Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium IVIember Libraries
http://www.archive.org/details/extremalquantitiOOcher
HB31
^
I
A
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
EXTREMAL QUANTILES AND VALUE-AT-RISK
Victor
Chemozhukov
Songzi
Du
Working Paper 07-01
May 24, 2006
Room
E52-251
50 Mennorial Drive
Cannbridge, MA 021 42
This paper can be downloaded without charge from the
Social Science Research Network
Paper Collection
http://ssrn.com/abstract=956433
at
„
.
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
EXTREMAL QUANTILES AND VALUE-AT-RISK
Victor
Chemozhukov
Songzi
Du
Working Paper 07-0'
May 24, 2006
Room
E52-251
50 Memorial Drive
Cambridge,
MA 021 42
This paper can be downloaded without charge from the
Social Science Research Network Paper Collection at
http://ssrn.com/abstract=956433
VASSACHUSETTS INSTH-UTE
OF TECHMOLOGV
j
JAN
1
8 2007
;J8RAR!ES
EXTREMAL QUANTILES AND VALUE-AT-RISK
VICTOR CHERNOZHUKOV AND SONGZI DU
Abstract. This
article looks at the
nomics, in particular value-at-risk.
theory and empirics of extremal quantiles in eco-
The theory
of extremes has gone through remarkable
developments and produced valuable empirical findings
cussion,
we put a
in the last
20 years.
particular focus on conditional extremal quantile models
which have applications
in
many
areas of economic analysis.
Examples
In the dis-
and methods,
of applications
include the analysis of factors of high risk in finance and risk management, the analysis
of socio-economic factors that contribute to extremely low infant birthweights, efficiency
analysis in industrial organization, the analysis of reservation rules in economic decisions,
and inference
Key Words:
JEL:
Date:
We
May
in structural auction models.
Extremes, Quantiles, Regiession, Value-at-risk, Extremal Bootstrap
C13, C14, C21, C41, C51, C53
24,
2006
.
thank Emily Gallagher, Greg Fischer, and Raymond Guiteras
for their help
and valuable comments.
extrea1al quantiles and value-at-risk
Introduction
1.
Some Basics. Let a real random
Fyd/) = Prob[Y < y]- A r-quantile
Fy^{t)] — T for some r G (0,1).^ The
is
is
Y
variable
1.1.
probability index r,
1
of
Y
have a continuous distribution function
number Fy
the
is
quantile function
Fy
<
viewed as a function of
the inverse of the distribution function Fy(j/).
The
quantile function
therefore a complete description of the distribution.
Let
X
be a vector of regressor variables.
conditional distribution function of
variable
Y
Y
= Prob[Y <
Let Fy-(y|x)
=
given A'
The
x.
< Fy
as a function of x
(r|x)jA
is
regression function
=
—
x]
The
t.
measure the
to
(r|a") is
is
the
effect of covariates
or a conditional r-quantile will be referred to as extremal
<
.15, or high,
>
r
.85.
number
The main
random
Fy'^{T\x) such
Without
use of the quantile
on outcomes, both
To
center and in the upper and lower tails of an outcome distribution.
either low, r
denote the
conditional quantile function Fy^{t\x) viewed
called the r-quantile regression function.
Fy
y\x]
conditional r-quantile of a
with a continuous conditional distribution function
that Prob[Y
is
such that Prob[Y
(t)
(t),
in the
this effect, a quantile
whenever the probability index r
loss of generality,
we
focus the discussion
on the low quantiles.
1.2.
Examples
as a Motivation. There are
many
applications of extremal quantiles in
economics, particularly of extremal conditional quantiles. Here we give a sample of these
applications as a motivation for
Example
1.
what
follows.
Conditional Value- at- Risk.
or explain very low conditional quantiles
Fy
^(t|A) of an institution's portfoho return, F,
tomorrow, using today's available information,
Typically, extremal quantiles
Fy ^(rjA) with
at-risk analysis is a daily activity for
Value-at-risk analysis seeks to forecast
r
=
X
(Chernozhukov and Umantsev 2001).
0.01
and
r
=
0.05 are of interest. Value-
banking and other financial institutions, as required by
the Securities and Exchange Commission and the Basle Committee on Banking Supervision.
As a
ized
risk
measure, value-at-risk
by Roy
(1952), in
is
motivated by the
the probability of the risk of a large loss
monly used
safety-first decision principle formal-
which one makes optimal decisions subject to the constraint that
in real-life financial
is
kept small. This and similar measures are com-
management, insurance, and actuarial science (Embrechts,
Kliippelberg, and Mikosch 1997).
Example
birthweights,
Determinants of Very Low Birthweights.
we may be interested in how smoking, absence of
2.
In the analysis of infant
prenatal care, and other
types of maternal behavior affect various birthweights (Abrevaya 2001). Of special interest,
however, are the very low quantiles, since low birthweights have been linked to subsequent
health problems. Chernozhukov (2006) provides an empirical study of extreme birthweights.
^More
generally, let Fy- '(r)
=
inf{i/
:
Fy(y) > t}.
VICTOR CHERNOZHUKOV AND SONGZI DU
2
Example
other factors,
of firms, the
important form of efficiency
economics industrial organization and regulation
production frontiers (Timmer 1971).
efficiency or
An
Probabilistic Production Frontiers.
3.
analj'sis in the
X, we
most
is
the determination of
Given cost of production and possibly
are interested in the highest production levels that only a small fraction
efficient firms,
can attain. These (nearly)
efficient
production levels can be
formally described by extremal quantile regression function, Fy^{t\X)
e
>
0;
so only an e-fraction of firms produce Fy7^(rjX) or more.
for
t G
[1
-
e,
and
1)
The models and methods
discussed in this article are highly pertinent for inference on the probabilistic frontiers.
Example 4- (S,s)-Rules and Other Approximate Reservation Rules in Economic Decisions. A related example is that of (5, s)-adjustment models, which arise as
optimal policies in many economic models (Arrow, Harris, and Marschak 1951). For example, the capital stock Z is adjusted up to the level S once it has depreciated to some low
level s. In terms of an econometric specification, we may think that the observed capital
stock satisfies the equation Z,; = s(X,) +
where Xi are covariates, and Vi is a disturbance
that is positive most of the time, i.e. Prob{vi > 0) is close to 1. Once the capital stock
Zj reaches the critical level, i.e. Zi < s(A',:), it is adjusted in the next period. We assume,
as in Caballero and Engel (1999), that when the disturbance Vi = Z, — s(A'j) is negative,
1',,
it
captures unobserved heterogeneity and small decision mistakes that are independent of
observed covariates Xj.^ In a given cross-section or time
series,
quently, so in fact data at or below the lower adjustment
band
small probability Prob{v,,
The lower-band
tion
up
<
hence
0);
adjustment
s(A'i) will
F^HtIX) = s{X) + F'^t)
will
occur infre-
be observed with a
e {0,Prob{v <
for r
0)].
function s{X) therefore coincides with the lower conditional quantile func-
to an additive constant.
A
similar
argument works
the upper-band function
for
S{X).
Example
price
5.
Structural Auction Models. In the standard specification of the
procurement auction where bidders hold independent valuations, the winning
Bi, satisfies the equation Bi
function and
/3(n.()
>
1
is
~
c(Xj)/?(r?,,)
-f-
>
e,, Cj
0,
a mark-up that approaches
approaches infinity (Donald and Paarsch 2002).
By
some small negative value
so that,
when
where
1
c(A'",;)
as the
is
it is
number
realistic to let
of bidders, n^,
is
the extreme
the disturbance
Cj
negative, these disturbances capture small
decision mistakes that are independent of included explanatory variables.
the quantile function satisfies
bid,
the efficient cost
construction, c(A'j)/?(?ii)
conditional Cjuantile function. In empirical analysis,
take
first-
In this case
Fg\T\X,n) = C{X)P{n) + F~\t), for r G (0, F[e <
make inference on c{X) and f3{n).
0]).
Quantile regression methods can be emploj^ed to
In this example, Z, could be any
monotone transformation
transformation gives an accelerated failure time model
for
of the stock variable.
the capital stock.
explore such specifications for empirical (S,s) models in detail.
For instance, the log
Caballero and Engel (1999)
EXTREMAL QUANTILES AND VALUE-AT-RISK
Organization of the Article. The
1.3.
rest of the article
is
3
organized as follows. Section 2
describes the basic model of extremal quantiles and extremal conditional quantiles. Section
and inference theory. Section 4 reviews the key empirical
3 describes basic estimation theory
applications and provides an illustrative example. Section 5 concludes.
Basic Models of Extremal Quantiles
2.
A Basic
2.1.
Model
sume that the
means that
of Extremal Quantiles. Towards discussing inference methods,
Y
distribution function of the response variable
behave approximately
tails
like
has Pareto-type
power functions.
Such
tails,
as-
which
prevalent in
tails are
economic data, as discovered by the prominent Italian econonometrician Vilfredo Pareto
Pareto-type
1895.''
tails
encompass or approximate a
in
rich variety of tail behavior, includ-
ing that of thick-tailed and thin-tailed distributions, having either
bounded
unbounded
or
support, and their mathematical theory in connection to extreme value theory has been
developed by Gnedenko (1943) and de Haan (1970).
F
Consider a random variable
end-point of the support of
the support of
Y
is
Fy7^(0)
has lower end-point Fj^{0)
function
Fu and
its
Y
U
and define a random variable
— oo, and U =
is
> — oo. The
= — oc>
— Fy
^(0),
if
,
ii
the lower
the lower end-point of
distribution function of U, denoted by
=
or Fj"'(0)
quantile function
1'
U = Y
as
The assumption
0.
Fy then
that the distribution
exhibit Pareto-type behavior in the tails can be
FJ^
formally stated as the following two equivalent conditions:
some
for
real
number
F~^{0), and L{t)
is
<^
Fuiii)
-
Fy\T)
-
7^ 0,
asu\Fj\0),
L{u)-ir'^^
asr\0,
L(r)T--'
where L{u)
The
A
^
<^
defined in (2.1) and (2.2)
absolute value
distribution
<
and a
|^|
Fy with
of the
EV
a nonparametric, slowly-varying function at
is
called the
is
support point
t
if
^
>
<^
1),
tlie
while setting
i^
extreme value (EV) index.
=
\
EV
and many
index ^
yields the
—
>
For example, the
others.
A
notation a
function u
1-^
~
6
means that ajh
L(u)
is
if
include stable
t-
Yjv and exhibits a wide range
Cauchy
distribution which has heavy
=
30 yields approximately the normal distribution which
—
1
^See Pareto (1964).
The
of
logarithmic function.
Distributions with ^
0.
distributions,
of tail behavior. In particular, setting v
=
and
The prime examples
index ^ measures heavy-tailedness of distributions.
distribution with u degrees of freedom has the
(with
— L
0.^
Pareto-tj^ae tails necessarily has a finite lower support point
infinite lower
distributions, Pareto distributions,
tails
(2.2)
a nonparametric slowly-varying function at
slowly-varying functions are the constant function L{y)
The number
(2.1)
'
>
as appropriate hmits are tal;en.
said to be slowly-varying at
if
limits [iO/iln^Ol
=
1
for ^i^Y
"t-
>
0.
VICTOR CHERNOZHUKOV AND SONGZI DU
4
has light
tails
(with
—
(,
On
1/30).
the other hand, distributions with
<
<,
include the
uniform, exponential, Weibull distributions, and others.
The assumption
variation assumption, as
Fu
tion
commonly done
is
=
It
extreme value theory.
m>
any
m
m~'^'^, for any
regular variation of quantile function
for
in
Distribution func-
said to be regularly varying at FJ" (0) with index of regular variation
is
lim \^-i,g>(Ft/(ym)/F[,i(y))
771"'',
be equivalently cast in terms of the regular
of Pareto-type tails can
Fj^
>
This condition
0.
equivalent to the
is
with index -^. limT\Q{Fy^ {Tm) / Fy^
at
=
(^
distribution functions. These distributions functions have exponentially light
in
this case explicitly.
(^,
including at ^
approximated by the case of
2.2.
A Basic
Model
^
chief examples.
that
it
=
tails,
0,
«
0.
inference theory for the case of
for
x'f3(T).
every x in the support of X.
aU r e
I=
Y
(O,??],
given
some
=
i^
we
statistics are
can be adecjuately
classical linear
X = x:
r?
G
This linear functional form
(2.3)
(0, 1],
is
flexible in the sense
has good approximation properties. Given the original regressor A'*, the
regressors
with the
simplify exposition,
of Extremal Conditional Quantiles. Consider the
Fy\T\x)
for
To
However, since the limit distribution of main
=
functional form for the conditional quantile function of
and
=
{t})
corresponds to the class of rapidly varying
normal and exponential distributions being the
continuous
if
0.
should be mentioned that the case of
do not discuss
—1/4
final set of
X to be used in estimation can be formed as a vector of approximating functions.
X may include power functions, splines, and other transformations of A'*.
For example,
The
linear functional
The
(2005).
following
form also provides computational convenience.
model
for the tails
The main assumption
is
and
its
generalizations were developed in Chernozhukov
that the response variable Y, transformed by
regression line, has regularly varying tails with
suppose there exists an auxiliary parameter
conditional end-point
or
— oo
a.s.
and
its
/i^
EV
index
c^.
some
auxiliary
Indeed, in addition to (2.3),
such that the disturbance
U = Y — X'P^
conditional quantile function F^^(t|x) satisfies
the following tail-equivalence relationship as r
\
0,
uniformly
for
x in the support of X:
F^\t\x)^F-\t\x)-x'i3,-^F~\t),
where F^^
is
(2.4)
a quantile function such that
F-Hr)^L{T)T-^,
where L{t)
is
has
"
.
a nonparametric slowly-varjdng function at
type behavior on the conditional law, while equation
0.
Equation
(2.5)
imposes Pareto-
(2.4) requires this behavior to hold
uniformly across conditioning values. Since this assumption only affects the
covariates to impact the extremal quantiles
(2..5)
and the central quantiles very
tails, it
allows
differently; the
EXTREMAL QUANTILES AND VALUE-AT-RISK
impact of covariates on extremal quantiles
approximated by
is
5
which could
(3^,
differ
sharply
from, for example, the impact on the median given by /3(l/2). Chernozhidrav (2005) provides
further generalizations of this model.
3.
Basic Estimation Methods
Estimates based on sample quantiles. Given
3.1.
T
observations {Yt,t
=
1,
...,T}, the
r-sample quantile can be obtained by solving the following optimization problem:
r
V
Fy' (r) e arg min
where Pt{u) =
Rubin (1964).
(r
—
< 0))u
\{u
Sample quantiles are
extreme order sequence
tT —
and
>
oo,
if
-
(3)
(3.6)
,
the asymmetric absolute deviation function of Fox and
also order statistics,
The sequence
quantile.
is
pr [Yt
and we
will refer to
T)
of quantile index-sample size pairs (r,
r
and a central
\
and rT
orrfer
-^ k
sequence
if
>
r
0,
is
tT
as the oi^der of r-
will
be said to be an
an intermediate order sequence
fixed
and
T—
oo.
>
Each
\
r
if
different type of
the sequence leads to different asymptotic approximations to the finite-sample distributions
of
sample quantiles. Extreme order sequences lead to non-normal (extreme- value)
butions (EV) that approximate the
finite
sample distributions of extremal (high and low)
much better than the normal distributions do. In
work much better than the normal distribution if tT < 30.
quantiles
3.1.1.
Extreme Order Quantiles. Consider an extreme-order
classical result
r
—
k/T, as
T
on the limit distribution
distri-
particular,
seciuence.
of order statistics:
for
EV
The
distributions
following
any integer
A:
>
is
1
the
and
—^ oo.
AriFyHr) - Fy^r)) -, T'^ -
fc-«,
(3.7)
+ 4,
(3..
/here
At^I/F^Hi/T),
and
(£'i,£^2!
)
is
Tfc
-
£!
+
...
an independent and identically distributed sequence of standard exponen-
tial variables.
Result (3.7) was obtained by Gnedenko (1943) under the assumption that Yi,Y2,--.
a sequence of independently and identically distributed
(3.7)
(i.i.d.)
random
variables.
is
Result
continues to hold for stationary weakly-dependent series, provided the probability
of extreme events occurring in clusters
is
negligible relative to the probability of a single
extreme event (Meyer 1973). The results have been generalized to more general time
processes (Leadbetter, Lindgi'en, and Rootzen 1983).
series
VICTOR CHERNOZHUKOV AND SONGZI DU
6
Result (3.7) gives an extreme value (EV) distribution as an approximation to the
sample distribution of Fj7^(r). The
EV
EV
the definition of the
distribution, are
distribution of the k-th order statistic
The EV
distribution
is
The
classical result
constant
make
At
is
is
characterized by the
is
known
if
(^
<
gamma random
as
may have
and has
L = {Fy\2T) -
Another way
to
~
One way
Lt~^ then one can estimate
,
F,Ti(r)))/(2-«
-
overcome the aforementioned
Zt{t)
m
>
1
up
to order
mk
At
distribution only depends on the
The
1/.J if
\
0.
problem
is
to
For instance,
methods described below and
is
to consider the asymptotics
mk
'
(3-9)
.
k
EV
index
(3.10)
.
,
it is
^,
completely a function of data. The limit
and
its
quantiles can be easily calculated
by simulation.
and tT —>
oo,
^ AT{Fy\T) -
Fy'ir))
-, A^
(o,
^,J\^, ^
(3.11)
,
Haan
where
At
gives a
normal asymptotic approximation to the finite-sample distribution of sample
defined as in (3.10).
This
result,
obtained by Dekkers and de
Fy^{t). The main condition for application of this distribution is that tT
samples, we may interpret this as requiring that tT > 30 at the minimum.
The normal approximation
better, because
As
under further regularity conditions,
Zt{t)
is
EV
>
^
Intermediate Order Quantiles. Consider next an intermediate order secjuence.
3.1.2.
ways
limit
variable.
an integer.
is
feasible in that
is
this
consistently.
^^ff
r~? — ~S'^
r"
->,
^
Fy\viT) ~ Fy\T)
r
The
gamma
i^,
Chernozhukov (2006):
^r=-^-T
analytically or
variables.
overcome
to
infeasibility
= AT(Fy\r) - FyHr))
such that
Here, the scaling factor
of
At
^ using
•^
for
entering
l)r-f).
of self-normalized extreme order quantiles, as in
where
index
F/j,
not feasible for purposes of inference on Fy^lr), since the scahng
additional strong assumptions in order to estimate
suppose that FJ" ^(r)
EV
significant (median) bias.
moments
finite
not easily estimable consistently.
is
Variables
therefore a transformation of a
not symmetric and
moments
distribution has finite
L by
distribution
can be estimated by one of the methods described below.
vifhich
finite-
it
(3.11)
does not
approximation (3.11) once k
is
fail
large.
is
(1989),
cjuantile
-^ oo. In finite
convenient, but extreme approximation (3.9)
when tT
-^ k
<
oo and
it
is al-
coincides with the normal
EXTREMAL QUANTILES AND VALUE-AT-RISK
3.1.3.
many
Extremal Bootstrap. In
cases
it is
following approach. Consider the sample of
7
convenient to implement inference using the
i.i.d.
variables:
{Yu...,Yt)-(^^,-J-^],
where
{S\,...,8t)
ated in this
an
is
way have
(3.12)
sequence of standard exponential variables.® Variables gener-
i.i.d.
quantile function:
Fy\r)^tMl^J^n^,
Observe that Fy7 ^(r)
—
1/.^
~
t~^/^, so condition (2.1)
the finite-sample distributions of Zt{t)
tribution of Zt{t) for the case
EV
the
limit (3.9)
when
and the normal
satisfied.
is
— AriFy^ {t) —
limit (3.11)
simulation can be done using the following algorithm:
< B, draw
suitable estimate
2.
^.
{Yi,...,Yt) as
Compute
for the case
the statistic ZT^iir)
=
AriFy'^
Use quantiles of the simulated sample {ZT,i{T),i < B)
EV
(3.27) holds exactly.
{t)
—
Fy^{t)).
for inference purposes.
statistics,
including estimators
index and extrapolation estimators.
Another method, developed
normalized quantile
However,
when
according to (3.27), replacing ^ with a
i.i.d.
This scheme could be used to estimate distributions of other
of the
dis-
we reproduce both
under extreme and intermediate sequences,
good finite-sample performance
The
i
propose to estimate
Fy^{t)) by the finite-sample
also guarantee
For each
We
the data follow (3.27). In this way,
and
1.
(3.13)
.
it
in
Chernozhukov (2006),
This method
statistic.
is
less
under more general conditions.
applies
is
based on subsampling the
self-
accurate than the extremal bootstrap.
It
should be noted, that the canonical
(nonparametric) bootstrap does not work in these settings (Bickel and R'eedman 1981).
3.1.4.
Confidence Intervals for
of Zt{t) be denoted by c{a).
Fy [t) and Bias Correction for Fy (r). Let the
The estimates of c{a) can be obtained using
a-quantile
either
EV
approximation, normal approximation, or the extremal bootstrap, also having replaced
with a suitable estimate. Denote the resulting estimates by
c{a.).
Then, the median
<^
bias-
corrected estimate and a%-confidence region for Fy'^{T) can be constructed as
.,.,.) -
Yj, defined in this
and Gumbell
3^
and
P,.M-31^.f.-(.)-^
(3.14)
way, follows generalized extreme value distribution, which nests the Frechet, Weibull,
distributions.
There are other
by uniform variables {Ui, ...,Ut)
,
in
possibilities, for
which case
Yt follows
example,
(fi,
...,
ifr) in (3.27)
can be replaced
the generalized Pareto distribution.
VICTOR CHERNOZHUKOV AND SONGZI DU
EstimMors of
due to Pickands
EV Index ^.
3.1.5.
tli.e
tor,
(1975), relies
There are two principal estimators. The
on the ratio of sample quantile spacings:
= -\n
^
estima-
first
In 2,
(3.15)
Fy\2T)-Fy'{T))
such that T
—
and tT ^> oo
>
T—
as
Another estimator, developed by
^
>
^
and tT —* oo
as
oo.
Under further
Hill (1975),
is
a
moments
T—
>
oo.
power law to the
Under further
tail data.
estimator:^
,3,,,
This estimator
The estimator can be motivated by a maximum
0.
regularity conditions,
£LmV^,-W)-
,-,
such that r
>
is
applicable only for the case of
likelihood
method that
fits
an exact
regularity conditions,
v^{i-o^dj^{o,e)
(3.18)
The methods for choosing r are described in Embrechts, Kliippelberg, and Mikosch
(1997). The variance of estimators decreases as r increases, but the bias (relative to the
true
£,)
goes up.
Another view on the choice of r
approximations, not
literal
is
descriptions of the data.
the following: statistical models are
In practice,
threshold r reflects that power laws with different values of ^
Therefore,
if
the interest
lies
in
making inference on Fy^{t)
fit
for a particular r,
sonable to use ^ constructed using the same r or most similar
that t'T
The
>
r'
£^
on the
tail regions.
it
seems rea-
subject to the condition
30.^
limit results
alternative,
dependence of
better different
above can be used for the construction of confidence regions.
we can apply extremal bootstrap
the quantiles of
Zj--
to statistic
Zt = \/tT(^ —
cj)
As an
to estimate
Given the estimated a-cjuantiles c{a), we can construct the median
bias-corrected estimate and Q%-confidence regions for
3.1.6.
Extrapolation Estimators.
cisely,
the following strategy
is
When
^.
very extreme ciuantiles cannot be estimated pre-
sensible: estimate less
extrapolate these estimates using the assumptions on
extreme
tail
ciuantiles reliably,
behavior stated
earlier.
and then
Dekkers
and de Haan (1989) developed the following extrapolation estimator:
F-\r,)
=^ldl^l^\F-'C2r) - Fy\T)\ + Fy^r),
Notation (i)- means (i)_
= —x
if
x
<
and [x)-
=
if
x
>
0.
o
The
latter condition requires that a sufficient
sample be available to estimate
^.
(3.19)
EXTREMAL QUANTILES AND VALUE-AT-RISK
where
Another useful estimator, which
Te <^ t.
is
vahd only
for the case of ^
>
0, is
the
following:
FyHre) = ire/Tr^-Fy\T),
where
Tg
<C
The above
r.
(3.20)
estimators have good properties provided the quantities on the
right-hand side of (3.19) and (3.20) are well estimated, which requires that
and that the
3.2.
1,
model be a good approximation of the underlying true
tail
Estimates based on sample regression quantiles. Given
,T},
the quantile regression estimate of Fy7^(r|a:)
Fy\T\x)=x'f3{T),
arg
/?(r) =.
is
large,
T observations
{Yt,Xt,t
—
given by:
T
min
tT be
tail.
pr {Yt
~ X[p)
(3.21)
,
where Pt(u) = (t - l{u < 0))u. Quantile regxession was introduced by Laplace (1818) for
the median case. Koenker and Bassett (1978) extended this formulation to other quantiles.
3.2.1.
Extreme Order Asymptotics. Chernozhukov (2005) derives asymptotic distributions
of regression quantiles under extreme order sequences. Consider the canonically-normalized
QR statistic
Zrik)
=
and the self-normalized
At{iS{t)
QR
-
/?(t)),
where
At = l/F-\l/T),
statistic
-,
ZT{k)=AT{l3{T)-i3{T)), where >^T
where TT{m. -
1)
>
d.
The
(3.22)
first statistic
(3.23)
X'iPimr) - P{t))
uses an infeasible canonical normalization, while
the second statistic uses a feasible normalization.
Then
as
rT —
>
A;
>
and
T
^ oo
ZT{T)^dZ^\k)-k~^,
(3.24)
Z<xi{k)
=
Zoo{k)
~ —argmin ~kE[xrz + Y^\xlz +
kE[Xrz + J2\xlz-T-'
argmin
r;'
{^
<
0)
(e>o)
i=l
where {ri,r2,...} :=
variables that
is
{^i,!?! +£'2,...}
independent of {Xi,
Zt{t)
The
results hold
and {81,82,
X-y, ...}.
}
is
an
Further, for any
iid
m
sequence of exponential
such that
k{m —
VkZ^{k)
E[X]'{ZUr,zk)-Z^\k)y
under the assmnption that the data come from either an
I)
>
d,
(3.25)
i.i.d.
sequence
or a stationary weakly-dependent sequence with extreme events satisfying a non-clustering
condition.
VICTOR CHERNOZHUKOV AND SONGZI DU
10
Related results
normalized statistics
for canonically
for the case
where
tT —
as
'
T—
>
oo
have been obtained by Knight (2001) and Portuoy and Jureckova (1999).
3.2.2.
Intermediate Order' Asymptotics. Chernozhukov (2005) shows that under intermedi-
ate order sequences, as r
\
and tT ^>
oo,
ZT{r)=AT{p{T)-P{T))-^,Af(^0,[E(XX')l'j--^l^-^y
where
At
is
(3.26)
defined as in (3.23). Like the result under extreme order sequences, this result
holds under the assumption that the data
come
either
from an
i.i.d.
sequence or from
a stationary weakly-dependent sequence with extreme events satisfying a non-clustering
condition.
3.2.3.
Extremal Bootstrap. In practice,
it is
convenient to implement inference by construct-
ing a bootstrap model that approximates the
model under the assumptions
tail
features of the true conditional quantile
of Section 2.2; then using this
butions of estimators of extreme quantiles and
tail
model
to simulate the distri-
parameters.
Consider the sample
where
(i?i, ...,£^r) is
an
i.i.d.
fixed set of observations
on
('f^^,X, V
^
{{Y,,X,),...,iYr,Xr))
....
{^l^^X^
(3.27)
,
sequence of standard exponential variables and (Xi,...,X7')
regi'essors that
we
have. Variable Yt generated in this
is
way has
the conditional quantile function
,,,.
,_i,
Fy^\T\X^)
Yt \- '---'
=
,.,..
.
X[(3{t)
'-<.-v-
hind
'
_^
^^''^
where,(r).(bMl^llI:i=-I,0,...,0)\
Observe that Fy [T\Xi)
—
1/^
~
t~^/^, so the model satisfies conditions (2.4) and
as does the true conditional quantile
the finite-sample distributions of Zt{t)
of Zt{t) in the case
the
EV
when data
(2.5),
model under our assumptions. Hence we can estimate
— Aj-{p{r)—P{T))
follows (3.27).
by the finite-sample distribution
In this simple way,
we can
replicate
both
approximation (3.26) and the normal approximation (3.24) and also guarantee good
finite-sample performance for the case
when
the model (3.27) holds.
The simulation can be
done using the following algorithm:
1.
For each
^.
2.
i
< B, draw data
Compute
according to (3.27), replacing ^ with a suitable estimate
the statistic Zt,i{t)
Use the empirical distribution
= At{P{t) —
of the simulated
P{t)).
sample (Zy
j(r),
i
< B)
for inference.
EXTREMAL QUANTILES AND VALUE-AT-RISK
11
This method can also be used to estimate distributions of estimators of the
EV
index and
extrapolation estimators described below.
As mentioned
before, there
another inference method proposed by Chernozhukov (2006)
is
which uses subsampling to estimate the distribution of self-normalized
method
less
is
accurate than the extremal bootstrap, but
it
statistic
applies under
Zt{t). This
more general
conditions.
3.2.4.
Confidence Intervals and Bias Corrected Estimates. Suppose we are interested in the
parameter
tp' (3{t)
for
some non-zero vector
Let the a-quantile of
)/'•
4''
Zt{t) be denoted by
Having replaced ^ with a suitable estimate, the estimates of c(q) can be obtained using
either EV approximation, normal approximation, or the extremal bootstrap. Denote the
c{a).
resulting estimates
by
The median-bias
cla).
corrected estimator and the Q;%-confidence
interval for 4''(3{t) can be constructed as
c(l/2)
^'/3(7
3. 2. .5.
At
Estimators of the
Pickands and
i'Pir)
EV Index ^.
The
Hill estimators.
The
first
At
tT
is
- a/2)
(3.29)
At
following estimators are regression analogs of the
estimator takes the form
In 2,
Fy\2T\X)
X
A>'P{t)
Fy'MX)-Fy\2r)\X)
^^-In
where
cjl
g(«/2)
and
the average value of
-^ DO
A'j.
(3.30)
Fy\T)\X)
Under additional
and
regularity conditions, as r
^
(3.31)
(2(2f-l)ln2)2^
The second
estimator, which
regularity
applicable
when
^
>
0,
takes the form:
Y:UHyt/Fy\r\Xt))Tt
conditions, as r \
and tT —
i
Under additional
is
=
>
(3.32)
oo
v^(e-o-d^v-(o.s^').
The
limit results
native approach
is
above can be used
(3.3.3)
for the construction of confidence regions.
to apply extremal bootstrap to statistic
Then we can
the quantiles of this statistic.
Zt
?(l/2)
^-^
,
- a.nd
^/^
alter-
to estimate
use estimated a-quantiles c{a) for constructing
the median bias-corrected estimate and a%-confidence regions for
e
'tT{^
An
?_ ?(l-Q./2) ^
TTT
^:
c{a/2)
v^
(3.34)
VICTOR CHERNOZHUKOV AND SONGZI DU
12
By analogy with
Extrapolation Estimators.
3.2.6.
estimators for
where
Tg <Si r.
Fy
where
(Te|x),
Fy\Te\x)
=
Fy\Te\x)
=
a very low value, can be constructed as
r^ is
^
the unconditional case, the extrapolation
~
^
^^'^^l
\Fy\mT\x) - Fy'{T\x))] + FyHrlx),
{Te/T)-^Fy'{T\x)
Note that the estimator
(3.36)
(^
is
>
(3.35)
(3.36)
0),
valid only in the case ^
given for the unconditional case apply here as well. Also,
>
0.
The comments
we can construct confidence regions
Fy ^(Te|x) based on extrapolation estimators. This can be done by applying the extremal
bootstrap to statistic Zt = ATiFy\Te\x)-Fy\Te\x)) where At = s/rf / X' (p{2T)-d{T)).
for
4.
4.1.
A
Empirical Applications: an Overview and an Illustration
Simple Overview. The
following review
is
not exhaustive by any means;
aims
it
to provide only a few quintessential references.
Extremal Unconditional Quantiles. As mentioned in Section
4.1.1.
come and wealth data
in 1895
2,
Pareto analyzed
and suggested that power laws accurately describe the
intail
data.
Pareto's discovery, although remarkably simple, had a profound elTect on both em-
pirics
and the theory
of extremes.
Zipf (1949), Mandelbrot (1963),
Fama
(1965), Praetz
(1972), Sen (1973), Jansen and de Vries (1991), Longin (1996), among others, gave
fur-
ther empirical evidence on the nature and prevalence of Pareto-type laws in economic data,
including city
incomes, and financial returns.
should be mentioned that
It
The
sizes,
theoretical
work
this aspect, the
in
many
of the early studies were highly informal in nature.
extreme value theory has opened paths
our knowledge, the
first
highly rigorous analysis of the
Jansen and de Vries (1991) estimate the
between
at-risk,
,^
=
for better analysis.
study of Jansen and de Vries (1991) can be singled out as
1/5 and ^
=
1/3.
EV
tail
Pi-om
gave, to
it
properties of financial returns.
indices for various
primary U.S. stocks to be
Using quantile extrapolation estimators to estimate value-
they also conclude that the 1987 market crash was not an
outlier.
Rather
it
was a
rare event, the magnitude of which could have been predicted using prior data. This study
stimulated numerous other studies that rigorously document the
tail
properties of economic
data (Embrechts, Kluppelberg, and Mikosch 1997).
4.1.2.
Extremal Conditional QuantUes. There has been considerably
tional methods.
will see active
of the topics
development
and
less
work on condi-
However, following recent theoretical advances we expect that
directions.
in the near future.
In
what
follows,
we merely
this area
highlight
some
EXTREMAL QUANTILES AND VALUE-AT-RISK
In
what might be the earhest example
13
of conditional quantile analysis, Quetelet (1871)
Remarkably, Quetelet's work
fitted various conditional quantile curves to age-height data.
included tabulations of very high and very low quantiles of heights as a function of age.
There
is
a great potential for the applications of extremal quantile regression methods in
similar problems. In a recent study,
Chernozhukov (2006) estimates the impact
and maternal behavior on extremely low birthweights
He
finds that the
1500 grams sharply
smoking
care
is
in the U.S., focusing
impact of these variables on birthweights
differs
in the
of
smoking
on black mothers.
ranges between
from their impact on the central birthweights.
and
2.50
For instance,
not correlated with extremal birthweights, while quality of prenatal medical
is
strongly linked to extremal birthweights.
Chu
Aigner and
(1968),
Timmer
and Aigner, Amemiya, and Poirier (1976)
(1971),
neered a large empirical literature on production frontiers.
A
pio-
major problem of the sub-
sequent empirical hterature has been the lack of statistical methods for construction of
The new methods
rehable estimates and confidence regions.
discussed in Section 3 solve
the problem, and should improve the rigor of the empirical work in this area.
There
is
a considerable appeal for the use of extremal conditional cjuantile methods in
An
auction models.
(2002),
who
important study that
illustrates the potential
analyze an empirical structural auction model.
support function of bids using extreme order
statistics for
is
by Donald and Paarsch
They estimate the
each covariate
cell,
conditional
then project
the estimated function onto a lower dimensional structural function implied by the model
via a minimum-distance method.
A
generalization of this approach
is
to
employ extremal
quantile regression for estimation of the (approximate) support function in the
Value-at-risk
tile
is
another potentially important area of applications of the extremal quan-
Chernozhukov and Umantsev (2001) apply these methods to the problem
regression.
forecasting value-at-risk of a major U.S.
quantiles, using
gions, using
An
some
of the
We
main questions
some
in
Chernozhukov
(2006).
The
consider a problem of forecasting conditional
of the questions asked in
Chernozhukov and Umantsev (2001)
with an improved methodology. To implement the analysis, we use algorithms written
R
re-
section below
of this study.
Example. Here we
revisit
of
company. They estimate extremal conditional
both ordinary and extrapolation methods, and implement confidence
Illustrative
value-at-risk.
oil
subsampling methods described
briefly revisits
4.2.
first stage.
language that rely on Koenker's (2006) quantreg package as the basic platform.
in
The
algorithms as well as the data set can be downloaded from www.mit.edu/~vchern/EQR/.
A
detailed description of the data set
The
given in Chernozhukov and
Umantsev
(2001).
use of near-extreme quantile regression for estimation of approximate support functions allows the
researcher to discard
Section
is
1.
some
outliers that
do not conform the model,
in the spirit of the discussion given in
VICTOR CHERNOZHUKOV AND SONGZI DU
14
We
estimate the following conditional quantile function for various low values of
Fy^\T\Xt)
where
Yt
We
Dow
first
graph of
X[(3{t)
=
/?o(r)
+
/3i(t)Xu
+ /?2(r)X,,2 + [h{T)Xt^z.
(4.37)
the daily (log) return on the stock of Occidental Petroleum, Xt^\
is
return on spot
on the
=
oil price,
Xt,2
—
Yt-i
is
the lagged
own
and Xt^z
return,
is
r:
is
the lagged
the lagged return
Jones Industrial Index. There are 2527 observations in the sample.
estimate and plot the function
(r, i) i-^
X[(3{t) in
shown
gives a
good picture
this function,
in Figure
1
,
time, indicating dates where the predicted risk
what causes these
is
(r, t)
space, with r
<
.2.5.
The
of the evolution of risk over
Let us next determine
especially high.
risk fluctuations.
0.25
Figure
Table
1
1.
The
fit
X[i3{t) as a function of time
reports the estimates of the coefficients of the model (4.37).
reports median bias-corrected estimates and
90%
primary determinant of the high
is
risk levels
is
The
table also
confidence regions, which were obtained
using the extremal bootstrap approach described in Section
larger
and
t
the market.
3.
The
The
results
show that the
further in the tail
the magnitude of the point estimate of the coefficient on the
the confidence region for this coefficient excludes zero even at r
=
DJI
.001.
we
go. the
return. Moreover,
EXTREMAL QUANTILES AND VALUE-AT-RISK
Table
1.
Estimate
Coefficients
Estimation Results
r
Lag Return
Oil Price
90%
Bias-Corrected
-0.08
Intercept
15
=
Conf. Region
.001
-0.08
[-0.11,-0.06]
0.12
0.01
[-0.19, 0.63]
-0.05
-0.05
[-0.42, 0.11]
0.71
0.73
DJI Return
r
—
[
0.05, 1.08]
.01
Intercept
-0.05
-0.05
[-0.05,-0.04]
Lag Return
-0.04
-0.06
[-0.16, 0.12]
Oil Price
-0.05
-0.06
[-0.17, 0.02]
0.49
0.50
DJI Return
r
=
[
0.24, 0.65]
.05
Intercept
-0.03
-0.03
[-0.03,-0.02]
Lag Return
-0.03
-0.04
[-0.10, 0.04]
Oil Price
0.01
0.01
[-0.04, 0.06]
DJI Return
0.29
0.30
[0.17, 0.38]
We next characterize
the estimates of
the
tail
EV index ^
90%
bias-corrected estimates, and
approach described
properties of the conditional quantile model. Table 2 reports
obtained using estimator (3.32).
The
table also reports
confidence regions, which were obtained using the nested
in Sections 3.2.3
and
3.2.5.
The
bias-corrected estimates tend to be
stable with respect to the start of the tail determined by probability index r.
of Table
2,
we
take \
«
1/4 to be the estimate of the
Table
r
=
T=
T
T
=
2.
Estimation Results
EV
for
EV
Index
90%
Bias-Correct. Estimate
0.24
0.22
[0.08, 0.34]
.01
0.23
0.17
[0.05, 0.25]
.025
0.32
0.24
[0.14, 0.30]
.05
0.35
0.23
[0.16, 0.27]
We
EV
index,
Fy^ (.0001[Xf) only once per about 30
Conf. Region
we can now estimate very extreme
set the risk level at
r
=
the basis
<^
Estimate
Having characterized the
On
index.
.005
extrapolation methods.
median
quantiles using
.0001, so that the return falls
years, a very rare,
below
extreme event. The extrapolated
estimates of .0001-quantile are obtained using equation (3.36) with r
=
.05
and ^
=
1/4.
VICTOR CHERNOZHUKOV AND SONGZI DU
16
The
resulting extrapolation
Fy^ {.000l\Xt) sharply
fit
The reason
obtained by quantile regression.
data that
likely contains
for this
simple: the ordinary
fit
fit
X[I3{t)
uses sample
no observations on the extreme events defined above. In sharp
contrast, the extrapolated
uses the
fit
tail
model and a
tail
model
reliably estimated conditional .05-
The
quantile to predict the magnitude of such events.
depends on whether the
is
from the ordinary
differs
quality of this prediction clearly
accurate.
is
Extrapolalated vs Ordinary Estimates of
Cond
Quantiles
in
o
d
1
I
1
"
M
"
i\
0)
M
c
'.
'
\
'
§
j_
2
d
j/ Y
' "x
I
5
o
q
'
B
N
'
\^
'/\'
'
I
V
'
A
I
/
*
\
^
'
\/
/
/
I
^
\
-'
'-y\j\-f\
1
1
'
1
'
'
'',
'
1
'l
1
g
'
Mr
1
11
1
I'll
^
5
''
1
'
\
\l
1
I
1
1l
c
ll
1
o
OJ
d
I
1
1
1
2480
2490
2500
2510
—
ordinary
- - -
extrap
fit
fit
2520
time
Figure
2.
Extrapolated and Ordinary Estimates of the Conditional .0001-Quantile.
5.
This
article
Conclusion
examines the theory and empirics of extremal quantiles
in economics.
The
theory of extremes provides a set of apphcable methods that have generated numerous
valuable empirical findings. There
conditional quantile methods.
for further empirical
The
is
equally promising scope for the use of the extremal
latter
methods are new - there are great opportunities
and theoretical developments.
Refere.nces
Abrevaya,
"The effect of demographics and maternal behavior on the distribution of birth
J. (2001):
outcomes," Empirical Economics, 26(1), 247-259.
EXTREMAL QUANTILES AND VALUE-AT-RISK
17
"On the estimation of produclion frontiers:
J., T. Amemiya, AND D. J. PoiRiER (1976):
likelihood estimation of the parameters of a discontinuous density function," Internal. Econom.
AlGNER, D.
maximum
Rev., 17(2), 377-396.
AlGNER, D.
J.,
AND
S. F.
Chu
Econom.ic Review, 58, 826-839.
Arrow, K., T. Harris, and J.
(1968):
"On Estimating the Industry Production Function," American
Marschak
(1951):
"Optimal Inventory Policy," Econornetrica,
19,
205-
272.
BiCKEL,
9,
P.,
and
D.
Freedman
(1981):
"Some Asymptotic Theory
for
the Bootstrap," Annats of Statistics,
1196-1217.
"Explaining Investment Dynamics in U.S. Manufacturing: a
R., and E. Engel (1999):
Generalized (S,s) Approach," Econornetrica, 67(4), 783-826.
Chernozhukov, V. (2005): "Extremal quantile regression," Ann. Statist., 33(2), 806-839.
(2006): "Inference for Extremal Conditional Quantile Models, with an Application to Birthweights,"
MIT, Department of Economics, Working Paper.
Chernozhukov, V., AND L. Umantsev (2001): "Conditional Value-at-Risk; Aspects of Modehng and
Estimation," Empirical Econoraics, 26(1), 271-293.
DE Haan, L. (1970): On Regular Variation and Its Applications to the Weak Convergence. Mathematical
Centre Tract 32, Mathematical Centre, Amsterdam, Holland.
Dekkers, a., and L. de Haan (1989): "On the estimation of the extreme- value index and large quantile
estimation," Ann. Statist, 17(4), 1795-1832.
Donald, S. C, and H. J. Paarsch (2002): "Superconsistent estimation and inference in structural
econometric models using extreme order statistics," J. Econometrics, 109(2), 305-340.
Embrechts, p., C. Kluppelberg, and T. Mikosch (1997): Modelling extremal events, vol. 33 of Applications of Mathematics. Springer- Verlag, Berlin.
Fama, E. F. (1965): "The Behavior of Stock Market Prices," Journal of Business, 38, 34-105.
Fox, M., and H. Rubin (1964): "Admissibility of quantile estimates of a single location parameter," Ann.
Math. Statist, 35, 1019-1030.
Gnedenko, B. (1943): "Sur la distribution limite du terme d' une serie aletoire," Ann. m.ath., 44, 423-453.
Hill, B. M. (1975): "A simple general approach to inference about the tail of a distribution," Ann. Statist.,
3(5), 1163-1174.
Jansen, D. W., and C. G. de Vries (1991): "On the Frequency of Large Stock Returns: Putting Booms
and Busts into Perspective," Review of Economics and Statistics, 73, 18-24.
Knight, K. (2001): "Limiting distributions of linear programming estimators," Extremes, 4(2), 87-103
Caballero,
(2002).
KoENKER,
Koenker,
R. (2006): Quantreg: Quantile Regression. R package version 3.90, http://www.r-project.org.
R., and G. S, Bassett (1978): "Regression Quantiles," Econometnca, 46, 33-50.
Laplace, P.-S. (1818): Theorie analytique des probabilites. Editions Jacques Gabay (1995), Paris.
Leadbetter, M. R., G. Lindgren, and H. Rootzen (1983): Extremes and related properties of random
sequences and processes. Springer- Verlag, New York-Berlin.
LONGIN, F. M. (1996): "The Asymptotic Distribution of Extreme Stock Market Returns," Journal of
Business, 69(3), 383-408.
Mandelbrot, M. (1963): "The Variation of Certain Speculative Prices," Journal of Business, 36,
Meyer, R. M. (1973): "A poisson-type limit theorem for mixing sequences of dependent "rare"
394-419.
events,"
Ann. Probability, 1, 480-483.
Pareto, v. (1964): Cours d'economie politique. Droz, Geneve.
Pickands, III, J. (1975): "Statistical inference using extreme order statistics," Ann. Statist, 3, 119-131.
Portnoy, S., and J. Jureckova (1999): "On extreme regression quantiles," Extremes, 2(3), 227-243
(2000).
Praetz, V. (1972): "The Distribution of Share Price Changes," Journal of Business, 45(1), 49-55.
QuETELET, a. (1871): Anthropometrie. Muquardt: Brussels.
Roy, a. D. (1952): "Safety First and the Holding of Assets," Econornetrica, 20, 431-449.
Sen, a. (1973): On Economic Inequality. Oxford University Press.
Timmer, C. p. (1971): "Using a Probabilistic Frontier Production Function to Measure Technical
ciency," Journal of Political Economy, 79, 776-794.
ZiPF, G. (1949): Human Behavior and the Principle of Last Effort. Cambridge, MA: Addison- Wesley.
Effi-
GOO^
3^1