Extreme expectile estimation for short-tailed data with an application to market risk assessment Abdelaati Daouiaa Simone A. Padoanb Gilles Stuperc

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Extreme expectile estimation for short-tailed data, with an

application to market risk assessment

Abdelaati Daouiaa, Simone A. Padoanb& Gilles Stupﬂerc

aToulouse School of Economics, University of Toulouse Capitole, France

bDepartment of Decision Sciences, Bocconi University, via Roentgen 1, 20136 Milano, Italy

cUniv Angers, CNRS, LAREMA, SFR MATHSTIC, F-49000 Angers, France

Abstract

The use of expectiles in risk management has recently gathered remarkable mo-

mentum due to their excellent axiomatic and probabilistic properties. In particular,

the class of elicitable law-invariant coherent risk measures only consists of expectiles.

While the theory of expectile estimation at central levels is substantial, tail estima-

tion at extreme levels has so far only been considered when the tail of the underlying

distribution is heavy. This article is the ﬁrst work to handle the short-tailed setting

where the loss (e.g. negative log-returns) distribution of interest is bounded to the

right and the corresponding extreme value index is negative. We derive an asymptotic

expansion of tail expectiles in this challenging context under a general second-order

extreme value condition, which allows to come up with two semiparametric estima-

tors of extreme expectiles, and with their asymptotic properties in a general model of

strictly stationary but weakly dependent observations. A simulation study and a real

data analysis from a forecasting perspective are performed to verify and compare the

proposed competing estimation procedures.

MSC 2010 subject classiﬁcations: 62G30, 62G32

Keywords: Expectiles, Extreme values, Second-order condition, Short tails, Weak

dependence

1 Introduction

The class of expectiles, introduced by Newey and Powell (1987), deﬁnes useful descriptors

ξτof the higher (τ≥1

2) and lower (τ≤1

2) regions of the distribution of a random

variable Xthrough the asymmetric least squares (ALS) minimization problem

ξτ= arg min

θ∈R

Eητ(X−θ)−ητ(X),(1.1)

where ητ(x) = |τ− {x≤0}|x2, with τ∈(0,1) and {·} being the indicator function.

Expectiles are well-deﬁned, ﬁnite and uniquely determined as soon the ﬁrst moment of X

arXiv:2210.02056v2 [math.ST] 19 Mar 2023

is ﬁnite. They generalize the mean ξ1/2=E(X) in the same way quantiles generalize the

median, thus deﬁning an ALS analog to quantiles. Indeed, Koenker and Bassett (1978)

showed that the τth quantile qτof Xsolves the asymmetric L1minimization problem

qτ∈arg min

θ∈R

E(%τ(X−θ)−%τ(X)),

where %τ(x) = |τ− {x≤0}||x|. Expectiles have received renewed attention for their

ability to quantify tail risk at least since the contribution of Taylor (2008). They depend

on the tail realizations of Xand their probability, while quantiles only depend on the

frequency of tail realizations, see Kuan et al. (2009). Most importantly, Ziegel (2016)

showed that expectiles are the sole coherent law-invariant measure of risk which is also

elicitable in the sense of Gneiting (2011), meaning that they abide by the intuitive diversi-

ﬁcation principle (Bellini et al.,2014) and that their prediction can be performed through

a straightforward principled backtesting methodology. These merits have motivated the

development of procedures for expectile estimation and inference over the last decade. A

key, but diﬃcult, question in any risk management setup is the estimation of the expectile

ξτat extreme levels, which grow to 1 as the sample size increases. This question was ﬁrst

tackled in Daouia et al. (2018,2020) under the assumption that the underlying distribu-

tion is heavy-tailed, that is, its distribution function tends to 1 algebraically fast. The

latest developments under this assumption have focused on, among others, bias reduc-

tion (Girard et al.,2022), accurate inference (Padoan and Stupﬂer,2022), and handling

more complex data in regression (Girard et al.,2021,2022) or time series (Davison et al.,

2022) setups.

The problem of estimating extreme expectiles outside of the set of heavy-tailed models

is substantially more complicated from a statistical standpoint. The contribution of the

present paper is precisely to build and analyze semiparametric extreme expectile estima-

tors in the challenging short-tailed model, in which the extreme value index (EVI) of the

underlying distributions is known to be negative. This requires employing a dedicated

extrapolation relationship for population extreme expectiles. Only Mao et al. (2015) have

initiated such a study at the population level when Xbelongs to the domain of attraction

of a Generalized Extreme Value distribution (GEV). Diﬀerently to Mao et al. (2015), we

work in the general semiparametric Generalized Pareto (GP) setting through a standard

second-order condition, which makes it possible to derive an asymptotic expansion of ex-

treme expectiles without resorting to an unnecessary restriction about the link between

the EVI and second-order parameter that featured in Mao et al. (2015). Based on this

asymptotic expansion, we present and study two diﬀerent extreme value estimators of

tail expectiles. The ﬁrst one builds upon the Least Asymmetrically Weighted Squares

(LAWS) estimator of expectiles, namely the empirical counterpart of ξτin (1.1), obtained

at intermediate levels τ=τn→1 with n(1 −τn)→ ∞ as the sample size n→ ∞. The

short-tail model assumption allows then to come up with an expectile estimator extrapo-

lated to the far tail at arbitrarily extreme levels τ= 1 −pnsuch that (1 −τn)/pn→ ∞

−0.6

−0.4

−0.2

0.0

0.2

0 50 100 150 200 250

Rolling window

EVI estimates

variable

Alibaba

Alphabet

Amazon

Apple

Bank of America

Goldman Sachs

Intel

JPMorgan Chase

Meta

Morgan Stanley

NASDAQ

Netflix

T. Rowe Price

Walmart

Figure 1: Maximum Likelihood estimates of the extreme value index over the resulting 253

successive rolling windows of 150 stationary data, obtained from 14 time series of weekly

logarithmic (loss) returns between 21st September 2014, and 12th June 2022.

as n→ ∞, in a semiparametric way reminiscent of how extreme quantiles are ﬁtted in

Section 4.3 of de Haan and Ferreira (2006). The second extrapolating estimator directly

relies on the asymptotic expansion of ξτthat involves its quantile analog qτ, the endpoint

q1≡ξ1and the EVI, by plugging in the GP quantile-based estimators of these tail quan-

tities. Our estimation theory is valid in a general setting of strictly stationary and weakly

dependent data satisfying reasonable mixing and tail dependence conditions. We explore

various theoretical and practical features of extreme expectile estimation in this setting,

and explain why this problem is statistically more diﬃcult than extreme quantile estima-

tion. In particular, an extreme expectile ξτis intrinsically less spread than its quantile

analog qτ, even at asymmetry levels τ≈1 where it remains much closer to the center of

the distribution than qτ. Consequently, any semiparametric procedure for extreme expec-

tile estimation should be expected to suﬀer at least from a worse bias than for extreme

quantile estimation.

Our focus on the problem of estimating extreme expectiles for bounded distributions is

motivated by the perhaps somewhat surprising ﬁnding that weekly returns of equities, used

in applications to circumvent the non-synchronicity of daily data, may have short-tailed

distributions. This is illustrated in Figure 1for 14 major companies and ﬁnancial institu-

tions, where the data consists of the loss returns (i.e. negative log-returns) on their weekly

equity price from 21st September 2014 to 12th June 2022, corresponding to 403 trading

weeks. The representative price is constructed by averaging daily closing prices within

the corresponding week. The nature of the upper tail of these loss returns is reﬂected by

the EVI of their distribution whose negative, zero or positive values indicate respectively

a distribution with short, light or Pareto-type tail. None of these three scenarios can be

excluded in practice for these 14 data examples, where the EVI is estimated on successive

rolling windows of length n= 150 using the GP distribution ﬁtted to exceedances over

a high threshold by means of the Maximum Likelihood (ML) method, with the optimal

threshold being chosen by the path stability procedure as described below in Section 4. It

is therefore important to construct an appropriate and fully data-driven estimation proce-

dure for the challenging scenario of short-tailed data. This problem also appears naturally

in production econometrics when analyzing the productivity of ﬁrms (Kokic et al.,1997).

All our methods and data have been incorporated into the Rpackage ExtremeRisks.

In Section 2, we explain in detail the short tail distributional assumption on X, state

our asymptotic expansion linking extreme expectiles and quantiles, construct our two

classes of extreme expectile estimators and study their asymptotic properties. A simulation

study examines their ﬁnite-sample performance in Section 3, and a time series of Bitcoin

data is analyzed in Section 4. The online Supplementary Material contains all the proofs

in Section A and further simulation results in Section B.

2 Main results

2.1 Connection between extreme expectiles and quantiles

Let F:x7→ P(X≤x) be the distribution function of the random variable of interest

Xand F= 1 −Fbe its survival function. Deﬁne the associated quantile function by

qτ= inf{x∈R|F(x)≥τ}and the tail quantile function Uby U(s) = q1−s−1,s > 1.

Diﬀerently from existing literature on extreme expectile estimation, we focus on the case

when the distribution of Xis short-tailed, or equivalently, when its EVI γis negative.

According to Theorem 1.1.6 on p.10 of de Haan and Ferreira (2006), this corresponds to

assuming that there is a positive function asuch that

∀z > 0,lim

s→∞

U(sz)−U(s)

a(s)=zγ−1

γ,with γ < 0.

This assumption can be informally rewritten as

∀z > 0, U(sz)≈U(s) + a(s)zγ−1

γwhen sis large. (2.1)

This means that extreme values of Xat the far tail (represented by U(sz)) can be achieved

by extrapolating in-sample large values (represented by U(s)) if the scale function a(s) and

the shape parameter γcan be consistently estimated. The theory of the resulting extreme

value estimators is usually developed under the following second-order reﬁnement of the

short-tailed model assumption above, which will be our main condition throughout (see

de Haan and Ferreira,2006, Equation (2.3.13) on p.45):

Condition C2(γ, a, ρ, A) There exist γ < 0, ρ≤0, a positive function a(·) and a measurable

function A(·) having constant sign and converging to 0 at inﬁnity such that, for all z > 0,

lim

s→∞

A(s)U(sz)−U(s)

a(s)−zγ−1

γ=Zz

vγ−1Zv

uρ−1dudv.

This condition enables one to control the bias incurred by using the approximation (2.1)

and represented by the function A. Under this condition, the right endpoint x?= sup{x∈

R|F(x)<1}of Xis necessarily ﬁnite (see de Haan and Ferreira,2006, Theorem 1.2.1

on p.19). This justiﬁes calling this model a short-tailed (or bounded) model.

Suppose now that E|min(X, 0)|<∞and that condition C2(γ, a, ρ, A) is satisﬁed, so

that E|X|<∞and expectiles of Xare well-deﬁned and ﬁnite. First, we motivate an

asymptotic expansion of extreme expectiles that will be instrumental in our subsequent

theory of extreme expectile estimation. Recall that the τth expectile ξτsatisﬁes

ξτ−E(X) = 2τ−1

1−τE((X−ξτ){X > ξτ}),(2.2)

see Equation (12) in Bellini et al. (2014). Writing E((X−x){X > x}) as an integral of

the quantiles of Xabove xand using condition C2(γ, a, ρ, A) justiﬁes the approximation

E((X−ξτ){X > ξτ})≈F(ξτ)a(1/F (ξτ))

1−γas τ↑1,

and therefore

lim

τ↑1

a(1/F (ξτ))F(ξτ)

1−τ= (x?−E(X))(1 −γ).(2.3)

The convergence a(s)/(x?−U(s)) → −γas s→ ∞ (see de Haan and Ferreira,2006,

Lemma 1.2.9 on p.22) then suggests

lim

τ↑1

(x?−ξτ)F(ξτ)

1−τ= (x?−E(X))(1 −γ−1).(2.4)

The approximations F(ξτ)/(1−τ)≈F(ξτ)/F (qτ)≈(x?−ξτ)−1/γ /(x?−qτ)−1/γ motivated

by the regular variation property of x7→ F(x?−1/x) (see de Haan and Ferreira,2006,

Theorem 1.2.1.2 on p.19) ﬁnally entail

lim

τ↑1

x?−ξτ

(x?−qτ)1/(1−γ)= [(x?−E(X))(1 −γ−1)]−γ/(1−γ).(2.5)

Consequently, extreme expectiles can be extrapolated from their quantile analogs in con-

junction with endpoint and EVI estimation. Analyzing the asymptotic properties of

the estimators built in this way will require quantifying the diﬀerence between the ra-

tio (x?−ξτ)/(x?−qτ)1/(1−γ)and its limit in (2.5). This is the focus of our ﬁrst main result

below.

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Extremeexpectileestimationforshort-taileddata,withanapplicationtomarketriskassessmentAbdelaatiDaouiaa,SimoneA.Padoanb&GillesStupercaToulouseSchoolofEconomics,UniversityofToulouseCapitole,FrancebDepartmentofDecisionSciences,BocconiUniversity,viaRoentgen1,20136Milano,ItalycUnivAngers,CNRS,LAREMA,SFRMA...

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Extreme expectile estimation for short-tailed data with an application to market risk assessment Abdelaati Daouiaa Simone A. Padoanb Gilles Stuperc

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