# Nonparametric Tests of Conditional Independence for Time Series

Xiaojun Song
Peking University Department of Business Statistics and Econometrics, Guanghua
School of Management and Center for Statistical Science, Peking University,
Beijing, 100871, China. E-mail<EMAIL_ADDRESS>Financial support from the
National Natural Science Foundation of China (Grant No. 71532001) is
acknowledged. Haoyu Wei
Peking University Guanghua School of Management, Peking University, Beijing,
China. Email<EMAIL_ADDRESS>
###### Abstract

We propose consistent nonparametric tests of conditional independence for time
series data. Our methods are motivated from the difference between joint
conditional cumulative distribution function (CDF) and the product of
conditional CDFs. The difference is transformed into a proper conditional
moment restriction (CMR), which forms the basis for our testing procedure. Our
test statistics are then constructed using the integrated moment restrictions
that are equivalent to the CMR. We establish the asymptotic behavior of the
test statistics under the null, the alternative, and the sequence of local
alternatives converging to conditional independence at the parametric rate.
Our tests are implemented with the assistance of a multiplier bootstrap. Monte
Carlo simulations are conducted to evaluate the finite sample performance of
the proposed tests. We apply our tests to examine the predictability of equity
risk premium using variance risk premium for different horizons and find that
there exist various degrees of nonlinear predictability at mid-run and long-
run horizons.

Keywords: Conditional CDFs; empirical processes; multiplier bootstrap;
nonparametric regression; time series.

JEL Classifications: C12; C14; C15.

##  1 Introduction

A variable $Y$ is said to be conditionally independent of $Z$ given $X$ if and
only if the conditional density of $Y$ given $Z$ and $X$ equals to the
conditional density of $Y$ given $X$, that is, $Z$ does not carry any
information about $Y$ once $X$ is given. Following David (1979), we write
$Y\bot Z|X$ to denote that $Y$ is independent of $Z$ given $X$. The hypotheses
$Y\bot Z|X$ is related to the hypothesis that $Y$ is independent of $Z$, i.e.
$Y\bot Z$ (unconditional independence), and the conditional mean independence
(CMI), $\mathrm{E}(Y|Z,X)=\mathrm{E}(Y|X)$.

The assumption of conditional independence plays an important role and is a
widely imposed one in both statistical and econometric literature. For
example, Markov property of a time series process, Granger non-causality, the
assumption of missing at random (MAR) and exogeneity all can be formulated as
a conditional independence restriction, see Hong et al. (2017) for motivating
examples of testing the conditional independence hypothesis in economics and
econometrics. However, in contrast to the many tests of unconditional
independence or CMI proposed in the context of independent and identically
distributed (i.i.d.) data, by far, not many tests are available for testing
conditional independence assumption with time series data. Nonparametric tests
for unconditional independence between random variables and/or vectors, and
nonparametric tests for serial independence are abundant, e.g. a nonparametric
test of Cramér-von Mises type first introduced by Hoeffding (1948), the
empirical distribution function-based tests of Blum et al. (1961), Skaug and
Tjøstheim (1993) for testing independence of raw data and Delgado and Mora
(2000) or Ghoudi et al. (2001) for testing serial independence of time series
or regression errors, the empirical characteristic function-based test of
Csörgő (1985), kernel smoothing-based tests like Rosenblatt (1975), Robinson
et al. (1991), and Hong and White (2005), and tests based on measures of
association and dependence between random variables and/or vectors such as
Bakirov et al. (2006), Székely et al. (2007) or Diks and Panchenko (2007). On
the other hand, among the available tests for conditional independence, the
majority is designed for i.i.d. data, e.g. Linton and Gozalo (2014) develop a
non-pivotal nonparametric test based on a generalization of the empirical
distribution function, Song et al. (2009) employs the Rosenblatt
transformation to obtain a distribution-free test for a different type of
conditional independence, Huang et al. (2010) proposes a test based on the
maximal nonlinear conditional correlation, and Huang et al. (2016) develop an
integrated conditional moment test. Tests suitable for time series data
include Su and White (2007, 2008, 2012, 2014), Bouezmarni et al. (2012) and
Wang et al. (2018), all of which are based on kernel smoothing. The exception
is Su and White (2012), who provide nonparametric tests for conditional
independence using local polynomial quantile regression.

In this paper we aim to further fill the gap of the literature and propose
consistent nonparametric tests based on an empirical process approach for
testing conditional independence which are applicable to time series data. Our
approach exploits a proper conditional moment restriction and is in the same
spirit with Delgado et al. (2001), which partially circumvents the “curse of
dimensionality” problem. Besides, in comparison with the existing tests based
on smoothing methods, our new tests are able to detect local alternatives
converging to the null at a parametric rate.

We introduce some notations for testing the conditional independence
hypothesis in a time series framework. Let $X_{t}$, $Y_{t}$ and $Z_{t}$ be
three generic random vectors with dimensions $d_{x}$, $d_{y}$ and $d_{z}$,
respectively. The null hypothesis of interest is that $Y_{t}$ is independent
of $Z_{t}$ conditional on $X_{t}$, i.e.,

$F_{Y,Z|X}(y,z|x)=F_{Y|X}(y|x)F_{Z|X}(z|x),$

for all $(x,y,z)\in\mathbb{R}^{d_{x}+d_{y}+d_{z}}$, where $F_{Y,Z|X}$,
$F_{Y|X}$ and $F_{Z|X}$ denote the conditional cumulative distribution
functions (CDFs).

The rest of the paper is as follows. In Section 2 we examine the testing
problem and provide the test statistics. Section 3 establishes the asymptotic
null distributions. Section 4 studies the consistency property of the test and
the asymptotic local power of the test under local alternatives. A bootstrap
procedure to implement the tests is proposed and formally justified in Section
5. In Section 6, we study the finite sample performance of our tests by means
of Monte Carlo simulations. Section 7 presents an empirical example of using
variance risk premium to predict equity risk premium. Finally, Section 8
concludes the paper. All proofs are collected in the online Appendix.

##  2 The testing procedure

Consider a $\mathbb{R}^{d_{x}+d_{y}+d_{z}}$-valued strictly stationary ergodic
time series process $\\{(X_{t},Y_{t},Z_{t})\\}$ defined on the probability
space $(\Omega,\mathcal{F},\mathbb{P})$, which satisfies the Markov’s property

$\mathrm{P}(Y_{t}\leq y,Z_{t}\leq z|\mathcal{A}_{t-1})=\mathrm{P}(Y_{t}\leq
y,Z_{t}\leq
z|W_{t})\,\,\,\text{a.s.}\,\,\,\forall(y,z)\in\mathbb{R}^{d_{y}+d_{z}},$ (1)

where $\mathcal{A}_{t-1}:=\sigma(\\{X_{s},Y_{s-1},Z_{s-1}\\}_{s=-\infty}^{t})$
with $\sigma(\cdot)$ the smallest sigma algebra,

$W_{t}=\\{(X^{\top}_{s},Y^{\top}_{s-1},Z^{\top}_{s-1})^{\top}\\}_{s=t-p+1}^{t}$

with $0<p<\infty$ an integer, and “⊤” denotes transpose. That is, the only
relevant information for explaining $(Y_{t},Z_{t})$ are the first $m$ lags of
$\left(X_{t+1},Y_{t},Z_{t}\right)$. Note that $W_{t}$ is a $d_{w}\times 1$
vector with $d_{w}=p(d_{x}+d_{y}+d_{z})$.

We propose a nonparametric test for the hypothesis that $Y_{t}$ and $Z_{t}$
are independent given $W_{t}$, i.e.,

$\text{H}_{0}:F_{Y,Z|W}(y,z|W_{t})=F_{Y|W}(y|W_{t})F_{Z|W}(z|W_{t})\quad\text{a.s.}\quad\forall\,(y,z)\in\mathbb{R}^{d_{y}+d_{z}}.$
(2)

The alternative hypothesis $\text{H}_{1}$ is the negation of $\text{H}_{0}$ in
(2). The alternative hypothesis $\text{H}_{1}$ consists of a broad class of
conditional dependence between $Y_{t}$ and $Z_{t}$ given $W_{t}$. That is,
conditional on $W_{t}$, $Y_{t}$ could depend on $Z_{t}$ through mean,
variance, skewness, kurtosis, or even higher moments. It is possible to have
situations where the dependence between $Y_{t}$ and $Z_{t}$ in lower moments
(e.g. mean or variance) does not exist, but it does exist in higher moments
(e.g. skewness or kurtosis). See Section 6 for some data generating processes
under $\text{H}_{1}$.

It is important to emphasize that the new formulation of testing conditional
independence assumption in (2) is attractive, since it partly circumvents the
problem of “curse of dimensionality” by conditioning on only $W_{t}$ in all
three conditional CDFs. Wang et al. (2018) has also exploited (2) to propose a
test based on the conditional characteristic functions. On the other hand,
existing tests are mainly based on testing
$F_{Y|W,Z}(y|W_{t},Z_{t})=F_{Y|W}(y|W_{t})$ a.s., which requires estimation of
conditional CDFs given both $W_{t}$ and $Z_{t}$, see e.g. Su and White (2007,
2008) and Bouezmarni et al. (2012) to name only a few.

Note that $F_{Y,Z|W}(y,z|W_{t})=\mathrm{E}[1(Y_{t}\leq y)1(Z_{t}\leq
z)|W_{t}]$ and $F_{Y|W}(y|W_{t})=\mathrm{E}[1(Y_{t}\leq y)|W_{t}]$, with
$1(\cdot)$ the indicator function. Thus, $\text{H}_{0}$ in (2) can be tested
by

$\text{H}_{0}:\mathrm{E}[1(Y_{t}\leq y)(1(Z_{t}\leq
z)-F_{Z|W}(z|W_{t}))|W_{t}]=0\quad\text{a.s.}\quad\forall\,(y,z)\in\mathbb{R}^{d_{y}+d_{z}}.$
(3)

The problem of testing (2) is in fact equivalent to testing a conditional
moment restriction (CMR) in (3), which has the advantage of conditioning on
only $W_{t}$. An important feature of the CMR stated in (3) is that it
involves all values of $(y,z)$. Since it has to hold for all $(y,z)$, we have
an infinite number of CMRs to be tested.

Fortunately, Stinchcombe and White (1998) give us a method to convert
conditional to unconditional moment restriction in a convenient way: testing
(3) is further equivalent to testing the following infinite number of
unconditional moment restrictions,

$\text{H}_{0}:\mathrm{E}\big{[}\varphi(W_{t},w)1(Y_{t}\leq
y)\big{(}1(Z_{t}\leq
z)-F_{Z|W}(z|W_{t})\big{)}\big{]}=0,\quad\forall\,(w,y,z)\in\mathcal{W}\times\mathbb{R}^{d_{y}+d_{z}},$
(4)

where $\mathcal{W}\subseteq\mathbb{R}^{d_{w}}$ is a proper chosen set with
typical choices $d_{w}=d_{x}+d_{y}+d_{z}$ or $d_{w}=d_{x}+d_{y}+d_{z}+1$, and
$\varphi$ is a generically comprehensively revealing (GCR) or comprehensively
revealing (CR) function. Examples of GCR functions include: (1)
$\varphi(W_{t},w)=\exp(\mathrm{i}w^{\top}W_{t})$; (2)
$\varphi(W_{t},\gamma)=\sin(w^{\top}W_{t})$, and examples of CR functions
include: (3) $\varphi(W_{t},\gamma)=1(W_{t}\leq w)$; (4)
$\varphi(W_{t},w)=1(\beta^{\top}W_{t}\leq\alpha)$ with
$w=(\alpha,\beta)^{\top}$. It is worthy to note that when $\varphi$ is GCR,
the deviations from the null hypothesis can be detected by essentially any
choice of $w\in\mathcal{W}$, where $\mathcal{W}$ can be chosen as any small
compact set with non-empty interior, whereas CR functions may require the set
$\mathcal{W}$ to be the whole Euclidean space to ensure the consistency of the
associate test. More discussion can be seen in Stinchcombe and White (1998)
and Su and White (2012). Hence in the following, we will always assume
$\mathcal{W}$ is a bounded space in $\mathbb{R}^{d_{w}}$. In addition, to
avoid the random denominator problem, we propose to test the following
modified version of (4),

$\text{H}_{0}:\mathrm{E}\big{[}\varphi(W_{t},w)1(Y_{t}\leq
y)\big{(}1(Z_{t}\leq
z)-F_{Z|W}(z|W_{t})\big{)}f_{W}(W_{t})\big{]}=0,\quad\forall\,(w,y,z)\in\mathcal{W}\times\mathbb{R}^{d_{y}+d_{z}},$
(5)

where $f_{W}(W_{t})$ is the density of $W_{t}$. As in Delgado et al. (2001),
the density-weighted formulation helps to avoid conveniently the random
denominator in the subsequent nonparametric estimation.

Given a sample
$\\{(W^{\top}_{t},Y^{\top}_{t},Z^{\top}_{t})^{\top}\\}_{t=1}^{n}$ of size $n$,
if $F_{Z|W}(z|W_{t})$ and $f_{W}(W_{t})$ were observable, test statistics
could be constructed using the (infeasible) empirical process

$\displaystyle
S^{0}_{n}(w,y,z)=\frac{1}{\sqrt{n}}\sum_{t=1}^{n}\varphi(W_{t},w)1(Y_{t}\leq
y)(1(Z_{t}\leq z)-F_{Z|W}(z|W_{t}))f_{W}(W_{t}).$

Then under $\text{H}_{0}$, $S^{0}_{n}(\cdot,\cdot,\cdot)\rightsquigarrow
S^{0}_{\infty}(\cdot,\cdot,\cdot)$, where $S^{0}_{\infty}(\cdot,\cdot,\cdot)$
is a zero mean Gaussian process with covariance kernel
$\mathrm{E}\left[S^{0}_{\infty}(w,y,z)S^{0}_{\infty}(w^{\prime},y^{\prime},z^{\prime})\right]$.

As $F_{Z|W}(z|W_{t})$ and $f_{W}(W_{t})$ are unobservable, testing procedures
based on $S^{0}_{n}(w,y,z)$ are not feasible. In this paper, we replace them
with $\widehat{F}_{Z|W}(z|W_{t})$ and $\widehat{f}_{W}(W_{t})$, where

$\widehat{F}_{Z|W}(z|W_{t})=\frac{\frac{1}{(n-1)h^{d_{w}}}\sum_{s=1,s\neq
t}^{n}1(Z_{s}\leq z)K\left(\frac{W_{t}-W_{s}}{h}\right)}{\hat{f}_{W}(W_{t})},$
$\widehat{f}_{W}(W_{t})=\frac{1}{(n-1)h^{d_{w}}}\sum_{s=1,s\neq
t}^{n}K\left(\frac{W_{t}-W_{s}}{h}\right),$

with $K(\cdot)$ and $h:=h_{n}\in\mathbb{R}^{+}$ the kernel function and
bandwidth, respectively. Define the feasible empirical process

$\displaystyle
S_{n}(w,y,z)=\frac{1}{\sqrt{n}}\sum_{t=1}^{n}\varphi(W_{t},w)1(Y_{t}\leq
y)(1(Z_{t}\leq z)-\widehat{F}_{Z|W}(z|W_{t}))\widehat{f}_{W}(W_{t}),$

which is algebraically equivalent to

$\frac{1}{\sqrt{n}(n-1)h^{d_{w}}}\sum_{t=1}^{n}\sum_{s=1,s\neq
t}^{n}K\left(\frac{W_{t}-W_{s}}{h}\right)\varphi(W_{t},w)1(Y_{t}\leq
y)(1(Z_{t}\leq z)-1(Z_{s}\leq z)).$

The above expression is a variant of $U$-processes considered by Delgado et
al. (2001) in an i.i.d. context. Note that the limiting distribution of
$S_{n}(w,y,z)$ will be different from that of $S^{0}_{n}(w,y,z)$ due to the
estimation of $F_{Z|W}(z|W_{t})$.

Test statistics are constructed based on suitable continuous functionals of
$S_{n}(w,y,z)$. A test statistic in the spirit of the Cramér-von Mises type is

$\displaystyle CvM_{n}=\int
S^{2}_{n}(w,y,z)\,dF_{n}(w,y,z)=n^{-1}\sum_{t=1}^{n}S_{n}^{2}(W_{t},Y_{t},Z_{t}),$
(6)

where $F_{n}(w,y,z)=n^{-1}\sum_{t=1}^{n}1(W_{t}\leq x)1(Y_{t}\leq
y)1(Z_{t}\leq z)$ is the empirical distribution function of
$(X_{t},Y_{t},Z_{t})$. Henceforth, an unspecified integral denotes integration
over the whole space. The Kolmogorov-Smirnov type test statistic basing on the
sup-norm is

$KS_{n}=\sup_{(w,y,z)\in\mathcal{W}\times\mathbb{R}^{d_{y}}\times\mathbb{R}^{d_{z}}}\left|S_{n}(w,y,z)\right|.$
(7)

In practice, one can compute $KS_{n}$ by simply taking the maximum over the
observations, i.e., $\widetilde{KS}_{n}=\max_{1\leq t\leq
n}|S_{n}(W_{t},Y_{t},Z_{t})|$.

Under $\text{H}_{0}$, test statistics $CvM_{n}$ and $KS_{n}$ converge in
distribution, while they diverge to infinity under $\text{H}_{1}$. We reject
the null hypothesis of conditional independence whenever they exceed certain
“large” values. Since the asymptotic null distributions of $CvM_{n}$ and
$KS_{n}$ depend on the data generating process in a complicated way, their
critical values are not readily available. To circumvent this problem, we
propose a bootstrap procedure to obtain the critical values of our tests in
Section 5.

##  3 Asymptotic null distributions

In this section, we will establish the asymptotic null distributions of our
test statistics $CvM_{n}$ and $KS_{n}$. We need to impose the following
assumptions, which are attached in Appendix A.

Let $\phi_{t}(y)=1(Y_{t}\leq y)-F_{Y|W}(y|W_{t})$,
$\epsilon_{t}(z)=1(Z_{t}\leq z)-F_{Z|W}(z|W_{t})$, and
$e_{t}(w,y,z)=\varphi(W_{t},w)\phi_{t}(y)\epsilon_{t}(z)f_{W}(W_{t})$. The
following theorem shows that $S_{n}(\cdot,\cdot,\cdot)$ converges weakly to a
Gaussian process under the null.

######  Theorem 1.

Suppose (1) and Assumption A1 - A7 in Appendix A hold. Then under the null

$S_{n}(\cdot,\cdot,\cdot)\rightsquigarrow S_{\infty}(\cdot,\cdot,\cdot),$

where $S_{\infty}(\cdot,\cdot,\cdot)$ is a zero mean Gaussian process with
covariance kernel
$\mathrm{E}\big{[}S_{\infty}(w,y,z)S_{\infty}(w^{\prime},y^{\prime},z^{\prime})\big{]}$.
Specially, if $(X_{t}^{\top},Y_{t}^{\top},Z_{t}^{\top})$ is IID sample, we
have

$\displaystyle\mathrm{E}\big{[}S_{\infty}(w,y,z),S_{\infty}(w^{\prime},y^{\prime},z^{\prime})\big{]}$
$\displaystyle=\mathrm{E}\big{[}e_{1}(w,y,z)e_{1}(w^{\prime},y^{\prime},z^{\prime})\big{]}$
$\displaystyle=\mathrm{E}[\varphi(W_{t},w)\varphi(W_{t},w^{\prime})\psi(y,y^{\prime};W_{1})\rho(z,z^{\prime};W_{1})f^{2}_{W}(W_{1})],$

where $\psi(y,y^{\prime};W)=F_{Y|W}(y\wedge
y^{\prime}|W)-F_{Y|W}(y|W)F_{Y|W}(y^{\prime}|W)$ and
$\rho(z,z^{\prime};W)=F_{Z|W}(z\wedge
z^{\prime}|W)-F_{Z|W}(z|W)F_{Z|W}(z^{\prime}|W)$.

######  Remark 1.

The term $F_{Y|W}(y|W_{t})$ in the definition of $e_{t}(w,y,z)$ reflects the
cost paid for replacing $F_{Z|W}(z|W_{t})$ with $\widehat{F}_{Z|W}(z|W_{t})$
in the infeasible process $S_{n}^{0}(w,y,z)$, a phenomenon known as the
“parameter estimation error”, leading to a covariance kernel different from
that of $S_{n}^{0}(w,y,z)$.

The asymptotic null distributions of test statistics $CvM_{n}$ and $KS_{n}$
are given in the corollary below.

######  Corollary 1.

Suppose (1) and Assumption A1 - A7 in Appendix A hold. Then under the null,

$\displaystyle CvM_{n}\rightsquigarrow
CvM_{\infty}:=\int_{\mathcal{W}\times\mathbb{R}^{d_{y}}\times\mathbb{R}^{d_{z}}}S^{2}_{\infty}(w,y,z)\,dF_{W,Y,Z}(w,y,z),$
$KS_{n}\rightsquigarrow
KS_{\infty}:=\sup_{(w,y,z)\in\mathcal{W}\times\mathbb{R}^{d_{y}}\times\mathbb{R}^{d_{z}}}\left|S_{\infty}(w,y,z)\right|,$

where $S_{\infty}(\cdot,\cdot,\cdot)$ is the Gaussian process defined in
Theorem 1.

##  4 Consistency and asymptotic local power

We investigate the consistency and asymptotic local power properties of test
statistics $CvM_{n}$ and $KS_{n}$ based on $S_{n}(w,y,z)$ under $\text{H}_{1}$
and under a sequence of local alternatives converging to $\text{H}_{0}$ at a
parametric rate $n^{-1/2}$.

The asymptotic behavior of $S_{n}(w,y,z)$ under $\text{H}_{1}$ is stated in
the next theorem.

######  Theorem 2.

Suppose (1) and Assumptions A1 - A7 in Appendix A hold.. Then under the
alternative, for each
$(w,y,z)\in\mathcal{W}\times\mathbb{R}^{d_{y}}\times\mathbb{R}^{d_{z}}$,

$n^{-1/2}S_{n}(w,y,z)\xrightarrow{P}\mathrm{E}\big{[}\varphi(W_{t},w)\big{(}F_{Y,Z|W}(y,z|W_{t})-F_{Y|W}(y|W_{t})F_{Z|W}(z|W_{t})\big{)}f_{W}(W_{t})\big{]}.$

Since
$\mathrm{E}\big{[}\varphi(W_{t},w)\big{(}F_{Y,Z|W}(y,z|W_{t})-F_{Y|W}(y|W_{t})F_{Z|W}(z|W_{t})\big{)}f_{W}(W_{t})\big{]}$
in a set with a positive Lebesgue measure under $\text{H}_{1}$, test
statistics $CvM_{n}$ and $KS_{n}$ will diverge to infinity and have asymptotic
power one against $\text{H}_{1}$.

To investigate the asymptotic local power properties of the tests, we
introduce the following sequence of local alternatives,

$\text{H}_{1n}:F_{Y,Z|W}(y,z|W_{t})=F_{Y|W}(y|W_{t})F_{Z|W}(z|W_{t})+n^{-1/2}\Delta(W_{t},y,z)\,\,\,\text{a.s.}\,\,\,\forall(y,z)\in\mathbb{R}^{d_{y}+d_{z}},$
(8)

where $\Delta(\cdot,\cdot,\cdot)$ is a non-constant measurable function,
satisfying $\Delta(W_{t},y,z)\neq 0$ a.s. for some $(y,z)$ and
$\Delta(W_{t},\infty,z)=\Delta(W_{t},y,\infty)=\Delta(W_{t},-\infty,z)=\Delta(W_{t},y,-\infty)=0$
a.s. to deliver a valid conditional CDF $F_{Y,Z|W}(y,z|W_{t})$ in (8). The
type of local alternatives in (8) is widely used in studying the asymptotic
local power properties of tests based on empirical processes.

In (8), the term $n^{-1/2}\Delta(W_{t},y,z)$ characterizes the departure of
the conditional joint CDF from the product of conditional marginal CDFs. In
particular, $\Delta(W_{t},y,z)$ specifies the direction of the departure,
while $n^{-1/2}$ specifies the rate at which the difference between
$F_{Y,Z|W}(y,z|W_{t})$ and $F_{Y|W}(y|W_{t})F_{Z|W}(z|W_{t})$ shrinks to zero.
Note that $n^{-1/2}$ is the fastest possible rate found in testing conditional
independence.

To derive the local power result, we need the extra assumption i.e. Assumption
A8 in Appendix A. The next theorem states the asymptotic behavior of
$S_{n}(w,y,z)$ under $\text{H}_{1n}$.

######  Theorem 3.

Suppose Assumption A1 - A8 in Appendix A hold. Then under the sequence of
local alternatives in (8),

$S_{n}(\cdot,\cdot,\cdot)\rightsquigarrow
S_{\infty}(\cdot,\cdot,\cdot)+G(\cdot,\cdot,\cdot),$

where $S_{\infty}(\cdot,\cdot,\cdot)$ is the Gaussian process defined in
Theorem 1.

Provided that the shift function
$G(w,y,z):=\int^{w}_{-\infty}\Delta(\bar{w},y,z)f^{2}_{W}(\bar{w})\,d\bar{w}\neq
0$ in a set with a positive Lebesgue measure under $\text{H}_{1n}$, test
statistics $CvM_{n}$ and $KS_{n}$ based on $S_{n}(x,y,z)$ will have non-
trivial local power against $\text{H}_{1n}$ converging to $\text{H}_{0}$ at a
parametric rate $n^{-1/2}$, the best rate known in testing conditional
independence.

######  Remark 2.

For local power analysis, tests based on empirical processes can achieve a
parametric rate $n^{-1/2}$, a rate much faster than those obtained from
smoothing-based nonparametric tests. For example, Su and White (2008) test
only has power against local alternatives at a rate $n^{-1/2}h^{-d/4}$ with
$d=d_{w}+d_{y}+d_{z}$, while tests proposed by Su and White (2007) and
Bouezmarni et al. (2012) have power against local alternatives at a rate
$n^{-1/2}h^{-(d_{w}+d_{z})/4}$. Wang et al. (2018) test is the only one that
has a better rate, which can detect local alternatives at a rate
$n^{-1/2}h^{-d_{w}/4}$, though still slower than $n^{-1/2}$. Nonetheless,
tests based on local smoothing are able to detect high frequency local
alternatives considered by Rosenblatt (1975), while our tests may not detect
such type of local alternatives.

The limiting distributions of $CvM_{n}$ and $KS_{n}$ under $\text{H}_{1n}$ is
stated in the next result, which is a direct consequence of the continuous
mapping theorem and Theorem 3.

######  Corollary 2.

Suppose (1) and Assumption A1 - A8 in Appendix A hold. Then under the sequence
of local alternatives in (8),

$\displaystyle
CvM_{n}\rightsquigarrow\int_{\mathcal{W}\times\mathbb{R}^{d_{y}}\times\mathbb{R}^{d_{z}}}\big{(}S_{\infty}(w,y,z)+G(w,y,z)\big{)}^{2}\,dF_{W,Y,Z}(w,y,z),$
$KS_{n}\rightsquigarrow\sup_{(w,y,z)\in\mathcal{W}\times\mathbb{R}^{d_{y}}\times\mathbb{R}^{d_{z}}}\left|S_{\infty}(w,y,z)+G(w,y,z)\right|,$

where $S_{\infty}(\cdot,\cdot,\cdot)$ and $G(\cdot,\cdot,\cdot)$ are defined
in Theorem 3.

Corollary 2 implies that under $\text{H}_{1n}$ in (8), the limiting
distributions of $CvM_{n}$ and $KS_{n}$ are no longer the same as in Corollary
1 and they shift in a non-trivial way. Consequently, our tests have non-
trivial power against the sequence of local alternatives in (8) converging to
the null at a parametric rate.

##  5 Bootstrap

Since the asymptotic null distributions of $CvM_{n}$ and $KS_{n}$ depend on
the underlying data generating process due to the complicated covariance
kernel of the limiting process $S_{\infty}(w,y,z)$ in Theorem 1, it is
difficult to tabulate the critical values for our tests. We propose a
bootstrap procedure, which is in the spirit of the multiplier bootstrap
suggested by Delgado et al. (2001). This bootstrap takes full advantage of the
asymptotic theory in Theorem 1, and is also easy to implement as it does not
have to compute new nonparametric estimates at each bootstrap replication.

Let $\widehat{\phi}_{t}(y)=1(Y_{t}\leq y)-\widehat{F}_{Y|W}(y|W_{t})$,
$\widehat{\epsilon}_{t}(z)=1(Z_{t}\leq z)-\widehat{F}_{Z|W}(z|W_{t})$, and
$\widehat{e}_{t}(w,y,z)=\varphi(W_{t},w)\widehat{\phi}_{t}(y)\widehat{\epsilon}_{t}(z)\widehat{f}_{W}(W_{t})$,
where $\widehat{F}_{Y|W}(y|W_{t})$ is defined with $1(Y_{s}\leq y)$ replacing
$1(Z_{s}\leq z)$ in $\widehat{F}_{Z|W}(z|W_{t})$. The bootstrap empirical
process of $S_{n}(w,y,z)$ is given by

$\displaystyle
S_{n}^{\ast}(w,y,z)=\frac{1}{\sqrt{n}}\sum_{t=1}^{n}\widehat{e}_{t}(w,y,z)v_{t},$

where $\\{v_{t}\\}_{t=1}^{n}$ is a sequence of i.i.d. random variables with
zero mean, unit variance, bounded support, and is independent of
$\\{(W^{\top}_{t},Y^{\top}_{t},Z^{\top}_{t})^{\top}\\}_{t=1}^{n}$. One popular
choice due to Mammen (1993) is the i.i.d. Bernoulli variates with probability
masses given by

$\mathrm{P}\left(v_{t}=\frac{1-\sqrt{5}}{2}\right)=\frac{1+\sqrt{5}}{2\sqrt{5}}\,\,\,\text{and}\,\,\,\mathrm{P}\left(v_{t}=\frac{1+\sqrt{5}}{2}\right)=\frac{-1+\sqrt{5}}{2\sqrt{5}}.$

See also Delgado et al. (2001) and Escanciano and Velasco (2006) for
applications of this choice.

In the next theorem the asymptotic validity the bootstrap procedure is
justified formally. We show that the bootstrapped process
$S^{\ast}_{n}(\cdot,\cdot,\cdot)$ converges weakly to the Gaussian process
$S_{\infty}(\cdot,\cdot,\cdot)$ in Theorem 1. As a result, we have convergence
in distributions of the bootstrapped test statistics $CvM^{\ast}_{n}$ and
$KS^{\ast}_{n}$, which are simply constructed by replacing $S_{n}(w,y,z)$ with
$S^{\ast}_{n}(w,y,z)$ in (6) and (7), respectively.

######  Theorem 4.

Suppose (1) and Assumption A1 - A8 in Appendix A hold. Then, under the null,
under the alternative, or under the sequence of local alternatives in (8),

$S_{n}^{\ast}(\cdot,\cdot,\cdot)\underset{\ast}{\overset{P}{\rightarrow}}S_{\infty}(\cdot,\cdot,\cdot),$

where $S_{\infty}(\cdot,\cdot,\cdot)$ is the Gaussian process defined in
Theorem 1, and $\underset{\ast}{\overset{P}{\rightarrow}}$ denotes weak
convergence in probability under the bootstrap law, i.e., conditional on
$\\{(W^{\top}_{t},Y^{\top}_{t},Z^{\top}_{t})^{\top}\\}_{t=1}^{n}$. In
addition, $CvM_{n}^{\ast}\underset{\ast}{\rightsquigarrow}CvM_{\infty}$ and
$KS_{n}^{\ast}\underset{\ast}{\rightsquigarrow}KS_{\infty}$ with
$CvM_{\infty}$ and $KS_{\infty}$ defined in Corollary 1.

Theorem 4 implies that the limiting behavior of $S_{n}(w,y,z)$ can be
approximated by that of $S^{\ast}_{n}(w,y,z)$. Thus, the bootstrap assisted
tests $CvM_{n}$ and $KS_{n}$ have a correct asymptotic level, are consistent
against $\text{H}_{1}$, and are able to detect local alternatives (8)
converging to the null at a parametric rate. In practice, we can obtain the
critical values of $CvM_{n}$ (and similarly for $KS_{n}$) as accurately as
desired by the following algorithm:
Step 1. Compute $S_{n}(W_{t},Y_{t},Z_{t})$, and get $CvM_{n}$
Step 2. Generate $\\{v_{t}\\}_{t=1}^{n}$ independently, compute
$S_{n}^{\ast}(W_{t},Y_{t},Z_{t})$, and get $CvM_{n}^{\ast}$.
Step 3. Repeat Step 2 $B$ times to have $\\{CvM_{n,b}^{\ast}\\}_{b=1}^{B}$,
and compute its empirical $(1-\alpha)$-th sample quantile
$CvM_{n}^{\ast\alpha}$ or bootstrapped $p$-value
$p_{n}^{\ast}=B^{-1}\sum_{b=1}^{B}1\left(CvM_{n,b}^{\ast}\geq CvM_{n}\right)$.
Rejects $\text{H}_{0}$ at the significance level $\alpha$ if
$CvM_{n}>CvM_{n}^{\ast\alpha}$ or if $p_{n}^{\ast}<\alpha$.

######  Remark 3.

The above bootstrap assisted procedure applies when (1) holds. If (1) is
violated, our proposed tests may suffer size distortion and power loss.
However, under the general dependence structure in the data, it is possible to
extend the block bootstrap (e.g., Buhlmann (1994)) to our context, the finite
sample performance of which will be investigated through simulations in the
next section.

##  6 Monte Carlo simulations

We carry out a set of Monte Carlo simulations to examine the finite sample
performance of the proposed test statistics $CvM_{n}$ and $KS_{n}$. To examine
the size performance, we consider the following four data generating processes
(DGPs):

$\displaystyle\text{(S1):}\quad Y_{t}$
$\displaystyle=\varepsilon_{1,t},Z_{t}=\varepsilon_{2,t},X_{t}=\varepsilon_{3,t}.$
$\displaystyle\text{(S2):}\quad Y_{t}$
$\displaystyle=0.5Y_{t-1}+\varepsilon_{1,t}.$ $\displaystyle\text{(S3):}\quad
Y_{t}$ $\displaystyle=0.5Y_{t-1}\exp{(-0.5Y^{2}_{t-1})}+\varepsilon_{1,t}.$
$\displaystyle\text{(S4):}\quad Y_{t}$
$\displaystyle=\sqrt{h_{1,t}}\varepsilon_{1,t},h_{1,t}=0.01+0.9h_{1,t-1}+0.05Y_{t-1}^{2},$
$\displaystyle Z_{t}$
$\displaystyle=\sqrt{h_{2,t}}\varepsilon_{2,t},h_{2,t}=0.01+0.9h_{2,t-1}+0.05Z_{t-1}^{2}.$

To examine the power performance, the following seven DGPs are considered:

$\displaystyle\text{(P1):}\quad Y_{t}$
$\displaystyle=0.5Y_{t-1}+0.5Z_{t-1}+\varepsilon_{1,t}.$
$\displaystyle\text{(P2):}\quad Y_{t}$
$\displaystyle=0.5Y_{t-1}+0.5Z_{t-1}^{2}+\varepsilon_{1,t}.$
$\displaystyle\text{(P3):}\quad Y_{t}$
$\displaystyle=0.5Y_{t-1}Z_{t-1}+\varepsilon_{1,t}.$
$\displaystyle\text{(P4):}\quad Y_{t}$
$\displaystyle=0.3+0.2\log(h_{t})+\sqrt{h_{t}}\varepsilon_{1,t},h_{t}=0.01+0.5Y_{t-1}^{2}+0.3Z_{t-1}^{2}.$
$\displaystyle\text{(P5):}\quad Y_{t}$
$\displaystyle=0.5Y_{t-1}+0.5Z_{t-1}\varepsilon_{1,t}.$
$\displaystyle\text{(P6):}\quad Y_{t}$
$\displaystyle=\sqrt{h_{t}}\varepsilon_{1,t},h_{t}=0.01+0.5Y_{t-1}^{2}+0.25Z_{t-1}^{2}.$
$\displaystyle\text{(P7):}\quad Y_{t}$
$\displaystyle=\sqrt{h_{1,t}}\varepsilon_{1,t},h_{1,t}=0.01+0.1h_{1,t-1}+0.4Y_{t-1}^{2}+0.5Z_{t-1}^{2}.$
$\displaystyle Z_{t}$
$\displaystyle=\sqrt{h_{2,t}}\varepsilon_{2,t},h_{2,t}=0.01+0.9h_{2,t-1}+0.05Z_{t-1}^{2}.$

Here and below, $\varepsilon_{1,t}$, $\varepsilon_{2,t}$ and
$\varepsilon_{3,t}$ are i.i.d. $N(0,1)$ and mutually independent, and $Z_{t}$
in (S2)-(S3) and (P1)-(P6) follows an AR(1) model:
$Z_{t}=0.5Z_{t-1}+\varepsilon_{2,t}$. These DGPs cover a wide range of both
linear and nonlinear time series processes. For (S1), $(Y_{t},Z_{t},X_{t})$ is
simply an i.i.d. sequence; we test $Y_{t}\bot Z_{t}|X_{t}$ such that
$W_{t}=X_{t}$. From (S2) to (P7), we test $Y_{t}\bot Z_{t-1}|Y_{t-1}$ such
that $W_{t}=Y_{t-1}$; that is, we are interested in checking whether $Z_{t-1}$
has any predictive ability in explaining $Y_{t}$ (in mean, variance, or higher
moments) after controlling the first lag of $Y_{t}$. Note that (S4) and (P7)
do not satisfy (1).

For each DGP, we first generate $n+500$ observations and then discard the
first 500 observations to minimize effects of initial values. The number of
Monte Carlo simulations is $2000$ and the bootstrap critical values are
obtained from $B=1000$ bootstrap replications. Four sample sizes, $n=100$,
$200$, $400$ and $800$, are considered. We only report results for the nominal
level of 5% and results for other levels are available upon request. We choose
the standard normal density $K(x)=(2\pi)^{-1/2}\exp{(-x^{2}/2)}$ as our
kernel. Bandwidth of the form $h=cn^{-1/3.5}$ is used and results with
$c=0.5$, $1.0$ and $1.5$ are reported to check the sensitivity of our tests to
different bandwidths. How to choose $h$ to maximize the performance in our
testing framework is beyond the scope of this paper and is left for future
research. In addition to the fixed bandwidth $h$, in the supplemental appendix
we provide more simulation results with a data-driven one $h=1.06\cdot
Std(X_{t})\cdot n^{-1/3.5}$.

Table (1) and (2) report respectively the empirical rejection rates of
$CvM_{n}$ and $KS_{n}$ under (S1)-(P7) using critical values obtained with the
multiplier bootstrap method proposed in Section 5. Both $CvM_{n}$ and $KS_{n}$
have acceptable empirical sizes in the cases of moderate sample sizes for
(S1)-(S3), with $KS_{n}$ having more accurate sizes than $CvM_{n}$. When (1)
fails in (S4), $CvM_{n}$ exhibits a severe size distortion for $n=100$ and
$n=200$, while $KS_{n}$ has less distortion; when sample size increases to
800, the two tests show improved sizes especially for $c=0.5$. For the
empirical power performance, when sample size is as small as $n=100$, both
$CvM_{n}$ and $KS_{n}$ are not very powerful against (P2)-(P7), with (P7) the
most difficult DGP to detect. They gain power rapidly as sample size
increases. Note that in (P7) (1) fails, which might be the reason for low
power in small samples. In summary, our simulation results indicate that
$KS_{n}$ preserves sizes better than $CvM_{n}$, while $CvM_{n}$ is more
powerful than $KS_{n}$; in addition, whether (1) holds will affect the
performance. The simulation results also show that the empirical sizes and
powers of $CvM_{n}$ and $KS_{n}$ are somehow sensitive to bandwidth choices in
very small samples; a general pattern is that larger $c$ tends to deliver a
higher power but produces a bigger size distortion. When $n=800$, the overall
performance is satisfactory.

——————————————

Tables 1 & 2 about here

——————————————

To investigate if we have improved performance when we take into account the
general dependence structure in the data, we also study the following block
multiplier bootstrap:

$S_{n}^{\ast
block}(w,y,z)=\frac{1}{\sqrt{n}}\sum_{t=1}^{n-L+1}\zeta_{t}\sum_{s=t}^{t+L-1}\widehat{e}_{t}(w,y,z),$

where $\\{\zeta_{t}\\}_{t=1}^{n-L+1}$ are i.i.d. $N(0,L^{-1})$ and independent
of the sample
$\\{(W_{t}^{\top},Y_{t}^{\top},Z_{t}^{\top})^{\top}\\}_{t=1}^{n}$, with
$L:=L_{n}$ the block length diverging to infinity at a slower rate than $n$ as
$n\to\infty$. Note that when $L=1$, $S_{n}^{\ast block}(w,y,z)$ reduces to the
multiplier bootstrap introduced in Section 5, the validity of which replies on
the restriction (1).

Table (3) and (4) report respectively the empirical rejection rates of
$CvM^{block}_{n}$ and $KS^{block}_{n}$ based on $S_{n}^{\ast block}(w,y,z)$.
It is expected that $CvM^{block}_{n}$ and $KS^{block}_{n}$ should work when
(1) does not hold and consequently the dependence structure in the data should
not be ignored. In implementing the block bootstrap, we have used $L=\lfloor
an^{1/4}\rfloor$ with $a=1$, $2$ and $4$ to check the sensitivity of
$CvM^{block}_{n}$ and $KS^{block}_{n}$ with respect to different block
lengths. To save space, we only report the results for bandwidth
$h=cn^{-1/3.5}$ with $c=1.0$. The results in Table (3) and (4) are based on
1000 simulations and 200 bootstrap replications. It is observed that
$CvM^{block}_{n}$ and $KS^{block}_{n}$ have less distorted empirical sizes
especially for (S4) than $CvM_{n}$ and $KS_{n}$ do. At the same time,
$CvM^{block}_{n}$ and $KS^{block}_{n}$ are not as powerful as $CvM_{n}$ and
$KS_{n}$ even for (P7). Like before, $KS^{block}_{n}$ preserves sizes better
than $CvM^{block}_{n}$, while $CvM^{block}_{n}$ is more powerful than
$KS^{block}_{n}$. Lastly, for given $h$, the tests also depend on the choice
of $L$; smaller $L$ tends to deliver a higher power but produces oversized
tests.

——————————————

Tables 3 & 4 about here

——————————————

##  7 An empirical study

We examine whether there exists nonlinear predictability of equity risk
premium using variance risk premium. The variance risk premium is defined as
the difference between the risk-neutral and objective expectations of realized
variance, where the risk-neutral expectation of variance is measured as the
end-of-month Volatility Index-squared de-annualized and the realized variance
is the sum of squared 5-minute log returns of the S&P 500 index over the
month.

There is a rich literature on the predictive power of variance risk premium
for the aggregate stock market returns, bond returns or exchange rate returns.
For example, Bollerslev et al. (2009) first discover that variance risk
premium is able to explain a nontrivial fraction of the time series variation
in post 1990 aggregate stock market returns, with high (low) premia predicting
high (low) future returns; Wang et al. (2013) find the empirical evidence
suggesting that the firm-level variance risk premium has a prominent
explanatory power for credit spreads in the presence of market- and firm-level
control variables; by defining a “global” variance risk premium, Bollerslev et
al. (2013) uncover stronger predictability of aggregate stock market returns
using variance risk premium across countries; while Della Corte et al. (2013)
investigate the predictive information content in foreign exchange volatility
risk premia for exchange rate returns and find that a portfolio that sells
currencies with high insurance costs and buys currencies with low insurance
costs generates sizeable out-of-sample returns and Sharpe ratios.

We use monthly aggregate S&P 500 composite index over the period January 1996
to September 2008. Our empirical analysis is based on the logarithmic return
on the S&P 500 in excess of the 3-month T-bill rate. Let $RP_{t+\tau}$ be the
risk premium $\tau$ months ahead and $VRP_{t}$ be the variance risk premium at
time $t$. In this empirical study, we take $\tau=$1, 3, 6, and 9 months. We
shall examine if the variance risk premium $VRP_{t}$ explains in a linear or
nonlinear way the risk premium $RP_{t+\tau}$ given the information $RP_{t}$,
which is equivalent to stating whether $VRP_{t}$ Granger causes $RP_{t}$ by
setting the lag order to $\tau$. To test for the presence of (nonlinear)
predictability of $VRP_{t}$, we consider to test

$\text{H}_{0}:\mathrm{P}\left\\{F(RP_{t+\tau},VRP_{t}|RP_{t})=F(RP_{t+\tau}|RP_{t})F(VRP_{t}|RP_{t})\right\\}=1$

against
$\text{H}_{1}:\mathrm{P}\left\\{F(RP_{t+\tau},VRP_{t}|RP_{t})=F(RP_{t+\tau}|RP_{t})F(VRP_{t}|RP_{t})\right\\}<1$.
That is, for a given horizon $\tau$, we test the conditional independence of
$RP_{t+\tau}$ and $VRP_{t}$ given $RP_{t}$, i.e. $RP_{t+\tau}\bot
VRP_{t}|RP_{t}$.

For the purpose of comparison, we also perform the popular linear causality
analysis in the literature. To this end, we consider the following linear
regression model:

$RP_{t+\tau}=\mu_{\tau}+\beta_{\tau}RP_{t}+\alpha_{\tau}VRP_{t}+\varepsilon_{t+\tau}.$
(9)

The hypothesis of interest is that VRP does not Granger cause RP for $\tau$
months ahead in a linear way, i.e. testing the null hypothesis
$\text{H}_{0}:\alpha_{\tau}=0$ against the alternative hypothesis
$\text{H}_{1}:\alpha_{\tau}\neq 0$. To test $\text{H}_{0}$, standard
$t$-statistic given by
$t_{\hat{\alpha}_{\tau}}=\hat{\alpha}_{\tau}/\hat{\sigma}_{\hat{\alpha}_{\tau}}$
will be calculated, where $\hat{\alpha}_{\tau}$ is the least squares estimator
of $\alpha_{\tau}$ and $\hat{\sigma}_{\hat{\alpha}_{\tau}}$ is the estimator
of its standard error $\sigma_{\hat{\alpha}_{\tau}}$. Moreover, to avoid the
impact of possible dependence in the residual terms
$\hat{\varepsilon}_{t+\tau}$ on our inference,
$\hat{\sigma}_{\hat{\alpha}_{\tau}}$ is calculated using the commonly used
heteroscedasticity autocorrelation consistent (HAC) robust variance estimator
suggested by Newey and West (1987).

Table 5 reports the testing results for Granger causality (e.g. nonlinear
predictability) from variance risk premium to risk premium, at four different
horizons, using our proposed tests $CvM_{n}$ and $KS_{n}$ as well as the
linear test. To check the robustness of our empirical findings, we also
include the block multiplier tests $CvM_{n}^{block}$ and $KS_{n}^{block}$,
where the block length is set to be $L=\lfloor an^{1/4}\rfloor$ with $a=2$.
Results for $a=1$ and $4$ are similar and hence are omitted. The
implementation of all bootstrap based tests is as demonstrated in the Monte
Carlo simulations part with the number of bootstrap replications $B=10,000$.
We have obtained encouraging findings from Table 5 that are relevant to both
empirical and theoretical studies of variance risk premium as a suitable
predictor for risk premium. Specifically, results from linear regression based
test in (9) clearly fail to reject the null hypothesis of no linear
predictability for one to six months (short-run) horizons; they only show weak
evidence of linear predictability until at the nine months (long-run) horizon
at the 5% significance level. On the other hand, using our tests, we find
convincing evidence that risk premium can be well predicted using variance
risk premium at both mid-run and long-run horizons. Testing results are not
sensitive to the bandwidth choices for six and nine months horizon, nor does
the block type tests. For the one month horizon, we may conclude that variance
risk premium has no predictive power, while the findings for three months
horizon tend to indicate the existence of predictability. Overall, we find
that there is a very high degree of predictability at horizons more than one-
month which could be attributed to the nonlinear predictive effect. Our
empirical evidence also indicates that caution is needed when interpreting
results based on the linear regression.

——————————————

Table 5 about here

——————————————

##  8 Conclusion

This paper proposes new consistent nonparametric tests of conditional
independence for time series data based on the empirical process method. The
asymptotic properties of the proposed tests under the null, the alternative,
and the sequence of local alternatives are investigated. To implement the test
in practice, a multiplier bootstrap procedure is suggested and its asymptotic
validity is formally justified. The test can be applied to testing for
conditional independence in a wide variety of nonparametric models. Using the
proposed test, we also study whether there exists some nonlinear
predictability of equity risk premium using variance risk premium.

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##  Appendix A A Assumptions

Denote $d:=d_{w}+d_{y}+d_{z}$.

  * A1

$\\{(X_{t}^{\top},Y_{t}^{\top},Z_{t}^{\top})^{\top}\\}$ is a strictly
stationary strong mixing process with mixing coefficients $\alpha(s)$ such
that $\sum_{s=1}^{\infty}s^{2m-1}\alpha(s)^{\eta/(2m+\eta)}<\infty$ for some
$m\in\mathbb{N}_{+}$ and $\eta>0$ such that

$m\geq\frac{d+\sqrt{d^{2}+4\eta
d}}{4}\vee\frac{(d+\eta-1)+\sqrt{(d+\eta-1)^{2}+4(d-2)\eta}}{4}.$

  * A2

$\varphi(\cdot,\cdot)$ is uniformly bounded on the support of $W_{t}$ and
$\mathcal{W}$, and $\varphi(w^{\prime},w)$ is continuous about
$w^{\prime}\in\mathcal{W}$. Besides, there exists some measurable function
$C_{\varphi}(\cdot)$ satisfying $\|C_{\varphi}\|_{2}<\infty$,

$\big{|}\varphi(W,w)-\varphi(W,w^{\prime})\big{|}\leq
C_{\varphi}(W)|w-w^{\prime}|^{\nu}$

with arbitrary $\nu\in(0,1]$.

  * A3

$K(\cdot)$ is a product kernel of $k(\cdot)$ such that
$K(u)=\prod_{j=1}^{d_{w}}k(u_{j})$, $\int u^{i}K(u)\,du=\delta_{0i}$ for
$i=0,1,\cdots,l-1$ and $\int u^{l}K(u)\,du\neq 0$, where $k(\cdot)$ is a
bounded, symmetric univariate kernel function and $\delta_{ij}$ is the delta
function taking value one when $i=j$ and zero otherwise.

  * A4

As $n\to\infty$, (i) $h\rightarrow 0$; (ii)
$nh^{2d_{w}}/\log^{\xi}n\rightarrow\infty$ and $nh^{2l}\rightarrow 0$ for some
positive $\xi>0$.

  * A5

For any $m\in\\{1,\ldots,n\\}$ and $\\{i_{1},\ldots
i_{m}\\}\subset\\{1,\ldots,n\\}$, the joint density for
$(W_{i_{1}}^{\top},\ldots,W_{i_{m}}^{\top})^{\top}$ is bounded.

  * A6

The density of $Y$ and $Z$ exist and is bounded.

  * A7

$F_{Y|W}(y|W)$ satisfy the following local Holder continuity: for any
$y,y^{\prime}\in\mathbb{R}^{d_{y}}$, there exist a distribution function
depending on $y,y^{\prime}$ such that

$\big{|}F_{Y|W}(y|W)-F_{Y|W}(y^{\prime}|W)\big{|}\leq
C_{F_{Y|W}}(W)\big{|}G_{y,y^{\prime}}(y)-G_{y,y^{\prime}}(y^{\prime})\big{|}^{\bar{\nu}}$

for some $\bar{\nu}>0$ satisfying

$\bar{\nu}>4\left(1+\frac{2}{\eta}\right)\left(\frac{d_{y}}{d_{y}+1}+\frac{d_{z}}{d_{z}+1}\right)$

with $\|C_{F_{Y|W}}(W)\|_{2}<\infty$. The same requirement is imposed on
$F_{Z|W}(z|W)$.

  * A8

(i) $\Delta(w,y,z)$ is uniformly bounded and is continuously differentiable
with respect to $w$ up to order $l$, and continuously differentiable with
respect to $y$ and $z$ (order $1$ is enough); (ii)
$n^{-1}\sum_{t=1}^{n}\varphi(W_{t},w)\Delta(W_{t},y,z)f_{W}(W_{t})\xrightarrow{\text{as}}G(w,y,z):=\mathrm{E}[\varphi(W_{1},w)\Delta(W_{1},y,z)f_{W}(W_{1})]$
uniformly in
$(w,y,z)\in\mathcal{W}\times\mathbb{R}^{d_{y}}\times\mathbb{R}^{d_{z}}$.

Remark on A1: Assumption A1 says that the higher dimension of the data is, the
stronger mixing condition we require!

Remark on A2: Assumption A2 says that we can use a very strong Holder property
in the bounded space to control the exponential increase in the covering
numbers cased by unbounded Euclidean space!

Remark on A7: Assumption A7 is not a strange assumption. As one can see, if we
ignore the d.f. $G_{y,y^{\prime}}$, Assumption A7 becomes:

$\big{|}F_{Y|W}(y|W)-F_{Y|W}(y^{\prime}|W)\big{|}\leq
C_{F_{Y|W}}(W)|y-y^{\prime}|^{\bar{\nu}},$

which is a very common assumption in various literature. However, since
$F_{Y|W}$ only value at $[0,1]$, the above assumption is meaningless when the
difference between $y$ and $y^{\prime}$ larger than $1$. To avoid this
problem, we introduce the function
$G_{y,y^{\prime}}:\mathbb{R}^{d_{y}}\rightarrow[0,1]$. Since
$G_{y,y^{\prime}}$ depends on $y$ and $y^{\prime}$, it is easy to see, the
relationship that $|F_{Y|W}(y|W)-F_{Y|W}(y^{\prime}|W)|\leq
C_{F_{Y|W}}(W)|G_{y,y^{\prime}}(y)-G_{y,y^{\prime}}(y^{\prime})|^{\bar{\nu}}$
can be easily satisfied if we didn’t ask further restrictions on
$G_{y,y^{\prime}}$. However, in order to calculate the covering (or
bracketing) number of the function class

$\mathcal{F}_{y,w}:=\big{\\{}F_{Y|W}(y|\cdot):\mathcal{W}\rightarrow[0,1]\,;\,y\in\mathbb{R}^{d_{y}}\big{\\}},$

some restrictions must be imposed on $G_{y,y^{\prime}}$. Here we choose it as
the distribution function (d.f.), any alternative is also possible. For
instance, we can assume $G_{y,y^{\prime}}$ is monotone when $d_{y}=1$.

##  Appendix B B Technical Lemmas

This section mainly serves for some part in Lemma C1 that says that the larger
dimension is, the more strong mixing condition requires. As well as the
important bracketing lemma for conditional distribution functions.

######  Lemma B1.

Let $\\{V_{i}\\}_{i=1}^{n}$ be a $v$-dimensional strong mixing process with
mixing coefficient $\alpha(\cdot)$. Let $F_{i_{1},\ldots,i_{m}}$ denote the
distribution function of $(V_{i_{1}},\ldots,V_{i_{m}})$. For any integer $m>1$
and integers $(i_{1},\ldots,i_{m})$ such that $1\leq
i_{1}<i_{2}<\cdots<i_{m}$, let $\theta$ be a Borel measurable function such
that
$\max\big{\\{}\int|\theta(v_{1},\ldots,v_{m})|^{1+\widetilde{\eta}}\,dF_{i_{1},\ldots,i_{j}}\,dF_{i_{j}+1,\ldots,i_{m}},\,\int|\theta(v_{1},\ldots,v_{m})|^{1+\widetilde{\eta}}\,dF_{i_{1},\ldots,i_{m}}\big{\\}}\leq
M_{n}$ for some $\widetilde{\eta}>0$. Then
$\big{|}\int\theta(v_{1},\ldots,v_{m})\,dF_{i_{1},\ldots,i_{m}}-\int\theta(v_{1},\ldots,v_{m})\,dF_{i_{1},\ldots,i_{j}}\,dF_{i_{j}+1,\ldots,i_{m}}\big{|}\leq
4M_{n}^{1/(1+\widetilde{\eta})}\alpha(i_{j+1}-i_{j})^{\widetilde{\eta}/(1+\widetilde{\eta})}$.

###### Proof.

See Lemma 2.1 in Sun and Chiang (1997). ∎

Now denote the collection of probability measures

$\displaystyle\mathcal{P}_{j}^{k}:=\bigg{\\{}P_{j}^{k}(V_{i_{1}},\ldots,V_{i_{k}}):=\prod_{s=1}^{j}P(\underline{V}_{s}):$
$\displaystyle\underline{V}_{s}\subset\\{V_{i_{1}},\ldots,V_{i_{k}}\\},$
$\displaystyle\bigcup_{s=1}^{j}\underline{V}_{s}=\\{V_{i_{1}},\ldots,V_{i_{k}}\\},\
\underline{V}_{s}\cap\underline{V}_{t}=\emptyset\text{ for all }s\neq
t\bigg{\\}},$

and we want to derive the property of the $U$-processes

$U_{n}:=\frac{1}{n^{2}}\sum_{1\leq i_{1}<i_{2}\leq
n}\phi(v_{i_{1}},v_{i_{2}})$

with $\int\phi(v_{1},v)\,dF_{v_{i}}(v)=\int\phi(v,v_{2})\,dF_{v_{i}}(v)=0$ for
all $i$. For any $m\in\mathbb{N}_{+}$, let $I_{m}=\\{i_{1},\ldots,i_{4m}\\}$,
and define

$\displaystyle M_{n1s}^{(m)}:=$ $\displaystyle\max_{1\leq i_{2k-1}<i_{2k}\leq
n,1\leq k\leq 2m\atop\text{ exactly }(4m+1)-s\text{ indices in }I_{m}\text{
are distinct }}\max_{1\leq j\leq(4m+1)-s}$
$\displaystyle\max_{P_{j}^{(4m+1)-s}\in\mathcal{P}_{j}^{(4m+1)-s}}\int\big{|}M(i_{1},\ldots,i_{4m})\big{|}^{1+\eta/(2m)}\,dP_{j}^{(4m+1)-s},\quad
s=1,2,\ldots,2m$ $\displaystyle M_{n2s}^{(m)}:=$ $\displaystyle\max_{1\leq
i_{2k-1}<i_{2k}\leq n,1\leq k\leq 2m\atop\text{ exactly }(4m-3)-s\text{
indices in }I_{m}\text{ are distinct }}\max_{1\leq j\leq(4m-3)-s}$
$\displaystyle\max_{P_{j}^{(4m-3)-s}\in\mathcal{P}_{j}^{(4m-3)-s}}\bigg{|}\int
M(i_{1},\ldots,i_{4m})\,dP_{j}^{(4m-3)-s}\bigg{|},\quad s=1,2,\ldots,2m-1.$

with $M(i_{1},\ldots,i_{4m})=\prod_{j=1}^{2m}\phi(v_{i_{2j-1}},v_{i_{2j}})$.

The following lemma is a extension for existent literature: Yoshihara (1976)
and Su and White (2012) show the case for $m=1$ and $m=2,3$ respectively.

######  Lemma B2.

If $\sum_{s=1}^{\infty}s^{2m-1}\alpha(s)^{\eta/(2m+\eta)}<\infty$ for some
$m\in\mathbb{N}_{+}$ and $\eta>0$, then

$\mathrm{E}U_{n}^{2m}=O\bigg{(}\sum_{s=1}^{2m}n^{-(2m-1)-s}\big{(}M_{n1s}^{(m)}\big{)}^{2m/(2m+\eta)}+\sum_{s=1}^{2m-1}n^{-(2m+1)-s}M_{n2s}^{(m)}\bigg{)}.$

###### Proof.

In the summation

$\mathrm{E}U_{n}^{2m}=n^{-4m}\sum_{j=1}^{2m}\sum_{1\leq i_{2j-1}<i_{2j}\leq
n}\phi(v_{i_{1}},v_{i_{2}})\cdots\phi(v_{i_{4m-1}},v_{i_{4m}}),$ (10)

there are $(4m-1)$ cases, denoted by $s=1,\ldots,4m-1$, in which are exactly
$(4m+1)-s$ distinct indices among $i_{1},\ldots,i_{4m}$. Let $1\leq
k_{1}\leq\ldots\leq k_{4m}\leq n$ be the permutation of $i_{1},\ldots,i_{4m}$
in non-decreasing order, and

$H(k_{1},\ldots,k_{4m}):=\phi(v_{i_{1}},v_{i_{2}})\cdots\phi(v_{i_{4m-1}},v_{i_{4m}}).$

For the first kind of case, i.e. $1\leq s\leq 2m$. We denote

$\underline{d}:=\min_{k_{i}<k_{j}}\big{(}k_{j}-k_{i}\big{)},\qquad\overline{d}:=\max_{k_{i}<k_{j}}\big{(}k_{j}-k_{i}\big{)}=\overline{d}(k_{1}^{*},k_{2}^{*}).$

By Lemma B1 with $\widetilde{\eta}=\eta/(2m)$, and noting that there are
exactly $(4m+1)-s$ distinct indices in case ($s$), we have

$\displaystyle\mathrm{E}U_{n(s)}$ $\displaystyle\leq
4\big{(}M_{n1s}^{(m)}\big{)}^{\frac{2m}{2m+\eta}}\underbrace{\sum\quad\cdots\quad\sum}_{\text{the
number of summation is exactly
}(4m+1)-s}\alpha^{\frac{\eta}{2m+\eta}}\big{(}\underline{d}\big{)}$
$\displaystyle\leq
4\big{(}M_{n1s}^{(m)}\big{)}^{\frac{2m}{2m+\eta}}\sum_{1\leq
k_{1}^{*}<k_{2}^{*}\leq
n}\overline{d}^{4m-1-s}\alpha^{\frac{\eta}{2m+\eta}}\big{(}\underline{d}\big{)}$
$\displaystyle\lesssim\big{(}M_{n1s}^{(m)}\big{)}^{\frac{2m}{2m+\eta}}n^{2m+1-s}\sum_{j=1}^{n}j^{2m-1}\alpha^{\frac{\eta}{2m+\eta}}(j),$

where the last step we use the fact
$\sum_{j=1}^{\infty}j^{2m-1}\alpha^{\frac{\eta}{2m+\eta}}(j)$ is finite.
Therefore, we have

$\mathrm{E}U_{n(s)}=O\Big{(}\big{(}M_{n1s}^{(m)}\big{)}^{\frac{2m}{2m+\eta}}n^{2m+1-s}\Big{)},\qquad
1\leq s\leq 2m.$

For the second kind of cases, i.e. $2m+1\leq s\leq 4m-1$, the expectation can
be calculated directly. Indeed, since the indices in pairs $(i_{2j-1},i_{2j})$
are different, $(4m+1)-s$ distinct ($s=2m+1,\ldots,4m-1$) indices means the
first ($i_{2j-1}$) or second ($i_{2j}$) must be all the same, and hence,

$\mathrm{E}U_{n(s)}\leq
M_{n2s}^{(m)}\underbrace{\sum\quad\cdots\quad\sum}_{\text{the number of
summation is exactly }(4m+1)-s}1=O\big{(}n^{(4m+1)-s}M_{n2s}^{(m)}\big{)}$

for $2m+1\leq s\leq 4m-1$. Then the result in the lemma follows. ∎

Write $a_{n}\lesssim b_{n}$ as there exists some positive constant $c$ free of
$n$ satisfying $a_{n}\leq cb_{n}$ for any $n\in\mathbb{N}$, and denote
$a_{n}\asymp b_{n}$ if $a_{n}\lesssim b_{n}$ and $b_{n}\lesssim a_{n}$. And
for notation simplicity, we denote
$\gamma=(w^{\top},y^{\top},z^{\top})^{\top}\in\mathbb{R}^{d}$ with
$d=d_{w}+d_{y}+d_{z}$.

######  Lemma B3.

Suppose Assumption A7 in Appendix A holds and $(Y,W)$ follow the joint law
$P$, then the bracketing number of the class

$\mathcal{F}:=\big{\\{}F_{Y|W}(y|\cdot):\mathcal{W}\rightarrow[0,1]\,;\,y\in\mathbb{R}^{d_{y}}\big{\\}}$

satisfies

$\log
N_{[\;]}\big{(}\varepsilon,\mathcal{F},L_{2}(P)\big{)}\lesssim\varepsilon^{-2d_{y}/(\bar{\nu}(d_{y}+1))}$

for any $\varepsilon\in(0,1]$

###### Proof.

When $\|C_{F_{Y|W}}\|_{2}=0$, the result is trivial. Now suppose
$\|C_{F_{Y|W}}\|_{2}\neq 0$. Note that

$\mathcal{F}\subseteq\big{\\{}F_{Y|W}(y|\cdot):\mathcal{W}\rightarrow[0,1]\text{
satisfying Assumption A7
};\,G_{y,y^{\prime}}:\mathbb{R}^{d_{y}}\rightarrow[0,1]\text{ is a
d.f.}\big{\\}}=:\overline{\mathcal{F}}.$

Then from the Assumption A7 by using a little extension of Theorem 2.7.11 in
Van Der Vaart and Wellner (1996) which says $\mathcal{F}=\\{f_{t}:t\in T\\}$
have property that $|f_{s}(x)-f_{t}(x)|\leq d(s,t)^{\bar{\nu}}F(x)$ for some
metric $d$ and fixed function $F(x)$, then for any norm $\|\cdot\|$,

$N_{[\;]}\big{(}2\varepsilon^{\bar{\nu}}\|F\|,\mathcal{F},\|\cdot\|\big{)}\leq
N(\varepsilon,T,d)$

with taking the norm $\|\cdot\|=\|\cdot\|_{2}$, we have

$\displaystyle\log N_{[\;]}\big{(}\varepsilon,\mathcal{F},L_{2}(P)\big{)}$
$\displaystyle\leq\log
N_{[\;]}\big{(}\varepsilon,\overline{\mathcal{F}},L_{2}(P)\big{)}$
$\displaystyle\leq\log
N\big{(}(2^{-1}\varepsilon/\|C_{F_{Y|W}}\|_{2})^{1/\bar{\nu}},\\{G_{y,y^{\prime}}:\mathbb{R}^{d_{y}}\rightarrow[0,1]\text{
is a d.f.}\\},L_{2}(P)\big{)}$
$\displaystyle\lesssim\varepsilon^{-\frac{2d_{y}}{\bar{\nu}(d_{y}+1)}}$

where the last $\lesssim$ comes from Example 1.6.4 in Wellner (2002). ∎

##  Appendix C C Main Proofs

Denote $\gamma:=(w^{\top},y^{\top},z^{\top})\in\mathbb{R}^{d}$.

######  Lemma C1.

Under Assumptions A1 - A7 in Appendix A and null hypothesis,

$\displaystyle\sup_{(w,y,z)\in\mathcal{W}\times\mathbb{R}^{d_{y}}\times\mathbb{R}^{d_{z}}}\left|S_{n}(w,y,z)-\bar{S}_{n}(w,y,z)\right|=o_{p}(1),$

with

$\displaystyle\bar{S}_{n}(w,y,z)=\frac{1}{\sqrt{n}}\sum_{t=1}^{n}\varphi(W_{t},w)\phi_{t}(y)\epsilon_{t}(z)f_{W}(W_{t}):=\frac{1}{\sqrt{n}}\sum_{t=1}^{n}e_{t}(w,y,z),$

where

$\displaystyle\phi_{t}(y)=1(Y_{t}\leq y)-F_{Y|W}(y|W_{t}),$

and

$\displaystyle\epsilon_{t}(z)=1(Z_{t}\leq z)-F_{Z|W}(z|W_{t}).$

###### Proof.

First, we note that $\mathcal{W}$ is bounded, whereas the value space for
$Y_{t}$ and $Z_{t}$ is the whole Euclidean space. So we consider the bounded
space

$\mathbb{B}_{n}:=\big{\\{}(y,z):|(y,z)|\leq\ell_{n}\big{\\}}\subseteq\mathbb{R}^{d_{y}+d_{z}}$

with some slowly increasing sequence $\ell_{n}\uparrow\infty$ we will specify
later. Now the set $\Gamma_{n}:=\mathcal{W}\times\mathbb{B}_{n}$ is bounded
and can be cover by
$\Gamma_{nk}:=\\{\gamma_{nk}:|\gamma-\gamma_{nk}|\leq\delta_{n},\,k=1,\ldots,N_{n}\\}$
with $\delta_{n}=n^{-1/2}(\log\log n)^{-\kappa}$ for some positive $\kappa>0$
and $N_{n}\asymp\ell_{n}^{d_{y}+d_{z}}\delta_{n}^{-d}$

$\sup_{\gamma\in\Gamma_{n}}|S_{n}(\gamma)|\leq\max_{1\leq k\leq
N_{n}}|S_{n}(\gamma_{nk})|+\max_{1\leq k\leq
N_{n}}\sup_{\gamma\in\Gamma_{n}}|S_{n}(\gamma)-S_{n}(\gamma_{nk})|.$ (11)

Define $\chi_{t}=(W^{\top}_{t},Y^{\top}_{t},Z^{\top}_{t})^{\top}$. We can
rewrite $S_{n}(\gamma)$ as a form of standard $U$-statistic. Introduce

$\displaystyle
U_{\gamma}(\chi_{t},\chi_{s})=\frac{1}{2h^{d_{w}}}K\left(\frac{W_{t}-W_{s}}{h}\right)\varphi(W_{t},w)1(Y_{t}\leq
y)(1(Z_{t}\leq z)-1(Z_{s}\leq z))$
$\displaystyle\qquad\qquad+\frac{1}{2h^{d_{w}}}K\left(\frac{W_{t}-W_{s}}{h}\right)\varphi(W_{s},w)1(Y_{s}\leq
y)(1(Z_{s}\leq z)-1(Z_{t}\leq z))$ $\displaystyle=$
$\displaystyle\frac{1}{2h^{d_{w}}}K\left(\frac{W_{t}-W_{s}}{h}\right)\big{(}\varphi(W_{t},w)1(Y_{t}\leq
y)-\varphi(W_{s},w)1(Y_{s}\leq y)\big{)}\big{(}1(Z_{t}\leq z)-1(Z_{s}\leq
z)\big{)}.$

Therefore, we have

$S_{n}(\gamma)=\frac{n}{n-1}\frac{1}{n^{3/2}}\sum_{1\leq s<t\leq
n}^{n}U_{\gamma}(\chi_{s},\chi_{t}).$

We reply on the $U$-statistic theory to study the asymptotic behavior of the
above $U$-statistic; that is, by Hoeffding decomposition, we have

$S_{n}(\gamma)=S^{(1)}_{n}(\gamma)+S^{(2)}_{n}(\gamma),$ (12)

where

$S^{(1)}_{n}(\gamma)=\frac{2}{\sqrt{n}}\sum_{s=1}^{n}\mathrm{E}U_{\gamma}(\chi,\chi_{t})\big{|}_{\chi=\chi_{s}}:=\frac{2}{\sqrt{n}}\sum_{s=1}^{n}U_{1\gamma}(\chi_{s}),$

and

$S^{(2)}_{n}(\gamma)=\frac{1}{n^{1/2}(n-1)}\sum_{1\leq s<t\leq
n}^{n}\Big{[}U_{\gamma}(\chi_{s},\chi_{t})-U_{1\gamma}(\chi_{s})-U_{1\gamma}(\chi_{t})\Big{]},$

where $F_{\chi}(\chi_{t})$ denotes the distribution of $\chi_{t}$. It is
straightforward to show that

$\displaystyle S^{(1)}_{n}(\gamma)=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{s=1}^{n}\int\frac{1}{h^{d_{w}}}K\left(\frac{W_{t}-W_{s}}{h}\right)\big{(}\varphi(W_{t},w)1(Y_{t}\leq
y)-\varphi(W_{s},w)1(Y_{s}\leq y)\big{)}$
$\displaystyle\times\big{(}1(Z_{t}\leq z)-1(Z_{s}\leq
z)\big{)}\,dF_{\chi}(\chi_{t})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{s=1}^{n}\int\frac{1}{h^{d_{w}}}K\left(\frac{W_{t}-W_{s}}{h}\right)\varphi(W_{t},w)1(Y_{t}\leq
y)1(Z_{t}\leq z)\,dF_{\chi}(\chi_{t})$
$\displaystyle-\frac{1}{\sqrt{n}}\sum_{s=1}^{n}\int\frac{1}{h^{d_{w}}}K\left(\frac{W_{t}-W_{s}}{h}\right)\varphi(W_{t},w)1(Y_{t}\leq
y)1(Z_{s}\leq z)\,dF_{\chi}(\chi_{t})$
$\displaystyle-\frac{1}{\sqrt{n}}\sum_{s=1}^{n}\varphi(W_{s},w)1(Y_{s}\leq
y)\int\frac{1}{h^{d_{w}}}K\left(\frac{W_{t}-W_{s}}{h}\right)1(Z_{t}\leq
z)\,dF_{\chi}(\chi_{t})$
$\displaystyle+\frac{1}{\sqrt{n}}\sum_{s=1}^{n}\varphi(W_{s},w)1(Y_{s}\leq
y)1(Z_{s}\leq
z)\int\frac{1}{h^{d_{w}}}K\left(\frac{W_{t}-W_{s}}{h}\right)\,dF_{\chi}(\chi_{t})$
$\displaystyle:=$ $\displaystyle\sum_{j=1}^{4}A_{nj},$

where in each integral, we regard $\chi_{t}$ as nonrandom variable with on
relation to $\chi_{s}$. Let us first calculate $A_{n1}$, which is equal to

$\displaystyle
A_{n1}=\frac{1}{\sqrt{n}}\sum_{s=1}^{n}\int\frac{1}{h^{d_{w}}}K\left(\frac{w^{\prime}-W_{s}}{h}\right)\varphi(w^{\prime},w)1(y^{\prime}\leq
y)1(z^{\prime}\leq z)\,dF(w^{\prime},y^{\prime},z^{\prime})$
$\displaystyle=\frac{1}{\sqrt{n}}\sum_{s=1}^{n}\frac{1}{h^{d_{w}}}\int
K\left(\frac{w^{\prime}-w}{h}\right)\varphi(w^{\prime},w)1(y^{\prime}\leq
y)1(z^{\prime}\leq
z)f_{W}(w^{\prime})f_{Y|W}(y^{\prime}|w^{\prime})f_{Z|W}(z^{\prime}|w^{\prime})\,dw^{\prime}\,dy^{\prime}\,dz^{\prime}$
$\displaystyle=\frac{1}{\sqrt{n}}\sum_{s=1}^{n}\int
K(u)\varphi(W_{s}+hu,w)1(y^{\prime}\leq y)1(z^{\prime}\leq z)$
$\displaystyle\qquad\qquad\qquad\times
f_{Y|W}(y^{\prime}|W_{s}+hu)f_{Z|W}(z^{\prime}|W_{s}+hu)f_{W}(W_{s}+hu)\,du\,dy^{\prime}\,dz^{\prime}$
$\displaystyle=\frac{1}{\sqrt{n}}\sum_{s=1}^{n}\varphi(W_{s}+hu,w)\int
1(y^{\prime}\leq y)f_{Y|W}(y^{\prime}|W_{s})\,dy^{\prime}\int 1(z^{\prime}\leq
z)f_{Z|W}(z^{\prime}|W_{s})\,dz^{\prime}f_{W}(W_{s})+O_{p}(\sqrt{n}h^{l})$
$\displaystyle=\frac{1}{\sqrt{n}}\sum_{s=1}^{n}\varphi(W_{s},w)F_{Y|W}(y|W_{s})F_{Z|W}(z|W_{s})f_{W}(W_{s})+o_{p}(1),$

where we apply the differentiable property of $f_{Y|W}(y|w)$, $f_{Z|W}(z|w)$,
and $f_{W}(w)$ in Assumption A2 and the higher order kernel in Assumption A3
and the bandwidth condition in Assumption A4, and use the fact
$F_{YZ|W}(y,z|w)=F_{Y|W}(y|w)F_{Z|W}(z|w)$ under the null of conditional
independence, so that
$dF_{\chi}(\chi^{\prime})=f_{Y|W}(y^{\prime}|w^{\prime})\,f_{Z|W}(z^{\prime}|w^{\prime})f_{W}(w^{\prime})\,dw^{\prime}\,dy^{\prime}\,dz^{\prime}$.
Similarly, we can find that

$\displaystyle
A_{n2}=-\frac{1}{\sqrt{n}}\sum_{s=1}^{n}\varphi(W_{s},w)F_{Y|W}(y|W_{s})1(Z_{s}\leq
z)f_{W}(W_{s})+o_{p}(1),$ $\displaystyle
A_{n3}=-\frac{1}{\sqrt{n}}\sum_{s=1}^{n}\varphi(W_{s},w)1(Y_{s}\leq
y)F_{Z|W}(z|W_{s})f_{W}(W_{s})+o_{p}(1),$

and

$\displaystyle
A_{n4}=\frac{1}{\sqrt{n}}\sum_{s=1}^{n}\varphi(W_{s},w)1(Y_{s}\leq
y)1(Z_{s}\leq z)f_{W}(W_{s})+o_{p}(1).$

Consequently,

$S^{(1)}_{n}(\gamma)=\frac{1}{\sqrt{n}}\sum_{s=1}^{n}\varphi(W_{s},w)\big{(}1(Y_{s}\leq
y)-F_{Y|W}(y|W_{s})\big{)}\big{(}1(Z_{s}\leq
z)-F_{Z|W}(z|W_{s})\big{)}f_{W}(W_{s})+o_{p}(1).$ (13)

Denote
$\psi_{\gamma}(\chi_{s},\chi_{t})=U_{\gamma}(\chi_{s},\chi_{t})-U_{1\gamma}(\chi_{s})-U_{1\gamma}(\chi_{t})$.
We next show that

$S^{(2)}_{n}(\gamma)=\frac{1}{\sqrt{n}(n-1)}\sum_{1\leq s<t\leq
n}^{n}\psi_{\gamma}(\chi_{s},\chi_{t})=\frac{1}{\sqrt{n}(n-1)}\sum_{t=s}^{n-1}\sum_{t=s+1}^{n}\psi_{\gamma}(\chi_{s},\chi_{t})=o_{p}(1),$

uniformly in $(w,y,z)$. For the $U$-process $n^{-1/2}S_{n}^{(2)}(\gamma)$,
define $M_{n1s}^{(m)}$ and $M_{n2s}^{(m)}$ as in Appendix B. For $s=1$, we
have all the probability measure in $\mathcal{P}_{j}^{(4m+1)-1}$ is separable.
Denote $\\{\chi_{t}^{*}\\}_{t=1}^{n}$ as the same series to
$\\{\chi_{t}\\}_{t=1}^{n}$ except $\chi_{s}^{*}\perp\chi_{t}^{*}$ for $s\neq
t$, then the integral in $M_{n11}^{(m)}$ is

$\displaystyle\int\big{|}M(i_{1},\ldots,i_{4m})\big{|}^{1+\eta/(2m)}\,dP_{j}^{4m}$
$\displaystyle=\prod_{j=1}^{2m}\int\big{|}\psi_{\gamma}(\chi_{i_{2j-1}},\chi_{i_{2j}})\big{|}^{1+\eta/(2m)}\,dP_{i_{2j-1}}\,dP_{i_{2j}}$
$\displaystyle=\prod_{j=1}^{2m}\mathrm{E}\big{|}\psi_{\gamma}(\chi_{i_{2j-1}}^{*},\chi_{i_{2j}}^{*})\big{|}^{1+\eta/(2m)}\leq\prod_{j=1}^{2m}\sqrt{\mathrm{E}\psi_{\gamma}^{2+\eta/m}(\chi_{i_{2j-1}}^{*},\chi_{i_{2j}}^{*})}.$

Now, for the second expectation in the above inequality,

$\mathrm{E}\psi_{\gamma}^{2+\eta/m}(\chi_{s}^{*},\chi_{t}^{*})\lesssim\mathrm{E}U_{\gamma}^{2+\eta/m}(\chi_{s}^{*},\chi_{t}^{*})+2\mathrm{E}U_{1\gamma}^{2+\eta/m}(\chi_{s}^{*}),$

with

$\displaystyle\mathrm{E}U_{\gamma}^{2+\eta/m}(\chi_{s}^{*},\chi_{t}^{*})$
$\displaystyle=\frac{1}{(2h^{d_{w}})^{2+\eta/m}}\int du\int
K^{2+\eta/m}\left(\frac{u-v}{h}\right)\big{(}\varphi(u,w)1(Y_{s}\leq y)$
$\displaystyle\qquad\qquad-\varphi(v,w)1(Y_{t}\leq
y)\big{)}^{2+\eta/m}\big{(}1(Z_{t}\leq z)-1(Z_{s}\leq
z)\big{)}^{2+\eta/m}\,dv$
$\displaystyle\lesssim\frac{1}{h^{(2+\eta/m)d_{w}}}\int du\int
K^{2+\eta/m}\left(\frac{u-v}{h}\right)\,dv\asymp h^{-\eta d_{w}/m},$

where the last step comes from the boundedness of the set $\mathcal{W}$, and

$\mathrm{E}U_{1\gamma}^{2+\eta/2}(\chi_{s}^{*})=\mathrm{E}\Big{[}\mathrm{E}\big{[}U_{\gamma}^{2+\eta/m}(\chi_{s}^{*},\chi_{t}^{*})\,|\,\chi_{t}^{*}\big{]}\Big{]}\asymp
h^{-\eta d_{w}/m},$

which implies
$\mathrm{E}\psi_{\gamma}^{2+\eta/m}(\chi_{s}^{*},\chi_{t}^{*})=O\big{(}h^{-\eta
d_{w}/m}\big{)}$. Hence

$\int\big{|}M(i_{1},\ldots,i_{4m})\big{|}^{1+\eta/(2m)}\,dP_{j}^{2m}\lesssim\prod_{j=1}^{2m}O\big{(}h^{-\eta
d_{w}/(2m)}\big{)}=O\big{(}h^{-\eta d_{w}}\big{)},$

and $M_{n11}^{(m)}=O\big{(}h^{-\eta d_{w}}\big{)}$. However, for other
$M_{n1s}^{(m)}$ ($s=2,\ldots,2m$), the integrals in them cannot be totally
separable. For fixed $2\leq s\leq 2m$, we denote
$\\{\mathscr{D}_{\ell}\\}_{\ell=1}^{(4m+1)-s}$ are subsets chosen from
$\\{i_{1},\ldots,i_{4m}\\}$ arbitrarily, then
$M_{n1s}^{(m)}\leq\max_{t<s}M_{n1t}^{(m)}$ as well as

$\displaystyle M_{n1s}^{(m)}$ $\displaystyle\leq\max_{i_{j},\,1\leq i_{j}\leq
4m}\max_{\mathscr{D}_{\ell},\,1\leq\ell\leq(4m+1)-s}\int\big{|}M(i_{1},\ldots,i_{4m})\big{|}^{1+\eta/(2m)}dP_{j}^{(4m+1)-s}$
(14) $\displaystyle\leq\max_{i_{j},\,1\leq i_{j}\leq
4m}\max_{\mathscr{D}_{\ell},\,1\leq\ell\leq(4m+1)-s}\Bigg{[}\prod_{|\mathscr{D}_{\ell_{1}}|=1}\int\prod_{i_{2j-1},i_{2j}\notin\mathscr{D}_{\ell_{2}}}\psi_{\gamma}^{2+\eta/m}(\chi_{i_{2j-1}},\chi_{i_{2j}})\,dP_{\mathscr{D}_{\ell_{1}}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\times\prod_{|\mathscr{D}_{\ell_{2}}|\geq
2}\int\prod_{i_{2j-1}\text{ or
}i_{2j}\in\mathscr{D}_{\ell_{2}}}\psi_{\gamma}^{2+\eta/m}(\chi_{i_{2j-1}},\chi_{i_{2j}})\,dP_{\mathscr{D}_{\ell_{2}}}\Bigg{]}^{1/2}.$

For finding the rate of the above integral, we first tackle each sub-sub-
integral the last part of the sub-integral. By changing of variables, it is
not hard to obtain that

$\displaystyle\int\prod_{i_{2j-1}\text{ or
}i_{2j}\in\mathscr{D}_{\ell_{2}}}\psi_{\gamma}^{2+\eta/m}(\chi_{i_{2j-1}},\chi_{i_{2j}})\,dP_{\mathscr{D}_{\ell_{2}}}$
$\displaystyle\leq\frac{1}{\big{(}h^{|\mathscr{D}_{\ell_{2}}|d_{w}}\big{)}^{2+\eta/m}}\int\prod_{a=1}^{|\mathscr{D}_{\ell_{2}}|}K^{2+\eta/m}\left(\frac{u_{i_{a}}-W_{i_{a}^{*}}}{h}\right)f_{W_{i_{1}^{*}},\ldots,W_{i_{|\mathscr{D}_{\ell_{2}}|}^{*}}}\big{(}u_{i_{1}},\ldots,u_{i_{|\mathscr{D}_{\ell_{2}}|}}\big{)}\,du$
$\displaystyle=\frac{1}{\big{(}h^{|\mathscr{D}_{\ell_{2}}|d_{w}}\big{)}^{2+\eta/m}}\times
O\big{(}h^{|\mathscr{D}_{\ell_{2}}|d_{w}}\big{)}=O\big{(}h^{-(1+\eta/m)|\mathscr{D}_{\ell_{2}}|d_{w}}\big{)},$

where

$\bigcup_{a=1}^{|\mathscr{D}_{\ell_{2}}|}\big{\\{}(i_{a},i_{a}^{*})\big{\\}}=\bigcup_{i_{2j-1}\text{
or }i_{2j}\in\mathscr{D}_{\ell_{2}}}\big{\\{}(i_{2j-1},i_{2j})\big{\\}},$

and the first ”$=$” is due to the boundedness of $\mathcal{W}$ as well as
Assumption A5. On the other hand, the order of the first sub-integral in (14)
is obvious due to the calculation of $M_{n11}^{(m)}$ given the correct order
of the second sub-integral:

$\displaystyle\prod_{|\mathscr{D}_{\ell_{1}}|=1}\int\prod_{i_{2j-1},i_{2j}\notin\mathscr{D}_{\ell_{2}}}\psi_{\gamma}^{2+\eta/m}(\chi_{i_{2j-1}},\chi_{i_{2j}})\,dP_{\mathscr{D}_{\ell_{1}}}$
$\displaystyle=\prod_{i_{2j-1},i_{2j}\notin\mathscr{D}_{\ell_{2}}}\int\psi_{\gamma}^{2+\eta/m}(\chi_{i_{2j-1}},\chi_{i_{2j}})\,dP_{i_{2j-1}}\,dP_{i_{2j}}=\prod_{i_{2j-1},i_{2j}\notin\mathscr{D}_{\ell_{2}}}\mathrm{E}\psi_{\gamma}^{2+\eta/m}(\chi_{i_{2j-1}}^{*},\chi_{i_{2j}}^{*})$
$\displaystyle=O\big{(}h^{-N(\mathscr{D}_{\ell_{1}})\eta d_{w}/m}\big{)},$

where $N(\mathscr{D}_{\ell_{1}})$ is the number of $\mathscr{D}_{\ell_{1}}$.
Combining these results, we immediately get from (14) that

$\displaystyle M_{n1s}^{(m)}$
$\displaystyle\leq\max_{\mathscr{D}_{\ell},\,1\leq\ell\leq(4m+1)-s}\sqrt{O\big{(}h^{-N(\mathscr{D}_{\ell_{1}})\eta
d_{w}/m}\big{)}\times
O\big{(}h^{-N(\mathscr{D}_{\ell_{2}})(1+\eta/m)|\mathscr{D}_{\ell_{2}}|d_{w}}\big{)}}$
(15)
$\displaystyle=\max_{\mathscr{D}_{\ell},\,1\leq\ell\leq(4m+1)-s}O\Big{(}h^{-d_{w}/2\cdot\left(N(\mathscr{D}_{\ell_{1}})\eta/m+(1+\eta/m)N(\mathscr{D}_{\ell_{2}})|\mathscr{D}_{\ell_{2}}|\right)}\Big{)},$

where $N(\mathscr{D}_{\ell_{1}})+N(\mathscr{D}_{\ell_{2}})=(4m+1)-s$ and
$N(\mathscr{D}_{\ell_{1}})+|\mathscr{D}_{\ell_{2}}|N(\mathscr{D}_{\ell_{2}})=4m$.
This is a integer optimization problem, it is obvious that different $m$ may
have different solutions. So we can only deal with the worst case. It is not
hard to see what we want is to maximize the power coefficients in (15), and
hence the worst case in asymptotic sense is

$|\mathscr{D}_{\ell_{2}}|=s,\qquad N(\mathscr{D}_{\ell_{1}})=4m-s,\qquad
N(\mathscr{D}_{\ell_{2}})=1.$

Therefore, the order of $M_{n1s}^{(m)}$ is

$M_{n1s}^{(m)}\leq\max_{1\leq t<s}O\big{(}M_{n1t}^{(m)}\big{)}\vee
O\Big{(}h^{-d_{w}(4\eta+s)/2}\Big{)}=O\Big{(}h^{-d_{w}(4\eta+s)/2}\Big{)},\qquad
s=2,3,\ldots,2m.$

To determine the order of $M_{n2s}^{(m)}$, we first use the inequality,

$\displaystyle\bigg{|}\int M(i_{1},\ldots,i_{4m})\,dP_{j}^{(4m-3)-s}\bigg{|}$
$\displaystyle\leq\bigg{|}\int
M_{2}(\mathscr{I}_{2})\,dP_{2j}^{(4m-3)-s}\int\big{|}M_{1}(\mathscr{I}_{1})\big{|}\,dP_{1j}^{(4m-3)-s}\bigg{|}$
$\displaystyle\leq\Bigg{|}\int
M_{2}(\mathscr{I}_{2})\,dP_{2j}^{(4m-3)-s}\sqrt{\int\big{|}M_{1}(\mathscr{I}_{1})\big{|}^{2}\,dP_{1j}^{(4m-3)-s}}\Bigg{|},$

where the $\mathscr{I}_{1}\cup\mathscr{I}_{2}=\\{i_{1},\ldots,i_{4m}\\}$,
$\mathscr{I}_{1}\cap\mathscr{I}_{2}=\emptyset$, and the following separable
property guarantees,

$\displaystyle M(i_{1},\ldots,i_{4m})=M_{1}(\mathscr{I}_{1})\times
M_{2}(\mathscr{I}_{2}),$ $\displaystyle
P_{j}^{(4m-3)-s}(\chi_{i_{1}},\ldots,\chi_{i_{8}})=P_{1j}^{(4m-3)-s}\big{(}(\chi_{j})_{j\in\mathscr{I}_{1}}\big{)}\times
P_{2j}^{(4m-3)-s}\big{(}(\chi_{j})_{j\in\mathscr{I}_{2}}\big{)},$

then by using the same notations and method as dealing with $M_{n1s}^{(m)}$,
we can get $M_{n2s}^{(m)}\leq\max_{t<s}M_{n2t}^{(m)}$ (where we define
$M_{n20}^{(m)}=1$), as well as

$\displaystyle M_{n2s}^{(m)}$ $\displaystyle\leq\max_{i_{j},\,1\leq i_{j}\leq
4m}\max_{\mathscr{D}_{\ell},\,1\leq\ell\leq(4m-3)-s}\Bigg{|}\prod_{|\mathscr{D}_{\ell_{1}}|=1}\prod_{i_{2k-1},i_{2k}\notin\mathscr{D}_{\ell_{2}}}\int\psi_{\gamma}(\chi_{i_{2k-1}},\chi_{i_{2k}})\,dP_{\mathscr{D}_{\ell_{1}}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\times\sqrt{\prod_{|\mathscr{D}_{\ell_{2}}|\geq
2}\int\prod_{i_{2k-1}\text{ or
}i_{2k}\in\mathscr{D}_{\ell_{2}}}\psi_{\gamma}^{2}(\chi_{i_{2k-1}},\chi_{i_{2k}})\,dP_{\mathscr{D}_{\ell_{2}}}}\,\Bigg{|}$
$\displaystyle\leq\max_{i_{j},\,1\leq i_{j}\leq
4m}\max_{\mathscr{D}_{\ell},\,1\leq\ell\leq(4m-3)-s}\bigg{|}O(1)\times\sqrt{\prod_{|\mathscr{D}_{\ell_{2}}|\geq
2}\frac{1}{(h^{|\mathscr{D}_{\ell_{2}}|d_{w}})^{2}}\cdot
O(h^{|\mathscr{D}_{\ell_{2}}|d_{w}})}\bigg{|}$
$\displaystyle=O\big{(}h^{-\widetilde{N}(\mathscr{D}_{\ell_{2}})|\mathscr{D}_{\ell_{2}}|d_{w}/2}\big{)},$

where the number of $\mathscr{D}_{\ell_{1}}$ and $\mathscr{D}_{\ell_{2}}$
satisfies

$\widetilde{N}(\mathscr{D}_{\ell_{1}})+\widetilde{N}(\mathscr{D}_{\ell_{2}})=(4m-3)-s,\qquad\widetilde{N}(\mathscr{D}_{\ell_{1}})+|\mathscr{D}_{\ell_{2}}|\widetilde{N}(\mathscr{D}_{\ell_{2}})=4m.$

Similarly, it is easy to see the asymptotically worst case is

$|\mathscr{D}_{\ell_{2}}|=s,\qquad\widetilde{N}(\mathscr{D}_{\ell_{1}})=4(m-1)-s,\qquad\widetilde{N}(\mathscr{D}_{\ell_{2}})=1.$

Therefore, the order of $M_{n2s}^{(m)}$ is

$M_{n2s}^{(m)}\leq\max_{0\leq t<s}O\big{(}M_{n1t}^{(m)}\big{)}\vee
O\big{(}h^{-(s+4)d_{w}/2}\big{)}=O\big{(}h^{-(s+4)d_{w}/2}\big{)},\qquad
s=1,\ldots,2m-1.$

Now, from Lemma B2,

$\mathrm{E}\big{[}n^{-1/2}S_{n}^{(2)}(\gamma)\big{]}^{2m}=O\bigg{(}\sum_{s=1}^{2m}n^{-(2m-1)-s}\big{(}M_{n1s}^{(m)}\big{)}^{2m/(2m+\eta)}+\sum_{s=1}^{2m-1}n^{-(2m+1)-s}M_{n2s}^{(m)}\bigg{)}.$

For the first summation,

$\displaystyle\sum_{s=1}^{2m}n^{-(2m-1)-s}\big{(}M_{n1s}^{(m)}\big{)}^{2m/(2m+\eta)}$
$\displaystyle\asymp n^{-2m}h^{-\eta
d_{w}\cdot\frac{2m}{2m+\eta}}+\sum_{s=2}^{2m}n^{-(2m-1)-s}h^{-\frac{d_{w}(4\eta+s)}{2}\cdot\frac{2m}{2m+\eta}}$
$\displaystyle=n^{-2m}h^{-\frac{2m\eta
d_{w}}{2m+\eta}}+n^{-2m-1}h^{-\frac{md_{w}(4\eta+2)}{2m+\eta}}.$

where we note that for $s<t$,

$\frac{n^{-(2m-1)-s}h^{-\frac{d_{w}(4\eta+s)}{2}\cdot\frac{2m}{2m+\eta}}}{n^{-(2m-1)-t}h^{-\frac{d_{w}(4\eta+t)}{2}\cdot\frac{2m}{2m+\eta}}}=\Big{(}nh^{\frac{md_{w}}{2m+\eta}}\Big{)}^{t-s}\gg\big{(}nh^{2d_{w}}\big{)}^{t-s}\rightarrow\infty.$

Similarly, for the second summation,

$\sum_{s=1}^{2m-1}n^{-(2m+1)-s}M_{n2s}^{(m)}\asymp\sum_{s=1}^{2m-1}n^{-(2m+1)-s}h^{-(s+4)d_{w}/2}=n^{-2(m+1)}h^{-5d_{w}/2}$

with

$\frac{n^{-(2m+1)-s}h^{-(s+4)d_{w}/2}}{n^{-(2m+1)-t}h^{-(t+4)d_{w}/2}}=\big{(}nh^{d_{w}/2}\big{)}^{t-s}\rightarrow\infty.$

for any $s<t$. Therefore, we obtain that

$\mathrm{E}\big{[}n^{-1/2}S_{n}^{(2)}(\gamma)\big{]}^{2m}=O\Big{(}n^{-2m}h^{-\frac{2m\eta
d_{w}}{2m+\eta}}+n^{-2m-1}h^{-\frac{md_{w}(4\eta+2)}{2m+\eta}}+n^{-2(m+1)}h^{-5d_{w}/2}\Big{)}.$
(16)

Furthermore, for any $\epsilon>0$, by Markov inequality

$\displaystyle\mathrm{P}\bigg{(}\max_{1\leq k\leq
N_{n}}\big{|}S_{n}^{(2)}(\gamma_{nk})\big{|}\geq\epsilon\bigg{)}$
$\displaystyle\leq N_{n}\max_{1\leq k\leq
N_{n}}\mathrm{P}\Big{(}\big{|}n^{-1/2}S_{n}^{(2)}(\gamma_{nk})\big{|}\geq
n^{-1/2}\epsilon\Big{)}$ $\displaystyle\leq\epsilon^{-4}N_{n}n^{m}\max_{1\leq
k\leq N_{n}}\mathrm{E}\big{[}n^{-1/2}S_{n}^{(2)}(\gamma_{nk})\big{]}^{2m}$
$\displaystyle\asymp\epsilon^{-4}\big{(}A_{1}+A_{2}+A_{3}\big{)}.$

From (16), we can precisely give the order of $A_{j}$, $j=1,2,3$. First, for
$A_{1}$,

$A_{1}=\ell_{n}^{d_{y}+d_{z}}n^{d/2}(\log\log)^{d\kappa}n\cdot
n^{-m}h^{-\frac{2m\eta
d_{w}}{2m+\eta}}=\frac{\ell_{n}^{d_{y}+d_{z}}(\log\log)^{\kappa
d}n}{\log^{(m-d/2)\xi}n}\left(\frac{nh^{2d_{w}\cdot\frac{m\eta}{(2m+\eta)(m-d/2)}}}{\log^{\xi}n}\right)^{-(m-d/2)}.$

Since by Assumption A1, we have

$m-d/2\geq 0,\qquad\frac{m\eta}{(2m+\eta)(m-d/2)}\leq 1,$

take

$\left(\frac{\log^{(m-d/2)\xi}n}{(\log\log)^{\kappa
d}n}\right)^{1/(d_{y}+d_{z})}\gg\ell_{n}\uparrow\infty,$

we have $A_{1}=o(1)$. Similarly, take $\ell_{n}$ are above, for $A_{2}$ and
$A_{3}$,

$\displaystyle A_{2}=\frac{\ell_{n}^{d_{y}+d_{z}}(\log\log)^{\kappa
d}n}{\log^{(m+1-d/2)\xi}n}\left(\frac{nh^{2d_{w}\cdot\frac{m(2\eta+1)}{(2m+\eta)(m+1-d/2)}}}{\log^{\xi}n}\right)^{-(m+1-d/2)}=o(1),$
$\displaystyle A_{3}=\frac{\ell_{n}^{d_{y}+d_{z}}(\log\log)^{\kappa
d}n}{\log^{(m+2-d/2)\xi}n}\left(\frac{nh^{2d_{w}\cdot\frac{5}{4(m+2-d/2)}}}{\log^{\xi}n}\right)^{-(m+2-d/2)}=o(1),$

given

$\frac{m(2\eta+1)}{(2m+\eta)(m+1-d/2)}\leq 1,\qquad\frac{5}{4(m+2-d/2)}\leq
1,$

by Assumption A1. Thus

$\max_{1\leq k\leq N_{n}}\big{|}S_{n}^{(2)}(\gamma_{nk})\big{|}=o_{p}(1).$
(17)

Next, for $\gamma_{nk}=(w_{nk},y_{nk},z_{nk})$, note that

$\displaystyle S_{n}(\gamma)-S_{n}(\gamma_{nk})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{n}(n-1)h^{d_{w}}}\sum_{1\leq s\neq t\leq
n}K\left(\frac{W_{t}-W_{s}}{h}\right)\left[-\varphi(W_{t},w)1(Y_{t}\leq
y)(1(Z_{t}\leq z)-1(Z_{t}\leq z_{nk}))\right.$
$\displaystyle+\varphi(W_{t},w)1(Y_{t}\leq y)(1(Z_{s}\leq z)-1(Z_{s}\leq
z_{nk}))$ $\displaystyle+\varphi(W_{t},w)(1(Y_{t}\leq y)-1(Y_{t}\leq
y_{nk}))(1(Z_{t}\leq z)-1(Z_{t}\leq z_{nk}))$
$\displaystyle-\varphi(W_{t},w)(1(Y_{t}\leq y)-1(Y_{t}\leq
y_{nk}))(1(Z_{s}\leq z)-1(Z_{s}\leq z_{nk}))$
$\displaystyle+(\varphi(W_{t},w)-\varphi(W_{t},w_{nk}))1(Y_{t}\leq
y)(1(Z_{t}\leq z)-1(Z_{t}\leq z_{nk}))$
$\displaystyle-(\varphi(W_{t},w)-\varphi(W_{t},w_{nk}))1(Y_{t}\leq
y)(1(Z_{s}\leq z)-1(Z_{s}\leq z_{nk}))$
$\displaystyle-(\varphi(W_{t},w)-\varphi(W_{t},w_{nk}))(1(Y_{t}\leq
y)-1(Y_{t}\leq y_{nk}))(1(Z_{t}\leq z)-1(Z_{t}\leq z_{nk}))$
$\displaystyle+(\varphi(W_{t},w)-\varphi(W_{t},w_{nk}))(1(Y_{t}\leq
y)-1(Y_{t}\leq y_{nk}))(1(Z_{s}\leq z)-1(Z_{s}\leq z_{nk}))$
$\displaystyle-\varphi(W_{t},w)(1(Y_{t}\leq y)-1(Y_{t}\leq
y_{nk}))(1(Z_{t}\leq z)-1(Z_{s}\leq z))$
$\displaystyle-(\varphi(W_{t},w)-\varphi(W_{t},w_{nk}))1(Y_{t}\leq
y)(1(Z_{t}\leq z)-1(Z_{s}\leq z))$
$\displaystyle\left.+(\varphi(W_{t},w)-\varphi(W_{t},w_{nk}))(1(Y_{t}\leq
y)-1(Y_{t}\leq y_{nk}))(1(Z_{t}\leq z)-1(Z_{s}\leq z))\right]$
$\displaystyle:=$ $\displaystyle\sum_{j=1}^{11}B_{nj}(\gamma;\gamma_{k}).$

We shall deal with $B_{nj}(\gamma;\gamma_{k})$ ($j=1,2\ldots,11$) separately.
By the uniform boundedness of indicator function and kernel function, the
absolute value of $B_{n1}(\gamma;\gamma_{k})$ is no bigger than

$\displaystyle\frac{1}{\sqrt{n}}\sum_{t=1}^{n}\varphi(W_{t},w)1(Y_{t}\leq
y)\left|1(Z_{t}\leq z)-1(Z_{t}\leq
z_{nk})\right|\frac{1}{(n-1)h^{d_{w}}}\sum_{s=1,\,s\neq
t}^{n}\left|K\left(\frac{W_{t}-W_{s}}{h}\right)\right|$ $\displaystyle\leq$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{t=1}^{n}\left|1(Z_{t}\leq z)-1(Z_{t}\leq
z_{nk})\right|O_{p}(1)$ $\displaystyle=$ $\displaystyle
o_{p}(\ell_{n}^{-1})\times O_{p}(1)=o_{p}(\ell_{n}^{-1}),$

by noting that the term $\frac{1}{(n-1)h^{d_{w}}}\sum_{s=1,\,s\neq
t}^{n}\left|K\left(\frac{W_{t}-W_{s}}{h}\right)\right|$ is $O_{p}(1)$
uniformly in $t$ by the LLN, and the term

$\frac{1}{\sqrt{n}}\sum_{t=1}^{n}|1(Z_{t}\leq z)-1(Z_{t}\leq
z_{nk})|\sim\sqrt{n}\mathrm{P}\big{(}z\leq Z_{1}\leq
z_{nk}\big{)}=O_{p}(\sqrt{n}\delta_{n})=o_{p}(\ell_{n}^{-1})$

by taking

$\left(\frac{\log^{(m-d/2)\xi}n}{(\log\log)^{\kappa
d}n}\right)^{1/(d_{y}+d_{z})}\wedge\log\log^{\kappa}n\gg\ell_{n}\uparrow\infty,$

and the density of $Z_{t}$ is bounded in Assumption A6. It clearly follows
that

$\max_{1\leq k\leq
N_{n}}\sup_{\gamma\in\Gamma_{nk}}\left|B_{n1}(\gamma;\gamma_{nk})\right|=o_{p}(\ell_{n}^{-1}).$

The absolute value of $B_{n2}(\gamma;\gamma_{nk})$ is no bigger than

$\displaystyle\frac{1}{\sqrt{n}}\sum_{s=1}^{n}\left|1(Z_{s}\leq z)-1(Z_{s}\leq
z_{nk})\right|\frac{1}{(n-1)h^{d_{w}}}\sum_{t=1,\,t\neq
s}^{n}\varphi(W_{t},w)1(Y_{t}\leq
y)\left|K\left(\frac{W_{t}-W_{s}}{h}\right)\right|$ $\displaystyle\lesssim$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{s=1}^{n}\left|1(Z_{s}\leq z)-1(Z_{s}\leq
z_{nk})\right|\frac{1}{(n-1)h^{d_{w}}}\sum_{t=1,\,t\neq
s}^{n}\left|K\left(\frac{W_{t}-W_{s}}{h}\right)\right|$ $\displaystyle=$
$\displaystyle o_{p}(\ell_{n}^{-1})\times O_{p}(1)=o_{p}(\ell_{n}^{-1}),$

leading to $\max_{1\leq k\leq
N_{n}}\sup_{\gamma\in\Gamma_{nk}}\left|B_{n2}(\gamma;\gamma_{k})\right|=o_{p}(\ell_{n}^{-1})$.
Similarly, $\max_{1\leq k\leq
N_{n}}\sup_{\gamma\in\Gamma_{nk}}\left|B_{nj}(\gamma;\gamma_{nk})\right|=o_{p}(1)$
for $j=3,4,\ldots,11$. Hence, $\max_{1\leq k\leq
N_{n}}\sup_{\gamma\in\Gamma_{nk}}|S_{n}(\gamma)-S_{n}(\gamma_{nk})|=o_{p}(\ell_{n}^{-1})$.
This result and (11), (12), (13) and (17) together imply that uniformly in
$\gamma\in\Gamma_{n}$,

$\displaystyle S_{n}(w,y,z)$ (18)
$\displaystyle=\frac{1}{\sqrt{n}}\sum_{s=1}^{n}1(W_{s}\leq
w)\big{(}(1(Y_{s}\leq y)-F_{Y|W}(y|W_{s})\big{)}\big{(}1(Z_{s}\leq
z)-F_{Z|W}(z|W_{s})\big{)}f_{W}(W_{s})+o_{p}(\ell_{n}^{-1})$
$\displaystyle=\bar{S}_{n}(w,y,z)+o_{p}(\ell_{n}^{-1})$

Finally, we note that

$\mathcal{W}\times\mathbb{R}^{d_{y}}\times\mathbb{R}^{d_{z}}=\bigcup_{n=1}^{\infty}\Gamma_{n}=\lim_{n\rightarrow\infty}\Gamma_{n},$

take $n\rightarrow\infty$, we obtain uniform result on
$\mathcal{W}\times\mathbb{R}^{d_{y}}\times\mathbb{R}^{d_{z}}$ in the lemma. ∎

Based on Lemma C1, now we can prove Theorem 1.

Proof of Theorem 1:

###### Proof.

By Lemma B.1, we have that, uniformly in $(w,y,z)$,

$S_{n}(w,y,z)=\frac{1}{\sqrt{n}}\sum_{t=1}^{n}e_{t}(w,y,z)+o_{p}(1)=:\bar{S}_{n}(w,y,z)+o_{p}(1).$

It suffices to show that $\bar{S}_{n}(\cdot,\cdot,\cdot)\rightsquigarrow
S_{\infty}(\cdot,\cdot,\cdot)$, where $S_{\infty}(\cdot,\cdot,\cdot)$ is
defined in Theorem 1. Define the pseudometric $\rho_{d}$ on $\mathbb{R}^{d}$:

$\rho_{d}((w,y,z),(w^{\prime},y^{\prime},z^{\prime})):=\\{\mathrm{E}|e_{t}(w,y,z)-e_{t}(w^{\prime},y^{\prime},z^{\prime})|^{2}\\}^{1/2}$

By Theorem 10.2 of Pollard (1990), this follows if we have (i) total
boundedness of a pseudometric space $(\mathbb{R}^{d},\rho_{d})$, (ii)
stochastic equicontinuity of $\\{\bar{S}_{n}(w,y,z),n\geq 1\\}$, and (iii)
finite dimensional (fidi) convergence.

Consider the class of functions

$\mathcal{F}:=\\{f_{(w,y,z)}:(w,y,z)\in\mathcal{W}\times\mathbb{R}^{d_{y}}\times\mathbb{R}^{d_{z}}\\},$

where
$f_{(w,y,z)}:\mathbb{R}^{d_{w}}\times\mathbb{R}^{d_{y}}\times\mathbb{R}^{d_{z}}\rightarrow\mathbb{R}$
is defined by

$\displaystyle
f_{(w,y,z)}(W_{t},Y_{t},Z_{t})=e_{t}(w,y,z)=\varphi(W_{t},w)\phi_{t}(y)\epsilon_{t}(z)f_{W}(W_{t}).$

Consider $N(\varepsilon,\mathcal{F},L_{2}(P))$ for $\varepsilon\in(0,1]$, i.e.
the covering number of $\mathcal{F}$. Since $f_{W}(\cdot)$ is bounded, it is
equivalent to covering the no-necessarily bounded set
$\Gamma=\mathcal{W}\times\mathcal{Y}\times\mathcal{Z}\subset\mathbb{R}^{d}$ by
the cover

$\Gamma_{k}(\varepsilon):=\Gamma_{k}^{(\mathcal{W})}(\varepsilon)\times\Gamma_{k}^{(\mathcal{Y})}(\varepsilon)\times\Gamma_{k}^{(\mathcal{Z})}(\varepsilon)=\big{\\{}\gamma_{k}=(w^{\top},y^{\top},z^{\top})^{\top}:k=1,\ldots,N_{\varepsilon}\big{\\}},$

which satisfies for any $\gamma\in\Gamma$, there exists at least one $k$ such
that

$\big{\|}\varphi(W_{t},w)\phi_{t}(y)\epsilon_{t}(z)-\varphi(W_{t},w)\phi_{t}(y)\epsilon_{t}(z)\big{\|}_{2}\leq\varepsilon.$

Obviously, the covering number satisfies

$N(\varepsilon,\mathcal{F},L_{2}(P))\asymp N_{\varepsilon}\leq
N\big{(}\varepsilon,\mathcal{F}_{w},L_{2}(P)\big{)}\times
N\big{(}\varepsilon,\mathcal{F}_{y},L_{2}(P)\big{)}\times
N\big{(}\varepsilon,\mathcal{F}_{z},L_{2}(P)\big{)},$

where

$\mathcal{F}_{w}:=\\{\varphi(W_{t},w):w\in\mathcal{W}\\},\quad\mathcal{F}_{y}:=\\{\phi_{t}(y):y\in\mathcal{Y}\\},\quad\mathcal{F}_{z}:=\\{\epsilon_{t}(z):z\in\mathcal{Z}\\},$

and the $P$ is the joint law for $(W_{t}^{\top},Y_{t}^{\top},Z_{t}^{\top})$.
Since $\mathcal{W}$ is bounded, we immediately have

$N\big{(}\varepsilon,\mathcal{F}_{w},L_{2}(P)\big{)}\lesssim\varepsilon^{-d_{w}/\nu}.$

For $N(\varepsilon,\mathcal{F}_{y},L_{2}(P))$, from Example 1.6.4 in Wellner
(2002),

$N\big{(}\varepsilon/2,\big{\\{}1(Y_{t}\leq
y):y\in\mathbb{R}^{d_{y}}\big{\\}},L_{2}(P)\big{)}\lesssim\varepsilon^{-2d_{y}};$

And from Lemma B3, we know that

$\displaystyle\log
N\big{(}\varepsilon/2,\big{\\{}F_{Y|W}(y|\cdot):\mathcal{W}\rightarrow[0,1]\,;\,y\in\mathbb{R}^{d_{y}}\big{\\}},L_{2}(P)\big{)}$
$\displaystyle\leq\log
N_{[\;]}\big{(}\varepsilon,\big{\\{}F_{Y|W}(y|\cdot):\mathcal{W}\rightarrow[0,1]\,;\,y\in\mathbb{R}^{d_{y}}\big{\\}},L_{2}(P)\big{)}\lesssim\varepsilon^{-\frac{2d_{y}}{\bar{\nu}(d_{y}+1)}}.$

Hence,

$\displaystyle N\big{(}\varepsilon,\mathcal{F}_{y},L_{2}(P)\big{)}$
$\displaystyle\leq N\big{(}\varepsilon/2,\big{\\{}1(Y_{t}\leq
y):y\in\mathbb{R}^{d_{y}}\big{\\}},L_{2}(P)\big{)}\times
N\big{(}\varepsilon/2,\big{\\{}F_{Y|W}(y|\cdot):\mathcal{W}\rightarrow[0,1]\,;\,y\in\mathbb{R}^{d_{y}}\big{\\}},L_{2}(P)\big{)}$
$\displaystyle\lesssim\varepsilon^{-2d_{y}}\exp\left\\{\varepsilon^{-\frac{2d_{y}}{\bar{\nu}(d_{y}+1)}}\right\\}$

Similarly, we have

$N\big{(}\varepsilon,\mathcal{F}_{z},L_{2}(P)\big{)}\lesssim\varepsilon^{-2d_{z}}\exp\left\\{\varepsilon^{-\frac{2d_{z}}{\bar{\nu}(d_{z}+1)}}\right\\}.$

As a consequence, for any $\varepsilon\in(0,1]$, the covering number of the
cover number of $\mathcal{F}$ satisfies

$N(\varepsilon,\mathcal{F},L_{2}(P))\lesssim\varepsilon^{-(\nu^{-1}d_{w}+2(d_{y}+d_{z}))}\cdot\exp\Big{\\{}\varepsilon^{-\frac{2}{\bar{\nu}}\left(\frac{d_{y}}{d_{y}+1}+\frac{d_{z}}{d_{z}+1}\right)}\Big{\\}}.$

Consequently,

$\int_{0}^{1}\epsilon^{-\frac{\eta}{2+\eta}}\log^{2}\mathcal{N}\big{(}\epsilon,\mathcal{F}_{1},L_{r}(P)\big{)}\,d\varepsilon\lesssim\int_{0}^{1}\varepsilon^{-\left(\frac{\eta}{2+\eta}+\frac{4}{\bar{\nu}}\left(\frac{d_{y}}{d_{y}+1}+\frac{d_{z}}{d_{z}+1}\right)\right)}\,d\varepsilon<\infty,$

with

$\frac{\eta}{2+\eta}+\frac{4}{\bar{\nu}}\left(\frac{d_{y}}{d_{y}+1}+\frac{d_{z}}{d_{z}+1}\right)<1,$

and $\sum_{j=1}^{\infty}j^{2m-1}\alpha(j)^{\frac{\eta}{2m+\eta}}<\infty$ by
Assumption A1 and A7. It follows that the conditions are satisfied by a
generalization of Theorem 2.2 of Andrews and Pollard (1994) (see p128 in
Andrews and Pollard (1994)). The fidi convergence follows by the Cramer-Wold
device and a central limit theorem for bounded random variables under strong
mixing conditions; see Corollary 5.1 in Hall and Heyde (1981). We are left to
demonstrate that the sample mean and covariance kernel converges to that of
the limiting Gaussian process $S_{\infty}(\cdot,\cdot,\cdot)$ under the Markov
property (2.1), i.e.,

$\displaystyle\mathrm{E}\big{[}\bar{S}_{n}(w,y,z)\big{]}$
$\displaystyle=\frac{1}{\sqrt{n}}\sum_{t=1}^{n}\mathrm{E}\big{[}e_{s}(w,y,z)\big{]}$
$\displaystyle=\frac{1}{\sqrt{n}}\sum_{t=1}^{n}\mathrm{E}\Big{[}1(W_{t}\leq
w)f_{W}(W_{t})\big{(}\mathrm{E}[1(Y_{t}\leq
y)\,|\,W_{t}]-F_{Y|W}(y|W_{t})\big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\quad\times\big{(}\mathrm{E}[1(Z_{t}\leq
z)\,|\,W_{t}]-F_{Z|W}(z|W_{t})\big{)}\Big{]}=0,$

and the covariance under IID case can be derived similarly. This completes the
proof of the theorem. ∎

Proof of Corollary 1:

###### Proof.

By continuous mapping theorem, see e.g. Theorem 5.1 in Billingsley (2013), and
the weak convergence of the process $S_{n}(w,y,z)$ in Theorem 1, we
immediately have the convergence of $KS_{n}$ to $KS_{\infty}$ in distribution,
e.g. Delgado and Escanciano (2007).

For the limiting null distribution of $CvM_{n}$, note that

$\displaystyle|CvM_{n}-CvM_{\infty}|$ $\displaystyle=$
$\displaystyle\left|\int S^{2}_{n}(w,y,z)\,dF_{n}(w,y,z)-\int
S^{2}_{\infty}(w,y,z)\,dF_{W,Y,Z}(x,y,z)\right|$ $\displaystyle\leq$
$\displaystyle\left|\int(S^{2}_{n}(w,y,z)-S^{2}_{\infty}(w,y,z))\,dF_{n}(w,y,z)\right|$
$\displaystyle+\left|\int
S^{2}_{\infty}(w,y,z)\,d(F_{n}(w,y,z)-F_{W,Y,Z}(w,y,z))\right|.$

The first term of the right-hand side of the above inequality is $o_{p}(1)$ by
Theorem 1. Glivenko-Cantelli’s Theorem yields that
$\sup_{(w,y,z)}|F_{n}(w,y,z)-F_{W,Y,Z}(w,y,z)|=o(1)$ a.s. Then, taking into
account that the trajectories of the limiting process
$S_{\infty}(\cdot,\cdot,\cdot)$ are bounded and continuous almost surely and
applying Helly-Bray Theorem (see p. 97 in Rao et al. (1973)) to each of these
trajectories, we obtain

$\left|\int
S^{2}_{\infty}(w,y,z)\,d(F_{n}(w,y,z)-F_{W,Y,Z}(w,y,z))\right|\rightarrow
0\,\,\,\text{a.s.}.$

This concludes the proof of the corollary. ∎

Proof of Theorem 2:

###### Proof.

We rewrite

$\displaystyle\frac{1}{\sqrt{n}}S_{n}(w,y,z)=$
$\displaystyle\frac{1}{n}\sum_{t=1}^{n}\varphi(W_{t},w)1(Y_{t}\leq
y)(1(Z_{t}\leq z)-F_{Z|W}(z|W_{t}))f_{W}(W_{t})$
$\displaystyle-\frac{1}{n}\sum_{t=1}^{n}\varphi(W_{t},w)1(Y_{t}\leq
y)(\widehat{F}_{Z|W}(z|W_{t})-F_{Z|W}(z|W_{t}))\widehat{f}_{W}(W_{t})$
$\displaystyle+\frac{1}{n}\sum_{t=1}^{n}\varphi(W_{t},w)1(Y_{t}\leq
y)(1(Z_{t}\leq z)-F_{Z|W}(z|W_{t}))(\widehat{f}_{W}(W_{t})-f_{W}(W_{t}))$
$\displaystyle:=$
$\displaystyle\frac{1}{\sqrt{n}}S_{n1}(w,y,z)+\frac{1}{\sqrt{n}}S_{n2}(w,y,z)+\frac{1}{\sqrt{n}}S_{n3}(w,y,z).$

From the consistency of the kernel estimators under Assumption A1 (since we at
least have $m\geq 1$ hold in Assumption A1 hold, then Su and White (2012) for
example immediately gives the consistency) and boundedness assumption for
$\varphi(\cdot,\cdot)$ and $\mathcal{W}$,

$\displaystyle\frac{1}{\sqrt{n}}S_{n2}(w,y,z)$
$\displaystyle=\frac{1}{n}\sum_{t=1}^{n}\varphi(W_{t},w)1(Y_{t}\leq y)\cdot
o_{p}(1)\cdot\big{(}f_{W}(W_{t})+o_{p}(1)\big{)}$
$\displaystyle=o_{p}(1)\cdot\mathrm{E}\big{[}\varphi(W_{t},w)1(Y_{t}\leq
y)f_{W}(W_{t})\big{]}=o_{p}(1).$

And similarly $n^{-1/2}S_{n3}(w,y,z)=o_{p}(1)$. It follows that

$\displaystyle\frac{1}{\sqrt{n}}S_{n}(w,y,z)=$
$\displaystyle\frac{1}{n}\sum_{t=1}^{n}\varphi(W_{t},w)1(Y_{t}\leq
y)(1(Z_{t}\leq z)-F_{Z|W}(z|W_{t}))f_{W}(W_{t})+o_{p}(1)$ $\displaystyle=$
$\displaystyle\mathrm{E}\Big{[}\varphi(W_{t},w)\big{(}(F_{Y,Z|W}(y,z|W_{t})-F_{Y|W}(y|W_{t})F_{Z|W}(z|W_{t})\big{)}f_{W}(W_{t})\Big{]}+o_{p}(1),$

where the last step is due to LLN and law of iterate expectations. Since

$\mathrm{E}\big{[}1(W_{t}\leq
w)(F_{Y,Z|W}(y,z|W_{t})-F_{Y|W}(y|W_{t})F_{Z|W}(z|W_{t}))f_{W}(W_{t})\big{]}\neq
0$

in a set with positive Lebesgue measure, the test statistics based on
$S_{n}(w,y,z)$, e.g. $CvM_{n}$ and $KS_{n}$, diverge to infinity under the
alternative. In particular, $n^{-1}CvM_{n}$ (cf. the proof of Corollary 1) and
$n^{-1/2}KS_{n}$ converge in probability to some positive values. This ends
the proof of the theorem. ∎

Proof of Theorem 3:

###### Proof.

Similar to Lemma C1, in this case, we also have

$S_{n}(\gamma)=S_{n}^{(1)}(\gamma)+S_{n}^{(2)}(\gamma)$

with $S_{n}^{(1)}(\gamma)=\sum_{j=1}^{4}A_{nj}$, where the definition of each
symbol is exactly the same as that in Lemma C1 until now. Now under
$\text{H}_{1n}$, the term $A_{n1}$ is

$\displaystyle A_{n1}$
$\displaystyle=\frac{1}{\sqrt{n}h^{d_{w}}}\sum_{s=1}^{n}\int
K\left(\frac{W_{t}-W_{s}}{h}\right)\varphi(W_{t},w)1(Y_{t}\leq y)1(Z_{t}\leq
z)\,dF_{\chi}(\chi_{t})$
$\displaystyle=\frac{1}{\sqrt{n}}\sum_{s=1}^{n}\int\int\int
K(u)\varphi(W_{s}+hu,w)1(y^{\prime}\leq y)1(z^{\prime}\leq z)$
$\displaystyle\times\bigg{[}f_{Y|W}(y^{\prime}|W_{s}+hu)f_{Z|W}(z^{\prime}|W_{s}+hu)+n^{-1/2}\frac{\partial^{2}\Delta(W_{s},y^{\prime},z^{\prime})}{\partial
y^{\prime}\partial
z^{\prime}}\bigg{]}f_{W}(W_{s}+hu)\,du\,dy^{\prime}\,dz^{\prime}$
$\displaystyle=\frac{1}{\sqrt{n}}\sum_{s=1}^{n}\varphi(W_{s},w)\int
1(y^{\prime}\leq y)f_{Y|W}(y^{\prime}|W_{s})\,dy^{\prime}\int 1(z^{\prime}\leq
z)f_{Z|W}(z^{\prime}|W_{s})\,dz^{\prime}f_{W}(W_{s})$
$\displaystyle+\frac{1}{n}\sum_{s=1}^{n}\varphi(W_{s},w)\int\int
1(y^{\prime}\leq y)1(z^{\prime}\leq
z)\frac{\partial^{2}\Delta(W_{s},y^{\prime},z^{\prime})}{\partial
y^{\prime}\partial
z^{\prime}}\,dy^{\prime}\,dz^{\prime}f_{W}(W_{s})+O_{p}(\sqrt{n}h^{l})$
$\displaystyle=\frac{1}{\sqrt{n}}\sum_{s=1}^{n}\varphi(W_{s},w)F_{Y|W}(y|W_{s})F_{Z|W}(z|W_{s})f_{W}(W_{s})$
$\displaystyle+\frac{1}{n}\sum_{s=1}^{n}\varphi(W_{s},w)\Delta(W_{s},y,z)f_{W}(W_{s})+o_{p}(1)$
$\displaystyle:=\frac{1}{\sqrt{n}}\sum_{s=1}^{n}\varphi(W_{s},w)F_{Y|W}(y|W_{s})F_{Z|W}(z|W_{s})f_{W}(W_{s})+G(w,y,z)+o_{p}(1),$

where the first step is due to
$f_{YZ|W}(y,z|W_{t})=f_{Y|W}(y|W_{t})f_{Z|W}(z|W_{t})+n^{-1/2}\partial^{2}\Delta(W_{t},y,z)/\partial
y\partial z$ under $\text{H}_{1n}$, while the last step follows by Assumption
A8 (ii). On the other hand, we can easily find that $A_{n2}$, $A_{n3}$ and
$A_{n4}$ stay the same as in Lemma C1. As a result, following similar
arguments of Lemma C1, we get

$\displaystyle S_{n}(w,y,z)=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{s=1}^{n}\varphi(W_{s},w)\big{(}1(Y_{s}\leq
y)-F_{Y|W}(y|W_{s})\big{)}\big{(}1(Z_{s}\leq
z)-F_{Z|W}(z|W_{s})\big{)}f_{W}(W_{s})$ $\displaystyle+G(w,y,z)+o_{p}(1),$

uniformly in
$(w,y,z)\in\mathcal{W}\times\mathbb{R}^{d_{y}}\times\mathbb{R}^{d_{z}}$ under
the local alternatives $\text{H}_{1n}$. By the proof of Theorem 1, under the
local alternatives $\text{H}_{1n}$,

$S_{n}(w,y,z)\rightsquigarrow
S_{\infty}(\cdot,\cdot,\cdot)+G(\cdot,\cdot,\cdot),$

which concludes the proof of the theorem. ∎

Proof of Theorem 4:

###### Proof.

We need to show that the process

$\displaystyle S_{n}^{\ast}(w,y,z)=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{t=1}^{n}\varphi(W_{t},w)(1(Y_{t}\leq
y)-\widehat{F}_{Y|W}(y|W_{t}))(1(Z_{t}\leq
z)-\widehat{F}_{Z|W}(z|W_{t}))\widehat{f}_{W}(W_{t})v_{t}$ $\displaystyle:=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{t=1}^{n}\widehat{e}_{t}(w,y,z)v_{t}$

(conditionally on the sample) has the same asymptotic fidis as the process
$S_{n}(w,y,z)$ in probability, total boundedness of the pseudometric space
$\big{(}\mathcal{W}\times\mathbb{R}^{d_{y}}\times\mathbb{R}^{d_{z}},\rho_{d}\big{)}$
and stochastic equicontinuity of $\\{S_{n}^{\ast}(w,y,z),n\geq 1\\}$, both in
probability. Recall that $\\{v_{t}\\}_{t=1}^{n}$ is a sequence of independent
random variables with zero mean, unit variance and is independent of the
original sample.

Define

$\displaystyle\bar{S}_{n}^{\ast}(w,y,z)=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{t=1}^{n}\varphi(W_{t},w)(1(Y_{t}\leq
y)-F_{Y|W}(y|W_{t}))(1(Z_{t}\leq z)-F_{Z|W}(z|W_{t}))f_{W}(W_{t})v_{t}$
$\displaystyle:=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{t=1}^{n}e_{t}(w,y,z)v_{t}.$

It suffices to prove that, uniformly in $(w,y,z)$, the process
$S_{n}^{\ast}(w,y,z)$ and the process $\bar{S}_{n}^{\ast}(w,y,z)$ are
asymptotically equivalent, i.e.

$\sup_{(w,y,z)\in\mathcal{W}\times\mathbb{R}^{d_{y}}\times\mathbb{R}^{d_{z}}}\left|S_{n}^{\ast}(w,y,z)-\bar{S}_{n}^{\ast}(w,y,z)\right|=o_{p}(1).$

To show this, note that

$\displaystyle S_{n}^{\ast}(w,y,z)-\bar{S}_{n}^{\ast}(w,y,z)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{t=1}^{n}\varphi(W_{t},w)\big{(}1(Y_{t}\leq
y)-\widehat{F}_{Y|W}(y|W_{t})\big{)}\big{(}1(Z_{t}\leq
z)-\widehat{F}_{Z|W}(z|W_{t})\big{)}(\widehat{f}_{W}(W_{t})-f_{W}(W_{t}))v_{t}$
$\displaystyle-$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{t=1}^{n}\varphi(W_{t},w)\big{(}\widehat{F}_{Y|W}(y|W_{t})-F_{Y|W}(y|W_{t})\big{)}(1(Z_{t}\leq
z)-F_{Z|W}(z|W_{t}))f_{W}(W_{t})v_{t}$ $\displaystyle-$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{t=1}^{n}\varphi(W_{t},w)\big{(}1(Y_{t}\leq
y)-F_{Y|W}(y|W_{t})\big{)}\big{(}\widehat{F}_{Z|W}(z|W_{t})-F_{Z|W}(z|W_{t})\big{)}f_{W}(W_{t})v_{t}$
$\displaystyle+$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{t=1}^{n}\varphi(W_{t},w)\big{(}\widehat{F}_{Y|W}(y|W_{t})-F_{Y|W}(y|W_{t})\big{)}\big{(}\widehat{F}_{Z|W}(z|W_{t})-F_{Z|W}(z|W_{t})\big{)}f_{W}(W_{t})v_{t}$
$\displaystyle:=$ $\displaystyle\sum_{j=1}^{4}D_{nj}.$

Following the uniform consistency of the kernel estimators and together with
the properties of $\\{v_{t}\\}_{t=1}^{n}$, it is easy to show that
$D_{nj}=o_{p}(1)$ ($j=1,2,3,4$), uniformly in
$(w,y,z)\in\mathcal{W}\times\mathbb{R}^{d_{y}}\times\mathbb{R}^{d_{z}}$.

So, we only have to focus on $\bar{S}_{n}^{\ast}(w,y,z)$. The fidi convergence
of $\bar{S}_{n}^{\ast}(w,y,z)$ follows from its expression and from the
Cramer-Wold device in conjunction with (1). The total boundedness and
stochastic equicontinuity of $\\{\bar{S}_{n}^{\ast}(w,y,z),n\geq 1\\}$ follow
from the same steps as in Theorem 1. The proof is finished. ∎

##  Appendix D D Simulation Tables

Table 1: Empirical rejection rates of $CvM_{n}$ with $h=cn^{-1/3.5}$ | DGPs
---|---
$c$ | (S1) | (S2) | (S3) | (S4) | (P1) | (P2) | (P3) | (P4) | (P5) | (P6) | (P7)
$n=100$
$0.5$ | 0.052 | 0.058 | 0.064 | 0.073 | 0.966 | 0.297 | 0.136 | 0.215 | 0.140 | 0.127 | 0.096
$1.0$ | 0.062 | 0.074 | 0.077 | 0.088 | 0.982 | 0.372 | 0.165 | 0.245 | 0.214 | 0.150 | 0.127
$1.5$ | 0.077 | 0.086 | 0.088 | 0.118 | 0.987 | 0.399 | 0.206 | 0.281 | 0.282 | 0.200 | 0.142
$n=200$
$0.5$ | 0.051 | 0.053 | 0.045 | 0.059 | 1.000 | 0.854 | 0.345 | 0.542 | 0.327 | 0.210 | 0.127
$1.0$ | 0.063 | 0.053 | 0.063 | 0.082 | 1.000 | 0.902 | 0.400 | 0.607 | 0.424 | 0.242 | 0.177
$1.5$ | 0.074 | 0.079 | 0.079 | 0.102 | 1.000 | 0.920 | 0.412 | 0.620 | 0.501 | 0.285 | 0.190
$n=400$
$0.5$ | 0.060 | 0.052 | 0.049 | 0.061 | 1.000 | 0.999 | 0.787 | 0.976 | 0.934 | 0.625 | 0.258
$1.0$ | 0.066 | 0.056 | 0.064 | 0.066 | 1.000 | 1.000 | 0.818 | 0.979 | 0.960 | 0.663 | 0.335
$1.5$ | 0.065 | 0.069 | 0.069 | 0.089 | 1.000 | 1.000 | 0.824 | 0.990 | 0.967 | 0.692 | 0.361
$n=800$
$0.5$ | 0.048 | 0.044 | 0.049 | 0.053 | 1.000 | 1.000 | 0.998 | 1.000 | 1.000 | 1.000 | 0.860
$1.0$ | 0.065 | 0.059 | 0.052 | 0.067 | 1.000 | 1.000 | 0.998 | 1.000 | 1.000 | 1.000 | 0.878
$1.5$ | 0.065 | 0.067 | 0.054 | 0.063 | 1.000 | 1.000 | 0.999 | 1.000 | 1.000 | 1.000 | 0.896
Table 2: Empirical rejection rates of $KS_{n}$ with $h=cn^{-1/3.5}$ | DGPs
---|---
$c$ | (S1) | (S2) | (S3) | (S4) | (P1) | (P2) | (P3) | (P4) | (P5) | (P6) | (P7)
$n=100$
$0.5$ | 0.042 | 0.044 | 0.035 | 0.055 | 0.800 | 0.236 | 0.096 | 0.231 | 0.146 | 0.115 | 0.072
$1.0$ | 0.047 | 0.047 | 0.048 | 0.063 | 0.862 | 0.290 | 0.104 | 0.270 | 0.200 | 0.130 | 0.102
$1.5$ | 0.047 | 0.055 | 0.046 | 0.066 | 0.889 | 0.325 | 0.141 | 0.269 | 0.255 | 0.156 | 0.117
$n=200$
$0.5$ | 0.042 | 0.036 | 0.040 | 0.049 | 0.999 | 0.667 | 0.197 | 0.534 | 0.361 | 0.217 | 0.122
$1.0$ | 0.049 | 0.038 | 0.054 | 0.056 | 0.996 | 0.718 | 0.219 | 0.587 | 0.397 | 0.243 | 0.158
$1.5$ | 0.051 | 0.052 | 0.059 | 0.072 | 0.997 | 0.730 | 0.232 | 0.587 | 0.459 | 0.274 | 0.169
$n=400$
$0.5$ | 0.047 | 0.043 | 0.042 | 0.051 | 1.000 | 0.986 | 0.462 | 0.938 | 0.779 | 0.510 | 0.237
$1.0$ | 0.055 | 0.044 | 0.051 | 0.056 | 1.000 | 0.989 | 0.467 | 0.944 | 0.816 | 0.527 | 0.275
$1.5$ | 0.055 | 0.051 | 0.054 | 0.073 | 1.000 | 0.997 | 0.501 | 0.958 | 0.839 | 0.545 | 0.289
$n=800$
$0.5$ | 0.052 | 0.051 | 0.042 | 0.055 | 1.000 | 1.000 | 0.856 | 1.000 | 0.999 | 0.936 | 0.575
$1.0$ | 0.058 | 0.052 | 0.043 | 0.063 | 1.000 | 1.000 | 0.873 | 1.000 | 0.995 | 0.954 | 0.616
$1.5$ | 0.056 | 0.057 | 0.048 | 0.054 | 1.000 | 1.000 | 0.878 | 1.000 | 1.000 | 0.960 | 0.635
Table 3: Empirical rejection rates of $CvM_{n}^{block}$ with $h=n^{-1/3.5}$ and $L=\lfloor an^{1/4}\rfloor$ | DGPs
---|---
$a$ | (S1) | (S2) | (S3) | (S4) | (P1) | (P2) | (P3) | (P4) | (P5) | (P6) | (P7)
$n=100$
$1$ | 0.052 | 0.053 | 0.069 | 0.082 | 0.977 | 0.318 | 0.129 | 0.240 | 0.199 | 0.147 | 0.116
$2$ | 0.037 | 0.049 | 0.056 | 0.068 | 0.970 | 0.285 | 0.118 | 0.203 | 0.179 | 0.172 | 0.112
$4$ | 0.039 | 0.036 | 0.036 | 0.058 | 0.896 | 0.175 | 0.061 | 0.156 | 0.169 | 0.134 | 0.099
$n=200$
$1$ | 0.058 | 0.060 | 0.052 | 0.070 | 1.000 | 0.878 | 0.373 | 0.575 | 0.409 | 0.241 | 0.155
$2$ | 0.054 | 0.058 | 0.062 | 0.057 | 1.000 | 0.852 | 0.305 | 0.553 | 0.412 | 0.223 | 0.177
$4$ | 0.037 | 0.054 | 0.052 | 0.068 | 1.000 | 0.838 | 0.254 | 0.515 | 0.396 | 0.249 | 0.115
$n=400$
$1$ | 0.059 | 0.071 | 0.049 | 0.064 | 1.000 | 1.000 | 0.769 | 0.973 | 0.937 | 0.641 | 0.312
$2$ | 0.053 | 0.061 | 0.067 | 0.073 | 1.000 | 0.998 | 0.726 | 0.976 | 0.955 | 0.662 | 0.319
$4$ | 0.047 | 0.042 | 0.062 | 0.062 | 1.000 | 0.999 | 0.706 | 0.959 | 0.896 | 0.613 | 0.301
$n=800$
$1$ | 0.046 | 0.056 | 0.047 | 0.058 | 1.000 | 1.000 | 0.993 | 1.000 | 1.000 | 0.998 | 0.828
$2$ | 0.040 | 0.048 | 0.057 | 0.057 | 1.000 | 1.000 | 0.992 | 1.000 | 1.000 | 0.998 | 0.829
$4$ | 0.046 | 0.056 | 0.058 | 0.066 | 1.000 | 1.000 | 0.991 | 1.000 | 1.000 | 0.992 | 0.813
Table 4: Empirical rejection rates of $KS_{n}^{block}$ with $h=n^{-1/3.5}$ and $L=\lfloor an^{1/4}\rfloor$ | DGPs
---|---
$a$ | (S1) | (S2) | (S3) | (S4) | (P1) | (P2) | (P3) | (P4) | (P5) | (P6) | (P7)
$n=100$
$1$ | 0.044 | 0.032 | 0.040 | 0.060 | 0.814 | 0.274 | 0.096 | 0.224 | 0.200 | 0.139 | 0.082
$2$ | 0.037 | 0.035 | 0.032 | 0.054 | 0.760 | 0.227 | 0.093 | 0.207 | 0.163 | 0.122 | 0.074
$4$ | 0.029 | 0.022 | 0.028 | 0.041 | 0.546 | 0.130 | 0.056 | 0.140 | 0.133 | 0.094 | 0.055
$n=200$
$1$ | 0.043 | 0.044 | 0.042 | 0.053 | 0.996 | 0.661 | 0.206 | 0.543 | 0.391 | 0.211 | 0.136
$2$ | 0.048 | 0.043 | 0.039 | 0.050 | 0.988 | 0.610 | 0.182 | 0.500 | 0.344 | 0.204 | 0.139
$4$ | 0.038 | 0.043 | 0.041 | 0.042 | 0.969 | 0.532 | 0.135 | 0.429 | 0.309 | 0.194 | 0.104
$n=400$
$1$ | 0.046 | 0.050 | 0.042 | 0.055 | 1.000 | 0.987 | 0.416 | 0.940 | 0.782 | 0.484 | 0.269
$2$ | 0.051 | 0.042 | 0.058 | 0.063 | 1.000 | 0.980 | 0.400 | 0.940 | 0.770 | 0.496 | 0.240
$4$ | 0.034 | 0.039 | 0.051 | 0.039 | 1.000 | 0.970 | 0.375 | 0.888 | 0.727 | 0.409 | 0.228
$n=800$
$1$ | 0.054 | 0.044 | 0.039 | 0.050 | 1.000 | 1.000 | 0.859 | 1.000 | 0.999 | 0.925 | 0.571
$2$ | 0.050 | 0.025 | 0.054 | 0.050 | 1.000 | 1.000 | 0.822 | 1.000 | 0.999 | 0.907 | 0.582
$4$ | 0.042 | 0.046 | 0.050 | 0.055 | 1.000 | 1.000 | 0.822 | 1.000 | 0.998 | 0.895 | 0.554
Table 5: Testing for nonlinear predicability from VRP to RP

Direction of Causality | $h=cn^{-1/3.5}$ | $CvM_{n}$ | $KS_{n}$ | $CvM^{block}_{n}$ | $KS^{block}_{n}$ | LIN
---|---|---|---|---|---|---
Horizon: One Month
| $c=0.5$ | 0.2267 | 0.3377 | 0.2298 | 0.3054 |
VRP$\rightarrow$RP | $c=1.0$ | 0.1711 | 0.1343 | 0.1764 | 0.1300 | 0.0017 (0.4172)
| $c=1.5$ | 0.1077 | 0.0585 | 0.1073 | 0.0564 |
Horizon: Three Months
| $c=0.5$ | 0.0640 | 0.0663 | 0.0642 | 0.0682 |
VRP$\rightarrow$RP | $c=1.0$ | 0.0141 | 0.0109 | 0.0202 | 0.0201 | 0.0016 (1.5009)
| $c=1.5$ | 0.0056 | 0.0055 | 0.0090 | 0.0142 |
Horizon: Six Months
| $c=0.5$ | 0.0000 | 0.0001 | 0.0011 | 0.0022 |
VRP$\rightarrow$RP | $c=1.0$ | 0.0000 | 0.0000 | 0.0002 | 0.0013 | 0.0002 (0.3329)
| $c=1.5$ | 0.0000 | 0.0000 | 0.0000 | 0.0027 |
Horizon: Nine Months
| $c=0.5$ | 0.0000 | 0.0000 | 0.0001 | 0.0004 |
VRP$\rightarrow$RP | $c=1.0$ | 0.0000 | 0.0000 | 0.0000 | 0.0004 | 0.0005 (2.0807)
| $c=1.5$ | 0.0000 | 0.0000 | 0.0000 | 0.0004 |

Note: For tests $CvM_{n}$, $KS_{n}$, $CvM_{n}^{block}$ and $KS_{n}^{block}$,
(block) multiplier bootstrap $p$-values are reported. For the block tests,
$L=\lfloor 2n^{1/4}\rfloor$ is used. LIN corresponds to the linear test, where
the least squares estimate $\hat{\alpha}_{\tau}$ and its corresponding
$t$-statistic $t_{\hat{\alpha}_{\tau}}$ (in parentheses) based on HAC robust
variance estimator, are reported.