Estimating Conditional Value-at-Risk with Nonstationary Quantile Predictive Regression Models

Estimating Conditional Value-at-Risk with Nonstationary Quantile Predictive Regression Models

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Introduction

Risk monitoring of financial time series as well as portfolio performance measures when obtaining optimal portfolios can lead to inaccurate results in the case of non-Gaussian (asymmetric) return distributions. Thus, in order to better measure downside risk in a non-Gaussian setting usually the risk measures of Value-at-Risk and Expected-Shortfall are used to quantify uncertainty (see, lin2023portfolio). Moreover, estimating risk measures such as the Conditional-Value-at-Risk, ($\mathsf{CoVaR}$), entails a generated regressor problem, which occurs when an explanatory variable is constructed using structural econometric methods. We propose an estimation method in nonstationary time series regression models with generated covariates, which has not seen much attention in the literature so far and thus requires the development of our own asymptotic theory.

Inferential procedures based on generated regressors are studied in linear and conditional quantile models as in chen2021quantile (see, pagan1984econometric, doran1983relative, hoffman1987two, oxley1993econometric, dufour2001finite and chen2023standard), although in the case of risk measures such as $\mathsf{VaR}$ and $\mathsf{CoVaR}$ a conditional quantile specification is required. The conventional approach for estimating these risk measures is to employ stationary quantile regressions (as in AB2016covar and hardle2016tenet). Several studies in the literature consider the estimation of risk measures using stationary time series environments either with parametric (e.g., see he2020inference) or semiparametric approaches (e.g., see, wang2016conditional). In this study, we extend the econometric specifications for the joint estimation of the pair $( \mathsf{VaR}, \mathsf{CoVaR} )$ as discussed in the study of katsouris2023statistical (see also katsouris2021optimal, katsouris2023quantile), to the case of nonstationary quantile predictive regression models (see, lee2016predictive), which implies using a local-to-unity parametrization for modelling the unknown form of persistence.

Specifically, katsouris2023statistical considers the statistical estimation for covariance-type matrices with tail forecasts using node-specific quantile predictive regressions, extending the framework of katsouris2021optimal who discusses identification and estimation issues of conditional quantile risk measures. However, both of these studies operate under the assumption of time series stationarity, thereby relying on OLS-based estimators (see, patton2019dynamic). Thus, a suitable estimation approach under the presence of both a generated covariates and persistent regressors, for the joint estimation of the risk measure pair of $( \mathsf{VaR}, \mathsf{CoVaR} )$, remains an open problem. Several studies in the literature develop estimation and inference methods in nonstationary predictive regressionss robust to the unknown form of persistence, with a notable approach being the IVX filtration of phillipsmagdal2009econometric. Motivated by the aforementioned issues, our research objective is to study the large-sample theory of a doubly IVX corrected estimator in nonstationary quantile predictive regression for the purpose of estimating risk measures. To the best of our knowledge, our approach that incorporates the time series properties of regressors (nonstationarity) when estimating risk measures in a low dimensional setting (small number of regressors relative to the sample size), is a novel contribution to the literature.

As a result, when developing robust estimation and inference methods in econometric environments with generated regressors, it is crucial to obtain analytical and computational feasible expressions for the asymptotic variance of these potentially semiparametric estimators. In particular, rilstone1996nonparametric investigates the aspect of generated covariate in a nonparameteric stationary time series environment. Moreover, hahn2013asymptotic study the asymptotic distribution of three-step estimators of a finite-dimensional parameter vector where the second step consists of one or more nonparametric regressions on a regressor that is estimated in the first step. Specifically, the authors using the path-derivative method of newey1994asymptotic, derive the contribution of the first-step estimator to the influence function is used to investigate the dual role that the first-step estimator plays in the second-step nonparametric regression. Moreover, bhattacharya2020quantile consider the aspects of estimation and inference in stationary quantile regression models with generated covariates. Recently, ghosh2022doubly develop a framework for semiparametric estimation where a doubly robust estimating function for a low-dimensional parameter is available, depending on two working models. Therefore, our framework proposes a computationally tractable two-step algorithm and an asymptotic theory analysis for the doubly IVX corrected estimator (which is found to robustify inference to the unknown form of persistence\footnote{The unknown form of persistence in time series data is modelled via the the local-to-unity parametrization introduced by the seminal study of phillips1987time, phillips1987towards who established a unified framework for unit root and nearly integrated processes in time series regression models.}), when estimating risk measures under the assumption of statistical elicitability (see, fissler2016higher).

Our practical two stage estimation procedure permits to obtain estimates for the forecasts of the pair $( \mathsf{VaR}, \mathsf{CoVaR} )$ based on the econometric specifications of AB2016covar (see also hardle2016tenet), but under the assumption of time series nonstationarity using the local-to-unity parametrization to capture nearly nonstationary regressors as in lee2016predictive and fan2019predictive. During the first-stage procedure using the nonstationary quantile predictive regression model (QPR) and the local-to-unity parametrized regressors, we obtain quantile dependent model estimates that correspond to the one-period ahead forecasted $\mathsf{VaR}$ risk measure. During the second stage estimation procedure, the generated regressor that corresponds to the $\mathsf{VaR}$ is employed as an additional covariate to a different QPR model which is employed to obtain the one-period ahead forecasted $\mathsf{CoVaR}$ risk measure. Furthermore, because of the well-known endogeneity problem (see, cai2014testing) that appears due to the correlation between the two innovation sequences of the predictive regression and the autoregressive equation, parameter estimation is achieved using the IVX filtration (see, phillipsmagdal2009econometric and kostakis2015Robust), which is robust to the unknown persistence properties of regressors. In particular, the IVX estimation approach is implemented to both stages of the procedure, thereby ensuring identification and joint estimation of the risk measures of $\mathsf{VaR}$ and $\mathsf{CoVaR}$\footnote{Although statistical comparisons between the assumption of stationary regressors against regressors generated via a LUR parametrization can be constructed more formally using goodness-of-fit testing (e.g., see chan2013marked); we leave these considerations for future research. On the other hand, more recently hoga2023estimation propose a framework for the estimation risk in extreme systemic risk forecasts for stationary time series data.}.

We focus on the IVX estimators in each estimation stage, since it is well-known to be robust to the abstract degree of persistence (e.g., see lee2016predictive). As a result, the doubly corrected IVX estimator (which is the IVX estimator obtained from the second step estimation procedure), is the main parameter of interest in terms of statistical inference and asymptotic theory. Specifically, we establish the asymptotic properties of the doubly corrected IVX estimator which verify the mixed Gaussianity property of the limiting distribution regardless of the degree of persistence in both estimation stages. Moreover, we consider a suitable correction to the expression of the asymptotic variance-covariance matrix in the second estimation step, which is adjusted in order to account for the first-step estimation error that produces the generated regressor under nonstationarity.

Regarding the statistical properties of parameters obtained from the first-stage procedure those are employed for the $\mathsf{VaR}$ risk measure forecast and in practice parameter consistency is not violated (see, AB2016covar) regardless of the LUR parametrization. In addition, the particular generated regressor that corresponds to fitted values of a stationary regressand can be treated as the stationary component of the vector of regressors used during the second stage procedure ($\mathsf{CoVaR}$). However, the generated regressors are included in the QPR model of the second estimation step with some sampling error, which introduces additional noise into the asymptotic variance-covariance matrix of the model coefficients. Therefore, the sampling error from the first stage contaminates the second stage estimation procedure (see, chen2021quantile). Furthermore, during the second-stage ($\mathsf{CoVaR}$ estimation), model estimates are obtained using the IVX estimator in which case the generated regressor is exogenously determined, this implies that it can only be assumed to be stationary (although is modelled as the same lag as the regressand). As a result, the usual way of calculating the QR variance-covariance matrix fails to account for the additional source of error and an adjustment is required. To sum-up, the generated regressor is a proxy of the estimated $\mathsf{CoVaR}$ when the nonstationary properties of regressors are included, which implies that this proxy is treated as a nonstochastic stationary regressor in a quantile predictive regression model with nonstationary regressors. Thus, accounting for the presence of generated regressor bias, inference is robustify against both the abstract degree of persistence that these nonstationary regressors exhibit as well as to the presence of any generated regressor bias.

Recall that, the conditional quantile function $F^{-1}_{ \textcolor{blue}{ y_{(j)t} } } \left( \uptau \big| \boldsymbol{x}_{t-1} , \textcolor{blue}{ y_{(i)t} } \right)$ is the $\mathsf{VaR}_{ (j) t} \left( \uptau \right)$ conditional on $\boldsymbol{x}_{t-1}$ and $\textcolor{blue}{ y_{(i)t} }$, where $F(\cdot)$ is the conditional distribution function and $F^{-1} (\cdot)$ is the conditional quantile function of the underline distribution. Then, by conditioning on $\textcolor{blue}{ y_{(i)t} } =\mathsf{VaR}_{ (i) t} \left( \uptau \right)$, we obtain the $\mathsf{CoVaR}_{ (j,i) t} \left( \uptau \right)$ defined as (see, AB2016covar and hardle2016tenet)

Equivalently, it holds that

Thus, we focus on estimating systemic risk measures conditioned on time series nonstationarity (persistent data) using the LUR parametrization. The nonstationary quantile regression specifications are employed for forecasting the pair $( \mathsf{VaR}, \mathsf{CoVaR} )$, based on the definitions of AB2016covar and the statistical elicitability property of these risk measures (see, patton2019dynamic, frongillo2021elicitation and fissler2016higher). Although, the loss function which characterizes M-estimators allows to obtain model estimates, thereby directly affecting the robustness of forecasts, we do not consider the estimation risk in systemic risk forecasts (see, dimitriadis2022characterizing, xu2022prediction, hoga2023estimation and katsouris2023statistical).

Main contributions

The use of endogenously generated instruments (IVX filter\footnote{The ideal IV has to meet two requirements: it is correlated with the endogenous regressor via an explainable and validated relationship (relevance relationship), yet uncorrelated with the structural error (exclusion restriction).}) is employed for the purpose of filtering out the unknown form of persistence in the regressors of each quantile predictive regression used in the two-stage estimation procedure.

Regarding identification issues of the risk measure pair $( \mathsf{VaR}, \mathsf{CoVaR})$ are discussed in several studies such as AB2016covar, hardle2016tenet, patton2019dynamic. Moreover, fissler2023backtesting mention that although there is no real-valued scoring function associated with risk measure pairs and propose to consider a $\mathbb{R}^2-$valued scoring function in the closely related context of forecast evaluation. Nevertheless, our estimation method does not rely on the score function estimation approach, thereby the statistical elicitability property of the risk measure pair facilitates econometric estimation. We also avoid using copula correction methods (see, yang2022addressing among others), since within the setting of nonstationary predictive regression models, the IVX filtration of phillipsmagdal2009econometric has been found to be robust to abstract degree of persistence. In addition to the proposed two-stage algorithm for the joint estimation of the risk measure pair, as well as asymptotic theory analysis of estimators and their covariance matrices, our framework allows to study statistical inference issues which we leave for future research\footnote{The related statistical inference aspects we refer to here is the subvector inference problem which is commonly examined in the linear instrumental variables regression model. In that case, our objective is to develop a powerful test such as the asymptotic null rejection probability is controlled uniformly over a parameter space. The inference problem consists of making inference on the coefficients of structural models, but in the context of systemic risk modelling. Such statistical problems usually give attention to aspects such as developing robust methods in the presence of weak instruments or powerful subvector testing in IV regressions under conditional heteroscedasticity; we give attention on the simultaneous modelling of risk measures which involve the use of a generated covariate. Therefore, drawing inference on individual coefficients of the parameter vector of the structural equation for estimating the $\mathsf{CoVaR}$ risk measure, motivates us to propose a subvector testing approach which is based on marginal conditional regression estimation and testing with nonstationary regressors (see, hsiao1997statistical and de2001nonlinear). }.

We propose a practical two-stage estimation procedure\footnote{We study a broad class of semiparametric problems, where a doubly robust estimating function $h_{\uptau}( U; \theta, \theta_1, \theta_2 )$, where $\uptau \in (0,1)$ for the parameter of interest $\theta$ is available as follows. Consider two working models denoted with $g_1( \boldsymbol{x}_{t-1} ; \theta_1 )$ and $g_{2|1} ( \boldsymbol{x}_{t-1} ; \theta_2 )$ for unknown functions $y_{1t} = g_1^*( \boldsymbol{x}_{t-1} )$ and $y_{2t} = g_2^*( \boldsymbol{x}_{t-1} )$. Furthermore, the estimating function (in our case it represents the tail estimates for the $\mathsf{CoVaR}$ risk measure), $h_{\uptau}( . ; . )$ is unbiased such that $\mathbb{E} \bigg[ h_{\uptau} \bigg( U; \theta, \theta_1, \theta_2 \bigg) \bigg] = 0$, when $\theta$ is set to the true value $\theta^{*}$, and either $\theta_1$ or $\theta_2$, but not necessarily both, is set to the true value $\theta_1^{*}$ or $\theta_2^{*}$ defined respectively such that $g_1^*( \boldsymbol{x}_{t-1} ) \equiv g_1( \boldsymbol{x}_{t-1} )$ or $g_2^*( \boldsymbol{x}_{t-1} ) \equiv g_2( \boldsymbol{x}_{t-1} )$ if model $g_1 ( \boldsymbol{x}_{t-1} ; \theta_1 )$ or $g_2 ( \boldsymbol{x}_{t-1} ; \theta_2 )$ is correctly specified. In general, doubly robust estimation using the estimating function $h_{\uptau}( U; \theta, \theta_1, \theta_2 )$ consists of two stages: some estimators $\left( \hat{\theta}_1, \hat{\theta}_2 \right)$ are first defined, and then $\hat{\theta}$ is defined by solving the estimating equation $\tilde{\mathbb{E} } \big[ h_{\uptau} \big( U; \theta, \hat{\theta}_1, \hat{\theta}_2 \big) \big] = 0$, where $\tilde{\mathbb{E} } [ \cdot ]$ denotes a sample average.} for estimating the model parameters of the quantile predictive regression with nonstationary regressors. Then, the second step employs the generated regressors in the form of fitted values, along with a different set of nonstationary regressors to a quantile predictive regression model. As a result, the doubly IVX corrected estimator, is the IVX-based estimator corresponding to the second step estimation procedure and is the main parameter of interest in terms of statistical inference and limit theory. Since the generated regressor is assumed to be estimated using a different functional form as well as nonstationary regressors, then any dependence structure between the generated regressors is captured by the variance covariance matrix of the second-step estimation procedure. Within our econometric environment the proposed two-stage (semiparametric) estimation method, depends on possibly nonstationary regressors, therefore relevant parameter restrictions on $\vartheta_1$ and $\vartheta_2$ (see, also katsouris2021optimal, katsouris2023statistical), which correspond to the "working" models, are necessary to ensure robust statistical inference.

For notational convenience consider the following two functional forms:

where $\xi ( \boldsymbol{x}_{t-1} )$ is $p \times 1$ vector of known functions such as $\xi ( \boldsymbol{x}_{t-1} ) = \big( 1, \boldsymbol{x}_{t-1}^{\top} \big)$, and $\boldsymbol{\beta}_1$ and $\boldsymbol{\beta}_2$ are vectors of unknown coefficients, which correspond to joint estimation as the two models in AB2016covar. Under time series nonstationarity, the two parameter vectors are estimated via the IVX filtration of phillipsmagdal2009econometric. Our implementation is, however, a plug-in estimation approach and a novel methodology to existing approaches discussed in the literature. Therefore, we consider the asymptotic theory analysis for $\big( \hat{\boldsymbol{\beta}}_1, \hat{\boldsymbol{\beta}}_2 \big)$ and $\widehat{\vartheta} \big( \hat{\boldsymbol{\beta}}_1, \hat{\boldsymbol{\beta}}_2 \big)$, where $\boldsymbol{\beta}_1$ and $\boldsymbol{\beta}_2$ are fixed-dimensional as the sample size $n$ grows. In particular, we assume that $\hat{\boldsymbol{\beta}}_1$ is consistent for $\boldsymbol{\beta}^{*}_1$ when the first working model is correctly specified, and $\hat{\boldsymbol{\beta}}_2$ is consistent for $\boldsymbol{\beta}^{*}_2$ if the second working model is correctly specified (absence of functional form misspecification).

Assume the existence of any two pair of random variables $\left\{ y_{1t}, y_{2t} \right\}$ and a set of predictors $\boldsymbol{x}_t$. Another way of viewing our econometric environment is to consider the example of two-countries (or firms across two countries) where we have a balanced set of $p$ nonstationary macroeconomic variables for both economies (or firms), respectively. In particular, let $\boldsymbol{x}_{1t}$ be the $( p \times 1)$ vector of domestic variables and $\boldsymbol{x}_{2t}$ be the $( p \times 1)$ vector of the same variables for the foreign country where each of these possible nonstationary processes have the following LUR parametrizations

Moreover, additionally to these nonstationary regressors that are equation-specific, a generated covariate is employed as proxy for the risk measure of $\mathsf{VaR}$ that represents a quantile risk event of the macroeconomic conditions of the foreign economy when predicting the current time conditions of the home economy. Our estimation approach combines both the IVX estimator of phillipsmagdal2009econometric (to filter out persistence present in the nonstationary regressors $\boldsymbol{x}_{1t}$ and $\boldsymbol{x}_{2t}$), as well as inspired by the residual augmentation approach of demetrescu2020residual we self-instrument the proxy variable (generated covariate) under time series nonstationarity (see, also demetrescu2014enhancing and breitung2015instrumental). As a result, when developing the asymptotic theory of the (long-run) quantile-dependent parameter we use a self-instrumentation approach for the generated regressors that correspond to the systemic risk proxy of each economy (as the joint estimation approach of AB2016covar and hardle2016tenet).

Estimation of the $\mathsf{CoVaR}$ risk measure is conditioned on the estimated $\mathsf{VaR}$, and relies on a quantile predictive regression, with both nearly integrated regressors and a generated regressor (proxy for estimated $\mathsf{VaR}$ under nonstationarity). Therefore, in the presence of this additional covariate, there are more regressors than available instruments. To overcome this issue, demetrescu2014enhancing propose a framework for inference in an augmented predictive regression. In that case, the author propose to instrument that additional covariate with the same process itself. We follow a similar approach in our setting although with major necessary modifications.

Estimation Algorithm

Our proposed two-stage estimation procedure is summarized as below:

Recall that the conditional quantile can be consistently estimated based on the quantile-dependent model parameter $\widehat{ \boldsymbol{\beta} }_2^{\star} (\uptau)$ such that $\widehat{Q}_{\uptau} ( \uptau | \boldsymbol{x}_{2t-1}, \widehat{y}^{\mathsf{VaR}}_{1t} ) = \tilde{\boldsymbol{x}}_{2t-1}^{\prime } \widehat{ \boldsymbol{\beta} }_2^{\star} (\uptau)$ (see, koenker1978regression). In a similar spirit the corresponding IVX estimators are obtained.

Therefore, one uses the estimates of $\boldsymbol{\theta}_j$, denoted by $\widehat{\boldsymbol{\theta}}_j$, in the first step, to obtain the generated regressor $\widehat{\boldsymbol{x}}_t$. Our aim is to demonstrate that our proposed methodology achieve rate and model double robustness simultaneously, provided that the parameter $\boldsymbol{\theta}$ satisfies certain regularity conditions. In the current study the doubly IVX corrected estimation approach corresponds to conditional quantile specification forms for nonstationary data.

The remained of this paper is organized as follows. In Section (ref), we introduce the econometric model and the main assumptions of our framework. In Section (ref) we present our proposed doubly corrected estimation method in the case of conditional mean functional forms while in Section (ref) discusses the conditional quantile case which corresponds to the estimation of the risk measures under time series nonstationarity. We conclude in Section (ref).

Throughout the paper, for any vector $\boldsymbol{a}$, we denote with $\left\lVert \boldsymbol{a} \right\rVert_1 = \sum_j | a_j |$ and with $\left\lVert \boldsymbol{a} \right\rVert_2 = \sqrt{ \sum_j | a_j |^2 }$. For any real arbitrary matrix $\boldsymbol{A}$, its Frobenius norm is defined by $\left\lVert \boldsymbol{A} \right\rVert = \sqrt{ \mathsf{trace} ( \boldsymbol{A}^{\prime} \boldsymbol{A} ) }$. The spectral norm of $\boldsymbol{A} $ is denoted by $\left\lVert \boldsymbol{A} \right\rVert_2$, such that, $\left\lVert \boldsymbol{A} \right\rVert_2 = \sqrt{ \lambda_1 ( \boldsymbol{A}^{\prime} \boldsymbol{A} ) }$, is maximum column sum norm is $\left\lVert \boldsymbol{A} \right\rVert_1 = \left\lVert \boldsymbol{A} \right\rVert_1 = \mathsf{max}_{ 1 \leq j \leq n} \sum_{i = 1}^m | A_{ij} |$ and its maximum row sum norm is $\left\lVert \boldsymbol{A} \right\rVert_{ \infty } = \left\lVert \boldsymbol{A} \right\rVert_{ \infty } = \mathsf{max}_{ 1 \leq i \leq n} \sum_{i = 1}^m | A_{ij} |$. The operator $\overset{P}{\to}$ denotes convergence in probability, and $\overset{D}{\to}$ denotes convergence in distribution.

Econometric Model and Assumptions

Main Assumptions

Consider the linear predictive regression model formulated as below

where $\boldsymbol{x}_{t} \in \mathbb{R}^p$ is a $p-$dimensional vector and $\boldsymbol{R}_n = \left( \boldsymbol{I}_p - \frac{ \boldsymbol{C}_p }{n^{\gamma}} \right)$ with $\gamma = 1$.

We begin by discussing the predictive regression with a conditional mean functional form, but for the purpose of estimating the risk measures pair $( \mathsf{VaR}, \mathsf{CoVaR} )$ a conditional quantile specification form is necessary. The asymptotic behaviour of estimators under moderate deviations from unity (e.g., on the explosive side) is beyond the scope of our study (e.g., see, phillips2007limit, buchmann2007asymptotic). Several studies consider estimation methods for unified inference in predictive regression models such as magdalinos2020least and magdalinos2022uniform. Furthermore, our estimation method differs in several ways to various respectful school of thoughts and approaches presented in the literature such as the two-step estimator of francq2015risk (see also beutner2024residual and francq2023local); the research perspective we follow for the development of our econometric environment is concisely explained in Section 1 (see, AB2016covar and hardle2016tenet).

Borrowing from the literature in the case when $\sigma_{uv} \neq 0$ (which is the case we consider), the exact OLS bias of $\hat{\beta}$ is computed from the predictive regression model, $\mathbb{E} \left[ \hat{\beta} - \beta \right] = \delta \mathbb{E} \left[ \hat{\rho} - \rho \right]$ where $\hat{\rho}$ is the OLS estimate of $\rho$ and $\delta := \sigma_{uv} / \sigma_v^2$ is the slope coefficient of in a regression of $u_t$ on $v_t$. Furthermore, since $\hat{\rho}$ is known to be downward biased in small-samples, and $( u_t, v_t )^{\prime}$ are typically strongly negatively contemporaneously correlated, the autoregressive OLS bias feeds into the small-sample distribution of $\hat{\beta}$ causing over-rejections of the null hypothesis of no predictability, $H_0: \beta = 0$. Moreover, notice even with the use of a possible finite-sample-bias correction on the OLS estimate, this reduces the noncentrality of the limiting distribution of the OLS t-statistic, but the distribution remains nonstandard in the near-integrated case.

Under Assumption (ref) a functional central limit theorem holds (phillips1992asymptotics)

which satisfies the following stochastic differential equation

Relevant notation can be found in several studies in the literature such as dickey1981likelihood, jansson2006optimal, mikusheva2012one and demetrescu2020residual among others. Furthermore, to avoid confusion regarding the properties of estimators based on a conditional mean functional form vis-a-vis a conditional quantile functional form, we include the relevant assumptions in this section. A similar approach of comparing estimation approaches across these two different modelling methods is presented by liu2023robust, although in our framework this is might be practical given the gap in the literature on the relevant econometric issues we aim to tackle. The identification of the risk measures is based on definitions (ref)-(ref).

Then, the conditional quantile function of $y_t$ denoted with $\mathsf{Q}_{y_t} \left( \uptau | \mathcal{F}_{t-1} \right)$, replaces the conditional mean function of the predictive regression which implies the following model specification

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

where the following probability bound holds uniformly (see for example kulperger2005high)

IVX filtration

A solution to the endogeneity problem discussed above, which provides robust statistical inference, is the IVX filtration proposed by phillipsmagdal2009econometric. The IVX instrumentation ensures instrument relevance while controlling for the abstract degree of persistence (see, kostakis2015Robust) and is constructed as below

The IVX filtration for $x_t$, is chosen so that $z_t$ corresponds to a mildly integrated process when the predictor $x_t$ is nearly integrated. Therefore, the IVX estimator of $\beta$ is found to have a slower convergence rate than the conventional OLS estimator under near integration, such that $n^{ \frac{1 + \gamma_z}{2} }$ (see, also katsouris2023quantile and duffy2021estimation). The asymptotic theory of the IVX estimator can be found to be mixed Gaussian irrespectively of the degree of endogeneity implied by $\delta$, leading to standard inference in $t $ and Wald tests. Moreover, it can be proved that under low persistence, the IVX estimator is asymptotically equivalent to the OLS procedure. Our main focus in this paper is the estimation of the risk measures of $\mathsf{VaR}$ and $\mathsf{CoVaR}$ using nearly integrated regressors, which is a novel aspect in the literature as conventional estimation approaches mainly consider that macroeconomic variables are stationary. Specifically, we start by considering the estimation of the $\mathsf{VaR}$ with a nearly integrated regressor. A vast literature has previously proposed robust methodologies for estimating the parameter of interest using the OLS approach and found evidence of significant size distortions when the model includes high persistence regressors.

Two-Stage Estimation Procedure

First-Stage Procedure

During the first stage of our procedure we obtain a consistent estimator of the parameter vector. We focus on the IVX estimator which is found to be robust to the abstract degree of persistence (nearly integrated regressors). The econometric model is the linear predictive regression as below

Denote with $\boldsymbol{\beta} (\uptau) = \left(\beta_{01}(\uptau), \boldsymbol{\beta}_{11}^{\prime} (\uptau) \right)^{\prime}$ and the corresponding IVX estimator with $\boldsymbol{\beta}^{ivx} (\uptau)$. The asymptotic behaviour of the IVX estimator that corresponds to the linear predictive regression model under abstract degree of regressors persistence is studied by kostakis2015Robust while the asymptotic properties of the IVX estimator for the quantile predictive regression model is studied by lee2016predictive. Since when estimating the risk measure pair $( \mathsf{VaR}, \mathsf{CoVaR})$, we use the conditional quantile distribution, i.e., to capture the effect of $\boldsymbol{x}_{t-1}$ on the conditional quantile of $\boldsymbol{y}_{t}$; however with the additional assumption of possibly nonstationary regressors (see, also koenker2004unit and xiao2009quantile), then estimation relies on the nonstationary quantile predictive regression.

Second-Stage Procedure

During the second stage of our procedure we consider a consistent estimator of the parameter vector for the set of regressors which includes the nonstationary regressors as well as the generated regressor from the first stage procedure. More precisely, the generated regressor in our study corresponds to the fitted values of the predictive regression model based on the IVX estimator obtained in the first stage procedure.

where $\boldsymbol{C}_{jp} = \mathsf{diag} \left\{ c_{j1},..., c_{jp} \right\}$ and $j \in \left\{ 1, 2 \right\}$. Define with $\widetilde{\boldsymbol{\beta}} (\uptau) = \big( \beta_{02}(\uptau), \boldsymbol{\beta}_{12}^{\prime}(\uptau), \delta (\uptau) \big)^{\prime}$ the parameter vector of the second stage procedure and the corresponding IVX estimator with $\widetilde{\boldsymbol{\beta}}^{ivx}(\uptau)$.

Specifically, we observe that this extended parameter vector includes both the nearly integrated regressors as well as the generated regressor from the first stage estimation step.

In this case, we need to develop the asymptotic distribution theory for both the OLS and IVX estimators when the generated regressor is included in the set of regressors in the second stage predictive regression model that corresponds to the econometric specification of the CoVaR risk measure. A modified IVX estimator is necessary to be developed in order to account for the presence of the particular effect in the setting of the quantile predictive regression models. The nuisance coefficient of persistence is defined such that $c_i > 0$ and is either $\gamma = 1$, which corresponds to near unit root regressors or $\gamma \in (0,1)$ that corresponds to mildly integrated regressors.

Therefore, the focus of this paper is the implementation of the above econometric environment and the development of the corresponding asymptotic theory in the case of the quantile predictive regression model. Consider the piecewise derivative of the loss function which is defined as below $ \psi_{\uptau} \left( u \right) = \uptau - \mathbf{1} \left( u < 0 \right)$. Then the innovation sequence of the quantile predictive regression model, $u_t ( \uptau ) = u_t - F_{u}^{-1} ( \uptau )$, and $F_{u}^{-1} ( \uptau )$ is the unconditional $\uptau$-quantile of $u_t$, where $\uptau \in (0,1)$ is a fixed quantile (see, lee2016predictive, fan2019predictive and katsouris2022asymptotic). Thus, the $\boldsymbol{\beta}^{ivx}$ estimator is obtained from the first stage procedure using the IVX filter\footnote{The reason that we apply the IVX instrumentation procedure only to the estimate from the first stage quantile predictive regression model rather to the corresponding generated covariate is to ensure that those fitted values can preserve their definition as the estimated Value-at-Risk but in our case adjusted based on the presence of persistent predictors and corrected accordingly using the IVX methodology.} for the model coefficients based on the nonstationary regressors which implies that the generated regressor is defined as

Then, to conduct inference we need to obtain a consistent estimator for the covariance matrix of $\tilde{ \boldsymbol{\beta} }_2^{ivx}$ using the usual "sandwich" formula given below (see, demetrescu2020residual).

Doubly Corrected Estimation Method: Conditional Mean Case

Residual Augmented IVX Estimation Example

We begin our asymptotic theory analysis by considering the framework for the residual augmented IVX estimation methodology previously examined in the literature (see, demetrescu2020residual). The IVX instrumentation provides a methodology such that the asymptotic distribution of the IVX-Wald based test statistic converges to a nuisance-parameter free distribution. Consider the feasible predictive regression defined with the following form

where $y^{\mu}_{t}$ and $x_{t-1}^{\mu}$ denote the demeaned variates, following the notation employed in several studies in the literature. Moreover, $\hat{v}_t$ represents a generated covariate which in our study corresponds to the estimated $\mathsf{VaR}$ under nonstationarity.

Generated Regressor in Nonstationary Linear Predictive Regression

Recall that the above expressions correspond to the model estimates of linear predictive regression model with conditional mean functional form, while the risk measures require to use the quantile predictive regression framework. In any case, to keep track of our derivations we begin our analysis using the linear predictive regression model and the framework of residual-based augmentation of demetrescu2020residual, but we replace the regressor $\hat{v}_t$ with the generated dependent variable, $\widehat{ \bar{y} }_{1,t}$, which is used as a proxy for $\mathsf{VaR}$ under time series nonstationarity

In compact form we have that

where $\boldsymbol{X}_{t-1} = \big( \bar{\boldsymbol{x}}_{2t-1} \ \widehat{ \bar{y} }_{1,t} \big)$ and $\boldsymbol{\Gamma} = \big( \boldsymbol{\beta} \ \delta \big)$. Moreover, define with $\boldsymbol{Z}_{t-1} = \big( \boldsymbol{z}_{2t-1} \ \widehat{ \bar{y} }_{1,t} \big)$ where $\boldsymbol{z}_{t-1}$ is the IVX instrument for $\boldsymbol{x}_{2t-1}$ and the generated regressor is self-instrumented.

The generated covariate $\widehat{ \bar{y} }_{1,t}$ corresponds to the estimated $\mathsf{VaR}$ under nonstationarity. In practise, these nonstationary predictors can be different than the nonstationary predictors corresponding to the regressand $\widehat{y}_{2,t}$ due to the proposed dependence structure. However, for simplicity we can also consider the case in which we use the same nonstationary regressors for both predictive regression models. In this section, we obtain derivations for the linear predictive regression model. However, when estimating the risk measures of $\mathsf{VaR}$ and $\mathsf{CoVaR}$, the nonstationary quantile predictive regression models are used which implies that all parameters are quantile-dependent for a fixed quantile $\uptau \in (0,1)$. Therefore, we are interested for the consistent estimation of the $\beta_2$ estimator (e.g., via IVX) as well as constructing hypothesis testing for: (i) only $\beta$ parameter (e.g., no predictability), (ii) only $\delta$ parameter (e.g., no presence of generated regressor) and (iii) linear restrictions on all the parameters of the predictive regression model that includes both nonstationary regressors and the generated regressor (e.g., univariate generated regressor).

The covariance estimator for $\tilde{ \boldsymbol{\beta} }_2^{ivx}$ is expressed as below

Therefore, the just-identified model for the combined IVX estimator of $\boldsymbol{\Gamma}$ is given by

Furthermore, notice that the nonstationary regressors $x_{t-1}$ and the generated regressor are orthogonal since the generated regressor can be considered as exogenous variable to the particular model, then we can compute the generated regressor augmented estimator and test statistic in two steps. Notice the constructed test statistic refers to the IVX estimator. Therefore, we focus on deriving the asymptotic distribution for the IVX estimator under the presence of the generated regressor in the quantile predictive regression model. The estimation procedure is implemented into two-stages, and our asymptotic theory analysis demonstrates that although the limit converges into a nonstandard distribution the particular limit is nuisance-parameter free.

The estimation step needs to account for two sources of errors, that is, the usaul estimation error in obtaining a consistent estimator that corresponds to the IVX instrumentation of the nearly integrated regressors, and the second source of error is the sampling error in generating the forecast for the VaR from the first stage of the process. More specifically, this implies that using a Bahadur representation of the QR-IVX estimator we need to determine the precise stochastic order of the remainder term when the generated regressor is included in the conditional quantile specification of the model. Although when the parameter vector from the first stage is obtained via the IVX filtration, it has the usual convergence rate that the IVX estimator has, an estimation error carries in the second-stage estimation procedure, which requires us to consider a suitable correction to the overall variance due to the presence of the generated regressor.

Moreover, we expect that the stochastic equicontinuity property to still hold regardless of the plug-in estimation approach and the presence of time series nonstationarity.

Estimation Procedure in Nonstationary Linear Predictive Regression Case

Recall the linear predictive regression model (for node 1) is given by $y_{1,t} = \boldsymbol{\beta}_1^{\prime} \boldsymbol{x}_{t-1} + u_{1,t}$ and $\boldsymbol{x}_t = \boldsymbol{R} \boldsymbol{x}_{t-1} + \boldsymbol{v}_{1,t}$. Then we estimate the following model

Moreover, by OLS-regressing $\bar{y}_{2,t}$ on $\widehat{ \bar{y} }_{1,t}$ such that $\bar{y}_{2,t} = \theta \widehat{ \bar{y} }_{1,t} + \eta_t$, where $\eta_t$ is a disturbance term, we construct the transformed dependent variable $\tilde{y}_{2,t}$ given by the following expression

Therefore, the IVX estimator during the second-stage estimation procedure ($\mathsf{CoVaR}$) is obtained by the following expression:

By rearranging we obtain that

The first term of the above expression is simplified as below

which holds when we have identical nonstationary regressors such that $\bar{\boldsymbol{x}}_{1,t-1} \equiv \bar{\boldsymbol{x}}_{2,t-1}$. Then,

Recall that $ \widehat{ \boldsymbol{\beta}}_1^{ivx}$ corresponds to the IVX estimator of the quantile predictive regression model for the $\mathsf{VaR}$ risk measure. From Lemma B4 of kostakis2015Robust it holds that

Therefore, by incorporating the convergence rate we have that

Thus, for the LUR regressor case we have that the second term of expression (ref) converges to

To see this, from the Appendix of KMS expression (27) gives that

where the stochastic covariance matrices are defined as below

Notice that by kostakis2015Robust we have that $\boldsymbol{\Psi}_{uu} = \left( \boldsymbol{\Omega}_{uu} + \displaystyle \int_0^1 \boldsymbol{J}^{\mu}_c d J^{\prime}_c \right)$ since we consider local unit root regressors. Therefore, similar to Theorem A (i) of KMS it holds that

which is a mixed Gaussian random variate since the covariance matrix is a function of the OU process. Next we investigate the asymptotic behaviour of the first term of expression (ref):

Specifically, assuming that node 1 and 2 have a set of nonstationary regressors that are generated from non-identical stochastic processes (i.e., not identical regressors), then it follows that

Regarding the asymptotic behaviour of the first term of the expression we have that the IVX estimator from the quantile predictive regression that corresponds to the VaR risk measure is

Moreover, the number of nonstationary regressors for both econometric specifications is the same although these two sets of regressors possibly have different nuisance parameters of persistence. The limiting distributions for their IVX estimators have the same dimensions such that

Rearranging the above expressions, we obtain the following limit result

In this paper we are not proposing any novel solution to the endogeneity issue in quantile predictive regressions models. We employ a well-investigated method in the literature that tackles the endogeneity problem, meaning it produces weak convergence results into standard asymptotic distributions (see, lee2016predictive, fan2019predictive). Alternative approaches that tackle the endogeneity problem and produce uniform valid inference regardless of the unknown persistence properties are proposed in the frameworks of cai2023new and liu2023unified. In contrast, our study provides a general framework and establish the asymptotic properties for quantile predictive regression models with a generated regressor, which is particularly useful when jointly estimating the risk measure pair of $( \mathsf{VaR}, \mathsf{CoVaR} )$ under the presence of time series nonstationarity.

In fact, regardless of the estimation method employed to robustify the quantile-based model parameters to the unknown persistence, when estimating the $\mathsf{CoVaR}$ as per AB2016covar and hardle2016tenet, under time series nonstationarity (LUR parametrization), similar econometric issues need to be tackled to ensure relevant statistical properties still hold.

Asymptotic Properties of Stage-2 Estimator

Next, consider for a moment the limiting distribution of $\left( \widehat{\delta}^{ols} - \delta \right)$ where $\bar{y}_{2,t} = \delta \widehat{ y }_{1,t} + \eta_t$. Therefore, the OLS estimator of $ \delta$ is given by $\widehat{ \delta }^{ols} = \displaystyle \left( \sum_{t=1}^n \widehat{y}^{2} _{1,t} \right)^{-1} \left( \sum_{t=1}^n \bar{y}_{2,t} \widehat{y}_{1,t} \right)$, where $\widehat{y}_{1,t} := \widehat{\boldsymbol{\beta}}_1^{\prime ivx} \boldsymbol{x}_{1,t-1}$ which implies that it can be expressed as below

However, $\bar{y}_{2,t} = \boldsymbol{\beta}_2^{\prime} \bar{\boldsymbol{x}}_{2,t-1} + \delta \widehat{y}_{1,t} + u_{2,t}$ which implies that

where

Therefore, we obtain that

Notice that the second term above becomes:

Thus, we have that

Furthermore, since the nonstationary regressors of node 1 are not correlated with the nonstationary regressors of node 2, then the term $\underset{ n \to \infty }{ \mathsf{plim} } \left( \sum_{t=1}^n u_{2,t} \boldsymbol{x}_{1,t-1}^{\prime} \right) = 0$, which implies that

Since the term $\left( \sum_{t=1}^n \widehat{y}^{2} _{1,t} \right)$ is always positive then we assume that $\underset{ n \to \infty }{ \mathsf{plim} } \left( \sum_{t=1}^n \widehat{y}^{2} _{1,t} \right)^{-1} = \mathcal{K}$ where $\mathcal{K}$ is some positive constant. Therefore, putting all related expressions together we obtain

The second term above corresponds to a squared mixed Gaussian distribution which gives a form of a $\chi^2$ distribution. Thus, the limiting distribution of (ref) is a linear combination of a $\chi^2$ distribution and a mixed Gaussian distribution, which is a Generalized $\chi^2$ distribution. Thus the $y_t$ inherits the properties of $x_t$, through a cointegrating relation, especially our goal is to have a stationary innovation sequence such that $u_t \sim I(0)$. Under the assumption of a consistent estimator for $\hat{\delta}^{ols}$ then we have convergence in probability to zero such that $\left( \hat{\delta}^{ols} - \delta \right) \overset{p}{\to} 0$. Thus, the first term of the expression asymptotically tends to zero (negligible) which implies that the second term converges into a mixed Gaussian distribution.

Doubly Corrected Estimation Method: Conditional Quantile Case

Estimation based on a Doubly IVX Corrected Procedure

In this section we consider the double corrected conditional quantile estimation methodology which is the main focus of our research study. The derivations presented in the previous sections were useful to shed light on some challenges we have to overcome to develop a robust framework in the proposed setting of quantile predictive regression models. In terms of estimation approach we focus on the QR estimators under nonstationarity which are adjusted using the IVX filtration. During the first stage of the estimation procedure in the same spirit as several studies in the literature related to the joint estimation of the risk measures of $( \mathsf{VaR}, \mathsf{CoVaR} )$ (see, AB2016covar, hardle2016tenet and patton2019dynamic among others), the $\mathsf{VaR}$ is estimated given a fixed quantile $\tau \in (0,1)$ level but using information on the nonstationary properties of regressors as given by the quantile predictive regression system below

Next, we focus on the required preliminary theory for establishing the consistency and asymptotic mixed Gaussianity of the doubly IVX corrected estimator $\boldsymbol{\beta}^{ivx} (\uptau)$. Before proceeding with our asymptotic theory analysis, it is worth mentioning some related studies on the aspect of generated regressor in quantile regressions. In particular, in comparison to the framework of bhattacharya2020quantile (see, also chen2021quantile), our generated covariate represents a proxy for $\mathsf{VaR}$ under time series nonstationarity. During the second-stage procedure this risk proxy is augmented within a nonstationary quantile predictive regression along with other nonstationary regressors. In addition, to ensure robustness to the abstract degree of persistence we employ the IVX estimator rather than the OLS estimator thereby avoiding the necessity to use the control function approach as a method for correcting size distortions under the null hypothesis of no predictability due to the presence of the nuisance parameter of persistence. Moreover, similar to the illustrative case of the doubly corrected estimation for the conditional mean case we present results for bivariate systems of quantile predictive regression models.

Our econometric environment can be extended to the multivariate case. In that case, we can develop asymptotic theory and estimation techniques for inference under two large-sample regimes such that $(a)$ with increasing time sample size, $n \to \infty$, and fixed network dimension, denoted by $m$, and $(b)$ with $m \to \infty$ and $n_m \to \infty$, where the temporal size depends on $m$. In other words, the case in which both indices tend to infinity has certain challenges when deriving the asymptotic properties or such complex tail dependent processes. To the best of our knowledge our study tackles the problem of general asymptotic inference with increasing dimension network time series models under nonstationarity which is a novel aspect to the literature. Suitable Bahadur-based representations for establishing asymptotic theory results are considered in several studies (see, portnoy2012nearly and wang2016conditional). Within our setting we have the additional complexity that the quantile-based estimators in each stage has to be robustify against the unknown persistence properties that appear due to the LUR parametrization of regressors (e.g., see lee2016predictive).

In relation to technical conditions, the notion of stochastic equicontinuity can be employed in nonstationary time series environments. These conditions are not uncommon in the cointegrating regression literature such as the restricted cointegrated regression model of saikkonen1995problems and the constrained least squares estimator proposed by nagaraj1991estimation (see, moon2002minimum). Furthermore, as pointed out by saikkonen1995problems, the classical stochastic equicontinuity conditions can be too strong in cointegrated regression models. In particular, due to the stationarity of the generated regressor we shall assume that the parameter space in the nonstationary quantile predictive regression model in the second-stage procedure corresponds to a short-run parameter while the coefficient of the nonstationary regressors corresponds to the long-run parameters of the model. This can be done by partitioning $\theta = ( \theta_1, \theta_2 )$ and considering a standardization matrix such that

where $\boldsymbol{D}_{1n}$ is $( k_1 \times k_1 )$ nonstochastic diagonal matrix whose diagonal elements are positive and increasing functions of $n$. Moreover, these SE conditions are considered to be sufficient conditions for establishing the weak convergence of such estimators while necessary conditions imply the inverse information matrix has eigenvalues bounded away from zero in probability. Therefore, in the case of integrated and cointegrated variables, saikkonen1995problems show that the standardized Hessian matrix $\displaystyle \boldsymbol{D}_{ \boldsymbol{\theta} n }^{-1} \frac{ \partial^2 \bar{Q}_n( \boldsymbol{\theta}_0 ) }{ \partial \boldsymbol{\theta} \partial \boldsymbol{\theta}^{\prime} } \boldsymbol{D}_{ \boldsymbol{\theta} n }^{-1}$ converges weakly and due to the strict exogeneity assumption, the weak limit is block diagonal between the parameters $\boldsymbol{\theta}_1$, $\psi$ and $\alpha$. The next step would be to show that the standardized Hessian satisfies an appropriate stochastic equicontinuity condition, that along with a suitable consistency result, makes it possible to obtain the limiting distribution. Roughly speaking, OLS-based residuals from the first-stage procedure depend on the nuisance parameter of persistence thus stochastic equicontinuity arguments need to be extended to a suitable probability space that accounts for dynamic misspecification (see, kasparis2012dynamic).

In a similar spirit as in the illustrative example above in which the an invertible representation of the ARDL exists, we consider that our proposed econometric specification which corresponds to an augmented quantile predictive regression model with generated regressors, could be modified accordingly using equivalent type of representations. We leave some derivations for a future study. On the other hand, we should emphasize again that the generated covariates we consider in our setting are allows to be possibly related to different observable variables in different functional forms. Nevertheless, these GR equations can be estimated separately (i.e., estimation of the main diagonal of the risk matrix in katsouris2021optimal), and thus any dependence between the GRs can be captured by the variance covariance matrix (this separation of the proxy risk measures of $\mathsf{VaR}$ is also reflected in the novel two-step estimation approach proposed by katsouris2021optimal).

Misspecification Analysis of $\mathsf{VaR}$-$\mathsf{CoVaR}$ risk matrix

In order to obtain some further insights in the implications of the theoretical aspects discussed in this paper, we briefly discuss the extension of the framework proposed by katsouris2023statistical into the econometric environment of this study. Suppose that $\boldsymbol{y}_{it}$ is generated by the true adjacency matrix $A$ (e.g., consider the novel $\mathsf{VaR}-\Delta\mathsf{CoVaR}$ risk matrix proposed by katsouris2021optimal as the network adjacency matrix). Related studies from the literature of network regression estimation and inference include zhu2017network, zhu2019network and zhu2020grouped among others).

Roughly speaking, the consistency result for the population counterpart of the risk matrix might not be held when the adjacency matrix $A$ is misspecified to be $A^{\star} = \left( a_{ij}^{\star} \right)$. Denote with $W^{\star}$ to be the row-normalized $A^{\star}$ and $X_{it}^{\star}$ to be the matrix of model regressors. Then, the estimator is

Define with

The conditions associated with the misspecified coefficients are listed below:

Denote with

Moreover, assuming that the amount of misspecification has a bounded convergence rate such that $d \left( W^{\star}, W \right) = o_p \left( \sqrt{N / T} \right)$, then we have that $\left( \hat{\boldsymbol{\beta}}^{\star}( \uptau ) - \boldsymbol{\beta}^{\star} ( \uptau ) \right) = o_p \left( \sqrt{N / T} \right)$. Consequently, it can be shown that when the misspecification amount is under control, that is, $d \left( W^{\star}, W \right) = \mathcal{O}_p \left( \sqrt{N / T} \right)$, the resulting estimator is still $\sqrt{NT}-$consistent.

A process $W_{NT} \left( \uptau \right)$ is said to be tight if and only if for any $\delta > 0$ there exists a compact set $E$ with

Moreover, we denote with $\zeta_{it} \left( \uptau, \dot{ \uptau} \right) := \psi_{ \uptau } \left( V_{it \uptau } \right) - \psi_{ \dot{ \uptau} } \left( V_{it \dot{ \uptau} } \right) $ for two quantile levels, $\uptau$ and $\dot{ \uptau}$. Then, since $\mathbb{E} \left[ \zeta_{it} \left( \uptau, \dot{ \uptau} \right) | \mathcal{F}_{t-1} \right] = 0$, using the Cauchy's inequality we can derive an appropriate bound

Therefore, it holds that

is non-zero only if (a) $i_1 = i_2, t_1 = t_2$ and $i_3 = i_4 \neq i_1, t_3 = t_4 \neq t_1$ or (b) $i_1 = i_2 = i_3 = i_4$ and $t_1 = t_2 = t_3 = t_4$. Then, it can be shown that

Notice that we have that $\mathbb{E} \left( \eta^{\top} X_{it} \right)^2 = \mathcal{O}_p(1)$ and $\mathbb{E} \left( \eta^{\top} X_{it} \right)^4 = \mathcal{O}_p(1)$. Moreover, it can be easily verified that $\mathbb{E} \left[ \zeta_{it}^2 \left( \uptau_1, \uptau_2 \right) \right] \leq \uptau_1 - \uptau_2$ and $\mathbb{E} \left[ \zeta_{it}^4 \left( \uptau_1, \uptau_2 \right) \right] \leq \uptau_1 - \uptau_2$. By combining the aforementioned results together we obtain

for some positive constant $C$. We can then conclude that the stochastic quantity $\xi_1 ( \uptau)$ converge weakly to a $( q + 1 )-$dimensional Brownian Bridge.

Conclusion

We study estimation and inference for nonstationary quantile predictive regression models when a generated covariate is required for the purposed of estimating pairs of risk measures under time series nontationarity. In We discuss the main estimation steps of the proposed modeling methodology and provide an asymptotic theory analysis for the associated quantile-dependent model estimators under regressors nonstationarity which is captured via the LUR parametrization.

We propose a two-stage estimation procedure, where the generated regressor that corresponds to the one-step ahead prediction of $\mathsf{VaR}$ based on persistent data is generated in the first stage of the estimation procedure. In the second-stage of the estimation procedure the two-stage estimator is employed to construct the one-step ahead prediction of $\mathsf{CoVaR}$ based on the estimated $\mathsf{VaR}$ and another set of persistent data not necessarily identical to the first-stage estimation procedure. We investigate the asymptotic properties of the two-stage estimator based on quantile predictive regression models. Moreover, we show that accounting for the presence of generated regressors when obtaining forecasts of the pair $( \mathsf{VaR}, \mathsf{CoVaR})$, based on the proposed two-stage estimation procedure, results in improved statistical testing after adjusting the variance-covariance matrix to account for the estimation error in the first-stage estimation procedure.

In a further study we consider the econometric identification and estimation of a novel SUR representation that corresponds to the system-specific quantile predictive regressions as proposed by katsouris2021optimal, katsouris2023quantile, katsouris2023statistical. Specifically, modelling tail interdependencies in a network using the SUR representation provides efficiency gains and permits to implement relevant statistical inference methods especially when regressors are assumed to be generated from a LUR process.

\paragraph{Conflicts of interest}

The author declares that there are no known conflicts of interest.

\paragraph{Data availability}

No data was used for the research described in this article.

\paragraph{Acknowledgements}

This article is based on the first chapter of my doctoral thesis which was submitted to the University of Southampton (see, katsouris2021optimal, katsouris2023statistical, katsouris2023quantile). I wish to thank Professor Dr. Jose Olmo, Professor Dr. Tassos Magdalinos and Dr. Jayeeta Bhattacharya from the Department of Economics, University of Southampton for helpful conversations and comments that significantly improve this paper as well as Professor Dr. Markku Lanne and Professor Dr. Mika Meitz from the Faculty of Social Sciences, University of Helsinki. I also sincerely thank Dr. Ji Hyung Lee for generously sharing the replication code for the IVX-QR procedure which has been adapted in this article. Financial support from the Research Council of Finland (grant 347986) is gratefully acknowledged. All remaining errors are my own responsibility.