Does It Exist Any Excess Skewness in GARCH Models? – a Comparison of Theoretical and White Noise Values

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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. 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.

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.

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.

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).

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.

∑ ̅

(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). 6

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)

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).

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.

{ } { } { } { } (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. 9

Table (3.1) Non-skewed models

Models Number of observations Innovation ( ) (length of the time series)

Model 1: 0.01 0.09 0.9 High Persistence Gaussian innovation

Model 2: 0.05 0.05 0.9 Medium Persistence Gaussian innovation

Model 3: 0.2 0.05 0.75 Low Persistence Gaussian innovation

Model 4: 0.01 0.09 0.9 High Persistence Generalized error innovation

Model 5: 0.05 0.05 0.9 Medium Persistence Generalized error innovation

Model 6: 0.2 0.05 0.75 Low Persistence Generalized error innovation

Model 7: 0.01 0.09 0.9 High Persistence Student’s t innovation

Model 8: 0.05 0.05 0.9 Medium Persistence Student’s t innovation

Model 9: 0.2 0.05 0. 75 Low Persistence Student’s t innovation

Table (3.2) Skewed models for Gaussian distributions

Model Number of observations Innovation ( ) Skewed (length of the time series)

Model 10: 0.01 0.09 0.9 0.5 High Persistence Gaussian innovation

Model 11: 0.01 0.09 0.9 0.8 High Persistence Gaussian innovation

Model 12: 0.01 0.09 0.9 0.9 High Persistence Gaussian innovation

Model 13: 0.05 0.05 0.9 0.5 Medium Persistence Gaussian innovation

Model 14: 0.05 0.05 0.9 0.8 Medium Persistence Gaussian innovation

Model 15: 0.05 0.05 0.9 0.9 Medium Persistence Gaussian innovation

Model 16: 0.2 0.05 0.75 0.5 Low Persistence Gaussian innovation

Model 17: 0.2 0.05 0.75 0.8 Low Persistence Gaussian innovation

Model 18: 0.2 0.05 0.75 0.9 Low Persistence Gaussian innovation

Table (3.3) Skewed models for Generalized Error distributions

Model Number of observations Innovation ( ) Skewed (length of the time series)

Model 19: 0.01 0.09 0.9 0.5 High Persistence Generalized error innovation

Model 20: 0.01 0.09 0.9 0.8 High Persistence Generalized error innovation

Model 21: 0.01 0.09 0.9 0.9 High Persistence Generalized error innovation

Model 22: 0.05 0.05 0.9 0.5 Medium Persistence Generalized error innovation

Model 23: 0.05 0.05 0.9 0.8 Medium Persistence Generalized error innovation

Model 24: 0.05 0.05 0.9 0.9 Medium Persistence Generalized error innovation

Model 25: 0.2 0.05 0.75 0.5 Low Persistence Generalized error innovation

Model 26: 0.2 0.05 0.75 0.8 Low Persistence Generalized error innovation

Model 27: 0.2 0.05 0.75 0.9 Low Persistence Generalized error innovation

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

Model Number of observations Innovation ( ) Skewed (length of the time series)

Model 28: 0.01 0.09 0.9 0.5 High Persistence Student’s t innovation

Model 29: 0.01 0.09 0.9 0.8 High Persistence Student’s t innovation

Model 30: 0.01 0.09 0.9 0.9 High Persistence Student’s t innovation

Model 31: 0.05 0.05 0.9 0.5 Medium Persistence Student’s t innovation

Model 32: 0.05 0.05 0.9 0.8 Medium Persistence Student’s t innovation

Model 33: 0.05 0.05 0.9 0.9 Medium Persistence Student’s t innovation

Model 34: 0.2 0.05 0.75 0.5 Low Persistence Student’s t innovation

Model 35: 0.2 0.05 0.75 0.8 Low Persistence Student’s t innovation

Model 36: 0.2 0.05 0.75 0.9 Low Persistence Student’s t innovation

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.

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 Mean Variance Skewness Kurtosis specification

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 4. High Persistence -0.001254693 0.03208321 -0.2261881 5.430798

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 - -

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.

Figure (4.1) Gaussian non-skewed innovation

4.1a Pdf of White Noise 4.1b Pdf of Low Persistence

density.default(x = white.noise) density.default(x = LPnormal1000)

0 0

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 N = 10000 Bandwidth = 0.01103 N = 10000 Bandwidth = 0.01106 4.1c Pdf of Medium Persistence 4.1d Pdf of High Persistence

density.default(x = MPnormal1000) density.default(x = HPnormal1000)

0 0

-0.4 -0.2 0.0 0.2 0.4 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

N = 10000 Bandwidth = 0.02085 N = 10000 Bandwidth = 0.01202

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.

Figure (4.2) Generalized error non-skewed innovation

4.2a Pdf of White Noise 4.2b Pdf of Low Persistence

density.default(x = white.noise) density.default(x = LPGED1000)

y y

t t

i i

s s

n n

e e

D D

0 0

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -0.2 0.0 0.2 0.4 N = 10000 Bandwidth = 0.01103 N = 10000 Bandwidth = 0.01136 4.2c Pdf of Medium Persistence 4.2d Pdf of High Persistence

density.default(x = MPGED1000) density.default(x = HPGED1000)

. 0

-0.4 -0.2 0.0 0.2 0.4 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 N = 10000 Bandwidth = 0.01222 N = 10000 Bandwidth = 0.02046

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).

Figure (4.3) Student t non-skewed innovation

4.3a Pdf of White Noise 4.3b Pdf of Low Persistence

density.default(x = white.noise) density.default(x = LPt1000)

. 0 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -10 0 10 20 N = 10000 Bandwidth = 0.01103 N = 10000 Bandwidth = 0.07302

4.3c Pdf of Medium Persistence 4.3d Pdf of High Persistence

density.default(x = MPt1000) density.default(x = HPt1000)

0 0

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

N = 10000 Bandwidth = 0.07338 N = 10000 Bandwidth = 0.08659

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).

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 Mean Variance Skewness Kurtosis specification

White Noise -0.0007761002 0.005975894 0.003566258 -0.01219913

Gaussian 12. High Persistence -0.1829159 0.03265746 -0.4769254 6.073339

15. Medium Persistence -0.1668421 0.007358689 -0.1041338 0.1703169

18. Low Persistence -0.1655104 0.006333658 -0.05672268 0.07896617

Generalized error 21. High Persistence -0.182112 0.03275647 -0.8332139 6.377269

24. Medium Persistence -0.1664451 0.007359994 -0.1181298 0.3120092

27. Low Persistence -0.1654759 0.006383742 -0.04840784 0.1130814

Student’s t 30. High Persistence -0.5678528 1.009676 -0.9205145 15.50167

33. Medium Persistence -0.5894995 1.057813 -1.731922 38.49673

36. Low Persistence -0.6116942 1.380634 -3.150867 58.58656 Expected Values 0 0.0059641256 - -

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.

Figure (4.4) Gaussian skewed innovation (0.9)

4.4a Pdf of White Noise 4.4b Pdf of Low Persistence skewed 0.9

density.default(x = white.noise) density.default(x = LPskew0.9normal1000)

y y

t t

i i

s s

n n

e e

D D

0 0

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 N = 10000 Bandwidth = 0.01103 N = 10000 Bandwidth = 0.01135 4.4c Pdf of Medium Persistence skewed 0.9 4.4d Pdf of High Persistence skewed 0.9

density.default(x = MPskew0.9normal1000) density.default(x = HPskew0.9normal1000)

0 0

-0.4 -0.2 0.0 0.2 -1.5 -1.0 -0.5 0.0 0.5 1.0

N = 10000 Bandwidth = 0.02044 N = 10000 Bandwidth = 0.01223

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.

Figure (4.5) Generalized error innovation (0.9)

4.5a Pdf of White Noise 4.5b Pdf of Low Persistence skewed 0.9

density.default(x = white.noise) density.default(x = LPskew0.9GED1000)

0 0

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

N = 10000 Bandwidth = 0.01103 N = 10000 Bandwidth = 0.01128

4.5c Pdf of Medium Persistence skewed 0.9 4.5d Pdf of High Persistence skewed 0.9

density.default(x = MPskew0.9GED1000) density.default(x = HPskew0.9GED1000)

y y

t t

i i

s s

n n

e e

D D

0 0

-0.6 -0.4 -0.2 0.0 0.2 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 N = 10000 Bandwidth = 0.01199 N = 10000 Bandwidth = 0.02054

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.

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

4.6a Pdf of White Noise 4.6b Pdf of Low Persistence skewed 0.9

density.default(x = white.noise) density.default(x = LPskew0.9t1000)

0 0

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -20 -10 0 10 20

N = 10000 Bandwidth = 0.01103 N = 10000 Bandwidth = 0.07269

4.6c Pdf of Medium Persistence skewed 0.9 4.6d Pdf of High Persistence skewed 0.9

density.default(x = MPskew0.9t1000) density.default(x = HPskew0.9t1000)

. 0 -10 -5 0 5 10 -15 -10 -5 0 5 10 15 N = 10000 Bandwidth = 0.08643 N = 10000 Bandwidth = 0.07247

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.

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. 24

References

Anderson, R. L. (1942). Distribution of the serial correlation coefficient. Annals of Mathematical Statistics. 13, page 1-13.

Bai, X. Russell, R. J. Tiao, C. G. (2003). Kurtosis of GARCH and stochastic volatility models with non-normal innovations. Journal of Econometrics. 114, page 349-360.

Bartlett, M. S. (1946). On the theoretical specifications of sampling properties of autocorrelated time series. Journal of the Royal Statistical Society. 8, page 27.

Bollerslev, T. (1986). Generalized Autoregressive Conditional Heteroskedasticity. Journal of Econometrics. 31, page 307-327.

Bollerslev, T. (2009). Glossary to ARCH (GARCH). Volatility and Time Series Econometrics: Essays in Honour of Robert F. Engle. Oxford. Oxford University press.

Breuer, T. Jandacka, M. (2010). Temporal Aggregation of GARCH Models: Conditional Kurtosis and Optimal Frequency. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=967824 (accessed 2014-05-20).

Casella, G. Berger L. R. (2002). Statistical inference, 2nd. ed. California. Duxbury advanced series.

Danielsson, J. (2011). Financial Risk Forecasting. West Sussex. John Wiley & sins Ltd.

Engel, F. R. (1982). Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation. Econometrica. 50(4), page 987-1007.

Galeano, P. Tsay, S. R. (2010). Shifts in Individual Parameters of GARCH Model. Journal of Financial Econometrics. 8(1), page 122-153.

Giller, L. G. (2005). A Generalized Error Distribution. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2265027 (accessed 2014-06-13).

Jondeau, E. Rockinger, M. (2003). Conditional volatility, skewness, and kurtosis: existence, persistence, and comovements. Journal of Economic Dynamic & Control. 27, page 169-1737.

Mantalos, P. (2010). Robust Critical Values for the Jarque-Bera Test for Normality. JIBS Working Papers. No. 8

Patton, J. A. (2004). On the Out-of-Sample Importance of Skewness and Asymmetric Dependence for Asset Allocation. Journal of Financial Econometrics. 2(1), page 130-168.

Pearson, E. S. (1931). Note on test for normality. Biometrika. 22(3/4), page 423-424.

Posedel, P. (2005). Properties and Estimation of GARCH (1,1) Model. Metodolosˇki zvezki. 2(2), page 243-257.

Quenouille, M. H. (1949). The joint distribution of serial correlation coefficients Annuals in Mathematical Statistics. 20, page 561- 571.

Schittenkopf, C. Dorffner, G. Dockner, J. E. Time Varying Conditional skewness and Kurtosis. http://www.wu.ac.at/fcs/downloads/time_varying_conditional_skewness_kurtosis.pdf (accessed 2014-05-28).

Tsay, R. (2005). Analysis of Financial Time Series. 2nd ed. New Jersey. John Wiley & Sons.

Appendix

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

Model number and Mean Variance Skewness Kurtosis specification

White Noise -0.0007761002 0.005975894 0.003566258 -0.01219913

Gaussian 1. High Persistence -0.8738785 0.04810066 -2.070663 9.285462

2. Medium Persistence -0.794225 0.009236015 -0.6022143 1.5013

3. Low Persistence -0.7868356 0.007286011 -0.2768702 0.272629

Generalized error 4. High Persistence -0.8702083 0.04736887 -2.034823 7.546166

5. Medium Persistence -0.7927382 0.009430885 -0.6593878 2.164074

6. Low Persistence -0.1654759 0.006383742 -0.04840784 0.1130814

Student’s t 21. High Persistence -2.524327 1.293032 -2.387546 11.00518

24. Medium Persistence -2.520729 1.558076 -3.608982 22.91821

27. Low Persistence -2.561594 2.06873 -4.781635 38.11213 Expected Values 0 0.0059641256 - -

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

Model number and Mean Variance Skewness Kurtosis specification

White Noise -0.0007761002 0.005975894 0.003566258 -0.01219913

Gaussian 1. High Persistence -0.371649 0.03435836 -1.091747 6.85677

2. Medium Persistence -0.3408329 0.007586963 -0.2219634 0.2604693

3. Low Persistence -0.3384017 0.00645697 -0.1193105 0.06948504

Generalized error 4. High Persistence -0.3716773 0.0354837 -1.369863 7.309607

5. Medium Persistence -0.3405644 0.007654437 -0.2721281 0.597686

6. Low Persistence -0.3382085 0.006496269 -0.1216246 0.1296202

Student’s t 21. High Persistence -1.131532 1.016336 -1.52028 15.04741

24. Medium Persistence -1.170541 1.124966 -2.717602 36.10767

27. Low Persistence -1.203768 1.485206 -4.238997 55.78478 Expected Values 0 0.0059641256 - -

Appendix (3) Gaussian skewed innovation (0.5)

3a Pdf of White Noise 3b Pdf of Low Persistence skewed 0.5

density.default(x = white.noise) density.default(x = LPskew0.5normal1000)

0 0

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -1.2 -1.0 -0.8 -0.6

N = 10000 Bandwidth = 0.01103 N = 10000 Bandwidth = 0.01196

3c Pdf of Medium Persistence skewed 0.5 3d Pdf of High Persistence skewed 0.5

density.default(x = MPskew0.5normal1000) density.default(x = HPskew0.5normal1000)

. 0

-1.4 -1.2 -1.0 -0.8 -0.6 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 N = 10000 Bandwidth = 0.01312 N = 10000 Bandwidth = 0.0234

Appendix (4) Gaussian skewed innovation (0.8)

4a Pdf of White Noise 4b Pdf of Low Persistence skewed 0.8

density.default(x = white.noise) density.default(x = LPskew0.8normal1000)

0 0

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -0.6 -0.4 -0.2 0.0

N = 10000 Bandwidth = 0.01103 N = 10000 Bandwidth = 0.01146

4c Pdf of Medium Persistence skewed 0.8 4d Pdf of High Persistence skewed 0.8

density.default(x = MPskew0.8normal1000) density.default(x = HPskew0.8normal1000)

i 5

. 0

-0.8 -0.6 -0.4 -0.2 0.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 N = 10000 Bandwidth = 0.01239 N = 10000 Bandwidth = 0.02105

Appendix (5) Generalized error skewed innovation (0.5)

5a Pdf of White Noise 5b Pdf of Low Persistence skewed 0.5

density.default(x = white.noise) density.default(x = LPskew0.5GED1000)

0 0

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -1.2 -1.0 -0.8 -0.6

N = 10000 Bandwidth = 0.01103 N = 10000 Bandwidth = 0.01219

5c Pdf of Medium Persistence skewed 0.5 5d Pdf of High Persistence skewed 0.5

density.default(x = MPskew0.5GED1000) density.default(x = HPskew0.5GED1000)

. 0

-1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 N = 10000 Bandwidth = 0.01337 N = 10000 Bandwidth = 0.02363

Appendix (6) Generalized error skewed innovation (0.8)

6a Pdf of White Noise 6b Pdf of Low Persistence skewed 0.8

density.default(x = white.noise) density.default(x = LPskew0.8GED1000)

0 0

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -0.6 -0.4 -0.2 0.0

N = 10000 Bandwidth = 0.01103 N = 10000 Bandwidth = 0.01147

6c Pdf of Medium Persistence skewed 0.8 6d Pdf of High Persistence skewed 0.8

density.default(x = MPskew0.8GED1000) density.default(x = HPskew0.8GED1000)

y y

t t

i i

s s

n n

e e

D D

0 0

-0.8 -0.6 -0.4 -0.2 0.0 -2.0 -1.5 -1.0 -0.5 0.0 0.5 N = 10000 Bandwidth = 0.01229 N = 10000 Bandwidth = 0.0209

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

7a Pdf of White Noise 7b Pdf of Low Persistence skewed 0.5

density.default(x = white.noise) density.default(x = LPskew0.5t1000)

0 0 -25 -20 -15 -10 -5 0 5 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 N = 10000 Bandwidth = 0.101 N = 10000 Bandwidth = 0.01103

7c Pdf of Medium Persistence skewed 0.5 7d Pdf of High Persistence skewed 0.5

density.default(x = MPskew0.5t1000) density.default(x = HPskew0.5t1000)

y y

t t

i i

s s

n n

e e

D D

0 0

. .

0 0

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

N = 10000 Bandwidth = 0.1023 N = 10000 Bandwidth = 0.1158

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

8a Pdf of White Noise 8b Pdf of Low Persistence skewed 0.8

density.default(x = white.noise) density.default(x = LPskew0.8t1000)

0 0

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -20 -10 0 10

N = 10000 Bandwidth = 0.01103 N = 10000 Bandwidth = 0.07826

8c Pdf of Medium Persistence skewed 0.8 8d Pdf of High Persistence skewed 0.8

density.default(x = MPskew0.8t1000) density.default(x = HPskew0.8t1000)

0 0

-20 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 N = 10000 Bandwidth = 0.07829 N = 10000 Bandwidth = 0.09119