Conditional Quantile Estimation for GARCH Models (Preliminary) Zhijie Xiaoy and Roger Koenkerz Boston College and University of Illinois March 6, 2008
Abstract Conditional quantile is an essential ingredient in various risk measures, and the GARCH process has proven to be highly successful in modelling …nancial data. In this paper, we study estimation of conditional quantiles for GARCH Models using quantile regressions. Quantile regression estimation of GARCH models is highly nonlinear. We propose a simple and e¤ective two-step approach of quantile regression estimation for linear GARCH time series. In the …rst step, we propose a time series sieve quantile regression approximation for the GARCH model by combining information over di¤erent quantiles; a second stage estimation for the GARCH model is then constructed based on the preliminary estimators. Asymptotic property of time series sieve quantile regression, combining quantile estimates, and quantile regression estimation with generated regressors are studied. These results are of independent interest and can be applied to other quantile regression applications. Monte Carlo results indicate that the proposed estimation method outperforms over existing quantile estimation methods.
We thank seminar participants at JSM2007, MIT, and Cass Conference in Econometrics for helpful comments and discussions. we also thank Chi Wan for excellent research assistance. y Department of Economics, Boston College, Chestnut Hill, MA 02467. Tel: 617-552-1709. Email:
[email protected]. z Department of Economics, University of Illinois, Champaign, Il, 61820. Email:
[email protected].
1
1
Introduction
Conditional quantiles play an essential role in risk measurement. Evaluation of Value-atRisk, as mandated in many current regulatory contexts, is explicitly a conditional quantile estimation problem. Closely related quantile-based concepts such as expected shortfall, conditional value at risk, and limited expected loss, are also intimately linked to quantile estimation, see, e.g., Artzner, Delbaen, Eber and Heath (1999), Wang (2000), Wu and Xiao (2002), Bassett, Koenker and G. Kordas (2004). The literature on estimating conditional quantiles is large. Many existing methods of quantile estimation in economics and …nance are based on the assumption that …nancial returns have normal (or conditional normal) distributions. Under the assumption of a conditionally normal returns distribution, the estimation of conditional quantiles is equivalent to estimating conditional volatility of returns. The massive literature on volatility modeling o¤ers a rich source of parametric methods of this type. However, there is accumulating evidence that …nancial time series, and returns distributions are not well approximated by Gaussian models. In particular, it is frequently found that market returns display negative skewness and excess kurtosis. Extreme realizations of returns can adversely effect the performance of estimation and inference designed for Gaussian conditions; this is particularly true of ARCH and GARCH models whose estimation of variances are very sensitive to large innovations. For this reason, research attention has recently shifted toward the development of more robust estimators of conditional quantiles. There is growing interest in non-parametric estimation of conditional quantiles; although local, nearest neighbor and kernel methods are somewhat limited in their ability to cope with more than one or two covariates. Other approaches to estimating VaR include the hybrid method of Boudoukh, Richardson and Whitelaw (1998), and methods based on extreme value theory see, e.g. Boos (1984), McNeil (1998), and Neftci (2000). Quantile regression as introduced by Koenker and Bassett (1978) is well suited to estimating conditional quantiles. Just as classical linear regression methods based on minimizing sums of squared residuals enable one to estimate models for conditional mean, quantile regression methods o¤er a mechanism for estimating models for the conditional quantiles.
1
These methods exhibit robustness to extreme shocks, and facilitate distribution-free inference. In recent years, quantile regression estimation for time-series models has gradually attracted more attention. Koenker and Zhao (1996) extended quantile regression to linear ARCH models where
t
=
0+ 1
jut 1 j+
+
q
jut q j ; and estimate conditional quantiles
of ut by a linear quantile regression of ut on (1; jut 1 j ;
; ut q ). However, evidence from
…nancial applications indicates that, comparing to the GARCH models, ARCH type of models can not parsimoniously capture the persistent in‡uence of long past shocks. Engle and Manganelli (2002) have proposed a large class of CaViaR models whose conditional quantiles themselves follow an autoregression. The statistical properties of such models are still somewhat unclear and are the subject of current research. In this paper, we study estimation of conditional quantiles for a class of GARCH models. The GARCH model originally proposed by Bollerslev (1986) has proven to be highly successful in modelling …nancial data, and is arguably the most widely used class of models in …nancial applications. Since Bollerslev (1986), a variety of GARCH models have been proposed by various researchers, including the EGARCH model of Nelson (1991) and linear GARCH model of Taylor (1986). In the original quadratic form of the GARCH model we say that : ut follows a GARCH(p; q) process if t "t ,
ut = where 2 t
=
0
+
2 1 t 1
+
+
2 p t p
+
2 1 ut 1
+
+
2 q ut q ,
and "t is an iid sequence of Gaussian random variables. As noted by Du¢ e and Pan (1997), maximum likelihood estimation of this form of the GARCH model has the potential disadvantage that it is overly sensitivity to extreme returns. For example, if we consider a market crash, extreme daily absolute returns may be 10 to 20 times “normal”daily ‡uctuation, so the quadratic form of GARCH model yields a return e¤ect which is 100 to 400 times the normal variance. This not only causes “overshooting”in volatility forecasting, but also carries this in‡uence far into the future. As an alternative, Taylor (1986)
2
suggested the following linear GARCH model: ut follows a GARCH(p; q) process if t
=
0
+
1 t 1
+
+
p t p
+
1
jut 1 j +
+
q
jut q j :
The quadratic GARCH model seems computationally more convenient than the linear GARCH model, but the linear GARCH may be more appropriate in modelling …nancial returns. The linear GARCH structure is less sensitive to extreme returns, but it is more di¢ cult to handle mathematically. However, the linear structure is well suited for quantile estimation. Quantile regression GARCH models are highly nonlinear and thus complicated to estimate. As will become apparent in our later discussion, the quantile estimation problem in GARCH models corresponds to a restricted nonlinear quantile regression and conventional nonlinear quantile regression techniques are not directly applicable, adding an additional challenge to the already complicated estimation problem. To circumvent these di¢ culties, we propose a robust and easy-to-implement two-step approach to estimating GARCH conditional quantiles based on quantile regression. In the …rst step, a sieve quantile regression approximation is estimated for multiple quantiles, and combined via minimum distance methods to obtain preliminary estimators for the parameters of the global GARCH model. The second step then focuses on the local behavior at the speci…c quantile and estimate the conditional quantile based on the …rst stage results. The proposed method is relatively easy to implement compared to other nonlinear estimation techniques in quantile regression and has good sampling performance in our simulation experiments. The methods that we employ to study the asymptotic behavior of our twostage procedure: combining information over quantiles via minimum distance estimation, and quantile regression with generated regressors are also of independent interest and applicable in other econometric and statistical applications. The remainder of the paper is organized as follows: We discuss the estimation of conditional quantiles in GARCH models and propose the two-stage estimation procedure in the next section; Section 3 studies the asymptotic behavior of the proposed estimators in each stage, including the sieve quantile estimation, the minimum distance estimation that combines information over various quantiles, and the proposed two-step estimator. 3
The results of a small Monte Carlo experiment are reported in Section 4.
2
Quantile Regression for Linear GARCH Models
We will consider the linear GARCH model: ut =
t
=
0
t
with
0
> 0; ( 1 ;
;
(1)
"t ; +
> q)
1 t 1
+
+
p t p
+
1
jut 1 j +
+
q
jut q j ;
(2)
2 X 1 xt xt : Dn = n t=m+1 t
2 Rm+1 , =
(8)
2
f" (F" ( ))
>
Dn 1
n
>
n(
)Dn 1 , and
(e ( )
( ))
10
n(
) N (0; 1) )=
1 n
Pn
t=m+1
xt x> t
2
(ut ).
(9)
3.2
Minimum Distance Estimation of Conditional Scale
Having estimated the truncated quantile autoregressions on a grid of ’s, we would now like to combine these estimates to obtain estimates of the conditional scale parameters, t.
This is accomplished most easily using minimum distance methods. Suppose that we estimate the m-th order quantile autoregression ! n m X X min ut 0 j jut j j t=m+1
at quantiles ( 1 ;
;
K ),
j=1
and obtain estimates e ( k ), k = 1;
; K:
Let e a0 = 1 in accordance with the identi…cation assumption. Denote a = [a1 ;
where g = [q1 ;
; qK ]> , ~ = e ( 1 )> ;
; am ; q1 ;
and (a) = g
(10)
=
; qK ]> and
h
q1 ; a1 q1 ; ; am q1 ; = [1; a1 ; a2 ;
;
; e(
> > K)
qK ; a1 qK ; ; am qK
,
i>
,
; am ]> . We consider the following estimator
for the vector a that combines information over the K quantile estimates based on the e t ( ) for j = 0; 1; 2; : : : : restrictions e ( ) = aej ( )Q e a = arg min (~ a
where An is a (K(m + 1))
(a))> An (~
(11)
(a)) ;
(K(m + 1)) positive de…nite matrix.
Thus, the two-step estimation for conditional quantiles of the linear GARCH model can be described in more detail as follows: Step 1 Estimate the following m-th order quantile autoregression (10) at quantiles ( 1 ; ;
K ),
and obtain e ( k ) = (e 0 ( k );
; e m ( k ))> , k = 1; 11
; K:
By setting e a0 = 1 and solving the minimum distance estimation problem (11), we
obtain an estimator for (a0 ;
;e am ). Thus
; am ), denoting it as (e a0 ;
estimated by
et = e a0 +
m X j=1
e aj jut j j :
Step 2 Run quantile regression of ut on zet = (1; et 1 ; min
X
>
(ut
t
the two-step estimator of ( )> = (
0(
);
1(
);
;
p(
);
can be
; jut q j)> by
; et p ; jut 1 j ;
zet );
t
(12)
1(
);
;
q(
)) are
then given by solution of (12), b( ), and the -th conditional quantile of ut can be estimated by
but ( jFt 1 ) = b( )> zet : Q
The asymptotic behavior of the …rst stage minimum distance estimator is described in the following Theorem. Theorem 2 Under assumptions S1 - S6, the minimum distance estimator e a solving (11) has the following asymptotic representation: p
n(b a
a0 ) =
G > An G
1
G > An
"
n 1 X p n t=m+1
Kt
Dn 1 xt
#
+ op (1)
where
G = g0
and
is an (m + 1)
. Jm ..IK
0
,
Kt
2
6 6 =6 4 2
6 6 Jm = 6 6 4
1
(ut
f" (F" 1 (
m
(ut
f" (F" 1 (
0 1 .. . 0
..
.
3
2 3 Q"t ( 1 ) ) 7 6 7 7 7 , g0 = 4 5; 5 ) K Q"t ( K ) K )) 1)
1)
0 0 .. . 1
3 7 7 7 7 5
m matrix and IK is an K-dimensional identity matrix.
12
When An is an identity matrix, "
g0> g0 > IK g0
G > An G = G > G =
Dn , Vxt = xt x> t , V
Alternatively, setting D = IK K
> Jm Jm > 0 Jm
n 1 X = V n t=m+1
Vxt ;
t
g0> IK > IK IK t
=
1
0
> 0
0
> Kt ,
Kt
=D
> Jm
#
:
and 1
KD
the optimal choice of A is given by 1
A= 1 K DG,
In this case, G> An G = G> D
3.3
=D
1 K D:
. Dn Jm ) .. (IK 2
and DG = (IK g0
Dn
0)
.
Asymptotic Distribution of the Second Stage Estimator
The limiting behavior of the second-stage estimator minimizing (12) is described in the following Theorem. Theorem 3 Under assumptions S1-S6, the two-step estimator b( ) based on (12) has the following asymptotic representation: p
n b( )
1
( ) =
where a = [a1 ; a2 ;
f" (F" 1 ( )) >
; am ] , =
1
p X
=E k Ck ,
h
(
zt zt > t
i
)
1 X p zt n t
(ut )
+
; jut
k m j)
1
p
n (e a
a) + op (1)
, and
Ck = E (jut
k=1
k 1j ;
zt
:
t
In particular, since the …rst stage estimation is based on (11) the above asymptotic repre. sentation can be rewritten, denoting Lm=K = Im ..0m K , as, p
n b( )
( )
=
1 f" (F" 1 ( )) 1
1
(
1 X p zt n t
Lm=K G> An G
+op (1). 13
1
G > An
)
(ut ) "
n 1 X p n t=m+1
Kt
Dn 1 x t
#
In the simple case where we estimate the …rst stage model at a single quantile , let e ( ) = (e 0 ( );
; e m ( ))> , by setting e a0 = 1 and solving the equations e ( ) =
e t ( ), we obtain the following estimator for (a0 ; aej ( )Q e a0 = 1; e a1 =
In this case, the estimator
e1( ) ; e0( )
et = e a0 +
in Step 1 has the following representation: 1 et = t + [e ( ) 0( )
m X j=1
( )] xt + Op
where
Pm
j=1
xt =
0(
j(
)
)
;e am =
e aj jut j j ;
m2 n
>
; am ):
; jut 1 j ;
=
em( ) ; e0( )
t
r
+ Op
; jut
!
mj
m n
+ Op
m2 n
;
and the two-stage estimator has the following simpli…ed asymptotic representation. Corollary 4 Under our assumptions S1 - S6, if we estimate the …rst stage model at same single quantile , the second stage quantile regression estimator b( ) based on (12) has the following Bahadur representation: p
n b( )
where R> =
( ) = 1 0( )
Pm
1 f" (F" 1 ( ))
j=1
j( 0(
) r ;r ; ) j 1
(
) 1 X > 1 p zt + R Dn xt (ut ) + op (1) n t h i P ; rm , and rj = pk=1 k E jut k j j ztt 1
Remark 2 We may compare the quantile regression estimator b( ) based on generated
regressors zet with the infeasible quantile regression estimator e( ) based on unobserved regressors zt . Note that the infeasible estimator e( ) has Bahadur representation: ) ( X p 1 1 1 p zt (ut ) + op (1): n e( ) ( ) = f" (F" 1 ( )) n t
Comparing it with the Bahadur representation of b( ) given in Corollary 1, we see that
the Bahadur representation (and thus the variance) of b( ) contains an additional term that arises from the preliminary estimation.
Remark 3 It is also possible to re-estimate
t
using the two-step parameter estimates
b( ) and use such an estimator to obtain …nal estimate of Qut ( jFt 1 ):
Remark 4 It is possible to iterate the above procedure to obtain improved estimator. 14
4
Monte Carlo Results
In this section, we report on some Monte Carlo experiments designed to examine the sampling performance of the proposed estimation procedures and compare them with existing methods. The data were generated from an GARCH(1,1) process with parameter values
0
= 0:1,
1
= 0:5,
1
= 0:3. We consider 3 di¤erent settings for the distribution
of "t : (i) i.i.d Normal; (ii) i.i.d. t(4) - student-t distribution with 4 degrees of freedom; (iii) skewed student-t distribution t(4; 0:5) recentered so that it has mean zero. The …rst design of "t actually has normal distribution and we expect the traditional methods based on normal assumption should be reasonable. The second design of "t has heavy tail distribution but is still symmetric. The third choice of distribution for "t is both heavy-tailed and asymmetric. For comparison purpose, we consider the following 5 estimation procedures: 1. RiskM: The conventional RiskMetrics method (RiskMetrics, 1996) that is widely used in …nancial applications for estimation of Value-at-Risk; 2. ARCH: Sieve ARCH quantile regression approximation 3. GARCH : The proposed two-step estimation method using information at the speci…ed quantile in the …rst step estimation. 4. GARCHc : The proposed two-step estimation method using information over multiple quantiles in the …rst step estimation. In particular, we estimate the sieve ARCH quantile regression at each percentile (
k
= k%, k = 1;
; 99:), and estimate the
GARCH parameters using the Minimum distance estimation (An = I) coupled with trimming to avoid the random denominator going to zero. 5. GARCH : The proposed estimation method using information at the speci…ed quantile in the …rst step estimation and iterate for potential improvements. Thus, following Step 1 in our procedure, we estimate a sieve quantile autoregression and obtain estimates of
t,
then we run quantile regression of ut based on the estimated
15
regressors and obtain the two-step estimator of ( )> = (
0(
);
1(
);
1(
)):
Estimate of parameters of the GARCH model can then be derived from the quantile regression estimates by solving 1(
) = 0( ) Re-compute the estimates of
t
1 0
;
1(
) = 0( )
1 0
;
0
1
= 1: 1
and iterate the process to convergence.
Sample sizes n = 100, n = 500, and n = 1000 are examined in the simulation, and number of repetition is 100. We estimate the 5% quantile. Table 1: Bias and MSE of the 5% Quantile Estimates ("t : i.i.d. Normal) Estimators n = 100 n = 500 n = 1000 RiskM Bias 0.00195438 0.00206543 0.00237998 MSE 0.0250305 0.02439572 0.0245549 ARCH
Bias MSE
GARCH
Bias MSE
GARCHc
Bias MSE
GARCH
Bias MSE
0.00980172 0.0902534
0.0018882 0.0162156
0.00060401 0.00549539
0.00263435 0.0451106
0.000718927 0.00757189
0.00025651 0.00244357
0.00284427 0.04621000
0.000673304 0.00693400
0.00018609 0.00157433
0.00123951 0.0243532
0.000647245 0.00548592
0.00016173 0.00106168
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Table 2: Bias and MSE of Quantile estimates ("t : i.i.d. student t(4)) Estimators n = 100 n = 1000 RiskM Bias 0.014847 0.00528312 MSE 0.0963872 0.0376036 ARCH
Bias MSE
GARCH
Bias MSE
GARCHc
Bias MSE
GARCH
Bias MSE
0.0127499 0.142131
0.000753013 0.00689912
0.00796561 0.0928502
0.000336906 0.0034666
0.0103258 0.0829553
0.000297569 0.00277122
0.00567884 0.0496881
0.00029151 0.00216489
Table 3: Bias and MSE of Quantile estimates ("t : i.i.d. skewed student-t) Estimators n = 100 n = 1000 RiskM Bias 0.0059688 0.00712428 MSE 0.0437164 0.0494451 ARCH
Bias MSE
GARCH
Bias MSE
GARCHc
Bias MSE
GARCH
Bias MSE
0.0261537 0.196904
0.00178461 0.0156443
0.00952587 0.0700994
0.00099653 0.00828989
0.00854402 0.0814387
0.000676337 0.00597404
0.00260169 0.0413836
0.000571442 0.00515963
The Tables show that the iterated quantile regression estimator is roughly comparable
17
to the RiskM estimator, at n = 100 and substantially outperforms RiskM at larger sample sizes. As the sample sizes increase, the proposed estimators all exhibit improved sampling properties, whereas RiskM performance is invariant to sample size.
18
5
Appendix: Proofs
19
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