#  ASYMPTOTIC THEORY FOR MODERATE DEVIATIONS FROM THE UNIT BOUNDARY
IN QUANTILE AUTOREGRESSIVE TIME SERIES††thanks: Article history: February
2022.

Christis Katsouris111Ph.D. Candidate, Department of Economics, University of
Southampton, Highfield Campus, Southampton, SO17 1BJ, UK. E-mail Address:
<EMAIL_ADDRESS>
Department of Economics, University of Southampton

(March 22, 2022)

Abstract

We 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 Phillips and Magdalinos,  (2007), we consider the limit theory for the
near-stationary and the near-explosive cases when the model is estimated with
a conditional quantile specification function and model parameters are
quantile-dependent. Additionally, a Bahadur-type representation and limiting
distributions based on the M-estimators of the model parameters are derived.
Specifically, 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.

##  Introduction

Moderate deviation principles from the unit boundary for quantile regression
models 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 Koenker and Xiao,  (2004, 2006) as
well as Koenker and Xiao,  (2002) who investigate estimation and inference
aspects for regression quantile models (see, also Hasan and Koenker,  (1997)).
On the contrary, studies for quantile autoregressive time series models that
consider moderate deviations within a unified framework allowing to
investigate the asymptotic behaviour of estimators with respect to different
modes 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 serial correlation coefficient when considering deviations from the unit
boundary in the quantile autoregressive model. More precisely, the present
paper builds on the framework proposed by Phillips and Magdalinos,  (2007)
(see, also Giraitis and Phillips,  (2006) and Huang et al.,  (2014)) that
corresponds to the linear autoregressive model under nonstationarity as well
as the study of Kong,  (2015) who develops related limit results for moderate
deviations in autoregressive models based on M-estimators.

According to Chan et al.,  (2006): "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
autocorrelation coefficient is expressed with respect to the local unit root
specification, what kind of approximation should be used for the distribution
of the test statistics?. To answer this question, Chan and Wei,  (1987) as
well as Phillips, 1987a ; Phillips, 1987b proposed the triangular array
framework, so-called nearly nonstationary AR(1) model, and established the
limiting distributions of the least squares estimator for $\theta_{n}$ under
the assumption that the conditional variance of the model $\varepsilon_{t}$ is
finite. Further results of the least squares estimate for the near-integrated
AR(1) model are presented by Chan,  (1988, 1990), Chan and Tran,  (1989),
Knight,  (1987), Rao,  (1978), Lai and Wei,  (1982), Cox and Llatas,  (1991),
Larsson,  (1995) as well as by Cavaliere,  (2002).

Additionally, the study of Hui et al.,  (2022) establishes the asymptotic
theory of the ordinary least squares estimator in the explosive first-order
Gaussian autoregressive process. Specifically, using a set of deviation
inequalities the authors obtain the limit theory of Cramér-type moderate
deviations for the explosive and mildly explosive autoregressive processes.
Thus, in this paper we focus on establishing the limit theory for moderate
deviations from the unit boundary for both the near-stationary and near-
explosive cases. Although, White,  (1958) obtained the limiting distribution
of an explosive serial correlation coefficient (see, also Mann and Wald,
(1943)), limit results for moderate deviations on the explosive side of unity
(e.g., mildly explosive case) were only recently established by the studies of
Phillips and Magdalinos,  (2007), Buchmann and Chan,  (2007) and Aue and
Horváth,  (2007).

Under the Cramér-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 Hui et al.,  (2022))

(1.1) $\displaystyle\underset{0\leq
x\leq\ell\lambda_{n}}{\mathsf{sup}}\left|\frac{1}{1-F_{n}(x)}\mathbb{P}\bigg{(}\nu_{n}\left(\hat{\theta}_{n}-\theta_{n}\right)\geq
x\bigg{)}-1\right|\to 0,$

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

(1.2)
$\displaystyle\mathbb{P}\bigg{(}\nu_{n}\left(\hat{\theta}_{n}-\theta_{n}\right)\geq
x\bigg{)}-F_{n}(x)\overset{p}{\to}0,\ \ \text{as}\ n\to\infty.$

Therefore, we consider 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). Specifically, due to the form
of the nonrandom sequence such that, $k_{n}\equiv n^{\gamma}$, the convergence
rate towards to unit boundary is slower than the sample size $n$. In this case
the autocorrelation coefficient approaches 1 at a rate slower than the usual
local alternative as $n$ goes to infinity. On the other hand, as $\gamma\to 1$
then $k_{n}\to n$ which encompasses the conventional local to unit root
specification. An alternative form, for the convergence rate include for
example the case when $k_{n}=\sqrt{n}$.

Furthermore to obtain limit results for model parameters based on the
M-estimators, we employ the Bahadur representation (see, Bahadur,  (1966))
that provides a mechanism to facilitate the asymptotic theory analysis since
it allows to obtain analytic expressions for quantile-dependent estimators by
approximating them with linear forms. Thus, when applied to the quantile
autoregressive model it permits to obtain joint convergence results. The limit
theory for an autoregressive model which includes a intercept and an
autocorrelation coefficient expressed using the local unit root specification
is studied by Liu and Liu,  (2018), however their framework differs from our
setting since we consider quantile-dependent parameters. Therefore, this paper
builds on and contributes to both the quantile autoregression literature as
well as to the literature of moderate deviations from the unit boundary.

###  Preliminary Theory

The asymptotic theory for moderate deviations from a unit root in
autoregressive models is examined in various studies such as by Fountis and
Dickey,  (1989), Jiang et al.,  (2015) who focus on the aspect of dependent
errors in AR(1) models and Yabe,  (2012) who obtain limit results for MA(1)
time series models222The MA(1) process is considered to be invertible if and
only if the root is away from the unit boundary. Moreover, estimators and test
statistics for non-invertible processes behave quite differently from an
invertible MA(1) time series model. In particular, the CLT holds for the
maximum likelihood estimator for an invertible process while for a non-
invertible process the CLT does not hold because the maximum likelihood
estimator has a probability mass at unity.. Furthermore, limit results for
moderate deviations for M-estimators and quantile processes333Related
theoretical aspects can be found in the book of Csörgő,  (1983) (see, also
Csorgo et al.,  (1986)). in autoregressive models are investigated in several
studies such as by Jurečková et al.,  (1988), Knight,  (1998), Mao and Guo,
(2019) as well as Kato,  (2009) where the author develops asymptotics for
Lasso quantile-dependent estimators. On the other hand, within the context of
quantile autoregressive models related studies which consider moderate
deviations from the unit boundary include Lucas,  (1995), Abadir and Lucas,
(2000), Ling and McAleer,  (2004), Chan et al.,  (2006) and Kong,  (2015).

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 1, as pointed out by Dickey and
Fuller,  (1979) (see, also Dickey and Fuller,  (1981)). More precisely, the
idea of developing asymptotics using the near-unit root specification is due
to the studies of Cavanagh,  (1985), Phillips, 1987a , Chan and Wei,  (1987).
In particular, Phillips and Magdalinos,  (2007) showed that
$\hat{\theta}_{n}-\theta_{n}$ has a $\sqrt{nk_{n}}$ rate of convergence and a
limit normal distribution when $c<0$,

(1.3)
$\displaystyle\sqrt{n\kappa_{n}}\left(\hat{\theta}_{n}-\theta_{n}\right)\overset{d}{\to}\mathcal{N}\big{(}0,-2c\big{)}$

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. Therefore, by expressing the AR(1) model with respect to the
canonical parameters of an exponential family we can obtain insights regarding
the asymptotic behaviour of related minimal sufficient statistics. The
particular literature is developed under the assumption of a stationary
autocorrelation 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 size444In other
words, when the autocorrelation coefficient is expressed using the moderate
deviations form then unit boundary then this can enable the analysis of unit
roots and explosive processes, which is necessary to link problems in
inference for unit roots to the statistical curvature. (see, Garderen,
(1999)). In addition, Jansson and Moreira,  (2006) using the local-unit-root
specification of the autocorrelation coefficient, study the properties of
predictive regression models under the assumption of persistence regressors
using differential geometry and sufficient statistics arguments to establish
the asymptotic theory of estimators and test statistics. Lastly, an important
aspect worth mentioning is the fact that several studies have demonstrated
that the nuisance parameter of persistence cannot be consistently estimated
(see, Phillips et al.,  (2001), Mikusheva,  (2012) 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. However, our asymptotic theory analysis which
focus on the asymptotic behaviour of the quantile-dependent parameter remains
valid despite the absence of such a desirable statistical property.

The main research objective of our paper is the asymptotic behaviour of the
quantile-dependent estimators. More precisely, we establish the asymptotic
theory of estimators in possibly nonstationary quantile autoregressive models
for the general near-integrated case assuming that $\theta_{n}\to 1$ with a
rate slower than $1/n$. In particular, Aue and Horváth,  (2007) 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 $\mathcal{O}(1/n)$.

The rest of the paper is organized as follows. Section 2, introduces the
framework of moderate deviation principles from the unit boundary in quantile
autoregressive processes as well as the main estimation methodology. Section
3, 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 4 presents a short Monte Carlo simulation study while Section 5 an
empirical application. Section 6 concludes.

##  Quantile Autoregression under the Moderate Deviations Framework

Consider the autoregressive AR(1) time series model with an intercept
expressed as below

(2.1) $\displaystyle y_{t}=\mu_{n}+\rho_{n,c}y_{t-1}+\varepsilon_{t},\ \ \ \ \
\text{for}\ \ t=1,...,n,\ \ \ \ $

with

(2.2) $\displaystyle\rho_{n,c}=\left(1+\frac{c}{k_{n}}\right),\ \text{with}\ \
c\in\mathbb{R}\ \ \text{and}\ \gamma\in(0,1).$

with innovations $\left(\varepsilon_{t}\right)_{t\in\mathbb{N}}$ an i.i.d
sequence of random variables from a common distribution function
$F_{\varepsilon}$ that satisfies regularity conditions for Lipschitz
continuity with zero mean and finite variance $\sigma_{\varepsilon}$.
Moreover, we denote by $\mathcal{F}_{t}$ the natural filtration
$\sigma\big{(}\varepsilon_{t},\varepsilon_{t-1},...\big{)}$ and with
$\mathbb{E}_{\mathcal{F}_{t}}\big{[}.\big{]}$ the corresponding conditional
expectation. Furthermore, the convergence rate of the autocorrelation
coefficient $\rho_{n,c}$ 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 2.1.

Denote with $y_{0n}=0$, 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 $n\to\infty$

(2.3)
$\displaystyle\frac{1}{n}\sum_{t=1}^{n}\mathbb{E}\left[\varepsilon_{t}^{2}\big{|}\mathcal{F}_{t-1}\right]$
$\displaystyle=1+o_{p}(1),$ (2.4)
$\displaystyle\frac{1}{n}\sum_{t=1}^{n}\mathbb{E}\left[\varepsilon_{t}^{2}\mathbf{1}\left\\{|\varepsilon_{t}\big{|}>n^{1/2}m\right\\}\big{|}\mathcal{F}_{t-1}\right]$
$\displaystyle=o_{p}(1),\ \ \text{for all}\ m>0,$

where $\mathcal{F}_{t}:=\sigma\big{(}\varepsilon_{s}:0\leq s\leq t\big{)}$.

Assumption 2.1 provides moment conditions and a uniform integrability
condition which induces a restriction on the tail behaviour of the underline
distribution of innovations. Furthermore, for the development of the
asymptotic theory 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. Then, asymptotic
approximations are obtained based on the Ornstein-Uhlenbeck processes.

######  Assumption 2.2.

The 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,

(2.5) $\displaystyle\mathbb{E}\left(\varepsilon_{1}\right)=0,\ \
\mathsf{Var}\left(\varepsilon_{1}\right)=\sigma^{2}_{\varepsilon},\ \ \
\text{and}\ \ \ \mathbb{E}\left|\varepsilon_{1}\right|^{2+m}<\infty,\
\text{for some}\ m>0.$

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 Lévy processes on
the space of functions $\mathcal{D}[0,1]$.

######  Lemma 2.1.

Suppose that $\varepsilon_{t}$ satisfies Assumption 2.2. Then as $n\to\infty$
it holds that

(2.6) $\displaystyle\left(\frac{1}{\nu_{n}}\sum_{t=1}^{\lfloor
nr\rfloor}\varepsilon_{t},\frac{1}{\nu_{n}^{2}}\sum_{t=1}^{\lfloor
nr\rfloor}\varepsilon_{t}^{2}\right)\Rightarrow\bigg{(}U_{\upalpha}(r),V_{\upalpha}(r)\bigg{)}$

where $\big{(}U_{\upalpha}(r),V_{\upalpha}(r)\big{)}$ is a Levy process in
$\mathcal{D}[0,1]^{2}$ with index $\upalpha\in(0,2)$.

######  Remark 2.1.

A general class of one-dimensional stochastic processes, are the so-called
Lévy processes. Similar to Wiener processes, Lévy processes have right
continuous paths with left limits, are initiated from the origin and both have
stationary and independent increments (Kyprianou,  (2014)). 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. More precisely, 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 Cramér,
(1951)).

Therefore, any asymptotic results followed by Lemma 2.1 coincide with the
standard random walk asymptotics for finite variance models. Specifically, in
this paper we focus in the case $\upalpha=2$. Then as $n\to\infty$ the
following limit result holds

(2.7) $\displaystyle
n\left(\hat{\rho}_{n,c}-1\right)\Rightarrow\frac{\displaystyle\int_{0}^{1}W(r)dW(r)}{\displaystyle\int_{0}^{1}W^{2}(r)dr}.$

Notice that 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
$-n^{\gamma}<c<0$. Therefore, 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 a natural OLS estimate of
the coefficient of persistence $c$, which is not consistent. Furthermore,
Mikusheva,  (2012) shows that

(2.8)
$\displaystyle\left(\hat{c}-c\right)=\frac{\displaystyle\int_{0}^{1}J_{c}(r)dW(r)}{\displaystyle\int_{0}^{1}J_{c}^{2}(r)dr}.$

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.

In general, 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 results of the paper 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. Furthermore, imposing assumptions
regarding the properties of the disturbance term $\varepsilon_{t}$ can imply
different asymptotic behaviour for estimators given certain modelling
conditions. According to Werker and Zhou,  (2022) 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. Therefore, the asymptotic theory of this paper is
established based on optimal test statistics due to the Gaussianity assumption
of the innovation processes. Relaxing the Gaussianity assumption requires to
apply semiparametric estimation methodologies that 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.
Considering the limit theory based on the innovation sequence being a
stationary time series under weak dependence, requires further regularity
conditions especially when the aim is to derive bounds and corresponding
convergence rates based on Berry-Esseen theorems (see, Jirak,  (2016) and
Lahiri and Sun,  (2009)). Further studies of the proposed econometric
environment with the use of moderate deviation principles are presented by
Penda et al.,  (2014) and Proïa,  (2020).

###  Quantile Conditional Estimation Methodology

In this Section we present a more detailed review of the estimation procedure
for the quantile autoregressive time series that accommodates the
specification of the autocorrelation coefficient with respect to moderate
deviations from the unit boundary. We first introduce the quantile estimation
method555A complete treatment of limit results for quantile regressions can be
found in the book of Koenker,  (2005). to obtain parameter estimates and then
establish the asymptotic theory of this estimator. Denote with $\mu(\uptau)$
and $\rho_{n}(\uptau)$ to be the $\uptau-$quantile dependent parameters, which
are determined based on a conditional quantile specification function

(2.9)
$\displaystyle\mathsf{Q}_{y_{t}}\left(\uptau|\mathcal{F}_{t-1}\right):=F^{-1}_{y_{t}|\boldsymbol{x}_{t-1}}(\uptau)\equiv\mu(\uptau)+\rho_{c}(\uptau)y_{t-1}.$

such that

(2.10) $\displaystyle
F_{y_{t}|\boldsymbol{x}_{t-1}}(\uptau):=\mathbb{P}\bigg{(}y_{t}\leq\mathsf{Q}_{y_{t}}\left(\uptau|\mathcal{F}_{t-1}\right)\big{|}\mathcal{F}_{t-1}\bigg{)}\equiv\uptau.$

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 Koenker and Bassett,  (1978) and Koenker and Portnoy,  (1987) the
quantile regression estimator is obtained via the following optimization
function

(2.11)
$\displaystyle\widehat{\boldsymbol{\vartheta}}_{n}\big{(}\uptau\big{)}:=\underset{\boldsymbol{\vartheta}(\uptau)}{\mathsf{arg\
min}}\
\sum_{t=1}^{n}\varrho_{\uptau}\bigg{(}y_{t}-\boldsymbol{\vartheta}(\uptau)^{\top}\boldsymbol{X}_{t}\bigg{)}.$

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
quantile regression optimization function and
$\widehat{\boldsymbol{\vartheta}}_{n}$ is the quantile-dependent estimator,
the main focus of our paper.

######  Assumption 2.3.

Suppose 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

(2.12)
$\displaystyle\xi:=\left|\frac{\partial}{\partial\theta}\mathbb{E}\bigg{[}\psi\bigg{(}u_{1}(\uptau)-\theta\bigg{)}\bigg{]}\right|_{\theta=0},\
\ \text{where}\ \xi\neq 0.$

where $\mathbb{E}|\psi\left(u_{1}(\uptau)\right)|^{2+m}<\infty$, for some
$m>0$.

Assumption 2.3 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 2.4 below.

######  Assumption 2.4.

The following conditions for the innovation sequence hold:

  * (i)

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 such that
$f_{\varepsilon_{t}(\uptau)}(0):=\mathbb{E}\left[f_{\varepsilon_{t}(\uptau),t-1}(0)\right]>0$
satisfies a Functional Central Limit Theorem (FCLT) expressed as below

(2.13) $\displaystyle\frac{1}{\sqrt{n}}\sum_{t=1}^{\lfloor
nr\rfloor}\left(f_{\varepsilon_{t}(\uptau),t-1}(0)-\mathbb{E}\left[f_{\varepsilon_{t}(\uptau),t-1}(0)\right]\right)\Rightarrow
B_{f_{\varepsilon_{t}(\uptau)}}(r),\ \text{with}\ r\in(0,1).$

  * (ii)

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, Neocleous and Portnoy,  (2008), Goh and Knight,
(2009) and Lee,  (2016)).

Assumption 2.4 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 specification, 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
(2.11) we apply the Taylor expansion to the check function, such that for a
given parameter $\boldsymbol{\delta}(\uptau)$ it holds that

(2.14)
$\displaystyle\varrho_{\uptau}\bigg{(}\varepsilon_{t}-\boldsymbol{\delta}(\uptau)^{\top}\boldsymbol{X}_{t}\bigg{)}=\varrho_{\uptau}(\varepsilon_{t})-\boldsymbol{\delta}(\uptau)^{\top}\psi(\varepsilon_{t})+\varphi_{t}\big{(}\boldsymbol{\delta}(\uptau)\big{)}.$

######  Remark 2.2.

Notice 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 $\rho=1$.

##  Asymptotic 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, Chan et al.,  (2006)) 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. More specifically,
using the local-to-unity specification in autoregressive processes helps to
overcome this gap by considering the limiting distribution for the whole
parameter space regardless of the existence of a limit singularity.

Firstly, using the local-to-unity asymptotics our aim is to derive a nuisance-
parameter-free limiting distribution which can facilitate statistical
inference. In particular, the form of the noncentrality parameter of the
$\chi^{2}-$distribution which is the limiting distribution of test statistics
can depend on the initial condition666Notice 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, Saxena and Alam,  (1982)). and the form of the
autoregressive parameter of the model (see, Theorem 2.2. of Hui et al.,
(2022)). Secondly, by decomposing the underline stochastic processes into
components which include a predictable quadratic variation (see, Magdalinos
and Phillips,  (2009)), 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.

######  Theorem 3.1 (Chan et al.,  (2006)).

Assume that Assumption 2.1-2.4 hold and that the autocorrelation coefficient
is expressed as $\rho_{n,c}=\left(1+\frac{c}{k_{n}}\right)$. Then, the
following limit result hold

(3.1)
$\displaystyle\boldsymbol{D}_{n}\left(\hat{\boldsymbol{\vartheta}}_{n}(\uptau)-\boldsymbol{\vartheta}(\uptau)\right)\overset{d}{\to}\frac{1}{f_{\varepsilon}\big{(}F_{\varepsilon}^{-1}(\varepsilon_{t}(\uptau))\big{)}}\Sigma^{-1}\left(W(\uptau;1),\int_{0}^{1}J(s)dW(\uptau,s)\right)^{\prime},$

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

(3.2)
$\displaystyle\boldsymbol{\Sigma}:=\int_{0}^{1}\big{(}1,J_{1}(s)\big{)}^{\prime}\big{(}1,J_{1}(s)\big{)}ds\equiv\begin{bmatrix}1&\displaystyle\int_{0}^{1}J_{1}(s)ds\\\
\displaystyle\int_{0}^{1}J_{1}(s)^{\prime}ds&\displaystyle\int_{0}^{1}J_{1}(s)J_{1}(s)^{\prime}ds\end{bmatrix}.$

Furthermore, it holds that

(3.3) $\displaystyle
n\bigg{(}\hat{\rho}_{n,c}(\uptau)-\rho_{n,c}(\uptau)\bigg{)}\overset{d}{\to}\frac{1}{f_{\varepsilon}\big{(}F_{\varepsilon}^{-1}(\varepsilon_{t}(\uptau))\big{)}}\frac{\displaystyle\int_{0}^{1}J_{1}(s)dW(\uptau,s)-W(\uptau,1)\int_{0}^{1}J_{1}(s)ds}{\displaystyle\int_{0}^{1}J^{2}_{1}(s)ds-\left(\int_{0}^{1}J_{1}(s)ds\right)^{2}}$

and with a suitable normalization it follows that

(3.4)
$\displaystyle\left\\{\sum_{t=1}^{n}y_{t-1}^{2}-\left(\sum_{t=1}^{n}y_{t-1}\right)^{2}\right\\}^{1/2}\bigg{(}\hat{\rho}_{n,c}(\uptau)-\rho_{n,c}(\uptau)\bigg{)}\overset{d}{\to}\mathcal{N}\left(0,\frac{\uptau(1-\uptau)}{f_{\varepsilon}^{2}\big{(}F_{\varepsilon}^{-1}(\varepsilon_{t}(\uptau))\big{)}}\right).$

###  Limit theory for near-stationary case

In a similar spirit as in the framework proposed by Phillips and Magdalinos,
(2007), in order to establish the limit theory of the autocorrelation
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 specification function, 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

(3.5) $\displaystyle\boldsymbol{D}_{n}=\begin{pmatrix}\sqrt{n}&0\\\
0&\sqrt{nk_{n}}\end{pmatrix},\ \ \ \boldsymbol{B}=\begin{pmatrix}1&0\\\
0&\sigma^{2}/(-2c)\end{pmatrix}$

where $k_{n}=n^{\gamma}$ and $\gamma\in(0,1)$.

######  Remark 3.1.

The $n^{-1/2}$ convergence rate corresponds to the model intercept while when
$k_{n}=n^{\gamma}$, then the autocorrelation 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 $c<0$.

######  Theorem 3.2.

Under Assumptions 2.2-2.4,

(3.6)
$\displaystyle\big{(}\hat{\mu}_{n},\hat{\rho}_{n,c}\big{)}^{\top}=\big{(}\mu,\rho_{n,c}\big{)}^{\top}+\frac{(\boldsymbol{B}\boldsymbol{D}_{n})^{-1}}{\xi}\sum_{t=1}^{n}\psi\big{(}\varepsilon_{t}\big{)}\boldsymbol{X}_{t}^{\top}+o_{p}(1).$

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

(3.7) $\displaystyle\begin{pmatrix}\hat{\mu}_{n}(\uptau)\\\
\hat{\rho}_{c,n}(\uptau)\end{pmatrix}=\begin{pmatrix}\mu_{n}(\uptau)\\\
\rho_{c,n}(\uptau)\end{pmatrix}+\frac{\left(\boldsymbol{B}\boldsymbol{D}_{n}\right)^{-1}}{f_{\varepsilon}\big{(}F_{\varepsilon}^{-1}(\uptau)\big{)}}\sum_{t=1}^{n}\bigg{(}\uptau-\mathbf{1}\left\\{\varepsilon_{t}\leq
F_{\varepsilon}^{-1}(\uptau)\right\\}\bigg{)}\begin{pmatrix}\displaystyle\frac{1}{\sqrt{n}}\\\
\\\ \displaystyle\frac{y_{t-1}}{\sqrt{nk_{n}}}\end{pmatrix}+o_{P}(1).$

where $f_{\varepsilon}(x)$ and $F_{\varepsilon}(x)$ denote the probability and
cumulative density functions of $\varepsilon_{1}$, respectively.

Theorem 3.2 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.

######  Theorem 3.3.

Under Assumptions 2.2-2.4,

(3.8)
$\displaystyle\boldsymbol{D}_{n}\left(\big{(}\hat{\mu}_{n},\hat{\rho}_{n,c}\big{)}-\big{(}\mu_{n},\rho_{n,c}\big{)}\right)\overset{d}{\to}\mathcal{N}\left(0,\frac{\boldsymbol{B}^{-1}\times\mathbb{E}\big{[}\psi^{2}(\varepsilon_{1})\big{]}}{\xi}\right).$

In particular, it follows that

  * (i)

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

(3.9)
$\displaystyle\frac{\hat{\rho}_{n,c}(\uptau)-\rho_{c}(\uptau)}{\sqrt{nk_{n}}}\overset{d}{\to}\mathcal{N}\left(0,\frac{-2c}{\sigma^{2}}\frac{\uptau(1-\uptau)}{f_{\varepsilon}^{2}\big{(}F_{\varepsilon}^{-1}(\uptau)\big{)}}\right).$

  * (ii)

If $\psi(\mathsf{u})=\mathsf{u}$, then

(3.10)
$\displaystyle\frac{\hat{\rho}_{n,c}(\uptau)-\rho_{c}(\uptau)}{\sqrt{nk_{n}}}\overset{d}{\to}\mathcal{N}\left(0,-2c\right).$

######  Remark 3.2.

The limit results given by Theorem 3.2 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 Pollard,  (1991) 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 $a_{n}$ 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

(3.11)
$\displaystyle\frac{\theta^{n}}{\theta^{2}-1}\left(\hat{\theta}_{n}-\theta\right)\Rightarrow\mathcal{C},\
\ \ \text{as}\ n\to\infty.$

More precisely, the seminal study of Anderson,  (1959) 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,
Magdalinos,  (2012)). Furthermore, in this direction, Phillips and Magdalinos,
(2007) consider autoregressive processes under the moderate deviations
framework by employing the local-to-unity specification 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 i.i.d innovations with finite second moments
the following least squares regression theory was proved

(3.12)
$\displaystyle\frac{1}{2c}n^{\gamma}\theta_{n}^{n}\left(\hat{\theta}_{n}-\theta\right)\Rightarrow\mathcal{C},\
\ \ \text{as}\ n\to\infty.$

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 Anderson,  (1959), 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, Phillips and Magdalinos,  (2007) 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.

(3.13) $\displaystyle\boldsymbol{D}_{n}=\begin{pmatrix}\sqrt{n}&0\\\
0&\rho_{n,c}^{n}k_{n}\end{pmatrix},\ \ \ \boldsymbol{B}=\begin{pmatrix}1&0\\\
0&\mathcal{Z}_{3}^{2}/(2c)\end{pmatrix}$

where $\mathcal{Z}_{3}$ is some normal random variable to be defined below.

######  Theorem 3.4.

Under the Assumptions 2.1-2.4 it holds that,

(3.14)
$\displaystyle\left(\hat{\mu},\hat{\rho}_{n,c}\right)^{\top}=\left(\mu,\rho_{n,c}\right)^{\top}+\frac{\left(\sum_{t=1}^{n}\boldsymbol{X}_{t}\boldsymbol{X}_{t}^{\top}\boldsymbol{D}_{n}\right)^{-1}}{\xi}\sum_{t=1}^{n}\psi\big{(}\varepsilon_{t}\big{)}\boldsymbol{X}_{t}^{\top}+o_{p}(1).$

######  Theorem 3.5.

Under Assumptions 2.1-2.4 it holds that,

(3.15)
$\displaystyle\boldsymbol{D}_{n}\big{(}\left(\hat{\mu},\hat{\rho}_{n,c}\right)-\left(\mu,\rho_{n,c}\right)\big{)}^{\top}\overset{d}{\to}\frac{1}{\xi}\boldsymbol{B}_{n}^{-1}\big{(}\mathcal{Z}_{1},\mathcal{Z}_{2}\mathcal{Z}_{3}\big{)}^{\top}.$

In particular, it follows that

  * (i)

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

(3.16)
$\displaystyle\frac{\hat{\rho}_{n,c}(\uptau)-\rho_{c}(\uptau)}{\rho_{n}k_{n}}\overset{d}{\to}\frac{2c}{f_{\varepsilon}\left(F_{\varepsilon}^{-1}(\uptau)\right)}\frac{\mathcal{Z}_{2}}{\mathcal{Z}_{3}}.$

  * (ii)

If $\psi(\mathsf{u})=\mathsf{u}$, then

(3.17)
$\displaystyle\frac{\hat{\rho}_{n,c}(\uptau)-\rho_{c}(\uptau)}{2c\rho_{n}^{n}k_{n}}\overset{d}{\to}\frac{\mathcal{Z}^{*}_{2}}{\mathcal{Z}^{*}_{3}}.$

Therefore, Theorem 3.5 verifies that we indeed obtain the equivalent
asymptotic theory results in comparison to the linear autoregressive time
series model. Specifically, the autocorrelation coefficient of the
nonstationary quantile autoregressive time series model converge into a Cauchy
random variate in the case of mildly explosive processes (see, also Aue and
Horváth,  (2007), Phillips and Magdalinos,  (2007), Magdalinos,  (2012) and
Lee,  (2018)).

######  Remark 3.3.

As we see from Theorem 3.5, part 2, the limiting distribution of the
normalized and centered estimator is Cauchy, similar to Theorem 4.3 of
Phillips and Magdalinos,  (2007). 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
White,  (1958) 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 3.5 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

(3.18) $\displaystyle y_{t}=\rho y_{t-1}+\varepsilon_{t},\ \ \ \ t=1,...,n,$

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, Dickey and Fuller,  (1979) 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, Abadir
and Lucas,  (2000)). 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, Dickey and Fuller,  (1979) and Buchmann and Chan,  (2007)).
Furthermore, 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
hypothesis777Specifically, Magdalinos,  (2007) consider the approximate
Bahadur slopes of test statistics by examining their asymptotic behaviour
under the alternative hypothesis..

The t-statistic for the null hypothesis $\mathbb{H}_{0}:\rho=\rho_{0}$ is
given by

(3.19) $\displaystyle
t_{\psi}=\frac{\displaystyle\bigg{(}\hat{\rho}_{n}(\uptau)-\rho_{n}(\uptau)\bigg{)}}{\displaystyle\left\\{\left(n^{-1}\sum_{t=1}^{n}\psi_{\uptau}\big{(}y_{t}-\hat{\rho}_{n}(\uptau)y_{t-1}\big{)}^{2}\right)\bigg{/}\left(n^{-1}\sum_{t=1}^{n}\psi^{\prime}_{\uptau}\big{(}y_{t}-\hat{\rho}_{n}(\uptau)y_{t-1}\big{)}\right)^{2}\right\\}^{1/2}}.$

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

##  Monte Carlo Simulations

In this section we investigate the finite-sample performance in terms of
standard deviations and coverage probabilities of the quantile estimate and
the least squares estimate are computed. We consider the autoregressive time
series model

(4.1) $\displaystyle y_{t}=\mu+\rho_{n}y_{t-1}+\varepsilon_{t},\ \ \ \
t=1,...,n$

where
$\varepsilon_{t}\overset{\textit{i.i.d}}{\sim}\mathcal{N}\big{(}0,1\big{)}$
and $\mu$ and $\rho$ the model parameters to be estimated. In particular,
since the autocorrelation coefficient is defined as
$\rho_{n,c}=\left(1+\frac{c}{n}\right)$, by definition of the nonstationary
autoregressive time series models, we begin by comparing the the performance
of the OLS-based estimator and the QR-based estimator for $\uptau=0.5$. The
least squares estimate is given by

(4.2)
$\displaystyle\hat{\rho}^{ols}_{n,c}=\frac{\displaystyle\sum_{t=1}^{n}y_{t}y_{t-1}}{\displaystyle\sum_{t=1}^{n}y_{t}^{2}}$

Therefore, we verify the simulation results presented in the study of Chan et
al.,  (2006) who show that the sample averages of the estimates and the
corresponding standard deviations of $\hat{\rho}_{n,c}^{qr}(\uptau)$ in
comparison to $\hat{\rho}_{n,c}^{ols}$ indicate to perform better in terms of
precision and standard errors. In particular, this improvement is shown to be
more apparent for finite-samples. To demonstrate this through simulations, we
estimate the empirical size of the $t-$test; the size of the test statistic
obtained from the simulated finite sample distribution and then compare with
the nominal size; the size that would be expected if the finite sample
distribution is perfectly approximated by the limiting distribution
(significance level $\alpha=0.05$). The empirical size under the null
hypothesis is presented below.

Table goes here.

##  An Empirical Application

We employ the R package developed by Koenker,  (2012) in order to obtain
estimates for the quantile autoregressive model. Furthermore, we compare the
time series behaviour of two cryptocurrencies which are considered to be among
the 10 largest cryptocurrencies with respect to market capitalization. We
focus on the sampling period 01 January 2018 to 20 March 2022. In particular,
several papers in the literature employ the framework proposed by Phillips et
al.,  (2015) to detect bubbles in cryptocurrencies which verify that the
bubble periods coincide with major events that affected the Bitcoin market. In
this paper we focus on comparing the performance of the quantile
autoregressive model with respect to the linear autoregressive time series
model as well as the testing for predictability using the Student’s
$t-$statistic applied to the autocorrelation coefficient of the model.

$0$$200$$400$$600$$800$$1{,}000$$1{,}200$$1{,}400$$1{,}600$$0$$2$$4$$6$$\cdot
10^{4}$timeEB

Figure 1 above shows the adjusted closing price (in US dollars) of the
cryptocurrencies Bitcoin and Ethereum for 1603 trading days between 01 January
2018 to 20 March 2022.

Table 1 shows the model estimates for the two time series of cryptocurrencies
Etherum and Bitcoin that we implement the quantile autoregressive model. To
begin with, the linear autoregressive model when with no model intercept has a
statistical significant and explosive autocorrelation coefficient while
similar results are obtained when we consider the quantile autoregressive
model around the median $(\uptau=0.5)$, which under the Gaussian innovation
assumption should be equivalent to the linear autoregressive model. On the
other hand, when considering the quantile autoregressive model at the upper
and lower tails of the underline distribution (e.g., $\uptau=0.05$ or
$\uptau=0.95$) then moderate deviations from the unit root on the explosive
side are more apparent especially when the model is fitted at the
$\uptau=0.95$ of the distribution. Therefore, overall there are statistical
significant evidence of explosive behaviour in the time series of
cryptocurrencies at the tails of the distribution.

Table 1: Estimation Results | Etherum
---|---
|  | $\hat{\theta}$ | $s.e(\hat{\theta})$ |  $t-$statistic |  | $\hat{\theta}$ | $s.e(\hat{\theta})$ |  $t-$statistic
linear | $\mu$ |  |  |  |  | 2.9759 | 2.5803 | 1.1530
$\rho$ | 1.0002 | 0.0012 | 813.6*** |  | 0.9990 | 0.0016 | 623.3***
QR $(\uptau=0.05)$ | $\mu$ |  |  |  |  | 0.7857 | 1.7359 | 0.4527
$\rho$ | 0.9222 | 0.0049 | 188.5*** |  | 0.9208 | 0.0066 | 140.5***
QR $(\uptau=0.5)$ | $\mu$ |  |  |  |  | -0.3981 | 0.6437 | -0.6184
$\rho$ | 1.0023 | 0.0024 | 424.5*** |  | 1.0029 | 0.0029 | 342.5***
QR $(\uptau=0.95)$ | $\mu$ |  |  |  |  | 0.7496 | 2.4964 | 0.3003
$\rho$ | 1.0845 | 0.0054 | 200.9*** |  | 1.0840 | 0.0071 | 153.5***
| Bitcoin
|  | $\hat{\theta}$ | $s.e(\hat{\theta})$ |  $t-$statistic |  | $\hat{\theta}$ | $s.e(\hat{\theta})$ |  $t-$statistic
linear | $\mu$ |  |  |  |  | 45.8416 | 38.1999 | 1.2000
$\rho$ | 1.0002 | 0.0010 | 1021*** |  | 0.9989 | 0.0014 | 689.2***
QR $(\uptau=0.5)$ | $\mu$ |  |  |  |  | 29.6022 | 41.0729 | 0.7207
$\rho$ | 0.9372 | 0.0035 | 270.6*** |  | 0.9350 | 0.0050 | 186.6***
QR $(\uptau=0.5)$ | $\mu$ |  |  |  |  | 9.9114 | 13.6279 | 0.7273
$\rho$ | 1.0012 | 0.0014 | 741.9*** |  | 1.0002 | 0.0020 | 495.5***
QR $(\uptau=0.5)$ | $\mu$ |  |  |  |  | 45.4248 | 66.6845 | 0.6812
$\rho$ | 1.0707 | 0.0041 | 260.1*** |  | 1.0679 | 0.0059 | 181.4***

Table 1 shows the model estimates for both the linear and quantile
autoregressive time series model given by $y_{t}=\mu+\rho
y_{t-1}+\varepsilon_{t}$, with $t=1,...,n$, based on the adjusted closing
price (in US dollars) of the cryptocurrencies Bitcoin and Ethereum for 1603
trading days between 01 January 2018 to 20 March 2022.

######  Remark 5.1.

Notice that for the quantile autoregressive time series the model estimates
are obtained using a bootstrap estimation procedure implemented as a build-in
option of the R package quantreg.

    tau <- 0.05

    ### Model 1: Etherum
    model.QR_etherum <- rq( e_t ~ e_lag, tau = tau )
    model.summary    <- summary( model.QR_etherum, se = "boot", bsmethod= "xy" )

    ### Model 2: Bitcoin
    model.QR_bitcoin <- rq( b_t  ~ b_lag, tau = tau )
    model.summary    <- summary( model.QR_bitcoin, se = "boot", bsmethod= "xy" )

##  Conclusion

In this paper we consider the asymptotic theory for moderate deviation from
the unit boundary in quantile autoregressive time series models. The moderate
deviation principles provide a framework for unifying the asymptotic theory by
investigating the limiting distribution discontinuities at certain regions of
the parameter space. Specifically, in this study we verify the limit results
obtained by Phillips and Magdalinos,  (2007) 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.

Furthermore, although we do not consider how the presence of serial
correlation can affect the limiting distributions under consideration, various
studies in the literature consider simple implementations of autocorrelation
robust tests such as in Jansson,  (2004) (see, also Vogelsang,  (1998)). In
particular, a good example is the OLS estimator in a linear regression model
with exogenous regressors and an autocorrelated error term. 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. In
particular, Kiefer et al.,  (2000), (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. Using higher-order
asymptotic theory, Jansson,  (2004) provides an analytical explanation of the
encouraging performance of the KVB procedure. 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 Xu and Pang,  (2018). Other aspects include
comparisons of the performance of $t-$tests with likelihood-based tests as in
Nielsen,  (2001).

##### Acknowledgements

I wish to thank the supervisory committee for helpful discussions and
encouragement throughout the PhD programme. All remaining errors are mine.

##### Funding

Financial support from the Vice Chancellor’s PhD studentship of the University
of Southampton is gratefully acknowledged.

##  Appendix A Mathematical Appendix

Recall that an $M-$estimator $\big{(}\hat{\mu},\hat{\rho}\big{)}$ of the
parameter vector $\big{(}\mu,\rho\big{)}$ is a minimizer for
$\mu\in\mathbb{R}$ and $\rho\in\mathbb{R}$ such that

(A.1) $\displaystyle
R_{n}(\mu,\rho)=\frac{1}{n}\sum_{t=1}^{n}\varrho_{\uptau}\left(y_{t}-\mu-\rho_{c}x_{t-1}\right).$

The formulation of the above expression covers the least squares regression
when $\rho_{\uptau}\left(\mathsf{u}\right)=\mathsf{u}/2$ and the quantile
regression such that
$\rho_{\uptau}\left(\mathsf{u}\right)=\left\\{\uptau\mathsf{u}\mathbf{1}\left(\mathsf{u}>0\right)-(1-\uptau)\mathsf{u}\mathbf{1}\left(\mathsf{u}<0\right)\right\\}$,
for some $0<\uptau<1$.

###  Main Results

####  Near-Stationary Case $(c<0)$

######  Lemma A.1.

Under Assumption 2.2, when $c<0$ then it holds that

(A.2) $\displaystyle\mathbb{P}\left(\underset{1\leq t\leq n}{\mathsf{max}}\
y_{t}^{2}\geq\lambda\right)\leq\frac{\mathbb{E}\left[y_{n}^{2}\right]}{\lambda^{2}},\
\text{for some}\ \lambda>0.$

###### Proof.

Notice that Lemma A.1 corresponds to the Kolmogorov, Doob maximal inequality
applied to the martingale sequence
$\big{(}y_{n},\mathcal{F}_{n}\big{)}_{n\in\mathbb{N}}$. The proof of Lemma A.1
can be easily obtained by considering the set
$\mathcal{S}_{s}=\left\\{y_{s}^{2}>\lambda,y_{j}\leq\lambda,j\leq s\right\\}$
and expanding the expression for the expectation
$\mathbb{E}\left[y_{n}^{2}\right]$. The particular result provides a
probability bound for the tails of the autoregressive time series which
inherits the properties of the stochastic difference equation. ∎

In addition to Lemma A.1, the following two properties hold:

  * •

If $c<0$, $\mathbb{E}\big{[}y_{n}^{2}\big{]}=\mathcal{O}(k_{n})$.

  * •

$\underset{1\leq t\leq n}{\mathsf{max}}\
\displaystyle\frac{y_{t}^{2}}{n}=o_{p}(1)$.

######  Lemma A.2.

Under Assumptions 2.2 \- 2.4, it holds that

(A.3)
$\displaystyle\sum_{t=1}^{n}\mathbb{E}_{\mathcal{F}_{t-1}}\bigg{[}\varphi_{nt}\big{(}\boldsymbol{\delta}(\uptau)\big{)}\bigg{]}\overset{p}{\to}\frac{1}{2}\xi\times\boldsymbol{\delta}^{\top}(\uptau)\boldsymbol{B}\boldsymbol{\delta}(\uptau).$

where

(A.4)
$\displaystyle\xi:=\left|\frac{\partial}{\partial\theta}\mathbb{E}\bigg{[}\psi\bigg{(}u_{1}(\uptau)-\theta\bigg{)}\bigg{]}\right|_{\theta=0}\
\ \ \text{and}\ \ \ \boldsymbol{B}=\begin{pmatrix}1&0\\\
0&\sigma^{2}/(-2c)\end{pmatrix}.$

###### Proof.

By rearranging expression (2.14), taking the conditional expectation and sum
over $1\leq t\leq n$, we obtain the following expression

(A.5)
$\displaystyle\sum_{t=1}^{n}\mathbb{E}_{\mathcal{F}_{t-1}}\bigg{[}\varphi_{nt}(\boldsymbol{\delta})\big{)}\bigg{]}=\sum_{t=1}^{n}\mathbb{E}_{\mathcal{F}_{t-1}}\bigg{[}\varrho_{\uptau}\left(\varepsilon_{t}-\boldsymbol{\delta}(\uptau)^{\top}\boldsymbol{X}_{t}\right)-\varrho_{\uptau}\left(\varepsilon_{t}\right)\bigg{]}.$

Using Knight,  (1998)’s identity we have that

(A.6)
$\displaystyle\varrho_{\uptau}(\mathsf{u}_{1}-\mathsf{u}_{2})-\varrho_{\uptau}(\mathsf{u}_{1})=\mathsf{u}_{2}\big{(}\uptau-\mathbf{1}\left\\{\mathsf{u}_{1}\leq
0\right\\}\big{)}+\mathsf{u}_{2}\int_{0}^{1}\bigg{[}\mathbf{1}\big{\\{}\mathsf{u}_{1}\leq\mathsf{u}_{2}s\big{\\}}-\mathbf{1}\big{\\{}\mathsf{u}_{1}\leq
0\big{\\}}\bigg{]}ds.$

which implies that we can decompose
$Z_{n}(\mathsf{u},\uptau)=Z_{n}^{(1)}(\mathsf{u},\uptau)+Z_{n}^{(2)}(\mathsf{u},\uptau)$.
Then, it can be proved that

(A.7)
$\displaystyle\sum_{t=1}^{n}\mathbb{E}_{\mathcal{F}_{t-1}}\bigg{[}\varphi_{nt}(\boldsymbol{\delta})\big{)}\bigg{]}=\frac{1}{2}\xi\times\boldsymbol{\delta}^{\top}\left(\sum_{t=1}^{n}\boldsymbol{X}_{t}\boldsymbol{X}_{t}^{\top}\right)\boldsymbol{\delta}+o_{p}(1).$

Furthermore, by Theorem 3.2 (a) of Phillips and Magdalinos,  (2007) it holds
that

(A.8)
$\displaystyle\frac{1}{nk_{n}}\sum_{t=1}^{n}y_{t-1}^{2}\overset{p}{\to}\frac{\sigma^{2}}{-2c}.$

Therefore, it suffices to show that

(A.9) $\displaystyle\frac{1}{n\sqrt{k_{n}}}\sum_{t=1}^{n}y_{t-1}=o_{p}(1).$

To see this, we consider the left side of the expression above such that

(A.10)
$\displaystyle\frac{(1-\rho_{n})}{n\sqrt{k_{n}}(1-\rho_{n})}\sum_{t=1}^{n}y_{t-1}=\frac{1}{n\sqrt{k_{n}}(1-\rho_{n})}\sum_{t=1}^{n}(1-\rho_{n})y_{t-1}=\frac{1}{n\sqrt{k_{n}}(1-\rho_{n})}\sum_{t=1}^{n}y_{t-1}-\rho_{n}y_{t-1}$

However, it holds that $y_{t}=\rho_{n}y_{t-1}+u_{t}$, and by rearranging we
have that $-\rho_{n}y_{t-1}=-(y_{t}-u_{t})$. Thus,

$\displaystyle\frac{(1-\rho_{n})}{n\sqrt{k_{n}}(1-\rho_{n})}\sum_{t=1}^{n}y_{t-1}$
$\displaystyle=\frac{1}{n\sqrt{k_{n}}(1-\rho_{n})}\sum_{t=1}^{n}\big{[}y_{t-1}-\left(y_{t}-u_{t}\right)\big{]}$
$\displaystyle=\frac{1}{n\sqrt{k_{n}}(c/k_{n})}\sum_{t=1}^{n}\left(y_{0}-y_{n}+\sum_{t=1}^{n}u_{t}\right)=o_{p}(1).$

which shows that
$\frac{1}{n\sqrt{k_{n}}}\sum_{t=1}^{n}y_{t-1}\overset{p}{\to}0$, converges in
probability to zero. ∎

######  Lemma A.3.

Under Assumptions 2.1-2.4, it holds that

(A.11)
$\displaystyle\sum_{t=1}^{n}\varphi_{nt}\big{(}\boldsymbol{\delta}(\uptau)\big{)}\overset{p}{\to}\frac{1}{2}\xi\times\boldsymbol{\delta}(\uptau)^{\top}\boldsymbol{B}\boldsymbol{\delta}(\uptau).$

###### Proof.

By Lemma A.2, we can show that

(A.12)
$\displaystyle\sum_{t=1}^{n}\bigg{(}\varphi_{nt}\big{(}\boldsymbol{\delta}(\uptau)\big{)}-\mathbb{E}_{t-1}\bigg{[}\varphi_{nt}\big{(}\boldsymbol{\delta}(\uptau)\big{)}\bigg{]}\bigg{)}=o_{p}(1).$

Define the set
$\mathcal{B}_{t}(\lambda):=\big{\\{}\frac{1}{n}y_{t-1}^{2}\leq\lambda\big{\\}}$
for some positive $\lambda\in\mathbb{R}$. Then, (A.12) becomes as below

(A.13)
$\displaystyle\sum_{t=1}^{n}\bigg{(}\varphi_{nt}(\boldsymbol{\delta})\mathbf{1}\big{\\{}\mathcal{B}_{t}(\lambda)\big{\\}}-\mathbb{E}_{t-1}\bigg{[}\varphi_{nt}(\boldsymbol{\delta})\mathbf{1}\big{\\{}\mathcal{B}_{t}(\lambda)\big{\\}}\big{)}\bigg{]}\bigg{)}=o_{p}(1).$

In particular, the expression
$\bigg{\\{}\varphi_{nt}(\boldsymbol{\delta})\mathbf{1}\big{\\{}\mathcal{B}_{t}(\lambda)\big{\\}}-\mathbb{E}_{t-1}\bigg{[}\varphi_{nt}(\boldsymbol{\delta})\mathbf{1}\big{\\{}\mathcal{B}_{t}(\lambda)\big{\\}}\bigg{]}\bigg{\\}}_{t=1}^{n}$
forms a martingale difference sequence, which by the Lenglart’s inequality
(see, Jacod and Shiryaev,  (2003)) it follows that

(A.14)
$\displaystyle\sum_{t=1}^{n}\mathbb{E}_{t-1}\bigg{[}\varphi_{nt}^{2}(\boldsymbol{\delta})\mathbf{1}\big{\\{}\mathcal{B}_{t}(\lambda)\big{\\}}\bigg{]}=o_{p}(1).$

Therefore, by definition of $\varphi_{nt}$ we have that

(A.15)
$\displaystyle\varphi_{nt}(\boldsymbol{\delta})=\boldsymbol{\delta}^{\top}\boldsymbol{X}_{t}\int_{0}^{1}\bigg{(}\psi(\tilde{\varepsilon}_{t})-\psi\left(\tilde{\varepsilon}_{t}-\varepsilon_{t}\boldsymbol{\delta}^{\top}\boldsymbol{X}_{t}\right)\bigg{)}d\varepsilon.$

Furthermore, by the non-decreasing property of $\psi(x)$ it holds that

(A.16)
$\displaystyle\sum_{t=1}^{n}\mathbb{E}_{t-1}\bigg{[}\varphi_{nt}^{2}(\boldsymbol{\delta})\mathbf{1}\big{\\{}\mathcal{B}_{t}(\lambda)\big{\\}}\bigg{]}\leq\underset{1\leq
t\leq
n}{\mathsf{max}}\mathbb{E}_{t-1}\bigg{[}\varphi_{nt}^{2}(\boldsymbol{\delta})\mathbf{1}\big{\\{}\mathcal{B}_{t}(\lambda)\big{\\}}\bigg{]}\times\left(\sum_{t=1}^{n}\boldsymbol{\delta}^{\top}\boldsymbol{X}_{t}\boldsymbol{X}_{t}^{\top}\boldsymbol{\delta}\right)=o_{p}(1).$

∎

######  Corollary A.1.

(A.17)
$\displaystyle\sum_{t=1}^{n}\bigg{(}\rho_{\uptau}\left(\tilde{\varepsilon}_{t}-\boldsymbol{\delta}^{\top}\boldsymbol{X}_{t}\right)-\rho_{\uptau}\left(\varepsilon_{t}\right)\bigg{)}=-\boldsymbol{\delta}^{\top}\sum_{t=1}^{n}\boldsymbol{X}_{t}\psi\left(\varepsilon_{t}\right)+\frac{1}{2}\xi\boldsymbol{\delta}^{\top}\boldsymbol{B}\boldsymbol{\delta}+R_{n}\left(\boldsymbol{\delta}\right)$

with $R_{n}(\boldsymbol{\delta})=o_{p}(1)$ for a fixed parameter vector
$\boldsymbol{\delta}$ and
$\underset{\left\lVert\boldsymbol{\delta}\right\rVert\leq
C}{\mathsf{max}}R_{n}(\boldsymbol{\delta})=o_{p}(1)$.

###### Proof.

Notice that the function $\varrho(\mathsf{u})$ is convex, therefore we can
apply the same argument as that in the proof of Theorem 1 in Pollard,  (1991)
and show that an equivalent solution to the optimization problem is given by
the following expression

(A.18)
$\displaystyle\hat{\gamma}=\sum_{t=1}^{n}\frac{1}{\xi}\boldsymbol{B}^{-1}\psi(\varepsilon_{t})\boldsymbol{X}_{t}^{\top}+o_{p}(1).$

∎

####  Near-Explosive Case $(c>0)$

######  Lemma A.4.

Consider that $y_{1},...,y_{n}$ are random variables generated from the
autoregressive process. Then, when $c>0$ it holds that

(A.19)
$\displaystyle\mathbb{E}\big{[}y_{n}^{2}\big{]}=o\left(\rho_{n}^{2n}k_{n}^{2}\right)$

In addition to Lemma A.4 the following two results hold

(A.20) $\displaystyle\underset{1\leq t\leq
n}{\mathsf{max}}\left\\{\frac{y_{t}^{2}}{\rho_{n,c}^{2n}k_{n}^{2}}\right\\}$
$\displaystyle=o_{p}(1)$ (A.21)
$\displaystyle\frac{1}{\sqrt{n}\rho_{n,c}^{n}k_{n}}\sum_{t=1}^{n}y_{t-1}$
$\displaystyle=o_{p}(1).$

######  Lemma A.5.

We consider the following two joint convergence results

  * (i).

Under Assumptions 2.1-2.4 it holds that

(A.22)
$\displaystyle\left(\frac{1}{\sqrt{n}}\sum_{t=1}^{n}\psi_{\uptau}\big{(}\varepsilon_{t}\big{)},\frac{1}{\sqrt{k_{n}}}\sum_{t=1}^{n}\rho_{n}^{t-(n-1)}\psi_{\uptau}\big{(}\varepsilon_{t}\big{)},\frac{1}{\sqrt{n}}\sum_{t=1}^{n}\rho_{n}^{-t}\varepsilon_{t}\right)^{\top}\overset{d}{\to}\bigg{(}\mathcal{Z}_{1},\mathcal{Z}_{2},\mathcal{Z}_{3}\bigg{)}^{\top}.$

where $\bigg{(}\mathcal{Z}_{1},\mathcal{Z}_{2},\mathcal{Z}_{3}\bigg{)}$ is a
Gaussian random vector with independent components and the finite variance
terms given by $\mathbb{E}\left[\psi_{\uptau}^{2}(\varepsilon_{1})\right]$,
$\frac{1}{2c}\mathbb{E}\left[\psi_{\uptau}^{2}(\varepsilon_{1})\right]$ and
$\sigma^{2}\big{/}(2c)$, respectively.

  * (ii).

Under Assumptions 2.1-2.4 it holds that

(A.23)
$\displaystyle\left(\frac{1}{\sqrt{n}}\sum_{t=1}^{n}\psi_{\uptau}\big{(}\varepsilon_{t}\big{)},\frac{1}{\rho_{n,c}^{n}k_{n}}\sum_{t=1}^{n}y_{t-1}\psi_{\uptau}\big{(}\varepsilon_{t}\big{)},\frac{1}{\rho_{n,c}^{2n}k_{n}^{2}}\sum_{t=1}^{n}y_{t-1}^{2}\right)^{\top}\overset{d}{\to}\bigg{(}\mathcal{Z}_{1},\mathcal{Z}_{2}\mathcal{Z}_{3},\frac{\mathcal{Z}_{3}^{2}}{2c}\bigg{)}^{\top}.$

###### Proof.

Recall that the difference equation with no model intercept, that is,
$y_{t}=\rho_{n,c}y_{t-1}+\tilde{\varepsilon}_{t}$ has a general solution of
the form
$y_{t}=\rho^{t}_{n,c}y_{0}+\sum_{j=1}^{t}\rho_{n,c}^{t-j}\tilde{\varepsilon}_{j}$.
Similarly, for $y_{t-1}$ by shifting the time index such that $t\mapsto t-1$,
then the equivalent general solution is given by the following expression

(A.24) $\displaystyle
y_{t-1}=\rho^{t-1}_{n,c}y_{0}+\sum_{j=1}^{t-1}\rho_{n,c}^{t-1-j}\tilde{\varepsilon}_{j}$

Thus, by substituting the above expression to the sample moment
$\sum_{t=1}^{n}y_{t-1}\psi_{\uptau}\big{(}\varepsilon_{t}\big{)}$ we obtain
that

(A.25)
$\displaystyle\frac{1}{\rho_{n,c}^{n}k_{n}}\sum_{t=1}^{n}y_{t-1}\psi_{\uptau}\big{(}\varepsilon_{t}\big{)}$
$\displaystyle=\frac{y_{0}}{\rho_{n,c}^{n}k_{n}}\sum_{t=1}^{n}\rho_{n,c}^{t-1}\psi_{\uptau}\big{(}\varepsilon_{t}\big{)}+\frac{1}{\rho_{n,c}^{n}k_{n}}\sum_{t=1}^{n}\left(\sum_{j=1}^{t-1}\rho_{n,c}^{t-j-1}\tilde{\varepsilon}_{j}\right)\psi_{\uptau}\big{(}\varepsilon_{t}\big{)}$

Then, since the first term of the above expression is asymptotically
negligible by splitting the inner summation of the last term we obtain that

$\displaystyle\frac{1}{\rho_{n,c}^{n}k_{n}}\sum_{t=1}^{n}y_{t-1}\psi_{\uptau}\big{(}\varepsilon_{t}\big{)}$
$\displaystyle=\frac{1}{\rho_{n,c}^{n}k_{n}}\sum_{t=1}^{n}\left(\sum_{j=1}^{n}\rho_{n,c}^{t-1-j}\tilde{\varepsilon}_{j}-\sum_{j=t}^{n}\rho_{n,c}^{t-1-j}\tilde{\varepsilon}_{j}\right)\psi_{\uptau}\big{(}\varepsilon_{t}\big{)}+o_{p}(1).$

Since,
$\sum_{j=t}^{n}\rho_{n,c}^{t-1-1}\tilde{\varepsilon}_{j}\overset{p}{\to}0$,
then it follows that

$\displaystyle\frac{1}{\rho_{n,c}^{n}k_{n}}\sum_{t=1}^{n}y_{t-1}\psi_{\uptau}\big{(}\varepsilon_{t}\big{)}$
$\displaystyle=\frac{1}{\rho_{n,c}^{n}k_{n}}\sum_{t=1}^{n}\left(\sum_{j=1}^{n}\rho_{n,c}^{t-1-j}\tilde{\varepsilon}_{j}\right)\psi_{\uptau}\big{(}\varepsilon_{t}\big{)}+o_{p}(1)$
$\displaystyle=\left(\frac{1}{\sqrt{k_{n}}}\sum_{t=1}^{n}\rho_{n,c}^{t-(n+1)}\psi_{\uptau}\big{(}\varepsilon_{t}\big{)}\right)\left(\frac{1}{\sqrt{k_{n}}}\sum_{t=1}^{n}\rho_{n,c}^{-t}\tilde{\varepsilon}_{t}\right)+o_{p}(1).$

Similarly, it holds that

(A.26)
$\displaystyle\frac{1}{\rho_{n,c}^{2n}k_{n}^{2}}\sum_{t=1}^{n}y_{t-1}^{2}=\frac{1}{2c}\left(\frac{1}{\sqrt{k_{n}}}\sum_{t=1}^{n}\rho_{n,c}^{-t}\tilde{\varepsilon}_{t}\right)+o_{p}(1).$

∎

######  Lemma A.6.

The following joint convergence results hold

  * (i).

Under Assumptions above we have that

$\displaystyle-\boldsymbol{\vartheta}_{n}(\uptau)^{\top}\sum_{t=1}^{n}\boldsymbol{X}_{t}\psi_{\uptau}\big{(}\varepsilon_{t}\big{)}+\frac{\xi}{2}\boldsymbol{\vartheta}_{n}(\uptau)^{\top}\left(\sum_{t=1}^{n}\boldsymbol{X}_{t}\boldsymbol{X}_{t}^{\top}\right)\boldsymbol{\vartheta}_{n}(\uptau)\overset{d}{\to}-\boldsymbol{\vartheta}_{n}(\uptau)\big{(}\mathcal{Z}_{1},\mathcal{Z}_{2}\mathcal{Z}_{3}\big{)}^{\top}+\frac{\xi}{2}\boldsymbol{\vartheta}_{n}(\uptau)^{\top}\boldsymbol{B}\boldsymbol{\vartheta}_{n}(\uptau)$

  * (ii).

Under Assumptions above we have that

$\displaystyle\sum_{t=1}^{n}\left[\varrho_{\uptau}\bigg{(}y_{t}-\boldsymbol{\vartheta}_{n}(\uptau)^{\top}\boldsymbol{X}_{t}\bigg{)}-\varrho_{\uptau}\left(\varepsilon_{t}\right)\right]=-\boldsymbol{\vartheta}_{n}(\uptau)^{\top}\sum_{t=1}^{n}\boldsymbol{X}_{t}\psi_{\uptau}\big{(}\varepsilon_{t}\big{)}+\frac{\xi}{2}\boldsymbol{\vartheta}_{n}(\uptau)^{\top}\left(\sum_{t=1}^{n}\boldsymbol{X}_{t}\boldsymbol{X}_{t}^{\top}\right)\boldsymbol{\vartheta}_{n}(\uptau)+R_{n}\bigg{(}\boldsymbol{\vartheta}_{n}(\uptau)\bigg{)}.$

with $R_{n}\bigg{(}\boldsymbol{\vartheta}_{n}(\uptau)\bigg{)}=o_{p}(1)$ for
fixed $\boldsymbol{\vartheta}_{n}(\uptau)$ and further

###### Proof.

In order to prove the uniformity condition we denote with

(A.27)
$\displaystyle\varphi\big{(}\boldsymbol{\vartheta}_{n}(\uptau)\big{)}=\frac{1}{2}\xi\times\boldsymbol{\vartheta}_{n}(\uptau)^{\top}\left(\sum_{t=1}^{n}\boldsymbol{X}_{t}\boldsymbol{X}_{t}^{\top}\right)\boldsymbol{\vartheta}_{n}(\uptau).$

Furthermore, we need to show that

(A.28)
$\displaystyle\underset{\left\lVert\boldsymbol{\vartheta}_{n}(\uptau)\right\rVert\leq
C}{\mathsf{sup}}\left|\sum_{t=1}^{n}\varphi_{nt}\big{(}\boldsymbol{\vartheta}_{n}(\uptau)\big{)}-\varphi\big{(}\boldsymbol{\vartheta}(\uptau)\big{)}\right|=o_{P}(1).$

Since $\sum_{t=1}^{n}\boldsymbol{X}_{t}\boldsymbol{X}_{t}^{\top}$ converges in
distribution, for any $\lambda>0$ there exists $M$ large enough, such that,

(A.29)
$\displaystyle\mathbb{P}\left(\underset{\left\lVert\boldsymbol{\vartheta}_{n}(\uptau)\right\rVert\leq
C}{\mathsf{sup}}\left|\sum_{t=1}^{n}\varphi_{nt}\big{(}\boldsymbol{\vartheta}_{n}(\uptau)\big{)}-\varphi\big{(}\boldsymbol{\vartheta}(\uptau)\big{)}\right|\mathbf{1}\left\\{\left\lVert\sum_{t=1}^{n}\boldsymbol{X}_{t}\boldsymbol{X}_{t}^{\top}\right\rVert>M\right\\}>\lambda/2\right)<\lambda^{*}/2.$

On the other hand, on
$\left\\{\left\lVert\sum_{t=1}^{n}\boldsymbol{X}_{t}\boldsymbol{X}_{t}^{\top}\right\rVert\right\\}$,
for any $\lambda>0$, there exists $\delta>0$, such that,

(A.30)
$\displaystyle\underset{\left\lVert\boldsymbol{\vartheta}\right\rVert\leq
C}{\mathsf{sup}}\bigg{|}\varphi(\gamma+\vartheta)-\varphi(\gamma)\bigg{|}\leq\lambda.$

Moreover, following the convexity Lemma of Pollard,  (1991), one can show that

(A.31)
$\displaystyle\mathbb{P}\left(\underset{\left\lVert\boldsymbol{\vartheta}_{n}(\uptau)\right\rVert\leq
C}{\mathsf{sup}}\left|\sum_{t=1}^{n}\varphi_{nt}\big{(}\boldsymbol{\vartheta}_{n}(\uptau)\big{)}-\varphi\big{(}\boldsymbol{\vartheta}(\uptau)\big{)}\right|\mathbf{1}\left\\{\left\lVert\sum_{t=1}^{n}\boldsymbol{X}_{t}\boldsymbol{X}_{t}^{\top}\right\rVert\leq
M\right\\}>\lambda/2\right)<\lambda^{*}/2.$

Therefore, the combination of the above yields the uniformity result of
interest. ∎

###  Supplementary Results

Following Koenker,  (2005), we consider that all parameters share the same
monotone behaviour with respect to the quantile level $\uptau\in(0,1)$. Then
the consistency of quantile dependent parameter can be deduced from the
monotonicity of the subgradient as well as a direct consequence of the uniform
convergence of the empirical distribution function and the Glivenko-Cantelli
Theorem. Therefore, the asymptotic behaviour of
$\sqrt{n}\left(\hat{\boldsymbol{\beta}}(\uptau)-\boldsymbol{\beta}(\uptau)\right)$
follows by considering the following objective function

(A.32)
$\displaystyle\mathcal{Z}_{n}(\delta)=\frac{1}{n}\sum_{t=1}^{n}\varrho_{\uptau}\left(\varepsilon_{t}-\boldsymbol{X}_{t}^{\top}\boldsymbol{\delta}/\sqrt{n}\right)-\varrho_{\uptau}\big{(}\varepsilon_{t}\big{)}$

where
$\varepsilon_{t}(\uptau)=y_{t}-\boldsymbol{X}_{t}^{\top}\boldsymbol{\beta}(\uptau)$.
The function $\mathcal{Z}_{n}(\boldsymbol{\delta})$ is convex, and is
minimized at
$\hat{\boldsymbol{\delta}}_{n}=\sqrt{n}\left(\hat{\boldsymbol{\beta}}(\uptau)-\boldsymbol{\beta}(\uptau)\right)$.
Therefore, we can show that the limit distribution of
$\hat{\boldsymbol{\delta}}_{n}$ can be determined by the asymptotic
distribution of the objective function $\mathcal{Z}_{n}(\boldsymbol{\delta})$.
Furthermore, it follows from the Lindeberg-Feller central limit theorem that
$\mathcal{Z}_{n}^{(1)}\overset{d}{\to}-\boldsymbol{\delta}^{\prime}\mathcal{W}$,
where
$\mathcal{W}\overset{d}{\to}\mathcal{N}\big{(}0,\uptau(1-\uptau)\boldsymbol{D}_{0}\big{)}$.
Then, following the proof of Theorem 4.1 of Koenker,  (2005) it holds that
(see also derivations in Knight,  (1998) and Kato,  (2009))

(A.33)
$\displaystyle\sum_{t=1}^{n}\mathbb{E}\left[\mathcal{Z}_{nt}^{(2)}(\boldsymbol{\delta})\right]\overset{d}{\to}\frac{1}{2}\boldsymbol{\delta}^{\top}\boldsymbol{D}_{1}\boldsymbol{\delta}.$

Therefore, it can be proved that

(A.34)
$\displaystyle\mathcal{Z}_{n}(\boldsymbol{\delta})\overset{d}{\to}\mathcal{Z}_{0}(\boldsymbol{\delta})\equiv-\boldsymbol{\delta}^{\top}\mathcal{W}+\frac{1}{2}\boldsymbol{\delta}^{\top}\boldsymbol{D}_{1}\boldsymbol{\delta}.$

The convexity of the limiting distribution of
$\mathcal{Z}_{0}(\boldsymbol{\delta})$ ensures that the uniqueness of the
minimizer

(A.35) $\displaystyle\hat{\delta}_{n}:=\underset{\delta}{\mathsf{arg\min}}\
\mathcal{Z}_{n}(\boldsymbol{\delta})\mapsto\hat{\delta}_{0}:=\underset{\boldsymbol{\delta}}{\mathsf{arg\min}}\
\mathcal{Z}_{0}(\boldsymbol{\delta})$

where
$\mathcal{Z}_{n}(\boldsymbol{\delta})=\mathcal{Z}_{n}^{(1)}(\boldsymbol{\delta})+\mathcal{Z}_{n}^{(2)}(\boldsymbol{\delta})$
and
$\mathcal{Z}_{0}(\boldsymbol{\delta})=-\boldsymbol{\delta}^{\top}\mathcal{W}+\frac{1}{2}\boldsymbol{\delta}^{\top}\boldsymbol{D}_{1}\boldsymbol{\delta}$
such that
$\hat{\boldsymbol{\delta}}_{n}=\sqrt{n}\left(\hat{\boldsymbol{\beta}}(\uptau)-\boldsymbol{\beta}(\uptau)\right)$.
A similar approach is followed in the study of Mao and Guo,  (2019) who prove
that for any $\uptau\in(0,1)$ the solution of the following expression is
obtained by
$\frac{\sqrt{n}}{\lambda(n)}\left(\widehat{\boldsymbol{\beta}}(\uptau)-\boldsymbol{\beta}(\uptau)\right)\in\underset{\mathsf{u}\in\mathbb{R}^{p}}{\mathsf{arg\
min}}\ \mathcal{Z}_{n}(\mathsf{u},\uptau)$.

Then, it follows that

$\displaystyle\bigg{|}\mathcal{Z}_{n}(\mathsf{u},\uptau)-G_{n}(\mathsf{u},\uptau)\bigg{|}$
$\displaystyle=\left|\mathcal{Z}_{n}^{(2)}(\mathsf{u},\uptau)-\frac{\mathsf{u}^{\top}\boldsymbol{D}\mathsf{u}}{2}f\left(F^{-1}(\uptau)\right)\right|$
$\displaystyle\leq\left|\mathbb{E}\left[\mathcal{Z}_{n}^{(2)}(\mathsf{u},\uptau)\right]-\frac{\mathsf{u}^{\top}\boldsymbol{D}\mathsf{u}}{2}f_{\varepsilon}\left(F_{\varepsilon}^{-1}(\uptau)\right)\right|+\bigg{|}\mathcal{Z}_{n}^{(2)}(\mathsf{u},\uptau)-\mathbb{E}\left[\mathcal{Z}_{n}^{(2)}(\mathsf{u},\uptau)\right]\bigg{|}$

where

(A.36)
$\displaystyle\mathcal{Z}_{n}^{(2)}(\mathsf{u},\uptau)=\frac{1}{\sqrt{n}\lambda(n)}\sum_{i=1}^{n}\int_{0}^{1}\left(\mathbf{1}\left\\{\varepsilon_{i}\leq
F_{\varepsilon}^{-1}(\uptau)+\frac{\lambda(n)\sigma_{in}^{-1}x_{in}^{\top}\mathsf{u}s}{\sqrt{n}}\right\\}-\mathbf{1}\left\\{\varepsilon_{i}\leq
F_{\varepsilon}^{-1}(\uptau)\right\\}\right)ds.$

Therefore, it can be shown that

(A.37)
$\displaystyle\underset{\uptau\in[\alpha,1-\alpha]}{\mathsf{sup}}\left|\mathbb{E}\left[\mathcal{Z}_{n}^{(2)}(\mathsf{u},\uptau)\right]-\frac{\mathsf{u}^{\top}\boldsymbol{D}\mathsf{u}}{2}f_{\varepsilon}\left(F_{\varepsilon}^{-1}(\uptau)\right)\right|\to
0,\ \ \text{as}\ n\to\infty$

Thus, for large $n$, we further have that

(A.38)
$\displaystyle\mathbb{P}\left(\underset{\uptau\in[\alpha,1-\alpha]}{\mathsf{sup}}\
\bigg{|}\mathcal{Z}_{n}(\mathsf{u},\uptau)-G_{n}(\mathsf{u},\uptau)\bigg{|}\geq\lambda\right)\leq\mathbb{P}\left(\underset{\uptau\in[\alpha,1-\alpha]}{\mathsf{sup}}\bigg{|}\mathcal{Z}^{(2)}_{n}(\mathsf{u},\uptau)-\mathbb{E}\left[\mathcal{Z}^{(2)}_{n}(\mathsf{u},\uptau)\right]\bigg{|}\geq\frac{\lambda}{2}\right)$

The following step is to prove that

(A.39) $\displaystyle\underset{n\to\infty}{\mathsf{lim\ sup}}\
\frac{1}{\lambda^{2}(n)}\mathsf{log}\left\\{\mathbb{P}\left(\underset{\uptau\in[\alpha,1-\alpha]}{\mathsf{sup}}\bigg{|}\mathcal{Z}^{(2)}_{n}(\mathsf{u},\uptau)-\mathbb{E}\left[\mathcal{Z}^{(2)}_{n}(\mathsf{u},\uptau)\right]\bigg{|}\geq\lambda\right)\right\\}=-\infty$

From the definition of $\mathcal{Z}^{(2)}_{n}(\mathsf{u},t)$, we obtain that

(A.40)
$\displaystyle\mathcal{Z}^{(2)}_{n}(\mathsf{u},\uptau)-\mathbb{E}\left[\mathcal{Z}^{(2)}_{n}(\mathsf{u},\uptau)\right]=\int_{0}^{1}\bigg{(}\mathcal{W}_{n}(\mathsf{u}s,\uptau)-\mathcal{W}_{n}(0,\uptau)\bigg{)}ds,$

where

$\displaystyle\mathcal{W}_{n}(r,\uptau)=\frac{\sum_{t=1}^{n}\boldsymbol{X}_{nt}^{\prime}\mathsf{u}}{\sqrt{n}\lambda(n)}\left[\mathbf{1}\left\\{\varepsilon_{i}\leq
F_{\varepsilon}^{-1}(\uptau)+\frac{\lambda(n)\sigma_{n}^{-1}\boldsymbol{X}_{nt}^{\prime}r}{\sqrt{n}}\right\\}-F_{\varepsilon}\left(F_{\varepsilon}^{-1}(\uptau)+\frac{\lambda(n)\sigma_{n}^{-1}\boldsymbol{X}_{nt}^{\prime}r}{\sqrt{n}}\right)\right]$

In practise we employ Theorem 1 from Knight,  (1998). Therefore, we have that

(A.41)
$\displaystyle\mathcal{Z}_{n}^{(1)}(\mathsf{u})=-\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\boldsymbol{x}_{i}\mathsf{u}\bigg{[}\mathbf{1}\big{(}\epsilon_{i}>0\big{)}-\mathbf{1}\big{(}\epsilon_{i}<0\big{)}\bigg{]}$

and

(A.42) $\displaystyle\mathcal{Z}_{n}^{(2)}(\mathsf{u})$
$\displaystyle=\frac{2a_{n}}{\sqrt{n}}\sum_{t=1}^{n}\int_{0}^{v_{n_{i}}}\boldsymbol{X}_{t}\mathsf{u}\bigg{[}\mathbf{1}\big{(}\epsilon_{i}<s\big{)}-\mathbf{1}\big{(}\epsilon_{i}<0\big{)}\bigg{]}ds$

where $v_{n}=\boldsymbol{X}_{t}^{\top}\mathsf{u}/a_{n}$. Then, by the
Lindeberg-Feller central limit theorem, for each $\mathsf{u}$ it holds that,

(A.43)
$\displaystyle\mathcal{Z}_{n}^{(1)}(\mathsf{u})\overset{d}{\to}-\mathsf{u}\boldsymbol{W}$

and the convergence in distribution holds for any finite collection of
$\mathsf{u}^{\prime}$s. For $\mathcal{Z}_{n}^{(2)}(\mathsf{u})$, we have that

(A.44)
$\displaystyle\mathcal{Z}_{n}^{(2)}(\mathsf{u})=\sum_{t=1}^{n}\mathbb{E}\left[\mathcal{Z}_{nt}^{(2)}(\mathsf{u})\right]+\sum_{t=1}^{n}\bigg{[}\mathcal{Z}_{nt}^{(2)}(\mathsf{u})-\mathbb{E}\left[\mathcal{Z}_{nt}^{(2)}(\mathsf{u})\right]\bigg{]}.$

We employ the above orthogonal decomposition when proving the limit results
for the estimator of the quantile autoregressive model for moderate deviations
from the unit boundary. Notice that the implementation of the corresponding
FCLT for the quantile-dependent innovation term is applicable for the i.i.d
innovation sequence assumption. An extension of the particular results to the
case in which innovations have serial correlation via the use of a linear
process representation for instance is also possible. Further applications
with suitable econometric conditions we could consider within our framework
are presented in the study of Doukhan and Louhichi,  (1999) who consider a
different type of weak dependence condition for time series models.

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