $\displaystyle\textbf{({ii})}\
\mathcal{SQ}^{ivx}_{n}(\lambda;\uptau_{0})$
$\displaystyle:=\underset{\lambda\in[0,1]}{\mathsf{sup}}\
\bigg{\|}\frac{1}{\sqrt{\uptau_{0}(1-\uptau_{0})}}\bigg{[}\hat{\mathcal{J}}_{n}\left(\lambda,\uptau_{0},\widehat{\boldsymbol{\beta}}^{ivx}_{n}(\uptau_{0})\right)-\lambda\hat{\mathcal{J}}_{n}\left(1,\uptau_{0},\widehat{\boldsymbol{\beta}}^{ivx}_{n}(\uptau_{0})\right)\bigg{]}\bigg{\|}_{\infty}$
$\displaystyle\Rightarrow\underset{\lambda\in[0,1]}{\mathsf{sup}}\
\big{\|}\boldsymbol{\mathcal{BB}}_{p}(\lambda)\big{\|}_{\infty},\ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{when}\
\upgamma_{x}=(0,\upgamma_{z})$ $\displaystyle\textbf{({iii})}\
\mathcal{SQ}^{ivz}_{n}(\lambda;\uptau_{0})$
$\displaystyle:=\underset{\lambda\in[0,1]}{\mathsf{sup}}\
\bigg{\|}\frac{1}{\sqrt{\uptau_{0}(1-\uptau_{0})}}\bigg{[}\hat{\mathcal{J}}_{n}\left(\lambda,\uptau_{0},\widehat{\boldsymbol{\beta}}^{ivz}_{n}(\uptau_{0})\right)-\lambda\hat{\mathcal{J}}_{n}\left(1,\uptau_{0},\widehat{\boldsymbol{\beta}}^{ivz}_{n}(\uptau_{0})\right)\bigg{]}\bigg{\|}_{\infty}$
$\displaystyle\Rightarrow\underset{\lambda\in[0,1]}{\mathsf{sup}}\
\big{\|}\boldsymbol{\mathcal{BB}}_{p}(\lambda)\big{\|}_{\infty},\ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{when}\ \upgamma_{x}=(0,1]$

where $\boldsymbol{\mathcal{BB}}_{p}(.)$ is a vector of $p$ independent
Brownian bridge processes171717Note that $\boldsymbol{\mathcal{BB}}_{p}(.)$ is
known as the square of a standardized tied-down Bessel process of order $p$.
on $\mathcal{D}_{\mathbb{R}^{p}}\left([0,1]\right)$,

$\displaystyle\mathbb{S}_{xx}:=\begin{bmatrix}1&\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)^{\prime}dr\\\
\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)dr&\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)\boldsymbol{J}_{c}(r)^{\prime}\end{bmatrix}_{{\color[rgb]{1,0,0}(p+1)\times(p+1)}}\
\text{with}\ \ 0<r<1$

where $\mathbb{S}_{xx}$ is a positive definite stochastic matrix,
$\mathbf{\mathcal{BB}}_{\psi_{\uptau}}(\lambda):=B_{\psi_{\uptau}}(\lambda)-\lambda
B_{\psi_{\uptau}}(1)$.

A necessary condition to apply weak convergence arguments that yields
invariance principles for partial sum processes in the Skorokhod space
$\mathcal{D}\left([0,1]\right)$ follows

$\displaystyle\underset{\lambda\in(0,1)}{\mathsf{sup}}\boldsymbol{D}_{n}^{-1}\bigg{|}\left[\hat{\mathcal{J}}_{n}\big{(}\lambda,\uptau_{0},\widehat{\boldsymbol{\beta}}_{n}(\uptau_{0})\big{)}-\lambda\hat{\mathcal{J}}_{n}\big{(}1,\uptau_{0},\hat{\boldsymbol{\beta}}_{n}(\uptau_{0})\big{)}\right]-\left[\mathcal{J}_{n}\big{(}\lambda,\uptau_{0},\boldsymbol{\beta}_{n}(\uptau_{0})\big{)}-\lambda\mathcal{J}_{n}\big{(}1,\uptau_{0},\boldsymbol{\beta}_{n}(\uptau_{0})\big{)}\right]\bigg{|}=o_{\mathbb{P}}(1).$

The proposed test statistics extend the fluctuation type tests studied by Qu,
(2008), to the nonstationary quantile predictive regression model of our
setting. We compare the instrumental variable based method to the classical
OLS approach for constructing the fluctuation test. Moreover, we employ the
IVZ estimator given by Proposition 1 (iii), which replaces the original
covariate vector with the constructed instruments as a post-estimation
correction method that permits to obtain further simplifications of asymptotic
terms. Our asymptotic theory analysis shows that the fluctuation type test
weakly converges into a Brownian bridge limit when the IVZ estimator is
employed and the same limit holds for both OLS and IVX based tests under mild
integratedness.

On the other hand, under high persistence the fluctuation type test based on
the OLS estimator is proved to have a weak convergence into a nonstandard and
nonpivotal asymptotic distribution.181818Similar results are proved by
Katsouris,  (2021) who propose structural break tests for the linear
predictive regression model. Within the particular framework, the author
proves that the asymptotic distribution of sup-Wald type statistics when the
regressors of the model exhibit high persistence are nonstandard (when the OLS
estimator is employed), since the asymptotic distribution of the test depend
on the unknown persistence coefficient. A similar asymptotic result holds for
the corresponding IVX based test statistic when $\upgamma_{x}=1$ such that

$\displaystyle\hat{\mathcal{J}}_{n}$
$\displaystyle\left(\lambda,\uptau_{0},\widehat{\boldsymbol{\beta}}^{ivx}_{n}(\uptau_{0})\right)-\lambda\hat{\mathcal{J}}_{n}\left(1,\uptau_{0},\widehat{\boldsymbol{\beta}}^{ivx}_{n}(\uptau_{0})\right)$
$\displaystyle\Rightarrow\sqrt{\uptau_{0}(1-\uptau_{0})}\times\boldsymbol{\Gamma}_{cxz}^{-1/2}\times\bigg{\\{}\mathcal{N}\displaystyle\bigg{(}\boldsymbol{0},\lambda\boldsymbol{V}_{cxz}\bigg{)}-\lambda\mathcal{N}\bigg{(}\boldsymbol{0},\boldsymbol{V}_{cxz}\bigg{)}\bigg{\\}}$
(3.25)

unless we assume that the exponent rate of persistence is such that
$\upgamma_{x}(0,\upgamma_{z})$.

Fluctuation type statistics have been previously examined as a detector for
parameter instability in studies such as Kuan and Chen,  (1994), Chu et al.,
(1996) and Leisch et al.,  (2000). More precisely, the statistical advantage
of these statistics lies in the fact that they utilize properties of the
maximum of Wiener processes (see Révész,  (1982)) and consequently weakly
convergence arguments as defined by Billingsley,  (1968) can be employed to
derive their asymptotic behaviour. Furthermore, these class of tests belong to
the same class as CUSUM type tests although in the latter case the test
function is constructed based on regression residuals (see Kulperger et al.,
(2005)).

Overall within the nonstationary quantile predictive regression model
framework of our study, we observe some important conclusions for the
implementation of fluctuation type tests as structural break detectors.
Firstly, the asymptotic distributionof the test statistic
$\mathcal{SQ}(\lambda;\uptau_{0})$ depends on the chosen estimator when
constructing the test function as seen from the limiting distributions when
constructing the test based on the two estimators, under high persistent
regressors. On the other, under the assumption of mildly integrated regressors
these test statistics weakly converge into a Brownian bridge type limit
regardless of the chosen estimator when constructing the test function.
Therefore, for the particular persistence class fluctuatuon type tests depends
only on the number of parameters subject to structural break since the
nuisance coefficient of persistence that captures the nonstationary properties
of predictors is filtered out. Secondly, these test statistics do not require
to estimate the sparsity coefficient
$f_{y|x}\big{(}F^{-1}_{y|x}(\uptau_{0})\big{)}$. According to Qu,  (2008) this
occurs since the subgradient, when evaluated at the true parameter value
$\boldsymbol{\beta}_{0}(\uptau_{0})$, does not depend on the distribution of
the errors.

Thus, conducting statistical inference with some prior information regarding
the presence of persistence regressors, using fluctuation type tests as
detectors is preferable to construct the test function based on the IVZ
estimator which can lead to conventional inference methods (e.g., using
tabulated critical values).

Next, we examine the self-normalized191919Specifically a relevant application
of self normalized statistics is the construction of confidence intervals for
model parameters as in the study of Shao,  (2010). We leave this aspect for
future research. property of Wald type tests by deriving the related
asymptotic theory. Due to the assumptions and conditions under which we
construct the proposed test statistics, to examine their limit distributions
stochastic equicontinuity arguments are necessary in proofs (see, Newey,
(1991)).

####  Wald type tests

We now introduce the Wald type tests based on the two estimation methodologies
which we focus on (OLS versus IVX based tests). The formulation of the model
under the null and under the alternative hypothesis can change the
interpretation in the notation we employ for model parameters. One approach is
to employ the formulations given by expressions (2.28)-(2.29). In that case,
the Wald type test is constructed for testing the null hypothesis that the
parameter vector $\boldsymbol{\upbeta}_{(2)}(\uptau_{0})$ which implies that
we are testing the null hypothesis that
$\boldsymbol{\beta}_{1}(\uptau_{0})-\boldsymbol{\beta}_{1}(\uptau_{0})=\boldsymbol{0}$.
However, one has to be careful when constructing the covariance matrix as the
regressors need to be adjusted accordingly. Furthermore, a second approach is
to construct the stacked regressors that correspond to the time series
observations from each of the two subsamples.

To simplify the notation, we employ the second approach and denote with
$\widehat{\boldsymbol{\beta}}_{1}(\lambda;\uptau_{0})$ the estimator of
$\boldsymbol{\beta}_{0}(\uptau_{0})$, using observations up to
$\kappa=\lfloor\lambda n\rfloor$ for some $0<\lambda<1$ and with
$\widehat{\boldsymbol{\beta}}_{2}(\lambda;\uptau_{0})$ the corresponding
parameter estimator based on the remaining observations in the sample.
Moreover, denote with $\tilde{\boldsymbol{X}}\equiv\big{[}\boldsymbol{X}_{1}\
\boldsymbol{X}_{2}\big{]}$ and $\boldsymbol{R}\equiv\big{[}\boldsymbol{I}_{p}\
-\boldsymbol{I}_{p}\big{]}$ the selection matrix, then
$\Delta\widehat{\boldsymbol{\beta}}_{n}(\lambda;\uptau_{0}):=\big{(}\widehat{\boldsymbol{\beta}}_{2}(\lambda;\uptau_{0})-\widehat{\boldsymbol{\beta}}_{1}(\lambda;\uptau_{0})\big{)}$.
Then, the Wald test for testing the null hypothesis that the two regimes have
equivalent parameter vectors, based on the OLS estimator and some unknown
break-point $\kappa=\lfloor\lambda n\rfloor$ is formulated as below

$\displaystyle\mathcal{W}_{n}(\lambda,\uptau_{0})=n\Delta\widehat{\boldsymbol{\beta}}_{n}(\lambda;\uptau_{0})^{\prime}\big{[}\widehat{\boldsymbol{V}}_{n}(\lambda;\uptau_{0})\big{]}^{-1}\Delta\widehat{\boldsymbol{\beta}}_{n}(\lambda;\uptau_{0})$
(3.26)

where $\widehat{\boldsymbol{V}}_{n}(\lambda;\uptau_{0})$ is a consistent
estimate of the limiting variance of
$\Delta\widehat{\boldsymbol{\beta}}_{n}(\lambda;\uptau_{0})$ under the null
hypothesis, $\mathcal{H}_{0}^{(A)}$, of no parameter instability for a fixed
quantile level $\uptau_{0}\in(0,1)$. The variance estimator is a key quantity
which will affect the robustness of Wald type tests and takes different forms
depending on the estimation method we employ when fitting the nonstationary
quantile predictive regression.

Consider the following limiting variance estimate

$\displaystyle\underset{n\to\infty}{\mathsf{plim}}\
\big{\\{}\widehat{\boldsymbol{V}}_{n}(\lambda;\uptau_{0})\big{\\}}\equiv\left[\frac{\uptau_{0}(1-\uptau_{0})}{\lambda(1-\lambda)}\right]\boldsymbol{\Omega}_{0},\
\ \ \ (\lambda,\uptau_{0})\in(0,1)\times(0,1),$ (3.27)

where
$\boldsymbol{\Omega}_{0}=\boldsymbol{H}_{0}^{-1}\boldsymbol{D}_{0}\boldsymbol{H}_{0}^{-1}$
is the unknown variance of the OLS-Wald test.

Furthermore, define with

$\displaystyle\boldsymbol{H}_{0}$
$\displaystyle=\underset{n\to\infty}{\mathsf{plim}}\
\frac{1}{n}\sum_{t=1}^{n}f_{y|x}\left(y_{t}|\boldsymbol{x}_{t-1}\right)\boldsymbol{x}_{t-1}^{\prime}\boldsymbol{x}_{t-1}$
(3.28) $\displaystyle\boldsymbol{D}_{0}$
$\displaystyle=\underset{n\to\infty}{\mathsf{plim}}\
\frac{1}{n}\sum_{t=1}^{n}\boldsymbol{x}_{t-1}^{\prime}\boldsymbol{x}_{t-1}$
(3.29)

where $f_{y|x}(.|\boldsymbol{x}_{t-1})$ and $F_{y|x}(.|\boldsymbol{x}_{t-1})$
are the conditional density and conditional cumulative distribution function
of $y_{t}$ respectively (see Goh and Knight,  (2009) and Aue et al.,  (2017)).

However, the particular form of the asymptotic variance only holds under the
assumption of stationarity in which case the variance estimator simplifies
since it does not depend on any nuisance parameters (such as the coefficients
of persistence) and thus equivalent matrix moments to the expressions in Qu,
(2008) (see also Andrews,  (1993)) hold. In our setting, we consider
alternative variance estimators based on both the chosen estimator and the
persistence class of regressors.

In any of the aforementioned cases, the supremum Wald test is defined as below

$\displaystyle\mathcal{SW}_{n}(\lambda;\uptau_{0}):=\underset{\lambda\in\Lambda_{\eta}}{\mathsf{sup}}\
\bigg{\\{}n\Delta\widehat{\boldsymbol{\beta}}_{n}(\lambda;\uptau_{0})^{\prime}\big{[}\widehat{\boldsymbol{V}}_{n}(\lambda;\uptau_{0})\big{]}^{-1}\Delta\widehat{\boldsymbol{\beta}}_{n}(\lambda;\uptau_{0})\bigg{\\}}$
(3.30)

In practise, a symmetric trimming coefficient is employed such that
$0<\eta<1/2$ which lead to the admissible set $\Lambda_{\eta}:=[\eta,1-\eta]$,
in order to ensure that the test statistics converge in distribution under the
null hypothesis. Therefore, we investigate the asymptotic behaviour of the OLS
based Wald test for the nonstationary quantile predictive regression model
given by (2.1)-(2.2) which encompasses the case of stationary regressors.
Under the assumption of stable regressors the asymptotic variance of the OLS-
Wald test is equivalent to the case when regressors are stationary and
ergodic.

Under the assumption of nonstationarity, the formulation of the OLS-Wald test
statistic requires to determine the asymptotic behaviour of the following
quantities

$\displaystyle\widehat{\boldsymbol{\beta}}_{1}^{ols}(\lambda;\uptau_{0})$
$\displaystyle=\left(\frac{1}{\kappa}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{x}_{t-1}\boldsymbol{x}_{t-1}^{\prime}\right)^{-1}\left(\frac{1}{\kappa}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{x}_{t-1}y_{t}\right)$ (3.31)
$\displaystyle\widehat{\boldsymbol{\beta}}_{2}^{ols}(\lambda;\uptau_{0})$
$\displaystyle=\left(\frac{1}{n-\kappa}\sum_{t=\lfloor\lambda
n\rfloor+1}^{n}\boldsymbol{x}_{t-1}\boldsymbol{x}_{t-1}^{\prime}\right)^{-1}\left(\frac{1}{n-\kappa}\sum_{t=\lfloor\lambda
n\rfloor+1}^{n}\boldsymbol{x}_{t-1}y_{t}\right)$ (3.32)

and
$\Delta\widehat{\boldsymbol{\beta}}^{ols}_{n}(\lambda;\uptau_{0})=\widehat{\boldsymbol{\beta}}_{2}^{ols}(\lambda;\uptau_{0})-\widehat{\boldsymbol{\beta}}_{1}^{ols}(\lambda;\uptau_{0})$,
for some $0<\lambda<1$ and $\uptau\in(0,1)$. Then, due to orthogonality of the
two set of regressors the covariance matrix simplifies into the following
expression:

$\displaystyle\widehat{\boldsymbol{V}}_{n}^{ols}(\lambda;\uptau_{0}):=\bigg{[}\boldsymbol{R}\big{(}\tilde{\boldsymbol{X}}^{\prime}\tilde{\boldsymbol{X}}\big{)}^{-1}\boldsymbol{R}^{\prime}\bigg{]}\equiv\bigg{[}\big{(}\boldsymbol{X}_{1}^{\prime}\boldsymbol{X}_{1}\big{)}^{-1}+\big{(}\boldsymbol{X}_{2}^{\prime}\boldsymbol{X}_{2}\big{)}^{-1}\bigg{]}$
(3.33)

###  Asymptotic Theory

As we discussed previously, the limit theory of the Wald type statistics for
both the OLS and IVX estimators seems more difficult than the limit results
for the fluctuation type tests, especially due to the dependence of regressors
and parameter estimates to the nuisance parameter of persistence. Therefore,
here we generalize the functionals introduced in Section 3.2 and 3.2 in order
to study their asymptotic properties which can alleviate the difficulty in
obtaining stochastic approximations under the presence of abstract degree of
persistence; simplifying this way derivations for their limit distributions.

####  OLS-Wald test statistic

We focus on the asymptotic theory for the OLS-Wald test statistic, which is
employed as a structural break detection for the nonstationary quantile
predictive regression model. In particular, we investigate the asymptotic
behaviour of the partial sum processes for the OLS based functionals we
introduced previously. For a general parameter vector
$\boldsymbol{b}\in\mathbb{R}^{p}$ we denote with
$S_{n}(\lambda,\uptau_{0},\boldsymbol{b})$ the partial sum given by the
following expression

$\displaystyle
S_{n}\big{(}\lambda,\uptau_{0},\boldsymbol{b}\big{)}=n^{-1/2}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{x}_{t-1}\psi_{\uptau}\big{(}y_{t}-\boldsymbol{x}_{t-1}^{\prime}\boldsymbol{b}\big{)}$
(3.34)

where $\psi_{\uptau}(\mathsf{u})$ is such that
$\psi_{\uptau}(\mathsf{u}):=\big{[}\uptau-\mathds{1}\left\\{\mathsf{u}\leq
0\right\\}\big{]}$. Therefore, $S_{n}(\lambda,\uptau_{0},\boldsymbol{b})$ is
written as

$\displaystyle
S_{n}\big{(}\lambda,\uptau_{0},\boldsymbol{b}\big{)}=n^{-1/2}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{x}_{t-1}\big{[}\uptau_{0}-\mathds{1}\big{\\{}y_{t}-\boldsymbol{x}_{t-1}^{\prime}\boldsymbol{b}\leq
0\big{\\}}\big{]}.$ (3.35)

Following conventional laws of invariance principles for i.i.d partial sums
the induced sequence of increments are tight within a suitable topological
space202020Related theory to weak convergence arguments of partial sum
processes can be found in various studies. For instance, Wang and Phillips,
(2012) redefine the innovation sequence of their model, in the context of
specification testing under nonstationarity, to a richer probability space
which contains a standard Brownian motion. To do this, a triangular
representation of the near unit process is employed in order to investigate
the asymptotic behaviour of the transformed functional with respect to this
triangular array. Although this would be an interesting way to represent our
functionals we avoid the introduction of triangular arrays which could be more
challenging to handle. and in fact they converge weakly to Gaussian processes.
Therefore, investigating the asymptotic behaviour and properties of these
functionals is useful for the development of the asymptotic theory of the
proposed test statistics as well as for other applications. In order to do
this, we consider centering the quantity
$\mathds{1}\big{\\{}y_{t}-\boldsymbol{x}_{t-1}^{\prime}\boldsymbol{b}\leq
0\big{\\}}$ at its expectation conditional on $\boldsymbol{x}_{t-1}$, instead
around the quantile level $\uptau_{0}$. Furthermore, since we assume that the
nonparametric functional given by expression (3.35) can be employed as a
stochastic process in $\mathcal{D}\left([0,1]\right)$, which is the
topological space of all right continuous functions with left limits then we
can derive an invariance principle for this partial sum process.

To simplify derivations for the asymptotic theory, and following Qu,  (2008),
we define the quantity
$\widetilde{S}_{n}\big{(}\lambda,\uptau_{0},\boldsymbol{b}\big{)}$ with the
expression below

$\displaystyle\widetilde{S}_{n}\big{(}\lambda,\uptau_{0},\boldsymbol{b}\big{)}=n^{-1/2}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{x}_{t-1}\big{[}F_{y|x}\big{(}\boldsymbol{x}_{t-1}^{\prime}\boldsymbol{b}\big{)}-\mathds{1}\big{\\{}y_{t}-\boldsymbol{x}_{t-1}^{\prime}\boldsymbol{b}\leq
0\big{\\}}\big{]}.$ (3.36)

where $F_{y|x}\big{(}\boldsymbol{x}_{t-1}^{\prime}\boldsymbol{b}\big{)}$ is
assumed to be monotonic. A necessary and sufficient condition for the
monotonicity property of the cumulative distribution function to hold is
presented by Lemma LABEL:lemma3A (see, also Lemma A1 in Qu,  (2008)).
Consequently, we obtain that

$\displaystyle
S_{n}\big{(}\lambda,\uptau_{0},\boldsymbol{b}\big{)}\equiv\widetilde{S}_{n}\big{(}\lambda,\uptau_{0},\boldsymbol{b}\big{)}+n^{-1/2}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{x}_{t-1}\big{[}\uptau_{0}-F_{y|x}\big{(}\boldsymbol{x}_{t-1}^{\prime}\boldsymbol{b}\big{)}\big{]}.$
(3.37)

Our research objective here is to establish the weak convergence argument that
holds for the random quantity
$S_{n}\left(\lambda,\uptau_{0},\boldsymbol{b}\right)$ on
$\left(\mathcal{D}[0,1]\right)^{2}$ by accommodating for the different
convergence rates with appropriate matrix normalizations according to the
estimator employed in each case. Moreover, the limit results of these
functionals212121Notice that the proposed functionals in this paper similar to
the framework of Qu,  (2008), clearly depend on the estimated parameter
vector. Therefore, in our setting the assumption of a nonstationary quantile
model contributes to some challenging asymptotic theory aspects, which we are
motivated to tackle. Moreover, we shall note that a related large stream of
literature considers functionals of estimated residuals with associated test
statistics such as CUSUM and CUSUM-square commonly employed in the change-
point literature. We avoid presenting the related literature here, as it
beyond our scope. can be utilized to show the following type of stochastic
convergence

$\displaystyle\sqrt{n}\left(\widehat{\boldsymbol{\beta}}^{ols}_{1}(\lambda;\uptau_{0})-\boldsymbol{\beta}_{0}(\uptau_{0})\right)=\mathcal{O}_{\mathbb{P}}(1),$
(3.38)

where $\widehat{\boldsymbol{\beta}}^{ols}_{1}(\lambda;\uptau_{0})$ is the
quantile regression OLS based estimator222222For instance, for the IVX
estimator based on observations of the full sample the following order of
convergence holds:
${\color[rgb]{0,0,1}n^{\frac{1+\upgamma_{x}}{2}}(\widehat{\boldsymbol{\beta}}^{ivx}_{n}(\uptau)-\boldsymbol{\beta}_{0}(\uptau))=\mathcal{O}_{\mathbb{P}}(1)}$,
which is proved by Corollary 2 in the Appendix of the paper (see also Theorem
3.1 in Lee,  (2016)). that corresponds to the subsample $1\leq
t\leq\lfloor\lambda n\rfloor$ for some $0<\lambda<1$ and $\uptau_{0}\in(0,1)$,
when the quantile regression has no model intercept. Intuitively, when the
model structure incorporates both intercept and slopes then the different
convergence rates of these coefficients due to the presence of nonstationarity
is accommodated with the use of embedded normalization matrices. Therefore,
for the remaining of this section, we suppose that the parameter vector is of
the form
$\boldsymbol{\theta}(\uptau)=\big{[}\alpha(\uptau),\boldsymbol{\beta}(\uptau)^{\prime}\big{]}^{\prime}$.

Consider the nonstationary quantile predictive regression (2.1)-(2.2) which
includes a model intercept. Then, testing for a structural break via a Wald
type formulation based on the OLS estimator implies to use the parameter
vector $\boldsymbol{\theta}(\uptau)$ instead of $\boldsymbol{\beta}(\uptau)$.

######  Proposition 2.

Under the null hypothesis $\mathcal{H}_{0}^{(A)}$ and given that Assumptions
1-2 hold, then the Wald type statistics weakly converge to the limit
distributions below

$\displaystyle\textbf{({i})}\ \
\mathcal{SW}^{ols}_{n}\big{(}\lambda;\uptau_{0}\big{)}$
$\displaystyle\Rightarrow\underset{\lambda\in[0,1]}{\mathsf{sup}}\ \frac{\
\big{\|}\boldsymbol{\mathcal{BB}}_{p+1}(\lambda)\
\big{\|}^{2}}{\lambda(1-\lambda)},\ \text{for}\ \upgamma_{x}\in(0,1)$
$\displaystyle\textbf{({ii})}\ \
\mathcal{SW}^{ols}_{n}\big{(}\lambda;\uptau_{0}\big{)}$
$\displaystyle\Rightarrow\underset{\lambda\in\Lambda_{\eta}}{\mathsf{sup}}\
\boldsymbol{\Delta}^{ols}_{0}\big{(}\lambda;\uptau_{0}\big{)}^{\prime}\big{[}\boldsymbol{\Sigma}^{-1}_{0}\big{(}\lambda;\uptau_{0}\big{)}\big{]}\boldsymbol{\Delta}^{ols}_{0}\big{(}\lambda;\uptau_{0}\big{)},\text{for}\
\upgamma_{x}=1$

where $\boldsymbol{\mathcal{BB}}_{p+1}(.)$ is a vector of $(p+1)$ independent
Brownian bridge processes on
$\mathcal{D}_{\mathbb{R}^{p+1}}\left([0,1]\right)$. Denote with

$\displaystyle\boldsymbol{\Sigma}^{-1}_{0}\big{(}\lambda;\uptau_{0}\big{)}$
$\displaystyle:={\color[rgb]{1,0,0}f_{u_{t}(\uptau)}(0)^{2}}\bigg{[}\mathbb{S}_{xx}(\lambda)-\mathbb{S}_{xx}(\lambda)\mathbb{S}^{-1}_{xx}(1)\mathbb{S}_{xx}(\lambda)\bigg{]}$
$\displaystyle\boldsymbol{\Delta}^{ols}_{0}\big{(}\lambda;\uptau_{0}\big{)}$
$\displaystyle:=\mathbb{S}^{-1}_{xx}(\lambda)\begin{bmatrix}\displaystyle
B_{\psi_{\uptau}}(\lambda)\\\
\displaystyle\int_{0}^{\lambda}\boldsymbol{J}_{c}(r)dB_{\psi_{\uptau}}\end{bmatrix}-\big{[}\mathbb{S}_{xx}(1)-\mathbb{S}_{xx}(\lambda)\big{]}^{-1}\begin{bmatrix}\displaystyle
B_{\psi_{\uptau}}(1)-B_{\psi_{\uptau}}(\lambda)\\\
\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)dB_{\psi_{\uptau}}-\int_{0}^{\lambda}\boldsymbol{J}_{c}(r)dB_{\psi_{\uptau}}\end{bmatrix}.$

where
$\boldsymbol{\Sigma}^{-1}_{0}\big{(}\lambda;\uptau_{0}\big{)}\in\mathbb{R}^{(p+1)\times(p+1)}$
and
$\boldsymbol{\Delta}^{ols}_{0}\big{(}\lambda;\uptau_{0}\big{)}\in\mathbb{R}^{(p+1)\times
n}$ since the model included both an intercept and slopes.

The proof of Proposition 2 can be found in the Appendix of the paper. Notice
that $\boldsymbol{\Sigma}^{-1}_{0}\big{(}\lambda;\uptau_{0}\big{)}$ represents
the weakly convergence result of the inverse of the covariance matrix of the
stochastic process
$\sqrt{n}\big{(}\hat{\boldsymbol{\upbeta}}_{(2)}(\uptau_{0})-\boldsymbol{\upbeta}_{(2)}(\uptau_{0})\big{)}$,
in which case $\mathcal{W}_{n}(\uptau_{0})$ is the Wald statistic for testing
the null hypothesis
$\mathcal{H}_{0}:\hat{\boldsymbol{\upbeta}}_{(2)}(\uptau_{0})=\boldsymbol{0}$.

####  IVX-Wald test statistic

The instrumentation methodology proposed by Phillips and Magdalinos,  (2009)
has been proved to be robust in filtering abstract degree of persistence in
predictive regression models (see also Phillips and Lee,  (2013, 2016)). Our
research objective in this section is to study the asymptotic behaviour of the
proposed structural break tests based on the endogenous instrumentation
procedure in nonstationary quantile predictive regressions. In particular, the
self-normalized sup IVX-Wald test function has been recently examined by
Katsouris,  (2021) as break detector for coefficients of linear predictive
regressions. Specifically, the supremum IVX-Wald test corresponds to the
maximum232323Further details regarding the formulation of Wald type tests and
asymptotic theory is presented in the seminal study of Andrews,  (1993). The
particular framework propose for structural change tests in linear regression
models under the assumption of stationary and ergodic time series. of a
sequence of test statistics constructed based on sequential sample splitting
locations such that $\kappa=\lfloor\lambda n\rfloor$ where
$\lambda\in\Lambda_{\eta}:=[\eta,1-\eta]$ with $0<\eta<1/2$. Furthermore, we
employ the dequantiled model structure and denote with
$\widehat{\boldsymbol{\beta}}_{1}^{ivx}(\lambda;\uptau_{0})$ and
$\widehat{\boldsymbol{\beta}}_{2}^{ivx}(\lambda;\uptau_{0})$ the IVX based
estimators for the two sub-samples occurred at each splitting step. Therefore,
these estimators are computed via the following expressions

$\displaystyle\widehat{\boldsymbol{\beta}}_{1}^{ivx}(\kappa;\uptau_{0})$
$\displaystyle=\left(\frac{1}{\kappa}\sum_{t=1}^{\kappa+j}\tilde{\boldsymbol{z}}_{1,t-1}\boldsymbol{x}_{1,t-1}^{\prime}\right)^{-1}\left(\frac{1}{\kappa}\sum_{t=1}^{\kappa+j}\tilde{\boldsymbol{z}}_{1,t-1}y_{t}\right),$
(3.39)
$\displaystyle\widehat{\boldsymbol{\beta}}_{2}^{ivx}(\kappa;\uptau_{0})$
$\displaystyle=\left(\frac{1}{n-\kappa}\sum_{t=\kappa+1+j}^{n}\tilde{\boldsymbol{z}}_{2,t-1}\boldsymbol{x}_{2,t-1}^{\prime}\right)^{-1}\left(\frac{1}{n-\kappa}\sum_{t=\kappa+1+j}^{n}\tilde{\boldsymbol{z}}_{2,t-1}y_{t}\right).$
(3.40)

where $\kappa=\lfloor\lambda n\rfloor$ for some $0<\lambda<1$ and the
indicator $j\in\left\\{0,...,(n-\kappa)\right\\}$ shows that a sequence of
parameter estimates is obtained by moving along all proportions within the
compact set $\Lambda_{\eta}=[\eta,1-\eta]$, to compute the maximum Wald
statistic. However, for notation convenience we drop the index notation
$(\kappa+j)$ and $(\kappa+1+j)$ which can be confused with notation used for
time-varying parameter estimates. Also,
$\boldsymbol{x}_{1,t-1}:=\boldsymbol{x}_{t-1}\mathds{1}\left\\{t\leq\kappa\right\\}$
and
$\boldsymbol{x}_{2,t-1}:=\boldsymbol{x}_{t-1}\mathds{1}\left\\{t>\kappa\right\\}$.
Furthermore, $\tilde{\mathbf{Q}}_{1}(\lambda;\uptau_{0})$ and
$\tilde{\mathbf{Q}}_{2}(\lambda;\uptau_{0})$ denotes the covariance matrices
which correspond to the two subsample parameter estimates and permits to
decompose the covariance matrix242424The decomposition of the covariance
matrix for the IVX-Wald statistic can be obtained using a formula for
inverting partitioned matrices. In particular, since
$\boldsymbol{Z}_{1}^{\prime}\boldsymbol{X}_{2}=\boldsymbol{Z}_{2}^{\prime}\boldsymbol{X}_{1}=0$
then the matrix inversion formula simplifies further, allowing us to obtain an
expression for the variance of the test. of the test with respect to each
regime

$\displaystyle\tilde{\mathbf{Q}}_{1}\big{(}\lambda;\uptau_{0}\big{)}=\left(\tilde{\boldsymbol{Z}}_{1}^{\prime}\boldsymbol{X}_{1}\right)^{-1}\left(\tilde{\boldsymbol{Z}}_{1}^{\prime}\tilde{\boldsymbol{Z}}_{1}\right)\left(\boldsymbol{X}_{1}^{\prime}\tilde{\boldsymbol{Z}}_{1}\right)^{-1}\
\
\tilde{\mathbf{Q}}_{2}\big{(}\lambda;\uptau_{0}\big{)}=\left(\tilde{\boldsymbol{Z}}_{2}^{\prime}\boldsymbol{X}_{2}\right)^{-1}\left(\tilde{\boldsymbol{Z}}_{2}^{\prime}\tilde{\boldsymbol{Z}}_{2}\right)\left(\boldsymbol{X}_{2}^{\prime}\tilde{\boldsymbol{Z}}_{2}\right)^{-1}$

Then, under the null hypothesis, $\mathcal{H}_{0}^{(A)}$, the sup IVX-Wald
statistic is formulated as

$\displaystyle\mathcal{SW}_{n}^{ivx}\big{(}\lambda;\uptau_{0}\big{)}:=\underset{\lambda\in\Lambda_{\eta}}{\mathsf{sup}}\
\left\\{\Delta\widehat{\boldsymbol{\beta}}_{n}^{ivx}\big{(}\lambda;\uptau_{0}\big{)}^{\prime}\bigg{[}\widehat{\boldsymbol{V}}_{n}^{ivx}\big{(}\lambda;\uptau_{0}\big{)}\bigg{]}^{-1}\Delta\widehat{\boldsymbol{\beta}}_{n}^{ivx}\big{(}\lambda;\uptau_{0}\big{)}\right\\}$
(3.41)

where
$\Delta\widehat{\boldsymbol{\beta}}_{n}^{ivx}\big{(}\lambda;\uptau_{0}\big{)}:=\left(\widehat{\boldsymbol{\beta}}_{1}^{ivx}(\lambda;\uptau_{0})-\widehat{\boldsymbol{\beta}}_{2}^{ivx}(\lambda;\uptau_{0})\right)$
and
$\widehat{\boldsymbol{V}}_{n}^{ivx}\big{(}\lambda;\uptau_{0}\big{)}:=\tilde{\mathbf{Q}}_{1}\big{(}\lambda;\uptau_{0}\big{)}+\tilde{\mathbf{Q}}_{2}\big{(}\lambda;\uptau_{0}\big{)}$.

######  Theorem 1.

Under the null hypothesis and given that Assumptions 1-2 hold, then the sup
IVX-Wald statistic weakly convergence to limit distribution below

$\displaystyle\mathcal{SW}_{n}^{ivx}\big{(}\lambda;\uptau_{0}\big{)}\Rightarrow\underset{\lambda\in\Lambda_{\eta}}{\mathsf{sup}}\bigg{\\{}\boldsymbol{\Delta}^{ivx}_{0}\big{(}\lambda;\uptau_{0}\big{)}^{\prime}\big{[}\boldsymbol{\Sigma}^{ivx}_{0}\big{(}\lambda;\uptau_{0}\big{)}\big{]}^{-1}\boldsymbol{\Delta}^{ivx}_{0}\big{(}\lambda;\uptau_{0}\big{)}\bigg{\\}}$
(3.42)

where $\Lambda_{\eta}:=[\eta,1-\eta]$ with $0<\eta<1/2$ and

$\displaystyle\boldsymbol{\Delta}^{ivx}_{0}\big{(}\lambda;\uptau_{0}\big{)}$
$\displaystyle:=\boldsymbol{W}_{p}(\lambda)-\boldsymbol{\Psi}_{c}(\lambda)\boldsymbol{W}_{p}(1)$
(3.43)
$\displaystyle\boldsymbol{\Sigma}^{ivx}_{0}\big{(}\lambda;\uptau_{0}\big{)}$
$\displaystyle:=\lambda\big{(}\boldsymbol{I}_{p}-\boldsymbol{\Psi}_{c}(\lambda)\big{)}\big{(}\boldsymbol{I}_{p}-\boldsymbol{\Psi}_{c}(\lambda)\big{)}^{\prime}+(1-\lambda)\boldsymbol{\Psi}_{c}(\lambda)\boldsymbol{\Psi}_{c}(\lambda)^{\prime}$
(3.44)

such that

$\boldsymbol{\Psi}_{c}(\lambda)=\begin{cases}\displaystyle\left(\lambda\boldsymbol{\Omega}_{xx}+\int_{0}^{\lambda}\boldsymbol{J}_{c}^{\mu}(r)d\boldsymbol{J}_{c}^{\prime}\right)\left(\boldsymbol{\Omega}_{xx}+\int_{0}^{1}\boldsymbol{J}^{\mu}_{c}(r)d\boldsymbol{J}_{c}^{\prime}\right)^{-1}&,\text{for}\
\upgamma_{x}=1\\\ \\\ \displaystyle\lambda\boldsymbol{I}_{p}&,\text{for}\
\upgamma_{x}\in(0,1)\end{cases}$

where $\boldsymbol{W}_{p}(.)$ is a $p-$dimensional standard Brownian motion,
$\boldsymbol{J}_{c}(\lambda)=\int_{0}^{\lambda}e^{(\lambda-s)\boldsymbol{C}_{p}}d\boldsymbol{B}(s)$
is an Ornstein-Uhkenbeck process and we denote with
$\boldsymbol{J}^{\mu}_{c}(\lambda)=\boldsymbol{J}_{c}(\lambda)-\int_{0}^{1}\boldsymbol{J}_{c}(s)ds$
and
$\boldsymbol{W}_{p}^{\mu}(\lambda)=\boldsymbol{W}_{p}(\lambda)-\int_{0}^{1}\boldsymbol{W}(s)ds$
the demeaned processes of $\boldsymbol{J}_{c}(\lambda)$ and
$\boldsymbol{W}_{p}(\lambda)$ respectively.

######  Remark 4.

Notice that inference on $\boldsymbol{\beta}_{n}^{ivx}(\uptau)$ critically
depends on the estimator of the covariance matrix
$\widehat{\boldsymbol{V}}_{n}^{ivx}(\lambda;\uptau_{0})$. Moreover, the
estimation of the sparsity coefficient does not affect the estimation accuracy
when constructing test statistics in quantile time series models due to the
self-normalized property of Wald type tests. On the other hand, the robust
estimation of the covariance matrix is ensured by employing fully modified
type of transformations as in the linear model (see, Kostakis et al.,
(2015)).

Theorem 1 presents the asymptotic distribution of the sup IVX-Wald test under
the null hypothesis of a single unknown break-point. Furthermore, it covers
some practical considerations that arise in empirical work especially with
respect to the persistence properties of regressors. As we can observe from
the asymptotic behaviour of the test for local unit root regressors (high
persistence), it converges to a nonstandard and nonpivotal distribution. On
the other hand, for mildly integrated regressors the test behaves in large
samples similar to the sup OLS-Wald test which weakly converge into a Brownian
bridge type of limit. In the former case, a comparison of the limit
distributions of the two tests does not necessarily indicate which test
statistic might have better performance in detecting structural breaks to the
coefficients of nonstationary quantile predictive regressions. To investigate
the particular aspect, we use simulation experiments where allow us to use a
suitable experimental design that accommodates these conditions. The proof of
Theorem 1 can be found in the Appendix of the paper.