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Structural Break Detection in Quantile Predictive Regression Models with Persistent Covariates
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Parts of this paper are derived from my Ph.D. thesis at the University of Southampton with title: \textcolor{blue}{"Aspects of Estimation and Inference for Predictive Regression Models"}. } \\ } }
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Time series predictability is a research question that sparked the development of various methodologies for estimation and inference in predictive regression models. Related studies include jansson2006optimal, mikusheva2007uniform, Phillips2014Confidence, cai2014testing, kostakis2015Robust, Kasparis2015nonparametric, gonzalo2012regime, gonzalo2017inferring, ren2019balanced, demetrescu2020testing, yang2020testing, georgiev2021extensions, andersen2021consistent, harvey2021simpleA as well as dou2021generalized among others\footnote{The problem of distorted statistical inference in predictive regression models under nearly integrated predictors has been also reported in previous studies such as elliott1994inference, elliott2011control, campbell2006efficient and references therein.}. The aforementioned frameworks operate under the general assumption of a stable relation between the predictant and the predictors of the model. However, the possible presence of parameter instability under these conditions require a different treatment (e.g., see pitarakis2017simple and georgiev2018testing). Thus, to derive limits for structural break tests suitable for predictive regressions, necessitate to employ certain regularity conditions and invariance principles of partial sum processes as in Phillips2007limit, PhillipsMagdal2009econometric, to obtain stochastic integral approximations.
Most of the current literature focuses on detecting structural break in the conditional mean of the distribution of $f ( y_t | \boldsymbol{x}_t)$ or $f ( y_t | \boldsymbol{x}_{t-1})$ where $\boldsymbol{x}_{t}$ is assumed to follow a near unit root process such as in cai2015testing, georgiev2018testing and Katsouris2021breaks, leaving the conditional quantile distribution vastly unexplored. Consequently, this paper addresses these issues by developing a framework suitable for structural break testing for quantile regressions under nonstationarity which has not seen much attention. Specifically, we build on related studies such as lee2016predictive and fan2019predictive who propose a framework for estimation and inference in quantile predictive regressions as well as the study of qu2008testing who focus on break testing procedures for regression quantiles. More precisely, the latter approach studies a linear quantile model with stationary covariates, while we consider the quantile predictive regression model with possibly nonstationary predictors as in lee2016predictive. Another related study is presented by aue2017piecewise in the context of piecewise quantile autoregressions. Therefore, proposing tests which bridge the gap between these two approaches is a further development of the current toolkit while the local-to-unity theory proposed by the seminal work of Phillips1987time, Phillips1987towards, Phillips1988testing can facilitate the asymptotic theory.
Consider the $\uptau-$th conditional quantile of $y_t$ which is defined as following
Specifically, using the conditional quantile function (ref) (see, kiefer1967bahadur) to be the underline functional form for the predictive regression model then the induced econometric environment permits to investigate the presence of quantile predictability.
As a result, the parameter vector of the predictive regression model is quantile dependent for some fixed quantile level within a compact set. In practise, the parameter vector of a quantile regression can be estimated as the unique solution of the following unconstrained optimization problem (see, koenker1987estimation and portnoy1991asymptotic)
where $\uprho_{\uptau} ( \mathsf{u} ) \equiv \mathsf{u} \left( \uptau - \mathds{1} \left\{ \mathsf{u} < 0 \right\} \right)$, is the check function as defined by Koenker1978regression, $\boldsymbol{b}$ is some parameter vector and $\boldsymbol{x}_{t-1}$ is the lagged regressor of the model. Our research interest is on the asymptotic behaviour of the parameter estimators of $\boldsymbol{b}$ under both the null hypothesis of no parameter instability as well as under the alternative hypothesis of structural break at an unknown break-point location within the full sample.
The structural break literature has various applications for different modelling conditions, however there are currently limited studies related to break testing in quantile time series models even under the assumption of stationarity and ergodicity such as in the seminal paper of andrews1993tests for linear models. Furthermore, when considering predictive regression models inference regarding structural break involves deriving nonstandard asymptotic theory due to the presence of the nuisance coefficient of persistence. More precisely, testing based on conventional estimation methods (such as OLS-based tests) has been proved to have size distortions for persistent regressors (e.g., see georgiev2018testing and Katsouris2021breaks). Our study is considered as a unified framework for structural break detection in quantile predictive regressions (see, lee2016predictive and fan2019predictive) which encompasses regressor properties such as high persistent, mildly integrated or stationary under certain parameter space restrictions on the persistence parameters. In particular, the proposed tests can statistically evaluate for breaks in the model coefficients at both fixed and multiple quantiles of the underline conditional quantile distribution. We investigate the asymptotic theory and implementation of both Wald type statistics as in andrews1993tests as well as fluctuation type statistics as in qu2008testing but within the setting of nonstationary quantile predictive regressions.
Our contributions are threefold. Firstly, we study the estimation problem of quantile predictive regressions with multiple predictors assumed to be generated as either near unit root or mildly integrated processes. We construct quantile predictability Wald statistics under the null of no predictability and derive the asymptotic distributions (as in lee2016predictive). Secondly, we propose a structural break testing procedure for quantile predictive regressions which permits testing for the presence of breaks at the tails. Thirdly, we examine the statistical performance of these tests for detecting parameter instability in the coefficients of quantile predictive regressions with extensive simulation experiments of empirical size.
Under the null hypothesis of no structural break the specific weakly convergence argument permits to establish convergence to stochastic integral approximations defined with respect to a two-parameter process; limit results employed to derive the asymptotic distributions of fluctuation type statistics. Similarly, for the Wald type statistics we employ the asymptotic theory developed by PhillipsMagdal2009econometric and lee2016predictive. Thus, we introduce invariance principles for partial sum processes of matrix moments based on these functionals to obtain asymptotic results for the proposed econometric environment.
Throughout the paper, we assume that all random elements are defined within a probability space denoted with the triple $\left( \Omega, \mathcal{F}, \mathbb{P} \right)$. All limits are taken as $n \to \infty$, where $n$ is the sample size. Denote with $\mathcal{D} \left( [0,1] \right)$ to be the set of functions on $[0,1]$ that are right continuous and have left limits, equipped with the Skorokhod metric. Then, the symbol $"\Rightarrow"$ is used to denote the weak convergence of the associated probability measures as $n \to \infty$. The symbol $\overset{ \mathcal{D} }{\to}$ and $\overset{ \mathbb{P} }{ \to }$ are employed to denote convergence in distribution and convergence in probability respectively. Moreover, we denote with $\mathcal{O}_{ \mathbb{P} }(.)$ and $o_{ \mathbb{P} }(.)$ the stochastic order of convergence in probability (see, billingsley1968convergence). In terms of vector and matrix notation, for any random matrix $X$, $\left\lVert X \right\rVert_p$ denotes the $L_p-$norm, that is, $\left\lVert X \right\rVert_p = \left( \mathbb{E} \left\lVert X \right\rVert^p \right)^{ 1 / p}$ where $p$ is some positive constant. The proposed test statistics are constructed based on the Euclidean norm. Moreover, we employ the term $\textit{stationary regressor}$ meaning a regressor of the predictive regression model which is generated from the local-to-unity specification that satisfies $\upgamma_x = 0$ and $|1 + c_i | < 1 \ \forall i$.
The paper is organized as follows. Section (ref) discusses the econometric environment of the quantile predictive regression model along with main assumptions, the estimation methodology as well as the testing hypotheses of interest. Section (ref) presents the testing procedure and asymptotic theory of structural break testing for a fixed quantile level. Section (ref) investigates the finite sample performance of the proposed tests via Monte Carlo experiments. Section (ref) illustrates the implementation of of the proposed testing procedures with an empirical application. Section (ref) concludes. Proofs of main limit results can be found in the Appendix (see Section (ref)).
Estimation and inference with quantile models has been proposed to the literature with the seminal work of Koenker1978regression and Koenker1982Robust. In particular, the uniform Bahadur-type representation established by koenker1987estimation as well as the representation of quantiles proposed by knight1989limit (see also koul1995autoregression) are commonly used to derive limit distributions for estimators and test statistics. Several papers in the literature follow these methodologies that examine related aspects to the proposed estimation environment and testing procedures.
Firstly, related literature to the problem of structural change for regression quantiles include the studies of su2008testing and qu2008testing. Both frameworks develop diagnostic tools for break detection in quantile regression models under the assumption of stationary covariates. In the former case the alternative hypothesis of a single break is formulated with respect to the magnitude of the break-point. In the latter case, the author also propose a multiple break point testing procedure. Additionally, furno2014quantile implements these quantile regression based statistics using the methodology proposed by chow1960tests which implies testing for break at a known location in the sample.
A different perspective is presented by hoga2018structural who consider detecting for breaks using tail dependence measures (see also hoga2017change\footnote{The framework proposed by hoga2017change considers a set of change point tests for the tail index of random variables and contributes to the literature of nonparametric modelling methods.}) based on an empirical estimator of extremal dependence. An extension of the method to an alternative hypothesis with multiple breaks is also examined. All aforementioned procedures correspond to structural break tests suitable for quantile models\footnote{Notice that the literature of break tests presented here differs from the literature of slope heterogeneity in quantile models. Related aspects to inference are koenker2001quantile, chernozhukov2005extremal, portnoy2012nearly and escanciano2018quantile which are of independent interest.} while in our study we focus on implementing testing procedures specifically for quantile predictive regression models in which the time series properties of predictors are modelled with the nuisance parameter of persistence. In particular, a separate autoregressive model with a local unit root coefficient matrix is used to model the unknown degree of persistence which implies that conventional approaches for deriving large sample theory are no longer valid. However, the fluctuation based tests implemented by qu2008testing for break detection in quantile models with stationary covariates provides a suitable econometric environment for investigating the effect of nonstationarity to the asymptotic distribution of the tests.
Secondly, asymptotic theory for quantile time series regressions has been examined by Koenker2002inference, Koenker2004unit, koenker2006quantile in the context of autoregressive model specifications and unit root testing\footnote{A unified framework for econometric inference with nearly integrated regressors and unit roots is proposed by the seminal work of Phillips1988Regression, Phillips1987towards, Phillips1987time. Further relevant literature includes the study of stock1994unit who discuss aspects related to testing with unit roots and structural breaks in time series models as well as Chapter 14 in davidson2000econometric that has additional derivations and examples.}. Further applications in the time series literature include testing procedures for threshold effects under the assumption of a conditional quantile function as in the studies of Galvao2011threshold, galvao2014testing and the case of nonstationary (nonlinear) quantile regressions by xiao2009quantile, cho2015quantile, li2016estimation and uematsu2019nonstationary. Moreover, kato2009asymptotics derives related limit results which are useful when considering the asymptotic behaviour of quantile estimators for a wide class of modelling approaches and econometric conditions. All these studies cover both stationary and nonstationary models however they operate under the null hypothesis of no parameter instability in model parameters. Therefore, our proposed testing procedure, is considered to be a novel contribution to the time series econometrics literature.
Thirdly, the proposed structural break testing methodology is closely related to the problem of unidentified parameters under the null hypothesis which is well-known in the econometrics and statistics literature as the Davies problem (see, davies1977hypothesis). The particular aspect which is relevant to parameter admissibility has been investigated in problems of estimation and testing such as in the studies of hansen1996inference, andrews1994optimal, pitarakis2004least and elliott2015nearly. A more recent approach to the unidentified parameter problem under the null is presented by mccloskey2017bonferroni\footnote{Specifically, mccloskey2017bonferroni proposes a framework for a set of flexible size-corrected critical values construction methods that lead to tests with correct asymptotic size and desirable power properties in testing problems with nuisance parameter under the null hypothesis.}. Therefore, to overcome this challenge, we employ the supremum operator when constructing Wald type statistics. Furthermore, a nonparametric estimation approach for quantile regressions such in the study of qu2015nonparametric, may require different moment conditions for estimation and inference which is beyond the scope of this paper.
Lastly, in terms of the model structure of the proposed modelling approach of the paper, we impose standard econometric assumptions and conditions in the literature of predictive regressions. More precisely, by imposing a standard martingale difference condition on the equation innovations $u_t$, this implies an orthogonality condition between the innovations and the model regressors, such that $\mathsf{Cov} \big( u_t, \boldsymbol{x}_{t} \big) = \mathbb{E} \big[ \boldsymbol{x}_{t} \mathbb{E} \left( u_t | \mathcal{F}_{t-1} \right) \big] = 0$. Furthermore, the model structure we follow does not allow for endogeneity even though the IVX instrumentation method implies the construction of endogenous instruments. Specifically, as explained by PhillipsMagdal2009econometric and kostakis2015Robust, the IVX filtration implies the construction of instrumental covariates based on information obtained only from the regressors of the model. According to WangPhillips2012specification the model structure would permit for the presence of endogeneity when the equation error could be serially dependent and cross-correlated under certain moment restrictions (see also yang2020testing\footnote{Specifically, yang2020testing propose a unified IVX-AR Wald statistic that accounts for serial correlation in the error terms of the linear predictive regression model. The IVX-AR estimator corrects the size distortions arising from serially correlated error under the presence of high persistence.}). However, the particular aspect can complicate the derivation of the limit distributions as additional considerations will be needed, such as to incorporate conditional heteroscedasticity, which is beyond the scope of our study and we leave as future research related to the proposed framework.
Based on the aforementioned challenges in the literature, we are motivated to develop an econometric framework for structural break testing in quantile predictive regressions with persistent covariates. Overall our research objectives are concentrated around two main pillars: (i) to introduce a new set of weak convergence results for certain partial sums of functionals which are based on either the OLS or the IVX estimators and involve nonstationary time series. Specifically, these functionals are instrumental for deriving the asymptotic behaviour of the proposed test statistics; and (ii) to demonstrate the practicality of a new set of structural break tests for detecting parameter instability.
Consider the (linear) predictive regression model
where $y_t \in \mathbb{R}^{n \times 1}$ is a scalar dependent variable and $\boldsymbol{x}_{t-1} \in \mathbb{R}^{n \times p}$ is a $p-$dimensional vector of predictors such that $\boldsymbol{x}_{t}$ is generated as a near unit root process (or local unit root process) with an autocorrelation coefficient matrix as defined by the studies of PhillipsMagdal2009econometric and kostakis2015Robust expressed as below
where $n$ is the sample size, and $\boldsymbol{C}_p = \mathsf{diag} \left\{ c_1,...,c_p \right\}$ is a $p \times p $ diagonal matrix with the coefficients of persistence $c_i$ for $i = 1,...,p$. We consider that the predictors of the model are allowed to belong only to one of the two degree of persistence as specified below
Let $\mathcal{F}_t$ denote the natural filtration, then for the error term of the predictive regression we assume that $\mathbb{E} \left( u_t | \mathcal{F}_{t-1} \right) = 0$ and $\mathbb{E} \left( u^2_t | \mathcal{F}_{t-1} \right) = \sigma^2_{uu}$. Specifically, the innovation structure of the predictive regression model allows to impose a linear process dependence for $\boldsymbol{v}_t $, with a conditionally homoscedastic martingale difference sequence condition such that
with necessary conditions for the linear process representation to hold given as below
Denote with $\boldsymbol{e}_t = \left( u_t, \boldsymbol{v}_t^{\prime} \right)^{\prime}$, then under regularity conditions, the following invariance principle (FCLT) holds (see Phillips1992asymptotics)
where $\boldsymbol{\Sigma}_{vv} \in \mathbb{R}^{ p \times p}$ is a positive definite covariance matrix and $0 < r < 1$.
Specifically, the individual components of the vector sequence $\boldsymbol{e}_t = \left( u_t, \boldsymbol{v}_t^{\prime} \right)^{\prime}$ have partial sums processes that weakly converge into their Brownian motion counterparts as below
where $\boldsymbol{B}(r) = \big( B_{u} (r), \boldsymbol{B}_{v} (r)^{\prime} \big)^{\prime}$ being a $(p \times 1)$ Brownian motion with long-run covariance matrix $\boldsymbol{\Sigma}_{ee}$, that is, a Gaussian vector process with almost surely continuous sample paths. More precisely, since $\boldsymbol{x}_t$ is an adapted process to $\mathcal{F}_t$ then in practise there exists a correlated vector Brownian motion $\boldsymbol{B}_n(r) = \big( B_{un} (r), \boldsymbol{B}_{vn} (r)^{\prime} \big)^{\prime}$ such that
on $\mathcal{D}\left( [0,1] \right)^2$ as $n \to \infty$, with covariance matrix as in (ref) implying joint convergence. Assuming the above conditions hold, then the following local to unity principle applies (as proposed by Phillips1987towards, Phillips1987time)
The functional $\boldsymbol{J}_c(r)$ represents the Ornstein-Uhlenbeck process\footnote{The OU process is a stationary Gaussian process with an autocorrelation function that decays exponentially over time. Moreover, the continuous time OU diffusion process has a unique solution and this property allows to approximate asymptotic terms for estimators and corresponding test statistics as a function of the nuisance parameter of persistence (see, perron1991continuous).} which is employed to derive stochastic integral approximations for the nonstationary predictive regression model (ref)-(ref). Denote with $\boldsymbol{K}_c(r) := \boldsymbol{\Sigma}_{vv} \boldsymbol{J}_c(r)$, where $\boldsymbol{K}_c(r)$ is a $p-$dimensional Gaussian process defined as $\boldsymbol{K}_c(r) = \displaystyle \int_0^r e^{(r-s) \boldsymbol{C}_p } d \boldsymbol{B}_v(s)$ is the solution of Black-Scholes differential equation $d \boldsymbol{K}_c(r) \equiv c \boldsymbol{K}_c(r) + d \boldsymbol{B}_v(r)$, with $\boldsymbol{K}_c(r) =0$ as the initial condition.
The specific autoregression matrix specification (ref) allows to examine other persistence properties such as unit root processes, when $c_i = 0$ for all $i \in \left\{ 1,...,p \right\}$, or explosive processes when $c_i > 0$. In this paper, we consider two types of nonstationarity, that is, the near unit or high persistent regressors and the mildly integrated regressors. Therefore, in both cases the coefficient of persistent, $c_i$, lies below the unit root boundary. Thus, the main difference between these two persistence classes is that the mildly integrated regressors have an exponent rate below the unit boundary, which implies that these regressors are less persistent than regressors generated from near unit root processes. Furthermore, one can consider extending our estimation and testing framework to the case of explosive regressors; we leave this aspect for future research.
Our main goal is to investigate the asymptotic theory and empirical implementation of structural break tests suitable for quantile predictive regression models. Therefore, we consider suitable modifications of the innovation structure that corresponds to the linear predictive regression model, following standard conditions and assumptions employed for quantile time series models, currently presented in literature. More precisely, the conditional quantile function of $y_t$ denoted with $\mathsf{Q}_{y_t} \left( \uptau | \mathcal{F}_{t-1} \right)$, replaces the conditional mean function of the predictive regression which implies the following model specification
such that $F_{ y_t | \boldsymbol{x}_{t-1} } (\uptau) := \mathbb{P} \big( y_t \leq \mathsf{Q}_{y_t} \left( \uptau | \mathcal{F}_{t-1} \right) \big| \mathcal{F}_{t-1} \big) \equiv \uptau$, where $\uptau \in (0,1)$ is some quantile level in the compact set $(0,1)$. Therefore, in order to define the innovation structure that corresponds to the quantile predictive regression, we employ the piecewise derivative of the loss function such that $\psi_{\uptau} ( \mathsf{u} ) = \big[ \uptau - \mathds{1} \left\{ \mathsf{u} < 0 \right\} \big]$. Consequently, this implies that $u_t (\uptau) := u_{t} - F^{-1}_{ u } (\uptau)$ where $F^{-1}_{ u } (\tau)$ denotes the unconditional $\uptau-$quantile of the error term $u_{t}$. Then, the corresponding invariance principle for the nonstationary quantile predictive regression model is formulated as below
Assumption (ref) (i) provides a standard weak convergence argument to a Brownian motion process that corresponds to the underline distribution generating the innovation sequence of the quantile predictive regression model (see, lee2016predictive and fan2019predictive). Furthermore, Assumption (ref) (ii) provides the weak convergence argument for the sparsity function of the model to its Brownian motion counterpart for some $0 < r < 1$.
We investigate the statistical properties of the estimation methodology in relation to the handling of the nuisance parameter of persistence. We derive the asymptotic distributions of the associated test statistics based on two optimization methods which are known to have different convergence rates in the time series predictability literature.
We consider the OLS based estimation using the check function $\uprho_{\uptau} ( . )$, which is common practise for optimization problems of quantile series. Specifically, the asymptotic behaviour of the quantity $\boldsymbol{\mathcal{E}}_n(\uptau) \equiv \sqrt{n} \big( \boldsymbol{\beta}_n(\uptau) - \boldsymbol{\beta}_0(\uptau) \big)$ is of interest. The traditional approach to asymptotics for $\hat{\boldsymbol{\beta}}(\uptau)$ is to employ a Bahadur representation which allows to decompose the expression into a Brownian bridge component and an error term (see, portnoy2012nearly and kato2009asymptotics). Furthermore, various studies are concerned with the determination of sharp error bounds for the specific error term. However in our setting, $\sqrt{n}-$consistent asymptotics do not always apply due to the presence of nonstationarity. Additionally, the chosen estimator affects the stochastic rates of convergence.
Similar to the linear predictive regression (a model with a conditional mean functional form), under the assumption of persistent regressors, the OLS estimator has been proved to be biased due to the presence of nuisance parameters (e.g., see campbell2006efficient), resulting to distorted statistical inference\footnote{Relevant studies in the literature which examine the asymptotic behaviour of standard $t-$tests under these conditions include Phillips2013predictive, Phillips2016robust, lee2016predictive, fan2019predictive, kostakis2015Robust and Kasparis2015nonparametric among others.}. Nevertheless, focusing on the two persistence properties we introduced previously, we derive its limit distribution which is useful when considering structural break tests under nonstationarity.
Denote with $\boldsymbol{ \theta } ( \uptau ) = \big[ \alpha (\uptau), \boldsymbol{\beta} (\uptau)^{\prime} \big]^{\prime} \in \mathbb{R}^{ (p+1) \times 1 }$ and $\boldsymbol{X}_{t-1} = \left( \boldsymbol{1}, \boldsymbol{x}_{t-1}^{\prime} \right)^{\prime} \in \mathbb{R}^{ (p+1) \times n }$, then the OLS based estimator is obtained by solving the following optimization problem
where $\uprho_{\uptau}( \mathsf{u} ) = \mathsf{u} \big( \uptau - \mathds{1} \left\{ \mathsf{u} < 0 \right\} \big)$ with $\uptau \in (0,1)$, represents the asymmetric quantile regression function. Following lee2016predictive, we use the normalization matrices below which are different according to the persistence properties of predictors such that
Then, Corollary (ref) summarizes the asymptotic distribution of the OLS-QR estimator (see also Theorem 2.1 lee2016predictive) for mildly integrated and high persistent regressors.
Consequently, the limiting joint distribution of the model intercept and slopes for the nonstationary quantile predictive regression model under the assumption of mildly integrated regressors, is a mixed normal of the form $\mathcal{MN} \big( 0, \boldsymbol{\Sigma}^{\star} \big)$, where
while under high persistence the asymptotic behaviour of $\boldsymbol{D}_n \left( \widehat{\boldsymbol{ \theta }}_n^{qr} \left( \uptau \right) - \boldsymbol{ \theta } \left( \uptau \right) \right)$ depends on functionals of OU processes which are more challenging to approximate, especially if one is interested to obtain sharp error bounds\footnote{In particular, the study of portnoy2012nearly obtains a near $\sqrt{n}-$consistent error bound by employing the "Hungarian construction" which requires to approximate the quantity $\boldsymbol{\mathcal{E}}_n(\uptau)$ using a Brownian bridge limit which converges to this non-zero Gaussian process with an appropriate rate of convergence. } as in portnoy2012nearly.
Furthermore, we assume that the sparsity coefficient can be consistently estimated with a fixed unbiased estimator for all $t$. This a strong assumption which can be regarded as a trade-off relation between the complicated objective function (i.e., non-differentiable and nonstationary) and the tractable error term of the model uematsu2019nonstationary. On the other hand, it permits to consider the limiting distributions when testing for a set of parameter restrictions. Thus, to overcome the problem of nonstandard statistical inference due to the presence of the nuisance coefficient of persistence, we employ the instrumental variable regression approach proposed by PhillipsMagdal2009econometric.
The estimator of PhillipsMagdal2009econometric performs reasonably well in finite samples and even performs better that the OLS counterpart (see georgiev2021extensions), demonstrating the robustness of the method in filtering out abstract degree of peristence when testing for linear restrictions in predictive regressions. In addition, the suggested instrumental variable approach is by definition neither spurious, since it is always correlated with the corresponding regressor, not a poor instrument, because the correlation of unit root processes tends to one asymptotically.
The endogenous instrumentation (IVX) procedure for predictive regression models\footnote{Notice that the related asymptotic theory which is robust to abstract degree of persistence and results to nuisance-parameter free inference was pioneered by Magdalinos2009limit in the context of cointegration models.} proposed by PhillipsMagdal2009econometric implies the use of a mildly integrated instrumental variable. The instrumented variable is constructed as below
where $\boldsymbol{C}_{z} = \mathsf{diag} \{ c_{z1},...,c_{zp} \}$ is a $p \times p$ diagonal matrix such that $c_{zj} < 0 \ \forall \ j \in \left\{ 1,..., p \right\}$ with $0 < \upgamma_z < 1$, where $\upgamma_z$ is the exponent rate of the persistence coefficient of the instrumental variable, such that $\upgamma_z \neq \upgamma_x$. The IVX filtering methodology transforms a possibly nonstationary autoregressive process that generates the set of predictors, $\boldsymbol{x}_t$, which encompasses both stable or unstable processes based on the behaviour of the local unit root coefficient, into a mildly integrated process which is less persistent than the endogenous variables. Another statistical property is the choice of the exponent rate for the coefficient of persistence that corresponds to the instrumental variable $c_{zj}$. Specifically, the econometric literature has documented a choice of $\upgamma_z$ close to 0.95 as a reasonable value with desirable finite-sample properties when constructing predictability tests (see, lee2016predictive, Phillips2016robust and kostakis2015Robust). Thus, to account for the different convergence rates due to nonstationarity and obtain the asymptotic distribution of the IVX-QR estimator we employ the following normalization matrices
where $\tilde{\boldsymbol{D} }_n = n^{\frac{ 1 + \upgamma_x \wedge \upgamma_z }{2}} \boldsymbol{I}_p$, such that $\upgamma_x \wedge \upgamma_z \equiv \mathsf{min} \left( \upgamma_x, \upgamma_z \right)$ which is identical for both the case of local unit root and mildly integrated regressors. Furthermore, we denote with $y_t (\uptau) := y_t - \alpha( \uptau ) + \mathcal{O}_{ \mathbb{P} } ( n^{- 1/ 2} )$ to be the zero-intercept QR dependent variable. The particular dequantiling procedure permits to reformulate the quantile model as $y_t (\uptau) = \boldsymbol{x}_{t-1}^{\prime} \beta( \uptau ) + u_t (\uptau )$ that simplifies the derivations for the asymptotics of the model estimator which is known to have different convergence rates when an intercept is included (e.g., see gonzalo2012regime, gonzalo2017inferring). Then, the IVX-QR estimator for the quantile regression, is defined by the following unconstrained optimization problem
where $h_t (.)$ is defined below such that $\psi_{ \uptau } ( \mathsf{u} ) := \big[ \mathds{1} \left\{ \mathsf{u} \leq 0 \right\} - \uptau \big]$ and $\psi_{ \uptau } ( \mathsf{u} ) \overset{ \mathsf{maps \ to} }{ \mapsto } \uprho_{\uptau}^{-1} ( \mathsf{u} )$
The minimization of expression (ref) leads to the following first order condition:
Furthermore, it can be proved that for both the cases of high persistent and mildly integrated predictors (LUR or MI) the asymptotic distribution of the IVX-QR estimator is identical as presented by Corollary (ref). Notice that for the interested reader the limit distribution for other classes of persistence (e.g., such as predictors which exhibit near stationary or mildy explosive persistence) can be found in Theorem 3.1 of lee2016predictive.
Analytic definitions of the covariance matrices $\boldsymbol{\Gamma}_{cxz}$ and $\boldsymbol{V}_{cxz}$ can be found in expressions (3.4) and (3.5) in lee2016predictive. Lastly, Lemma (ref) presents the asymptotic behaviour of the self-normalized Wald statistic based on the IVX-QR estimator (see, Proposition 3.1 in lee2016predictive) for both the cases of LUR and MI regressors.
Furthermore, the above result can be generalized when testing for a set of linear restrictions under the null hypothesis, $\mathcal{H}_0 : \boldsymbol{R} \boldsymbol{\beta} ( \uptau ) = \boldsymbol{q} ( \uptau )$ where $\boldsymbol{R}$ is a $r \times p$ known matrix and $\boldsymbol{q} ( \uptau )$ is a prespecified vector. The corresponding asymptotic distribution for the IVX-Wald statistic for the quantile predictive regression is given by the following expression
where $\chi^2_r$ denotes the chi-square random variate with $r$ degrees of freedom such that $\mathbb{P} \left( \chi^2 \geq \chi^2_{ r ; \upalpha } \right) = \upalpha$, where $0 < \upalpha < 1$ denotes the fixed significance level.
In this section we present the testing problem of interest in this paper. More precisely, we consider two type of testing hypotheses, that is: (i) testing for structural break for a fixed quantile level $\uptau \in (0,1)$ and (ii) testing for a structural break multiple quantile levels. For both testing hypotheses we operate under the assumption of a single structural break at an unknown location. Notice, that similar formulations can be found in the study of qu2008testing, however the focus of our study in the quantile predictive regression model with possibly nonstationary predictors in the context of return predictability literature.
\paragraph{Testing Hypothesis A.}
The first testing hypothesis of interest is concerned with testing for structural break in a pre-specified quantile with the null and alternative hypothesis given as below
\textcolor{blue}{for a fixed $\uptau \in (0,1)$}, where $\kappa = \floor{ \lambda n }$ the unknown break-point with $\lambda \in (0,1)$.
\paragraph{Testing Hypothesis B.}
The second testing hypothesis of interest is concerned with testing for structural break across multiple quantiles, that is, quantiles contained in a set $\mathcal{T}_{\iota}$, with the null and alternative hypothesis given as below
\textcolor{blue}{for some $\uptau \in \mathcal{T}_{\iota}$}, where $\kappa = \floor{ \lambda n }$ the unknown break-point with $\lambda \in (0,1)$.
where $\boldsymbol{\beta}_0 ( \uptau )$ is the value of the true population parameter under the null hypothesis of no parameter instability.
Both Testing Hypothesis A and B summarize the modelling environment under the null as well as under the alternative hypothesis. More precisely, under the null hypothesis the parameter vector is taken to be constant throughout the sample such that $\boldsymbol{\beta}_t(\uptau) \equiv \boldsymbol{\beta}_0(\uptau)$ for $t = 1,...,n$ where $\boldsymbol{\beta}_0(\uptau)$ is the unknown quantile dependent regression parameter. Therefore, in practise we are interested in testing the null hypothesis that $\boldsymbol{\beta}_t(\uptau)$ remains constant, that is, $\boldsymbol{\beta}_t(\uptau) = \boldsymbol{\beta}_0(\uptau)$ for all $t$ against the alternative that the quantile dependent parameter vector $\boldsymbol{\beta}_t(\uptau)$ has a single structural break at an unknown location within the full sample, resulting to two regimes\footnote{Notice that the two regimes we refer to here, are not equivalent to testing methodologies proposed in studies such as gonzalo2012regime, gonzalo2017inferring and galvao2014testing in which emphasis is given to testing the null hypothesis of linearity based on the presence of no threshold effect.}. Under the alternative hypothesis:
where $\mathcal{F}_t$ denotes the $\sigma-$field generated by $\left\{ \boldsymbol{x}_{t-1}, \boldsymbol{x}_{t-2},..., \right\}$. Therefore, it is convenient to write the hypotheses with a different formulation. Denote with $\boldsymbol{\upbeta}_{(1)}(\uptau) = \boldsymbol{\beta}_1(\uptau)$ and $\boldsymbol{\upbeta}_{(2)}(\uptau) = \boldsymbol{\beta}_2(\uptau) - \boldsymbol{\beta}_1(\uptau)$. Furthermore, to construct the model so that it can capture the magnitude of the structural break we denote with $\boldsymbol{\mathcal{X}}_{t-1} = \big( \boldsymbol{x}_{t-1}^{\prime}, \boldsymbol{x}_{t-1}^{\prime} \mathds{1} \left\{ t > \kappa \right\} \big)^{\prime}$ and $\boldsymbol{\vartheta}(\uptau) = \big( \boldsymbol{\upbeta}_{(1)}(\uptau)^{\prime}, \boldsymbol{\upbeta}_{(2)}(\uptau)^{\prime} \big)^{\prime}$. Thus, we express the null hypothesis as following
Then, the alternative hypothesis can be formulated as below
Furthermore, since we consider the nonstationary quantile predictive regression model without an intercept, following the formulations given by expressions (ref) and (ref) for the null and alternative hypotheses respectively, then we define the quantile dependent estimator as the optimization problem below
Therefore, with the above formulation of the estimator $\widehat{\boldsymbol{ \vartheta }}_n ( \uptau, \lambda )$ is the quantile dependent regression estimator when we employ $\boldsymbol{\mathcal{X}}_{t-1}$ to be the model predictor variables. Specifically, when the $\mathcal{H}_0^{(A)}$ is true, under suitable regularity conditions, $\widehat{\boldsymbol{ \vartheta }}_2 ( \lambda, \uptau )$ converges in probability to $\boldsymbol{0}$ for each $( \lambda, \uptau ) \in \Lambda_{\eta} \times \mathcal{T}_{\iota}$. On the other hand, when $\mathcal{H}_1^{(A)}$ is true, $\widehat{\boldsymbol{ \vartheta }}_2 ( \lambda ; \uptau_0 )$ converges in probability to $\boldsymbol{\upbeta}_{(2)}(\uptau_0) = \big( \boldsymbol{\beta}_2(\uptau_0) - \boldsymbol{\beta}_1(\uptau_0) \big) \neq \boldsymbol{0}$. In summary, since the quantile level $\uptau_0$ especially for Testing Hypothesis B is unknown a prior, then it is reasonable to reject $\mathcal{H}_0$ when the magnitude of $\widehat{\boldsymbol{ \vartheta }}_2 ( \lambda, \uptau )$ is suitable large for some $( \lambda, \uptau ) \in \Lambda_{\eta} \times \mathcal{T}_{\iota}$. Thus, an example of a suitable test statistic to test whether $\mathcal{H}_0$ against the alternative hypothesis $\mathcal{H}_1$ is to employ the supremum of the Wald process.
In general, a Wald type statistic has the following form
where $\boldsymbol{V}_n ( \lambda; \uptau_0 )$ is the asymptotic covariance matrix of the stochastic process $\sqrt{n} \hat{\boldsymbol{\upbeta}}_{(2)}(\uptau_0)$, under the null hypothesis. However, since the covariance matrix that corresponds to the population regression parameters is in practise unknown, is replaced by a suitable consistent estimate that holds under the null hypothesis of no structural break in the quantile predictive regression model. Obviously within the aforementioned structural break setting which is our main research focus in the paper, the break-point location is not identified under the null hypothesis as we explained in the introduction.
Moreover, additionally to Remark (ref), the Testing Hypotheses of interest A and B correspond to two different test functions such that Testing Hypothesis A requires to formulate test statistics by employing the supremum funcitonal while Testing Hypothesis B requires to construct test statistics with the use of the double supremum functional. Intuitively, we are interested to examine both hypotheses since it might be the case that A is not rejected, that is, there are no statistical evidence of the presence of a structural break for a given quantile level $\uptau_0 \in (0,1)$, while when employing Testing Hypothesis B, it could be the case that the test statistic provides statistical evidence of rejecting the null hypothesis, implying that a structural break still exist at some other quantile level not the one which is kept fixed, within the set (0,1).
Next, we focus on the structural break testing procedures for the quantile regression model with regressors generated as near unit root processes based on two test statistics.
We consider structural break tests for a fixed quantile level, say $\uptau_0 \in (0,1)$. Consider the subgradient\footnote{The required convexity arguments for obtaining estimators based on the conditional quantile function are based on the convexity lemma result presented by pollard1991asymptotics. Furthermore, related results are presented by koenker1987estimation.} $S_n \big( \lambda, \uptau_0 , \boldsymbol{b} \big)$, based on the subsample $1 \leq t \leq \kappa$
where $\boldsymbol{b}$ corresponds to an estimator of the parameter vector $\boldsymbol{\beta} ( \uptau_0 )$ which encompasses both the OLS and IVX estimators under suitable parametrizations.
The continuous function $\psi_{ \uptau } ( . )$ is defined as $\psi_{ \uptau } ( \mathsf{u} ) = \big[ \uptau_0 - \mathds{1} \left\{ \mathsf{u} \leq 0 \right\} \big]$ and $\kappa = \floor{ \lambda n }$ denotes the unknown break-point location implying a break fraction $\lambda \equiv \underset{ n \to \infty }{ \mathsf{lim} } \kappa/n$ such that $\lambda \in \Lambda_{\eta}:= [ \eta, 1 - \eta]$ is a compact set. Thus, under the null hypothesis of no structural break with stationary and ergodic regressors, the quantity $\psi_{ \uptau } \big( y_t - \boldsymbol{x}_{t-1}^{\prime} \boldsymbol{\beta} ( \uptau_0 ) \big)$ is a pivotal statistic. In particular, since $\boldsymbol{x}_{t-1}^{\prime} \boldsymbol{\beta} ( \uptau_0 )$ is equal to the conditional $\uptau-$quantile of $y_t$ given $\boldsymbol{x}_{t-1}$, then the random variables $\mathds{1} \left\{ y_1 \leq \boldsymbol{x}_{t-1}^{\prime} \boldsymbol{\beta} ( \uptau_0 ) \right\},..., \mathds{1} \left\{ y_n \leq \boldsymbol{x}_{t-1}^{\prime} \boldsymbol{\beta} ( \uptau_0 ) \right\}$ are independent Bernoulli trials with success probability $\uptau$ (see, galvao2014testing), which implies a sequence of random variables with mean zero and variance $\uptau_0 ( 1 - \uptau_0 )$. A similar result should hold in our setting of possibly nonstationary regressors. Furthermore, denote with $\boldsymbol{X} = \left( x_1^{\prime} ,..., x_n^{\prime} \right)^{\prime}$ and define the following auxiliary quantity
Furthermore, replacing the unknown parameter vector $\boldsymbol{\beta}_0 ( \uptau_0 )$ with the quantile dependent regression estimator based on the full sample, under the null hypothesis of no structural break in the model, the quantity given by (ref) can be formulated as below
for some $0 < \lambda < 1$ and $\uptau_0 \in (0,1)$.
The use of fluctuation type statistics provide a way for statistical inference regarding the presence of structural breaks in model coefficients (leisch2000monitoring). Intuitively for these class of tests we consider the asymptotic behaviour of the corresponding empirical processes to decide whether to accept or reject the null. In particular, the fluctuation type test converges to a nondegenerate limiting distribution under the null hypothesis, since the random quantity given by expression (ref) is essentially governed by the invariance principle under the null. In practise, when the quantile dependent parameters exhibit no structural break in the sample, then $\widehat{ \boldsymbol{ \beta} }_n( \uptau_0 )$ is a consistent estimator and as a result, $\hat{\mathcal{J}}_n \big( \lambda, \uptau_0, \widehat{ \boldsymbol{ \beta} }_n( \uptau_0 ) \big)$ has the same stochastic order as its population counterpart. On the other hand, when the null hypothesis is false, the underline stochastic process exhibit excessive fluctuations. Specifically, under the alternative hypothesis, model parameters have a break at some unknown location in the sample, which implies that $\widehat{ \boldsymbol{\beta} }_n( \uptau_0 )$ will differ significantly from the true value for some sub-sample and the estimated residuals will have high fluctuations (beyond the usual increments of a Wiener process) resulting to falsely rejecting the null due to a large value of the statistic qu2008testing.
The focus of the proposed econometric environment in this paper is the structural break detection in the model parameters of quantile predictive regression with possibly nonstationary regressors, under the assumption that these are generated as near unit root processes. Intuitively, the two persistence classes (mildly integrated and near unit root) we consider encompasses moderate deviations from the unit boundary similar to the case of near integrated (see Phillips1988Regression). Therefore, both the value (and sign) of the coefficient of persistence as well as its exponent rate\footnote{Practically, these are nuisance parameters however via Monte Carlo simulations we can choose suitable values for $c_i$ and $\upgamma_x$ in order to simulate these experimental conditions and thus evaluate the finite-sample performance of the test statistics with high persistence or mildly integrated regressors.} (a tuning parameter), determine the asymptotic behaviour of functionals based on these near unit root process. As a result, the chosen estimator can affect the asymptotic theory of the proposed test statistics as well as the corresponding functionals which we examine separately below.
Within the proposed econometric environment which corresponds to the modelling of nonstationary quantile time series models, regressors are assumed to follow a local unit root process. Thus, we expect that OLS based functionals will dependent on the nuisance coefficient of persistence\footnote{Notice that this is the standard inference problem in the predictability literature. Further details regarding the bias (nonstandard distortion) occurred in predictability tests (i.e., $t-$tests) in quantile predictive regression models can be found in the study of lee2016predictive. In our study, we aim to compare both the OLS as well as the instrumental variable approach of PhillipsMagdal2009econometric.}. Furthermore, due to the presence of both a model intercept and the set of nonstationary regressors, we also need to modify the functionals given by expressions (ref)-(ref) in order to account for the different convergence rates.
Therefore, to obtain equivalent representations to the quantity $S_n \big( \lambda, \uptau_0, \boldsymbol{b} \big)$, we consider the corresponding partial sum process of the functional $\boldsymbol{K}_{nx} \big( \uptau_0, \boldsymbol{\theta}^{ols}_n( \uptau_0 ) \big)$ as given by Definition (ref). We obtain the limit result for these functionals based on the full sample and then focus on deriving invariance principles for the corresponding partial sum processes for the two estimators under examination (see, Section (ref)). More precisely, the functionals given by Definition (ref) correspond to a quantile regression ordinary least squares estimator and are employed when the asymptotic behaviour of the OLS based test statistics is concerned (see also Lemma A1 in lee2016predictive).
A key observation is that under the null hypothesis of no parameter instability these functional converge to a nondegenerate limit distribution. Corollary (ref) demonstrates the asymptotic distributions of the functionals given by Definition (ref).
In this Section, we derive the asymptotic distribution of the IVX based functionals which are useful to obtain the asymptotic behaviour of the proposed structural break tests under the assumption of nonstationary regressors in the model. We employ the embedded normalization version of the instruments such that $\tilde{ \boldsymbol{Z} }_{t-1,n} := \tilde{ \boldsymbol{D} }_n^{-1} \tilde{ \boldsymbol{z} }_{t-1}$.
Moreover, the definition of the moment matrix $\boldsymbol{\Gamma}_{cxz}$ is presented by lee2016predictive via expression (3.4) which is the corresponding asymptotic limit given by expression (20) in PhillipsMagdal2009econometric as given below
The proofs of Corollary (ref) and (ref) can be found in the Appendix of the paper. Note that the stochastic convergence of these functional holds for large sample size, $n \to \infty$, and the existence of well-defined moment matrices with negligible higher-order terms. Therefore, to facilitate the development of the asymptotic theory we define the following empirical process for some parameter vector $\boldsymbol{b} \in \mathbb{R}^p$ such that
where $\uptau \in (0,1)$ and $0 < \upgamma_x < 1$. In particular, the empirical process $\boldsymbol{G}_n \left( \uptau, \boldsymbol{b} \right)$ is consider stochastically $\varrho-$equicontinuous over $\mathcal{T}_{\iota} \times B$, such that for any $\epsilon > 0$,
where $[ \mathcal{\delta} ] := \big\{ ( \uptau_1, \boldsymbol{b}_1 ), ( \uptau_2, \boldsymbol{b}_2 ) \in \left( \mathcal{T} \times B \right)^2 : \ \varrho \big( ( \uptau_1, \boldsymbol{b}_1 ), ( \uptau_2, \boldsymbol{b}_2 ) \big) < \delta \big\}$.
An additional condition for convergence in probability for the empirical process is imposed by Lemma (ref), which can be employed to derive the convergence rate of the IVX estimator for the nonstationary quantile predictive regression model.
More precisely, Lemma (ref) provides a simplified way to derive the convergence limit for the IVX-QR estimator (see, also lee2016predictive) that ensures consistent estimation of the model parameters for the quantile predictive regression model. A related study to our setting with detailed derivations for nonstandard inference problems, (Wald type statistics), for nonstationary quantile regressions is presented in the study of goh2009nonstandard. Overall, the asymptotic theory of this paper aims to combine unit root asymptotics with empirical process methods. Specifically, we employ a two-parameter empirical process that converges weakly to a two-parameter Brownian motion. Therefore, our asymptotic distributions involve stochastic integrals with respect to this two-parameter process.
To obtain the asymptotic distributions of the test statistics, we consider the asymptotic behaviour of the partial sum processes of the functionals defined in the previous section. We focus in the case of nonstationary regressors which are either high persistent or mildly integrated (see, Section (ref) for definitions and kostakis2015Robust). Moreover, since we derive and compare the limit distributions of structural break tests based on the chose estimation methodology, we derive invariance principles that correspond to each of these two estimators. Therefore, we define with
where $u_t ( \uptau_0 ) = \big( y_t - \boldsymbol{X}_{t-1}^{\prime} \boldsymbol{\theta}^{ols}_n( \uptau_0 ) \big)$ for $\uptau_0 \in (0,1)$, which can be determined uniquely, making the mapping $\psi_{ \uptau } ( \mathsf{u} ) \mapsto \uprho_{\uptau}^{-1} ( \mathsf{u} )$ one-to-one and well-defined. Moreover, we denote with $\boldsymbol{X}_{t-1} = \big( \boldsymbol{1}, \boldsymbol{x}^{\prime}_{t-1} \big)^{\prime}$ the regressors and $\boldsymbol{\theta} (\uptau_0)= \big( \alpha(\uptau_0), \boldsymbol{\beta}^{\prime}(\uptau_0) \big)^{\prime}$ the parameters.
Recall that for mildly integrated regressors it holds that (see, Corollary (ref))
Similarly, we can show that $S_{nx}^{ols} \big( \lambda, \uptau , \boldsymbol{\theta}^{ols}_n ( \uptau_0 ) \big) \Rightarrow S_{x} \big( \lambda, \uptau_0 , \boldsymbol{\theta}_0( \uptau_0 ) \big)$ as $n \to \infty$, where
\color{black} for some $0 < \lambda < 1$. Then, for the corresponding IVX based functional it holds that
\color{black} where $\tilde{ \boldsymbol{Z} }_{t-1,n} := \tilde{ \boldsymbol{D} }_n^{-1} \tilde{ \boldsymbol{z} }_{t-1}$ since we employ the corresponding dequantiled model.
Consider the functionals given by Definition (ref), then when we employ the OLS estimator for a model with mildly integrated regressors, $\upgamma_x \in (0,1)$, the following limit result holds
since the term $\frac{1}{ n^{ 1 + \upgamma_x } } \sum_{t=1}^{ \floor{\lambda n} } \boldsymbol{x}_{t-1} \boldsymbol{x}_{t-1}^{\prime} \overset{ \mathbb{P} }{ \to } \lambda \boldsymbol{V}_{xx}$ converges in probability. A similar limit result holds for the IVZ based functional such that $\hat{\mathcal{J}}^{ivz}_{nx} \big( \lambda, \uptau_0, \widehat{ \boldsymbol{\theta} }^{ivz}_n( \uptau_0 ) \big) \Rightarrow \sqrt{\uptau_0 ( 1 - \uptau_0 ) } \times \mathcal{N} \big( \boldsymbol{0}, \lambda \boldsymbol{I}_p \big)$, regardless of whether the regressors exhibit high persistence. On the other hand, the OLS based functional has a nonstandard limit distribution with regressors of high persistence. In the case of the IVX functional one needs to consider the limit result for these two classes of persistence separately. These conjectures are summarized and proved by Proposition (ref) in the next section where we formalize the test statistics.
We consider as detectors two types of test statistics commonly employed in the literature related to structural break testing methodologies. The first type of test corresponds to the fluctuation type statistic studied by qu2008testing specifically for a quantile regression model, while the second type of test corresponds to the Wald statistic proposed by the seminal paper of andrews1993tests for the linear regression model. Both test statistics utilize the supremum functional since the underline assumptions allow for a structural break for the coefficients of the nonstationary quantile predictive regression model at an unknown break-point location within the full sample.
Therefore, the null hypothesis of interest (e.g., see (ref)) is formulated as below
where $\boldsymbol{\theta}^{(j)}_n ( \lambda ; \uptau_0 ) = \big( \alpha^{(j)}_n(\lambda ; \uptau_0), \boldsymbol{\beta}^{(j)}_n(\lambda ; \uptau_0)^{\prime} \big)^{\prime}$, for $j \in \left\{ 1, 2 \right\}$ and the location of the break-point is denoted with $\kappa = \floor{ \lambda n }$ for some $0 < \lambda < 1$. Specifically, the implementation of structural break tests for the purpose of detecting parameter instability in nonstationary quantile predictive regressions is a novel aspect in the literature. To facilitate for the development of large sample theory, Assumption (ref) presents necessary conditions relating the matrix moments to the quantile structure of the model.
Assumption (ref) (a) and (b) are standard convergence in probability limits for the nonstationary quantile predictive regression model. Assumption (ref) (c) is employed for the convergence of the weighted empirical process $n^{- 1 / 2} \sum_{t =1 }^{ \floor{ \lambda n } } \boldsymbol{x}_{t-1} \big( \uptau_0 - \mathds{1} \left\{ F_{y | x } ( y_t ) \leq \uptau_0 \right\} \big)$. Furthermore, Assumption (ref) (d) ensures stochastic equicontinuity (see, Chapter 2 in van1996WeakConergence) of the sequential empirical process based on estimated quantile regression residuals, which is needed to establish weak convergence of the tests (see bai1996testing). Moreover, in the case of possibly nonstationary regressors, since standard quantile regression estimators follow a locally uniform weak convergence (see De2006extreme) then invariance principles hold uniformly for $\lambda \in (0,1)$.
Specifically, since we assume that the true break point is unknown, we need to search over all possible candidate subsets within the full sample. Furthermore, according to qu2008testing recentering $\hat{\mathcal{J}}_n \big( \lambda, \uptau_0, \widehat{ \boldsymbol{ \beta} }_n( \uptau_0 ) \big)$ by the quantity $\lambda \hat{\mathcal{J}}_n \big( 1 , \uptau_0, \widehat{ \boldsymbol{ \beta} }_n( \uptau_0 ) \big)$ often yields better finite sample performance. Such considerations lead to the following test statistic:
where $\left\lVert . \right\rVert_{ \infty }$ is the $\mathsf{sup}-$norm such that for a generic vector $\mathsf{z} = \left( \mathsf{z}_1,..., \mathsf{z}_p \right)$ implies that $\left\lVert . \right\rVert_{ \infty } := \mathsf{max} \left( | \mathsf{z}_1 |, ..., | \mathsf{z}_p | \right)$ (see, Koenker2002inference).
We focus on the implementation of two different estimation methodologies. Therefore, to investigate the practical use of the proposed fluctuation type test for structural break detection in the nonstationary quantile predictive regression model, we consider the asymptotic distribution of the test statistics according to the estimator employed to construct the test function. Thus, Proposition (ref) summarizes the formulations of the test according to the estimation methodology employed for a fixed quantile level $\uptau_0 \in (0,1)$.
A necessary condition to apply weak convergence arguments that yields invariance principles for partial sum processes in the Skorokhod space $\mathcal{D} \left( [0,1] \right)$ follows
The proposed test statistics extend the fluctuation type tests studied by qu2008testing, to the nonstationary quantile predictive regression model of our setting. We compare the instrumental variable based method to the classical OLS approach for constructing the fluctuation test. Moreover, we employ the IVZ estimator given by Proposition (ref) (iii), which replaces the original covariate vector with the constructed instruments as a post-estimation correction method that permits to obtain further simplifications of asymptotic terms. Our asymptotic theory analysis shows that the fluctuation type test weakly converges into a Brownian bridge limit when the IVZ estimator is employed and the same limit holds for both OLS and IVX based tests under mild integratedness.
On the other hand, under high persistence the fluctuation type test based on the OLS estimator is proved to have a weak convergence into a nonstandard and nonpivotal asymptotic distribution.\footnote{Similar results are proved by Katsouris2021breaks who propose structural break tests for the linear predictive regression model. Within the particular framework, the author proves that the asymptotic distribution of sup-Wald type statistics when the regressors of the model exhibit high persistence are nonstandard (when the OLS estimator is employed), since the asymptotic distribution of the test depend on the unknown persistence coefficient.} A similar asymptotic result holds for the corresponding IVX based test statistic when $\upgamma_x = 1$ such that
unless we assume that the exponent rate of persistence is such that $\upgamma_x (0, \upgamma_z)$.
Fluctuation type statistics have been previously examined as a detector for parameter instability in studies such as kuan1994implementing, chu1996monitoring and leisch2000monitoring. More precisely, the statistical advantage of these statistics lies in the fact that they utilize properties of the maximum of Wiener processes (see revesz1982increments) and consequently weakly convergence arguments as defined by billingsley1968convergence can be employed to derive their asymptotic behaviour. Furthermore, these class of tests belong to the same class as CUSUM type tests although in the latter case the test function is constructed based on regression residuals (see kulperger2005high).
Overall within the nonstationary quantile predictive regression model framework of our study, we observe some important conclusions for the implementation of fluctuation type tests as structural break detectors. Firstly, the asymptotic distributionof the test statistic $\mathcal{SQ}( \lambda ; \uptau_0 )$ depends on the chosen estimator when constructing the test function as seen from the limiting distributions when constructing the test based on the two estimators, under high persistent regressors. On the other, under the assumption of mildly integrated regressors these test statistics weakly converge into a Brownian bridge type limit regardless of the chosen estimator when constructing the test function. Therefore, for the particular persistence class fluctuatuon type tests depends only on the number of parameters subject to structural break since the nuisance coefficient of persistence that captures the nonstationary properties of predictors is filtered out. Secondly, these test statistics do not require to estimate the sparsity coefficient $f_{y | x } \big( F^{-1}_{y | x } ( \uptau_0 ) \big)$. According to qu2008testing this occurs since the subgradient, when evaluated at the true parameter value $\boldsymbol{\beta}_0( \uptau_0 )$, does not depend on the distribution of the errors.
Thus, conducting statistical inference with some prior information regarding the presence of persistence regressors, using fluctuation type tests as detectors is preferable to construct the test function based on the IVZ estimator which can lead to conventional inference methods (e.g., using tabulated critical values).
Next, we examine the self-normalized\footnote{Specifically a relevant application of self normalized statistics is the construction of confidence intervals for model parameters as in the study of shao2010self. We leave this aspect for future research.} property of Wald type tests by deriving the related asymptotic theory. Due to the assumptions and conditions under which we construct the proposed test statistics, to examine their limit distributions stochastic equicontinuity arguments are necessary in proofs (see, newey1991uniform).
We now introduce the Wald type tests based on the two estimation methodologies which we focus on (OLS versus IVX based tests). The formulation of the model under the null and under the alternative hypothesis can change the interpretation in the notation we employ for model parameters. One approach is to employ the formulations given by expressions (ref)-(ref). In that case, the Wald type test is constructed for testing the null hypothesis that the parameter vector $\boldsymbol{\upbeta}_{(2)}(\uptau_0)$ which implies that we are testing the null hypothesis that $\boldsymbol{\beta}_1(\uptau_0) - \boldsymbol{\beta}_1(\uptau_0) = \boldsymbol{0}$. However, one has to be careful when constructing the covariance matrix as the regressors need to be adjusted accordingly. Furthermore, a second approach is to construct the stacked regressors that correspond to the time series observations from each of the two subsamples.
To simplify the notation, we employ the second approach and denote with $\widehat{ \boldsymbol{\beta} }_{1} ( \lambda ; \uptau_0 )$ the estimator of $\boldsymbol{\beta}_0 ( \uptau_0 )$, using observations up to $\kappa = \floor{ \lambda n }$ for some $0 < \lambda < 1$ and with $\widehat{ \boldsymbol{\beta} }_{2} ( \lambda ; \uptau_0 )$ the corresponding parameter estimator based on the remaining observations in the sample. Moreover, denote with $\tilde{\boldsymbol{X}} \equiv \big[ \boldsymbol{X}_1 \ \boldsymbol{X}_2 \big]$ and $\boldsymbol{R} \equiv \big[ \boldsymbol{I}_p \ - \boldsymbol{I}_p \big]$ the selection matrix, then $\Delta \widehat{ \boldsymbol{\beta} }_n ( \lambda ; \uptau_0 ) := \big( \widehat{ \boldsymbol{\beta} }_{2} ( \lambda ; \uptau_0 ) - \widehat{ \boldsymbol{\beta} }_{1} ( \lambda ; \uptau_0 ) \big)$. Then, the Wald test for testing the null hypothesis that the two regimes have equivalent parameter vectors, based on the OLS estimator and some unknown break-point $\kappa = \floor{ \lambda n }$ is formulated as below
where $\widehat{\boldsymbol{V}}_n( \lambda ; \uptau_0 )$ is a consistent estimate of the limiting variance of $\Delta \widehat{ \boldsymbol{\beta} }_n ( \lambda ; \uptau_0 )$ under the null hypothesis, $\mathcal{H}_0^{(A)}$, of no parameter instability for a fixed quantile level $\uptau_0 \in (0,1)$. The variance estimator is a key quantity which will affect the robustness of Wald type tests and takes different forms depending on the estimation method we employ when fitting the nonstationary quantile predictive regression.
Consider the following limiting variance estimate
where $\boldsymbol{\Omega}_0 = \boldsymbol{H}_0^{-1} \boldsymbol{D}_0 \boldsymbol{H}_0^{-1}$ is the unknown variance of the OLS-Wald test.
Furthermore, define with
where $f_{y | x } ( . | \boldsymbol{x}_{t-1} )$ and $F_{y | x } ( . | \boldsymbol{x}_{t-1} )$ are the conditional density and conditional cumulative distribution function of $y_t$ respectively (see goh2009nonstandard and aue2017piecewise).
However, the particular form of the asymptotic variance only holds under the assumption of stationarity in which case the variance estimator simplifies since it does not depend on any nuisance parameters (such as the coefficients of persistence) and thus equivalent matrix moments to the expressions in qu2008testing (see also andrews1993tests) hold. In our setting, we consider alternative variance estimators based on both the chosen estimator and the persistence class of regressors.
In any of the aforementioned cases, the supremum Wald test is defined as below
In practise, a symmetric trimming coefficient is employed such that $0 < \eta < 1 / 2$ which lead to the admissible set $\Lambda_{\eta} := [ \eta, 1 - \eta ]$, in order to ensure that the test statistics converge in distribution under the null hypothesis. Therefore, we investigate the asymptotic behaviour of the OLS based Wald test for the nonstationary quantile predictive regression model given by (ref)-(ref) which encompasses the case of stationary regressors. Under the assumption of stable regressors the asymptotic variance of the OLS-Wald test is equivalent to the case when regressors are stationary and ergodic.
Under the assumption of nonstationarity, the formulation of the OLS-Wald test statistic requires to determine the asymptotic behaviour of the following quantities
and $\Delta \widehat{ \boldsymbol{\beta} }^{ols}_n ( \lambda ; \uptau_0 ) = \widehat{ \boldsymbol{\beta} }_{2}^{ols} ( \lambda ; \uptau_0 ) - \widehat{ \boldsymbol{\beta} }_{1}^{ols} ( \lambda ; \uptau_0 )$, for some $0 < \lambda < 1$ and $\uptau \in (0,1)$. Then, due to orthogonality of the two set of regressors the covariance matrix simplifies into the following expression:
As we discussed previously, the limit theory of the Wald type statistics for both the OLS and IVX estimators seems more difficult than the limit results for the fluctuation type tests, especially due to the dependence of regressors and parameter estimates to the nuisance parameter of persistence. Therefore, here we generalize the functionals introduced in Section (ref) and (ref) in order to study their asymptotic properties which can alleviate the difficulty in obtaining stochastic approximations under the presence of abstract degree of persistence; simplifying this way derivations for their limit distributions.
We focus on the asymptotic theory for the OLS-Wald test statistic, which is employed as a structural break detection for the nonstationary quantile predictive regression model. In particular, we investigate the asymptotic behaviour of the partial sum processes for the OLS based functionals we introduced previously. For a general parameter vector $\boldsymbol{b} \in \mathbb{R}^p$ we denote with $S_n ( \lambda, \uptau_0 , \boldsymbol{b} )$ the partial sum given by the following expression
where $\psi_{ \uptau } ( \mathsf{u} )$ is such that $ \psi_{ \uptau } ( \mathsf{u} ) := \big[ \uptau - \mathds{1} \left\{ \mathsf{u} \leq 0 \right\} \big]$. Therefore, $S_n ( \lambda, \uptau_0, \boldsymbol{b} )$ is written as
Following conventional laws of invariance principles for $\textit{i.i.d}$ partial sums the induced sequence of increments are tight within a suitable topological space\footnote{Related theory to weak convergence arguments of partial sum processes can be found in various studies. For instance, WangPhillips2012specification redefine the innovation sequence of their model, in the context of specification testing under nonstationarity, to a richer probability space which contains a standard Brownian motion. To do this, a triangular representation of the near unit process is employed in order to investigate the asymptotic behaviour of the transformed functional with respect to this triangular array. Although this would be an interesting way to represent our functionals we avoid the introduction of triangular arrays which could be more challenging to handle.} and in fact they converge weakly to Gaussian processes. Therefore, investigating the asymptotic behaviour and properties of these functionals is useful for the development of the asymptotic theory of the proposed test statistics as well as for other applications. In order to do this, we consider centering the quantity $\mathds{1} \big\{ y_t - \boldsymbol{x}_{t-1}^{\prime} \boldsymbol{b} \leq 0 \big\}$ at its expectation conditional on $\boldsymbol{x}_{t-1}$, instead around the quantile level $\uptau_0$. Furthermore, since we assume that the nonparametric functional given by expression (ref) can be employed as a stochastic process in $\mathcal{D} \left( [0,1] \right)$, which is the topological space of all right continuous functions with left limits then we can derive an invariance principle for this partial sum process.
To simplify derivations for the asymptotic theory, and following qu2008testing, we define the quantity $\widetilde{S}_n \big( \lambda, \uptau_0, \boldsymbol{b} \big)$ with the expression below
where $F_{y | x } \big( \boldsymbol{x}_{t-1}^{\prime} \boldsymbol{b} \big)$ is assumed to be monotonic. A necessary and sufficient condition for the monotonicity property of the cumulative distribution function to hold is presented by Lemma (ref) (see, also Lemma A1 in qu2008testing). Consequently, we obtain that
Our research objective here is to establish the weak convergence argument that holds for the random quantity $S_n \left( \lambda, \uptau_0, \boldsymbol{b} \right)$ on $\left( \mathcal{D} [0,1] \right)^2$ by accommodating for the different convergence rates with appropriate matrix normalizations according to the estimator employed in each case. Moreover, the limit results of these functionals\footnote{Notice that the proposed functionals in this paper similar to the framework of qu2008testing, clearly depend on the estimated parameter vector. Therefore, in our setting the assumption of a nonstationary quantile model contributes to some challenging asymptotic theory aspects, which we are motivated to tackle. Moreover, we shall note that a related large stream of literature considers functionals of estimated residuals with associated test statistics such as CUSUM and CUSUM-square commonly employed in the change-point literature. We avoid presenting the related literature here, as it beyond our scope.} can be utilized to show the following type of stochastic convergence
where $\widehat{\boldsymbol{\beta}}^{ols}_{1} ( \lambda ; \uptau_0 )$ is the quantile regression OLS based estimator\footnote{For instance, for the IVX estimator based on observations of the full sample the following order of convergence holds: $\textcolor{blue}{ n^{ \frac{1 + \upgamma_x }{2} } ( \widehat{ \boldsymbol{\beta} }^{ivx}_{n} ( \uptau ) - \boldsymbol{\beta}_0 ( \uptau ) ) = \mathcal{O}_{ \mathbb{P} }(1)}$, which is proved by Corollary (ref) in the Appendix of the paper (see also Theorem 3.1 in lee2016predictive).} that corresponds to the subsample $1 \leq t \leq \floor{ \lambda n}$ for some $0 < \lambda < 1$ and $\uptau_0 \in (0,1)$, when the quantile regression has no model intercept. Intuitively, when the model structure incorporates both intercept and slopes then the different convergence rates of these coefficients due to the presence of nonstationarity is accommodated with the use of embedded normalization matrices. Therefore, for the remaining of this section, we suppose that the parameter vector is of the form $\boldsymbol{ \theta } ( \uptau ) = \big[ \alpha (\uptau), \boldsymbol{\beta} (\uptau)^{\prime} \big]^{\prime}$.
Consider the nonstationary quantile predictive regression (ref)-(ref) which includes a model intercept. Then, testing for a structural break via a Wald type formulation based on the OLS estimator implies to use the parameter vector $\boldsymbol{\theta}(\uptau)$ instead of $\boldsymbol{\beta}(\uptau)$.
The proof of Proposition (ref) can be found in the Appendix of the paper. Notice that $\boldsymbol{\Sigma}^{-1} _0 \big( \lambda; \uptau_0 \big)$ represents the weakly convergence result of the inverse of the covariance matrix of the stochastic process $\sqrt{n} \big( \hat{\boldsymbol{\upbeta}}_{(2)}(\uptau_0) - \boldsymbol{\upbeta}_{(2)}(\uptau_0) \big)$, in which case $\mathcal{W}_n(\uptau_0)$ is the Wald statistic for testing the null hypothesis $\mathcal{H}_0: \hat{\boldsymbol{\upbeta}}_{(2)}(\uptau_0) = \boldsymbol{0}$.
The instrumentation methodology proposed by PhillipsMagdal2009econometric has been proved to be robust in filtering abstract degree of persistence in predictive regression models (see also Phillips2013predictive, Phillips2016robust). Our research objective in this section is to study the asymptotic behaviour of the proposed structural break tests based on the endogenous instrumentation procedure in nonstationary quantile predictive regressions. In particular, the self-normalized sup IVX-Wald test function has been recently examined by Katsouris2021breaks as break detector for coefficients of linear predictive regressions. Specifically, the supremum IVX-Wald test corresponds to the maximum\footnote{Further details regarding the formulation of Wald type tests and asymptotic theory is presented in the seminal study of andrews1993tests. The particular framework propose for structural change tests in linear regression models under the assumption of stationary and ergodic time series.} of a sequence of test statistics constructed based on sequential sample splitting locations such that $\kappa = \floor{ \lambda n }$ where $\lambda \in \Lambda_{\eta} := [ \eta, 1 - \eta]$ with $0 < \eta < 1/2$. Furthermore, we employ the dequantiled model structure and denote with $\widehat{\boldsymbol{\beta}}_{1}^{ivx} ( \lambda; \uptau_0 )$ and $\widehat{\boldsymbol{\beta}}_{2}^{ivx} ( \lambda; \uptau_0 )$ the IVX based estimators for the two sub-samples occurred at each splitting step. Therefore, these estimators are computed via the following expressions
where $\kappa = \floor{ \lambda n }$ for some $0 < \lambda < 1$ and the indicator $j \in \left\{ 0,..., (n-\kappa) \right\}$ shows that a sequence of parameter estimates is obtained by moving along all proportions within the compact set $\Lambda_{\eta} = [ \eta, 1 - \eta]$, to compute the maximum Wald statistic. However, for notation convenience we drop the index notation $(\kappa + j)$ and $(\kappa + 1 + j)$ which can be confused with notation used for time-varying parameter estimates. Also, $\boldsymbol{x}_{1,t-1}:= \boldsymbol{x}_{t-1} \mathds{1} \left\{ t \leq \kappa \right\}$ and $\boldsymbol{x}_{2,t-1}:= \boldsymbol{x}_{t-1} \mathds{1} \left\{ t > \kappa \right\}$. Furthermore, $\tilde{\mathbf{Q}}_1 ( \lambda; \uptau_0 )$ and $\tilde{\mathbf{Q}}_2 ( \lambda; \uptau_0 )$ denotes the covariance matrices which correspond to the two subsample parameter estimates and permits to decompose the covariance matrix\footnote{The decomposition of the covariance matrix for the IVX-Wald statistic can be obtained using a formula for inverting partitioned matrices. In particular, since $\boldsymbol{Z}_1^{\prime} \boldsymbol{X}_2 = \boldsymbol{Z}_2^{\prime} \boldsymbol{X}_1 = 0$ then the matrix inversion formula simplifies further, allowing us to obtain an expression for the variance of the test.} of the test with respect to each regime
Then, under the null hypothesis, $\mathcal{H}_0^{(A)}$, the sup IVX-Wald statistic is formulated as
Theorem (ref) presents the asymptotic distribution of the sup IVX-Wald test under the null hypothesis of a single unknown break-point. Furthermore, it covers some practical considerations that arise in empirical work especially with respect to the persistence properties of regressors. As we can observe from the asymptotic behaviour of the test for local unit root regressors (high persistence), it converges to a nonstandard and nonpivotal distribution. On the other hand, for mildly integrated regressors the test behaves in large samples similar to the sup OLS-Wald test which weakly converge into a Brownian bridge type of limit. In the former case, a comparison of the limit distributions of the two tests does not necessarily indicate which test statistic might have better performance in detecting structural breaks to the coefficients of nonstationary quantile predictive regressions. To investigate the particular aspect, we use simulation experiments where allow us to use a suitable experimental design that accommodates these conditions. The proof of Theorem (ref) can be found in the Appendix of the paper.
Some important implications follow from Theorem (ref). More precisely, our asymptotic theory analysis confirms some of the conclusions drawn in similar studies. For instance, hanson2002tests and seo1998tests (see also georgiev2018testing) demonstrated that testing for structural breaks with integrated regressors based on the OLS estimation method converges to a nonstandard and nonpivotal limiting distribution. Furthermore, although the IVX-Wald statistic is proved to be robust to abstract degree of persistence when testing for parameter restrictions, within the structural break testing framework due to the presence of the unknown break-point location, the sup IVX-Wald test similar to the OLS counterpart is coverages to a nonpivotal limiting distribution, which is not Brownian bridge (as defined in the stationary case) even though it is still tied down.
Nevertheless, some interesting further simplifications occur; for instance under the assumption of mildly integrated regressors, it can be easily proved that the limiting distribution of the sup IVX-Wald test converges to a normalized Brownian bridge limit. This occurs due to the asymptotic matrix moments such that, for $0 < \upgamma_x < 1$ it holds that $\sum_{t=1}^{ n } \boldsymbol{x}_{t-1} \tilde{\boldsymbol{z}}_{t-1}^{\prime} \Rightarrow - \boldsymbol{\Omega}_{xx} \boldsymbol{C}_z^{-1}$ by expression (20) in PhillipsMagdal2009econometric. Thus, it also holds that $\sum_{t=1}^{\floor{ \lambda n } } \boldsymbol{x}_{t-1} \tilde{\boldsymbol{z}}_{t-1}^{\prime} \Rightarrow - \lambda \boldsymbol{\Omega}_{xx} \boldsymbol{C}_z^{-1}$, which implies that $\boldsymbol{\Psi}_c (\lambda) = \lambda \boldsymbol{I}_p$. Furthermore, we also consider the limiting distribution of the sup IVZ-Wald test.
In summary, in this section we show that the limiting distribution of the sup IVX-Wald test statistic under high persistence is nonstandard and nonpivotal. Furthermore, the particular limit result simplifies when regressors in the model are assumed to be mildly integrated resulting to weakly convergence into a Brownian bridge type limit. On the other hand, when we construct the test statistic based on the IVZ estimator then the sup IVZ-Wald test converges into a Brownian bridge type of limit regardless of the degree of persistence. Lastly, for a known break-point Wald type tests converge to a nuisance-parameter free limiting distribution, simplifying this way statistical inference. The proof of Corollary (ref) can be found in the Appendix.
Another important aspect we consider for the development of the asymptotic theory of the paper, is the classical result of Huber for models with nonstandard conditions such as quantile regression models. More specifically, the first order condition (FOC) defined as the right derivative of the objective function plays a key role in deriving the asymptotic theory of estimators for the quantile model. In particular, we can show that the parameter vector estimator solves these FOC and then apply a Bahadur representation for the estimator. All these results hold with almost surely convergence in large samples.
Practically, it is unclear how well the asymptotic theory can provide reliable reference and guidance in finite samples when applied to time series data since usually they can exhibit abstract degree of persistence. However, under the assumption that regressors incorporated in the quantile predictive regression model are generated by near unit root processes for which their asymptotic behaviour is well-understood, then our test statistics can provide an indication regarding the ability of the testing procedures in detecting structural breaks in coefficients of nonstationary quantile predictive regression models. Thus, to investigate the finite sample performance of the proposed tests for their adequacy in detecting parameter instability we focus on the empirical size simulation results as well as on asymptotic power analysis under relevant sequence of local alternatives.
We conduct a number of Monte Carlo experiments to evaluate the performance of the limit distribution of the Wald and fluctuation type statistics against the conventional $\chi^2$ asymptotic approximation. We simulate the following data generating process
where $\boldsymbol{x}_t = \big( x_{1t}, x_{1t}, x_{1t} \big)^{\prime}$, $\boldsymbol{v}_t = \big( v_{1t}, v_{1t}, v_{1t} \big)^{\prime}$ and $\varrho_n ( c_j, \upgamma_x ) = \left( 1 + \displaystyle \frac{c_j}{ n^{ \upgamma_x }} \right) $. Then, the innovation sequence of the model, denoted with $\boldsymbol{e}_t = \left( u_t, \boldsymbol{v}_t \right)^{\prime}$ is generated such that $\boldsymbol{e}_t \sim \mathcal{N} \left( \boldsymbol{0}_{(p+1) \times 1}, \boldsymbol{\Sigma}_{ee} \right)$, where $\boldsymbol{\Sigma}_{ee}$ is an $( p+1 ) \times (p+1)$ positive-definite covariance matrix with a pre-specified variance-covariance structure given as below
where the matrix $\boldsymbol{\Sigma}_{vv}$ is of full rank $p$, resulting to a nonsingular matrix $\boldsymbol{\Sigma}_{ee}$.
The coefficients of persistence are such that $c_j \in \left\{ -1, -2, -5 \right\}$ and $\upgamma_x$ is defined to be $\upgamma_x = 1$ to simulate near unit root predictors and $\upgamma_x = 0.75$ to simulate mildly integrated predictors. Moreover, we consider different sample size such that $n \in \left\{ 250, 500, 750, 1000 \right\}$. Under the null hypothesis of no parameter instability we use the following parameters $\alpha = 1$, $\beta_1 = 0.25, \beta_2 = 0.75, \beta_ 3 = -0.50$ and construct the proposed test statistics with significance level $\upalpha = 5 \%$. For the IVX instrumentation we use $c_z = 1$ and $\upgamma_z = 0.95$.
In particular, to construct the test statistics the simulated pair $\left\{ y_t, \boldsymbol{x}_t \right\}_{t=1}^n$ is formulated:
where $\kappa = \floor{ \lambda n }$ and the search over all possible subsets occurs for values of $\lambda \in \Lambda_{\eta}$. Denote with $\boldsymbol{\theta}^{(j)} ( \uptau) = \left( \alpha^{(j)} ( \uptau ), \beta_1^{(j)} ( \uptau ), \beta_2^{(j)} ( \uptau ), \beta_3^{(j)} ( \uptau ) \right)^{\prime}$ with $j \in \left\{ 1, 2 \right\}$ to be the quantile dependent parameter vector of each of the two regimes, for a fixed quantile $\uptau_0$ that belongs in the compact set\footnote{Notice that the compact set $\mathcal{T}_{\eta}$ falls strictly within the unit interval to allow the conditional distribution to have an unbounded support.} $\mathcal{T}_{\eta}$ such that $0 < \eta < \uptau_0 < 1 - \eta < 1$.
Then, the testing hypothesis of interest is formulated as below
with a fixed quantile $\uptau_0 \in \mathcal{T}_{\eta} := [ \eta, 1 - \eta ]$, for an unknown break-point location $\kappa= \floor{ \lambda n}$ where $0 < \lambda < 1$ and a significance level $\upalpha = 5 \%$.
One important aspect for the correct implementation of the fluctuation based tests (which involves the estimation of a subgradient) is that no nuisance parameter is needed (in the case when the LUR process is not employed). More precisely, for the Wald type statistics, we need a consistent estimate of the variance-covariance matrix $\boldsymbol{\Omega}_0$. In particular, it requires estimating the following matrix\footnote{Notice that the estimation of the sparsity function will affect the finite-sample performance of the test statistics if not consistently estimated.}
A discussion regarding methodologies for estimating the matrix given by expression (ref) can be found in qu2008testing. On the other hand, when we implement the quantile predictive regression model with the presence of persistence covariates, then we will need to examine the asymptotic properties of the above covariance estimator since it will be a function of the unknown coefficient of persistence.
In summary, in this paper we take the position than when using Wald type statistics as structural break detectors in nonstationary quantile predictive regression models, the chosen estimator will affect the limit distribution of the tests and thus its finite-sample performance, especially under the presence of high persistence regressors. In particular, we have proved that when selecting the OLS estimator then the limit distribution is nonstandard and nonpivotal making inference challenging since critical values can be constructed only with the use of bootstrap-based methodologies. On the other hand, when the IVZ estimator is chosen then the limit distribution is proved to be nuisance-parameter free regardless of the persistence properties driving the behaviour of regressors employed when estimating the quantile predictive regression model.
Another relevant aspect to investigate is the sensitivity of the proposed test statistics to the break-point location within the full sample. Therefore, for instance we are interested to examine whether the proposed tests have roughly equal sensitivity to a break occurring early or late in the sample (see, also leisch2000monitoring). We expect that the break-point location will not affect the power performance of the proposed test statistics especially due to the fact that we do not operate within a sequential monitoring scheme in which case parameter estimates and functionals are updated continuously.
The simulation results of the Monte Carlo experiments focus on the performance of the tests by obtaining the empirical size and empirical power. In particular, we evaluate the performance of the tests at different quantiles by employing the test statistics that correspond to the fixed quantile level. More precisely, this allow us to check for structural breaks at the median, or at the upper and lower quantiles for example, thus observing the presence of parameter instability at different levels of the predictant with respect to persistent predictors. Our simulation experiments verify the empirical and theoretical results observed by Katsouris2021breaks in the case of the linear predictive regression model, that is, a trend of over-rejecting the null hypothesis when the OLS estimator is employed when constructing structural break tests\footnote{Notice that the concept of spurious break is discussed by hansen2000testing who emphasize that instability in the exogenous variables can cause over-rejection in the standard OLS-based tests.}.
Overall, the finite-sample results reflect the main conjectures presented in the asymptotic theory of the paper and appear to be reasonable for practical use in testing for structural breaks in the coefficients of nonstationary quantile predictive regression models, especially with persistent regressors and endogeneity\footnote{Notice that exogeneity plays an important role in dealing with non-stationary variables. More specifically, in Chapter of banerjee1993co it is mentioned that dynamic regression equations in which the conditioning is on weakly or strongly exogenous variables (for the parameter of interest) provide asymptotically unbiased estimates.}. In practise, in those cases in which the exact $\upalpha-$level critical values for $0 < \upalpha < 1$ depend on the unknown parameters of persistence, we employ bootstrap based resampling methods for inference purposes. Therefore, we can observe that these near unit root processes driving the regressors of the model, can affect the ability of the proposed structural break tests for detecting parameter instability in quantile predictive regression models. Specifically, in the simulation study of WangPhillips2012specification, the authors mention that serial dependence can affect power. Furthermore, the lower long-run signal strength in the regressor tends to reduce disciminatory power.
Our empirical application is concerned with the monitoring of the US housing price index returns (HPI). Using macroeconomic variables with predictive regression models has been demonstrated in various studies. For instance, the empirical study of paye2012deja verifies the episodic predictability conclusions documented in the literature such as in gonzalo2012regime, gonzalo2017inferring (see also demetrescu2020testing). The author finds statistical evidence of predictability in relation to countercyclical macroeconomic events when forecasting volatility using predictive regressions with macroeconomic covariates. Furthermore, atanasov2020consumption investigate the impact of consumption fluctuations on predictability of expected returns using the IVX filter. Overall, our empirical study focuses in the implementation of the proposed test statistics\footnote{Notice, that a different stream of literature proposes testing procedures for detecting market exuberance and bubble effects. Our empirical application is concerned with the detection of structural breaks in the data based on the nonstationary quantile predictive regression.}; thus our research goal here is to test for structural breaks in the relation between the predictant and the predictors at various quantile levels of the underline data specific distribution.
For the empirical application of our study, we utilize the dataset of yang2020testing that includes the US housing price index returns along with ten common macroeconomic variables. Specifically, the HPI covers more transactions and longer time interval, and thus can well represent the trend of the national-wide housing price such as the housing bubble collapsed during the 2007 subprime mortgage crisis. Furthermore, based on the HPI the authors obtain the quarterly growth rate of the housing price and use this rate as the dependent variable. More precisely, the ten macroeconomic variables are collected from FRED, and all data are quarterly between 1975:Q1 and 2018:Q2.
We begin our analysis by applying standard unit root tests to the predictors\footnote{Notice that we have $t = 1,..., 174$ time series observations which correspond to quarterly economic indicators and macroeconomic variables.} employed for the quantile predictive regression model. Furthermore, we test each individual predictor separately for the presence of parameter instability using a toolkit of various structural break tests commonly employed in the literature. In particular, testing for breaks in housing price indices has been previously studied by canarella2012unit. However, testing for quantile predictability as well as testing for breaks in nonstationary quantile regressions is a novel aspect not previously examined in the literature.
Secondly, we implement the joint IVX-Wald statistic under the null hypothesis that all slope coefficients simultaneously equal to zero\footnote{We consider rejections of the null hypothesis at significance level $5\%$ to match the rejection probabilities employed in the simulation study of the paper.}. More precisely, the particular hypothesis correspond to the null hypothesis of no quantile predictability. Thus, formulating the model in this manner allow us to investigate whether there is a stable relation between regressand and regressors at a specific quantile level\footnote{In particular, the empirical study presented by lee2016predictive demonstrates statistical evidence of predictive ability using the nonstationary qunatile predictive regression model, at some specific quantiles of stock returns such as at lower or upper quantiles while on the other hand evidence of predictability disappear at the median of the conditional distribution of stock returns. } $\uptau_0 \in (0,1)$.
Thirdly, we implement the joint IVX-Wald statistic under the null hypothesis that at least two of the slope coefficients have no structural break throughout the sample.
In this paper we develop a framework for structural break detection for nonstationary quantile predictive regression models, under the null hypothesis of no structural break\footnote{Notice that we avoid to explicitly use the terminology of a null hypothesis of “stationarity” versus an alternative hypothesis of “non-stationarity” (see pitarakis2014joint and Kwiatkowski1992testing). Specifically, within our setting these two terms are interpreted in relation to the persistence properties of model predictors rather that with respect to the parameter constancy of coefficients in terms of temporal dependence. Related limit theory and conditions relevant to temporal dependence specifically for quantile and tail empirical processes can be found in Chapter 5 of De2006extreme.}. A major challenge when deriving the asymptotic theory of these structural break tests is to obtain nuisance-parameter free limit distributions which is not a trivial task due to the stochastic approximation terms that depend on nuisance parameters, such as higher order covariance terms as functions of the coefficients of persistence that capture the time series properties of regressors. More precisely, we establish the asymptotic distributions for both Wald type (i.e., as in andrews1993tests) and fluctuation type tests (i.e., as in qu2008testing) with respect to two different estimation methods, that is, the OLS and IVX estimators\footnote{Notice in the study of Phillips1988asymptotic the authors demonstrate the asymptotic equivalence of OLS and GLS based estimators in regression models with integrated regressors.}. Our test statistics show to have good finite-sample properties as shown by the Monte Carlo experiments in which we obtain the empirical size and power.
Firstly, we verify that indeed the self-normalization of Wald type statistics when testing the null hypothesis of no predictability in quantile predictive regressions results to a nuisance-free distribution, that is, ensuring their pivotal property (as also proved by lee2016predictive for abstract degree of persistence). Secondly, we demonstrate that the limit distribution of the proposed test statistics for structural break detection is not depending on the particular choice of the estimator of the quantile predictive regression model under mildly integrated; however under high persistence the choice of the estimator alters the limit theory due since different weakly convergence arguments apply. Furthermore, keeping the quantile level fixed versus testing for breaks across multiple quantile levels requires to consider extending the limit result into the two-parameter Gaussian process, for the latter case. Moreover, bootstrap-based methodologies can be applied when the limit distribution is nonstandard, allowing to infer regarding the presence of structural breaks under these conditions (e.g., high persistence).
Further research worth mentioning includes to extend the current framework proposed in this paper, for quantile predictability tests robust to parameter instability. The specific application has important implications, from both the theoretical as well as the empirical perspective, especially under nonstationarity, since currently the common practise in the literature is the proposition of methods that investigate these two aspects separately. Additionally, our framework can be extended to the alternative hypothesis of multiple structural breaks as in qu2008testing as well as within a multivariate setting such that the framework proposed by the study of qu2007estimating.