# Joint parametric specification checking of conditional mean and volatility
in time series models with martingale difference innovations

Kilani Ghoudi1,111Email<EMAIL_ADDRESS>Funding is provided in part by
the United Arab Emirates University UPAR grant. Naâmane
<EMAIL_ADDRESS><EMAIL_ADDRESS>and
Mohamed Chaouch3,333Email<EMAIL_ADDRESS>
1College of Business and Economics, United Arab Emirates University.
2 CY Cergy Paris Université, Laboratoire AGM, UMR 8088 du CNRS.
F-95000 Cergy, France.
3Department of Mathematics, Statistics and Physics, Qatar University

###### Abstract

Using cumulative residual processes, we propose joint goodness-of-fit tests
for conditional means and variances functions in the context of nonlinear time
series with martingale difference innovations. The main challenge comes from
the fact the cumulative residual process no longer admits, under the null
hypothesis, a distribution-free limit. To obtain a practical solution one
either transforms the process in order to achieve a distribution-free limit or
approximates the non-distribution free limit using a numerical or a re-
sampling technique. Here the three solutions will be considered. It is shown
that the proposed tests have nontrivial power against a class of root-n local
alternatives, and are suitable when the conditioning information set is
infinite-dimensional, which allows including models like autoregressive
conditional heteroscedastic stochastic models with dependent innovations. The
approach presented assumes only certain conditions on the first- and second-
order conditional moments, without imposing any autoregression model. The test
procedures introduced are compared with each other and with other competitors
in terms of their power using a simulation study and a real data application.
These simulations have shown that the statistical powers of tests based on re-
sampling or numerical approximation of the original statistics are in general
slightly better than those based on a martingale transformation of the
original process.

Key words: Autoregression, conditional mean, conditional variance, cumulative
residual process, heteroscedasticity, martingale transform, re-sampling,
interest rate, martingale differences, nonlinear times series, parametric
specification.

Subject Classifications: 62F10, 62F05, 62J02, 62M10, 60J15.

##  1 Introduction

A great deal of the data in econometrics, engineering and natural sciences
occurs in the form of time series, where the data are naturally dependent and
the volatility is, in general, function of the past. One then expects better
forecast results if additional information allowing the conditional variance
to depend upon the past realizations is supposed. One of the earliest
development in this area is the work of Engle, (1982) who introduced the so-
called autoregressive conditional heteroscedastic (ARCH) model, which has been
extended in a number of directions. The most important of these is the
extension designed to include moving-average parts, namely the generalized
ARCH (GARCH) model introduced by Bollerslev, (1986). These models are often
used to parameterize conditional heteroscedasticity that appears in many
financial time-series such as exchange rates and stock return. In
applications, GARCH models have been specified for data at different
frequencies assuming that the rescaled innovations are independent and
identically distributed (i.i.d.). Drost and Nijman, (1993) pointed out that,
the common assumption in applications that rescaled innovations are
independent is disputable, since it depends upon the available data frequency.
They classified GARCH models into three categories: Strong GARCH requires that
rescaled innovations are independent, semi-strong GARCH assumes that rescaled
innovations are martingale difference sequences, the weak form, where the
martingale difference sequence assumption is relaxed. The weak-GARCH
representation has been used previously, for instance, by Gonçalves and White,
(2004), were the authors showed, through simulations, the effects of
misspecification when the true model is a GARCH with an innovation term that
follows an AR(1) process. Kristensen and Rahbek, (2005) also noted the
relevance of this model. Dahl and Iglesias, (2007) provided an empirical
application showing that this process has empirical relevance.

Diagnostic tests are integral part of any modelling exercise. Several time
series models are given by specifying conditional mean and conditional
variance functions. Testing the correct specification of these quantities is
of major importance in the model validation. A great deal of tests proposed in
the literature focuses on testing either the mean function or the volatility
function for time series, but usually not both. Escanciano, (2008, 2010)
discussed joint tests for parametric form of the mean and volatility
functions. He also argued that if the mean is misspecified, tests of
volatility functions are usually misleading.

This paper derives joint goodness-of-fit tests for the conditional means and
variances functions for strictly stationary ergodic time series
$\\{X_{i},i\geq 0\\}$ with martingale difference innovations
$X_{i}-\mathbb{E}\\{X_{i}|{\cal F}_{i-1}\\}$, where ${\cal F}_{i}$ is the
$\sigma$-field generated by the observations obtained up to time $i$, without
imposing any type of autoregressive model. The martingale hypothesis is very
important in economics theory, for instance, dynamic equilibrium approaches to
macroeconomics have imposed martingale restrictions on numerous time series of
interest (see, e.g., Durlauf, (1991) for more discussions on the martingale
hypothesis arising in other contexts of economic theory). The assumption of
martingale difference innovations considered here is more general than the
standard assumption of i.i.d innovations as it allows some dependence
structure in the innovations. The framework we are considering here is
suitable for cases in which the conditioning set is infinite-dimensional and
may be used for models that do not necessary satisfy Markov property,
particularly, semi-parametric models, where the conditional mean and the
conditional variance of $X_{i}$ given ${\cal F}_{i-1}$ have parametric forms.
This includes most processes usually used for modeling financial time series
with dependent innovations, such as GARCH, ARMA-GARCH, exponential and
threshold autoregressive processes with GARCH errors.

Goodness-of-fit tests for parametric and semiparametric hypotheses of the
regression function have been considered in the literature, with emphasis on
i.i.d innovations, see for instance Stute, (1997) who presented non-parametric
full-model checks for regression based on the limiting law of the residual
marked process, see also González-Manteiga and Crujeiras, (2013) for a survey
on the topic. Escanciano et al., (2018) discussed a general methodology for
constructing nonparametric/semiparametric asymptotically distribution-free
tests about regression models for possibly dependent data. Similar study
investigated the autoregressive function in time series models, see, e.g.,
Diebolt, (1990), McKeague and Zhang, (1994), Laïb, (1999), Koul and Stute,
(1999). In the context of time series with martingale difference innovations,
Stute et al., (2006) provided non-parametric tests based on residual cusums
for testing the autoregressive function in higher-order time-series models,
and Escanciano and Mayoral, (2010) proposed data-driven smooth asymptotically
distribution-free tests based on the principal components of certain marked
empirical processes for testing the martingale difference hypothesis of a
possibly non-linear time series.
Testing hypotheses about the conditional variance function of regression
models are investigated by many authors in the literature. Wang and Zhou,
(2005) considered a nonparametric diagnostic test for checking the constancy
of the conditional variance function, in a nonparametric regression model,
where the co-variables are fixed design points, without assuming a known
parametric form for the conditional mean function. Dette et al., (2007)
proposed a test procedure for testing the parametric form of the conditional
variance in the common nonparametric regression model. Koul and Song, (2010)
discusses the problem of fitting a parametric model to the conditional
variance function in a class of heteroscedastic regression models. Their test
is based on the supremum of the Khmaladze type martingale transformation of a
certain partial sum process of calibrated squared residuals. The proposed
statistical test is shown to be consistent against a large class of fixed
alternatives and to have nontrivial asymptotic power against a class of
nonparametric local alternatives. Recently, Pardo-Fernández et al., (2015)
have proposed several nonparametric statistical tests for checking whether the
conditional variances are equal in $k\geq 2$ location-scale regression models.
Their procedure is based on the comparison of the error distributions under
the null hypothesis of equality of variances functions. Polonik and Yao,
(2008) propose two tests for testing multivariate volatility functions using
minimum volume sets and inverse regression. Their tests are based on
cumulative sums coupled with either minimum volume sets or inverse regression
ideas.

Tests of conditional variance functions in time series context were also
previously considered in the literature. In particular, Auestad and Tjøstheim,
(1990), focused on kernel estimate of the one step lagged conditional mean and
variance functions for the purpose of identifying common linear models such as
threshold and exponential autoregressive. Diebolt, (1990) established the
consistency of regressogram type estimators of the conditional mean and
conditional variance functions. He deduced nonparametric goodness-of-fit tests
for known form of these functions. Chen and An, (1997) proposed a Kolmogorov-
Smirnov type statistic to test the homoscedasticity hypothesis, when the
observations are assumed to be strongly mixing. Their test uses only a
subsample which induces a loss of information and power. Ngatchou-Wandji,
(2002) presented a procedure, based on marked empirical process, for testing
the goodness-of-fit of the conditional variance function of a Markov model of
order one.

For time series with martingale difference innovations Laïb and Chebana,
(2011) considered a class of nonlinear semi-parametric models and established
the local asymptotic normality for cumulative residual process. They also
derived an efficient simultaneous test for testing the conditional mean and
the conditional variance functions. Laïb and Louani, (2002) provided a test of
conditional homoscedasticity hypothesis of the one-step forecast error in the
context of first-order AR-ARCH model. Their works was extended by Laïb, (2003)
for the context of time series with martingale difference innovations. The
author established the asymptotic of the cumulative residual process and
developed a test for homoscedasticity when the innovations are independent of
the past of ${\cal F}_{i-1}$. Chen et al., (2015) developed two tests for
parametric volatility function of a diffusion model, with i.i.d. innovations,
based on Khamaladze’s martingale transformation. Their tests use the
structural properties of the diffusion process and do not require the
estimation or the specification of the drift function. Escanciano, (2008)
proposed a class of joint and marginal spectral diagnostic tests for
parametric conditional means and variances of time series models. The proposed
tests are not distribution-free and the author introduced a bootstrap
procedure that should be used to implement these tests. Escanciano, (2010)
constructed asymptotic distribution-free joint specification tests, that can
be applied in many financial and economic time series including GARCH and
ARMA-GARCH models. These tests are based on carefully weighted residual
empirical process. The weights are chosen in a way to insure that the weighted
empirical process of residuals admits a distribution free limit. It is shown
that the proposed tests generalize those of Wooldridge, (1990). Note however
that, the performance of the constructed tests strongly depend on the choice
of the weights.

In this paper we develop a joint test for parametric form specification of the
conditional mean and variance functions when innovations satisfy the
martingale difference hypothesis and are allowed to depend on the past
$\sigma$-field. The main challenge comes from the fact the marked cumulative
residual process no longer admits a distribution-free limit. To obtain a
practical solution one either transforms the process in order to achieve a
distribution-free limit or approximates the non-distribution free limit using
numerical or re-sampling techniques. Here, three solutions, based on Khmaladze
transform of cumulative residual process, a multiplier bootstrap re-sampling
procedure, and a numerical approximation of the limiting distribution, are
considered and their finite sample performance is compared with Escanciano,
(2010)’s procedure.

Though Khmaladze transform of the cumulative residual process was considered
in previous works, such as Chen et al., (2015) or Stute et al., (1998), the
setting addressed here is more general and does not assume any explicit data-
generating model or independence of innovations. It may be applied for several
non-linear time series models with martingale innovations. The second added
value of this paper is that it compares numerically the three classical
procedures used for constructing joint tests based on the cumulative residual
process. The numerical study particularly revealed that the Khmaladze
transform is slightly less performant than re-sampling and numerical
approximation techniques.

The rest of the paper is organize as follows: Section 2 defines the problem
and states some preliminaries results. Section 3 defines and establishes the
asymptotic behavior of the martingale transformation applied to the cumulative
residual process. Section 4 introduces the marginals as well as the joint
parametric specification Cramér von-Mises-type tests statistics for the
conditional mean and conditional variance functions. Subsection 4.1 presents
the martingale transform based test statistics. A numerical approximation
procedure for the asymptotic distribution of the test statistics based on the
original process is given in Subsection 4.2. A re-sampling algorithm for test
statistics based on the original cumulative residual process is also detailed
in Subsection 4.3. A comparison between these statistical procedures, via
simulations, is outlined in Section 5 and an application to real data is given
in Section 6. A conclusion summarizing our findings is given in Section 7. All
proofs are provided in the Appendix.

##  2 Assumptions and Main results

Let $\\{(X_{i},Z_{i}),i\in\mathbb{Z}\\}$ be a strictly stationary ergodic
defined on the probability space $(\Omega,{\cal A},\mathbb{P})$. The random
variables $X_{i}$’s are real-valued with common continuous distribution
function ${F}$. For each $i\geq 1$ we let
$I_{i-1}=(X_{i-1},X_{i-2},\ldots,Z_{i})$ denote the past information at time
$i$. We let ${\cal F}_{i}=\sigma(I_{0},I_{1},\ldots,I_{i})$ denotes
$\sigma$-field generated by $I_{0},\ldots,I_{i}$. The purpose is to verify if
the conditional mean $\mu(I_{i-1})=\mathbb{E}\left(X_{i}|{\cal
F}_{i-1}\right)$ almost surely (a.s.) and the conditional variance
$\sigma^{2}(I_{i-1})=\mathrm{Var}\left(X_{i}|{\cal F}_{i-1}\right)$ a.s. of
$X_{i}$ given ${\cal F}_{i-1}$ satisfy, respectively, the following relations
$\mu(I_{i-1})=m(\theta_{0},I_{i-1})$ a.s. and
$\sigma^{2}(I_{i-1})=\sigma^{2}(\theta_{0},I_{i-1})$ a.s., where
$m(\theta,\cdot)$ and $\sigma^{2}(\theta,\cdot)$ are
$\mathcal{F}_{i-1}$-measurable known parametric functions that depend on a
finite dimensional vector of parameters $\theta\in\mathbb{R}^{d}$ for $d\geq
1$, and assumed to be finite with probability one. More precisely, we are
interested in examining hypotheses stating that

$\displaystyle\ (H_{0})$ $\displaystyle\mu(I_{i-1})=m(\theta_{0},I_{i-1})\
\mbox{a.s. and }\sigma^{2}(I_{i-1})=\sigma^{2}(\theta_{0},I_{i-1})\
\mbox{a.s.}\ \mbox{ versus }$ (2.1) $\displaystyle(H_{1})$
$\displaystyle\quad\mu(I_{i-1})\neq m(\theta,I_{i-1})\mbox{ or
}\sigma^{2}(I_{i-1})\neq\sigma^{2}(\theta_{0},I_{i-1}).$

We assume throughout this manuscript that for the true value of $\theta$,
denoted $\theta_{0}$ belongs to the interior of some compact subset
$\mathbf{\Xi}\subset\mathbb{R}^{d}$. We also assume that the sequence of
innovations $\\{X_{i}-m(\theta_{0},I_{i-1}):\,\ i\geq 0\\}$ is a sequence of
martingale differences with respect to ${\cal F}_{i}$, that is
$X_{i}-m({\theta_{0}},I_{i-1})$ is ${\cal F}_{i}$-measurable and
$\mathbb{E}\left[\left(X_{i}-m({\theta_{0}},I_{i-1})\right)\
|{\mathcal{F}}_{i-1}\right]=0\ \ \text{a.s.}$ This condition combined with the
fact that the set of information $I_{i}$ has an infinite dimensional will
allow to consider non markovian processes such as the ARMA-GARCH process and
all others. In practice the set $I_{i}$ is not observable and may be
estimated, see Remark 2.2 and Escanciano, (2010) for more discussion.

To test the specification of the conditional mean function we will use
$W^{1}_{\theta}(X,I):=X-m({\theta},I)$ and to test for the conditional
variance function we will use
$W^{2}_{\theta}(X,I):=(X-m({\theta},I))^{2}-\sigma^{2}({\theta},I)$. Following
Escanciano, 2007a , Escanciano, (2008), Koul and Stute, (1999), Laïb, (2003)
and Ngatchou-Wandji, (2002) we introduce the following cumulative empirical
residuals processes

$\displaystyle\mathbb{D}_{n}^{k}(x)$ $\displaystyle=$ $\displaystyle
n^{-1/2}\sum_{i=1}^{n}W^{k}_{\theta_{0}}(X_{i},I_{i-1})1\kern-1.99997pt{\rm
l}{\\{X_{i-1}\leq x\\}},\ \ x\in\mathbb{R}\mbox{ and }k=1,2.$
$\displaystyle\widehat{\mathbb{D}}_{n}^{k}(x)$ $\displaystyle=$ $\displaystyle
n^{-1/2}\sum_{i=1}^{n}W^{k}_{\theta_{n}}(X_{i},I_{i-1})1\kern-1.99997pt{\rm
l}{\\{X_{i-1}\leq x\\}},\ \ x\in\mathbb{R},\mbox{ and }k=1,2$

where $1\kern-1.99997pt{\rm l}(A)$ is the indicator function of the set $A$,
$\theta_{n}$ is consistent estimators of $\theta$. Note that, under $H_{0}$,
$\mathbb{E}\\{W^{k}_{\theta_{0}}(X_{i},I_{i-1})|\mathcal{F}_{i-1}\\}=0$ for
$k=1,2$. To test specification of the conditional mean and variance jointly,
we introduce the bivariate processes $\mathbb{D}_{n}$ and
$\widehat{\mathbb{D}}_{n}$ defined by
$\mathbb{D}_{n}(x):=(\mathbb{D}_{n}^{1}(x),\mathbb{D}_{n}^{2}(x))^{\top}$ and
$\widehat{\mathbb{D}}_{n}(x):=(\widehat{\mathbb{D}}_{n}^{1}(x),\widehat{\mathbb{D}}_{n}^{2}(x))^{\top}$,
where the script $B^{\top}$ stands for the transpose of the matrix $B$. Note
that letting
$W_{\theta}(X,I):=\\{W^{1}_{\theta}(X,I),W^{2}_{\theta}(X,I)\\}^{\top}$ one
sees that

$\mathbb{D}_{n}(x)=n^{-1/2}\sum_{i=1}^{n}W_{\theta_{0}}(X_{i},I_{i-1})1\kern-1.99997pt{\rm
l}{\\{X_{i-1}\leq x\\}}$

and

$\widehat{\mathbb{D}}_{n}(x)=n^{-1/2}\sum_{i=1}^{n}W_{\theta_{n}}(X_{i},I_{i-1})1\kern-1.99997pt{\rm
l}{\\{X_{i-1}\leq x\\}}.$

######  Remark 2.1.

Comparing the empirical process $\widehat{\mathbb{D}}_{n}^{k}$ with that used
in Escanciano, (2010), one notice that $\widehat{\mathbb{D}}_{n}^{k}$ is a
marked empirical process based on $X_{i-1}$ with marks/weights given by
$W_{\theta_{n}}^{k}$ while the process used in Escanciano, (2010) is based on
the residuals $\widehat{\epsilon}_{i}$, with weights carefully estimated from
data.

######  Remark 2.2.

As pointed in Escanciano, (2010), the past information $I_{i-1}$ is not
completely observable and needs to be estimated by $\hat{I}_{i-1}$. Such
estimation usually involves replacing the unknown initial state $I_{0}$ by
some quantity $\hat{I}_{0}$. One must insure that replacing $I_{i-1}$ by
$\hat{I}_{i-1}$ does not affect the asymptotic behaviour of the process
$\widehat{\mathbb{D}}_{n}$. One easily verifies that if for $k=1,2$,

$n^{-1/2}\sum_{i=1}^{n}\mathbb{E}\left\\{\sup_{\theta\in\mathbf{\Xi}}|W^{k}_{\theta}(X_{i},\hat{I}_{i-1})-W^{k}_{\theta}(X_{i},I_{i-1})|\right\\}=o(1)$

then the asymptotic behaviour of $\widehat{\mathbb{D}}_{n}$ is not altered
when $I_{i-1}$ is replaced by $\hat{I}_{i-1}$. We will assume that such
condition holds and will use $I_{i-1}$ in the definition of
$\widehat{\mathbb{D}}_{n}$ throughout the manuscript. The discussion in
Escanciano, (2010) shows that this condition holds true for ARMA-GARCH models
in particular.

The limiting laws of $\mathbb{D}_{n}^{1}$ and $\widehat{\mathbb{D}}_{n}^{1}$
are given in Koul and Stute, (1999) and Escanciano, 2007a . The limiting
behavior of $\mathbb{D}_{n}^{2}$ is given in Laïb, (2003), who also obtained
the limit law of $\widehat{\mathbb{D}}_{n}^{2}$ in the special case
$\sigma^{2}(\cdot)=\zeta^{2}$ with $\zeta^{2}\in(0,\infty)$. The next section
studies the asymptotic for the bivariate process $\widehat{\mathbb{D}}_{n}$ .

###  2.1 Limiting law of the process $\widehat{\mathbb{D}}_{n}$ under the null
hypothesis

The following notations are used in the rest of the paper. $\|v\|$ denotes the
Euclidian norm of the vector $v$ and for any matrix $A$,
$\|A\|=\sup_{v:\;\|v\|=1}\\{\|Av\|\\}$ is the associated matrix norm for the
matrix $A$. For any bounded function $f$, let
$\|f\|=\sup_{x}\left|f(x)\right|$ and for any $(x,y)\in\mathbb{R}^{2}$, set
$x\wedge y=\min(x,y)$ and $x\vee y=\max(x,y)$.

The asymptotic results are stated under the following assumptions:
Assumption A1: For each $k=1,2$,

  1. 1.

$\mathbb{E}\left(|W^{k}_{\theta_{0}}(X_{i},I_{i-1})|^{2}\right)<\infty$.

  2. 2.

$\lim_{n\to\infty}\mathbb{E}\left\\{W^{k}_{\theta_{0}}(X_{i},I_{i-1})1\kern-1.99997pt{\rm
l}\\{|W^{k}_{\theta_{0}}(X_{i},I_{i-1})|>\delta\sqrt{n}\\}\right\\}=0$ for any
real $\delta>0$.

  3. 3.

$K_{k}(x)=\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}\left(W^{k}_{\theta_{0}}(X_{i},I_{i-1})^{2}1\kern-1.99997pt{\rm
l}\\{X_{i-1}\leq x\\}\right)$ is non-decreasing continuous function of $x$.

  4. 4.

$\mathbb{E}\left(W^{k}_{\theta_{0}}(X_{i},I_{i-1})^{2}1\kern-1.99997pt{\rm
l}\\{x\leq X_{i-1}\leq
y\\}|\mathcal{F}_{i-2}\right)=C_{k,i}|K_{k}(y)-K_{k}(x)|$ with $C_{k,i}$ such
that $\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}|C_{k,i}|=O(1)$.

Assumption A2: Under $(H_{0})$, the estimators $\theta_{n}$ of $\theta_{0}$ is
such that

$n^{1/2}(\theta_{n}-\theta_{0})=n^{-1/2}\sum_{i=1}^{n}\phi^{\star}(X_{i},I_{i-1},\theta_{0})+o_{\mathbb{P}}(1)$

where $\phi^{\star}$ is an $\mathbb{R}^{d}$-valued measurable function
satisfying $\mathbb{E}\left(\phi^{\star}(X_{i},I_{i-1},\theta_{0})|{\cal
F}_{i-1}\right)=0$ a.s. and

$\Sigma_{0}=\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}\left(\phi^{\star}(X_{i},I_{i-1},\theta_{0})\phi^{\star}(X_{i},I_{i-1},\theta_{0})^{\top}\right)$

exists and is positive definite.
Assumption A3: For $k=1,2,$ let $\dot{W}^{k}_{\theta^{\star}}(X_{i},I_{i-1})$
denote the gradient of $W^{k}_{\theta}(X_{i},I_{i-1})$ with respect to the
components of $\theta$ evaluated at a fixed point $\theta^{\star}$. Assume
that $\mathbb{E}\|\dot{W}^{k}_{\theta_{0}}(X_{i},I_{i-1}))\|\leq C<\infty$ and

$\left\|W^{k}_{\theta}(X_{i},I_{i-1})-W^{k}_{\theta_{0}}(X_{i},I_{i-1})-(\theta-\theta_{0})^{\top}\dot{W}^{k}_{\theta_{0}}(X_{i},I_{i-1})\right\|\leq\|\theta-\theta_{0}\|M_{1}(X_{i},I_{i-1})\lambda_{1}(\|\theta-\theta_{0}\|),$

where $M_{1}$ and $\lambda_{1}$ are positive functions satisfying
$\mathbb{E}(M_{1}(X_{i},I_{i-1})\leq C<\infty$ and $\lambda_{1}(t)$ goes to
zero as $t\to 0$.

Note that Assumption (A1) is an adaptation of conditions (A–D) of Theorem 1 of
Escanciano, 2007b  to the context of stationary and ergodic sequence. As
argued in Escanciano, 2007b  these are among the weakest conditions to ensure
the weak convergence of marked empirical processes. Assumption (A2) and (A3)
are commonly used to ensure the convergence of
$\sqrt{n}(\theta_{n}-\theta_{0})$ and to validate the expansion of the process
$\widehat{\mathbb{D}}_{n}$. These assumptions hold for most commonly used
models and estimation procedures, see, for example Koul and Stute, (1999);
Escanciano, 2007a ; Escanciano, (2010) for discussion and details.

From now on let $\mathbf{D}(\overline{\mathbb{R}})$ denotes the space of
càdlàg functions and for $x,y\in\mathbb{R}$ define

$\displaystyle
K(x,y)=\rm{Cov}(\mathbb{D}_{n}(\mathbf{x}),\mathbb{D}_{n}(\mathbf{y}))=\left(\begin{array}[]{ll}K_{1}(x\wedge
y)&K_{1,2}(x\wedge y)\\\ K_{1,2}(x\wedge y)&K_{2}(x\wedge
y)\end{array}\right),$ (2.4)

where $K_{1}$ and $K_{2}$ are given in Assumption (A1) and for
$x\in\mathbb{R}$

$\displaystyle K_{1,2}(x)$ $\displaystyle=$
$\displaystyle\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}\left[W^{1}_{\theta_{0}}(X_{i},I_{i-1})W^{2}_{\theta_{0}}(X_{i},I_{i-1})1\kern-1.99997pt{\rm
l}\left(X_{i-1}\leq x\right)\right].$ (2.5)

Let $\mathbb{D}$ denotes the Gaussian process with covariance function $K$
defined above. The following result summarizes the weak convergence of the
processes ${\mathbb{D}}_{n}$ and $\widehat{\mathbb{D}}_{n}$.

######  Theorem 2.3.

If Assumption (A1) holds then, under $H_{0}$, ${\mathbb{D}}_{n}$ converges
weakly to ${{\mathbb{D}}}.$ If Assumptions (A1)-(A3) hold true, then, under
$H_{0}$, $\widehat{\mathbb{D}}_{n}$ converges weakly to
${\widehat{\mathbb{D}}},$ where ${\widehat{\mathbb{D}}}$ is a centered
Gaussian process given by

$\displaystyle\widehat{\mathbb{D}}(x)=\mathbb{D}(x)-\Gamma_{\theta_{0}}^{\top}(x)\Theta$
(2.6)

where $\Theta$ is a multivariate normal with mean zero and covariance
$\Sigma_{0}$ and the function
$\Gamma_{\theta_{0}}(x)=(\Gamma^{1}_{\theta_{0}}(x),\Gamma^{2}_{\theta_{0}}(x))$
with

$\Gamma^{1}_{\theta_{0}}(x)=\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}\left[\dot{m}(\theta_{0},I_{i-1})1\kern-1.99997pt{\rm
l}\\{X_{i-1}\leq x\\}\right]$

and

$\Gamma^{2}_{\theta_{0}}(x)=\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}\left\\{\dot{\sigma}^{2}(\theta_{0},I_{i-1})1\kern-1.99997pt{\rm
l}\\{X_{i-1}\leq x\\}\right\\}.$

The covariance function of $\widehat{\mathbb{D}}$ is given by

$\displaystyle{\mathcal{K}}(x,y)={K}(x,y)-\Gamma^{\top}_{\theta_{0}}(x)G(y)-G^{\top}(x)\Gamma_{\theta_{0}}(y)+\Gamma^{\top}_{\theta_{0}}(x)\Sigma_{0}\Gamma_{\theta_{0}}(y),$
(2.7)

where $G(x)=\mathrm{Cov}(\Theta,\mathbb{D})=(G^{1}(x),G^{2}(x))$ with

$G^{k}(x)=\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}\left[W^{k}_{\theta_{0}}(X_{i},I_{i-1})\phi^{\star}(X_{i},I_{i-1},\theta_{0})1\kern-1.99997pt{\rm
l}\left(X_{i-1}\leq x\right)\right]\mbox{ for }k=1,2.$

######  Remark 2.4.

The limiting covariance function ${\mathcal{K}}$ is in general complicated,
therefore classical statistics based on the process $\widehat{\mathbb{D}}_{n}$
do not admit distribution-free limits. To overcome this, one usually uses one
of the following approaches. The first approach consists in transforming the
process $\widehat{\mathbb{D}}_{n}$ in a such way to achieve a distribution-
free limit and then using the transformed process to define test statistics.
While the second approach uses $\widehat{\mathbb{D}}_{n}$ to construct test
statistics and then adopts either a numerical approximation or a re-sampling
technique to estimate the non-distribution free limit. Both approaches will be
discussed and compared in the rest of this manuscript. The transformation is
discussed in Section 3 while the numerical and re-sampling approximations are
outlined in Section 4.

###  2.2 Limiting law of the process $\widehat{\mathbb{D}}_{n}$ under local
alternatives

In this section we establish the limiting behavior of the process
$\widehat{\mathbb{D}}_{n}$ under local alternatives $H_{A}$ defined as
follows.

$H_{A}:\;\mu(I_{i-1})=m(\theta_{0},I_{i-1})+a_{1}(I_{i-1})/\sqrt{n}\mbox{ and
}\sigma^{2}(I_{i-1})=\sigma^{2}(\theta_{0},I_{i-1})+a_{2}(I_{i-1})/\sqrt{n},$

where $a_{1}$ and $a_{2}$ are some non-zero functions indicating the direction
of departure from the null hypothesis. We assume that
$\mathbb{E}|a_{k}(I_{i-1})|<\infty$ for $k=1,2$. The following assumption is
needed to establish the limiting behavior under $H_{A}$.
Assumption (L1) Under $H_{A}$, the estimator $\theta_{n}$ satisfies

$n^{1/2}(\theta_{n}-\theta_{0})=n^{-1/2}\sum_{i=1}^{n}\phi^{\star}(X_{i},I_{i-1},\theta_{0})+\xi_{A}+o_{\mathbb{P}}(1),$

where $\xi_{A}\in\mathbb{R}^{d}$ is a random/nonrandom vector and
$\phi^{\star}$ is as in Assumption (A2).

The following Theorem shows that our tests are able to detect local
alternatives of the type described by $H_{A}$.

######  Theorem 2.5.

Under $H_{A}$, if Assumptions A1, A3 and L1 hold, the process
$\widehat{\mathbb{D}}_{n}$ converges weakly to
$\widehat{\mathbb{D}}+\Psi_{A}(x)-\Gamma^{\top}_{\theta_{0}}(x)\xi_{A},$ where
$\Psi_{A}(x)=\mathbb{E}\left\\{\left(\begin{array}[]{l}a_{1}(I_{i-1})\\\
a_{2}(I_{i-1})\end{array}\right)1\kern-1.99997pt{\rm l}\\{X_{i-1}\leq
x\\}\right\\}$.

##  3 Khmaladze transform of the process $\widehat{\mathbb{D}}_{n}$

This section presents a martingale transformation of the Khmaladze type that
will be applied to the process $\widehat{\mathbb{D}}_{n}$ in order to achieve
a distribution-free limit for classical test statistics. It worth mentioning
that the Khmaladze transformation will ensure that the limit of the
transformed process is Brownian motion.

To define such transformation, assume the functions
$\Gamma_{\theta_{0}}^{k}(x)$, for $k=1,2$, can be written as

$\Gamma_{\theta_{0}}^{k}(x)=\int_{-\infty}^{x}g_{k}(t)K_{k}(dt),$ (3.1)

where $g_{k}(t)=(g_{1,k}(t),\ldots,g_{d,k}(t))^{\top}$. Let the matrix $A_{k}$
for $k=1,2$ be defined as
$A_{k}(x)=\int_{x}^{\infty}g_{k}(u)g_{k}^{\top}(u){K_{k}}(du)$. Assume that
there exists $x_{0}<\infty$ such $A_{k}(x_{0})$ is non-singular. Note that it
follows from the definition of $A_{k}(x)$ that $A_{k}(x)-A_{k}(x_{0})$ is non-
negative definite for all $x\leq x_{0}$ implying that $A_{k}(x)$ is invertible
for all $x\leq x_{0}$ whenever $A_{k}(x_{0})$ is invertible.

Following the concept in Khmaladze, (1988), for any function $\mathbf{f}$
consider the linear transformations $T^{k}(\mathbf{f})$ for $k=1,2$ given by

$\displaystyle(T^{k}{\mathbf{f}})(x)={\mathbf{f}}(x)-\int_{-\infty}^{x}(g_{k}(y))^{\top}A_{k}^{-1}(y)\left[\int_{y}^{\infty}g_{k}(z){\mathbf{f}}(dz)\right]{K_{k}}(dy),$
(3.2)

for all $x\leq x_{0}$. The Khmaladze transform of the cumulative residual
process considered here is an application of the transformation $T^{1}$ and
$T^{2}$ to the processes $\widehat{\mathbb{D}}^{1}_{n}$ and
$\widehat{\mathbb{D}}^{2}_{n}$. It is formally defined by

$T(\widehat{\mathbb{D}}_{n}(x))=\left(\begin{array}[]{l}T^{1}(\widehat{\mathbb{D}}^{1}_{n}(x))\\\
T^{2}(\widehat{\mathbb{D}}^{2}_{n}(x))\end{array}\right)\\\
=\left(\begin{array}[]{l}\widehat{\mathbb{D}}_{n}^{1}(x)-\int_{-\infty}^{x}(g_{1}(y))^{\top}A_{1}^{-1}(y)\left[\int_{y}^{\infty}g_{1}(z)\widehat{\mathbb{D}}_{n}^{1}(dz)\right]{K_{1}}(dy)\\\
\widehat{\mathbb{D}}_{n}^{2}(x)-\int_{-\infty}^{x}(g_{2}(y))^{\top}A_{2}^{-1}(y)\left[\int_{y}^{\infty}g_{2}(z)\widehat{\mathbb{D}}_{n}^{2}(dz)\right]{K_{2}}(dy)\end{array}\right).$
(3.3)

Easy manipulations show that for $k=1,2$

$T^{k}(\widehat{\mathbb{D}}^{k}_{n}(x))=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}W^{k}_{\theta_{n}}(X_{i},I_{i-1})\biggl{[}1\kern-1.99997pt{\rm
l}{\left\\{X_{i-1}\leq x\right\\}}\\\
-\int_{-\infty}^{\min(x,X_{i-1})}(g_{k}(y))^{\top}A_{k}^{-1}(y){K_{k}}(dy)g_{k}(X_{i-1})\biggr{]}.$
(3.4)

Observe also that $T^{k}(\widehat{\mathbb{D}}^{k}_{n})(x)$ cannot be used in
practice to build test statistics since it depends on several unknown
quantities, namely, $g_{k}$ and $K_{k}$. Replacing $g_{k}$ and $K_{k}$ by
their consistent estimates $\hat{g}_{k}$ and $\hat{K}_{k}$ respectively, one
obtains an empirical version of $T^{k}(\widehat{\mathbb{D}}^{k}_{n})$ defined
as follows

$T^{k}_{n}(\widehat{\mathbb{D}}^{k}_{n}(x))=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}W^{k}_{\theta_{n}}(X_{i},I_{i-1})\biggl{[}1\kern-1.99997pt{\rm
l}{\left\\{X_{i-1}\leq x\right\\}}\\\
-\int_{-\infty}^{\min(x,X_{i-1})}(\hat{g}_{k}(y))^{\top}\hat{A}_{k}^{-1}(y){\hat{K}_{k}}(dy)\hat{g}_{k}(X_{i-1})\biggr{]},$

where
$\hat{A}_{k}(x)=\int_{x}^{\infty}\hat{g}_{k}(u)\hat{g}_{k}^{\top}(u){\hat{K}_{k}}(du)$.
The Following assumptions are needed to ensure the convergence of
$T_{n}(\widehat{\mathbb{D}}_{n})=(T^{1}_{n}(\widehat{\mathbb{D}}^{1}_{n}),T^{2}_{n}(\widehat{\mathbb{D}}^{2}_{n}))^{\top}$.

Assumption K1: Assume there exists $x_{0}<\infty$ such that $A_{k}(x_{0})$ is
invertible for $k=1,2$.
Assumption K2: Assume that
$\mathbb{E}\|W_{\theta_{0}}^{k}(X_{i},I_{i-1})g_{k}(X_{i-1})\|\\}$,
$\mathbb{E}\\{M_{1}(X_{i},I_{i-1})\|g_{k}(X_{i-1})\|\\}$,
$\mathbb{E}\\{M_{2}(X_{0})\|g(X_{0})\|\\}$, $\mathbb{E}\\{\|g(X)\|\\}$,
$\mathbb{E}\\{\|g(X)g(X)^{\top}\|W_{\theta_{0}}^{k}(X_{i},I_{i-1})^{2}\\}$ and
$\mathbb{E}\\{\|g(X)g(X)^{\top}\|\\}$ are all finite.
Assumption K3: Assume that $\|\hat{g}_{k}-g_{k}\|$ converges to zero in
probability, and that
$\sup_{y\in\mathbb{R}}\left\|\int_{y}^{\infty}(g_{k}(z)-\hat{g}_{k}(z)){\mathbb{D}}_{n}(dz)\right\|$
converges to zero in probability.
Assumption K4: For each $k=1,2$ and for each $j=1,\ldots,d$,

  1. 1.

$\mathbb{E}\left(|W^{k}_{\theta_{0}}(X_{i},I_{i-1})g_{j,k}(X_{i-1})|^{2}\right)<\infty$.

  2. 2.

$\lim_{n\to\infty}\mathbb{E}\left\\{|W^{k}_{\theta_{0}}(X_{i},I_{i-1})g_{j,k}(X_{i-1})|^{2}1\kern-1.99997pt{\rm
l}\\{|W^{k}_{\theta_{0}}(X_{i},I_{i-1})g_{j,k}(X_{i-1})|>\delta\sqrt{n}\\}\right\\}=0$,
for any $\delta>0$.

  3. 3.

$\bar{K}_{j,k}(x)=\mathbb{E}\left(W^{k}_{\theta_{0}}(X_{i},I_{i-1})^{2}g_{j,k}(X_{i-1})^{2}1\kern-1.99997pt{\rm
l}\\{X_{i-1}\leq x\\}\right)$ is non-decreasing continuous function of $x$.

  4. 4.

$\mathbb{E}\left(W^{k}_{\theta_{0}}(X_{i},I_{i-1})^{2}g_{j,k}(X_{i-1})^{2}1\kern-1.99997pt{\rm
l}\\{x\leq X_{i-1}\leq
y\\}|\mathcal{F}_{i-2}\right)=C^{j}_{k,i}|\bar{K}_{j,k}(y)-\bar{K}_{j,k}(x)|$
with $C^{j}_{k,i}$ such that
$\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}|C^{j}_{k,i}|=O(1)$.

The next result establishes the weak convergence of
$T_{n}(\widehat{\mathbb{D}}_{n})$.

######  Theorem 3.1.

If assumptions A1–A4 and K1–K4 are satisfied then for $k=1,2$

$T_{n}^{k}(\widehat{\mathbb{D}}_{n}^{k})\rightsquigarrow{\mathbb{D}^{k}}\quad\text{in}\quad\mathbf{D}((-\infty,x_{0}]),$

where $\mathbb{D}^{k}=\mathbb{W}_{k}\circ K_{k}$ is the $k$-th component in
the process $\mathbb{D}$ defined in Theorem 2.3.

######  Remark 3.2.

Note that the process $\mathbb{D}$ has marginal $\mathbb{W}_{1}\circ K_{1}$
and $\mathbb{W}_{2}\circ K_{2}$ where $\mathbb{W}_{1}$ and $\mathbb{W}_{2}$
are Brownian motions. In general $\mathbb{W}_{1}$ and $\mathbb{W}_{2}$ are not
independent. The distribution of $\int\|\mathbb{D}\|^{2}$ will only be
distribution free if $\mathbb{W}_{1}$ and $\mathbb{W}_{2}$ are independent.
This defeats the purpose of the martingale transformation in general. Though
the above results clearly show that the transformation provides a
distribution-free limit for testing the mean or the variance function
separately. The transformation is only useful for joint testing of conditional
mean and variance functions only in the case of independent components of the
bivariate process $\mathbb{D}$. This was the case in the work of Chen et al.,
(2015). In general, the components $\mathbb{D}^{1}$ and $\mathbb{D}^{2}$ will
be independent if $K_{12}=0$ which translates to a condition on the third
moment of $\varepsilon_{i}=X_{i}-m(\theta,I_{i-1})$. In particular, if
$\mathbb{E}(\varepsilon_{i}^{3}|\mathcal{F}_{i-1})=0$ a.s. then the components
$\mathbb{D}^{1}$ and $\mathbb{D}^{2}$ are independent. This holds true for all
time series models that assume that the errors are i.i.d provided that we add
a condition stating that the error is symmetric or just has zero third
conditional moment.

######  Remark 3.3.

Note that the conditions imposed on the estimate $g_{n}$ are similar to those
used in Bai, (2003) and Bai and Ng, (2001). Note also that the
$\|\hat{g}_{k}-g\|$ converges to zero in probability can be weakened to
$\int\|\hat{g}_{k}(x)-g(x)\|^{2}F(dx)$ converges to zero in probability. But
for simplicity, the proof is presented with the stronger assumption. The
second condition on $\hat{g}_{k}$, in Assumption (K3), is usually verified
using the structure of the estimator and the nature of the process
$\mathbb{D}_{n}$, for details see the discussion in Bai, (2003).

##  4 Test statistics

Test statistics for the hypothesis $(H_{0})$ are obtained by considering
continuous functional $\mathcal{G}$ of the process $\widehat{\mathbb{D}}_{n}$
or of its Khmaladze transform $T_{n}(\widehat{\mathbb{D}}_{n})$. That is,
given a continuous functional $\mathcal{G}$ a test statistic can be obtained
using $\mathcal{G}\left(\widehat{\mathbb{D}}_{n}\right)$ or
$\mathcal{G}\left(T_{n}(\widehat{\mathbb{D}}_{n})\right)$. The continuous
mapping theorem provides the asymptotic behavior of
$\mathcal{G}\left(\widehat{\mathbb{D}}_{n}\right)$ and
$\mathcal{G}\left(T_{n}(\widehat{\mathbb{D}}_{n})\right)$. The most commonly
used functional, in practice, are of the Cramér-von-Mises or Kolmogorov-
Smirnov type. For simplicity, the rest of this paper only focuses on Cramér-
von-Mises type statistics, but the approaches discussed apply to any
continuous functional. To be specific, the following marginal Cramér-von-Mises
test statistics

$\displaystyle S_{n}^{k}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\left[\widehat{\mathbb{D}}_{n}^{k}(x)\right]^{2}F_{n}(dx)$
$\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}\left\\{1-F_{n}(X_{i-1}\vee
X_{j-1})\right\\}W^{k}_{\theta_{n}}(X_{i},I_{i-1})W^{k}_{\theta_{n}}(X_{j},I_{j-1})$

for $k=1,2$, will be used here. Clearly, the statistics
$\\{S_{n}^{k}\\}_{k=1,2}$ given in (4) converge in distribution to
$S^{k}:=\int_{-\infty}^{\infty}[\widehat{\mathbb{D}}^{k}(x)]^{2}F(dx)$ which
is equivalent to $\int_{0}^{1}[\widehat{\mathbb{D}}^{k}(F^{-1}(u))]^{2}du$.

The statistic ${S}_{n}^{1}$ can be used to test the conditional mean function
and the statistic ${S}_{n}^{2}$ can be used to test the conditional variance
function. To conduct a joint test of mean and variance functions one must
combine ${S}_{n}^{1}$ and ${S}_{n}^{2}$. To combine both statistics, Chen et
al., (2015) used ${S}_{n}^{1}+{S}_{n}^{2}$ and
$\max({S}_{n}^{1},{S}_{n}^{2})$. Such combinations were adequate since in his
case both statistics had the same limiting distribution. In our context, this
is not true in general, hence to combine these test statistics we will need a
more general approach such as the ones discussed in Ghoudi et al., (2001) and
Genest and Rémillard, (2004). To be specific, define
$S_{n}^{\star}=S_{n}^{1}/L_{1}+S_{n}^{2}/L_{2}$,
$S_{n}^{\bullet}=-2[\ln(P_{v}((S_{n}^{1}))+\ln(P_{v}(S_{n}^{2}))]$ and
$S_{n}^{\circ}=\max(S_{n}^{1}/L_{1},S_{n}^{2}/L_{2})$, where
$L_{k}=\mathbb{E}(S^{k})=\int K_{k}(x)F(dx)$ and $P_{v}(S^{k}_{n})$ denotes
the P-value of the statistics $S^{k}_{n}$. According to Fisher, (1950, pp.
99-101), when $S^{1}$ and $S^{2}$ are independent $S_{n}^{\bullet}$ converges
to Chi-square distribution with $4$ degrees of freedom and provides the
optimal way of combining $S^{1}_{n}$ and $S^{2}_{n}$. It will be seen, in the
simulation and application sections, that $S_{n}^{\star}$ and $S_{n}^{\circ}$
are easier to compute and have similar power to $S_{n}^{\bullet}$.

###  4.1 Martingale transform based test statistics

Next, we repeat a similar approach using the transformed process
$T_{n}(\widehat{\mathbb{D}}_{n})$ yielding

$\displaystyle\widetilde{S}_{n}^{k}=\int_{-\infty}^{\infty}\frac{\left(T_{n}^{k}(\widehat{\mathbb{D}}_{n}^{k})(x)\right)^{2}}{(\gamma^{k}_{n})^{2}}{\hat{K}_{k}}(dx)=\frac{1}{n(\gamma^{k}_{n})^{2}}\sum_{i=1}^{n}\left(T_{n}^{k}(\widehat{\mathbb{D}}_{n}^{k})(X_{i-1})\right)^{2}W^{k}_{\theta_{n}}(X_{i},I_{i-1})^{2},$
(4.2)

where

$\displaystyle\hat{K}_{k}(x)$ $\displaystyle=$
$\displaystyle\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}W^{k}_{\theta_{n}}(X_{i},I_{i-1})^{2}1\kern-1.99997pt{\rm
l}\left(X_{i-1}\leq x\right),\ x\in\mathbb{R}\mbox{ and }k=1,2.$ (4.3)

and
$\gamma_{n}^{k}=\lim_{x\to\infty}\hat{K}_{k}(x)=\frac{1}{n}\sum_{i=1}^{n}W^{k}_{\theta_{n}}(X_{i},I_{i-1})^{2}$.
Observe that Theorem 3.1 and the continuous mapping Theorem imply that, under
the null hypothesis, $\tilde{S}_{n}^{k}$ converges in distribution to
$\tilde{S}^{k}:=\int_{0}^{1}{\mathbb{W}_{k}(u)^{2}}du$, where
${\mathbb{W}_{k}}$ is a standard Brownian motion. According to Shorack and
Wellner, (1986) (page 748), for $\widetilde{S}_{n}^{k}$ with $k=1,2$, the
limiting critical values for 90%, 95% and 99% are 1.2, 1.657 and 2.8
respectively.

Since $\tilde{S}_{n}^{1}$ and $\tilde{S}_{n}^{2}$ admit the same limiting
distribution, there is no need for the normalization by $L_{1}$ and $L_{2}$ in
the definition of the combined statistics. Therefore, the combined statistics
using the transformed process are defined as follows:
$\tilde{S}_{n}^{\star}=\tilde{S}_{n}^{1}+\tilde{S}_{n}^{2}$,
$\tilde{S}_{n}^{\bullet}=-2[\ln(P_{v}((\tilde{S}_{n}^{1}))+\ln(P_{v}(\tilde{S}_{n}^{2}))]$
and $\tilde{S}_{n}^{\circ}=\max(\tilde{S}_{n}^{1},\tilde{S}_{n}^{2})$.

The statistics $\tilde{S}_{n}^{\star}$, $\tilde{S}_{n}^{\bullet}$ and
$\tilde{S}_{n}^{\circ}$ will only be distribution-free if $\tilde{S}^{1}$ and
$\tilde{S}^{2}$ are independent. As argued earlier, this is the case if the
covariance term $K_{12}=0$ or more precisely if
$\mathbb{E}\left\\{(X_{i}-m(\theta_{0},I_{i-1}))^{3}|\mathcal{F}_{i-1}\right\\}=0$
a.s.

As mentioned above, the statistics $\\{S_{n}^{k}\\}_{k=1,2}$ given in (4)
converge weakly towards
$S^{k}=\int_{0}^{1}[\widehat{\mathbb{D}}^{k}(F^{-1}(u))]^{2}du$. Since the
process $\widehat{\mathbb{D}}^{k}$ has a complicated covariance, which depends
on the unknown distribution $F(\cdot)$ and some unknown parameters, the
p-values of the test statistic $S_{n}^{k}$ need to be approximated using
either numerical techniques or re-sampling algorithms. A numerical
approximation of the limiting distribution, using the approach introduced by
Deheuvels and Martynov, (1996), is outlined in Subsection 4.2 and a re-
sampling algorithm based on multipliers bootstrap is described in Subsection
4.3.

###  4.2 Numerical approximation of the asymptotic distribution of the test
statistics

To approximate the limiting distribution of the statistics
$\\{S_{n}^{k}\\}_{k=1,2}$ and $S_{n}^{\star}$ we follow the numerical
integration procedure given by Deheuvels and Martynov, (1996). We will only
outline how the procedure applies to $S_{n}^{\star}$ since the application to
$\\{S_{n}^{k}\\}_{k=1,2}$ is similar and much simpler. The main idea consists
in using numerical integration to approximate
$S^{\star}=\int_{0}^{1}\left[\widehat{\mathbb{D}}^{1}(F^{-1}(u))^{2}/L_{1}+\widehat{\mathbb{D}}^{2}(F^{-1}(u))^{2}/L_{2}\right]du$
by the quadratic form
$Q_{m}=\sum_{k=1}^{m}a_{k}\left[\widehat{\mathbb{D}}^{1}(F^{-1}(u_{k}))^{2}/L_{1}+\widehat{\mathbb{D}}^{2}(F^{-1}(u_{k}))^{2}/L_{2}\right]$,
where $u_{k}$ are the quadrature nodes, $a_{k}$ are the quadrature
coefficients and $m$ is the number of quadrature points. Because
$\widehat{\mathbb{D}}$ is a Gaussian process, $Q_{m}$ is a quadratic form of
$(2m)$ normal random variables. Imhof’s characteristic function inversion
procedure (Imhof, (1961)) is used to compute the distribution function of
$Q_{m}$. When applying the procedure to $S_{n}^{1}$ or $S_{n}^{2}$, $Q_{m}$
will be a quadratic form with $(m)$ normal random variables. Note that in the
computation of the characteristic function the covariance operator
$\widehat{\mathcal{K}}$ is replaced by its empirical estimate obtained by
replacing $K$, $\Gamma_{\theta_{0}}$, $G$ and $\Sigma_{0}$ by their consistent
estimators $\hat{K}$, $\hat{\Gamma}$, $\hat{G}$ and $\hat{\Sigma}_{0}$, where
$\hat{K}(t)$ is a symmetric two by two matrix whose diagonal entries are
$\hat{K}_{k}$ for $k=1,2,$ and whose off diagonal entry $\hat{K}_{12}$ is
given by

$\hat{K}_{12}(t)=\frac{1}{n}\;\sum_{i=1}^{n}W_{\theta_{n}}^{1}(X_{i},I_{i-1})W_{\theta_{n}}^{2}(X_{i},I_{i-1})1\kern-1.99997pt{\rm
l}{\\{X_{i-1}\leq t\\}},$

and where

$\hat{G}(t)=\frac{1}{n}\
\sum_{i=1}^{n}W_{\theta_{n}}(X_{i},I_{i-1})\phi^{\star}(X_{i},I_{i-1},\theta_{n})1\kern-1.99997pt{\rm
l}{\\{X_{i-1}\leq t\\}},$
$\hat{\Gamma}(t)=\frac{1}{n}\sum_{i=1}^{n}\dot{W}_{\theta_{n}}(X_{i},I_{i-1})1\kern-1.99997pt{\rm
l}{\\{X_{i-1}\leq t\\}}$

and $\hat{\Sigma}_{0}=\frac{1}{n}\
\sum_{i=1}^{n}\phi^{\star}(X_{i},I_{i-1},\theta_{n})\phi^{\star}(X_{i},I_{i-1},\theta_{n})^{\top}.$
We also used, in the simulation and application sections, the simplest
quadratures, namely $a_{k}=1/m$ and $u_{k}=k/m$.

For the statistics $S_{n}^{\bullet}$ is computed after obtaining the p-values
of statistics $S_{n}^{1}$ and $S_{n}^{2}$. The p-value of $S_{n}^{\bullet}$
are obtained form the Chi-square distribution which is only valid when $S^{1}$
and $S^{2}$ are independent. The independence argument also allows for easy
computation of the p-values of the statistics $S_{n}^{\circ}$. If the
independence argument is not valid, then the computation for both
$S_{n}^{\bullet}$ and $S_{n}^{\circ}$ are more complex and requires the
numerical approximation of the joint distribution of $S_{n}^{1}/L_{1}$ and
$S_{n}^{2}/L_{2}$.

###  4.3 Multipliers bootstrap for the test statistics

Another approach to approximate the limiting distributions of
$S^{1}_{n},S_{n}^{2},S_{n}^{\star},S_{n}^{\bullet}$ and $S_{n}^{\circ}$
consists in using a re-sampling algorithm. To define such algorithm, let $B$
be a positive integer denoting the number of bootstrap samples. For
$b=1,\dots,B$, let $\left(Z_{1,b},\dots,Z_{n,b}\right)$ be a sequence of
independent identically distributed random variables with mean zero and
variance one independent of the sigma field generated by the $X_{i}$’s. The
multipliers bootstrap technique applied to the empirical process
$\mathbb{D}_{n}$, is defined as follows

$\displaystyle\mathbb{D}_{n,b}^{\star}(x):=\frac{1}{\sqrt{n}}\;\sum_{i=1}^{n}Z_{i,b}\biggl{\\{}W_{\theta_{0}}(X_{i},I_{i-1})1\kern-1.99997pt{\rm
l}{\\{X_{i-1}\leq
x\\}}-\Gamma^{\top}_{\theta_{0}}(x)\phi^{\star}(X_{i},I_{i-1},\theta_{0})\biggr{\\}}.$
(4.4)

Observe that the process $\mathbb{D}_{n}^{\star k}(x)$ defined in (4.4)
depends on unknown quantities, namely, $\theta$ and $\Gamma_{\theta}$. A plug-
in estimator of $\mathbb{D}_{n}^{\star k}(x)$ is obtained by replacing
$\theta$ and $\Gamma_{\theta}$ by their consistent estimators $\theta_{n}$ and
${\Gamma}_{\theta_{n}}(x)$ yielding

$\displaystyle\check{\mathbb{D}}_{n,b}(x):=\frac{1}{\sqrt{n}}\;\sum_{i=1}^{n}Z_{i,b}\biggl{\\{}W_{\theta_{n}}(X_{i},I_{i-1})1\kern-1.99997pt{\rm
l}{\\{X_{i-1}\leq
x\\}}-\Gamma^{\top}_{\theta_{n}}(x)\phi^{\star}(X_{i},I_{i-1},\theta_{n})\biggr{\\}}.$
(4.5)

The following extra assumptions are needed to establish the weak convergence
of $\check{\mathbb{D}}_{n}^{k}$.
Assumption M1: Assume that there exists $\delta>0$ such that for all $\theta$
satisfying $\|\theta-\theta_{0}\|\leq\delta$ one has
$\|\dot{W}^{k}_{\theta}(X_{i},I_{i-1})-\dot{W}^{k}_{\theta_{0}}(X_{i},I_{i-1})\|\leq\|\theta-\theta_{0}\|M_{3}(X_{i},I_{i-1})$
where $M_{3}$ is a positive function satisfying
$\mathbb{E}(M_{3}(X_{i},I_{i-1}))\leq C<\infty$.
Assumption M2: Assume that there exists $\delta>0$ such that for all $\theta$
satisfying $\|\theta-\theta_{0}\|\leq\delta$ one has
$\|\phi^{\star}(X_{i},I_{i-1},\theta)-\phi^{\star}(X_{i},I_{i-1},\theta_{0})-(\theta-\theta_{0})^{\top}\dot{\phi}^{\star}(X_{i},I_{i-1},\theta_{0})\|\leq\|\theta-\theta_{0}\|\lambda_{3}(\|\theta-\theta_{0}\|)M_{4}(X_{i},I_{i-1})$
where $M_{4}$ and $\lambda_{3}$ are positive functions satisfying
$\mathbb{E}(M_{4}(X_{i},I_{i-1}))\leq C<\infty$ and $\lim_{t\to
0}\lambda_{3}(t)=0$.
Assumption M3: Assume that
$E\|\dot{\phi}^{\star}(X_{i},I_{i-1},\theta_{0})\dot{\phi}^{\star}(X_{i},I_{i-1},\theta_{0})^{\top}\|\leq
C<\infty$.
The next result summarizes the asymptotic behavior of the bootstrapped process
$\check{\mathbb{D}}_{n,b}$.

######  Theorem 4.1.

Suppose that assumptions A1–A4 and M1–M3 hold true, then for $b=1,\ldots,B$,
$\check{\mathbb{D}}_{n,b}$ converge weakly to independent copies of
$\widehat{\mathbb{D}}=\mathbb{D}-\Gamma^{\top}_{\theta_{0}}\Theta$.

Note that the bootstrapped version of the statistic $S^{k}_{n}$ for $k=1,2$ is
given by

$S_{n,b}^{k}:=\int_{-\infty}^{\infty}\check{\mathbb{D}}_{n,b}^{k}(x)^{2}F_{n}(dx)=\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}Z_{i,b}Z_{j,b}\mathcal{M}^{k}_{i,j}$

where

$\mathcal{M}^{k}_{i,j}=W^{k}_{\theta_{n}}(X_{i},I_{i-1})W^{k}_{\theta_{n}}(X_{j},I_{j-1})(1-F_{n}(X_{i-1}\vee
X_{j-1}))\\\
-W^{k}_{\theta_{n}}(X_{i},I_{i-1})\phi^{\star}(X_{j},I_{j-1},\theta_{n})^{\top}\mathcal{L}^{k}_{n}(X_{i-1})\\\
-W^{k}_{\theta_{n}}(X_{j},I_{j-1})\phi^{\star}(X_{i},I_{i-1},\theta_{n})^{\top}\mathcal{L}^{k}_{n}(X_{j-1})+\phi^{\star}(X_{i},I_{i-1},\theta_{n})^{\top}\mathcal{N}\phi^{\star}(X_{i},I_{i-1},\theta_{n}),$

where
$\mathcal{L}_{n}^{k}(t):=\int_{t}^{\infty}\Gamma^{k}_{\theta_{n}}(u)F_{n}(du)$
and
$\mathcal{N}:=\int\Gamma^{k}_{\theta_{n}}(u)\Gamma^{k}_{\theta_{n}}(u)^{\top}F_{n}(du)$.

Let $L_{n,k}=\sum_{i=1}^{n}\hat{K}_{k}(X_{i})/n$ for $k=1,2$. To obtain a
bootstrapped version of $S_{n}^{\star}$ and $S_{n}^{\circ}$ one uses
$S_{n,b}^{\star}=S_{n,b}^{1}/L_{n,1}+S_{n,b}^{2}/L_{n,2}$ and
$S_{n,b}^{\circ}=\max\\{S_{n,b}^{1}/L_{n,1},S_{n,b}^{2}/L_{n,2}\\}$
respectively.

To apply the multipliers bootstrap procedure to approximate the p-value of any
of the Cramér-von Mises statistics $S^{1}_{n}$,$S^{2}_{n}$, $S^{\star}_{n}$ or
$S^{\circ}_{n}$ one proceeds according to the following algorithm. The
algorithm is explained in the context of $S^{1}_{n}$ but works in the same
manner for any of the statistics mentioned above.

  * •

Estimate $\theta$ by $\theta_{n}$ and compute the test statistic $S^{1}_{n}$.

  * •

For $b\in\\{1,\dots,B\\}$

    * –

Generate $(Z_{1,b},\ldots,Z_{n,b})$ as sequence of i.i.d normal $(0,1)$ random
variables.

    * –

Compute the bootstrapped Cramér-von Mises statistics $S_{n,b}^{1}$ for
$b=1,\ldots,B$.

  * •

Estimate the $p$-value, $Pv_{1}$ of $S^{1}_{n}$ by
$Pv_{1}=\frac{1}{B}\sum_{b=1}^{B}1\kern-1.99997pt{\rm
l}{\\{S_{n,b}^{1}>S^{1}_{n}\\}}$.

The statistics $S_{n}^{\bullet}=-2\ln(Pv_{1})-2\ln(Pv_{2})$ where $Pv_{k}$ for
$k=1,2,$ are the estimated $p$-values given by the above procedure. The
$p$-value of the statistics $S_{n}^{\bullet}$ are obtained using the Chi-
square distribution with $4$ degrees of freedom.

######  Remark 4.2.

Note that $\theta$ is estimated only once and the matrix $\mathcal{M}$ is only
computed once which make the multipliers Bootstrap extremely fast to
implement. This is much faster than the classical parametric bootstrap which
requires re-estimating the parameters for each bootstrap iteration. Moreover,
Theorem 4.1 and the continuous mapping Theorem imply that
$\left(S_{n,b}^{1},\dots,S_{n,B}^{1}\right)$ converges in law to independent
copies of $S^{1}$.

##  5 Finite sample performance

This section presents several simulation experiments carried out to assess the
power of the proposed test statistics. The first experiment, described in
Subsection 5.1, is designed to assess the fit of an ARCH(1) model. The second
experiment, outlined in Subsection 5.2, assess the power of our test
statistics for detecting departure from GARCH(1,1) model. The third experiment
given in Subsection 5.3 studies the behavior of the test when fitting and
AR(1)-GARCH(1,1) model. The fourth experiment, presented in Subsection 5.4
outlines the finding of applying our test procedures to the case of stochastic
differential equation models. Two sub-experiments are considered, in the first
we test if the model has a linear drift and a constant diffusion while in the
second we test if the model has linear drift and a diffusion proportional of
square root of series.

###  5.1 Testing ARCH(1) model

The purpose of the simulation experiment considered here is to test whether
the data is generated from pure ARCH(1) model or not. That is we wish to test

$\displaystyle(H_{0}):$ $\displaystyle\mu(\cdot)=0\mbox{ and
}\sigma^{2}(\cdot)=\alpha_{0}+\alpha_{1}X_{t-1}^{2}\ \quad\mbox{ versus }$
(5.1) $\displaystyle(H_{1}):$ $\displaystyle\quad\mu(\cdot)\neq 0\mbox{ or
}\sigma^{2}(\cdot)\neq\alpha_{0}+\alpha_{1}X_{t-1}^{2}.$

The data is generated according to one of the following models,
$X_{t}=\sqrt{h_{t}}\;\epsilon_{t},$ where $\epsilon_{t}$ are independent and
identically distributed $N(0,1)$ random variables and

  * ($M_{0}$)

$h_{t}=1.1+0.5X_{t-1}^{2}$

  * ($M_{1}$)

$h_{t}=1.1+0.5X_{t-1}^{2}+0.5X_{t-1}$

  * ($M_{2}$)

$h_{t}=1.1+0.5X_{t-1}^{2}+0.5\;\mbox{sign}(X_{t-1})$

  * ($M_{3}$)

$h_{t}=1.1+0.5X_{t-1}^{2}+X_{t-1}$

  * ($M_{4}$)

$h_{t}=1.1+0.5X_{t-1}^{2}+\mbox{sign}(X_{t-1}).$

A pure ARCH(1) model is fitted to the data. Then tests described in Section 4
are applied using the $\phi^{\star}$ corresponding to maximum likelihood
estimator of the ARCH parameters $\alpha_{0}$ and $\alpha_{1}$. We generated
series of length $n=100,300$ following models $M_{0}$–$M_{4}$. The results
given in Table 1 summarize a Monte-Carlo simulation study with 2000
replications of tests with $5\%$ significance levels. Note that $M_{0}$
corresponds to the null hypothesis and as shown in the table, the test
maintains its level quite well. Tests based on Khamaladze transform are in
general a bit less powerful than those based on the original process. Table 1
also shows that $S_{n}^{1}$ and $\tilde{S}_{n}^{1}$ have no detection power in
this context. This is expected since there is no change in the mean function
for all the alternative considered here. The power of the combined statistics
$S_{n}^{\star},S_{n}^{\circ}$ and $S_{n}^{\bullet}$ (or
$\tilde{S}_{n}^{\star},\tilde{S}_{n}^{\circ}$ and $\tilde{S}_{n}^{\bullet}$ )
are similar in general. The powers obtained using the multipliers procedure
and those derived from numerical approximation are very close. This shows that
both techniques provide very good estimation of p-values of these test
statistics.

Table 1:  Percentage of rejection of the null hypothesis (ARCH(1) model) when the data are generated according to models $M_{0}$ to $M_{4}$ and $n=100$ and $300$.  |  | Transformation technique | Multipliers procedure | Numerical approximation
---|---|---|---|---
n | DGP | $\tilde{S}_{n}^{1}$ | $\tilde{S}_{n}^{2}$ | $\tilde{S}_{n}^{\star}$ | $\tilde{S}_{n}^{\circ}$ | $\tilde{S}_{n}^{\bullet}$ | $S_{n}^{1}$ | $S_{n}^{2}$ | $S_{n}^{\star}$ | $S_{n}^{\circ}$ | $S_{n}^{\bullet}$ | $S_{n}^{1}$ | $S_{n}^{2}$ | $S_{n}^{\star}$ | $S_{n}^{\circ}$ | $S_{n}^{\bullet}$
| $(M_{0})$ | 3.7 | 4.2 | 3.9 | 4.0 | 4.0 | 4.3 | 3.6 | 3.7 | 3.8 | 4.0 | 4.7 | 3.3 | 3.6 | 3.8 | 3.8
| $(M_{1})$ | 4.5 | 20.8 | 15.0 | 14.5 | 15.1 | 5.3 | 21.2 | 12.0 | 10.8 | 14.0 | 5.5 | 18.4 | 11.6 | 10.2 | 13.3
100 | $(M_{2})$ | 4.3 | 20.9 | 14.9 | 14.9 | 14.8 | 4.8 | 15.2 | 9.5 | 7.9 | 11.8 | 5.2 | 14.2 | 9.0 | 7.6 | 11.3
| $(M_{3})$ | 3.8 | 43.4 | 30.7 | 32.9 | 29.4 | 5.2 | 61.0 | 42.2 | 44.7 | 44.5 | 5.3 | 60.4 | 42.6 | 44.5 | 43.1
| $(M_{4})$ | 3.1 | 47.4 | 35.1 | 35.9 | 34.3 | 3.6 | 43.5 | 24.8 | 24.1 | 32.5 | 3.9 | 42.6 | 24.5 | 24.0 | 31.0
| $(M_{0})$ | 3.9 | 5.5 | 4.6 | 5.0 | 4.7 | 4.4 | 5.0 | 4.5 | 4.5 | 4.8 | 4.4 | 4.6 | 4.3 | 4.7 | 4.4
| $(M_{1})$ | 3.7 | 45.3 | 35.9 | 37.0 | 35.1 | 4.7 | 57.7 | 44.3 | 44.5 | 45.3 | 4.9 | 57.8 | 44.3 | 45.1 | 44.3
300 | $(M_{2})$ | 3.8 | 40.2 | 31.7 | 31.6 | 30.5 | 4.6 | 38.0 | 23.8 | 22.2 | 29.3 | 4.1 | 37.9 | 23.6 | 21.2 | 27.6
| $(M_{3})$ | 3.5 | 63.1 | 54.6 | 56.4 | 53.9 | 4.3 | 92.1 | 88.9 | 89.8 | 87.0 | 4.2 | 92.2 | 89.3 | 90.0 | 86.9
| $(M_{4})$ | 3.6 | 68.0 | 59.2 | 60.4 | 58.4 | 4.3 | 76.0 | 64.6 | 64.7 | 66.9 | 4.4 | 75.9 | 64.7 | 64.9 | 66.7

###  5.2 Testing GARCH(1,1) model

This section presents the result of a simulation study in which we assess the
power of the tests, discussed in this manuscript, in detecting departure from
a GARCH(1,1) model. That is we wish to test is the mean and variance functions
are those of a GARCH(1,1) or not. We use the same settings as in Experiment 2
of Escanciano, (2010). More precisely, we generated the data from an
AR(1)-GARCH(1,1) model with autoregressive coefficient $a_{1}$ varying from
$-0.9$ to $0.9$. We fitted a GARCH(1,1) and recorded the percentage of
rejection. As in all simulation reported in this manuscript we used $2000$
Monte-Carlo simulation iterations. For this experiment we used a sample size
$n=100$ an in Escanciano, (2010). The results are reported in Table 2. Note
that $a_{1}=0$ corresponds to the null hypothesis in this case. Table 2 shows
that the $5\%$ level is respected quite well in general. It also shows that
the type of alternative considered here is mainly detected by the contribution
of component $S_{n}^{1}$ or $\tilde{S}_{n}^{1}$ to the combined statistics.
This makes all combined statistics quite power in detecting such alternatives.
The component $S_{n}^{2}$ and $\tilde{S}_{n}^{2}$ have no power against this
type of alternatives. This is expected since these components were designed to
detect change in the variance function. In this context statistics based on
the transformed process seem to be a bit more powerful than those based on the
original process. Comparing our results to Figure 1 of Escanciano, (2010), we
notice that the power of the tests presented here is slightly better.

Table 2:  Percentage of rejection of the null hypothesis (GARCH(1,1) model) when the data are generated according to AR(1)-GARCH(1,1) with autoregressive coefficient $a_{1}$.  |  | Transformation technique | Multipliers procedure | Numerical approximation
---|---|---|---|---
n | $a_{1}$ | $\tilde{S}_{n}^{1}$ | $\tilde{S}_{n}^{2}$ | $\tilde{S}_{n}^{\star}$ | $\tilde{S}_{n}^{\circ}$ | $\tilde{S}_{n}^{\bullet}$ | $S_{n}^{1}$ | $S_{n}^{2}$ | $S_{n}^{\star}$ | $S_{n}^{\circ}$ | $S_{n}^{\bullet}$ | $S_{n}^{1}$ | $S_{n}^{2}$ | $S_{n}^{\star}$ | $S_{n}^{\circ}$ | $S_{n}^{\bullet}$
| -0.9 | 100 | 1.3 | 100 | 100 | 100 | 99.4 | 2.5 | 96.2 | 96.2 | 99.1 | 99.4 | 1.9 | 95.9 | 96.0 | 99.2
| -0.7 | 100 | 3.0 | 100 | 100 | 100 | 98.7 | 1.5 | 96.4 | 96.9 | 97.7 | 98.7 | 1.2 | 96.2 | 96.7 | 97.8
| -0.5 | 98.9 | 2.9 | 97.8 | 97.9 | 97.7 | 93.4 | 2.4 | 85.3 | 87.5 | 86.9 | 93.3 | 2.1 | 84.4 | 87.0 | 86.5
| -0.3 | 76.3 | 2.9 | 63.8 | 66.0 | 62.3 | 59.6 | 3.1 | 43.9 | 45.6 | 46.3 | 59.7 | 2.4 | 42.5 | 45.1 | 45.2
100 | 0.0 | 6.3 | 2.9 | 4.4 | 4.2 | 4.1 | 3.6 | 4.1 | 3.4 | 3.4 | 4.1 | 3.5 | 4.0 | 2.9 | 3.2 | 3.8
| 0.3 | 68.1 | 2.1 | 54.2 | 57.9 | 52.4 | 42.4 | 2.2 | 26.8 | 31.2 | 26.7 | 42.4 | 1.5 | 25.4 | 30.1 | 24.7
| 0.5 | 99.0 | 2.5 | 96.8 | 97.3 | 96.3 | 80.7 | 1.4 | 67.7 | 72.4 | 65.6 | 81.4 | 1.5 | 67.4 | 72.0 | 65.3
| 0.7 | 100 | 1.5 | 100 | 100 | 100 | 87.9 | 0.5 | 79.8 | 83.9 | 77.8 | 87.9 | 0.4 | 79.8 | 83.6 | 77.5
| 0.9 | 100 | 1.0 | 100 | 100 | 100 | 80.6 | 0.3 | 62.1 | 69.6 | 69.8 | 80.9 | 0.2 | 62.0 | 68.5 | 69.9
| -0.9 | 100 | 2.9 | 100 | 100 | 100 | 100 | 4.1 | 99.8 | 99.8 | 99.9 | 100 | 3.9 | 99.8 | 99.8 | 99.9
| -0.7 | 100 | 4.7 | 100 | 100 | 100 | 100 | 3.8 | 99.9 | 100 | 100 | 100 | 3.1 | 100 | 100 | 100
| -0.5 | 100 | 4.9 | 100 | 100 | 100 | 99.7 | 3.1 | 99.3 | 99.5 | 99.5 | 99.7 | 3.0 | 99.3 | 99.5 | 99.5
| -0.3 | 99.3 | 4.6 | 98.4 | 98.6 | 98.2 | 95.6 | 3.1 | 91.5 | 92.9 | 90.9 | 95.8 | 2.7 | 91.8 | 92.9 | 91.0
300 | 0.0 | 4.8 | 4.1 | 4.7 | 4.4 | 4.5 | 4.4 | 4.4 | 4.2 | 4.6 | 4.5 | 4.4 | 4.3 | 4.0 | 4.2 | 4.2
| 0.3 | 99.2 | 3.1 | 97.8 | 98.5 | 97.5 | 94.3 | 2.5 | 88.8 | 92.1 | 88.3 | 94.5 | 2.6 | 89.3 | 91.5 | 88.4
| 0.5 | 100 | 2.9 | 100 | 100 | 100 | 99.6 | 2.2 | 99.3 | 99.5 | 99.4 | 99.6 | 2.0 | 99.3 | 99.5 | 99.4
| 0.7 | 100 | 2.0 | 100 | 100 | 100 | 99.7 | 1.3 | 99.5 | 99.6 | 99.5 | 99.7 | 0.8 | 99.4 | 99.5 | 99.5
| 0.9 | 100 | 0.5 | 100 | 100 | 100 | 96.6 | 0.3 | 90.7 | 94.1 | 93.1 | 96.6 | 0.2 | 90.6 | 93.6 | 92.9

###  5.3 Testing AR(1)-GARCH(1,1) model

In this section we present the result of a study testing the null hypothesis
that the mean and variance functions are those of an AR(1)-GARCH(1,1) model.
The same five alternatives considered in Experiment 3 of Escanciano, (2010)
are used. To be specific the data is generated according to one of these
alternatives, and AR(1)-GARCH(1,1) model is fitted to the data. The test
statistics proposed in this manuscript are applied and the percentage of
rejection, in a $2000$ replicates Monte-Carlo simulation, is reported in Table
3. The alternative considered are defined as follows:

A0:

The null hypothesis AR(1)-GARCH(1,1) model:
$X_{t}=0.02+0.02X_{t-1}+\varepsilon_{t}$ where
$\varepsilon_{t}=\sqrt{h_{t}}u_{t}$ with
$h_{t}=0.08+0.1\varepsilon_{t-1}^{2}+0.85h_{t-1}$ and $u_{t}$ i.i.d Normal
random variables with mean zero and variance one.

A1:

ARMA(1,1)-GARCH(1,1) model:
$X_{t}=0.02+0.02X_{t-1}+0.5\varepsilon_{t-1}+\varepsilon_{t}$ and
$\varepsilon_{t}$ is as in model A0.

A2:

TAR model: $X_{t}=0.6X_{t-1}+\varepsilon_{t}$ if $X_{t-1}\leq 1$ and
$X_{t}=-0.5X_{t-1}+\varepsilon_{t}$ if $X_{t-1}>1$ with $\varepsilon_{t}$ is
as in model A0.

A3:

EGARCH(1,1) model: $X_{t}=\sqrt{h_{t}}u_{t}$ where
$\ln(h_{t})=0.025+0.5\ln(h_{t-1})+0.25(|u_{t-1}|-\sqrt{2/\pi})-0.8u_{t-1}$ and
$u_{t}$ is as in model A0.

A4:

Bilinear model: $X_{t}=0.6X_{t-1}+0.7u_{t-1}X_{t-2}+u_{t}$ where $u_{t}$ is as
in model A0.

A5:

Nonlinear Moving average model: $X_{t}=0.8u_{t-1}^{2}+u_{t}$ where $u_{t}$ is
as in model A0.

Table 3:  Percentage of rejection of the null hypothesis (AR(1)-GARCH(1,1) model) when the data are generated according to A0–A5.  |  | Transformation technique | Multipliers procedure | Numerical approximation
---|---|---|---|---
n | DGP | $\tilde{S}_{n}^{1}$ | $\tilde{S}_{n}^{2}$ | $\tilde{S}_{n}^{\star}$ | $\tilde{S}_{n}^{\circ}$ | $\tilde{S}_{n}^{\bullet}$ | $S_{n}^{1}$ | $S_{n}^{2}$ | $S_{n}^{\star}$ | $S_{n}^{\circ}$ | $S_{n}^{\bullet}$ | $S_{n}^{1}$ | $S_{n}^{2}$ | $S_{n}^{\star}$ | $S_{n}^{\circ}$ | $S_{n}^{\bullet}$
| A0 | 4.6 | 3.9 | 4.6 | 4.5 | 4.9 | 2.6 | 5.4 | 4.4 | 5.3 | 4.5 | 2.3 | 5.3 | 4.5 | 5.1 | 4.2
| A1 | 3.6 | 4.1 | 3.4 | 3.8 | 3.6 | 1.4 | 4.3 | 3.2 | 3.5 | 2.9 | 1.3 | 3.8 | 2.9 | 3.4 | 2.8
| A2 | 100 | 5.8 | 99.8 | 99.9 | 99.8 | 77.8 | 31.9 | 61.6 | 47.2 | 78.8 | 77.7 | 31.9 | 63.1 | 47.3 | 78.4
300 | A3 | 6.2 | 6.0 | 7.1 | 6.8 | 7.1 | 55.5 | 51.6 | 71.8 | 64.9 | 78.1 | 55.5 | 52.0 | 71.7 | 64.9 | 76.9
| A4 | 68.3 | 5.3 | 59.0 | 60.3 | 57.7 | 13.6 | 27.6 | 30.5 | 27.2 | 32.2 | 13.0 | 27.4 | 31.1 | 27.3 | 31.7
| A5 | 100 | 15.4 | 100 | 100 | 100 | 91.3 | 18.1 | 87.2 | 87.4 | 89.0 | 91.4 | 18.2 | 87.7 | 86.7 | 89.4
| A0 | 6.2 | 3.4 | 4.8 | 4.7 | 4.7 | 3.5 | 4.4 | 3.8 | 4.1 | 3.9 | 3.2 | 4.7 | 3.9 | 4.5 | 3.8
| A1 | 4.0 | 4.9 | 3.7 | 3.8 | 3.6 | 2.2 | 5.0 | 4.2 | 4.5 | 3.3 | 2.2 | 4.7 | 3.8 | 4.4 | 3.1
| A2 | 100 | 11.2 | 100 | 100 | 100 | 98.2 | 63.7 | 96.0 | 91.0 | 98.4 | 98.1 | 63.6 | 96.1 | 91.0 | 98.4
600 | A3 | 4.8 | 13.7 | 10.8 | 11.2 | 10.6 | 80.1 | 69.6 | 92.6 | 90.5 | 94.6 | 80.7 | 70.0 | 92.6 | 90.4 | 94.8
| A4 | 88.4 | 6.4 | 84.0 | 84.4 | 83.3 | 30.7 | 41.7 | 51.6 | 44.0 | 56.6 | 29.5 | 41.1 | 50.8 | 43.3 | 55.9
| A5 | 100 | 40.9 | 100 | 100 | 100 | 97.6 | 50.2 | 97.5 | 97.2 | 97.9 | 97.6 | 51.0 | 97.5 | 97.3 | 97.8

Comparing with Table 2 in Escanciano, (2010), one notice that combined tests
introduced in this manuscript are more powerful than the tests considered in
Escanciano, (2010). The only exception being the case of alternative A1:
ARMA(1,1)-GARCH(1,1) where our tests were not able to detect such alternative
while those in Escanciano, (2010) had reasonable power. We also notice that
tests based on the transformed process are bit more powerful in the case of
alternatives A2, A4 and A5. On the other hand these tests fall way behind in
the case A3. This seems to concord with the observations from the first
simulation experiment where tests based on the transformed process were a bit
more powerful in detecting a change in the mean function. In fact, for the
three alternatives A2, A4 and A5 there is a change in the conditional mean
function. While in the case of alternative A3, the conditional mean function
is still that of an AR(1) while the change occurred in the conditional
variance function.

###  5.4 Testing stochastic differential equation models

In this part, we are interested in testing whether the data fits a specified
stochastic differential equation (SDE) model. In finance continuous time
models are widely used to study the dynamic of some financial products such as
asset prices, interest rate and bonds. Continuous-time modelling in finance
was introduced by Bachelier, (1900) on the theory of speculation and really
started with Merton, (1970) seminal work. Since then several models were
developed which can be formulated by the following general SDE

$\displaystyle dX_{t}=\mu_{\theta}(X_{t})dt+\sigma(X_{t})dW_{t},$ (5.2)

where $W_{t}$ is a standard Brownian motion. The drift $\mu_{\theta}(\cdot)$
and the diffusion $\sigma^{2}(\cdot)$ are known functions. Several well-known
models in financial econometrics, including Black and Scholes, (1973),
Vasicek, (1977), Cox et al., (1980), Chan et al., (1992) and Ait-Sahalia,
(1996), among others, can be written under the form (5.2) with a specific form
of drift and diffusion functions. Below the list of SDE models considered
here.

  * N1:

Vasicek: $dX_{t}=3(10-X_{t})dt+5dW_{t},\quad x_{0}=0.03$

  * N2:

Hyperbolic: $dX_{t}=5\frac{X_{t}}{\sqrt{1+X_{t}^{2}}}dt+5dW_{t}\quad x_{0}=3$

  * N3:

CIR: $dX_{t}=(1+4.5X_{t})dt+0.75\sqrt{X_{t}}dW_{t},\quad x_{0}=3$

  * N4:

CKLS1: $dX_{t}=1.5(1-X_{t})dt+1.5X_{t}^{0.8}dW_{t},\quad x_{0}=5$

  * N5:

CKLS2: $dX_{t}=1.5(1-X_{t})dt+0.5X_{t}^{1.5}dW_{t},\quad x_{0}=5$

  * N6:

Aït-Sahalia:
$dX_{t}=(1+15X_{t}+0.25X_{t}^{-1}-2X_{t}^{2})dt+0.5X_{t}^{1.5}dW_{t},\quad
x_{0}=5$

In practice, the diffusion process $\\{X_{t}\\}$ is observed at instants
$\\{t=i\Delta|i=0,\dots,n\\}$, where $\Delta>0$ is generally small, but fixed
as $n$ increases. For instance the series could be observed hourly, daily,
weekly or monthly. Therefore, we may model these discretely observed
measurements using time series models. In fact, despite the fact that (5.2) is
written in a continuous-time form, one often uses the following Euler
discretization scheme to get

$\displaystyle
X_{t+\Delta}-X_{t}=\mu_{\theta}(X_{t})\Delta+\sigma(X_{t})\left(W_{t+\Delta}-W_{t}\right),\quad
t=0,\Delta,2\Delta,\dots,$ (5.3)

as an approximation that facilitates computational and theoretical derivation.
The accuracy of such Euler discretisation is studied in Jacod and Protter,
(1998). The purpose of this simulation study is to generate processes
satisfying the SDEs described by models N1–N6 above, then use the procedures
described in Section 4 to test whether the data fits a specific type of SDE
models. Two types of null hypotheses will be considered. In the first sub-
experiment we test if the data fits a Vasicek type SDE model, namely we test
if $\mu$ is of the form $a+bX_{t-1}$ for some parameters
$(a,b)\in\mathbb{R}^{2}$ and that $\sigma=c$ for some parameter $c>0$. Note
that only model N1 belongs to this class of models. In the second sub-
experiment the null hypothesis considered is a CIR model where $\mu$ is of the
form $a+bX_{t-1}$ and $\sigma$ of the form $c\sqrt{X_{t-1}}$. For this second
sub-experiment, N3 belongs to the null hypothesis.

We suppose the process is observed over the time interval $[0,T]$ and $n$
corresponds to the number of instants where the process is observed. The
sampling mesh in such case is $\Delta=T/n$. In order to assess the sensitivity
of our test to the sampling frequency from the underlying process, we consider
$T=1$ and $\Delta=0.002,0.005$ and $0.01$, corresponding to a total sampling
instants $n=500,200$ and $100$ respectively. Such framework supposes that
collecting more observations means shortening the time interval between
successive existing observations, not lengthening the time period over which
data are recorded.
The data is generated according to one of SDE models listed above. The
parameters are estimated, after discretization, using the maximum pseudo-
likelihood method (see Aït-Sahalia, (2002)). Table 4 and 5 report the
percentage of rejection of the two null hypotheses considered. The results are
obtained using $1000$ Monte-Carlo replications for different values of
sampling mesh $\Delta$.

For the first sub-experiment, Table 4 shows that the tests are quite powerful
in detecting change in the diffusion term. Alternatives N3–N6 are rejected
with high probability even for $n=100$. The three combined statistics are more
or less similar in term of their detection power. As expected the statistics
$S_{n}^{1}$ or $\tilde{S}_{n}^{1}$ do no detect change in the diffusion term.
All statistics fail to detect the alternative N2 even for large sample sizes.

For the second sub-experiment, reported in Table 5, tests based on the
original process and using either the multipliers procedure or the numerical
approximation do respect their levels more appropriately than those based on
the transformed process. The three combined statistics have the power to
detect all alternatives considered here with N5 and N6 being easier to detect.
This could be explained by the large change in the diffusion functions between
N5 or N6 and the null hypothesis in this case N3.

Table 4: Percentage of rejection of the null hypothesis $(H_{0})$ (Vasicek type Model) for different sample sizes $n=100,200$ and $500$. |  | Transformation technique | Multipliers procedure | Numerical approximation
---|---|---|---|---
n | DGP | $\tilde{S}_{n}^{1}$ | $\tilde{S}_{n}^{2}$ | $\tilde{S}_{n}^{\star}$ | $\tilde{S}_{n}^{\circ}$ | $\tilde{S}_{n}^{\bullet}$ | $S_{n}^{1}$ | $S_{n}^{2}$ | $S_{n}^{\star}$ | $S_{n}^{\circ}$ | $S_{n}^{\bullet}$ | $S_{n}^{1}$ | $S_{n}^{2}$ | $S_{n}^{\star}$ | $S_{n}^{\circ}$ | $S_{n}^{\bullet}$
| N1 | 6.1 | 4.1 | 4.7 | 4.6 | 5.1 | 6.1 | 4.1 | 3.8 | 4.1 | 4.7 | 5.2 | 3.7 | 3.5 | 3.6 | 4
| N2 | 5.1 | 5.9 | 5.2 | 4.9 | 5.3 | 6.6 | 4.2 | 4.2 | 4.4 | 5.9 | 5.8 | 4.1 | 4.3 | 4.1 | 4.5
100 | N3 | 2.4 | 95 | 84.8 | 87.8 | 82.5 | 4.9 | 98.4 | 98.3 | 98.4 | 94.1 | 4.9 | 98.6 | 98.3 | 98.6 | 93.3
| N4 | 6.1 | 84.4 | 73.1 | 76.1 | 70.9 | 4 | 90.7 | 90 | 90.7 | 79.4 | 3.7 | 90.8 | 90.2 | 90.8 | 77.3
| N5 | 9.2 | 92.4 | 82.8 | 82.9 | 81.9 | 4.9 | 91.5 | 90.6 | 91.5 | 80.7 | 5.1 | 92.3 | 91.5 | 92.2 | 79.9
| N6 | 11.3 | 88.9 | 80.3 | 79.3 | 78.6 | 10.2 | 91.1 | 89.4 | 91.1 | 77.9 | 9.3 | 91.3 | 90 | 91.3 | 75.9
| N1 | 5.2 | 5.5 | 5.4 | 5.2 | 5.2 | 6.9 | 4.7 | 4.5 | 5.2 | 5.4 | 7.6 | 4.6 | 4 | 4.8 | 4.7
| N2 | 5.9 | 6.1 | 4.9 | 5.6 | 4.8 | 6.5 | 3.9 | 3.8 | 3.9 | 5.4 | 6.3 | 3.9 | 4.1 | 4 | 4.7
200 | N3 | 4.1 | 100 | 99.8 | 99.9 | 99.5 | 5.6 | 100 | 100 | 100 | 99.6 | 5.7 | 100 | 100 | 100 | 99.7
| N4 | 7.5 | 96.5 | 94.7 | 94.4 | 94.1 | 4.1 | 98.2 | 98.2 | 98.2 | 96.1 | 3.9 | 98.2 | 98.1 | 98.2 | 95.8
| N5 | 10.7 | 99.5 | 97.8 | 98.7 | 97.4 | 5.1 | 98.9 | 98.7 | 98.9 | 95.9 | 4.8 | 98.7 | 98.6 | 98.7 | 96.8
| N6 | 11.4 | 99.4 | 98.3 | 98.7 | 97.8 | 6 | 99.6 | 99.6 | 99.6 | 97 | 6.3 | 99.6 | 99.5 | 99.6 | 97.3
| N1 | 4.9 | 5.5 | 5.3 | 5.4 | 6.4 | 8.2 | 4.8 | 5.6 | 5.1 | 7.7 | 8.3 | 4.3 | 5.5 | 4.7 | 6.3
| N2 | 5.1 | 6.4 | 5.6 | 5.7 | 5.8 | 7.1 | 4.3 | 4.6 | 4.4 | 5.3 | 6.8 | 4.4 | 4.6 | 4.5 | 4.7
500 | N3 | 4.1 | 100 | 100 | 100 | 100 | 5.5 | 100 | 100 | 100 | 100 | 5.2 | 100 | 100 | 100 | 100
| N4 | 8.6 | 100 | 100 | 100 | 100 | 6.4 | 100 | 100 | 100 | 100 | 6 | 100 | 100 | 100 | 100
| N5 | 11.8 | 100 | 100 | 100 | 100 | 4.4 | 100 | 99.8 | 99.9 | 99.9 | 4.4 | 100 | 99.8 | 99.8 | 99.9
| N6 | 13.7 | 100 | 100 | 100 | 100 | 7.6 | 100 | 100 | 100 | 99.9 | 7.8 | 100 | 100 | 100 | 99.9
Table 5: Percentage of rejection of the null hypothesis $(H_{0})$ (CIR type Model) for different sample sizes $n=100,200$ and $500$. |  | Transformation technique | Multipliers procedure | Numerical approximation
---|---|---|---|---
n | DGP | $\tilde{S}_{n}^{1}$ | $\tilde{S}_{n}^{2}$ | $\tilde{S}_{n}^{\star}$ | $\tilde{S}_{n}^{\circ}$ | $\tilde{S}_{n}^{\bullet}$ | $S_{n}^{1}$ | $S_{n}^{2}$ | $S_{n}^{\star}$ | $S_{n}^{\circ}$ | $S_{n}^{\bullet}$ | $S_{n}^{1}$ | $S_{n}^{2}$ | $S_{n}^{\star}$ | $S_{n}^{\circ}$ | $S_{n}^{\bullet}$
| N1 | 4.8 | 50.1 | 38.9 | 39.6 | 38.9 | 4.4 | 30.9 | 27.1 | 30.5 | 23.2 | 3.6 | 31.4 | 26 | 29.7 | 20.7
| N2 | 4.9 | 63.3 | 53.9 | 53.5 | 52.9 | 4.3 | 42.2 | 37.5 | 41.3 | 30.4 | 4 | 42.3 | 36 | 40.4 | 28.9
100 | N3 | 4.1 | 10.2 | 7.3 | 8.2 | 6.4 | 4.5 | 3.9 | 3.8 | 4.2 | 4.9 | 4 | 4.9 | 3.8 | 4.5 | 4.4
| N4 | 5.4 | 10.5 | 9.4 | 7 | 10.1 | 3.8 | 24.6 | 19.2 | 23.6 | 16.5 | 3.3 | 25.5 | 19.2 | 23.8 | 16.3
| N5 | 10.6 | 54 | 40.7 | 35.7 | 41.7 | 5.8 | 70.1 | 65.9 | 69.7 | 52.2 | 5.3 | 71.9 | 66.1 | 70.7 | 51.7
| N6 | 10.2 | 51.1 | 42.9 | 37.5 | 44 | 7.6 | 53.8 | 50.5 | 53.7 | 39.5 | 6.4 | 55.2 | 50 | 55 | 37.6
| N1 | 5.4 | 71.2 | 62.2 | 63.6 | 61 | 5.5 | 55.7 | 50.8 | 52.7 | 45.1 | 4.7 | 54.9 | 49.6 | 52.1 | 44.4
| N2 | 4.9 | 85.5 | 79.9 | 80.6 | 79.1 | 5.8 | 77.6 | 70.9 | 75.8 | 64.3 | 5.2 | 76.1 | 70 | 74.7 | 63.9
200 | N3 | 5.9 | 9.8 | 8.5 | 9.6 | 8.2 | 5.3 | 4.7 | 4.5 | 4.5 | 5.3 | 5.7 | 4.6 | 4.4 | 4.1 | 4.5
| N4 | 6.3 | 24.1 | 19.3 | 17.5 | 19.7 | 3.9 | 51.4 | 46.6 | 49.6 | 39.7 | 4.2 | 50.4 | 45.5 | 49.1 | 38.5
| N5 | 11.7 | 88.2 | 79.2 | 77.5 | 79.0 | 5.1 | 95.4 | 95.1 | 95.3 | 89.0 | 4.6 | 95.4 | 95.1 | 95.4 | 88.3
| N6 | 12.2 | 84.3 | 75.7 | 73.3 | 74.8 | 7.1 | 87.7 | 87.0 | 87.7 | 76.9 | 6.8 | 88.1 | 86.8 | 88.1 | 77.6
| N1 | 7.6 | 94.3 | 91.6 | 91.8 | 90.9 | 6.7 | 88.4 | 85.2 | 87 | 84.2 | 6.9 | 89.1 | 85.6 | 86.9 | 83.5
| N2 | 7.1 | 98.4 | 97.1 | 96.8 | 97.3 | 4.6 | 96.3 | 94.7 | 96 | 93 | 4.4 | 96.5 | 94.5 | 96 | 92.2
500 | N3 | 4.8 | 9.3 | 7.8 | 8 | 7.6 | 4.6 | 4.3 | 3.9 | 4.3 | 3.9 | 4.5 | 3.8 | 3.4 | 3.8 | 4
| N4 | 7.5 | 64.8 | 56.3 | 54.5 | 55.9 | 4.3 | 82.1 | 81.3 | 81.9 | 75.9 | 4.3 | 82 | 81.5 | 81.6 | 75.7
| N5 | 13.0 | 99.9 | 99.4 | 99.3 | 99.3 | 6.0 | 100.0 | 99.7 | 99.8 | 99.8 | 5.7 | 99.7 | 99.7 | 99.7 | 99.6
| N6 | 14.9 | 99.7 | 99.7 | 99.4 | 99.6 | 9.2 | 99.8 | 99.7 | 99.8 | 99.5 | 8.7 | 99.8 | 99.7 | 99.8 | 99.5

##  6 Application to interest rate data

This section presents as application the testing methodology to interest rate
dynamics. Monthly values of interest rate for a maturity of 1 months (% per
year) from July 1964 to April 1989 are considered. The monthly interest rate
$(r_{t})$ dynamic is displayed in Figure 1. Figure 2 is a scatter plot of
monthly changes in interest rate against the previous month’s rate and it
clearly shows an obvious heteroscedasticity, with the range of changes
increasing significantly as the level of interest rates increases. An
augmented Dickey-Fuller test of non stationarity of interest rate data is
performed and the non stationarity hypothesis is rejected, with p-value equal
to 0.01, for 1 to 12 months lagged series.

Figure 1: Interest rate for a maturity of 1 months (% per year) from July 1964
to April 1989.  Figure 2: Monthly changes in interest rate against interest
rate on preceding month.

The aim here is to verify if the interest rate data described above fits a
selected diffusion model. The following models will be considered as
candidates

  * (D1)

Vasicek: $dr_{t}=(\alpha+\beta r_{t})dt+\sigma dW_{t}$

  * (D2)

Hyperbolic: $dr_{t}=\alpha\frac{r_{t}}{\sqrt{1+r_{t}^{2}}}dt+\sigma dW_{t}$

  * (D3)

Aït-Sahalia 1:
$dr_{t}=(\alpha_{0}+\alpha_{1}r_{t}+\alpha_{2}r_{t}^{-1}+\alpha_{3}r_{t}^{2})dt+\sigma
dW_{t}$

  * (D4)

CIR: $dr_{t}=(\alpha+\beta r_{t})dt+\sigma\sqrt{r_{t}}dW_{t}$

  * (D5)

CKLS 1: $dr_{t}=\kappa(\alpha-r_{t})dt+\sigma r_{t}^{0.8}dW_{t}$

  * (D6)

CKLS 2: $dr_{t}=\kappa(\alpha-r_{t})dt+\sigma r_{t}^{1.5}dW_{t}$

  * (D7)

Aït-Sahalia 2:
$dr_{t}=(\alpha_{0}+\alpha_{1}r_{t}+\alpha_{2}r_{t}^{-1}+\alpha_{3}r_{t}^{2})dt+\sigma
r_{t}^{1.5}dW_{t}$

The procedure consists in fitting the interest rate data to each of the models
outlined above and then apply the statistics described of Section 4 to see how
good is the model fit. Table 6 reports the p-values resulting from applying
the statistics discussed in Section 4 after fitting each of the models D1–D7
to the monthly interest rate data. As expected, all models with constant
conditional volatility (D1-D3) are clearly rejected even when we considered
different drift functions.

Table 6: P-values of the test statistics applied to different models (D1–D7) fitted to the monthly interest rate data. | Test | D1 | D2 | D3 | D4 | D5 | D6 | D7
---|---|---|---|---|---|---|---|---
| $\tilde{S}_{n}^{1}$ | 0.1432 | 0.1108 | 0.3589 | 0.1548 | 0.1574 | 0.1779 | 0.5179
| $\tilde{S}_{n}^{2}$ | 0.0001 | 0.0001 | 0.0000 | 0.0018 | 0.0096 | 0.7873 | 0.7937
Transformation technique | $\tilde{S}_{n}^{\star}$ | 0.0001 | 0.0002 | 0.0002 | 0.0028 | 0.0128 | 0.3911 | 0.7779
| $\tilde{S}_{n}^{\circ}$ | 0.0001 | 0.0002 | 0.0001 | 0.0036 | 0.0191 | 0.3241 | 0.7676
| $\tilde{S}_{n}^{\bullet}$ | 0.0001 | 0.0002 | 0.0002 | 0.0026 | 0.0113 | 0.4153 | 0.7765
| $S_{n}^{1}$ | 0.2840 | 0.3080 | 0.5130 | 0.5370 | 0.6410 | 0.8170 | 0.9850
| $S_{n}^{2}$ | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0020 | 0.6080 | 0.5920
Multipliers procedure | $S_{n}^{\star}$ | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0020 | 0.7800 | 0.8350
| $S_{n}^{\circ}$ | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0020 | 0.7060 | 0.6680
| $S_{n}^{\bullet}$ | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0098 | 0.8443 | 0.8976
| $S_{n}^{1}$ | 0.2566 | 0.2893 | 0.5424 | 0.5302 | 0.6522 | 0.8286 | 0.9868
| $S_{n}^{2}$ | 0.0000 | 0.0001 | 0.0000 | 0.0001 | 0.0010 | 0.6077 | 0.6126
Numerical approximation | $S_{n}^{\star}$ | 0.0000 | 0.0002 | 0.0000 | 0.0002 | 0.0019 | 0.7812 | 0.8343
| $S_{n}^{\circ}$ | 0.0000 | 0.0001 | 0.0000 | 0.0001 | 0.0010 | 0.7169 | 0.6908
| $S_{n}^{\bullet}$ | 0.0001 | 0.0003 | 0.0001 | 0.0008 | 0.0056 | 0.8490 | 0.9088

Models D4 and D5 are also rejected, implying that the rate in the diffusion
function of the monthly interest rate is not $0.5$ or $0.8$. On the other
hand, all tests cannot reject the D6 and D7 implying that the diffusion of the
interest rate is proportional to $X_{t}^{1.5}$. This conclusion is in perfect
concordance with existing findings in the literature (see for instance Aït-
Sahalia, (1999)). In fact, the rate of $1.5$ was the recommended choice in
Ait-Sahalia, (1996), among others, in a study of the same interest rate data.

##  7 Conclusion

Though Khmaldze matringale transform is quite popular as it simplifies the
asymptotic behavior, its finite sample behavior is in general far from being
the best compared to re-sampling or numerical approximation of the original
statistics. We also notice that even the level of tests based on Khmaladze
transform was sensitive to the estimation of the function $g$, defined in
(3.1). In our case, the estimation function $g$ used a non-parametric estimate
of the derivative of the function $\Gamma_{\theta}$ and we noticed that the
level was sensitive to the bandwidth parameter utilized in the estimation. The
level of the tests obtained by numerical approximation or re-sampling of the
original statistics are stable and close to their target values. Tests, using
the original cumulative residual process, are in general more powerful than
the ones based on martingale transformation. Moreover, as mentioned earlier,
p-values of the combined statistics based on the transformed process are only
computable when the components $\mathbb{D}^{1}$ and $\mathbb{D}^{2}$ are
independent. Whereas p-values for the multipliers procedure for both
$S_{n}^{\star}$ and $S_{n}^{\circ}$ are obtained in the same manner whether
the components $\mathbb{D}^{1}$ and $\mathbb{D}^{2}$ are independent or not.
For the approximation technique, the algorithm described in the manuscript
provides the p-values of $S_{n}^{\star}$ whether the components
$\mathbb{D}^{1}$ and $\mathbb{D}^{2}$ are independent or not. For
$S_{n}^{\circ}$ we used the independence of $\mathbb{D}^{1}$ and
$\mathbb{D}^{2}$ to simplify our computation, but the process can be
generalized to the dependent case by approximating the distribution of the
maximum of two dependent quadratic forms of normal random variables. The
computations involved in the approximation techniques are extremely fast. As
mentioned in the paper, the computation involved in the multipliers bootstrap
are quite fast since the parameters are estimated only once and that each
bootstrap iteration just requires generating $n$ i.i.d normal mean zero and
variance one random variables. The combined statistics $S_{n}^{\star}$,
$S_{n}^{\circ}$ and $S_{n}^{\bullet}$ have similar power behavior in general,
but $S_{n}^{\star}$ is much easier to compute and its p-values are easier to
obtain. We, therefore, recommend practitioners to use tests based on the
original process more often with a multipliers procedure or a numerical
approximation technique. Among these statistics $S_{n}^{\star}$ would be the
easiest to implement. The tests introduced in this manuscript are in general a
bit more powerful than those in Escanciano, (2010), but there are
alternatives, like A1, where tests in Escanciano, (2010) clearly outperform
those discussed here.

The procedures discussed here generalize to the case of multivariate time
series where $X_{i}\in\mathbb{R}^{d}$. The procedure based on the multipliers
bootstrap would be the easiest to generalize. In fact, it will not require any
modification it suffices to adjust the definition of the process and the
functions $\phi^{\star}$ and $\Gamma_{\theta}$ to the multivariate case. The
generalization of the numerical approximation and that of the martingale
transform would need more technical work.

##  Appendix A Proofs

We start by proving the next Lemma which is used repeatedly in the proofs. It
establishes a uniform law of large number result needed to show that the
convergence is uniform in $x$ for both theorems (3.1) and (4.1).

######  Lemma A.1.

If $\chi_{i}$, $i=1\ldots,n$ is strictly stationary and ergodic series
satisfying $\mathbb{E}|\chi_{1}|<\infty$ and if $X_{i}$ is strictly stationary
and ergodic series then, we have

  1. i)

$\sup_{x\in\mathbb{R}}\left|\frac{1}{n}\sum_{i=1}^{n}\chi_{i}1\kern-1.99997pt{\rm
l}\\{X_{i-1}\leq x\\}-\mathbb{E}[\chi_{1}1\kern-1.99997pt{\rm l}\\{X_{0}\leq
x\\}]\right|$ converges to zero almost surely,

  2. ii)

If $g_{k}$ and $A_{k}$ satisfy the conditions of Theorem 3.1 then

$\sup_{x\leq
x_{0}}\biggl{\|}\frac{1}{n}\sum_{i=1}^{n}\chi_{i}\int_{-\infty}^{\min(x,X_{i-1})}(g_{k}(y))^{\top}A_{k}^{-1}(y)K_{k}(dy)\\\
-\mathbb{E}\biggl{[}\chi_{1}\int_{-\infty}^{\min(x,X_{0})}(g_{k}(y))^{\top}A_{k}^{-1}(y)K_{k}(dy)\biggr{]}\biggr{\|}$

converges to zero almost surely for $k=1,2$ and for every $x_{0}$ satisfying
$A_{k}(x_{0})$ invertible.

###### Proof.

The proof of i) and ii) are quite similar. Here only the proof of ii) is
given. For i) one repeats the same steps. One also can see that i) is similar
to 4.1 in Koul and Stute, (1999). Though ii) can be deduced from of ULLN of
Andrews, (1992) if the set of $x$’s is compact. Since this is not the case
here, a direct proof using a Glivenko-Cantelli type argument shall be given
next. First recall that $A_{k}(x)$ is invertible for all $x\leq x_{0}$
whenever $A_{k}(x_{0})$ is invertible and that one can easily verify that
$\|A_{k}^{-1}(x)\|\leq\|A_{k}^{-1}(x_{0})\|<\infty$. One also sees that

$\displaystyle\left\|\int_{-\infty}^{\min(x,X_{0})}(g_{k}(y))^{\top}A_{k}^{-1}(y)K_{k}(dy)\right\|\leq\left[\int_{-\infty}^{x_{0}}\|(g_{k}(y))^{\top}\|K_{k}(dy)\right]\|A^{-1}(x_{0})\|<\infty,$

since $g_{k}$ satisfies the condition of Theorem 3.1. The LLN for stationary
ergodic sequence yields the almost sure convergence of ii) for every fixed
$x\leq x_{0}$. To prove that the convergence is uniform in $x\leq x_{0}$ one
uses a Glivenko-Cantelli type argument applied using
$\eta^{\star}(x)=\mathbb{E}\left[|\chi_{1}|\int_{-\infty}^{\min(x,X_{0})}\|(g_{k}(y))^{\top}A_{k}^{-1}(y)\|K_{k}(dy)\right]$.
Note that $\eta^{\star}$ is a continuous increasing function and for all
$x\leq x_{0}$ one has $\eta^{\star}(x)\leq\eta^{\star}(x_{0})<\infty$.
Therefore for every $\epsilon>0$ there exist a finite partition
$-\infty=t_{0}<t_{1}<\ldots t_{k}=x_{0}$ such that
$0\leq\eta^{\star}(t_{j+1})-\eta^{\star}(t_{j})\leq\epsilon$. To ease
presentation, let

$\eta_{n}(x)=\frac{1}{n}\sum_{i=1}^{n}\chi_{i}\int_{-\infty}^{\min(x,X_{i-1})}(g_{k}(y))^{\top}A_{k}^{-1}(y)K_{k}(dy)$

and

$\eta(x)=\mathbb{E}\left[\chi_{1}\int_{-\infty}^{\min(x,X_{0})}(g_{k}(y))^{\top}A_{k}^{-1}(y)K_{k}(dy)\right].$

Note that for any $x\leq x_{0}$ there exists $j<k$ such that $t_{j}<x\leq
t_{j+1}$. Clearly,
$\|\eta_{n}(x)-\eta(x)\|\leq\|\eta_{n}(x)-\eta_{n}(t_{j})\|+\|\eta_{n}(t_{j})-\eta(t_{j})\|+\|\eta(t_{j})-\eta(x)\|.$
Using its definition one sees that

$\displaystyle\|\eta_{n}(x)-\eta_{n}(t_{j})\|$ $\displaystyle\leq$
$\displaystyle\biggl{|}\frac{1}{n}\sum_{i=1}^{n}|\chi_{i}|\int_{\min(t_{j},X_{i-1})}^{\min(t_{j+1},X_{i-1})}\|(g_{k}(y))^{\top}A_{k}^{-1}(y)\|K_{k}(dy)$
$\displaystyle\quad-\eta^{\star}(t_{j+1})+\eta^{\star}(t_{j})\biggr{|}+|\eta^{\star}(t_{j+1})-\eta^{\star}(t_{j})|.$

One also easily verifies that
$\|\eta(t_{j})-\eta(x)\|\leq|\eta^{\star}(t_{j+1})-\eta^{\star}(t_{j})|$.
Therefore

$\sup_{x\leq
x_{0}}\left|\eta_{n}(x)-\eta(x)\right|\leq\|\eta_{n}(t_{j})-\eta(t_{j})\|+2\epsilon+\max_{j=0,\ldots,k}\biggl{|}\frac{1}{n}\sum_{i=1}^{n}|\chi_{i}|\\\
\times\int_{\min(t_{j},X_{i-1})}^{\min(t_{j+1},X_{i-1})}\|(g_{k}(y))^{\top}A_{k}^{-1}(y)\|K_{k}(dy)-\eta^{\star}(t_{j+1})+\eta^{\star}(t_{j})\biggr{|}.$

The pointwise LLN and the fact that $k$ is finite implies that the first and
last terms in the above inequality converge almost surely to zero. Since
$\epsilon$ was arbitrary, one concludes that $\sup_{x\leq
x_{0}}\left|\eta_{n}(x)-\eta(x)\right|$ converges to zero almost surely. ∎

###  A.1 Proof of Theorem 2.3

The weak convergence of $\mathbb{D}_{n}$ follows from Theorem 1 in Escanciano,
2007b . For $\widehat{\mathbb{D}}_{n}$, direct manipulations show that

$\displaystyle\widehat{\mathbb{D}}_{n}(x)=\mathbb{D}_{n}(x)-\Gamma^{\top}_{\theta_{0}}(x)\sqrt{n}(\theta_{n}-\theta_{0})+B_{n,1}(x)+B_{n,2}(x)$

where

$B_{n,1}(x)=\left[\frac{1}{n}\sum_{i=1}^{n}\dot{W}^{\top}_{\theta_{0}}(X_{i},I_{i-1})1\kern-1.99997pt{\rm
l}\\{X_{i-1}\leq
x\\}-\Gamma^{\top}_{\theta_{0}}(x)\right]\sqrt{n}(\theta_{n}-\theta_{0}),$

and

$B_{n,2}(x)=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left[W_{\theta_{n}}(X_{i},I_{i-1})-W_{\theta_{0}}(X_{i},I_{i-1})-\dot{W}^{\top}_{\theta_{0}}(X_{i},I_{i-1})(\theta_{n}-\theta_{0})\right]1\kern-1.99997pt{\rm
l}\\{X_{i-1}\leq x\\}.$

Note that $\|B_{n,1}\|$ converges in probability to zero by Lemma A.1, the
definition of $\Gamma_{\theta_{0}}$ and the fact that
$\sqrt{n}(\theta_{n}-\theta_{0})=O_{\mathbb{P}}(1)$. By Assumption (A3), the
term $\|B_{n,2}\|$ is bounded by
$\sqrt{n}\|\theta_{n}-\theta_{0}\|\lambda_{1}(\|\theta_{n}-\theta_{0}\|)\frac{1}{n}\sum_{i=1}^{n}M_{1}(X_{i},I_{i-1})$
which converges to zero in probability by the LLN and the fact that
$\sqrt{n}\|\theta_{n}-\theta_{0}\|=O_{\mathbb{P}}(1)$. Therefore,
$\widehat{\mathbb{D}}_{n}(x)$ is asymptotically equivalent to
${\mathbb{D}}_{n}(x)-\Gamma^{\top}_{\theta_{0}}(x)\sqrt{n}(\theta_{n}-\theta_{0})$.
Calling on Assumption (A2), one verifies that
$\sqrt{n}(\theta_{n}-\theta_{0})$ is tight and converges to $\Theta$ and that
$(\mathbb{D}_{n},\sqrt{n}(\theta_{n}-\theta_{0}))$ converge jointly to
$(\mathbb{D},\Theta)$. Hence $\widehat{\mathbb{D}}_{n}$ converges to
$\widehat{\mathbb{D}}={\mathbb{D}}-\Gamma^{\top}_{\theta_{0}}(x)\Theta$.
Straightforward computations show the covariance function of
$\widehat{\mathbb{D}}$ is precisely given by (2.7).∎

###  A.2 Proof of Theorem 2.5

The proof follows the same approach as that of Theorem 2.3. Precisely, one
writes

$\displaystyle\widehat{\mathbb{D}}_{n}(x)=\mathbb{D}_{n}(x)+B_{n,0}(x)-B_{n,1}(x)+B_{n,2}(x)$

where

$B_{n,0}(x)=\frac{1}{n}\sum_{i=1}^{n}\left(\begin{array}[]{l}a_{1}(I_{i-1})\\\
a_{2}(I_{i-1})\end{array}\right)1\kern-1.99997pt{\rm l}\\{X_{i-1}\leq x\\},$
$B_{n,1}(x)=\left[\frac{1}{n}\sum_{i=1}^{n}\dot{W}^{\top}_{\theta_{0}}(X_{i},I_{i-1})1\kern-1.99997pt{\rm
l}\\{X_{i-1}\leq x\\}\right]\sqrt{n}(\theta_{n}-\theta_{0}),$

and

$B_{n,2}(x)=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left[W_{\theta_{n}}(X_{i},I_{i-1})-W_{\theta_{0}}(X_{i},I_{i-1})-\dot{W}^{\top}_{\theta_{0}}(X_{i},I_{i-1})(\theta_{n}-\theta_{0})\right]1\kern-1.99997pt{\rm
l}\\{X_{i-1}\leq x\\}.$

$B_{n,0}$ converges uniformly to $\Psi_{A}(x)$ by Lemma A.1. Assumption (L1)
and Lemma A.1, yield that $B_{n,1}$ converges to
$\Gamma^{\top}_{\theta_{0}}(x)(\Theta+\xi_{A})$. From Assumption (A3), one
concludes that the term $B_{n,2}$ is uniformly bounded by
$\sqrt{n}\|\theta_{n}-\theta_{0}\|\lambda_{1}(\|\theta_{n}-\theta_{0}\|)\frac{1}{n}\sum_{i=1}^{n}M_{1}(X_{i},I_{i-1})$
which goes to zero in probability since
$\mathbb{E}\\{M_{1}(X_{i},I_{i-1})\\}<\infty$ and by Assumption (L3),
$\sqrt{n}\|\theta_{n}-\theta_{0}\|=O_{\mathbb{P}}(1)$ and
$\|\theta_{n}-\theta_{0}\|=o_{\mathbb{P}}(1)$.∎

###  A.3 Proof of Theorem 3.1

The proof of Theorem (3.1) is as follows. First, in Lemma A.2 we establish,
for $k=1,2,$ that $T_{n}^{k}(\widehat{\mathbb{D}}_{n}^{k})$ is asymptotically
equivalent to $T^{k}(\widehat{\mathbb{D}}_{n})$. Second, using the continuous
mapping theorem, one concludes that $T^{k}(\widehat{\mathbb{D}}_{n}^{k})$
converges to $T^{k}(\widehat{\mathbb{D}}^{k})$. Then the proof is completed by
showing that $T^{k}(\widehat{\mathbb{D}}^{k})$ is equal to
$T^{k}(\mathbb{D}^{k})$ which has the same law as $\mathbb{D}^{k}$.

######  Lemma A.2.

Under the assumptions of Theorem 3.1, $\sup_{x\leq
x_{0}}\|T_{n}^{k}(\widehat{\mathbb{D}}_{n}^{k})(x)-T^{k}(\widehat{\mathbb{D}}_{n}^{k})(x)\|$
converges to zero in probability for $k=1,2$.

###### Proof.

To prove the Lemma, observe that

$T_{n}^{k}(\widehat{\mathbb{D}}_{n}^{k})(x)-T^{k}(\widehat{\mathbb{D}}_{n}^{k})(x)=\int_{-\infty}^{x}g_{k}^{\top}(y)A_{k}^{-1}(y)\left(\int_{y}^{\infty}g_{k}(z){\widehat{\mathbb{D}}_{n}^{k}(dz)}\right)K_{k}(dy)\\\
-\int_{-\infty}^{x}\hat{g}_{k}^{\top}(y)\hat{A}_{k}^{-1}(y)\left(\int_{y}^{\infty}\hat{g}_{k}(z){\widehat{\mathbb{D}}_{n}^{k}(dz)}\right)\hat{K}_{k}(dy).$

First, one establishes that $\psi_{n}^{k}$ is tight and that
$\sup_{y}|\psi_{n}^{k}(y)-\widehat{\psi}_{n}^{k}(y)|$ converges to zero in
probability, where
$\psi_{n}^{k}(y)=\int_{y}^{\infty}g_{k}(z){\widehat{\mathbb{D}}_{n}^{k}(dz)}$
and
$\widehat{\psi}_{n}^{k}(y)=\int_{y}^{\infty}\hat{g}_{k}(z){\widehat{\mathbb{D}}_{n}^{k}(dz)}$.
For the tightness of $\psi_{n}^{k}$, set
$\psi_{n}^{0,k}(y)=\int_{y}^{\infty}g_{k}(z){\mathbb{D}}_{n}^{k}(dz)=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}W^{k}_{\theta_{0}}(X_{i},I_{i-1})g_{k}(X_{i-1})1\kern-1.99997pt{\rm
l}\\{X_{i-1}>y\\}$. Note that $\psi_{n}^{0,k}$ is a marked empirical process
its tightness follows, from Escanciano, 2007b  and Assumption (K4). Next,
using the same decomposition as in the proof of Theorem 2.3, one sees that

$\sup_{x\in\mathbb{R}}\left|\psi_{n}^{k}(x)-\psi_{n}^{0,k}(x)+\frac{1}{n}\sum_{i=1}^{n}g_{k}(X_{i-1})\dot{W}_{\theta_{0}}^{k}(X_{i},I_{i-1})^{\top}1\kern-1.99997pt{\rm
l}\\{X_{i-1}>x\\}\sqrt{n}(\theta_{n}-\theta)\right|=o_{\mathbb{P}}(1).$

The above with tightness of $\psi_{n}^{0,k}$ and assumptions (A3-A4, K4) imply
the tightness of $\psi_{n}^{k}$. For
$\sup_{y}|\psi_{n}^{k}(y)-\widehat{\psi}_{n}^{k}(y)|$, the decomposition in
the proof of Theorem 2.3, shows that

$\displaystyle\sup_{y}|\psi_{n}^{k}(y)-\widehat{\psi}_{n}^{k}(y)|\leq\left\|\int_{y}^{\infty}(\hat{g}_{k}(y)-g_{k}(y))\mathbb{D}_{n}^{k}(dy)\right\|$
$\displaystyle+\left\|\frac{1}{n}\sum_{i=1}^{n}(\hat{g}_{k}(X_{i-1})-{g}_{k}(X_{i-1}))\dot{W}_{\theta_{0}}^{k}(X_{i},I_{i-1})^{\top}1\kern-1.99997pt{\rm
l}\\{X_{i-1}>x\\}\sqrt{n}(\theta_{n}-\theta)\right\|+o_{\mathbb{P}}(1),$
$\displaystyle\leq$
$\displaystyle\left\|\int_{y}^{\infty}(\hat{g}_{k}(y)-g_{k}(y))\mathbb{D}_{n}^{k}(dy)\right\|+\|\sqrt{n}(\theta_{n}-\theta)\|\|\hat{g}_{k}-g_{k}\|\frac{1}{n}\sum_{i=1}^{n}\|\dot{W}_{\theta_{0}}^{k}(X_{i},I_{i-1})\|+o_{\mathbb{P}}(1)$

which converges to zero by assumptions and the LLN.

Next, it will be shown that $\sup_{x}\|\hat{A}_{k}(x)-A_{k}(x)\|$ converges to
zero in probability. Observe that

$\|\hat{A}_{k}(x)-A_{k}(x)\|=\left\|\int_{x}^{\infty}g_{k}(t)g_{k}^{\top}(t)K_{k}(dt)-\int_{x}^{\infty}\hat{g}_{k}(t)\hat{g}_{k}^{\top}(t)\hat{K}_{k}(dt)\right\|\\\
\leq\left\|\int_{x}^{\infty}g_{k}(t)g_{k}^{\top}(t)(K_{k}(dt)-\hat{K}_{k}(dt))\right\|+\left\|\int_{x}^{\infty}\left[g_{k}(t)g_{k}^{\top}(t)-\hat{g}_{k}(t)\hat{g}_{k}^{\top}(t)\right]\hat{K}_{k}(dt)\right\|.$

The first term above is equal

$\|\frac{1}{n}\sum_{i=1}^{n}g_{k}(X_{i-1})g_{k}^{\top}(X_{i-1})W^{k}_{\theta_{n}}(X_{i},I_{i-1})1\kern-1.99997pt{\rm
l}\\{X_{i-1}>x\\}-\int_{x}^{\infty}g_{k}(t)g_{k}^{\top}(t)K_{k}(dt)\|$

which converges to almost surely zero by Lemma (A.1) and Assumption (K2). The
second term is bounded by
$(\|\hat{g}_{k}-g_{k}\|+\|g_{k}\|)\|\hat{g}_{k}-g_{k}\|\int_{-\infty}^{\infty}\hat{K}_{k}(dt)$
which goes to zero in probability by assumptions.
As pointed earlier, for all $x\leq x_{0}$ one has $A_{k}(x)-A_{k}(x_{0})$ non-
negative definite implying that $A_{k}(x)$ is invertible whenever
$A_{k}(x_{0})$ is invertible and that
$\|A_{k}^{-1}(x)\|\leq\|A_{k}^{-1}(x_{0})\|<\infty$. Using the above result
for $\hat{A}_{k}-A_{k}$ and classical algebraic manipulations one also easily
see that $\sup_{x\leq
x_{0}}\|\hat{A}_{k}^{-1}(x)-A_{k}^{-1}(x)\|\stackrel{{\scriptstyle\mathbb{P}}}{{\longrightarrow}}0$.
To complete the proof of the Lemma A.2 note that

$\displaystyle
T_{n}^{k}(\widehat{\mathbb{D}}_{n}^{k})(x)-T^{k}(\widehat{\mathbb{D}}_{n}^{k})(x)=\int_{-\infty}^{x}g_{k}^{\top}(y)A_{k}^{-1}(y)\psi_{n}^{k}(y)K_{k}(dy)-\int_{-\infty}^{x}\hat{g}_{k}^{\top}(y)\hat{A}_{k}^{-1}(y)\widehat{\psi}_{n}^{k}(y)\hat{K}_{k}(dy).$

Direct computations show that

$T_{n}^{k}(\widehat{\mathbb{D}}_{n}^{k})(x)-T^{k}(\widehat{\mathbb{D}}_{n}^{k})(x)=\int_{-\infty}^{x}g_{k}^{\top}(y)A_{k}^{-1}(y)\psi_{n}(y)(K_{k}(dy)-\hat{K}_{k}(dy))\\\
+\int_{-\infty}^{x}\left[g_{k}^{\top}(y)A_{k}^{-1}(y)\psi_{n}^{k}(y)-\hat{g}_{k}^{\top}(y)\hat{A}_{k}^{-1}(y)\hat{\psi}_{n}^{k}(y)\right]\hat{K}_{k}(dy).$

Hence

$\sup_{x\leq
x_{0}}|T_{n}^{k}(\widehat{\mathbb{D}}_{n}^{k})(x)-T^{k}(\widehat{\mathbb{D}}_{n}^{k})(x)|\leq\sup_{x\leq
x_{0}}\left\|\int_{-\infty}^{x}g_{k}^{\top}(y)A_{k}^{-1}(y)\psi_{n}(y)(K_{k}(dy)-\hat{K}_{k}(dy))\right\|\\\
+\sup_{x\leq
x_{0}}\left\|\int_{-\infty}^{x}\left[g_{k}^{\top}(y)A_{k}^{-1}(y)\psi_{n}^{k}(y)-\hat{g}_{k}^{\top}(y)\hat{A}_{k}^{-1}(y)\hat{\psi}_{n}^{k}(y)\right]\hat{K}_{k}(dy)\right\|.$

Using Lemma (4.1) in Koul and Stute, (1999) and the fact that $\psi_{n}$ is
tight, one concludes that the first term in the above goes to zero in
probability. Term 2 is bounded by $\sum_{\ell=1}^{3}\check{C}_{\ell,n}$ where

$\check{C}_{1,n}=\|\widehat{\psi}_{n}^{k}-\psi_{n}^{k}\|\sup_{x\leq
x_{0}}\|A_{k}^{-1}(x)\|\int_{-\infty}^{\infty}\|{g}_{k}^{\top}(y)\|\hat{K}_{k}(dy),$
$\check{C}_{2,n}=\sup_{x\leq
x_{0}}\|\hat{A}_{k}^{-1}(x)-A_{k}^{-1}(x)\|(\|\hat{\psi}_{n}^{k}-\psi_{n}^{k}\|+\|\psi_{n}^{k}\|)\int_{-\infty}^{\infty}\|\hat{g}_{k}^{\top}(y)\|\hat{K}_{k}(dy)$

and

$\check{C}_{3,n}=\sup_{x\leq
x_{0}}(\|\hat{A}_{k}^{-1}(x)-A_{k}^{-1}(x)\|+\|A_{k}^{-1}(x)\|)(\|\hat{\psi}_{n}^{k}-\psi_{n}^{k}\|+\|\psi_{n}^{k}\|)\|\hat{g}_{k}^{\top}-g_{k}^{\top}\|\int_{-\infty}^{\infty}\hat{K}_{k}(dy).$

Since $\psi_{n}^{k}$ is tight and $\|\psi_{n}^{k}\|=O_{\mathbb{P}}(1)$, the
proof is complete by noting that, using the assumptions and the above results,
$\check{C}_{\ell,n}$ for $\ell=1,\ldots,3$ converge to zero in probability.

The transformation $T^{k}$ is linear and continuous which implies that
$T^{k}(\widehat{\mathbb{D}}_{n}^{k})$ converges to
$T^{k}(\widehat{\mathbb{D}}^{k})=T^{k}(\mathbb{D}^{k}-\Theta^{\top}\Gamma_{\theta_{0}}^{k})=T^{k}(\mathbb{D}^{k})-\Theta^{\top}T^{k}(\Gamma_{\theta_{0}}^{k})=T^{k}(\mathbb{D}^{k})$
since direct computations show that $T^{k}(\Gamma_{\theta_{0}}^{k})=0$.
Straightforward computations, similar to those in Stute et al., (1998), enable
us to verify that $T^{k}(\mathbb{D}^{k})$ has the same distribution as
$\mathbb{D}^{k}=\mathbb{W}_{k}\circ K_{k}$ where $\mathbb{W}_{k}$ is standard
Brownian motion. ∎

###  A.4 Proof of Theorem 4.1

Theorem 4.1 will be shown by establishing parts $(a)$ and $(b)$ below

  * $(a)$

$\check{\mathbb{D}}_{n,b}$ is asymptotically equivalent to
$\mathbb{D}_{n,b}^{\star}$

  * $(b)$

$\mathbb{D}_{n,b}^{\star}$ converge in Law to independent copies of
$\widehat{\mathbb{D}}.$

Proof of $(a)$:
Straightforward computations show that
$\check{\mathbb{D}}^{k}_{n}(x)-\mathbb{D}_{n}^{\star
k}(x)=\sum_{k=1}^{5}R_{n,k}(x)$, where
$R_{n,1}(x)=\left(\frac{1}{n}\sum_{i=1}^{n}Z_{i,b}\dot{W}^{\top}_{\theta_{0}}(X_{i},I_{i-1})1\kern-1.99997pt{\rm
l}_{\\{X_{i-1}\leq x\\}}\right)\sqrt{n}(\theta_{n}-\theta_{0}),$
$R_{n,2}(x)=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}Z_{i,b}\left[{W}_{\theta_{n}}(X_{i},I_{i-1})-{W}_{\theta_{0}}(X_{i},I_{i-1})-\dot{W}^{\top}_{\theta_{0}}(X_{i},I_{i-1})(\theta_{n}-\theta_{0})\right]1\kern-1.99997pt{\rm
l}_{\\{X_{i-1}\leq x\\}},$
$R_{n,3}(x)=\left[\Gamma_{\theta_{n}}(x)-\Gamma_{\theta_{0}}(x)\right]^{\top}\left(\frac{1}{\sqrt{n}}\sum_{i=1}^{n}Z_{i,b}\phi^{\star}(X_{i},X_{i-1},\theta_{0})\right),$
$R_{n,4}(x)=\Gamma^{\top}_{\theta_{n}}(x)\frac{1}{\sqrt{n}}\sum_{i=1}^{n}Z_{i,b}\dot{\phi}^{\star}(X_{i},X_{i-1},\theta_{0})^{\top}(\theta_{n}-\theta_{0}),$
and
$R_{n,5}(x)=\Gamma^{\top}_{\theta_{n}}(x)\frac{1}{\sqrt{n}}\sum_{i=1}^{n}Z_{i,b}\biggl{[}\phi^{\star}(X_{i},X_{i-1},\theta_{n})-\phi^{\star}(X_{i},X_{i-1},\theta_{0})-\dot{\phi}^{\star}(X_{i},X_{i-1},\theta_{0})^{\top}(\theta_{n}-\theta_{0})\biggr{]}.$
The term $R_{n,1}$ converges to zero in probability by Lemma A.1, term
$R_{n,2}$ converges in probability to zero uniformly for all $x\in\mathbb{R}$
following the same steps as for the term $B_{n,2}$ in the proof of Theorem
2.3. By Assumption (M1),
$\sup_{x}\|\Gamma_{\theta_{n}}(x)-\Gamma_{\theta_{0}}(x)\|\leq\|\theta_{n}-\theta_{0}\|C$
which goes to zero in probability. One also sees that
$\sup_{x}\|\Gamma_{\theta_{0}}(x)\|\leq\mathbb{E}\|\dot{W}_{\theta_{0}}(X_{i},I_{i-1})\|<\infty$.
Next, since
$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}Z_{i,b}\phi^{\star}(X_{i},X_{i-1},\theta_{0})=O_{\mathbb{P}}(1)$
by the multiplier central limit theorem, then the term $R_{n,3}$ converge
uniformly to zero in probability.

$\|R_{n,4}\|\leq\|\theta_{n}-\theta_{0}\|\|\frac{1}{\sqrt{n}}\sum_{i=1}^{n}Z_{i,b}\dot{\phi}^{\star}(X_{i},X_{i-1},\theta_{0})^{\top}\|(\|\Gamma_{\theta_{n}}-\Gamma_{\theta_{0}}\|+\|\Gamma_{\theta_{0}}\|)=o_{\mathbb{P}}(1)$

since
$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}Z_{i,b}\dot{\phi}^{\star}(X_{i},X_{i-1},\theta_{0})=O_{\mathbb{P}}(1)$
by Assumption (M3). Finally, by Assumption (M2),

$\|R_{n,5}\|\leq\sqrt{n}\|\theta_{n}-\theta_{0}\|\lambda_{3}(\|\theta_{n}-\theta_{0}\|)\left(\frac{1}{n}\sum_{i=1}^{n}|Z_{i,b}|M_{4}(X_{i},X_{i-1})\right)\times(\|\Gamma_{\theta_{n}}-\Gamma_{\theta_{0}}\|+\|\Gamma_{\theta_{0}}\|)=o_{\mathbb{P}}(1).$

Combining these results, yields
$\sup_{x}|\check{\mathbb{D}}^{n,b}(x)-\mathbb{D}_{n,b}{\star}(x)|=o_{\mathbb{P}}(1)$
which completes the proof of part $(a)$.
Proof of $(b)$:
One notices that the multiplier central limit theorem yields the weak
convergence of $\mathbb{D}_{n,b}^{\star}$. It just remain to show that the
asymptotic covariance operator of the process $\mathbb{D}_{n,b}^{\star}$ is
the same as that of the limiting process $\widehat{\mathbb{D}}.$

$\mbox{Cov}\left(\mathbb{D}_{n,b}^{\star}(x),\mathbb{D}_{n,b}^{\star}(y)\right)=\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}\mathbb{E}(V_{i}(x)V_{j}(y))\mathbb{E}(Z_{i,b}Z_{j,b}),$

where $V_{i}(x)={W}_{\theta_{0}}(X_{i},I_{i-1})1\kern-1.99997pt{\rm
l}{\\{X_{i-1}\leq
x\\}}-\phi^{\star}(X_{i},I_{i-1},\theta_{0})^{\top}\Gamma_{\theta_{0}}(x).$
One has only to consider the terms with $i=j$ in the above sums, since for
$i\neq j,\mathbb{E}\left\\{Z_{j,b}Z_{i,b}\right\\}=0$. Recalling that
$\mathbb{E}([Z_{i,b}]^{2})=1$, one sees that the above covariance reduces to
$\mbox{Cov}\left(\mathbb{D}_{n}^{\star k}(x),\mathbb{D}_{n}^{\star
k}(y)\right)=\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}\left(V_{i}(x)V_{i}(y)\right)=\mathbb{E}\left(V_{1}(x)V_{1}(y)\right).$
Straightforward computations show that

$\mathbb{E}\left(V_{1}(x)V_{1}(y)\right)={K}(x,y)-\Gamma^{\top}_{\theta_{0}}(x)G(y)-G^{\top}(x)\Gamma_{\theta_{0}}(y)+\Gamma^{\top}_{\theta_{0}}(x)\Sigma_{0}\Gamma_{\theta_{0}}(y),$

which is the same as the covariance of the process $\widehat{\mathbb{D}}(x)$.
∎

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