Predicting extreme Value at Risk: Nonparametricquantile regression with refinements from extreme
value theory
Julia Schaumburg∗
December 9, 2011

Abstract
A framework is introduced allowing to apply nonparametric quantile regression
to Value at Risk (VaR) prediction at any probability level of interest. A monotonizeddouble kernel local linear estimator is used to estimate moderate (1%) conditional
quantiles of index return distributions. For extreme (0.1%) quantiles, nonparametricquantile regression is combined with extreme value theory. The abilities of the proposed estimators to capture market risk are investigated in a VaR prediction study
with empirical and simulated data. Possibly due to its flexibility, the out-of-sample
forecasting performance of the new model turns to be superior to competing models.
Keywords: Value at Risk, nonparametric quantile regression, risk management, extreme value statistical applications, monotonization∗

Humboldt-Universität zu Berlin, School of Business and Economics, Institute of Statistics and Econometrics, Chair of Econometrics, Spandauer Str. 1, 10178 Berlin, Germany, email: julia.schaumburg@wiwi.huberlin.de, phone +49-(0)30-2093-5603, fax +49-(0)30-2093-.

1

1

INTRODUCTION

1Introduction

1

In bank regulation, the effectiveness of capital requirements in preventing funding shortfalls rests upon the estimation accuracy of market risk measures, the most widely usedof which is Value at Risk (VaR). According to the Market Risk Amendment to the BaselII Capital Accord of 2004, issued by the Bank for International Settlements, VaR is to be
calculated daily, using a ’99th percentile, one-tailed confidence interval’.
Not only against the background of the financial turbulences during the crisis 2007-2009,
there is a practical need for VaR models that are rich enough to capture the dynamics ofquickly changing market environments, while being parsimonious at the same time inorder to avoid overfitting. In this context, kernel-based quantile regression is a natural
choice. Being fully nonparametric, the approach does not require any structural assumptions, neither on the form of the VaR function nor on the shape of the loss distribution.
Furthermore, the primary interest is to predict quantiles accurately, not to estimate structural parameters, so that we do not lose interpretability by employing nonparametricestimation methods.
Until now, however, nonparametric quantile regression has not been widely used in risk
management practice. One possible reason is that precisely because no structure is imposed on the regression function, the speed of convergence is slower than in parametric
regression settings, and generally more observations are needed for accurate estimation.
Therefore, when considering nonparametric quantile regression for VaR prediction, one
has to address the problem of data sparseness at the tails of the loss distribution, whichgets more severe when considering quantiles corresponding to extreme probabilities.
This paper proposes a framework allowing to operationalize nonparametric quantile regression as a VaR estimation tool for both moderate and extreme probabilities. We include only past returns as regressor, which is sufficient, as a maximum of information
can be exploited and any kinds of nonlinearities are captured within the modelling approach. In addition, the model is simple, overfitting is avoided and problems arising fromlimited observations are minimized. A double kernel-type estimator is used to smooth
distortions. Additionally, monotonization by rearrangement is applied at the boundary of the estimated function. Finally, in order to predict VaR corresponding to extreme

1

INTRODUCTION

2probability levels (such as 0.1%), the peaks over threshold method is incorporated andapplied to the standardized nonparametric quantile residuals, resulting in an estimatorthat combines nonparametric quantile regression and extreme value theory (EVT).
Several studies have compared the forecast performances of different VaR models, see,among others, Kuester et al. (2006), Manganelli and Engle (2001) and Nieto and Ruiz
(2008). They take a broad variety of models into account, but nonparametric quantile
regression as a tool for VaR estimation is rarely included. There are three exceptions weare aware of: Cai and Wang (2008) suggest to estimate VaR and Expected Shortfall using
a new nonparametric VaR estimator, combining the Weighted Nadaraya Watson (WNW)
estimator of Cai (2002) and the Double Kernel Local Linear (DKLL) estimator of Yu andJones (1998). In the empirical section, however, only 5% quantile curves are estimated andno forecasts are computed. Chen and Tang (2005) investigate nonparametric VaR estimation, when no regressors are present. Taylor (2008) proposes to combine double kernel
quantile regression with exponential smoothing of the dependent variable in the time domain. 1% and 99% VaRs are predicted from the model along with some benchmarks, but
extreme quantiles are not considered. Although the Basel Committee on Banking Supervision asks banks to report VaR for a 10-day holding period, we focus on one-day-ahead
forecasts, which is in line with the literature.
In contrast to the studies already available in the literature, we present a method that allows to nonparametrically estimate VaR corresponding to any probability that might beof practical interest. We estimate, predict and backtest 1% and 0.1% VaRs for four sets ofindex returns and a simulated time series. Our focus is on gains to loosening assumptions
in comparison to existing VaR models. Therefore, in the empirical application, we choose
to benchmark our model against the most flexible parametric VaR models, the Conditional Autoregressive Value at Risk (CAViaR) models of Engle and Manganelli (2004).
They allow to set up different linear and nonlinear specifications, including the lagged
VaR estimate as a regressor, and they do not rely on distributional assumptions.
We find that although the CAViaR models obtain almost perfect in-sample fits in the caseof 1% VaR, the Double Kernel Local Linear (DKLL) estimator of Yu and Jones (1998) often
outperforms them in terms of out-of-sample backtesting results, especially when the estimation period is short relative to the forecasting period. Results are promising for 0.1%

2

QUANTILE REGRESSION APPROACHES TO VAR ESTIMATION3

VaR as well. The superiority of the EVT-refined DKLL estimator over the plain DKLL
estimator is shown in a small simulation study. It turns out that, especially when theestimation window is small relative to the forecasting period, the extreme value theoryrefined nonparametric model predicts extreme VaRs very accurately.
Section 2 outlines the basic setup of conditional quantile models, before describing CAViaR
models. The Double Kernel Local Linear estimator used in the following is presented in
Section 3. Furthermore, the incorporation of extreme value theory into the model is explained. The investigated data sets and the backtesting method are summarized in Section 4. The empirical results on 1% and 0.1% VaR estimation are stated in Section 5. In
Section 6, the performance of the EVT-refined nonparametric model is further assessed
via a small simulation study. Section 7 concludes.

2

Quantile regression approaches to VaR estimation

2.1

Conditional quantiles

Let {Yt }nt=1 be a strictly stationary time series of portfolio returns and let Xt be a ddimensional vector of regressors. The pth conditional quantile of Yt , denoted by qp (x), is
defined as
qp (x) = inf {y ∈ R : F (y|x) ≥ p} ≡ F −1 (p|x),(2.1)

or, equivalently, as the argument that solvesmin E p − I(Yt < q(Xt )) Yt − q(Xt ) |Xt = x ,qp (x)(2.2)

where I(A) denotes the indicator function on some set A. Both formulations are widelyused in the literature. In the seminal paper by Koenker and Bassett (1978) a sample equivalent of (2.2) where q(Xt ) = X0t β, also including the special case Xt = 1, is established. β
is a d-dimensional vector of unknown parameters. The linear quantile model is extended
to conditionally heteroskedastic processes in Koenker and Zhao (1996). In Engle andManganelli (2004) conditional autoregressive quantile functions are estimated using (2.2)
with q(Xt ) possibly being nonlinear in parameters, see Section 2.2 for some examples.
In a number of papers, localized kernel versions of (2.2) are estimated, leading to a non-2

QUANTILE REGRESSION APPROACHES TO VAR ESTIMATION

4

parametric fit: Yu and Jones (1997) compare the goodness of fit of local constant and local
linear models. On the other hand, Cai (2002), Yu and Jones (1998), Cai and Wang (2008)
propose nonparametric methods to estimate the distribution function in (2.1), which, in asecond step, is inverted. Section 3.1 contains more details on these approaches. Wu et al.
(2007) model (2.1) without regressors, and Chernozhukov and Umantsev (2001) operationalize a linear version of (2.1).
Following the convention of expressing VaR as a positive number, it is defined as
V aRpt (·) = −qpt (·),
where qpt is the quantile of the return distribution corresponding to probability p, at timet. V aRpt denotes a generic VaR measure which may depend on x and/or a vector ofparameters β. To simplify notation, index t is suppressed in contexts where it does notcause confusion.

2.2

Conditional Autoregressive VaR (CAViaR) Models

The class of Conditional Autoregressive Value at Risk (CAViaR) models, first introduced
by Engle and Manganelli (2004), is used to benchmark the forecast performance of thenonparametric VaR estimators considered here. Several comparison studies have doneso, for example Kuester et al. (2006) or Taylor (2008). CAViaR models are dynamic VaRmodels describing the quantile of a random variable at time t, e.g. the return on a financial portfolio, as possibly nonlinear function of its own lags and, in addition, of a vectorof observable variables, Xt :V aRpt (β) = β0 +

r1
X
i=1

βi V aRpt−i (β) +r2X

βj f (Xt−j ),j=1where d = r1 + r2 + 1 is the dimension of β, the parameter vector that solves
n

min
β


1 X
p − I Yt < −V aRpt (β) (Yt + V aRpt (β)).
n

(2.3)

t=1

A straightforward choice for Xt is lagged returns. Following the original article, the specifications used here include the first lagged value of V aRp (·) and the first lagged value of

2

QUANTILE REGRESSION APPROACHES TO VAR ESTIMATION

5

Yt , therefore Xt = Yt−1 .
Well-known stylized facts on asset returns are, firstly, that they exhibit volatility clustering. It carries over to VaR: if high variation is observed in returns of the recent past, it
is likely to continue, and risk is therefore high as well. Secondly, quantiles (or volatility)
might react differently according to the sign of past returns. This possibility is capturedby the Asymmetric Slope specification
V aRpt (β) = β1 + β2 V aRpt−1 (β) + β3 (Yt−1 )+ + β4 (Yt−1 )− ,(2.4)

where (x)+ = max(x, 0) and (x)− = − min(x, 0), but not by the Indirect GARCH(1,1)
specification
V aRpt (β) =q2 .
β1 + β2 (V aRpt−1 )2 (β) + β3 Yt−1

(2.5)

On the other hand, the Asymmetric Slope CAViaR imposes a piecewise linear structure on VaR, although the true functional form might be nonlinear. As pointed out inKuester et al. (2006), financial returns may also have an autoregressive (AR) mean, which
is neglected by the CAViaR specifications, which artificially set the mean return equal tozero. For these reasons we combine the positive features of the above, by allowing for
nonlinearity and asymmetric effects of past returns, and additionally incorporate an AR
mean, by introducing an alternative specification, called Indirect Autoregressive Threshold GARCH (AR-TGARCH(1,1)) CAViaR:
1
2
V aRpt (β) = β1 Yt−1 + β2 + β3 (V aRpt−1 )2 (β) + β4 Yt−1
+ β5 (Yt−1 )2 I(Yt−1 < 0) 2 , (2.6)

Including the AR term introduces the possibility for a nonzero autoregressive mean,
asymmetry is present if β5 6= 0 and the square root allows for a nonlinear functionalform.
For estimating the parameters of the CAViaR models, an algorithm similar to the one
proposed in the original paper is applied, see Engle and Manganelli (2004). A grid searchis conducted by generating a large number of random vectors, the dimension of whichcorresponds to the number of model parameters. The five vectors which lead to the lowest values of the objective function (2.3) are selected and fed into a simplex optimization

3

NONPARAMETRIC QUANTILE REGRESSION WITH REFINEMENTS FROM EXTREME VALUE THE

algorithm. The final parameter vector is chosen to be the one minimizing (2.3). Our new
AR-TGARCH specification fits into this procedure.

3

Nonparametric quantile regression with refinements from extreme value theory

3.1

Double Kernel Local Linear VaR regression

In general, estimating nonparametric models requires large amounts of data. Since VaR
corresponds to a quantile at the tail of the return distribution, suitable nonparametricquantile estimators should be able to deal with areas where data are sparse. Therefore,
from the variety of nonparametric quantile estimators, the Double Kernel Local Linear(DKLL) estimator of Yu and Jones (1998) is chosen for the VaR application, because it
localize the data in both x- and y-direction, which leads to smoother estimates. For moredetails, regularity assumptions and asymptotic properties, see the original article by Yu
and Jones (1998). The Weighted Double Kernel Local Linear estimator of Cai and Wang(2008), a Nadaraya Watson type estimator, forms an alternative to the DKLL estimator.
In a small simulation study, which is not reported here, the DKLL estimator performed
slightly better at design points at the boundary of the support of the data. Therefore, wechose it for our application.
For notational convenience, observations {(Xt , Yt )}nt=1 are assumed to be drawn fromunderlying bivariate distribution F (x, y) with density f (x, y). The extension to the multivariate case is straightforward, but requires more tedious notation. The estimator is defined as inverse of a conditional distribution function as in (2.1). Throughout this section,
quantiles of return distributions are discussed, so that VaR corresponds to the negative
quantile.
A generic nonparametric method of estimating a conditional distribution F (y|x) isF̌ (y|x) =nXwt (x)I(Yt ≤ y),

(3.1)

t=1

where I(·) is again an indicator function and the weights wt (x) are positive and sum
up to one. Choosing equal weights w = 1/n yields the empirical distribution function.

3

NONPARAMETRIC QUANTILE REGRESSION WITH REFINEMENTS FROM EXTREME VALUE THE

Using instead a kernel function with bandwidth parameter h, in the following sometimes
abbreviated by Kh (·) = h1 K(·/h), which is often chosen to be a symmetric probability
density function, results in the Nadaraya Watson estimator of a conditional distribution

F̌N W (y|x) =nX

K (x − Xt )Pn hI(Yt ≤ y),
Kh (x − Xt )
t=1
t=1 |
{z
}

(3.2)

wt (x)

see for example Li and Racine (2007). It attaches a smooth set of weights to the data, andis known to be monotone increasing and bounded between zero and one. However, it
suffers from boundary distortion, as shown by Fan and Gijbels (1996). They advocate theuse of local polynomial estimators, the simplest of which is the local linear estimator.
One way to reduce distortions that arise due to a limited number of observations is tosmooth not only the observations of the regressor variable Xt , but also the observationsof the dependent variable Yt . This requires the introduction of a second symmetric kernel
Wh2 (·). Its kernel distribution, which is defined by
Z y
−∞


Wh2 (Yt − u)du = Ω

y − Yt
h2


(3.3)

,

with h2 being the bandwidth parameter, can be viewed as a smooth, differentiable version of the indicator function.
As a next step, the conditional distribution value of y is approximated by a linear Taylor
expansion around x. The estimate F̃ (y|x) = β̂0 is obtained from

2
n  
X
y − YtΩ− β0 − β1 (Xt − x) Kh1 (x − Xt ) ,β0 ,β1h2(β̂0 , βˆ1 ) = arg min

(3.4)

t=1

where h1 > h2 . Solving for β̂0 yields the explicit expression for the conditional distribution function estimator,nX

K (x − Xt ) [S2 − (x − Xt )S1 ]Pn h1ΩF̃ (y|x) =K (x − Xt ) [S2 − (x − Xt )S1 ]
t=1 | t=1 h1{z}
wt (x)

y − Yt
h2


,

(3.5)

3

NONPARAMETRIC QUANTILE REGRESSION WITH REFINEMENTS FROM EXTREME VALUE THE

where
Sl =

n
X
i=1


K

x − Xth1

(x − Xi )l ,l = 1, 2.

(3.5) is a version of (3.1) where the kernel distribution function Ω(·) in (3.3) replaces theindicator. The DKLL quantile estimator q̃p (x), the sample analogue to (2.1), is then defined bynoq̃p (x) = inf y ∈ < : F̃ (y|x) ≥ p ≡ F̃ −1 (p|x).

(3.6)

with F̃ from (3.5). In finite samples, F̃ (y|x) might not always be monotonically increasing. In such cases, however, the inverse is not defined. Yu and Jones (1998) suggest thefollowing implementation scheme: For q̃1/2 (x), any value satisfying (3.6) is chosen; forp > 1/2, the largest, and for p < 1/2, the smallest solutions to (3.6) are taken as quantile
estimates.
In this paper, a stronger procedure is applied, avoiding to delete estimated values. Chernozhukov et al. (2009) show that any nonmonotone estimate of a monotone function canbe improved in terms of common metrics, such as the Lp -norm, by simple rearranging.
In an earlier work, Dette et al. (2006) propose a similar method of smoothed rearrangements. For the case of a monotone increasing (decreasing) function, the point estimates
are sorted in ascending (descending) order. Making use of these theoretical results, nonmonotone distribution estimates are rearranged before inverting. In the present context
of monotonizing the estimated distribution function, a further effect is that quantile crossing is circumvented. Estimated values greater than one are discarded.

3.2

Refining nonparametric quantile regression with extreme value theory

For extreme quantiles, usually very few data points are available, so that fully nonparametric regression does not yield reliable estimates. Extreme value theory (EVT) is analternative to model extreme quantiles. In the following a method of incorporating extreme value theory into CAViaR models, which was introduced by Manganelli and Engle(2001), is adapted to obtain 0.1% VaR estimates from a nonparametric model.

3

NONPARAMETRIC QUANTILE REGRESSION WITH REFINEMENTS FROM EXTREME VALUE THE

The strategy is to first calculate the standardized quantile residuals,
ˆtp2
Yt − q̂pt 2
Yt== t − 1.
q̂pt 2q̂pt 2q̂p2(3.7)

p denotes the (very low) probability of interest, and p2 corresponds to a moderately low
probability for which the quantile can be estimated nonparametrically, for example p2 =0.01 or p2 = 0.05. McNeil and Frey (2000) employ a similar technique to estimate 1% VaR
from a GARCH residual series. An EVT-augmented nonparametric kernel distribution
estimator is also considered by MacDonald et al. (2011), who show consistency of their
method via Bayesian inference.
Reformulating the definition of the pth quantile of portfolio returns in terms of the p2 th
quantile yields

P [Yt < qpt ] = P Yt < qpt 2 − qpt 2 + qpt
#
"
qpt
Yt
= P t − 1 > t − 1 = p.
qp 2
qp 2
The inequality sign is switched assuming that qpt 2 is a negative number. Let
zp ≡

qpt
−1
qpt 2

denote the (1 − p)th quantile of the standardized residuals. It is then estimated by thepeaks over threshold (POT) method, though it can be estimated by other methods, suchas the Hill estimator, as well. A number of applications employ the POT method to forecast extreme quantiles; for a selection of applications and an investigation of its finite
sample properties see El-Arouia and Diebolt (2002). An estimate for the pth return quantile can be expressed by means of ẑp and q̂pt 2 :q̂pt− 1 = ẑpq̂pt 2⇔q̂pt = q̂pt 2 (ẑp + 1).

(3.8)

t

Again, V[
aRp = −q̂pt . In the remainder of this section, the underlying extreme value arguments, which is used to obtain ẑp in (3.8), is discussed very briefly, following Embrechts
et al. (1997).

3

NONPARAMETRIC QUANTILE REGRESSION WITH REFINEMENTS FROM EXTREME VALUE THE

Large observations which exceed a high threshold can be approximated reasonably wellby the Generalized Pareto Distribution (GPD) with distribution functionGξ,β (x) = −(1 + ξx/β)1/ξ for ξ 6= 0
 1 − ex/β

(3.9)

for ξ = 0

with shape parameter ξ and scale parameter β > 0. The support is x ≥ 0 when ξ ≥ 0
and 0 ≤ x ≤ − βξ if ξ < 0. The parameters can be consistently estimated if the thresholdexceedances are independent, regardless of the true underlying distribution, see Smith(1987). In general, given a high threshold u and a random variable Y , the probability ofY exceeding u at most by x is given by

Fu (x) = P [Y − u ≤ x|Y > u] =F (x + u) − F (u).
1 − F (u)(3.10)

Balkema and de Haan (1974) and Pickands (1975) show that for a large class of distribution functions F it is possible to find a positive function β(u) such that

lim

sup

u→y0 0≤x<y0 −u

Fu (x) − Gξ,β(u) (x) = 0,

(3.11)

with y0 corresponding to the right endpoint of F . Rearranging (3.10) and using Fu (·) ≈Gξ,β (·), it holds that
1 − F (u + x) ≈ [1 − F (u)][1 − Gξ,β (x)].
Then, 1 − Gξ,β (x) can be obtained by estimating the GPD parameters by maximum likelihood. Let Nu denote the number of exceedances over threshold u. A common way of
estimating S(u) := 1 − F (u) is to use the empirical distribution function Nnu . Substituting
the estimates,
Nu\S(u+ x) =n − 1
ξ̂
x
ˆ
1+ξ
.
β̂

(3.12)The quantile can be estimated by inverting (3.12), employing a change of variables y =u + x and fixing the distribution value at the probability of interest: F (y) = p. Therefore,

3

NONPARAMETRIC QUANTILE REGRESSION WITH REFINEMENTS FROM EXTREME VALUE THE

the quantile estimator q̂p is obtained from


− 1
ξ̂
y−u
Nu
ˆ
1+ξ
1−p=
n
β̂
"
#
−ξ̂
n
β̂
⇔ q̂p = u +(1 − p)−1 · .
Nuξˆ

(3.13)

In general, extreme value methods require the underlying data to be i.i.d. Although computing standardized residuals in (3.7) should remove most of the dynamics, one cannot
eliminate the possiblitity of remaining autocorrelation. However, under some conditions
on the dependence structure (see e.g. Drees (2003) for details), the relationship between
the limiting distributions of the maxima of a dependent but strictly stationary sequence,(Yt )t∈N say, and a white noise sequence (Yet )t∈N with the same distribution function F
may be described by the so-called extremal index θ ∈ (0, 1]. If the distribution of normalized threshold exceedances in the sequence (Yet ) converges to an extreme value distribution G(x), as in (3.11), then it can be shown that the equally normalized exceedances of(Yt )t∈N converge to Gθ (x), see Embrechts et al. (1997). Thus, the same limiting extreme
value distribution may be used, while changing only the normalization parameters.
Intuitively, if our sequence of standardized residuals possesses an extremal index whichis < 1, then its extremal behavior is asymptotically the same as that of a shorter white
noise sequence with the same distribution. However, it might still be interesting to findout about the extent of deviation from white noise for a given data set. The extremalindex may be estimated by the so-called Runs Method where θ is computed as the conditional probability that a threshold exceedance is followed by a run of r non-exceedances.
The idea is that periods in between clusters of exceedances are longer than periods between independent exceedances (for details see Embrechts et al. (1997), Chapter 8). The
higher the clustering tendency, the fewer runs will be present. The estimator is
Pn−r
I(At )b.
θ = Pn t=1
t=1 I(Yt > u)

(3.14)

where I(·) is again the indicator function and

At = {Yt > u, Yt+1 ≤ u, ..., Yt+r ≤ u}

(3.15)

4

DATA AND BACKTESTING METHOD

12

Drees (2003) shows that a wide variety of financial time series, including ARMA and
ARCH processes, may be estimated by the POT maximum likelihood estimator which
we use here. He finds that the only difference to i.i.d. data is an increased variance of thequantile estimator, but this drawback does not affect our results, as our goal is forecasting
accuracy, which is checked via backtesting.

4

Data and backtesting method

We analyze four data sets of daily index returns, DAX, FTSE 100 (FTSE), EuroSTOXX 50(EuroSTOXX) and S&P 500 (S&P). The longest available time series of each are used tocompute in-sample fits. The common end date of the in-sample period is 28/02/2003.
We predict VaR for the subsequent 1000 days, until the end of 2006 (29/12/2006). Asa second step, we take the same forecast period, but additionally include 300 days tocheck whether model performances worsen when the data contains the beginning of thefinancial crisis. The 1300-day forecast period ends on 22/02/2008. Table 4 summarizes
some features of the data.

start dateend in-sample period
no. of observations
mean
median0.5% quantile
99.5% quantile
skewnesskurtosisDAX
03/10/1966
28/02/2003
9500
0.020
0.000
-4.1073.686-0.422
11.158FTSE
03/01/1984
28/02/2003
4998
0.026
0.020
-3.4703.258-0.794
13.571EuroSTOXX
02/01/1987
28/02/2003
4215
0.021
0.057
-4.9244.157-0.326
8.414

S&P
31/07/1970
28/02/2003
8500
0.028
0.007
-3.0093.239-1.46839.075

Table 4.1: Data summary for the time series used in the long estimation period.
The reason for using a rather long estimation period relative to the forecast horizon, isthat our aim is to assess and compare the quality of the nonparametric model in capturing market risk, using as much information as possible. We are aware that in real life
risk management, available time series are typically much shorter. Therefore, in Section
5.2.2, we additionally estimate our models using only the last 1000 data points, from30/04/1999 to 28/02/2003 and forecast 1% VaRs for the subsequent 200 and 1000 days.

4

DATA AND BACKTESTING METHOD

13

Realizations of quantiles cannot be observed. Therefore, backtesting of the models iscarried out using the dynamic quantile (DQ) out-of-sample test developed in Engle andManganelli (2004) to test and compare the performance of VaR models. From the varietyof alternatives, we choose this particular test, firstly because it is the standard test to compare CAViaR and other models, and secondly, because it is based on a moment condition
rather than an augmented regression, where VaR excess indicators are modelled as function of some information set. Escanciano and Olmo (2010) show that many popular VaR
backtests do not take estimation risk into account, which may lead to incorrect limiting
distributions of the test statistic. However, in the context of Theorem 2 in their paper,
they also point out that the out-of-sample DQ test does not suffer from this drawback.
Define the binary variable
Hitt ≡ I(Yt < −V aRpt ) − p.

If the chosen model is correct,
E[Hitt |Ωt ] = 0,

(4.1)

where Ωt is any information known up to time t. Thus, VaR is estimated correctly, if independently for each day of the forecasting period, the probability of exceeding it equals
p. Note that this also implies that Hitt is uncorrelated with its own lagged values. Let Zt
denote a K-vector of variables potentially related to Hitt , and let Z denote the (N × K)matrix stacking the values of Zt , where N is the number of observations in the forecast
period. Then, the moment condition in (4.1) can be checked using the test statistic

DQ =Hit0 Z(Z0 Z)−1 Z0 Hit,p(1 − p)(4.2)

where Hit is a vector containing all values of Hitt . Under the null hypothesis that (4.1)
holds, DQ is asymptotically χ2 distributed with K degrees of freedom. In analogy to Engle and Manganelli (2004), Kuester et al. (2006) and Taylor (2008), we include a constant,four lagged values of Hitt and the current VaR estimate in the information set.
It can be seen immediately that for our information set, the DQ test statistic requires atleast one out-of-sample VaR exceedance in order to be defined. Otherwise, the lagged

5

APPLICATION TO STOCK INDEX RETURNS

14

values of Hitt would cause multicollinearity in the matrix Z. However, when considering extreme VaR, as we do in Section 5.3, it might well be that there are no exceedances
within the forecasting period (note that the correct number of exceedances for 1000 forecasts of 0.1% VaR is 1). Therefore, in order to be able to compare the different models
while taking the possibility of no exceedances into account, we employ the test proposedby Kupiec (1995) which is an unconditional test of the correctness of the achieved shareof VaR exceedances. Define the indicator variableIt ≡ I(Yt < −V aRpt ).

The idea of the Kupiec test is to check whether E[It ] = p, in which case the number ofexceedances

exN =

n+N
X

It

t=n+1

has a binomial distribution with parameters N and p. Under the null hypothesis of correct coverage, the corresponding likelihood ratio statistic

LRKup = 2 log






exN N −exN  exN exN
− 2 log (1 − p)N −exN pexN .
1−
NNis asymptotically χ2 distributed with one degree of freedom.

5

Application to stock index returns

5.1

Monotonicity of quantile curves

When forecasting from a nonparametric model, one has to balance two effects occurringat the boundary areas: The support from which predictions of the dependent variable
can be computed is limited to the range in which the estimated function is located. Thismeans that for outlying lagged returns, which are not in the support of the estimatedcurve, no forecasts for VaR exist. On the other hand, often only few data points are available at boundary areas, so that outliers have more influence and the resulting curve may
show distortions. Therefore, one has to decide carefully about the range of the grid at

5

APPLICATION TO STOCK INDEX RETURNS

15

which the function is evaluated, balancing possible distortions against a limited range ofregressor values to compute forecasts from.
We estimate the time-varying conditional 1% VaRs of DAX, EuroSTOXX, S&P and FTSE
using the DKLL estimator. Due to its double smoothing property, distortions are eased
and quantile curves are smoother. Additionally, we make use of the monotonization
method proposed by Chernozhukov et al. (2009). Whenever curves are not monotonically decreasing on the left of the minimum and monotonically increasing on the right,
estimated values are rearranged in descending and ascending order, respectively. To illustrate possible changes in the in-sample fit, Figure 5.1 shows the original as well as therearranged 1% VaR curves of DAX and EuroSTOXX. Both curves cover 99% of the data.

7
6
VaR
5
4
3

3

4

5

VaR

6

7

8

Rearranged DKLL estimate of 1% DAX VaR8

DKLL estimate of 1% DAX VaR

−2

0

2

4

−4

0

2

4

DKLL estimate of 1% Eurostoxx VaR

Rearranged DKLL estimate of 1% Eurostoxx VaR

5.5
5.0
4.5
3.5
3.0

3.0

3.5

4.0

4.5

VaR

5.0

5.5

6.0

lagged return

4.0

VaR

−2

lagged return

6.0

−4

−4

−2

0
lagged return

2

4

−4

−2

0

2

4

lagged return

Figure 5.1: Original and rearranged DKLL estimates of 1% conditional DAX VaR curve(upper panel) EuroSTOXX VaR curve (lower panel).

5

APPLICATION TO STOCK INDEX RETURNS

16

Table 5.1 compares backtesting results on original and rearranged DKLL fits, on a forecasthorizon of 1000 days. In-sample and out-of-sample coverages are aimed to be as close
as possible to the underlying probability, in this case 1%. The P -value of the LR test
described in Section 4 expresses the highest significance level at which the variables in theinformation set are jointly significant. Therefore, a larger P -value indicates that the null
hypothesis of independent VaR exceedances is more likely not to be rejected, suggestingthat a model is more adequate

in-sample (%)
out-of-sample (%)
DQ P -value

DAX
DKLL orig.
0.78
1.00
0.182

in-sample (%)
out-of-sample (%)
DQ P -value

EuroSTOXXDKLL orig.
0.81
0.50
0.859

DKLL rearr.
0.78
1.00
0.182

FTSE
DKLL orig.
0.94
0.40
0.614

DKLL rearr.
0.95
0.50
0.830

DKLL rearr.
0.81
0.50
0.859

S&P500DKLL orig.
0.97
0.30
0.544

DKLL rearr.
0.97
0.30
0.544

Table 5.1: DQ test results for original and rearranged DKLL models as well as in-sample
and out-of sample share of VaR exceedances (in percent). The forecast period is 1000
observations.
The theoretical results from Chernozhukov et al. (2009), that rearranging weakly improves estimation, are confirmed by our empirical results: Whenever values in the columns
are different, they are superior for the rearranged estimates. In-sample and out-of-sample
coverages are closer to 1% in case of the FTSE return series. Furthermore, the LR test P value substantially increases, indicating that the null hypothesis of the LR test is ’further
away’ from rejection than in the case of the original DKLL model. Whenever we mention
results for the DKLL estimator in the following, it refers to the rearranged version.

5.2

Comparing 1% VaR predictions

5.2.1

Long estimation period

Table 5.2 lists backtest results of the CAViaR models and the rearranged DKLL estimates
which are obtained from the large data set data set described in Table 4. Generally, the5

APPLICATION TO STOCK INDEX RETURNS

17

in-sample exceedance shares achieved by all three CAViaR specifications are very close
to the underlying probability 1%. In contrast, the DKLL estimator has a slight tendency
to overestimate VaR, leading to in-sample coverages below 1%. The news impact curves
shown in Figure 5.2 reveal that the AR-TGARCH CAViaR specification resembles thenonparametric VaR estimate better than the other two models.

3.5
2.0

2.5

3.0

VaR

4.0

4.5

5.0

S&P 500: AR−TGARCH CAViaR vs. DKLL

−4

−2

0

2

4

lagged returnS&P 500: GARCH CAViaR

VaR

2.6
2.2

2.2

2.4

2.4

2.6

VaR

2.8

2.8

3.0

3.0

3.2

S&P 500: Asymm. Slope CAViaR

−4

−2

0
lagged return

2

4

−4

−2

0

2

4

lagged return

Figure 5.2: News impact curves, i.e. reactions of S&P VaR to different magnitudes oflagged returns (’news’), of TGARCH(1,1) CAViaR (lower curve in the upper panel) together with DKLL estimate (upper curve in the upper panel), of Asymmetric Slope and
GARCH(1,1) CAViaR.
In terms of out-of-sample forecasting, on the other hand, results differ among the four indices. In predicting DAX VaR, the CAViaR models perform quiete poorly on both forecasthorizons. Out-of-sample coverages are too high, and the DQ test P -values raise doubtthat the models are able to generate conditionally independent VaR exceedances and5

APPLICATION TO STOCK INDEX RETURNS

18

correct coverage. In contrast, the DKLL estimate achieves more accurate out-of-sample
exceedance rates in predicting VaR over both 1000 and 1300 days, and the test results
suggest that the model is appropriate at least for the shorter forecast horizon.

in-sample
out-of-sample (1000)
out-of-sample (1300)
DQ P -value (1000)
DQ P -value (1300)
in-sample
out-of-sample (1000)
out-of-sample (1300)
DQ P -value (1000)
DQ P -value (1300)
in-sample
out-of-sample (1000)
out-of-sample (1300)
DQ P -value (1000)
DQ P -value (1300)
in-sample
out-of-sample (1000)
out-of-sample (1300)
DQ P -value (1000)
DQ P -value (1300)

Asymm. Slope GARCH
DAX
1.01
1.01
1.50
1.50
1.54
1.54
0.054∗
0.011∗∗
0.043∗∗
0.009∗∗∗
FTSE
1.02
1.00
0.60
0.60
1.08
1.16
∗∗∗
0.005
0.007∗∗∗
∗∗
0.040
0.104
EuroSTOXX
1.04
0.99
0.70
0.80
0.76
0.92
0.970
0.058∗
0.980
0.015∗∗
S&P
1.01
1.00
0.30
0.30
0.92
1.00
0.547
0.285
0.042∗∗
0.000∗∗∗

AR-TGARCH

DKLL

1.04
1.50
1.61
0.054∗
0.039∗∗

0.78
1.00
1.07
0.182
0.040∗∗

1.00
0.60
1.08
0.007∗∗∗
0.059∗

0.94
0.50
1.00
0.830
0.011∗∗

0.99
0.80
0.92
0.057∗
0.015∗∗

0.81
0.50
0.62
0.859
0.031∗∗

0.97
0.30
0.69
0.547
0.079∗

0.97
0.30
1.15
0.544
0.008∗∗∗

Table 5.2: Backtesting results for 1% VaR models. In-sample and out-of-sample exceedance probabilities in %. Considered forecast horizons are 1000 and 1300 observations. Models which are rejected by the DQ test are marked with ∗ for rejection on 10%,∗∗ on 5% and ∗∗∗ on 1% significance level.
In the case of FTSE VaR, results are mixed: The DKLL estimator overestimates VaR in the
short prediction period (coverage is 0.5%), where, however, it yields a good fit according
to the backtest. For 1300 forecasts, it achieves the correct coverage, but the DQ P -value
drops sharply. The CAViaR models, on the other hand, are strongly rejected in the shorthorizon, but show a better performance in the longer one.
The picture for EuroSTOXX is somewhat similar to that of FTSE, except that the resultsfrom the CAViaR models now differ more strongly among each other, and the Asymmetric Slope CAViaR beats all the other models considered. The P -value obtained by the

5

APPLICATION TO STOCK INDEX RETURNS19

DKLL estimator again drops when moving from the short to the longer forecast horizon,
but it is still above the P -values of GARCH and AR-TGARCH CAViaR, which, on theother hand, perform better in terms of out-of-sample coverages.
The results for S&P VaR show a different structure. Although all coverages within the
short prediction period are low, the DQ test indicates adequate out-of-sample fits. For
the extended horizon, all P -values drop, such that the GARCH CAViaR and the DKLL
models are even rejected at a 1% significance level. One possible reason is that the additionally included observations exhibit some dynamics which are not well captured bythese models, leading to a clustering of VaR exceedances. Interestingly, the AR-TGARCH
is least affected by this effect.
The AR-TGARCH CAViaR model does not outperform the other two CAViaR models andthe DKLL model systematically, but its results are less varying: For both in-sample andout-of-sample forecast horizons, its coverage and backtest results are often better than
the results of one of the two others. We attribute this finding to the fact that the model
combines the features of Asymmetric Slope and Indirect GARCH specification, and it istherefore more universally applicable.
Summing up, the out-of-sample VaR prediction results produced by the fully nonparametric DKLL estimator are satisfactory except for the extended forecast horizon in thecase of the S&P. The CAViaR models are strong competitors, however, they have thedrawback that it is not possible to detect one parameterization that systematically dominates others. As it is often the case with parametric models, the question remains which
one to choose in practical applications. The DKLL model, on the other hand, outperforms at least one of the CAViaR models in most cases, and is therefore the most robust
alternative.

5.2.2

Short estimation period

In real life risk management, time series available for the estimation of VaR models are
rarely as long as the ones we investigate in the previous section. For this reason, we
repeat the estimation using only 1000 observations, i.e. roughly the last four years up to
28/02/2003, and forecast VaRs for both the subsequent 200 and 1000 days. Table 5.3 lists
the backtesting results.

5

APPLICATION TO STOCK INDEX RETURNS

in-sample
out-of-sample (200)
out-of-sample (1000)
DQ P -value (200)
DQ P -value (1000)
in-sample
out-of-sample (200)
out-of-sample (1000)
DQ P -value (200)
DQ P -value (1000)
in-sample
out-of-sample (200)
out-of-sample (1000)
DQ P -value (200)
DQ P -value (1000)
in-sample
out-of-sample (200)
out-of-sample (1000)
DQ P -value (200)
DQ P -value (1000)

Asymm. Slope GARCH
DAX
1.10
1.10
2.50
1.50
0.70
0.80
0.409
0.839
0.603
0.061∗
FTSE
1.00
1.00
1.00
1.00
0.70
0.60
0.810
1.000
0.020∗∗
0.008∗∗∗EuroSTOXX
1.10
1.00
1.50
1.00
0.40
0.40
0.985
1.000
0.674
0.652
S&P
1.00
1.00
1.00
0.50
0.20
0.10
0.536
0.742
0.206
0.110

20
AR-TGARCH

DKLL

0.90
2.50
0.80
0.410
0.671

0.80
0.50
0.10
0.967
0.195

1.10
1.00
0.60
0.975
0.007∗∗∗

0.70
0.50
0.10
0.997
0.225

1.10
1.50
0.50
0.992
0.713

1.10
1.00
0.20
1.000
0.356

1.10
1.00
0.20
0.390
0.210

0.90
0.50
0.10
0.975
0.212

Table 5.3: Backtesting results for 1% VaR models which are estimated using only 1000
observations. In-sample and out-of-sample exceedance probabilities in %. Considered
forecast horizons are 200 and 1000 observations. Models which are rejected by the DQ
test are marked with ∗ for rejection on 10%, ∗∗ on 5% and ∗∗∗ on 1% significance level.
The good performance of the DKLL estimator carries over to the short estimation period.
Although VaR estimates are again too conservative in particular for the longer forecasting
period, the null hypothesis of valid moment conditions tested by the DQ test is neverrejected even on a 10% significance level. The CAViaR models also overestimate VaR for
the subsequent 1000 observations, and additionally, they are rejected by the DQ test in
the case of FTSE. Based on these results, it can be said that the DKLL estimator is also
applicable when the estimation period is rather short, and keeps yielding reliable VaR
forecasts even for the distant future.

5

APPLICATION TO STOCK INDEX RETURNS

5.3

Comparing extreme VaR predictions

21

Following the procedure described in Section 3.2, standardized residuals are computedfrom the rearranged DKLL estimate and the time-varying 0.1% quantile of time series
Yt is calculated according to (3.8). The underlying ’moderate’ probability is chosen to
be 1%. Similarly, VaR estimates obtained from the EVT-augmented CAViaR models arecomputed, following Manganelli and Engle (2001). As mentioned at the end of Section

0.00

ACF
0.05

0.10

0.15

ACF of stand. residuals: FTSE

−0.05

−0.05

0.00

ACF
0.05

0.10

0.15

ACF of stand. residuals: DAX

0

10

20
Lag

30

40

0

10

15 20
Lag

25

30

35

0.00

ACF
0.05

0.10

0.15

ACF of stand. residuals: S&P

−0.05

−0.05

0.00

ACF
0.05

0.10

0.15

ACF of stand. residuals: EuroSTOXX

5

0

5

10

15 20
Lag

25

30

35

0

10

20
Lag

30

40

Figure 5.3: Autocorrelation functions of the standardized nonparametric residuals for thefour indices. The dashed lines are 95% confidence intervals.
3.2, the data should be checked for dependence before applying extreme value methods.
Figure 5.3 shows autocorrelation functions (ACFs) for the standardized residuals fromthe nonparametric model, together with 95% confidence intervals. Although the magnitude of the autocorrelation is not very high, for some lags, the confidence intervals are
exceeded. Therefore, we carry out Ljung-Box tests on independence based on 20 lags,
the results of which imply significant autocorrelation on a 1% confidence level for all

5

APPLICATION TO STOCK INDEX RETURNS

22four considered models and all model specifications. Fitting simple AR models to thestandardized residuals, however, removes the autocorrelation entirely, see Table 5.3. The
corresponding results for the three CAViaR specifications are very similar, and thus, notshown here. They are available upon request. Since McNeil et al. (2005) state that ARMA
processes with innovations drawn from fat-tailed distributions exhibit values of the extremal index θ < 1, we also estimate the extremal indices using the Runs Method (3.14)
described in Section 3.2 and report them in the last column of Table 5.3. The parameter r,
corresponding to the length of runs, was set to 30 for all three indices, after finding thatthe estimated θb was very robust with respect to plausible choices of r. As McNeil et al.
(2005) point out, the distribution of the maximum of n dependent data points with extremal index θ can be approximated by the associated i.i.d. series with nθ observations.
Given the large number of data points in all our samples, we conclude that the loss inaccuracy due to dependence in the standardized residuals is not too severe, so that we
can apply the proposed method to estimate the 0.1%-VaRs.

DAX
FTSE
EuroSTOXX
S&P

LB P -value
0.000
0.000
0.001
0.000

LB P -value for AR residuals
0.529
0.203
0.357
0.224

AR lag order
7
9
9
4

θb
0.80
0.91
0.85
0.88

Table 5.4: The first column reports the outcomes (P -values) of the Ljung-Box (LB) test on
independence of the standardized residuals from the DKLL model. The null hypothesisof independence is always rejected on a 1% confidence level. The second column contains the LB test results after fitting an autoregressive (AR) model to the standardized
residuals. In all cases, the null hypothesis cannot be rejected. The selected lag order isreported in the third column. The forth column contains estimates of the extremal index
θ.
Table 5.5 contains both in-sample and out-of-sample shares of 0.1% VaR exceedances forthe four considered models. Only the long estimation period is considered. However,the simulation study in Section 6 contains a discussion of results from the nonparametricmodel for extreme quantiles, based on a shorter space of time. Due to the occurrence ofno VaR exceedances within the prediction period, we use the Kupiec test instead of the
DQ test for backtesting. It checks the correctness of the achieved unconditional coverage
via a likelihood ratio approach, which is based on the Bernoulli likelihood, see Section 4.
In contrast to the outcomes of the 1% VaR analysis, in-sample VaR exceedance shares

5

APPLICATION TO STOCK INDEX RETURNS

in-sample
out-of-sample(1000)
out-of-sample(1300)
Kupiec P -value (1000)
Kupiec P -value (1300)
in-sample
out-of-sample(1000)
out-of-sample(1300)
Kupiec P -value (1000)
Kupiec P -value (1300)
in-sample
out-of-sample(1000)
out-of-sample(1300)
Kupiec P -value (1000)
Kupiec P -value (1300)
in-sample
out-of-sample(1000)
out-of-sample(1300)
Kupiec P -value (1000)
Kupiec P -value (1300)

Asymm. Slope GARCH
DAX
0.11
0.17
0.00
0.00
0.15
0.15
0.157
0.157
0.570
0.570
FTSE
0.14
0.16
0.10
0.20
0.15
0.38
1.000
0.379
0.570
0.014∗∗EuroSTOXX
0.19
0.26
0.10
0.20
0.23
0.31
1.000
0.379
0.203
0.058∗
S&P
0.20
0.22
0.10
0.00
0.15
0.08
1.000
0.157
0.570
0.784

23
AR-TGARCH

DKLL

0.09
0.00
0.15
0.157
0.570

0.13
0.10
0.23
1.000
0.203

0.14
0.10
0.15
1.000
0.570

0.16
0.10
0.15
1.000
0.570

0.31
0.20
0.31
0.379
0.058∗

0.24
0.00
0.15
0.157
0.570

0.19
0.00
0.08
0.157
0.784

0.12
0.00
0.00
0.157
0.107

Table 5.5: Backtesting results for 0.1% VaR models. In-sample and out-of-sample exceedance probabilities in %. Considered forecast horizons are 1000 and 1300 observations. Models which are rejected by the DQ test are marked with ∗ for rejection on 10%,∗∗ on 5% and ∗∗∗ on 1% significance level.
achieved by the DKLL estimator are now less conservative, but instead always slightly
higher than the target probability 0.1%. On the other hand, out-of sample coverage andbacktest results are remarkably good especially for DAX and EuroSTOXX, where it shows
best results on one of the two considered forecast horizon, but also for DAX, where its
coverages are very close to 0.1%. Concerning S&P, the Asymmetric Slope CAViaR modelyields the most accurate fit, except for the in-sample exceedance rate, which is closer to
0.1% in the case of the DKLL model. According to the Kupiec test, the differences to thenominal coverage are rarely significant, the only exception being the GARCH and ARTGARCH CAViaRs when predicting 1300 days of EuroSTOXX VaR, and GARCH CAViaR
for the extended forecast period of FTSE VaR.
It was pointed out in Kuester et al. (2006), that for comparison of VaR prediction strate-

5

APPLICATION TO STOCK INDEX RETURNS

24

gies, the focus should not be limited to one or two probability levels, but one should takea range of quantiles into account when deciding which model is the best.
FTSE

0.4

DKLL & EVT
ASYM. CAV. & EVT
GARCH CAV. & EVT

DKLL & EVT
ASYM. CAV. & EVT
GARCH CAV. & EVT

−0.8

−0.6

−0.6

−0.4

−0.4

−0.2

−0.2

0.0

0.0

0.2

0.2

0.4

DAX

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.8

1.0

0.8

1.0

S&P

0.4

DKLL & EVT
ASYM. CAV. & EVT
GARCH CAV. & EVT

DKLL & EVT
ASYM. CAV. & EVT
GARCH CAV. & EVT

−0.4

−0.2

−0.2

0.0

0.0

0.2

0.2

0.4

EuroSTOXX

0.6

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

Figure 5.4: Coverage results in % for 0.1%-1% ranges of estimated EVT-augmented index VaRs. The evaluated forecast horizon is 1300 days. Nominal coverages are on thehorizontal axis, and the lines correspond to the differences of nominal and estimatedVaR exceedance shares (in %). The closer they are to zero, the better the (unconditional)
model fit.
We adapt their graphical representation of coverage accuracy in Figure 5.4 for VaR levels
between 0.1% and 1%. For the sake of clarity, the AR-TGARCH CAViaR, which usually
showed results that were similar to one of the other two CAViaR models, is not includedin the graph. It turns out that the Asymmetric Slope CAViaR is the stronger competitor
for the DKLL model, as in three out of four cases, the lines corresponding to its differences to the nominal coverages are closer to zero than those corresponding to the GARCH
CAViaR. In some ranges, e.g. 0.1%-0.5% FTSE VaR, the DKLL model clearly yields a very

6

SIMULATION: COMPARING DKLL AND EVT-REFINED VAR

25

good fit. In other cases, such as DAX, all three models do not hit the correct coverages.
The DKLL estimator, however, goes head to head with the CAViaR models, while sometimes even beating them.

6

Simulation: Comparing DKLL and EVT-refined VaRThis section is devoted to the question of whether it is sensible to refine the nonparametric VaR estimator with extreme value methods, instead of using the plain version evenfor extreme VaR estimation. One would expect that especially for small data sets, theEVT extrapolation into the far tails of return distributions yields more stable results thanestimating the tail quantiles directly. In order to check this, we carry out a small simulation study to complement the empirical results from the previous sections. As the goal
is to assess the relative accuracies of DKLL and EVT-refined DKLL estimators, we do not
additionally include the CAViaR models.
To the FTSE time series, we fit a GARCH(1,1) model with t-distributed error terms. It has
the following form:
2
Yt = µ + σt t , σt2 = ω + α2t−1 + βσt−1,  t ∼ tν .

(6.1)

Parameter estimates are listed in Table 6.1.
µ
b
0.054

ω
b
0.015

α
b
0.083

βb
0.904

νb
10

Table 6.1: Estimated GARCH parameters from the FTSE return series with 4998 observations. This estimated model is used in the simulation.
From the estimated model, a time series of 13000 observations is simulated. To obtain asetup which is realistic with respect to usual data availability, only 2000 observations areused to estimate the models, and 0.1% VaR predictions are computed over two forecasthorizons, N =5000 and N =10000. The advantages of simulated data are that they allow
for much longer horizons, and that the return quantile functions

qpt = σt Fp−1 (t ),7

CONCLUSION

26

where Fp−1 (t ) denotes the p-quantile of the error term distribution, can be computed because the input parameters are known. This allows us to compare the estimated VaRs totheir true counterparts. Table 6.2 shows coverages, mean squared errors, mean absolute
errors and median absolute errors in-sample and out-of-sample for both models.

DKLL
EVT-DKLLDKLL
EVT-DKLL

in-sample: n=2000\\
cov. M
SE M
AE M\
ed.AE
0.001 1.049 0.753
0.52
0.001 0.605 0.579
0.429

out-of-sample: N =5000\\
cov. M
SE M
AE M\
ed.AE
0.009 3.673 1.359
0.9
0.003 2.455 1.067
0.649

out-of-sample: N =10000\\
cov. M
SE M
AE M\
ed.AE
0.008 3.379 1.253
0.787
0.004 2.281 0.981
0.578

Table 6.2: Coverages and different loss functions from comparing the estimated 0.1%
VaRs with the true quantile function. Cov. stands for coverage, MSE for mean squared
error, MAE for mean absolute error and Med. AE for median absolute error.
Throughout, the EVT-augmented DKLL model yields lower losses and better coverages
than the plain DKLL model. In order to robustify this result, we repeated the simulation
for GARCH parameters estimated from EuroSTOXX data, and using different numbersof in-sample observations. All these results, which are available on request, lead to theconclusion that that the combination of standardized nonparametric residuals and extreme value theory is a valuable complement to the rearranged DKLL estimator, which
we suggest to use for estimating moderately low quantiles.

7

Conclusion

In this paper, we propose a way to estimate and predict conditional Value at Risk from a
nonparametric model. We consider probabilities that are of practical interest for financial
institutions. For external market risk reporting, 1% portfolio VaRs have to be estimatedon a daily basis. Internal risk management sometimes requires to take into account evenmore extreme probabilities such as 0.1%. Although typically very few observations areavailable in the extreme tails, models to be used should be flexible and rest upon as few
structural assumptions as possible. We suggest to use nonparametric quantile regression, more specifically, a rearranged Double Kernel Local Linear VaR estimator as well

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CONCLUSION27

as a version of the latter augmented by extreme value theory. Both are applied to different index return time series. Forecast performances are benchmarked against the widelyused CAViaR models. Although these also perform well in many occations, none of theconsidered specifications systematically dominates the others. In constrast to them, nonparametric regression circumvents the issue of choosing the appropriate parametrization.
Backtesting results from the evaluation of real as well as simulated data examples lead tothe conclusion that the fully nonparametric and the EVT-refined nonparametric models
do not only outperform the parametric alternatives in a considerable number of situations, but that they can be used to predict VaR of any probability level of interest, evenwhen the estimation period is of moderate size. In recent years, computing power has
increased to such an extent that fully nonparametric models come at little more computation cost than other models that rely on more restrictive assumptions. From the results inthis paper, however, we conclude that the gains on the additional flexibility are substantial and nonparametric quantile regression with EVT refinements should be considered
as a practical alternative for estimating and forecasting VaR.

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CONCLUSION

28

Acknowledgements
I would like to thank Melanie Schienle and Nikolaus Hautsch for their instructive andhelpful comments on this project. The support from the Deutsche Forschungsgemeinschaft via SFB 649 ’Ökonomisches Risiko’, Humboldt-Universität zu Berlin, is gratefully
acknowledged.

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