Some important implications follow from Theorem 1. More precisely, our
asymptotic theory analysis confirms some of the conclusions drawn in similar
studies. For instance, Hanson,  (2002) and Seo,  (1998) (see also Georgiev et
al.,  (2018)) demonstrated that testing for structural breaks with integrated
regressors based on the OLS estimation method converges to a nonstandard and
nonpivotal limiting distribution. Furthermore, although the IVX-Wald statistic
is proved to be robust to abstract degree of persistence when testing for
parameter restrictions, within the structural break testing framework due to
the presence of the unknown break-point location, the sup IVX-Wald test
similar to the OLS counterpart is coverages to a nonpivotal limiting
distribution, which is not Brownian bridge (as defined in the stationary case)
even though it is still tied down.

Nevertheless, some interesting further simplifications occur; for instance
under the assumption of mildly integrated regressors, it can be easily proved
that the limiting distribution of the sup IVX-Wald test converges to a
normalized Brownian bridge limit. This occurs due to the asymptotic matrix
moments such that, for $0<\upgamma_{x}<1$ it holds that
$\sum_{t=1}^{n}\boldsymbol{x}_{t-1}\tilde{\boldsymbol{z}}_{t-1}^{\prime}\Rightarrow-\boldsymbol{\Omega}_{xx}\boldsymbol{C}_{z}^{-1}$
by expression (20) in Phillips and Magdalinos,  (2009). Thus, it also holds
that $\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{x}_{t-1}\tilde{\boldsymbol{z}}_{t-1}^{\prime}\Rightarrow-\lambda\boldsymbol{\Omega}_{xx}\boldsymbol{C}_{z}^{-1}$,
which implies that $\boldsymbol{\Psi}_{c}(\lambda)=\lambda\boldsymbol{I}_{p}$.
Furthermore, we also consider the limiting distribution of the sup IVZ-Wald
test.

######  Corollary 5.

Under the null hypothesis, $\mathcal{H}_{0}^{(A)}$ and suppose that
Assumptions 1-2 hold, then the sup IVZ-Wald statistic weakly convergence to
the asymptotic distribution below

$\displaystyle\mathcal{SW}^{ivz}_{n}\big{(}\lambda;\uptau_{0}\big{)}$
$\displaystyle\Rightarrow\underset{\lambda\in\Lambda_{\eta}}{\mathsf{sup}}\
\frac{\ \big{\|}\boldsymbol{\mathcal{BB}}_{p}(\lambda)\
\big{\|}^{2}}{\lambda(1-\lambda)},\ \ \text{for}\ \ 0<\upgamma_{x}\leq 1.$
(3.45)

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

In summary, in this section we show that the limiting distribution of the sup
IVX-Wald test statistic under high persistence is nonstandard and nonpivotal.
Furthermore, the particular limit result simplifies when regressors in the
model are assumed to be mildly integrated resulting to weakly convergence into
a Brownian bridge type limit. On the other hand, when we construct the test
statistic based on the IVZ estimator then the sup IVZ-Wald test converges into
a Brownian bridge type of limit regardless of the degree of persistence.
Lastly, for a known break-point Wald type tests converge to a nuisance-
parameter free limiting distribution, simplifying this way statistical
inference. The proof of Corollary 5 can be found in the Appendix.

Another important aspect we consider for the development of the asymptotic
theory of the paper, is the classical result of Huber for models with
nonstandard conditions such as quantile regression models. More specifically,
the first order condition (FOC) defined as the right derivative of the
objective function plays a key role in deriving the asymptotic theory of
estimators for the quantile model. In particular, we can show that the
parameter vector estimator solves these FOC and then apply a Bahadur
representation for the estimator. All these results hold with almost surely
convergence in large samples.

##  Monte Carlo Experiments

Practically, it is unclear how well the asymptotic theory can provide reliable
reference and guidance in finite samples when applied to time series data
since usually they can exhibit abstract degree of persistence. However, under
the assumption that regressors incorporated in the quantile predictive
regression model are generated by near unit root processes for which their
asymptotic behaviour is well-understood, then our test statistics can provide
an indication regarding the ability of the testing procedures in detecting
structural breaks in coefficients of nonstationary quantile predictive
regression models. Thus, to investigate the finite sample performance of the
proposed tests for their adequacy in detecting parameter instability we focus
on the empirical size simulation results as well as on asymptotic power
analysis under relevant sequence of local alternatives.

###  Experimental Design

We conduct a number of Monte Carlo experiments to evaluate the performance of
the limit distribution of the Wald and fluctuation type statistics against the
conventional $\chi^{2}$ asymptotic approximation. We simulate the following
data generating process

$\displaystyle y_{t}$
$\displaystyle=\alpha+\sum_{j=1}^{3}\beta_{j}x_{j,t-1}+u_{t},\ \ \ \ 1\leq
t\leq n$ (4.1) $\displaystyle\boldsymbol{x}_{t}$
$\displaystyle=\begin{bmatrix}\varrho_{n}(c_{j},\upgamma_{x})&0&0\\\
0&\varrho_{n}(c_{j},\upgamma_{x})&0\\\
0&0&\varrho_{n}(c_{j},\upgamma_{x})\end{bmatrix}\boldsymbol{x}_{t-1}+\boldsymbol{v}_{t}$
(4.2)

where $\boldsymbol{x}_{t}=\big{(}x_{1t},x_{1t},x_{1t}\big{)}^{\prime}$,
$\boldsymbol{v}_{t}=\big{(}v_{1t},v_{1t},v_{1t}\big{)}^{\prime}$ and
$\varrho_{n}(c_{j},\upgamma_{x})=\left(1+\displaystyle\frac{c_{j}}{n^{\upgamma_{x}}}\right)$.
Then, the innovation sequence of the model, denoted with
$\boldsymbol{e}_{t}=\left(u_{t},\boldsymbol{v}_{t}\right)^{\prime}$ is
generated such that
$\boldsymbol{e}_{t}\sim\mathcal{N}\left(\boldsymbol{0}_{(p+1)\times
1},\boldsymbol{\Sigma}_{ee}\right)$, where $\boldsymbol{\Sigma}_{ee}$ is an
$(p+1)\times(p+1)$ positive-definite covariance matrix with a pre-specified
variance-covariance structure given as below

$\displaystyle\boldsymbol{\Sigma}_{ee}=\begin{bmatrix}\sigma^{2}_{uu}\ &\
\boldsymbol{\rho}^{\prime}\\\ \boldsymbol{\rho}\ &\
\boldsymbol{\Sigma}_{vv}\end{bmatrix}$ (4.3)

where the matrix $\boldsymbol{\Sigma}_{vv}$ is of full rank $p$, resulting to
a nonsingular matrix $\boldsymbol{\Sigma}_{ee}$.

The coefficients of persistence are such that
$c_{j}\in\left\\{-1,-2,-5\right\\}$ and $\upgamma_{x}$ is defined to be
$\upgamma_{x}=1$ to simulate near unit root predictors and $\upgamma_{x}=0.75$
to simulate mildly integrated predictors. Moreover, we consider different
sample size such that $n\in\left\\{250,500,750,1000\right\\}$. Under the null
hypothesis of no parameter instability we use the following parameters
$\alpha=1$, $\beta_{1}=0.25,\beta_{2}=0.75,\beta_{3}=-0.50$ and construct the
proposed test statistics with significance level $\upalpha=5\%$. For the IVX
instrumentation we use $c_{z}=1$ and $\upgamma_{z}=0.95$.

In particular, to construct the test statistics the simulated pair
$\left\\{y_{t},\boldsymbol{x}_{t}\right\\}_{t=1}^{n}$ is formulated:

$\displaystyle
y_{t}=\left(\alpha^{(1)}(\uptau)+\sum_{j=1}^{3}\beta^{(1)}_{j}(\uptau)x_{j,t-1}\right)\mathds{1}\big{\\{}t\leq\kappa\big{\\}}+\left(\alpha^{(2)}(\uptau)+\sum_{j=1}^{3}\beta^{(2)}_{j}(\uptau)x_{j,t-1}\right)\mathds{1}\big{\\{}t>\kappa\big{\\}}+u_{t}$

where $\kappa=\lfloor\lambda n\rfloor$ and the search over all possible
subsets occurs for values of $\lambda\in\Lambda_{\eta}$. Denote with
$\boldsymbol{\theta}^{(j)}(\uptau)=\left(\alpha^{(j)}(\uptau),\beta_{1}^{(j)}(\uptau),\beta_{2}^{(j)}(\uptau),\beta_{3}^{(j)}(\uptau)\right)^{\prime}$
with $j\in\left\\{1,2\right\\}$ to be the quantile dependent parameter vector
of each of the two regimes, for a fixed quantile $\uptau_{0}$ that belongs in
the compact set252525Notice that the compact set $\mathcal{T}_{\eta}$ falls
strictly within the unit interval to allow the conditional distribution to
have an unbounded support. $\mathcal{T}_{\eta}$ such that
$0<\eta<\uptau_{0}<1-\eta<1$.

Then, the testing hypothesis of interest is formulated as below

$\displaystyle\mathcal{H}^{(A)}_{0}:\boldsymbol{\theta}_{n}^{(1)}(\uptau)=\boldsymbol{\theta}_{n}^{(2)}(\uptau)\
\ \ \text{versus}\ \ \
\mathcal{H}^{(A)}_{1}:\boldsymbol{\theta}_{n}^{(1)}(\uptau)\neq\boldsymbol{\theta}_{n}^{(2)}(\uptau)$
(4.4)

with a fixed quantile $\uptau_{0}\in\mathcal{T}_{\eta}:=[\eta,1-\eta]$, for an
unknown break-point location $\kappa=\lfloor\lambda n\rfloor$ where
$0<\lambda<1$ and a significance level $\upalpha=5\%$.

###  Implementation

One important aspect for the correct implementation of the fluctuation based
tests (which involves the estimation of a subgradient) is that no nuisance
parameter is needed (in the case when the LUR process is not employed). More
precisely, for the Wald type statistics, we need a consistent estimate of the
variance-covariance matrix $\boldsymbol{\Omega}_{0}$. In particular, it
requires estimating the following matrix262626Notice that the estimation of
the sparsity function will affect the finite-sample performance of the test
statistics if not consistently estimated.

$\displaystyle\boldsymbol{H}_{0}=\underset{n\to\infty}{\mathsf{plim}}\left(n^{-1}\sum_{t=1}^{n}f_{y|x}\big{(}F^{-1}_{y|x}\left(\uptau\right)\big{)}\boldsymbol{x}_{t-1}\boldsymbol{x}_{t-1}^{\prime}\right)^{-1}$
(4.5)

A discussion regarding methodologies for estimating the matrix given by
expression (4.5) can be found in Qu,  (2008). On the other hand, when we
implement the quantile predictive regression model with the presence of
persistence covariates, then we will need to examine the asymptotic properties
of the above covariance estimator since it will be a function of the unknown
coefficient of persistence.

In summary, in this paper we take the position than when using Wald type
statistics as structural break detectors in nonstationary quantile predictive
regression models, the chosen estimator will affect the limit distribution of
the tests and thus its finite-sample performance, especially under the
presence of high persistence regressors. In particular, we have proved that
when selecting the OLS estimator then the limit distribution is nonstandard
and nonpivotal making inference challenging since critical values can be
constructed only with the use of bootstrap-based methodologies. On the other
hand, when the IVZ estimator is chosen then the limit distribution is proved
to be nuisance-parameter free regardless of the persistence properties driving
the behaviour of regressors employed when estimating the quantile predictive
regression model.

Another relevant aspect to investigate is the sensitivity of the proposed test
statistics to the break-point location within the full sample. Therefore, for
instance we are interested to examine whether the proposed tests have roughly
equal sensitivity to a break occurring early or late in the sample (see, also
Leisch et al.,  (2000)). We expect that the break-point location will not
affect the power performance of the proposed test statistics especially due to
the fact that we do not operate within a sequential monitoring scheme in which
case parameter estimates and functionals are updated continuously.

###  Simulation Results

The simulation results of the Monte Carlo experiments focus on the performance
of the tests by obtaining the empirical size and empirical power. In
particular, we evaluate the performance of the tests at different quantiles by
employing the test statistics that correspond to the fixed quantile level.
More precisely, this allow us to check for structural breaks at the median, or
at the upper and lower quantiles for example, thus observing the presence of
parameter instability at different levels of the predictant with respect to
persistent predictors. Our simulation experiments verify the empirical and
theoretical results observed by Katsouris,  (2021) in the case of the linear
predictive regression model, that is, a trend of over-rejecting the null
hypothesis when the OLS estimator is employed when constructing structural
break tests272727Notice that the concept of spurious break is discussed by
Hansen,  (2000) who emphasize that instability in the exogenous variables can
cause over-rejection in the standard OLS-based tests..

Overall, the finite-sample results reflect the main conjectures presented in
the asymptotic theory of the paper and appear to be reasonable for practical
use in testing for structural breaks in the coefficients of nonstationary
quantile predictive regression models, especially with persistent regressors
and endogeneity282828Notice that exogeneity plays an important role in dealing
with non-stationary variables. More specifically, in Chapter of Banerjee et
al.,  (1993) it is mentioned that dynamic regression equations in which the
conditioning is on weakly or strongly exogenous variables (for the parameter
of interest) provide asymptotically unbiased estimates.. In practise, in those
cases in which the exact $\upalpha-$level critical values for $0<\upalpha<1$
depend on the unknown parameters of persistence, we employ bootstrap based
resampling methods for inference purposes. Therefore, we can observe that
these near unit root processes driving the regressors of the model, can affect
the ability of the proposed structural break tests for detecting parameter
instability in quantile predictive regression models. Specifically, in the
simulation study of Wang and Phillips,  (2012), the authors mention that
serial dependence can affect power. Furthermore, the lower long-run signal
strength in the regressor tends to reduce disciminatory power.

##  Empirical Application

Our empirical application is concerned with the monitoring of the US housing
price index returns (HPI). Using macroeconomic variables with predictive
regression models has been demonstrated in various studies. For instance, the
empirical study of Paye,  (2012) verifies the episodic predictability
conclusions documented in the literature such as in Gonzalo and Pitarakis,
(2012, 2017) (see also Demetrescu et al.,  (2020)). The author finds
statistical evidence of predictability in relation to countercyclical
macroeconomic events when forecasting volatility using predictive regressions
with macroeconomic covariates. Furthermore, Atanasov et al.,  (2020)
investigate the impact of consumption fluctuations on predictability of
expected returns using the IVX filter. Overall, our empirical study focuses in
the implementation of the proposed test statistics292929Notice, that a
different stream of literature proposes testing procedures for detecting
market exuberance and bubble effects. Our empirical application is concerned
with the detection of structural breaks in the data based on the nonstationary
quantile predictive regression.; thus our research goal here is to test for
structural breaks in the relation between the predictant and the predictors at
various quantile levels of the underline data specific distribution.

###  Data Description

For the empirical application of our study, we utilize the dataset of Yang et
al.,  (2020) that includes the US housing price index returns along with ten
common macroeconomic variables. Specifically, the HPI covers more transactions
and longer time interval, and thus can well represent the trend of the
national-wide housing price such as the housing bubble collapsed during the
2007 subprime mortgage crisis. Furthermore, based on the HPI the authors
obtain the quarterly growth rate of the housing price and use this rate as the
dependent variable. More precisely, the ten macroeconomic variables are
collected from FRED, and all data are quarterly between 1975:Q1 and 2018:Q2.

  * •

CPI: Consumer price index with all items less shelter for all urban consumers
(Index 1982 to 1984 = 100).

  * •

DEF: The implicit price deflator of the gross domestic product (Index 2012 =
100).

  * •

GDP: $\%-$Change of the gross domestic product from the preceding period.

  * •

INC: $\%-$Change of the real disposable personal income from the quarter one
year ago.

  * •

IND: The industrial production index (Index 2012 = 100). An economic indicator
that measures real output for all U.S. located facilities manufacturing,
mining, and electric, and gas utilities (excluding those in U.S. territories).

  * •

INT: The effective federal funds rate. The interest rate at which depository
institutions trade federal funds (balances held at FRBs) with each other
overnight.

  * •

INV: The shares of the residential fixed investment in the gross domestic
product. Gross private domestic investment is a critical component of gross
domestic product as it provides an indicator of the future productive capacity
of the economy. Residential investment represents expenditures on residential
structures and residential equipment that is owned by landlords and rented to
tenants.

  * •

MOG: 30-year mortgage rate. It represents contract interest rates on
commitments for fixed-rate first mortgages.

  * •

RES: The total reserve balances maintained with the Federal Reserve banks.

  * •

UNE: The civilian unemployment rate. It represents the number of unemployed as
a percentage of the labor force.

###  Data Analysis

We begin our analysis by applying standard unit root tests to the
predictors303030Notice that we have $t=1,...,174$ time series observations
which correspond to quarterly economic indicators and macroeconomic variables.
employed for the quantile predictive regression model. Furthermore, we test
each individual predictor separately for the presence of parameter instability
using a toolkit of various structural break tests commonly employed in the
literature. In particular, testing for breaks in housing price indices has
been previously studied by Canarella et al.,  (2012). However, testing for
quantile predictability as well as testing for breaks in nonstationary
quantile regressions is a novel aspect not previously examined in the
literature.

Secondly, we implement the joint IVX-Wald statistic under the null hypothesis
that all slope coefficients simultaneously equal to zero313131We consider
rejections of the null hypothesis at significance level $5\%$ to match the
rejection probabilities employed in the simulation study of the paper.. More
precisely, the particular hypothesis correspond to the null hypothesis of no
quantile predictability. Thus, formulating the model in this manner allow us
to investigate whether there is a stable relation between regressand and
regressors at a specific quantile level323232In particular, the empirical
study presented by Lee,  (2016) demonstrates statistical evidence of
predictive ability using the nonstationary qunatile predictive regression
model, at some specific quantiles of stock returns such as at lower or upper
quantiles while on the other hand evidence of predictability disappear at the
median of the conditional distribution of stock returns.
$\uptau_{0}\in(0,1)$.

Thirdly, we implement the joint IVX-Wald statistic under the null hypothesis
that at least two of the slope coefficients have no structural break
throughout the sample.

##  Conclusion

In this paper we develop a framework for structural break detection for
nonstationary quantile predictive regression models, under the null hypothesis
of no structural break333333Notice that we avoid to explicitly use the
terminology of a null hypothesis of “stationarity” versus an alternative
hypothesis of “non-stationarity” (see Pitarakis,  (2014) and Kwiatkowski et
al.,  (1992)). Specifically, within our setting these two terms are
interpreted in relation to the persistence properties of model predictors
rather that with respect to the parameter constancy of coefficients in terms
of temporal dependence. Related limit theory and conditions relevant to
temporal dependence specifically for quantile and tail empirical processes can
be found in Chapter 5 of De Haan and Ferreira,  (2006).. A major challenge
when deriving the asymptotic theory of these structural break tests is to
obtain nuisance-parameter free limit distributions which is not a trivial task
due to the stochastic approximation terms that depend on nuisance parameters,
such as higher order covariance terms as functions of the coefficients of
persistence that capture the time series properties of regressors. More
precisely, we establish the asymptotic distributions for both Wald type (i.e.,
as in Andrews,  (1993)) and fluctuation type tests (i.e., as in Qu,  (2008))
with respect to two different estimation methods, that is, the OLS and IVX
estimators343434Notice in the study of Phillips and Park,  (1988) the authors
demonstrate the asymptotic equivalence of OLS and GLS based estimators in
regression models with integrated regressors.. Our test statistics show to
have good finite-sample properties as shown by the Monte Carlo experiments in
which we obtain the empirical size and power.

Firstly, we verify that indeed the self-normalization of Wald type statistics
when testing the null hypothesis of no predictability in quantile predictive
regressions results to a nuisance-free distribution, that is, ensuring their
pivotal property (as also proved by Lee,  (2016) for abstract degree of
persistence). Secondly, we demonstrate that the limit distribution of the
proposed test statistics for structural break detection is not depending on
the particular choice of the estimator of the quantile predictive regression
model under mildly integrated; however under high persistence the choice of
the estimator alters the limit theory due since different weakly convergence
arguments apply. Furthermore, keeping the quantile level fixed versus testing
for breaks across multiple quantile levels requires to consider extending the
limit result into the two-parameter Gaussian process, for the latter case.
Moreover, bootstrap-based methodologies can be applied when the limit
distribution is nonstandard, allowing to infer regarding the presence of
structural breaks under these conditions (e.g., high persistence).

Further research worth mentioning includes to extend the current framework
proposed in this paper, for quantile predictability tests robust to parameter
instability. The specific application has important implications, from both
the theoretical as well as the empirical perspective, especially under
nonstationarity, since currently the common practise in the literature is the
proposition of methods that investigate these two aspects separately.
Additionally, our framework can be extended to the alternative hypothesis of
multiple structural breaks as in Qu,  (2008) as well as within a multivariate
setting such that the framework proposed by the study of Qu and Perron,
(2007).

##### Acknowledgements

This paper is part of my Ph.D thesis at the Department of Economics of the
University of Southampton. I am deeply indebted to my academic advisors Jose
Olmo and Tassos Magdalinos as well as to Jean-Yves Pitarakis for helpful
discussions and constructive feedback. I am also grateful for given the chance
to participate to various econometrics seminars and conferences during the
Ph.D candidature. The author declares no conflicts of interests.

##  APPENDIX

We provide detailed mathematical derivations for the proofs of main asymptotic
theory. Relevant references are Phillips and Magdalinos,  (2009) and Qu,
(2008).

###  Proofs of asymptotic theory results

For $\boldsymbol{b}\in\mathbb{R}^{p}$ we define (see, Lee,  (2016) and Galvao
et al.,  (2014)) the empirical process:

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

such that $\left(\uptau,\boldsymbol{b}\right)\in\mathcal{T}_{\iota}\times
B\mapsto\boldsymbol{G}_{n}\left(\uptau,\boldsymbol{b}\right)$ is
stochastically equicontinous for any $\epsilon>0$.

####  Proof of Corollary 2

The consistency of the OLS quantile estimator is derived by Koenker and Xiao,
(2006). Here, we prove the convergence rate for the IVX-QR estimator by
employing standard methods from from the literature of extremum estimators and
the regularity conditions imposed by Bickel,  (1975). Thus, we aim to show
that

$\displaystyle\left(\widehat{\boldsymbol{\beta}}_{n}^{ivx-
qr}\left(\uptau\right)-\boldsymbol{\beta}\left(\uptau\right)\right)=\mathcal{O}_{\mathbb{P}}\left(n^{-(1+\upgamma_{z})/2}\right).$
(7.1)

###### Proof.

For the remaining of the proof we denote with
$\widehat{\boldsymbol{\beta}}_{n}^{ivx-
qr}(\uptau):=\widehat{\boldsymbol{\beta}}_{n}^{\star}(\uptau)$ to simplify
notation. Then, consider the estimator distance such that
$\widehat{\boldsymbol{\mathcal{E}}}_{n}^{\star}(\uptau)=\left(\widehat{\boldsymbol{\beta}}_{n}^{\star}(\uptau)-\boldsymbol{\beta}^{\star}(\uptau)\right)$,
which implies that by evaluating $\boldsymbol{G}_{n}(\uptau,\boldsymbol{b})$
at $\boldsymbol{b}=\widehat{\boldsymbol{\mathcal{E}}}_{n}^{\star}(\uptau)$ we
obtain the following expression

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

Next, we apply the result353535The norm $\left\lVert.\right\rVert$ represents
the Euclidean norm, i.e., $\left\lVert
x\right\rVert=\left(\sum_{i=1}^{p}x_{i}^{2}\right)^{1/2}$ for
$x\in\mathbb{R}^{p}$. given by Lemma 2, which implies that for some constant
$\mathcal{C}_{1}$

$\displaystyle\mathsf{sup}\big{\\{}\big{\|}\boldsymbol{G}_{n}\left(\uptau,\boldsymbol{b}\right)-\boldsymbol{G}_{n}\left(\uptau,\boldsymbol{0}\right)\big{\|}:\left\lVert\boldsymbol{b}\right\rVert\leq
n^{(1+\upgamma_{x})/2}\mathcal{C}_{1}\big{\\}}=o_{\mathbb{P}}(1).$ (7.2)

Moreover, for $\boldsymbol{b}=\boldsymbol{0}$ the following empirical process
holds

$\displaystyle\boldsymbol{G}_{n}(\uptau,\boldsymbol{0})=n^{-\frac{(1+\upgamma_{z})}{2}}\sum_{t=1}^{n}\tilde{\boldsymbol{z}}_{t-1}\bigg{\\{}\psi_{\uptau}\big{(}u_{t}\left(\uptau\right)\big{)}-\mathbb{E}_{\mathcal{F}_{t-1}}\left[\psi_{\uptau}\big{(}u_{t}\left(\uptau\right)\big{)}\right]\bigg{\\}}$
(7.3)

Therefore, the argument of Lemma 2 is expanded as below

$\displaystyle\boldsymbol{G}_{n}(\uptau,\boldsymbol{b})-\boldsymbol{G}_{n}(\uptau,\boldsymbol{0})$
$\displaystyle=n^{-\frac{(1+\upgamma_{z})}{2}}\sum_{t=1}^{n}\tilde{\boldsymbol{z}}_{t-1}\big{\\{}\psi_{\uptau}\big{(}u_{t}\left(\uptau\right)-\boldsymbol{x}_{t-1}^{\prime}\widehat{\boldsymbol{\mathcal{E}}}_{n}^{\star}(\uptau)\big{)}-\mathbb{E}_{\mathcal{F}_{t-1}}\left[\psi_{\uptau}\big{(}u_{t}\left(\uptau\right)-\boldsymbol{x}_{t-1}^{\prime}\widehat{\boldsymbol{\mathcal{E}}}_{n}^{\star}(\uptau)\big{)}\right]\big{\\}}$
$\displaystyle\ \ \ \
-n^{-\frac{(1+\upgamma_{z})}{2}}\sum_{t=1}^{n}\tilde{\boldsymbol{z}}_{t-1}\big{\\{}\psi_{\uptau}\big{(}u_{t}\left(\uptau\right)\big{)}-\mathbb{E}_{\mathcal{F}_{t-1}}\left[\psi_{\uptau}\big{(}u_{t}\left(\uptau\right)\big{)}\right]\big{\\}}$
$\displaystyle=n^{-\frac{(1+\upgamma_{z})}{2}}\sum_{t=1}^{n}\tilde{\boldsymbol{z}}_{t-1}\psi_{\uptau}\big{(}u_{t}\left(\uptau\right)-\boldsymbol{x}_{t-1}^{\prime}\widehat{\boldsymbol{\mathcal{E}}}_{n}^{\star}(\uptau)\big{)}{\color[rgb]{1,0,0}\overset{\mathbb{P}}{\to}0}$
$\displaystyle\ \ \ \
-n^{-\frac{(1+\upgamma_{z})}{2}}\sum_{t=1}^{n}\tilde{\boldsymbol{z}}_{t-1}\mathbb{E}_{\mathcal{F}_{t-1}}\left[\psi_{\uptau}\big{(}u_{t}\left(\uptau\right)-\boldsymbol{x}_{t-1}^{\prime}\widehat{\boldsymbol{\mathcal{E}}}_{n}^{\star}(\uptau)\big{)}\right]$
$\displaystyle\ \ \ \
-n^{-\frac{(1+\upgamma_{z})}{2}}\sum_{t=1}^{n}\tilde{\boldsymbol{z}}_{t-1}\psi_{\uptau}\big{(}u_{t}\left(\uptau\right)\big{)}+n^{-\frac{(1+\upgamma_{z})}{2}}\sum_{t=1}^{n}\mathbb{E}_{\mathcal{F}_{t-1}}\left[\psi_{\uptau}\big{(}u_{t}\left(\uptau\right)\big{)}\right]{\color[rgb]{1,0,0}=0}$

Taking the absolute value since the result holds for the Euclidean norm we
obtain

$\displaystyle\bigg{|}\boldsymbol{G}_{n}(\uptau,\boldsymbol{b})-\boldsymbol{G}_{n}(\uptau,\boldsymbol{0})\bigg{|}=n^{-\frac{(1+\upgamma_{z})}{2}}\sum_{t=1}^{n}\left\\{\tilde{\boldsymbol{z}}_{t-1}\psi_{\uptau}\big{(}u_{t}\left(\uptau\right)\big{)}+\tilde{\boldsymbol{z}}_{t-1}\mathbb{E}_{\mathcal{F}_{t-1}}\left[\psi_{\uptau}\big{(}u_{t}\left(\uptau\right)-\boldsymbol{x}_{t-1}^{\prime}\widehat{\boldsymbol{\mathcal{E}}}_{n}^{\star}(\uptau)\big{)}\right]\right\\}+o_{\mathbb{P}}(1).$

Similarly, with the embedded normalization matrices an equivalent expression
holds

$\displaystyle\bigg{|}\boldsymbol{G}_{n}(\uptau,\boldsymbol{b})-\boldsymbol{G}_{n}(\uptau,\boldsymbol{0})\bigg{|}=\sum_{t=1}^{n}\left\\{\tilde{\boldsymbol{Z}}_{t-1,n}\psi_{\uptau}\big{(}u_{t}\left(\uptau\right)\big{)}+\tilde{\boldsymbol{Z}}_{t-1,n}\mathbb{E}_{\mathcal{F}_{t-1}}\left[\psi_{\uptau}\big{(}u_{t}\left(\uptau\right)-\boldsymbol{X}_{t-1,n}^{\prime}\widehat{\boldsymbol{\mathcal{E}}}_{n}^{\star}(\uptau)\big{)}\right]\right\\}+o_{\mathbb{P}}(1).$

Then, the conditional expectation
$\mathbb{E}_{\mathcal{F}_{t-1}}\left[\psi_{\uptau}\left(u_{t}(\uptau)-\boldsymbol{x}_{t-1}^{\prime}\widehat{\boldsymbol{\mathcal{E}}}_{n}^{\star}(\uptau)\right)\right]$
can be expanded around the point
$\boldsymbol{\mathcal{E}}(\uptau)=\boldsymbol{0}$ using the first-order taylor
expansion since the term
$\widehat{\boldsymbol{\beta}}^{\star}(\uptau)\boldsymbol{x}_{t-1}$ is strictly
monotonic, uniformly on
$\left\\{\left\lVert\boldsymbol{x}_{t-1}\right\rVert\leq\delta\right\\}$ where
$\iota\leq\uptau\leq 1-\iota$ for some $\delta>0$ (see, Theorem 1 of Neocleous
and Portnoy,  (2008)). Hence, we have that

$\displaystyle\mathbb{E}_{\mathcal{F}_{t-1}}\left[\psi_{\uptau}\left(u_{t}(\uptau)-\boldsymbol{x}_{t-1}^{\prime}\widehat{\boldsymbol{\mathcal{E}}}_{n}^{\star}(\uptau)\right)\right]$
$\displaystyle\equiv\mathbb{E}_{\mathcal{F}_{t-1}}\big{[}\psi_{\uptau}\big{(}u_{t}(\uptau)-\boldsymbol{x}_{t-1}^{\prime}\boldsymbol{\mathcal{E}}_{n}^{\star}(\uptau)\big{)}\big{]}\bigg{|}_{\boldsymbol{\mathcal{E}}(\uptau)=0}$
$\displaystyle\ \ \ \ \ \
+\frac{\partial\mathbb{E}_{\mathcal{F}_{t-1}}\big{[}\psi_{\uptau}\big{(}u_{t}(\uptau)-\boldsymbol{x}_{t-1}^{\prime}\boldsymbol{\mathcal{E}}(\uptau)\big{)}\big{]}}{\partial\boldsymbol{\mathcal{E}}(\uptau)}\bigg{|}_{\boldsymbol{\mathcal{E}}(\uptau)=0}\times\widehat{\boldsymbol{\mathcal{E}}}^{\star}_{n}(\uptau)+o_{\mathbb{P}}\left(\widehat{\boldsymbol{\mathcal{E}}}^{\star}_{n}(\uptau)\right).$

Note that $\boldsymbol{\mathcal{E}}(\uptau)=0$ implies that
$\boldsymbol{\beta}\left(\uptau\right)=\boldsymbol{\beta}^{\star}\left(\uptau\right)$.
Moreover, by definition of $\psi_{\uptau}(.)$ and by applying standard results
for the conditional expectation operator we obtain that

$\displaystyle\mathbb{E}_{\mathcal{F}_{t-1}}\big{[}\psi_{\uptau}\big{(}u_{t}(\uptau)-\boldsymbol{\mathcal{E}}(\uptau)^{\prime}\boldsymbol{x}_{t-1}\big{)}\big{]}$
$\displaystyle=\uptau-\mathbb{E}_{\mathcal{F}_{t-1}}\big{[}\mathds{1}\big{\\{}u_{t}(\uptau)<\boldsymbol{\mathcal{E}}(\uptau)^{\prime}\boldsymbol{x}_{t-1}\big{\\}}\big{]}$
$\displaystyle=\uptau-\int_{\infty}^{\boldsymbol{x}_{t-1}^{\prime}\boldsymbol{\mathcal{E}}(\uptau)}f_{u_{t}(\uptau),t-1}(s)ds$
(7.4)

Hence, by differentiating expression (7.1.1) around the neighbourhood of
$\boldsymbol{\mathcal{E}}(\uptau)$ we get

$\displaystyle\frac{\partial\mathbb{E}_{\mathcal{F}_{t-1}}\big{[}\psi_{\uptau}\big{(}u_{t}(\uptau)-\boldsymbol{x}_{t-1}^{\prime}\boldsymbol{\mathcal{E}}(\uptau)\big{)}\big{]}}{\partial\boldsymbol{\mathcal{E}}(\uptau)}\bigg{|}_{\boldsymbol{\mathcal{E}}(\uptau)=0}=-\boldsymbol{x}_{t-1}^{\prime}f_{u_{t}(\uptau),t-1}(0)$
(7.5)

Therefore, it holds that

$\displaystyle\mathbb{E}_{\mathcal{F}_{t-1}}\left[\psi_{\uptau}\left(u_{t}(\uptau)-\boldsymbol{x}_{t-1}^{\prime}\right)\right]=-\boldsymbol{x}_{t-1}^{\prime}f_{u_{t}(\uptau),t-1}(0)\widehat{\boldsymbol{\mathcal{E}}}^{\star}_{n}(\uptau)+o_{\mathbb{P}}(1)$
(7.6)

Next substituting the limit given by (7.6) into the original expansion for the
term
$\big{|}\boldsymbol{G}_{n}(\uptau,\boldsymbol{b})-\boldsymbol{G}_{n}(\uptau,\boldsymbol{0})\big{|}$
as well as by replacing $\boldsymbol{x}_{t-1}$ with the corresponding embedded
normalization matrix $\boldsymbol{X}_{t-1,n}$ we obtain the expression

$\displaystyle\bigg{|}$
$\displaystyle\boldsymbol{G}_{n}(\uptau,\boldsymbol{b})-\boldsymbol{G}_{n}(\uptau,\boldsymbol{0})\bigg{|}$
$\displaystyle\equiv\boldsymbol{K}_{nz}\big{(}\uptau,\boldsymbol{\beta}^{\star}(\uptau)\big{)}-\sum_{t=1}^{n}f_{u_{t}(\uptau),t-1}(0)\tilde{\boldsymbol{Z}}_{t-1,n}\boldsymbol{X}_{t-1}^{\prime}n^{-\frac{1+\upgamma_{z}}{2}}\left(\widehat{\boldsymbol{\beta}}_{n}^{\star}(\uptau)-\boldsymbol{\beta}^{\star}(\uptau)\right)+o_{\mathbb{P}}(1)$
(7.7)

where

$\displaystyle\boldsymbol{K}_{nz}\big{(}\uptau,\boldsymbol{\beta}^{\star}(\uptau)\big{)}:=\boldsymbol{D}_{n}^{-1}\sum_{t=1}^{n}\tilde{\boldsymbol{Z}}_{t-1}\psi_{\uptau}\big{(}u_{t}(\uptau)\big{)}$
(7.8)

Moreover, we define with

$\displaystyle\boldsymbol{M}_{nz}\big{(}\uptau,\boldsymbol{\beta}^{\star}(\uptau)\big{)}:=\sum_{t=1}^{n}f_{u_{t}(\uptau),t-1}(0)\tilde{\boldsymbol{Z}}_{t-1,n}\tilde{\boldsymbol{X}}_{t-1,n}^{\prime}\
$ (7.9)

Next, by noting that from Lemma 2 it holds that
$\mathsf{sup}\big{\\{}\left\lVert\boldsymbol{G}_{n}(\uptau,\boldsymbol{b})-\boldsymbol{G}_{n}(\uptau,\boldsymbol{0})\right\rVert\big{\\}}=o_{\mathbb{P}}(1)$,
then by rearranging expression (7.1.1) we obtain that

$\displaystyle
n^{\frac{1+\upgamma_{z}}{2}}\big{(}\widehat{\boldsymbol{\beta}}^{\star}_{n}(\uptau)-\boldsymbol{\beta}^{\star}(\uptau)\big{)}=\bigg{[}\boldsymbol{M}_{nz}\big{(}\uptau,\boldsymbol{\beta}^{\star}(\uptau)\big{)}\bigg{]}^{-1}\boldsymbol{K}_{nz}\big{(}\uptau,\boldsymbol{\beta}^{\star}(\uptau)\big{)}+o_{\mathbb{P}}(1)$
(7.10)

which proves the result given by expression (7.1). In summary, we proved that
$\widehat{\boldsymbol{\beta}}_{n}^{\star}\left(\uptau\right)$ is a consistent
estimator of $\boldsymbol{\beta}^{\star}\left(\uptau\right)$ with convergence
rate $\sqrt{n}\sqrt[\upgamma_{z}]{n}$ where $\upgamma_{z}\in(0,1)$.
Furthermore, using the result given by expression (7.10) we can prove the
limit results summarized in Corollary 2 using the asymptotic results presented
in Corollary 3. ∎

######  Remark 5.

Notice that for the proof of Corollary 3, we use the fact that

$\displaystyle\frac{1}{\sqrt{n}}\sum_{t=1}^{n}\psi_{\uptau}\big{(}u_{t}(\uptau)\big{)}\overset{\mathbb{P}}{\to}\mathcal{N}\big{(}0,\uptau(1-\uptau)\big{)}\
\ \text{for some}\ \uptau\in(0,1),$ (7.11)

since due to the structure of the model the quantile regression induced
innovation term
$\psi_{\uptau}\big{(}u_{t}(\uptau)\big{)}\sim\mathsf{mds}\big{(}0,\uptau(1-\uptau)\big{)}$,
i.e., it has a covariance which depends on the quantile $\uptau$.

####  Proof of Corollary 3

Part (i)

We begin by considering the limiting distribution of the functional,

$\displaystyle\boldsymbol{K}_{nx}\left(\uptau,\boldsymbol{\theta}^{ols}_{n}(\uptau)\right):=\boldsymbol{D}_{n}^{-1}\sum_{t=1}^{n}\boldsymbol{X}_{t-1}\psi_{\uptau}\big{(}u_{t}(\uptau)\big{)}\Rightarrow\boldsymbol{K}_{x}\left(\uptau,\boldsymbol{\theta}^{ols}(\uptau)\right)$
(7.12)
$\displaystyle\boldsymbol{K}_{x}\left(\uptau,\boldsymbol{\theta}^{ols}(\uptau)\right)\equiv\begin{cases}\begin{bmatrix}B_{\psi_{\uptau}}(1)\\\
\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)dB_{\psi_{\uptau}}\end{bmatrix}&\
\ \ \textit{LUR},\\\ \\\
\mathcal{N}\displaystyle\left(\boldsymbol{0},\uptau(1-\uptau)\times\begin{bmatrix}1&\boldsymbol{0}^{\prime}\\\
\boldsymbol{0}&\boldsymbol{V}_{xx}\end{bmatrix}\right)&\ \ \
\textit{MI}.\end{cases}$

###### Proof.

Mildly Integrated: (MI)

By expanding the expression for
$\boldsymbol{K}_{nx}\big{(}\uptau,\boldsymbol{\theta}^{ols}_{n}(\uptau)\big{)}$,
for the mildly integrated regressors case we obtain the following expression

$\displaystyle\boldsymbol{K}_{nx}\left(\uptau,\boldsymbol{\theta}^{ols}_{n}(\uptau)\right)$
$\displaystyle=\boldsymbol{D}_{n}^{-1}\sum_{t=1}^{n}\boldsymbol{X}_{t-1}\psi_{\uptau}\big{(}u_{t}(\uptau)\big{)}$
$\displaystyle=\begin{bmatrix}\displaystyle\frac{1}{\sqrt{n}}&\boldsymbol{0}^{\prime}\\\
\boldsymbol{0}&\displaystyle
n^{-\frac{1+\upgamma_{x}}{2}}\boldsymbol{I}_{p}\end{bmatrix}_{{\color[rgb]{1,0,0}(p+1)\times(p+1)}}\times\sum_{t=1}^{n}\begin{bmatrix}\boldsymbol{1}\\\
\boldsymbol{x}^{\prime}_{t-1}\end{bmatrix}_{{\color[rgb]{1,0,0}(p+1)\times
n}}\times\psi_{\uptau}\big{(}u_{t}(\uptau)\big{)}$
$\displaystyle=\begin{bmatrix}\displaystyle\frac{1}{\sqrt{n}}\sum_{t=1}^{n}\psi_{\uptau}\big{(}u_{t}(\uptau)\big{)}\otimes\boldsymbol{1}^{\prime}_{{\color[rgb]{1,0,0}(1\times
n)}}\\\ \\\ \displaystyle
n^{-\frac{1+\upgamma_{x}}{2}}\sum_{t=1}^{n}\boldsymbol{x}_{t-1}\psi_{\uptau}\big{(}u_{t}(\uptau)\big{)}\otimes\boldsymbol{I}_{p}\end{bmatrix}_{{\color[rgb]{1,0,0}(p+1)\times
n}}\Rightarrow\mathcal{N}_{p+1}\displaystyle\left(\boldsymbol{0},\uptau(1-\uptau)\times\begin{bmatrix}1&\boldsymbol{0}^{\prime}\\\
\boldsymbol{0}&\boldsymbol{V}_{xx}\end{bmatrix}\right).$

Since the two scalar processes
$\displaystyle\bigg{\\{}\frac{1}{\sqrt{n}}\sum_{t=1}^{n}\psi_{\uptau}\big{(}u_{t}(\uptau)\big{)}\bigg{\\}}$
and
$\displaystyle\bigg{\\{}n^{-\frac{1+\upgamma_{x}}{2}}\sum_{t=1}^{n}\boldsymbol{x}_{t-1}\psi_{\uptau}\big{(}u_{t}(\uptau)\big{)}\bigg{\\}}$
are uncorrelated with negligible higher-order moment terms, thus mutually
independent Gaussian with zero mean and variance 1 and and
$\boldsymbol{V}_{xx}$ respectively. Therefore, a joint convergence to a
Gaussian random variate holds due to their conditional independence. To see
this, consider the following limit

$\displaystyle
n^{-\frac{1+\upgamma_{x}}{2}}\sum_{t=1}^{n}\boldsymbol{x}_{t-1}\psi_{\uptau}\big{(}u_{t}(\uptau)\big{)}\otimes\boldsymbol{I}_{p}\Rightarrow\mathcal{N}\big{(}\boldsymbol{0},\uptau(1-\uptau)\boldsymbol{V}_{xx}\big{)}$
(7.13)

when $0<\upgamma_{x}<1$.

######  Remark 6.

Further details regarding these derivations can be found in the proof of
Theorem 1 in Xiao,  (2009). Notice that although the proof of Theorem 1 in
Xiao,  (2009) corresponds to the framework of quantile cointegrating
regression model, after appropriate modifications we can obtain the limit
given by expression (7.13). Furthermore, joint convergence of these two terms
holds to a Gaussian random variable with mean vector zero and covariance
matrix determined by the covariance of each individual term.

Additionally, the following invariance principle for the corresponding partial
sum process holds for the exponent rate $0<\upgamma_{x}<1$ such that

$\displaystyle n^{-\frac{1+\upgamma_{x}}{2}}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{x}_{t-1}\psi_{\uptau}\big{(}u_{t}(\uptau)\big{)}\otimes\boldsymbol{I}_{p}\Rightarrow\mathcal{N}\big{(}\boldsymbol{0},\uptau(1-\uptau)\lambda\boldsymbol{V}_{xx}\big{)}.$
(7.14)

which is useful when deriving the convergence limit for the OLS based
functionals.

Local Unit Root: (LUR)

$\displaystyle\boldsymbol{K}_{nx}\left(\uptau,\boldsymbol{\theta}^{ols}_{n}(\uptau)\right)$
$\displaystyle=\boldsymbol{D}_{n}^{-1}\sum_{t=1}^{n}\boldsymbol{X}_{t-1}\psi_{\uptau}\big{(}u_{t}(\uptau)\big{)}$
$\displaystyle=\begin{bmatrix}\displaystyle\frac{1}{\sqrt{n}}&\boldsymbol{0}^{\prime}\\\
\boldsymbol{0}&\displaystyle\frac{1}{n}\boldsymbol{I}_{p}\end{bmatrix}_{{\color[rgb]{1,0,0}(p+1)\times(p+1)}}\times\sum_{t=1}^{n}\begin{bmatrix}\boldsymbol{1}^{\prime}_{{\color[rgb]{1,0,0}(1\times
n)}}\\\ \boldsymbol{x}_{t-1}\end{bmatrix}_{{\color[rgb]{1,0,0}(p+1)\times
n}}\times\psi_{\uptau}\big{(}u_{t}(\uptau)\big{)}$
$\displaystyle=\begin{bmatrix}\displaystyle\frac{1}{\sqrt{n}}\sum_{t=1}^{n}\psi_{\uptau}\big{(}u_{t}(\uptau)\big{)}\otimes\boldsymbol{1}^{\prime}\\\
\\\
\displaystyle\frac{1}{n}\sum_{t=1}^{n}\boldsymbol{x}_{t-1}\psi_{\uptau}\big{(}u_{t}(\uptau)\big{)}\otimes\boldsymbol{I}_{p}\end{bmatrix}_{{\color[rgb]{1,0,0}(p+1)\times
n}}\Rightarrow\begin{bmatrix}B_{\psi_{\uptau}}(1)\\\
\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)dB_{\psi_{\uptau}}\end{bmatrix}_{{\color[rgb]{1,0,0}(p+1)\times
n}}$

where
$\boldsymbol{x}_{t}=\left(\boldsymbol{I}_{p}+\frac{\boldsymbol{C}_{p}}{n}\right)\boldsymbol{x}_{t-1}+\boldsymbol{v}_{t}$,
for $1\leq t\leq n$, since $\boldsymbol{X}_{t-1}=\big{[}\boldsymbol{1}\
\boldsymbol{x}_{t-1}^{\prime}\big{]}^{\prime}$.

Therefore, for deriving the limit result above the following weakly
convergence arguments can be applied

$\displaystyle\frac{1}{\sqrt{n}}\sum_{t=1}^{n}\psi_{\uptau}\big{(}u_{t}(\uptau)\big{)}\Rightarrow
B_{\psi_{\uptau}}(1)\ \ \ \text{and}\ \ \
\frac{1}{\sqrt{n}}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\psi_{\uptau}\big{(}u_{t}(\uptau)\big{)}\Rightarrow
B_{\psi_{\uptau}}(\lambda)$ (7.15)

for some $0<\lambda<1$ and $\uptau\in(0,1)$, where $B_{\psi_{\uptau}}(1)$ is a
standard Brownian motion that corresponds to the error function
$\psi_{\uptau}\big{(}u_{t}(\uptau)\big{)}$.

######  Remark 7.

Overall, in the case of LUR regressors (i.e., under high persistence) the
convergence to Brownian motion functionals occurs due to the different
normalization rates employed for these terms. Furthermore, since the local
unit root coefficient is a general case that covers abstract degrees of
persistence, then in practise we have convergence to correlated Brownian
motions.

Part (ii)

Next, we consider the limiting distribution of the functional,
$\boldsymbol{L}_{nx}\big{(}\uptau,\boldsymbol{\theta}^{ols}_{n}(\uptau)\big{)}$
for the two persistence classes separately as explained below.

Local Unit Root: (LUR)

We obtain the following expression

$\displaystyle\boldsymbol{L}_{nx}\big{(}\uptau,\boldsymbol{\theta}^{ols}_{n}(\uptau)\big{)}$
$\displaystyle\overset{\mathsf{def}}{=}\boldsymbol{D}_{n}^{-1}\left[\sum_{t=1}^{n}f_{u_{t}(\uptau),t-1}(0)\boldsymbol{X}_{t-1}\boldsymbol{X}_{t-1}^{\prime}\right]\boldsymbol{D}_{n}^{-1}$
$\displaystyle=\begin{bmatrix}\displaystyle\frac{1}{\sqrt{n}}&0\\\
0&\displaystyle\frac{1}{n}\boldsymbol{I}_{p}\end{bmatrix}\begin{bmatrix}\displaystyle\sum_{t=1}^{n}{\color[rgb]{1,0,0}f_{u_{t}(\uptau),t-1}(0)}&\displaystyle\sum_{t=1}^{n}{\color[rgb]{1,0,0}f_{u_{t}(\uptau),t-1}(0)}\boldsymbol{x}_{t-1}^{\prime}\\\
\displaystyle\sum_{t=1}^{n}{\color[rgb]{1,0,0}f_{u_{t}(\uptau),t-1}(0)}\boldsymbol{x}_{t-1}&\displaystyle\sum_{t=1}^{n}{\color[rgb]{1,0,0}f_{u_{t}(\uptau),t-1}(0)}\boldsymbol{x}_{t-1}\boldsymbol{x}_{t-1}^{\prime}\end{bmatrix}\begin{bmatrix}\displaystyle\frac{1}{\sqrt{n}}&0\\\
0&\displaystyle\frac{1}{n}\boldsymbol{I}_{p}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\displaystyle\frac{1}{\sqrt{n}}\sum_{t=1}^{n}{\color[rgb]{1,0,0}f_{u_{t}(\uptau),t-1}(0)}&\displaystyle\frac{1}{\sqrt{n}}\sum_{t=1}^{n}{\color[rgb]{1,0,0}f_{u_{t}(\uptau),t-1}(0)}\boldsymbol{x}_{t-1}^{\prime}\\\
\displaystyle\frac{1}{n}\sum_{t=1}^{n}{\color[rgb]{1,0,0}f_{u_{t}(\uptau),t-1}(0)}\boldsymbol{x}_{t-1}\otimes\boldsymbol{I}_{p}&\displaystyle\frac{1}{n}\sum_{t=1}^{n}{\color[rgb]{1,0,0}f_{u_{t}(\uptau),t-1}(0)}\boldsymbol{x}_{t-1}\boldsymbol{x}_{t-1}^{\prime}\otimes\boldsymbol{I}_{p}\end{bmatrix}\begin{bmatrix}\displaystyle\frac{1}{\sqrt{n}}&0\\\
0&\displaystyle\frac{1}{n}\boldsymbol{I}_{p}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\displaystyle\frac{1}{n}\sum_{t=1}^{n}{\color[rgb]{1,0,0}f_{u_{t}(\uptau),t-1}(0)}&\displaystyle\frac{1}{n}\frac{1}{\sqrt{n}}\sum_{t=1}^{n}{\color[rgb]{1,0,0}f_{u_{t}(\uptau),t-1}(0)}\boldsymbol{x}_{t-1}^{\prime}\otimes\boldsymbol{I}_{p}\\\
\displaystyle\frac{1}{n}\frac{1}{\sqrt{n}}\sum_{t=1}^{n}{\color[rgb]{1,0,0}f_{u_{t}(\uptau),t-1}(0)}\boldsymbol{x}_{t-1}\otimes\boldsymbol{I}_{p}&\displaystyle\frac{1}{n^{2}}\sum_{t=1}^{n}{\color[rgb]{1,0,0}f_{u_{t}(\uptau),t-1}(0)}\boldsymbol{x}_{t-1}\boldsymbol{x}_{t-1}^{\prime}\otimes\boldsymbol{I}_{p}\end{bmatrix}$
$\displaystyle\Rightarrow
f_{u_{t}(\uptau)}(0)\times\begin{bmatrix}1&\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)^{\prime}\\\
\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)&\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)\boldsymbol{J}_{c}(r)^{\prime}\end{bmatrix}$

Since, the following convergence in probability holds

$\displaystyle\frac{1}{n}\sum_{t=1}^{n}{\color[rgb]{1,0,0}f_{u_{t}(\uptau),t-1}(0)}\overset{\mathbb{P}}{\to}\mathbb{E}\big{[}{\color[rgb]{1,0,0}f_{u_{t}(\uptau),t-1}(0)}\big{]}=:f_{u_{t}(\uptau)}(0)$
(7.16)

Moreover, recall that for the case of LUR regressors it holds that

$\displaystyle\boldsymbol{D}_{n}\big{(}\widehat{\boldsymbol{\theta}}_{n}^{qr}(\uptau)-\boldsymbol{\theta}^{qr}(\uptau)\big{)}$
$\displaystyle=\begin{bmatrix}\displaystyle\sqrt{n}&0\\\ 0&\displaystyle
n\boldsymbol{I}_{p}\end{bmatrix}\begin{pmatrix}\widehat{\alpha}_{n}^{qr}(\uptau)-\alpha(\uptau)\\\
\widehat{\boldsymbol{\beta}}_{n}^{qr}(\uptau)-\boldsymbol{\beta}(\uptau)\end{pmatrix}$
$\displaystyle\equiv\left\\{\boldsymbol{D}_{n}^{-1}\left[\sum_{t=1}^{n}f_{u_{t}(\uptau),t-1}(0)\boldsymbol{X}_{t-1}\boldsymbol{X}_{t-1}^{\prime}\right]\boldsymbol{D}_{n}^{-1}\right\\}^{-1}\times\left\\{\boldsymbol{D}_{n}^{-1}\sum_{t=1}^{n}\boldsymbol{X}_{t-1}\psi_{\uptau}\big{(}u_{t}(\uptau)\big{)}\right\\}$
$\displaystyle\Rightarrow\left\\{f_{u_{t}(\uptau)}(0)\times\begin{bmatrix}1&\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)^{\prime}\\\
\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)&\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)\boldsymbol{J}_{c}(r)^{\prime}\end{bmatrix}\right\\}^{-1}\times\begin{bmatrix}B_{\psi_{\uptau}}(1)\\\
\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)dB_{\psi_{\uptau}}\end{bmatrix}$

which follows by an application of the continuous mapping theorem to the first
term of the expression above. Notice that in the case of persistent
regressors, the limiting distribution of the normalized quantile OLS based
estimator is nonstandard and nonpivotal.

Mildly Integrated: (MI)

We obtain the following expression

$\displaystyle\boldsymbol{L}_{nx}\big{(}\uptau,\boldsymbol{\theta}^{ols}_{n}(\uptau)\big{)}$
$\displaystyle\overset{\mathsf{def}}{=}\boldsymbol{D}_{n}^{-1}\left[\sum_{t=1}^{n}f_{u_{t}(\uptau),t-1}(0)\boldsymbol{X}_{t-1}\boldsymbol{X}_{t-1}^{\prime}\right]\boldsymbol{D}_{n}^{-1}$
$\displaystyle=\begin{bmatrix}\displaystyle\frac{1}{n}\sum_{t=1}^{n}{\color[rgb]{1,0,0}f_{u_{t}(\uptau),t-1}(0)}&\displaystyle\frac{1}{n^{1+\frac{\upgamma_{x}}{2}}}\sum_{t=1}^{n}{\color[rgb]{1,0,0}f_{u_{t}(\uptau),t-1}(0)}\boldsymbol{x}_{t-1}^{\prime}\otimes\boldsymbol{I}_{p}\\\
\displaystyle\frac{1}{n^{1+\frac{\upgamma_{x}}{2}}}\sum_{t=1}^{n}{\color[rgb]{1,0,0}f_{u_{t}(\uptau),t-1}(0)}\boldsymbol{x}_{t-1}\otimes\boldsymbol{I}_{p}&\displaystyle\frac{1}{n^{1+\upgamma_{x}}}\sum_{t=1}^{n}{\color[rgb]{1,0,0}f_{u_{t}(\uptau),t-1}(0)}\boldsymbol{x}_{t-1}\boldsymbol{x}_{t-1}^{\prime}\otimes\boldsymbol{I}_{p}\end{bmatrix}$
$\displaystyle\Rightarrow
f_{u_{t}(\uptau)}(0)\times\begin{bmatrix}1&\boldsymbol{0}^{\prime}\\\
\boldsymbol{0}&\boldsymbol{V}_{xx}\end{bmatrix}$ (7.17)

Since it holds that,

$\displaystyle\frac{1}{n^{1+\frac{\upgamma_{x}}{2}}}\sum_{t=1}^{n}\boldsymbol{x}_{t-1}\overset{\mathbb{P}}{\to}\boldsymbol{0}\
\ \ \ \text{and}\ \ \ \
\frac{1}{n^{1+\upgamma_{x}}}\sum_{t=1}^{n}\boldsymbol{x}_{t-1}\boldsymbol{x}_{t-1}^{\prime}\overset{\mathbb{P}}{\to}\boldsymbol{V}_{xx}\
\ \ \text{when}\ \upgamma_{x}\in(0,1),$

where the second limit follows by Lemma B3 in the Appendix of Kostakis et al.,
(2015). Therefore, for the case of mildly integrated regressors in the model
we obtain

$\displaystyle\boldsymbol{D}_{n}\big{(}\widehat{\boldsymbol{\theta}}_{n}^{qr}(\uptau)-\boldsymbol{\theta}^{qr}(\uptau)\big{)}$
$\displaystyle=\begin{bmatrix}\displaystyle\sqrt{n}&0\\\ 0&\displaystyle
n^{\frac{1+\upgamma_{x}}{2}}\boldsymbol{I}_{p}\end{bmatrix}\begin{pmatrix}\widehat{\alpha}_{n}^{qr}(\uptau)-\alpha(\uptau)\\\
\widehat{\boldsymbol{\beta}}_{n}^{qr}(\uptau)-\boldsymbol{\beta}(\uptau)\end{pmatrix}$
$\displaystyle\equiv\left\\{\boldsymbol{D}_{n}^{-1}\left[\sum_{t=1}^{n}f_{u_{t}(\uptau),t-1}(0)\boldsymbol{X}_{t-1}\boldsymbol{X}_{t-1}^{\prime}\right]\boldsymbol{D}_{n}^{-1}\right\\}^{-1}\times\left\\{\boldsymbol{D}_{n}^{-1}\sum_{t=1}^{n}\boldsymbol{X}_{t-1}\psi_{\uptau}\big{(}u_{t}(\uptau)\big{)}\right\\}$
$\displaystyle\Rightarrow\left\\{f_{u_{t}(\uptau)}(0)\times\begin{bmatrix}1&\boldsymbol{0}^{\prime}\\\
\boldsymbol{0}&\boldsymbol{V}_{xx}\end{bmatrix}\right\\}^{-1}\times\mathcal{N}\left(0,\uptau(1-\uptau)\times\begin{bmatrix}1&\boldsymbol{0}^{\prime}\\\
\boldsymbol{0}&\boldsymbol{V}_{xx}\end{bmatrix}\right)$
$\displaystyle=\frac{1}{f_{u_{t}(\uptau)}(0)}\times\begin{bmatrix}1&\boldsymbol{0}^{\prime}\\\
\boldsymbol{0}&\boldsymbol{V}_{xx}^{-1}\end{bmatrix}\times\mathcal{N}\left(0,\uptau(1-\uptau)\times\begin{bmatrix}1&\boldsymbol{0}^{\prime}\\\
\boldsymbol{0}&\boldsymbol{V}_{xx}\end{bmatrix}\right)$
$\displaystyle\equiv\mathcal{N}\left(0,\frac{\uptau(1-\uptau)}{f_{u_{t}(\uptau)}(0)^{2}}\times\begin{bmatrix}1&\boldsymbol{0}^{\prime}\\\
\boldsymbol{0}&\boldsymbol{V}^{-1}_{xx}\end{bmatrix}\right)$ (7.18)

by an application of the continuous mapping theorem to the first term. ∎

######  Remark 8.

Notice also that the limit results given by Corollary 3 can be used to prove
the limiting distribution provided by Corollary 1 of the paper. This can be
done, by employing the trick presented in the proof of Theorem 1 in Xiao,
(2009). In particular, by linearizing the optimization function in terms of an
arbitrary centered quantity
$\boldsymbol{D}_{n}\left(\widehat{\boldsymbol{\theta}}_{n}(\uptau)-\boldsymbol{\theta}(\uptau)\right)$.
Thus, using the convexity lemma we take the distributional limit of the
linearized part and then minimize to get the desired expression as in (7.10).

####  Proof of Corollary 4

###### Proof.

For the IVX based estimation of the quantile regression model, we use the
dequantile procedure proposed by Lee,  (2016). Thus,
$y_{t}(\uptau)=y_{t}-\alpha(\uptau)+\mathcal{O}_{\mathbb{P}}(n^{-1/2})$.
Furthermore, we employ the following embedded normalization matrices

$\displaystyle\tilde{\boldsymbol{Z}}_{t-1,n}:=\tilde{\boldsymbol{D}}_{n}\tilde{\boldsymbol{z}}_{t-1}\
\ \ \ \text{and}\ \ \ \
\tilde{\boldsymbol{X}}_{t-1,n}:=\tilde{\boldsymbol{D}}_{n}\boldsymbol{x}_{t-1},\
\ \ \text{where}\ \
\tilde{\boldsymbol{D}}_{n}:=n^{\frac{1+(\upgamma_{x}\wedge\upgamma_{z})}{2}}\boldsymbol{I}_{p}$
(7.19)

Part (i)

The limit holds regardless of the stochastic dominance of the exponent rates
$\upgamma_{x}$ and $\upgamma_{z}$

$\displaystyle\boldsymbol{K}_{nz}\big{(}\uptau,\boldsymbol{\beta}^{ivx}_{n}(\uptau)\big{)}$
$\displaystyle:=\sum_{t=1}^{n}\tilde{\boldsymbol{Z}}_{t-1}\psi_{\uptau}\big{(}u_{t}(\uptau)\big{)}\equiv
n^{\frac{1+(\gamma_{x}\wedge\gamma_{z})}{2}}\sum_{t=1}^{n}\tilde{\boldsymbol{z}}_{t-1}\psi_{\uptau}\big{(}u_{t}(\uptau)\big{)}\otimes\boldsymbol{I}_{p}$
$\displaystyle\Rightarrow\mathcal{N}\big{(}\boldsymbol{0},\uptau(1-\uptau)\boldsymbol{V}_{cxz}\big{)}.$
(7.20)

Part (ii)

$\displaystyle\boldsymbol{M}_{nz}\big{(}\uptau,\boldsymbol{\beta}^{ivx}_{n}(\uptau)\big{)}:=\sum_{t=1}^{n}f_{u_{t}(\uptau),t-1}(0)\tilde{\boldsymbol{Z}}_{t-1,n}\tilde{\boldsymbol{Z}}_{t-1,n}^{\prime}\Rightarrow
f_{u_{t}(\uptau)}(0)\times\boldsymbol{V}_{cxz}$ (7.21)

which can be easily shown, since
$\sum_{t=1}^{n}\tilde{\boldsymbol{z}}_{t-1}\tilde{\boldsymbol{z}}_{t-1}^{\prime}\overset{\mathbb{P}}{\to}\boldsymbol{V}_{cxz}$.

Part (iii)

Then,
$\tilde{\boldsymbol{Z}}_{t-1,n}\tilde{\boldsymbol{X}}_{t-1,n}^{\prime}\equiv\tilde{\boldsymbol{D}}_{n}\tilde{\boldsymbol{z}}_{t-1}\tilde{\boldsymbol{x}}_{t-1}^{\prime}\tilde{\boldsymbol{D}}_{n}^{\prime}$.
Therefore, the limit result follows as below

$\displaystyle\boldsymbol{M}_{nz}\big{(}\uptau,\boldsymbol{\beta}^{ivx}_{n}(\uptau)\big{)}$
$\displaystyle:=\left[\sum_{t=1}^{n}f_{u_{t}(\uptau),t-1}(0)\tilde{\boldsymbol{Z}}_{t-1,n}\tilde{\boldsymbol{X}}_{t-1,n}^{\prime}\right]\equiv{\color[rgb]{1,0,1}\tilde{\boldsymbol{D}}_{n}}\left[\sum_{t=1}^{n}f_{u_{t}(\uptau),t-1}(0)\tilde{\boldsymbol{z}}_{t-1}\tilde{\boldsymbol{x}}_{t-1}^{\prime}\right]{\color[rgb]{1,0,1}\tilde{\boldsymbol{D}}_{n}^{\prime}}$
$\displaystyle\Rightarrow f_{u_{t}(\uptau)}(0)\times\boldsymbol{\Gamma}_{cxz}$

Since, a convergence in probability holds
$\frac{1}{n}\sum_{t=1}^{n}{\color[rgb]{1,0,0}f_{u_{t}(\uptau),t-1}(0)}\overset{\mathbb{P}}{\to}\mathbb{E}\big{[}{\color[rgb]{1,0,0}f_{u_{t}(\uptau),t-1}(0)}\big{]}=:f_{u_{t}(\uptau)}(0)$.
Moreover, we have that

$\displaystyle\tilde{\boldsymbol{D}}_{n}$
$\displaystyle\big{(}\widehat{\boldsymbol{\beta}}_{n}^{ivx-
qr}(\uptau)-\boldsymbol{\beta}(\uptau)\big{)}$
$\displaystyle\equiv\left\\{{\color[rgb]{1,0,1}\tilde{\boldsymbol{D}}_{n}}\left[\sum_{t=1}^{n}f_{u_{t}(\uptau),t-1}(0)\tilde{\boldsymbol{z}}_{t-1}\boldsymbol{x}_{t-1}^{\prime}\right]{\color[rgb]{1,0,1}\tilde{\boldsymbol{D}}_{n}^{\prime}}\right\\}^{-1}\times\left\\{{\color[rgb]{1,0,1}\tilde{\boldsymbol{D}}_{n}}\sum_{t=1}^{n}\tilde{\boldsymbol{z}}_{t-1}\psi_{\uptau}\big{(}u_{t}(\uptau)\big{)}\right\\}$
$\displaystyle\Rightarrow\bigg{\\{}f_{u_{t}(\uptau)}(0)\times\boldsymbol{\Gamma}_{cxz}\bigg{\\}}^{-1}\times\mathcal{N}\big{(}\boldsymbol{0},\uptau(1-\uptau)\boldsymbol{V}_{cxz}\big{)}$
$\displaystyle\equiv\frac{1}{f_{u_{t}(\uptau)}(0)}\times\boldsymbol{\Gamma}_{cxz}^{-1}\times\mathcal{N}\big{(}\boldsymbol{0},\uptau(1-\uptau)\boldsymbol{V}_{cxz}\big{)}$
$\displaystyle=\mathcal{N}\left(\boldsymbol{0},\frac{\uptau(1-\uptau)}{f_{u_{t}(\uptau)}(0)^{2}}\boldsymbol{\Gamma}_{cxz}^{-1}\boldsymbol{V}_{cxz}\left(\boldsymbol{\Gamma}_{cxz}^{-1}\right)^{\prime}\right)\equiv\mathcal{N}\left(\boldsymbol{0},\frac{\uptau(1-\uptau)}{f_{u_{t}(\uptau)}(0)^{2}}\left(\boldsymbol{\Gamma}_{cxz}\boldsymbol{V}_{cxz}^{-1}\boldsymbol{\Gamma}_{cxz}^{\prime}\right)^{-1}\right).$

The above result proves the Gaussian random variable limit given by Corollary
2 which holds for both the cases of local unit root and mildly integrated
regressors in the quantile predictive regression model. Furthermore, in the
case we employ the alternative IVX-QR estimator proposed by Lee,  (2016) (IVZ
estimator); in which case the set of nonstationary regressors,
$\boldsymbol{x}_{t-1}$, is replaced by the mildly integrated instruments,
$\tilde{\boldsymbol{z}}_{t-1}$, we obtain the following limit result

$\displaystyle\tilde{\boldsymbol{D}}_{n}\big{(}\widehat{\boldsymbol{\beta}}_{n}^{ivz-
qr}(\uptau)-\boldsymbol{\beta}(\uptau)\big{)}$
$\displaystyle\Rightarrow\bigg{\\{}f_{u_{t}(\uptau)}(0)\times\boldsymbol{V}_{cxz}\bigg{\\}}^{-1}\times\mathcal{N}\big{(}\boldsymbol{0},\uptau(1-\uptau)\boldsymbol{V}_{cxz}\big{)}$
$\displaystyle=\frac{1}{f_{u_{t}(\uptau)}(0)}\times\boldsymbol{V}_{cxz}^{-1}\times\mathcal{N}\big{(}\boldsymbol{0},\uptau(1-\uptau)\boldsymbol{V}_{cxz}\big{)}$
$\displaystyle=\mathcal{N}\left(\boldsymbol{0},\frac{\uptau(1-\uptau)}{f_{u_{t}(\uptau)}(0)}\boldsymbol{V}_{cxz}^{-1}\right)$
(7.22)

since,
$\sum_{t=1}^{n}f_{u_{t}(\uptau),t-1}(0)\tilde{\boldsymbol{Z}}_{t-1,n}\tilde{\boldsymbol{Z}}_{t-1,n}^{\prime}\Rightarrow
f_{u_{t}(\uptau)}(0)\times\boldsymbol{V}_{cxz}$, which is nuisance-parameter
free for both the case of local unit root or mildly integrated regressors in
the model. ∎

######  Remark 9.

Notice that in a standard time series quantile regression with stationary
covariates it holds that the regression $\uptau-$quantile is asymptotically
normal, with

$\displaystyle\sqrt{n}\left(\widehat{\boldsymbol{\beta}}_{n}-\boldsymbol{\beta}(\uptau)\right)\overset{d}{\to}\mathcal{N}\big{(}\boldsymbol{0},\uptau(1-\uptau)\boldsymbol{D}_{1}^{-1}(\uptau)\boldsymbol{D}_{0}\boldsymbol{D}_{1}^{-1}(\uptau)\big{)}.$
(7.23)

with an appropriate defined covariance matrix which is a function of the
moments of the underline error distribution (see, Goh and Knight,  (2009)).
Therefore, we can clearly see that under nonstationarity the covariance matrix
of the Gaussian random variate is stochastic due to the presence of the
nuisance parameter of persistence.

######  Remark 10.

Notice that the objective function for the setting of the nonstationary
quantile predictive regression model, becomes globally convex in the
parameter. Hence, the method based on the convexity lemma by Pollard (1991) is
applicable. Therefore, our proofs for the asymptotic theory of test of
parameter restrictions is based on the asymptotic theory framework proposed by
Xiao,  (2009). Consequently, the limit results for the linear parameter
restrictions can be employed when constructing the parameter specific
restrictions that correspond to structural break tests.

####  Alternative IVZ-QR estimator

Following the framework proposed by Lee,  (2016) we also consider the limiting
distribution of the IVX-QR estimator when the original persistent regressors
are replaced by the instrumental variables in the optimization function. The
particular approach is convenient as it significantly reduces the
computational time by avoiding the nonconvex optimization procedure given by
expression (2.20) which requires to use a grid search with several local
optima. More specifically, we consider

$\displaystyle\widehat{\boldsymbol{\gamma}}_{n}^{ivx-
qr}\left(\uptau\right):=\underset{\boldsymbol{\gamma}\in\mathbb{R}^{p}}{\mathsf{arg\
min}}\
\sum_{t=1}^{n}\uprho_{\tau}\big{(}y_{t}(\uptau)-\tilde{\boldsymbol{z}}_{t-1}^{\prime}\boldsymbol{\gamma}\big{)}.$
(7.24)

######  Corollary 6.

Under the null hypothesis $\mathcal{H}_{0}:\boldsymbol{\beta}(\uptau)=0$, it
holds that

$\displaystyle\tilde{\boldsymbol{D}}_{n}\big{(}\widehat{\boldsymbol{\gamma}}_{n}^{ivx-
qr}(\uptau)-\boldsymbol{\beta}(\uptau)\big{)}\Rightarrow\mathcal{N}\left(\boldsymbol{0},\frac{\displaystyle\uptau(1-\uptau)}{\displaystyle
f_{u_{t}(\uptau)}(0)^{2}}\boldsymbol{V}_{cxz}^{-1}\right)$ (7.25)

both for near unit root and mildly integrated predictors, where
$\tilde{\boldsymbol{D}}_{n}=n^{\frac{1+\upgamma_{x}\wedge\upgamma_{z}}{2}}\boldsymbol{I}_{p}$.

######  Lemma 3.

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

$\displaystyle\frac{\widehat{f_{u_{t}(\uptau)}}(0)^{2}}{\uptau(1-\uptau)}\left(\widehat{\boldsymbol{\gamma}}^{ivx-
qr}_{n}(\uptau)-\boldsymbol{\beta}(\uptau)\right)^{\prime}\big{(}\tilde{\boldsymbol{Z}}^{\prime}\tilde{\boldsymbol{Z}}\big{)}^{-1}\left(\widehat{\boldsymbol{\gamma}}^{ivx-
qr}_{n}(\uptau)-\boldsymbol{\beta}(\uptau)\right)\Rightarrow\chi^{2}_{p}$
(7.26)

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

Therefore, Lemma 3 provides a uniform inference limit result which allows to
easily obtain critical values since is nuisance-parameter free. Furthermore,
if we are interested to test for example the predictability of a specific
subgroup among the predictors, say
$\mathcal{H}_{0}:\beta_{1}(\uptau)=\beta_{2}(\uptau)=0$, then the formulation
of the Wald statistic with the linear restrictions matrix can be employed. In
the particular example, the restrictions matrix takes the form
$\boldsymbol{R}=\big{[}\boldsymbol{I}_{2},\boldsymbol{0}_{2\times(p-2)}\big{]}$.
Then, generalizing the specific example for testing a set of linear
restrictions, implies that the null hypothesis is formulated as
$\mathcal{H}_{0}:\boldsymbol{R}\boldsymbol{\beta}(\uptau)=0$ where
$\boldsymbol{R}$ is a $r\times p$ known restriction matrix.

Then, the limiting distribution for the IVX-Wald statistic for the quantile
predictive regression is given by the following expression

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

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

####  Proof of Lemma 1

We have that

$\displaystyle\mathcal{W}^{ivx-
qr}_{n}(\uptau)=\frac{\widehat{f_{u_{t}(\uptau)}}(0)^{2}}{\uptau(1-\uptau)}\left(\widehat{\boldsymbol{\beta}}^{ivx-
qr}_{n}(\uptau)-\boldsymbol{\beta}(\uptau)\right)^{\prime}\bigg{(}\boldsymbol{X}^{\prime}\boldsymbol{P}_{\tilde{\boldsymbol{Z}}}\boldsymbol{X}\bigg{)}\left(\widehat{\boldsymbol{\beta}}^{ivx-
qr}_{n}(\uptau)-\boldsymbol{\beta}(\uptau)\right)\Rightarrow\chi^{2}_{p}$
(7.27)

where

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

Moreover, we use the embedded normalization matrices such that

$\displaystyle\left(\sum_{t=1}^{n}f_{u_{t}(\uptau),t-1}(0)\right)\times\left(\sum_{t=1}^{n}\tilde{\boldsymbol{Z}}_{t-1,n}\boldsymbol{X}^{\prime}_{t-1,n}\right)\equiv\left(\sum_{t=1}^{n}f_{u_{t}(\uptau),t-1}(0)\tilde{\boldsymbol{Z}}_{t-1,n}\boldsymbol{X}^{\prime}_{t-1,n}\right)\overset{\mathbb{P}}{\to}f_{u_{t}(\uptau)}(0)\boldsymbol{\Gamma}_{cxz}$

and the fact that
$\displaystyle\left(\sum_{t=1}^{n}\boldsymbol{z}_{t-1}\tilde{\boldsymbol{z}}^{\prime}_{t-1}\right)\overset{\mathbb{P}}{\to}\boldsymbol{V}_{cxz}$.

###### Proof.

From Corollary 2 we have that

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

Denote the consistent sample estimator of $\widehat{f_{u_{t}(\uptau)}}(0)$,
with $f_{u_{t}(\uptau)}(0)$ where

$\displaystyle\widehat{f_{u_{t}(\uptau)}}(0)=\sum_{t=1}^{n}f_{u_{t}(\uptau),t-1}(0)$
(7.28)

Therefore, the following asymptotic convergence result follows

$\displaystyle\bigg{(}\boldsymbol{X}^{\prime}\boldsymbol{P}_{\tilde{\boldsymbol{Z}}}\boldsymbol{X}\bigg{)}\Rightarrow\big{[}f_{u_{t}(\uptau)}(0)\boldsymbol{\Gamma}_{cxz}\big{]}\times\boldsymbol{V}_{cxz}^{-1}\times\big{[}f_{u_{t}(\uptau)}(0)\boldsymbol{\Gamma}_{cxz}\big{]}^{\prime}\equiv
f_{u_{t}(\uptau)}(0)^{2}\big{(}\boldsymbol{\Gamma}_{cxz}\boldsymbol{V}_{cxz}^{-1}\boldsymbol{\Gamma}_{cxz}^{\prime}\big{)}$

Thus, we obtain

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

$\displaystyle\mathcal{W}^{ivx-qr}_{n}(\uptau)$
$\displaystyle\Rightarrow\frac{f_{u_{t}(\uptau)}(0)^{2}}{\uptau(1-\uptau)}\frac{\uptau(1-\uptau)}{f_{u_{t}(\uptau)}(0)^{2}}\times\bigg{[}\mathcal{N}\left(0,\big{(}\boldsymbol{\Gamma}_{cxz}\boldsymbol{V}^{-1}_{cxz}\boldsymbol{\Gamma}_{cxz}^{\prime}\big{)}^{-1}\right)\bigg{]}^{\prime}$
$\displaystyle\ \
\times\big{(}\boldsymbol{\Gamma}_{cxz}\boldsymbol{V}_{cxz}^{-1}\boldsymbol{\Gamma}_{cxz}^{\prime}\big{)}^{-1}\times\bigg{[}\mathcal{N}\left(0,\big{(}\boldsymbol{\Gamma}_{cxz}\boldsymbol{V}^{-1}_{cxz}\boldsymbol{\Gamma}_{cxz}^{\prime}\big{)}^{-1}\right)\bigg{]}$
$\displaystyle=\mathcal{N}(0,1)\left[\big{(}\boldsymbol{\Gamma}_{cxz}\boldsymbol{V}^{-1}_{cxz}\boldsymbol{\Gamma}_{cxz}^{\prime}\big{)}^{-1/2}\right]^{\prime}\times\big{(}\boldsymbol{\Gamma}_{cxz}\boldsymbol{V}_{cxz}^{-1}\boldsymbol{\Gamma}_{cxz}^{\prime}\big{)}\times\left[\big{(}\boldsymbol{\Gamma}_{cxz}\boldsymbol{V}^{-1}_{cxz}\boldsymbol{\Gamma}_{cxz}^{\prime}\big{)}^{-1/2}\right]\mathcal{N}(0,1)$
$\displaystyle=\big{[}\mathcal{N}(0,1)\big{]}^{2}\equiv\chi^{2}_{2}.$

which is a standard $\chi^{2}-$distribution with $2$ degrees of freedom. ∎

####  Proof of Proposition 1

###### Proof.

Part (i)

First we consider the limit expression for the case of mildly integrated
regressors

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

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

$\displaystyle\mathcal{SQ}^{ols}_{n}(\lambda;\uptau_{0})\Rightarrow\underset{\lambda\in[0,1]}{\mathsf{sup}}\
\big{\|}\boldsymbol{\mathcal{BB}}_{p}(\lambda)\big{\|}_{\infty}.$ (7.29)

which is a nuisance-parameter free distribution that holds under the null
hypothesis. In summary, suppose that the data are generated by the quantile
predictive regression model and Assumptions 1-2 are satisfied. Then, under the
null null hypothesis $\mathcal{H}_{0}^{(A)}$ the fluctuation type statistics
weakly converge to the limiting distribution given by expression (7.29) for
mildly integrated regressors for some unknown break-point location
$0<\lambda<1$.

Second, for local unit root regressors (high persistence) then, following
limit holds

$\displaystyle\mathcal{L}$
$\displaystyle:=\hat{\mathcal{J}}_{n}\left(\lambda,\uptau_{0},\widehat{\boldsymbol{\theta}}^{ols}_{n}(\uptau_{0})\right)-\lambda\hat{\mathcal{J}}_{n}\left(1,\uptau_{0},\widehat{\boldsymbol{\theta}}^{ols}_{n}(\uptau_{0})\right)$
$\displaystyle=\left(\boldsymbol{D}_{n}^{-1}\left[\sum_{t=1}^{n}\boldsymbol{X}^{\prime}_{t-1}\boldsymbol{X}_{t-1}\right]\boldsymbol{D}_{n}^{-1}\right)^{-1/2}\left\\{\boldsymbol{D}_{n}^{-1}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{X}_{t-1}\psi_{\uptau}\big{(}u_{t}(\uptau_{0})\big{)}-\lambda\boldsymbol{D}_{n}^{-1}\sum_{t=1}^{n}\boldsymbol{X}_{t-1}\psi_{\uptau}\big{(}u_{t}(\uptau_{0})\big{)}\right\\}$
$\displaystyle\Rightarrow\begin{bmatrix}1&\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)^{\prime}\\\
\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)&\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)\boldsymbol{J}_{c}(r)^{\prime}\end{bmatrix}^{-1/2}\times\left\\{\begin{bmatrix}B_{\psi_{\uptau}}(\lambda)\\\
\displaystyle\int_{0}^{\lambda}\boldsymbol{J}_{c}(r)dB_{\psi_{\uptau}}\end{bmatrix}-\lambda\begin{bmatrix}B_{\psi_{\uptau}}(1)\\\
\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)dB_{\psi_{\uptau}}\end{bmatrix}\right\\}$
$\displaystyle\equiv\begin{bmatrix}1&\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)^{\prime}\\\
\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)&\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)\boldsymbol{J}_{c}(r)^{\prime}\end{bmatrix}^{-1/2}\times\begin{bmatrix}B_{\psi_{\uptau}}(\lambda)-\lambda
B_{\psi_{\uptau}}(1)\\\
\displaystyle\int_{0}^{\lambda}\boldsymbol{J}_{c}(r)dB_{\psi_{\uptau}}-\lambda\int_{0}^{1}\boldsymbol{J}_{c}(r)dB_{\psi_{\uptau}}\end{bmatrix}$
$\displaystyle\equiv\begin{bmatrix}1&\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)^{\prime}_{{\color[rgb]{1,0,0}(1\times
p)}}\\\
\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)_{{\color[rgb]{1,0,0}(p\times
1)}}&\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)\boldsymbol{J}_{c}(r)^{\prime}_{{\color[rgb]{1,0,0}(p\times
p)}}\end{bmatrix}^{-1/2}_{{\color[rgb]{1,0,0}(p+1)\times(p+1)}}\times\begin{bmatrix}\mathbf{\mathcal{BB}}_{\psi_{\uptau}}(\lambda)_{{\color[rgb]{1,0,0}(1\times
n)}}\\\
\displaystyle\mathbf{\mathcal{JB}}_{\psi_{\uptau}}(\lambda)_{{\color[rgb]{1,0,0}(p\times
n)}}\end{bmatrix}_{{\color[rgb]{1,0,0}(p+1)\times n}}$ (7.30)

where
$\mathbf{\mathcal{BB}}_{\psi_{\uptau}}(\lambda):=B_{\psi_{\uptau}}(\lambda)-\lambda
B_{\psi_{\uptau}}(1)$ and
$\displaystyle\mathbf{\mathcal{JB}}_{\psi_{\uptau}}(\lambda):=\int_{0}^{\lambda}\boldsymbol{J}_{c}(r)dB_{\psi_{\uptau}}-\lambda\int_{0}^{1}\boldsymbol{J}_{c}(r)dB_{\psi_{\uptau}}$.
Thus, under the null hypothesis the OLS based functional with local unit root
regressors converges into a nonstandard and nonpivotal limiting distribution.

Part (ii)

The limit expression of the fluctuation type test based on the IVX estimator
is given as

$\displaystyle\mathcal{L}$
$\displaystyle:=\hat{\mathcal{J}}_{n}\left(\lambda,\uptau_{0},\widehat{\boldsymbol{\beta}}^{ivx}_{n}(\uptau_{0})\right)-\lambda\hat{\mathcal{J}}_{n}\left(1,\uptau_{0},\widehat{\boldsymbol{\beta}}^{ivx}_{n}(\uptau_{0})\right)$
$\displaystyle=\left(\sum_{t=1}^{n}\tilde{\boldsymbol{Z}}_{t-1,n}\boldsymbol{X}^{\prime}_{t-1,n}\right)^{-1/2}\left\\{\sum_{t=1}^{\lfloor\lambda
n\rfloor}\tilde{\boldsymbol{Z}}_{t-1,n}\psi_{\uptau}\big{(}u_{t}(\uptau_{0})\big{)}-\lambda\sum_{t=1}^{n}\tilde{\boldsymbol{Z}}_{t-1,n}\psi_{\uptau}\big{(}u_{t}(\uptau_{0})\big{)}\right\\}$
$\displaystyle=\left(\tilde{\boldsymbol{D}}_{n}^{-1}\left[\sum_{t=1}^{n}\tilde{\boldsymbol{z}}_{t-1}\boldsymbol{x}^{\prime}_{t-1}\right]\tilde{\boldsymbol{D}}_{n}^{-1}\right)^{-1/2}\left\\{\tilde{\boldsymbol{D}}_{n}^{-1}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\tilde{\boldsymbol{z}}_{t-1}\psi_{\uptau}\big{(}u_{t}(\uptau_{0})\big{)}-\lambda\tilde{\boldsymbol{D}}_{n}^{-1}\sum_{t=1}^{n}\tilde{\boldsymbol{z}}_{t-1}\psi_{\uptau}\big{(}u_{t}(\uptau_{0})\big{)}\right\\}$
$\displaystyle\Rightarrow\boldsymbol{\Gamma}_{cxz}^{-1/2}\times\bigg{\\{}\mathcal{N}\displaystyle\bigg{(}\boldsymbol{0},\uptau_{0}(1-\uptau_{0})\lambda\boldsymbol{V}_{cxz}\bigg{)}-\lambda\mathcal{N}\bigg{(}\boldsymbol{0},\uptau_{0}(1-\uptau_{0})\boldsymbol{V}_{cxz}\bigg{)}\bigg{\\}}$
$\displaystyle=\sqrt{\uptau_{0}(1-\uptau_{0})}\times\boldsymbol{\Gamma}_{cxz}^{-1/2}\times\bigg{\\{}\mathcal{N}\displaystyle\bigg{(}\boldsymbol{0},\lambda\boldsymbol{V}_{cxz}\bigg{)}-\lambda\mathcal{N}\bigg{(}\boldsymbol{0},\boldsymbol{V}_{cxz}\bigg{)}\bigg{\\}}\equiv\sqrt{\uptau_{0}(1-\uptau_{0})}\big{[}\boldsymbol{W}_{p}(\lambda)-\lambda\boldsymbol{W}_{p}(1)\big{]}.$

provided that $\boldsymbol{\Gamma}_{cxz}\equiv\boldsymbol{V}_{xx}$, which
applies when $\upgamma_{x}\in(0,\upgamma_{z})$. Proposition 1 (ii) shows that
the limiting distribution of the fluctuation type test is nonstandard in
general when the IVX estimator is employed since we employ limit results which
hold for both LUR and MI regressors. However, when the coefficient of
persistence of regressors has an exponent rate with an absolute value less
than the exponent rate of the mildly integrated instruments, then the
asymptotic covariance matrix of the Gaussian variant has a simpified form due
to the stochastic dominance property of these covariance matrices.

Part (iii)

We obtain the following limit result which holds for both LUR and MI
regressors

$\displaystyle\mathcal{L}$
$\displaystyle:=\hat{\mathcal{J}}^{ivz}_{n}\left(\lambda,\uptau_{0},\widehat{\boldsymbol{\beta}}^{ivz}_{n}(\uptau_{0})\right)-\lambda\hat{\mathcal{J}}^{ivz}_{n}\left(1,\uptau_{0},\widehat{\boldsymbol{\beta}}^{ivz}_{n}(\uptau_{0})\right)$
$\displaystyle=\big{(}\tilde{\boldsymbol{Z}}^{\prime}\tilde{\boldsymbol{Z}}\big{)}^{-1/2}\left\\{\tilde{\boldsymbol{D}}_{n}^{-1}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\tilde{\boldsymbol{z}}_{t-1}\psi_{\uptau}\left(y_{t}-\tilde{\boldsymbol{z}}_{t-1}^{\prime}\widehat{\boldsymbol{\beta}}^{ivz}_{n}(\uptau_{0})\right)-\lambda\tilde{\boldsymbol{D}}_{n}^{-1}\sum_{t=1}^{n}\tilde{\boldsymbol{z}}_{t-1}\psi_{\uptau}\left(y_{t}-\tilde{\boldsymbol{z}}_{t-1}^{\prime}\widehat{\boldsymbol{\beta}}^{ivz}_{n}(\uptau_{0})\right)\right\\}$
$\displaystyle=\left(\tilde{\boldsymbol{D}}_{n}^{-1}\left[\sum_{t=1}^{n}\tilde{\boldsymbol{z}}_{t-1}\tilde{\boldsymbol{z}}_{t-1}^{\prime}\right]\tilde{\boldsymbol{D}}_{n}^{-1}\right)^{-1/2}\left\\{\tilde{\boldsymbol{D}}_{n}^{-1}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\tilde{\boldsymbol{z}}_{t-1}\psi_{\uptau}\big{(}y_{t}-\tilde{\boldsymbol{z}}_{t-1}^{\prime}\boldsymbol{\beta}_{0}(\uptau_{0})\big{)}-\lambda\tilde{\boldsymbol{D}}_{n}^{-1}\sum_{t=1}^{n}\boldsymbol{x}_{t-1}\psi_{\uptau}\big{(}y_{t}-\tilde{\boldsymbol{z}}_{t-1}^{\prime}\boldsymbol{\beta}_{0}(\uptau_{0})\big{)}\right\\}$
$\displaystyle\Rightarrow\boldsymbol{V}_{cxz}^{-1/2}\times\bigg{\\{}\mathcal{N}\bigg{(}\boldsymbol{0},\uptau_{0}(1-\uptau_{0})\lambda\boldsymbol{V}_{cxz}\bigg{)}-\lambda\mathcal{N}\bigg{(}\boldsymbol{0},\uptau_{0}(1-\uptau_{0})\boldsymbol{V}_{cxz}\bigg{)}\bigg{\\}}$
$\displaystyle=\boldsymbol{V}_{cxz}^{-1/2}\times\sqrt{\uptau_{0}(1-\uptau_{0})}\times\bigg{\\{}\mathcal{N}\bigg{(}\boldsymbol{0},\lambda\boldsymbol{V}_{cxz}\bigg{)}-\lambda\mathcal{N}\bigg{(}\boldsymbol{0},\boldsymbol{V}_{cxz}\bigg{)}\bigg{\\}}$
$\displaystyle=\sqrt{\uptau_{0}(1-\uptau_{0})}\
\boldsymbol{V}_{cxz}^{-1/2}\times\boldsymbol{V}_{cxz}^{1/2}\times\bigg{\\{}\mathcal{N}\big{(}\boldsymbol{0},\lambda\boldsymbol{I}_{p}\big{)}-\lambda\mathcal{N}\big{(}\boldsymbol{0},\boldsymbol{I}_{p}\big{)}\bigg{\\}}$
$\displaystyle=\sqrt{\uptau_{0}(1-\uptau_{0})}\bigg{[}\boldsymbol{W}_{p}(\lambda)-\lambda\boldsymbol{W}_{p}(1)\bigg{]},\
\ \text{since}\
\boldsymbol{V}_{cxz}^{-1/2}\times\boldsymbol{V}_{cxz}^{1/2}=\boldsymbol{I}_{p}.$

which implies that

$\displaystyle\frac{1}{\sqrt{\uptau_{0}(1-\uptau_{0})}}\bigg{[}\mathcal{J}_{n}\big{(}\lambda,\uptau_{0},\widehat{\boldsymbol{\beta}}^{ivz}_{n}(\uptau_{0})\big{)}-\lambda\mathcal{J}_{n}\big{(}1,\uptau_{0},\widehat{\boldsymbol{\beta}}^{ivz}_{n}(\uptau_{0})\big{)}\bigg{]}\Rightarrow\frac{\sqrt{\uptau_{0}(1-\uptau_{0})}}{\sqrt{\uptau_{0}(1-\uptau_{0})}}\bigg{[}\boldsymbol{W}_{p}(\lambda)-\lambda\boldsymbol{W}_{p}(1)\bigg{]}$

Thus, for some $0<\lambda<1$ and $\uptau_{0}\in(0,1)$ it holds that

$\displaystyle\mathcal{SQ}^{ivz}_{n}(\lambda;\uptau_{0})\Rightarrow\underset{\lambda\in[0,1]}{\mathsf{sup}}\
\big{\|}\boldsymbol{\mathcal{BB}}_{p}(\lambda)\big{\|}_{\infty}$ (7.31)

Overall, it seems that the fluctuation type statistics are non-pivotal, at the
first glance, for all estimators and across the two persistence classes, due
to the dependence of their limiting distributions on the nuisance coefficient
of persistence; appearing in the estimation of the covariance matrix
$\boldsymbol{V}_{cxz}$ of the Gaussian variant that the corresponding partial
sum processes converge to. However, the IVZ based statistic for both types of
persistence induce an approximation which weakly converges into a Brownian
bridge type limit. In practise, when we utilize the IVZ estimator (see also
Theorem 3.2 and Proposition 3.2 in Lee,  (2016)) then the limiting
distribution of the moment matrix between the covariates and the residuals of
the model simplifies, and thus the overall limit is nuisance-parameter free.
Furthermore, a similar limit for the IVX based test hold in the case of mildly
integrated regressors. A nonstandard limit distribution appears in the case of
high persistence (e.g., LUR) for both the OLS and IVX based test statistics. ∎

####  Proof of Proposition 2

We need to prove the following limit result

$\displaystyle\mathcal{SW}^{ols}_{n}\big{(}\lambda;\uptau_{0}\big{)}$
$\displaystyle\Rightarrow\underset{\lambda\in[0,1]}{\mathsf{sup}}\ \frac{\
\big{\|}\boldsymbol{\mathcal{BB}}_{p+1}(\lambda)\
\big{\|}^{2}}{\lambda(1-\lambda)}$ (7.32)

where the exponent rate that captures the degree of persistence for the
original regressors can be $\upgamma_{x}=1$ or $\upgamma_{x}\in(0,1)$. In
particular to prove the Brownian Bridge limit, we need to derive the limiting
distribution of the term
$\mathcal{J}_{nx}\big{(}\lambda,\uptau_{0},\widehat{\boldsymbol{\theta}}^{ols}_{n}(\uptau_{0})\big{)}-\lambda\mathcal{J}_{nx}\big{(}1,\uptau_{0},\widehat{\boldsymbol{\theta}}^{ols}_{n}(\uptau_{0})\big{)}$.

Using Definition 3, for the case of mildly integrated regressors,
$\upgamma_{x}\in(0,1)$, it holds that

$\displaystyle\mathcal{J}^{ols}_{nx}\big{(}\lambda,\uptau_{0},\widehat{\boldsymbol{\theta}}^{ols}_{n}(\uptau_{0})\big{)}$
$\displaystyle:=\big{(}\boldsymbol{X}^{\prime}\boldsymbol{X}\big{)}^{-1/2}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{X}_{t-1}\psi_{\uptau}\left(y_{t}-\boldsymbol{X}_{t-1}^{\prime}\widehat{\boldsymbol{\theta}}^{ols}_{n}(\uptau_{0})\right)$
$\displaystyle\Rightarrow\sqrt{\uptau_{0}(1-\uptau_{0})}\times\bigg{[}\boldsymbol{W}_{p+1}(\lambda)-\lambda\boldsymbol{W}_{p+1}(1)\bigg{]}.$
(7.33)

where $\boldsymbol{W}_{p+1}(.)$ is a $p-$vector of independent Wiener
processes and the convergence holds because
$\big{\\{}\boldsymbol{x}_{t-1}\psi_{\uptau}\big{(}y_{t}-\boldsymbol{x}_{t-1}^{\prime}\boldsymbol{\beta}_{0}(\uptau_{0})\big{)}\big{\\}}$
is a sequence of martingale differences under the null. The particular limit
is derived in the proof of Proposition 1.

Part (i)

###### Proof.

Let
$\boldsymbol{X}_{t-1}=\big{(}\boldsymbol{1},\boldsymbol{x}^{\prime}_{t-1}\big{)}^{\prime}$
the design matrix and
$\boldsymbol{\theta}(\uptau_{0})=\big{(}\alpha(\uptau_{0}),\boldsymbol{\beta}^{\prime}(\uptau_{0})\big{)}^{\prime}$
to be the parameter vector. Then, to derive the asymptotic distribution of the
OLS-Wald test we employ the following functional

$\displaystyle\hat{S}_{nx}^{ols}\big{(}\lambda,\uptau_{0},\widehat{\boldsymbol{\theta}}^{ols}_{n}(\uptau_{0})\big{)}:=\boldsymbol{D}_{n}^{-1}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{X}_{t-1}\psi_{\uptau}\big{(}u_{t}(\uptau_{0})\big{)},\ \
\text{for some}\ \ 0<\lambda<1,$ (7.34)

where
$u_{t}(\uptau_{0})=\big{(}y_{t}-\boldsymbol{X}_{t-1}^{\prime}\widehat{\boldsymbol{\theta}}^{ols}_{n}(\uptau_{0})\big{)}$
with $\uptau_{0}\in(0,1)$. Then, the OLS based estimator for the subsample
$1\leq t\leq\lfloor\lambda n\rfloor$ denoted with
$\widehat{\boldsymbol{\theta}}^{ols}_{1}(\lambda;\uptau_{0})$ is given by

$\displaystyle\widehat{\boldsymbol{\theta}}^{ols}_{1}(\lambda;\uptau_{0})$
$\displaystyle=\left(\frac{1}{k}\sum_{t=1}^{\lfloor\lambda
n\rfloor}{\color[rgb]{1,0,0}f_{u_{t}(\uptau),t-1}(0)}\boldsymbol{X}_{t-1}\boldsymbol{X}_{t-1}^{\prime}\right)^{-1}\left(\frac{1}{k}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{X}_{t-1}y_{t}\right)$ (7.35)

Therefore, it holds that

$\displaystyle\bigg{\|}\hat{S}_{nx}^{ols}\left(\lambda,\uptau_{0},\widehat{\boldsymbol{\theta}}^{ols}_{1}(\lambda;\uptau_{0})\right)\bigg{\|}$
$\displaystyle\leq\bigg{\|}\ \boldsymbol{D}_{n}^{-1}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{X}_{t-1}\bigg{[}\mathds{1}\big{\\{}y_{t}=\boldsymbol{X}_{t-1}^{\prime}\widehat{\boldsymbol{\theta}}^{ols}_{1}(\lambda;\uptau_{0})\big{\\}}\bigg{]}\
\bigg{\|}$ $\displaystyle\leq(p+1)D_{n}^{-1}\underset{1\leq i\leq
n}{\mathsf{max}}\left\lVert\boldsymbol{X}_{t-1}\right\rVert\overset{\mathbb{P}}{\to}0.$
(7.36)

which implies that
$\hat{S}_{nx}^{ols}\left(\lambda,\uptau_{0},\widehat{\boldsymbol{\theta}}^{ols}_{1}(\lambda;\uptau_{0})\right)=o_{\mathbb{P}}(1)$.

Moreover, for the estimator of the first subsample we obtain the following
expression

$\displaystyle\boldsymbol{D}_{n}\left(\widehat{\boldsymbol{\theta}}_{1}^{ols}(\lambda;\uptau_{0})-\boldsymbol{\theta}_{0}(\uptau_{0})\right)$
$\displaystyle\overset{p}{\to}\bigg{\\{}{\color[rgb]{1,0,0}f_{u_{t}(\uptau)}(0)}\lambda\mathbb{V}_{xx}\bigg{\\}}^{-1}\times
S_{nx}^{ols}\big{(}\lambda,\uptau_{0},\boldsymbol{\theta}_{0}(\uptau_{0})\big{)}+o_{\mathbb{P}}(1)$
$\displaystyle\equiv{\color[rgb]{1,0,0}\frac{1}{f_{u_{t}(\uptau)}(0)}}\frac{1}{\lambda}\mathbb{V}_{xx}^{-1}\times\mathcal{N}\displaystyle\left(\boldsymbol{0},\uptau_{0}(1-\uptau_{0})\lambda\begin{bmatrix}1&\boldsymbol{0}^{\prime}\\\
\boldsymbol{0}&\boldsymbol{V}_{xx}\end{bmatrix}\right)$
$\displaystyle={\color[rgb]{1,0,0}\frac{1}{f_{u_{t}(\uptau)}(0)}}\frac{1}{\lambda}\sqrt{\uptau_{0}(1-\uptau_{0})}\mathbb{V}_{xx}^{-1}\times\begin{bmatrix}1&\boldsymbol{0}^{\prime}\\\
\boldsymbol{0}&\boldsymbol{V}_{xx}\end{bmatrix}^{1/2}\times\mathcal{N}\displaystyle\big{(}\boldsymbol{0},\lambda\big{)}$
$\displaystyle={\color[rgb]{1,0,0}\frac{1}{f_{u_{t}(\uptau)}(0)}}\frac{1}{\lambda}\sqrt{\uptau_{0}(1-\uptau_{0})}\times\mathbb{V}_{xx}^{-1/2}\times\boldsymbol{W}_{p+1}(\lambda).$
(7.37)

since it holds that
$S_{x}^{ols}\big{(}\lambda,\uptau_{0},\boldsymbol{\theta}(\uptau_{0})\big{)}\equiv\mathcal{N}\displaystyle\left(\boldsymbol{0},\uptau_{0}(1-\uptau_{0})\lambda\begin{bmatrix}1&\boldsymbol{0}^{\prime}\\\
\boldsymbol{0}&\boldsymbol{V}_{xx}\end{bmatrix}\right)$. Similarly,

$\displaystyle\boldsymbol{D}_{n}\left(\widehat{\boldsymbol{\theta}}^{ols}_{2}(\lambda;\uptau_{0})-\boldsymbol{\theta}_{0}(\uptau_{0})\right)$
$\displaystyle\overset{p}{\to}\bigg{\\{}{\color[rgb]{1,0,0}f_{u_{t}(\uptau)}(0)}(1-\lambda)\mathbb{V}_{xx}\bigg{\\}}^{-1}\times
S_{nx}^{ols}\big{(}\lambda,\uptau_{0},\boldsymbol{\theta}_{0}(\uptau_{0})\big{)}+o_{\mathbb{P}}(1)$
$\displaystyle\equiv{\color[rgb]{1,0,0}\frac{1}{f_{u_{t}(\uptau)}(0)}}\frac{1}{1-\lambda}\sqrt{\uptau_{0}(1-\uptau_{0})}\times\mathbb{V}_{xx}^{-1/2}\times\bigg{[}\boldsymbol{W}_{p+1}(1)-\boldsymbol{W}_{p+1}(\lambda)\bigg{]}.$
(7.38)

Therefore, combining (7.1.7) and (7.1.7) we obtain the following expression

$\displaystyle\boldsymbol{D}_{n}\left[\Delta\widehat{\boldsymbol{\theta}}^{ols}_{n}\left(\lambda;\uptau_{0}\right)\right]\Rightarrow-{\color[rgb]{1,0,0}\frac{1}{f_{u_{t}(\uptau)}(0)}}\frac{1}{\lambda(1-\lambda)}\sqrt{\uptau_{0}(1-\uptau_{0})}\times\mathbb{V}_{xx}^{-1/2}\times\big{[}\boldsymbol{W}_{p+1}(\lambda)-\lambda\boldsymbol{W}_{p+1}(1)\big{]}.$

Moreover, the convergence of the covariance matrix follows as below

$\displaystyle\underset{n\to\infty}{\mathsf{plim}}\widehat{\boldsymbol{V}}^{ols}_{n}(\lambda;\uptau_{0})\equiv\uptau_{0}(1-\uptau_{0})\times\bigg{\\{}\underset{n\to\infty}{\mathsf{plim}}\widehat{\boldsymbol{V}}^{ols}_{1n}(\lambda;\uptau_{0})+\underset{n\to\infty}{\mathsf{plim}}\widehat{\boldsymbol{V}}^{ols}_{2n}(\lambda;\uptau_{0})\bigg{\\}}$
(7.39)

Therefore, it holds that

$\displaystyle\underset{n\to\infty}{\mathsf{plim}}$
$\displaystyle\widehat{\boldsymbol{V}}^{ols}_{1n}(\lambda;\uptau_{0})$ (7.40)
$\displaystyle=\left\\{\underset{n\to\infty}{\mathsf{plim}}{\color[rgb]{0,0,1}\tilde{\boldsymbol{L}}_{nx}\left(\uptau_{0},\boldsymbol{\theta}_{n}^{ols}(\uptau_{0})\right)}\right\\}^{-1}\left\\{\underset{n\to\infty}{\mathsf{plim}}\boldsymbol{D}_{n}^{-1}\left[\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{X}_{t-1}\boldsymbol{X}_{t-1}^{\prime}\right]\boldsymbol{D}_{n}^{-1}\right\\}\left\\{\underset{n\to\infty}{\mathsf{plim}}{\color[rgb]{0,0,1}\tilde{\boldsymbol{L}}_{nx}\left(\uptau_{0},\boldsymbol{\theta}_{n}^{ols}(\uptau_{0})\right)}\right\\}^{-1}$
$\displaystyle=\left\\{\underset{n\to\infty}{\mathsf{plim}}{\color[rgb]{0,0,1}\boldsymbol{D}_{n}^{-1}\left[\sum_{t=1}^{\lfloor\lambda
n\rfloor}f_{u_{t}(\uptau),t-1}(0)\boldsymbol{X}_{t-1}\boldsymbol{X}_{t-1}^{\prime}\right]\boldsymbol{D}_{n}^{-1}}\right\\}^{-1}\left\\{\underset{n\to\infty}{\mathsf{plim}}\boldsymbol{D}_{n}^{-1}\left[\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{X}_{t-1}\boldsymbol{X}_{t-1}^{\prime}\right]\boldsymbol{D}_{n}^{-1}\right\\}$
$\displaystyle\times\left\\{\underset{n\to\infty}{\mathsf{plim}}{\color[rgb]{0,0,1}\boldsymbol{D}_{n}^{-1}\left[\sum_{t=1}^{\lfloor\lambda
n\rfloor}f_{u_{t}(\uptau),t-1}(0)\boldsymbol{X}_{t-1}\boldsymbol{X}_{t-1}^{\prime}\right]\boldsymbol{D}_{n}^{-1}}\right\\}^{-1}$
$\displaystyle\equiv\bigg{\\{}{\color[rgb]{1,0,0}f_{u_{t}(\uptau)}(0)}\lambda\mathbb{V}_{xx}\bigg{\\}}^{-1}\times\bigg{\\{}\lambda\mathbb{V}_{xx}\bigg{\\}}\times\bigg{\\{}{\color[rgb]{1,0,0}f_{u_{t}(\uptau)}(0)}\mathbb{V}_{xx}\bigg{\\}}^{-1}$
$\displaystyle={\color[rgb]{1,0,0}\frac{1}{f_{u_{t}(\uptau)}(0)^{2}}}\frac{1}{\lambda}\mathbb{V}_{xx}^{-1}$
(7.41)

Similarly, we obtain that

$\displaystyle\underset{n\to\infty}{\mathsf{plim}}$
$\displaystyle\widehat{\boldsymbol{V}}^{ols}_{2n}(\lambda;\uptau_{0})\Rightarrow{\color[rgb]{1,0,0}\frac{1}{f_{u_{t}(\uptau)}(0)^{2}}}\frac{1}{1-\lambda}\mathbb{V}_{xx}^{-1}$
(7.42)

Thus,

$\displaystyle\underset{n\to\infty}{\mathsf{plim}}\widehat{\boldsymbol{V}}^{ols}_{n}(\lambda;\uptau_{0})\equiv{\color[rgb]{1,0,0}\frac{\uptau_{0}(1-\uptau_{0})}{f_{u_{t}(\uptau)}(0)^{2}}}\mathbb{V}_{xx}^{-1}\left\\{\frac{1}{\lambda}+\frac{1}{1-\lambda}\right\\}={\color[rgb]{1,0,0}\frac{1}{f_{u_{t}(\uptau)}(0)^{2}}}\frac{\uptau_{0}(1-\uptau_{0})}{\lambda(1-\lambda)}\mathbb{V}_{xx}^{-1}$
(7.43)

which implies that

$\displaystyle\mathcal{W}_{n}^{ols}\left(\lambda;\uptau_{0}\right)$
$\displaystyle:=\boldsymbol{D}_{n}\bigg{\\{}\Delta\widehat{\boldsymbol{\theta}}^{ols}_{n}(\lambda;\uptau_{0})\bigg{\\}}^{\prime}\bigg{[}\widehat{\boldsymbol{V}}_{n}(\lambda;\uptau_{0})\bigg{]}^{-1}\bigg{\\{}\Delta\widehat{\boldsymbol{\theta}}^{ols}_{n}(\lambda;\uptau_{0})\bigg{\\}}$
$\displaystyle\Rightarrow{\color[rgb]{1,0,0}\frac{1}{f_{u_{t}(\uptau)}(0)^{2}}}\frac{\uptau_{0}(1-\uptau_{0})}{\left[\lambda(1-\lambda)\right]^{2}}\big{[}\boldsymbol{W}_{p+1}(\lambda)-\lambda\boldsymbol{W}_{p+1}(1)\big{]}^{\prime}\left(\mathbb{V}_{xx}^{-1/2}\right)^{\prime}\times\left\\{{\color[rgb]{1,0,0}f_{u_{t}(\uptau)}(0)^{2}}\frac{\lambda(1-\lambda)}{\uptau_{0}(1-\uptau_{0})}\mathbb{V}_{xx}\right\\}$
$\displaystyle\ \ \ \ \ \ \ \
\times\mathbb{V}_{xx}^{-1/2}\big{[}\boldsymbol{W}_{p+1}(\lambda)-\lambda\boldsymbol{W}_{p+1}(1)\big{]}$
$\displaystyle\equiv\frac{1}{\lambda(1-\lambda)}\big{[}\boldsymbol{W}_{p+1}(\lambda)-\lambda\boldsymbol{W}_{p+1}(1)\big{]}^{\prime}\big{[}\boldsymbol{W}_{p+1}(\lambda)-\lambda\boldsymbol{W}_{p+1}(1)\big{]}.$
(7.44)

Hence, we have that

$\displaystyle\mathcal{SW}^{ols}_{n}(\lambda;\uptau_{0})$
$\displaystyle:=\underset{\lambda\in\Lambda_{\eta}}{\mathsf{sup}}\
\boldsymbol{D}_{n}\left\\{\Delta\widehat{\boldsymbol{\theta}}^{ols}_{n}(\lambda;\uptau_{0})\right\\}^{\prime}\times\bigg{[}\widehat{\boldsymbol{V}}_{n}(\lambda;\uptau_{0})\bigg{]}^{-1}\times\left\\{\Delta\widehat{\boldsymbol{\theta}}^{ols}_{n}(\lambda;\uptau_{0})\right\\}$
$\displaystyle\Rightarrow\underset{\lambda\in\Lambda_{\eta}}{\mathsf{sup}}\
\frac{\big{[}\boldsymbol{W}_{p+1}(\lambda)-\lambda\boldsymbol{W}_{p+1}(1)\big{]}^{\prime}\big{[}\boldsymbol{W}_{p+1}(\lambda)-\lambda\boldsymbol{W}_{p+1}(1)\big{]}}{\lambda(1-\lambda)}$
$\displaystyle\equiv\underset{\lambda\in\Lambda_{\eta}}{\mathsf{sup}}\
\frac{\big{\|}\boldsymbol{\mathcal{BB}}_{p+1}(\lambda)\big{\|}^{2}}{\lambda(1-\lambda)},\
\text{for}\ \upgamma_{x}\in(0,1),$ (7.45)

where $\boldsymbol{\mathcal{BB}}_{p+1}(\lambda)$ is a Brownian Bridge process,
which holds for the case of mildly integrated regressors, that is,
$\upgamma_{x}\in(0,1)$ and holds even under the presence of model intercept.
Next, we provide of some auxiliary derivations employed for Part (i). ∎

###### Proof.

$\displaystyle\widehat{\boldsymbol{\theta}}^{ols}_{1}(\lambda;\uptau_{0})$
$\displaystyle=\left(\frac{1}{\kappa}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{X}_{t-1}\boldsymbol{X}_{t-1}^{\prime}\right)^{-1}\left(\frac{1}{\kappa}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{X}_{t-1}y_{t}\right)$
$\displaystyle=\left(\frac{1}{\kappa}\sum_{t=1}^{\lfloor\lambda
n\rfloor}{\color[rgb]{1,0,0}f_{u_{t}(\uptau),t-1}(0)}\boldsymbol{X}_{t-1}\boldsymbol{X}_{t-1}^{\prime}\right)^{-1}\left(\frac{1}{\kappa}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{X}_{t-1}\big{[}\boldsymbol{X}_{t-1}^{\prime}\boldsymbol{\theta}_{0}(\uptau_{0})+\psi_{\uptau}\big{(}u_{t}(\uptau_{0})\big{)}\big{]}\right)$
$\displaystyle=\boldsymbol{\theta}_{0}(\uptau_{0})+\left(\frac{1}{\kappa}\sum_{t=1}^{\lfloor\lambda
n\rfloor}{\color[rgb]{1,0,0}f_{u_{t}(\uptau),t-1}(0)}\boldsymbol{X}_{t-1}\boldsymbol{X}_{t-1}^{\prime}\right)^{-1}\left(\frac{1}{\kappa}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{X}_{t-1}\psi_{\uptau}\big{(}u_{t}(\uptau_{0})\big{)}\right)+o_{\mathbb{P}}(1).$

Thus,

$\displaystyle\boldsymbol{D}_{n}\left(\widehat{\boldsymbol{\theta}}^{ols}_{1}(\lambda;\uptau_{0})-\boldsymbol{\theta}_{0}(\uptau_{0})\right)=\left(\boldsymbol{D}_{n}^{-1}\left[\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{X}_{t-1}\boldsymbol{X}_{t-1}^{\prime}\right]\boldsymbol{D}_{n}^{-1}\right)^{-1}\left(\boldsymbol{D}_{n}^{-1}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{X}_{t-1}\psi_{\uptau}\big{(}u_{t}(\uptau_{0})\big{)}\right)+o_{\mathbb{P}}(1).$

which implies that

$\displaystyle\boldsymbol{D}_{n}\left(\widehat{\boldsymbol{\theta}}^{ols}_{1}(\lambda;\uptau_{0})-\boldsymbol{\theta}_{0}(\uptau_{0})\right)$
$\displaystyle\Rightarrow\big{\\{}\lambda\mathbb{V}_{xx}\big{\\}}^{-1}\times\mathcal{N}\displaystyle\big{(}\boldsymbol{0},\uptau_{0}(1-\uptau_{0})\lambda\mathbb{V}_{xx}\big{)}$
$\displaystyle\equiv\frac{1}{\lambda}\sqrt{\uptau_{0}(1-\uptau_{0})}\mathbb{V}_{xx}^{-1}\times\mathbb{V}_{xx}^{1/2}\times\boldsymbol{W}_{p+1}(\lambda)$
$\displaystyle=\frac{1}{\lambda}\sqrt{\uptau_{0}(1-\uptau_{0})}\times\mathbb{V}_{xx}^{-1/2}\times\boldsymbol{W}_{p+1}(\lambda).$
(7.46)

Recall that

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

where
$u_{t}(\uptau_{0})=\big{(}y_{t}-\boldsymbol{X}_{t-1}^{\prime}\boldsymbol{\theta}_{n}(\uptau_{0})\big{)}$.
Thus,
$S_{nx}^{ols}\big{(}\lambda,\uptau_{0},\boldsymbol{\theta}^{ols}_{n}(\uptau_{0})\big{)}\Rightarrow\mathcal{N}\displaystyle\left(\boldsymbol{0},\uptau_{0}(1-\uptau_{0})\lambda\begin{bmatrix}1&\boldsymbol{0}^{\prime}\\\
\boldsymbol{0}&\boldsymbol{V}_{xx}\end{bmatrix}\right)$. Denote with
$\mathbb{V}_{xx}:=\begin{bmatrix}1&\boldsymbol{0}^{\prime}\\\
\boldsymbol{0}&\boldsymbol{V}_{xx}\end{bmatrix}$, which implies that

$\displaystyle
S_{nx}^{ols}\big{(}\lambda,\uptau_{0},\boldsymbol{\theta}^{ols}_{n}(\uptau_{0})\big{)}\Rightarrow\mathcal{N}\displaystyle\big{(}\boldsymbol{0},\uptau_{0}(1-\uptau_{0})\lambda\mathbb{V}_{xx}\big{)}\equiv\sqrt{\uptau_{0}(1-\uptau_{0})}\times\mathbb{V}_{xx}^{-1/2}\times\boldsymbol{W}_{p+1}(\lambda).$

Similarly, we can prove that

$\displaystyle\boldsymbol{D}_{n}\left(\widehat{\boldsymbol{\theta}}^{ols}_{2}(\lambda;\uptau_{0})-\boldsymbol{\theta}_{0}(\uptau_{0})\right)$
$\displaystyle\Rightarrow\frac{1}{1-\lambda}\sqrt{\uptau_{0}(1-\uptau_{0})}\times\mathbb{V}_{xx}^{-1/2}\times\big{[}\boldsymbol{W}_{p+1}(1)-\boldsymbol{W}_{p+1}(\lambda)\big{]}.$

Practically, the above results can be deduced from Assumption 2 $(b)$ such
that

$\displaystyle\underset{r\in[0,\lambda]}{\mathsf{sup}}\left|\frac{1}{n^{1+\upgamma_{x}}}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{x}_{t-1}\boldsymbol{x}_{t-1}^{\prime}-r\mathbb{E}\big{[}\boldsymbol{x}_{t-1}\boldsymbol{x}_{t-1}^{\prime}\big{]}\right|=o_{\mathbb{P}}(1),\
\text{as}\ n\to\infty.$ (7.47)

where
$\mathbb{E}\big{[}\boldsymbol{x}_{t-1}\boldsymbol{x}_{t-1}^{\prime}\big{]}\equiv\boldsymbol{V}_{xx}$,
which implies that
$\displaystyle\frac{1}{n^{1+\upgamma_{x}}}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{x}_{t-1}\boldsymbol{x}_{t-1}^{\prime}\overset{p}{\to}\lambda\boldsymbol{V}_{xx}$.
Similarly,

$\displaystyle\underset{r\in(\lambda,1]}{\mathsf{sup}}\left|\frac{1}{n^{1+\upgamma_{x}}}\sum_{t=\lfloor\lambda
n\rfloor+1}^{n}\boldsymbol{x}_{t-1}\boldsymbol{x}_{t-1}^{\prime}-r\mathbb{E}\big{[}\boldsymbol{x}_{t-1}\boldsymbol{x}_{t-1}^{\prime}\big{]}\right|=o_{\mathbb{P}}(1),\
\text{as}\ n\to\infty.$ (7.48)

which implies that
$\displaystyle\frac{1}{n^{1+\upgamma_{x}}}\sum_{t=\lfloor\lambda
n\rfloor+1}^{n}\boldsymbol{x}_{t-1}\boldsymbol{x}_{t-1}^{\prime}\overset{p}{\to}(1-\lambda)\boldsymbol{V}_{xx}$.

∎

Part (ii)

Next, we investigate the asymptotic behaviour of the OLS-Wald test statistic
in the case of local unit root regressors (i.e., high persistent). To minimize
complexity of notation for the derivations of this proof we denote with

$\displaystyle\mathbb{S}_{xx}:=\begin{bmatrix}1&\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)^{\prime}\\\
\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)&\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)\boldsymbol{J}_{c}(r)^{\prime}\end{bmatrix}\
\ \text{and}\ \
\mathbb{S}_{xx}(\lambda):=\begin{bmatrix}\lambda&\displaystyle\int_{0}^{\lambda}\boldsymbol{J}_{c}(r)^{\prime}\\\
\displaystyle\int_{0}^{\lambda}\boldsymbol{J}_{c}(r)&\displaystyle\int_{0}^{\lambda}\boldsymbol{J}_{c}(r)\boldsymbol{J}_{c}(r)^{\prime}\end{bmatrix}$
(7.49)

Moreover, it holds that
$\displaystyle\left(\boldsymbol{D}_{n}^{-1}\left[\sum_{t=1}^{n}\boldsymbol{X}_{t-1}\boldsymbol{X}_{t-1}^{\prime}\right]\boldsymbol{D}_{n}^{-1}\right)\Rightarrow\mathbb{S}_{xx}$
and

$\displaystyle\left(\boldsymbol{D}_{n}^{-1}\left[\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{X}_{t-1}\boldsymbol{X}_{t-1}^{\prime}\right]\boldsymbol{D}_{n}^{-1}\right)\Rightarrow\begin{bmatrix}\lambda&\displaystyle\int_{0}^{\lambda}\boldsymbol{J}_{c}(r)^{\prime}\\\
\displaystyle\int_{0}^{\lambda}\boldsymbol{J}_{c}(r)&\displaystyle\int_{0}^{\lambda}\boldsymbol{J}_{c}(r)\boldsymbol{J}_{c}(r)^{\prime}\end{bmatrix},\text{when}\
\upgamma_{x}=1.$ (7.50)

###### Proof.

Therefore, for LUR regressors, $\upgamma_{x}=1$, we have that

$\displaystyle\boldsymbol{D}_{n}\left(\widehat{\boldsymbol{\theta}}^{ols}_{1}(\lambda;\uptau_{0})-\boldsymbol{\theta}_{0}(\uptau_{0})\right)=\left(\boldsymbol{D}_{n}^{-1}\left[\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{X}_{t-1}\boldsymbol{X}_{t-1}^{\prime}\right]\boldsymbol{D}_{n}^{-1}\right)^{-1}\left(\boldsymbol{D}_{n}^{-1}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{X}_{t-1}\psi_{\uptau}\big{(}u_{t}(\uptau_{0})\big{)}\right)+o_{\mathbb{P}}(1)$

which implies that

$\displaystyle\boldsymbol{D}_{n}\left(\widehat{\boldsymbol{\theta}}^{ols}_{1}(\lambda;\uptau_{0})-\boldsymbol{\theta}_{0}(\uptau_{0})\right)$
$\displaystyle\Rightarrow\displaystyle\mathbb{S}_{xx}^{-1}(\lambda)\times\begin{bmatrix}B_{\psi_{\uptau}}(\lambda)\\\
\displaystyle\int_{0}^{\lambda}\boldsymbol{J}_{c}(r)dB_{\psi_{\uptau}}\end{bmatrix}$
(7.51)
$\displaystyle\boldsymbol{D}_{n}\left(\widehat{\boldsymbol{\theta}}^{ols}_{2}(\lambda;\uptau_{0})-\boldsymbol{\theta}_{0}(\uptau_{0})\right)$
$\displaystyle\Rightarrow\big{(}\displaystyle\mathbb{S}_{xx}(1)-\mathbb{S}_{xx}(\lambda)\big{)}^{-1}\times\begin{bmatrix}B_{\psi_{\uptau}}(1)-B_{\psi_{\uptau}}(\lambda)\\\
\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)dB_{\psi_{\uptau}}-\displaystyle\int_{0}^{\lambda}\boldsymbol{J}_{c}(r)dB_{\psi_{\uptau}}\end{bmatrix}$
(7.52)

Therefore, combining (7.51) and (7.52) we obtain the following expression

$\displaystyle\boldsymbol{D}_{n}\left[\Delta\widehat{\boldsymbol{\theta}}^{ols}_{n}\left(\lambda;\uptau_{0}\right)\right]$
$\displaystyle\Rightarrow\begin{bmatrix}\displaystyle\big{[}\mathbb{S}_{xx}(1)-\mathbb{S}_{xx}(\lambda)\big{]}^{-1}\times\bigg{\\{}B_{\psi_{\uptau}}(1)-B_{\psi_{\uptau}}(\lambda)\bigg{\\}}-\mathbb{S}^{-1}_{xx}(\lambda)\times
B_{\psi_{\uptau}}(\lambda)\\\
\displaystyle\big{[}\mathbb{S}_{xx}(1)-\mathbb{S}_{xx}(\lambda)\big{]}^{-1}\times\bigg{\\{}\int_{0}^{1}\boldsymbol{J}_{c}(r)dB_{\psi_{\uptau}}-\int_{0}^{\lambda}\boldsymbol{J}_{c}(r)dB_{\psi_{\uptau}}\bigg{\\}}-\mathbb{S}^{-1}_{xx}(\lambda)\times\int_{0}^{\lambda}\boldsymbol{J}_{c}(r)dB_{\psi_{\uptau}}\end{bmatrix}$
$\displaystyle\equiv\big{[}\mathbb{S}_{xx}(1)-\mathbb{S}_{xx}(\lambda)\big{]}^{-1}\times\begin{bmatrix}\displaystyle
B_{\psi_{\uptau}}(1)-B_{\psi_{\uptau}}(\lambda)\\\
\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)dB_{\psi_{\uptau}}-\int_{0}^{\lambda}\boldsymbol{J}_{c}(r)dB_{\psi_{\uptau}}\end{bmatrix}-\mathbb{S}^{-1}_{xx}(\lambda)\times\begin{bmatrix}\displaystyle
B_{\psi_{\uptau}}(\lambda)\\\
\displaystyle\int_{0}^{\lambda}\boldsymbol{J}_{c}(r)dB_{\psi_{\uptau}}\end{bmatrix}$
(7.53)

Then, the OLS-Wald test for testing the null hypothesis of no parameter
instability in the nonstationary quantile predictive regression model at an
unknown break-point location $\kappa=\lfloor\lambda n\rfloor$ is given by the
following expression

$\displaystyle\mathcal{W}_{n}^{ols}\left(\lambda;\uptau_{0}\right):=\uptau_{0}(1-\uptau_{0})\boldsymbol{D}_{n}\bigg{\\{}\Delta\widehat{\boldsymbol{\theta}}^{ols}_{n}(\lambda;\uptau_{0})\bigg{\\}}^{\prime}\bigg{[}\widehat{\boldsymbol{V}}_{n}(\lambda;\uptau_{0})\bigg{]}^{-1}\bigg{\\{}\Delta\widehat{\boldsymbol{\theta}}^{ols}_{n}(\lambda;\uptau_{0})\bigg{\\}}$
(7.54)

Furthermore, we study the limiting variance of the OLS-Wald test under the
null hypothesis. In particular, the convergence of the covariance matrix
follows as below

$\displaystyle\underset{n\to\infty}{\mathsf{plim}}\ $
$\displaystyle\bigg{\\{}\widehat{\boldsymbol{V}}^{ols}_{1n}(\lambda;\uptau_{0})\bigg{\\}}$
$\displaystyle\equiv\left\\{\underset{n\to\infty}{\mathsf{plim}}\
{\color[rgb]{0,0,1}\boldsymbol{L}_{nx}\left(\uptau_{0},\boldsymbol{\theta}_{n}^{ols}(\uptau_{0})\right)}\right\\}^{-1}\left\\{\underset{n\to\infty}{\mathsf{plim}}\
\boldsymbol{D}_{n}^{-1}\left[\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{X}_{t-1}\boldsymbol{X}_{t-1}^{\prime}\right]\boldsymbol{D}_{n}^{-1}\right\\}\left\\{\underset{n\to\infty}{\mathsf{plim}}\
{\color[rgb]{0,0,1}\boldsymbol{L}_{nx}\left(\uptau_{0},\boldsymbol{\theta}_{n}^{ols}(\uptau_{0})\right)}\right\\}^{-1}$
$\displaystyle=\left\\{\underset{n\to\infty}{\mathsf{plim}}\
{\color[rgb]{0,0,1}\boldsymbol{D}_{n}^{-1}\left[\sum_{t=1}^{\lfloor\lambda
n\rfloor}f_{u_{t}(\uptau),t-1}(0)\boldsymbol{X}_{t-1}\boldsymbol{X}_{t-1}^{\prime}\right]\boldsymbol{D}_{n}^{-1}}\right\\}^{-1}\times\left\\{\underset{n\to\infty}{\mathsf{plim}}\
\boldsymbol{D}_{n}^{-1}\left[\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{X}_{t-1}\boldsymbol{X}_{t-1}^{\prime}\right]\boldsymbol{D}_{n}^{-1}\right\\}$
$\displaystyle\times\left\\{\underset{n\to\infty}{\mathsf{plim}}\
{\color[rgb]{0,0,1}\boldsymbol{D}_{n}^{-1}\left[\sum_{t=1}^{\lfloor\lambda
n\rfloor}f_{u_{t}(\uptau),t-1}(0)\boldsymbol{X}_{t-1}\boldsymbol{X}_{t-1}^{\prime}\right]\boldsymbol{D}_{n}^{-1}}\right\\}^{-1}$
$\displaystyle\equiv\bigg{\\{}{\color[rgb]{1,0,0}f_{u_{t}(\uptau)}(0)}\mathbb{S}_{xx}(\lambda)\bigg{\\}}^{-1}\times\mathbb{S}_{xx}(\lambda)\times\bigg{\\{}{\color[rgb]{1,0,0}f_{u_{t}(\uptau)}(0)}\mathbb{S}_{xx}(\lambda)\bigg{\\}}^{-1}$
$\displaystyle={\color[rgb]{1,0,0}\frac{1}{f_{u_{t}(\uptau)}(0)^{2}}}\mathbb{S}_{xx}^{-1}(\lambda).$
(7.55)

and

$\displaystyle\underset{n\to\infty}{\mathsf{plim}}\
\bigg{\\{}\widehat{\boldsymbol{V}}^{ols}_{2n}(\lambda;\uptau_{0})\bigg{\\}}\Rightarrow{\color[rgb]{1,0,0}\frac{1}{f_{u_{t}(\uptau)}(0)^{2}}}\bigg{[}\mathbb{S}_{xx}(1)-\mathbb{S}_{xx}(\lambda)\bigg{]}^{-1}.$
(7.56)

Therefore,

$\displaystyle\underset{n\to\infty}{\mathsf{plim}}\
\bigg{\\{}\widehat{\boldsymbol{V}}^{ols}_{n}(\lambda;\uptau_{0})\bigg{\\}}\Rightarrow{\color[rgb]{1,0,0}\frac{1}{f_{u_{t}(\uptau)}(0)^{2}}}\bigg{\\{}\mathbb{S}_{xx}^{-1}(\lambda)+\bigg{[}\mathbb{S}_{xx}(1)-\mathbb{S}_{xx}(\lambda)\bigg{]}^{-1}\bigg{\\}}.$
(7.57)

We can also simplify further the term
$\bigg{\\{}\underset{n\to\infty}{\mathsf{plim}}\
\widehat{\boldsymbol{V}}^{ols}_{n}(\lambda;\uptau_{0})\bigg{\\}}^{-1}={\color[rgb]{1,0,0}f_{u_{t}(\uptau)}(0)^{2}}\bigg{[}\mathbb{S}_{xx}(\lambda)-\mathbb{S}_{xx}(\lambda)\mathbb{S}^{-1}_{xx}(1)\mathbb{S}_{xx}(\lambda)\bigg{]}$.
Denote with

$\displaystyle\boldsymbol{\Delta}^{ols}_{0}\big{(}\lambda;\uptau_{0}\big{)}:=\big{[}\mathbb{S}_{xx}(1)-\mathbb{S}_{xx}(\lambda)\big{]}^{-1}\times\begin{bmatrix}\displaystyle
B_{\psi_{\uptau}}(1)-B_{\psi_{\uptau}}(\lambda)\\\
\displaystyle\int_{0}^{1}\boldsymbol{J}_{c}(r)dB_{\psi_{\uptau}}-\int_{0}^{\lambda}\boldsymbol{J}_{c}(r)dB_{\psi_{\uptau}}\end{bmatrix}-\mathbb{S}^{-1}_{xx}(\lambda)\times\begin{bmatrix}\displaystyle
B_{\psi_{\uptau}}(\lambda)\\\
\displaystyle\int_{0}^{\lambda}\boldsymbol{J}_{c}(r)dB_{\psi_{\uptau}}\end{bmatrix}$

Then, it follows that

$\displaystyle\mathcal{W}_{n}^{ols}\left(\lambda;\uptau_{0}\right)$
$\displaystyle:=\boldsymbol{D}_{n}\bigg{\\{}\Delta\widehat{\boldsymbol{\theta}}^{ols}_{n}(\lambda;\uptau_{0})\bigg{\\}}^{\prime}\bigg{[}\widehat{\boldsymbol{V}}^{ols}_{n}(\lambda;\uptau_{0})\bigg{]}^{-1}\bigg{\\{}\Delta\widehat{\boldsymbol{\theta}}^{ols}_{n}(\lambda;\uptau_{0})\bigg{\\}}$
$\displaystyle\Rightarrow\boldsymbol{\Delta}^{ols}_{0}\big{(}\lambda;\uptau_{0}\big{)}^{\prime}\big{[}\boldsymbol{\Sigma}^{-1}_{0}\big{(}\lambda;\uptau_{0}\big{)}\big{]}\boldsymbol{\Delta}^{ols}_{0}\big{(}\lambda;\uptau_{0}\big{)}$
$\displaystyle\equiv{\color[rgb]{1,0,0}f_{u_{t}(\uptau)}(0)^{2}}\bigg{\\{}\boldsymbol{\Delta}^{ols}_{0}\big{(}\lambda;\uptau_{0}\big{)}\bigg{\\}}^{\prime}\bigg{[}\mathbb{S}_{xx}(\lambda)-\mathbb{S}_{xx}(\lambda)\mathbb{S}^{-1}_{xx}(1)\mathbb{S}_{xx}(\lambda)\bigg{]}\bigg{\\{}\boldsymbol{\Delta}^{ols}_{0}\big{(}\lambda;\uptau_{0}\big{)}\bigg{\\}}.$

∎

Therefore, the asymptotic distribution of the sup OLS-Wald test statistic is
nonpivotal and nonstandard and has the above analytical expression.
Furthermore, we verify that a trivial aspect such as the inclusion of a model
intercept can complicate the asymptotic theory of the structural break test
since the model intercept and the slopes are known to have different rates of
convergence.

####  Proof of Theorem 1

$\displaystyle\mathcal{SW}_{n}^{ivx}\big{(}\lambda;\uptau_{0}\big{)}\Rightarrow\underset{\lambda\in\Lambda_{\eta}}{\mathsf{sup}}\bigg{\\{}\boldsymbol{\Delta}_{0}\big{(}\lambda;\uptau_{0}\big{)}^{\prime}\big{[}\boldsymbol{\Sigma}_{0}\big{(}\lambda;\uptau_{0}\big{)}\big{]}^{-1}\boldsymbol{\Delta}_{0}\big{(}\lambda;\uptau_{0}\big{)}\bigg{\\}}$
(7.58)

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

$\displaystyle\boldsymbol{\Delta}_{0}\big{(}\lambda;\uptau_{0}\big{)}$
$\displaystyle:=\boldsymbol{W}_{p}(\lambda)-\boldsymbol{\Psi}_{c}(\lambda)\boldsymbol{W}_{p}(1)$
(7.59) $\displaystyle\boldsymbol{\Sigma}_{0}\big{(}\lambda;\uptau_{0}\big{)}$
$\displaystyle:=\lambda\big{(}\boldsymbol{I}_{p}-\boldsymbol{\Psi}_{c}(\lambda)\big{)}\big{(}\boldsymbol{I}_{p}-\boldsymbol{\Psi}_{c}(\lambda)\big{)}^{\prime}+(1-\lambda)\boldsymbol{\Psi}_{c}(\lambda)\boldsymbol{\Psi}_{c}(\lambda)^{\prime}$
(7.60)

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.

###### Proof.

We consider the limiting distribution of the sup IVX-Wald test for the
nonstationary quantile predictive regression model. An analytic expression for
the limiting variance of the IVX-Wald test statistic,
$\widehat{\boldsymbol{V}}_{n}^{ivx}\big{(}\lambda;\uptau_{0}\big{)}:=\tilde{\mathbf{Q}}_{1}\big{(}\lambda;\uptau_{0}\big{)}+\tilde{\mathbf{Q}}_{2}\big{(}\lambda;\uptau_{0}\big{)}$,
can be obtained by formulating the test statistic with respect to the linear
restrictions matrix, under the null hypothesis of no structural break for a
fixed quantile $\uptau_{0}\in(0,1)$. Furthermore, by applying weakly
convergence arguments to the particular expression (with zero off-diagonal
terms) holds due to the orthogonality property of the regressors of the two
regimes with respect to the unknown break-location $\kappa=\lfloor\lambda
n\rfloor$, $0<\lambda<1$.

$\displaystyle\widehat{\boldsymbol{V}}_{n}^{ivx}\big{(}\lambda;\uptau_{0}\big{)}$
$\displaystyle:=\bigg{[}\boldsymbol{I}_{p}-\boldsymbol{I}_{p}\bigg{]}\begin{bmatrix}\left(\boldsymbol{Z}_{1}^{\prime}\boldsymbol{X}_{1}\right)^{-1}&\boldsymbol{0}\\\
\boldsymbol{0}&\left(\boldsymbol{Z}_{2}^{\prime}\boldsymbol{X}_{2}\right)^{-1}\\\
\end{bmatrix}\begin{bmatrix}\boldsymbol{Z}_{1}^{\prime}\boldsymbol{Z}_{1}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{Z}_{2}^{\prime}\boldsymbol{Z}_{2}\\\
\end{bmatrix}\begin{bmatrix}\left(\boldsymbol{X}_{1}^{\prime}\boldsymbol{Z}_{1}\right)^{-1}&\boldsymbol{0}\\\
\boldsymbol{0}&\left(\boldsymbol{X}_{2}^{\prime}\boldsymbol{Z}_{2}\right)^{-1}\\\
\end{bmatrix}\begin{bmatrix}\boldsymbol{I}_{p}\\\
-\boldsymbol{I}_{p}\end{bmatrix}$
$\displaystyle\equiv\bigg{[}\tilde{\mathbf{Q}}_{1}\big{(}\lambda;\uptau_{0}\big{)}+\tilde{\mathbf{Q}}_{2}\big{(}\lambda;\uptau_{0}\big{)}\bigg{]}$
(7.61)

where

$\displaystyle\tilde{\mathbf{Q}}_{1}\big{(}\lambda;\uptau_{0}\big{)}$
$\displaystyle:=\left(\tilde{\boldsymbol{Z}}_{1}^{\prime}\boldsymbol{X}_{1}\right)^{-1}\left(\tilde{\boldsymbol{Z}}_{1}^{\prime}\tilde{\boldsymbol{Z}}_{1}\right)\left(\boldsymbol{X}_{1}^{\prime}\tilde{\boldsymbol{Z}}_{1}\right)^{-1}$
(7.62) $\displaystyle\tilde{\mathbf{Q}}_{2}\big{(}\lambda;\uptau_{0}\big{)}$
$\displaystyle:=\left(\tilde{\boldsymbol{Z}}_{2}^{\prime}\boldsymbol{X}_{2}\right)^{-1}\left(\tilde{\boldsymbol{Z}}_{2}^{\prime}\tilde{\boldsymbol{Z}}_{2}\right)\left(\boldsymbol{X}_{2}^{\prime}\tilde{\boldsymbol{Z}}_{2}\right)^{-1}$
(7.63)
$\displaystyle\tilde{\boldsymbol{D}}_{n}\widehat{\boldsymbol{V}}_{n}^{ivx}\big{(}\lambda;\uptau_{0}\big{)}\tilde{\boldsymbol{D}}_{n}$
$\displaystyle=\tilde{\boldsymbol{D}}_{n}\left[\tilde{\mathbf{Q}}_{1}\big{(}\lambda;\uptau_{0}\big{)}+\tilde{\mathbf{Q}}_{2}\big{(}\lambda;\uptau_{0}\big{)}\right]\tilde{\boldsymbol{D}}_{n}$
(7.64)

Therefore, by employing the corresponding normalized versions it holds that

$\displaystyle\tilde{\mathbf{Q}}_{1}\big{(}\lambda;\uptau_{0}\big{)}$
$\displaystyle:=\left(\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{X}_{t-1,n}\tilde{\boldsymbol{Z}}_{t-1,n}^{\prime}\right)^{-1}\left(\sum_{t=1}^{\lfloor\lambda
n\rfloor}\tilde{\boldsymbol{Z}}_{t-1,n}\tilde{\boldsymbol{Z}}_{t-1,n}^{\prime}\right)\left(\sum_{t=1}^{\lfloor\lambda
n\rfloor}\tilde{\boldsymbol{Z}}_{t-1,n}\boldsymbol{X}_{t-1,n}^{\prime}\right)^{-1}$
$\displaystyle=\left(\tilde{\boldsymbol{D}}_{n}^{-1}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{x}_{t-1}\tilde{\boldsymbol{z}}_{t-1}^{\prime}\tilde{\boldsymbol{D}}_{n}^{-1}\right)^{-1}\left(\tilde{\boldsymbol{D}}_{n}^{-1}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\tilde{\boldsymbol{z}}_{t-1}\tilde{\boldsymbol{z}}_{t-1}^{\prime}\tilde{\boldsymbol{D}}_{n}^{-1}\right)\left(\tilde{\boldsymbol{D}}_{n}^{-1}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\tilde{\boldsymbol{z}}_{t-1}\boldsymbol{x}_{t-1}^{\prime}\tilde{\boldsymbol{D}}_{n}^{-1}\right)^{-1}$
$\displaystyle\equiv\tilde{\boldsymbol{D}}_{n}\left(\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{x}_{t-1}\tilde{\boldsymbol{z}}_{t-1}^{\prime}\right)^{-1}\left(\sum_{t=1}^{\lfloor\lambda
n\rfloor}\tilde{\boldsymbol{z}}_{t-1}\tilde{\boldsymbol{z}}_{t-1}^{\prime}\right)\left(\sum_{t=1}^{\lfloor\lambda
n\rfloor}\tilde{\boldsymbol{z}}_{t-1}\boldsymbol{x}_{t-1}^{\prime}\right)^{-1}\tilde{\boldsymbol{D}}_{n}$
(7.65)

Moreover, we denote the weakly convergence of the following moment matrices as
below

$\displaystyle\tilde{\boldsymbol{D}}_{n}^{-1}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\boldsymbol{x}_{t-1}\tilde{\boldsymbol{z}}_{t-1}^{\prime}\tilde{\boldsymbol{D}}_{n}^{-1}\Rightarrow\boldsymbol{\Gamma}_{cxz}(\lambda)\
\ \ \ \text{and}\ \ \ \
\tilde{\boldsymbol{D}}_{n}^{-1}\sum_{t=1}^{n}\boldsymbol{x}_{t-1}\tilde{\boldsymbol{z}}_{t-1}^{\prime}\tilde{\boldsymbol{D}}_{n}^{-1}\Rightarrow\boldsymbol{\Gamma}_{cxz}(1)$
(7.66)

By applying weak convergence arguments we obtain that

$\displaystyle\tilde{\boldsymbol{D}}_{n}\times\tilde{\mathbf{Q}}_{1}\big{(}\lambda;\uptau_{0}\big{)}\times\tilde{\boldsymbol{D}}_{n}$
$\displaystyle\Rightarrow\boldsymbol{\Gamma}_{cxz}(\lambda)^{-1}\big{(}\lambda\boldsymbol{V}_{cxz}\big{)}\big{(}\boldsymbol{\Gamma}_{cxz}(\lambda)^{\prime}\big{)}^{-1}$
$\displaystyle\equiv\lambda\left(\boldsymbol{\Gamma}_{cxz}(\lambda)\boldsymbol{V}_{cxz}^{-1}\boldsymbol{\Gamma}_{cxz}(\lambda)^{\prime}\right)^{-1}$
(7.67)

Similarly, it holds that

$\displaystyle\tilde{\mathbf{Q}}_{2}\big{(}\lambda;\uptau_{0}\big{)}$
$\displaystyle:=\left(\sum_{t=\lfloor\lambda
n\rfloor+1}^{n}\boldsymbol{X}_{t-1,n}\tilde{\boldsymbol{Z}}_{t-1,n}^{\prime}\right)^{-1}\left(\sum_{t=\lfloor\lambda
n\rfloor+1}^{n}\tilde{\boldsymbol{Z}}_{t-1,n}\tilde{\boldsymbol{Z}}_{t-1,n}^{\prime}\right)\left(\sum_{t=\lfloor\lambda
n\rfloor+1}^{n}\tilde{\boldsymbol{Z}}_{t-1,n}\boldsymbol{X}_{t-1,n}^{\prime}\right)^{-1}$
$\displaystyle=\left(\tilde{\boldsymbol{D}}_{n}^{-1}\sum_{t=\lfloor\lambda
n\rfloor+1}^{n}\boldsymbol{x}_{t-1}\tilde{\boldsymbol{z}}_{t-1}^{\prime}\tilde{\boldsymbol{D}}_{n}^{-1}\right)^{-1}\left(\tilde{\boldsymbol{D}}_{n}^{-1}\sum_{t=\lfloor\lambda
n\rfloor+1}^{n}\tilde{\boldsymbol{z}}_{t-1}\tilde{\boldsymbol{z}}_{t-1}^{\prime}\tilde{\boldsymbol{D}}_{n}^{-1}\right)\left(\tilde{\boldsymbol{D}}_{n}^{-1}\sum_{t=\lfloor\lambda
n\rfloor+1}^{n}\tilde{\boldsymbol{z}}_{t-1}\boldsymbol{x}_{t-1}^{\prime}\tilde{\boldsymbol{D}}_{n}^{-1}\right)^{-1}$
$\displaystyle\equiv\tilde{\boldsymbol{D}}_{n}\left(\sum_{t=\lfloor\lambda
n\rfloor+1}^{n}\boldsymbol{x}_{t-1}\tilde{\boldsymbol{z}}_{t-1}^{\prime}\right)^{-1}\left(\sum_{t=\lfloor\lambda
n\rfloor+1}^{n}\tilde{\boldsymbol{z}}_{t-1}\tilde{\boldsymbol{z}}_{t-1}^{\prime}\right)\left(\sum_{t=\lfloor\lambda
n\rfloor+1}^{n}\tilde{\boldsymbol{z}}_{t-1}\boldsymbol{x}_{t-1}^{\prime}\right)^{-1}\tilde{\boldsymbol{D}}_{n}$

Thus, it follows that

$\displaystyle\tilde{\boldsymbol{D}}_{n}\times\tilde{\mathbf{Q}}_{2}\big{(}\lambda;\uptau_{0}\big{)}\times\tilde{\boldsymbol{D}}_{n}$
$\displaystyle\Rightarrow\big{(}\boldsymbol{\Gamma}_{cxz}(1)-\boldsymbol{\Gamma}_{cxz}(\lambda)\big{)}^{-1}\big{(}(1-\lambda)\boldsymbol{V}_{cxz}\big{)}\left[\big{(}\boldsymbol{\Gamma}_{cxz}(1)-\boldsymbol{\Gamma}_{cxz}(\lambda)\big{)}^{\prime}\right]^{-1}$
$\displaystyle\equiv(1-\lambda)\bigg{\\{}\big{(}\boldsymbol{\Gamma}_{cxz}(1)-\boldsymbol{\Gamma}_{cxz}(\lambda)\big{)}\boldsymbol{V}_{cxz}^{-1}\big{(}\boldsymbol{\Gamma}_{cxz}(1)-\boldsymbol{\Gamma}_{cxz}(\lambda)\big{)}^{\prime}\bigg{\\}}^{-1}$
(7.68)

Therefore, for the limiting variance of the sup IVX-Wald test we have that

$\displaystyle\tilde{\boldsymbol{D}}_{n}\widehat{\boldsymbol{V}}_{n}^{ivx}\big{(}\lambda;\uptau_{0}\big{)}\tilde{\boldsymbol{D}}_{n}$
$\displaystyle=\tilde{\boldsymbol{D}}_{n}\left[\tilde{\mathbf{Q}}_{1}\big{(}\lambda;\uptau_{0}\big{)}+\tilde{\mathbf{Q}}_{2}\big{(}\lambda;\uptau_{0}\big{)}\right]\tilde{\boldsymbol{D}}_{n}$
$\displaystyle=\bigg{\\{}\lambda\boldsymbol{\Gamma}_{cxz}(\lambda)^{-1}\boldsymbol{V}_{cxz}\boldsymbol{\Gamma}_{cxz}^{\prime}(\lambda)^{-1}+(1-\lambda)\big{(}\boldsymbol{\Gamma}_{cxz}(1)-\boldsymbol{\Gamma}_{cxz}(\lambda)\big{)}^{-1}\boldsymbol{V}_{cxz}\big{(}\boldsymbol{\Gamma}^{\prime}_{cxz}(1)-\boldsymbol{\Gamma}_{cxz}^{\prime}(\lambda)\big{)}^{-1}\bigg{\\}}$

Furthermore, it holds that

$\displaystyle\tilde{\boldsymbol{D}}_{n}\left(\widehat{\boldsymbol{\beta}}^{ivx}_{1}(\lambda;\uptau_{0})-\boldsymbol{\beta}_{0}(\uptau_{0})\right)$
$\displaystyle\Rightarrow\boldsymbol{\Gamma}^{-1}_{cxz}(\lambda)\times\mathcal{N}\displaystyle\big{(}\boldsymbol{0},\uptau_{0}(1-\uptau_{0})\lambda\boldsymbol{V}_{cxz}\big{)}$
$\displaystyle\equiv\sqrt{\uptau_{0}(1-\uptau_{0})}\times\boldsymbol{\Gamma}^{-1}_{cxz}(\lambda)\times\boldsymbol{V}_{cxz}^{1/2}\times\boldsymbol{W}_{p}(\lambda).$
(7.69)
$\displaystyle\tilde{\boldsymbol{D}}_{n}\left(\widehat{\boldsymbol{\beta}}_{2}^{ivx}(\lambda;\uptau_{0})-\boldsymbol{\beta}_{0}(\uptau)\right)$
$\displaystyle\Rightarrow\bigg{(}\boldsymbol{\Gamma}_{cxz}(1)-\boldsymbol{\Gamma}_{cxz}(\lambda)\bigg{)}^{-1}\times\mathcal{N}\displaystyle\big{(}\boldsymbol{0},\uptau_{0}(1-\uptau_{0})(1-\lambda)\boldsymbol{V}_{cxz}\big{)}$
$\displaystyle\equiv\sqrt{\uptau_{0}(1-\uptau_{0})}\times\bigg{(}\boldsymbol{\Gamma}_{cxz}(1)-\boldsymbol{\Gamma}_{cxz}(\lambda)\bigg{)}^{-1}\times\boldsymbol{V}_{cxz}^{1/2}\times\bigg{[}\boldsymbol{W}_{p}(1)-\boldsymbol{W}_{p}(\lambda)\bigg{]}.$
(7.70)

which implies that

$\displaystyle\tilde{\boldsymbol{D}}_{n}\left(\widehat{\boldsymbol{\beta}}_{2}^{ivx}(\lambda;\uptau_{0})-\widehat{\boldsymbol{\beta}}_{1}^{ivx}(\lambda;\uptau_{0})\right)\Rightarrow-\sqrt{\uptau_{0}(1-\uptau_{0})}\boldsymbol{V}_{cxz}^{1/2}\bigg{\\{}{\color[rgb]{1,0,0}\mathcal{A}}\boldsymbol{W}_{p}(\lambda)-{\color[rgb]{1,0,0}\mathcal{B}}\boldsymbol{W}_{p}(1)\bigg{\\}}$
(7.71)

where

$\displaystyle{\color[rgb]{1,0,0}\mathcal{A}}=\left\\{\bigg{(}\boldsymbol{\Gamma}_{cxz}(1)-\boldsymbol{\Gamma}_{cxz}(\lambda)\bigg{)}^{-1}+\boldsymbol{\Gamma}^{-1}_{cxz}(\lambda)\right\\}\
\ \ \text{and}\ \ \
{\color[rgb]{1,0,0}\mathcal{B}}=\bigg{(}\boldsymbol{\Gamma}_{cxz}(1)-\boldsymbol{\Gamma}_{cxz}(\lambda)\bigg{)}^{-1}$
(7.72)

Therefore, putting all the above together the expression can be simplified as
below

$\displaystyle\mathcal{SW}_{n}^{ivx}\big{(}\lambda;\uptau_{0}\big{)}\Rightarrow\uptau_{0}(1-\uptau_{0})$
$\displaystyle\times\bigg{(}\boldsymbol{W}_{p}(\lambda)-\boldsymbol{\Gamma}_{cxz}(\lambda)\boldsymbol{\Gamma}^{-1}_{cxz}(1)\boldsymbol{W}_{p}(1)\bigg{)}^{\prime}$
$\displaystyle\times\bigg{\\{}\lambda\big{(}\boldsymbol{I}_{p}-\boldsymbol{\Gamma}_{cxz}(\lambda)\boldsymbol{\Gamma}^{-1}_{cxz}(1)\big{)}\big{(}\boldsymbol{I}_{p}-\boldsymbol{\Gamma}_{cxz}(\lambda)\boldsymbol{\Gamma}^{-1}_{cxz}(1)\big{)}^{\prime}+(1-\lambda)\left(\boldsymbol{\Gamma}_{cxz}(\lambda)\boldsymbol{\Gamma}^{-1}_{cxz}(1)\right)\left(\boldsymbol{\Gamma}_{cxz}(\lambda)\boldsymbol{\Gamma}^{-1}_{cxz}(1)\right)^{\prime}\bigg{\\}}^{-1}$
$\displaystyle\times\bigg{(}\boldsymbol{W}_{p}(\lambda)-\boldsymbol{\Gamma}_{cxz}(\lambda)\boldsymbol{\Gamma}^{-1}_{cxz}(1)\boldsymbol{W}_{p}(1)\bigg{)}.$

Denote with

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

Thus, in summary, the sup IVX-Wald statistic can be formulated as below

$\displaystyle\mathcal{SW}_{n}^{ivx}\big{(}\lambda;\uptau_{0}\big{)}$
$\displaystyle\Rightarrow\underset{\lambda\in\Lambda_{\eta}}{\mathsf{sup}}\bigg{\\{}\boldsymbol{\Delta}_{0}\big{(}\lambda;\uptau_{0}\big{)}^{\prime}\big{[}\boldsymbol{\Sigma}_{0}\big{(}\lambda;\uptau_{0}\big{)}\big{]}^{-1}\boldsymbol{\Delta}_{0}\big{(}\lambda;\uptau_{0}\big{)}\bigg{\\}}$
$\displaystyle\boldsymbol{\Delta}_{0}\big{(}\lambda;\uptau_{0}\big{)}$
$\displaystyle:=\boldsymbol{W}_{p}(\lambda)-\boldsymbol{\Psi}_{c}(\lambda)\boldsymbol{W}_{p}(1)$
$\displaystyle\boldsymbol{\Sigma}_{0}\big{(}\lambda;\uptau_{0}\big{)}$
$\displaystyle:=\lambda\big{(}\boldsymbol{I}_{p}-\boldsymbol{\Psi}_{c}(\lambda)\big{)}\big{(}\boldsymbol{I}_{p}-\boldsymbol{\Psi}_{c}(\lambda)\big{)}^{\prime}+(1-\lambda)\boldsymbol{\Psi}_{c}(\lambda)\boldsymbol{\Psi}_{c}(\lambda)^{\prime}$

∎

##### Proof of Expression (7.1.8):

###### Proof.

We consider the dequantiled model with $p$ regressors which can be either
mildly integrated or high persistent. Then, by employing the embedded
normalization matrices we can derive analytic expressions for the IVX
estimators of the two subsamples.

$\displaystyle\widehat{\boldsymbol{\beta}}_{1}^{ivx}(\lambda;\uptau_{0})$
$\displaystyle=\left(\frac{1}{\kappa}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\tilde{\boldsymbol{Z}}_{t-1,n}\boldsymbol{X}_{t-1,n}^{\prime}\right)^{-1}\left(\frac{1}{\kappa}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\tilde{\boldsymbol{Z}}_{t-1,n}y_{t}(\uptau)\right)$
$\displaystyle=\left(\frac{1}{\kappa}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\tilde{\boldsymbol{Z}}_{t-1,n}\boldsymbol{X}_{t-1,n}^{\prime}\right)^{-1}\left(\frac{1}{\kappa}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\tilde{\boldsymbol{Z}}_{t-1,n}\left[\boldsymbol{X}_{t-1,n}^{\prime}\boldsymbol{\beta}_{0}(\uptau_{0})\
+\psi_{\uptau}\big{(}u_{t}(\uptau_{0})\big{)}\right]\right)$
$\displaystyle=\boldsymbol{\beta}_{0}(\uptau_{0})+\left(\frac{1}{\kappa}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\tilde{\boldsymbol{Z}}_{t-1,n}\boldsymbol{X}_{t-1,n}^{\prime}\right)^{-1}\left(\frac{1}{\kappa}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\tilde{\boldsymbol{Z}}_{t-1,n}\psi_{\uptau}\big{(}u_{t}(\uptau_{0})\big{)}\right)+o_{\mathbb{P}}(1)$

Therefore, it holds that

$\displaystyle{\color[rgb]{1,0,0}\tilde{\boldsymbol{D}}_{n}}\left(\widehat{\boldsymbol{\beta}}_{1}^{ivx}(\lambda;\uptau_{0})-\boldsymbol{\beta}_{0}(\uptau)\right)=\left({\color[rgb]{1,0,0}\tilde{\boldsymbol{D}}_{n}}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\tilde{\boldsymbol{Z}}_{t-1,n}\boldsymbol{X}_{t-1,n}^{\prime}\right)^{-1}\left({\color[rgb]{1,0,0}\tilde{\boldsymbol{D}}_{n}}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\tilde{\boldsymbol{Z}}_{t-1,n}\psi_{\uptau}\big{(}u_{t}(\uptau_{0})\big{)}\right)+o_{\mathbb{P}}(1)$

Notice that it holds that

$\displaystyle\left(\sum_{t=1}^{\lfloor\lambda
n\rfloor}\tilde{\boldsymbol{Z}}_{t-1,n}\boldsymbol{X}_{t-1,n}^{\prime}\right)^{-1}$
$\displaystyle=\left(\sum_{t=1}^{\lfloor\lambda
n\rfloor}{\color[rgb]{0,0,1}\tilde{\boldsymbol{D}}^{-1}_{n}}\tilde{\boldsymbol{z}}_{t-1}\left({\color[rgb]{0,0,1}\tilde{\boldsymbol{D}}^{-1}_{n}}\boldsymbol{x}_{t-1}\right)^{\prime}\right)^{-1}$
$\displaystyle=\left({\color[rgb]{0,0,1}\tilde{\boldsymbol{D}}^{-1}_{n}}\left[\sum_{t=1}^{\lfloor\lambda
n\rfloor}\tilde{\boldsymbol{z}}_{t-1}\boldsymbol{x}_{t-1}^{\prime}\right]{\color[rgb]{0,0,1}\tilde{\boldsymbol{D}}^{-1}_{n}}\right)^{-1}$
$\displaystyle=\left({\color[rgb]{0,0,1}\frac{1}{n^{1+(\upgamma_{x}\wedge\upgamma_{z})}}\otimes\boldsymbol{I}_{p}}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\tilde{\boldsymbol{z}}_{t-1}\boldsymbol{x}_{t-1}^{\prime}\right)^{-1}\Rightarrow\boldsymbol{\Gamma}^{-1}_{cxz}(\lambda)$
(7.73)

For the second term of the expression for
${\color[rgb]{1,0,0}\tilde{\boldsymbol{D}}_{n}}\left(\widehat{\boldsymbol{\beta}}_{1}^{ivx}(\lambda;\uptau_{0})-\boldsymbol{\beta}_{0}(\uptau)\right)$
it holds that

$\displaystyle\left(\sum_{t=1}^{\lfloor\lambda
n\rfloor}\tilde{\boldsymbol{Z}}_{t-1,n}\psi_{\uptau}\big{(}u_{t}(\uptau_{0})\big{)}\right)$
$\displaystyle=\left({\color[rgb]{0,0,1}\tilde{\boldsymbol{D}}_{n}^{-1}}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\tilde{\boldsymbol{z}}_{t-1}\psi_{\uptau}\big{(}u_{t}(\uptau_{0})\big{)}\right)$
$\displaystyle=\left({\color[rgb]{0,0,1}n^{-\frac{1+(\upgamma_{x}\wedge\upgamma_{z})}{2}}\otimes\boldsymbol{I}_{p}}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\tilde{\boldsymbol{z}}_{t-1}\psi_{\uptau}\big{(}u_{t}(\uptau_{0})\big{)}\right)$
$\displaystyle\Rightarrow\mathcal{N}\bigg{(}\boldsymbol{0},\uptau_{0}(1-\uptau_{0})\lambda\boldsymbol{V}_{cxz}\bigg{)}$
(7.74)

which implies that

$\displaystyle{\color[rgb]{1,0,0}\tilde{\boldsymbol{D}}_{n}}\left(\widehat{\boldsymbol{\beta}}^{ivx}_{1}(\lambda;\uptau_{0})-\boldsymbol{\beta}_{0}(\uptau_{0})\right)$
$\displaystyle\Rightarrow\boldsymbol{\Gamma}^{-1}_{cxz}(\lambda)\times\mathcal{N}\displaystyle\big{(}\boldsymbol{0},\uptau_{0}(1-\uptau_{0})\lambda\boldsymbol{V}_{cxz}\big{)}$
$\displaystyle\equiv\sqrt{\uptau_{0}(1-\uptau_{0})}\times\boldsymbol{\Gamma}^{-1}_{cxz}(\lambda)\times\boldsymbol{V}_{cxz}^{1/2}\times\boldsymbol{W}_{p}(\lambda).$
(7.75)

where $\boldsymbol{\Gamma}_{cxz}(\lambda)$ corresponds to the weakly
convergence argument of the partial sum process expression such that

$\displaystyle\boldsymbol{\Gamma}_{cxz}(\lambda)\equiv\begin{cases}\displaystyle-\boldsymbol{C}_{z}^{-1}\left(\lambda\boldsymbol{\Omega}_{xx}+\int_{0}^{\lambda}\boldsymbol{J}_{c}(r)d\boldsymbol{J}_{c}^{\prime}\right)&\
\text{if}\ \upgamma_{x}=1,\\\
-\lambda\boldsymbol{C}_{z}^{-1}\bigg{(}\boldsymbol{\Omega}_{xx}+\boldsymbol{C}_{p}\boldsymbol{V}_{xx}\bigg{)}&\
\text{if}\ \upgamma_{x}\in(\upgamma_{z},1),\\\ \lambda\boldsymbol{V}_{xx}&\
\text{if}\ \upgamma_{x}\in(0,\upgamma_{z}).\end{cases}$ (7.76)

Following similar arguments expression (7.1.8) follows, and the rest of the
arguments for the proof of Theorem 1. ∎

Known break-point:

###### Proof.

For a known-break point the asymptotic term that corresponds to the case of
high persistence, $\upgamma_{x}=1$, can simplify further, when $\lambda$ is
fixed say $\lambda\equiv\lambda_{0}$, such that

$\displaystyle\boldsymbol{\Gamma}_{cxz}(\lambda_{0})$
$\displaystyle\equiv\displaystyle-\boldsymbol{C}_{z}^{-1}\left(\lambda_{0}\boldsymbol{\Omega}_{xx}+\int_{0}^{\lambda_{0}}\boldsymbol{J}_{c}(r)d\boldsymbol{J}_{c}^{\prime}\right)$
$\displaystyle=\displaystyle-\lambda_{0}\boldsymbol{C}_{z}^{-1}\left(\boldsymbol{\Omega}_{xx}+\int_{0}^{1}\boldsymbol{J}_{c}(r)d\boldsymbol{J}_{c}^{\prime}\right)\equiv\lambda_{0}\boldsymbol{\Gamma}_{cxz}(1).$
(7.77)

Thus, further generalizations occur based on the formulation of Theorem 1,
which cover both the case of mildly integrated and persistent regressors,
demonstrating the convergence to a nuisance-free limiting distribution. In
particular,

$\displaystyle\tilde{\mathbf{Q}}_{1}\big{(}\lambda_{0};\uptau_{0}\big{)}$
$\displaystyle\Rightarrow\big{\\{}\lambda_{0}\boldsymbol{\Gamma}_{cxz}\big{\\}}^{-1}\times\big{\\{}\lambda_{0}\boldsymbol{V}_{cxz}\big{\\}}\times\big{\\{}\lambda_{0}\boldsymbol{\Gamma}_{cxz}^{\prime}\big{\\}}^{-1}$
$\displaystyle=\left(\frac{1}{\lambda_{0}}\boldsymbol{\Gamma}_{cxz}^{-1}\boldsymbol{V}_{cxz}\left(\boldsymbol{\Gamma}_{cxz}^{\prime}\right)^{-1}\right)\equiv\bigg{(}\lambda_{0}\boldsymbol{\Gamma}_{cxz}\boldsymbol{V}_{cxz}^{-1}\boldsymbol{\Gamma}_{cxz}^{\prime}\bigg{)}^{-1}$
(7.78)
$\displaystyle\tilde{\mathbf{Q}}_{2}\big{(}\lambda_{0};\uptau_{0}\big{)}$
$\displaystyle\Rightarrow\big{\\{}(1-\lambda_{0})\boldsymbol{\Gamma}_{cxz}\big{\\}}^{-1}\times\big{\\{}(1-\lambda_{0})\boldsymbol{V}_{cxz}\big{\\}}\times\big{\\{}(1-\lambda_{0})\boldsymbol{\Gamma}_{cxz}^{\prime}\big{\\}}^{-1}$
$\displaystyle=\left(\frac{1}{1-\lambda_{0}}\boldsymbol{\Gamma}_{cxz}^{-1}\boldsymbol{V}_{cxz}\left(\boldsymbol{\Gamma}_{cxz}^{\prime}\right)^{-1}\right)\equiv\bigg{(}(1-\lambda_{0})\boldsymbol{\Gamma}_{cxz}\boldsymbol{V}_{cxz}^{-1}\boldsymbol{\Gamma}_{cxz}^{\prime}\bigg{)}^{-1}$
(7.79)

Therefore, it follows that

$\displaystyle\widehat{\boldsymbol{V}}_{n}^{ivx}\big{(}\lambda_{0};\uptau_{0}\big{)}\Rightarrow\left\\{\bigg{(}\lambda_{0}\boldsymbol{\Gamma}_{cxz}\boldsymbol{V}_{cxz}^{-1}\boldsymbol{\Gamma}_{cxz}^{\prime}\bigg{)}^{-1}+\bigg{(}(1-\lambda_{0})\boldsymbol{\Gamma}_{cxz}\boldsymbol{V}_{cxz}^{-1}\boldsymbol{\Gamma}_{cxz}^{\prime}\bigg{)}^{-1}\right\\}$
(7.80)

and by denoting with
$\mathbb{S}_{cxz}:=\big{(}\boldsymbol{\Gamma}_{cxz}\boldsymbol{V}_{cxz}^{-1}\boldsymbol{\Gamma}_{cxz}^{\prime}\big{)}$
we obtain

$\displaystyle\widehat{\boldsymbol{V}}_{n}^{ivx}\big{(}\lambda_{0};\uptau_{0}\big{)}\Rightarrow\bigg{\\{}\big{(}\lambda_{0}\mathbb{S}_{cxz}\big{)}^{-1}+\big{(}(1-\lambda_{0})\mathbb{S}_{cxz}\big{)}^{-1}\bigg{\\}}\equiv\bigg{\\{}\frac{1}{\lambda_{0}}+\frac{1}{1-\lambda_{0}}\bigg{\\}}\times\mathbb{S}_{cxz}^{-1}$
(7.81)

Then, the expression for the IVX-Wald test statistic given by

$\displaystyle\mathcal{W}_{n}^{ivx}\big{(}\lambda_{0};\uptau_{0}\big{)}\Rightarrow\bigg{\\{}\boldsymbol{\Delta}_{0}\big{(}\lambda_{0};\uptau_{0}\big{)}^{\prime}\big{[}\boldsymbol{\Sigma}_{0}\big{(}\lambda_{0};\uptau_{0}\big{)}\big{]}^{-1}\boldsymbol{\Delta}_{0}\big{(}\lambda_{0};\uptau_{0}\big{)}\bigg{\\}}$
(7.82)

can be simplified further by combining the asymptotic convergence results
above

$\displaystyle\Delta\widehat{\boldsymbol{\beta}}^{ivx}_{n}\big{(}\lambda_{0};\uptau_{0}\big{)}$
$\displaystyle\Rightarrow\boldsymbol{\Delta}_{0}\big{(}\lambda_{0};\uptau_{0}\big{)}:=\frac{1}{\lambda_{0}(1-\lambda_{0})}\sqrt{\uptau_{0}(1-\uptau_{0})}\boldsymbol{\Gamma}_{cxz}^{-1}\times\boldsymbol{V}_{cxz}^{1/2}\times\big{[}\boldsymbol{W}_{p}(\lambda_{0})-\lambda\boldsymbol{W}_{p}(1)\big{]}.$
$\displaystyle\widehat{\boldsymbol{V}}_{n}^{ivx}\big{(}\lambda_{0};\uptau_{0}\big{)}$
$\displaystyle\Rightarrow\boldsymbol{\Sigma}_{0}\big{(}\lambda_{0};\uptau_{0}\big{)}:=\frac{1}{\lambda_{0}(1-\lambda_{0})}\times\mathbb{S}_{cxz}^{-1}.$

Therefore, we obtain

$\displaystyle\mathcal{W}_{n}^{ivx}\big{(}\lambda_{0};\uptau_{0}\big{)}\Rightarrow$
$\displaystyle\bigg{\\{}\frac{\uptau_{0}(1-\uptau_{0})}{\lambda_{0}(1-\lambda_{0})}\big{[}\boldsymbol{W}_{p}(\lambda_{0})-\lambda\boldsymbol{W}_{p}(1)\big{]}^{\prime}\left(\boldsymbol{V}_{cxz}^{1/2}\right)\left(\boldsymbol{\Gamma}_{cxz}^{-1}\right)^{\prime}\mathbb{S}_{cxz}\boldsymbol{\Gamma}_{cxz}^{-1}\boldsymbol{V}_{cxz}^{1/2}\big{[}\boldsymbol{W}_{p}(\lambda_{0})-\lambda\boldsymbol{W}_{p}(1)\big{]}\bigg{\\}}$
(7.83)

Consider the formulation for the covariance matrix such that