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Cautions on Tail Index Regressions and a Comparative Study with Extremal Quantile Regression
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Tail index estimation has been widely applied in modeling extreme events, such as natural environmental and financial market extremes; see, for example, de_Haan_Ferreira and references within. Building on the seminal paper of Hill1975, subsequent work has sought to model the tail index as a function of covariates, either parametrically (e.g., HallTajvidi and WangTsai2009), semiparametrically (e.g., LiLengYou2022), or nonparametrically (e.g., 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
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 (ref) much more general than formulations without it; see de_Haan_Ferreira and 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:
As a comparison, we investigate this rank condition for the extremal quantile regression, pioneered by Chernozhukov2005. The framework in 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) formalizes the results for tail index regressions. Section (ref) investigates the extremal quantile regression. Sections (ref) and (ref) provide small sample evidences by simulations and an empirical illustration, respectively. Section (ref) concludes. All proofs, technical lemmas and figures are collected in the Appendix.
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.
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.
For model ((ref)), 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 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
converges to zero even if $\mathbb{E}(XX')$ is non-singular. That is, $\bar{\varSigma}_{w_{n}}$ becomes nearly singular as $w_{n}$ grows.
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
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 (ref) 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, (ref) 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 $X.$
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 $L(y;x).$
Theorem (ref) 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) shows that the result applies to $X_{1}$ as well:
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.
For $X_{1}=\log Z$, Lemma (ref) shows
Therefore, the minimum eigenvalue of $\bar{\varSigma}_{w_{n}}$, defined in ((ref)), 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). 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). Apply Theorem (ref) and the result in ((ref)), we obtain:
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 $w\to\infty$.
In this section, we provide additional conditions and a new proof showing that the main result in 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
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 WangTsai2009. The results are qualitatively unchanged if $L(y;x)$ is not constant.
Theorem (ref) has the same form as the main theorem in 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) directly.
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.
For convenience, suppose that
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) 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 (ref) 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
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
for some positive constant $C$. The inequality above, together with (ref), 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.
Suppose $X=\left[1,X_{1},X_{2}\right]$. The semiparametric framework in 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 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)). 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.
The nonparametric tail index regression in 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.
The case of right-tail behavior of $Y$ conditional on $X$ with unbounded $Y$ corresponds to the type 2 tail in Chernozhukov2005. We present the results in this setting. The results can be generalized to the type 3 tail with bounded support in Chernozhukov2005.
It is assumed in Chernozhukov2005 that
and
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)) and ((ref)), the key difference is where $X$ enters. $k\left(X\right)$ in ((ref)) 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)). Apparently, $X$ has a stronger effect on the tail distribution in ((ref)) than in ((ref)). 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.
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 Chernozhukov2005, ((ref)) implies that
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)) in WangTsai2009 implies
The expressions in ((ref)) and ((ref)) confirm the observation in the previous section: $X$ has a stronger effect on the tail behavior of $Y$ in the framework of WangTsai2009 than in Chernozhukov2005.
Assuming $L(u)=1$, ((ref)) and ((ref)) are equivalent to
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:
with
We have the following result for the conditional density $f\left(X_{1},X_{2}|Y>w\right).$
The mode of the right-hand side of ((ref)) 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)). This is in sharp contrast with the implication in ((ref)) 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.
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)) 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 $X=x$:
We label this design as “DGP1M-Tail-Index”. In the extremal quantile regression setting, $(Y,X)$ are generated according to
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 $X=x$:
The extremal quantile regression setting follows
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) and (ref). 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
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)) 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 $Y\geq Q_{0.99}\left(Y\right)$.
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, ChernozhukovHandbook, deHaan03072021, and 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, EinmahlEtal2014 find that the tail index varies over time, whereas 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
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). 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 EinmahlEtal2014 and 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).
The qualitative pattern mirrors that in Figure (ref): 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.
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.