\left\{
\begin{array}{ll}
\mathcal{MN} \displaystyle  \left( 0, \frac{ \tau (1 - \tau) }{ f_{u_{0\tau}}(0)^2  } \psi_{cxz}^{-1} V_{cxz} \left( \psi_{cxz}^{-1} \right)^{\prime} \right)
& \text{for (LUR) and (MI)}
\\
\\
\mathcal{MN} \displaystyle  \left( 0, \frac{ \tau (1 - \tau) }{ f_{u_{0\tau}}(0)^2  }  \left( \widetilde{V}_{xx} \right)^{-1}  \right)
& \text{for (ME)}  \\
\end{array}
\right.
\end{align*}

The above Theorem shows that we obtain a mixed normal limiting distribution regardless of the persistence properties of the predictors (see, Theorem 3.1 in [92]).

We aim to show that

\begin{align}
\left( \hat{\beta}_1^{\text{IVX-QR}} \left( \tau  \right) - \beta_1 \left( \tau  \right)  \right) = \mathcal{O}_p \left(  n^{ - (1 + \delta ) / n } \right)
\end{align}

Consider the estimator distance measure such as

\begin{align}
\hat{ \mathcal{E} } ( \tau ) = \left( \hat{\beta}_1 (\tau) - \beta_1 (\tau) \right)
\end{align}

and note that
$\mathbb{E} \left[ \psi_{\tau} \left( u_{0t\tau} - \hat{ \mathcal{E} }( \tau )^{\prime} x_{t-1} \right) \right]$ which can be expanded around $\mathcal{E}( \tau )  = 0$ such that $\beta = \beta \left( \tau \right)$.

Then we obtain that

\begin{align*}
&n^{ - \frac{(1 + \delta ) }{2} } \sum_{t=1}^n \tilde{z}_{t-1} \left\{ \psi_{\tau} \left( u_{0t\tau} - \left( \hat{\beta}_1 (\tau) - \beta_1 (\tau) \right)^{\prime} x_{t-1} \right) \right\}
\\
&= n^{ - \frac{(1 + \delta ) }{2} } \sum_{t=1}^n \tilde{z}_{t-1} \left\{ \psi_{\tau} \left( u_{0t\tau} - \hat{ \mathcal{E} } ( \tau )^{\prime} x_{t-1} \right)  - \mathbb{E}_{t-1} \left[ \psi_{\tau} \left( u_{0t\tau} - \hat{ \mathcal{E} } ( \tau )^{\prime} x_{t-1} \right)  \right] - \psi_{\tau} \left( u_{0t\tau} \right)  \right\}
\\
&+  n^{ - \frac{(1 + \delta ) }{2} } \sum_{t=1}^n \tilde{z}_{t-1} \mathbb{E}_{t-1} \big[ \psi_{\tau} \left( u_{0t\tau} \right) \big]
\end{align*}

In other words, we can prove that

\begin{align}
G_{\tau, n} + \sum_{t=1}^n f_{ u_{0t \tau}, t-1 } (0) \tilde{Z}_{t-1} X_{t-1,n}^{\prime} n^{ (1 + \delta) / 2 } \big( \hat{\beta}_{1} ( \tau )  -  \beta_{1}( \tau ) \big)
\end{align}

Therefore, we have that

\begin{align}
n^{ \frac{1 + \delta}{2} } \big( \hat{\beta}_{1} ( \tau )  -  \beta_{1}( \tau ) \big) = \left( M_{ \beta_n(\tau) } \right)^{-1} G_n ( \tau ) + o_p(1),
\end{align}

§.§ Derivations on Quantile Cointegrating regression

§.§.§ Quantile regression on cointegrated time series

Consider the following partial sum process

\begin{align}
Y_n(r) = \frac{1}{ \omega^{*}_{\psi} } n^{- 1/ 2} \sum_{j=1}^{ \floor{nr} } \psi_{\uptau} \left( \epsilon_{j \uptau} \right),
\end{align}

where $\omega^{*}_{\psi}$ is the long-run variance of $\psi_{\uptau} \left( \epsilon_{j \uptau} \right)$ and the partial sum process follows an invariance principle and converges weakly to a standard Brownian motion $W(r)$. Choosing a continuous functional $h(.)$ that measures the fluctuation of $Y_n(r)$, then a robust test for cointegration can be constructed based on $h \big( Y_n(r) \big)$. By the continuous mapping theorem, under regularity conditions and the null of cointegration,

\begin{align}
h \big( Y_n(r) \big) \Rightarrow h \big( W(r) \big).
\end{align}

In principle, any metric that measures the fluctuations in $Y_n(r)$ is a natural candidate for the functional $h$. The classical KS typr or CvM type measures are of particular interest. A robust test for cointegration can then be constructed based on

\begin{align}
\widehat{Y}_n(r) = \frac{1}{ \widehat{ \omega }^{*}_{\psi} } n^{- 1/ 2} \sum_{j=1}^{ \floor{nr} } \psi_{\uptau} \left( \widehat{ \epsilon }_{j \uptau} \right),
\end{align}

where $\widehat{ \omega }^{*}_{\psi}$ is a consistent estimator of $\omega^{*}_{\psi}$.

Consider the following cointegration model

\begin{align}
y_t = \alpha + \beta^{\prime} \boldsymbol{x}_t + u_t \equiv \boldsymbol{\theta}^{\prime} \boldsymbol{z}_t + u_t
\end{align}

where $\boldsymbol{x}_t$ is a $k-$dimensional vector of integrated regressors, $\boldsymbol{z}_t = \left( 1, \boldsymbol{x}_t^{\prime} \right)^{\prime}$ and $u_t$ is mean zero stationary innovation sequence. The quantile regression estimator of the cointegrating vector can be obtained by solving the following (where $p = k +1$)

\begin{align}
\widehat{ \boldsymbol{\theta} } ( \uptau ) = \underset{ \boldsymbol{\theta}  \in \mathbb{R}^{p+1} }{  \text{arg min} } \sum_{t=1}^n \rho_{ \uptau } \left( y_t - \boldsymbol{z}_t^{\prime} \boldsymbol{\theta} \right),
\end{align}

where $\rho_{ \uptau } (u) = u \big( \uptau - \mathbf{1} \left\{  \mathsf{u} < 0  \right\} \big)$.

In order to derive the limit distribution of the quantile regression estimator of the cointegrating vector we follow the approach as presented in [140]. Let $f(.)$ and $F_{y|x}(.)$  be the pdf and the CDF of $u_t$ and denote the inverse of the check function with $\psi_{ \uptau } (u) = \big[ \uptau - \mathbf{1} \left\{  u < 0  \right\} \big]$. The following assumptions set the background theory in order to support the econometric analysis of this section.

§.§.§ Main Assumptions

Let $\boldsymbol{v}_t = \Delta \boldsymbol{x}_t$. Then, $\boldsymbol{w}_t = ( u_t, \boldsymbol{v}_t^{\prime} )^{\prime}$ is a zero-mean, stationary sequence of $(k+1)-$dimensional random vectors. The partial sums of the vector process $\big( \psi_{\uptau} \big( u_t ( \uptau ) \big), \boldsymbol{v}_t \big)$ follow a multivariate invariance principle as expressed below

\begin{align}
n^{-1/2} \sum_{t=1}^{ \floor{ nr} }
\begin{bmatrix}
\psi_{\uptau} \big( u_t ( \uptau ) \big) \\
\boldsymbol{v}_t
\end{bmatrix}
\Rightarrow
\begin{bmatrix}
B_{ \psi } ( r ) \\
\boldsymbol{B}_v ( r )
\end{bmatrix}
\mathcal{BM} \big( \boldsymbol{0}, \boldsymbol{\Omega}_{ww} \big)
\end{align}

where $\boldsymbol{\Omega}_{ww}$ is the covariance matrix of the Brownian motion $\left(  B_{ \psi_{\uptau} } ( r ),  \boldsymbol{B}_v ( r )^{\prime} \right)^{\prime}$.

The distribution function of $u_t$, $F_{y|x}( \mathsf{u} )$, has a continuous density $f( \mathsf{u} )$, with $f(\mathsf{u}) > 0$ on $\left\{ \mathsf{u} : 0 < F( \mathsf{u} ) < 1 \right\}$.

The conditional distribution function $F_{t-1}( \mathsf{u} ) = \mathbb{P} \big( u_t < \mathsf{u} | u_{t-j} , j \geq 1  \big) $ has a derivative $f_{t-1} ( \mathsf{u} )$, almost surely, and $f_{t-1}(s_n)$ is uniformly integrable for any sequence $s_n \mapsto F^{-1} ( \uptau )$, and $\mathbb{E} \left( f_{t-1}^{\delta} \left(  F_{y|x}^{-1} ( \uptau ) \right)  \right) < \infty$ for some $\delta > 1$.

Furthermore, we partition $\boldsymbol{\Omega}_{ww}$ as below

\begin{align}
\boldsymbol{\Omega}_{ww}
\begin{bmatrix}
\omega_{\psi}^2  &  \boldsymbol{\Omega}_{\psi v}
\\
\boldsymbol{\Omega}_{ v\psi } & \boldsymbol{\Omega}_{vv}
\end{bmatrix}
\end{align}

Note that the asymptotic behaviour of $n^{-1} \sum_{t=1}^n \boldsymbol{x}_t \psi_{ \uptau } \big( u_t ( \uptau ) \big)$. Under Assumption (<ref>) the following asymptotic result holds

\begin{align}
n^{-1} \sum_{t=1}^n \boldsymbol{x}_t \psi_{ \uptau } \big( u_t ( \uptau ) \big) \Rightarrow \int_0^1 \boldsymbol{B}_v d\boldsymbol{B}_{ \psi_{ \uptau } } + \lambda_{ v \psi_{ \uptau } }
\end{align}

where $\lambda_{ v \psi_{ \uptau } } $ is the one-sided long-run covariance between $\boldsymbol{v}_t$ and  $\psi_{ \uptau } \big( u_t ( \uptau ) \big)$.

Due to the nonstationarity in $\boldsymbol{x}_t$, the two model parameters in the parameter vector $\widehat{ \boldsymbol{\theta} } ( \uptau ) = \left( \hat{ \boldsymbol{\alpha} }(\uptau), \hat{ \boldsymbol{\beta} }(\uptau)^{\prime}  \right)^{\prime}$ have different rates of convergence. In particular, the estimate of the cointegrating vector $\hat{ \boldsymbol{\beta} }(\uptau)^{\prime}$ converges at rate $n$, while the intercept $\hat{ \boldsymbol{\alpha} }(\uptau)$ converges at rate $\sqrt{n}$. Thus, in order to account for the different convergence rates among the model intercept  and the estimate of the cointegrating vector we use the normalization matrix $\mathcal{ \boldsymbol{D} }_n = \mathsf{diag} \left( \sqrt{n} , n \boldsymbol{I}_k \right)$, where $\boldsymbol{I}_k$ is the $k \times k$ identity matrix. The limit distribution of the quantile regression estimator for the cointegration model is given by the Theorem <ref>.

Under Assumptions (<ref>) - (<ref>)

\begin{align}
\mathcal{ \boldsymbol{D} }_n \left( \widehat{ \boldsymbol{\theta} } ( \uptau ) - \boldsymbol{\theta} ( \uptau )  \right)
\Rightarrow \frac{1}{ f \big( F_{y|x}^{-1} ( \uptau) \big) } \left[ \int_0^1 \widetilde{ \boldsymbol{B} }_v \widetilde{ \boldsymbol{B} }_v^{\prime} \right]^{-1} \times \left[ \int_0^1 \widetilde{ \boldsymbol{B }}_v (r) d \boldsymbol{B}_{ \psi_{\uptau}} + \Delta_{v \psi_{\uptau} }  \right],
\end{align}

where $\widetilde{ \boldsymbol{B} } = \big( 1, \boldsymbol{B}_v(r)^{\prime} \big)^{\prime}$, and $\Delta_{v \psi} = \left( 0, \lambda_{ v \psi}^{\prime} \right)^{\prime}$. In particular, we obtain that

\begin{align}
\label{limit.beta}
n \left( \widehat{ \boldsymbol{\beta} } ( \tau ) - \boldsymbol{\beta} \right) \Rightarrow \frac{1}{ f \left( F_{y|x}^{-1} ( \tau) \right)} \left[ \int_0^1 \boldsymbol{B}_v^{\mu} \boldsymbol{B}_v^{ \prime \mu }  \right]^{-1} \times \left[ \int_0^1  \boldsymbol{B}_v^{\mu}(r) d\boldsymbol{B}_{\psi}  +  \lambda_{ v \psi} \right]
\end{align}

where $\boldsymbol{B}_v^{\mu}(r) = \boldsymbol{B}_v(r) - r \boldsymbol{B}_v(1)$ is a $k-$dimensional demeaned Brownian motion.

Note that the quantile regression estimator is consistent at the usual $\mathcal{O}(n)$ rate, however similar to OLS, the quantile regression estimator suffers from second order bias $( \lambda_{ v \psi} )$ which is induced from the correlation between the set of regressors $x_t$ and the residual $u_t$. Furthermore, note that the Brownian motions $\boldsymbol{B}_v(r)$ and $\boldsymbol{B}_{ \psi_{\uptau} } ( r )$ are generally correlated as long as $\boldsymbol{\Omega}_{\psi_{\uptau} v} \neq 0$. Moreover, for both stationary and nonstationary time series regression, the limit distribution in (<ref>) depends on the sparsity function $1 / f \left( F_{y|x}^{-1} ( \uptau ) \right)$. In particular, in the special case when $\Delta_{ v \psi } = 0$ and $\boldsymbol{\Omega}_{\psi_{\uptau} v} = 0$, that is, $x_t$ and $u_t$ are independent, then the limit distribution given by (<ref>) is a mixed normal.

§.§.§ A fully-modified quantile regression estimator

In this section, we are interested to propose a robust econometric framework for robust estimation and inference for the quantile regression in the case of a cointegrating model. Firstly, we note that the asymptotic behaviour of quantile regression-based  inference procedures depends on the limit distribution of $\widehat{ \boldsymbol{\theta} } ( \tau )$. However, as shown by Theorem <ref>, the limiting processes $\boldsymbol{B}_v ( r )$  and $B_{ \psi_{\uptau} } ( r )$ are correlated Brownian motions whenever contemporaneous correlation between $v_t$ and $\psi_{ \uptau } \big( u_t( \uptau ) \big)$ exists.

Moreover, super-consistency, $\widehat{ \boldsymbol{\theta} } ( \uptau )$ is second-order biased and the miscentering effect in the limit distribution is reflected in $\Delta_{v \psi}$. Consequently, the distribution of the test based on the quantile regression residual will be dependent on nuisance parameters. Therefore, in order to restore the asymptotic nuisance parameter free property of the inference procedure, we need to modify the original quantile regression estimator so that we obtain a mixed normal limit distribution. Usually in the literature, there are two approaches one can take to achieve the restoration of the the asymptotic nuisance parameter free property. For example, one approach is the implementation of a nonparametric fully-modification of the original quantile regression estimator, and a second approach is the parametrically augmented quantile regression implementation using leads and lags. In order to develop a fully-modified quantile cointegrating regression estimator one can follow the approach proposed by [114].

We first decompose the asymptotic distribution given by (<ref>) into the following two components:

\begin{align}
\frac{1}{ f \left( F_{y|x}^{-1} ( \uptau ) \right) } \left[ \int_0^1 \boldsymbol{B}_v^{\mu} \boldsymbol{B}_v^{\prime \mu}  \right]^{-1} &\times \left[ \int_0^1  \boldsymbol{B}_v^{\mu} d\boldsymbol{B}_{\psi . v}  \right], \text{and}
\\
\frac{1}{ f \left( F_{y|x}^{-1} ( \uptau ) \right) } \left[ \int_0^1 \boldsymbol{B}_v^{\mu} \boldsymbol{B}_v^{\prime \mu}  \right]^{-1} &\times \left[ \int_0^1  \boldsymbol{B}_v^{\mu} d\boldsymbol{B}_{v}^{\prime} \boldsymbol{\Omega}_{vv}^{-1} \boldsymbol{\Omega}_{v \psi} + \lambda_{v \psi }  \right]
\end{align}

\begin{align}
B_{\psi . v} &= B_{\psi} ( r ) - \Omega_{\psi v } \Omega_{vv}^{-1} B_v (r)
\\
\omega^2_{ \psi . v} &= \omega^2_{ \psi} - \Omega_{\psi v} \Omega_{vv}^{-1} \Omega_{v \psi}
\end{align}

such that $B_{\psi . v}$ is a Brownian motion with variance $\omega^2_{ \psi . v}$ as defined above. Moreover, note that $\boldsymbol{B}_{\psi . v}$ is independent of $\boldsymbol{B}_v(r)$ and the first term in the above decomposition, $\left[ \int_0^1 \boldsymbol{B}_v^{\mu} \boldsymbol{B}_v^{\prime \mu}  \right]^{-1} \left[ \int_0^1 \boldsymbol{B}_v^{\mu} d\boldsymbol{B}_{\psi . v}  \right]$, is a mixed Gaussian variate. The basic idea of fully-modification on $\widehat{ \boldsymbol{\beta} } ( \uptau )$ is to construct a nonparametric correction to remove the second term in the above decomposition.

Therefore, to facilitate the nonparametric correction, we consider the following kernel estimators of $\Omega_{vv}$, $\Omega_{v \psi}$, $\lambda_{v \psi}$,  $\lambda_{vv}$, such that

\begin{align}
\widehat{\lambda}_{v \psi} &= \sum_{ h = 0}^M k \left( \frac{h}{M} \right) \Sigma_{v \psi } (h ), \ \ \ \ \ \widehat{\lambda}_{vv} = \sum_{ h = 0}^M k \left( \frac{h}{M} \right) \Sigma_{vv} (h),
\\
\widehat{\Omega}_{v \psi} &= \sum_{ h = -M}^M k \left( \frac{h}{M} \right) \Sigma_{v \psi } (h ), \ \ \ \ \ \widehat{\Omega}_{vv} = \sum_{ h = -M}^M k \left( \frac{h}{M} \right) \Sigma_{vv} (h),
\end{align}

where $k(.)$ is the kernel function defined on the interval $[-1,1]$ with $k(0) = 1$, and $M$ the bandwidth parameter satisfying the property that $M \to \infty$ and $M / n \to 0$. The sample covariance matrices are

\begin{align}
\boldsymbol{\Sigma}_{ v\psi_{\uptau} } (h) =  \frac{1}{n} \sum_{t=1}^{t+h = n } v_t \psi_{\tau} \left( \hat{u}_{t+h, \tau} \right), \ \ \ \ \boldsymbol{\Sigma}_{vv} (h) =  \frac{1}{n} \sum_{t=1}^{t+h = n } \boldsymbol{v}_t \boldsymbol{v}_{t+h}^{\prime}
\end{align}

We define the following nonparametric fully modified quantile regression estimators as below
\begin{align}
\widehat{ \boldsymbol{\theta} } ( \uptau )^{+} &=
\begin{pmatrix}
\widehat{ \alpha } ( \uptau )
\\
\widehat{ \beta } ( \uptau )^{+}
\end{pmatrix}
\\
\widehat{ \boldsymbol{\beta} } ( \uptau )^{+} &=  \widehat{ \boldsymbol{\beta} } ( \uptau ) - \frac{1}{ f_{y|x} \left( \widehat{ F_{y|x}^{-1} ( \uptau)  }\right) } \left[ \sum_{t=1}^n \boldsymbol{x}_t \boldsymbol{x}_t^{\prime} \right]^{-1} \times \left[ \sum_{t=1}^n \boldsymbol{x}_t \boldsymbol{v}_t ^{\prime} \widehat{\boldsymbol{\Omega} }_{vv}^{-1} \widehat{\Omega}_{v \psi} + \widehat{\lambda}_{v \psi}^{+}  \right]
\end{align}

and $\widehat{\lambda}_{v \psi}^{+} = \widehat{\lambda}_{v \psi} - \widehat{\lambda}_{v v} \widehat{\Omega}_{vv}^{-1} \widehat{\Omega}_{v\psi}   $.

Similar to the fully modified OLS estimators, the fully modified quantile regression estimator of the cointegrating vector has a mixed normal distribution in the limit.

Under Assumptions (<ref>)-(<ref>)

\begin{align}
\mathcal{ \boldsymbol{D} }_n \left( \widehat{ \boldsymbol{\theta} } ( \uptau )^{+} - \widehat{ \boldsymbol{\theta} } ( \uptau ) \right) \Rightarrow \frac{1}{ f_{y|x} \left( F_{y|x}^{-1} ( \uptau)\right) } \left[ \int_0^1 \widetilde{ \boldsymbol{B} }_v \widetilde{ \boldsymbol{B} }_v^{\prime} \right]^{-1} \int_0^1 \widetilde{ \boldsymbol{B} }_v d\boldsymbol{B}_{ \psi.v }
\\
\equiv \mathcal{MN} \left( 0, \frac{ \omega^2_{ \psi.v } }{ f_{y|x} \left(  F_{y|x}^{-1} ( \uptau) \right)^2} \left[ \int_0^1 \widetilde{\boldsymbol{B}}_v \widetilde{\boldsymbol{B}}_v^{\prime} \right]^{-1} \right).
\end{align}

In particular, the cointegrating vector has the following asymptotic distribution

\begin{align}
n \left( \widehat{ \boldsymbol{\beta} } ( \uptau )^{+} - \widehat{ \boldsymbol{\beta}  } ( \uptau ) \right) \Rightarrow \frac{1}{ f_{y|x} \left( F_{y|x}^{-1} ( \uptau ) \right) } \left[ \int_0^1 \boldsymbol{B}_v^{\mu} \boldsymbol{B}_v^{\prime \mu} \right]^{-1} \int_0^1 \boldsymbol{B}_v^{\mu} d\boldsymbol{B}_{ \psi.v }
\\
\equiv \mathcal{MN} \left( 0, \frac{ \omega^2_{ \psi.v } }{ f_{y|x} \left(  F_{y|x}^{-1} ( \uptau) \right)^2} \left[ \int_0^1 \boldsymbol{B}_v^{\mu} \boldsymbol{B}_v^{\prime \mu} \right]^{-1} \right).
\end{align}

Note that the fully modified quantile regression estimator and the resulting asymptotic mixed normal asymptotic distribution facilitates statistical inference (such as the use of the Wald test) based on quantile cointegrating regression. Therefore, the classical inference problem of linear restrictions on the cointegrating vector $\beta$ is such that $\mathbb{H}_0: \mathcal{R} \boldsymbol{\beta} = r$, where $\mathcal{R}$ denotes an $(q \times k)$ matrix of linear restrictions and $r$ denotes a $q-$ dimensional vector.

Thus, under the null hypothesis, and the assumptions of Theorem <ref>, we have that

\begin{align}
\frac{ f_{y|x} \left(  F_{y|x}^{-1} ( \uptau) \right)  }{ \omega_{\psi.v} } \left[  \mathcal{ \boldsymbol{R} } \left( \int_0^1 \boldsymbol{B}_v^{\mu} \boldsymbol{B}_v^{\prime \mu} \right)^{-1} \mathcal{\boldsymbol{R}}^{\prime} \right]^{-1 / 2} n \left( \mathcal{\boldsymbol{R}} \widehat{ \boldsymbol{\beta} } ( \uptau )^{+} - r \right) \Rightarrow \mathcal{N} \big( \boldsymbol{0}, \boldsymbol{I}_q \big)
\end{align}

where $\mathcal{N} \big( \boldsymbol{0}, \boldsymbol{I}_q \big)$ represents a $q-$dimensional standard Normal distribution.

Define with $\mathcal{ \boldsymbol{Q} }_x = \sum_{t=1}^n ( \boldsymbol{x}_t - \bar{\boldsymbol{x}} ) ( \boldsymbol{x}_t - \bar{\boldsymbol{x}} )^{\prime}$. Then, a the classical Wald test in the case of quantile cointegrating regression can be constructed via the following expression

\begin{align}
\mathcal{W}_n \left( \tau \right) = \frac{ f_{y|x} \left(  \widehat{ F_{y|x}^{-1} ( \uptau)} \right)  }{ \widehat{ \omega}_{\psi.v} } \left( \mathcal{\boldsymbol{R}} \widehat{ \boldsymbol{\beta} } ( \uptau )^{+} - r \right)^{ \prime } \left[ \mathcal{ \boldsymbol{R} }  \mathcal{ \boldsymbol{Q} }_x^{-1} \mathcal{ \boldsymbol{R} } \right]^{-1} \left( \mathcal{\boldsymbol{R}} \widehat{ \boldsymbol{\beta} } ( \uptau )^{+} - r \right),
\end{align}

where $f_{y|x} \left(  \widehat{ F_{y|x}^{-1} ( \uptau)} \right)$ and $\widehat{ \omega}_{\psi.v}$ are consistent estimators of $f_{y|x} \left( F_{y|x}^{-1} ( \uptau) \right)$ and $\omega_{\psi.v}$ respectively. Then, the asymptotic distribution of the Wald statistic is summarized via the following Theorem.

Under the assumptions of Theorem <ref> and the linear restriction under the null hypothesis $\mathbb{H}_0$, we have that

\begin{align*}
\mathcal{W}_n \Rightarrow \chi_q^2,
\end{align*}

where $\chi_q^2$ is a centred Chi-square random variable with $q-$degrees of freedom.

We consider the derivations of the proof of Theorem 1 in [140]. The following results is useful in developing asymptotics for the regression quantile estimates: For $\mathsf{u} \neq 0$, we have that

\begin{align}
\label{trick}
\rho_{\uptau} ( \mathsf{u} - \mathsf{v} ) - \rho_{\uptau}(\mathsf{u}) &=
- \mathsf{v} \psi_{\uptau}(\mathsf{u}) + ( \mathsf{u} - \mathsf{v} ) \bigg[ \mathbf{1} \big\{ \mathsf{v}  < \mathsf{u}  < 0 \big\} - \mathbf{1} \big\{ 0 < \mathsf{u} < \mathsf{v} \big\}   \bigg]
\\
\psi_{\uptau}( \mathsf{u} ) &= \big[ \tau - \mathbf{1} \big\{ \mathsf{u} < 0  \big\} \big], \ \ u_{t} (\uptau) = u_t - F^{-1} ( \tau )  \ \ \text{and} \ \  Q_{ u_t (\uptau) } = 0.
\end{align}

Moreover, we denote with $\boldsymbol{\theta}( \uptau ) := \big( \alpha + F^{-1}( \uptau ), \boldsymbol{\beta} \big)^{\prime}$, then $u_t (\uptau)  = y_t - \boldsymbol{\theta} ( \uptau )^{\prime} \boldsymbol{z}_t$. Furthermore, let $\hat{ \boldsymbol{v} } = \boldsymbol{D}_n  \left( \widehat{ \boldsymbol{\theta} }( \uptau) - \boldsymbol{\theta}( \uptau) \right)$, where $\boldsymbol{D}_n = \mathsf{diag} \left( \sqrt{n}, n,...n \right)$. Then, we obtain

\begin{align}
\rho_{\uptau} \left( y_t - \widehat{ \boldsymbol{\theta} }( \uptau)^{\prime} \boldsymbol{z}_t  \right)
\rho_{\uptau} \left( u_t (\uptau) - \left( \boldsymbol{D}_n^{-1} \hat{ \boldsymbol{v} } \right)^{\prime} \boldsymbol{z}_t \right).
\end{align}

Denote with

\begin{align}
\mathcal{Z}_n( \boldsymbol{v} ) = \sum_{t=1}^n \left[  \rho_{\uptau} \left(  u_t (\uptau) - \left( \boldsymbol{D}_n^{-1} \boldsymbol{v} \right)^{\prime} \boldsymbol{z}_t  \right)  - \rho_{\uptau} \big( u_t (\uptau) \big) \right]
\end{align}

which is a convex random function.

Then, the minimization of $\mathcal{Z}_n(\boldsymbol{v})$ is equivalent to the minimization of the original optimization problem, which implies that if $\hat{\boldsymbol{v}}$ is the minimizer of $\mathcal{Z}_n(\boldsymbol{v})$, then we have that $\hat{\boldsymbol{v}} = \boldsymbol{D}_n  \left( \widehat{ \boldsymbol{\theta} }( \uptau) - \boldsymbol{\theta} ( \uptau) \right)$. Furthermore, the convexity of $\mathcal{Z}_n( .)$ implies that $\hat{\boldsymbol{v}}$ converges in distribution to the minimizer of $\mathcal{Z}(.)$. In general, $u_t$ and $\Delta \boldsymbol{x}_t$ are correlated and thus $B_{\psi}$ and $\boldsymbol{B}_v$ are correlated Brownian motions. Using the invariance principle given by Assumption <ref> and the property given by expression (<ref>), then the objective function of the minimization problem can be written as below

\begin{align*}
\mathcal{Z}_n( \boldsymbol{v} )
\sum_{t=1}^n \bigg[  \rho_{\uptau} \left(  u_t (\uptau) - \left( \boldsymbol{D}_n^{-1} \boldsymbol{v} \right)^{\prime} \boldsymbol{z}_t  \right)  - \rho_{\uptau} \big( u_t (\uptau) \big) \bigg]
\nonumber
\\
&\ = - \sum_{t=1}^n  \left( \boldsymbol{D}_n^{-1} \boldsymbol{v} \right)^{\prime} \boldsymbol{z}_t  \psi_{\uptau} \big( u_t (\uptau) \big)
\nonumber
\\
&\ \ \ \ \ + \sum_{t=1}^n  \left( u_t (\uptau) - \left( \boldsymbol{D}_n^{-1} \boldsymbol{v} \right)^{\prime} \boldsymbol{z}_t \right)  \bigg[ \mathbf{1}\left\{ \left( \boldsymbol{D}_n^{-1} \boldsymbol{v} \right)^{\prime} \boldsymbol{z}_t < u_t (\uptau) < 0 \right\} - \mathbf{1}\left\{ 0 < u_t (\uptau) < \left( \boldsymbol{D}_n^{-1} \boldsymbol{v} \right)^{\prime} \boldsymbol{z}_t \right\}   \bigg]
\end{align*}

Thus, under Assumptions <ref> - <ref>, we have that

\begin{align}
\boldsymbol{D}_n^{-1} \sum_{t=1}^n \boldsymbol{z}_t \psi_{ \uptau } \big( u_t ( \uptau ) \big)
\begin{bmatrix}
\displaystyle n^{-1 / 2}  \sum_{t=1}^n \psi_{ \uptau } \big( u_t ( \uptau ) \big)
\\
\displaystyle  n^{-1}  \sum_{t=1}^n \boldsymbol{x}_t \psi_{ \uptau } \big( u_t ( \uptau ) \big)
\end{bmatrix}
\Rightarrow
\begin{bmatrix}
\displaystyle \int_0^1 d \boldsymbol{B}_{ \psi_{ \uptau } }
\\
\displaystyle \int_0^1 \boldsymbol{B}_{x} d \boldsymbol{B}_{ \psi_{ \uptau } } + \lambda_{x \psi}
\end{bmatrix}
\end{align}

Next, we examine the limits of the following expressions

\begin{align}
\sum_{t=1}^n  \left( u_t ( \uptau )  -  \boldsymbol{v}^{ \prime } \boldsymbol{D}_n^{-1} \boldsymbol{z}_t \right) &\mathbf{1}\left\{ 0 < u_t ( \uptau )  < \boldsymbol{v}^{ \prime } \boldsymbol{D}_n^{-1} \boldsymbol{z}_t \right\}  \
\\
\sum_{t=1}^n  \left( u_t ( \uptau )  - \boldsymbol{v}^{ \prime } \boldsymbol{D}_n^{-1} \boldsymbol{z}_t \right) &\mathbf{1}\left\{ \boldsymbol{v}^{ \prime } \boldsymbol{D}_n^{-1} \boldsymbol{z}_t < u_t ( \uptau )  < 0 \right\}  \
\end{align}

Note that we denote with $\boldsymbol{v} = \boldsymbol{D}_n \left( \boldsymbol{\theta} - \boldsymbol{\theta} ( \uptau ) \right)$, and partition $\boldsymbol{v}$ and $ \boldsymbol{\theta} ( \uptau )$ comfortable with $\boldsymbol{z}_t = (1, \boldsymbol{x}_t^{\prime})^{\prime}$,

\begin{align*}
v =
\begin{bmatrix}
\\
\end{bmatrix},
\ \  \
\boldsymbol{\theta} ( \uptau )
\begin{bmatrix}
\alpha( \uptau )
\\
\boldsymbol{\beta}( \uptau )
\end{bmatrix}
\end{align*}

Then, for convenience of the asymptotic analysis, we denote with

\begin{align}
W_n( v) = \sum_{t=1}^n \big( \boldsymbol{v}^{ \prime } \boldsymbol{D}_n^{-1} \boldsymbol{z}_t - u_t ( \uptau ) \big) \mathbf{1} \left\{ 0 < u_t ( \uptau ) < \boldsymbol{v}^{ \prime } \boldsymbol{D}_n^{-1} \boldsymbol{z}_t  \right\}.
\end{align}

Furthermore, to avoid technical problems in taking conditional expectations, we consider truncation of $\boldsymbol{v}^{ \prime } \boldsymbol{D}_n^{-1} \boldsymbol{z}_t$ at some finite number $m > 0$ and denote with

\begin{align}
W_{nm} (v) &= \sum_{t=1}^n \xi_{tm} (v)
\\
\xi_{tm} (v) &= \big( \boldsymbol{v}^{ \prime } \boldsymbol{D}_n^{-1} \boldsymbol{z}_t  - u_t ( \uptau ) \big) \mathbf{1} \left\{ 0 < u_t ( \uptau ) < \boldsymbol{v}^{ \prime } \boldsymbol{D}_n^{-1} \boldsymbol{z}_t \right\} \mathbf{1} \left\{ \boldsymbol{v}^{ \prime } \boldsymbol{D}_n^{-1} \boldsymbol{z}_t \leq m \right\}.
\end{align}

Moreover, we denote the information set up to time $t$ as $\mathcal{F}_{t-1} = \sigma \big\{ u_{t-j}, v_{t-j+1}, j \geq 1 \big\}$, then $z_t \in \mathcal{F}_{t-1}$. We further define with

\begin{align*}
\bar{\xi}_{tm} (v) &= \mathbb{E} \big[ \left( \boldsymbol{v}^{ \prime } \boldsymbol{D}_n^{-1} \boldsymbol{z}_t  - u_t ( \uptau ) \right)  \mathbf{1} \big\{ 0 < u_t ( \uptau ) < \boldsymbol{v}^{ \prime } \boldsymbol{D}_n^{-1} \boldsymbol{z}_t \big\} \times \mathbf{1} \big\{  \boldsymbol{v}^{ \prime } \boldsymbol{D}_n^{-1} \boldsymbol{z}_t \leq m \big| \mathcal{F}_{t-1} \big\} \big]
\\
\bar{W}_{nm} (v) &= \sum_{t=1}^n \bar{\xi}_{tm} (v),
\end{align*}

Then, we have that $\big\{ \xi_{tm} (v) - \bar{\xi}_{tm} (v) \big\}$ is a martingale difference sequence. Therefore, we have that

\begin{align*}
\bar{W}_{nm} (v)
\sum_{t=1}^n \mathbb{E} \big[ \left( \boldsymbol{v}^{ \prime } \boldsymbol{D}_n^{-1} \boldsymbol{z}_t  - u_t ( \uptau ) \right)  \mathbf{1} \big\{ 0 < u_t ( \uptau ) < \boldsymbol{v}^{ \prime } \boldsymbol{D}_n^{-1} \boldsymbol{z}_t \big\} \times \mathbf{1} \big\{  \boldsymbol{v}^{ \prime } \boldsymbol{D}_n^{-1} \boldsymbol{z}_t \leq m \big| \mathcal{F}_{t-1} \big\} \big]
\end{align*}

By replacing with $u_t ( \uptau ) = F^{-1} (u) - u_t$. Moreover, denote with

* $\textcolor{red}{ \boldsymbol{a} := \big[ \boldsymbol{v}^{ \prime } \boldsymbol{D}_n^{-1} \boldsymbol{z}_t + F^{-1} (\uptau) \big] \times \mathbf{1} \left\{ \boldsymbol{v}^{ \prime } \boldsymbol{D}_n^{-1} \boldsymbol{z}_t \leq m \right\} }$

* $\textcolor{blue}{ \boldsymbol{b} := F^{-1} (\uptau) \leq s \leq \big[ \boldsymbol{v}^{ \prime } \boldsymbol{D}_n^{-1} \boldsymbol{z}_t + F^{-1} (\uptau) \big] \times \mathbf{1} \left\{ \boldsymbol{v}^{ \prime } \boldsymbol{D}_n^{-1} \boldsymbol{z}_t \leq m \right\} }$

Then, we obtain the following result

\begin{align*}
\bar{W}_{nm} (v)
\sum_{t=1}^n \mathbb{E} \big[ \left( \boldsymbol{v}^{ \prime } \boldsymbol{D}_n^{-1} \boldsymbol{z}_t  - \textcolor{red}{ F^{-1} (u) - u_t} \right)  \mathbf{1} \big\{ 0 < \textcolor{red}{u_t} < \boldsymbol{v}^{ \prime } \boldsymbol{D}_n^{-1} \boldsymbol{z}_t + \textcolor{red}{ F^{-1} (u) } \big\} \times \mathbf{1} \big\{  \boldsymbol{v}^{ \prime } \boldsymbol{D}_n^{-1} \boldsymbol{z}_t \leq m \big| \mathcal{F}_{t-1} \big\} \big]
\\
\sum_{t=1}^n \int_{ F^{-1} ( \uptau ) }^{ \textcolor{red}{ \boldsymbol{a} } } \times \left[ \int_r^{ \textcolor{red}{ \boldsymbol{a} } } ds \right] f_{t-1} (r) dr
\\
\sum_{t=1}^n \int_{ \textcolor{blue}{ \boldsymbol{b} } } \times \int_{ F^{-1} ( \uptau ) \leq r \leq s } f_{t-1} (r) dr ds
\\
\sum_{t=1}^n \int_{ F^{-1} ( \uptau ) }^{ \textcolor{red}{ \boldsymbol{a} } } \big[ s - F^{-1} ( \uptau ) \big] \times \left[ \frac{ F_{t-1}(s) - F_{t-1} \big( F^{-1} ( \uptau ) \big) }{ s - F^{-1} ( \uptau ) } \right] ds.
\end{align*}

Furthermore, notice that the probability density function $f_{t-1} ( s_n )$ is uniformly integrable for any sequence $s_n \to F^{-1} ( \uptau )$, then we have

\begin{align*}
\bar{W}_{nm} (v)
\sum_{t=1}^n \int_{ F^{-1} ( \uptau ) }^{ \textcolor{red}{ \boldsymbol{a} } } \big[ s - F^{-1} ( \uptau ) \big] f_{t-1} \big( F^{-1} ( \uptau ) \big) ds + o_{ \mathbb{P} } (1)
\\
\sum_{t=1}^n f_{t-1} \big( F^{-1} ( \uptau ) \big) \times \left\{ \frac{ \big[ s - F^{-1} ( \uptau ) \big]^2 }{2} \bigg|_{ F^{-1} ( \uptau ) }^{ \textcolor{red}{ \boldsymbol{a} } } \right\} + o_{\mathbb{P}} (1)
\end{align*}

Therefore, we have that

\begin{align*}
\bar{W}_{nm} (v)
\frac{1}{2} \sum_{t=1}^n f_{t-1} \big( F^{-1} ( \uptau ) \big) \big[ \boldsymbol{v}^{ \prime } \boldsymbol{D}_n^{-1} \boldsymbol{z}_t \big]^2 \times \mathbf{1} \left\{ \boldsymbol{v}^{ \prime } \boldsymbol{D}_n^{-1} \boldsymbol{z}_t \leq m \right\} + o_{\mathbb{P}}(1)
\\
\frac{1}{2n} \sum_{t=1}^n f_{t-1} \big( F^{-1} ( \uptau ) \big) \boldsymbol{v}^{ \prime } \big[ \sqrt{n} \boldsymbol{D}_n^{-1} \boldsymbol{z}_t \boldsymbol{z}_t^{\prime} \boldsymbol{D}_n^{-1} \sqrt{n} \big] \times \boldsymbol{v} \times \mathbf{1} \left\{ \boldsymbol{v}^{ \prime } \boldsymbol{D}_n^{-1} \boldsymbol{z}_t \leq m \right\} + o_{\mathbb{P}}(1).
\end{align*}

Furthermore, notice that by stationarity of the sparsity function $f_{t-1} \big( F^{-1} ( \uptau ) \big)$ , we have

\begin{align}
\underset{ 0 \leq r \leq 1 }{ \mathsf{sup} } \left| \frac{1}{ n^{1 - \epsilon} } \sum_{t=1}^{ \floor{nr} } \big[ f_{t-1} \left( F^{-1} ( \uptau ) \right) \big] - f \left( F^{-1} ( \uptau ) \right) \right| \overset{ p }{ \to } 0
\end{align}

for some $\epsilon > 0$. Thus, it follows that

\begin{align*}
\bar{W}_{nm} (v) \Rightarrow \frac{1}{2} f \big( F^{-1} (\uptau)   \big) \boldsymbol{v}^{\prime}
\left\{
\int_0^1
\begin{bmatrix}
1 &  \displaystyle \int_0^1 \boldsymbol{B}_x
\\
\displaystyle \int_0^1 \boldsymbol{B}_x & \displaystyle \int_0^1 \boldsymbol{B}_x \boldsymbol{B}_x^{\prime}
\end{bmatrix} \times
\mathbf{1} \big\{ 0 < \mathsf{v}_1 + \mathsf{v}_2 \boldsymbol{B}_x (s) \leq m \big\}
\right\} \boldsymbol{v} := \eta_m
\end{align*}

We now follow similar arguments as in [118] by noting that

\begin{align}
\big( \boldsymbol{v}^{ \prime } \boldsymbol{D}_n^{-1} \boldsymbol{z}_t \big) \times \mathbf{1} \big\{ 0 \leq \boldsymbol{v}^{ \prime } \boldsymbol{D}_n^{-1} \boldsymbol{z}_t \leq m \big\} \overset{ p }{ \to } 0
\end{align}

which holds uniformly in $t$. Therefore, we obtain that

\begin{align*}
\sum_{t=1}^n \mathbb{E} \big[ \xi_{tm} (v)^2 | \mathcal{F}_{t-1} \big] \leq \mathsf{max} \bigg\{  \big( \boldsymbol{v}^{ \prime } \boldsymbol{D}_n^{-1} \boldsymbol{z}_t \big) \times \mathbf{1} \big\{ 0 \leq \boldsymbol{v}^{ \prime } \boldsymbol{D}_n^{-1} \boldsymbol{z}_t \leq m \big\} \bigg\} \times \sum_{t=1}^n \bar{\xi}_{tm} (v) \overset{ p }{ \to } 0.
\end{align*}

Therefore, the following summation of martingale difference sequence

\begin{align}
\sum_{t=1}^n \big[ \xi_{tm} (v) - \bar{\xi}_{tm} (v) \big]
\end{align}

converges to zero in probability. Notice that by the asymptotic equivalence lemma, the limit distribution of $\sum_{t=1}^n \xi_{tm} (v)$ is the same as that of $\sum_{t=1}^n \bar{\xi}_{tm} (v)$ such that,

\begin{align}
W_{nm} ( v ) \Rightarrow \eta_{m}
\end{align}

Let $m \to \infty$, we have that

\begin{align}
\eta_{m} \Rightarrow \frac{1}{2} f \big( F^{-1} \uptau ) \big) \boldsymbol{v}^{\prime} \big[ \int_0^1 \boldsymbol{B}_z \boldsymbol{B}_z^{\prime} \big] \boldsymbol{v} \mathbf{1} \big\{  \boldsymbol{v}^{\prime} \boldsymbol{B}_z (s) > 0 \big\} = \eta,
\end{align}

where $\boldsymbol{B}_z = \left( 1, \boldsymbol{B}_x (s)^{\prime}  \right)$.

Next, we show that

\begin{align}
\underset{ m \to \infty }{ \mathsf{lim} } \underset{ n \to \infty }{ \mathsf{lim \ sup} } \ \mathbb{P} \big( \big| W_n(v) - W_{nm} (v) \big| \geq \epsilon \big) = 0.
\end{align}

In particular, the above result holds because of the following

\begin{align*}
\mathbb{P}
\big( \big| W_n(v) - W_{nm} (v) \big| \geq 0 \big)
\mathbb{P} \left[ \sum_{t=1}^n \big( \boldsymbol{v}^{ \prime } \boldsymbol{D}_n^{-1} \boldsymbol{z}_t - u_t (\uptau) \big) \times \mathbf{1} \big\{ 0 < u_{t}(\uptau) < \boldsymbol{v}^{\prime} \boldsymbol{D}_n^{-1} \boldsymbol{z}_t \big\} \times \mathbf{1} \big\{ \boldsymbol{v}^{\prime} \boldsymbol{D}_n^{-1} \boldsymbol{z}_t > m \big\} > 0 \right]
\\
\leq \mathbb{P} \left[ \bigcup_{t=1}^n \big\{ \boldsymbol{v}^{\prime} \boldsymbol{D}_n^{-1} \boldsymbol{z}_t  > m \big\} \right]
\mathbb{P} \big[ \underset{ t }{ \mathsf{max} } \big\{ \boldsymbol{v}^{\prime} \boldsymbol{D}_n^{-1} \boldsymbol{z}_t \big\} > m \big],
\end{align*}

\begin{align}
\underset{ m \to \infty }{ \mathsf{lim} } \mathbb{P} \left[ \underset{ 1 \leq t \leq 1 }{ \mathsf{sup} } \boldsymbol{v}^{\prime} \boldsymbol{D}_n^{-1} \boldsymbol{z}_t > m  \right] = 0.
\end{align}

Then, notice that by [9] we have that

\begin{align}
W_{nm} (v) \Rightarrow \eta,
\end{align}

which implies that

\begin{align*}
&\sum_{t=1}^n \left[ \big( \boldsymbol{D}_n^{-1} \boldsymbol{v}^{\prime} \big)^{\prime} \boldsymbol{z}_t - u_t(\uptau) \right] \mathbf{1} \left\{ 0 < u_t(\uptau) < \big( \boldsymbol{D}_n^{-1} \boldsymbol{v}^{\prime} \big)^{\prime} \boldsymbol{z}_t \right\}
\Rightarrow
\frac{1}{2} f \big( F^{-1} (v) \big) \boldsymbol{v}^{\prime} \int_0^1 \boldsymbol{B}_z \boldsymbol{B}_z^{\prime} \boldsymbol{v}
\end{align*}

Similarly, we can show that

\begin{align*}
\sum_{t=1}^n \big( u_t (\uptau) - \big( \boldsymbol{D}_n^{-1} \boldsymbol{v}^{\prime} \big)^{\prime} \boldsymbol{z}_t  \big) \mathbf{1} \big\{ \big( \boldsymbol{D}_n^{-1} \boldsymbol{v}^{\prime} \big)^{\prime} \boldsymbol{z}_t < u_t(\uptau) < 0 \big\}
\Rightarrow
\frac{1}{2} f \big( F^{-1} (v) \big) \boldsymbol{v}^{\prime} \left[ \int_0^1 \boldsymbol{B}_z \boldsymbol{B}_z^{\prime} \right] \boldsymbol{v} \mathbf{1} \big\{ \boldsymbol{v}^{\prime} \boldsymbol{B}_z (s) < 0 \big\}.
\end{align*}

Therefore, we have that

\begin{align*}
&\sum_{t=1}^n \big( u_t(\uptau) - \big( \boldsymbol{D}_n^{-1} \boldsymbol{v} \big)^{\prime} \boldsymbol{z}_t \big) \times \bigg[ \mathbf{1} \big\{ \big( \boldsymbol{D}_n^{-1} \boldsymbol{v} \big)^{\prime} \boldsymbol{z}_t  <  u_t(\uptau) < 0 \big\} - \mathbf{1} \big\{ 0  <  u_t(\uptau) < \big( \boldsymbol{D}_n^{-1} \boldsymbol{v} \big)^{\prime} \boldsymbol{z}_t \big\} \bigg]
\\
f \big( F^{-1} (\uptau) \big) \boldsymbol{v}^{\prime}
\begin{bmatrix}
1 & \displaystyle \int_0^1 \boldsymbol{B}_x
\\
\displaystyle \int_0^1 \boldsymbol{B}_x & \displaystyle \int_0^1 \boldsymbol{B}_x \boldsymbol{B}_x^{\prime}
\end{bmatrix}
\boldsymbol{v}.
\end{align*}

As a result we obtain that

\begin{align*}
\mathcal{Z}_n (v)
\sum_{t=1}^n \left[ \rho_{\uptau} \big( u_t(\uptau) - \big( \boldsymbol{D}_n^{-1} \boldsymbol{v} \big)^{\prime} \boldsymbol{z}_t \big) - \rho_{\uptau} \big( u_t(\uptau) \big) \right]
\sum_{t=1}^n \big( \boldsymbol{D}_n^{-1} \boldsymbol{v} \big)^{\prime} \boldsymbol{z}_t \psi_{\uptau} \big( u_t (\uptau) \big)
\\
&\ \ +
\sum_{t=1}^n \big( u_t(\uptau) - \big( \boldsymbol{D}_n^{-1} \boldsymbol{v} \big)^{\prime} \boldsymbol{z}_t \big) \big) \times \bigg[ \mathbf{1} \bigg\{ \big( \boldsymbol{D}_n^{-1} \boldsymbol{v} \big)^{\prime} \boldsymbol{z}_t  <  u_t(\uptau) < 0 \bigg\} - \mathbf{1} \bigg\{ 0  <  u_t(\uptau) < \big( \boldsymbol{D}_n^{-1} \boldsymbol{v} \big)^{\prime} \boldsymbol{z}_t \bigg\} \bigg]
\\
- \boldsymbol{v}^{\prime}
\begin{bmatrix}
\displaystyle \int_0^1 d \boldsymbol{B}_{ \psi_{\uptau} }
\\
\displaystyle \int_0^1  \boldsymbol{B}_x d \boldsymbol{B}_{ \psi_{\uptau} } + f \big( F^{-1} ( \uptau) \big) \boldsymbol{v}^{\prime}
\end{bmatrix}
f \big( F^{-1} (\uptau) \big)
\boldsymbol{v}^{\prime}
\begin{bmatrix}
1 & \displaystyle \int_0^1 \boldsymbol{B}_x
\\
\displaystyle \int_0^1 \boldsymbol{B}_x & \displaystyle \int_0^1 \boldsymbol{B}_x \boldsymbol{B}_x^{\prime}
\end{bmatrix}
\boldsymbol{v}
:= \mathcal{Z}(v).
\end{align*}

Therefore, by the convexity Lemma of [118] and arguments presented by [80], notice that the functional $\mathcal{Z}_n(v)$ is minimized at $\widehat{v} = \boldsymbol{D}_n \big( \widehat{\boldsymbol{\theta}}(\uptau) - \boldsymbol{\theta}(\uptau) \big)$ while $\mathcal{Z}(v)$ is minimized at

\begin{align}
\frac{1}{2 f \big( F^{-1} (\uptau) \big)}
\begin{bmatrix}
1 & \displaystyle \int_0^1 \boldsymbol{B}_x
\\
\displaystyle \int_0^1 \boldsymbol{B}_x & \displaystyle \int_0^1 \boldsymbol{B}_x \boldsymbol{B}_x^{\prime}
\end{bmatrix}^{-1}
\times
\begin{bmatrix}
\displaystyle \int_0^1 d \boldsymbol{B}_{ \psi_{\uptau} }
\\
\displaystyle \int_0^1 \boldsymbol{B}_x d \boldsymbol{B}_{ \psi_{\uptau} }  + \lambda_{x \psi_{\uptau} }
\end{bmatrix}
\end{align}

by Lemma A of [78] we have that

\begin{align}
\boldsymbol{D}_n \big( \widehat{ \boldsymbol{\theta}} (\uptau) - \boldsymbol{\theta}(\uptau) \big)
\Rightarrow
\frac{1}{2 f \big( F^{-1} (\uptau) \big)}
\begin{bmatrix}
1 & \displaystyle \int_0^1 \boldsymbol{B}_x
\\
\displaystyle \int_0^1 \boldsymbol{B}_x & \displaystyle \int_0^1 \boldsymbol{B}_x \boldsymbol{B}_x^{\prime}
\end{bmatrix}^{-1}
\times
\begin{bmatrix}
\displaystyle \int_0^1 \boldsymbol{B}_{ \psi_{\uptau} }
\\
\displaystyle \int_0^1 \boldsymbol{B}_x d \boldsymbol{B}_{ \psi_{\uptau} }  + \lambda_{x \psi_{\uptau} }
\end{bmatrix}.
\end{align}

Therefore, by Theorem 1, the asymptotic distribution of the random measure $n \big( \widehat{\beta}(\uptau) - \beta(\uptau) \big)$ can be expressed as below

\begin{align*}
\frac{1}{ f \big( F^{-1} (\uptau) \big) } \left[ \int_0^1 \boldsymbol{B}_v^{\mu} \boldsymbol{B}_v^{\mu \prime} \right]^{-1} \left( \int_0^1 \boldsymbol{B}^{\mu}_v  d \boldsymbol{B}_{ \psi_{\uptau}. v } \right)
\frac{1}{ f \big( F^{-1} (\uptau) \big) } \left[ \int_0^1 \boldsymbol{B}_v^{\mu} \boldsymbol{B}_v^{\mu \prime} \right]^{-1}  \bigg[ \boldsymbol{B}_v^{\mu} d \boldsymbol{B}_v^{\prime} \boldsymbol{\Omega}_{vv}^{-1} \boldsymbol{\Omega}_{v \psi_{\uptau} } + \lambda_{ v \psi_{\uptau} } \bigg].
\end{align*}

Furthermore, we have that

\begin{align*}
\big[ \widehat{\boldsymbol{\Omega}}_{vv}, \ \ \ \widehat{\boldsymbol{\Omega}}_{v \psi_{\uptau} }, \ \ \ \widehat{\lambda}_{v \psi_{\uptau} }, \ \ \ \widehat{\lambda}_{vv} \big]
\end{align*}

are consistent estimates of $\boldsymbol{\Omega}_{vv}$, $\boldsymbol{\Omega}_{v \psi_{\uptau} }$, $\lambda_{v \psi_{\uptau} }$ and $\lambda_{vv}$. Thus, we obtain the following weak convergence
\begin{align*}
\boldsymbol{D}_n \left( \widehat{\boldsymbol{\theta}} (\uptau)^{+} - \boldsymbol{\theta}(\uptau) \right)
\begin{pmatrix}
\sqrt{n} \big[ \widehat{\alpha}(\uptau) - \alpha(\uptau) \big]
\\
\\
n \big[ \widehat{\boldsymbol{\beta}}^{+}(\uptau) - \boldsymbol{\beta}(\uptau) \big]
\end{pmatrix}
\\
\begin{pmatrix}
\sqrt{n} \big[ \widehat{\alpha}(\uptau) - \alpha(\uptau) \big]
\\
\\
n \big[ \widehat{\boldsymbol{\beta}} (\uptau) - \boldsymbol{\beta}(\uptau) \big]
\end{pmatrix}
\begin{bmatrix}
\\
\\
- \frac{ \displaystyle 1}{ \displaystyle f \big( \widehat{F^{-1} ( \uptau )}  \big) } \left[ \displaystyle \frac{1}{n^2} \sum_{t=1}^n \boldsymbol{x}_t \boldsymbol{x}_t^{\prime} \right]^{-1} \left[ \displaystyle \frac{1}{n} \sum_{t=1}^n \boldsymbol{x}_t\boldsymbol{v}_t^{\prime} \widehat{\boldsymbol{\Omega}}_{vv}^{-1}  \widehat{\boldsymbol{\Omega}}_{v \psi_{\uptau} } + \widehat{\lambda}_{v \psi_{\uptau} }^{+}  \right]
\end{bmatrix}
\\
\frac{ \displaystyle 1}{ \displaystyle f \big( \widehat{F^{-1} ( \uptau )}  \big) } \left[ \int_0^1 \widetilde{\boldsymbol{B}}_v \widetilde{\boldsymbol{B}}_v^{\prime} \right]^{-1} \left( \int_0^1 \widetilde{\boldsymbol{B}}_v  d \boldsymbol{B}_{ \psi_{\uptau}. v } \right)
\end{align*}

§.§.§ Inference in quantile cointegration models

The asymptotic distribution of $\widehat{\beta}( \uptau )$ in quantile cointegrating regressions is mixture normal. Another interesting problem in the quantile cointegration regression model is the hypothesis test on constancy of the cointegrating vector $\boldsymbol{\beta} ( \uptau ) = \bar{\boldsymbol{\beta} }$ , over $\uptau \in \mathcal{T}_{\iota}$, where $\bar{ \boldsymbol{\beta} }$ is a vector of unknown constants. A natural preliminary candidate for testing constancy of the cointegrating vector is a standardized version of $\left( \widehat{ \boldsymbol{\beta} } - \bar{\boldsymbol{\beta}} \right)$. Under the null hypothesis, we have that

\begin{align}
n \left( \widehat{ \boldsymbol{\beta} } ( \uptau ) - \bar{\boldsymbol{\beta}} \right) \Rightarrow \frac{1}{ f_{\epsilon} \big( F_{\epsilon}^{-1} ( \uptau ) \big)} \left[ \int_0^1 \boldsymbol{B}_v^{\mu} \boldsymbol{B}_v^{\mu \prime} \right]^{-1} \times \left[ \int_0^1 \boldsymbol{B}_v^{\mu} d \boldsymbol{B}^{*}_{ \psi_{\uptau} } \right]
\end{align}

Denote with $\widehat{\boldsymbol{\beta}}$ as a preliminary estimator of  $\bar{ \boldsymbol{\beta} }$  and consider the following process

\begin{align}
\widehat{V}_n ( \uptau ) = n \left( \widehat{ \boldsymbol{\beta} } ( \uptau ) - \bar{\boldsymbol{\beta}} \right)
\end{align}

Then, under the null hypothesis $\mathcal{H}_0: \boldsymbol{\beta} (\uptau) = \bar{\boldsymbol{\beta}}$ it holds that

\begin{align}
\underset{ \uptau \in \mathcal{T}_{\iota}  }{ \mathsf{sup} } \ \left| \widehat{V}_n ( \uptau ) \right| \Rightarrow \underset{ \uptau \in \mathcal{T}_{\iota}  }{ \mathsf{sup} } \left| \frac{1}{ f_{\epsilon} \big( F_{\epsilon}^{-1} ( \uptau ) \big)} \left[ \int_0^1 \boldsymbol{B}_v^{\mu} \boldsymbol{B}_v^{\mu \prime} \right]^{-1} \times \int_0^1 \boldsymbol{B}_v^{\mu} d \big\{ B^{*}_{ \psi_{\uptau} }  - f_{\epsilon} \big( F_{\epsilon}^{-1} ( \uptau ) \big) B^{*}_{\epsilon}  \big\} \right|
\end{align}

where $B^{*}_{\epsilon}$ is the Brownian motion limit for the partial sum process of $\epsilon_t$. Therefore, we may test varying-coefficient behaviour based on the KS statistic. Notice that for the matrix $\boldsymbol{V}_{xx} =
\displaystyle \int_{0}^{\infty} e^{r \boldsymbol{C}_p} \boldsymbol{\Omega}_{xx} e^{r \boldsymbol{C}_p} dr$, and thus by applying integration by parts we can obtain the following formula $\boldsymbol{C} \boldsymbol{V}_{xx} + \boldsymbol{V}_{xx} \boldsymbol{C} = - \boldsymbol{\Omega}_{xx}$ (see, for example expression (49) in [102]).

In general, lets suppose we have that

\begin{align*}
u_t (\uptau) = y_t - \alpha_0 (\uptau) - g_t(\beta)
\end{align*}

Define with
such that $u_t(\uptau) = u_t - F^{-1}(\uptau)$. Then, for the error terms of the quantile predictive regression model the following invariance principles hold

\begin{align}
U_n^{\psi} ( \lambda, \uptau )
\frac{1}{ \sqrt{n} } \sum_{t=1}^{ \floor{ \lambda n } } \psi_{\uptau} \big( u_t (\uptau) \big)
\ \ \
V_n( \lambda )
\frac{1}{ \sqrt{n} } \sum_{t=1}^{ \floor{ \lambda n } } v_{t+1}
\end{align}

In addition, suppose that the following conditions hold:

(i) $\bigg\{ \psi_{\uptau} \big( u_t (\uptau) \big), \mathcal{F}_t \bigg\}$ is a martingale difference sequence.

(ii) $\left\{ x_t \right\}$ is adapted to the filtration $\mathcal{F}_{t-1}$ and, for all $0 < \lambda < 1$, then the vector $\big( U_n^{\psi} ( \lambda, \uptau ) , V_n ( \lambda ) \big)$ converges weakly to a two-dimensional vector Brownian motion $\big( U^{\psi} ( \lambda, \uptau ), V ( \lambda ) \big)$ with a covariance matrix given by the following expression

\begin{align}
\lambda
\begin{bmatrix}
\omega_{\psi}(\uptau)^2 & \omega_{\psi v}(\uptau)
\\
\omega_{v \psi}(\uptau) & \omega_{v}(\uptau)^2
\end{bmatrix}
\end{align}

Based on the martingale difference sequence assumption on $u_t$, it follows that $U_n^{\psi} ( \lambda, \uptau ) \overset{ d }{ \to } U^{\psi} ( \lambda, \uptau )$, where $U^{\psi} ( \lambda, \uptau )$ is viewed as a Brownian motion with variance $\lambda \omega_{\psi}(\uptau)^2 = \lambda \uptau ( 1 - \uptau)$ for a fixed value of $\uptau$. Thus, for each fixed pair $( \lambda, \uptau )$, it holds that $U^{\psi} ( \lambda, \uptau )$ is distributed as $\mathcal{N} \big( 0, \lambda \omega_{\psi}(\uptau)^2  \big)$. A standard assumption in the nonstationary time series analysis is that $V_n$ converge weakly jointly with $U_n^{\psi}$ to a vector Brownian motion. Notice that for the development of the asymptotic theory we keep the argument $\uptau$ fixed (see, also [25]). Another example is the framework proposed by [25] who discuss quantile cointegration in ADL models. In that case, it holds

\begin{align}
\frac{1}{n} \sum_{t=1}^n \psi_{\uptau} \big( u_t(\uptau) \big) \boldsymbol{X}_t
\Rightarrow
\int_0^1 \boldsymbol{B}_w(r) d \boldsymbol{B}_{\psi}( r, \uptau)
\end{align}

§.§ Structural Break Testing for  Quantile Regressions

Let $F_{t-1} = \mathbb{P} \big( y_t < . | \mathcal{F}_{t-1} \big)$ be the conditional distribution function of $y_t$ given $\mathcal{F}_{t-1}$ and denote by $f_{t-1}$ its derivative. Under stationarity and if $f_{t-1}$ is uniformly integrable on $\mathcal{X} = \left\{ x : 0 < F(x) < 1 \right\}$, then

\begin{align}
\boldsymbol{\Sigma}^{- 1 / 2} n^{1/ 2} \left[ \hat{\theta} (.) - \theta(.) \right] \overset{ \mathcal{D} }{ \to } \boldsymbol{B}_{p+1} (.), \ \ \text{as} \ n \to \infty,
\end{align}

where $\boldsymbol{\Sigma} = \boldsymbol{\Omega}_1^{-1} \boldsymbol{\Omega}_0 \boldsymbol{\Omega}_1^{-1}$ with $\boldsymbol{\Omega}_0 = \mathbb{E} \left( \boldsymbol{X}_t \boldsymbol{X}_t^{\prime} \right)$ and $\boldsymbol{\Omega}_1 =   \underset{ n \to \infty  }{ \mathsf{lim} } \sum_{t=1}^n f_{t-1} \left\{ F_{t-1}^{-1} \left( \uptau \right) \right\} \boldsymbol{X}_t \boldsymbol{X}_t^{\prime}$. Moreover, $\big( \boldsymbol{B}_{p+1}(\uptau) : \uptau \in [0,1] \big)$ is a standard $( p + 1)-$dimensional Brownian bridge.

Therefore, one may be interested to develop a framework for detecting multiple breaks in quantile piecewise regression models or quantile cointegrated regressions. For instance if the number of break points $m$ is given, then estimating their locations and the $(m + 1)$ piecewise quantile autoregressive models at a specific quantile $\uptau \in (0,1)$ can be done via solving

\begin{align}
\underset{ \theta ( \uptau ), \mathcal{K} }{ \mathsf{min} } \sum_{ j=1 }^{ m+1 } \sum_{ t = k_{j-1} + 1 }^{ k_j } \uprho_{ \uptau } \big( y_t - X_{j,t}^{\prime} \theta_j ( \uptau ) \big).
\end{align}

Further related studies on structural break testing for quantile regressions include [123] and [42] within a stationary time series environment and [75] within a nonstationary time series environment. Although both of these streams of literature correspond to structural break testing on quantile-dependent coefficients within a full-sample period. On the other hand, a different application would correspond to an implementation of the monitoring framework within based on a quantile regression, in the case of risk measures (see the recent study by  [66]).

The next two sections cover briefly key results from two additional relevant topics for quantile regression models, that is,  (i) estimation in high dimensional quantile regressions and, (ii) specification testing in quantile regressions. Therefore, developing powerful tests for the correct specification[Recall that an omnibus test is a statistical test for which the alternative hypothesis is a negation of the null hypothesis. For example, the particular family of tests allows to test the monotonicity of conditional moments. On the other hand, a different approach in the literature, that is, the unconditional backtesting can be decomposed into an estimation risk component and a model risk component which implies that under correct specification of the parametric VaR model, almost surely, the model risk vanishes (see, [39]). In contract, under misspecification, the model risk does not vanish and has a negligible effect on the unconditional test.] of parametric conditional quantiles over a possibly continuous range of quantiles and under general conditions on the underlying data-generating process is a good model validity practice. The main idea of regression model misspecification is explained by  [131] as below:

Let $\mathcal{S} :=  \left\{ f(., \theta ) : \mathbb{R}^k \to \mathbb{R} | \theta \in \Theta \right\}$, $ \Theta \subset \mathbb{R}^p, p \in \mathbb{N}$. The model is correctly specified for $\mathbb{E} \left( Y | X \right)$ when $f( X , \theta_0 )$ is a version of $\mathbb{E} \left( Y | X \right)$ for some $\theta_0 \in \Theta$. For instance, $\hat{\theta}_n$ can be a quantile regression estimator. Denote with $\epsilon:= Y - f(X, \theta^{*} )$, the correct specification of $\mathcal{S}$ for $\mathbb{E} \left( Y | X \right)$ is equivalent to $e := \mathbb{E} \left( \epsilon | X \right) = 0$, almost surely. Under suitable conditions on $Y$ and $f$, $e$ is an integrable function of $X$, that is, $e \in L^p ( X ) := L^p ( \Omega, \sigma(X), P )$ for some $p \in [1, \infty]$.

§ HIGH DIMENSIONAL QUANTILE REGRESSION APPLICATIONS

§.§ High Dimensional Quantile Regression

Following the framework of [6] consider a response variable $y$ and $p-$dimensional covariates $x$ such that the $u-$th conditional quantile function of $y$ given $x$ is

\begin{align}
F_{ y_i | x_i }^{-1} ( u| x_i ) = x^{\prime} \beta(u), \ \ \ \beta (u) \in \mathbb{R}^p \ \ \ \text{for all} \ \ u \in \mathcal{U},
\end{align}

where $\mathcal{U} \subset (0,1)$ is a compact set of quantile indices. Recall that the $u-$th conditional quantile $F_{ y_i | x_i }^{-1} ( u| x_i )$ is the inverse of the conditional distribution function $F_{ y_i | x_i }^{-1} ( y | x_i )$ of $y_i$ given $x_i$. Furthermore, we consider the case where the dimension $p$ of the model is large, possibly much larger than the available sample size $n$, but the true model $\beta (u)$ has a sparse support

\begin{align}
T_u = \mathsf{support} \left(  \beta(u)  \right) = \left\{ j \in \left\{ 1,..., p \right\} : \left| \beta_j(u) \right| > 0 \right\}
\end{align}

having only $s_u \leq s \leq n / \mathsf{log} ( n \cup p )$ nonzero components for all $u \in \mathcal{U}$. Therefore, the corresponding population coefficient $\beta (u)$ is known to minimize the criterion function

\begin{align}
\mathcal{Q}_u := \mathbb{E} \big[ \rho_u \left( y - x^{\prime} \beta \right) \big]
\end{align}

In other words, given a random sample $\left\{ (y_1, x_1),..., (y_n, x_n)     \right\}$, the quantile regression estimator of $\beta(u)$ is defined as a minimizer of the empirical analoge given by

\begin{align}
\widehat{\mathcal{Q}}_u := \mathbb{E} \left[ \rho_u \left( y - x_i^{\prime} \beta \right) \right].
\end{align}

The main challenge of the statistical problem under examination is that in high-dimensional settings, particularly when $p \geq n$, ordinary quantile regression is generally inconsistent, which motivates the use of penalization in order to remove all, or at least nearly all, regressors whose populations coefficients are zero, thereby possibly restoring consistency. Then, the $\ell_1-$penalized quantile regression estimator $\widehat{\beta}(u)$ is a solution to the following optimization problem:

\begin{align}
\underset{ \beta in \mathbb{R}^p  }{ \mathsf{min} }   \ \widehat{\mathcal{Q}}_u (\beta) + \frac{ \lambda \sqrt{u (1 - u)} }{n} \sum_{j=1}^p \widehat{\sigma}_j \left| \beta_j  \right|
\end{align}

where $\hat{\sigma}^2_j = \mathbb{E}_n \big[ x_{ij}^2 \big]$, which ensures that the conditional variance of the error term has a bounded variance and thus excluding infitine variance cases as the related distribution function of the innovation sequence generating the data mechanism under examination. To show the result of consistency, it suffices to show that for any $\epsilon > 0$, there exists a sufficiently large $C$ such that

\begin{align}
\mathbb{P} \left( \underset{ \norm{c}_2 = C  }{ \mathsf{inf} } \mathcal{Q}_n^{qr} \big( \beta ( \uptau ) + a_n c \big) > \mathcal{Q}_n^{qr} \big( \beta ( \uptau ) \big) \right) \geq 1 - \epsilon.
\end{align}

In other words, this inequality implies that with probability at least $1 - \epsilon$, there is a local minimizer $\tilde{\beta} ( \uptau )$ within the shrinking ball $\big\{  \beta ( \uptau ) + a_n c,  \norm{c}_2 = C \big\}$ such that $\norm{ \tilde{\beta} ( \uptau ) - \beta ( \uptau ) }_2 = \mathcal{O}_p ( a_n )$. Therefore, the proof can be obtained by showing that the following term is positive

\begin{align*}
\mathcal{Q}_n^{qr} \big( \beta ( \uptau ) + a_n c \big) - \mathcal{Q}_n^{qr} \big( \beta ( \uptau ) \big) = \sum_{t=1}^n \rho_{ \uptau } \big( u_t ( \uptau ) - x_{t-1}^{\prime} a_n c \big) - \sum_{t=1}^n \rho_{ \uptau } \big( u_t ( \uptau ) \big)
\end{align*}

\begin{align*}
Var ( I_3 )
Var \left( \sum_{t=1}^n \int_0^{ x_{t-1}^{\prime} a_n c } \bigg[ \mathbf{1} \big( u_t ( \uptau ) \leq s \big) -  \mathbf{1} \big( u_t ( \uptau ) \leq 0 \big) \bigg] ds \right)
\\
\mathbb{E} \left[ \sum_{t=1}^n \int_0^{ x_{t-1}^{\prime} a_n c } \bigg[ \mathbf{1} \big( u_t ( \uptau ) \leq s \big) -  \mathbf{1} \big( u_t ( \uptau ) \leq 0 \big) \bigg] ds  \right]
\\
a_n^2 \sum_{t=1}^n c^{\prime} \mathbb{E} \big[ x_{t-1} x_{t-1}^{\prime} \big] c + 2 a_n^2 \sum_{t=2}^n \sum_{k=1}^{t-1} \mathbb{E} \big[ \left| x_{t-1}^{\prime} c \right| \left| x_{t-1}^{\prime} c \right| \big]
\\
&\equiv V_{3,1} + V_{3,2}.
\end{align*}

Therefore, using the similar arguments in $I_1$, we have that

\begin{align}
V_{3,1} \leq a_n^2 C^2 \frac{ \bar{c}_{A(n,p)} }{ c_f } \frac{ n(n+1) }{2} = \mathcal{O}_p \left( a_n^2 \bar{c}_{A(n,p)} n^2 \right).
\end{align}

Consequently, by the Cauchy-Schwarz inequality, and $t > k$, we have that

\begin{align}
V_{3,2} \leq 2 a_n^2 \sum_{t=2}^n \sum_{k=1}^{ t - 1 } \sqrt{ \mathbb{E} \left[ \left( x_{t-1}^{\prime} x_{t-1}  \right) \right] } \sqrt{ \mathbb{E} \left[ \left( x_{t-1}^{\prime} x_{t-1}  \right) \right] }
\end{align}

\begin{align*}
\mathbb{E}  \left| I_1 \right| ^2
a_n^2 \sum_{t=1}^n c^{\prime} M_n^{\prime} \mathbb{E} \left[ \psi_{ \uptau } \big( u_t ( \uptau ) \big)^2 \tilde{x}_{t-1} \tilde{x}_{t-1}^{\prime}  \right] M_n c
2 a_n^2 \sum_{t=2}^n \sum_{k=1}^{t-1} c^{\prime} M_n^{\prime} \mathbb{E} \left[ \psi_{ \uptau } \big( u_t ( \uptau ) \big)^2 \tilde{x}_{t-1} \tilde{x}_{t-1}^{\prime}  \right] M_n c
\\
a_n^2 \sum_{t=1}^n c^{\prime} M_n^{\prime} \mathbb{E} \left[ \psi_{ \uptau } \big( u_t ( \uptau ) \big)^2 \tilde{x}_{t-1} \tilde{x}_{t-1}^{\prime}  \right] M_n c
\\
a_n^2 c^{\prime} M_n^{\prime} \sum_{t=1}^n \mathbb{E} \left[ \psi_{ \uptau } \big( u_t ( \uptau ) \big)^2 \tilde{x}_{t-1} \tilde{x}_{t-1}^{\prime}  \right] M_n c
\\
a_n^2  n^2_{ \bar{c}_{ B(n,p) }  } C^2
\end{align*}

Further applications of high dimensional quantile time series regression models are studied by [5] while in a cross-sectional setting relevant frameworks are proposed by [64], [15], [62], [93] and  [150].

We consider the framework of  [93] who develops asymptotic theory for Complete Subset Averaging in quantile regressions.

Denote with (see, page 11 in [93])

\begin{align}
z_{(m,k)} = x^{\prime}_{(m,k)} \Theta^{*}_{(m,k)} - \mathbb{E} \left[   x^{\prime}_{(m,k)} \Theta^{*}_{(m,k)} \right]
\end{align}

Depending on the dependence structure of $z_{(m,k)}$ a corresponding uniform law of large numbers hold. For example, consider the following maximal inequality for $\delta > 0$,

\begin{align*}
\mathbb{P} \left( \underset{ 1 \leq k \leq K }{ \text{max} } \left| M^{-1} \sum_{m=1}^M z_{(m,k)} \right| > \delta \right)
K \underset{ 1 \leq k \leq K }{ \text{max} } \ \mathbb{P} \left(  \left| M^{-1} \sum_{m=1}^M z_{(m,k)} \right| > \delta \right)
\\
\frac{K}{M} \underset{ 1 \leq k \leq K }{ \text{max} }  \frac{ \displaystyle \mathbb{E} \left[ \sum_{m=1}^M z_{m,k} \right]^2 }{ \displaystyle  M \delta^2 }
\end{align*}

where the second line holds from the Markov inequality. Moreover, since $K / M = o_p(1)$, a sufficient condition for the uniform convergence is $\text{max}_{ 1 \leq k \leq K} \mathbb{E} \left[ \sum_{m=1}^M z_{m,k} \right]^2 / M = O_p(1)$. Therefore, if $z_{(m,k)}$ is covariance stationary over $m$ for all $k$, then the sufficient condition becomes the absolute summability condition $\text{max}_{ 1 \leq k \leq K} \sum_{ j = 0}^{ \infty } \big| \mathbb{E} \left[ z_{m,k} z_{m+j,k} \right] \big| < \infty$.

Denote with $v_{i(m,k)} := \norm{ x_{i(m,k)} } - \mathbb{E} \left[ x_{i(m,k)} \right]$. Notice that $\text{Var} \left( v_{i(m,k)}  \right) \leq CK$ for some generic constant $C > 0$. Let $e_n := \left( nMK^2 \right)^{1 / 4}$. We have that

\begin{align*}
\mathbb{P} \left( A_2 \geq 2 \epsilon \right)
\mathbb{P} \left(   \underset{ 1 \leq k \leq K }{ \text{max} } \  \underset{ 1 \leq m \leq M }{ \text{max} } \ \left| \frac{1}{n} \sum_{i=1}^n v_{i(m,k)}   \right| \geq 2 \epsilon \right)
\\
\mathbb{P} \left(   \underset{ 1 \leq k \leq K }{ \text{max} } \  \underset{ 1 \leq m \leq M }{ \text{max} } \  \frac{1}{n} \sum_{i=1}^n \left| v_{i(m,k)}   \right| \geq 2 \epsilon \right)
\\
\mathbb{P} \left(   \underset{ 1 \leq k \leq K }{ \text{max} } \  \underset{ 1 \leq m \leq M }{ \text{max} } \  \frac{1}{n} \sum_{i=1}^n \left| v_{i(m,k)} \right| \mathbf{1} \left\{ \left| v_{i(m,k)} \right| \leq e_n \right\} \geq \epsilon \right)
\\
&+ \mathbb{P} \left(   \underset{ 1 \leq k \leq K }{ \text{max} } \  \underset{ 1 \leq m \leq M }{ \text{max} } \  \frac{1}{n} \sum_{i=1}^n \left| v_{i(m,k)} \right| \mathbf{1} \left\{ \left| v_{i(m,k)} \right| > e_n \right\} \geq \epsilon \right)
\equiv A_{21} + A_{22}.
\end{align*}

By Boole's Bernestein inequalities we have that

\begin{align*}
KM \underset{ 1 \leq k \leq K }{ \text{max} } \ \underset{ 1 \leq m \leq M }{ \text{max} } \mathbb{P} \left(   \frac{1}{n} \sum_{i=1}^n \left| v_{i(m,k)} \right| \mathbf{1} \left\{ \left| v_{i(m,k)} \right| \leq e_n \right\} \geq \epsilon \right)
\\
&\leq 2KM \text{exp} \left\{ - \frac{n \epsilon^2 }{ 2CK + 2 \epsilon e_n / 3} \right\}
\end{align*}

Some further applications of quantile time series regressions are discussed in [44].

§.§ Smoothed Quantile Regression with Large-Scale Inference

Following the framework of [62], consider a univariate response variable $y \in \mathbb{R}$ and a $p-$dimensional covariate vector, the primary goal here is to learn the effect of $\boldsymbol{x}$ on the distribution of $y$. Let $F_{y| \boldsymbol{x}}$ be the conditional distribution function of y given  $\boldsymbol{x}$. The dependence between $y$ and $\boldsymbol{x}$ is then fully characterized by the conditional quantile functions of $y$ given $\boldsymbol{x}$, denoted as $F^{-1}_{y| \boldsymbol{x}}$, for $0 < \uptau < 1$.

Consider a linear quantile regression model at a given $\uptau \in (0,1)$, that is, the $\uptau-$the conditional quantile function is

\begin{align}
F^{-1}_{y| \boldsymbol{x}} = \langle \boldsymbol{x}, \boldsymbol{\beta}_0(\uptau) \rangle
\end{align}

where $\boldsymbol{\beta}_0(\uptau) = \big( \beta_1^{*},..., \beta_p^{*} \big)^{\prime} \in \mathbb{R}^p$ is the true quantile regression coefficient.

§.§.§ Smoothed Estimation Equation and Convolution-type smoothing

Let $Q( \boldsymbol{\beta} ) = \mathbb{E} \big[ \widehat{Q}(\boldsymbol{\beta}) \big]$ be the population quantile loss function.  Under mild conditions, $Q(.)$ is twice differentiable and strongly convex in a neighbourhood of $\boldsymbol{\beta}$ with Hessian matrix such that

\begin{align}
\boldsymbol{J} := \nabla^2 Q( \boldsymbol{\beta}^{*} ) = \mathbb{E} \big[ f_{\boldsymbol{\epsilon | \boldsymbol{x} } }(0) \boldsymbol{x}  \boldsymbol{x}^{\prime} \big], \ \ \text{where} \ \ \epsilon = y - \langle \boldsymbol{x}, \boldsymbol{\beta}^{*} \rangle
\end{align}

is the random noise and $f_{\boldsymbol{\epsilon | \boldsymbol{x} } }(.)$ is the conditional density of $\epsilon$ given $\boldsymbol{x}$.

On the other hand, by the first-order condition the population parameter $\boldsymbol{0}$ satisfies the moment condition

\begin{align}
\nabla Q( \boldsymbol{\beta} ) = \mathbb{E} \big[ \uptau - \mathbf{1} \left\{ y < \boldsymbol{x}^{\prime} \boldsymbol{\beta} \right\} \big] \big|_{ \boldsymbol{\beta} = \boldsymbol{\beta}_0}
\end{align}

[see, [62]]
For a standard QR with fixed design, Theorem 2 in Belloni et al (2019) we write below:

\begin{align}
\sqrt{n} \left( \widehat{\boldsymbol{\beta}}_h - \boldsymbol{\beta}_0   \right)
\boldsymbol{J}_h^{-1} \boldsymbol{U} + \boldsymbol{r}
\end{align}

\begin{align}
\boldsymbol{U}
\frac{1}{ \sqrt{n} } \sum_{i=1}^n \big[ \uptau - \mathcal{K}_h ( - \epsilon_i ) \big] \boldsymbol{x}_i
\mathbb{E} \big[ \uptau - \mathcal{K}_h ( - \epsilon_i ) \big] \boldsymbol{x}_i
\\
\norm{ \boldsymbol{r} }_2 &= \mathcal{O}_p \left( p^{3/4} \zeta_p ( \mathsf{log} n )^{1/2} n^{- 1 / 4} \right)
\end{align}

such that $\zeta_p = \underset{ \boldsymbol{x} \in \mathcal{X} }{ \mathsf{sup} } \norm{ \boldsymbol{x} }_2$. From an asymptotic perspective, the QR estimator has the advantage of being (conditionally) pivotal asymptotically.

Moreover, the Bahadur representation can be used to establish the limiting distribution of the estimator or its functionals. We consider a fundamental statistical inference problem for testing the linear hypothesis $\mathcal{H}_0: \langle \boldsymbol{a}, \boldsymbol{b}_0 \rangle = 0$, where $\boldsymbol{a} \in \mathbb{R}^p$ is a deterministic vector that satisfies that defines a linear functional of interest. It is then natural to consider a test statistic that depends on $\sqrt{n} \langle \boldsymbol{a}, \widehat{\boldsymbol{b}}_h \rangle$. Based on the non-asymptotic result (finite sample) of the following theorem, we establish a Berry-Esseen bound for the linear projection of the conquer estimator.

Assume that the conditions in Theorem 3.2 hold. Then,

\begin{align*}
\Delta_{n,p}(h)
\underset{ x \in \mathbb{R}, \boldsymbol{a} \in \mathbb{R}^p }{ \mathsf{sup} } \ \left| \mathbb{P} \left( \sqrt{n} \sigma_h^{-1} \langle \boldsymbol{a},  \widehat{\boldsymbol{b}}_h -  \boldsymbol{b}_0 \rangle \leq x \right) - \Phi(x) \right| \leq \frac{ p + \mathsf{log}(n) }{ (nh)^{1/2} } + n^{1/2} h^2,
\end{align*}

where $\sigma_h^2 = \sigma_h^2(\boldsymbol{a}) = \boldsymbol{a}^{\prime} \boldsymbol{J}_h^{-1} \mathbb{E} \big[ \big\{ \mathcal{K}_h ( - \epsilon ) - \uptau \big\}^2 \boldsymbol{x} \boldsymbol{x}^{\prime} \big] \boldsymbol{J}_h^{-1} \boldsymbol{a}$, where $\Phi(.)$ denotes the standard normal distribution function. Moreover, it holds that

\begin{align}
\underset{ \boldsymbol{a} \in \mathbb{R}^p }{ \mathsf{sup} } \ \left|  \frac{ \sigma_h^2 (\boldsymbol{a}) }{ \boldsymbol{a}^{\prime} \boldsymbol{J}_h^{-1} \boldsymbol{\Sigma} \boldsymbol{J}_h^{-1} \boldsymbol{a}^{\prime} } - \uptau( 1 - \uptau) \right|
= \mathcal{O}(h), \ \ \text{as} \ h \to 0.
\end{align}

An advantage of convolution smoothing is that it facilitates conditional density estimation for the quantile regression process. Assume that $Q_y ( \uptau | \boldsymbol{\mathcal{X}} ) = F^{-1}_{y| \boldsymbol{\mathcal{X}} } = \langle \boldsymbol{\mathcal{X}}, \boldsymbol{\beta}_0 (\uptau) \rangle$ for all $\uptau \in (0,1)$. Under mild regularity conditions,

\begin{align}
q_y \big( \uptau | \boldsymbol{\mathcal{X}} \big)
\displaystyle \frac{ \partial  }{ \partial \uptau } Q_y ( \uptau | \boldsymbol{\mathcal{X}} )
\frac{1}{ f_{ y | \boldsymbol{\mathcal{X}} } \big( \langle \boldsymbol{\mathcal{X}}, \boldsymbol{\beta}_0 (\uptau) \rangle     \big) }
= \boldsymbol{0}.
\end{align}

In particular, the inverse conditional density function plays an important role in, for example, the study of quantile treatment effects through modelling inverse prosperity scores. Therefore, by the linear conditional quantile model assumption, we have that

\begin{align}
\displaystyle \frac{ \partial }{ \partial \uptau } Q_y ( \uptau | \boldsymbol{\mathcal{X}} )
\langle \boldsymbol{\mathcal{X}}, \frac{ \partial }{ \partial \uptau } \boldsymbol{\beta}_0 (\uptau) \rangle
\end{align}

Recall that the conquer estimator $\hat{\boldsymbol{\beta}}_h(\uptau)$ satisfies the first-order condition such that $\nabla \widehat{Q}_h \big( \widehat{\boldsymbol{\beta}}_h(\uptau) \big) = \boldsymbol{0}$.  Therefore, taking the partial derivative with respect to $\uptau$ on both sides, it follows by the chain rule that

\begin{align}
\frac{ \partial }{ \partial \uptau } \widehat{\boldsymbol{\beta}}_h(\uptau)
\bigg\{ \nabla^2 \widehat{Q}_h \big( \widehat{\boldsymbol{\beta}}_h(\uptau) \big) \bigg\}^{-1} \frac{1}{n} \sum_{i=1}^n \boldsymbol{\mathcal{X}}_i
\nonumber
\\
\left\{ \frac{1}{n} \sum_{i=1}^n K_h \big( y_i - \boldsymbol{\mathcal{X}}_i^{\prime} \widehat{\boldsymbol{\beta}}_h(\uptau) \big) \boldsymbol{\mathcal{X}}_i  \boldsymbol{\mathcal{X}}_i^{\prime}  \right\}^{-1} \frac{1}{n} \sum_{i=1}^n \boldsymbol{\mathcal{X}}_i
\end{align}

Consequently, the inverse functions $1 / f_{y_i | \boldsymbol{x}_i} \big( \boldsymbol{x}_i \boldsymbol{\beta}_0(\uptau) \big)$ can be directly estimated by $\boldsymbol{x}_i^{\prime} \frac{ \partial \widehat{\boldsymbol{\beta}}_h(\uptau) }{  \partial \uptau }$.

§ SPECIFICATION TESTING IN QUANTILE REGRESSION MODELS

§.§ Specification testing for parametric conditional quantile functional form

In this section, we follow carefully the framework proposed by [41] which corresponds to the class of consistent parametric specification tests for dynamic conditional quantiles. A recent relevant framework is proposed by [67].
The conditional moment restriction test approach considers directly the model estimates (for the corresponding model functional form we impose), which is a more flexible method allowing to consider e.g., different estimation methods (parametric versus nonparametric such as the NW estimation method or the nonparametric series estimation).

Let $W_{t-1} = \left( Y_{t-1}, Y_{t-2}, Z^{\prime}_{t-1}, Z^{\prime}_{t-2}   \right)$ be the set of variables included in the quantile regression model for estimating the CoVaR and VaR risk measures. Assuming that the conditional distribution of $Y_t$ given $W_{t-1}$ is continuous, we define the $a-$th conditional VaR of $Y_t$ given $W_{t-1}$ as the $\mathcal{F}_{t-1}-$measurable function $q_{\alpha}\left( W_{t-1} \right)$ satisfying the following probability statement

\begin{align}
\label{statement1}
\mathbb{P} \bigg(  q_{\alpha}\left( W_{t-1} \right) \big| W_{t-1} \bigg) = \alpha \ \ \textit{almost surely} \ , \alpha \in (0,1) \ \forall \ t \in \mathbb{Z}.
\end{align}

For parametric VaR inference, one assumes the existence of a parametric family of functions

\begin{align*}
\mathcal{M} = \bigg\{ m_{\alpha} \left( ., \theta \right) : \theta \in \Theta \subset \mathbb{R}^p \bigg\}
\end{align*}

and proceeds to make VaR forecasts using the model $\mathcal{M}$. Statistical inference is based on the crucial assumption that $q_{\alpha} \in \mathcal{M}$, which implies that there exists some $\theta_0 \in \Theta$ such that $m_{\alpha} \left( W_{t-1}, \theta_0 \right) = q_{\alpha} \left( W_{t-1} \right)$ almost surely. For instance, in parametric model the nuisance parameter $\theta_0$ belongs to $\Theta$, with $\Theta$ a compact set in a Euclidean space $\mathbb{R}^p$. More specifically, parametric VaR models are a commonly used methodology in the literature, since the the functional form $m_{\alpha} \left( W_{t-1}, \theta_0 \right)$ along with the parameter $\theta_0$ provides a parsimonious representation of the VaR risk measure based on the available information set of the decision maker at the particular time period.

According to [41] a direct implication of the almost surely convergence statement given by expression (<ref>) is that a parametric VaR model $m_{\alpha} \left( W_{t-1}, \theta_0 \right)$ is correctly specified if and only if the following condition holds:

\begin{align}
\label{statement2}
\mathbf{E} \big( \mathcal{E}_{t, \alpha} \left( \theta_0 \right) \big| W_{t-1}   \big) = \alpha \ \ \textit{almost surely} \ \ \text{for some} \ \theta_0 \in \Theta,
\end{align}

where we define $\mathcal{E}_{t, \alpha} \left( \theta \right) := \mathbf{1} \left\{ Y_t \leq  m_{\alpha} \left( ., \theta \right) \right\}$, with $\theta \in \Theta$ and $\textbf{1} (A)$ is the indicator function implying that $\textbf{1} \left\{ A \right\} \equiv 1$ , if the event A occurs and 0 otherwise. In other words, the conditional moment condition implying testing that the almost surely condition given by expression (<ref>) holds, which is the main purpose of using specification testing for statistical validity of conditional quantiles as the ones proposed by  [41] and [67] (see, also [32]).

§.§.§ Tests statistics and Asymptotic Theory

Here we present some examples which demonstrate the choice of the family of models required for the parametric estimation of the VaR. A suitable model is the linear quantile regression (LQR)

\begin{align}
m \big( \mathcal{I}_{t-1}, \theta_0( \alpha ) \big)
\equiv
m \big(  W_{t-1}, \theta_0( \alpha ) \big)
\end{align}

Following the framework of [41] we can obtain some insights regarding the specification testing methodology. In particular, the proposed test statistics are based on the fact that $q \in \mathcal{M}$ is characterized by the infinite set of conditional moment restrictions given as below

\begin{align}
\label{statement3}
\mathbb{E} \bigg[ \mathbf{1}\bigg\{ Y_t \leq m \big( \mathcal{I}_{t-1}, \theta_0 (\alpha) \big) \bigg\}  - \alpha \bigg| \mathcal{I}_{t-1} \bigg]
= 0 \ \ \textit{almost surely} \ \text{for} \ \ \theta_0 \in \Theta \subset \mathbb{R}^p.
\end{align}

Note that the moment condition given by expression (<ref>) is equivalent to the condition of expression (<ref>). Therefore, the test statistic and hypothesis testing of interest is

\begin{align}
\mathbb{H}_0 : \mathbb{E} \bigg[ \Psi_{\alpha} \bigg( Y_t \leq m \big( \mathcal{I}_{t-1}, \theta_0 (\alpha) \big) \bigg)  - \alpha \bigg| \mathcal{I}_{t-1} \bigg]
= 0 \ \ \textit{a.s} \ \text{for some} \ \theta_0
\end{align}

against the nonparametric alternatives given by

\begin{align}
\mathbb{H}_1: \mathbb{P} \bigg( \mathbb{E} \bigg[ \Psi_{\alpha} \bigg( Y_t \leq m \big( \mathcal{I}_{t-1}, \theta_0 (\alpha) \big) \bigg)  - \alpha \bigg| \mathcal{I}_{t-1} \bigg] \neq 0  \bigg) > 0 \ \ \textit{a.s} \ \forall \ \theta \left( \alpha \right) \in \Theta \subset \mathbb{R}^p
\end{align}

where $\Psi \left( \epsilon \right) = \mathbf{1} \left( \epsilon \leq 0 \right) - \alpha$. Therefore, to simplify notation we denote with $\Psi_{ \alpha, t } \left( \theta  \right) \equiv \Psi_{ \alpha } \big( Y_t - m \left( \mathcal{I}_{t-1}, \theta  \right)  \big)$ and $m_{t-1} \left( \theta \right) \equiv m \left( \mathcal{I}_{t-1}, \theta \right)$. Thus, under the null hypothesis, and assuming that a continuity condition for $m( . )$ holds, then $m_{t-1} \left( \theta_0 \right)$ is identified as the $\alpha-$th quantile of the conditional distribution of $Y_t$ given $\mathcal{I}_{t-1}$, for all $\alpha$. Then, to characterize $\mathbb{H}_0$ by the infinite number of unconditional moment restrictions we write

\begin{align}
\mathbb{H}_0 : \mathbb{E} \bigg[ \Psi_{\alpha,t} \left( \theta_0 \right) \text{exp} \left( i x^{\prime} \mathcal{I}_{t-1} \right) \bigg] = 0 \ \ \forall x \in \mathbb{R}^d, \ \text{for some} \ \theta_0 \in \Theta.
\end{align}

Therefore, given a sample $\left\{ \left( Y_t, \mathcal{I}_{t-1}^{\prime} \right)^{\prime}: 1 \leq t \leq n \right\}$ and a parameter value $\theta$, one can consider the following quantile-marked empirical process indexed by $x \in \mathbb{R}^d$

\begin{align}
S_n \left( x, \alpha, \theta \right) := n^{-1/2} \sum_{t=1}^n \Psi_{\alpha, t} \ \text{exp} \left( i x^{\prime} \mathcal{I}_{t-1} \right).
\end{align}

in order to construct a formal specification testing procedure based on the functional form proposed by [41].
A commonly used estimator for the unknown parameter $\theta_0$ is the quantile regression estimator, which can be obtained as the solution of the corresponding minimization problem with $\rho_{\alpha} \left( \epsilon \right) = - \Psi_{ \alpha } \left( \epsilon \right) \epsilon $.

Some key points here include the fact that the parametric approach although is considered to be a robust estimation methodology when one utilizes the correct distributional assumption, it also requires careful use of complex analysis arguments to establish the asymptotic theory of the conditional quantile specification test. On the other hand, a specification testing procedure based on a semiparametric estimation of moment conditions as in the seminal paper of  [106] could be possible but in that case additional regularity conditions are necessary for estimation and inference purposes (see, for example [135]). In particular, [67] use a direct estimation approach for the conditional expectation function estimated by series regression in order to construct a consistent conditional quantile specification test (via the classical Bierens test) based on an empirical process for a continuum of instrumented unconditional moments.

§.§.§ Asymptotic null distribution

We briefly discuss the main results of the asymptotic theory proposed by [41]. More precisely, in order to establish the limit distribution of the quantile-marked empirical process, under the null hypothesis, $\mathbb{H}_0$, one needs the following notation. Define the family of conditional distributions $F_x( y ) : = \mathbb{P} \big( Y_t \leq y | \mathcal{I}_{t-1} = x \big)$. Moreover, let

\begin{align}
\epsilon_{t, \alpha} := Y_t - q_{\alpha} \left( \mathcal{I}_{t-1} \right) \ \ \ \text{and} \ \ \ e_t \big( \theta \big) \equiv e_t \left( \theta (\alpha ) \right) := Y_t - m \big( \mathcal{I}_{t-1}, \theta(\alpha) \big)
\end{align}

where $\left\{ ( Y_t, Z_t^{\prime} )^{\prime} : t \in \mathbb{Z} \right\}$ is a strictly stationary and ergodic process. Then, under the null hypothesis, $\mathbb{H}_0$, we have that $\left\{ \Psi_{\alpha, t} \left( \theta_0  \right), \mathcal{F}_t \right\}$ is a mds  $\forall$  $\alpha \in \Pi$, such that, $\Pi \subset [0,1]$. Furthermore, it holds that the finite-dimensional distribution of $R_n$ converge to those of a multivariate normal distribution with mean a zero mean vector and variance-covariance matrix given by the covariance function as expressed below

\begin{align}
K_{ \infty } \left( v_1, v_2 \right) = ( \alpha_1 \land \alpha_2 - \alpha_1 \alpha_2 ) \mathbb{E} \big[ \text{exp} \left(  i (x_1 - x_2 ) \mathcal{I}_{0} \right) \big]
\end{align}

where $v_1 = ( x_1^{\prime}, \alpha_1 )$ and $v_2 = ( x_2^{\prime}, \alpha_1 )$ represent generic elements of $\Pi$.

The estimator $\theta_n$ satisfies that $\mathbb{P} \left( \theta_n \in \mathcal{B} \right) \to 1$, as $n \to \infty$, and the following asymptotic expansion under $\mathbb{H}_0$, uniformly in $\alpha \in \mathcal{G}$ and $\theta_0 \in \mathcal{B}$, which implies

\begin{align}
Q_n ( \alpha )
&= \sqrt{n} \left( \theta_n ( \alpha ) - \theta_0 ( \alpha ) \right)
= \frac{1}{ \sqrt{n} } \sum_{t=1}^n \mathcal{L}_{\alpha} \left( Y_{t}, \mathcal{I}_{t-1}, \theta_0 ( \alpha ) \right) + o_p(1).
\end{align}

where $l_{ \alpha }( . )$ is such that $\mathbb{E} \big[ l_{ \alpha } \big( Y_1, I_0, \theta_0 ( \alpha ) \big) \big] = 0$ and

\begin{align}
\mathcal{L}_{ \alpha } \left( \theta_0 ( \alpha ) \right) = \mathbb{E} \bigg[ l_{ \alpha } \big( Y_1, I_0, \theta_0 ( \alpha ) \big) l_{ \alpha }^{\prime} \big( Y_1, I_0, \theta_0 ( \alpha ) \big) \bigg]
\end{align}

exists and is positive definite.

Moreover, the following condition holds

\begin{align}
\mathbb{E} \bigg[ l_{ \alpha } \bigg( Y_t, I_{t-1}, \theta_0 ( \alpha ) \bigg) \Psi_{\alpha} \bigg( Y_s - m ( \mathcal{I}_{s-1}, \theta_0 ( \alpha ) ) \bigg) \bigg] = 0, \ \ \text{if} \ \ t \neq s.
\end{align}

Then, $Q_n ( \alpha )$ converges weakly to a Gaussian process $Q(.)$ with zero mean and covariance function

\begin{align}
K_Q ( \alpha_1, \alpha_2 ) = \underset{ n \to \infty }{ \text{lim} } \ \frac{1}{n} \sum_{t=1}^n \sum_{s=1}^n \ \mathbb{E} \bigg[ l_{ \alpha_1 } \big( Y_1, \mathcal{I}_{t-1}, \theta_0 ( \alpha_1 ) \big) \times l_{ \alpha_2 } \big( Y_s, \mathcal{I}_{s-1}, \theta_0 ( \alpha_2 ) \big) \bigg]
\end{align}

The estimator $\theta_n ( \alpha )$ satisfies the following asymptotic expansion under $\mathbb{H}_{1,n}$, uniformly in $\alpha$, such that

\begin{align}
\sqrt{n} \big( \theta_n ( \alpha ) - \theta_0 ( \alpha ) \big) = \xi_{ \alpha } ( \alpha ) + \frac{1}{ \sqrt{n} } \sum_{t=1}^n l_{ \alpha } \big( Y_t, \mathcal{I}_{t-1}, \theta_0 ( \alpha ) \big) + o_p( 1 ).
\end{align}

§.§.§ CvM test statistic

Denote with $\mathcal{T}_m := \left\{ \alpha_j \right\}_{j = 1}^{ \alpha}$ the points in the grid which represent a set of different quantile levels, with $\alpha_1 < ... < \alpha_m$. Let $W_{\text{exp}}$ be the $n \times n$ matrix with $w_{\text{exp}, t, s} = \text{exp} \left( -\frac{1}{2} \left| \mathcal{I}_{t-1} - \mathcal{I}_{s-1} \right|^2 \right)$ and let $\psi$ be the $n \times n$ matrix with typical elements given by $\psi_{ij} = \psi_{\alpha j } \left(  Y_i - m \left( \mathcal{I}_{i-1}, \theta_n  \right) \right)$. Hence, the CvM test statistic is computed as below

\begin{align}
\label{cvm.statistic}
\text{CvM}_n = \frac{1}{mn} \sum_{j=1}^m \psi_{.j}^{\prime} W_{\text{exp}} \psi_{.j}
\end{align}

where $\psi_{.j}$ denotes the $j-$th column of $\Psi$, with $m$ fixed points such as $\left\{ \alpha_j \right\}_{j = 1}^{ \alpha}$ are deterministic.

The CvM test proposed by [41] is constructed by first constructing the $\Psi$ matrix with the quantile residuals of the model (for a general functional form) on each column, over different quantile levels alpha. That is, we evaluate for example the quantile regression model with the quantile level being in the interval e.g., $[0.1, 0.9]$. Then, the CvM statistic given by (<ref>) which is a quadratic form functional with covariance kernel defined by $W_{\text{exp}}$, which is constructed based on scanning the information set (e.g. the predictors we include in the model). This is the unconditional moment restriction approach proposed in the literature. In particular, estimation of the CoVaR risk measure require to use a generated regressor. Testing for the correct specification of a dynamic quantile process is crucial for the robust identification of the region which represents risk measures such as the CoVaR (see, [132]). Correct specification of a conditional quantile model implies that a particular conditional moment is equal to zero. Regarding the inclusion of nearly-integrated predictors, relevant research questions include: (i). How are the particular fluctuation residual based test is affected when the predictors are generated separately with an AR(1) model which allows additional features such as different types of persistence, serial correlation, heteroscedasticity, and (ii). How does the $W_{\text{exp}}$ function changes when we allow for the AR(1) specification.

§.§ Misspecification testing in Quantile Regression

A different perspective is presented in the framework proposed by [45] which in practise it corresponds to a misspecification testing methodology in conditional quantile regression models based on a GMM estimation approach. Specifically, consider the linear quantile regression model (QR) as

\begin{align}
Q_{Y|X} ( \tau | X ) = X^{\top} \beta( \tau ),
\end{align}

where $Y \in \mathcal{Y} \subset \mathbb{R}$ is the outcome variable, $X \in \mathcal{X} \subset \mathbb{R}^K$ is a $K-$dimensional vector of covariates and $Q_{Y|X}$ is the conditional quantile function of $Y$ given $X$ and $\tau \in (0,1)$ is the fixed quantile. Thus, a quantile functional form such as the slope vector or a subset of its components can be written as below

\begin{align}
\beta(\tau) = \mathsf{g} ( \theta, \tau )
\end{align}

Moreover, by imposing the above restriction on the first $K_1$ components of $\beta$ such that

\begin{align}
Q_{Y|X} ( \tau | X ) = X_1^{\top} \mathsf{g}_1 (\theta, \tau) + X_2^{\top} \mathsf{\beta}_2 \mathsf{g}_2 (\theta, \tau)
\end{align}

where $X_1 \in \mathbb{R}^{K_1}$ are the first components of $X$ and $X_2 \in \mathbb{R}^{K_2}$ are the remaining components. Therefore, if $X = ( X_1, X_2^{\top} )^{\top}$, where $X_1$ is a scalar variable, we can compute $\widehat{\theta}$ by minimizing the weighted distance between the QR estimator $\beta_1 (\tau)$ and parametrization $\mathsf{g}_1(\theta, \tau)$ using the following objective function

\begin{align}
\int_{ \mathcal{T} } \norm{ \widehat{\beta}_1 (\tau) - \mathsf{g}_1(\theta, \tau ) }^2 \widehat{W}_1 (\tau) d \tau.
\end{align}

Then, [45] impose the following assumptions to study the limiting properties of $\widehat{\theta}_{GMM}$.

A1. The data $\left\{ Y_i, X_i \right\}_{i=1}^n$ are independent and identically distributed.

A2. $Y|X=x$ has a density with respect to the Lebesgue measure for all $x \in \mathcal{X}$, the support of $X$. Moreover, it holds that, the matrix $ J(\tau)$ is invertible, where

\begin{align}
J(\tau) = \mathbb{E} \left[ f_{Y|X} \left( X^{\top} \beta(\tau) | X \right) X X^{\top} \right]
\end{align}

First we show the uniform convergence of $\widehat{Q}_n (\theta)$ to $Q(\theta)$. Notice that

\begin{align*}
\underset{ \theta \in \Theta }{ \mathsf{sup} } \left|  \widehat{Q}_n( \theta ) - Q( \theta ) \right| &\leq \underset{ \theta \in \Theta }{ \mathsf{sup} } \ \int_{ \mathcal{T} } \left| \widehat{W}(\tau) - W(\tau) \right| \left| \widehat{\beta}(\tau) - \mathsf{g}( \theta, \tau ) \right|^2 d \tau
\\
\underset{ \theta \in \Theta }{ \mathsf{sup} } \ \int_{\mathcal{T} } \left| W(\tau) \right|
\left|  \big| \widehat{\beta}(\tau) - \mathsf{g}( \theta, \tau ) \big|^2 - \big| \beta(\tau) - \mathsf{g} (\theta, \tau ) \big|^2  \right| d \tau
\\
\underset{ \tau \in \mathcal{T} }{ \mathsf{sup} } \ \left| \widehat{W}(\tau) - W(\tau) \right| \underset{ ( \theta, \tau ) \in \Theta \times \mathcal{T} }{ \mathsf{sup} } \big| \widehat{\beta}(\tau) - \mathsf{g}( \theta, \tau ) \big|^2 \int_{\mathcal{T} } \boldsymbol{1} . d \tau
\\
\int_{ \mathcal{T} } \left| W(\tau) \right| d \tau \ \times \underset{ \tau \in \mathcal{T} }{ \mathsf{sup} }  \left| \widehat{\beta}(\tau) - \beta (\tau) \right| \left(  \underset{ \tau \in \mathcal{T} }{ \mathsf{sup} }  \left| \widehat{\beta}(\tau) - \beta (\tau) \right| + \underset{ ( \theta, \tau ) \in \Theta \times \mathcal{T} }{ \mathsf{sup} } \left| \mathsf{g} (\theta_0, \tau ) - \mathsf{g} ( \theta, \tau ) \right| \right).
\end{align*}

Next, we establish the asymptotic normality of the MD-QR estimator. In particular, by the first order condition, the estimator $\widehat{\theta}_{MD}$ satisfies the following condition

\begin{align}
\int_{ \mathcal{T} } \mathsf{g} \left( \widehat{\theta}_{MD}, \tau \right)^{\top} \widehat{W}(\tau) \left( \widehat{\beta}(\tau) - \mathsf{g} \left( \widehat{\theta}_{MD}, \tau \right) \right) d \tau = 0.
\end{align}

Moreover, using a Taylor expansion of $\mathsf{g} \left( \widehat{\theta}_{MD}, \tau \right)$ around $\theta_0$ we obtain

\begin{align*}
\int_{ \mathcal{T} } \mathsf{g}_{\theta} \left( \widehat{\theta}_{MD}, \tau \right)^{\top} \widehat{W}(\tau) \left( \widehat{\beta}(\tau) - \mathsf{g} \left( \theta_0, \tau \right) \right) d \tau
\left( \int_{ \mathcal{T} } \mathsf{g}_{\theta} \left( \widehat{\theta}_{MD}, \tau \right)^{\top} \widehat{W}(\tau) \mathsf{g}_{\theta} \left( \bar{\theta}, \tau \right) d \tau \right) \left( \widehat{\theta}_{MD} - \theta_0 \right) = 0.
\end{align*}

where $\bar{\theta}$ is a line segment between $\widehat{\theta}_{MD}$ and $\theta_0$. Therefore, we obtain that

\begin{align*}
\widehat{\theta}_{MD} - \theta_0 =  \left( \int_{ \mathcal{T} } \mathsf{g}_{\theta} \left( \widehat{\theta}_{MD}, \tau \right)^{\top} \widehat{W}(\tau) \mathsf{g}_{\theta} \left( \bar{\theta}, \tau \right) d \tau \right)^{-1} \left( \int_{ \mathcal{T} } \mathsf{g}_{\theta} \left( \widehat{\theta}_{MD}, \tau \right)^{\top} \widehat{W}(\tau) \left( \widehat{\beta}(\tau) - \mathsf{g} \left( \theta_0, \tau \right) \right) d \tau    \right)
\end{align*}

By Theorem 3 in [3], it follows that $\sqrt{n} \left( \widehat{\beta}(\tau) - \beta (\tau) \right) \overset{d}{\to} \boldsymbol{Z}_{\beta}( \tau)$, is a mean-zero Gaussian process in $\ell^{\infty} ( \mathcal{T}, \boldsymbol{R}^K )$ which is the set of bounded functions that map $\mathcal{T}$ into $\mathbb{R}^K$ with covariance kernel $\Sigma ( \tau, \tau^{\prime} )$. Therefore, the parametric rate $\sqrt{n}$ times the above term, converges in distribution to

\begin{align}
\int_{ \mathcal{T} } \mathsf{g}_{\theta} \left( \widehat{\theta}_{MD}, \tau \right)^{\top} \widehat{W}(\tau) \textcolor{blue}{ \sqrt{n} } \left( \widehat{\beta}(\tau) - \mathsf{g} \left( \theta_0, \tau \right) \right) d \tau \overset{d}{\to} \int_{\mathcal{T}} \mathsf{g}_{\theta} (\theta_0, \tau )^{\top} W(\tau) \boldsymbol{Z}_{\beta}(\tau) d \tau \equiv \boldsymbol{Z},
\end{align}

by the continuous mapping theorem for functionals. Since $\boldsymbol{Z}$ is a linear functional of $\boldsymbol{Z}_{\beta}$, a mean-zero Gaussian process, is also mean-zero and Gaussian. Its variance is equal to

\begin{align}
\mathbb{E} \big[ \boldsymbol{Z} \boldsymbol{Z}^{\top} \big] = \int_{\mathcal{T}^2 } \mathsf{g}_{\theta} (\theta_0, \tau )^{\top} W(\tau) \mathbb{E} \left[ \boldsymbol{Z}_{\beta}(\tau) \boldsymbol{Z}_{\beta}(\tau^{\prime})^{\top} \right] W(\tau^{\prime} )  \mathsf{g}_{\theta} (\theta_0, \tau ) d \tau d \tau^{\prime} =: V_{MD} (\theta_0)
\end{align}

§.§.§ Minimum Distance QR Estimator

Moreover, [45] propose a minimum distance quantile regression (MD-QR) estimator as an alternative to GMM estimators. The MD-QR is defined by minimizing the integrated weighted distance, over a subset of quantiles $\mathcal{T}$, between the standard QR estimator and the parametric model for some of the coefficients of interest. Let $\widehat{\beta}(\tau)$ denote the standard QR estimator. The MD-QR estimator of $\theta$

\begin{align}
\widehat{\theta}_{MD} = \underset{ \theta \in \Theta }{ \mathsf{arg min} } \ \int_{\mathcal{T} } \left( \widehat{\beta}(\tau) - \mathsf{g}(\theta, \tau) \right)^{\top} \widehat{W}(\tau) \left( \widehat{\beta}(\tau) - \mathsf{g}(\theta, \tau) \right) d \tau,
\end{align}

where $\widehat{W}(\tau) \in \mathbb{R}^{K \times K}$ is a weight function that converges in probability to a positive-definite valued function $W(\tau)$. Notice that if one is interested about the correct specification of the function $\mathsf{g} (.,.)$ which is used to model some of the components of $\beta(\tau)$. Since we are using the GMM objective function, hence a simple specification test can be applied.

As $n \to \infty$, $\widehat{\theta}_{MD}$ is a consistent estimator of $\theta_0$and it holds that

\begin{align}
\sqrt{n} \left( \widehat{\theta}_{MD} - \theta_0 \right) &\overset{d }{\to} \mathcal{N} \big( 0, H(\theta_0)^{-1} V_{MD}(\theta_0) H(\theta_0)^{-1} \big)
\\
H(\theta_0) &:= \int_{ \mathcal{\theta}_0 } \mathsf{g} (\theta_0, \tau )^{\top} W(\tau) \mathsf{g} (\theta_0, \tau ) d \tau
\end{align}

\begin{align}
\textcolor{blue}{ \sqrt{n} }
\begin{pmatrix}
\widehat{\beta}(\tau) - \beta(\tau)
\\
\widehat{\theta}_{MD} - \theta_0
\end{pmatrix}
\overset{d}{\to}
\begin{pmatrix}
\boldsymbol{Z}_{\beta}(\tau)
\\
\displaystyle H(\theta_0)^{-1} \int_{\mathcal{T} } \mathsf{g}_{\theta}(\theta_0, u)^{\top} W(u) \boldsymbol{Z}_{\beta}(\tau) du
\end{pmatrix},
\end{align}

and by the delta method, we have that

\begin{align}
\textcolor{blue}{ \sqrt{n} }
\begin{pmatrix}
\widehat{\beta}(\tau) - \beta(\tau)
\\
\mathsf{g} \left( \widehat{\theta}_{MD}, \tau \right)  -  \mathsf{g} \left( \theta_0, \tau \right)
\end{pmatrix}
\overset{d}{\to}
\begin{pmatrix}
\boldsymbol{Z}_{\beta}(\tau)
\\
\displaystyle \mathsf{g} \left( \theta_0, \tau \right)^{\top} H(\theta_0)^{-1} \int_{\mathcal{T} } \mathsf{g}_{\theta}(\theta_0, u)^{\top} W(u) \boldsymbol{Z}_{\beta}(\tau) du
\end{pmatrix}.
\end{align}

Therefore, we can write the normalized objective function as below

\begin{align*}
&n \widehat{Q}_n \left( \widehat{\theta}_{MD} \right)
\int_{ \mathcal{T} } \left[ \textcolor{red}{\sqrt{n}} \left( \widehat{\beta}(\tau) - \mathsf{g} \left( \widehat{\theta}_{MD}, \tau \right) \right)  \right]^{\top} \widehat{W}(\tau) \left[ \textcolor{red}{\sqrt{n}} \left( \widehat{\beta}(\tau) - \mathsf{g} \left( \widehat{\theta}_{MD}, \tau \right) \right)  \right]
\\
\int_{ \mathcal{T} } \left[ \textcolor{red}{\sqrt{n}} \left( \widehat{\beta}(\tau) - \mathsf{g} \left( \widehat{\theta}_{MD}, \tau \right) \right) - \textcolor{blue}{ \big( \beta(\tau) - \mathsf{g} (\theta_0, \tau)  \big) } \right]^{\top} \widehat{W}(\tau) \left[ \textcolor{red}{\sqrt{n}} \left( \widehat{\beta}(\tau) - \mathsf{g} \left( \widehat{\theta}_{MD}, \tau \right) \right) - \textcolor{blue}{ \big( \beta(\tau) - \mathsf{g} (\theta_0, \tau)  \big) } \right] d \tau
\end{align*}

since under correct specification it holds that $\textcolor{blue}{\beta(\tau) \equiv \mathsf{g}(\theta_0, \tau)}$ for all $\tau \in \mathcal{T}$. Therefore, by an application of the functional delta method and by the consistency of $\widehat{W}(\tau)$ for $W(\tau)$, we obtain

\begin{align*}
n \widehat{Q}_n \left( \widehat{\theta}_{MD} \right)
\int_{\mathcal{T}} \left( \boldsymbol{Z}_{\beta}(\tau) - \mathsf{g}_{\theta}(\theta_0, \tau)^{\top} H(\theta_0)^{-1} \int_{\mathcal{T}} \mathsf{g}_{\theta} (\theta_0, u )^{\top} W(u) \boldsymbol{Z}_{\beta}(u) du \right)^{\top} W(\tau)
\\
&\ \ \ \ \ \times \left( \boldsymbol{Z}_{\beta}(\tau) - \mathsf{g}_{\theta}(\theta_0, \tau)^{\top} H(\theta_0)^{-1} \int_{\mathcal{T}} \mathsf{g}_{\theta} (\theta_0, u )^{\top} W(u) \boldsymbol{Z}_{\beta}(u) du \right) d \tau
\\
&= \int_{\mathcal{T} } \widetilde{\boldsymbol{Z}}(\tau)^{\top} W(\tau) \widetilde{\boldsymbol{Z}}(\tau).
\end{align*}

Recall that correct specification means correct (dynamic) specification of the conditional mean, that is, $\mathbb{E} \big[ Y_t | X_t = x_t \big] = m_t \big( x_t, \theta_0 \big)$, for some $\theta_0 \in \Theta$. More specifically, a conditional mean test is designed to detect departures from the hypothesis $\mathbb{H}_0: \mathbb{E} \big[ Y_t | X_t  \big] = m_t \big( X_t, \theta_0 \big)$. Define with $u_t ( \theta ) \equiv Y_t - m_t ( \Theta )$ be the $1 \times K$ residual function, and let $u_t^{0} = u_t ( \theta_0 )$ be the true residuals under the null hypothesis, $\mathbb{H}_0$. Suppose that the random vector $( Y_t, X_{t1}, X_{t2}, ... , X_{tk} )$ is continuously distributed with joint probability density $f_{Y,X} (.,.)$ and conditional probability density $f_{Y|X} (.|x)$ for $Y_t$ given $X_t = x$. Then, the conditional quantile $q_{\theta} ( Y_t | X_t )$ satisfies $\displaystyle \int_{- \infty}^{ q_{\theta ( Y_t | X_t )}} f_{ Y | X } ( y | X_t ) dy - \theta = 0$ (see, [77]).

§.§ Specification testing in Quantile Cointegrating Regressions

A relevant framework is proposed by [133]. In particular, their model specification testing approach is constructed by a comparison of the nonparametric estimator of $\mathsf{g} ( x,z )$ and the corresponding parametric estimator of $\mathsf{g}_0 ( x, z ; \theta_0 )$ under the null hypothesis. The following test statistic is used

\begin{align}
R_n = \int \int \left[ \widehat{\mathsf{g}}(x,z) - \mathsf{g}_0 (x,z; \widehat{\theta}_n )      \right]^2 \pi_1 (x) \pi_2 (z) p_n(x,z) dx dz
\end{align}

where $\pi_1 (x)$ and $\pi_2 (z)$ are positive integrable weight functions and the parametric quantile estimator $\hat{\theta}_n$

\begin{align}
\hat{\theta}_n = \underset{ \theta \in \Theta }{ \mathsf{arg min} } \sum_{t=1}^n \rho_{\tau} \left[ y_t - \mathsf{g}_0 \left( x_t, z_t ; \theta \right) \right]
\end{align}

and $p_n( x, z )$ is a weighting function. Furthermore, due to the presence of nonstationary regressors as well as the kernel density estimators involved makes deriving the asymptotic theory quite challenging. Therefore, we consider an alternative test statistic to $R_n$ which is a corresponding smooth version. Recall that the quantile estimator $\widehat{\mathsf{g}}(x,z)$ of $\mathsf{g}(x,z)$ such that

\begin{align}
\widehat{\mathsf{g}}(x,z) = \mathsf{g}(x,z) + \frac{ \displaystyle \sum_{t=1}^n K_1 \left( \frac{x_t - x}{h_1} \right) K_2 \left( \frac{z_t - z}{h_2} \right) \psi_{\tau} (u_t) f^{-1} (0) }{  \displaystyle \sum_{t=1}^n K_1 \left( \frac{x_t - x}{h_1} \right) K_2 \left( \frac{z_t - z}{h_2} \right) } + o_p \left( \nu_n^{-1} \right),
\end{align}

where $\psi_{\tau} = \tau - \mathbf{1} \left\{ t < 0 \right\}$, and there exists a smooth version $\widetilde{\mathsf{g}}_0 \left( x, z; \theta \right)$ of $\mathsf{g}_0 \left( x, z; \theta \right)$ such that

\begin{align}
\widetilde{\mathsf{g}}_0 \left( x, z; \theta \right) \equiv \frac{ \displaystyle \sum_{t=1}^n K_1 \left( \frac{x_t - x}{h_1} \right) K_2 \left( \frac{z_t - z}{h_2} \right) \mathsf{g}_0 \left( x_t, z_t; \theta \right) }{  \displaystyle \sum_{t=1}^n K_1 \left( \frac{x_t - x}{h_1} \right) K_2 \left( \frac{z_t - z}{h_2} \right) } = \mathsf{g}_0 \left( x_t, z_t; \theta \right) + o_p \left( \nu_n^{-1} \right)
\end{align}

Therefore, a smoothed version of $R_n$, with  $\mathsf{g}_0 (x,z; \widehat{\theta}_n )$ replaced by $\widetilde{\mathsf{g}}_0 (x,z; \widehat{\theta}_n )$ is

\begin{align}
R_n = \int \int \left[ \widehat{\mathsf{g}}(x,z) - \widetilde{\mathsf{g}}_0 (x,z; \widehat{\theta}_n )      \right]^2 \pi_1 (x) \pi_2 (z) p_n(x,z) dx dz
\end{align}

and, if we let $p_n( x,z ) = \left\{ \sum_{t=1}^n K_1 \left( \frac{x_t - x}{h_1} \right) K_2 \left( \frac{z_t - z}{h_2} \right) \right\}^2$. Define with

\begin{align}
\mathcal{S}_n := \int_{- \infty}^{ + \infty } \int_{- \infty}^{ + \infty } \left\{  \sum_{k=1}^n \psi_{\tau} (u_k) K_1 \left( \frac{x_k - x}{ h_1 } \right) K_2 \left( \frac{z_k - z}{ h_2 } \right)  \right\}^2 \pi_1(x) \pi_2(z) dxdz,
\end{align}

Thus, $\frac{ d_n }{n h_1 h_2} \mathcal{S}_n$ converges in distribution to a local time of an Ornstein-Uhlenbeck process.

§.§ Specification testing in Nonparametric Cointegrating Regressions

In this Section, we present the main results of the framework proposed by [137] who construct a statistical methodology for parametric specification testing in cointegrating models (see, also  [70]).

§.§.§ Specification test in Nonlinear Nonstationary Cointegrating Regression

Consider the nonlinear cointegrating regression model given by

\begin{align}
y_{t+1} = f(x_t) + u_{t+1}, \ \ \ t= \left\{ 1,...,n \right\}.
\end{align}

where $u_t$ is a stationary error process, and $x_t$ is a nonstationary regressor. The testing hypothesis of interest is expressed as below

\begin{align}
\mathbb{H}_0: f(x) = f(x, \theta), \ \ \theta \in \Omega_0
\end{align}

To test the null hypothesis, we use the following kernel-smoothed test statistic

\begin{align}
S_n = \sum_{s,t=1, s \neq t}^n \hat{u}_{t+1} \hat{u}_{s+1} K  \left( \frac{x_t - x_s}{h} \right)
\end{align}

involving the parametric regression residuals $\hat{u}_{t+1} = y_{t-1} - f( x_t, \hat{\theta} )$, where $K(x)$ is a nonnegative real kernel function and $h$ the window bandwidth satisfying $h \equiv h_n \to 0$ as $n \to \infty$. Note that the difficulty for developing an asymptotic theory for $S_n$ stems from the presence of the kernel weights $K \left( (x_t - x_s ) / h \right)$. The behaviour of these weights depends on the self intersection properties of $x_t$ in the sample. Therefore, to establish asymptotics for $S_n$, we need to account for the related limit theory which involves self-intersection local time of a Gaussian process.
Let $\left\{ \epsilon_t \right\}_{ t \in \mathbb{Z} }$ be a sequence of i.i.d continuous random variables with $E \epsilon_0 = 0$ and $E \epsilon^2_0 = 1$, and with a characteristic function $\phi(t)$. Then, the regressor $x_t$

\begin{align}
x_t = \rho x_{t-1} + \eta_t, \ \ \ x_0 = 0, \ \ \rho = \left( 1 + \frac{\kappa}{n} \right), \ \ \ 1 \leq t \leq n,
\end{align}

where $\kappa$ is some constant and $\eta_t = \sum_{k=0}^{\infty} \phi_k \epsilon_{t-k}$, with $\phi \equiv \sum_{j=0}^{ \infty } \phi_j \neq 0$ and the following summability condition $\sum_{j=0}^{ \infty } j^{1+ \delta} | \phi_j | < \infty$ holds for some $\delta > 0$.

The following conditions hold:

(i). Let $\left\{ u_t, \mathcal{F}_{t} \right\}_{ t \geq 1 }$, where $\mathcal{F}_{t}$ is a sequence of increasing $\sigma-$fields which is independent of $\epsilon_j$, $j \geq t + 1$, forms a martingale difference sequence satisfying  $E \left( u^2_{t+1} | \mathcal{F}_{t} \right) \to \sigma^2 > 0$, as $t \to \infty$ and $\text{sup}_{t \geq 1} E \left( | u_{t+1}|^4 | \mathcal{F}_{t} \right) < \infty$.

(ii).$x_t$ is adapted to $\mathcal{F}_t$, and there exists a correlated vector Brownian motion $(W,V)$ such that the following joint weakly convergence holds

\begin{align}
\left( \frac{1}{\sqrt{n} } \sum_{j=1}^{\floor{nt} } \epsilon_j , \frac{1}{\sqrt{n} } \sum_{j=1}^{\floor{nt} } u_{j+1}   \right) \Rightarrow \big( W(t), V(t) \big)
\end{align}

on $\mathcal{D}[0,1]^2$ as $n \to \infty$.

§.§.§ Specification testing in Nonparametric  Predictive Regression models

These class of specification testing methodologies as in the study of [36] (among others) compare the functional forms of the parametric against the alternative hypothesis of a correctly specified nonparametric functional form (currently the literature corresponds to the conditional mean functional form but extensions to a conditional quantile functional form can potentially be possible).

Parametric Regression

Consider the OLS estimator of $\left( \mu, \gamma \right)$ of the regression model given by

\begin{align}
\label{parametric}
y_t = \mu + \gamma f \left( x_{t-1} \right) + u_t
\end{align}

where the unknown model parameters are obtained as below:

\begin{align}
\left( \hat{\mu}, \hat{\gamma} \right )
\underset{ \left( \mu ,  \gamma  \right) }{ \text{ argmin}} \sum_{t=1}^n \big[ y_t - \mu - \gamma f \left( x_{t-1} \right) \big]^2
\end{align}

where $f$ has a known nonlinear functional form. The single regressor included in the nonparametric predictive regression model is assumed to be predetermined or $\mathcal{F}_{t-1}-$measurable. Then, [36] show that if $x_t$ is stationary, the OLS estimator is asymptotically normal, whereas in the case that $x_t(n)$ exhibits strong dependence structure, then the OLS estimator has a non-standard limiting distribution. Specifically, the authors assume a linear array of the form

\begin{align}
x_t(n) = \sum_{j=0}^{t-1} \phi_j(n) v_{t-j}, \ \ \ \text{where} \ \ \ v_t = \sum_{i=0}^{\infty} c_i \xi_{t-i}
\end{align}

where $\left\{ \xi_t \right\}_{ t \in \mathbb{Z} }$ satisfies the following assumption.

Let $\left\{ \xi_t \right\}_{ t \in \mathbb{Z} }$ be a sequence of innovation terms, then

(i) $\xi_t$ is i.i.d with $\mathbf{E} ( \xi_1 ) = 0$ and Var$(\xi_1) = \sigma^2_{\xi} < \infty$.

(ii) $\xi_1$ has an absolute continuous distribution, and a characteristic function $\psi_{\xi} ( \lambda )$ that satisfies $\int_{ \mathbb{R}} | \psi_{\xi} ( \lambda )^{\theta}| d \lambda < \infty$, for some $\theta \in \mathbb{N}$.

Furthermore, based on the local-to-unity parametrization  [36] consider in their framework two distinct classes of persistence processes, that is, nearly integrated (NI) processes and mildly integrated (MI) processes. Both MI and NI processes can be defined in terms of triangular arrays

\begin{align}
x_t(n) = \left( 1 - \frac{1}{\kappa_n}  \right) x_{t-1} (n) + v_t, \ x_0(n) = 0.
\end{align}

where $v_t$ is a stationary process and $\kappa_n > 0$ with $\kappa_n < \infty$, so that the autoregressive coefficient becomes approximates to unity as $n$ grows.
Both NI and MI processes describe highly persistent autoregressive processes, which have a root in the vicinity of unity. As a matter of fact both processes have been extensively used to study the behaviour of inferential procedures under local departures from unit roots and robust inferential procedures (e.g., see [116] and [90]). The crucial difference between NI and MI processes concerns the assumed growth rate of the sequence $\kappa_n$. In particular, NI processes are defined by $\kappa_n / n \to c \neq 0$, with the consequence that $n^{-1 / 2 } x_{ \floor{ n r}}$ converges weakly to an Ornstein-Uhlenbeck procees. On the other hand, MI processes have $\kappa_n / n \to 0$, which implies that the process $x_t(n)$ moves closer to the neighbourhood of stationarity, which implies that FCLT cannot be used to derive the asymptotics of functionals of these processes [36].

Nonparametric regression

Consider the nonparametric estimation of the predictive regression

\begin{align}
y_t = m \left( x_{t-1} \right) + u_t
\end{align}

The unknown function is estimated nonparametrically via the Nadaraya-Watson (NW) estimator

\begin{align}
\hat{m} ( x )
\frac{ \displaystyle \sum_{t=2}^n K_{th} (x) y_t }{ \displaystyle \sum_{t=2}^n K_{th} (x)  }
\end{align}

and the corresponding local linear (LL) estimator is expressed as

\begin{align}
\begin{bmatrix}
\tilde{\mu} (x)
\\
\tilde{\gamma} (x)
\end{bmatrix}
\underset{ ( \mu, \gamma ) }{ \text{argmin}  } \sum_{t=2}^n \big[ y_t - \mu - \gamma \left( x_{t-1} - x \right)  \big]^2 K_{th} \left( x \right)
\end{align}

where $K_{th} (x) := K \left( \frac{x_{t-1} - x}{ h_n }  \right)$. Then, the testing hypothesis of interest is that the true regression function $m$ belongs to a certain parametric family. More specifically, the null hypothesis is formulated as

\begin{align}
\label{hypothesis}
\mathbb{H}_0: m(x) = \mu + \gamma f(x), \ \ \text{for some} \ \ ( \mu, \gamma) \in \mathbb{R}^2 \ \ \text{and all} \ \ x \in \mathbb{R}
\end{align}

where $f$ is a known function. The alternative hypothesis implies that no such $\mu$ and $\gamma$ exist.

In other words, the specification test proposed by [36] allows to test the null hypothesis of parametric fit under the presence of NI regressor, which closely resembles that of [55], who propose a studentized U-statistic formed of kernel-weighted OLS regression residuals. The specification test proposed by [36] tests the null hypothesis $\mathbb{H}_0$ by comparing the parametric OLS and the corresponding nonparametric (kernel) estimates of the $m$ function. In particular, the model specified in (<ref>) can be estimated nonparametrically at each $x$ as below

\begin{align}
\widetilde{m}_f( x) := \text{argmin} \ \underset{ \gamma \in \mathbb{R}}{ \text{min} } \sum_{t=1}^n \bigg\{  y_t  - \mu - \gamma \big[ g(x_{t-1} - g(x)  \big] \bigg\}^2 K_{th } (x)
\end{align}

Denote with $\left( \hat{\mu}, \hat{\gamma} \right)$ the OLS estimates of $\left( \mu, \gamma \right)$, then the statistical distance corresponds to a comparison between the fit provided by the parametric and local nonparametric estimates of the model via an ensemble of $t$statistics of the following form

\begin{align}
\tilde{t} ( x; \hat{\mu}, \hat{\gamma} ) := \left[  \frac{ \sum_{t=2}^n K_{th} (x)  }{  \hat{\sigma}_u^2 (x) Q_{11} } \right]^{1 / 2} \bigg[   \widetilde{m}_f( x) - \hat{\mu} - \hat{\gamma} f(x) \bigg]
\end{align}

where $\hat{\sigma}_u^2 (x): = \bigg(  \sum_{t=2}^n K_{th} (x) \bigg)^{-1} \sum_{t=2}^n \big[ y_t - \hat{\mu} - \hat{\gamma} f(x_{t-1}  \big] K_{th} (x)$. Therefore, under the null hypothesis of no misspecification both the parametric and nonparametric estimators converge to identical limits and so for each $x \in \mathbb{R}$, the following limit is obtained

\begin{align}
\tilde{t} \big( x; \hat{\mu}, \hat{\gamma} \big) :=  \tilde{t} \big( x; \mu, \gamma \big) + o_p(1) \overset{ d }{ \to } \mathcal{N} (0,1).
\end{align}

§.§.§ Specification testing in Nonlinear Cointegrating Regression with nonlinear dependence

Following [38] who consider specification testing in the context of nonlinear cointegrating regressions with nonlinear dependence. Specifically, the near-epoch dependence condition when imposed on the error term of the regression model is considered to impose a form of nonlinear dependence in time series regressions models.

(Strong Mixing) Let $\left\{ v_t \right\}$ be a sequence of random variables. Let $\mathcal{F}_s^t \equiv \sigma \left( \nu_s,...., \nu_t \right)$ be the generated $\sigma-$algebra. Define the $\alpha-$mixing coefficients such that

\begin{align}
\alpha_m = \underset{ t }{ \mathsf{sup} }  \left|  P( G \cup F ) - P (G) P(F) \right|
\end{align}

The process $\left\{ v_t \right\}$ is said to be strong mixing if $\alpha_m \to 0$ as $m \to \infty$. Moreover, if $\alpha_m \leq m^{- \alpha}$, we say that $\left\{ v_t \right\}$ is a strong mixing of size $- \alpha$.

Let $\left\{ w_t \right\}$ be a sequence of random variables with $\mathbb{E} \left[ w_t^2 \right] < \infty$ for all $t$. Then, it is said that $\left\{ w_t \right\}$ is NED on the underlying sequence $\left\{ v_t \right\}$ of size-$\alpha$ if $\phi(n)$ is of size $-\alpha$, where $\phi(n)$ is

\begin{align}
\mathsf{sup} \norm{ w_t - \mathbb{E}_{t-n}^{t+n} (w_t) }_2 \equiv \phi(n)
\end{align}

where $\mathbb{E}_{t-n}^{t+n} (w_t) = \mathbb{E} \left( w_t | \nu_{t-n},..., \nu_{t+n} \right)$ and $\norm{.}_2$ is the $L_2$ norm of a random vector.

A sequence $\left\{ w_t \right\}$ is $I(0)$ if it is NED on an underlying $\alpha-$mixing sequence $\left\{ v_t \right\}$ but the sequence $\left\{ x_t \right\}$ given by $x_t = \sum_{s=1}^t w_s$ is not NED on $\left\{ v_t \right\}$. In this case, we say that $x_t$ is $I(1)$.

Two $I(1)$ sequences $\left\{ y_t \right\}$ and $\left\{ x_t \right\}$ are (linearly) cointegrated with cointegrating vector $[ 1, - \beta ]$, if $y_t - \beta x_t$ is NED on a particular $\alpha-$mixing sequence but $y_t - \delta_{12} x_t$ is not NED for $\delta_{12} \neq \beta$.

Consider the nonlinear correction model (NEC) for the vector $X_t = \left( y_t, x_t \right)^{\prime}$, given by the following expression

\begin{align}
\Delta X_t = \Psi_1 \Delta X_{t-1} + F ( X_{t-1} ) + v_t
\end{align}

Assume that

(a) Let $v_t = \left( v_{y_t},  v_{x_t} \right)^{\prime}$ is $\alpha-$mixing of size $-s/(s-2)$ for $s > 2$.

(b) $\sum_t v_t$ is not NED on $\alpha-$mixing sequence.

In particular, the above conditions ensure that

(i) $\Delta X_t$ and $Z_t$ are simultaneously NED on the $\alpha-$mixing sequence $( v_t, u_t )$, where $u_t = v_{y,x} - \beta^{\prime} v_{x,t}$,

(ii) $X_t$ is $I(1)$.

According to [38], the above theorem provides sufficient conditions for cointegrated variables to be generated by a nonlinear error correction model. Furthermore, the fully modified OLS and FM-OLS estimation approach involves a two-step procedure in which in the first step the following econometric specification $\Delta y_t = \psi_1 \Delta x_t + \gamma z_{t-1} + v_t$, is estimated by OLS. Moreover, in the second step semiparametric corrections are made for the serial correlations of the residuals $z_t$ and for the endogeneity of the $x-$regressors. Under general conditions the fully modified estimator is asymptotically efficient. A particular case of the more general form of a nonlinear parametric cointegration is of the form $y_t = f( x_t, \beta ) + v_t$, where $x_t$ is a $( p \times 1 )$ vector of $I(1)$ regressors, $v_t$ is a zero-mean stationary error term and $f( x_t, \beta )$ a smooth function of the process $x_t$, known up to the finite-dimensional parameter vector $\beta$. Thus, in the classical asymptotic theory, the properties of estimators of $\beta$, i.e., the nonlinear least squares estimator (NLSE), depend on the specific class of functions where $f( x_t, \beta )$ belongs. The rate of convergence of the NLSE is class-specific and in some cases involves random scaling.

§ APPENDIX

§.§ Elements on Gaussian Approximation of Suprema of Empirical Processes

Following the framework of [23] who consider the problem of approximating suprema of empirical processes by a sequence of suprema of Gaussian processes. To formulate the problem, let $X_1,..., X_n$ be an i.i.d random variables taking values in a measurable space $( S, \mathbb{S} )$ with common distribution $\mathcal{P}$. Suppose there is a sequence $\mathcal{F}_n$ of classes of measurable functions $S \mapsto \mathbb{R}$, and consider the empirical process indexed by $\mathcal{F}_n$:

\begin{align}
\label{problem1}
\mathbb{G}_n f = \frac{1}{ \sqrt{n} } \sum_{i=1}^n \big( f(X_i) - \mathbb{E} \left[ f(X_i) \right]  \big), \ \ f \in \mathcal{F}_n.
\end{align}

More specifically, [23] consider the problem of approximating $Z_n = \mathsf{sup}_{f \in \mathcal{F}_n } \mathbb{G}_n f$ by a sequence of random variables $\widetilde{Z}_n$ equal in distribution to $\mathsf{sup}_{f \in \mathcal{F}_n } B_n f$, where each $B_n$ is a centered Gaussian process indexed by $\mathcal{F}_n$ with covariance function $\mathbb{E} \left[ B_n(f) B_n(g) \right] = \mathsf{Cov} \left( f(X_1), g(X_1) \right)$ for all $f,g \in \mathcal{F}_n$.
We look for conditions under which there exists a sequence of such random variables $\widetilde{Z}_n$ with

\begin{align}
\left| Z_n - \widetilde{Z}_n \right| = \mathcal{O}_{\mathbb{P} } ( r_n)
\end{align}

where $r_n \to 0$ as $n \to \infty$ is a sequence of constants.

A related but different problem is that of approximating whole empirical processes by a sequence of Gaussian processes in the sup-norm. This problem is more difficult than (<ref>). Indeed,  (<ref>) is implied if there exists a sequence of versions of $B_n$ such that

\begin{align}
\label{problem2}
\norm{ \mathbb{G}_n - B_n }_{ \mathcal{F}_n } := \underset{ f \in \mathcal{F}_n }{ \mathsf{sup} } \left| ( \mathbb{G}_n - B_n ) f    \right| = \mathcal{O}_{ \mathbb{P } } (r_n).
\end{align}

There is a large literature on the latter problem. Notably, [89] (henceforth, abbreviated as KMT) proved that $\norm{ \mathbb{G}_n - B_n }_{ \mathcal{F} } = \mathcal{O}_{a.s} \left( n^{- 1 / 2} \text{log} n \right)$. The KMT construction is a powerful tool in addressing the problem (<ref>), but when applied to general empirical processes, it typically requires strong conditions on classes of functions and distributions.

There exists $c_1, c_2 > 0$ such that for every small $\epsilon > 0$ there exists $\delta (\epsilon) > 0$ such that for all $h > 0$ and all $s \in T$, it holds that

\begin{align}
\mathbb{P} \left( \underset{ t \in T: \rho(s,t) \leq \delta ( \epsilon ) }{ \mathsf{sup} } \ U(t) \leq h e^{ c_1 \epsilon^2 } \ \bigg| \ \underset{ t \in T: \rho(s,t) \leq \delta ( \epsilon ) }{ \mathsf{inf} } \ U(t) \leq h    \right) \geq 1 - c_2 \epsilon^2.
\end{align}

Moreover, if $N_{T, \delta}$ is the minimal number of closed $\delta-$balls needed to cover $T$, then it holds that

\begin{align*}
\int_0^1 \sqrt{ \mathsf{log} N_{T, \delta ( \epsilon )  }  } < \infty.
\end{align*}

§.§ Elements on Stochastic Processes under Nonlinear Dependence

In the econometrics and statistics literature the main ways to define the dependence structure of innovation sequences in time series regression models is using martigale differences and mixing processes. However, a function of a mixing sequence, which depends on an infinite number of lags and/or leads of the sequence, is not generally mixing (see, [121]). In particular, a concept which reflects weak dependence is the near-epoch dependence. The main idea is that although $\left\{ X_t \right\}$ may not be mixing, if it depends almost entirely on the near epoch of a mixing sequence $\left\{ V_t \right\}$, it will often have properties permitting the application of limit theorems.

Let $p > 0$ and $\left\{ X_n, n \geq 1 \right\}$ will be called an $L_p-$near-epoch dependent $L_p-$NED on $\left\{ V_n, n \geq 1 \right\}$, if there exist sequence $\left\{ d_n \right\}$ and $\nu (m)$ of non-negative constants, where $\nu (m) \to 0$ as $m \to \infty$ such that for $n \geq 1$ and $m \geq 0$,

\begin{align}
\norm{ X_n - \mathbb{E}_{n-m}^{n + m} X_n }_p \leq \nu (m) d_n.
\end{align}

hen, we can $\left\{ X_n, n \geq 1 \right\}$ an $L_p-$NED sequence of size $-\lambda$ if $\nu (m)$ in Definition <ref> is of size $- \lambda$, that is, $\nu (m) = \mathcal{O} \left( m^{-\lambda-\epsilon} \right)$, for some $\epsilon > 0$.

Notice that Definition <ref> above can be adjusted accordingly to correspond to the innovation sequences of the structural cointegrating regression model. In the literature a maximal inequality on $p-$th moment, $p > 2$, under weaker dependent sizes was established. Moreover, applying this inequality, we focus on deriving a CLT for strong $L_p-$NED under the same moment conditions but weaker size requirement for dependence ([121])).

Let $p > 0$ and $\left\{ X_n, n \geq 1 \right\}$ will be called an $L_p-$near-epoch dependent $L_p-$NED on $\left\{ V_n, n \geq 1 \right\}$, if there exist sequence $\left\{ d_n \right\}$ and $\nu (m)$ of non-negative constants, where $\nu (m) \to 0$ as $m \to \infty$ such that for all $k > 0$, $n \geq 1$ and $m \geq 0$, it holds that

\begin{align*}
\norm{ \sum_{t=k+1}^{k+n} X_t  - \mathbb{E} \left[ \sum_{t=k+1}^{k+n} X_t \bigg| \mathcal{F}_{k+n+m} \right] }_p
\left\{ \left| \mathbb{E} \left[  \sum_{t=k+1}^{n+k} X_t  - \mathbb{E} \left( \sum_{t=k+1}^{k+n} X_t \bigg| \mathcal{F}_{k+n+m} \right) \right] \right| \right\}^{1/p}
\\
&\leq \nu (m) \times \left(  \sum_{j=1}^n d_{k+j}^2 \right)^{1/2}.
\end{align*}

In order to facilitate the development of the asymptotic theory in our study our aim is to impose the NED conditions given by [100] which incorporates a more general dependence structure in time series regression models than the classical $\text{i.i.d}$ assumption of error terms, to the innovation structure of the nonparametric cointegrating regression model such that

\begin{align}
\nu_p(m) = \mathbb{E} \left[ y_t - y_t^{(m)} \right] + \left\{ \mathbb{E} \left[ x_t - x_t^{(m)} \right] \right\}^p.
\end{align}

Furthermore, a fundamental result which is essential for developing the asymptotic behaviour of estimators and test statistics is to establish the weak convergence of partial sum processes of stationary sequences (i.e., the innovation sequences of the structural cointegrating regression) under the conditions of near-epoch dependence to  functionals of Brownian motion. Specifically, [28] establishes conditions for the FCLT in nonlinear and semiparametric time series processes to hold, that is, demonstrates that such sums of random variables under NED have a Gaussian limit distribution. The main idea here is that since we can prove that under the condition of NED the partial sum processes of the corresponding stochastic process weakly converges into a Brownian motion functional, then our aim is to investigate whether a similar result applies for the weakly convergence of the joint functional of partial sum processes under NED conditions. Although NED conditions can be considered as a form of $m-$dependence (e.g., see [7]), the main challenging task in our setting is that we consider cointegrating regression models with nonstationary regressors and this requires to determine the exact form auxiliary limit functionals will have.

§.§.§ Adaptation

A natural case to consider when the sequence $\left\{ X_t, \mathcal{F}_t \right\}$ is adapted, such that $\mathbb{E} \left[ X_t | \mathcal{F}_t \right] = X_t$ almost surely. This has the simple implication that the process is causal, if the progression of the sequence is identified with the passage of time, then under the adaptation assumption it is driven by present and past shocks alone. This is a natural assumption in the context of econometric time series modelling, implying that knowledge of future shocks cannot improve the prediction of $X_t$ once its history is known. In the mixingale context, it means that the norm is equal to zero for every $m \geq 0$, whereas the NED-defining inequality becomes

\begin{align}
\norm{ X_t - \mathbb{E} \left( X_t | \mathcal{F}_{t-m}^t \right) } \leq d_t \nu_m
\end{align}

Uniform integrability implies that

\begin{align}
\underset{ n,t }{ \mathsf{sup} } \ \mathbb{E} \left[  \left| \frac{X_{nt} }{ c_{nt} } \right|^p \mathbf{1}_{ \left\{  \left| \frac{X_{nt}}{ c_{nt} } \right| > M  \right\} } \right] \to 0 \ \ \ \text{as} \ \ M \to \infty.
\end{align}

Therefore, for $\varepsilon > 0$ there is a constant $B_{\varepsilon} < \infty$ such that

\begin{align}
\underset{ n,t }{ \mathsf{sup} } \  \left\{ \frac{ \norm{ X_{nt} \mathbf{1}_{ \left\{ \left| X_{nt} \right| > B_{\varepsilon} c_{nt} \right\} } }_p }{ c_{nt} } \right\} \leq \varepsilon.
\end{align}

Define with $Y_{nt} = X_{nt} \mathbf{1}_{ \left\{ \left| X_{nt} \right| \leq B_{\varepsilon} c_{nt} \right\} }$ and $Z_{nt} = X_{nt} - Y_{nt}$. Then, since $\mathbb{E} \left( X_{nt} | \mathcal{F}_{n,t-1} \right) = 0$,

\begin{align}
X_{nt} = Y_{nt} - \mathbb{E} \left[ Y_{nt} | \mathcal{F}_{n,t-1} \right] + Z_{nt} - \mathbb{E} \left[ Z_{nt} | \mathcal{F}_{n,t-1} \right].
\end{align}

By the Minkowski inequality it holds that

\begin{align}
\norm{ \sum_{t=1}^{k_n} X_{nt} }_p \leq \norm{ \sum_{t=1}^{k_n} \big( Y_{nt} - \mathbb{E} \left[  Y_{nt} | \mathcal{F}_{n,t-1} \right]  \big) }_p
\norm{ \sum_{t=1}^n \big( Z_{nt} - \mathbb{E} \left[ Z_{nt} |  \mathcal{F}_{n,t-1} \right] \big) }_p.
\end{align}

§.§.§ FCLT for Strong NED Random Variables

Let $\left\{ X_n, n \geq 1 \right\}$ be a sequence of random variables with $\mathbb{E} \left[ X_n \right] = 0$ and $\mathbb{E} \left[ \left| X_n \right|^p \right] \leq M < \infty$ , for some $p > 2$. Moreover, suppose that $\left\{ X_n, n \geq 1 \right\}$ is strong $L_p-$NED on a $\phi-$mixing sequence $\left\{ V_n, n \geq 1 \right\}$ with

\begin{align}
\phi(n) = \mathcal{O} \left( \left( \mathsf{log}(n) \right) ^{ \frac{1}{ p ( 1 + \delta/ 2 )} } \right)
\end{align}

and $\left\{ d_n \right\}$ and $\nu(m)$ satisfying

\begin{align}
\underset{ n \to \infty }{ \mathsf{lim sum  } }  \  \underset{ k \geq 0 }{ \mathsf{sup} } \ \left\{ \frac{1}{n} \sum_{j=1}^n d^2_{j+k} \right\} = B
\end{align}

and $\sigma_n^2 := \mathbb{E} \left[ S_n^2 \right] \to \infty$, as $n \to \infty$, where $S_n = \sum_{j=1}^n X_j$. Then it holds that

\begin{align}
S_n / \sigma_n \to \mathcal{N} (0,1).
\end{align}

Moreover, we consider the limit behaviour of the partial sums of $X_{nt}$ where

\begin{align}
X_n(r) = \sum_{t=1}^{ \floor{nr} } X_{nt}, \ \ \ \text{for} \ r \in [0,1],
\end{align}

where $X_{nt} = X_t / \sigma_n$.

Let $\left\{ X_n, n \in \mathbb{Z} \right\}$ be a $\phi-$mixing sequence and $X \in L_p \left( \mathcal{F}_{-\infty}^k \right)$ and $Y \in L_q \left( \mathcal{F}_{k + n}^{\infty} \right)$, with $p > 1$ and $q > 1$. Then,

\begin{align}
\big| \mathbb{E} \left( XY \right) - \mathbb{E} (X) \mathbb{E} (Y)  \big| \leq 2 \phi (n)^{1/p} \norm{X}_p \norm{Y}_q
\end{align}

Let $\left\{ X_n, n \geq 1 \right\}$ and assume that

\begin{align}
\psi ( r ) := \underset{ n \to \infty }{ \mathsf{lim} } \mathbb{E} \left[ X_n(r) \right]^2
\end{align}

exists for all $r \in [0,1]$.
Then $X_n ( . ) \to X(.)$, where $X(.)$ is a Gaussian process having almost surely continuous sample paths and independent increments (see, [121]).

Then, the main step for proving the particular result is to obtain

\begin{align}
\norm{  \sum_{ s = k_n( \theta_{j-1} ) + 1 }^{ k_n ( \theta_j + \delta ) } X_{ns} }_2 \leq   CD \big( ... \big) \sigma_n^{-2}
\end{align}

Therefore, by the Cauchy-Schwarz inequality and the definition of strong $L_p-$NED, we apply the Remark 2 such that we can bound the following expression

\begin{align}
I_2 \leq \left\{ \mathbb{E} \bigg[ Z_{ m_{i-1} } ( n_i ) \right] - \mathbb{E}_{ m_{i-1} + 1 - m }^{ m_{i-1} + n_i + m } \left[  Z_{ m_{i-1} } ( n_i ) \right]  \bigg\} \times \left\{ \mathbb{E} \left[  Z_{ m_{j-1}  (\delta ) } \big( n_j ( \delta ) \big) \right]^2 \right\}^{1/2}
\end{align}

Some remarks based on the limit results established above are:

(i). The basic idea is to consider that we approximate the pair $\left\{ \left( y_t, x_t \right)  \right\}$ by the $\alpha-$mixing process $\left\{ \left( y_t^{(m)}, x_t^{(m)} \right) \right\}$. Notice that this approximation even though is not changing the nonstationary properties as these are captured by the structural regression model, it does affect the asymptotic theory when considering the local linear fitting estimation approach. In particular, we consider that the structural cointegrating regression is generated by an $\alpha-$mixing $m-$dependent pair of stochastic processes under NED conditions (see,  [76]).

(ii). Thus, in order to demonstrate that the joint weak convergence holds for these functionals under NED one needs to extend the result given by Theorem 1 in the paper of Jin and Zhengyan, which corresponds only to one coordinate, to the case in which we have joint weak convergence of these functionals for both coordinates. Notice that these two coordinates do not necessarily have to correspond to both random variables of the $m-$dependent pair of random variables $( y_t, x_t)$.

Roughly speaking, a structural cointegrating regression under conditions of fading memory; since the NED condition is assumed to be a form of weak dependence such that the underline stochastic processes have fading long-term memory due to less interactions (e.g., see [127]) and weakening dependence which can affect the degree of cointegration in the long-run. Exactly for this reason, the local linear fitting method (see, [100], [95]) is a suitable approach for estimation and inference purposes within such a setting.

Consider the process below

\begin{align}
h_t^m = \sum_{j= -m}^m \theta_j V_{t-j}
\end{align}

Then, the function above is different from  $E( X_t | \mathcal{F}_{t-m}^{t+m} )$ unless $\left\{ V_t \right\}$ is an independent process, but is also an $L_p-$approximator for $X_t$ since

\begin{align}
\norm{ X_t - h_t^m }_p = \norm{  \sum_{j=m+1}^{\infty} \big( \theta_j V_{t-j} + \theta_{-j} V_{t+j} \big) }_p \leq d_t \nu_m
\end{align}

Therefore, in contrast to the property of "long memory"; which implies that shocks have persistent effect resulting to a slow rate of decay of the autocorrelation function with lag or by a divergence of the power spectrum near the origin - NED conditions its a form of weak dependence with memory that fades away. Therefore, in order to estimate the cointegrating regression model under NED we consider the local linear fitting approach. The main challenge however is to employ correctly adjusted invariance principles for NED sequences to the corresponding brownian motion functionals.

The stationary process $\left\{ \right( Y_t, \boldsymbol{X}_t \left) \right\}_{t=1}^n$ is said to be $L_{p}-$NED on $\left\{ \varepsilon_t \right\}$ if

\begin{align}
\nu_{p}(m) = \mathbb{E} \left|  Y_t - Y_t^{(m)} \right|^{p}  +   \mathbb{E} \norm{ \boldsymbol{X}_t - \boldsymbol{X}_t^{(m)} }^{p}  \to 0,
\end{align}

as $m \to \infty$. Then, the $\nu_{p}(m)$ are called the stability coefficients of order $p$ for the joint stochastic process $\left\{ \left( Y_t, \boldsymbol{X}_t \right) \right\}_{t=1}^n$ under the presence of Near-epoch dependence (NED).

Let $\mathcal{F}_{- \infty}^n$ and $\mathcal{F}_{\infty}^{n + k}$ be the $\sigma-$fields generated by $\left\{ \varepsilon_t, t \leq n \right\}$ and $\left\{ \varepsilon_t, t > n + k \right\}$, respectively. Moreover, define with

\begin{align}
\alpha (k) = \underset{ A \in \mathcal{F}_{- \infty}^n, B \in \mathcal{F}_{\infty}^{n + k} }{ \mathsf{sup} } \ \left|  \mathbb{P} \left( AB \right) - \mathbb{P} \left(A \right) \mathbb{P} \left( B \right) \right| \to 0, \ \ \text{as} \ n \to \infty.
\end{align}

Then, the stationary sequence $\left\{ \varepsilon_t, t = 0,  \pm 1,...  \right\}$ is said to be $\alpha-$mixing and the $\alpha(k)$ is termed mixing coefficient. Thus, $\left\{ \right( Y_t^{(m)}, \boldsymbol{X}_t^{(m)} \left) \right\}$ is an $\alpha-$mixing process with mixing coefficient

\begin{align}
\alpha_m(k) \leq
\begin{cases}
\alpha (k - m) &, k \geq m + 1,
\\
1 & k \geq m.
\end{cases}
\end{align}

In other words NED is a property of the mapping rather of the random variables themselves (see, [29] and [122]). The NED condition fully determines the restrictions on the memory of the observed process. In other words, the econometrician can assume independent shocks, although this is principally for tractability of asymptotic theory analysis derivations. Notice that the class of NED functions of mixing processes includes the cases in which infinite distributed lag functions of strong-mixing processes are strong mixing  such as ARMA processes with mixing innovations. Moreover, a strong approximation result for heterogenous martingale difference arrays is given by  [28] and [31]. The particular structure generalizes the independent data assumption towards a general setting with serial dependency and heterogeneity.

Furthermore, as per the discussion of [91], when the stochastic process $\left\{ X_t \right\}$ is a mixingale and $X_t$ is measurable with respect to the filtration $\mathcal{F}_t$, we have an adapted mixingale and the condition of $\norm{ X_t - \mathbb{E} \left( X_t | \mathcal{F}_{t+m}  \right) }_2$ holds. The mixingale condition implies that the term $\norm{ \mathbb{E} \left( X_t | \mathcal{F}_{t+m} \right) }_2$ converges to zero as $m \to \infty$, which is a form of asymptotic martigale difference property.

Special cases of mixingale processes include independent sequences, $\phi-$, $\rho-$ and $\alpha-$mixing processes, finite and certain infinite order moving average processes and sequences of NED functions of infinite histories of mixing processes. Therefore, an invariance principle for $\left\{ X_n \right\}$ to hold the degree of dependence among $X_n$'s is controlled. In other words, the controlling sequence $\psi_k$ has to decay to zero sufficiently fact. Let $S_{n,k} = \sum_{j=1}^k X_{n,j}$.

Then, $W_n(t)$ as defined below corresponds to a standard Brownian motion.

\begin{align}
W_{n}(t) = \frac{ S_{n, \floor{nt} } }{  \sqrt{n} \sigma }
\end{align}

Similarly, an invariance principle for triangular arrays of mixingales holds.

Let $\left\{ \left( X_{n,j}, \mathcal{F}_{n,j} \right) \right\}$ be an array of mixingales satisfying the NED conditions with $\left\{  d_m \right\}$ of size $- \frac{1}{2}$. If $\left\{ X_{n,j}, j = 1,2,...  \right\}$ is uniformly integrable, then $\left\{ W_n \right\}$ is tight and

\begin{align}
\norm{ \mathbb{E}_{k-m} \left[  \frac{ \left( S_{n,k+d} - S_{n,k} \right)^2 }{ d} - \sigma^2    \right] }_1 \to 0
\end{align}

as $\mathsf{min} \left( m, k, n, d \right) \to \infty$, then $W_n$ converges weakly to $W$ on $D[0,1]$.