# Pivotal Test Statistic for Nonparametric Cointegrating Regression Functions

Sepideh Mosaferi 111Corresponding author; E-mail address:
<EMAIL_ADDRESS>
Sepideh Mosaferi is a Doctoral Candidate with the Department of Statistics and
Statistical Laboratory (E-mail: mosaferi@iastate.edu), and Mark S. Kaiser is a
Professor with the Department of Statistics and Statistical Laboratory
(E-mail<EMAIL_ADDRESS>at Iowa State University, 2438 Osborn Dr.,
Ames, IA 50011-1090, USA. and Mark S. Kaiser
Department of Statistics, Iowa State University

###### Abstract

This article focuses on cointegrating regression models in which covariate
processes exhibit long range or semi-long range memory behaviors, and may
involve endogeneity in which covariate and response error terms are not
independent. We assume semi-long range memory is produced in the covariate
process by tempering of random shock coefficients. The fundamental properties
of long memory processes are thus retained in the covariate process. We modify
a test statistic proposed for the long memory case by Wang and Phillips (2016)
to be suitable in the semi-long range memory setting. The limiting
distribution is derived for this modified statistic and shown to depend only
on the local memory process of standard Brownian motion. Because, unlike the
original statistic of Wang and Phillips (2016), the limit distribution is
independent of the differencing parameter of fractional Brownian motion, it is
pivotal. Through simulation we investigate properties of nonparametric
function estimation for semi-long range memory cointegrating models, and
consider behavior of both the modified test statistic under semi-long range
memory and the original statistic under long range memory. We also provide a
brief empirical example.

Keywords: Fractional differencing parameter; pivotal test statistic; standard
Brownian motion; simulation studies; tempered process.

JEL classification codes: C01, C12, C14, C15, C22.

##  Introduction

Statistical inference for nonlinear cointegrating regression models have been
intensively studied in the past decade. These models have been applied to
problems in financial markets, stock prices, heavy traffic, and energy markets
(e.g., Fasen (2013) and references therein). Let

$y_{k}=f(x_{k})+u_{k},\qquad k=1,\ldots,N,$ (1)

be a nonlinear cointegrating regression model with regressor process $x_{k}$,
an error process $u_{k}$ and an unknown real function $f(\cdot)$. Much of the
asymptotic theory in the literature has assumed strict exogeneity in which the
regressor $x_{k}$ is assumed to be uncorrelated with the regression error
$u_{k}$ (e.g., Karlsen et al. (2007), Cai et al. (2009), Wang and Phillips
(2009a), and Wang (2014)). Often, the covariate $x_{k}$ is also assumed to be
short memory process.

Extending this framework by allowing the regressor $x_{k}$ to be driven by
long memory innovations and permitting correlation with $u_{k}$ (so-called
endogeneity) has received less attention and produces some technical problems
that have not been completely resolved. Recently, Wang and Phillips (2016)
established a limit theory for nonparametric nonstationary regression that
allows for both endogeneity and long memory. The test of Wang and Phillips
(2016) is a modification of a test statistic given by Hardle and Mammen (1993)
for the random sample case. The test was also used in Gao et al. (2012) for a
nonlinear cointegrating model with a martingale error structure and no
endogeneity.

The limit theory of Wang and Phillips (2016) was then used to develop a model
specification test for parametric forms of regression functions of the type
$f(x)=g(x,\theta_{0})$ where $g(\cdot,\theta_{0})$ represents a parametric
family of functions with unknown parameter $\theta_{0}\in\Theta$. These
authors assumed that $\Theta$ was a compact subspace of $\mathbb{R}^{m}$ for
some finite $m$. The specification test of Wang and Phillips (2016) involved
local alternative models $f(x)=g(x,\theta_{0})+\rho_{N}m(x)$, where $\rho_{N}$
is a sequence of constants measuring local deviations from the null hypothesis
of $f(x)=g(x,\theta_{0})$, and $m(x)$ is a real function.

A limitation of the test under long memory and endogeneity is that it is not
pivotal, and its limit distribution depends on the value of the fractional
differencing parameter in the long memory $x_{k}$ process. The motivation for
the work reported here was that the applicability of the Wang and Phillips
(2016) test statistic might be improved if one employes the idea of tempering
and assumes that there are semi-long memory input shocks to the regressor
process $x_{k}$. We describe this idea in detail and list the consequences,
which are related to the asymptotic theory for nonparametric cointegrating
regression functions.

We now summarize the main contributions of this paper. First, we demonstrate
asymptotic properties of kernel estimators of the unknown regression function
$f(x)$ in (1). Then we consider limit theory for the test statistic of Wang
and Phillips (2016) under the assumption that the regressor process involves
semi-long memory input shocks. We show that the limit theory of this test
statistic under semi-long memory involves the local time of standard Browninan
motion, so that the limit distribution in this case does not involve the
fractional differencing parameter $d$. These findings extend the theory
developed in Wang and Phillips (2009a, b, 2016), making the methods more
applicable because the statistic is pivotal under semi-long memory. In
addition, we investigate the properties of regression function estimators and
the performance of the test statistic using simulation and present an
empirical example related to the Carbon Kuznets curve previously investigated
in a number of works (e.g., Müller-Fürstenberger and Wagner (2007), Wang et
al. (2018)).

The remainder of the article is organized as follows. The general model and
notation is described in Section 2. In Section 3, we provide some limit
behaviors, which are essential for developing the limit theory presented in
following sections. In Section 4, we introduce nonlinear cointegrating
regression models under the assumption that the regressor process involves
strongly tempered shocks. We develop a consistent estimator of the regression
function and establish its limit distribution. In Section 5, we present a test
for parametric forms of the regression function, show that it is based on a
pivotal quantity, and consider its power under local alternatives. Section 6
contains the results of simulation studies for the regression function and
proposed test statistic. Section 7 contains an empirical example, where the
fractional differencing parameter is greater than $1/2$. Section 8 contains
concluding remarks and mentions possible directions for future work. The
proofs of all technical results are contained in the Appendix.

##  The cointegrating regression context

The models we will consider throughout this article are of the general form
(1) in which the regressor process $x_{k}$ is a sum of input shocks that
result in either a long memory or a semi-long memory process. In the long
memory case we take, for $k=1,\ldots,N$,

$\displaystyle x_{k}$ $\displaystyle=$ $\displaystyle\sum_{s=1}^{k}X_{d}(s),$
$\displaystyle X_{d}(s)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}b_{d}(j)\zeta(s-j),$ (2)

where $\zeta(k)$ is an independent and identically distributed (i.i.d.) noise
such that $\mathbb{E}\zeta(0)=0$ and $\mathbb{E}\zeta^{2}(0)=1$. The
coefficients $b_{d}(j)$ regularly vary at infinity as $j^{d-1}$, viz.,

$b_{d}(j)\ \sim\ \frac{c_{d}}{\Gamma(d)}\,j^{d-1},\quad j\to\infty,\quad
c_{d}\neq 0,\quad d\neq 0,$

where $0<d<1/2$ is the fractional differencing parameter.

In contrast to this long memory process, a semi-long memory setting contains
strongly tempered shocks as,

$\displaystyle x_{k}$ $\displaystyle=$
$\displaystyle\sum_{s=1}^{k}X_{d,\lambda}(s),$ $\displaystyle
X_{d,\lambda}(s)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}e^{-\lambda j}b_{d}(j)\zeta(s-j),$ (3)

where $\zeta(k)$ is the same as for the long memory case. In (2),
$\lambda\equiv\lambda_{N}>0$ is a sample size dependent parameter and
satisfies the following main assumption:

  * •

Main Assumption: The tempering parameter $\lambda\to 0$ and
$N\lambda\to\infty$ as $N\to\infty$.

We note that the tempering of the coefficients in expression (2) extends the
range of fractional differencing parameter $d$ from $(0,1/2)$ to $(0,\infty)$.
This means, in contrast with the stationary long memory time series
$X_{d}(.)$, the semi-long memory time series $X_{d,\lambda}(.)$ forms a
stationary process for $d$ belonging to any positive value on the real line.
The strongly tempered process $\eqref{SLMx}$ belongs to the general class of
stochastic processes called tempered linear processes, see Sabzikar and
Surgailis (2018). These processes have a semi-long memory property in the
sense that their autocovariance functions initially resemble that of a long
memory process but eventually decay fast at an exponential rate. One special
case of such processes is the autoregressive tempered fractionally integrated
moving average ARTFIMA$(p,d,\lambda,q)$ process, which has been studied by
Meerschaert et al. (2014) and Sabzikar et al. (2019).

Throughout the paper, we denote $C,C_{1},C_{2},...$ as generic constants which
may differ at each appearance. We use $\rightarrow_{P}$ for convergence in
probability, $\Rightarrow$ for weak convergence of the associated probability
measures, $\rightarrow_{D}$ for convergence in distribution, and $=_{D}$ for
equivalence in distribution. For any two functions $f$ and $g$, $f\asymp g$
means $C_{1}\leq f/g\leq C_{2}$. We use $||x||=\max_{i}|x_{i}|$ for vector
$x=(x_{i})$, $\wedge$ for the minimum between two numbers, and a.s. for almost
surely. Finally, i.i.d. and f.d.d. mean independent and identically
distributed and finite dimensional distribution, respectively.

##  Initial results

Throughout this section, set $d_{N}:=[\mathbb{E}(x_{N}^{2})]^{1/2}$. Under the
semi-long memory setting, the asymptotic form of $d_{N}$ is given by
$d_{N}\sim\sqrt{N}/\lambda^{d}$ as $N\rightarrow\infty$. Then,

$\frac{x_{N}}{d_{N}}:=\frac{\lambda^{d}}{\sqrt{N}}\sum_{k=1}^{N}X_{d,\lambda}(k)=\frac{1}{\sqrt{N}}\sum_{k=1}^{N}\zeta(k)+o_{P}(1),$
(4)

see expression (A.7) in Sabzikar et al. (2020). For any $t\in[0,1]$, the
following weak convergence applies on $D[0,1]$:

$\frac{x_{\lfloor Nt\rfloor}}{d_{N}}\Rightarrow B(t),$ (5)

where $\lfloor Nt\rfloor$ is the floor function, and $B(t)$ is the standard
Brownian motion.

######  Remark 3.1.

Under the long memory case, $d_{N}\sim N^{d+1/2}$ as $N\rightarrow\infty$ and
$x_{\lfloor Nt\rfloor}/d_{N}\Rightarrow B_{d+1/2}(t)$, where $B_{d+1/2}(t)$ is
fractional Brownian motion with parameter $d+1/2$. This makes the process
$B_{d+1/2}(t)$ depend on the unknown fractional differencing parameter $d$.
This is the reason the test statistic of Wang and Phillips (2016) is not
pivotal under long memory. On the other hand, the process $B(t)$ given in
expression (5) is free of the unknown parameter $d$, and this will result in
the test statistic being pivotal under semi-long memory.

Consider $x_{k,N}:=x_{k}/d_{N}$ for $1\leq k\leq N$ to be a triangular array.
A function of $x_{k,N}$ that will be used in the sequel takes the form of a
sample average of functions of $x_{k,N}$,

$S_{N}:=\frac{c_{N}}{N}\sum_{k=1}^{N}g(c_{N}x_{k,N}),$

where $g(x)$ is a bounded function such that
$\int_{\mathbb{R}}|g(x)|dx<\infty$ and $c_{N}:=d_{N}/h$ where
$c_{N}\rightarrow\infty$ and $c_{N}/N\rightarrow 0$. The bandwidth parameter
is denoted by $h$ and satisfies $h\equiv h_{N}\rightarrow 0$ as
$N\rightarrow\infty$. Functions in the form of $S_{N}$ commonly arise in
nonlinear cointegrating regressions, which could be the kernel function $K(.)$
or its squared $K^{2}(.)$; see Karlsen and Tjøstheim (2001), Karlsen et al.
(2007), and Wang and Phillips (2009a).

The limit behavior of $S_{N}$ is important in order to establish the limit
behavior of the kernel estimate of $f(x)$ in the nonparametric regression
context. The limit distribution of $S_{N}$ is as follows

$S_{N}\rightarrow_{D}\int_{\mathbb{R}}g(x)dxL_{B}(1,0),$ (6)

where $L_{B}(t,s)$ is the local time of standard Brownian motion $B(t)$ at the
spatial point $s$; see Proposition A.9 part (i) to follow. When the function
$g$ is a kernel density, the integral in expression (6) is unity, and the
limit is then the local time of $B$ at the origin; see Jeganathan (2004) and
Wang and Phillips (2009a) for some of the related results.

######  Definition 3.1.

The local time process of a stochastic process $G(x)$ is defined as

$L_{G}(t,s)=\lim\limits_{\epsilon\rightarrow
0}\frac{1}{2\epsilon}\int_{0}^{t}I\\{|G(r)-s|\leq\epsilon\\}dr,$

where $(t,s)\in\mathbb{R}_{+}\times\mathbb{R}$.

Under long memory case, the local time given in expression (6) is in the form
of $L_{B_{d+1/2}}(t,s)$, and therefore depends on the unknown parameter $d$.
This complicates the limit theory of the associated test statistic; see Remark
3.1.

##  Nonparametric regression function estimation

Assume model (1) where $f(\cdot)$ is an unknown real regression function. To
induce endogeneity, let $\eta_{k}=(\xi_{k},\epsilon_{k})^{\prime}$ be a
sequence of i.i.d. random vectors with $\mathbb{E}(\eta_{0})=0$ and
$\mathbb{E}(\eta_{0}\eta_{0}^{\prime})=\Sigma$ where

$\Sigma=\begin{pmatrix}1&\mathbb{E}(\xi_{0}\epsilon_{0})\\\
\mathbb{E}(\epsilon_{0}\xi_{0})&\mathbb{E}(\epsilon_{0}^{2})\end{pmatrix},$
(7)

and take $u_{k}$ in (1) to be $u_{k}=\sum_{j=0}^{\infty}\psi_{j}\eta_{k-j}$
for $\psi_{j}=(\psi_{j1},\psi_{j2})$. Now, assume $\eta_{0}$ and $\psi_{j}$
satisfy the following assumption.

######  Assumption 4.1.

$\sum_{j=0}^{\infty}\psi_{j}\neq 0$,
$\sum_{j=0}^{\infty}j^{1/4}(|\psi_{j1}|+|\psi_{j2}|)<\infty$, and
$\mathbb{E}||\eta_{0}||^{\alpha}<\infty$ for $\alpha>2$.

Therefore,
$\mathbb{E}(u_{0}^{2})=\sum_{j=0}^{\infty}\psi_{j}\Sigma\psi_{j}^{\prime}$ and
$cov(u_{k},x_{k})\neq 0$. Additionally, assume the characteristic function
$\varphi(t)$ of $\xi_{0}$ satisfies
$\int_{\mathbb{R}}(1+|t|)|\varphi(t)|dt<\infty,$ which ensures the smoothness
in the corresponding kernel density (see, Wang and Phillips (2016)).

The Nadaraya-Watson regression estimator of $f(x)$ is

$\hat{f}(x)=\frac{\sum_{k=1}^{N}y_{k}K_{h}(x_{k}-x)}{\sum_{k=1}^{N}K_{h}(x_{k}-x)},$
(8)

where $K_{h}(s)=h^{-1}K(s/h)$, and $K(\cdot)$ is a non-negative bounded
continuous function. In order to establish the asymptotic behavior for the
kernel estimate $f(x)$, $\\{x_{k,N}\\}_{k\geq 1,N\geq 1}$ needs to be a strong
smooth array (see Proposition A.1). Now, we impose the following two
assumptions on the kernel $K(.)$:

######  Assumption 4.2.

$K(x)$ is a non-negative bounded continuous function satisfying
$\int_{\mathbb{R}}K(x)dx=1$ and $\int_{\mathbb{R}}|\hat{K}(x)|dx<\infty$,
where $\hat{K}(x)=\int_{\mathbb{R}}e^{ixt}K(t)dt$.

######  Assumption 4.3.

For a given $x$, there exists a real positive function $f_{1}(s,x)$ and
$\gamma\in(0,1]$ such that when $h$ is sufficiently small, $|f(hy+x)-f(x)|\leq
h^{\gamma}f_{1}(y,x)\,\forall y\in\mathbb{R},$ and
$\int_{\mathbb{R}}K(s)[f_{1}(s,x)+f_{1}^{2}(s,x)]ds<\infty$.

The following theorem is the main result on the Nadaraya-Watson kernel
estimator (8).

######  Theorem 4.1.

Under Assumptions 4.1–4.3 and for any $h$ satisfying
$\sqrt{N}\lambda^{d}h\rightarrow\infty$ and
$\sqrt{N}\lambda^{d}h^{1+2\gamma}\rightarrow 0$, we have

$\Big{\\{}\sqrt{N}\lambda^{d}h\Big{\\}}^{1/2}\Big{(}\hat{f}(x)-f(x)\Big{)}\rightarrow_{D}\frac{d_{0}}{\sqrt{d_{1}}}N(0,1)\,L_{B}^{-1/2}(1,0),$
(9)

where $d_{0}^{2}=\mathbb{E}(u^{2}_{0})\int_{\mathbb{R}}K^{2}(s)ds$,
$d_{1}=\int_{\mathbb{R}}K(s)ds$, and $N(0,1)$ denotes a standard normal which
is independent from the local time of Brownian motion $L_{B}(1,0)$. If we
normalize the limit form (9), we have:

$\Big{\\{}h\sum_{k=1}^{N}K_{h}(x_{k}-x)\Big{\\}}^{1/2}\Big{(}\hat{f}(x)-f(x)\Big{)}\rightarrow_{D}N(0,\sigma^{2}),$
(10)

where $\sigma^{2}=d_{0}^{2}/d_{1}$; see also expression (A.6) given in the
Appendix.

######  Remark 4.1.

The local time given in Theorem 4.1 is the local time of standard Brownian
motion, which is independent of any unknown parameters. This is a direct
consequence of tempering the time series in the regressor $x_{k}$ (see Remark
3.1). In this regard, the result of Theorem 4.1 is different from Theorem 2.1
in Wang and Phillips (2016), which involves the local time process
$L_{B_{d+1/2}}(1,0)$, which is related to a fractional Brownain motion with
parameter $d+1/2$.

The condition of Theorem 4.1 that $\sqrt{N}\lambda^{d}h\rightarrow\infty$ is
required for the consistency of $\hat{f}(x)$. The normalized limit (10) is
pivotal upon the estimation of $\mathbb{E}(u^{2}_{0})$ by

$\hat{\sigma}^{2}_{N}=\frac{\sum_{k=1}^{N}[y_{k}-\hat{f}(x_{k})]^{2}K_{h}(x_{k}-x)}{\sum_{k=1}^{N}K_{h}(x_{k}-x)}.$
(11)

Expression (11) is appropriate for use in interval estimation and other
inferential procedures. To investigate the bias of $\hat{f}(x)$, we make use
of Assumption 4.1 but replace Assumptions 4.2 and 4.3 for some $p\geq 2$ with
the following stronger assumptions:

######  Assumption 4.4.

$\int_{\mathbb{R}}y^{i}K(y)dy=0$ for $i=1,...,p-1$ and
$\int_{\mathbb{R}}y^{p}K(y)dy\neq 0$. Also, $K(x)$ has a compact support which
is twice continuously differentiable on $\mathbb{R}$.

######  Assumption 4.5.

For a fixed value of $x$, $f(x)$ has a continuous $p+1$ derivative in a small
neighbourhood of $x$. For $h$ satisfying
$\sqrt{N}\lambda^{d}h\rightarrow\infty$ and
$\sqrt{N}\lambda^{d}h^{2(p+1)+1}\rightarrow 0$, we have

$\displaystyle\Big{\\{}\sqrt{N}\lambda^{d}h\Big{\\}}^{1/2}\Bigg{[}\hat{f}(x)-f(x)-\frac{h^{p}f^{(p)}(x)}{p!}\int_{\mathbb{R}}y^{p}K(y)dy\Bigg{]}\rightarrow_{D}N(0,\sigma^{2})L_{B}^{-1/2}(1,0),\quad\text{and}$
$\displaystyle\Big{\\{}h\sum_{k=1}^{N}K_{h}(x_{k}-x)\Big{\\}}^{1/2}\Bigg{[}\hat{f}(x)-f(x)-\frac{h^{p}f^{(p)}(x)}{p!}\int_{\mathbb{R}}y^{p}K(y)dy\Bigg{]}\rightarrow_{D}N(0,\sigma^{2}).$

The proof of these follow from a Taylor series expansion, and the details are
similar to the ones given in Theorem 2.4 of Wang (2014). Hence, we omit them.
For any $h$ satisfying $\sqrt{N}\lambda^{d}h\rightarrow\infty$ as
$h\rightarrow 0$, we have $\hat{f}(x)\rightarrow_{P}f(x)$. Therefore,
$\hat{f}(x)$ is a consistent estimator of $f(x)$.

##  Specification test for the regression function

When there is no reason to believe that $f(x)$ in (1) follows a particular
parametric form, the use of a nonparametric estimator is attractive, but
nonparametric estimators generally have a slow convergence rate in comparison
with the parametric estimators. It is often possible to determine a plausible
parametric regression function, and we then might wish to conduct a test of
the posited parametric specification, formulated as a hypothesis as,

$H_{0}:f(x)=g(x,\theta_{0}),$ (12)

where $\theta_{0}$ in (12) is a vector of unknown parameters that belong to a
compact and convex space $\Omega_{0}$. Tests of (12) have been previously
considered by Hardle and Mammen (1993), Horowitz and Spokoiny (2001), Gao et
al. (2009), Wang and Phillips (2012), and Wang and Phillips (2016) under
different assumptions on the data generating mechanism. Indeed, Gao et al.
(2009) and Wang and Phillips (2012) considered kernel-smoothed U statistic of
the form $\sum_{j,k=1,j\neq k}^{N}\hat{u}_{k}\hat{u}_{j}K[(x_{k}-x_{j})/h]$
with $\hat{u}_{k}=y_{k}-g(x_{k},\hat{\theta}_{N})$.

The asymptotics for U statistics are hard to be extended to the case of
endogenous regressors. Thus, Wang and Phillips (2016) modified a test
statistic suggested previously by Hardle and Mammen (1993) for the
consideration of endogenous regressors with long memory. In that case, the
test statistic is not pivotal and its limit theory and convergence rate both
depend on the memory parameter $d$ through the local time of fractional
Brownian motion $L_{B_{d+1/2}}(t,s)$, which complicates its use in actual
problems. In this section, we consider the statistic under an assumption of
semi-long memory input shocks to the regressors $x_{k}$. Both the test
statistic of Wang and Phillips (2016) and our modification for use with semi-
long memory regressors are based on the statistic,

$T_{N}:=\int_{\mathbb{R}}\Big{\\{}\sum_{k=1}^{N}K\Big{[}\frac{x_{k}-x}{h}\Big{]}[y_{k}-g(x_{k},\hat{\theta}_{N})]\Big{\\}}^{2}\pi(x)dx.$
(13)

For estimating $\hat{\theta}_{N}$, we use a non-linear least squares method by
setting $Q_{N}(\theta)=\sum_{k=1}^{N}(y_{k}-g(x_{k},\theta))^{2}$ and
minimizing it over $\theta\in\Theta$ as follows
$\hat{\theta}_{N}=\text{argmin}_{\theta\in\Theta}Q_{N}(\theta).$ Additionally,
$\pi(x)$ in (13) is a positive integrable weight function with a compact
support. To develop the asymptotic theory of $T_{N}$ in the context of semi-
long memory regressors and endogeneity, we provide three assumptions as
follows.

######  Assumption 5.1.

$K(x)\pi(x)$ has a compact support such that $\int_{\mathbb{R}}K(x)dx=1$ and
$|K(x)-K(y)|\leq C|x-y|$ whenever $|x-y|$ is sufficiently small.

######  Assumption 5.2.

There exist $g_{1}(x)$ and $g_{2}(x)$ such that for each
$\theta,\theta_{0}\in\Omega_{0}$, $|g(x,\theta)-g(x,\theta_{0})|\leq
C||\theta-\theta_{0}||g_{1}(x)$, and for some $0<\beta\leq 1$,
$|g_{1}(x+y)-g_{1}(x)|\leq C|y|^{\beta}g_{2}(x)$, whenever $y$ is sufficiently
small. Additionally,
$\int_{\mathbb{R}}[1+g_{1}^{2}(x)+g^{2}_{2}(x)]\pi(x)dx<\infty$ holds.

######  Assumption 5.3.

Under $H_{0}$,
$||\hat{\theta}_{N}-\theta_{0}||=o_{P}\Big{(}\\{\sqrt{N}\lambda^{d}h\\}^{-1/2}\Big{)}$.

As noted in Wang and Phillips (2016), Assumption 5.2 covers a wide range of
functions $g(x,\theta)$ and $\pi(x)$. Typical examples of $g(x,\theta)$
include $\theta e^{x}/(1+e^{x})$, $e^{-\theta|x|}$, $e^{-\theta x^{2}}$,
$e^{\theta|x|}/(1+e^{\theta|x|})$, $\theta\log|x|$, $(x+\theta)^{2}$, etc.
Note that assumptions 5.1–5.3 may be compared with those imposed by Wang and
Phillips (2016). We now consider the asymptotic behavior of $T_{N}$ in (13)
and its convergence rate in the semi-long memory setting.

######  Theorem 5.1.

Let Assumptions 4.1, and 5.1–5.3 hold. Then, under $H_{0}$, we have

$T_{\lambda,d}:=\frac{1}{\sqrt{N}\lambda^{d}h}T_{N}\rightarrow_{D}d_{(0)}^{2}L_{B}(1,0),$
(14)

where
$d_{(0)}^{2}=\mathbb{E}(u^{2}_{0})\int_{\mathbb{R}}K^{2}(s)ds\int_{\mathbb{R}}\pi(x)dx$
and $N^{1/2-\delta_{0}}\lambda^{d}h\rightarrow\infty$ for a small enough
$\delta_{0}$.

######  Remark 5.1.

(i) Under semi-long memory for $d\in\mathbb{R}_{+}$, the limiting null
distribution of (14) is free of the unknown fractional differencing parameter
$d$, therefore it is a pivotal quantity. (ii) Under long memory for
$d\in(0,1/2)$, the test statistic and limit distribution given by Wang and
Phillips (2016) are
$T_{N,d}:=(d_{N}/Nh)T_{N}\rightarrow_{D}d^{2}_{(0)}L_{B_{d+1/2}}(1,0)$, where
$d_{N}\sim N^{d+1/2}$ as $N\rightarrow\infty$, and $T_{N}$ is as given in
(13). The limiting null distribution of this version of the test statistic
relies on $d$ through the fractional Brownian motion process $B_{d+1/2}$.
(iii) Under short memory for $d=0$ and $d_{N}\sim\phi\sqrt{N}$, the test
statistic and limit distribution are:
$T_{N,0}=(\phi/\sqrt{N}h)T_{N}\rightarrow_{D}d^{2}_{(0)}L_{B}(1,0)$. (iv) Note
that the limiting null distribution of test statistic (14) formulated under
semi-long memory is similar to the short memory case. In neither case does the
limiting distribution depend on $d$.

######  Remark 5.2.

The role of tempering in the normalizing term
$\\{\sqrt{N}\lambda^{d}h\\}^{-1}$ in (14) is intermediate; i.e., under long
memory and in the presence of tempering parameter $\lambda$, the rate of
convergence of $\\{\sqrt{N}\lambda^{d}h\\}^{-1}\rightarrow 0$ is faster
compared to the case without a tempering parameter. On the other hand, under
short memory and in the presence of tempering parameter $\lambda$, the rate of
convergence of $\\{\sqrt{N}\lambda^{d}h\\}^{-1}\rightarrow 0$ is slower
compared to the case without a tempering parameter.

To verify that a test has nontrivial power, one can assess the null hypothesis
(12) against a local alternative

$H_{A}:f(x)=g(x,\theta)+\rho_{N}m(x),\quad\text{for}\,\,x\in\mathbb{R}.$ (15)

Here, $\rho_{N}$ is a sequence of numerical constants which measures the local
deviation from the null hypothesis. Also in (15), $m(x)$ is a real function
free of $\theta$ without lying in the span of $g(x,\theta)$ and its derivative
functions. To make $m(x)$ smooth enough under $H_{A}$ for the sake of
asymptotic power development, we give the following two assumptions.

######  Assumption 5.4.

(i) There exist $m_{1}(x)$ and $\gamma\in(0,1]$ such that for any $y$
sufficiently small, $|m(x+y)-m(x)|\leq C|y|^{\gamma}m_{1}(x)$ holds. (ii) Let
$\int_{\mathbb{R}}[1+m^{2}(x)+m^{2}_{1}(x)]\pi(x)dx<\infty$ and
$\int_{\mathbb{R}}m^{2}(x)\pi(x)dx>0$ hold. (iii) The function $m(x)$ is not
an element of the space spanned by $g(x,\theta_{0})$ and its derivative
functions.

######  Assumption 5.5.

Under $H_{A}$,
$||\hat{\theta}_{N}-\theta_{0}||=o_{P}\Big{(}\\{\sqrt{N}\lambda^{d}h\\}^{-1/2}\Big{)}$.

######  Theorem 5.2.

Let Assumptions 4.1, 5.1–5.2, and 5.4–5.5 hold. Then, under $H_{A}$, we have

$\lim\limits_{N\rightarrow\infty}P\Big{(}T_{\lambda,d}\geq T_{0}\Big{)}=1,$

where $T_{0}$ is positive, and $h\rightarrow 0$ satisfying
$N^{1/2-\delta_{0}}\lambda^{d}h\rightarrow\infty$ for a small enough
$\delta_{0}$ and any $\rho_{N}$ satisfying
$N^{1/2}\lambda^{d}h\rho_{N}^{2}\rightarrow\infty$.

######  Remark 5.3.

Based on Theorem 5.2, the test statistic $T_{\lambda,d}$ has nontrivial power
against the local alternatives in the form of (15) whenever
$\rho_{N}\rightarrow 0$ at a rate that is slower than
$\\{\sqrt{N}\lambda^{d}h\\}^{-1/2}$ as
$\\{\sqrt{N}\lambda^{d}h\\}^{-1}\rightarrow 0$. Since $m(x)$ moves faster for
a nonstationary process $x_{k}$, it is natural to expect a higher power under
the alternative $H_{A}$ in (15). Proof of Theorem 5.2 ensures the divergence
of normalized test statistic $T_{\lambda,d}$ and test consistency under
$H_{A}$; see Remarks 3.2 and 3.3 in Wang and Phillips (2016) for a related
discussion.

##  Simulation studies

In this section, we illustrate the theory of the previous sections through the
use of several Monte Carlo studies. We first examine properties of
nonparametric function estimators and then turn our attention toward behavior
of specification tests for hypothesized parametric regression functions.

###  Regression function estimation

To examine the behavior of regression function estimators, we generated data
sets from a model with the form of (1), where we incorporate $\sigma$ as
follows

$y_{k}=f(x_{k})+\sigma u_{k}.$ (16)

The regressor process $x_{k}$ is defined for the long memory setting (LM) in
(2) and for the semi-long memory setting (SLM) in (2). Let $u_{k}=\psi
u_{k-1}+\epsilon_{k}$ with $\psi=0.25$ and
$\mathbb{E}(\xi_{k}\epsilon_{k})=:\rho_{\xi,\epsilon}$ (c.f., (7)). Also, let
$\rho_{\xi,\epsilon}=0.5$ and $\sigma=0.2$ in (16). We consider the following
regression function, which was also used by Wang and Phillips (2016),

$f(x)=\sum_{j=1}^{\infty}\frac{(-1)^{j+1}\sin(j\pi x)}{j^{2}}.$ (17)

We set the sample size as $N=1000$, and the number of Monte Carlo replications
as $R=10,000$.

We use the estimator (8) with an Epanechnikov kernel in the form of
$K(u)=0.75(1-u^{2})1_{\\{|u|\leq 1\\}}$. We take the bandwidth
$h=\\{N^{-1/3},N^{-1/4},N^{-1/5},N^{-1/6}\\}$ and the tempering parameter
$\lambda$ to be
$\\{N^{-1/3}(\text{SLM1}),N^{-1/4}(\text{SLM2}),N^{-1/5}(\text{SLM3}),N^{-1/6}(\text{SLM4})\\}$
under the assumption of $N\lambda\rightarrow\infty$. In Table 1, we give the
Monte Carlo approximations of bias, standard deviation (Std) and root of mean
squared error (RMSE) of $\hat{f}(x)$ for the true regression function (17)
over the interval $[0,1]$. The values of Table 1 result from computing bias,
Std, and RMSE at 100 points equally spaced between 0 and 1 and then averaging.
Errors of the Monte Carlo approximations of Table 1 were generally less than
$0.003$, and were nearly uniformly smaller for SLM settings than for the LM
case. Bias is small and similar for both LM and SLM settings and across values
of $d$. Values for Std are uniformly smaller for SLM than for LM processes,
increase somewhat as $d$ increases, and are larger for the larger bandwidth
used. As a result, values for RMSE follow the same pattern, being smaller for
SLM, smaller for smaller $d$, and smaller for shorter bandwidth.

To construct point-wise confidence intervals for the regression function
$f(x)$, we use the limit distribution given in (10). An asymptotic
$100(1-\alpha)\%$ level confidence interval for $f(x)$ is then given by

$\hat{f}(x)\pm
z_{\alpha/2}\Bigg{\\{}\frac{\hat{\sigma}^{2}_{N}\int_{\mathbb{R}}K^{2}(s)ds}{\sum_{k=1}^{N}K[\frac{x_{k}-x}{h}]\int_{\mathbb{R}}K(s)ds}\Bigg{\\}}^{1/2},$
(18)

where $\hat{\sigma}^{2}_{N}$ is given in expression (11). The observed
coverage probabilities of confidence intervals are given in Figure 1 as well
as Table 2 for $(1-\alpha)=0.95$. Observe that, as the bandwidth increases,
the coverage probabilities become closer to the nominal level of $95\%$. Also,
as the tempering parameter $\lambda$ increases, the coverage probabilities
increases.

Table 1: Comparison of Bias, Std and RMSE for $\hat{f}(x)$ between LM and SLM processes under the assumption of $\rho_{\xi,\epsilon}=0.5$. Criterion | Regressor | Bandwidth | $d=0$ | $d=0.1$ | $d=0.2$ | $d=0.3$ | $d=0.4$
---|---|---|---|---|---|---|---
| LM | $N^{-1/3}$ | 0.007 | 0.007 | 0.007 | 0.005 | 0.000
|  | $N^{-1/5}$ | 0.046 | 0.046 | 0.046 | 0.043 | 0.043
Bias | SLM1 | $N^{-1/3}$ | 0.007 | 0.008 | 0.007 | 0.007 | 0.008
|  | $N^{-1/5}$ | 0.046 | 0.047 | 0.047 | 0.047 | 0.047
| SLM3 | $N^{-1/3}$ | 0.007 | 0.007 | 0.007 | 0.007 | 0.006
|  | $N^{-1/5}$ | 0.046 | 0.047 | 0.047 | 0.047 | 0.046
| LM | $N^{-1/3}$ | 0.114 | 0.127 | 0.139 | 0.150 | 0.157
|  | $N^{-1/5}$ | 0.125 | 0.147 | 0.165 | 0.184 | 0.200
Std | SLM1 | $N^{-1/3}$ | 0.114 | 0.121 | 0.128 | 0.135 | 0.140
|  | $N^{-1/5}$ | 0.125 | 0.134 | 0.144 | 0.154 | 0.164
| SLM3 | $N^{-1/3}$ | 0.114 | 0.120 | 0.123 | 0.129 | 0.132
|  | $N^{-1/5}$ | 0.125 | 0.131 | 0.138 | 0.142 | 0.148
| LM | $N^{-1/3}$ | 0.116 | 0.129 | 0.141 | 0.152 | 0.160
|  | $N^{-1/5}$ | 0.151 | 0.173 | 0.191 | 0.209 | 0.226
RMSE | SLM1 | $N^{-1/3}$ | 0.116 | 0.123 | 0.130 | 0.137 | 0.143
|  | $N^{-1/5}$ | 0.151 | 0.161 | 0.170 | 0.179 | 0.189
| SLM3 | $N^{-1/3}$ | 0.116 | 0.122 | 0.125 | 0.131 | 0.134
|  | $N^{-1/5}$ | 0.151 | 0.157 | 0.165 | 0.168 | 0.174

In Figure 1, we compare the coverage probabilities between LM and SLM settings
for different values of $d$. Coverage probabilities based on the SLM process
remain close to the nominal level across values of $d$, but coverage
probabilities for the LM process drift further away from the nominal level as
$d$ increases. Additionally, we demonstrate the width of confidence intervals
in Figure 2 as well as Table 2. We observe that, under LM, changes in $d$
affect the width of confidence intervals, with wider intervals resulting from
larger $d$. In contrast, under SLM, changes in $d$ have relatively little
influence on the width of computed intervals. In Table 2 it can be seen that
as bandwidth increases, the widths of confidence intervals decrease and this
holds for both LM and SLM settings and across all levels of the differencing
parameter $d$. As the tempering parameter $\lambda$ increases, the confidence
interval becomes shorter.

|
---|---
Figure 1: Comparison of coverage probabilities for the regression function $f(x)$ between LM (left) and SLM (right) cases under the assumption of $\rho_{\xi,\epsilon}=0.5$ for different values of $d$. We assume the bandwidth is $h=N^{-1/3}$ and the tempering parameter for the SLM is $\lambda=N^{-1/6}$. |
---|---
Figure 2: Comparison of empirical confidence intervals for the regression function $f(x)$ between LM (left) and SLM (right) cases under the assumption of $\rho_{\xi,\epsilon}=0.5$ for different values of $d$. We assume the bandwidth is $h=N^{-1/3}$ and the tempering parameter for the SLM is $\lambda=N^{-1/6}$. Table 2: Comparison of coverage probabilities and expected lengths of confidence intervals between LM and SLM processes for $f(x)$ given some selected points of $x$ on the interval [0,1], where we assume $\rho_{\xi,\epsilon}=0.5$. The values in the parentheses are the expected lengths of confidence intervals. $x$ | Regressor | Bandwidth | $d=0$ | $d=0.1$ | $d=0.2$ | $d=0.3$ | $d=0.4$
---|---|---|---|---|---|---|---
| LM | $N^{-1/3}$ | 0.854 | 0.775 | 0.677 | 0.588 | 0.479
|  |  | (1.053) | (1.188) | (1.297) | (1.431) | (1.558)
|  | $N^{-1/5}$ | 0.926 | 0.880 | 0.823 | 0.760 | 0.664
|  |  | (0.829) | (0.992) | (1.125) | (1.261) | (1.371)
$0.25$ | SLM1 | $N^{-1/3}$ | 0.854 | 0.828 | 0.790 | 0.755 | 0.706
|  |  | (1.053) | (1.117) | (1.194) | (1.273) | (1.336)
|  | $N^{-1/5}$ | 0.926 | 0.913 | 0.893 | 0.876 | 0.849
|  |  | (0.829) | (0.907) | (0.973) | (1.048) | (1.132)
| SLM3 | $N^{-1/3}$ | 0.854 | 0.837 | 0.818 | 0.797 | 0.768
|  |  | (1.053) | (1.101) | (1.161) | (1.218) | (1.231)
|  | $N^{-1/5}$ | 0.926 | 0.918 | 0.906 | 0.896 | 0.882
|  |  | (0.829) | (0.885) | (0.932) | (0.976) | (1.020)
| LM | $N^{-1/3}$ | 0.870 | 0.786 | 0.696 | 0.597 | 0.484
|  |  | (1.687) | (1.924) | (2.109) | (2.286) | (2.406)
|  | $N^{-1/5}$ | 0.945 | 0.907 | 0.860 | 0.806 | 0.703
|  |  | (1.286) | (1.537) | (1.779) | (1.998) | (2.206)
$0.50$ | SLM1 | $N^{-1/3}$ | 0.870 | 0.838 | 0.806 | 0.7693 | 0.724
|  |  | (1.687) | (1.818) | (1.943) | (2.081) | (2.186)
|  | $N^{-1/5}$ | 0.945 | 0.933 | 0.917 | 0.902 | 0.881
|  |  | (1.286) | (1.398) | (1.521) | (1.629) | (1.767)
| SLM3 | $N^{-1/3}$ | 0.870 | 0.851 | 0.832 | 0.807 | 0.786
|  |  | (1.687) | (1.768) | (1.860) | (1.944) | (2.010)
|  | $N^{-1/5}$ | 0.945 | 0.936 | 0.930 | 0.918 | 0.910
|  |  | (1.286) | (1.359) | (1.472) | (1.506) | (1.596)
| LM | $N^{-1/3}$ | 0.867 | 0.779 | 0.687 | 0.591 | 0.483
|  |  | (1.926) | (2.165) | (2.353) | (2.646) | (2.852)
|  | $N^{-1/5}$ | 0.935 | 0.897 | 0.843 | 0.783 | 0.681
|  |  | (1.326) | (1.580) | (1.818) | (2.056) | (2.226)
$0.75$ | SLM1 | $N^{-1/3}$ | 0.867 | 0.836 | 0.799 | 0.762 | 0.721
|  |  | (1.926) | (2.059) | (2.187) | (2.349) | (2.397)
|  | $N^{-1/5}$ | 0.935 | 0.921 | 0.906 | 0.885 | 0.866
|  |  | (1.326) | (1.452) | (1.565) | (1.668) | (1.812)
| SLM3 | $N^{-1/3}$ | 0.867 | 0.850 | 0.829 | 0.805 | 0.779
|  |  | (1.926) | (1.996) | (2.098) | (2.200) | (2.301)
|  | $N^{-1/5}$ | 0.935 | 0.928 | 0.919 | 0.910 | 0.895
|  |  | (1.326) | (1.411) | (1.488) | (1.559) | (1.633)

Table 2. (continued) $x$ | Regressor | Bandwidth | $d=0$ | $d=0.1$ | $d=0.2$ | $d=0.3$ | $d=0.4$
---|---|---|---|---|---|---|---
| LM | $N^{-1/3}$ | 0.796 | 0.701 | 0.596 | 0.501 | 0.397
|  |  | (0.855) | (0.974) | (1.032) | (1.160) | (1.240)
|  | $N^{-1/5}$ | 0.875 | 0.833 | 0.777 | 0.718 | 0.623
|  |  | (0.989) | (1.183) | (1.377) | (1.568) | (1.733)
$0.95$ | SLM1 | $N^{-1/3}$ | 0.796 | 0.753 | 0.716 | 0.672 | 0.629
|  |  | (0.855) | (0.902) | (0.978) | (1.027) | (1.053)
|  | $N^{-1/5}$ | 0.875 | 0.860 | 0.842 | 0.822 | 0.804
|  |  | (0.989) | (1.085) | (1.165) | (1.256) | (1.374)
| SLM3 | $N^{-1/3}$ | 0.796 | 0.773 | 0.744 | 0.721 | 0.690
|  |  | (0.855) | (0.907) | (0.921) | (0.957) | (1.009)
|  | $N^{-1/5}$ | 0.875 | 0.870 | 0.858 | 0.847 | 0.831
|  |  | (0.989) | (1.045) | (1.118) | (1.161) | (1.234)

###  Performance of test

The test statistic $T_{\lambda,d}$ has asymptotic reference distribution given
in Theorem 5.1, and the statistic $T_{N,d}$ has the distribution provided by
Wang and Phillips (2016). Both of these limiting distributions are of complex
form, and are not easy to use in practice. In the case of a LM assumption for
the regressor variables $x_{k}$, the limit distribution of $T_{N,d}$ depends
on the differencing parameter $d$. The limit distribution of both statistics
involve the term $d^{2}_{(0)}$, which does not have a simple form and depends
on $x$ through $\hat{\sigma}^{2}_{N}$.

Under appropriate conditions, we can approximate the sampling distribution of
test statistics through the use of subsampling Politis and Romano (1994),
which can make the testing procedure more feasible for use with real
applications. This allows us to be able to study the propoerties of test
statistic under both long memory and semi-long memory input shocks to the
regressor. The crux of the method is to recompute the test statistic on
smaller blocks or “subsamples” of the observed data to construct the empirical
distribution of the subsample test estimates, and thus find the critical
values.

We use blocks of size $b$ of consecutive observations as subsamples. Define
the following vectors for each block, $x_{b,t}:=(x_{t},...,x_{t+b-1})$ and
$y_{b,t}:=(y_{t},...,y_{t+b-1})$ for $t=1,...,N-b+1$. Based on the subsample
$x_{b,t}$ and $y_{b,t}$, the test statistic is:

$T_{b,t}:=\int_{\mathbb{R}}\Big{\\{}\sum_{a=t}^{t+b-1}K\Big{[}\frac{x_{b,a}-x}{h}\Big{]}[y_{b,a}-g(x_{b,a},\hat{\theta}_{b,t})]\Big{\\}}^{2}\pi(x)dx,$

where $\hat{\theta}_{b,t}=\hat{\theta}_{b}(x_{b,t},y_{b,t})$ is a parametric
estimator of $\theta_{0}$ based on the information in block $t$. Then, the
subsampling estimator $\hat{F}_{\lambda,b}(x)$ of the distribution function
$F_{\lambda}(x)$ with
$F_{\lambda}(x)=\text{Prob}_{P}\\{(\sqrt{N}\lambda^{d}h)^{-1}T_{N}\leq x\\}$
can be defined as

$\hat{F}_{\lambda,b}(x)=\frac{1}{N-b+1}\sum_{t=1}^{N-b+1}1\Big{\\{}\frac{1}{\sqrt{b}\lambda_{b}^{d}h_{b}}T_{b,t}\leq
x\Big{\\}},$

where $\lambda_{b}$ and $h_{b}$ are both dependent on the block size $b$.
Under an assumption of long memory input shocks to the regressors, we remove
the tempering parameter.

In this section, we conduct a Monte Carlo to find the empirical size of the
test statistic based on subsamples. The design is similar to the previous
Monte Carlo exercise of Section 6.1. To keep the computational cost feasible,
we assume the sample size is $N=500$ and the number of replications from the
Monte Carlo is $R=2000$. We use several block sizes,
$b=\\{[0.5N^{0.5}],[N^{0.5}],[2N^{0.5}],[4N^{0.5}]\\}$. The null hypothesis to
be tested can be stated as,

$H_{0}:f(x,\theta)=\theta_{0}+\theta_{1}x.$ (19)

The true values of $(\theta_{0},\theta_{1})$ are set to $(0,1)$. For
nonparametric function estimation we used a Gaussian kernel with $\pi(x)=1$
for $-100\leq x\leq 100$ and $0$ otherwise such that Assumption 5.1 holds.

Monte Carlo distributions of test statistics for both LM and SLM under the
null hypothesis are given in Figure 3. We observe that the histograms under
SLM are almost overlapped unlike the histograms under LM. In order to study
the empirical size of test, we generate data under the null hypothesis $H_{0}$
and find the proportion of times where $T_{\lambda,d}$ falls above the
($1-\alpha$)-th quantiles of its subsampling distribution, where $\alpha$ is
the nominal significance level of Type I error. The results are given in Table
3.

We observe that the size of the test is not maintained. The Type I error rate
is large for all the cases and both test statistics, although under SLM
regressors, the values are smaller. Additionally, as bandwidth $h$ and
tempering paremeter $\lambda$ increase, the values of the size of the test
become smaller, but are still not close to the nominal level. When $d=0$, the
values of the size of the test between LM and SLM processes are equal. Since
the values of empirical size of the test are significantly large, it is
meaningless to study the power of the test with subsampling technique, even
though the power is expected to be large based on Remark 5.3. Whether the
cause of this poor performance is due to a subtle scaling issue or some other
problem is an open question. If some modification of one or both of the
statsitics would lead to tests with the ability to maintain size, then the
power of these procedures would become an issue worth addressing.

Table 3: Empirical test sizes. The given values under $d$ correspond to the 4 block sizes in the manuscript; i.e. $\\{[0.5N^{0.5}],[N^{0.5}],[2N^{0.5}],[4N^{0.5}]\\}$. Regressor |  |  |  |  |  |
---|---|---|---|---|---|---
Bandwidth | Levels | $d=0$ | $d=0.1$ | $d=0.2$ | $d=0.3$ | $d=0.4$
LM | $0.01$ | $\\{0.939,0.884,$ | $\\{0.998,0.994,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$
|  | $0.872,0.850\\}$ | $0.991,0.987\\}$ | $1.000,0.998\\}$ | $1.000,1.000\\}$ | $1.000,1.000\\}$
$N^{-1/3}$ | $0.05$ | $\\{1.000,0.998,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$
|  | $0.976,0.933\\}$ | $1.000,0.997\\}$ | $1.000,1.000\\}$ | $1.000,1.000\\}$ | $1.000,1.000\\}$
| $0.10$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$
|  | $0.999,0.981\\}$ | $1.000,1.000\\}$ | $1.000,1.000\\}$ | $1.000,1.000\\}$ | $1.000,1.000\\}$
| $0.01$ | $\\{0.888,0.786,$ | $\\{0.990,0.968,$ | $\\{1.000,0.999,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$
|  | $0.722,0.635\\}$ | $0.946,0.908\\}$ | $0.996,0.989\\}$ | $1.000,0.999\\}$ | $1.000,1.000\\}$
$N^{-1/5}$ | $0.05$ | $\\{0.997,0.970,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$
|  | $0.900,0.808\\}$ | $0.994,0.965\\}$ | $1.000,0.998\\}$ | $1.000,0.999\\}$ | $1.000,1.000\\}$
| $0.10$ | $\\{1.000,0.997,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$
|  | $0.975,0.901\\}$ | $1.000,0.987\\}$ | $1.000,1.000\\}$ | $1.000,1.000\\}$ | $1.000,1.000\\}$
SLM1 | $0.01$ | $\\{0.939,0.884,$ | $\\{0.984,0.959,$ | $\\{0.996,0.987,$ | $\\{1.000,0.996,$ | $\\{1.000,1.000,$
|  | $0.872,0.850\\}$ | $0.952,0.943\\}$ | $0.985,0.983\\}$ | $0.997,0.998\\}$ | $1.000,1.000\\}$
$N^{-1/3}$ | $0.05$ | $\\{1.000,0.998,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$
|  | $0.976,0.933\\}$ | $0.996,0.981\\}$ | $0.998,0.993\\}$ | $1.000,1.000\\}$ | $1.000,1.000\\}$
| $0.10$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$
|  | $0.999,0.981\\}$ | $1.000,0.996\\}$ | $1.000,0.999\\}$ | $1.000,1.000\\}$ | $1.000,1.000\\}$
| $0.01$ | $\\{0.888,0.786,$ | $\\{0.946,0.882,$ | $\\{0.983,0.951,$ | $\\{0.996,0.984,$ | $\\{0.999,0.993,$
|  | $0.722,0.635\\}$ | $0.829,0.772\\}$ | $0.925,0.889\\}$ | $0.969,0.954\\}$ | $0.989,0.984\\}$
$N^{-1/5}$ | $0.05$ | $\\{0.997,0.970,$ | $\\{1.000,0.993,$ | $\\{1.000,0.999,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$
|  | $0.900,0.808\\}$ | $0.958,0.890\\}$ | $0.988,0.950\\}$ | $0.997,0.984\\}$ | $0.999,0.995\\}$
| $0.10$ | $\\{1.000,0.997,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$
|  | $0.975,0.901\\}$ | $0.993,0.957\\}$ | $0.998,0.984\\}$ | $1.000,0.996\\}$ | $1.000,1.000\\}$
SLM3 | $0.01$ | $\\{0.939,0.884,$ | $\\{0.972,0.937,$ | $\\{0.990,0.969,$ | $\\{0.995,0.987,$ | $\\{0.998,0.994,$
|  | $0.872,0.850\\}$ | $0.927,0.912\\}$ | $0.959,0.957\\}$ | $0.981,0.977\\}$ | $0.992,0.993\\}$
$N^{-1/3}$ | $0.05$ | $\\{1.000,0.998,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$
|  | $0.976,0.933\\}$ | $0.992,0.965\\}$ | $0.995,0.985\\}$ | $0.997,0.991\\}$ | $1.000,0.997\\}$
| $0.10$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$
|  | $0.999,0.981\\}$ | $1.000,0.993\\}$ | $1.000,0.997\\}$ | $1.000,1.000\\}$ | $1.000,1.000\\}$
| $0.01$ | $\\{0.888,0.786,$ | $\\{0.925,0.845,$ | $\\{0.961,0.906,$ | $\\{0.983,0.951,$ | $\\{0.991,0.971,$
|  | $0.722,0.635\\}$ | $0.799,0.720\\}$ | $0.860,0.809\\}$ | $0.924,0.877\\}$ | $0.958,0.934\\}$
$N^{-1/5}$ | $0.05$ | $\\{0.997,0.970,$ | $\\{0.999,0.990,$ | $\\{1.000,0.997,$ | $\\{1.000,0.999,$ | $\\{1.000,1.000,$
|  | $0.900,0.808\\}$ | $0.938,0.861\\}$ | $0.973,0.913\\}$ | $0.986,0.945\\}$ | $0.993,0.974\\}$
| $0.10$ | $\\{1.000,0.997,$ | $\\{1.000,0.999,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$ | $\\{1.000,1.000,$
|  | $0.975,0.901\\}$ | $0.988,0.935\\}$ | $0.996,0.963\\}$ | $0.998,0.982\\}$ | $1.000,0.992\\}$

|
---|---
Figure 3: Comparison of Monte Carlo histograms of test statistic under the
null hypothesis $H_{0}$ between LM (left) and SLM (right) cases under the
assumption of $\rho_{\xi,\epsilon}=0.5$ for different values of $d$. We assume
the bandwidth is $h=N^{-1/3}$ and the tempering parameter for the SLM is
$\lambda=N^{-1/6}$.

##  Application: Carbon Kuznets Curve

The Carbon Kuznets Curve (CKC) relates the per capita $\text{CO}_{2}$
emissions of countries to their per capita GDP. Basically, the CKC hypothesis
states that there is an inverted-U shaped relationship between economic
activities and per capita $\text{CO}_{2}$ emissions, where the upward slope of
the CKC can be explained by the increase in natural resources depletion as
economic activities grow, and the downward slope of the CKC results from a
reduction in the emission of air pollutants as the country continues to
develop technological advance and stricter regulatory policies; see Chan and
Wang (2015) and references therein.

It is believed that the GDP variable has nonstationary behavior over time, and
the assumption of exogeneity may not hold because of measurement error and
other sources of errors; see Wang et al. (2018) for a related discussion. In
the usual CKC formulation, the logarithm of per capita $\text{CO}_{2}$
emissions and GDP are assumed to be integrated processes, and there exists a
nonlinear link function, see Müller-Fürstenberger and Wagner (2007). Here, we
consider 3 countries (Belgium, Denmark, and France), which are shown in Figure
4 and are given in Chan and Wang (2015) and Piaggio and Padilla (2012). The
annual data set is from 1950 to 2008 and contains 59 observations. The
$\text{CO}_{2}$ emission data come from the Carbon Dioxide Information
Analysis Center (Boden et al. (2009)), and the GDP data come from Maddison
(2003).

The range of our data do not exactly match with those of Chan and Wang (2015)
as they modified the data set by transformations. We fit an ARTFIMA model with
$p=q=0$ to the data set using the artfima package from the statistical
software R developed by Sabzikar et al. (2019). The Whittle estimators of
fitted parameters $d$ and $\lambda$ for $\log(x_{k})$ and $\log(y_{k})$ of 3
countries are given in Table 4. Also, we fit an ARFIMA model with $p=q=0$ to
the data. The values of MSE for fitting ARFIMA model and ARTFIMA model are
given in Table 4.

We observe that the MSEs based on fitting the ARTFIMA model are much smaller
than fitting the ARFIMA model. Thus, we use the test statistic under the
assumption of semi-long memory input shocks to the regressors given in (14).
All the values of $d$ given in Table 4 are substantially greater than $0.5$,
and our proposed test statistic that assumes a semi-long memory regressor
process is applicable for $d>0.5$. Note that this may be a larger problem for
the test statistic given by (Wang and Phillips, 2016) where $d$ has to belong
to the interval $(0,0.5)$.

Table 4: The Whittle estimators of fitted parameters from an ARTFIMA model for 3 countries as well as the MSE of fitting an ARFIMA model versus an ARTFIMA model. |  |  |  | MSE
---|---|---|---|---
Country | Variable | $\lambda$ | $d$ | ARFIMA | ARTFIMA
Belgium | $\log(x_{k})$ | $0.129$ | $1.086$ | $0.977$ | $0.411$
| $\log(y_{k})$ | $0.118$ | $1.000$ | $0.235$ | $0.026$
Denmark | $\log(x_{k})$ | $0.154$ | $1.142$ | $0.858$ | $0.350$
| $\log(y_{k})$ | $0.124$ | $1.128$ | $1.415$ | $0.286$
France | $\log(x_{k})$ | $0.138$ | $1.093$ | $0.976$ | $0.395$
| $\log(y_{k})$ | $0.071$ | $1.133$ | $1.013$ | $0.188$
---
Figure 4: Plots of $\log(\text{CO}_{2})$ versus $\log(\text{GDP})$ for
Belgium, Denmark, and France based on the observed values (top) and selected
bandwidths (bottom).

The plots in Figure 4 (top) exhibit some of the quadratic behavior anticipated
by the theory of CKC curves. We formulate the following hypotheses:

$\displaystyle H_{1}:z_{k}$
$\displaystyle=\theta_{0}+\theta_{1}e_{k}+u_{k},\quad$ $\displaystyle
H_{2}:z_{k}$
$\displaystyle=\theta_{0}+\theta_{1}e_{k}+\theta_{2}e_{k}^{2}+u_{k},$

where $y_{k}$ and $x_{k}$ are per capita emission of $\text{CO}_{2}$ and GDP
in year $k=1,\ldots,59$, $z_{k}=\log(y_{k})$, and $e_{k}=\log(x_{k})$. We set
the bandwidth at $h=\\{1/\sqrt{N},1/N\\}$ and give the related plots in Figure
4 (bottom). We consider the Gaussian kernel with $\pi(x)=1$ for $-100\leq
x\leq 100$ and $0$ otherwise. We use the subsampling technique described in
Section 6.2 to find the p-values for the given hypotheses. We consider the
following block sizes: $b=\\{[2N^{0.5}],[4N^{0.5}],[6N^{0.5}]\\}$.

The corresponding p-values for both the bandwidths are displayed in Table 5.
For most block sizes and both hypothesized models the p-values are very small,
rejecting both straight line and quadratic models. When the block size is
increased to the largest size considered, the quadratic model fares better
under the assumption of $h=1/N$, as does the straight line model for France
only, which is not clear given the scatterplots of Figure 4 (top). Overall, we
conclude that the test has poor performance, this time in an actual
application.

Table 5: Approximate p-values based on subsampling distribution. |  | p-value
---|---|---
|  | Belgium | Denmark | France
$h$ | Block Size | $H_{1}$ | $H_{2}$ | $H_{1}$ | $H_{2}$ | $H_{1}$ | $H_{2}$
$1/\sqrt{N}$ | $[2N^{0.5}]$ | $<0.0001$ | $<0.0001$ | $<0.0001$ | $<0.0001$ | $<0.0001$ | $<0.0001$
| $[4N^{0.5}]$ | $<0.0001$ | $<0.0001$ | $<0.0001$ | $<0.0001$ | $<0.0001$ | $<0.0001$
| $[6N^{0.5}]$ | $<0.0001$ | $0.300$ | $<0.0001$ | $<0.0001$ | $<0.0001$ | $<0.0001$
$1/N$ | $[2N^{0.5}]$ | $<0.0001$ | $<0.0001$ | $<0.0001$ | $<0.0001$ | $<0.0001$ | $<0.0001$
| $[4N^{0.5}]$ | $<0.0001$ | $0.225$ | $<0.0001$ | $<0.0001$ | $<0.0001$ | $<0.0001$
| $[6N^{0.5}]$ | $<0.0001$ | $0.800$ | $<0.0001$ | $0.600$ | $0.500$ | $0.300$

##  Conclusion

In this paper, we have given a pivotal test statistic for the non-parametric
cointegrating regression function, which extends the earlier work of Wang and
Phillips (2016). The limit theory for the test involves local time of standard
Brownian motion and is free of any unknown parameters. This development has
relied on the idea of tempering (in the strong sense) such that there are
semi-long memory input shocks to the regressor. In addition, the tempering of
the coefficients in the regressors can also extend the range of fractional
differencing parameter $d$ from $(0,1/2)$ to any positive value on the real
line.

We adopted a strategy of subsampling to approximate the reference distribution
of our test statistic, as well as that of Wang and Phillips (2016), which
increases the applicability of tests based on these statistics. The existence
of limiting distributions for the statistics is an important part of ensuring
that a subsampling approach is justified. Our work has also indicated that,
for the most part, tests based on these statistics are unable to maintain
size, which calls their use in applications into question. Additional
investigation is required for a modified test statistic such that it can have
the correct size. This is beyond the scope of this article, and hence we leave
it for future work.

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## Appendix A: Technical Details and Proofs

### A.1 Preliminaries and Lemmas

We can decompose $x_{k}$ as follows:

$\displaystyle x_{k}\equiv\sum_{j=1}^{k}X_{d,\lambda}(j)$
$\displaystyle=\sum_{j=1}^{k}\sum_{i=-\infty}^{j}\xi(i)\phi_{N}(j-i),\quad\phi_{N}(k):=e^{-\lambda
k}b_{d}(k)$
$\displaystyle=x_{s}+\sum_{j=s+1}^{k}\sum_{i=-\infty}^{s}\xi(i)\phi_{N}(j-i)+\sum_{j=s+1}^{k}\sum_{i=s+1}^{j}\xi(i)\phi_{N}(j-i)$
$\displaystyle=:x^{*}_{s,k}+x^{\prime}_{s,k},\qquad s<k,$

such that
$x^{*}_{s,k}=x_{s}+\sum_{j=s+1}^{k}\sum_{i=-\infty}^{s}\xi(i)\phi_{N}(j-i)$
and it depends on $\\{\cdots,\xi(s-1),\xi(s)\\}$, and
$x^{\prime}_{s,k}=\sum_{j=1}^{k-s}\sum_{i=1}^{j}\xi(i+s)\phi_{N}(j-i).$ If we
define $a_{N}:=\sum_{j=0}^{N}\phi_{N}(j)$, then we can show $a_{N}\asymp
O(\lambda^{-d})$ as $N\lambda\rightarrow\infty$ and recall that $\lambda$ is a
sample size dependent parameter; see Lemma A.5 part (b) in Sabzikar et al.
(2020).

######  Definition A.1.

Let $\Omega_{N}(\eta)\equiv\\{(l,k):\eta N\leq k\leq(1-\eta)N,k+\eta N\leq
l\leq N\\}$ for all $0\leq k<l\leq N$ and $N\geq 1$, where $0<\eta<1$. A
random array $\\{x_{k,N}\\}_{k\geq 1,N\geq 1}$ is called strong smooth if
there exists a sequence of constants $d_{l,k,N}>0$ and a sequence of
$\sigma$-fields $\mathcal{F}_{k,N}$ (define
$\mathcal{F}_{0,N}=\sigma\\{\phi,\Omega\\}$, the trivial $\sigma$-field) such
that

  * (a)

For some $m_{0}>0$ and $C>0$,
$\inf_{(l,k)\in\Omega_{N}(\eta)}d_{l,k,N}\geq\eta^{m_{0}}/C$ as
$N\rightarrow\infty$,

    * (i)

$\lim_{\eta\rightarrow
0}\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{l=(1-\eta)N}^{N}(d_{l,0,N})^{-1}=0$,

    * (ii)

$\lim_{\eta\rightarrow 0}\lim_{N\rightarrow\infty}\frac{1}{N}\max_{0\leq
k\leq(1-\eta)N}\sum_{l=k+1}^{k+\eta N}(d_{l,k,N})^{-1}=0$, and

    * (iii)

$\limsup_{N\rightarrow\infty}\frac{1}{N}\max_{0\leq k\leq
N-1}\sum_{l=k+1}^{N}(d_{l,k,N})^{-1}<\infty$.

  * (b)

$x_{k,N}$ is adapted to $\mathcal{F}_{k,N}$, and conditional on
$\mathcal{F}_{k,N}$, $(x_{l,N}-x_{k,N})/d_{l,k,N}$ has a density
$h_{l,k,N}(x)$ which is uniformly bounded by a constant $K$ and

$\displaystyle\lim_{\kappa\rightarrow
0}\lim_{N\rightarrow\infty}\sup_{(l,k)\in\Omega_{N}[\kappa^{1/(2m_{0})}]}\sup_{|u|\leq\kappa}|h_{l,k,N}(u)-h_{l,k,N}(0)|=0.$

######  Lemma A.1.

Let’s define $\eta_{i}=(\xi_{i},\epsilon_{i})^{\prime}$ where
$\mathbb{E}(\eta_{0})=0$ and $\mathbb{E}||\eta_{0}||^{2}<\infty$. Let
$\mathcal{F}_{s}=\sigma(\eta_{s},\eta_{s-1},...)$, $\Lambda(.)$ is a real
function of its components where $\mathbb{E}(\Lambda(\eta_{1}))=0$, and $g(x)$
is a bounded function such that $\int_{\mathbb{R}}|g(x)|dx<\infty$.

  * (i)

For a constant $H_{0}$, we have

$\displaystyle\sup_{x}\mathbb{E}\Big{|}\sum_{k=1}^{N}g\Big{[}\frac{x_{k}-x}{h}\Big{]}\Big{|}^{m}\leq
H_{0}^{m}(m+1)!\,\Big{(}\sqrt{N}\lambda^{d}h\Big{)}^{m},\quad m\geq 1.$ (A.1)

  * (ii)

As $\\{\sqrt{N}\lambda^{d}h\\}^{-1}\rightarrow 0$ and
$N\lambda\rightarrow\infty$, we have

$\displaystyle\sup\limits_{x}\mathbb{E}\Big{|}\sum_{k=1}^{N}\Lambda(\eta_{k-j})g\Big{[}\frac{x_{k}-x}{h}\Big{]}\Big{|}^{2}$
$\displaystyle\leq
C\,\mathbb{E}\Lambda^{2}(\eta_{1})\sqrt{N}\lambda^{d}h\,\Big{\\{}1+h\sqrt{j}\Big{\\}}.$
(A.2)

  * (iii)

If we define $u_{k}:=\sum_{j=0}^{\infty}\psi_{j}\eta_{k-j}$, then

$\displaystyle\mathbb{E}\Big{|}\sum_{k=1}^{N}u_{k}\,g\Big{(}\frac{x_{k}}{h}\Big{)}\Big{|}^{2}\leq
C\,\mathbb{E}||\eta_{1}||^{2}\sqrt{N}\lambda^{d}h.$ (A.3)

  * (iv)

If we define
$u_{k,m_{0}}:=\sum_{j=m_{0}}^{\infty}\psi_{j}\Lambda^{\prime}_{k-j}$ where
$\psi_{j}=(\psi_{1j},\psi_{2j})$ and
$\Lambda_{i}=(\Lambda_{1}(\eta_{i}),\Lambda_{2}(\eta_{i}))$, then we have

$\displaystyle\sup_{x}\mathbb{E}\Big{|}\sum_{k=1}^{N}u_{k,m_{0}}\,g\Big{[}x_{k}-\frac{x}{h}\Big{]}\Big{|}^{2}\leq
C\,\mathbb{E}||\Lambda_{1}||^{2}\sqrt{N}\lambda^{d}h\,\Big{\\{}\sum_{j=m_{0}}^{\infty}j^{1/4}(|\psi_{1j}|+|\psi_{2j}|)\Big{\\}}^{2}.$
(A.4)

###### Proof.

For the proof of expression (A.1), see Lemma 5.1. of Chan and Wang (2014). To
prove (A.2), note that some of the details are very similar to the ones given
for the proof of expression (8.11) in Lemma 8.2. of Wang and Phillips (2016),
and therefore we only give an outline for the different parts as follows. Let

$\displaystyle\Delta_{N}$
$\displaystyle\equiv\Big{|}\sum_{k=1}^{N}\Lambda(\eta_{k-j})g\Big{[}\frac{x_{k}-x}{h}\Big{]}\Big{|}^{2}\leq
2\Big{|}\sum_{k=A_{0}}^{N}\Lambda(\eta_{k-j})g\Big{[}\frac{x_{k}-x}{h}\Big{]}\Big{|}^{2}+C\Big{(}\sum_{k=1}^{A_{0}}|\Lambda(\eta_{k-j})|\Big{)}^{2}$
$\displaystyle=2\Big{(}\sum_{k=A_{0}}^{N}\sum_{l=1,|k-l|<A_{0}}^{N}+2\sum_{k=A_{0}}^{N-1}\sum_{l=k+A_{0}}^{N}\Big{)}\Lambda(\eta_{k-j})\Lambda(\eta_{l-j})$
$\displaystyle\quad\times
g\Big{[}\frac{x_{k}-x}{h}\Big{]}g\Big{[}\frac{x_{l}-x}{h}\Big{]}+C\Big{(}\sum_{k=1}^{A_{0}}|\Lambda(\eta_{k-j})|\Big{)}^{2}$
$\displaystyle=\Delta_{1N}+\Delta_{2N}+\Delta_{3N},$

where $A_{0}$ is a positive constant. Then, we have

$\displaystyle\sup\limits_{x}\mathbb{E}|\Delta_{1N}|$ $\displaystyle\leq
C\,\mathbb{E}\Lambda^{2}(\eta_{1})\sqrt{N}\lambda^{d}h,$
$\displaystyle\sup\limits_{x}\mathbb{E}|\Delta_{2N}|$ $\displaystyle\leq
C\,\mathbb{E}\Lambda^{2}(\eta_{1})h^{2}\sum_{k=A_{0}}^{N-1}d_{k}^{-1}\Big{(}\sum_{l=k+A_{0}}^{N\land(k+j)}d_{l-k}^{-1}+\sum_{k=0}^{j}|\phi_{N}(k)|\sum_{l=k+j}^{N}d_{l-k}^{-2}\Big{)}$
$\displaystyle\leq
C\,\mathbb{E}\Lambda^{2}(\eta_{1})\sqrt{N}\lambda^{d}h^{2}\Big{(}\frac{j}{d_{j}}+\sum_{k=0}^{j}|\phi_{j}(k)|\sum_{l=k+j}^{N}\frac{\lambda_{l-k}^{2d}}{l-k}\Big{)}$
$\displaystyle\leq
C\,\mathbb{E}\Lambda^{2}(\eta_{1})\sqrt{N}\lambda^{d}h^{2}\Big{(}\frac{j}{d_{j}}+\sum_{k=0}^{j}|\phi_{j}(k)|\Big{)}$
$\displaystyle\leq
C\,\mathbb{E}\Lambda^{2}(\eta_{1})\sqrt{N}\lambda^{d}h^{2}\Big{(}\frac{\lambda_{j}^{d}}{\sqrt{j}}j+\lambda_{j}^{-d}\Big{)}$
$\displaystyle\leq
C\,\mathbb{E}\Lambda^{2}(\eta_{1})\sqrt{N}\lambda^{d}h^{2}\sqrt{j}\quad\text{for}\quad
j\geq 1,\quad\text{and}$ $\displaystyle\sup\limits_{x}\mathbb{E}|\Delta_{3N}|$
$\displaystyle\leq C\,\mathbb{E}\Lambda^{2}(\eta_{1}).$

Eventually, we can conclude expression (A.2). Expression (A.3) easily follows
from expression (A.2), and expression (A.4) can be followed by a similar path
from the proof of expression (8.13) in Lemma 8.2. of Wang and Phillips (2016).
∎

### A.2 Proofs of Theorems

Proof of Theorem 4.1. First, in order to establish the asymptotic distribution
of $[\hat{f}(x)-f(x)]$, we decompose it as follows:

$\displaystyle\hat{f}(x)-f(x)=\frac{\sum_{k=1}^{N}u_{k}K[(x_{k}-x)/h]}{\sum_{k=1}^{N}K[(x_{k}-x)/h]}+\frac{\sum_{k=1}^{N}[f(x_{k})-f(x)]K[(x_{k}-x)/h]}{\sum_{k=1}^{N}K[(x_{k}-x)/h]}.$
(A.5)

Then, only the first term in expression (A.5) determines the asymptotic
distribution of $[\hat{f}(x)-f(x)]$ as the second term is negligible under
mild conditions. To prove the self-normalized expression (10), we let
$\Theta_{1N}:=\sum_{k=1}^{N}u_{k}K[\frac{x_{k}-x}{h}]$,
$\Theta_{2N}:=\sum_{k=1}^{N}(f(x_{k})-f(x))K[\frac{x_{k}-x}{h}]$, and
$\Theta_{3N}:=1/\sum_{k=1}^{N}K[\frac{x_{k}-x}{h}]$. Now, we have

$\displaystyle\Big{\\{}h\sum_{k=1}^{N}K_{h}(x_{k}-x)\Big{\\}}^{1/2}\Big{(}\hat{f}(x)-f(x)\Big{)}$

$\displaystyle\qquad=\frac{\sum_{k=1}^{N}u_{k}K[(x_{k}-x)/h]}{\\{\sum_{k=1}^{N}K[(x_{k}-x)/h]\\}^{1/2}}+\frac{\sum_{k=1}^{N}[f(x_{k})-f(x)]K[(x_{k}-x)/h]}{\\{\sum_{k=1}^{N}K[(x_{k}-x)/h]\\}^{1/2}}$
$\displaystyle\qquad=\frac{\Theta_{1N}}{\Theta_{3N}^{-1/2}}+\Theta_{2N}\sqrt{\Theta_{3N}}=\frac{\\{\sqrt{N}\lambda^{d}h\\}^{-1/2}\,\Theta_{1N}}{\\{\sqrt{N}\lambda^{d}h\\}^{-1/2}\,\Theta_{3N}^{-1/2}}+\Theta_{2N}\,\,\sqrt{\Theta_{3N}}$
$\displaystyle\qquad\rightarrow_{D}\frac{d_{0}\,N(0,1)\,L_{B}^{1/2}(1,0)}{\sqrt{d_{1}}\,L_{B}^{1/2}(1,0)}=\frac{d_{0}}{\sqrt{d_{1}}}N(0,1)=:N(0,\sigma^{2}),$
(A.6)

where $\sigma^{2}=d_{0}^{2}/d_{1}$. Note that we have used the fact that
$\Theta_{2N}\sqrt{\Theta_{3N}}\rightarrow_{P}0$ from Proposition A.8. Also,
the last line in (A.6) comes from (a) Slutsky’s theorem, (b)
$\\{\sqrt{N}\lambda^{d}h\\}^{-1}\sum_{k=1}^{\lfloor
Nt\rfloor}K[\frac{x_{k}-x}{h}]$
$\rightarrow_{D}d_{1}\,L_{B}(t,0)$, and (c) the following result:

$\displaystyle\Big{\\{}\sqrt{N}\lambda^{d}h\Big{\\}}^{-1/2}\sum_{k=1}^{N}K\Big{[}\frac{x_{k}-x}{h}\Big{]}u_{k}\rightarrow_{D}d_{0}\,N(0,1)\,L_{B}^{1/2}(1,0).$
(A.7)

The convergence in distribution in (A.7) comes from Proposition A.9 part (iv).
Expression (10) easily implies expression (9) and the fact that

$\displaystyle\Big{\\{}\sqrt{N}\lambda^{d}h\Big{\\}}^{1/2}\Big{(}\hat{f}(x)-f(x)\Big{)}$
$\displaystyle=\Big{\\{}\sqrt{N}\lambda^{d}h\Big{\\}}^{1/2}\frac{\\{h\sum_{k=1}^{N}K_{h}(x_{k}-x)\\}^{1/2}(\hat{f}(x)-f(x))}{\\{h\sum_{k=1}^{N}K_{h}(x_{k}-x)\\}^{1/2}}$
$\displaystyle\rightarrow_{D}\frac{d_{0}}{\sqrt{d_{1}}}\,L_{B}^{-1/2}(1,0)\,N(0,1),$

thus the proof is complete. $\square$

Proof of Theorem 5.1. First, $T_{N}$ can be decomposed into three terms under
$H_{0}$, i.e. $y_{k}=g(x_{k},\theta_{0})+u_{k}$, as follows

$\displaystyle T_{N}$
$\displaystyle=\int_{\mathbb{R}}\Big{\\{}\sum_{k=1}^{N}K\Big{[}\frac{x_{k}-x}{h}\Big{]}\Big{[}g(x_{k},\theta_{0})+u_{k}-g(x_{k},\hat{\theta}_{N})\Big{]}\Big{\\}}^{2}\pi(x)dx$
$\displaystyle=T_{N}^{\text{I}}+T_{N}^{\text{II}}+T_{N}^{\text{III}},\quad\text{such
that}$ $\displaystyle T_{N}^{\text{I}}$
$\displaystyle:=\int_{\mathbb{R}}\Big{\\{}\sum_{k=1}^{N}K\Big{[}\frac{x_{k}-x}{h}\Big{]}u_{k}\Big{\\}}^{2}\pi(x)dx,$

$\displaystyle T_{N}^{\text{II}}$
$\displaystyle:=\int_{\mathbb{R}}\Big{\\{}\sum_{k=1}^{N}K\Big{[}\frac{x_{k}-x}{h}\Big{]}\Big{(}g(x_{k},\theta_{0})-g(x_{k},\hat{\theta}_{N})\Big{)}\Big{\\}}^{2}\pi(x)dx,\quad\text{and}$
$\displaystyle T_{N}^{\text{III}}$
$\displaystyle:=2\int_{\mathbb{R}}\Big{\\{}\sum_{1\leq k<j\leq
N}K\Big{[}\frac{x_{k}-x}{h}\Big{]}K\Big{[}\frac{x_{j}-x}{h}\Big{]}u_{k}\Big{(}g(x_{j},\theta_{0})-g(x_{j},\hat{\theta}_{N})\Big{)}\Big{\\}}\pi(x)dx.$

Based on Assumption 5.2, there exists $g_{1}(x)$ such that for each
$\theta,\theta_{0}\in\Omega_{0}$, we have $|g(x,\theta)-g(x,\theta_{0})|\leq
C||\theta-\theta_{0}||g_{1}(x)$. Therefore,

$\displaystyle T_{N}^{\text{II}}\leq
C||\hat{\theta}_{N}-\theta_{0}||^{2}\int_{\mathbb{R}}\Big{\\{}\sum_{k=1}^{N}K\Big{[}\frac{x_{k}-x}{h}\Big{]}g_{1}(x_{k})\Big{\\}}^{2}\pi(x)dx=o_{P}\Big{(}\sqrt{N}\lambda^{d}h\Big{)}.$
(A.8)

Expression (A.8) comes from (a) assumption 5.3,
$||\hat{\theta}_{N}-\theta_{0}||=o_{P}(\\{\sqrt{N}\lambda^{d}h\\}^{-1/2})$ and
(b) the following result:

$\displaystyle\Big{\\{}\sqrt{N}\lambda^{d}h\Big{\\}}^{-2}\int_{\mathbb{R}}\Big{\\{}\sum_{k=1}^{N}K\Big{[}\frac{x_{k}-x}{h}\Big{]}m(x_{k})\Big{\\}}^{2}\pi(x)dx\rightarrow_{D}\frac{d_{(1)}^{2}}{d_{1}}L_{B}^{2}(1,0),$
(A.9)

where $d^{2}_{(1)}=\int_{\mathbb{R}}m^{2}(x)\pi(x)dx\int_{\mathbb{R}}K(s)ds$
and $d_{1}=\int_{\mathbb{R}}K(s)ds$. Expression (A.9) can be obtained by a
similar path from the expression (7.12) in Proposition 7.3 of Wang and
Phillips (2016). Also, note that

$\displaystyle\sum_{1\leq k<j\leq
N}\tilde{u}_{1k}\tilde{u}_{1j}\Big{\\{}\int_{\mathbb{R}}K\Big{[}\frac{x_{k}-x}{h}\Big{]}K\Big{[}\frac{x_{j}-x}{h}\Big{]}\pi(x)dx\Big{\\}}=o_{P}\Big{(}\sqrt{N}\lambda^{d}h\Big{)},$
(A.10)

which can be followed by a similar path from the proof of expression (8.18) in
Lemma 8.3 of Wang and Phillips (2016). Therefore, we can conclude
$T_{N}^{\text{III}}=o_{P}(\sqrt{N}\lambda^{d}h)$. Now, it suffices to show
$\\{\sqrt{N}\lambda^{d}h\\}^{-1}T_{N}^{\text{I}}\rightarrow_{D}d^{2}_{(0)}L_{B}(1,0)$.
For a fixed $A>0$, we define

$\displaystyle\hat{\eta_{i}}$ $\displaystyle=\eta_{i}\,I(||\eta_{i}||\leq
A),\quad\tilde{\eta}_{i}=\hat{\eta}_{i}-\mathbb{E}(\hat{\eta}_{i}),\quad\mathring{\eta}_{i}=\eta_{i}-\tilde{\eta}_{i},\quad\text{and}$
$\displaystyle\tilde{u}_{k}$
$\displaystyle=\sum_{j=0}^{\infty}\psi_{j}\tilde{\eta}_{k-j},\quad\mathring{u}_{k}=\sum_{j=0}^{\infty}\psi_{j}\mathring{\eta}_{k-j}.$

From expression (A.4) of Lemma A.1, we have

$\displaystyle\mathbb{E}\int_{\mathbb{R}}\Big{\\{}\sum_{k=1}^{N}K\Big{[}\frac{x_{k}-x}{h}\Big{]}\mathring{u}_{k}\Big{\\}}^{2}\pi(x)dx\leq
C\,\mathbb{E}||\mathring{\eta}_{1}||^{2}\,\sqrt{N}\lambda^{d}h\leq
C\,\mathbb{E}||\eta_{1}||^{2}I(||\eta_{1}||>A)\sqrt{N}\lambda^{d}h,$

where $\mathbb{E}(\mathring{\eta}_{1})=0$. Note that
$\mathbb{E}||\eta_{1}||^{2}I(||\eta_{1}||>A)\rightarrow 0$ as
$A\rightarrow\infty$ and $u_{k}=\tilde{u}_{k}+\mathring{u}_{k}$, therefore it
suffices to show

$\displaystyle\Big{\\{}\sqrt{N}\lambda^{d}h\Big{\\}}^{-1}T_{N}^{\text{I}}:=\Big{\\{}\sqrt{N}\lambda^{d}h\Big{\\}}^{-1}\int_{\mathbb{R}}\Big{\\{}\sum_{k=1}^{N}K\Big{[}\frac{x_{k}-x}{h}\Big{]}\tilde{u}_{k}\Big{\\}}^{2}\pi(x)dx\rightarrow_{D}d^{2}_{(0)}L_{B}(1,0).$

Let $\tilde{u}_{1k}=\sum_{j=0}^{m_{0}}\psi_{j}\tilde{\eta}_{k-j}$ and
$\tilde{u}_{2k}=\tilde{u}_{k}-\tilde{u}_{1k}$. Then, we have
$T_{N}^{\text{I}}=T^{\text{I}(a)}_{N}+T^{\text{I}(b)}_{N}+T^{\text{I}(c)}_{N}$
where

$\displaystyle T^{\text{I}(a)}_{N}$
$\displaystyle=\int_{\mathbb{R}}\Big{\\{}\sum_{k=1}^{N}K\Big{[}\frac{x_{k}-x}{h}\Big{]}\tilde{u}_{1k}\Big{\\}}^{2}\pi(x)dx,$
$\displaystyle T^{\text{I}(b)}_{N}$
$\displaystyle=\int_{\mathbb{R}}\Big{\\{}\sum_{k=1}^{N}\Big{[}\frac{x_{k}-x}{h}\Big{]}\tilde{u}_{2k}\Big{\\}}^{2}\pi(x)dx,\quad\text{and}$
$\displaystyle T^{\text{I}(c)}_{N}$
$\displaystyle=2\int_{\mathbb{R}}\Big{\\{}\sum_{1\leq k<j\leq
N}K\Big{[}\frac{x_{k}-x}{h}\Big{]}K\Big{[}\frac{x_{j}-x}{h}\Big{]}\tilde{u}_{1k}\tilde{u}_{2k}\Big{\\}}\,\pi(x)dx.$

Now based on expression (A.4) in Lemma A.1, we have

$\displaystyle\mathbb{E}(T^{\text{I}(b)}_{N})$ $\displaystyle\leq
C\,\sup_{x}\mathbb{E}\Big{\\{}\sum_{k=1}^{N}K\Big{[}\frac{x_{k}-x}{h}\Big{]}\tilde{u}_{2k}\Big{\\}}^{2}$
$\displaystyle\leq
C\,\mathbb{E}||\eta_{1}||^{2}\sqrt{N}\lambda^{d}h\,\Big{\\{}\sum_{k=m_{0}}^{\infty}k^{1/4}(|\psi_{1k}|+|\psi_{2k}|)\Big{\\}}^{2},\quad\text{as}\quad
N\rightarrow\infty\quad\text{and}\quad m_{0}\rightarrow 0.$

Under Assumption 4.1,
$\sum_{k=m_{0}}^{\infty}k^{1/4}(|\psi_{1k}|+|\psi_{2k}|)<\infty$. Therefore,
we conclude $T^{\text{I}(b)}_{N}=o_{P}(\sqrt{N}\lambda^{d}h)$. Based on
expression (A.10), $T^{\text{I}(c)}_{N}=o_{P}(\sqrt{N}\lambda^{d}h)$. We can
furtherly write $T^{\text{I}(a)}_{N}$ as

$\displaystyle T^{\text{I}(a)}_{N}$
$\displaystyle=\int_{\mathbb{R}}\Big{\\{}\sum_{k=1}^{N}K\Big{[}\frac{x_{k}-x}{h}\Big{]}\tilde{u}_{1k}\Big{\\}}^{2}\pi(x)dx=\mathbb{E}(\tilde{u}_{10}^{2})\sum_{k=1}^{N}\int_{\mathbb{R}}K^{2}\Big{[}\frac{x_{k}-x}{h}\Big{]}\pi(x)dx$
$\displaystyle\quad+\sum_{k=1}^{N}(\tilde{u}^{2}_{1k}-\mathbb{E}(\tilde{u}_{1k}^{2}))\int_{\mathbb{R}}K^{2}\Big{[}\frac{x_{k}-x}{h}\Big{]}\pi(x)dx$

$\displaystyle\qquad\quad+2\sum_{1\leq k<j\leq
N}\tilde{u}_{1k}\tilde{u}_{1j}\int_{\mathbb{R}}K\Big{[}\frac{x_{k}-x}{h}\Big{]}K\Big{[}\frac{x_{j}-x}{h}\Big{]}\pi(x)dx$
$\displaystyle\qquad=R^{\text{I}(a)}_{N}+R^{\text{II}(a)}_{N}+R^{\text{III}(a)}_{N}.$

From

$\displaystyle\sum_{k=1}^{N}\Big{(}\tilde{u}^{2}_{1k}-\mathbb{E}(\tilde{u}^{2}_{1k})\Big{)}\int_{\mathbb{R}}K^{2}\Big{[}\frac{x_{k}-x}{h}\Big{]}\pi(x)dx=O_{P}\Big{(}\\{\sqrt{N}\lambda^{d}h\\}^{1/2}\Big{)},$

which can be followed by a similar path from the expression (8.17) in Lemma
8.3 of Wang and Phillips (2016) and expression (A.10), we can conclude
$|R^{\text{II}(a)}_{N}|+|R^{\text{III}(a)}_{N}|=o_{P}(\sqrt{N}\lambda^{d}h)$.
Eventually, by virtue of
$\mathbb{E}(\tilde{u}_{10}^{2})\rightarrow\mathbb{E}(\tilde{u}_{0}^{2})$, we
only need to prove

$\displaystyle\Big{\\{}\sqrt{N}\lambda^{d}h\Big{\\}}^{-1}R^{\text{I}(a)}_{N}\rightarrow_{D}\mathbb{E}(\tilde{u}_{0}^{2})\int_{\mathbb{R}}K^{2}(x)dx\int_{\mathbb{R}}\pi(s)ds\,L_{B}(1,0).$
(A.11)

If we define
$\int_{\mathbb{R}}g_{h}(y)dy=\int_{\mathbb{R}}K^{2}(z)dz\int_{\mathbb{R}}\pi(x)dx$
such that $|g_{h}(y)|\leq C\int_{\mathbb{R}}|\pi(x)|dx<\infty$, then (A.11)
could be followed from the proof of Theorem 2.1 in Wang and Phillips (2009a),
and we omit it. Therefore, the proof is complete. $\square$

Proof of Theorem 5.2. The proof is similar to the proof of Theorem 3.2 in Wang
and Phillips (2016). Therefore, we give an outline. We first decompose $T_{N}$
under $H_{A}$, i.e. $y_{k}=g(x_{k},\theta_{0})+\rho_{N}m(x_{k})+u_{k}$ as
follows:

$\displaystyle T_{N}$
$\displaystyle=\int_{\mathbb{R}}\Big{\\{}\sum_{k=1}^{N}K\Big{[}\frac{x_{k}-x}{h}\Big{]}\Big{[}g(x_{k},\theta_{0})+\rho_{N}m(x_{k})+u_{k}-g(x_{k},\hat{\theta}_{N})\Big{]}\Big{\\}}^{2}\pi(x)dx$
$\displaystyle=T_{N}^{\text{I}}+T_{N}^{\text{II}}+T_{N}^{\text{III}}+2\rho_{N}T_{N}^{\text{IV}}+\rho_{N}^{2}T_{N}^{\text{V}},$

where

$\displaystyle T_{N}^{\text{IV}}$
$\displaystyle:=\int_{\mathbb{R}}\Big{\\{}\sum_{1\leq k<j\leq
N}K\Big{[}\frac{x_{k}-x}{h}\Big{]}K\Big{[}\frac{x_{j}-x}{h}\Big{]}m(x_{k})\Big{[}u_{j}-\Big{(}g(x_{j},\hat{\theta}_{N})-g(x_{j},\theta_{0})\Big{)}\Big{]}\Big{\\}}\,\pi(x)dx,$
$\displaystyle T_{N}^{\text{V}}$
$\displaystyle:=\int_{\mathbb{R}}\Big{\\{}\sum_{k=1}^{N}K\Big{[}\frac{x_{k}-x}{h}\Big{]}m(x_{k})\Big{\\}}^{2}\pi(x)dx,$

such that
$|T_{N}^{\text{IV}}|\leq[T_{N}^{\text{I}}+T_{N}^{\text{II}}+T_{N}^{\text{III}}]^{1/2}\times[T_{N}^{\text{V}}]^{1/2}$
by Hölder’s inequality. The remained terms have been already defined in the
proof of Theorem 5.1. From the proof of Theorem 5.1,
$T_{N}^{\text{I}}+T_{N}^{\text{II}}+T_{N}^{\text{III}}=O_{P}(\sqrt{N}\lambda^{d}h)$.
Following Proposition A.9 part (iv),
$\\{\sqrt{N}\lambda^{d}h\\}^{-1}\sum_{k=1}^{N}K(\frac{x_{k}}{h})$
$\rightarrow_{D}L_{B}(1,0)$ under the assumption of
$\int_{\mathbb{{R}}}K(x)dx=1$. Therefore, from continuous mapping theorem and
expression (A.1), we have
$\\{\sqrt{N}\lambda^{d}h\\}^{-2}T_{N}^{\text{V}}\rightarrow_{D}d_{(1)}^{2}\,L_{B}^{2}(1,0)$,
where $d_{(1)}^{2}=\int_{\mathbb{R}}m^{2}(x)\pi(x)dx$, and
$P(0<L^{2}_{B}(1,0)<\infty)=1$. We conclude that for any $T_{0}>0$ and as
$N\rightarrow\infty$, the following holds

$\displaystyle\lim\limits_{N\rightarrow\infty}P\Big{(}\\{\sqrt{N}\lambda^{d}h\\}^{-1}T_{N}\geq
T_{0}\Big{)}\geq\lim\limits_{N\rightarrow\infty}P\Big{(}\\{\sqrt{N}\lambda^{d}h\\}^{-2}T_{N}^{\text{V}}\geq\epsilon_{N}^{-1/4}\Big{)}=1,$

under the assumption of
$\epsilon_{N}:=\sqrt{N}\lambda^{d}h\rho_{N}^{2}\rightarrow\infty$.

Note that
$\\{\sqrt{N}\lambda^{d}h\\}^{-2}T_{N}^{V}\rightarrow_{D}d_{(1)}^{2}\,L_{B}^{2}(1,0)>0$
a.s., which confirms the divergence of $\\{\sqrt{N}\lambda^{d}h\\}^{-1}T_{N}$
and the test consistency under $H_{A}$. $\square$

### A.3 Propositions and Their Proofs

######  Proposition A.1.

Let $x_{k,N}=x_{k}/d_{N}$. A random array $\\{x_{k,N}\\}_{k\geq 1,N\geq 1}$ is
called strong smooth.

###### Proof.

We verify this proposition from the Definition A.1. Assume
$\\{x_{k,N}\\}_{k\geq 1,N\geq 1}$ is a random triangular array such that
$x_{0,N}\equiv 0$. Let $d_{l,k}\equiv\lambda_{l-k}^{-d}\,(l-k)^{1/2}$,
therefore

$\displaystyle
d_{l,k,N}\equiv\frac{d_{l,k}}{d_{N}}\propto\Big{(}\frac{\lambda_{l-k}}{\lambda_{N}}\Big{)}^{-d}\,\sqrt{\frac{l-k}{N}},\qquad
N\lambda_{N}\rightarrow\infty.$

Recall that $\lambda_{N}\equiv\lambda>0$ is a sample size dependent parameter.

  * (a)

$\inf_{(l,k)\in\Omega_{N}(\eta)}d_{l,k,N}\equiv\inf_{(l,k)\in\Omega_{N}(\eta)}(\lambda_{l-k}/\lambda_{N})^{-d}\sqrt{(l-k)/N}\geq\sqrt{\frac{\eta
N}{N}}\lambda_{\eta N/N}^{d}>0$.

(i) $\displaystyle\lim_{\eta\rightarrow
0}\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{l=(1-\eta)N}^{N}\Big{\\{}\Big{(}\frac{\lambda_{l}}{\lambda_{N}}\Big{)}^{-d}\sqrt{l/N}\Big{\\}}^{-1}$
$\displaystyle\quad\leq\lim_{\eta\rightarrow
0}\lim_{N\rightarrow\infty}\frac{1}{\sqrt{N}\lambda_{N}^{d}}\int_{(1-\eta)N}^{N}\frac{\lambda_{l}^{d}}{\sqrt{l}}dl$
$\displaystyle\quad\leq\lim_{\eta\rightarrow
0}\lim_{N\rightarrow\infty}\frac{1}{\sqrt{N}\lambda_{N}^{d}}\Big{\\{}\sqrt{N}\lambda_{N}^{d}-\sqrt{(1-\eta)N}\lambda_{(1-\eta)N}^{d}\Big{\\}}=0$
(ii) $\displaystyle\lim_{\eta\rightarrow
0}\lim_{N\rightarrow\infty}\frac{1}{N}\max_{0\leq
k\leq(1-\eta)N}\sum_{l=k+1}^{k+\eta
N}\Big{\\{}\Big{(}\frac{\lambda_{l-k}}{\lambda_{N}}\Big{)}^{-d}\sqrt{(l-k)/N}\Big{\\}}^{-1}$
$\displaystyle\quad\leq\lim_{\eta\rightarrow
0}\lim_{N\rightarrow\infty}\frac{1}{\sqrt{N}\lambda_{N}^{d}}\max\limits_{0\leq
k\leq(1-\eta)N}\int_{k+1}^{k+\eta N}\frac{\lambda_{l-k}^{d}}{\sqrt{l-k}}dl$

  * $\displaystyle=\lim_{\eta\rightarrow 0}\lim_{N\rightarrow\infty}\frac{1}{\sqrt{N}\lambda_{N}^{d}}\Big{\\{}\sqrt{\eta N}\lambda_{\eta N}^{d}-1\Big{\\}}=0$ (iii) $\displaystyle\limsup\limits_{N\rightarrow\infty}\frac{1}{N}\max\limits_{0\leq k\leq N-1}\sum\limits_{l=k+1}^{N}\Big{\\{}\Big{(}\frac{\lambda_{l-k}}{\lambda_{N}}\Big{)}^{-d}\sqrt{(l-k)/N}\Big{\\}}^{-1}$ $\displaystyle\quad\leq\limsup_{N\rightarrow\infty}\frac{1}{\sqrt{N}\lambda_{N}^{d}}\max\limits_{0\leq k\leq N-1}\int_{k+1}^{N}\frac{\lambda_{l-k}^{d}}{\sqrt{l-k}}dl$ $\displaystyle\quad\leq\limsup\limits_{N\rightarrow\infty}\frac{1}{\sqrt{N}\lambda_{N}^{d}}\max\limits_{0\leq k\leq N-1}\Big{\\{}\sqrt{N-k}\lambda_{N-k}^{d}-1\Big{\\}}$ $\displaystyle\quad=\limsup\limits_{N\rightarrow\infty}\Big{\\{}1-\frac{1}{\sqrt{N}\lambda_{N}^{d}}\Big{\\}}=1<\infty.$

  * (b)

If we define
$f_{l,k,N}(t)=\mathbb{E}\Big{(}e^{it[x_{l,N}-x_{k,N}]/d_{l,k,N}}\Big{)}$, then

$\displaystyle\sup\limits_{x\in\mathbb{R}}\Big{|}h_{l,k,N}(x)-n(x)\Big{|}\leq
C\int_{\mathbb{R}}\Big{|}f_{l,k,N}(t)-e^{-t^{2}/2}\Big{|}dt\rightarrow 0,$

where $n(x)=e^{-x^{2}/2}\big{/}\sqrt{2\pi}$ is the density of standard normal
(see proof of Corollary 2.2 in Wang and Phillips (2009a) for a similar
argument). All the conditions given in Definition A.1 hold. Thus, ${x_{k,N}}$
is a strong smooth array. The alternative way to prove this proposition is the
following. Firstly, note that

$\displaystyle\frac{\lambda_{N}^{d}}{\sqrt{N}}\sum_{k=1}^{N}X_{d,\lambda_{N}}(k)=\frac{1}{\sqrt{N}}\sum_{k=1}^{N}\zeta(k)+o_{P}(1),$
(A.12)

where $\\{\zeta(j)\\}_{j\in\mathbb{Z}}$ is a sequence of i.i.d. random
variables with $\mathbb{E}(\zeta_{0})=0$ and $\mathbb{E}(\zeta^{2}_{0})=1$;
see expression (A.7) in Sabzikar et al. (2020). Secondly, the triangular array
$N^{-1/2}\sum_{k=1}^{N}\xi(k)$ satisfies the strong smooth conditions; see
Wang and Phillips (2009a). This together with (A.12) completes the proof of
Proposition A.1. ∎

######  Proposition A.2.

Let

$\displaystyle S_{N}(t)=\frac{1}{[\sqrt{N}h]^{1/2}}\sum\limits_{k=1}^{\lfloor
Nt\rfloor}u_{k}K\Big{[}\frac{x_{k}-x}{h}\Big{]},\quad\text{and}\quad\psi_{N}(t)=\frac{1}{\sqrt{N}h}\sum\limits_{k=1}^{\lfloor
Nt\rfloor}u^{2}_{k}K^{2}\Big{[}\frac{x_{k}-x}{h}\Big{]}.$

For any fixed $0\leq t\leq 1$, $S_{N}(t)$, $S^{2}_{N}(t)$ and $\psi_{N}(t)$
are uniformly integrable.

###### Proof.

The proof is similar to the Proposition 7.3 in Wang and Phillips (2009b);
hence, we only give an outline. If we let

$\displaystyle\psi_{N}^{\prime\prime}(t)=\frac{1}{\sqrt{N}h}\sum_{k=1}^{\lfloor
Nt\rfloor}K^{2}\Big{[}\frac{x_{k}-x}{h}\Big{]}\mathbb{E}(u_{k}^{2}),$

for $0\leq t\leq 1$ and define $r_{y}(t):=K^{2}(\frac{y}{h}+t)$, then
$\sup_{N}\mathbb{E}[\psi^{\prime\prime}_{N}(t)]^{2}<\infty$ as follows

$\displaystyle\mathbb{E}[\psi^{\prime\prime}_{N}(t)]^{2}$
$\displaystyle=\mathbb{E}\Big{\\{}\frac{1}{\sqrt{N}h}\sum_{k=1}^{\lfloor
Nt\rfloor}K^{2}\Big{[}\frac{x_{k}-x}{h}\Big{]}\mathbb{E}(u_{k}^{2})\Big{\\}}^{2}$
$\displaystyle\leq\frac{C}{Nh^{2}}\Big{\\{}\sum_{k=1}^{N}\mathbb{E}K^{4}\Big{[}\frac{x_{k}-x}{h}\Big{]}+2\sum_{1\leq
k<l\leq
N}\mathbb{E}\Big{\\{}K^{2}\Big{[}\frac{x_{k}-x}{h}\Big{]}K^{2}\Big{[}\frac{x_{l}-x}{h}\Big{]}\Big{\\}}\Big{\\}}$
$\displaystyle\leq\frac{C}{Nh^{2}}\Big{\\{}\sum_{k=1}^{N}\sup_{y}\mathbb{E}r^{2}_{y}\Big{(}\frac{x^{\prime}_{0,k}}{h}\Big{)}+2\sum_{1\leq
k<l\leq
N}\sup_{y}\mathbb{E}r_{y}\Big{(}\frac{x^{\prime}_{0,k}}{h}\Big{)}\sup_{y}\mathbb{E}r_{y}\Big{(}\frac{x^{\prime}_{k,j}}{h}\Big{)}\Big{\\}}$
$\displaystyle\leq\frac{C}{Nh^{2}}\Big{\\{}\sum_{k=1}^{N}\frac{h}{\sqrt{k}}+2\sum_{1\leq
k<l\leq N}\frac{h^{2}}{\sqrt{k}\sqrt{l-k}}\Big{\\}}<\infty,$

uniformly on $N$. The last line could be followed by a similar path from the
proof of Lemma 7.1 part (a) in Wang and Phillips (2009b). Note that
$\sup_{0\leq t\leq
1}\mathbb{E}|\psi_{N}(t)-\psi_{N}^{\prime\prime}(t)|^{2}=o(1)$, which can be
verified by a similar argument given in Proposition A.6. These implies
$\sup_{N}\mathbb{E}[\psi_{N}(t)]^{2}<\infty$, and therefore $\psi_{N}(t)$ is
uniformly integrable. Furtherly, we can write

$\displaystyle|\psi_{N}(t)-S^{2}_{N}(t)|=\frac{2}{\sqrt{N}h}\sum\limits_{1\leq
k<l\leq\lfloor
Nt\rfloor}u_{k}u_{l}K\Big{[}\frac{x_{k}-x}{h}\Big{]}K\Big{[}\frac{x_{l}-x}{h}\Big{]},$

which implies $\sup_{0\leq t\leq 1}\mathbb{E}|\psi_{N}(t)-S^{2}_{N}(t)|=o(1)$
and can be verified by a similar argument given in Proposition A.6. Therefore,
$S^{2}_{N}(t)$ is uniformly integrable, which implies $S_{N}(t)$ is uniformly
integrable. ∎

######  Proposition A.3.

Let $\Theta_{1N}:=\sum_{k=1}^{N}u_{k}K[\frac{x_{k}-x}{h}]$. Then,
$\Theta_{1N}=O_{P}(\\{\sqrt{N}h\\}^{1/2})$.

###### Proof.

Based on the Proposition A.2, uniformly in $N$, we have
$\mathbb{E}S^{2}_{N}(1)\leq C$, where $C$ is a generic constant. Therefore,
based on the Markov’s inequality

$\displaystyle
P(\Theta_{1N}\geq\\{\sqrt{N}h\\}^{1/2})\leq\frac{\mathbb{E}(\Theta^{2}_{1N})}{\sqrt{N}h}=\mathbb{E}S^{2}_{N}(1)\leq
C.$

As a result, $\Theta_{1N}=O_{P}(\\{\sqrt{N}h\\}^{1/2})$. ∎

######  Proposition A.4.

Let $\Theta_{2N}:=\sum_{k=1}^{N}[f(x_{k})-f(x)]K[(x_{k}-x)/h]$. Then,
$\Theta_{2N}=O_{P}(Nh^{\gamma+1}/d_{N})$, where $d_{N}=\sqrt{N}/\lambda^{d}$.

###### Proof.

$\displaystyle\mathbb{E}|\Theta_{2N}|$
$\displaystyle\leq\sum_{k=1}^{N}\mathbb{E}\Big{\\{}|f(x_{k,N}d_{N})-f(x)|K\Big{[}\frac{x_{k,N}d_{N}-x}{h}\Big{]}\Big{\\}}$
$\displaystyle=\sum_{k=1}^{N}\int_{\mathbb{R}}\Big{\\{}|f(d_{N}d_{k,0,N}y)-f(x)|\times
K\Big{(}\frac{d_{N}d_{k,0,N}y}{h}-\frac{x}{h}\Big{)}\Big{\\}}h_{k,0,N}(y)dy$
$\displaystyle\leq\frac{h}{d_{N}}\sum_{k=1}^{N}\frac{1}{d_{k,0,N}}\int_{\mathbb{R}}\\{|f(hy+x)-f(x)|K(y)\\}dy$
$\displaystyle\leq\frac{Nh^{\gamma+1}}{d_{N}}\frac{1}{N}\sum_{k=1}^{N}\frac{1}{d_{k,0,N}}\int_{\mathbb{R}}K(s)f_{1}(s,x)ds\leq
A\frac{Nh^{\gamma+1}}{d_{N}},$

where $A$ is a constant and $\gamma\in(0,1]$. The last line of proof follows
from Assumption 4.3 and note that we have considered $x_{k,N}=d_{k,0,N}y$ and
$d_{N}d_{k,0,N}y=hy+x$. Therefore, $\Theta_{2N}=O_{P}(Nh^{\gamma+1}/d_{N})$. ∎

######  Proposition A.5.

Let $\Theta_{3N}:=\\{\sum_{k=1}^{N}K[(x_{k}-x)/h]\\}^{-1}$. Then,
$\Theta_{3N}=o_{P}(\\{\sqrt{N}\lambda^{d}h\\}^{-1})$.

###### Proof.

From Proposition A.4, we can easily conclude this under the assumption of
$\sqrt{N}\lambda^{d}h\rightarrow\infty$. ∎

######  Proposition A.6.

Assume $h\rightarrow 0$ and $\sqrt{N}\lambda^{d}h\rightarrow\infty$. Let

$\displaystyle T_{iN}(t):=\\{\sqrt{N}\lambda^{d}h\\}^{-1}\sum_{k=1}^{\lfloor
Nt\rfloor}K^{i}\Big{[}\frac{x_{k}-x}{h}\Big{]}\quad\text{and}\quad
T^{\prime}_{iN}(t):=\\{\sqrt{N}\lambda^{d}h\\}^{-1}\sum_{k=1}^{\lfloor
Nt\rfloor}K^{2}\Big{[}\frac{x_{k}-x}{h}\Big{]}u^{2}_{k}.$

Then, $T_{iN}(t)\rightarrow_{D}d_{i}L_{B}(t,0)$ and
$T^{\prime}_{iN}(t)\rightarrow_{D}d_{0}^{2}L_{B}(t,0)$ for $i=1,2$, where
$d_{i}=\int_{\mathbb{R}}K^{i}(s)ds$ and
$d_{0}^{2}=\mathbb{E}(u^{2}_{0})\int_{\mathbb{R}}K^{2}(s)ds$.

###### Proof.

Let’s define $x_{k,N}:=\hat{\xi}_{N}(k/N)$ for $1\leq k\leq N$ such that
$\hat{\xi}_{N}(t)\Rightarrow B(t)$ on $D[0,1]$ following Theorem 4.3 in
Sabzikar and Surgailis (2018). There exists an equivalent process
$\hat{\xi}_{N}(k/N)$ of $\xi_{N}(k/N)$; i.e.
$\hat{\xi}_{N}(k/N)=_{D}\xi_{N}(k/N)$ such that $\sup_{0\leq t\leq
1}|\hat{\xi}_{N}(t)-B(t)|=o_{P}(1)$. Now, apply part (i) of Proposition A.9 by
setting $c_{N}=\frac{d_{N}}{h}$ and $g(t)=K^{i}[t-\frac{x}{h}]$ for $i=1,2$;
then we have

$\displaystyle\sup\limits_{0\leq t\leq
1}\Big{|}\\{\sqrt{N}\lambda^{d}h\\}^{-1}\sum_{k=1}^{\lfloor
Nt\rfloor}K^{i}\Big{[}\frac{d_{N}\hat{\xi}_{N}(k/N)-x}{h}\Big{]}-L_{B}(t,0)\int_{\mathbb{R}}K^{i}(s)ds\Big{|}\rightarrow_{P}0,$
(A.13)

as $c_{N}\rightarrow\infty$, $\frac{c_{N}}{N}\rightarrow 0$, and
$N\rightarrow\infty$. This together with the fact that
$\hat{\xi}_{N}(\frac{k}{N})=_{D}\xi_{N}(\frac{k}{N})=\frac{x_{k}}{d_{N}}$ for
$1\leq k\leq N$ implies for $N\geq 1$, $T_{iN}(t)$ is f.d.d. for
$d_{i}L_{B}(t,0)$. Additionally, $T_{iN}(t)$ is tight for $N\geq 1$ following
Proposition A.7, i.e.

$\displaystyle\max\limits_{1\leq k\leq
N}\Big{|}K^{i}\Big{[}\frac{x_{k}-x}{h}\Big{]}\Big{|}=o_{P}\Big{(}\sqrt{N}\lambda^{d}h\Big{)},$

under the assumption of $\sup_{0\leq t\leq 1}|x_{\lfloor
Nt\rfloor,N}-B(t)|=o_{P}(1)$. Therefore,

$\displaystyle T_{iN}(t)\rightarrow_{D}d_{i}\,L_{B}(t,0)\quad\text{for}\quad
i=1,2.$

Now, in order to prove
$T^{\prime}_{iN}(t):=\\{\sqrt{N}\lambda^{d}h\\}^{-1}\sum_{k=1}^{\lfloor
Nt\rfloor}K^{2}[\frac{x_{k}-x}{h}]u^{2}_{k}\rightarrow_{D}d_{0}^{2}\,L_{B}(t,0)$,
we generally define

$\displaystyle
T_{iN}^{\prime\prime}(t):=\\{\sqrt{N}\lambda^{d}h\\}^{-1}\sum_{k=1}^{\lfloor
Nt\rfloor}K^{2}\Big{[}\frac{x_{k}-x}{h}\Big{]}\mathbb{E}u^{2}_{k},$

and need to show $\sup_{0\leq t\leq
1}\mathbb{E}|T^{\prime}_{iN}(t)-T^{\prime\prime}_{iN}(t)|^{2}=o(1)$. Based on
the Preliminary given in Section A.1, $x_{k}=x^{*}_{0,k}+x^{\prime}_{0,k}$ for
$s=0$, we have:

$\displaystyle\mathbb{E}\Big{[}\Big{|}T^{\prime}_{iN}(t)-T^{\prime\prime}_{iN}(t)\Big{|}^{2}\Big{|}\xi_{0},\xi_{-1},...\Big{]}$
$\displaystyle\quad\leq\Big{\\{}\frac{1}{\sqrt{N}\lambda^{d}h}\Big{\\}}^{2}\sup\limits_{y,1\leq
M\leq
N}\mathbb{E}\Big{[}\sum_{k=1}^{M}K^{2}\Big{[}\frac{y+x^{\prime}_{0,k}}{h}\Big{]}(u^{2}_{k}-\mathbb{E}(u^{2}_{k}))^{2}\Big{]}^{2}\quad\text{a.s.}$
$\displaystyle\quad\leq\Big{\\{}\frac{1}{\sqrt{N}\lambda^{d}h}\Big{\\}}^{2}\sup\limits_{y}\Big{[}\sum_{k=1}^{N}\mathbb{E}r^{2}\Big{(}\frac{x^{\prime}_{0,k}}{h}\Big{)}g^{2}(u_{k})+2\sum\limits_{1\leq
k<l\leq
N}\Big{|}\mathbb{E}r\Big{(}\frac{x^{\prime}_{0,k}}{h}\Big{)}r\Big{(}\frac{x^{\prime}_{0,l}}{h}\Big{)}g(u_{k})g_{1}(u_{l})\Big{|}\Big{]}$

as $g(t)=t^{2}-\mathbb{E}u^{2}_{k}$, $g_{1}(t)=t^{2}-\mathbb{E}u^{2}_{l}$, and
$r(t)=K^{2}(\frac{y}{h}+t)$. Now based on the last part of proof of
proposition 7.2 in Wang and Phillips (2009b) and as
$\\{\sqrt{N}\lambda^{d}h\\}^{-1}\rightarrow 0$, we have

$\displaystyle\mathbb{E}\Big{[}\Big{|}T^{\prime}_{iN}(t)-T^{\prime\prime}_{iN}(t)\Big{|}^{2}|\xi_{0},\xi_{-1},\cdots\Big{]}\rightarrow
0\qquad\text{a.s.}$

Therefore, $T^{\prime}_{iN}(t)$ and $T^{\prime\prime}_{iN}(t)$ have the same
f.d.d.. Since $\mathbb{E}u^{2}_{k}=\mathbb{E}u^{2}_{m_{0}}$ for $k\geq m_{0}$
and $T^{\prime\prime}_{iN}(t)\rightarrow_{D}d_{0}^{2}L_{B}(t,0)$ from the
modification of expression (A.11), we can easily show that
$T^{\prime}_{iN}(t)$ is tight from Proposition A.7. Therefore,
$T^{\prime}_{iN}(t)\rightarrow_{D}d_{0}^{2}L_{B}(t,0)$ on $D[0,1]$. ∎

######  Proposition A.7.

$S_{N}(t)=\\{\sqrt{N}h\\}^{-1/2}\sum_{k=1}^{\lfloor
Nt\rfloor}u_{k}K[\frac{x_{k}-x}{h}]$ is tight on $D[0,1]$.

###### Proof.

According to Theorem 4 of Billingsley (1974), we need to show

$\displaystyle\max\limits_{1\leq k\leq
N}\Big{|}u_{k}K\Big{[}\frac{x_{k}-x}{h}\Big{]}\Big{|}=o_{P}\Big{(}\\{\sqrt{N}h\\}^{1/2}\Big{)}.$
(A.14)

The proof follows from Proposition 7.4 in Wang and Phillips (2009b), and
therefore we give an outline. There exists a sequence of
$\alpha_{N}(\nu,\kappa)$ such that $\lim_{\kappa\rightarrow
0}\limsup_{N\rightarrow\infty}\alpha_{N}(\nu,\kappa)=0$ for each $\nu>0$ and
$0\leq t_{1}\leq t_{2}\leq\cdots\leq t_{m}\leq t\leq 1$ where
$t-t_{m}\leq\kappa$. We have

$\displaystyle
P\Big{[}\Big{|}S_{N}(t)-S_{N}(t_{m})\Big{|}\geq\nu\Big{|}S_{N}(t_{1})...S_{N}(t_{m})\Big{]}\leq\alpha_{N}(\nu,\kappa)\quad\text{a.s.}$

Therefore, we need to show

$\displaystyle\sup\limits_{|t-t_{m}|\leq\kappa}P\Big{(}\Big{|}\sum_{k=\lfloor
Nt_{m}\rfloor+1}^{\lfloor
Nt\rfloor}u_{k}K\Big{[}\frac{x_{k}-x}{h}\Big{]}\Big{|}\geq\nu\,N^{1/2}\lambda^{-d}\Big{|}\xi_{\lfloor
Nt_{m}\rfloor},\xi_{{\lfloor
Nt_{m}\rfloor}-1},\cdots,\xi_{1}\Big{)}\leq\alpha_{N}(\nu,\kappa).$

We may choose $\alpha_{N}(\nu,\kappa)$ such that

$\displaystyle\alpha_{N}(\nu,\kappa)$
$\displaystyle:=\frac{1}{\nu^{2}}\frac{1}{\sqrt{Nh^{2}}}\sup\limits_{y,0\leq
t\leq\kappa}\mathbb{E}\Big{\\{}\sum_{k=1}^{\lfloor
Nt\rfloor}u_{k}K\Big{[}\frac{y+x^{\prime}_{0,k}}{h}\Big{]}\Big{\\}}^{2}$
$\displaystyle\leq\frac{1}{\nu^{2}}\frac{1}{\sqrt{Nh^{2}}}\sup\limits_{y}\sum_{k=1}^{\lfloor
N\kappa\rfloor}\mathbb{E}\Big{\\{}u^{2}_{k}K^{2}\Big{[}\frac{y+x^{\prime}_{0,k}}{h}\Big{]}\Big{\\}}$
$\displaystyle\quad+\frac{1}{\nu^{2}}\frac{1}{\sqrt{Nh^{2}}}\sup\limits_{y}2\sum\limits_{1\leq
k<l\leq\lfloor
N\kappa\rfloor}\Big{|}\mathbb{E}\Big{\\{}u_{k}u_{l}K\Big{[}\frac{y+x^{\prime}_{0,k}}{h}\Big{]}K\Big{[}\frac{y+x^{\prime}_{0,l}}{h}\Big{]}\Big{\\}}\Big{|}.$

Following expression (A.2) in Lemma A.1, we can show

$\displaystyle\alpha_{N}(\nu,\kappa)\leq\frac{C}{\nu^{2}\sqrt{Nh^{2}}}\times\frac{\lfloor
N\kappa\rfloor h\lambda^{d}}{(\lfloor
N\kappa\rfloor)^{1/2}}\Big{[}1+h\sqrt{j}\Big{]}=\frac{C}{\nu^{2}\sqrt{N}}\times(\lfloor
N\kappa\rfloor)^{1/2}\lambda^{d}\Big{[}1+h\sqrt{j}\Big{]},$

which yields $\lim_{\kappa\rightarrow
0}\limsup_{N\rightarrow\infty}\alpha_{N}(\nu,\kappa)=0$. Note that

$\displaystyle\max_{1\leq k\leq
N}\Big{|}u_{k}K\Big{[}\frac{x_{k}-x}{h}\Big{]}\Big{|}\leq\Big{\\{}\sum_{j=1}^{N}u_{j}^{4}K^{4}\Big{[}\frac{x_{j}-x}{h}\Big{]}\Big{\\}}^{1/4}.$

Following Lemma A.1, one can show that
$\mathbb{E}u_{j}^{4}K^{4}[\frac{x_{j}-x}{h}]\leq Ch/\sqrt{j}$, which concludes
(A.14). As a result, $S_{N}(t)$ is tight. ∎

######  Proposition A.8.

Let $\Theta_{2N}:=\sum_{k=1}^{N}(f(x_{k})-f(x))K[\frac{x_{k}-x}{h}]$ and
$\Theta_{3N}:=1/\sum_{k=1}^{N}K[\frac{x_{k}-x}{h}]$. Then,
$\Theta_{2N}\sqrt{\Theta_{3N}}\rightarrow_{P}0$.

###### Proof.

First, note that

$\displaystyle\Theta_{2N}\sqrt{\Theta_{3N}}=O_{P}\Big{(}\frac{Nh^{\gamma+1}}{d_{N}}\Big{)}o_{P}\Big{(}\\{\sqrt{N}\lambda^{d}h\\}^{-1/2}\Big{)}=o_{P}\Big{(}(N/d_{N})^{1/2}h^{\gamma+1-1/2}\Big{)},$

which is equal to $o_{P}((N/d_{N})^{1/2}h^{\gamma+1/2})$. Under assumptions
$Nh^{2\gamma+1}/d_{N}\rightarrow 0$ and $h\rightarrow 0$ as
$N\rightarrow\infty$, we conclude
$\Theta_{2N}\sqrt{\Theta_{3N}}\rightarrow_{P}0$. ∎

######  Proposition A.9.

Assume $g(x)$ is a bounded function such that
$\int_{\mathbb{R}}|g(x)|dx<\infty$. Let $c_{N}:=\frac{d_{N}}{h}$ such that
$c_{N}\rightarrow\infty$ and $\frac{c_{N}}{N}\rightarrow 0$. Then, we have

  * (i)

$\frac{c_{N}}{N}\sum_{k=1}^{N}g(c_{N}x_{k,N})\rightarrow_{D}\int_{\mathbb{R}}g(x)dx\,L_{B}(1,0)$,

  * (ii)

$\frac{d_{N}}{N}\sum_{k=1}^{N}|g(x_{k})|(1+|u_{k}|)=O_{P}(1)$,

  * (iii)

$(\frac{d_{N}}{N})^{1/2}\sum_{k=1}^{N}g(x_{k})u_{k}=O_{P}(1)$, and

  * (iv)

For $\sqrt{N}\lambda^{d}h\rightarrow\infty$, we have

$\displaystyle\Big{\\{}(\sqrt{N}\lambda^{d}h)^{-1/2}\sum_{k=1}^{N}K[(x_{k}-x)/h]u_{k},(\sqrt{N}\lambda^{d}h)^{-1}\sum_{k=1}^{N}g\Big{[}\frac{x_{k}-x}{h}\Big{]}\Big{\\}}$
$\displaystyle\quad\rightarrow_{D}\Big{\\{}d_{0}N(0,1)L_{B}^{1/2}(1,0),\int_{\mathbb{R}}g(s)dsL_{B}(1,0)\Big{\\}}\quad\text{on}\quad
D^{2}[0,1].$

###### Proof.

Proof of (i) is similar to Theorem 2.1 in Wang and Phillips (2009a) and Lemma
7 of Jeganathan (2004). Proof of (ii) follows from expression (7.2) in
Proposition 7.1 of Wang and Phillips (2016), and proof of (iii) follows from
expression (A.3) in Lemma A.1. The proof of (iv) is similar to the proof of
expression (3.8) in Wang and Phillips (2009b) with some minor modifications. ∎