Asymptotic Theory for Unit Root Moderate Deviations in Quantile Autoregressions and Predictive Regressions

Extracted main text — title through conclusion, appendix excluded. This is what our citation measures are computed over, published so the extraction can be checked by eye.

77,062 characters · 14 sections · 135 citation commands

Rendered from LaTeX for readability, not typeset faithfully. Citation keys are highlighted; maths is left as source; figures, tables and equation environments are summarised rather than reproduced; unrecognised commands are greyed out so nothing is silently dropped. Email addresses are removed.

\pagenumbering{roman}

abstractWe establish the asymptotic theory in quantile autoregression when the model parameter is specified with respect to moderate deviations from the unit boundary such that $\rho_n = \left( 1 + \frac{c}{k_n} \right)$ where $\left( k_n \right)_{ n \in \mathbb{N} }$ is a nonrandom sequence that diverges at a rate slower than the sample size $n$. Then, extending the framework proposed by Phillips2007limit, we consider the limit theory for the near-stationary and the near-explosive cases when the model is estimated with a conditional quantile functional form and model parameters are quantile-dependent. A Bahadur-type representation and limiting distributions based on the M-estimators of the model parameters are derived. We show that the serial correlation coefficient converges in distribution to a ratio of two independent random variables. Monte Carlo simulations illustrate the finite-sample performance of the estimation procedure under investigation. \\ Keywords: Quantile autoregressive model, moderate deviations, local-to-unity, near-integrated processes, explosive processes, bahadur representation. \\ JEL classification: C22

\setcounter{page}{1} \pagenumbering{arabic}

Introduction

Moderate deviation principles from the unit boundary for quantile autoregressions are commonly employed when considering the limit distribution of quantile-dependent parameters under regressors nonstationarity. In particular, the development of asymptotic theory for nonstationary quantile time series models has been pioneered by the studies of koenker2004unit, koenker2006quantile as well as koenker2002inference who investigate estimation and inference aspects for regression quantile models (see, also hasan1997robust). Specifically, studies for quantile autoregressive regressions that consider moderate deviations within a unified framework allowing to investigate the asymptotic behaviour of estimators with respect to different regimes of stability such as stable, unstable and explosive processes has seen less attention in the literature. Therefore, our main objective is to use the moderate deviation principles in order to derive the limiting distribution of the autoregressive coefficient when considering deviations from the unit boundary under a conditional quantile functional form. The present paper builds on the framework proposed by Phillips2007limit (see, also giraitis2006uniform and huang2014limit) that corresponds to the linear autoregressive model under nonstationarity as well as the study of kong2015m and wang2022asymptotics who develop limit theory for moderate deviations in autoregressive models based on M-estimators.

In this line of literature, the two relevant research questions are summarized by chan2006quantile:

quotation"The study of the unit root AR(1) model has been actively pursued by statisticians and econometricians alike, and a related question that needs to be addressed is what happens to the limiting distribution of the test statistics when the autoregressive parameter $\theta_n$ is close to the unit boundary? Consequently, when the autoregression coefficient is expressed with respect to the local-to-unity parametrization, what kind of approximation should be used for the distribution of the test statistics?.

The seminal studies of chan1987asymptotic1 and phillips1987time, phillips1987towards tackle exactly these nonstandard statistical problems via their triangular array framework. The framework of the nearly nonstationary AR(1) model allows to establish the limiting distributions of the least squares estimator for $\theta_n$ under the assumption that the conditional variance of the model is finite. Moreover, the properties of the least squares estimator when the true stochastic process is nearly integrated are investigated by chan1988parameter, chan1990inference, chan1989first, knight1987rate, rao1978asymptotic, lai1982least, cox1991maximum, larsson1995asymptotic, cavaliere2002bounded, Buchmann2007asymptotic, Phillips2007limit and duffy2021estimation among others. Furthermore, the asymptotic theory for moderate deviations from a unit root in autoregressive models is presented by fountis1989testing, jiang2015moderate who focus on the aspect of dependent errors in AR(1) models and yabe2012limiting who obtain limit results for MA(1) time series models.

On the other hand, the properties of nonstationary autoregressive models for the case of an explosive autoregressive coefficient is also of interest. In particular, white1958limiting obtained the limit distribution of an explosive serial correlation coefficient (see, also mann1943statistical). Limit theory for moderate deviations on the explosive side of unity (e.g., mildly explosive case) were developed by Buchmann2007asymptotic, aue2007limit, magdalinos2012mildly, arvanitis2018mildly, oh2018mildly, lee2018limit, proia2020moderate, yu2021inference, hirukawa2021asymptotic, liu2022mildly. Recently, Jian2022 derived the limit theory for the ordinary least squares estimator in the explosive first-order Gaussian autoregressive process using a set of deviation inequalities\footnote{The authors obtain the limit theory of Cram\'er-type moderate deviations for the explosive and mildly explosive autoregressive processes.}. In particular, aue2007limit develop the limit theory for the serial correlation coefficient in the mildly explosive case with moderate deviations from the unit boundary. Thus, the model parameter satisfies $\theta_n \to 1 \ \ \text{and} \ \ n \left( \theta_n - 1 \right) \to \infty$, as $n \to \infty$, while for large $n$, $\theta_n > 1$ diverges away from unity but not with the usual convergence rate $\mathcal{O} ( 1 / n)$.

Under the Cram\'er-type moderate deviations framework, there exists positive sequences $\nu_n$ and $\lambda_n$ tending to infinity such that for every $\ell > 0$ as $n \to \infty$ it holds that (e.g., see Jian2022)

align[align omitted — 201 chars of source]

where $F_n (x) $ is the distribution function, satisfying for all $x \in \mathbb{R}$

align[align omitted — 154 chars of source]

We focus on the neighbourhood near the unit boundary, which can be approaching unity from below (near-stationary or near-integrated) or approaching unity from above (near-explosive). Due to the form of the nonrandom sequence such that, $k_n \equiv n^{\gamma}$, the convergence rate towards unity is slower than the sample size $n$. In this case the autoregression coefficient approaches 1 at a rate slower than the usual local alternative as $n$ goes to infinity. When, as $\gamma \to 1$ then $k_n \to n$ which encompasses the conventional local-to-unity parametrization. An alternative form for the convergence rate include the case when $k_n = \sqrt{n}$. To obtain limit results for model parameters based on M-estimation, we employ the Bahadur representation (see, bahadur1966note) that provides a mechanism to facilitate the asymptotic theory, which allows to obtain analytic expressions for quantile-dependent estimators by approximating them with linear forms. Although we do not consider how the presence of serial correlation can affect the limiting distributions, various studies in the literature consider simple implementations of autoregression robust tests as in jansson2004error (see, also vogelsang1998trend). Similar implementations can be considered within the modelling environment of quantile regression models especially of those with possibly nonstationary autoregressive processes with serial correlated innovation terms. kiefer2000simple, (KVB), demonstrate that the properties of Wald-type statistics can be ameliorated if an inconsistent covariance matrix estimator is used and the critical values are adjusted to accommodate the randomness of the matrix employed in the standardization.

Literature Review

Regression asymptotics with roots at or near unity are typically carried out by using autoregressive models with fixed coefficients and then testing for the autoregressive parameter being equal to one, as pointed out by dickey1979distribution (see, also dickey1981likelihood). The idea of developing asymptotics using the local-to-unity parametrization is due to the studies of Cavanagh1985LUR, phillips1987time, chan1987asymptotic1. Phillips2007limit showed that $\left( \hat{\theta}_n - \theta_n \right)$ has a $\sqrt{n k_n}$ rate of convergence and a limit normal distribution when $c < 0$ for $\theta_n = \left( 1 + \frac{c}{n} \right)$,

align[align omitted — 120 chars of source]

Thus, we are interested to establish a martingale central limit theorem for a normalized version of $\sum_{t=1}^n y_{t-1} u_t$ which can give rise to a Gaussian asymptotic distribution for the normalized and centered least squares estimator specifically for the quantile autoregressive model.

A different stream of literature considers a representation of the autoregressive model based on the exponential family with specific canonical parameter. In that case, by expressing the AR(1) model with respect to the canonical parameters of an exponential family one can establish the asymptotic behaviour of related minimal sufficient statistics\footnote{In particular, jansson2006optimal using the local-to-unity parametrization of the autoregression coefficient study the properties of predictive regressions under the assumption of persistence regressors using differential geometry and sufficient statistics arguments to establish the asymptotic theory of estimators and test statistics.} (see, jansson2006optimal). The particular literature is developed under the assumption of a stationary autoregression coefficient such that $| \theta | < 1$ for $y_t = \theta y_{t-1} + u_t$. Furthermore, it has been argued that the Efron curvature depends heavily on the AR parameter, especially near the boundary of the parameter space, and increasingly so with increasing sample size\footnote{Using the local-to-unity parametrization allows the analysis of unit roots and explosive processes, which is necessary to link problems in inference for unit roots to the statistical curvature.} (see, garderen1999exact). Specifically, this implies that when the true parameter value is explosive rather than being stationary, then the asymptotic theory of estimators and test statistics based on OLS estimation are driven by the distribution of the innovations $u_t$. On the other hand, comparing the cases of Gaussian innovations against non-Gaussian innovations (e.g., heavy tailed errors) and an explosive autoregressive parameter then the asymptotic theory of the OLS estimator in these two cases will not necessarily be identical.

Various studies in the literature consider limit results for moderate deviations for M-estimators and quantile processes\footnote{Related theoretical aspects can be found in the book of csorgHo1983quantile (see, also csorgo1986weighted).} in autoregressive models include among others jurevckova1988moderate, knight1998limiting, mao2019moderate as well as kato2009asymptotics who develops asymptotics for Lasso quantile-dependent estimators. Limit theory for moderate deviations from the unit root in the context of quantile autoregressive models are established in the studies of lucas1995unit, abadir2000quantiles, ling2004regression, koenker2004unit, chan2006quantile, kong2015m, zhou2015quantile, wang2022asymptotics and fu2022cqr.

Another important aspect worth mentioning, is the fact that several studies demonstrated that the nuisance parameter of persistence cannot be consistently estimated (see, phillips2001estimate, mikusheva2012one among others). Similarly, when considering the quantile autoregressive model, the availability of a consistent estimator for the unknown coefficient of persistence, $c$, still remains a challenging issue. The limit theory for an autoregressive model which includes a intercept and an autoregression coefficient expressed using the local-to-unity parametrization is studied by liu2018limit and hwang2009asymptotic, however their framework differs from our setting since we consider quantile-dependent parameters. This article, builds on and contributes to both the quantile autoregression literature as well as to the literature of moderate deviations from the unit boundary. We focus on establishing the limit theory for moderate deviations from the unit boundary for both the near-stationary and near-explosive cases in quantile autoregression and quantile predictive regressions. Our main objective is to develop a unified framework for the asymptotic behaviour of the quantile-dependent estimators across the whole spectrum of nonstationarity regimes, including the general near-integrated case assuming that $\theta_n \to 1$ with a rate slower than $1/n$.

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 (billingsley1968convergence).

The rest of the paper is organized as follows. Section (ref), introduces the framework of moderate deviation principles from the unit boundary in quantile autoregressive processes. Section (ref), demonstrates the main results of the paper, that is, the limit theory for the near-integrated and near-explosive cases for the quantile autoregression. Section (ref) presents a short Monte Carlo simulation study while Section (ref) an empirical application. Section (ref) concludes.

Moderate Deviations from the Unit Boundary Framework

Main Terminology, Definitions and Assumptions

Consider the first order autoregressive process with expressed as below

align[align omitted — 98 chars of source]

with a possible sample-size dependent autoregressive root $\theta_n$ and $\left( u_t \right)_{ t \in \mathbb{N} }$ is the innovation sequence\footnote{The innovations $\left( u_t \right)_{ t \in \mathbb{N} }$ is an i.i.d sequence of random variables from a common distribution function $F_{u}$ that satisfies regularity conditions for Lipschitz continuity with zero mean and finite variance $\sigma_{u}$. }.

Moreover, we allow for the presence of a non-zero intercept by considering the process

align[align omitted — 95 chars of source]

where $x_{0t}$ represents the solution of the autoregressive process with $\mu_x = 0$ and $X_0 = 0$, the initial condition of the recursion. Furthermore, since we are interested about the asymptotic stability of the autoregressive process near the unit boundary we allows a local-to-unity parametrization as

align[align omitted — 192 chars of source]

The convergence rate of the autoregression coefficient $\theta_{n}$ is such that $k_n := n^{\gamma}$ where the exponent rate is defined such that $\gamma \in (0,1)$. Specifically, the given convergence rate implies that the $\left( k_n \right)_{ n \in \mathbb{N} }$ sequence increases to infinity at a slower rate than the sample size such that $k_n = o(n)$ as $n \to \infty$.

assumption[Persistence] Consider the autoregressive process $X_t = \theta_n X_{t-1} + \varepsilon_t$ with $\left( \varepsilon_t \right)_{ t \in \mathbb{N}}$. Then, consider the following limit to determine the degree of persistence for the processes \begin{align} \zeta_n := \underset{ n \to + \infty }{ \mathsf{lim} } n \left( \theta_n - 1 \right) \to \zeta \end{align} \begin{itemize} • nearly stable processes: if $\left( \theta_n \right)_{ n \in \mathbb{N} }$ is such that $\zeta = - \infty$ and it holds that $\theta_n \to | \theta | < 1$. • \textit{nearly unstable} processes: if $\left( \theta_n \right)_{ n \in \mathbb{N} }$ is such that $\zeta \equiv c \in \mathbb{R}$ and it holds that $\theta_n \to \theta = 1$. • \textbf{\textit{nearly explosive}} processes: if $\left( \theta_n \right)_{ n \in \mathbb{N} }$ is such that $\zeta = + \infty$ and it holds that $\theta_n \to | \theta | > 1$. \end{itemize}

assumptionDenote with $x_{0n} = 0$, almost surely, for all $n \in \mathbb{N}$. Then, the innovations $\left( \varepsilon_t \right)_{ t \in \mathbb{N} }$ forms a sequence of martingale differences such that as $n \to \infty$ \begin{align} \frac{1}{n} \sum_{ t = 1}^n \mathbb{E} \left[ \varepsilon_t^2 \big| \mathcal{F}_{t-1} \right] &= 1 + o_p(1), \\ \frac{1}{n} \sum_{ t = 1}^n \mathbb{E} \left[ \varepsilon_t^2 \mathbf{1} \left\{ | \varepsilon_t \big| > n^{1/2} m \right\} \big| \mathcal{F}_{t-1} \right] &= o_p(1), \ \ for all \ m > 0, \end{align} where $\mathcal{F}_t := \sigma \big( \varepsilon_s: 0 \leq s \leq t \big)$.

figure[figure omitted — 126 chars of source]

remarkAssumption (ref) provides moment conditions and a uniform integrability condition which induces a restriction on the tail behaviour of the underline distribution of innovations. For asymptotic theory analysis purposes related conditions are imposed to the noise sequences $\varepsilon_t$ such that the corresponding partial sum processes lie in the domain of attraction of functionals of Brownian motions. Thus asymptotic approximations are obtained based on the Ornstein-Uhlenbeck process.

Extending and verifying the existence of Cauchy limiting distribution theory in mildly explosive autoregression under various conditions on the innovation structure (stationary conditional heteroscedastic errors, anti-persistence errors - lui2021mildly, non-Gaussian errors, serially correlated errors - lui2018mild, errors with possibly infinite variance - liu2021quantile and wang2022asymptotics, mixing innovations - oh2018mildly and liu2022mildly) doesn't necessarily imply the presence of a unified theory across the spectrum of nonstationarity as per Assumption 1. In other words, the fact that there is a limiting distribution discontinuity across these different regions of the parameter space of $\theta_n$, despite that the separate Gaussian and Cauchy limit results are found to be robust against different properties in the innovation structure, still requires considering whether a unified distribution-free inference technique can be applied regardless of: (i) the presence of a model intercept, (ii) the innovation sequence properties and (iii) the region of the parameter space.

Furthermore, based on extensive empirical applications using the dataset of hardle2016tenet, where cross-sectional specific autoregressive and predictive regressions were fitted on stock returns with predictors the macroeconomic variable of the dataset, we have observed that in cases where the estimated autoregressive coefficient is above the unit boundary (i.e., on the mildly explosive side), the estimation method for the model coefficient of the predictive regression model remains unchanged if one uses the existing IVX instrumentation approach proposed by Magdalinos2009limit (see, also kostakis2015Robust). This observation motivated us to consider an alternative instrumentation procedure when the estimated autoregressive coefficient is within that region is explosive.

More precisely, the novel instrumentation procedure proposed by Magdalinos2022uniform does exactly that, while theoretical and empirical illustrations are given for the linear autoregressive and predictive regression models. Specifically, the asymptotic theory of estimators, test statistics and corresponding confidence intervals can be constructed using a common distribution without imposing additional assumptions. These properties are found to hold specifically for autoregressive and predictive regression models based on a conditional mean functional form while the purpose of this paper is to extend those to the case of a conditional quantile functional form.

The class P.1 implies that $\theta_n$ is near to the unit boundary under there is no local-to-unity parametrization, thus the limit tends to $- \infty$. For the class P.2 we have that $\theta_n \to 1$ when specified as $\theta_n = \left( 1 + \frac{c}{k_n} \right)$, where $k_n = n^{\gamma}$ with $\gamma = 0$, $\gamma = 1$ or $\gamma \in (0,1)$. Thus, when $\theta_n = \left( 1 + \frac{c}{k_n} \right)$,

align[align omitted — 282 chars of source]

Assume that $\textcolor{blue}{c < 0}$ then it holds that

enumerate• If $k_n = n^{\gamma}$ with $\gamma = 1$ $( k_n = n )$, then $\textcolor{blue}{\zeta \equiv c \in \mathbb{R} }$, which falls in the class of nearly-unstable processes (near-nonstationary). • If $k_n = n^{\gamma}$ with $\gamma \in (0,1)$, then $\textcolor{blue}{\zeta = - \infty}$. In this case, although we are in the region of the "black circle" within the "blue zone" then we have a \textcolor{blue}{mildly integrated} process which falls in the class of nearly-stable processes (near-stationary). • If $k_n = n^{\gamma}$ with $\gamma = 0$ $( k_n = 1 )$, then $\textcolor{blue}{\zeta = - \infty}$ which falls in the class of nearly-stable processes (near-stationary).

Assume that $\textcolor{red}{c > 0}$ then it holds that

enumerate• If $k_n = n^{\gamma}$ with $\gamma = 1$ $( k_n = n )$, then $\textcolor{red}{\zeta \equiv c \in \mathbb{R} }$, which falls in the class of nearly-unstable processes (near-nonstationary). • If $k_n = n^{\gamma}$ with $\gamma \in (0,1)$, then $\textcolor{red}{\zeta = + \infty}$. In this case, although we are in the region of the "black circle" within the "red zone" then we have a \textcolor{red}{mildly explosive} process which falls in the class of nearly-explosive processes. • If $k_n = n^{\gamma}$ with $\gamma = 0$ $( k_n = 1 )$, then $\textcolor{red}{\zeta = + \infty}$ which falls in the class of nearly-explosive processes.

Assume that $c = 0$ then we have a pure unit root process.

Persistence Classes:

itemize• Pure Stationary Processes: $\rho_n \to \rho \in (-1,1)$. \ • Near Unit Root Processes: $\rho_n \to \rho = 1$ where $\rho_n$ is specified as $\rho_n = \left( 1 + \frac{c}{k_n} \right)$, $k_n = n^{\gamma}$. For all the case below we can employe the original IVX instrument. \begin{itemize} • \textcolor{blue}{Near-stationary processes}: $c <0$, $\gamma = 1$ (near-integrated) \begin{align} \underset{ n \to + \infty }{ \text{lim} } \rho_n := \underset{ n \to + \infty }{ \text{lim} } \left( 1 + \frac{c}{n} \right) = 1 + \underset{ n \to + \infty }{ \text{lim} } \left\{ \frac{c}{n} \right\} = \textcolor{blue}{ - \infty }, \ \text{when} \ c < 0. \end{align} • \textbf{\textcolor{blue}{Mildly-integrated processes}}: $c <0$, $\gamma \in (0,1)$ \begin{align} \underset{ n \to + \infty }{ \text{lim} } \rho_n := \underset{ n \to + \infty }{ \text{lim} } \left( 1 + \frac{c}{n} \right) = 1 + \underset{ n \to + \infty }{ \text{lim} } \left\{ \frac{c}{n} \right\} = \textcolor{blue}{ - \infty }, \ \text{when} \ c < 0. \end{align} • \textbf{\textcolor{blue}{Stationary processes}}: $c < 0$, $\gamma = 0$ \begin{align} \underset{ n \to + \infty }{ \text{lim} } \rho_n := \underset{ n \to + \infty }{ \text{lim} } \left( 1 + c \right) = ( 1 + c ) \in \mathbb{R} \end{align} • \textbf{\textcolor{red}{Near-explosive processes}}: $c > 0$, $\gamma = 1$ \begin{align} \underset{ n \to + \infty }{ \text{lim} } \rho_n := \underset{ n \to + \infty }{ \text{lim} } \left( 1 + \frac{c}{n} \right) = 1 + \underset{ n \to + \infty }{ \text{lim} } \left\{ \frac{c}{n} \right\} = \textcolor{red}{ + \infty}, \ \text{when} \ c > 0. \end{align} • \textbf{\textcolor{red}{Mildly-explosive process}}: $c > 0$, $\gamma \in (0,1)$ \begin{align} \underset{ n \to + \infty }{ \text{lim} } \rho_n := \underset{ n \to + \infty }{ \text{lim} } \left( 1 + \frac{c}{n} \right) = 1 + \underset{ n \to + \infty }{ \text{lim} } \left\{ \frac{c}{n} \right\} = \textcolor{red}{+ \infty}, \ \text{when} \ c > 0. \end{align} • \textbf{\textcolor{red}{Stationary process}}: $c > 0$, $\gamma = 0$ \begin{align} \underset{ n \to + \infty }{ \text{lim} } \rho_n := \underset{ n \to + \infty }{ \text{lim} } \left( 1 + c \right) = ( 1 + c ) \in \mathbb{R} \end{align} • \textbf{Pure Unit Root processes}: $c = 0$ regardless if $\gamma = 1$ or $\gamma \in (0,1)$ \begin{align} \underset{ n \to + \infty }{ \text{lim} } \rho_n := \underset{ n \to + \infty }{ \text{lim} } \left( 1 + \frac{0}{n} \right) = 1 \in \mathbb{R}. \end{align} \end{itemize} \ • \textbf{Pure Explosive Processes:} $\rho_n \to \rho > 1$. Thus, in this case to construct the IVX instrument we need to convert the process to be mildly explosive, i.e., so that is within the red region.

exampleConsider the classification of nonstationarity given by benke2021nearly. \begin{align} X_k = \beta X_{k-1} + \varepsilon_k, \ \ \ X_0 = 0. \end{align} \begin{itemize} • In the case that $|\beta | < 1$, the sequence $\left\{ \widehat{\beta}_n \right\}_{ n \in \mathbb{N} }$ is asymptotically normal such that \begin{align} \sqrt{n} \left( \widehat{\beta}_n - \beta \right) \overset{d}{\to} \mathcal{N} \big( 0, 1 - \beta^2 \big), \ \ \ as \ \ \ n \to \infty. \end{align} • In the case that $| \beta | = 1$, then the sequence $\left\{ \widehat{\beta}_n \right\}_{ n \in \mathbb{N} }$ has a limit distribution that depends on functionals of Brownian motion such that \begin{align} n \left( \widehat{\beta}_n - \beta \right) \overset{d}{\to} \frac{ \int_0^1 W(r) dW(r) }{ \int_0^1 W(r)^2 dr },\ \ \ as \ \ \ n \to \infty. \end{align} • A sequence $\left( X^{ \beta^{(n)} } \right)_{ k \in \mathbb{N} }$ for $n \in \mathbb{N}$ of an AR(1) process is called nearly unstable if $\beta^{(n)} \to \beta$ as $n \to \infty$, where $| \beta | = 1$. Moreover, in the case that $\beta = 1$ and $\beta^{(n)} = \left( 1 + \frac{c}{n} \right)$ with some $c \in \mathbb{R}$, \begin{align} n \left( \widehat{\beta}_n^{ (n) } - \beta^{(n)} \right) \overset{d}{\to} \frac{ \int_0^1 J(r) dW(r) }{ \int_0^1 J(r)^2 dr }, \ \ \ as \ n \to \infty. \end{align} \end{itemize} such that $\left\{ J_{ c } (r) \right\}_{ r \in [0,1] }$ is a continuous AR(1) process, that is, an OU process defined as a unique strong solution of the SDE such that \begin{align} \begin{cases} dJ_{c} (r) &= c J_{c} (r) dr + dW(r), \ \ \ r \in [0,1] \\ Y_{(c)} (r) &= 0 \end{cases} \end{align} Furthermore, if we consider $c$ to be a parameter instead of $\beta_{(n)}$ then the LSE of $c = n \left( \beta^{(n)} - 1 \right)$ is given by the following expression: \begin{align} \hat{c}_n &= n \left( \widehat{\beta}_n^{(n)} - 1 \right) = n \left( \beta_n^{(n)} - \beta^{(n)} \right) + c, \\ \widehat{c}_n &\overset{d}{\to} \frac{ \int_0^1 J_c(r) dW(r) }{ \int_0^1 J_c(r)^2 dr } + c = \frac{ \int_0^1 J_c(r) dJ_c (r) }{ \int_0^1 J_c(r)^2 dr }, \end{align} where the limit distribution above turns out to be the maximum likelihood estimator (MLE) of the parameter $c$ for the expression that gives the stochastic differential equation based on a sample $\left\{ J_{ c } (r) \right\}_{ r \in [0,1] }$.

exampleConsider the following processes: \begin{align} x_t &= \mu + X_t \\ X_t &= \rho X_{t-1} + \varepsilon_t \end{align} which implies that \begin{align} x_t = \mu \big( 1 - \rho_n \big) + \rho_n x_{t-1} + \varepsilon_t \end{align} Moreover, denote with \begin{align*} x_t^{\mu} = \big( x_t - \bar{x}_t \big), \ \ \bar{x}_t = \frac{1}{n} \sum_{t=1}^n x_t, \ \ x_{t-1}^{\mu} = \big( x_{t-1} - \bar{x}_{t-1} \big), \ \ \bar{x}_{t-1} = \frac{1}{n} \sum_{t=1}^n x_{t-1}. \end{align*} Equivalently, for $\mu \neq 0$, we consider the utoregressive model $x_t^{\mu} = \rho_n x_{t-1}^{\mu} + \varepsilon_t^{\mu}$, which implies \begin{align} \hat{\rho}_n = \left( \sum_{t=1}^n x_{t-1}^{\mu} \right)^{-1} \left( \sum_{t=1}^n x_{t-1}^{\mu} x_{t}^{\mu} \right) , \ \ \ \hat{\varepsilon}_t^{\mu} = x_{t}^{\mu} - \hat{\rho}_n x_{t-1}^{\mu} \end{align}

Therefore in this article we focus on the quantile autoregressive model with a model intercept the particular modeling framework motivate us to investigate the interplay between near-zero intercept\footnote{Notice that since the initial condition of the autoregressive process corresponds to the boundary condition of an ordinary differential equation problem, then the stochastic solution, that is, the limiting distribution under equilibrium conditions will depend on the initial condition (see, saxena1982estimation).} and stochastic persistence by deriving the asymptotic distribution.

assumptionThe distribution $F_{\varepsilon}$ is in the domain of attraction of a stable law indexed with $\upalpha \in (0,2)$, which has a strictly positive density. When $\upalpha = 2$ then it corresponds to the case of innovation sequences from a distribution function with finite variance. Furthermore, \begin{align} \mathbb{E} \left( \varepsilon_1 \right) = 0, \ \ \mathsf{Var} \left( \varepsilon_1 \right) = \sigma^2_{\varepsilon }, \ \ \ and \ \ \ \mathbb{E} \left| \varepsilon_1 \right|^{2+m} < \infty, \ for some \ m > 0. \end{align}

When $c = 0$ then the AR(1) model is a random walk model with stable innovations. Therefore, under the assumption that $c = 0$, which implies the presence of an integrated process the asymptotic behaviour of the ordinary least squares estimator can be obtained based on Brownian motion functionals. Define with $U_{\upalpha}(r)$ and $V_{\upalpha}(r)$ to be L\'evy processes on the space of functions $\mathcal{D}[0,1]$.

lemmaSuppose that $\varepsilon_t$ satisfies Assumption (ref). Then as $n \to \infty$ it holds that \begin{align} \left( \sum_{t=1}^{ \floor{nr} } \frac{\varepsilon_t}{a_n}, \sum_{t=1}^{ \floor{nr} } \frac{\varepsilon_t^2}{a_n^2} \right) \Rightarrow \bigg( U_{\upalpha} (r), V_{\upalpha}(r) \bigg) \end{align} where $\big( U_{\upalpha} (r), V_{\upalpha}(r) \big)$ is a Levy process in $\mathcal{D}[0,1]^2$ with index $\upalpha \in (0,2)$.

remarkA general class of one-dimensional stochastic processes, are the so-called L\'evy processes. Similar to Wiener processes, L\'evy processes have right continuous paths with left limits, are initiated from the origin and both have stationary and independent increments (kyprianou2014fluctuations). Under the i.i.d innovation assumption it can be shown that $V_{\upalpha} (r) = \int_0^r \left( dU_{\upalpha} (s) \right)^2 = \big[ U_{\alpha}, U_{\alpha} \big]_{ r \in [0,1] }$ which is the quadratic variation of the Levy process $U_{\upalpha} (r)$. Furthermore, $V_{\alpha} (r)$ is a stochastic integral. When $\upalpha = 2$, which corresponds to the finite variance case, it holds that $ U_{\upalpha} (r) \equiv W(r)$ for some $0 \leq r \leq 1$, the standard Brownian motion, and $V_{\upalpha} (r) = \big[ W, W \big]_r = r$ (see, also cramer1951contribution).

Any asymptotic results followed by Lemma (ref) coincide with the standard random walk asymptotics for finite variance models (we focus in the case $\upalpha = 2$). Then as $n \to \infty$, it holds that

align[align omitted — 142 chars of source]

Since $\rho_n = \left( 1 + \frac{c}{n^{\gamma}}\right)$ under the stationary condition which implies that $0 < \rho_n < 1$ then it holds that $- n^{\gamma} < c < 0$. The OLS estimate $\hat{\rho}$ is $n-$consistent, that is, $n \left( \hat{\rho} - \rho \right)$ has a nondegenerate limit distribution depending on $c$, while $\hat{\mu}$ is $\sqrt{n}-$asymptotically normal. As a result, $\hat{c} = - n( 1 - \hat{\rho} )$ is the OLS estimate of the coefficient of persistence $c$, which is not consistent. mikusheva2012one shows

align[align omitted — 127 chars of source]

The particular asymptotic result demonstrates the well-known conjecture that the OLS estimate of the nuisance parameter of persistence, $c$, is not consistent. In fact, $\hat{c}$ is asymptotically highly biased to the left, thus the estimated model looks more stationary that it actually is. Roughly speaking, the assumption regarding the dependence structure for the disturbance term can affect the limit theory of estimators. In particular, serial correlation in the errors induces an asymptotic bias for $\hat{\rho}_n$ and contributes to the bias of the Gaussian limiting distribution. For the asymptotic theory we assume that $\mathcal{D}[0,1]$ is endowed with the Skorokhod topology such that any partial sum processes are measurable for the associated Borel $\sigma-$algebra under the absence of serial correlation. Imposing assumptions regarding the properties of the disturbance term $\varepsilon_t$ can imply different asymptotic behaviour for estimators given certain modelling conditions. According to werker2022semiparametric the usual procedures established in the literature thus far are based on the assumption of Gaussian innovations and, while their validity has been established under weak assumptions, the asymptotic power of all these procedures cannot go beyond the Gaussian power envelope. Relaxing the Gaussianity assumption requires to apply semiparametric estimation methodologies is beyond the scope of our study. Thus, the i.i.d innovation sequence assumption with finite variance, simplifies the representation of the necessary regularity conditions.

Quantile Conditional Estimation

In this Section we discuss in more details the estimation procedure for the quantile autoregressive time series that accommodates the parametrization of the autoregression coefficient with respect to moderate deviations from the unit boundary. We first introduce the quantile estimation method\footnote{A complete treatment of limit results for quantile regressions can be found in the book of Koenker2005.} to obtain parameter estimates and then establish the asymptotic theory of this estimator for the autoregressive model denoted with $y_t = \mu + \rho y_{t-1} + u_t$ (with a slight change of notation).

Denote with $\mu( \uptau )$ and $\rho_n ( \uptau )$ to be the $\uptau-$quantile dependent parameters, which are determined based on a conditional quantile functional form as below:

align[align omitted — 353 chars of source]

for some $\uptau \in (0,1)$, where $\uptau$ denotes the quantile level within a compact set $(0,1)$.

Denote the parameter vector with $\boldsymbol{\vartheta} ( \uptau ) = \big( \mu ( \uptau ), \rho_c ( \uptau ) \big)^{\top}$ and $\boldsymbol{X}_t = \boldsymbol{D}_n^{-1} \big( 1, y_{t-1} \big)$, where $\boldsymbol{D}_n$ is the normalization matrix which includes the different convergence rates for the model intercept vis-a-vis the slope coefficient. Then, from koenker1978regression and koenker1987estimation the quantile regression estimator is obtained via the following optimization function

align[align omitted — 276 chars of source]

such that $\widehat{ \boldsymbol{\vartheta} }_n \big( \uptau \big) \equiv \boldsymbol{D}_n \left( \hat{\mu}_n ( \uptau ) - \mu ( \uptau ), \hat{\rho}_{n,c} ( \uptau ) - \rho_{c}( \uptau ) \right)$. Moreover, we denote with $\psi(\mathsf{u})$ to be the left derivative of $\varrho(\mathsf{u})$. In particular, when $\psi( \mathsf{u} ) := \mathsf{u}$ is the identity function then the $\widehat{ \boldsymbol{\vartheta} }_n$ corresponds to the least squares estimator, while when $\psi(\mathsf{u}) := ( 1 - \uptau )$ for $\mathsf{u} \leq 0$ and $\psi(\mathsf{u}) = \uptau$ for $\mathsf{u} > 0$, then it corresponds to the optimization function and $\widehat{ \boldsymbol{\vartheta} }_n$ is the quantile-dependent estimator.

assumptionSuppose that $\mathbb{E} \big[ \psi ( u_t ( \uptau ) ) \big] = 0$ and consider the random variable which corresponds to the first derivative around some parameter $\theta \in \mathbb{R}$ such that \begin{align} \xi := \left| \frac{ \partial }{ \partial \theta } \mathbb{E} \bigg[ \psi \bigg( u_1 ( \uptau ) - \theta \bigg) \bigg] \right|_{ \theta = 0 }, \ \ where \ \xi \neq 0. \end{align} where $\mathbb{E}| \psi \left( u_1 ( \uptau ) \right) |^{2 + m} < \infty$, for some $m > 0$.

Assumption (ref) ensures that the first derivative is Lipschitz continuous and bounded which corresponds to the first derivative for the expectation of the check function as a random variable evaluated within the neighbourhood of the true parameter vector $\theta = 0$. Furthermore, due to the fact that the quantile autoregressive model we consider in this paper corresponds to a possibly nonstationary time series model, then the asymptotic theory of estimators and corresponding test statistics depends on Brownian motion functionals as introduced with Assumption (ref) below.

assumptionThe following conditions for the innovation sequence hold: \begin{itemize} • The sequence of stationary conditional probability distribution functions denoted with $\big\{ f_{ \varepsilon_t (\uptau), t-1}(.) \big\}$ evaluated at zero with a non-degenerate mean function $f_{ \varepsilon_t (\uptau) }(0) := \mathbb{E} \left[ f_{ \varepsilon_t (\uptau), t-1}(0) \right] > 0$, that satisfies a Functional Central Limit Theorem (FCLT) expressed as below \begin{align} \frac{1}{ \sqrt{n} } \sum_{t=1}^{ \floor{nr} } \left( f_{ \varepsilon_t (\uptau), t-1}(0) - \mathbb{E} \left[ f_{ \varepsilon_t (\uptau), t-1}(0) \right] \right) \Rightarrow B_{ f_{ \varepsilon_t (\uptau) } } (r), \ with \ r \in (0,1). \end{align} • For each $t$ and $\uptau \in (0,1)$, $f_{ \varepsilon_t (\uptau), t-1}(.)$ is uniformly bounded away from zero with a corresponding conditional distribution function $F_t(.)$ which is absolutely continuous with respect to Lebesgue measure on $\mathbb{R}$ (see, neocleous2008monotonicity, goh2009nonstandard and lee2016predictive). \end{itemize}

remarkAssumption (ref) gives necessary and sufficient conditions for a functional central limit theorem to hold for the corresponding innovation sequence based on the conditional quantile functional form, which is instrumental for deriving the asymptotic behaviour of the quantile-dependent estimators under nonstationarity based on Brownian motion functionals.

Therefore, to obtain the model estimates based on the optimization problem (ref) we apply the Taylor expansion to the check function, such that for a given parameter $\boldsymbol{\delta} ( \uptau )$ it holds that

align[align omitted — 295 chars of source]

remarkNotice that for instance the t-ratio for $\rho_{n,c}$ is defined by $\sqrt{ \sum_{t=1}^n y_{t-1}^2 } \left( \hat{\rho}_{n,c} - \rho_c \right)$, thus to obtain the limiting distribution of the $t-$test we need to obtain an asymptotic expression for the normalized centered estimator $\left( \hat{\rho}_{n,c} - \rho_c \right)$. Furthermore, for sequences such that $\underset{ n \to \infty }{ \mathsf{lim} } n \left( 1 - \rho_n \right) = 0$, the nearly unstable model behaves asymptotically like the strictly unstable model in which case $\rho = 1$.

Large-Sample Theory

In this section we present the main asymptotic theory results while detailed proofs can be found in the Appendix of the paper. Some important aspects worth emphasizing again is that while the case of near-integrated (NI) processes, such that $c < 0 $ and $\gamma = 1$, has been considered before in quantile autoregressive time series (see, chan2006quantile) as well as the case of mildly integrated (MI) such that $c < 0 $ and $\gamma \in (0,1)$, the mildly explosive (ME) such that $c > 0 $ and $\gamma \in (0,1)$ and the explosive case, such that $c > 0 $ and $\gamma =1$ has not been widely explored before. In particular, all aforementioned cases which correspond to different regions of the parameter space, overcome the singularity problem; the region that the underline stochastic process does not have a solution. Using the local-to-unity parametrization in autoregressive processes overcomes this gap by considering the limiting distribution\footnote{The form of the noncentrality parameter of the $\chi^2-$distribution depends on the initial condition and the form of the autoregressive parameter of the model (see, Theorem 2.2. of Jian2022). } for the whole parameter space regardless of the existence of a limit singularity.

Using the local-to-unity asymptotics our aim is to derive a nuisance-parameter-free limiting distribution which can facilitate statistical inference. Therefore, by decomposing the underline stochastic processes into components which include a predictable quadratic variation, allows us to obtain a self-normalized martingale sequence, which is especially useful when deriving the limiting distribution of Wald-type statistics. In other words, Wald statistics constructed with a variance estimator which induced by the predictable quadratic variation ensures that the self-normalization property holds. When asymptotic theory of the autoregressive parameter that corresponds to a particular region of the parameter space is nonstandard, then this implies that the limit distribution of the MLE of the model can be represented as the MLE of a parameter of a process satisfying the stochastic differential equation for the OU process (see, benke2021nearly). Thus, the limiting distributions of the model estimator and the corresponding test statistic are functions of the local-to-unity parameter which is not consistenty estimable since $\left( \hat{c} - c \right) = \mathcal{O}_p (1)$.

Although in this article we only consider point inference procedures, in the case of interval inference, that is, when is concerned with the construction of confidence interval for unknown model parameters then several studies discuss the possibility of obtaining uniform inference procedures when the nonstationary autoregressive model is expressed using the local-to-unity parametrization (see, mikusheva2007uniform, phillips2014confidence and Magdalinos2022uniform). In this article we focus on bridging the gap in the robust and uniform inference literature on quantile autoregressions and quantile predictive regression models by considering that the instrumentation procedure depends on the region of the parameter space. In other words, although the original IVX method proposed by PM2009econometric and kostakis2015Robust it is found to be robust in LUR moderate deviations from the unit boundary, it is clear than in explosive and mildly explosive regimes the same IVX instrumentation might not work so well.

theorem[chan2006quantile] Assume that Assumption (ref)-(ref) hold and that the autoregression coefficient is expressed as $\rho_{n,c} = \left( 1 + \frac{c}{k_n} \right)$. Then, the following limit result hold \begin{align} \boldsymbol{D}_n \left( \hat{ \boldsymbol{\vartheta} }_n ( \uptau ) - \boldsymbol{\vartheta} ( \uptau ) \right) &\overset{ d }{ \to } \frac{1}{ f_{\varepsilon} \big( F_{\varepsilon}^{-1} ( \varepsilon_t ( \uptau ) ) \big) } \Sigma^{-1} \left( W ( \uptau ; 1 ) , \int_0^1 J(s) d W( \uptau, s ) \right)^{\prime}, \\ n \bigg( \hat{\rho}_{n,c}( \uptau ) - \rho_{n,c} ( \uptau ) \bigg) &\overset{ d }{ \to } \frac{1}{ f_{\varepsilon} \big( F_{\varepsilon}^{-1} ( \varepsilon_t ( \uptau ) ) \big) } \frac{ \int_0^1 J_1(s) dW( \uptau, s ) - W(\uptau, 1 ) \int_0^1 J_1(s) ds }{ \int_0^1 J^2_1(s) ds - \left( \int_0^1 J_1(s) ds \right)^2 } \end{align} where $\boldsymbol{D}_n = \mathsf{diag} \left( \sqrt{n}, n \right)$ and $\boldsymbol{\vartheta} ( \uptau ) = \big( \mu( \uptau ), \rho_{n,c} ( \uptau ) \big)$ such that \begin{align} \boldsymbol{\Sigma} := \int_0^1 \big( 1, J_1(s) \big)^{\prime} \big( 1, J_1(s) \big) ds \equiv \begin{bmatrix} 1 & \int_0^1 J_1(s) ds \\ \int_0^1 J_1(s)^{\prime} ds & \int_0^1 J_1(s) J_1(s)^{\prime} ds \end{bmatrix}. \end{align}

align[align omitted — 332 chars of source]

Limit theory for near-stationary case

In a similar spirit as in the framework proposed by Phillips2007limit, in order to establish the limit theory of the autoregression coefficient within our modelling environment, we consider the asymptotic behaviour of the sample moments that appear in the quantile-dependent estimator separately. However, in contrast to the ordinary least squares estimation, when the model parameters are estimated using the conditional quantile functional form, we employ standard approximation methods (such as the Bahadur representation) from the quantile regression literature to obtain analytical expressions for the quantities of interest. Specifically, in the near-stationary case, which implies that $c < 0$, the limit to the unit boundary is approached from the left of the triangular array. Furthermore, due to the different convergences rate of the model intercept versus the slope parameter we employ the normalization matrices $\boldsymbol{D}_n$ and $\boldsymbol{B}$ as defined below

align[align omitted — 179 chars of source]

where $k_n = n^{\gamma}$ and $\gamma \in (0,1)$. The $n^{-1 / 2 }$ convergence rate corresponds to the model intercept while when $k_n = n^{\gamma}$, then the autoregression parameter of the model has a convergence rate of $n^{ - \frac{1 + \gamma}{2} }$ which is also the rate of convergence that corresponds to a mildly integrated process. Furthermore, in empirical applications in practise we do not know a prior whether the expression $\sqrt{n} \left( \hat{\rho}_n - \rho \right)$ is positive or negative. However, since we do not partition the parameter space accordingly, the asymptotic theory mainly focus on the near-integrated case and does not cover the mildly explosive or pure explosive since $c < 0$.

theoremUnder Assumptions (ref)-(ref), \begin{align} \big( \hat{\mu}_n, \hat{\rho}_{n,c} \big)^{\top} = \big( \mu, \rho_{n,c} \big)^{\top} + \frac{ ( \boldsymbol{B} \boldsymbol{D}_n )^{-1} }{ \xi } \sum_{t=1}^n \psi \big( \varepsilon_t \big) \boldsymbol{X}_t^{\top} + o_p(1). \end{align} In particular, when $\psi( \mathsf{u} ) = \big( \uptau - \mathbf{1} \left\{ \mathsf{u} \leq 0 \right\} \big)$ corresponds to the quantile regression and therefore the above expression reduces to \begin{align} \begin{pmatrix} \hat{\mu}_n ( \uptau ) \\ \hat{\rho}_{c,n} ( \uptau ) \end{pmatrix} = \begin{pmatrix} \mu_n ( \uptau ) \\ \rho_{c,n} ( \uptau ) \end{pmatrix} + \frac{ \left( \boldsymbol{B} \boldsymbol{D}_n \right)^{-1} }{ f_{\varepsilon} \big( F_{\varepsilon}^{-1} ( \uptau ) \big) } \sum_{t=1}^n \bigg( \uptau - \mathbf{1} \left\{ \varepsilon_t \leq F_{\varepsilon}^{-1} ( \uptau ) \right\} \bigg) \begin{pmatrix} \frac{1}{\sqrt{n}} \\ \\ \frac{ y_{t-1} }{\sqrt{n k_n}} \end{pmatrix} + o_P(1). \end{align} where $f_{\varepsilon}(x)$ and $F_{\varepsilon}(x)$ denote the probability and cumulative density functions of $\varepsilon_1$, respectively.

remarkTheorem (ref) provides a Bahadur representation for the parameter vector of the quantile autoregressive time series which includes a model intercept and a slope. In particular for M-regressions a necessary requirement for the functional form is to include a model intercept which can be different than zero. Moreover, the given limit results are employed to derive the asymptotic behaviour of model parameters based on moderate deviations from the unit boundary on the stationary region as summarized by the next theorem. Then, the robust estimation of the sparsity coefficient which depends on the kernel density function can be improve the accuracy of the quantile-dependent model estimates.

theoremUnder Assumptions (ref)-(ref), \begin{align} \boldsymbol{D}_n \left( \big( \hat{\mu}_n, \hat{\rho}_{n,c} \big) - \big( \mu_n, \rho_{n,c} \big) \right) \overset{ d }{ \to } \mathcal{N} \left( 0, \frac{ \boldsymbol{B}^{-1} \times \mathbb{E} \big[ \psi^2 ( \varepsilon_1 ) \big] }{ \xi } \right). \end{align} In particular, it follows that \begin{itemize} • If $\psi( \mathsf{u} ) = \big( \uptau - \mathbf{1} \left\{ \mathsf{u} \leq 0 \right\} \big)$, and the pdf $f(\mathsf{u})$ of $\varepsilon_1$ exists and satisfies $f_{\varepsilon} \big( F_{\varepsilon}^{-1} (\uptau) \big) > 0$, then \begin{align} \frac{ \hat{\rho}_{n,c} ( \uptau ) - \rho_{c} ( \uptau ) }{ \sqrt{n k_n} } \overset{ d }{ \to } \mathcal{N} \left( 0, \frac{-2c}{ \sigma^2 } \frac{\uptau(1 -\uptau)}{ f_{\varepsilon}^2 \big( F_{\varepsilon}^{-1} (\uptau) \big) } \right). \end{align} • If $\psi( \mathsf{u} ) = \mathsf{u}$, then \begin{align} \frac{ \hat{\rho}_{n,c} ( \uptau ) - \rho_{c} ( \uptau ) }{ \sqrt{n k_n} } \overset{ d }{ \to } \mathcal{N} \left( 0, -2c \right). \end{align} \end{itemize}

remarkThe limit results given by Theorem (ref) summarize the joint asymptotic behaviour of the model intercept and slope from moderate deviations from the unit boundary on the stationary region. In order to prove the above asymptotic results, we employ standard arguments introduced by pollard1991asymptotics for optimization of convex function relevant to quantile processes. Specifically, by the convexity lemma, if the finite-dimensional distributions of $\Omega_n (v)$ converge weakly to those of $\Omega (v)$, and $\Omega (v)$ has a unique minimum, then the convexity of $\Omega_n (v)$ implies that $\hat{v}$ converges in distribution to the minimizer of $\Omega (v)$. In other words, $\hat{\rho}_n( \uptau )$ is shown to be weakly consistent, thus to prove that the estimator is asymptotically normally distributed we restrict the spaces $\mathcal{B}$ to shrinking neighbourhoods around the true value of the parameter $\rho ( \uptau )$ in order to avoid possible local minima. To do this, we can define the restricted space $\mathcal{B}_{a} = \big\{ \rho_n \in \mathcal{B} \left\lVert \beta - \beta ( \uptau ) \right\rVert \leq a_n \big\}$ where $\left\{ a_n \right\}$ is some positive sequence.

Limit theory for near-explosive case

The near-explosive case corresponds to the nuisance parameters of persistence $c >0$ and the exponent rate $\gamma \in (0,1)$ or $\gamma = 1$. In particular for the linear autoregressive process $y_t = \theta y_{t-1} + \varepsilon_t$ and an explosive root such that $| \theta | > 1$, a Cauchy limit theory can be derived for the OLS estimator $\hat{\theta}$ as

align[align omitted — 143 chars of source]

More precisely, the seminal study of anderson1959asymptotic provides examples demonstrating that central limit theory does not apply and the asymptotic distribution of the least squares estimator depends by the distributional assumptions imposed on the innovations which makes inference procedures specifically for purely explosive autoregressions more challenging (see, magdalinos2012mildly). Furthermore, in this direction, Phillips2007limit consider autoregressive processes under the moderate deviations framework by employing the local-to-unity parametrization for the autoregression coefficient such that $\theta_n = \left( 1 + \frac{c}{n^{\gamma} } \right), \gamma \in (0,1)$. Therefore, in this case under the assumption of $\textit{i.i.d}$ innovations with finite second moments the following least squares regression theory was proved

align[align omitted — 143 chars of source]

On the other hand, for the pure explosive root case such that $| \theta | > 1$ then, the limit distribution of $\big( \hat{\theta} - \theta \big)$ is standard Cauchy if it is normalized with $\theta^n / ( 1 - \theta^2 )$. However, the limit distribution depends on the distribution of the noise, as was pointed out by anderson1959asymptotic, and hence no central limit theorem applies on the explosive side. Moreover, from empirical data financial applications it can be observed that the parameter $\theta$ tends to 1 with increasing sample size.

To accommodate this observation, $\theta = \theta_n$ is allowed to depend on $n$, the number of observations, such that $\theta_n \to 1$ as $n \to \infty$. The process is then referred to as near-integrated. Depending on whether $\theta_n < 1$ or $\theta_n > 1$, it is called near-stationary or mildly explosive. Furthermore, Phillips2007limit investigated the general parameter case in the near-integrated setting assuming that $\theta_n \to 1$ with a rate slower than $1 / n$, the so-called moderate deviations from unity. All aforementioned approaches operate under the assumption of a finite variance along with independent, identically distributed or weakly dependent errors. However, it can be proved that the serial coefficient $\hat{\theta}_n - \theta_n$ has, under a suitable normalization, a limit that consists of a fraction of two independent strictly stable random variables.

Therefore, specifically for the quantile autoregressive time series model we consider in our study we employ the following normalization matrices.

align[align omitted — 191 chars of source]

where $\mathcal{Z}_3$ is some normal random variable to be defined below.

theoremUnder the Assumptions (ref)-(ref) it holds that, \begin{align} \left( \hat{\mu}, \hat{\rho}_{n,c} \right)^{\top} = \left( \mu, \rho_{n,c} \right)^{\top} + \frac{ \left( \sum_{t=1}^n \boldsymbol{X}_t \boldsymbol{X}_t^{\top} \boldsymbol{D}_n \right)^{-1} }{ \xi } \sum_{t=1}^n \psi \big( \varepsilon_t \big) \boldsymbol{X}_t^{\top} + o_p(1). \end{align}

theoremUnder Assumptions (ref)-(ref) it holds that, \begin{align} \boldsymbol{D}_n \big( \left( \hat{\mu}, \hat{\rho}_{n,c} \right) - \left( \mu, \rho_{n,c} \right) \big)^{\top} \overset{ d }{ \to } \frac{1}{ \xi } \boldsymbol{B}_n^{-1} \big( \mathcal{Z}_1, \mathcal{Z}_2 \mathcal{Z}_3 \big)^{\top}. \end{align} In particular, it follows that \begin{itemize} • If $\psi( \mathsf{u} ) = \big( \uptau - \mathbf{1} \left\{ \mathsf{u} \leq 0 \right\} \big)$, and the pdf $f(\mathsf{u})$ of $\varepsilon_1$ exists and satisfies $f_{\varepsilon} \big( F_{\varepsilon}^{-1} (\uptau) \big) > 0$, then \begin{align} \frac{ \hat{\rho}_{n,c} ( \uptau ) - \rho_{c} ( \uptau ) }{ \rho_n k_n } \overset{ d }{ \to } \frac{2c}{ f_{\varepsilon} \left( F_{\varepsilon}^{-1} ( \uptau ) \right) } \frac{ \mathcal{Z}_2 }{ \mathcal{Z}_3 }. \end{align} • If $\psi( \mathsf{u} ) = \mathsf{u}$, then \begin{align} \frac{ \hat{\rho}_{n,c} ( \uptau ) - \rho_{c} ( \uptau ) }{ 2 c \rho_n^n k_n } \overset{ d }{ \to } \frac{ \mathcal{Z}^{*}_2 }{ \mathcal{Z}^{*}_3 }. \end{align} \end{itemize}

Therefore, Theorem (ref) verifies that we indeed obtain the equivalent asymptotic theory results in comparison to the linear autoregressive time series model. Specifically, the autoregression coefficient of the nonstationary quantile autoregressive time series model converge into a Cauchy random variate in the case of mildly explosive processes (see, also aue2007limit, Phillips2007limit, magdalinos2012mildly and lee2018limit).

remarkAs we see from Theorem (ref), part 2, the limiting distribution of the normalized and centered estimator is Cauchy, similar to Theorem 4.3 of Phillips2007limit. As a matter of fact when we replace $\rho_n$ by $\rho_{n,c} = \left( 1 + \frac{c}{k_n} \right)$ we obtain that $\left( \rho^2 - 1 \right) = \frac{ 2c }{ k_n } \big[ 1 + o(1) \big]$. Hence, we see that the normalizations in the Theorem above and the expression derived by white1958limiting are asymptotically equivalent as $n \to \infty$. Furthermore, the asymptotic theory for the case of moderate deviations from the unity boundary is not restricted to Gaussian processes. More specifically, the Cauchy limit result applied for $\rho_{n,c} = \left( 1 + \frac{c}{k_n} \right)$ and innovations $\varepsilon_t$ with finite second moment (e.g., innovations with stable law of attraction). On the other hand, the main difference between the mildly explosive processes given by Theorem (ref) above and explosive autoregressions with $| \rho | > 1$, occurs due to the different convergence rates of these two cases. In particular, in the case of mildly explosive processes we define the convergence rate such that $k_n = n^{\gamma}$ for some $\gamma \in (0,1)$ while for the case of moderately explosive processes we define with $k_n = n^{\gamma}$ for some $\gamma > 1$.

Testing Linear Hypotheses

Consider the autoregressive model

align[align omitted — 70 chars of source]

such that $\rho \in [-1,1]$, is within the stationary region. Then, the usual testing hypothesis of interest is such that, $\mathbb{H}_0: \rho = \rho_0$. In particular, dickey1979distribution showed that the finite sample distribution for $\rho$ in the neighbourhood of unity is very close to the asymptotic unit root case, under the assumption that the error term $\epsilon_t$ is normally distributed with finite variance.

Statistical inference for M-estimators of possibly nonstationary time series models (local-to-unit root) is a nonstandard problem due to the presence of nuisance parameters in the limiting distributions of test statistics. However, indeed one of the advantages of M-estimators is that are considered to be robust to outliers since they have a bounded influence function (see, abadir2000quantiles). Considering now specifically the case of unit root such that $| \rho | = 1$, the asymptotic distribution of the t-statistic denoted by $\mathcal{T}_n ( \hat{\rho} )$ can be represented by functionals of Wiener processes (see, dickey1979distribution and Buchmann2007asymptotic). Thus, the asymptotic distribution of the $t-$statistic based on $M-$estimators, denoted by $\mathcal{T}_{\psi ( \uptau ) } ( \hat{\rho} )$ depends on the nuisance parameter $\delta$, that is, the correlation between the innovations $\left\{ \epsilon_t \right\}$ and the pseudo-score function $\psi( \epsilon_t )$ that is employed to define the $M-$estimator. On the other hand, $t-$statistics based on $M-$estimators lead to a reduction in asymptotic MSE relative to LSE for local alternatives to the unit-root null hypothesis.

The t-statistic for the null hypothesis $\mathbb{H}_0: \rho = \rho_0$ is given by

align[align omitted — 368 chars of source]

where $\psi^{\prime}_{\uptau} (.)$ denotes the first derivative of the function $\psi_{\uptau} (.)$. \color{black}

Asymptotic theory for Quantile Predictive Regression

In this section we unify the theory for the quantile predictive regression model while presenting the corresponding asymptotic theory for the quantile autoregressive process we introduced in the previous section with our novel pruned-based endogenous instrumentation approach.

Model specification and assumptions

Consider the following predictive regression model

align[align omitted — 97 chars of source]

Various studies in the literature consider inference methodologies for nearly-integrated processes. Specifically for the conditional quantile functional form lee2016predictive propose a framework for inference in quantile predictive regression models using the IVX instrumentation of PM2009econometric. However, robust inference in the near-explosive region for both the quantile autoregression and the predictive regression has not been examined previously in the literature.

In particular, OLS-based inference on $\beta$ for nearly-explosive processes and specifically purely-explosive suffers from the same problem as OLS-based inference on $\rho_n$, with standard inference applying only under $\textit{i.i.d}$ Gaussian innovations $\varepsilon_t$. Therefore, the inference procedure proposed in this paper for the quantile-dependent coefficient of $\beta$, in the quantile predictive regression model can accommodate regressons with time series properties along the entire spectrum of autoregressive procceses. Although our setting considers a model with only one regressor, therefore further theory is needed in the case of multiple regressors that correspond to either the same or different persistence class. Specifically, one can establish its asymptotic validity uniformly over the autoregressive regime and regardless of the distributional assumptions of the innovations $\varepsilon_t$ and $u_t$. Robust and uniform inference in quantile predictive regression models has been studied by maynard2023inference, lee2016predictive, fan2019predictive, cai2022new and more recently liu2023unified.

The main idea with the proposed endogenously generated instrumentation is to decompose the the two mutually disjoint parameter space regions and then develop an asymptotic theory for both these regions. In other words, the main intuition behind this is that although the dependence structure of innovation sequences is not affected by considering two different regions of the parameter space, it does however have an effect on the asymptotic behaviour of the corresponding estimators which bridge the gap in both the explosive as well as the near-nonstationary cases. The proposed instrumentation method is based on the framework of Magdalinos2022uniform and combines the nearly stable with the nearly explosive processes (near stationary/near explosive), thereby implying an adaptive and uniform inference technique for quantile autoregressions and quantile predictive regression models regardless of the peristence properties of regressors or whether the autoregressive and predictive equations include a model intercept.

Pruned-based endogenous instrumentation approach

In this paper we propose a novel pruned-based endogenous instrumentation approach based on the original instrumentation methodology proposed in the paper of PM2009econometric (see, also kostakis2015Robust). The procedure we propose is similar to the IV instrument proposed in the recent paper of \textcolor{blue}{Magdalinos and Petrova (2022)}, although our motivation is to unify inference in autoregressions when considering a conditional quantile functional form, that is, is applied specifically to quantile autoregressive processes. We call our novel estimator IVX-P which can be employed in either linear or quantile conditional functional forms with univariate or multivariate regressors. In the literature such estimators mainly were concerned with Stein-type estimators, however our study is the first to consider an estimator which is obtained using properties of the underline stochastic processes both within the admissible parameter space as well as outside the usual parameter space.

Thus, the proposed pruned-based instrumental estimator implies a data driven instrument selection such that $\textcolor{blue}{ \mathcal{B}_n } = \mathbf{1} \left\{ n \left( \hat{\vartheta}_n^{ols} - 1 \right) \leq 0 \right\} $. More specifically, the chosen $\rho_{nz}$ is such that

align[align omitted — 322 chars of source]

where $\kappa_{1n}, \kappa_{2n} \to \infty$ with $\kappa_{1n} / n \to 0$ and $\kappa_{2n} / n \to 0$. Then, the combined instrument process is

align[align omitted — 235 chars of source]

In other words, we have an orthogonal decomposition such that $z_t = z_{1t} \mathbf{1} \big\{ \textcolor{blue}{ \mathcal{B}_n } \big\} + z_{2t} \mathbf{1} \big\{ \textcolor{red}{ \mathcal{B}_n^{c} } \big\}$ where

align[align omitted — 182 chars of source]

Therefore, for all cases it holds that

align[align omitted — 282 chars of source]

where $\tilde{z}_t^{IVX-P}$ represents the IVX pruned-based estimator. A key result that we are aiming to illustrate is the Asymptotic Mixed Gaussianity (AMG) property of the IVX-P estimator.

Instrument Construction

Successful instrumentation based on a combined near-stationary/near-explosive process requires statistical information separating the near-stationary autoregressive class from the near-explosive class asymptotically. In other words, the advantage of the inference procedure proposed in this paper over existing procedures is that it is valid for any $\rho_n \to \rho \in ( 0, + \infty )$, which includes all three parameter regions of interest of empirical interest. \color{black} Practical implementation of our instrumentation procedure requires a choice for $\varphi_{1n}$ and $\varphi_{2n}$. Specifically, choosing $\left( \varphi_{1n} \right)_{ n \in \mathbb{N} }$ and $\left( \varphi_{2n} \right)_{ n \in \mathbb{N} }$ with $n \left( \varphi_{1n} - 1 \right) \to \infty$ as $n \to \infty$ and $\varphi_{2n} \to 1$ with $n \left( \varphi_{1n} - 1 \right) \to + \infty$.

align[align omitted — 154 chars of source]

reduces to the problem of selecting the values for $\gamma_1$ and $\gamma_2$.

We construct the instrumentation procedure based on a min-max optimality guarantee. In other words, we have that $\tilde{z}_{1t}$ can be asymptotically approximated by a near-stationary process such that

align[align omitted — 98 chars of source]

In particular, when $\rho_n$ is closer to 1 that $\varphi_{1n}$ then $\tilde{z}_{1t}$ reduces asymptotically to the original process $x_t$. Furthermore, the instrument $\tilde{z}_{2t}$ is always approximated by a mildly explosive process such that

align[align omitted — 87 chars of source]

Therefore, from the above decomposition we see that sample moments involving the near-stationary instrument $\tilde{z}_{1t}$ will contribute asymptotically when the original process $x_t$ belongs to the persistence classes P.1-P.2 (i.e., near stationary and near-nonstationary), whearas sample moments involving the mildly-explosive instrument $\tilde{z}_{2t}$ will make an asymptotic contribution for autoregressions that belong to persistence classes P.2-P.3.

Moreover, under Assumption 4 we denote the autocovariance function and long-run variance of $\left( u_t \right)$ by $\gamma_u( . )$ and $\omega^2 = \sum_{ k = - \infty }^{ + \infty } \gamma_u ( k ) = C(1)^2 \sigma^2$ respectively and let

align[align omitted — 163 chars of source]

Thus, we have that $\rho_N \to \rho$ and $\Gamma = \underset{ n \to \infty }{ \text{lim} } \Gamma_n$ exists by the dominated convergence theorem since $\sum_{ k = 1 }^{ \infty } \left| \gamma_u (k) \right| < + \infty$. Notice that when $\rho = 1, \Gamma = \sum_{k=1}^{+\infty} \gamma_u (k)$ is the one-sided long-run covariance of $\left( u_t \right)_{ t \in \mathbb{N} }$. Denote with $W(t)$ to be the standard Brownian motion on $[0,1]$ and $B(t) = \omega W(t)$ and define the Ornstein-Uhlenbeck processes below

align[align omitted — 105 chars of source]

Conclusion

In this paper we consider the asymptotic theory for moderate deviation from the unit boundary in quantile autoregressive and quantile predictive regression models. Using the moderate deviation principles we unify the asymptotic theory with a modified endogenous instrumentation procedure, without inducing limiting distribution discontinuities at certain regions of the parameter space. Specifically, in this study we verify the limit results obtained by Phillips2007limit in the case of the linear autoregressive time series model. In particular, for both the case of near-stationary and near-explosive roots we establish the asymptotic theory of the quantile-dependent estimator which converges into a nuisance-parameter free limiting distribution.

An extension of our framework in the regions which unifies all cases such as being in the unstable region with nearly stable or unstable processes such as the explosive and pure explosive processes, is an aspect of ongoing research that the author is actively undertaking. Further research aspects worth mentioning include the investigation of the asymptotic behaviour of quantile autoregressive models when a structural break occurs at an unknown break-point location. A relevant study using moderate deviations principles when testing for structural breaks include the framework proposed by xu2018limit as is presented by katsouris2023structural.