1. Modified Kolmogorov-Smirnov
Goodness-of-Fit Test for GARCH
Models with Mixture of Normal
Innovations
Dong Zhang
Supervisor: Hao Yu
Department of Statistical and Actuarial Sciences
University of Western Ontario
August 8, 2013
2. Abstract
In many financial fields, the GARCH model plays an important
role in modeling the return rates of securities, such as stocks. In the
meantime, the Kolmogorov-Smirnov (KS) test is one of the common
way to test whether the innovation is from a specific distribution. In
this project we use the modified KS statistic to test if the innovation
of a GARCH model is from a mixture of normal distribution instead
of a standard normal distribution. Simulation study is conducted
to study the size and power of the KS and the modified KS tests.
4. Chapter 1
Introduction
1.1 Mixture of Normal Distributions
In probability and statistics, a mixture distribution is the probability distribu-
tion of a random variable whose values can be interpreted as being derived in the
following way from an underlying set of other random variables: In cases where
each of the underlying random variables is continuous, the outcome variable will
also be continuous and its probability density function is sometimes referred to
as a mixture density. The cumulative distribution function (and the probability
density function if it exists) can be expressed as a convex combination (i.e. a
weighted sum, with non-negative weights that sum to 1) of other distribution
functions and density functions. The individual distributions that are combined
to form the mixture distribution are called the mixture components, and the
probabilities (or weights) associated with each component are called the mix-
ture weights.
In this project, I use the mixture of two independent normal distributions. In
other words,we have a random variable X and the observations are X1, . . . , Xn.
Then we can see
X ∼ N(µ1, σ2
1) with probability λ,
and
X ∼ N(µ2, σ2
2) with probability 1 − λ.
2
5. 1.2 ARCH & GARCH Model
ARCH model (Engle, 1982) [1] is short for AutoRegressive Conditional Het-
eroskedasticity model, it is often used to characterize and model observed time
series in econometrics. In economic fields, the variances of residuals of many
time series, such as the log return of one stock, are not constant. Instead they
are related the previous error terms. Generally, the ARCH model is defined as
t = σtηt
σ2
t = α0 + α1
2
t−1 + · · · + αq
2
t−q = α0 +
q
i=1
αi
2
t−i,
where t denotes the error terms, i.e. the return residuals which are with respect
to a mean process. Generally η comes from a standardised normal distribution
N ∼ (0, 1). But in this project, we let η come from the mixture of two indepen-
dent normal distributions. And σ denotes a time-dependent standard deviation
of the variance of the residuals.
Then in 1986, Bollerslev [2] introduced the Generalized AutoRegressive Condi-
tional Heteroskedasticity (GARCH) model. Comparing with the ARCH model,
the GARCH model has the additional σ2
t component. Its function is
t = σtηt
σ2
t = α0+α1
2
t−1+· · ·+αq
2
t−q+β1σ2
t−1+· · ·+βpσ2
t−p = α0+
q
i=1
αi
2
t−i+
p
i=1
βiσ2
t−i.
In this project, GARCH(1,1) model will be selected to use, i.e.
t = σtηt
σ2
t = α0 + α1
2
t−1 + β1σ2
t−1.
1.3 ARMA-GARCH model
ARMA model is short for Autoregressive-moving-average model. Its notation
is ARMA(p,q), which refers to the model with p autoregressive terms and q
moving-average terms. This model contains the AR(p) and MA(q) models,
3
6. Xt = c + t +
p
i=1
φiXt−i +
q
i=1
θi t−i.
In the ARMA-GARCH model, both the conditional expectation and conditional
variance of the current value are depending on the past. They can be thought
of as ARMA models with GARCH errors.
In this project, we will use the AR(1)-GARCH(1,1) model with mean 0, i.e.,
Xt = φ1Xt−1 + t
t = σtηt
σ2
t = α0 + α1
2
t−1 + β1σ2
t−1.
1.4 Kolmogorov-Smirnov Goodness-of-fit Test
Generally, the Kolmogorov - Smirnov test is a nonparametric test for the equal-
ity of continuous, one-dimensional probability distributions that can be used to
compare a sample with a reference probability distribution. Firstly, F(x) is the
theoretical cumulative distribution function (CDF) and Fn(x) is the empirical
CDF (ECDF) for n independent identically distributed (iid) observations Xi .
It is defined as
Fn(x) =
1
x
n
i=1
IXi≤x.
The IXi≤x denotes the indicator function. When Xi ≤ x, it will be 1; otherwise
it will be 0. And the Xi denotes a set of observation. Then the KS statistic is
defined as
KS = sup
−∞<x<∞
| Fn(x) − F(x) |,
where supx is the supremum of the set of distances. According to the Glivenko
- Cantelli theorem, if the sample comes from the distribution F(x), then the KS
statistic converges to 0 almost surely. Furthermore, under null hypothesis that
4
7. the sample comes from the hypothesized distribution F(x), as n approaches
to infinity, the KS statistic which multiplies
√
n converges in distribution to
sup0<t<1 | B(t) |, Where B(t) is the Brownian bridge. i.e.
√
nKS
n→∞
−−−−→ sup
0<t<1
| B(t) |
1.5 The distribution used in this project
1.5.1 Normal distribution
The normal (or Gaussian) distribution is a continuous probability distribution
with the notation N ∼ (µ, σ2
). Its density function is
f(x) =
1
σ
√
2π
e−
(x−µ)2
2σ2
.
1.5.2 Students’t-distribution
Student’s t-distribution has the probability density function given by
f(t) =
Γ(ν+1
2 )
√
νπΓ(ν
2 )
(1 +
t2
ν
)− ν+1
2 ,
where ν is the number of degrees of freedom and Γ is the gamma function. Its
notation is t(ν). And its cumulative distribution function is
1
2
+ xΓ(
ν + 1
2
) ×
2F1(1
2 , ν+1
2 ; 3
2 ; −x2
ν )
√
πνΓ(ν
2 )
,
where 2F1 is the hypergeometric function.
1.5.3 Generalized Error Distribution
The Generalized Error Distribution is a symmetrical unimodal member of the
exponential family.[2] Its probability distribution function, F(x), is given by
dF(x | µ, σ, κ) =
e− 1
2 | x−µ
σ |
1
κ
2κ+1σΓ(κ + 1)
dx.
5
8. In this project, I use the standardization of all three distributions, i.e., for the
Generalized error distribution and Normal distribution, let µ = 0 and σ = 1,
and for the Students’t-distribution, ν−2
2 t(ν) is the standardized form.
6
9. Chapter 2
Methodology
2.1 Objective
The main objective of this project is to test if the innovation of GARCH model
comes from the mixture of normal distribution, which is
η ∼ MN(λ, µ1, σ2
1, µ2, σ2
2).
In order to complete this objective, in this project the one of the most common
goodness-of-fit tests – KS test will be used. In addition to the original KS
test, the modified KS (MKS) test will also be introduced. After getting the
two tests, we can conduct the simulation study for different sample sizes and
three alternative distributions - Normal distribution, Student’s t-distribution
and Generalized error distribution to calculate the rejection rete(power).
2.2 Modified Kolmogorov-Smirnov goodness-of-
fit Test
According to the study of J. Kawczak, R. Kulperger, and H. Yu [2], the original
KS test will have undesired power and size when dealing with the innovation
of GARCH model with mean included. Furthermore, Yu [3] has shown that
the modified KS test is valid for GARCH residuals regardless of whether the
mean is included or not. So here we introduce the modified KS test in order to
overcome this problem. In order to get the modified one, we should first define
7
10. the standardization for the data(residuals), i.e. for a process {ηt, t = 1 · · · n}
˙ηt =
ηt − ¯η
ση
t
,
where ˙ηt denotes the standardized ηt, ¯η denotes the sample mean of ηt, ση
t
denotes the standard deviation of ηt. After that, we can calculate the modified
KS statistic, i.e.
MKS = sup
−∞<x<∞
| F ˙ηt
n (x) − F(x) | .
Here Yu [3] has also shown that, as the sample size n increases to infinity, under
the null hypothesis, the empirical process based on the standardized residuals
will converge to the empirical process based on the standardized innovations, by
assuming the parameter estimator being
√
n consistent. Therefore, the model
parameters under the null hypothesis will not be influenced by the distribution
of modified KS test statistic. Based on this result, we can calculate the critical
value of the modified KS test by simulating the empirical process based on the
standardized innovations. Then use the similar method as we calculate the KS
statistic, we can get the modified KS statistic. And if the test statistic is too
large, we will reject the null hypothesis.
8
11. Chapter 3
Simulation Study
3.1 iid case
Under the iid case, we can calculate the critical value of KS and MKS tests.
Firstly we create the data which comes from the mixture of normal distribution
MN ∼ (0.75, −0.08, 1.2, 0.24, 0.19), which is the null hypothesis in this project,
then replicate the KS and MKS tests 100000 times and get the distributions for
both tests. The 90%, 95%, 99% quantiles will be the critical values.
Critical Value
Quantile type n=100 n=200 n=500 n=1000 n=2000
90% KS 1.161894 1.180175 1.194088 1.203638 1.208870
MKS 0.8749453 0.8898157 0.9043869 0.9118516 0.9174947
95% KS 1.296875 1.314373 1.326634 1.337371 1.343331
MKS 0.9587413 0.9728855 0.9880932 0.9957414 1.0018253
99% KS 1.564381 1.584370 1.593493 1.606207 1.610528
MKS 1.1260738 1.1424772 1.1590036 1.1656956 1.1708214
Here n denotes the sample size, KS and MKS denote the KS statistic and mod-
ified KS statistic respectively.
After getting the critical value, we can do the simulation study to calculate
the size and power. The algorithm is
9
12. 1. Simulate the data according to the specific hypothesis.
2. Use these data to calculate the KS and MKS statistics respectively. Repeat
this procedure 10000 times and get the rejection rate using the critical
value in the above.
First we should do the mixture of normal distribution self test to make sure
that whether the KS and MKS tests work well. Here is the table for mixture of
normal distribution rejection rate.
Mixture of normal distribution rejection rate
(times of replication=10000)
sample size type 90% 95% 99%
100 KS 0.0972 0.0498 0.0098
MKS 0.1028 0.0509 0.0094
200 KS 0.0979 0.0500 0.0103
MKS 0.1050 0.0530 0.0105
500 KS 0.1011 0.0515 0.0113
MKS 0.0979 0.0482 0.0082
1000 KS 0.0971 0.0481 0.0109
MKS 0.0982 0.0496 0.0104
2000 KS 0.1014 0.0484 0.0093
MKS 0.0999 0.0524 0.0107
The plot of it is (here we select 95% confident level)
10
13. From this table we can easily see that the KS and MKS tests are both valid.
So we calculate the rejection rates for three alternative distributions. First ta-
ble is normal distribution rejection rate, the subsequent ones are t distribution
rejection rate and generalized error distribution rejection rate
Normal distribution rejection rate
(times of replication=10000)
sample size type 90% 95% 99%
100 KS 0.2200 0.1326 0.0415
MKS 0.3045 0.1967 0.0593
200 KS 0.3266 0.2067 0.0675
MKS 0.5151 0.3762 0.1518
500 KS 0.6187 0.4582 0.2122
MKS 0.8808 0.7844 0.5298
1000 KS 0.9075 0.8022 0.5046
MKS 0.9944 0.9852 0.9248
2000 KS 0.9987 0.9939 0.9240
MKS 1.0000 0.9999 0.9995
11
16. From these plots, we can see that the MKS test always have more power than
KS test, and as the sample sizes get larger, the rejection rates of all three al-
ternatives get higher. When analysing these plots separately, we know that the
normal distribution rejection rate and generalized error distribution rejection
rate have the similar trends, and the t distribution has less power than the nor-
mal and generalized error distribution.
When we fix the sample size (eg, 500), we can generate the following plot. Here
we choose 95% confident level. Let H0 denotes the null hypothesis - the innova-
tion comes from the mixture of normal distribution MN(0.75, −0.080, 1.2, 0.24, 0.19),
and H1, H2, H3 denote the innovation comes from the normal distribution, t
distribution and generalized error distribution respectively.
14
17. We can easily get that when sample size is fixed, the normal distribution and
generalized error distribution have the similar power, and both the power of the
two distributions are higher than that of the Students’t-distribution.
3.2 GARCH case
When it comes to the GARCH case, since we generate the innovation of GARCH
model, ηt, which is also independent and identically distributed from the mix-
ture of normal distribution, we can use the same critical value we calculated for
the iid case.
The algorithm of GARCH case is slightly different from the iid case, it has
four steps,
1. Simulate ηt according to the specific hypothesis.
2. Use these ηt to compute the GARCH(1,1) model with the parameters
α0 = 0.025,α1 = 0.25,β1 = 0.5
3. Use the function ’garchFit’ from R package ’fGarch’ to calculate the resid-
uals ˆηt.
15
18. 4. calculate the KS and MKS statistics respectively. Repeat this procedure
10000 times and get the rejection rate using the critical value in the above.
In GARCH case, there are two methods to calculate the residuals ˆηt - with mean
included and without mean included.That is, when the mean is included, the
GARCH(1,1) model is like,
t = µ + σtηt
σ2
t = α0 + α1
2
t−1 + β1σ2
t−1,
when the mean is not included, t = σtηt. Here we deal with the ’without mean
included’ case at first. The same with the iid case, first we do the mixture of
normal distribution self test to make sure whether the KS and MKS tests work
well. The table of the mixture of normal distribution rejection rate is
Mixture of normal distribution rejection rate for GARCH
model
(times of replication=10000)
sample size type 90% 95% 99%
100 KS 0.0936 0.0456 0.0090
MKS 0.1035 0.0530 0.0099
200 KS 0.0944 0.0481 0.0103
MKS 0.0959 0.0493 0.0099
500 KS 0.0986 0.0488 0.0130
MKS 0.1022 0.0518 0.0106
1000 KS 0.1023 0.0514 0.0106
MKS 0.1110 0.0552 0.0106
2000 KS 0.1084 0.0538 0.0122
MKS 0.1096 0.0506 0.0082
From these rejection rates we can see that both KS and MKS tests work well
when the mean isn’t included.
Next we calculate the three alternative distributions rejection rates. The same
as the iid case, first table is normal distribution rejection rate, the subsequent
ones are t distribution rejection rate and generalized error distribution rejection
rate.
16
19. Normal distribution rejection rate for GARCH model
(times of replication=10000)
sample size type 90% 95% 99%
100 KS 0.2170 0.1226 0.0370
MKS 0.3602 0.2384 0.0752
200 KS 0.3214 0.2074 0.0694
MKS 0.5812 0.4236 0.1854
500 KS 0.6250 0.4550 0.2042
MKS 0.9168 0.8416 0.6052
1000 KS 0.9358 0.8340 0.5010
MKS 0.9986 0.9950 0.9458
2000 KS 0.9998 0.9978 0.9442
MKS 1.0000 1.0000 1.0000
t distribution rejection rate for GARCH model
(times of replication=10000)
sample size type 90% 95% 99%
100 KS 0.1528 0.0910 0.0258
MKS 0.2978 0.1986 0.0676
200 KS 0.2070 0.1264 0.0410
MKS 0.4218 0.3014 0.1324
500 KS 0.3580 0.2470 0.1058
MKS 0.6804 0.5634 0.3506
1000 KS 0.5906 0.4518 0.2310
MKS 0.9082 0.8446 0.6546
2000 KS 0.8896 0.7750 0.5358
MKS 0.9936 0.9862 0.9416
GED distribution rejection rate for GARCH model
(times of replication=10000)
17
20. sample size type 90% 95% 99%
100 KS 0.2228 0.1314 0.0388
MKS 0.3656 0.2334 0.0688
200 KS 0.3310 0.2148 0.0758
MKS 0.5790 0.4282 0.1882
500 KS 0.6282 0.4586 0.2050
MKS 0.9220 0.8528 0.6140
1000 KS 0.9324 0.8234 0.5078
MKS 0.9980 0.9936 0.9538
2000 KS 0.9994 0.9962 0.9364
MKS 1.0000 1.0000 1.0000
Now we deal with the ’with mean included’ case. The table of the mixture of
normal self test rejection rate is
Mixture of normal distribution rejection rate for GARCH
model (include mean)
(times of replication=10000)
sample size type 90% 95% 99%
100 KS 0.0090 0.0016 0.0000
MKS 0.1010 0.0500 0.0090
200 KS 0.0116 0.0038 0.0002
MKS 0.1004 0.0500 0.0118
500 KS 0.0090 0.0026 0.0000
MKS 0.1032 0.0500 0.0104
1000 KS 0.0120 0.0034 0.0000
MKS 0.1058 0.0500 0.0106
2000 KS 0.0136 0.0028 0.0002
MKS 0.0978 0.0484 0.0094
From this table, we can see that when the mean is included, the MKS test still
works while the KS test fails.
The next is the tables of the alternative distribution rejection rates. First ta-
ble is normal distribution rejection rate, the subsequent ones are t distribution
rejection rate and generalized error distribution rejection rate.
18
21. Normal distribution rejection rate for GARCH model
(include mean)
(times of replication=10000)
sample size type 90% 95% 99%
100 KS 0.0594 0.0190 0.0016
MKS 0.3690 0.2362 0.0730
200 KS 0.1522 0.0624 0.0072
MKS 0.5770 0.4280 0.1856
500 KS 0.5074 0.3042 0.0754
MKS 0.9218 0.8518 0.6108
1000 KS 0.9064 0.7696 0.3770
MKS 0.9976 0.9918 0.9492
2000 KS 0.9994 0.9946 0.9396
MKS 1.0000 1.0000 0.9998
t distribution rejection rate for GARCH model (include
mean)
(times of replication=10000)
sample size type 90% 95% 99%
100 KS 0.0670 0.0268 0.0054
MKS 0.2990 0.2048 0.0748
200 KS 0.1154 0.0578 0.0096
MKS 0.4070 0.2934 0.1276
500 KS 0.2844 0.1664 0.0430
MKS 0.6764 0.5656 0.3462
1000 KS 0.5784 0.4146 0.1560
MKS 0.9100 0.8474 0.6598
2000 KS 0.9006 0.7966 0.5246
MKS 0.9936 0.9856 0.9392
Generalize error distribution rejection rate for GARCH
model (include mean)
19
22. (times of replication=10000)
sample size type 90% 95% 99%
100 KS 0.0670 0.0268 0.0054
MKS 0.2990 0.2048 0.0748
200 KS 0.1406 0.0614 0.0070
MKS 0.5684 0.4256 0.1796
500 KS 0.5048 0.3060 0.0820
MKS 0.9200 0.8466 0.6228
1000 KS 0.9130 0.7846 0.3964
MKS 0.9984 0.9924 0.9462
2000 KS 0.9998 0.9972 0.9374
MKS 1.0000 1.0000 0.9996
Based these tables, we can do some plots in order to make the outcomes more
observable.
First is the plot which combines the 95% confident level rejection rates of the
mixture of normal distributions with mean included and without mean included.
20
23. From this plot we can easily see that when the mean is not included, both KS
and MKS tests work. But when the mean is included, the KS test fails and
MKS test still works.
Next three plots are based on three alternative distribution rejection rates. All
contain both the ’with mean included’ and ’without mean included’ case.
21
24. From these plots, we can see that the MKS test always have more power than
KS test, and as the sample sizes get larger, the rejection rates of all three al-
ternatives get higher. When analysing these plots separately, we know that the
22
25. normal distribution rejection rate and generalized error distribution rejection
rate have the similar trends, and the t distribution has less power than the nor-
mal and generalized error distribution.
Then we also fix the sample size(500), then we get the next plot.
From this plot we can the same results we get in iid case. Besides that, we can
see that when the mean is included, the original KS test fails. Furthermore, the
original KS test which is without mean included always has more power than
that which is with mean included.
3.3 ARMA-GARCH case
Now we extend the GARCH case to ARMA-GARCH case. The algorithm of
the ARMA-GARCH case is similar with that of GARCH case.
1. Simulate ηt according to the specific hypothesis.
2. Use these ηt to compute the ARMA(1)-GARCH(1,1) model with the pa-
rameters AR: φ1 = 0.5; GARCH: α0 = 0.025, α1 = 0.25, β1 = 0.5.
3. Use the function ’garchFit’ from R package ’fGarch’ to calculate the resid-
uals ˆηt.
23
26. 4. calculate the KS and MKS statistics respectively. Repeat this procedure
10000 times and get the rejection rate using the critical value in the above.
The same with the GARCH case, we first calculate the mixture of normal dis-
tribution rejection rate. Here we only consider the ’with mean included’ case.
Next are the table and plot(95% confident level) of that.
Mixture of normal distribution rejection rate for
ARMA-GARCH model (include mean)
(times of replication=10000)
sample size type 90% 95% 99%
100 KS 0.0105 0.0031 0.0001
MKS 0.1012 0.0504 0.0106
200 KS 0.0104 0.0030 0.0000
MKS 0.1008 0.0544 0.0116
500 KS 0.0100 0.0014 0.0000
MKS 0.1002 0.0516 0.0094
1000 KS 0.0122 0.0026 0.0002
MKS 0.1008 0.0502 0.0090
2000 KS 0.0114 0.0038 0.0006
MKS 0.0960 0.0462 0.0098
24
27. From this plot we can easily see that when ARMA case is included, the modified
KS test still works while the original KS test fails. Then the next tables and
plots show us the rejection rates for normal distribution, Students’t-distribution
and generalized error distribution respectively.
Normal distribution rejection rate for ARMA-GARCH
model (include mean)
(times of replication=10000)
sample size type 90% 95% 99%
100 KS 0.0544 0.0162 0.0014
MKS 0.3274 0.2187 0.0678
200 KS 0.1364 0.0588 0.0048
MKS 0.5566 0.4086 0.1742
500 KS 0.4830 0.2814 0.0634
MKS 0.9146 0.8352 0.5770
1000 KS 0.8994 0.7634 0.3762
MKS 0.9976 0.9908 0.9404
2000 KS 0.9998 0.9966 0.9360
MKS 1.0000 1.0000 0.9996
T distribution rejection rate for ARMA-GARCH model
(include mean)
(times of replication=10000)
sample size type 90% 95% 99%
100 KS 0.0656 0.0262 0.0040
MKS 0.2920 0.1918 0.0710
200 KS 0.1162 0.0612 0.0098
MKS 0.4072 0.2852 0.1292
500 KS 0.2784 0.1664 0.0486
MKS 0.6758 0.5684 0.3428
1000 KS 0.5660 0.4056 0.1608
MKS 0.9066 0.8432 0.6534
2000 KS 0.8972 0.7948 0.5238
MKS 0.9944 0.9862 0.9446
25
28. Generalized error distribution rejection rate for
ARMA-GARCH model (include mean)
(times of replication=10000)
sample size type 90% 95% 99%
100 KS 0.0528 0.0148 0.0010
MKS 0.3380 0.2118 0.0620
200 KS 0.1290 0.0526 0.0046
MKS 0.5478 0.3832 0.1646
500 KS 0.4942 0.2784 0.0646
MKS 0.9150 0.8472 0.6102
1000 KS 0.9002 0.7686 0.3682
MKS 0.9974 0.9906 0.9432
2000 KS 0.9996 0.9964 0.9368
MKS 1.0000 1.0000 1.0000
These plots are all under 95% confident level.
26
29. When we fix the sample size (500), then the plot we generate is
27
30. From these plots, we can see that the rejection rates for three alternatives in
ARMA-GARCH case are almost the same with those in GARCH case, so we
can get the same conclusion as we get in GARCH case.
28
31. Chapter 4
Conclusions and Future
work
In this project, we want to test whether the innovation of GARCH model comes
from the mixture of normal distribution MN ∼ (0.75, −0.08, 1.2, 0.24, 0.19),
which is the null hypothesis. Then we use KS test and modified KS test to
complete this objective. After getting the critical value, we can conduct the
simulation study for different sample sizes and three alternative distributions -
Normal distribution, Student’s t-distribution and Generalized error distribution
to calculate the rejection rate. Here are three main conclusions:
1. The original KS test fails when mean is included and has poor power under
the alternatives.
2. The modified KS test works regardless of whether the mean is included or
not and has relative high power under the alternatives
3. When sample size is fixed, the normal distribution and generalized er-
ror distribution have the similar power, and both the power of the two
distributions are higher than that of the Students’t-distribution.
All the results we mention before is based on the known parameters - the pa-
rameters of mixture of normal distribution are fixed and set up at first. But
for real data such as the return rate of a stock, we don’t know the parameters
when we fit the data in to a mixture of normal distribution. So further study is
needed in this regard.
29
32. Chapter 5
Acknowledgement
I would like the express the deepest appreciation to my supervisor - Dr. Hao
Yu. Without his patient help and guidance I couldn’t finish this project. It’s
also a pleasure for me to be under his supervision.
30
33. Bibliography
[1] R. Engle. Autoregressive conditional heteroscedasticity with estimates of the
variance of United Kingdom inflation. Econometrica 50:987 - 1007, 1982.
[2] T. Bollerslev. Generalized autoregressive conditional heteroskedasticity.
Journal of Econometrics 31:307 - 327,1986.
[3] J. Kawczak, R. Kulperger, and H. Yu. The empirical distribution function
and partial sum process of residuals from a stationary ARCH with drift
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