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Cautions on Tail Index Regressions and a Comparative Study with Extremal Quantile Regression
\title{Cautions on Tail Index Regressions and a Comparative Study with Extremal
Quantile Regression\thanks{This paper was previously circulated under the title \textquotedblleft Cautions
on Tail Index Regressions\textquotedblright . We thank Yonghui Zhang
for helpful discussions.}}
\author{{Thomas T. Yang}\thanks{Corresponding author. The Australian National University, Canberra,
ACT 0200, Australia. Email: \protect\protect\protect\protect\protect\protect\protect\protect\protect\protect\protect\protect\protect\protect\protect\protect\protect\href{http://[email removed]}{[email removed]}.} \\
{\normalsize{}{}{}{}{}Australian National University}}
\date{\today}
\maketitle
\onehalfspacing
\begin{abstract}
We re-visit tail the index regressions framework. For linear specifications,
we find that the usual full rank condition can fail because conditioning
on extreme outcomes causes regressors to degenerate to constants.
Taking this into account, we provide additional regular conditions
and establish its asymptotics in this irregular setup. For more general
specifications, the conditional distribution of the covariates in
the tails concentrates on the values at which the tail index is minimized.
Such issue does not exist for the extremal quantile regression framework,
where the tail index is assumed constant. Simulations support these
findings. Using daily S\&P 500 returns, we find that the extremal
quantile regression framework appears more suitable than tail-index
regression with respect to the tail rank condition.
\vspace{4mm}
\medskip{}
\par\noindent {\small{\em Keywords\/}: Tail Index Regression, Extremal Quantile Regression, Rank
Condition, Irregular Convergence Rate}
\vspace{2cm}
\end{abstract}
\setstretch{1.6}
\section{Introduction}
Tail index estimation has been widely applied in modeling extreme
events, such as natural environmental and financial market extremes;
see, for example, \citet{de_Haan_Ferreira} and references within.
Building on the seminal paper of \citet{Hill1975}, subsequent work
has sought to model the tail index as a function of covariates, either
parametrically (e.g., \citet{HallTajvidi} and \citet{WangTsai2009}),
semiparametrically (e.g., \citet{LiLengYou2022}), or nonparametrically
(e.g., \citet{deHaan03072021}). The literature is big, we just name
a few important and representative developments. In this paper, we
caution some issue with tail index regressions. We emphasize that
our purpose is not to criticize the existing literature, as these
works are valuable and important contributions. Rather, our goal is
to highlight issues that researchers should be aware of when applying
tail index regressions.
To set the stage, suppose we observe $(Y,X)$, where $X$ enters the
tail index function of $Y$. Following the literature, we assume
\begin{equation}
\bar{F}(y\mid x)=1-F(y\mid x)=y^{-\alpha(x)}L(y;x),\quad\text{for }y\geq w,\label{eq:surviveCDF}
\end{equation}
where $F(y\mid x)$ is the cumulative distribution function of $Y$
conditional on $X=x,$ and $L(y;x)$ is a slowly varying function
that satisfies $L(yt;x)/L(y;x)\to1$ for any $t>0$ as $y\to\infty$.
The inclusion of $L(y;x)$ makes the distributional assumption in
\eqref{eq:surviveCDF} much more general than formulations without
it; see \citet{de_Haan_Ferreira} and \citet{WangTsai2009} for discussion.
Because of the presence of $L(y;x)$, the maximum likelihood estimation
based on the approximation, $\bar{F}(y\mid x)\approx Cy^{-\alpha(x)}\textrm{ for some }C>0,$
contains a bias term, which vanishes only as $w\to\infty$. Therefore,
as the sample size grows, one must take a larger threshold $w$ and
restrict the analysis to samples with $Y>w$ in order to eliminate
the bias asymptotically.
We find that the full rank condition on $\mathbb{E}(XX'|Y>w)$ is
likely to fail asymptotically in tail index regressions as $w\rightarrow\infty$.
A similar phenomenon arises in semiparametric and nonparametric tail
index regressions. The intuition follows from Bayes’ theorem: conditional
on large values of $Y$, the distribution of $X$ degenerates toward
the point $x$ where $\alpha(x)$ attains its minimum, because extreme
events are more likely to occur when $\alpha(x)$ is smaller. Since
only observations with $Y>w$ are retained and the threshold $w\to\infty$
as the sample size grows, the effective variation in $X$ diminishes,
leading to a failure of the full rank condition.
Specifically, the main results on tail index regressions are as follows:
\begin{enumerate}
\item In the parametric case where $\alpha(X)=\exp\left(X'\theta^{*}\right)$
and the usage of the $\exp$ function is to ensure the index is always
positive, we show that $\mathbb{E}(XX'\mid Y>w)$ becomes nearly singular
as $w$ grows large, under fairly general conditions, even when $\mathbb{E}(XX')$
is nonsingular. As a consequence, the convergence rate of the parametric
estimator in \citet{WangTsai2009} is slower than originally anticipated.
We provide conditions under which the nearly singular $\mathbb{E}(XX'\mid Y>w)$
can still be accommodated, and establish the corresponding asymptotic
properties.
\item In semiparametric cases where $\alpha(X)$ is a general smooth function
of $X$, we show that the conditional density $f(x\mid Y>w)$ converges
to zero for all $x$ except at points where $\alpha(x)$ attains its
minimum, even though the unconditional density $f(x)$ is uniformly
bounded away from zero. This property may undermine the estimation
of the nonparametric component in semiparametric models.
\item The nonparametric settings are robust to this issue due to the fact
that only local observations are used and $\alpha(\cdot)$ behaves
like a constant locally.
\end{enumerate}
As a comparison, we investigate this rank condition for the extremal
quantile regression, pioneered by \citet{Chernozhukov2005}. The framework
in \citet{Chernozhukov2005} assumed the conditional survival function
satisfies
\[
\bar{F}\left(y|x\right)=k\left(x\right)\cdot y^{-\alpha}L\left(y\right),
\]
where $k\left(x\right)$ is some positive function, and $L\left(y\right)$
is a slowly varying function. The key difference from tail index regressions
is $\alpha$ being a constant, and the heterogeneity comes solely
from $k\left(x\right)$. We find that the rank condition issue on
$\mathbb{E}(XX'|Y>w)$ does not show up for extremal quantile regression
for very general $k\left(x\right),$ although $X|\left(Y>w\right)$
appears more often around the mode of $k\left(x\right)$. This highlights
the root of the rank condition issue: a varying tail index $\alpha$.
A set of small-scale simulation studies confirms these findings. We
further study extreme daily losses (negative returns) of the S\&P
500 over a long time horizon. As losses become more extreme, we find
that the occurrence of such events becomes more concentrated around
major financial crises, but the conditional variance of the timing
of these extreme events does not seem to degenerate. This evidence
suggests that, for modeling daily S\&P 500 returns over long horizons,
a specification with a constant tail index and a time-varying $k$
is more appropriate.
The rest of the paper is structured as follows. Section~\ref{sec:2}
formalizes the results for tail index regressions. Section \ref{sec:compare}
investigates the extremal quantile regression. Sections \ref{sec:Simulations}
and \ref{sec:Empirical-Illustration} provide small sample evidences
by simulations and an empirical illustration, respectively. Section~\ref{sec:Conclusion}
concludes. All proofs, technical lemmas and figures are collected
in the Appendix.
\textit{Notation.} $\log\left(\cdot\right)$ stands for the natural
logarithm. For a vector $x$, $\left\Vert x\right\Vert $ denotes
the $L_{2}$ norm, and for a matrix $A,$ $\left\Vert A\right\Vert $
denotes the Frobenius norm. $\rho_{\min}(\cdot)$ and $\rho_{\max}(\cdot)$
denote the minimum and the maximum eigenvalues of a matrix, respectively.
$\mathbb{I}_{p}$ denotes the $p\times p$ identity matrix. For a
positive definite matrix $A=LL'$, we write $A^{1/2}=L$. For the
deterministic series $\{a_{n},b_{n}\}_{n=1}^{\infty}$, we denote
$a_{n}\propto b_{n}$ if $0<C_{1}\leq\liminf_{n\rightarrow\infty}\left\vert a_{n}/b_{n}\right\vert \leq\limsup_{n\rightarrow\infty}\left\vert a_{n}/b_{n}\right\vert \leq C_{2}<\infty$
for some constants $C_{1}$ and $C_{2}$ , $a_{n}\ll b_{n}$ if $a_{n}=o(b_{n})$,
and $a_{n}\gg b_{n}$ if $b_{n}\ll a_{n}$. $\overset{P}{\rightarrow}$
and $\overset{d}{\rightarrow}$ denote convergence in probability
and distribution, respectively. $C$ denotes some generic positive
constants that may vary from line to line.
\section{Models and the Issue}
\label{sec:2}
We investigate the right tail behavior of $Y$ conditional on $X.$
For an easier illustration, we assume that $Y$ is unbounded from
right conditional on any $X,$ and $Y$ is heavy tail distributed.
\subsection{The Linear Model}
For model (\ref{eq:surviveCDF}), \citet{WangTsai2009} assumed that
$\alpha(X)=\exp\left(X'\theta^{*}\right)$, and observations are $(x_{i},y_{i}),\,i=1,2,\ldots,n$,
i.i.d. across $n$. For finite samples, the truncation parameter is
allowed to depend on $n$, denoted by $w_{n}$. They proposed estimating
$\theta^{*}$ by minimizing the approximate negative log-likelihood
function:
\[
\hat{\theta}=\arg\min_{\theta}\sum_{i=1}^{n}\Bigl\{\exp(x_{i}'\theta)\,\log\!\left(\frac{y_{i}}{w_{n}}\right)-x_{i}'\theta\Bigr\}\,I(y_{i}>w_{n}),
\]
where $I(\cdot)$ is the indicator function.
A key condition in \citet{WangTsai2009} for identification of $\theta^{*}$
is that the Gram matrix
\[
\hat{\varSigma}_{w_{n}}=\frac{1}{n_{0}}\sum_{i=1}^{n}x_{i}x_{i}'I(y_{i}>w_{n}),\quad\text{with }n_{0}=\sum_{i=1}^{n}I(y_{i}>w_{n}),
\]
is non-singular. When deriving the asymptotic properties of $\hat{\theta}$,
they implicitly assume that the minimum eigenvalue of $\hat{\varSigma}_{w_{n}}$
is uniformly bounded away from zero as $n\rightarrow\infty$, so that
\[
\hat{\theta}-\theta^{*}=O_{P}(n_{0}^{-1/2}),
\]
see, for example, their proof of Theorem 4.
We will show that under fairly general conditions, the minimum eigenvalue
of the population counterpart
\begin{equation}
\bar{\varSigma}_{w_{n}}=\mathbb{E}\left(\left.XX'\right|Y>w_{n}\right)\label{eq:Sigma_w_bar}
\end{equation}
converges to zero even if $\mathbb{E}(XX')$ is non-singular. That
is, $\bar{\varSigma}_{w_{n}}$ becomes nearly singular as $w_{n}$
grows.
\subsection{The Issue of the Rank Condition}
We start with the simple regression case where there is only one regressor
along with the intercept, $X=(1,X_{1})$. To simplify notation, let
\begin{equation}
\alpha(X)=\exp\left(X_{1}\right),\label{eq:alpha}
\end{equation}
where the intercept term is 0, and $X_{1}$ has a compact support
on $\left[\underline{u}_{x},\bar{u}_{x}\right]$ with $\bar{u}_{x}>\underline{u}_{x}$.
Note that \eqref{eq:alpha} is equivalent to
\[
\alpha(X)=\exp\left(\theta_{0}^{*}+\theta_{1}^{*}\tilde{X}_{1}\right),
\]
where $\theta_{0}^{*}=\underline{u}_{x},\theta_{1}^{*}=\bar{u}_{x}-\underline{u}_{x}$
and $\tilde{X}_{1}=(X_{1}-\underline{u}_{x})\left/\left(\bar{u}_{x}-\underline{u}_{x}\right)\right.$,
and the support of $\tilde{X}_{1}$ is $[0,1]$. Thus, \eqref{eq:alpha}
is some normalization of a general linear index.
For illustration, we write
\[
Z=\exp\left(X_{1}\right).
\]
We will show the properties of $Z$ first, then extend the results
to $X.$
\begin{assumption}\label{A:1} The support of $X_{1}$ is finite,
that is, $-\infty<\underline{u}_{x}<\bar{u}_{x}<\infty$. The density
function of $X_{1}$ is bounded and bounded away from zero: $0<\underline{c}_{x}\leq f(x_{1})\leq\bar{c}_{x}<\infty.$
\end{assumption}
Denote $\underline{u}=\exp\left(\underline{u}_{x}\right)$ and $\bar{u}=\exp\left(\bar{u}_{x}\right)$,
then the support of $Z$ is $\left[\underline{u},\bar{u}\right]$,
and the distribution of $Z$ is bounded and bounded away from zero;
specifically, there exist constants such that
\[
0<\underline{c}\leq f(z)\leq\bar{c}<\infty\textrm{ for }z\in[\underline{u},\bar{u}].
\]
For simplicity of analysis, we assume $L(y;x)=1$. The results will
not change much with a more general $L(y;x).$
\begin{theorem}\label{TH:1} Suppose Assumption \ref{A:1} holds
and $Y$ follows the conditional distribution in \eqref{eq:surviveCDF}
with $L=1$. Then,
\[
f(z\mid Y>w)\leq\frac{\bar{c}}{\underline{c}}\frac{w^{-(z-\underline{u})}\log w}{1-w^{-(\bar{u}-\underline{u})}},\quad z\in[\underline{u},\bar{u}],
\]
and as $w\to\infty$,
\begin{align*}
\mathbb{E}(Z\mid Y>w) & \to\underline{u}\textrm{ \textrm{ and}},\\
\operatorname{Var}(Z\mid Y>w) & \to0.
\end{align*}
\end{theorem}
Theorem \ref{TH:1} shows that $Z$ degenerates to its lower bound
as $w\to\infty$.
Note that $X_{1}=\log Z.$ The conditional density of $X_{1}$ takes
a similar form by the change of variable method. Lemma \ref{LE:1}
shows that the result applies to $X_{1}$ as well:
\begin{align}
\mathbb{E}(X_{1} & \mid Y>w)\rightarrow\log\left(\underline{u}\right)=\underline{u}_{x}\textrm{ and}\nonumber \\
\textrm{Var}(X_{1} & \mid Y>w)\rightarrow0.\label{eq:var->0}
\end{align}
In the special case where $Z$ is uniformly distributed or equivalently
$\alpha(X)=\exp\left(X_{1}\right)$ is uniformly distributed, we are
able to derive the rate at which $\operatorname{Var}(Z\mid Y>w)$
converges to zero.
\begin{corollary}\label{Coro:1} Suppose the assumptions in Theorem
\ref{TH:1} hold and $Z$ is uniformly distributed on $[\underline{u},\bar{u}]$.
Then, as $w\to\infty$,
\begin{align*}
\mathbb{E}(Z\mid Y>w) & =\underline{u}+\frac{1}{\log w}+o\!\left(\frac{1}{\log w}\right)\textrm{ \textrm{ and}},\\[6pt]
\operatorname{Var}(Z\mid Y>w) & =\frac{1}{(\log w)^{2}}+o\!\left(\frac{1}{(\log w)^{2}}\right).
\end{align*}
\end{corollary}
For $X_{1}=\log Z$, Lemma \ref{LE:2} shows
\begin{equation}
\frac{1}{\bar{u}^{2}}\textrm{Var}\left(Z\mid Y>w\right)\leq\textrm{Var}\left(X_{1}\mid Y>w\right)\leq\frac{1}{\underline{u}^{2}}\textrm{Var}\left(Z\mid Y>w\right).\label{eq:varX1bound}
\end{equation}
Therefore, the minimum eigenvalue of $\bar{\varSigma}_{w_{n}}$, defined
in (\ref{eq:Sigma_w_bar}), is proportional to $\textrm{Var}\left(X_{1}\mid Y>w\right)$
and is at the rate of $1/(\log w)^{2}.$
Now consider the case with multiple regressors, $X=(1,X_{1},X_{2},\ldots,X_{p})$.
Suppose we are in the most favorable scenario for the rank condition
of the Gram matrix: the regressors are mutually independent, $X_{1}\perp X_{2}\perp\cdots\perp X_{p}$,
and each $X_{j}$ satisfies the support and density conditions in
Assumption \ref{A:1}. In addition, assume
\[
\alpha(X)=\exp\left(X_{1}+X_{2}+\cdots+X_{p}\right).
\]
Define $\tilde{X}_{1}=X_{1}+X_{2}+\cdots+X_{p}$. Clearly, $\tilde{X}_{1}$
also satisfies Assumption \ref{A:1}. Apply Theorem \ref{TH:1} and
the result in (\ref{eq:var->0}), we obtain:
\begin{corollary}\label{Coro:2} Suppose $X_{1},X_{2},\ldots,X_{p}$
satisfy Assumption \ref{A:1} and are mutually independent. Let $\underline{\tilde{u}}_{x_{1}}=\inf\tilde{X}_{1}$.
Then, as $w\to\infty$,
\begin{align*}
\mathbb{E}(\tilde{X}_{1}\mid Y>w) & \to\underline{\tilde{u}}_{x_{1}}\textrm{\textrm{ and}},\\[6pt]
\operatorname{Var}(\tilde{X}_{1}\mid Y>w) & \to0.
\end{align*}
\end{corollary}
In other words, $(1,X_{1},X_{2},\ldots,X_{p})$ becomes nearly collinear
as $w\to\infty$, since $\tilde{X}_{1}=X_{1}+X_{2}+\cdots+X_{p}$
behaves like a constant. Thus, under fairly general conditions, $\bar{\varSigma}_{w}$
degenerates to a singular matrix as $w\to\infty$.
\subsection{A Remedy of the Asymptotics}
In this section, we provide additional conditions and a new proof
showing that the main result in \citet{WangTsai2009} continues to
hold, albeit with a slower convergence rate and some extra condition.
We continue to assume $X=(1,X_{1},X_{2},\ldots,X_{p})$. We show that
as long as the eigenvalues of
\[
\bar{\varSigma}_{w_{n}}=\mathbb{E}\!\left(XX'|Y>w_{n}\right),
\]
the population counterpart of $\hat{\varSigma}_{w_{n}}$, do not decay
too quickly, the estimator $\hat{\theta}$ remains consistent and
asymptotically normal. For simplicity, we assume that the minimum
and maximum eigenvalues of $\bar{\varSigma}_{w_{n}}$ converge to
zero at the same rate. That is, there exist positive finite constants
$\underline{B}$ and $\bar{B}$ that do not depend on $n$, and a
sequence $\{a_{n}\}$, such that
\begin{equation}
\underline{B}a_{n}^{-1}\;\leq\;\rho_{\min}\!\left(\bar{\varSigma}_{w_{n}}\right)\;\leq\;\rho_{\max}\!\left(\bar{\varSigma}_{w_{n}}\right)\;\leq\;\bar{B}a_{n}^{-1},\label{eq:varying_rank}
\end{equation}
with $a_{n}\to\infty\quad\text{as }w_{n}\to\infty.$ We can straightforwardly
generalize the results to the case where the eigenvalues tend to zero
at different rates, yet with more tedious notation.
To simplify the analysis, we assume $L(y;x)=1$, so that we do not
need to account for the bias term in \citet{WangTsai2009}. The results
are qualitatively unchanged if $L(y;x)$ is not constant.
\begin{theorem}\label{TH:2} Suppose $L=1$. $\mathbb{E}\!\left[\|X\|^{2+\delta}\mid Y>w_{n}\right]$
is uniformly bounded for some $\delta\geq2$. The rank condition \eqref{eq:varying_rank}
holds. Let $(x_{i},y_{i}),i=1,2,\ldots,n$, be i.i.d. across $n$.
In addition, $a_{n}$ satisfies that $a_{n}^{2}/n_{0}\to0$, where
$n_{0}=\sum_{i=1}^{n}I(y_{i}>w_{n})$. Then
\[
\sqrt{n_{0}}\,\hat{\varSigma}_{w_{n}}^{1/2}(\hat{\theta}-\theta^{*})\;\overset{d}{\longrightarrow}\;N(0,\mathbb{I}_{p}).
\]
\end{theorem}
Theorem \ref{TH:2} has the same form as the main theorem in \citet{WangTsai2009},
but with two key differences. First, the convergence rate of $\hat{\theta}$
is $\sqrt{n_{0}/a_{n}}$, which is slower than $\sqrt{n_{0}}$. Second,
we require the crucial condition $a_{n}^{2}/n_{0}\to0$. Since $a_{n}$
generally increases while $n_{0}$ decreases as $w_{n}\to\infty$,
this condition is satisfied as long as $w_{n}$ does not grow too
quickly. This implies that practitioners should use effectively more
observations for estimation; in other words, adopt a relatively smaller
choice of $w_{n}$.
In practice, there is no need to know or estimate $a_{n}$; inference
can be conducted using the asymptotic distribution in Theorem \ref{TH:2}
directly.
\subsection{Semi/Non-parametric Tail Index Regression}
The problem is more severe in the semiparametric case, particularly
concerning the nonparametric component within the semiparametric framework.
We first derive the conditional density of $X$ in the tail, and then
present the results regarding the semiparametric regression.
\subsubsection*{Density of $X$ on the Tail of $Y$}
For convenience, suppose that
\begin{equation}
\alpha(X)=X_{1},\label{eq:alphax_semiparametric}
\end{equation}
and we wish to estimate $\alpha(X)$ nonparametrically. The support
of $X_{1}$ is $[\underline{u}_{x},\bar{u}_{x}]$ with $\underline{u}_{x}>0$.
Assume the density of $X_{1}$ is bounded and bounded away from zero.
This is the most favorable scenario for nonparametric estimation.
Theorem \ref{TH:1} implies that
\[
f(x_{1}\mid Y>w)\;\leq\;C\frac{w^{-(x_{1}-\underline{u}_{x})}\log w}{1-w^{-(\bar{u}_{x}-\underline{u}_{x})}},\quad x_{1}\in[\underline{u}_{x},\bar{u}_{x}],
\]
for some $C>0.$ This shows that the conditional density of $X_{1}$
converges to zero for all $x_{1}\in(\underline{u}_{x},\bar{u}_{x}]$,
i.e., for all values in the support of $X_{1}$ except the minimum,
even under the best-case scenario. The speed of decay becomes faster
as $x_{1}$ moves farther away from its minimum. The problem can be
exacerbated in the presence of bias, where $L\left(y;x\right)$ is
not a constant. The bias term will dominate more easily as $x_{1}$
moves away from its minimum.
The discussion under condition \eqref{eq:alphax_semiparametric} can
be generalized. Suppose we have a general nonlinear $\alpha(X)$.
Define
\[
Z=\alpha(X).
\]
Assume $\alpha(x)$ is continuously differentiable, with $\alpha'(x)$
bounded and bounded away from zero, and the density of $X$ bounded
and bounded away from zero. For any value of $z$, there exist only
finitely many $x$ such that $\alpha(x)=z$. Then $Z$ also has bounded
support, and
\begin{equation}
f(z)=\sum_{\alpha(x)=z}\frac{f(x)}{|\alpha'(x)|},\label{eq:fzfx}
\end{equation}
which is well defined for all $z$ in the support of $Z$. It is straightforward
to see that $f(z)$ is also bounded and bounded away from zero. If
we denote the support of $Z$ as $[\underline{u},\bar{u}]$, then
applying the previous result we obtain
\begin{equation}
f(z\mid Y>w)\;\leq\;C\,\frac{w^{-(z-\underline{u})}\log w}{1-w^{-(\bar{u}-\underline{u})}},\quad z\in[\underline{u},\bar{u}],\label{eq:fz|y}
\end{equation}
for some positive constant $C$. The inequality above, together with
\eqref{eq:fzfx}, implies that $f(x\mid Y>w)$ converges to zero whenever
$\alpha(x)\neq\underline{u}$, with faster decay the further $\alpha(x)$
lies above its minimum.
\subsubsection*{Semiparametric Regression}
Suppose $X=\left[1,X_{1},X_{2}\right]$. The semiparametric framework
in \citet{LiLengYou2022} can be written as
\[
\alpha(X)=\theta_{0}^{*}+\theta_{1}^{*}X_{1}+g^{*}\left(X_{2}\right),
\]
where $g^{*}$ is some unknown function. In an attempt to estimate
the nonparametric component together with the parametric component,
the initial step (Step 1) in \citet{LiLengYou2022} approximated the
nonparametric part using sieves, allowing both components to be estimated
simultaneously. That is,
\[
\min_{\theta,g_{n}}\sum_{i=1}^{n}\Bigl\{\exp\left[\theta_{0}+\theta_{1}x_{1i}+g_{n}\left(x_{2i}\right)\right]\,\log\!\left(\frac{y_{i}}{w_{n}}\right)-\theta_{0}-\theta_{1}x_{1i}-g_{n}\left(x_{2i}\right)\Bigr\}\,I(y_{i}>w_{n}),
\]
where $g_{n}$ is an approximation of $g^{*}$ using sieves (e.g.,
B-splines). Thus, the above mimic a parametric tail index regression.
We caution that this may create issues for the nonparametric component,
$g^{*}\left(X_{2}\right)$, due to the degenerate density function
in (\ref{eq:fz|y}). Specifically, most of the extreme observations
are likely to be concentrated around the point where the minimum of
$\alpha$ is attained. Consequently, $g^{*}$ is weakly identified
and estimated imprecisely for$X_{2}$ values away from the minimizer.
\subsubsection*{Nonparametric Regression}
The nonparametric tail index regression in \citet{deHaan03072021}
is robust to the issue we identified, because it relies only on local
data for estimation and $\alpha(x)$ behaves approximately like a
constant locally. In addition, they adopt order statistics to decide
which observations to include for local estimation. Clearly, order
statistics reflect the value of the index.
\section{A Comparison with Extremal Quantile Regression}
\label{sec:compare}
The case of right-tail behavior of $Y$ conditional on $X$ with unbounded
$Y$ corresponds to the type 2 tail in \citet{Chernozhukov2005}.
We present the results in this setting. The results can be generalized
to the type 3 tail with bounded support in \citet{Chernozhukov2005}.
\subsection{The Model}
It is assumed in \citet{Chernozhukov2005} that
\begin{equation}
Y=\beta\left(X\right)+U,\label{eq:quantile_model}
\end{equation}
and
\begin{equation}
\bar{F}_{U}(u\mid x)=1-F_{U}\left(u|x\right)=k\left(x\right)\cdot u^{-\alpha}L(u),\label{eq:surviveCDF1}
\end{equation}
where $\alpha$ is a positive constant, $\beta\left(\cdot\right)$
and $k\left(\cdot\right)$ are the location and scale functions, respectively;
$F_{U}\left(u|x\right)$ is the cumulative distribution function of
$U$ conditional on $X=x$; and $L(y)$ is a slowly varying function
that satisfies $L(yt)/L(y)\to1$ for any $t>0$ as $y\to\infty$.
From (\ref{eq:surviveCDF}) and (\ref{eq:surviveCDF1}), the key difference
is where $X$ enters. $k\left(X\right)$ in (\ref{eq:surviveCDF1})
affects the cumulative distribution function (CDF) as a scale function,
and the tail index $\alpha$ does not depend on $X$. On the other
hand, $X$ enters the tail index as $\alpha(X)$ in (\ref{eq:surviveCDF}).
Apparently, $X$ has a stronger effect on the tail distribution in
(\ref{eq:surviveCDF}) than in (\ref{eq:surviveCDF1}). For example,
changing $k\left(x\right)=2$ to $k\left(x\right)=4$ doubles the
density at the tail while the tail decays to zero at the same rate,
but changing $\alpha(x)$ from 2 to 4 leads to $y^{-2}$ becoming
$y^{-4}$, which are completely different tail behaviors. This observation
provides intuition for the different behaviors of $X$ conditional
on $Y$ in the tails under these two frameworks.
\subsection{The Quantiles}
We present the implications of the tail CDF for the extremal $\tau$-th
quantile of $Y$ ($\tau$ close to 1). Since $L$ is slowly varying
at the right tail, we set $L\left(y;x\right)$ and $L\left(y\right)$
equal to 1 as an approximation to simplify the analysis.
For \citet{Chernozhukov2005}, (\ref{eq:surviveCDF1}) implies that
\begin{equation}
Q_{Y}\left(\tau|X\right)\approx\beta\left(X\right)+\left[k\left(X\right)\right]^{1/\alpha}\left(1-\tau\right)^{-1/\alpha},\label{eq:quantile1}
\end{equation}
where $Q_{Y}\left(\tau|x\right)$ denotes the $\tau$-th quantile
of $Y$ conditional on $X=x$, and ``$\approx$'' holds by setting
$L\left(y\right)=1$.
As a parallel, (\ref{eq:surviveCDF}) in \citet{WangTsai2009} implies
\begin{equation}
Q_{Y}\left(\tau|X\right)\approx\left(1-\tau\right)^{-1/\alpha(X)}.\label{eq:quantile2}
\end{equation}
The expressions in (\ref{eq:quantile1}) and (\ref{eq:quantile2})
confirm the observation in the previous section: $X$ has a stronger
effect on the tail behavior of $Y$ in the framework of \citet{WangTsai2009}
than in \citet{Chernozhukov2005}.
\subsection{Different Behaviors of $X$ on the Tail of $Y$}
\label{sec:3}
Assuming $L(u)=1$, (\ref{eq:quantile_model}) and (\ref{eq:surviveCDF1})
are equivalent to
\begin{equation}
Y=\beta\left(X\right)+k\left(X\right)^{1/\alpha}\tilde{U},\textrm{ and }\bar{F}_{\tilde{U}}(\tilde{u}\mid x)=\tilde{u}^{-\alpha}.\label{eq:extremal_rewrite}
\end{equation}
We simply write $\beta\left(X\right)$ as $X_{1}$ and $k\left(X\right)^{1/\alpha}$
as $X_{2}$, both of which are scalars. In view of the above, the
following representation is rather general:
\begin{equation}
Y=X_{1}+X_{2}\cdot\tilde{U}\label{eq:quantile_equivalent}
\end{equation}
with
\begin{equation}
X_{2}>0\textrm{ and }\bar{F}_{\tilde{U}}\left(\tilde{u}|X_{1},X_{2}\right)=\tilde{u}^{-\alpha},\alpha>1.\label{eq:equivalent_condition}
\end{equation}
We have the following result for the conditional density $f\left(X_{1},X_{2}|Y>w\right).$
\begin{theorem}\label{TH:density_quantile} Suppose the model in
(\ref{eq:quantile_equivalent}) with condition (\ref{eq:equivalent_condition})
holds. In addition, $\left(X_{1},X_{2}\right)$ takes values in $\left[\underline{u}_{x_{1}},\bar{u}_{x_{1}}\right]\times\left[\underline{u}_{x_{2}},\bar{u}_{x_{2}}\right]$,
$\underline{u}_{x_{2}}>0$, and the density of $\left(X_{1},X_{2}\right)$
is uniformly bounded and bounded away from zero. Then,
\begin{equation}
f\left(x_{1},x_{2}|Y>w\right)\propto\frac{\left(\alpha+1\right)x_{2}^{\alpha}}{\left(\bar{u}_{x_{1}}-\underline{u}_{x_{1}}\right)\left(\bar{u}_{x_{2}}^{\alpha+1}-\underline{u}_{x_{2}}^{\alpha+1}\right)}\textrm{ as }w\rightarrow\infty,\label{eq:conditional_den_x1x2|w}
\end{equation}
$\textrm{for }\left(x_{1},x_{2}\right)\in\left[\underline{u}_{x_{1}},\bar{u}_{x_{1}}\right]\times\left[\underline{u}_{x_{2}},\bar{u}_{x_{2}}\right].$
\end{theorem}
The mode of the right-hand side of (\ref{eq:conditional_den_x1x2|w})
is the maximum of $X_{2}$ (or, equivalently, $k\left(X\right)$),
which fits the intuition. Clearly, $\left(X_{1},X_{2}\right)|Y>w$
does not degenerate as $w\rightarrow\infty$, due to the limiting
result in (\ref{eq:conditional_den_x1x2|w}). This is in sharp contrast
with the implication in (\ref{eq:fz|y}) that the density converges
to zero except at the point where $\alpha$ reaches its minimum. As
such, we conclude that the rank condition issue does not exist for
the extremal quantile regression framework. The intuition can be seen
from the previous two subsections: $X$ has a much stronger impact
on the tail behavior of $Y$ in the tail index regression framework
than in the extremal quantile regression framework.
\section{Numerical Verification}
\label{sec:Simulations}
We check the rank condition for these two frameworks by means of simulations.
We consider two data-generating processes (DGPs). In the first DGP,
$\alpha\left(X\right)$ has a single global minimum in the tail index
regression setting, and $k\left(X\right)$ in (\ref{eq:surviveCDF1})
has a single global maximum in the extremal quantile regression setting.
We refer to this design as DGP1M, where 1M stands for ``one mode''.
In the second DGP, $\alpha\left(X\right)$ has four global minima,
and $k\left(X\right)$ has four global maxima. We refer to this design
as DGP4M. The second DGP is designed to mimic situations in which
extreme risks are elevated over more than one period.
Specifically, the tail index regression setting in DGP1M follows
\[
X\sim\textrm{Uniform}\left(\left[0,1\right]\right),
\]
and $Y$ follows the following CDF given $X=x$:
\begin{align*}
F\left(y|X=x\right) & =\left\{ \begin{array}{cc}
1-y^{-1.5-10x} & \textrm{ for }y\geq0,\\
0 & \textrm{ for }y<0.
\end{array}\right.
\end{align*}
We label this design as ``DGP1M-Tail-Index''. In the extremal quantile
regression setting, $(Y,X)$ are generated according to
\begin{align*}
X & \sim\textrm{Uniform}\left(\left[0,1\right]\right),\\
Y & =X+\left(11.5-10X\right)U,\textrm{ with }U\sim\left|\textrm{Student }t\left(4\right)\right|.
\end{align*}
We correspondingly label this design as ``DGP1M-Extremal-Quantile''.
The mode of $f\left(X|Y>w\right)$ is $X=0$ for both settings.
In DGP4M, the tail index regression setting follows
\[
X\sim\textrm{Uniform}\left(\left[0,1\right]\right),
\]
and $Y$ follows the following CDF given $X=x$:
\begin{align*}
F\left(y|X=x\right) & =\left\{ \begin{array}{cc}
1-y^{-6.5+5\cos\left(20x\right)} & \textrm{ for }y\geq0,\\
0 & \textrm{ for }y<0.
\end{array}\right.
\end{align*}
The extremal quantile regression setting follows
\begin{align*}
X & \sim\textrm{Uniform}\left(\left[0,1\right]\right),\\
Y & =X+\left[6.5+5\cos\left(20x\right)\right]U,\textrm{ with }U\sim\left|\textrm{Student }t\left(4\right)\right|.
\end{align*}
The modes of $f\left(X|Y>w\right)$ are $X=0,\pi/10,\pi/5,3\pi/10$
for both settings. We similarly label these two designs as ``DGP4M-Tail-Index''
and ``DGP4M-Extremal-Quantile''.
Although $f\left(X|Y>w\right)$ can be solved analytically, the derivation
is rather involved. To better visualize the results, we draw $10^{9}$
i.i.d.\ observations for each DGP and estimate the density of
\[
f\left(X\mid Y\geq Q_{\tau}\left(Y\right)\right),\textrm{ for }\tau=0.9,0.95,0.99,0.995.
\]
Here, $Q_{\tau}\left(Y\right)$ is estimated by the $\tau$-th quantile
of the sampled $Y$, and $f$ is obtained using a histogram density
estimator. Specifically, we define bins
\[
\mathbb{B}_{j}=\left(\left(j-1\right)h,jh\right],\quad j=1,2,\ldots,\left\lfloor 1/h\right\rfloor ,
\]
and
\[
\hat{f}_{h}\left(x\mid Y\geq Q_{\tau}\left(Y\right)\right)=\frac{1}{10^{9}\tau h}\sum_{i:y_{i}\geq Q_{\tau}\left(Y\right)}1\left(x_{i}\in\mathbb{B}_{j}\right),\textrm{ for }x\in\mathbb{B}_{j}.
\]
We set $h=0.01$. Since the number of sampled observations is extremely
large, the estimated density is very close to the true density. As
a result, we conduct each simulation only once.
We report the estimated densities in Figures \ref{fig:DPG1M} and
\ref{fig:DPG4M}. For DGP1M, we also report the variance of $X$ conditional
on $Y\geq Q_{\tau}\left(Y\right)$. For DGP4M, instead of reporting
$\textrm{Var}\left(X\mid Y\geq Q_{\tau}\left(Y\right)\right)$, we
report
\begin{align}
& \textrm{Var}_{4\textrm{M}}\left(X\mid Y\geq Q_{\tau}\left(Y\right)\right)\nonumber \\
\equiv\; & \textrm{Var}\left(X\mid Y\geq Q_{\tau}\left(Y\right),X<\frac{\pi}{20}\right)+\textrm{Var}\left(X\mid Y\geq Q_{\tau}\left(Y\right),\frac{\pi}{20}\leq X<\frac{3\pi}{20}\right)\nonumber \\
& +\textrm{Var}\left(X\mid Y\geq Q_{\tau}\left(Y\right),\frac{3\pi}{20}\leq X<\frac{\pi}{4}\right)+\textrm{Var}\left(X\mid Y\geq Q_{\tau}\left(Y\right),X\geq\frac{\pi}{4}\right).\label{eq:var4M}
\end{align}
This modification is motivated by the fact that, under DGP4M, the
distribution of $X$ is concentrated around four modes at $X=0,\pi/10,\pi/5,3\pi/10$.
As a result, $\textrm{Var}\left(X\mid Y\geq Q_{\tau}\left(Y\right)\right)$
fails to capture the local variation around each mode. The cutoffs
are chosen as the midpoints between adjacent modes so that each term
in (\ref{eq:var4M}) measures the variation of $X$ in a neighborhood
of a single mode.
The different behaviors of $X$ conditional on the tail of $Y$ under
the varying tail index (tail index regression framework) and the constant
tail index (extremal quantile regression framework) are evident. The
density of $X$ quickly degenerates as $\tau$ increases toward one
(or, in plain words, as $Y$ becomes more extreme) for DGP1M-Tail-Index
and DGP4M-Tail-Index. This behavior is reflected in the conditional
variance of $X$, which decreases rapidly as $\tau$ increases: the
conditional variance of $X$ is only about 4\% of the unconditional
variance of $X$ at $\tau=0.995$ for both DGP1M-Tail-Index and DGP4M-Tail-Index.
In contrast, for DGP1M-Extremal-Quantile and DGP4M-Extremal-Quantile,
although $X$ also concentrates around the modes, the conditional
density of $X$ stabilizes and does not degenerate. Moreover, the
variance of $X$ conditional on $Y\geq Q_{0.995}\left(Y\right)$ is
about 35\% and 20\% of the unconditional variance of $X$ for DGP1M-Extremal-Quantile
and DGP4M-Extremal-Quantile, respectively, changing little from the
variance conditional on $Y\geq Q_{0.99}\left(Y\right)$.
\section{An Empirical Illustration}
\label{sec:Empirical-Illustration}
Stock returns provide a natural setting for the application of extreme-value
methods. Both tail index regressions and extremal quantile regression
have been applied in this context; see, for example, \citet{ChernozhukovHandbook},
\citet{deHaan03072021}, and \citet{NICOLAUJoE2023}, and the references
therein.
We study daily returns of the S\&P 500 index from January 1, 1929
to December 31, 2024, yielding a total of $T=24{,}114$ observations.
The daily return $Y$ is defined as the log difference of the index
between two consecutive trading days. Our objective is to investigate
which framework is more reasonable for modeling the left tail (losses)
of returns. From a modeling perspective, this question reduces to
whether the tail index $\alpha$ is constant over time $t$. The existing
literature offers mixed evidence. For example, \citet{EinmahlEtal2014}
find that the tail index varies over time, whereas \citet{deHaan03072021}
cannot reject the hypothesis of a constant tail index for daily S\&P
500 returns over the period 1988--2012.
We examine this issue by studying the density and variance of $t$
conditional on extreme negative returns. We normalize time $t$ to
the interval $\left(0,1\right]$; for instance, the first observation
corresponds to $t=1/24114$ and the last to $t=1$. As shown in the
simulation studies, the analysis is complicated by the possible presence
of multiple modes. We therefore treat major financial crises in the
United States stock market as potential modes of the conditional density
when calculating conditional variances.
We consider two mode specifications. The first specification has four
modes: the 1929 Wall Street Crash ($t=0$), the 1987 Black Monday
crash ($t=0.61$), the 2008 Global Financial Crisis ($t=0.83$), and
the 2020 COVID-19 crash ($t=0.958$). The second specification includes
two additional modes, yielding six modes in total: the 1973 oil shock
and stagflation crisis ($t=0.46$) and the 2000 dot-com crash ($t=0.75$).
We consider conditional sets $\{Y<Q_{\tau}(Y)\}$ with $\tau=0.1,0.05,0.01,$
and $0.005$. The conditional variance is defined as
\begin{align*}
\textrm{Var}_{K\textrm{M}}\left(t\mid Y<Q_{\tau}(Y)\right) & =\textrm{Var}\left(t\mid Y<Q_{\tau}(Y),\,t<c_{1}\right)+\sum_{l=1}^{K-2}\textrm{Var}\left(t\mid Y<Q_{\tau}(Y),\,c_{l}\le t<c_{l+1}\right)\\
& \quad+\textrm{Var}\left(t\mid Y<Q_{\tau}(Y),\,t\ge c_{K-1}\right),
\end{align*}
where $K=4,6$, and $c_{l}$ denotes the midpoint between two adjacent
modes. For example, $c_{1}=(0+0.61)/2$ in the four-mode setting and
$c_{1}=(0+0.46)/2$ in the six-mode setting, with the remaining cutoffs
defined analogously.
The conditional density is estimated using a histogram estimator.
Define bins
\[
\mathbb{B}_{j}=\left(\left(j-1\right)h,jh\right],\quad j=1,2,\ldots,1/h,\quad\textrm{with }h=0.01,
\]
and
\[
\hat{f}_{h}\left(t\mid Y<Q_{\tau}(Y)\right)=\frac{1}{T\tau h}\sum_{t:y_{t}\le Q_{\tau}(Y)}1\left(t\in\mathbb{B}_{j}\right),\quad\textrm{for }t\in\mathbb{B}_{j}.
\]
The estimated densities and conditional variances for both mode specifications
are reported in Figure \ref{fig:sp500_f1}. The effective number of
observations used in the estimation is $T\tau$, which is denoted
as $N_{\textrm{observations}}$ in the figures. As $\tau$ decreases,
the timing of extreme events becomes increasingly concentrated around
the identified modes. This pattern indicates that either $\alpha(t)$,
$k(t)$, or both vary over time.
Examining the conditional variances, we find that the conditional
density does not degenerate as rapidly as in the tail index regression
framework with a varying $\alpha(t)$ observed in the simulation studies.
However, the conditional variance also does not stabilize as in the
extremal quantile regression framework. Consequently, the empirical
evidence is somewhat mixed. Nevertheless, $\textrm{Var}_{4\textrm{M}}\left(t\mid Y\ge Q_{0.005}(Y)\right)$
and $\textrm{Var}_{6\textrm{M}}\left(t\mid Y\ge Q_{0.005}(Y)\right)$
remain at 59 percent and 30 percent of the unconditional variance,
respectively, suggesting that the conditional distribution does not
degenerate. Based on this evidence, we view the extremal quantile
regression framework, that is, a constant $\alpha$ with a time-varying
$k(t)$, as more suitable.
We next examine the subperiod from January 1, 1988 to December 31,
2012, studied by \citet{EinmahlEtal2014} and \citet{deHaan03072021},
for which $T=6{,}302$. We normalize $t$ to $\left(0,1\right]$ as
before and consider a two-mode specification: the 2000 dot-com crash
($t=0.5$) and the 2008 Global Financial Crisis ($t=0.83$). The conditional
density and the two-mode conditional variance, $\textrm{Var}_{2\textrm{M}}\left(t\mid Y<Q_{\tau}(Y)\right)$,
are computed as before. The results are reported in Figure \ref{fig:sp500_f2}.
The qualitative pattern mirrors that in Figure \ref{fig:sp500_f1}:
either $\alpha(t)$, $k(t)$, or both vary over time, while the evidence
regarding the more appropriate framework remains mixed. However, $\textrm{Var}_{2\textrm{M}}\left(t\mid Y\ge Q_{\tau}(Y)\right)/\textrm{Var}_{2\textrm{M}}(t)$
equals 59 percent, 46 percent, 41 percent, and 26 percent for $\tau=0.1,0.05,0.01,$
and $0.005$, respectively. Given that $N_{\textrm{observations}}$
is only 32 for $\tau=0.005$, focusing on $\tau=0.1,0.05,$ and $0.01$
provides clearer support for the extremal quantile regression framework.
We conclude by noting that a limitation of this empirical illustration
is that the results rely on ad hoc choices of modes.
\section{Conclusion}
\label{sec:Conclusion}
In this paper, we identify some issue with the rank condition and
the conditional density of explanatory variables in tail-index regressions.
These issues arise because the estimation sample is not random but
is instead correlated with the explanatory variables. We rectify the
asymptotics by taking this issue into account for the linear tail
index model. Such issue does not exist for extremal quantile regression
framework because explanatory variables do not impact the tail behavior
as much as that in tail index regressions framework. Such distinct
behaviors provide a simple diagnostic tool to determine which framework
is more suitable.