Does it exist any excess skewness in GARCH models?752348/FULLTEXT01.pdf · higher moments (skewness...

Örebro University

School of Business and Economics

Statistics, advanced level thesis, 15 hp

Supervisor: Panagiotis Mantalos

Examiner: Per-Gösta Andersson

Spring 2014

Does it exist any excess skewness in

GARCH models?

–

A comparison of theoretical and White Noise values

Åsa Grek

890727

Abstract

GARCH models are widely used to estimate the volatility. According to previous studies

higher moments (skewness and kurtosis) has an impact on how well the GARCH models can

capture the volatility sufficiently. There is not any previous study, which investigate if there is

excess skewness of the GARCH process. So the aim in this thesis is to investigate the

skewness and see if it differs from the theoretical values for the normal distribution. If the

values differ it can be an indication of excess skewness. A Monte Carlo simulation conducted

the results, where different GARCH models were generated. The mean, variance, skewness

and kurtosis were calculated, compared to theoretical values and White Noise values. The

result shows that there is an affect on all the mean, variance, skewness and kurtosis.

Keywords: GARCH, moments, skewness, kurtosis, White Noise, High Persistence, Medium

Persistence, Low Persistence

Table of content

1. Introduction 1

1.1 Background and aim 1

1.2 Previous studies 2

1.3 Disposition 3

2. Theory 4

2.1 GARCH (Generalized autoregressive conditional heteroskedasticity) model 4

2.2 GARCH with Generalized Error Distribution (GED) (innovation) 5

2.3 GARCH with Student’s t distribution (innovation) 5

2.4 Centred moments about the mean 5

2.5 White Noise 7

2.4 Kurtosis of the GARCH 7

3. Method 9

3.1 Data generating process 9

3.2 Monte Carlo simulation 14

4. Analysis and Results 15

5. Discussion and conclusion 24

References 25

Appendix 27

1

1. Introduction

1.1 Background

In finance it is important to estimate the volatility (the conditional standard deviation of asset

return), but there is usually problem to estimate it correctly due to the skewness and kurtosis.

The asset is assumed to follow a Gaussian distribution (i.e. normal distribution) and then the

skewness is equal to zero and kurtosis is equal to three. This leads to uncertainty problem

because sometimes the asset seems to follow other distributions. In case when the skewness

do not have the same values as if the asset has a normal distribution (see section 2.4) (Tsay,

2005).

Often the attention has been on the estimation of the mean and the standard deviation since

skewness and kurtosis have valuable information about the volatility and thereby the risk of

holding the asset (Schittenkopf, Dorffner and Dockner).

Patton (2004) found that knowledge of higher moments (skewness and kurtosis) make a

significant better forecast on assets, both statistically and economically. As skewness and

kurtosis clearly have an effect on volatility then they are vital measurements for Value-at-

Risk (VaR) (Jondeau and Rockinger, 2003).

But even if the higher moments are important for forecasting the volatility, few studies have

been done in this field.

1.2 Previous studies

The first study that examined and derived the formula for kurtosis in GARCH models were

Bai, Russell and Tiao in 2003 (se section 2.6). According to them other distributions than the

Gaussian includes leptokurtosis (excess kurtosis), which exists in volatility clustering. They

found that this affects the kurtosis on the whole time series and the effect is symmetrical.

2

According to Galeano and Tsay (2010) asset returns often have high excess kurtosis, but even

when GARCH models with heavy tailed innovations are used the volatility is not sufficiently

caught.

Posedel (2005) examine the properties of the GARCH (1,1) and find out that the model is

heavy tailed but asymptotic normal. Posedel also examine the higher moments and found a

restricted conditional kurtosis of the standard deviation in the model.

Jondeau and Rockinger (2003) used a GARCH model with Student’s t distribution and then

calculated the skewness and kurtosis. The skewness and kurtosis were compared with the

maximum theoretical values (see part 2.4) and it seemed like the GARCH model allowed for

varying moments. They applied the model on real data and discovered that skewness exists

more frequently compared to kurtosis.

Breuer and Jandacka (2010) investigated the properties of the aggregated distributions of a

GARCH procedure. They observed that as the time series became longer the closer the

kurtosis became to three (i.e. converged to the normal distribution) or converged into infinity.

From these studies it can be observed that there is an information gap in how the skewness for

GARCH models behaves. Because the assets are assume to follow a normal distribution, to be

able to capture the movements of the assets (volatility) then the GARCH models has also to

follow a normal distribution. So the aim of this thesis is to generate different GARCH models

and try to see if the skewness differs from the theoretical value and to the values of White

Noise. The study will use GARCH models with different persistence (movement) and

different innovation (distribution) terms. The results will be conducted by using Monte Carlo

simulation. According to the Central Limit Theorem (CLT) the skewness should converge to

the normal distribution and the skewness mean and variance should be close to the theoretical

values (see part 2.4). Otherwise if the skewness differs from the values for the theoretical

values it indicates that there is excess skewness.

3

1.3 Disposition

The next chapter is an introduction to the GARCH model and the different innovations that

will be used in the Monte Carlo simulation. The third part is the method, where Data

Generating Process (DGP) and the steps in the Monte Carlo simulation are explained. The

fourth section is the result where some of the simulation results are shown. The fifth and the

final part is discussion and conclusion, where the results are discussed and the conclusions are

drawn.

4

2. Theory

2.1 GARCH (Generalized Autoregressive Conditional

Heteroskedasticity) models

Bollerslev (1986) created the GARCH model in 1986 as a transformation to the ARCH

(Autoregressive Conditional Heteroskedasticity) model, which was first created by Engle in

1982. The GARCH (m,s) model is written as:

,

∑

∑

(2.1)

Where the ARCH parameters are and the GARCH parameter is . The restrictions is

>0, ≥ 0, ≥ 0 and ∑ , which makes the conditional variance is

positive and increases ( ). If then the model becomes an ARCH (m) model (Tsay,

2005).

For the GARCH (m,s) model with Gaussian distribution , but if the GARCH

has a Generalized Error distribution (GED) or Student’s t distribution, follows an GED or a

Student’s t distribution (Bollerslev, 2009).

The GARCH (1,1) can be written as

,

(2.2)

Where (Tsay, 2005).

5

2.2 GARCH models with Generalized Error Distribution (GED)1

(innovation)

The GARCH models can have different distribution (innovation) terms, as in equation (2.1)

and (2.2) the first part where for GARCH with Gaussian

distribution is for GARCH with Generalized error distribution with d degrees of

freedom. The degrees of freedom are by default two (Bollerslev, 2009).

2.3 GARCH models with Student’s t distribution (innovation)

As in the previous section the GARCH with student’s t distribution (innovation) is when

, where where is degrees of freedom. By default the degree of freedom

is usually four (Bollerslev, 2009).

2.4 Central moments about the mean

The jth central moment (moment about the mean) for continuous distribution is defined by:

[ ] ∫

(2.3)

where the first centred moment is the mean and the second is the variance .

Skewness is

and the kurtosis is

.

The j:th sample moment is

1 The Generalized error distribution is also known as the generalized normal distribution and generalized

Gaussian distribution. The distribution is centred round zero and it is a member of the exponential family of

distribution. See Giller (2005) for more information.

6

∑

(2.4)

To estimate the sample skewness and kurtosis one can use:

∑

(2.5)

∑

(2.6)

which are consistent estimates for skewness and kurtosis (Danielsson, 2011) and

asymptotically normal distributed (Pearson, 1931). When the sample has a normal

distribution2 (or in this case normal innovation) then the sample distribution have movements

(2.7)

(2.8)

where is the sample size (Pearson, 1931).

2 For the normal distribution the skewness is equal to zero and kurtosis is equal to three (Casella and

Berger, 2002).

7

2.5 White Noise

To be able to estimate the skewness and kurtosis for a Normal (i.e. Gaussian) distribution a

White Noise process is used. A sample from a White Noise process is asymptotically normal

with mean zero and variance , as the sample size gets large. According to Anderson

(1942), Bartlett (1946) and Quenouille (1949) the White Noise can be used to evaluate

properties because it is asymptotic normally distributed.

2.6 Kurtosis of GARCH models

As in equation (2.2) the GARCH (1,1) model is:

,

Where >0, ≥ 0, ≥ 0, and .

With . Where is the excess kurtosis

of (Tsay, 2005).

This implies according to Bai, Russell and Tiao (2003) that the expected value and variance

of the GARCH (1,1) is:

⁄

(2.9)

Then the excess kurtosis of the GARCH (1,1) model is

(2.10)

8

If the GARCH (1,1) model has a Gaussian distribution ( ) then the excess kurtosis is

written as

(2.11)

The excess kurtosis can only exist if and > 0 if then

(Tsay, 2005).

9

3. Method

3.1 Data generating process

To construct the study, recall that the GARCH (1,1) is written as (equation (2.2))

,

where the parameters are generated by using different GARCH process. High Persistence

(HP) [0.01, 0.09, 0.9] generates a high volatility GARCH process. Medium Persistence (MP)

[0.05, 0.05, 0.9] generates a medium volatility GARCH process. And finally Low Persistence

(LP) [0.2, 0.05, 0.75] generates a low volatility GARCH process (Mantalos, 2010).

All of these GARCH series will be simulated with four different lengths; 300, 500, 1000 and

3000.

3

{ }

{ }

{ }

{ } (2.12)

Using the skewed GARCH models, with skewness of 0.5, 0.8 and 0.9, will also simulate these

models.

(2.13)

The models are stated in Table (3.1), Table (3.2), Table (3.3) and Table (3.4).

A White Noise process was also generated for all lengths; 300, 500, 1000 and 3000.

3 See part 2.2 and 2.3.

10

Table (3.1) Non-skewed models

Models

Number of observations

(length of the time series)

Innovation ( )

Model 1:

High Persistence

Gaussian innovation

0.01 0.09 0.9

Model 2:

Medium Persistence

Gaussian innovation

0.05

0.05

0.9

Model 3:

Low Persistence

Gaussian innovation

0.2

0.05

0.75

Model 4:

High Persistence

Generalized error

innovation

0.01

0.09

0.9

Model 5:

Medium Persistence

Generalized error

innovation

0.05

0.05

0.9

Model 6:

Low Persistence

Generalized error

innovation

0.2

0.05

0.75

Model 7:

High Persistence

Student’s t innovation

0.01

0.09

0.9

Model 8:

Medium Persistence


0.05

0.05

0.9

Model 9:

Low Persistence


0.2

0.05

0. 75

11

Table (3.2) Skewed models for Gaussian distributions

Model



Innovation ( ) Skewed

Model 10:

High Persistence

Gaussian innovation

0.01

0.09

0.9

0.5

Model 11:

High Persistence

Gaussian innovation

0.01 0.09 0.9 0.8

Model 12:

High Persistence

Gaussian innovation

0.01 0.09 0.9 0.9

Model 13:

Medium Persistence

Gaussian innovation

0.05 0.05 0.9 0.5

Model 14:

Medium Persistence

Gaussian innovation

0.05 0.05 0.9 0.8

Model 15:

Medium Persistence

Gaussian innovation

0.05 0.05 0.9 0.9

Model 16:

Low Persistence

Gaussian innovation

0.2 0.05 0.75 0.5

Model 17:

Low Persistence

Gaussian innovation

0.2 0.05 0.75 0.8

Model 18:

Low Persistence

Gaussian innovation

0.2 0.05 0.75 0.9

12

Table (3.3) Skewed models for Generalized Error distributions

Model




Model 19:

High Persistence

Generalized error innovation

0.01

0.09

0.9

0.5

Model 20:

High Persistence


0.01 0.09 0.9 0.8

Model 21:

High Persistence


0.01 0.09 0.9 0.9

Model 22:

Medium Persistence


0.05 0.05 0.9 0.5

Model 23:

Medium Persistence


0.05 0.05 0.9 0.8

Model 24:

Medium Persistence


0.05 0.05 0.9 0.9

Model 25:

Low Persistence


0.2 0.05 0.75 0.5

Model 26:

Low Persistence


0.2 0.05 0.75 0.8

Model 27:

Low Persistence


0.2 0.05 0.75 0.9

13

Table (3.4) Skewed models for Student’s t distributions

Model




Model 28:

High Persistence


0.01

0.09

0.9

0.5

Model 29:

High Persistence


0.01 0.09 0.9 0.8

Model 30:

High Persistence


0.01 0.09 0.9 0.9

Model 31:

Medium Persistence


0.05 0.05 0.9 0.5

Model 32:

Medium Persistence


0.05 0.05 0.9 0.8

Model 33:

Medium Persistence


0.05 0.05 0.9 0.9

Model 34:

Low Persistence


0.2 0.05 0.75 0.5

Model 35:

Low Persistence


0.2 0.05 0.75 0.8

Model 36:

Low Persistence


0.2 0.05 0.75 0.9

14

3.2 Monte Carlo simulation

The structure of the Monte Carlo simulation was done to assess the distribution for the

skewness of the GARCH processes. The simulation was done according to the following

steps.

i. Simulate a White Noise process with zero autocorrelation.

ii. Calculate the skewness for the simulated White Noise.

iii. Calculate the moments: mean, variance, skewness and kurtosis for the 10 000 skewed

results.

iv. Simulate the GARCH model with specifications from the tables above with

.

v. Calculate the skewness for the simulated GARCH returns.

vi. Repeat it times.

vii. Calculate the moments: mean, variance, skewness and kurtosis for the 10 000 skewed

results.

viii. Graph the skewed results for sample size of 1000.

15

4. Analysis and Results

The results of the Monte Carlo simulation are presented in the tables below (4.1 and 4.2) and

some of the additional results are in Appendix (1) and Appendix (2). The first row is the

result of the simulations for the white noise process and in the end of the table are the

expected theoretical values for the mean and variance of estimated skewness according to

Pearson (1931) when the sample has a normal distribution (see section 2.4). According to the

Central Limit Theorem (CLT), even if the sample is not normally distributed the sample

should convert, as sample size gets larger to a normal distribution. This means GARCH

processes should converge to the theoretical values if there does not exist any excess

skewness.

The model number is the numbers from the data generating process.

Table (4.1) Results from non-skewed innovation for 1000 observations

Model number and

specification

Mean Variance Skewness Kurtosis

White Noise

-0.0007761002

0.005975894

0.003566258

-0.01219913

Gaussian

1. High Persistence -0.001065014 0.03309817 -0.1648432 5.732647

2. Medium Persistence -0.0002650263 0.00724599 0.005958856 0.1737693

3. Low Persistence -0.0003496688 0.006200227 0.01853256 0.0592591

Generalized error


5. Medium Persistence -0.0002107584

0.007334852 -0.01888934 0.1088943

6. Low Persistence -0.0002358025 0.00636376 -0.006219489 0.03979624

Student’s t

7. High Persistence -0.003639477 1.05002 1.051148 24.77947

8. Medium Persistence -0.005656082 1.146918 2.207344 57.71248

9. Low Persistence -0.006388693 1.512463 1.968903 70.60844

Expected Values 0 0.0059641256 - -

16

It is observed from the table above that the White Noise process is closest to the theoretical

value when it comes to the mean and variance. So, therefore the White Noise values for the

skewness and kurtosis will be set as reference values. If the GARCH process should follow a

normal distribution (i.e. non excess skewness) then the skewness should be close to the

theoretical- and reference values.

Some models seem to converge closely to the theoretical values. The model, which is closest

to the theoretical values and the White Noise value, is model 6 (Low Persistence GARCH

with Generalized error innovation). The model furthest away is model 9 (Low Persistence

GARCH with Student’s t innovation), which indicates that there can be is some excess

skewness.

It is observed from the table above that the skewed values of the GARCH process has a

higher skewness than it should have if the GARCH process is normally distributed, especially

high is the skewness for the Student’s t innovation term. For High Persistence GARCH with

Gaussian and Generalized error innovations it seems like there is an effect on the variance,

skewness and kurtosis values.

17

Figure (4.1) Gaussian non-skewed innovation

4.1a Pdf of White Noise

4.1b Pdf of Low Persistence

4.1c Pdf of Medium Persistence

4.1d Pdf of High Persistence

In Figure (4.1) are the pdfs of the GARCH process with non-skewed innovation. It is

observable that Low Persistence has a similar curve as the White Noise. One can see that the

High Persistence is spikier and has longer tails then the others.

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

01

23

45

density.default(x = white.noise)

N = 10000 Bandwidth = 0.01103

De

nsity

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.30

12

34

5

density.default(x = LPnormal1000)

N = 10000 Bandwidth = 0.01106

De

nsity

-0.4 -0.2 0.0 0.2 0.4

01

23

4

density.default(x = MPnormal1000)

N = 10000 Bandwidth = 0.01202

De

nsity

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

0.0

0.5

1.0

1.5

2.0

2.5

density.default(x = HPnormal1000)

N = 10000 Bandwidth = 0.02085

De

nsity

18

Figure (4.2) Generalized error non-skewed innovation





The High Persistence in Figure (4.2) appears to differ from the other graphs. It has a more

pointed top and longer tails than the others, but there is not any obvious sign that the

skewness should differ from the expected value for the mean (zero).

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

01

23

45


N = 10000 Bandwidth = 0.01103

De

nsity

-0.2 0.0 0.2 0.4

01

23

45

density.default(x = LPGED1000)

N = 10000 Bandwidth = 0.01136

De

nsity

-0.4 -0.2 0.0 0.2 0.4

01

23

4

density.default(x = MPGED1000)

N = 10000 Bandwidth = 0.01222

De

nsity

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

0.0

0.5

1.0

1.5

2.0

2.5

density.default(x = HPGED1000)

N = 10000 Bandwidth = 0.02046

De

nsity

19

Figure (4.3) Student t non-skewed innovation





In Table (4.1) it is visible that the behaviour of the GARCH with Student’s t innovations are

different from the other distributions and from Figure (4.3) it is observable that both Low-,

Medium- and High Persistence are more spikey and have longer tails towards right. It gives

the impression that the pdfs are skewed, which is consistent with Table (4.1).

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

01

23

45


N = 10000 Bandwidth = 0.01103

De

nsity

-10 0 10 20

0.0

0.2

0.4

0.6

0.8

density.default(x = LPt1000)

N = 10000 Bandwidth = 0.07302

De

nsity

-10 -5 0 5 10 15 20

0.0

0.2

0.4

0.6

0.8

density.default(x = MPt1000)

N = 10000 Bandwidth = 0.07338

De

nsity

-10 -5 0 5 10 15 20

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

density.default(x = HPt1000)

N = 10000 Bandwidth = 0.08659

De

nsity

20

In Table (4.2) are some of the skewed results from the Monte Carlo simulation.

Table (4.2) Results from skewed (0.9) innovation for 1000 observations

Model number and

specification


White Noise

-0.0007761002

0.005975894

0.003566258

-0.01219913

Gaussian


15. Medium Persistence -0.1668421 0.007358689 -0.1041338 0.1703169

18. Low Persistence -0.1655104 0.006333658 -0.05672268 0.07896617

Generalized error



27. Low Persistence -0.1654759 0.006383742 -0.04840784 0.1130814

Student’s t



36. Low Persistence -0.6116942 1.380634 -3.150867 58.58656


As it appears from the table above that Low Persistence GARCH with both Gaussian and

Generalized error innovation (models 18 and 27) has the best convergence to the theoretical

and reference values (White Noise). But the difference is bigger now when skewed

distributions are used, the mean has become larger. It indicates that there is excess skewness.

The model, which is clearly worst in converge is model 36 where it is skewed to left (-3.15)

and has a high kurtosis (58.59).

Compared to the reference value, which has a slightly skewness towards right, all of the

GARCH models are skewed to the left. The Student’s t distribution is mostly skewed,

compared to Gaussian and Generalized error distribution. It seems like for Gaussian and

Generalized error distribution, that the more GARCH effect (high persistence) is the more

skewed the model become, the more likely it is that there is excess skewness.

21

Figure (4.4) Gaussian skewed innovation (0.9)


4.4b Pdf of Low Persistence skewed 0.9

4.4c Pdf of Medium Persistence skewed 0.9

4.4d Pdf of High Persistence skewed 0.9

From Figure (4.4) it seems like the Gaussian innovation terms are skewed toward the left.

And the stronger the GARCH effect (high persistence) is the more skewed the pdfs are. It

indicates that there is excess skewness, and it becomes bigger the higher the persistence the

model has.

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

01

23

45


N = 10000 Bandwidth = 0.01103

De

nsity

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2

01

23

45

density.default(x = LPskew0.9normal1000)

N = 10000 Bandwidth = 0.01135

De

nsity

-0.4 -0.2 0.0 0.2

01

23

4

density.default(x = MPskew0.9normal1000)

N = 10000 Bandwidth = 0.01223

De

nsity

-1.5 -1.0 -0.5 0.0 0.5 1.0

0.0

0.5

1.0

1.5

2.0

2.5

density.default(x = HPskew0.9normal1000)

N = 10000 Bandwidth = 0.02044

De

nsity

22

Figure (4.5) Generalized error innovation (0.9)





As observed from Figure (4.5) it shows that all the Generalized error models are skewed

towards right compared to the White Noise process – which indicates that the GARCH

process is not normal distributed for its skewed values. High Persistence GARCH is the most

skewed, which is corresponding to the values in Table (4.2). This point towards that there

occurs excess skewness.

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

01

23

45


N = 10000 Bandwidth = 0.01103

De

nsity

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2

01

23

45

density.default(x = LPskew0.9GED1000)

N = 10000 Bandwidth = 0.01128

De

nsity

-0.6 -0.4 -0.2 0.0 0.2

01

23

45

density.default(x = MPskew0.9GED1000)

N = 10000 Bandwidth = 0.01199

De

nsity

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0

0.0

0.5

1.0

1.5

2.0

2.5

density.default(x = HPskew0.9GED1000)

N = 10000 Bandwidth = 0.02054

De

nsity

23

Figure (4.6) Student’s t skewed innovation (0.9)





In Figure (4.6) it is shown that all GARCH with Student’s t innovation is skewed to left and

has long tails, which correspond to Table (4.2). The stronger the GARCH effect is the shorter

the tails become. The difference from the non-skewed pdfs is that the models with Student’s t

innovation are skewed to right, but it seems to be excess skewness.

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

01

23

45


N = 10000 Bandwidth = 0.01103

De

nsity

-20 -10 0 10 20

0.0

0.2

0.4

0.6

0.8

density.default(x = LPskew0.9t1000)

N = 10000 Bandwidth = 0.07269

De

nsity

-15 -10 -5 0 5 10 15

0.0

0.2

0.4

0.6

0.8

density.default(x = MPskew0.9t1000)

N = 10000 Bandwidth = 0.07247

De

nsity

-10 -5 0 5 10

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

density.default(x = HPskew0.9t1000)

N = 10000 Bandwidth = 0.08643

De

nsity

24

5. Discussion and conclusion

According to Pearson (1931) the skewed values should converge to the theoretical values if

the GARCH process is asymptotically normally distributed (see section 2.4). One problem is

that the theoretical values only exist for mean and variance and it was important to have a

reference value for the skewness and kurtosis as well. Therefore the White Noise was used

(see part 2.5).

If the GARCH process did not have any effect on the skewness then the saved skewed values

should converge. The result shows that it seems like Gaussian and Generalized error

innovations for Low- and Medium Persistence converge better then the High Persistence. But

for Student’s t innovation it was the opposite, where Low Persistence did the worst in

converge.

Another finding is that the convergence for the Student’s t was worst than for the other two

distributions of the innovation. Bai, Russell and Tiao (2003) derived the excess kurtosis for

the GARCH models and found that the excess kurtosis was smaller for Gaussian innovations

than the others. It appears in the results, which shows that there is a larger effect from the

GARCH process on the skewness if it has a Student’s t innovation.

The result indicates that the skewed GARCH process does not have a normal distribution for

the skewness, though the values differ from the expected values. The non-skewed GARCH

process did converge better but it seems like there is still an effect. This result is the same for

the other sample sizes, and the different sample sizes have not been included due to the size of

the data4. From the data it appears that there is an excess skewness for GARCH processes and

it differ depending on which distribution the innovation term has.

4 The other results can be given if anyone requires it.

25

References

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(accessed 2014-05-20).

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http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2265027 (accessed 2014-06-13).

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http://papers.ssrn.com/sol3/papers.cfm?abstract_id=967824

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Posedel, P. (2005). Properties and Estimation of GARCH (1,1) Model. Metodolosˇki zvezki.

2(2), page 243-257.

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Schittenkopf, C. Dorffner, G. Dockner, J. E. Time Varying Conditional skewness and

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27

Appendix

Appendix (1) Results form skewed (0.5) innovation for 1000 observations

Model number and

specification


White Noise

-0.0007761002

0.005975894

0.003566258

-0.01219913

Gaussian



3. Low Persistence -0.7868356 0.007286011 -0.2768702 0.272629

Generalized error



6. Low Persistence -0.1654759 0.006383742 -0.04840784 0.1130814

Student’s t



27. Low Persistence -2.561594 2.06873 -4.781635 38.11213


28

Appendix (2) Results form skewed (0.8) innovation for 1000 observations

Model number and

specification


White Noise

-0.0007761002

0.005975894

0.003566258

-0.01219913

Gaussian



3. Low Persistence -0.3384017 0.00645697 -0.1193105 0.06948504

Generalized error



6. Low Persistence -0.3382085 0.006496269 -0.1216246 0.1296202

Student’s t



27. Low Persistence -1.203768 1.485206 -4.238997 55.78478


29

Appendix (3) Gaussian skewed innovation (0.5)

3a Pdf of White Noise

3b Pdf of Low Persistence skewed 0.5

3c Pdf of Medium Persistence skewed 0.5

3d Pdf of High Persistence skewed 0.5

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

01

23

45


N = 10000 Bandwidth = 0.01103

De

nsity

-1.2 -1.0 -0.8 -0.6

01

23

4


N = 10000 Bandwidth = 0.01196

De

nsity

-1.4 -1.2 -1.0 -0.8 -0.6

01

23

4


N = 10000 Bandwidth = 0.01312

De

nsity

-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0

0.0

0.5

1.0

1.5

2.0

2.5


N = 10000 Bandwidth = 0.0234

De

nsity

30

Appendix (4) Gaussian skewed innovation (0.8)





-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

01

23

45


N = 10000 Bandwidth = 0.01103

De

nsity

-0.6 -0.4 -0.2 0.0

01

23

45


N = 10000 Bandwidth = 0.01146

De

nsity

-0.8 -0.6 -0.4 -0.2 0.0

01

23

4


N = 10000 Bandwidth = 0.01239

De

nsity

-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0


N = 10000 Bandwidth = 0.02105

De

nsity

31

Appendix (5) Generalized error skewed innovation (0.5)





-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

01

23

45


N = 10000 Bandwidth = 0.01103

De

nsity

-1.2 -1.0 -0.8 -0.6

01

23

4


N = 10000 Bandwidth = 0.01219

De

nsity

-1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4

01

23

4


N = 10000 Bandwidth = 0.01337

De

nsity

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5

0.0

0.5

1.0

1.5

2.0

2.5


N = 10000 Bandwidth = 0.02363

De

nsity

32

Appendix (6) Generalized error skewed innovation (0.8)





-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

01

23

45


N = 10000 Bandwidth = 0.01103

De

nsity

-0.6 -0.4 -0.2 0.0

01

23

45


N = 10000 Bandwidth = 0.01147

De

nsity

-0.8 -0.6 -0.4 -0.2 0.0

01

23

4


N = 10000 Bandwidth = 0.01229

De

nsity

-2.0 -1.5 -1.0 -0.5 0.0 0.5

0.0

0.5

1.0

1.5

2.0

2.5


N = 10000 Bandwidth = 0.0209

De

nsity

33

Appendix (7) Student’s t skewed innovation (0.5)





-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

01

23

45


N = 10000 Bandwidth = 0.01103

De

nsity

-25 -20 -15 -10 -5 0 5

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7


N = 10000 Bandwidth = 0.101

De

nsity

-20 -15 -10 -5 0 5

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7


N = 10000 Bandwidth = 0.1023

De

nsity

-10 -5 0 5

0.0

0.1

0.2

0.3

0.4

0.5

0.6


N = 10000 Bandwidth = 0.1158

De

nsity

34

Appendix (8) Student’s t skewed innovation (0.8)





-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

01

23

45


N = 10000 Bandwidth = 0.01103

De

nsity

-20 -10 0 10

0.0

0.2

0.4

0.6

0.8


N = 10000 Bandwidth = 0.07826

De

nsity

-20 -15 -10 -5 0 5 10 15

0.0

0.2

0.4

0.6

0.8


N = 10000 Bandwidth = 0.07829

De

nsity

-15 -10 -5 0 5 10

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7


N = 10000 Bandwidth = 0.09119

De

nsity

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