# Extremal Dependence-Based Specification Testing of Time Series

Yannick Hoga
Faculty of Economics and Business Administration
University of Duisburg-Essen  The author is grateful to seminar participants
at CREST and Erasmus University Rotterdam for valuable comments and
suggestions, in particular Christian Francq, Jeroen Rombouts, Jean-Michel
Zakoïan and Chen Zhou. The author would also like to thank Christoph Hanck and
Till Massing for their careful reading of an earlier version of this
manuscript. This work was supported by the German Research Foundation (DFG)
under Grant HO 6305/1-1.

###### Abstract

We propose a specification test for conditional location–scale models based on
extremal dependence properties of the standardized residuals. We do so
comparing the left-over serial extremal dependence—as measured by the pre-
asymptotic tail copula—with that arising under serial independence at
different lags. Our main theoretical results show that the proposed
Portmanteau-type test statistics have nuisance parameter-free asymptotic
limits. The test statistics are easy to compute, as they only depend on the
standardized residuals, and critical values are likewise easily obtained from
the limiting distributions. This contrasts with extant tests (based, e.g., on
autocorrelations of squared residuals), where test statistics depend on the
parameter estimator of the model and critical values may need to be
bootstrapped. We show that our test performs well in simulations. An empirical
application to S&P 500 constituents illustrates that our test can uncover
violations of residual serial independence that are not picked up by standard
autocorrelation-based specification tests, yet are relevant when the model is
used for, e.g., risk forecasting.

Keywords: Location–scale models, Serial extremal dependence, Specification
test, Tail copula

##  1 Motivation

The dynamics of many economic and financial time series can be successfully
captured by location–scale models, that allow for mean and variance changes in
the conditional distribution. The benchmark models for incorporating mean
changes are ARMA models, and GARCH-type processes have become the standard
volatility models, since the seminal work of Engle (1982) and Bollerslev
(1986). The main aim of this paper is to derive tests that check whether the
fitted location–scale model adequately captures the time variation in the mean
and variance of the conditional distribution. As is common for specification
tests, a rejection points to a problem in the mean or variance dynamics, but
does not establish which of the two.111A rare exception is the work of Francq
_et al._ (2006). While our tests may also pick up misspecification in the
conditional mean, they are first and foremost designed to detect misspecified
volatility dynamics. Thus, our main focus in the following will be on
volatility models.

Testing for correctly specified volatility models is of high practical
relevance. This is because volatility models are widely used to predict risk
measures, such as the Value-at-Risk (VaR) and the Expected Shortfall (ES);
see, e.g., McNeil and Frey (2000); Chan _et al._ (2007); Hoga (2019). Risk
measure forecasts are key inputs in risk management procedures of financial
institutions, due to regulatory requirements in the Basel framework of the
Basel Committee on Banking Supervision (2019). The Basel framework penalizes
sustained underpredictions of risk by imposing higher capital requirements. On
the other hand, if risk forecasts are too high, too much capital is put aside
as a buffer against large losses. In both cases of over- and underprediction,
some portion of the capital can no longer earn premiums, leading to foregone
profits. Thus, accurate risk forecasts from correctly specified volatility
models are important. Consequently, diagnostic tests should be routinely
applied to the chosen volatility specification.

To this end, several specification tests are available. These tests typically
verify whether the model residuals are independent, identically distributed
(i.i.d.). However, standard tests of the i.i.d. property—such as Box and
Pierce (1970) or Ljung and Box (1978) tests—cannot be applied ‘as usual’,
since the residuals are only estimated. Li and Mak (1994) were the first to
propose a Portmanteau-type test for the autocorrelations of squared residuals
that corrects for this fact in (conditionally Gaussian) location–scale models.
Berkes _et al._ (2003) and Ling and Li (1997) extend the applicability of the
Li and Mak (1994) test to more general GARCH and $N(0,1)$–FARIMA–GARCH models,
respectively. Similarly, Fisher and Gallagher (2012) consider weighted
Portmanteau statistics applied to ARCH residuals. Hidalgo and Zaffaroni (2007)
develop spectral-based tests for ARCH($\infty$) series. The limiting
distributions for all these proposed test statistics have only been derived
for the quasi-maximum likelihood (QML) estimator. Moreover, practical
application of these tests is complicated by the fact that the limiting
distributions depend on nuisance parameters induced by parameter estimation.
Valid tests thus require consistent estimators of the nuisance parameters
(that have to be computed on a case-by-case basis) or involved bootstrap-based
procedures to compute critical values. By construction, all these tests
diagnose the complete serial dependence structure of the standardized
residuals.

Here, instead of considering the complete dependence structure, we take a
different tack by focusing on the left-over serial extremal dependence in the
standardized residuals. When diagnosing the complete dependence structure, the
effect of any remaining residual extremal dependence may be ‘washed-out’.
However, overlooked serial dependence in the extremes may be very harmful as
it invalidates, e.g., the model’s risk forecasts, such as VaR and ES forecasts
(Su _et al._ , 2021+). As pointed out above, misspecified risk forecasts are
penalized under the Basel framework. Hence, a separate diagnostic for the
extremes is desirable. In empirical work, Davis _et al._ (2012) and Davis _et
al._ (2013) use the pre-asymptotic (PA) extremogram as a diagnostic. For
instance, to assess the GARCH fit for FTSE returns, Davis _et al._ (2012) show
that the PA-extremogram of the standardized residuals exhibits no signs of
extremal dependence. However, such a model check lacks theoretical
justification so far, because the central limit theory for extremal dependence
measures has not been developed for model residuals. It is the main
theoretical contribution of this paper to do so.

Specifically, we base our tests on the PA-tail copula (Schmidt and
Stadtmüller, 2006) of the (absolute values of the) standardized residuals. The
tail copula—i.e., the limit of the PA-tail copula—has been used by, e.g.,
Bücher _et al._ (2015) to detect structural changes in tail dependence. The
PA-tail copula is closely related to the PA-extremogram of Davis _et al._
(2012). However, for our purposes, working with the PA-tail copula gives rise
to nuisance parameter-free limiting distributions, which would not be the case
for the PA-extremogram.

It turns out that—unlike the above mentioned traditional specification
tests—our tests are easy to apply: We only require the standardized residuals
to compute the test statistics. In particular, it does not matter for our
tests which parameter estimator (QML estimator, Self-Weighted QML estimator,
Least Squares, etc.) was used in fitting the volatility model. This gives our
tests wide practical appeal.

Testing for any left-over serial extremal dependence in the residuals directs
the power of our specification tests to the tails. Thus, they are more
sensitive to any remaining dependence in the tails. This may be important
when, say, a GARCH model captures well the volatility dynamics in the body of
the distribution, but not so much in the extremes (see also the simulations
for such an example). It is exactly for these cases that our tests are
designed. Thus, the diagnostic tests developed here are not meant to replace
standard tests based (e.g.) on the autocorrelations of the squared residuals,
but rather to supplement them.

We illustrate the good size and power of our tests in simulations.
Specifically, we show that—as predicted by theory—in sufficiently large
samples the size is indeed unaffected by parameter estimation effects. We also
demonstrate that empirically relevant misspecifications can be detected more
easily using our extremal dependence-based tests instead of autocorrela-tion-
based tests.

The empirical application shows that for the S&P 500 components an
autocorrelation-based test can often not detect any significant departures
from a fitted GARCH-type model. However, in many of these cases, applying our
tests suggests some statistically significant serial extremal dependence left
over in the residuals. This indicates that the fitted GARCH-type processes
capture well the volatility changes in the center of the distribution, but not
so much in the tails. This result is consistent with Engle and Manganelli
(2004), who conclude from fitting their CAViaR models at different risk levels
that ‘our findings suggest that the process governing the behavior of the
tails might be different from that of the rest of the distribution.’ Such a
dynamic is particularly worrisome when using GARCH-type models to forecast
risk measures, such as VaR and ES, far out in the tail.

The remainder of the paper proceeds as follows. In Section 2, we present our
model setup and propose our Portmanteau-type tests for serial extremal
dependence. Section 3 explores size and power of our tests in finite samples,
while Section 4 illustrates the benefits of our test for real data. Finally,
Section 5 concludes. The supplementary Appendix contains all proofs and
additional simulations.

##  2 Main Results

###  2.1 Location–Scale Model

Denote by $Y_{1},\ldots,Y_{n}$ the observations of interest to which some
parametric model will be fitted. For instance, the $Y_{1},\ldots,Y_{n}$ may
denote log-returns on some speculative asset. We define the $\sigma$-field
generated by $Y_{t},Y_{t-1},\ldots$ and some exogenous (possibly multivariate)
variables $\bm{x}_{t},\bm{x}_{t-1},\ldots$ by
$\mathcal{F}_{t}=\sigma(Y_{t},Y_{t-1},\ldots;\bm{x}_{t},\bm{x}_{t-1},\ldots)$.
We assume the $Y_{t}$ to follow the parametric conditional location–scale
model

$Y_{t}=\mu_{t}(\bm{\theta}^{\circ})+\sigma_{t}(\bm{\theta}^{\circ})\varepsilon_{t},$
(1)

where $\bm{\theta}^{\circ}\in\bm{\varTheta}$ is the true parameter vector from
the parameter space $\bm{\varTheta}\subset\mathbb{R}^{m}$ ($m\in\mathbb{N}$),
$\mu_{t}(\bm{\theta}^{\circ})$ and $\sigma_{t}(\bm{\theta}^{\circ})$ are the
$\mathcal{F}_{t-1}$-measurable conditional mean and conditional standard
deviation, and $\varepsilon_{t}$ is i.i.d. with mean zero and unit variance
(written: $\varepsilon_{t}\overset{\text{i.i.d.}}{\sim}(0,1)$), independent of
$\mathcal{F}_{t-1}$.

In the following, we require a smoothness condition on the tail of the
$|\varepsilon_{t}|$, whose distribution function (d.f.) we denote by
$F(\cdot)$.

######  Assumption 1.

The d.f. $F(\cdot)$ of the $|\varepsilon_{t}|$ satisfies:

  1. (i)

The upper endpoint of $F(\cdot)$ is infinite.

  2. (ii)

There exists some constant $C_{F}>0$ such that $F(\cdot)$ is differentiable
with density $f(\cdot)$ on $[C_{F},\infty)$.

  3. (iii)

$\lim_{x\to\infty}xf(x)/[1-F(x)]=\alpha\in(0,\infty)$.

Assumption 1 is known as one of the von Mises’ conditions (see, e.g., de Haan
and Ferreira, 2006, Rem. 1.1.15) and ensures a sufficiently well-behaved tail
of the $|\varepsilon_{t}|$. This is needed to justify the replacement of
$|\varepsilon_{t}|$ in our test statistic with the standardized residuals. In
their autocorrelation-based diagnostic test, Berkes _et al._ (2003) impose a
similar regularity condition; see in particular their equation (2.4).
Assumption 1 is only needed to ensure the validity of Lemma 1 in Appendix C.1.
Thus, it may be replaced by any other assumption on the tail decay, as long as
the conclusions of Lemma 1 remain valid.

Next, we require a $\sqrt{n}$-consistent estimator of the unknown
$\bm{\theta}^{\circ}$:

######  Assumption 2.

There exists an estimator $\widehat{\bm{\theta}}$ satisfying
$n^{1/2}|\widehat{\bm{\theta}}-\bm{\theta}^{\circ}|=O_{\operatorname{P}}(1)$,
as $n\to\infty$.

Root-$n$ consistent estimators are available under various regularity
conditions for ARMA–GARCH and GARCH-type models; see, e.g., Berkes and Horváth
(2004), Francq and Zakoïan (2004), Pan _et al._ (2008), Francq and Thieu
(2019).

Once the parameters have been estimated, volatility
$\widehat{\sigma}_{t}=\widehat{\sigma}_{t}(\widehat{\bm{\theta}})$, the
conditional mean $\widehat{\mu}_{t}=\widehat{\mu}_{t}(\widehat{\bm{\theta}})$
and the standardized residuals
$\widehat{\varepsilon}_{t}=\widehat{\varepsilon}_{t}(\widehat{\bm{\theta}})=[Y_{t}-\widehat{\mu}_{t}(\widehat{\bm{\theta}})]/\widehat{\sigma}_{t}(\widehat{\bm{\theta}})$
can be computed. Note that $\mu_{t}(\bm{\theta})$ and
$\sigma_{t}(\bm{\theta})$ may depend on the infinite past of $Y_{t}$ (and,
possibly, $\bm{x}_{t}$) and, hence, can only be approximated via
$\widehat{\mu}_{t}(\bm{\theta})$ and $\widehat{\sigma}_{t}(\bm{\theta})$ using
some initial values; see, e.g., the truncated recursion (A.2) in Appendix A.

For our final assumption, we define the neighborhood of the true
$\bm{\theta}^{\circ}$ as

$N_{n}(\eta)=\left\\{\bm{\theta}\ \mathrel{\mathop{\mathchar 58\relax}}\
n^{1/2}|\bm{\theta}-\bm{\theta}^{\circ}|\leq\eta\right\\},\qquad\eta>0,$

where $|\cdot|$ is some norm on $\bm{\varTheta}$.

######  Assumption 3.

For any $\eta>0$, there exist r.v.s $m_{t}\mathrel{\mathop{\mathchar
58\relax}}=m_{n,t}(\eta)\geq 0$ and $s_{t}\mathrel{\mathop{\mathchar
58\relax}}=s_{n,t}(\eta)\geq 0$ with
$\max_{t=\ell_{n},\ldots,n}m_{t}=o_{\operatorname{P}}(1)$ and
$\max_{t=\ell_{n},\ldots,n}s_{t}=o_{\operatorname{P}}(1)$ for any
$\ell_{n}\leq n$ with $\ell_{n}\to\infty$, such that with probability
approaching 1, as $n\to\infty$,

$|\varepsilon_{t}|(1-s_{t})-m_{t}\leq|\widehat{\varepsilon}_{t}(\bm{\theta})|\leq|\varepsilon_{t}|(1+s_{t})+m_{t}$

for all $t=\ell_{n},\ldots,n$ and all $\bm{\theta}\in N_{n}(\eta)$.

Assumption 3 imposes a uniform approximability of the innovations
$|\varepsilon_{t}|$ by the $|\widehat{\varepsilon}_{t}(\bm{\theta})|$ in a
$n^{-1/2}$-neighborhood of the true parameter. However, Assumption 3 does not
require the first $(\ell_{n}-1)$ innovations to be approximable. This is
convenient as the first few residuals are often imprecise, due to
initialization effects caused by using artificial initial values in the mean
and variance recursions. The variables $m_{t}$ and $s_{t}$ bound the error in
the $\widehat{\varepsilon}_{t}(\bm{\theta})$ associated with approximating the
conditional mean and conditional variance, respectively. Of course, Assumption
3 is a high-level condition that must be verified for each specific model on a
case-by-case basis. In Appendix A, we verify Assumption 3 for two popular
model classes, namely APARCH and ARMA–GARCH models.

###  2.2 The Tail Copula and its Estimators

The survival copula at lag $d$ of the $\\{|\varepsilon_{t}|\\}$ from Section
2.1 is

$\overline{C}^{(d)}(u,v)=\operatorname{P}\big{\\{}|\varepsilon_{t}|>F^{\leftarrow}(1-u),\
|\varepsilon_{t-d}|>F^{\leftarrow}(1-v)\big{\\}},\qquad 0\leq u,v\leq 1,$

where $F^{\leftarrow}(\cdot)$ denotes the left-continuous inverse of
$F(\cdot)$. Following Schmidt and Stadtmüller (2006), the (upper) tail copula
is the directional derivative of the survival copula at the origin with
direction $(x,y)$, i.e.,

$\Lambda^{(d)}(x,y)=\lim_{s\to\infty}s\overline{C}^{(d)}(x/s,y/s),$

where the limit is assumed to exist. Thus, the tail copula describes the
serial extremal dependence structure of the $|\varepsilon_{t}|$ at different
lags $d$. For $x=y=1$, the tail copula simplifies to the tail dependence
coefficient of Sibuya (1960). In this case, it also coincides with the most
popular version of the extremogram due to Davis and Mikosch (2009).

Let $k=k(n)$ be an intermediate sequence satisfying $k\to\infty$ and $k/n\to
0$ as $n\to\infty$. Then, replacing $s$ with $n/k\to\infty$ and also replacing
population quantities with empirical counterparts, one may estimate the tail
copula non-parametrically via

$\widetilde{\Lambda}_{n}^{(d)}(x,y)=\frac{1}{k}\sum_{t=d+1}^{n}I_{\\{|\varepsilon_{t}|>|\varepsilon|_{(\lfloor
kx\rfloor+1)},\ |\varepsilon_{t-d|}>|\varepsilon|_{(\lfloor
ky\rfloor+1)}\\}},$

where $|\varepsilon|_{(1)}\geq\ldots\geq|\varepsilon|_{(n)}$ denote the order
statistics, and $\lfloor\cdot\rfloor$ rounds to the nearest smaller integer.

In the case of interest here, where the $\varepsilon_{t}$ from (1) are i.i.d.,
the tail copula can easily shown to be identically zero, i.e.,
$\Lambda^{(d)}(x,y)\equiv 0$. However, even when there is strong (non-
extremal) dependence between $|\varepsilon_{t}|$ and $|\varepsilon_{t-d}|$, we
have $\Lambda^{(d)}(x,y)\equiv 0$. For instance, this is the case when the
dependence between the two variables is governed by a Gaussian copula for any
correlation parameter $\rho\in(-1,1)$ (Heffernan, 2000). Hence, while the
relationship may be strong in non-extremal regions, it is non-existent in the
limit as measured by the tail copula. Likewise, Hill (2011a) shows that the
tail copula is identically zero if $\varepsilon_{t}$ follows a serially
dependent stochastic volatility model. These are well-known to display weaker
extremal dependence than GARCH-type models (Davis and Mikosch, 2009; Davis _et
al._ , 2013). Thus, the tail copula cannot adequately discriminate between
independent and dependent variables, which would result in compromised power
of a specification test based on the tail copula.

For this reason, we prefer to base our test on what we term the PA-tail
copula:

$\Lambda_{n}^{(d)}(x,y)=\frac{n}{k}\operatorname{P}\left\\{|\varepsilon_{t}|>b\Big{(}\frac{n}{kx}\Big{)},\
|\varepsilon_{t-d}|>b\Big{(}\frac{n}{ky}\Big{)}\right\\},$ (2)

where $b(x)=F^{\leftarrow}(1-1/x)$. Now, the PA-tail copula is
$\Lambda_{n}^{(d)}(x,y)=(k/n)xy$ for serially independent $\varepsilon_{t}$,
but $\Lambda_{n}^{(d)}(x,y)\neq(k/n)xy$ when $\varepsilon_{t}$ follows a
stochastic volatility model (Hill, 2011a) or when
$(|\varepsilon_{t}|,|\varepsilon_{t-d}|)^{\prime}$ possesses a Gaussian
copula. Thus, the PA-tail copula captures finer differences in pre-asymptotic
levels of extremal dependence than the tail copula, ensuring that our test has
power against more subtle misspecifications. For $x=y=1$, the PA-tail copula
is the perhaps most popular version of the PA-extremogram, due to Davis and
Mikosch (2009). Even though the PA-tail copula differs from the tail copula,
it may also be estimated via $\widetilde{\Lambda}_{n}^{(d)}(x,y)$.

###  2.3 A Portmanteau-Type Test for Residual Extremal Dependence

In specification testing, one is interested in verifying whether observations
$Y_{1},\ldots,Y_{n}$ are generated by some parametric model (1). Formally, one
would like to test

$H_{0}\ \mathrel{\mathop{\mathchar 58\relax}}\ \\{Y_{t}\\}_{t=1,2,\ldots}\
\text{are generated by the parametric model \eqref{eq:ls model}.}$

Typically, this is done by fitting the specific model (1) to the
$Y_{1},\ldots,Y_{n}$. Under $H_{0}$, the residuals should then be
approximately i.i.d. This is verified by testing some implication of the
i.i.d. property. For instance, Ljung–Box-type tests focus on the implication
that the autocorrelation function is identically zero. As pointed out in the
Motivation, we test the implication of $H_{0}$ that the residuals display no
serial extremal dependence. To this end, note that the PA-tail copula in (2)
equals $(k/n)xy$ under serial independence. Hence, we test the implication
that $\Lambda_{n}^{(d)}(x,y)=(k/n)xy$ for $d=1,2,\ldots$.

However, for purposes of model checking, the estimator
$\widetilde{\Lambda}_{n}^{(d)}(x,y)$ from the previous subsection is
infeasible, as the $\varepsilon_{t}$ are unobserved. Hence, we rely on the
feasible counterpart

$\widehat{\Lambda}_{n}^{(d)}(x,y)=\frac{1}{k}\sum_{t=d+1}^{n}I_{\\{|\widehat{\varepsilon}_{t}|>|\widehat{\varepsilon}|_{(\lfloor
kx\rfloor+1)},\
|\widehat{\varepsilon}_{t-d}|>|\widehat{\varepsilon}|_{(\lfloor
ky\rfloor+1)}\\}}.$

With this estimator, we can detect deviations from
$\Lambda_{n}^{(d)}(x,y)=(k/n)xy$ using the Portmanteau-type test statistic

$\mathcal{P}_{n}^{(D)}(x,y)=\frac{n}{xy}\sum_{d=1}^{D}\Big{[}\widehat{\Lambda}_{n}^{(d)}(x,y)-(k/n)xy\Big{]}^{2},$

where $D\in\mathbb{N}$ is some fixed user-specified integer. Once again, the
fact that we compare $\widehat{\Lambda}_{n}^{(d)}(x,y)$ with the null
hypothetical value of the PA tail copula $(k/n)xy$ (and not with the null
hypothetical value of the tail copula, i.e., $0$) gives our test power in
situations, where there is extremal independence (as measured by the tail
copula), but not independence (as measured by the PA tail copula).

In choosing $D$, there is the usual trade-off. Using a small $D$ possibly
leads to undetected misspecifications at higher lags, resulting in a loss of
power. However, choosing $D$ too large, the estimates
$\widehat{\Lambda}_{n}^{(D)}(x,y)$ may be based on too few observations, thus
distorting size. We explore the choice of $D$ in detail in the simulations in
Appendix E. There, we show that $D=5$ typically leads to a good balance
between size and power.

Regarding the sequence $k$, we impose

######  Assumption 4.

For $n\to\infty$, the sequence $k$ satisfies $k\to\infty$, $k/n\to 0$, and
$k/\sqrt{n}\to\infty$.

Consistent with the notion of extremal dependence, the requirement that
$k/n\to 0$ ensures that only a vanishing fraction of the residuals is used in
estimation. On the other hand, sufficiently many extremes need to be used, as
specified by $k/\sqrt{n}\to\infty$. In the related context of estimating the
tail event correlation

$r_{n}^{(d)}=\frac{n}{k}\left[\operatorname{P}\big{\\{}|\varepsilon_{t}|>b(n/k),\
|\varepsilon_{t-d}|>b(n/k)\big{\\}}-\operatorname{P}\big{\\{}|\varepsilon_{t}|>b(n/k)\big{\\}}\operatorname{P}\big{\\{}|\varepsilon_{t-d}|>b(n/k)\big{\\}}\right],$

Hill (2011b, Sec. 5.2) even requires $k/n^{2/3}\to\infty$. Hence, assuming
$k/\sqrt{n}\to\infty$ may be regarded as a rather mild requirement.

The following is our first main theoretical result:

######  Theorem 1.

Suppose Assumptions 1–4 are met under $H_{0}$. Then, for any $x>0$ and $y>0$,
as $n\to\infty$,

$\mathcal{P}_{n}^{(D)}(x,y)\overset{d}{\longrightarrow}\chi^{2}_{D},$

where $\chi^{2}_{D}$ denotes the $\chi^{2}$-distribution with $D$ degrees of
freedom.

We defer the proof of Theorem 1 to Appendix C. The proof reveals that the same
$\chi^{2}$-limit as in Theorem 1 is obtained, if the
$\widehat{\varepsilon}_{t}$’s are replaced with the true $\varepsilon_{t}$’s
in $\mathcal{P}_{n}^{(D)}(x,y)$. Thus, the effects of parameter estimation in
$\widehat{\varepsilon}_{t}=\widehat{\varepsilon}_{t}(\widehat{\bm{\theta}})$
vanish asymptotically. In the context of residual-based estimation of
univariate extremal quantities (such as the tail index or large quantiles),
such vanishing estimation effects have been observed before by, e.g., Chan _et
al._ (2007), Hill (2015), and Hoga (2019). The explanation is that the
convergence rate of the parameter estimator is faster than that of the tail
quantity, leading to vanishing parameter estimation effects.

For $x=y=1$ the PA-tail copula is the most prominent version of Davis and
Mikosch’s (2009) PA-extremogram. Thus, the choice $x=y=1$ is the leading case
for applications of $\mathcal{P}_{n}^{(D)}(x,y)$. However, other choices of
$x$ and $y$ may lead to different test results. Thus, the next section derives
a test that combines the evidence across different values of $x$ and $y$. This
may also lead to more powerful specification tests, as we will see in the
simulations.

###  2.4 A Functional Portmanteau-Type Test

To obtain powerful functional versions of $\mathcal{P}_{n}^{(D)}(x,y)$, we
need to integrate it over a suitable area in the $(x,y)^{\prime}$-space.
Observe that $\widehat{\Lambda}_{n}^{(d)}(x,y)$ estimates
$\Lambda_{n}^{(d)}(x,y)$, which converges to the tail copula
$\Lambda^{(d)}(x,y)$ (if it exists) for $n\to\infty$. From Theorem 1 (ii) of
Schmidt and Stadtmüller (2006), we know that the tail copula is homogeneous,
i.e., it satisfies $\Lambda^{(d)}(tx,ty)=t\Lambda^{(d)}(x,y)$ for all $t>0$.
This suggests that $\widehat{\Lambda}_{n}^{(d)}(tx,ty)\approx
t\widehat{\Lambda}_{n}^{(d)}(x,y)$, which implies that integrating over a
larger space than some sphere $S_{c,\
\mathinner{\\!\left\lVert\cdot\right\rVert}}\mathrel{\mathop{\mathchar
58\relax}}=\\{(x,y)^{\prime}\in[0,\infty)^{2}\ \mathrel{\mathop{\mathchar
58\relax}}\ \mathinner{\\!\left\lVert(x,y)^{\prime}\right\rVert}=c\\}$ does
not provide us with more information against the null. Here, $c>0$ is a
constant and $\mathinner{\\!\left\lVert\cdot\right\rVert}$ some norm on
$\mathbb{R}^{2}$. A typical choice in extreme value theory is the
$\mathinner{\\!\left\lVert\cdot\right\rVert}_{1}$-norm. Since—as pointed out
above—putting $x=y=1$ is a natural choice and
$\mathinner{\\!\left\lVert(1,1)^{\prime}\right\rVert}_{1}=2$, we fix $c=2$
and, thus, integrate over the sphere $S_{2,\
\mathinner{\\!\left\lVert\cdot\right\rVert}_{1}}$. Using the
$\mathinner{\\!\left\lVert\cdot\right\rVert}_{1}$-norm in defining the sphere
also has the added advantage of giving a limiting distribution in Theorem 2 in
terms of simple Brownian bridges.

These considerations lead to the functional Portmanteau-type test statistic

$\mathcal{F}_{n}^{(D)}=n\sum_{d=1}^{D}\int_{[\iota,1-\iota]}\Big{[}\widehat{\Lambda}_{n}^{(d)}(2-2z,2z)-\frac{k}{n}(2-2z)2z\Big{]}^{2}\,\mathrm{d}z,$

where $\iota\in(0,1/2)$ is some typically small constant chosen by the
practitioner. The parameter $\iota$ serves to bound the $z$-values in the
integral away 0 and 1, where our non-parametric estimator
$\widehat{\Lambda}_{n}^{(d)}(2-2z,2z)$ may be unreliable as very extreme
quantile thresholds are considered, above which hardly any observations lie.
We mention that there is a trade-off between size and power involved in
choosing $\iota$. A large value of $\iota$ leads to a smaller area being
integrated over, thus leading to compromised power, yet estimates tend to be
more stable, thus improving size. However, unreported simulations show that
the differences in size and power are not large. In our numerical experiments,
we choose $\iota=0.1$ and compute the integral in $\mathcal{F}_{n}^{(D)}$
using standard numerical integration techniques, which allow to approximate
integrals with any required accuracy.

######  Theorem 2.

Suppose Assumptions 1–4 are met under $H_{0}$. Then, as $n\to\infty$,

$\mathcal{F}_{n}^{(D)}\overset{d}{\longrightarrow}4\sum_{d=1}^{D}\int_{[\iota,1-\iota]}B_{d}^{2}(z)\,\mathrm{d}z,$

where $\\{B_{d}(\cdot)\\}_{d=1,\ldots,D}$ are mutually independent Brownian
bridges.

Theorem 2 is proven in Appendix D. While the limiting distribution in Theorem
2 is non-standard, it is free of nuisance parameters. Thus, critical values
can be computed to any desired degree of precision by simulating sufficiently
often from the limit. Table 1 shows some selected critical values for
$\iota=0.1$ and different numbers of lags $D$.222The critical values in Table
1 are based on 4,000,000 replications of a Brownian bridge simulated on
100,000 grid points.

###### Remark 1.

In the spirit of Anderson and Darling (1952), it is also possible to consider
a weighted version of the functional test statistic à la

$\mathcal{WF}_{n}^{(D)}=n\sum_{d=1}^{D}\int_{[\iota,1-\iota]}\psi(z)\Big{[}\widehat{\Lambda}_{n}^{(d)}(2-2z,2z)-\frac{k}{n}(2-2z)2z\Big{]}^{2}\,\mathrm{d}z\overset{d}{\longrightarrow}4\sum_{d=1}^{D}\int_{[\iota,1-\iota]}\psi(z)B_{d}^{2}(z)\,\mathrm{d}z,$
(3)

where $\psi(z)\geq 0$ is a weight function satisfying some regularity
conditions (e.g., continuity). The weight function allows to put more weight
on certain $z$-values; e.g., $\phi(z)=z(1-z)$ would accentuate $z$ that are
further away from the boundaries. We refrain from exploring such
possibilities. However, we mention that for $\psi(z)\equiv 1/4$, $\iota=0$ and
$D=1$, the limit in (3) equals the limit of the well-known Cramér–von Mises
goodness-of-fit test. For $\psi(z)=1/[z(1-z)]$, we obtain the limit of the
Anderson–Darling test statistic.

$\alpha$ | $D$
---|---
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
10% | 1.340 | 2.336 | 3.231 | 4.077 | 4.896 | 5.694 | 6.477 | 7.249 | 8.011 | 8.766
5% | 1.791 | 2.890 | 3.859 | 4.765 | 5.636 | 6.480 | 7.306 | 8.117 | 8.916 | 9.705
1% | 2.905 | 4.178 | 5.273 | 6.286 | 7.248 | 8.178 | 9.082 | 9.964 | 10.832 | 11.683
Table 1: Critical values $c_{\alpha}^{(D)}$ for significance level
$\alpha\in(0,1)$ and $D$ for $\iota=0.1$ in the limiting distribution in
Theorem 2.

##  3 Simulations

Here, we investigate the size and power of the specification tests based on
$\mathcal{P}_{n}^{(D)}$ (with the standard choice $x=y=1$) and
$\mathcal{F}_{n}^{(D)}$, and compare the results with those of classical
Ljung–Box tests based on the autocorrelations of the squared residuals. We do
so for different samples sizes $n\in\\{500,\ 1000,\ 2000\\}$ and different
significance levels $\alpha\in\\{1\%,\ 5\%,\ 10\%\\}$. Furthermore, we pick
$\iota=0.1$. Other choices of $\iota$ (e.g., $\iota=0.05$ or even
$\iota=0.01$) do not materially change the results. All simulations use R
version 4.0.3 (R Core Team, 2020) and results are based on 10,000
replications.

The choice of $D$ is common to all tests and we pick $D=5$ in the following.
This choice is supported by simulations in Appendix E, where we investigate
the influence of $D$ on the tests. Here, we instead explore the effect of
different $k$’s on our test statistics $\mathcal{P}_{n}^{(D)}$ and
$\mathcal{F}_{n}^{(D)}$. In applications of extreme value methods the choice
of $k$ is often a tricky issue, because the results may be very sensitive to
the specific value of $k$ (de Haan and Ferreira, 2006). Therefore, it is of
interest to investigate the robustness of our test with respect to $k$. A very
popular choice in applications of extreme value theory is to follow
DuMouchel’s (1983) rule by setting $k=\lfloor 0.1\cdot n\rfloor$ (Quintos _et
al._ , 2001; McNeil and Frey, 2000; Mancini and Trojani, 2011; Chavez-Demoulin
_et al._ , 2014). However, at least asymptotically, this is not a valid choice
(here and elsewhere) since it fails the Assumption 4 requirement that $k$ be a
vanishing fraction of $n$ ($k/n\to 0$). Nonetheless, its widespread use
suggests some merit in finite samples. Thus, we consider $k=\lfloor\varrho
n^{0.99}\rfloor$ for $\varrho\in[0.05,\,0.15]$, where the choice
$\varrho=0.11$ roughly corresponds to DuMouchel’s (1983) rule. Note that
$k=\lfloor\varrho n^{0.99}\rfloor$ satisfies both Assumption 4 requirements.

For simplicity, we only consider model (1) with zero conditional mean,
$\mu_{t}(\bm{\theta}^{\circ})\equiv 0$. We do so, because unmodeled dynamics
in the tails of the conditional distribution are most likely caused by
misspecified volatility dynamics in practice. Throughout, we simulate from
APARCH(1,1) models under the null of correct specifications, so that in
particular our high-level Assumption 3 is satisfied; see Appendix A.

We compare our tests with classical Ljung–Box tests when volatility dynamics
are misspecified (Section 3.1) and for non-i.i.d. $\varepsilon_{t}$ (Section
3.2). In both cases, we estimate our models using the rugarch package
(Ghalanos, 2020).

###  3.1 Misspecified Volatility

We generate time series from an APARCH model with an exogenous covariate
$x_{t}$ in the volatility equation. Specifically, we simulate from the
APARCH–X(1,1) model with $\delta^{\circ}=1$ given by

$\displaystyle Y_{t}$
$\displaystyle=\sigma_{t}(\bm{\theta}^{\circ})\varepsilon_{t},\qquad\varepsilon_{t}\overset{\text{i.i.d.}}{\sim}(0,1)$
$\displaystyle\sigma_{t}(\bm{\theta}^{\circ})$
$\displaystyle=\omega^{\circ}+\alpha_{+,1}^{\circ}(Y_{t-1})_{+}+\alpha_{-,1}^{\circ}(Y_{t-1})_{-}+\beta_{1}^{\circ}\sigma_{t-1}(\bm{\theta}^{\circ})+\pi_{1}^{\circ}x_{t-1}.$
(4)

Here, the $\varepsilon_{t}$ follow a (standardized) Student’s $t$-distribution
with $4.1$ degrees of freedom to ensure
$\operatorname{E}|\varepsilon_{t}^{4}|<\infty$, which is required for
$\sqrt{n}$-consistent QML estimation in Assumption 2 (Francq and Thieu, 2019).
The exogenous $x_{t}$ are stationary with

$\displaystyle x_{t}$ $\displaystyle=\exp(z_{t}),$ $\displaystyle z_{t}$
$\displaystyle=0.9\cdot z_{t-1}+e_{t},\qquad
e_{t}\overset{\text{i.i.d.}}{\sim}N(0,1).$

For the parameters, we take $\bm{\theta}^{\circ}\mathrel{\mathop{\mathchar
58\relax}}=\bm{\theta}^{\circ}_{s}\mathrel{\mathop{\mathchar
58\relax}}=(\omega^{\circ},\alpha_{+,1}^{\circ},\alpha_{-,1}^{\circ},\beta_{1}^{\circ},\pi_{1}^{\circ})^{\prime}=(0.046,0.027,0.092,0.843,0)^{\prime}$
(to compute size) and $\bm{\theta}^{\circ}\mathrel{\mathop{\mathchar
58\relax}}=\bm{\theta}^{\circ}_{p}\mathrel{\mathop{\mathchar
58\relax}}=(0.046,0.027,0.092,0.843,0.089)^{\prime}$ (to compute power).333The
data-generating process is taken from Francq and Thieu (2019, Sec. 3.1), who
obtained the parameters in (4) from fitting an APARCH–X model to Boeing
returns. Irrespective of the choice of $\bm{\theta}^{\circ}$, we estimate an
APARCH(1,1) model (i.e., we estimate (4) imposing $\pi_{1}^{\circ}=0$) via
QML. Thus, when $\bm{\theta}^{\circ}=\bm{\theta}^{\circ}_{s}$ we fit a
correctly specified model (yielding size), yet when
$\bm{\theta}^{\circ}=\bm{\theta}^{\circ}_{p}$ the fitted model is misspecified
(yielding power). In the latter case, the exogenous variable serves to
introduce some unmodeled dynamics in the variance equation, with infrequent
outbursts of $z_{t}$ inducing some dependence in the tails of the
residuals.444The second-to-top panel of Figure 6 in Appendix E shows a
representative trajectory of $z_{t}$. Hence, our simulation setup is well-
suited to $\mathcal{P}_{n}^{(D)}$ and $\mathcal{F}_{n}^{(D)}$ and, at the same
time, relevant in practice as exogenous volatility shocks are often
empirically plausible; see Han and Kristensen (2014) for references.

For $v=10$, we generate $\\{Y_{t}\\}_{t=-v+1,\ldots,n}$ from (4). Since
volatility estimates $\widehat{\sigma}_{t}(\widehat{\bm{\theta}})$ may be
imprecise for the first few $t$ due to initialization effects in the variance
equation (see also Hall and Yao, 2003), we discard the first $v$ standardized
residuals and only consider
$\\{\widehat{\varepsilon}_{t}=Y_{t}/\widehat{\sigma}_{t}(\widehat{\bm{\theta}})\\}_{t=1,\ldots,n}$
in our test statistics $\mathcal{P}_{n}^{(D)}$ and $\mathcal{F}_{n}^{(D)}$.

Since we consider $k=\lfloor\varrho n^{0.99}\rfloor$ with
$\varrho\in[0.05,\,0.15]$, the $k$’s are between $[23,70]$ / $[46,139]$ /
$[92,278]$ for $n=500$ / $n=1000$ / $n=2000$. These values may be regarded as
being sufficiently small relative to the sample size for extreme value methods
to be applicable.

For purposes of comparison, we also include the results of a classical
Ljung–Box test (with test statistic denoted by $\operatorname{LB}_{n}^{(D)}$)
based on the first $D$ autocorrelations of the squared residuals. We employ
the corrected Ljung–Box test statistic of Carbon and Francq (2011), which
ensures that $\operatorname{LB}_{n}^{(D)}$ is asymptotically
$\chi_{D}^{2}$-distributed. Note that this correction is specific to APARCH
models estimated via a Gaussian QML estimator.

Figure 1: Rejection frequencies at the 5%-level under the null and alternative
for $\mathcal{P}_{n}^{(D)}$-test (black), $\mathcal{F}_{n}^{(D)}$-test (red)
and $\operatorname{LB}_{n}^{(D)}$-test (green) for $D=5$ and $k=\lfloor\varrho
n^{0.99}\rfloor$ with $\varrho\in[0.05,\,0.15]$. Nominal level of 5% indicated
by the dotted horizontal line. Dashed vertical line indicates value of
$k=\lfloor 0.11\cdot n^{0.99}\rfloor$. Results under the null (alternative)
are for model (4) with $\bm{\theta}^{\circ}=\bm{\theta}^{\circ}_{s}$
($\bm{\theta}^{\circ}=\bm{\theta}^{\circ}_{p}$).

Figure 1 displays the rejection frequencies of all tests at the 5%-level. Both
the $\mathcal{P}_{n}^{(D)}$-test (black) and the $\mathcal{F}_{n}^{(D)}$-test
(red) have almost identical size, which—as the sample size increases—converges
rapidly to the nominal level for all $k$. Larger differences in power between
the two tests only emerge for large $n$. We also see that size and power are
reasonably stable across different choices of $k$, suggesting that test
results are quite robust to the particular value of $k$. This is encouraging
given that extreme value methods can be very sensitive to the choice of the
cutoff. Our approximation $k=\lfloor 0.11\cdot n^{0.99}\rfloor$ to DuMouchel’s
(1983) rule—indicated by the dashed vertical lines in Figure 1—yields good
results in terms of both size and power. Note that for $n=500$, where at first
sight our heuristic does not lead to optimal power, size and power decrease
with increasing $k$, such that the size-corrected power of our proposal for
the choice of $k$ appears very close to optimal.

Finally, we compare our two tests with the $\operatorname{LB}_{n}^{(D)}$-test
(green), which is of course independent of $k$. We find that—despite rejecting
more often under the null—the $\operatorname{LB}_{n}^{(D)}$-test rejects less
often under the alternative. Hence, the size-corrected power of our tests is
even larger than the differences in the right-hand panels of Figure 1 suggest.

###  3.2 Misspecified Innovations

Standard location–scale models as in (1) can only dynamically model the
conditional mean and the conditional variance. However, higher-order features
of the conditional distribution—such as skewness and kurtosis—may sometimes
change as well for real data (Hansen, 1994; Jondeau and Rockinger, 2003; Bali
_et al._ , 2008). Here, we assess how our tests perform relative to a standard
Ljung–Box test in such a setting. To account for estimation effects, we use
the version of the Ljung–Box test based on Theorem 8.2 of Francq and Zakoïan
(2010). Since no confusion can arise, we denote the corresponding test
statistic by $\operatorname{LB}_{n}^{(D)}$, which again has a standard
$\chi_{D}^{2}$-limit.

To carry out the comparison, we consider the following GARCH(1,1) model

$\displaystyle Y_{t}$
$\displaystyle=\sigma_{t}(\bm{\theta}^{\circ})\varepsilon_{t},\qquad\varepsilon_{t}\sim(0,1),$
$\displaystyle\sigma_{t}^{2}(\bm{\theta}^{\circ})$
$\displaystyle=\omega^{\circ}+\alpha^{\circ}Y_{t-1}^{2}+\beta^{\circ}\sigma_{t-1}^{2}(\bm{\theta}^{\circ})$
(5)

with $\bm{\theta}^{\circ}\mathrel{\mathop{\mathchar
58\relax}}=(\omega^{\circ},\alpha^{\circ},\beta^{\circ})^{\prime}=(0.046,0.127,0.843)^{\prime}$.
We assume that $\varepsilon_{t}$ is distributed according to Hansen’s (1994)
skewed $t$-distribution (written: $\varepsilon_{t}\sim st_{\lambda,\eta}$),
i.e., it has density

$f_{\lambda,\eta}(x)=bc\Bigg{[}1+\frac{1}{\eta-2}\Big{(}\frac{bx+a}{1+\operatorname{sgn}(x+a/b)\lambda}\Big{)}^{2}\Bigg{]}^{-\frac{\eta+1}{2}},\qquad
x\in(-\infty,\infty),\ \eta\in(2,\infty),\ \lambda\in(-1,1),$

where

$a=4\lambda c\frac{\eta-2}{\eta-1},\qquad b^{2}=1+3\lambda^{2}-a^{2},\qquad
c=\frac{\Gamma((\eta+1)/2)}{\sqrt{\pi(\eta-2)}\Gamma(\eta/2)}.$

For $\lambda=0$, $f_{\lambda,\eta}(\cdot)$ reduces to the standard
$t$-distribution with degrees of freedom equal to $\eta$. Jondeau and
Rockinger (2003, Appendix A) show that $f_{\lambda,\eta}(\cdot)$ has zero mean
and unit variance. In their equations (2) and (3), they also derive formulas
for the skewness and kurtosis as functions of $\lambda$ and $\eta$.

To introduce time variation in the skewness and kurtosis, we let
$\lambda=\lambda_{t}$ and $\eta=\eta_{t}$ vary over time. To do so, we use
Model M4 of Jondeau and Rockinger (2003, Table 1). That is, we let

$\displaystyle\widetilde{\eta}_{t}$
$\displaystyle=a_{1}+b_{1}Y_{t-1}+c_{1}\widetilde{\eta}_{t-1},$
$\displaystyle\widetilde{\lambda}_{t}$
$\displaystyle=a_{2}+b_{2}Y_{t-1}+c_{2}\widetilde{\lambda}_{t-1},$

and restrain $\eta_{t}$ and $\lambda_{t}$ to the intervals $(2,30)$ and
$(-1,1)$ by the logistic transformations
$\eta_{t}=g_{(2,30)}(\widetilde{\eta}_{t})$ and
$\lambda_{t}=g_{(-1,1)}(\widetilde{\lambda}_{t})$, where
$g_{(L,U)}(x)=L+(U-L)/[1+\exp(-x)]$. Under the null, we set
$(a_{1},b_{1},c_{1})^{\prime}=(-3,0,0)^{\prime}$ and
$(a_{2},b_{2},c_{2})^{\prime}=(-1,0,0)^{\prime}$, leading to constant
$\eta_{t}\equiv 28.67...$ and $\lambda_{t}\equiv 0.45...$. Thus, the
$\varepsilon_{t}$ are serially independent and $H_{0}$ is satisfied. Under the
alternative, we introduce time-variation in the skewness and kurtosis by using
non-zero $b_{i}$ and $c_{i}$ ($i=1,2$). Specifically, we let
$(a_{1},b_{1},c_{1})^{\prime}=(-3,-6,0.6)^{\prime}$ and
$(a_{2},b_{2},c_{2})^{\prime}=(-1,-2,0.6)^{\prime}$. Thus, while
$\varepsilon_{t}\sim(0,1)$, the $\varepsilon_{t}$ have time-varying skewness
and kurtosis and, thus, are not i.i.d., as required under the null.

To estimate the model parameters, we again use Gaussian QML. Escanciano (2009,
Theorem 2) shows that the QML estimator is asymptotically normal under both
the null and the alternative.

Figure 2: Rejection frequencies at the 5%-level under the null and alternative
for $\mathcal{P}_{n}^{(D)}$-test (black), $\mathcal{F}_{n}^{(D)}$-test (red)
and $\operatorname{LB}_{n}^{(D)}$-test (green) for $D=5$ and $k=\lfloor\varrho
n^{0.99}\rfloor$ with $\varrho\in[0.05,\,0.15]$. Nominal level of 5% indicated
by the dotted horizontal line. Dashed vertical line indicates value of
$k=\lfloor 0.11\cdot n^{0.99}\rfloor$. Results under the null (alternative)
are for model (5) with $(a_{1},b_{1},c_{1})^{\prime}=(-3,0,0)^{\prime}$ and
$(a_{2},b_{2},c_{2})^{\prime}=(-1,0,0)^{\prime}$
($(a_{1},b_{1},c_{1})^{\prime}=(-3,-6,0.6)^{\prime}$ and
$(a_{2},b_{2},c_{2})^{\prime}=(-1,-2,0.6)^{\prime}$).

Figure 2 displays the results. For all tests, size is very close to the
nominal level, such that power is directly comparable. While for the
$\operatorname{LB}_{n}^{(D)}$-test power is low and even decreases below the
nominal level for $n=2000$, the power of our tests is high and approaches one.
The huge difference in power may be explained as follows. The
$\operatorname{LB}_{n}^{(D)}$-test only looks at the autocorrelations of the
squared residuals $\widehat{\varepsilon}_{t}^{2}$. However, the
$\varepsilon_{t}^{2}$ satisfy
$\operatorname{E}[\varepsilon_{t}^{2}\mid\mathcal{F}_{t-1}]=1$, such
that—despite being serially dependent—they have zero autocorrelations. Hence,
the power of the $\operatorname{LB}_{n}^{(D)}$-test is seriously impaired.
Comparing the $\mathcal{P}_{n}^{(D)}$\- and the $\mathcal{F}_{n}^{(D)}$-test,
we find that the latter has higher power for small $k$, yet the difference
becomes negligible for larger $k$. For these two tests, the results are now
more sensitive to the choice of $k$ than in Figure 1. Yet, our suggestion
$k=\lfloor 0.11\cdot n^{0.99}\rfloor$ once again leads to good size and power.

To summarize the simulation results, we find that size and power of our tests
may depend somewhat on the choice of $k$. However, setting $k=\lfloor
0.11\cdot n^{0.99}\rfloor$, which approximates DuMouchel’s (1983) rule, gives
good results—irrespective of the type of misspecification. Although our tests
work well for $k=\lfloor 0.11\cdot n^{0.99}\rfloor$, in practice we recommend
to apply the tests for several values of $k$ as a ‘robustness check’ (see
Figure 5 in the empirical application for an example), much like results for
standard Ljung–Box tests are also routinely reported for several lags. Our
tests also depend on the number of included lags and, although we find $D=5$
to lead to good results in Appendix E, we again recommend—if possible—to
report results for different $D$. We generally find that the
$\mathcal{F}_{n}^{(D)}$-test is more powerful than the
$\mathcal{P}_{n}^{(D)}$-test, yet both have markedly higher power than the
$\operatorname{LB}_{n}^{(D)}$-test for the types of misspecifications
considered here. As we have argued, these misspecifications may be empirically
plausible, such that our tests should have substantial merit in practice. We
investigate this next.

##  4 Diagnosing S&P 500 Constituents

Consider the 500 components of the Standard & Poor’s 500 (S&P 500) as of
31/5/2021. We apply the $\mathcal{F}_{n}^{(D)}$\- and
$\operatorname{LB}_{n}^{(D)}$-test to the log-returns of each of the
constituents. We do so for the recommended values $D=5$, $k=\lfloor 0.11\cdot
n^{0.99}\rfloor$ and $\iota=0.1$. For brevity, we do not report results for
$\mathcal{P}_{n}^{(D)}$, which showed slightly inferior performance to
$\mathcal{F}_{n}^{(D)}$ in the simulations. We consider the 495 components of
the S&P 500 for which we have complete data for our sample period from
1/1/2010 to 31/12/2019. We split the 10-year sample into an ‘in-sample’ period
of 8 years (used for specification testing) and an ‘out-of-sample’ period of 2
years (used for backtesting VaR forecasts). The goal of our analysis is
twofold. First, we want to illustrate that (similarly as in the simulations)
cases may arise where $\mathcal{F}_{n}^{(D)}$ warrants a rejection, but
$\operatorname{LB}_{n}^{(D)}$ does not. Our second goal is to show that in
these cases, a rejection by $\mathcal{F}_{n}^{(D)}$ is not spurious, but—on
the contrary—indicative of ‘out-of-sample’ forecast failure of the model.

Specifically, we fit an APARCH(1,1) model without covariates as in (4) (i.e.,
with $\pi_{1}^{\circ}=0$) based on the ‘in-sample’ period, and use the fitted
model for one-step-ahead VaR forecasting in the ‘out-of-sample’ part. The VaR
at level $\theta$ is the loss in $(t+1)$ that is only exceeded with
probability $\theta$ given the current state of the market (embodied by some
information set $\mathcal{F}_{t}$). Formally, $\operatorname{VaR}_{t}$ is the
$\mathcal{F}_{t}$-measurable random variable satisfying

$\operatorname{P}\\{Y_{t+1}\leq\operatorname{VaR}_{t}\mid\mathcal{F}_{t}\\}=\theta.$

It immediately follows from the multiplicative structure in (4) that
$\operatorname{VaR}_{t}=\sigma_{t+1}\operatorname{VaR}_{\varepsilon}$, where
$\operatorname{VaR}_{\varepsilon}$ is the (unconditional) $\theta$-quantile of
the $\varepsilon_{t}$. Thus, we can easily compute the ‘out-of-sample’ VaR
forecasts from the volatility forecasts of the fitted model (say,
$\widehat{\sigma}_{t+1}$) and an estimate of
$\operatorname{VaR}_{\varepsilon}$ from the standardized residuals of the ‘in-
sample’ period. Then, we backtest the VaR forecasts using the DQ test of Engle
and Manganelli (2004).555We use the DQ test as implemented in the GAS package
(Ardia _et al._ , 2020) with 4 lags.

$\operatorname{LB}_{n}^{(D)}$ | $\mathcal{F}_{n}^{(D)}$ | Total | DQ-test
---|---|---|---
0 | 0 | 181 | 60
0 | 1 | 122 | 69
1 | 0 | 113 | 31
1 | 1 | 79 | 38
Table 2: Results of specification and DQ tests carried out at 5%-level.

As we run two specification tests, there are four potential outcomes because
the $\operatorname{LB}_{n}^{(D)}$\- and the $\mathcal{F}_{n}^{(D)}$-test can
either accept (indicated by a 0 in Table 2) or reject (indicated by a 1 in
Table 2). The column ‘Total’ in Table 2 indicates the total number of times
each of the four cases occurs for the 495 S&P 500 stocks. In 181 out of 495
cases, none of the tests reject. For these 181 stocks, the ‘out-of-sample’ VaR
forecasts are rejected 60 times by the DQ-test. One reason for this rather
large number of rejections may be the long out-of-sample period of 2 years.
Typically, models are re-estimated at least quarterly (Ardia and Hoogerheide,
2014), yet we only fitted the model once based on the ‘in-sample’ data.
Nonetheless, the rather long out-of-sample period of 2 years (yielding roughly
500 daily observations) is needed for the DQ-test to have reasonable power.
Table 2 further shows that for 122 stocks, $\mathcal{F}_{n}^{(D)}$ but not
$\operatorname{LB}_{n}^{(D)}$ leads to a rejection. The fact that for these
122 stocks the DQ test rejects in 69 cases suggests some unmodeled dynamics in
these stocks, that were picked up by our $\mathcal{F}_{n}^{(D)}$-test, yet not
the Ljung–Box test. When both tests agree in rejecting a model (which happens
in 79 cases), the model produces inadequate VaR forecasts 38 times. For the
remaining 113 stocks, only $\operatorname{LB}_{n}^{(D)}$ rejects the null, yet
the DQ-test rejects only 31 times.

Figure 3: Top: Cisco returns ($Y_{t}$) in black and estimated volatility
($\widehat{\sigma}_{t}$) in red. Bottom: Standardized residuals
($\widehat{\varepsilon}_{t}$).

Overall, we find that in most of the 198 cases when the DQ-test rejects, at
least one of the specification test signals problems in advance. Of these 138
cases, 107 cases are identified by our $\mathcal{F}_{n}^{(D)}$-test, yet the
$\operatorname{LB}_{n}^{(D)}$-test rejects in only 69 of these cases. This
suggests that our extreme value-based tests provide complementary information
to more classical specification tests.

Figure 4: Top: Estimates $\widehat{\rho}_{n}^{(d)}$ of the autocorrelation at
lag $d$ for the squared standardized residuals; bottom: Estimates
$\widehat{\Lambda}_{n}^{(d)}(1,1)$ for the standardized residuals
$\widehat{\varepsilon}_{t}$.

Next, we illustrate one of the cases where $\operatorname{LB}_{n}^{(D)}$ does
not reject, yet $\mathcal{F}_{n}^{(D)}$ does. We do so for Cisco log-returns,
shown in the top panel of Figure 3. Volatility estimates are superimposed in
red. During our in-sample period from 2010-2017 there are only mild signs of
volatility clustering. A cursory inspection of the residuals in the lower
panel suggests that the model successfully captures this. This is also
confirmed by the plots of the squared residual autocorrelations in the top
part of Figure 4, which shows that autocorrelations up to lag 10 are
insignificant. However, focusing on the extremes, we find that the
$\widehat{\Lambda}_{n}^{(d)}(1,1)$-estimates in Figure 4 show strong
(positive) serial extremal dependence at all lags. This may be due to
occasional exogenous shocks to the share price. For instance, the largest drop
on 11/11/2010 followed a profit warning of Cisco. This and other outliers
apparent from the plot of the $Y_{t}$ in Figure 3 seem to have been
unpredictable from past information in $\mathcal{F}_{t-1}$, thus violating the
modeling paradigm in (1). Similarly as for model (4) in the simulations, such
exogenous shocks may have caused the serial extremal dependence in the
standardized residuals.

Figure 5: $\mathcal{F}_{n}^{(D)}$ (top) and $\mathcal{P}_{n}^{(D)}$ (top) for
Cisco returns as a function of $k$. Horizontal red lines indicate respective
5%-critical values.

The results for the S&P 500 components of this section are for $k=\lfloor
0.11\cdot n^{0.99}\rfloor$. For a single time series one may want to assess
the robustness of the result with respect to $k$. To do so, it is common in
extreme value theory to plot the results as a function of $k$. We illustrate
this for the Cisco returns in Figure 5. There, the test statistics
$\mathcal{F}_{n}^{(D)}$ and $\mathcal{P}_{n}^{(D)}$ are plotted as functions
of $k$. As in the simulations, the test result is quite robust to $k$ with any
reasonable choice leading to a rejection.

##  5 Conclusion

We propose two specification tests based on the tail copula of time series
residuals. By relying on the tail copula, our tests direct the power to any
remaining serial extremal dependence and, thus, complement the evidence by
more classical Ljung–Box-type tests. The test statistics are easy to compute
and their limiting distributions are free of nuisance parameters. Thus, our
tests are simple to implement and enjoy broad applicability. This contrasts
with more classical diagnostic tests, which require involved bootstrap
procedures for valid inference and/or depend on the specific estimator used to
fit the model, hence, limiting practical applications. Simulations demonstrate
the good size of our proposals. They also show that when a misspecified model
captures well the serial dependence in the body of the distribution but not in
the tail, our tests have higher power than Ljung–Box tests. The misspecified
models considered in the simulations are realistic in that they display
exogenous shocks in the volatility equation or unmodeled higher-order dynamics
in the innovations, both of which being empirically relevant sources of
misspecification (Hansen, 1994; Han and Kristensen, 2014). We also exemplify
the better detection properties of our tests on S&P 500 constituents, where a
rejection of our test is a more reliable indicator of poor out-of-sample risk
predictions. Thus, our tests help to identify unsuitable risk forecasting
models in advance, which is economically desirable under the Basel framework
(Basel Committee on Banking Supervision, 2019). Consequently, by focusing on
the extremes, our tests are useful complements to standard specification
tests, where valuable information on the serial extremal dependence may be
‘washed-out’.

SUPPLEMENTARY MATERIAL

Title:

Appendix containing verification of Assumption 3 for APARCH and ARMA–GARCH
models (Appendices A and B), the proofs of Theorems 1–2 (Appendices C and D),
and additional simulations (Appendix E). (pdf-file)

R code:

File containing the R code to reproduce the simulation study and the empirical
application. (zipped file)

## References

  * Anderson and Darling (1952) Anderson TW, Darling DA. 1952. Asymptotic theory of certain ”goodness of fit” criteria based on stochastic processes. _The Annals of Mathematical Statistics_ 23: 193–212.
  * Ardia _et al._ (2020) Ardia D, Boudt K, Catania L. 2020. _Generalized Autoregressive Score Models in R: The GAS Package_. R package version 0.3.3.
  * Ardia and Hoogerheide (2014) Ardia D, Hoogerheide LF. 2014. GARCH models for daily stock returns: Impact of estimation frequency on value-at-risk and expected shortfall forecasts. _Economics Letters_ 123: 187–190.
  * Bali _et al._ (2008) Bali TG, Mo H, Tang Y. 2008. The role of autoregressive conditional skewness and kurtosis in the estimation of conditional VaR. _Journal of Banking & Finance_ 32: 269–282.
  * Basel Committee on Banking Supervision (2019) Basel Committee on Banking Supervision. 2019. _Basel Framework_. Basel: Bank for International Settlements (http://www.bis.org/basel_framework/index.htm?export=pdf).
  * Berkes and Horváth (2004) Berkes I, Horváth L. 2004. The efficiency of the estimators of the parameters in GARCH processes. _Annals of Statistics_ 32: 633–655.
  * Berkes _et al._ (2003) Berkes I, Horváth L, Kokoszka P. 2003. Asymptotics for GARCH squared residual correlations. _Econometric Theory_ 19: 515–540.
  * Bickel and Wichura (1971) Bickel PJ, Wichura MJ. 1971. Convergence criteria for multiparameter stochastic processes and some applications. _The Annals of Statistics_ 42: 1656–1670.
  * Bollerslev (1986) Bollerslev T. 1986. Generalized autoregressive conditional heteroskedasticity. _Journal of Econometrics_ 31: 307–327.
  * Box and Pierce (1970) Box GEP, Pierce DA. 1970. Distribution of residual autocorrelations in autoregressive-integreted moving average time series models. _Journal of the American Statististical Association_ 65: 1509–1526.
  * Bücher _et al._ (2015) Bücher A, Jäschke S, Wied D. 2015. Nonparametric tests for constant tail dependence with an application to energy and finance. _Journal of Econometrics_ 187: 154–168.
  * Carbon and Francq (2011) Carbon M, Francq C. 2011. Portmanteau goodness-of-fit test for asymmetric power GARCH models. _Austrian Journal of Statistics_ 40: 55–64.
  * Chan _et al._ (2007) Chan NH, Deng SJ, Peng L, Xia Z. 2007. Interval estimation of value-at-risk based on GARCH models with heavy-tailed innovations. _Journal of Econometrics_ 137: 556–576.
  * Chavez-Demoulin _et al._ (2014) Chavez-Demoulin V, Embrechts P, Sardy S. 2014. Extreme-quantile tracking for financial time series. _Journal of Econometrics_ 181: 44–52.
  * Davidson (1994) Davidson J. 1994. _Stochastic Limit Theory_. Oxford: Oxford University Press.
  * Davis and Mikosch (2009) Davis RA, Mikosch T. 2009. The extremogram: A correlogram for extreme events. _Bernoulli_ 15: 977–1009.
  * Davis _et al._ (2012) Davis RA, Mikosch T, Cribben I. 2012. Towards estimating extremal serial dependence via the bootstrapped extremogram. _Journal of Econometrics_ 170: 142–152.
  * Davis _et al._ (2013) Davis RA, Mikosch T, Zhao Y. 2013. Measures of serial extremal dependence and their estimation. _Stochastic Processes and their Applications_ 123: 2575–2602.
  * de Haan and Ferreira (2006) de Haan L, Ferreira A. 2006. _Extreme Value Theory_. New York: Springer.
  * Ding _et al._ (1993) Ding Z, Granger CWJ, Engle RF. 1993. A long memory property of stock market returns and a new model. _Journal of Empirical Finance_ 1: 83–106.
  * DuMouchel (1983) DuMouchel WH. 1983. Estimating the stable index $\alpha$ in order to measure tail thickness: A critique. _The Annals of Statistics_ 11: 1019–1031.
  * Engle (1982) Engle RF. 1982. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. _Econometrica_ 50: 987–1007.
  * Engle and Manganelli (2004) Engle RF, Manganelli S. 2004. CAViaR: Conditional autoregressive value at risk by regression quantiles. _Journal of Business & Economic Statistics_ 22: 367–381.
  * Escanciano (2009) Escanciano JC. 2009. Quasi-maximum likelihood estimation of semi-strong GARCH models. _Econometric Theory_ 25: 561–570.
  * Fisher and Gallagher (2012) Fisher TJ, Gallagher CM. 2012. New weighted Portmanteau statistics for time series goodness of fit testing. _Journal of the American Statistical Association_ 107: 777–787.
  * Francq _et al._ (2006) Francq C, Roy R, Zakoïan JM. 2006. Diagnostic checking in ARMA models with uncorrelated errors. _Journal of the American Statistical Association_ 100: 532–544.
  * Francq and Thieu (2019) Francq C, Thieu LQ. 2019. QML inference for volatility models with covariates. _Econometric Theory_ 35: 37–72.
  * Francq and Zakoïan (2004) Francq C, Zakoïan JM. 2004. Maximum likelihood estimation of pure GARCH and ARMA–GARCH processes. _Bernoulli_ 10: 605–637.
  * Francq and Zakoïan (2010) Francq C, Zakoïan JM. 2010. _GARCH Models: Structure, Statistical Inference and Financial Applications_. Chichester: Wiley.
  * Ghalanos (2020) Ghalanos A. 2020. _rugarch: Univariate GARCH models_. R package version 1.4-2.
  * Glosten _et al._ (1993) Glosten LR, Jagannathan R, Runkle DE. 1993. On the relation between the expected value and the volatility of the nominal excess return on stocks. _The Journal of Finance_ 48: 1779–1801.
  * Hall and Yao (2003) Hall P, Yao Q. 2003. Inference in ARCH and GARCH models with heavy-tailed errors. _Econometrica_ 71: 285–317.
  * Han and Kristensen (2014) Han H, Kristensen D. 2014. Asymptotic theory for the QMLE in GARCH-X models with stationary and nonstationary covariates. _Journal of Business & Economic Statistics_ 32: 416–429.
  * Hansen (1994) Hansen BE. 1994. Autoregressive conditional density estimation. _International Economic Review_ 35: 705–730.
  * Heffernan (2000) Heffernan JE. 2000. A directory of coefficients of tail dependence. _Extremes_ 3: 279–290.
  * Hidalgo and Zaffaroni (2007) Hidalgo J, Zaffaroni P. 2007. A goodness-of-fit test for ARCH($\infty$) models. _Journal of Econometrics_ 141: 973–1013.
  * Hill (2011a) Hill JB. 2011a. Extremal memory of stochastic volatility with an application to tail shape inference. _Journal of Statistical Planning and Inference_ 141: 663–676.
  * Hill (2011b) Hill JB. 2011b. Tail and nontail memory with applications to extreme value and robust statistics. _Econometric Theory_ 27: 844–884.
  * Hill (2015) Hill JB. 2015. Tail index estimation for a filtered dependent time series. _Statistica Sinica_ 25: 609–629.
  * Hoga (2019) Hoga Y. 2019. Confidence intervals for conditional tail risk measures in ARMA–GARCH models. _Journal of Business & Economic Statistics_ 37: 613–624.
  * Jondeau and Rockinger (2003) Jondeau E, Rockinger M. 2003. Conditional volatility, skewness, and kurtosis: Existence, persistence, and comovements. _Journal of Economic Dynamics & Control_ 27: 1699–1737.
  * Kim and Lee (2016) Kim L, Lee S. 2016. On the tail index inference for heavy-tailed GARCH-type innovations. _Annals of the Institute of Statistical Mathematics_ 68: 237–267.
  * Li and Mak (1994) Li WK, Mak TK. 1994. On the squared residual autocorrelations in non-linear time series with conditional heteroskedasticity. _Journal of Time Series Analysis_ 15: 627–636.
  * Ling and Li (1997) Ling S, Li WK. 1997. On fractionally integrated autoregressive moving-average time series models with conditional heteroscedasticity. _Journal of the American Statistical Association_ 92: 1184–1194.
  * Ling and McAleer (2002) Ling S, McAleer M. 2002. Necessary and sufficient conditions for the GARCH($r,s$) and Asymmetric Power GARCH($r,s$) models. _Econometric Theory_ 18: 722–729.
  * Ljung and Box (1978) Ljung GM, Box GEP. 1978. On a measure of lack of fit in time series models. _Biometrika_ 65: 297–303.
  * Mancini and Trojani (2011) Mancini L, Trojani F. 2011. Robust value at risk prediction. _Journal of Financial Econometrics_ 9: 281–313.
  * McNeil and Frey (2000) McNeil AJ, Frey R. 2000. Estimation of tail-related risk measures for heteroscedastic financial time series: An extreme value approach. _Journal of Empirical Finance_ 7: 271–300.
  * Pan _et al._ (2008) Pan J, Wang H, Tong H. 2008. Estimation and tests for power-transformed and threshold GARCH models. _Journal of Econometrics_ 142: 352–378.
  * Quintos _et al._ (2001) Quintos C, Fan Z, Phillips PCB. 2001. Structural change tests in tail behaviour and the Asian crisis. _Review of Economic Studies_ 68: 633–663.
  * R Core Team (2020) R Core Team. 2020. _R: A Language and Environment for Statistical Computing_. R Foundation for Statistical Computing, Vienna, Austria.
  * Resnick and Stǎricǎ (1997) Resnick S, Stǎricǎ C. 1997. Smoothing the Hill estimator. _Advances in Applied Probability_ 29: 271–293.
  * Schmidt and Stadtmüller (2006) Schmidt R, Stadtmüller U. 2006. Nonparametric estimation of tail dependence. _Scandinavian Journal of Statistics_ 33: 307–335.
  * Shao (1993) Shao QM. 1993. Almost sure invariance principles for mixing sequences of random variables. _Stochastic Processes and their Applications_ 48: 319–334.
  * Sibuya (1960) Sibuya M. 1960. Bivariate extreme statistics. _Annals of the Institute of Statistical Mathematics_ 11: 195–210.
  * Su _et al._ (2021+) Su Q, Qin Z, Peng L, Qin G. 2021+. Efficiently backtesting conditional value-at-risk and conditional expected shortfall. _Journal of the American Statistical Association(https://doi.org/10.1080/01621459.2020.1763804)_ : 1–12.
  * White (2001) White H. 2001. _Asymptotic Theory for Econometricians_. San Diego: Academic Press, First edn.
  * Zakoïan (1994) Zakoïan JM. 1994. Threshold heteroskedastic models. _Journal of Economic Dynamics and Control_ 18: 931–955.

##  Appendix A APARCH and ARMA–GARCH Models

We first verify Assumption 3 for the asymmetric power ARCH (APARCH) model of
Ding _et al._ (1993); see also Pan _et al._ (2008). To that end, consider

$\displaystyle Y_{t}$
$\displaystyle=\sigma_{t}(\bm{\theta}^{\circ})\varepsilon_{t},\qquad\varepsilon_{t}\overset{\text{i.i.d.}}{\sim}(0,1),$
$\displaystyle\sigma_{t}^{\delta^{\circ}}(\bm{\theta}^{\circ})$
$\displaystyle=\omega^{\circ}+\sum_{j=1}^{p}\left\\{\alpha_{+,j}^{\circ}(Y_{t-j})_{+}^{\delta^{\circ}}+\alpha_{-,j}^{\circ}(Y_{t-j})_{-}^{\delta^{\circ}}\right\\}+\sum_{j=1}^{q}\beta_{j}^{\circ}\sigma_{t-j}^{\delta^{\circ}}(\bm{\theta}^{\circ}),$
(A.1)

where $y_{+}=\max\\{y,0\\}$, $y_{-}=\max\\{-y,0\\}$, $\omega^{\circ}>0$,
$\delta^{\circ}>0$, and $\alpha_{+,j}^{\circ}$, $\alpha_{-,j}^{\circ}$ and
$\beta_{j}^{\circ}$ are non-negative. Note that
$\mu_{t}(\bm{\theta}^{\circ})\equiv 0$ here. As the likelihood is typically
quite flat in the $\delta^{\circ}$-direction, we follow Francq and Thieu
(2019) and assume the power $\delta^{\circ}$ to be a fixed known
constant.666Francq and Thieu (2019, Sec. 2.5) provide some theoretically-
backed guidance on the choice of $\delta^{\circ}$ in practice. Thus, the
vector of the true unknown parameters is
$\bm{\theta}^{\circ}=(\omega^{\circ},\alpha_{+,1}^{\circ},\ldots,\alpha_{+,p}^{\circ},\alpha_{-,1}^{\circ},\ldots,\alpha_{-,p}^{\circ},\beta_{1}^{\circ},\ldots,\beta_{q}^{\circ})^{\prime}$.
A generic parameter vector from the parameter space
$\bm{\varTheta}\subset(0,\infty)\times[0,\infty)^{d-1}$ ($d=2p+q+1$) is
denoted by

$\bm{\theta}=(\omega,\alpha_{+,1},\ldots,\alpha_{+,p},\alpha_{-,1},\ldots,\alpha_{-,p},\beta_{1},\ldots,\beta_{q})^{\prime}.$

The APARCH($p,q$) model in (A.1) nests several popular GARCH variants. We
obtain Zakoïan’s (1994) TARCH model for $\delta^{\circ}=1$, and the GJR–GARCH
of Glosten _et al._ (1993) for $\delta^{\circ}=2$. When $\delta^{\circ}=2$ and
$\alpha_{+,j}^{\circ}=\alpha_{-,j}^{\circ}$, (A.1) is the classic GARCH
specification of Bollerslev (1986).

As volatility depends on the infinite past in the APARCH model, we rely on the
truncated recursion

$\widehat{\sigma}_{t}^{\delta^{\circ}}(\bm{\theta})=\omega+\sum_{j=1}^{p}\left\\{\alpha_{+,j}(Y_{t-j})_{+}^{\delta^{\circ}}+\alpha_{-,j}(Y_{t-j})_{-}^{\delta^{\circ}}\right\\}+\sum_{j=1}^{q}\beta_{j}\widehat{\sigma}_{t-j}^{\delta^{\circ}}(\bm{\theta}),\qquad
t\geq 1,$ (A.2)

with initial values $Y_{1-p}=\ldots=Y_{0}=0$ and
$\widehat{\sigma}_{1-q}^{\delta^{\circ}}(\bm{\theta})=\ldots=\widehat{\sigma}_{0}^{\delta^{\circ}}(\bm{\theta})=0$.

######  Theorem 3.

Suppose for the APARCH model in (A.1) that

  1. (a)

$\min\\{\omega^{\circ},\alpha_{+,1}^{\circ},\ldots,\alpha_{+,p}^{\circ},\alpha_{-,1}^{\circ},\ldots,\alpha_{-,p}^{\circ},\beta_{1}^{\circ},\ldots,\beta_{q}^{\circ}\\}>0$,
$\sum_{i=1}^{q}\beta_{i}^{\circ}<1$;

  2. (b)

the top Lyapunov exponent $\gamma$ (Francq and Thieu, 2019, p. 41) satisfies
$\gamma<0$.

Suppose further that Assumption 2 is satisfied. Then, Assumption 3 holds for
$\widehat{\varepsilon}_{t}(\bm{\theta})=Y_{t}/\widehat{\sigma}_{t}(\bm{\theta})$
with $\widehat{\sigma}_{t}(\bm{\theta})$ defined in (A.2).

Appendix B contains the proof of Theorem 3. Condition (b) ensures strict
stationarity of the APARCH model, while (a) bounds the true parameters away
from the boundary of the parameter space to guarantee the approximability of
the innovations.

Next, we consider the ARMA–GARCH model

$Y_{t}=\sum_{j=1}^{\overline{p}}\phi_{j}^{\circ}Y_{t-j}+X_{t}-\sum_{j=1}^{\overline{q}}\vartheta_{j}^{\circ}X_{t-j},$
(A.3)

where the $X_{t}$ follow the GARCH model

$\displaystyle X_{t}$
$\displaystyle=\sigma_{t}(\bm{\theta}^{\circ})\varepsilon_{t},\qquad\varepsilon_{t}\overset{\text{i.i.d.}}{\sim}(0,1),$
$\displaystyle\sigma_{t}^{2}(\bm{\theta}^{\circ})$
$\displaystyle=\omega^{\circ}+\sum_{j=1}^{p}\alpha_{j}^{\circ}X_{t-j}^{2}+\sum_{j=1}^{q}\beta_{j}^{\circ}\sigma_{t-j}^{2}(\bm{\theta}^{\circ}).$
(A.4)

Here,
$\bm{\theta}^{\circ}=(\phi_{1}^{\circ},\ldots,\phi_{\overline{p}}^{\circ},\vartheta_{1}^{\circ},\ldots,\vartheta_{\overline{q}}^{\circ},\omega^{\circ},\alpha_{1}^{\circ},\ldots,\alpha_{p}^{\circ},\beta_{1}^{\circ},\ldots,\beta_{q}^{\circ})^{\prime}$
is the true parameter vector contained is some parameter space
$\bm{\varTheta}=\bm{\varTheta}_{\operatorname{ARMA}}\times\bm{\varTheta}_{\operatorname{GARCH}}$,
where
$\bm{\varTheta}_{\operatorname{ARMA}}\subset\mathbb{R}^{\overline{p}+\overline{q}}$
and
$\bm{\varTheta}_{\operatorname{GARCH}}\subset(0,\infty)\times[0,\infty)^{p+q}$.
One can show that the model can be rewritten in the form (1) with
$\mu_{t}(\bm{\theta}^{\circ})=\sum_{j=1}^{\overline{p}}\phi_{j}^{\circ}Y_{t-j}-\sum_{j=1}^{\overline{q}}\vartheta_{j}^{\circ}[Y_{t-j}-\mu_{t-j}(\bm{\theta}^{\circ})]$.
We estimate the GARCH errors for a generic parameter vector
$\bm{\theta}\in\bm{\varTheta}$ via

$\widehat{X}_{t}(\bm{\theta})=Y_{t}-\sum_{j=1}^{\overline{p}}\phi_{j}Y_{t-j}+\sum_{j=1}^{\overline{q}}\vartheta_{j}\widehat{X}_{t-j}(\bm{\theta}),$

where we artificially set $\widehat{X}_{t}(\bm{\theta})=Y_{t}=0$ for $t\leq
0$. Based on the $\widehat{X}_{t}(\bm{\theta})$, we approximate volatility by
the truncated recursion

$\widehat{\sigma}_{t}^{2}(\bm{\theta})=\omega+\sum_{j=1}^{p}\alpha_{j}\widehat{X}_{t-j}^{2}(\bm{\theta})+\sum_{j=1}^{q}\beta_{j}\widehat{\sigma}_{t-j}^{2}(\bm{\theta}),$
(A.5)

where $\widehat{\sigma}_{t}^{2}(\bm{\theta})=0$ for $t\leq 0$.

######  Theorem 4.

Suppose for the ARMA–GARCH model in (A.3) that

  * (a)

the polynomials $\phi(z)=1-\sum_{j=1}^{\overline{p}}\phi_{j}^{\circ}z^{j}$ and
$\vartheta(z)=1-\sum_{j=1}^{\overline{q}}\vartheta_{j}^{\circ}z^{j}$ have no
common roots and no roots on the unit circle;

  * (b)

the GARCH parameters satisfy
$\sum_{j=1}^{p}\alpha_{j}^{\circ}+\sum_{j=1}^{q}\beta_{j}^{\circ}<1$;

  * (c)

$\operatorname{E}|X_{t}|^{2+\delta}<\infty$ for some $\delta>0$.

Suppose further that Assumption 2 is satisfied. Then, Assumption 3 holds for
$\widehat{\varepsilon}_{t}(\bm{\theta})=\widehat{X}_{t}(\bm{\theta})/\widehat{\sigma}_{t}(\bm{\theta})$
with $\widehat{\sigma}_{t}(\bm{\theta})$ defined in (A.5).

The proof of Theorem 4 is also in Appendix B. Assumption (a) is a standard
stationarity, invertibility and identifiability condition for model (A.3).
Assumption (b) ensures a strictly stationary solution to the GARCH recurrence
in (A.4) with $\operatorname{E}|X_{t}|^{2}<\infty$. Thus, (c) is only a mild
additional requirement, for which Theorem 2.1 in Ling and McAleer (2002)
provides necessary and sufficient conditions.

##  Appendix B Proofs of Theorems 3 and 4

All $o_{(\operatorname{P})}$\- and $O_{(\operatorname{P})}$-symbols in
Appendices B–D are to be understood with respect to $n\to\infty$.

###### Proof of Theorem 3:.

To avoid superscripts, we put
$h_{t}(\bm{\theta})=\sigma_{t}^{\delta^{\circ}}(\bm{\theta})$,
$h_{t}=h_{t}(\bm{\theta}^{\circ})=\sigma_{t}^{\delta^{\circ}}(\bm{\theta}^{\circ})$
and
$\widehat{h}_{t}(\bm{\theta})=\widehat{\sigma}_{t}^{\delta^{\circ}}(\bm{\theta})$,
where $\widehat{\sigma}_{t}^{\delta^{\circ}}(\bm{\theta})$ is defined in (A.2)
in the main paper. Hence, $\widehat{h}_{t}(\bm{\theta})$ satisfies

$\widehat{h}_{t}(\bm{\theta})=\omega+\sum_{j=1}^{p}\left\\{\alpha_{+,j}(Y_{t-j})_{+}^{\delta^{\circ}}+\alpha_{-,j}(Y_{t-j})_{-}^{\delta^{\circ}}\right\\}+\sum_{j=1}^{q}\beta_{j}\widehat{h}_{t-j}(\bm{\theta}),\qquad
t\geq 1,$

for initial values $Y_{1-p}=\ldots=Y_{0}=0$ and
$\widehat{h}_{1-q}(\bm{\theta})=\ldots=\widehat{h}_{0}(\bm{\theta})=0$. Thus,
while $h_{t}(\bm{\theta})$ denotes the true volatility (raised to the power of
$\delta^{\circ}$) if $\bm{\theta}$ were the true parameter vector,
$\widehat{h}_{t}(\bm{\theta})$ approximates $h_{t}(\bm{\theta})$ using
artificial initial values. Recalling that $h_{t}=h_{t}(\bm{\theta}^{\circ})$,
write

$\displaystyle\widehat{\varepsilon}_{t}(\bm{\theta})$
$\displaystyle=\frac{Y_{t}}{\widehat{\sigma}_{t}(\bm{\theta})}=\varepsilon_{t}\frac{h_{t}^{1/\delta^{\circ}}}{[\widehat{h}_{t}(\bm{\theta})]^{1/\delta^{\circ}}}$
$\displaystyle=\varepsilon_{t}\Big{[}1+\frac{h_{t}(\bm{\theta})-\widehat{h}_{t}(\bm{\theta})}{\widehat{h}_{t}(\bm{\theta})}\Big{]}^{1/\delta^{\circ}}\Big{[}1+\frac{h_{t}-h_{t}(\bm{\theta})}{h_{t}(\bm{\theta})}\Big{]}^{1/\delta^{\circ}}.$
(B.1)

Note that the quantities in square brackets are positive almost surely and,
hence, the power is well defined for $\delta^{\circ}>0$. Finally, define
$N_{n}^{-}(\eta)=N_{n}(\eta)\setminus\\{\bm{\theta}^{\circ}\\}$.

We require two results that follow directly from Lemma 13 and its proof in Kim
and Lee (2016). First, for

$\Delta_{n,t}=\sup_{\bm{\theta}\in
N_{n}^{-}(\eta)}\left|\frac{h_{t}-h_{t}(\bm{\theta})}{|\bm{\theta}-\bm{\theta}^{\circ}|h_{t}(\bm{\theta})}\right|,$

it holds that

$\max_{t=1,\ldots,n}\frac{\Delta_{n,t}}{n^{1/2}}=o_{\operatorname{P}}(1).$
(B.2)

Second, there exists $r_{0}\in[0,1)$, such that for sufficiently large $n$

$\sup_{\bm{\theta}\in
N_{n}(\eta)}\left|\frac{h_{t}(\bm{\theta})-\widehat{h}_{t}(\bm{\theta})}{\widehat{h}_{t}(\bm{\theta})}\right|\leq
r_{0}^{t}V_{0},\qquad t=1,\ldots,n,$ (B.3)

where $V_{0}=V_{0}(\eta)\geq 0$ with
$\operatorname{E}|V_{0}|^{\nu_{0}}<\infty$ for some $\nu_{0}>0$. We have
$\max_{t=\ell_{n},\ldots,n}r_{0}^{t}V_{0}=r_{0}^{\ell_{n}}V_{0}=o_{\operatorname{P}}(1)$.

Consider the two right-hand side factors in (B.1). Observe that

$\sup_{\bm{\theta}\in
N_{n}(\eta)}\left|\frac{h_{t}-h_{t}(\bm{\theta})}{h_{t}(\bm{\theta})}\right|\leq\sup_{\bm{\theta}\in
N_{n}^{-}(\eta)}\big{\\{}n^{1/2}|\bm{\theta}-\bm{\theta}^{\circ}|\big{\\}}\Delta_{n,t}/n^{1/2}\leq\eta\Delta_{n,t}/n^{1/2}.$

Set $s_{1t}=r_{0}^{t}V_{0}$ and $s_{2t}=\eta\Delta_{n,t}/n^{1/2}$, whence
$\max_{t=\ell_{n},\ldots,n}s_{1t}=o_{\operatorname{P}}(1)$ and
$\max_{t=\ell_{n},\ldots,n}s_{2t}=o_{\operatorname{P}}(1)$. Then, for
sufficiently large $n$,

$\displaystyle 1-\underline{s}_{t}$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=[1-\min\\{s_{1t},1\\}]^{1/\delta^{\circ}}[1-\min\\{s_{2t},1\\}]^{1/\delta^{\circ}}$
$\displaystyle\leq\Big{[}1+\frac{h_{t}(\bm{\theta})-\widehat{h}_{t}(\bm{\theta})}{\widehat{h}_{t}(\bm{\theta})}\Big{]}^{1/\delta^{\circ}}\Big{[}1+\frac{h_{t}-h_{t}(\bm{\theta})}{h_{t}(\bm{\theta})}\Big{]}^{1/\delta^{\circ}}$
$\displaystyle\leq[1+s_{1t}]^{1/\delta^{\circ}}[1+s_{2t}]^{1/\delta^{\circ}}=\mathrel{\mathop{\mathchar
58\relax}}1+\overline{s}_{t}.$

Put $s_{t}=\max\\{\underline{s}_{t},\ \overline{s}_{t}\\}$. Then,
$\max_{t=\ell_{n},\ldots,n}s_{t}=o_{\operatorname{P}}(1)$ and

$1-s_{t}\leq\Big{[}1+\frac{h_{t}(\bm{\theta})-\widehat{h}_{t}(\bm{\theta})}{\widehat{h}_{t}(\bm{\theta})}\Big{]}^{1/\delta^{\circ}}\Big{[}1+\frac{h_{t}-h_{t}(\bm{\theta})}{h_{t}(\bm{\theta})}\Big{]}^{1/\delta^{\circ}}\leq
1+s_{t}$

holds with probability approaching 1 (w.p.a. 1), as $n\to\infty$. Hence, the
conclusion follows with our choice of $s_{t}$ and $m_{t}\equiv 0$. ∎

###### Proof of Theorem 4:.

Denoting true volatility by
$\sigma_{t}^{2}=\sigma_{t}^{2}(\bm{\theta}^{\circ})$, we write

$\displaystyle\widehat{\varepsilon}_{t}(\bm{\theta})$
$\displaystyle=\frac{\widehat{X}_{t}(\bm{\theta})}{\widehat{\sigma}_{t}(\bm{\theta})}=\frac{X_{t}}{\widehat{\sigma}_{t}(\bm{\theta})}+\frac{\widehat{X}_{t}(\bm{\theta})-X_{t}}{\widehat{\sigma}_{t}(\bm{\theta})}$
$\displaystyle=\varepsilon_{t}\Big{[}1+\frac{\sigma_{t}^{2}(\bm{\theta})-\widehat{\sigma}_{t}^{2}(\bm{\theta})}{\widehat{\sigma}_{t}^{2}(\bm{\theta})}\Big{]}^{1/2}\Big{[}1+\frac{\sigma_{t}^{2}-\sigma_{t}^{2}(\bm{\theta})}{\sigma_{t}^{2}(\bm{\theta})}\Big{]}^{1/2}+\frac{\widehat{X}_{t}(\bm{\theta})-X_{t}}{\widehat{\sigma}_{t}(\bm{\theta})}.$
(B.4)

We again set $N_{n}^{-}(\eta)=N_{n}(\eta)\setminus\\{\bm{\theta}^{\circ}\\}$.
Kim and Lee (2016, p. 264) show that for $\bm{\theta}\in N_{n}(\eta)$

$\frac{|\sigma_{t}^{2}-\sigma_{t}^{2}(\bm{\theta})|}{\sigma_{t}^{2}(\bm{\theta})}\leq|\bm{\theta}-\bm{\theta}^{\circ}|\Pi_{1,n,t}^{*}+|\bm{\theta}-\bm{\theta}^{\circ}|^{2}\Pi_{2,n,t}^{*}$
(B.5)

for some $\Pi_{i,n,t}^{*}=\Pi_{i,n,t}^{*}(\eta)\geq 0$ ($i=1,2$), both
independent of $\bm{\theta}$. Furthermore, define
$X_{t}(\bm{\theta})=\sigma_{t}(\bm{\theta})\varepsilon_{t}$ and set

$\Pi_{3,n,t}=\Pi_{3,n,t}(\eta)=\sup_{\bm{\theta}\in
N_{n}^{-}(\eta)}\frac{1}{|\bm{\theta}-\bm{\theta}^{\circ}|}\frac{|X_{t}(\bm{\theta})-X_{t}|}{\widehat{\sigma}_{t}(\bm{\theta})}.$

By Equation (A.26) in Hoga (2019), there exists $v_{0}>2$ such that

$\limsup_{n\to\infty}\operatorname{E}|\Pi_{1,n,t}^{*}|^{v_{0}}<\infty,\quad\limsup_{n\to\infty}\operatorname{E}|\Pi_{2,n,t}^{*}|^{v_{0}/2}<\infty,\quad\limsup_{n\to\infty}\operatorname{E}|\Pi_{3,n,t}|^{v_{0}}<\infty$
(B.6)

uniformly in $t\in\mathbb{N}$. From this it follows that

$\max_{t=\ell_{n},\ldots,n}\Pi_{1,n,t}^{*}/\sqrt{n}=o_{\operatorname{P}}(1),\quad\max_{t=\ell_{n},\ldots,n}\Pi_{2,n,t}^{*}/n=o_{\operatorname{P}}(1),\quad\max_{t=\ell_{n},\ldots,n}\Pi_{3,n,t}/\sqrt{n}=o_{\operatorname{P}}(1).$
(B.7)

We only show the first claim in (B.7), as the others follow along similar
lines. Using subadditivity, Markov’s inequality and (B.6), we get

$\displaystyle\operatorname{P}\Big{\\{}\max_{t=\ell_{n},\ldots,n}\Pi_{1,n,t}^{*}/\sqrt{n}\geq\epsilon\Big{\\}}$
$\displaystyle=\operatorname{P}\Big{\\{}\bigcup_{t=\ell_{n},\ldots,n}\big{\\{}\Pi_{1,n,t}^{*}/\sqrt{n}\geq\epsilon\big{\\}}\Big{\\}}$
$\displaystyle\leq\sum_{t=\ell_{n}}^{n}\operatorname{P}\big{\\{}\Pi_{1,n,t}^{*}/\sqrt{n}\geq\epsilon\big{\\}}$
$\displaystyle\leq\sum_{t=\ell_{n}}^{n}\epsilon^{-v_{0}}n^{-v_{0}/2}\operatorname{E}|\Pi_{1,n,t}^{*}|^{v_{0}}=o(1).$

By Lemma A 1 in Hoga (2019), there exist $r\in(0,1)$ and r.v.s $V_{t}\geq 0$
with $\sup_{t\in\mathbb{N}}\operatorname{E}|V_{t}|^{v}<\infty$ for some $v>0$,
such that for sufficiently large $n$,

$\sup_{\bm{\theta}\in
N_{n}(\eta)}\max\big{\\{}|\widehat{X}_{t}(\bm{\theta})-X_{t}(\bm{\theta})|,\
|\widehat{\sigma}_{t}^{2}(\bm{\theta})-\sigma_{t}^{2}(\bm{\theta})|\big{\\}}\leq
r^{t}V_{t}\qquad\text{for all }t\in\mathbb{N}.$ (B.8)

We easily obtain that
$\max_{t=\ell_{n},\ldots,n}r^{t}V_{t}=o_{\operatorname{P}}(1)$, since

$\displaystyle\operatorname{P}\Big{\\{}\max_{t=\ell_{n},\ldots,n}r^{t}V_{t}\geq\epsilon\Big{\\}}$
$\displaystyle=\operatorname{P}\Big{\\{}\bigcup_{t=\ell_{n},\ldots,n}\big{\\{}r^{t}V_{t}\geq\epsilon\big{\\}}\Big{\\}}$
$\displaystyle\leq\sum_{t=\ell_{n}}^{n}\operatorname{P}\big{\\{}r^{t}V_{t}\geq\epsilon\big{\\}}$
$\displaystyle\leq\sum_{t=\ell_{n}}^{n}\epsilon^{-v}(r^{t})^{v}\operatorname{E}[V_{t}^{v}]$
$\displaystyle\leq
C\Big{\\{}\sup_{t=\ell_{n},\ldots,n}\operatorname{E}[V_{t}^{v}]\Big{\\}}\sum_{t=\ell_{n}}^{n}(r^{t})^{v}=o(1),$

where the last equality follows from properties of the geometric series.

Equipped with these results, we now consider the terms on the right-hand side
of (B.4) separately. First, by (A.5) and the fact that $\omega^{\circ}>0$, we
have for sufficiently large $n$ that
$\widehat{\sigma}_{t}^{2}(\bm{\theta})\geq\underline{\omega}>0$ for all
$\bm{\theta}\in N_{n}(\eta)$ and all $t$. Combine this with (B.8) to obtain

$\displaystyle\max_{t=\ell_{n},\ldots,n}\frac{|\sigma_{t}^{2}(\bm{\theta})-\widehat{\sigma}_{t}^{2}(\bm{\theta})|}{\widehat{\sigma}_{t}^{2}(\bm{\theta})}$
$\displaystyle\leq\underline{\omega}^{-1}\max_{t=\ell_{n},\ldots,n}r^{t}V_{t}=\mathrel{\mathop{\mathchar
58\relax}}\max_{t=\ell_{n},\ldots,n}s_{1t}=o_{\operatorname{P}}(1),$ (B.9)
$\displaystyle\max_{t=\ell_{n},\ldots,n}\frac{|\widehat{X}_{t}(\bm{\theta})-X_{t}|}{\widehat{\sigma}_{t}(\bm{\theta})}$
$\displaystyle\leq\max_{t=\ell_{n},\ldots,n}\Big{\\{}\underline{\omega}^{-1/2}|\widehat{X}_{t}(\bm{\theta})-X_{t}(\bm{\theta})|+\frac{|X_{t}(\bm{\theta})-X_{t}|}{\widehat{\sigma}_{t}(\bm{\theta})}\Big{\\}}$
$\displaystyle\leq\max_{t=\ell_{n},\ldots,n}\Big{\\{}\underline{\omega}^{-1/2}r^{t}V_{t}+\sup_{\bm{\theta}\in
N_{n}^{-}(\eta)}\big{\\{}\sqrt{n}|\bm{\theta}-\bm{\theta}^{\circ}|\big{\\}}\Pi_{3,n,t}/\sqrt{n}\Big{\\}}$
$\displaystyle\leq\max_{t=\ell_{n},\ldots,n}\Big{\\{}\underline{\omega}^{-1/2}r^{t}V_{t}+\eta\Pi_{3,n,t}/\sqrt{n}\Big{\\}}$
$\displaystyle=\mathrel{\mathop{\mathchar
58\relax}}\max_{t=\ell_{n},\ldots,n}m_{t}=o_{\operatorname{P}}(1).$

Finally, we have from (B.5) and (B.7) that

$\displaystyle\max_{t=\ell_{n},\ldots,n}\frac{|\sigma_{t}^{2}-\sigma_{t}^{2}(\bm{\theta})|}{\sigma_{t}^{2}(\bm{\theta})}$
$\displaystyle\leq\max_{t=\ell_{n},\ldots,n}\Big{\\{}\sup_{\bm{\theta}\in
N_{n}^{-}(\eta)}\big{\\{}\sqrt{n}|\bm{\theta}-\bm{\theta}^{\circ}|\big{\\}}\Pi_{1,n,t}^{*}/\sqrt{n}+\sup_{\bm{\theta}\in
N_{n}^{-}(\eta)}\big{\\{}n|\bm{\theta}-\bm{\theta}^{\circ}|^{2}\big{\\}}\Pi_{2,n,t}^{*}/n\Big{\\}}$
$\displaystyle\leq\max_{t=\ell_{n},\ldots,n}\Big{\\{}\eta\Pi_{1,n,t}^{*}/\sqrt{n}+\eta^{2}\Pi_{2,n,t}^{*}/n\Big{\\}}$
$\displaystyle=\mathrel{\mathop{\mathchar
58\relax}}\max_{t=\ell_{n},\ldots,n}s_{2t}=o_{\operatorname{P}}(1).$ (B.10)

Combining (B.9) and (B.10), we get

$\displaystyle 1-\underline{s}_{t}$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=[1-\min\\{s_{1t},1\\}]^{1/2}[1-\min\\{s_{2t},1\\}]^{1/2}$
$\displaystyle\leq\Big{[}1+\frac{\sigma_{t}^{2}(\bm{\theta})-\widehat{\sigma}_{t}^{2}(\bm{\theta})}{\widehat{\sigma}_{t}^{2}(\bm{\theta})}\Big{]}^{1/2}\Big{[}1+\frac{\sigma_{t}^{2}-\sigma_{t}^{2}(\bm{\theta})}{\sigma_{t}^{2}(\bm{\theta})}\Big{]}^{1/2}$
$\displaystyle\leq[1+s_{1t}]^{1/2}[1+s_{2t}]^{1/2}=\mathrel{\mathop{\mathchar
58\relax}}1+\overline{s}_{t}.$

Hence, for $s_{t}=\max\\{\underline{s}_{t},\ \overline{s}_{t}\\}$ with
$\max_{t=\ell_{n},\ldots,n}s_{t}=o_{\operatorname{P}}(1)$, it holds w.p.a. 1,
as $n\to\infty$, that

$1-s_{t}\leq\Big{[}1+\frac{\sigma_{t}^{2}(\bm{\theta})-\widehat{\sigma}_{t}^{2}(\bm{\theta})}{\widehat{\sigma}_{t}^{2}(\bm{\theta})}\Big{]}^{1/2}\Big{[}1+\frac{\sigma_{t}^{2}-\sigma_{t}^{2}(\bm{\theta})}{\sigma_{t}^{2}(\bm{\theta})}\Big{]}^{1/2}\leq
1+s_{t}.$

Thus, the conclusion follows with the above choices of $m_{t}$ and $s_{t}$. ∎

##  Appendix C Proof of Theorem 1

Let $C>0$ denote a large constant that may change from line to line. For
notational brevity, we put

$U_{t}=|\varepsilon_{t}|,\quad\widehat{U}_{t}=|\widehat{\varepsilon}_{t}|,\quad\widehat{U}_{(i)}=|\widehat{\varepsilon}|_{(i)}.$

We denote the survivor function of $U_{t}$ by
$\overline{F}(\cdot)=1-F(\cdot)$, where $F(\cdot)$ is from Assumption 1. For
$\bm{x}=(x,y)^{\prime}\in(0,\infty)^{2}$ and
$\bm{\xi}=(\xi_{1},\xi_{2})^{\prime}\in\mathbb{R}^{2}$, define

$\displaystyle M_{n}(\bm{\xi})$
$\displaystyle=M_{n}(\bm{x},\bm{\xi})=\frac{1}{k}\sum_{t=d+1}^{n}I_{\big{\\{}U_{t}>e^{\xi_{1}/\sqrt{k}}b(\frac{n}{kx}),\
U_{t-d}>e^{\xi_{2}/\sqrt{k}}b(\frac{n}{ky})\big{\\}}},$
$\displaystyle\widehat{M}_{n}(\bm{\xi})$
$\displaystyle=\widehat{M}_{n}(\bm{x},\bm{\xi})=\frac{1}{k}\sum_{t=d+1}^{n}I_{\big{\\{}\widehat{U}_{t}>e^{\xi_{1}/\sqrt{k}}b(\frac{n}{kx}),\
\widehat{U}_{t-d}>e^{\xi_{2}/\sqrt{k}}b(\frac{n}{ky})\big{\\}}}.$

If not specified otherwise, we assume the conditions of Theorem 1 to hold for
the following lemmas and propositions, even though some of them may hold under
a subset of the assumptions of Theorem 1.

The proof of Theorem 1 requires the following four propositions, which are
proved in Subsection C.1 of this Appendix. The first proposition derives an
(infeasible) estimate of the PA-tail copula for the (unobserved) innovations.
The limit theory developed by Schmidt and Stadtmüller (2006) cannot be applied
to prove the next proposition, since it only applies to tail dependent
sequences of random variables. Recall for the following that
$b(x)=F^{\leftarrow}(1-1/x)$.

######  Proposition 1.

For i.i.d. $U_{t}$ with d.f. $F(\cdot)$ satisfying Assumption 1 (ii), it holds
that, as $n\to\infty$,

$\sqrt{\frac{n}{xy}}\begin{pmatrix}\frac{1}{k}\sum_{t=2}^{n}\Big{[}I_{\left\\{U_{t}>b(\frac{n}{kx}),\
U_{t-1}>b(\frac{n}{ky})\right\\}}-\Big{(}\frac{k}{n}\Big{)}^{2}xy\Big{]}\\\
\vdots\\\ \frac{1}{k}\sum_{t=D+1}^{n}\Big{[}I_{\left\\{U_{t}>b(\frac{n}{kx}),\
U_{t-D}>b(\frac{n}{ky})\right\\}}-\Big{(}\frac{k}{n}\Big{)}^{2}xy\Big{]}\\\
\end{pmatrix}\overset{d}{\longrightarrow}N(\bm{0},\bm{I}_{D\times D}),$

where $\bm{I}_{D\times D}$ denotes the $(D\times D)$-identity matrix, and
$\bm{0}$ is a $(D\times 1)$-vector of zeros.

In the following, we show that in the indicators in Proposition 1 we can
replace the $U_{t}$ with the $\widehat{U}_{t}$, and the $b(n/[kz])$ by the
empirical analog $\widehat{U}_{(\lfloor kz\rfloor+1)}$ ($z\in\\{x,y\\}$)
without changing the limit. Then, Theorem 1 follows immediately from an
application of the continuous mapping theorem. The following three
propositions serve to justify these replacements.

######  Proposition 2.

For any $K>0$, it holds that

$\sup_{\bm{\xi}\in[-K,K]^{2}}|M_{n}(\bm{\xi})-M_{n}(\bm{0})|=o_{\operatorname{P}}(n^{-1/2}).$

######  Proposition 3.

For any $K>0$, it holds that

$\sup_{\bm{\xi}\in[-K,K]^{2}}|\widehat{M}_{n}(\bm{\xi})-M_{n}(\bm{\xi})|=o_{\operatorname{P}}(n^{-1/2}).$

######  Proposition 4.

For any $x>0$, it holds that

$|\xi_{n}(x)|=\left|\sqrt{k}\log\Bigg{(}\frac{\widehat{U}_{(\lfloor
kx\rfloor+1)}}{b(n/[kx])}\Bigg{)}\right|=O_{\operatorname{P}}(1).$

Equipped with these four propositions, we proceed to prove Theorem 1.

###### Proof of Theorem 1:.

We show that for each $d\in\\{1,\ldots,D\\}$,

$\Bigg{|}\frac{1}{k}\sum_{t=d+1}^{n}I_{\\{U_{t}>b(\frac{n}{kx}),\
U_{t-d}>b(\frac{n}{ky})\\}}-\frac{1}{k}\sum_{t=d+1}^{n}I_{\\{\widehat{U}_{t}>\widehat{U}_{(\lfloor
kx\rfloor+1)},\ \widehat{U}_{t-d}>\widehat{U}_{(\lfloor
ky\rfloor+1)}\\}}\Bigg{|}=o_{\operatorname{P}}(n^{-1/2}).$ (C.1)

Combining this with Proposition 1, the desired convergence follows from Lemma
4.7 of White (2001) and the continuous mapping theorem (e.g., White, 2001,
Theorem 7.20). Decompose

$\displaystyle M_{n}(\bm{0})$
$\displaystyle=\frac{1}{k}\sum_{t=d+1}^{n}I_{\\{U_{t}>b(\frac{n}{kx}),\
U_{t-d}>b(\frac{n}{ky})\\}}$
$\displaystyle=\frac{1}{k}\sum_{t=d+1}^{n}I_{\\{U_{t}>b(\frac{n}{kx}),\
U_{t-d}>b(\frac{n}{ky})\\}}-\frac{1}{k}\sum_{t=d+1}^{n}I_{\\{U_{t}>e^{\xi_{1}/\sqrt{k}}b(\frac{n}{kx}),\
U_{t-d}>e^{\xi_{2}/\sqrt{k}}b(\frac{n}{ky})\\}}$ $\displaystyle\hskip
11.38092pt+\frac{1}{k}\sum_{t=d+1}^{n}I_{\\{U_{t}>e^{\xi_{1}/\sqrt{k}}b(\frac{n}{kx}),\
U_{t-d}>e^{\xi_{2}/\sqrt{k}}b(\frac{n}{ky})\\}}-\frac{1}{k}\sum_{t=d+1}^{n}I_{\\{\widehat{U}_{t}>e^{\xi_{1}/\sqrt{k}}b(\frac{n}{kx}),\
\widehat{U}_{t-d}>e^{\xi_{2}/\sqrt{k}}b(\frac{n}{ky})\\}}$
$\displaystyle\hskip
11.38092pt+\frac{1}{k}\sum_{t=d+1}^{n}I_{\\{\widehat{U}_{t}>e^{\xi_{1}/\sqrt{k}}b(\frac{n}{kx}),\
\widehat{U}_{t-d}>e^{\xi_{2}/\sqrt{k}}b(\frac{n}{ky})\\}}$
$\displaystyle=o_{\operatorname{P}}(n^{-1/2})+\widehat{M}_{n}(\bm{\xi}),$
(C.2)

where the $o_{\operatorname{P}}(n^{-1/2})$-term is uniform on
$\bm{\xi}\in[-K,K]^{2}$ by Propositions 2 and 3.

Define $\bm{\xi}_{n}=\big{(}\xi_{n}(x),\ \xi_{n}(y)\big{)}^{\prime}$, where
$\xi_{n}(z)=\sqrt{k}\log\big{(}\widehat{U}_{(\lfloor
kz\rfloor+1)}/b(n/[kz])\big{)}$ ($z\in\\{x,y\\}$) as in Proposition 4. Then,
$\widehat{M}_{n}(\bm{\xi}_{n})=\frac{1}{k}\sum_{t=d+1}^{n}I_{\\{\widehat{U}_{t}>\widehat{U}_{(\lfloor
kx\rfloor+1)},\ \widehat{U}_{t-d}>\widehat{U}_{(\lfloor ky\rfloor+1)}\\}}$. To
show (C.1) or, equivalently, that
$\widehat{M}_{n}(\bm{\xi}_{n})-M_{n}(\bm{0})=o_{\operatorname{P}}(n^{-1/2})$
we have to prove that for any $\varepsilon>0$ and $\delta>0$

$\operatorname{P}\left\\{\sqrt{n}|\widehat{M}_{n}(\bm{\xi}_{n})-M_{n}(\bm{0})|>\varepsilon\right\\}\leq\delta$

for sufficiently large $n$. By Proposition 4, we can choose $K>0$ such that
$\operatorname{P}\\{|\bm{\xi}_{n}|>K\\}\leq\delta/2$ for all sufficiently
large $n$. Furthermore, by (C.2),
$\operatorname{P}\\{\sqrt{n}\sup_{\bm{\xi}\in[-K,K]^{2}}|\widehat{M}_{n}(\bm{\xi})-M_{n}(\bm{0})|>\varepsilon\\}\leq\delta/2$
for large $n$. Using these two results, we obtain

$\displaystyle\operatorname{P}\left\\{\sqrt{n}|\widehat{M}_{n}(\bm{\xi}_{n})-M_{n}(\bm{0})|>\varepsilon\right\\}$
$\displaystyle\leq\operatorname{P}\Big{\\{}|\bm{\xi}_{n}|>K\Big{\\}}+\operatorname{P}\Big{\\{}\sqrt{n}\sup_{\bm{\xi}\in[-K,K]^{2}}|\widehat{M}_{n}(\bm{\xi})-M_{n}(\bm{0})|>\varepsilon,\
|\bm{\xi}_{n}|\leq K\Big{\\}}$
$\displaystyle\leq\frac{\delta}{2}+\frac{\delta}{2}=\delta,$

i.e., (C.1). The conclusion follows. ∎

###  C.1 Proofs of Propositions 1–4

###### Proof of Proposition 1:.

For $d\in\\{1,\ldots,D\\}$, we define

$\displaystyle Z_{n,t}^{(d)}$
$\displaystyle=Z_{n,t}^{(d)}(x,y)=\begin{cases}0,&t=1,\ldots,d,\\\
\frac{n}{k}\left[I_{\left\\{U_{t}>b(\frac{n}{kx}),\
U_{t-d}>b(\frac{n}{ky})\right\\}}-\operatorname{P}\left\\{U_{t}>b\Big{(}\frac{n}{kx}\Big{)},\
U_{t-d}>b\Big{(}\frac{n}{ky}\Big{)}\right\\}\right],&t>d,\end{cases}$
$\displaystyle\bm{Z}_{n,t}$
$\displaystyle=\big{(}Z_{n,t}^{(1)},\ldots,Z_{n,t}^{(D)}\big{)}^{\prime}.$

Note that by Assumption 1 (ii)

$\operatorname{P}\left\\{U_{t}>b\Big{(}\frac{n}{kx}\Big{)},\
U_{t-d}>b\Big{(}\frac{n}{ky}\Big{)}\right\\}=\operatorname{P}\left\\{U_{t}>b\Big{(}\frac{n}{kx}\Big{)}\right\\}\operatorname{P}\left\\{U_{t-d}>b\Big{(}\frac{n}{ky}\Big{)}\right\\}=\left(\frac{k}{n}\right)^{2}xy$

for sufficiently large $n$.

Let $\bm{\lambda}=(\lambda_{1},\ldots,\lambda_{D})^{\prime}\in\mathbb{R}^{D}$
be any vector with $\bm{\lambda}^{\prime}\bm{\lambda}=1$. Then, by the
Cramér–Wold device (e.g., White, 2001, Proposition 5.1) it suffices to show
that
$n^{-1/2}\sum_{t=1}^{n}\bm{\lambda}^{\prime}\bm{Z}_{n,t}\overset{d}{\longrightarrow}\sqrt{xy}\bm{\lambda}^{\prime}\bm{Z}$
for $\bm{Z}\sim N(\bm{0},\bm{I}_{D\times D})$. To do so, we verify the
conditions of Theorem 5.20 of White (2001).

Since the $\bm{\lambda}^{\prime}\bm{Z}_{n,t}$ are (row-wise) $D$-dependent,
the array $\\{\bm{\lambda}^{\prime}\bm{Z}_{n,t}\\}$ is obviously $\phi$-mixing
of any rate (White, 2001, Sec. 3). Furthermore,
$\operatorname{E}[\bm{\lambda}^{\prime}\bm{Z}_{n,t}]=0$ and

$\displaystyle\operatorname{E}[(\bm{\lambda}^{\prime}\bm{Z}_{n,t})^{2}]$
$\displaystyle=\operatorname{Var}(\bm{\lambda}^{\prime}\bm{Z}_{n,t})=\operatorname{Var}\Big{(}\sum_{d=1}^{D}\lambda_{d}Z_{n,t}^{(d)}\Big{)}$
$\displaystyle=\sum_{d=1}^{D}\lambda_{d}^{2}\operatorname{Var}(Z_{n,t}^{(d)})+\sum_{c\neq
d}\lambda_{c}\lambda_{d}\operatorname{Cov}(Z_{n,t}^{(c)},\ Z_{n,t}^{(d)}).$

Consider the variances and covariances separately. For the variance we get for
$t>d$

$\displaystyle\operatorname{Var}\big{(}Z_{n,t}^{(d)}\big{)}$
$\displaystyle=\frac{n^{2}}{k^{2}}\operatorname{Var}\Big{(}I_{\left\\{U_{t}>b(\frac{n}{kx}),\
U_{t-d}>b(\frac{n}{ky})\right\\}}\Big{)}$
$\displaystyle=\frac{n^{2}}{k^{2}}\Big{(}\frac{k}{n}\Big{)}^{2}xy\left[1-\Big{(}\frac{k}{n}\Big{)}^{2}xy\right]$
$\displaystyle=xy+o(1),$ (C.3)

where the second line exploits that $I_{\left\\{U_{t}>b(\frac{n}{kx}),\
U_{t-d}>b(\frac{n}{ky})\right\\}}$ is Bernoulli-distributed with success
probability $p=(k/n)^{2}xy$ and, hence,

$\operatorname{Var}\Big{(}I_{\left\\{U_{t}>b(\frac{n}{kx}),\
U_{t-d}>b(\frac{n}{ky})\right\\}}\Big{)}=\Big{(}\frac{k}{n}\Big{)}^{2}xy\left[1-\Big{(}\frac{k}{n}\Big{)}^{2}xy\right].$

Using similar arguments, we obtain

$\displaystyle\operatorname{Cov}\big{(}Z_{n,t}^{(c)},Z_{n,t}^{(d)}\big{)}$
$\displaystyle=O\left(\frac{n^{2}}{k^{2}}\operatorname{P}\left\\{U_{r}>b\Big{(}\frac{n}{kx}\Big{)},\
U_{s}>b\Big{(}\frac{n}{ky}\Big{)},\
U_{t}>b\Big{(}\frac{n}{kz}\Big{)}\right\\}\right)$
$\displaystyle=O\Big{(}\frac{n^{2}}{k^{2}}\frac{k^{3}}{n^{3}}\Big{)}=O\Big{(}\frac{k}{n}\Big{)}=o(1),$
(C.4)

where $r,s,t$ are pairwise distinct. Overall, we have
$\operatorname{E}[(\bm{\lambda}^{\prime}\bm{Z}_{n,t})^{2}]\leq 2<\infty$ for
sufficiently large $n$.

Finally, exploiting $D$-dependence of $\bm{\lambda}^{\prime}\bm{Z}_{n,t}$
again,

$\displaystyle\overline{\sigma}_{n}^{2}$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Var}\Big{(}n^{-1/2}\sum_{t=1}^{n}\bm{\lambda}^{\prime}\bm{Z}_{n,t}\Big{)}$
$\displaystyle=\frac{1}{n}\sum_{t=1}^{n}\operatorname{Var}(\bm{\lambda}^{\prime}\bm{Z}_{n,t})+\frac{1}{n}\sum_{s\neq
t}\operatorname{Cov}(\bm{\lambda}^{\prime}\bm{Z}_{n,s},\bm{\lambda}^{\prime}\bm{Z}_{n,t})$
$\displaystyle=\frac{1}{n}\sum_{t=1}^{n}\operatorname{Var}(\bm{\lambda}^{\prime}\bm{Z}_{n,t})+\frac{2}{n}\sum_{s=1}^{n}\sum_{t=s+1}^{\min\\{s+D,n\\}}\operatorname{Cov}(\bm{\lambda}^{\prime}\bm{Z}_{n,s},\bm{\lambda}^{\prime}\bm{Z}_{n,t}).$
(C.5)

It is easy to check that the sum on the right-hand side of (C.5) vanishes,
because the covariance terms are of order $o(1)$ by (C.4). For the sum
involving the variances, we get using (C.3), (C.4) and
$\bm{\lambda}^{\prime}\bm{\lambda}=1$ that for $t>D$

$\displaystyle\operatorname{Var}(\bm{\lambda}^{\prime}\bm{Z}_{n,t})$
$\displaystyle=\operatorname{Var}\Big{(}\sum_{d=1}^{D}\lambda_{d}Z_{n,t}^{(d)}\Big{)}$
$\displaystyle=\sum_{d=1}^{D}\lambda_{d}^{2}\operatorname{Var}\big{(}Z_{n,t}^{(d)}\big{)}+\sum_{c\neq
d}\lambda_{c}\lambda_{d}\operatorname{Cov}\big{(}Z_{n,t}^{(c)},Z_{n,t}^{(d)}\big{)}$
$\displaystyle=\sum_{d=1}^{D}\lambda_{d}^{2}[xy+o(1)]+o(1)=xy+o(1).$

Overall, we obtain that $\overline{\sigma}_{n}^{2}=xy+o(1)\longrightarrow xy$,
as $n\to\infty$.

Thus, we may apply Theorem 5.20 of White (2001), yielding that
$n^{-1/2}\sum_{t=1}^{n}\bm{\lambda}^{\prime}\bm{Z}_{n,t}\overset{d}{\longrightarrow}\sqrt{xy}\bm{\lambda}^{\prime}\bm{Z}$.
This concludes the proof. ∎

The following lemma bounds the variation in the tail of the $U_{t}$, and is
used throughout the proofs.

######  Lemma 1.

Suppose Assumption 1 holds. Then, for any $\nu\in(-1,\infty)$ there exists
$\widetilde{\nu}$ between $1$ and $\nu$, such that

$\displaystyle\overline{F}\Big{(}(1+\nu)b\Big{(}\frac{n}{kx}\Big{)}\Big{)}$
$\displaystyle=\frac{kx}{n}\Big{[}1-\frac{\alpha\nu}{(1+\widetilde{\nu})^{1+\alpha}}+\nu
o(1)\Big{]},$ where the $o(1)$-term vanishes uniformly on compact $\nu$-sets
in $(-1,\infty)$ and on compact $x$-sets in $(0,\infty)$. In particular,
$\displaystyle\overline{F}\Big{(}e^{\xi/\sqrt{k}}b\Big{(}\frac{n}{kx}\Big{)}\Big{)}$
$\displaystyle=\frac{kx}{n}\Bigg{[}1-\frac{\alpha\xi}{\sqrt{k}}+o\Big{(}\frac{1}{\sqrt{k}}\Big{)}\Bigg{]}$

uniformly on compact $\xi$-sets in $\mathbb{R}$ and on compact $x$-sets in
$(0,\infty)$.

Lemma 1 is proved in Appendix C.2.

###### Proof of Proposition 2:.

We only consider the supremum over $\bm{\xi}\in[-K,0]^{2}$; that over the
other three quadrants can be dealt with similarly. Observe that

$\displaystyle\sup_{\bm{\xi}\in[-K,0]^{2}}|M_{n}(\bm{\xi})-M_{n}(\bm{0})|$
$\displaystyle\leq M_{n}(-(K,K)^{\prime})-M_{n}(\bm{0})$
$\displaystyle=\frac{1}{k}\sum_{t=d+1}^{n}\Big{[}I_{\\{U_{t}>e^{-K/\sqrt{k}}b(\frac{n}{kx}),\
U_{t-d}>e^{-K/\sqrt{k}}b(\frac{n}{ky})\\}}-I_{\\{U_{t}>b(\frac{n}{kx}),\
U_{t-d}>b(\frac{n}{ky})\\}}\Big{]}$ $\displaystyle=\mathrel{\mathop{\mathchar
58\relax}}\frac{1}{k}\sum_{t=d+1}^{n}I_{1t},$

where $I_{1t}\in\\{0,1\\}$. Markov’s inequality implies

$\operatorname{P}\left\\{\sqrt{n}\sup_{\bm{\xi}\in[-K,0]^{2}}|M_{n}(\bm{\xi})-M_{n}(\bm{0})|>\varepsilon\right\\}\leq\frac{\varepsilon^{-4}n^{2}}{k^{4}}\operatorname{E}\Big{[}\sum_{t=d+1}^{n}I_{1t}\Big{]}^{4}.$
(C.6)

Due to $d$-dependence, the $I_{1t}$ are trivially $\rho$-mixing, and thus we
may apply Lemma 2.3 of Shao (1993) (for, in his notation, $q=4$) to deduce
that

$\operatorname{E}\Big{[}\sum_{t=d+1}^{n}I_{1t}\Big{]}^{4}\leq
n^{2}C\big{\\{}\operatorname{E}[I_{1t}^{2}]\big{\\}}^{2}+nC\operatorname{E}[I_{1t}^{4}].$
(C.7)

Since $I_{1t}\in\\{0,1\\}$ and, hence,
$\operatorname{E}[I_{1t}^{4}]=\operatorname{E}[I_{1t}^{2}]=\operatorname{E}[I_{1t}]$,
it suffices to compute $\operatorname{E}[I_{1t}]$. Using Lemma 1, we get that

$\displaystyle\operatorname{E}[I_{1t}]$
$\displaystyle=\operatorname{P}\Big{\\{}U_{t}>e^{-K/\sqrt{k}}b\Big{(}\frac{n}{kx}\Big{)}\Big{\\}}\operatorname{P}\Big{\\{}U_{t-d}>e^{-K/\sqrt{k}}b\Big{(}\frac{n}{ky}\Big{)}\Big{\\}}-\operatorname{P}\Big{\\{}U_{t}>b\Big{(}\frac{n}{kx}\Big{)}\Big{\\}}\operatorname{P}\Big{\\{}U_{t-d}>b\Big{(}\frac{n}{ky}\Big{)}\Big{\\}}$
$\displaystyle=\frac{k^{2}}{n^{2}}xy\Big{[}1+\frac{\alpha
K}{\sqrt{k}}+o\Big{(}\frac{1}{\sqrt{k}}\Big{)}\Big{]}^{2}-\frac{k^{2}}{n^{2}}xy$
$\displaystyle=\frac{k^{2}}{n^{2}}xy\Big{[}1+\frac{2\alpha
K}{\sqrt{k}}+o\Big{(}\frac{1}{\sqrt{k}}\Big{)}-1\Big{]}$ $\displaystyle\leq
C\frac{k^{3/2}}{n^{2}}xy,$

whence, from (C.7),

$\operatorname{E}\Big{[}\sum_{t=d+1}^{n}I_{1t}\Big{]}^{4}\leq
C\frac{k^{3}}{n^{2}}+C\frac{k^{3/2}}{n}.$

Together with (C.6), this implies that

$\operatorname{P}\Big{\\{}\sqrt{n}\sup_{\bm{\xi}\in[-K,0]^{2}}|M_{n}(\bm{\xi})-M_{n}(\bm{0})|>\varepsilon\Big{\\}}=O\Big{(}\frac{1}{k}+\frac{n}{k^{5/2}}\Big{)}=o(1)$

by Assumption 4. Hence, the conclusion follows. ∎

Define

$\displaystyle A_{t}(\bm{\xi},\bm{\theta})$
$\displaystyle=I_{\left\\{\widehat{U}_{t}(\bm{\theta})>e^{\xi_{1}/\sqrt{k}}b(n/[kx]),\
\widehat{U}_{t-d}(\bm{\theta})>e^{\xi_{2}/\sqrt{k}}b(n/[ky])\right\\}},$
$\displaystyle A_{t}(\bm{\xi})$
$\displaystyle=I_{\left\\{U_{t}>e^{\xi_{1}/\sqrt{k}}b(n/[kx]),\
U_{t-d}>e^{\xi_{2}/\sqrt{k}}b(n/[ky])\right\\}},$ $\displaystyle
A_{t}(\bm{\xi},\eta,\eta_{0})$
$\displaystyle=I_{\left\\{U_{t}[1+\eta_{0}s_{t}]+\eta_{0}m_{t}>e^{\xi_{1}/\sqrt{k}}b(n/[kx]),\
U_{t-d}[1+\eta_{0}s_{t-d}]+\eta_{0}m_{t-d}>e^{\xi_{2}/\sqrt{k}}b(n/[ky])\right\\}},\qquad\eta_{0}\in\\{-1,1\\},$
$\displaystyle B_{t}(\bm{\xi},\eta,\eta_{0})$
$\displaystyle=B_{t}(\bm{x},\bm{\xi},\eta,\eta_{0})=A_{t}(\bm{\xi},\eta,\eta_{0})-A_{t}(\bm{\xi}),$
(C.8)

where $m_{t}=m_{n,t}(\eta)\geq 0$ and $s_{t}=s_{n,t}(\eta)\geq 0$ are from
Assumption 3.

The proof of Proposition 3 requires two further preliminary lemmas, which are
both proven in Appendix C.2.

######  Lemma 2.

Let $\eta>0$. Then, w.p.a. 1, as $n\to\infty$,

$A_{t}(\bm{\xi},\eta,-1)\leq A_{t}(\bm{\xi},\bm{\theta})\leq
A_{t}(\bm{\xi},\eta,1)$

for all $\bm{\theta}\in N_{n}(\eta)$ and $t=\ell_{n},\ldots,n$.

######  Lemma 3.

Let $\eta>0$, $\eta_{0}\in\\{-1,1\\}$. Then, for any $K>0$,

$\sup_{\bm{\xi}\in[-K,K]^{2}}\left|\frac{1}{k}\sum_{t=d+1}^{n}B_{t}(\bm{\xi},\eta,\eta_{0})\right|=o_{\operatorname{P}}(n^{-1/2}).$

###### Proof of Proposition 3:.

Let $\eta>0$ and $\ell_{n}\to\infty$ with $\ell_{n}=o(k/\sqrt{n})$ as
$n\to\infty$. Then, due to Lemma 2, we get that w.p.a. 1, as $n\to\infty$,

$B_{t}(\bm{\xi},\eta,-1)=A_{t}(\bm{\xi},\eta,-1)-A_{t}(\bm{\xi})\leq
A_{t}(\bm{\xi},\bm{\theta})-A_{t}(\bm{\xi})\leq
A_{t}(\bm{\xi},\eta,1)-A_{t}(\bm{\xi})=B_{t}(\bm{\xi},\eta,1)$

for all $\bm{\theta}\in N_{n}(\eta)$ and $t=\ell_{n},\ldots,n$. Hence, from
Lemma 3,

$\sup_{\bm{\theta}\in
N_{n}(\eta)}\sup_{\bm{\xi}\in[-K,K]^{2}}\frac{1}{k}\left|\sum_{t=\ell_{n}}^{n}\left\\{A_{t}(\bm{\xi},\bm{\theta})-A_{t}(\bm{\xi})\right\\}\right|=o_{\operatorname{P}}(n^{-1/2}).$

Using additionally that $\ell_{n}/k=o(n^{-1/2})=o(1)$, we also have that
(since $|A_{t}(\bm{\xi},\bm{\theta})-A_{t}(\bm{\xi})|\leq 1$)

$\sup_{\bm{\theta}\in
N_{n}(\eta)}\sup_{\bm{\xi}\in[-K,K]^{2}}\frac{1}{k}\left|\sum_{t=d+1}^{n}\left\\{A_{t}(\bm{\xi},\bm{\theta})-A_{t}(\bm{\xi})\right\\}\right|=o_{\operatorname{P}}(n^{-1/2}).$

Because by Assumption 2, $\widehat{\bm{\theta}}\in N_{n}(\eta)$ w.p.a. 1 as
$n\to\infty$ followed by $\eta\to\infty$, we get

$\sup_{\bm{\xi}\in[-K,K]^{2}}\frac{1}{k}\left|\sum_{t=d+1}^{n}\left\\{A_{t}(\bm{\xi},\widehat{\bm{\theta}})-A_{t}(\bm{\xi})\right\\}\right|=o_{\operatorname{P}}(n^{-1/2}).$

This, however, is just the conclusion. ∎

###### Proof of Proposition 4:.

The result follows immediately from Proposition 8, which is proven in Appendix
D.1. ∎

###  C.2 Proofs of Lemmas 1–3

###### Proof of Lemma 1:.

By de Haan and Ferreira (2006, Theorem 1.1.11 and Theorem B.1.10), we have
that for any $\widetilde{\widetilde{\nu}}\in(-1,\infty)$, any $\varepsilon>0$
and any $\delta>0$

$\left|\frac{\overline{F}\Big{(}(1+\widetilde{\widetilde{\nu}})b\Big{(}\frac{n}{kx}\Big{)}\Big{)}}{\overline{F}\Big{(}b\Big{(}\frac{n}{kx}\Big{)}\Big{)}}-(1+\widetilde{\widetilde{\nu}})^{-\alpha}\right|\leq\varepsilon\max\\{(1+\widetilde{\widetilde{\nu}})^{-\alpha-\delta},(1+\widetilde{\widetilde{\nu}})^{-\alpha+\delta}\\}$
(C.9)

for sufficiently large $n$. Use the mean value theorem to deduce that there is
some $\widetilde{\nu}$ between $1$ and $\nu$, such that

$\displaystyle\overline{F}\Big{(}b\Big{(}\frac{n}{kx}\Big{)}\Big{)}$
$\displaystyle-\overline{F}\Big{(}(1+\nu)b\Big{(}\frac{n}{kx}\Big{)}\Big{)}$
$\displaystyle=F\Big{(}(1+\nu)b\Big{(}\frac{n}{kx}\Big{)}\Big{)}-F\Big{(}b\Big{(}\frac{n}{kx}\Big{)}\Big{)}$
$\displaystyle=f\Big{(}(1+\widetilde{\nu})b\Big{(}\frac{n}{kx}\Big{)}\Big{)}\left[(1+\nu)b\Big{(}\frac{n}{kx}\Big{)}-b\Big{(}\frac{n}{kx}\Big{)}\right]$
$\displaystyle=\frac{(1+\widetilde{\nu})b\Big{(}\frac{n}{kx}\Big{)}f\Big{(}(1+\widetilde{\nu})b\Big{(}\frac{n}{kx}\Big{)}\Big{)}}{1-F\Big{(}(1+\widetilde{\nu})b\Big{(}\frac{n}{kx}\Big{)}\Big{)}}\cdot\frac{(1+\nu)b\Big{(}\frac{n}{kx}\Big{)}-b\Big{(}\frac{n}{kx}\Big{)}}{(1+\widetilde{\nu})b\Big{(}\frac{n}{kx}\Big{)}}\cdot\frac{\overline{F}\Big{(}(1+\widetilde{\nu})b\Big{(}\frac{n}{kx}\Big{)}\Big{)}}{\overline{F}\Big{(}b\Big{(}\frac{n}{kx}\Big{)}\Big{)}}\cdot\overline{F}\Big{(}b\Big{(}\frac{n}{kx}\Big{)}\Big{)}$
$\displaystyle=\big{[}\alpha+o(1)\big{]}\frac{1+\nu-1}{1+\widetilde{\nu}}\big{[}(1+\widetilde{\nu})^{-\alpha}+o(1)\big{]}\frac{kx}{n}$
$\displaystyle=\big{[}1+o(1)\big{]}\Bigg{[}\frac{\alpha\nu}{(1+\widetilde{\nu})^{1+\alpha}}+\nu
o(1)\Bigg{]}\frac{kx}{n}$
$\displaystyle=\Bigg{[}\frac{\alpha\nu}{(1+\widetilde{\nu})^{1+\alpha}}+\nu
o(1)\Bigg{]}\frac{kx}{n},$ (C.10)

where the fourth step uses that the convergence

$\frac{\overline{F}\Big{(}(1+\widetilde{\nu})b\Big{(}\frac{n}{kx}\Big{)}\Big{)}}{\overline{F}\Big{(}b\Big{(}\frac{n}{kx}\Big{)}\Big{)}}-(1+\widetilde{\nu})^{-\alpha}\longrightarrow
0$

is uniform in $\widetilde{\nu}$ and uniform in $x$ by (C.9). By continuity of
$F(\cdot)$ in the tail (see Assumption 1 (ii)), (C.10) implies

$\displaystyle\overline{F}\Big{(}(1+\nu)b\Big{(}\frac{n}{kx}\Big{)}\Big{)}$
$\displaystyle=\overline{F}\Big{(}b\Big{(}\frac{n}{kx}\Big{)}\Big{)}-\Big{[}\frac{\alpha\nu}{(1+\widetilde{\nu})^{1+\alpha}}+\nu
o(1)\Big{]}\frac{kx}{n}$
$\displaystyle=\frac{kx}{n}\Big{[}1-\frac{\alpha\nu}{(1+\widetilde{\nu})^{1+\alpha}}+\nu
o(1)\Big{]}.$

To prove the second statement, it suffices to recall the Taylor expansion
$e^{\xi/\sqrt{k}}=1+\xi/\sqrt{k}+o(1/\sqrt{k})$ and apply a Taylor expansion
to $(1+\widetilde{\nu})^{1+\alpha}$. ∎

###### Proof of Lemma 2:.

We have to show that w.p.a. 1, as $n\to\infty$,

$\displaystyle I_{\left\\{U_{t}[1-s_{t}]-m_{t}>e^{\xi_{1}/\sqrt{k}}b(n/[kx]),\
U_{t-d}[1-s_{t-d}]-m_{t-d}>e^{\xi_{2}/\sqrt{k}}b(n/[ky])\right\\}}$
$\displaystyle\hskip 28.45274pt\leq
I_{\left\\{\widehat{U}_{t}(\bm{\theta})>e^{\xi_{1}/\sqrt{k}}b(n/[kx]),\
\widehat{U}_{t-d}(\bm{\theta})>e^{\xi_{2}/\sqrt{k}}b(n/[ky])\right\\}}$
$\displaystyle\hskip 28.45274pt\leq
I_{\left\\{U_{t}[1+s_{t}]+m_{t}>e^{\xi_{1}/\sqrt{k}}b(n/[kx]),\
U_{t-d}[1+s_{t-d}]+m_{t-d}>e^{\xi_{2}/\sqrt{k}}b(n/[ky])\right\\}}$

for all $\bm{\theta}\in N_{n}(\eta)$ and $t=\ell_{n},\ldots,n$. However, this
follows immediately from Assumption 3. ∎

###### Proof of Lemma 3:.

Let $\eta_{0}=1$; the case $\eta_{0}=-1$ can be dealt with similarly. Define

$w_{t}=I_{\left\\{m_{t}\leq\varepsilon_{0},\ m_{t-d}\leq\varepsilon_{0},\
s_{t}\leq\varepsilon_{0},\
s_{t-d}\leq\varepsilon_{0}\right\\}},\qquad\varepsilon_{0}>0.$

If $w_{t}=1$, there exists $\nu=\nu(\varepsilon_{0})>0$, such that

$1-\nu/2<(1+s_{t})^{-1}\leq 1\qquad\text{and}\qquad
1-\nu/2<(1+s_{t-d})^{-1}\leq 1.$

Furthermore, $b(x)\to\infty$ for $x\to\infty$ by Assumption 1, such that

$1-\nu/2<1-e^{K/\sqrt{k}}\varepsilon_{0}/b\big{(}n/[k\max\\{x,\,y\\}]\big{)}\leq
1$

for sufficiently large $n$. Then, for $\bm{\xi}\in[-K,0]^{2}$ (the other
quadrants can be dealt with similarly) and sufficiently large $n$,

$\displaystyle
0\leq\frac{\sqrt{n}}{k}\sum_{t=\ell_{n}}^{n}w_{t}B_{t}(\bm{\xi},\eta,\eta_{0})$
$\displaystyle=\frac{\sqrt{n}}{k}\sum_{t=\ell_{n}}^{n}w_{t}\Big{[}I_{\Big{\\{}U_{t}>e^{\xi_{1}/\sqrt{k}}(1+s_{t})^{-1}\big{(}1-e^{-\xi_{1}/\sqrt{k}}\frac{m_{t}}{b(n/[kx])}\big{)}b(n/[kx]),}$
$\hskip
85.35826pt{}_{U_{t-d}>e^{\xi_{2}/\sqrt{k}}(1+s_{t-d})^{-1}\big{(}1-e^{-\xi_{2}/\sqrt{k}}\frac{m_{t-d}}{b(n/[ky])}\big{)}b(n/[ky])\Big{\\}}}$
$\displaystyle\hskip 233.88168pt-
I_{\left\\{U_{t}>e^{\xi_{1}/\sqrt{k}}b(n/[kx]),\
U_{t-d}>e^{\xi_{2}/\sqrt{k}}b(n/[ky])\right\\}}\Big{]}$
$\displaystyle\leq\frac{\sqrt{n}}{k}\sum_{t=\ell_{n}}^{n}\Big{[}I_{\left\\{U_{t}>(1-\nu)b(n/[kx]),\
U_{t-d}>(1-\nu)b(n/[ky])\right\\}}-I_{\left\\{U_{t}>b(n/[kx]),\
U_{t-d}>b(n/[ky])\right\\}}\Big{]}$ $\displaystyle=\mathrel{\mathop{\mathchar
58\relax}}\frac{\sqrt{n}}{k}\sum_{t=\ell_{n}}^{n}I_{2t}.$

Use Markov’s inequality and Lemma 2.3 of Shao (1993), to obtain that

$\displaystyle\operatorname{P}\Big{\\{}\frac{\sqrt{n}}{k}\sum_{t=\ell_{n}}^{n}I_{2t}>\varepsilon\Big{\\}}$
$\displaystyle\leq\varepsilon^{-4}\frac{n^{2}}{k^{4}}\operatorname{E}\Big{[}\sum_{t=\ell_{n}}^{n}I_{2t}\Big{]}^{4}$
$\displaystyle=C\frac{n^{2}}{k^{4}}\Big{\\{}n^{2}\big{\\{}\operatorname{E}[I_{2t}^{2}]\big{\\}}^{2}+n\operatorname{E}[I_{2t}^{4}]\Big{\\}}.$
(C.11)

Similarly as for $I_{1t}$ in the proof of Proposition 2, we obtain using Lemma
1 that for sufficiently large $n$,

$\displaystyle\operatorname{E}\big{[}I_{2t}\big{]}$
$\displaystyle=\operatorname{P}\left\\{U_{t}>(1-\nu)b(n/[kx])\right\\}\operatorname{P}\left\\{U_{t-d}>(1-\nu)b(n/[ky])\right\\}$
$\displaystyle\hskip
170.71652pt-\operatorname{P}\left\\{U_{t}>b(n/[kx])\right\\}\operatorname{P}\left\\{U_{t-d}>b(n/[ky])\right\\}$
$\displaystyle=\frac{k^{2}}{n^{2}}xy\Big{[}1+\frac{\alpha\nu}{(1+\widetilde{\nu})^{1+\alpha}}+\nu
o(1)\Big{]}^{2}-\frac{k^{2}}{n^{2}}xy$ $\displaystyle\leq
C\frac{k^{2}}{n^{2}}\nu.$

Thus, the right-hand side of (C.11) can be bounded by

$C\frac{n^{2}}{k^{4}}\Big{[}n^{2}\frac{k^{4}}{n^{4}}+n\frac{k^{2}}{n^{2}}\Big{]}\nu=O(1)\nu.$
(C.12)

For a suitable choice of $\varepsilon_{0}$ in $w_{t}$, $\nu$ in (C.12) can be
chosen arbitrarily close to zero. Thus, the expression on the right-hand side
can be made arbitrarily small, yielding that

$\sup_{\bm{\xi}\in[-K,0]^{2}}\frac{\sqrt{n}}{k}\Bigg{|}\sum_{t=\ell_{n}}^{n}w_{t}B_{t}(\bm{\xi},\eta,\eta_{0}=1)\Bigg{|}=o_{\operatorname{P}}(1).$

By Assumption 3, $w_{\ell_{n}}=\ldots=w_{n}=1$ w.p.a. 1, as $n\to\infty$.
Thus, we obtain

$\sup_{\bm{\xi}\in[-K,0]^{2}}\frac{\sqrt{n}}{k}\Bigg{|}\sum_{t=\ell_{n}}^{n}B_{t}(\bm{\xi},\eta,\eta_{0}=1)\Bigg{|}=o_{\operatorname{P}}(1).$
(C.13)

By choosing $\ell_{n}\to\infty$ with $\ell_{n}=o(k/\sqrt{n})$ (recall
Assumption 4), we also have by boundedness of the
$B_{t}(\bm{\xi},\eta,\eta_{0}=1)$ that

$\sup_{\bm{\xi}\in[-K,0]^{2}}\frac{\sqrt{n}}{k}\Bigg{|}\sum_{t=d+1}^{\ell_{n}-1}B_{t}(\bm{\xi},\eta,\eta_{0}=1)\Bigg{|}\leq\frac{\sqrt{n}}{k}\ell_{n}=o(1).$
(C.14)

Combining (C.13) and (C.14) gives the desired conclusion. ∎

##  Appendix D Proof of Theorem 2

The proof of Theorem 2 is structured similarly as that of Theorem 1. It
requires four preliminary propositions, whose proofs are relegated to Appendix
D.1. We follow the same notational conventions as in Appendix C. Furthermore,
we define $D[0,1]^{2}$ to be the space of $\mathbb{R}^{D}$-valued functions on
$[0,1]\times[0,1]$ that are continuous from above, with limits from below (see
Bickel and Wichura, 1971, Sec. 3).

######  Proposition 5.

For i.i.d. $U_{t}$ with d.f. $F(\cdot)$ satisfying Assumption 1 (ii), it holds
that, as $n\to\infty$,

$\bm{M}_{n}(x,y)=\sqrt{n}\begin{pmatrix}\frac{1}{k}\sum_{t=2}^{n}\Big{[}I_{\left\\{U_{t}>b(\frac{n}{kx}),\
U_{t-1}>b(\frac{n}{ky})\right\\}}-\Big{(}\frac{k}{n}\Big{)}^{2}xy\Big{]}\\\
\vdots\\\ \frac{1}{k}\sum_{t=D+1}^{n}\Big{[}I_{\left\\{U_{t}>b(\frac{n}{kx}),\
U_{t-D}>b(\frac{n}{ky})\right\\}}-\Big{(}\frac{k}{n}\Big{)}^{2}xy\Big{]}\\\
\end{pmatrix}\overset{d}{\longrightarrow}\bm{B}(x,y)\quad\text{in}\
D[0,1]^{2},$

where $\bm{B}=(B_{1},\ldots,B_{D})^{\prime}$ is a $D$-dimensional Brownian
sheet with independent components, i.e., a zero-mean Gaussian process with
$\operatorname{Cov}\big{(}\bm{B}(x_{1},y_{1}),\bm{B}(x_{2},y_{2})\big{)}=\min(x_{1},x_{2})\min(y_{1},y_{2})\bm{I}_{D\times
D}$.

For the proof of Theorem 2, we have to justify the replacement of $U_{t}$ by
$\widehat{U}_{t}$, and $b(\frac{n}{kz})$ by $\widehat{U}_{(\lfloor
kz\rfloor+1)}$ ($z\in\\{x,y\\}$) in the indicators in Proposition 5. The
following three propositions serve that purpose.

######  Proposition 6.

For any $K>0$, it holds that

$\sup_{\bm{\xi}\in[-K,K]^{2}}\sup_{\bm{x}\in[0,K]^{2}}|M_{n}(\bm{x},\bm{\xi})-M_{n}(\bm{x},\bm{0})|=o_{\operatorname{P}}(n^{-1/2}).$

######  Proposition 7.

For any $K>0$, it holds that

$\sup_{\bm{\xi}\in[-K,K]^{2}}\sup_{\bm{x}\in[0,K]^{2}}|\widehat{M}_{n}(\bm{x},\bm{\xi})-M_{n}(\bm{x},\bm{\xi})|=o_{\operatorname{P}}(n^{-1/2}).$

######  Proposition 8.

For any $0<\iota<K<\infty$, it holds that

$\sup_{x\in[\iota,K]}|\xi_{n}(x)|=\sup_{x\in[\iota,K]}\left|\sqrt{k}\log\Bigg{(}\frac{\widehat{U}_{(\lfloor
kx\rfloor+1)}}{b(n/[kx])}\Bigg{)}\right|=O_{\operatorname{P}}(1).$

These four propositions allow us to prove Theorem 2.

###### Proof of Theorem 2:.

The outline of the proof is similar to that of Theorem 1. We show that for
each $d\in\\{1,\ldots,D\\}$,

$\sup_{\bm{x}=(x,y)^{\prime}\in[\iota,1]^{2}}\Bigg{|}\frac{1}{k}\sum_{t=d+1}^{n}I_{\\{U_{t}>b(\frac{n}{kx}),\
U_{t-d}>b(\frac{n}{ky})\\}}-\frac{1}{k}\sum_{t=d+1}^{n}I_{\\{\widehat{U}_{t}>\widehat{U}_{(\lfloor
kx\rfloor+1)},\ \widehat{U}_{t-d}>\widehat{U}_{(\lfloor
ky\rfloor+1)}\\}}\Bigg{|}=o_{\operatorname{P}}(n^{-1/2}).$ (D.1)

Combining this with Proposition 5, Lemma 4.7 of White (2001) and the
continuous mapping theorem (e.g., White, 2001, Theorem 7.20) gives us

$\mathcal{F}_{n}^{(D)}\overset{d}{\longrightarrow}\sum_{d=1}^{D}\int_{[\iota,1-\iota]}W_{d}^{2}(2-2z,2z)\,\mathrm{d}z,$

where $\\{W_{d}(\cdot,\cdot)\\}_{d=1,\ldots,D}$ are mutually independent
Brownian sheets, i.e., zero-mean Gaussian processes with covariance function
$\operatorname{Cov}\big{(}W_{d}(x_{1},y_{1}),W_{d}(x_{2},y_{2})\big{)}=\min(x_{1},x_{2})\min(y_{1},y_{2})$.
Computing the covariance function of $\\{W_{d}(2-2z,2z)\\}_{z\in[0,1]}$, we
find that
$\\{W_{d}(2-2z,2z)\\}_{z\in[0,1]}\overset{d}{=}\\{2B_{d}(z)\\}_{z\in[0,1]}$,
where $B_{d}(\cdot)$ is a standard Brownian bridge. Due to this,

$\sum_{d=1}^{D}\int_{[\iota,1-\iota]}W_{d}^{2}(2-2z,2z)\,\mathrm{d}z\overset{d}{=}4\sum_{d=1}^{D}\int_{[\iota,1-\iota]}B_{d}^{2}(z)\,\mathrm{d}z$

and, hence, the limit is as claimed.

So it remains to show (D.1). Decompose

$\displaystyle M_{n}(\bm{x},\bm{0})$
$\displaystyle=\frac{1}{k}\sum_{t=d+1}^{n}I_{\\{U_{t}>b(\frac{n}{kx}),\
U_{t-d}>b(\frac{n}{ky})\\}}$
$\displaystyle=\frac{1}{k}\sum_{t=d+1}^{n}I_{\\{U_{t}>b(\frac{n}{kx}),\
U_{t-d}>b(\frac{n}{ky})\\}}-\frac{1}{k}\sum_{t=d+1}^{n}I_{\\{U_{t}>e^{\xi_{1}/\sqrt{k}}b(\frac{n}{kx}),\
U_{t-d}>e^{\xi_{2}/\sqrt{k}}b(\frac{n}{ky})\\}}$ $\displaystyle\hskip
11.38092pt+\frac{1}{k}\sum_{t=d+1}^{n}I_{\\{U_{t}>e^{\xi_{1}/\sqrt{k}}b(\frac{n}{kx}),\
U_{t-d}>e^{\xi_{2}/\sqrt{k}}b(\frac{n}{ky})\\}}-\frac{1}{k}\sum_{t=d+1}^{n}I_{\\{\widehat{U}_{t}>e^{\xi_{1}/\sqrt{k}}b(\frac{n}{kx}),\
\widehat{U}_{t-d}>e^{\xi_{2}/\sqrt{k}}b(\frac{n}{ky})\\}}$
$\displaystyle\hskip
11.38092pt+\frac{1}{k}\sum_{t=d+1}^{n}I_{\\{\widehat{U}_{t}>e^{\xi_{1}/\sqrt{k}}b(\frac{n}{kx}),\
\widehat{U}_{t-d}>e^{\xi_{2}/\sqrt{k}}b(\frac{n}{ky})\\}}$
$\displaystyle=o_{\operatorname{P}}(n^{-1/2})+\widehat{M}_{n}(\bm{x},\bm{\xi}),$
(D.2)

where the $o_{\operatorname{P}}(n^{-1/2})$-term is uniform on
$\bm{\xi}\in[-K,K]^{2}$ and $\bm{x}\in[\iota,1]^{2}$ by Propositions 6 and 7.

Define $\bm{\xi}_{n}\mathrel{\mathop{\mathchar
58\relax}}=\bm{\xi}_{n}(\bm{x})\mathrel{\mathop{\mathchar
58\relax}}=\big{(}\xi_{n}(x),\ \xi_{n}(y)\big{)}^{\prime}$, such that
$\widehat{M}_{n}(\bm{x},\bm{\xi}_{n})=\frac{1}{k}\sum_{t=d+1}^{n}I_{\\{\widehat{U}_{t}>\widehat{U}_{(\lfloor
kx\rfloor+1)},\ \widehat{U}_{t-d}>\widehat{U}_{(\lfloor ky\rfloor+1)}\\}}$.
Then, to show (D.1) or, equivalently, that
$\sup_{\bm{x}\in[\iota,1]^{2}}|\widehat{M}_{n}(\bm{x},\bm{\xi}_{n})-M_{n}(\bm{x},\bm{0})|=o_{\operatorname{P}}(n^{-1/2})$,
we have to prove that for any $\varepsilon>0$ and $\delta>0$

$\operatorname{P}\left\\{\sqrt{n}\sup_{\bm{x}\in[\iota,1]^{2}}|\widehat{M}_{n}(\bm{x},\bm{\xi}_{n})-M_{n}(\bm{x},\bm{0})|>\varepsilon\right\\}\leq\delta$

for sufficiently large $n$. By Proposition 8, we can choose $K>0$ such that
$\operatorname{P}\\{\sup_{\bm{x}\in[\iota,1]^{2}}|\bm{\xi}_{n}|>K\\}\leq\delta/2$
for sufficiently large $n$. Furthermore, by (D.2),
$\operatorname{P}\\{\sqrt{n}\sup_{\bm{\xi}\in[-K,K]^{2}}\sup_{\bm{x}\in[\iota,1]^{2}}|\widehat{M}_{n}(\bm{x},\bm{\xi})-M_{n}(\bm{x},\bm{0})|>\varepsilon\\}\leq\delta/2$
for large $n$. Using these two results, we obtain

$\displaystyle\operatorname{P}$
$\displaystyle\Bigg{\\{}\sqrt{n}\sup_{\bm{x}\in[\iota,1]^{2}}|\widehat{M}_{n}(\bm{x},\bm{\xi}_{n})-M_{n}(\bm{x},\bm{0})|>\varepsilon\Bigg{\\}}$
$\displaystyle\leq\operatorname{P}\Bigg{\\{}\sup_{\bm{x}\in[\iota,1]^{2}}|\bm{\xi}_{n}|>K\Bigg{\\}}+\operatorname{P}\Bigg{\\{}\sqrt{n}\sup_{\bm{\xi}\in[-K,K]^{2}}\sup_{\bm{x}\in[\iota,1]^{2}}|\widehat{M}_{n}(\bm{x},\bm{\xi})-M_{n}(\bm{x},\bm{0})|>\varepsilon,\
\sup_{\bm{x}\in[\iota,1]^{2}}|\bm{\xi}_{n}|\leq K\Bigg{\\}}$
$\displaystyle\leq\frac{\delta}{2}+\frac{\delta}{2}=\delta,$

i.e., (D.1). The conclusion follows. ∎

###  D.1 Proofs of Propositions 5–8

###### Proof of Proposition 5:.

The corollary on p. 1664 of Bickel and Wichura (1971) shows that to prove the
claim, it suffices to show convergence of the finite-dimensional distributions
and tightness. First, to prove convergence of the finite-dimensional
distributions, we have to show that for all pairwise distinct
$(x_{i},y_{i})\in[0,1]^{2}$, $i=1,\ldots,m$,

$\big{(}\bm{M}_{n}(x_{1},y_{1}),\ldots,\bm{M}_{n}(x_{m},y_{m})\big{)}^{\prime}\overset{d}{\longrightarrow}\big{(}\bm{B}(x_{1},y_{1}),\ldots,\bm{B}(x_{m},y_{m})\big{)}^{\prime}.$
(D.3)

Second, to prove tightness it suffices by Theorem 26.23 of Davidson (1994) to
do so for each of the marginals separately. To that end, denote the components
of $\bm{M}_{n}(x,y)$ by $M_{n}^{(d)}(x,y)$ $(d=1,\ldots,D)$ and define

$\displaystyle B$ $\displaystyle=(x_{1},x_{2}]\times(y_{1},y_{2}],$
$\displaystyle M_{n}^{(d)}(B)$
$\displaystyle=M_{n}^{(d)}(x_{2},y_{2})-M_{n}^{(d)}(x_{1},y_{2})-M_{n}^{(d)}(x_{2},y_{1})+M_{n}^{(d)}(x_{1},y_{1}).$

By their Theorem 3 and the subsequent comment together with their Equation
(3), Bickel and Wichura (1971) show that for tightness it suffices to prove
that for $d\in\\{1,\ldots,D\\}$

$\displaystyle\operatorname{E}\Big{[}\big{\\{}M_{n}^{(d)}(B_{1})\big{\\}}^{2}\big{\\{}M_{n}^{(d)}(B_{2})\big{\\}}^{2}\Big{]}$
$\displaystyle\leq K(x_{2}-x_{1})^{2}(y_{2}-y)(y-y_{1}),$ (D.4)
$\displaystyle\operatorname{E}\Big{[}\big{\\{}M_{n}^{(d)}(B_{3})\big{\\}}^{2}\big{\\{}M_{n}^{(d)}(B_{4})\big{\\}}^{2}\Big{]}$
$\displaystyle\leq K(x-x_{1})(x_{2}-x)(y_{2}-y_{1})^{2},$ (D.5)

where, for $0\leq x_{1}<x\leq x_{2}\leq 1$ and $0\leq y_{1}<y\leq y_{2}\leq
1$,

$\displaystyle B_{1}$ $\displaystyle=(x_{1},x_{2}]\times(y_{1},y],\quad
B_{2}=(x_{1},x_{2}]\times(y,y_{2}],$ $\displaystyle B_{3}$
$\displaystyle=(x_{1},x]\times(y_{1},y_{2}],\quad
B_{4}=(x,x_{2}]\times(y_{1},y_{2}].$

We first show convergence of the finite-dimensional distributions, i.e.,
(D.3). We do so for $m=2$; the case $m>2$ being only notationally more
complicated. By the Cramér–Wold device (Davidson, 1994, Thm. 25.5) it suffices
to show that, as $n\to\infty$,

$\bm{\lambda}_{1}^{\prime}\bm{M}_{n}(x_{1},y_{1})+\bm{\lambda}_{2}^{\prime}\bm{M}_{n}(x_{2},y_{2})\overset{d}{\longrightarrow}\bm{\lambda}_{1}^{\prime}\bm{B}(x_{1},y_{1})+\bm{\lambda}_{2}^{\prime}\bm{B}(x_{2},y_{2})$

for any
$\bm{\lambda}=(\bm{\lambda}_{1}^{\prime},\bm{\lambda}_{2}^{\prime})^{\prime}=(\lambda_{1,1},\ldots,\lambda_{1,D},\lambda_{2,1},\ldots,\lambda_{2,D})^{\prime}$
with $\bm{\lambda}^{\prime}\bm{\lambda}=1$. The proof of this convergence is
similar to that of Proposition 1, and uses Theorem 5.20 of White (2001). We
re-define

$\displaystyle\overline{\sigma}_{n}^{2}$
$\displaystyle=\operatorname{Var}\Big{(}\bm{\lambda}_{1}^{\prime}\bm{M}_{n}(x_{1},y_{1})+\bm{\lambda}_{2}^{\prime}\bm{M}_{n}(x_{2},y_{2})\Big{)}$
$\displaystyle=\operatorname{Var}\Big{(}\bm{\lambda}_{1}^{\prime}\bm{M}_{n}(x_{1},y_{1})\Big{)}+\operatorname{Var}\Big{(}\bm{\lambda}_{2}^{\prime}\bm{M}_{n}(x_{2},y_{2})\Big{)}+2\operatorname{Cov}\Big{(}\bm{\lambda}_{1}^{\prime}\bm{M}_{n}(x_{1},y_{1}),\
\bm{\lambda}_{2}^{\prime}\bm{M}_{n}(x_{2},y_{2})\Big{)}.$ (D.6)

From the proof of Proposition 1, we get that

$\operatorname{Var}\Big{(}\bm{\lambda}_{i}^{\prime}\bm{M}_{n}(x_{i},y_{i})\Big{)}=\sum_{d=1}^{D}\lambda_{i,d}^{2}x_{i}y_{i}+o(1)\qquad(i=1,2).$
(D.7)

Thus, it remains to compute the covariance term. Recall from the proof of
Proposition 1 that

$Z_{n,t}^{(d)}(x,y)=\begin{cases}0,&t=1,\ldots,d,\\\
\frac{n}{k}\Big{[}I_{\left\\{U_{t}>b(\frac{n}{kx}),\
U_{t-d}>b(\frac{n}{ky})\right\\}}-\Big{(}\frac{k}{n}\Big{)}^{2}xy\Big{]},&t>d.\end{cases}$

With this

$\displaystyle\operatorname{Cov}\Big{(}\bm{\lambda}_{1}^{\prime}\bm{M}_{n}(x_{1},y_{1}),\
\bm{\lambda}_{2}^{\prime}\bm{M}_{n}(x_{2},y_{2})\Big{)}$ $\displaystyle\hskip
28.45274pt=\operatorname{Cov}\Big{(}\sqrt{n}\sum_{d_{1}=1}^{D}\frac{1}{n}\sum_{t=1}^{n}\lambda_{1,d_{1}}Z_{n,t}^{(d_{1})}(x_{1},y_{1}),\
\sqrt{n}\sum_{d_{2}=1}^{D}\frac{1}{n}\sum_{t=1}^{n}\lambda_{2,d_{2}}Z_{n,t}^{(d_{2})}(x_{2},y_{2})\Big{)}$
$\displaystyle\hskip
28.45274pt=\sum_{d_{1}=1}^{D}\sum_{d_{2}=1}^{D}\frac{1}{n}\operatorname{Cov}\Big{(}\sum_{t=1}^{n}\lambda_{1,d_{1}}Z_{n,t}^{(d_{1})}(x_{1},y_{1}),\
\sum_{t=1}^{n}\lambda_{2,d_{2}}Z_{n,t}^{(d_{2})}(x_{2},y_{2})\Big{)}.$ (D.8)

For $d_{1}\neq d_{2}$, we obtain that

$\displaystyle\frac{1}{n}\operatorname{Cov}$
$\displaystyle\Big{(}\sum_{t=1}^{n}\lambda_{1,d_{1}}Z_{n,t}^{(d_{1})}(x_{1},y_{1}),\
\sum_{t=1}^{n}\lambda_{2,d_{2}}Z_{n,t}^{(d_{2})}(x_{2},y_{2})\Big{)}$
$\displaystyle=\frac{1}{n}\sum_{s=1}^{n}\sum_{t=1}^{n}\lambda_{1,d_{1}}\lambda_{2,d_{2}}\operatorname{Cov}\Big{(}Z_{n,s}^{(d_{1})}(x_{1},y_{1}),\
Z_{n,t}^{(d_{2})}(x_{2},y_{2})\Big{)}$
$\displaystyle=\frac{1}{n}\Bigg{\\{}\sum_{t=1}^{n}\lambda_{1,d_{1}}\lambda_{2,d_{2}}\operatorname{Cov}\Big{(}Z_{n,t}^{(d_{1})}(x_{1},y_{1}),\
Z_{n,t}^{(d_{2})}(x_{2},y_{2})\Big{)}$ $\displaystyle\hskip
28.45274pt+\Bigg{(}\sum_{s-t=d_{1}}+\sum_{t-s=d_{2}}+\sum_{t-s=d_{2}-d_{1}}\Bigg{)}\lambda_{1,d_{1}}\lambda_{2,d_{2}}\operatorname{Cov}\Big{(}Z_{n,s}^{(d_{1})}(x_{1},y_{1}),\
Z_{n,t}^{(d_{2})}(x_{2},y_{2})\Big{)}\Bigg{\\}}$ $\displaystyle=O(k/n)=o(1)$

by a relation similar to (C.4). This shows that the right-hand side of (D.8)
equals

$\displaystyle\sum_{d=1}^{D}\frac{1}{n}$
$\displaystyle\operatorname{Cov}\Big{(}\sum_{t=1}^{n}\lambda_{1,d}Z_{n,t}^{(d)}(x_{1},y_{1}),\
\sum_{t=1}^{n}\lambda_{2,d}Z_{n,t}^{(d)}(x_{2},y_{2})\Big{)}+o(1)$
$\displaystyle=\sum_{d=1}^{D}\frac{1}{n}\sum_{t=1}^{n}\lambda_{1,d}\lambda_{2,d}\operatorname{Cov}\Big{(}Z_{n,t}^{(d)}(x_{1},y_{1}),\
Z_{n,t}^{(d)}(x_{2},y_{2})\Big{)}$ $\displaystyle\hskip
28.45274pt+\sum_{d=1}^{D}\frac{1}{n}\sum_{|t-s|=d}\lambda_{1,d}\lambda_{2,d}\operatorname{Cov}\Big{(}Z_{n,s}^{(d)}(x_{1},y_{1}),\
Z_{n,t}^{(d)}(x_{2},y_{2})\Big{)}+o(1)$
$\displaystyle=\sum_{d=1}^{D}\lambda_{1,d}\lambda_{2,d}\min(x_{1},x_{2})\min(y_{1},y_{2})+o(1),$

where the last line follows from (C.4) and the fact that

$\displaystyle\operatorname{Cov}$
$\displaystyle\Big{(}Z_{n,t}^{(d)}(x_{1},y_{1}),\
Z_{n,t}^{(d)}(x_{2},y_{2})\Big{)}=\operatorname{E}\big{[}Z_{n,t}^{(d)}(x_{1},y_{1})Z_{n,t}^{(d)}(x_{2},y_{2})\big{]}-\operatorname{E}\big{[}Z_{n,t}^{(d)}(x_{1},y_{1})\big{]}\operatorname{E}\big{[}Z_{n,t}^{(d)}(x_{2},y_{2})\big{]}$
$\displaystyle=\Big{(}\frac{n}{k}\Big{)}^{2}\operatorname{P}\Big{\\{}U_{t}>b\big{(}n/[k\min(x_{1},x_{2})]\big{)},\
U_{t-d}>b\big{(}n/[k\min(y_{1},y_{2})]\big{)}\Big{\\}}-\Big{(}\frac{n}{k}\Big{)}^{2}\Big{(}\frac{k}{n}\Big{)}^{2}x_{1}y_{1}\Big{(}\frac{k}{n}\Big{)}^{2}x_{2}y_{2}$
$\displaystyle=\min(x_{1},x_{2})\min(y_{1},y_{2})+o(1).$

Overall, we get that

$\operatorname{Cov}\Big{(}\bm{\lambda}_{1}^{\prime}\bm{M}_{n}(x_{1},y_{1}),\
\bm{\lambda}_{2}^{\prime}\bm{M}_{n}(x_{2},y_{2})\Big{)}=\sum_{d=1}^{D}\lambda_{1,d}\lambda_{2,d}\min(x_{1},x_{2})\min(y_{1},y_{2})+o(1).$

Plugging this and (D.7) into (D.6) gives

$\overline{\sigma}_{n}^{2}=\sum_{d=1}^{D}\big{[}\lambda_{1,d}^{2}x_{1}y_{1}+\lambda_{2,d}^{2}x_{2}y_{2}+2\lambda_{1,d}\lambda_{2,d}\min(x_{1},x_{2})\min(y_{1},y_{2})\big{]}+o(1).$

The limit of $\overline{\sigma}_{n}^{2}$ then is easily seen to equal

$\displaystyle\operatorname{Var}$
$\displaystyle\big{(}\bm{\lambda}_{1}^{\prime}\bm{B}(x_{1},y_{1})+\bm{\lambda}_{2}^{\prime}\bm{B}(x_{2},y_{2})\big{)}$
$\displaystyle=\operatorname{Var}\big{(}\bm{\lambda}_{1}^{\prime}\bm{B}(x_{1},y_{1})\big{)}+\operatorname{Var}\big{(}\bm{\lambda}_{2}^{\prime}\bm{B}(x_{2},y_{2})\big{)}+2\operatorname{Cov}\big{(}\bm{\lambda}_{1}^{\prime}\bm{B}(x_{1},y_{1}),\
\bm{\lambda}_{2}^{\prime}\bm{B}(x_{2},y_{2})\big{)}$
$\displaystyle=\sum_{d=1}^{D}\lambda_{1,d}^{2}\operatorname{Var}\big{(}B_{d}(x_{1},y_{1})\big{)}+\sum_{d=1}^{D}\lambda_{2,d}^{2}\operatorname{Var}\big{(}B_{d}(x_{1},y_{1})\big{)}$
$\displaystyle\hskip
142.26378pt+2\operatorname{Cov}\Big{(}\sum_{d=1}^{D}\lambda_{1,d}B_{d}(x_{1},y_{1}),\
\sum_{d=1}^{D}\lambda_{2,d}B_{d}(x_{2},y_{2})\Big{)}$
$\displaystyle=\sum_{d=1}^{D}\big{[}\lambda_{1,d}^{2}x_{1}y_{1}+\lambda_{2,d}^{2}x_{2}y_{2}+2\lambda_{1,d}\lambda_{2,d}\min(x_{1},x_{2})\min(y_{1},y_{2})\big{]},$

as desired. The remaining conditions of Theorem 5.20 of White (2001) can be
checked easily and, hence, (D.3) follows.

It remains to show tightness of the marginals. We only show (D.4), because
(D.5) can be proved in exactly the same manner. Set

$\displaystyle G_{t}$
$\displaystyle=I_{\Big{\\{}U_{t}\in\big{(}b(\frac{n}{kx_{2}}),\
b(\frac{n}{kx_{1}})\big{]},\ U_{t-d}\in\big{(}b(\frac{n}{ky}),\
b(\frac{n}{ky_{1}})\big{]}\Big{\\}}}-\Big{(}\frac{k}{n}\Big{)}^{2}(x_{2}-x_{1})(y-y_{1})$
$\displaystyle=I_{\Big{\\{}U_{t}\in\big{(}b(\frac{n}{kx_{2}}),\
b(\frac{n}{kx_{1}})\big{]},\ U_{t-d}\in\big{(}b(\frac{n}{ky}),\
b(\frac{n}{ky_{1}})\big{]}\Big{\\}}}-p_{G},$ $\displaystyle H_{t}$
$\displaystyle=I_{\Big{\\{}U_{t}\in\big{(}b(\frac{n}{kx_{2}}),\
b(\frac{n}{kx_{1}})\big{]},\ U_{t-d}\in\big{(}b(\frac{n}{ky_{2}}),\
b(\frac{n}{ky})\big{]}\Big{\\}}}-\Big{(}\frac{k}{n}\Big{)}^{2}(x_{2}-x_{1})(y_{2}-y)$
$\displaystyle=I_{\Big{\\{}U_{t}\in\big{(}b(\frac{n}{kx_{2}}),\
b(\frac{n}{kx_{1}})\big{]},\ U_{t-d}\in\big{(}b(\frac{n}{ky_{2}}),\
b(\frac{n}{ky})\big{]}\Big{\\}}}-p_{H}.$

It is easy to check that

$\displaystyle M_{n}^{(d)}(B_{1})$
$\displaystyle=M_{n}^{(d)}(x_{2},y)-M_{n}^{(d)}(x_{1},y)-M_{n}^{(d)}(x_{2},y_{1})+M_{n}^{(d)}(x_{1},y_{1})$
$\displaystyle=\frac{\sqrt{n}}{k}\sum_{t=d+1}^{n}G_{t},$ $\displaystyle
M_{n}^{(d)}(B_{2})$ $\displaystyle=\frac{\sqrt{n}}{k}\sum_{t=d+1}^{n}H_{t}.$

Thus, we can write

$\displaystyle\operatorname{E}$
$\displaystyle\Big{[}\big{\\{}M_{n}^{(d)}(B_{1})\big{\\}}^{2}\big{\\{}M_{n}^{(d)}(B_{2})\big{\\}}^{2}\Big{]}$
$\displaystyle=\frac{n^{2}}{k^{4}}\sum_{t_{1}=d+1}^{n}\sum_{t_{2}=d+1}^{n}\sum_{t_{3}=d+1}^{n}\sum_{t_{4}=d+1}^{n}\operatorname{E}[G_{t_{1}}G_{t_{2}}H_{t_{3}}H_{t_{4}}]$
$\displaystyle=\frac{n^{2}}{k^{4}}\left\\{\sum_{t=d+1}^{n}\operatorname{E}[G_{t}G_{t}H_{t}H_{t}]+\sum_{t_{1}\neq
t_{2}}\operatorname{E}[G_{t_{1}}^{2}H_{t_{2}}^{2}]+2\sum_{t_{1}\neq
t_{2}}\operatorname{E}[G_{t_{1}}H_{t_{1}}G_{t_{2}}H_{t_{2}}]\right\\}+R_{n}.$
(D.9)

Here, the first term in curly brackets on the right-hand side of (D.9)
collects those terms that would arise for serially independent
$\\{(G_{t},H_{t})^{\prime}\\}$, whereas the remainder $R_{n}$ collects those
terms that arise due to the $d$-dependence of $\\{(G_{t},H_{t})^{\prime}\\}$.
Exploiting the fact that $G_{t}$ and $H_{t}$ are centered Bernoulli-
distributed random variables with respective success probabilities $p_{G}$ and
$p_{H}$, the first term on the right-hand side of (D.9) reduces to (with
$\widetilde{n}=n-d$)

$\displaystyle\frac{n^{2}}{k^{4}}$
$\displaystyle\left\\{\widetilde{n}\operatorname{E}[G_{1}^{2}H_{1}^{2}]+\widetilde{n}(\widetilde{n}-1)\operatorname{E}[G_{1}^{2}]\operatorname{E}[H_{1}^{2}]+2\widetilde{n}(\widetilde{n}-1)\operatorname{E}^{2}[G_{1}H_{1}]\right\\}$
$\displaystyle=\frac{n^{2}}{k^{4}}\left\\{\widetilde{n}[p_{G}^{2}p_{H}+p_{G}p_{H}^{2}+p_{G}^{2}p_{H}^{2}]+\widetilde{n}(\widetilde{n}-1)[p_{G}p_{H}(1-p_{G})(1-p_{H})]+2\widetilde{n}(\widetilde{n}-1)p_{G}^{2}p_{H}^{2}\right\\}$
$\displaystyle\leq K(x_{2}-x_{1})^{2}(y_{2}-y)(y-y_{1}).$

Lengthy but straightforward calculations establish a similar bound for
$R_{n}$. In sum, we obtain

$\operatorname{E}\Big{[}\big{\\{}M_{n}^{(d)}(B_{1})\big{\\}}^{2}\big{\\{}M_{n}^{(d)}(B_{2})\big{\\}}^{2}\Big{]}\leq
K(x_{2}-x_{1})^{2}(y_{2}-y)(y-y_{1}).$

This completes the proof. ∎

###### Proof of Proposition 6:.

Consider the supremum over $\bm{\xi}\in[-K,0]^{2}$; the cases where $\bm{\xi}$
lies in the other quadrants can be dealt with similarly. By definition of
$M_{n}(\cdot,\cdot)$, it holds that

$\sup_{\bm{\xi}\in[-K,0]^{2}}\sup_{\bm{x}\in[0,K]^{2}}|M_{n}(\bm{x},\bm{\xi})-M_{n}(\bm{x},\bm{0})|\leq\sup_{\bm{x}\in[0,K]^{2}}|M_{n}(\bm{x},-(K,K)^{\prime})-M_{n}(\bm{x},\bm{0})|.$
(D.10)

Let $\rho>0$. Assume without loss of generality that $K/\rho\in\mathbb{N}$.
Put

$\mathcal{L}=\left\\{(l_{1},l_{2})^{\prime}\in\mathbb{N}_{0}^{2}\
\mathrel{\mathop{\mathchar 58\relax}}\
(l_{1},l_{2})^{\prime}\in[0,K/\rho]^{2}\right\\}.$

Then, bound the right-hand side of (D.10) by

$\displaystyle\max_{(l_{1},l_{2})^{\prime}\in\mathcal{L}}|M_{n}((l_{1},l_{2})^{\prime}\rho,-(K,K)^{\prime})-M_{n}((l_{1},l_{2})^{\prime}\rho,\bm{0})|$
$\displaystyle\hskip
56.9055pt+\sup_{|\bm{x}_{1}-\bm{x}_{2}|\leq\rho}\big{|}\big{\\{}M_{n}(\bm{x}_{1},-(K,K)^{\prime})-M_{n}(\bm{x}_{1},\bm{0})\big{\\}}-\big{\\{}M_{n}(\bm{x}_{2},-(K,K)^{\prime})-M_{n}(\bm{x}_{2},\bm{0})\\}\big{|}$
$\displaystyle\leq\max_{(l_{1},l_{2})^{\prime}\in\mathcal{L}}|M_{n}((l_{1},l_{2})^{\prime}\rho,-(K,K)^{\prime})-M_{n}((l_{1},l_{2})^{\prime}\rho,\bm{0})|+\sup_{|\bm{x}_{1}-\bm{x}_{2}|\leq\rho}|M_{n}(\bm{x}_{1},\bm{0})-M_{n}(\bm{x}_{2},\bm{0})|$
$\displaystyle\hskip
56.9055pt+\sup_{|\bm{x}_{1}-\bm{x}_{2}|\leq\rho}|M_{n}(\bm{x}_{1},-(K,K)^{\prime})-M_{n}(\bm{x}_{2},-(K,K)^{\prime})|$
$\displaystyle=\mathrel{\mathop{\mathchar 58\relax}}A_{1n}+B_{1n}+C_{1n}.$

Here, $A_{1n}$ is the maximum of the function
$\bm{x}\mapsto|M_{n}(\bm{x},-(K,K)^{\prime})-M_{n}(\bm{x},\bm{0})|$ on the
grid $\mathcal{L}$, and $B_{1n}+C_{1n}$ bound the function’s maximum absolute
increase over a grid quadrangle.

By subadditivity and Proposition 2, we obtain that

$\displaystyle\operatorname{P}\\{\sqrt{n}A_{1n}\geq\varepsilon\\}$
$\displaystyle=\operatorname{P}\Big{\\{}\cup_{(l_{1},l_{2})^{\prime}\in\mathcal{L}}\big{\\{}\sqrt{n}\big{|}M_{n}((l_{1},l_{2})^{\prime}\rho,-(K,K)^{\prime})-M_{n}((l_{1},l_{2})^{\prime}\rho,\bm{0})\big{|}\geq\varepsilon\big{\\}}\Big{\\}}$
$\displaystyle\leq\sum_{(l_{1},l_{2})^{\prime}\in\mathcal{L}}\operatorname{P}\Big{\\{}\sqrt{n}\big{|}M_{n}((l_{1},l_{2})^{\prime}\rho,-(K,K)^{\prime})-M_{n}((l_{1},l_{2})^{\prime}\rho,\bm{0})\big{|}\geq\varepsilon\Big{\\}}=o(1).$

Thus, $A_{1n}=o_{\operatorname{P}}(n^{-1/2})$.

Since $B_{1n}=o_{\operatorname{P}}(n^{-1/2})$ can be shown similarly as
$C_{1n}=o_{\operatorname{P}}(n^{-1/2})$, we only prove the former. Define

$\widetilde{\mathcal{L}}=\left\\{(l_{1},l_{2})^{\prime}\in\mathbb{N}_{0}^{2}\
\mathrel{\mathop{\mathchar 58\relax}}\
[l_{1},l_{1}+2]\times[l_{2},l_{2}+2]\subset[0,K/\rho]^{2}\right\\}.$

Use a monotonicity argument to deduce that

$B_{1n}\leq\max_{(l_{1},l_{2})^{\prime}\in\widetilde{\mathcal{L}}}\big{|}M_{n}((l_{1}+2,l_{2}+2)^{\prime}\rho,\bm{0})-M_{n}((l_{1},l_{2})^{\prime}\rho,\bm{0})\big{|}.$

Put

$I_{3t}\mathrel{\mathop{\mathchar
58\relax}}=I_{\Big{\\{}U_{t}>b\big{(}\frac{n}{k[l_{1}+2]\rho}\big{)},\
U_{t-d}>b\big{(}\frac{n}{k[l_{2}+2]\rho}\big{)}\Big{\\}}}-I_{\Big{\\{}U_{t}>b\big{(}\frac{n}{kl_{1}\rho}\big{)},\
U_{t-d}>b\big{(}\frac{n}{kl_{2}\rho}\big{)}\Big{\\}}}.$

For fixed $(l_{1},l_{2})^{\prime}\in\widetilde{\mathcal{L}}$, use Markov’s
inequality to obtain that

$\displaystyle\operatorname{P}$
$\displaystyle\left\\{\sqrt{n}\big{|}M_{n}((l_{1}+2,l_{2}+2)^{\prime}\rho,\bm{0})-M_{n}((l_{1},l_{2})^{\prime}\rho,\bm{0})\big{|}\geq\varepsilon\right\\}$