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International Journal of Mathematics Trends and Technology (IJMTT) - Volume 65 Issue 3 - March 2019

ISSN: 2231 – 5373 http://www.ijmttjournal.org Page 72

Misspecification of Generalized

Autoregressive Score Models: Monte Carlo

Simulations and Applications

Oluwagbenga T. Babatunde#1, OlaOluwa S. Yaya*2and Damola M. Akinlana#3

#1Department of Statistics, University of Nigeria, Nigeria

*2Department of Statistics, University of Ibadan, Nigeria

#3Department of Mathematics and Statistics, University of South Florida, USA

Abstract

The specification and misspecification of a new class of volatility model that is robust to jumps and outliers

is investigated via Monte Carlo experiment and real life examples. The class includes the Generalized

Autoregressive Score (GAS) model derived from the classical Generalized Autoregressive Conditional

Heteroscedasticity (GARCH) model. The Exponential GAS (EGAS) and Asymmetric Exponential GAS (AEGAS)

models form the variants of the GAS model. Using three different levels of volatility persistence and GARCH

probability distributions which are Normal (N), Student-t (T) and Skewed-Student-t (SKT), with estimates of

Akaike Information Criterion (AIC) and kurtosis as criteria, we obtained useful information for studying the

misspecification and tail behaviour of the newly proposed volatility model. The results of the Monte Carlo

experiment, the crude oil and gas prices showed that the best misspecified model for AEGAS-SKT and EGAS-T

is EGAS-SKT.

Keywords - Misspecification, Volatility Persistence, Monte Carlo, Generalized Autoregressive Score

I. INTRODUCTION

The Generalized Autoregressive Score (GAS) class is a variant of Generalized Autoregressive Conditional

Heteroscedasticity (GARCH) model of [1]developed for capturing jumps/outliers effects in the returns series.

Following [2], the classical GARCH model is not robust to capturing these abnormalities; hence the GAS class

variants were proposed in [2], [3].

The driving mechanism of the GAS models and its variant is the scaled score of the likelihood function, and this

makes the model class unique among other earlier proposed volatility models. It combines the ability to capture

asymmetry with occasional jumps detection. The GAS model encompasses other well-known models such as

the Generalized Autoregressive Conditional Heteroscedasticity (GARCH), the Autoregressive Conditional

Duration (ACD), the Autoregressive Conditional Intensity (ACI), and the single source of error models. In

addition, the GAS specification provides a wide range of new observation driven models. Examples include

observation driven analogues of unobserved components time series models, multivariate point process models

with time-varying parameters and pooling restrictions, new models for time-varying copula functions, and

models for time-varying higher order moments. Based on these appealing properties of this new model, we were

therefore motivated in investigating further the model class.

The literatures [4], [5] define volatility and volatility clustering in stocks, and following these definitions,

several parametric volatility models have been developed. The first is the Autoregressive Conditional

Heteroscedasticity (ARCH) model earlier proposed in[6] and the generalized version as GARCH model, has

gained many applications in empirical financial time series literature. (see[1], [7]). These literatures have

extended to studying the asymmetric behaviour and jumps in stocks and other asset prices. Different asymmetric

robust volatility models have also been applied. The jump behaviour of stocks has recently been studied, and

nonparametric approaches to detecting jumps have been applied (see[8]). Jump robust volatility model is

introduced in[2], [3]. There, the authors proposed the Generalized Autoregressive Score (GAS) models and two

variants, the Exponential GAS (EGAS) and Asymmetric Exponential GAS (AEGAS) models for predicting the

conditional volatility with occasional jumps. As a result of newness of this model, there are fewer applications

so far, though small sample properties have been investigated in[2], [9], [10], [11], there is need to study the

property of this model class using simulation approach, with emphasis on the fitness ability and returns

distributions. The fitness ability is achieved by the estimates of information criterion and the tail effects

achieved by the estimates of the kurtosis.

The aim of this paper is to investigate misspecification of GAS models and its variants using Monte Carlo

simulation approach. The work is extended to real life crude oil and natural gas prices. Literature has shown that

these financial time series data display series of jumps over the historical years (see[12], [13]). Hence, they

serve as good applicable examples in this paper.

International Journal of Mathematics Trends and Technology (IJMTT) - Volume 65 Issue 3 - March 2019

ISSN: 2231 – 5373 http://www.ijmttjournal.org Page 73

The rest of the paper is structured as follows: Section 2 reviews literature on the volatility modelling and model

misspecification. Section 3 presents the volatility models as well as misspecification testing approach. Section 4

presents the Monte Carlo experiment and results, while Section 5 renders the concluding remarks.

II. REVIEW OF LITERATURE

A framework for the construction and analysis of misspecification tests for GARCH models was developed

in[14]and new asymptotically valid and locally optimal tests of asymmetry and nonlinearity were also proposed.

It was argued that the asymmetry test of[15] and nonlinearity test of[16] are neither asymptotically valid (since

they ignore asymptotically non-negligible estimation effects) nor locally optimal (since they ignore the recursive

nature of the conditional variance structure). The framework of[14]encompasses conditional mean specification

estimated by the Ordinary Least Squares (OLS), Nonlinear Least Squares (NLS) or Quasi Maximum Likelihood

(QML) method, and that the GARCH misspecification tests can be asymptotically sensitive to unconsidered

misspecification of the conditional mean. The Monte Carlo results indicate that the new tests are very powerful

when compared with the previous tests proposed by[15], [16].

The properties of the GARCH (1,1) model and the assumptions on the parameter space under which the process

is stationary was studied in depth in[17]. In particular, the ergodicity and strong stationarity for the conditional

variance (squared volatility) of the process was proved. He showed under which conditions higher order

moments of the GARCH (1,1) process exist and concluded that GARCH processes are heavy tailed. The impact

of misspecification on the innovations in fitting GARCH (1,1) models through a Monte Carlo approach was

investigated and showed that an incorrect specification of the innovations together with the reduction of the

parameter space to the weak stationarity region, could give rise to a spurious Integrated GARCH (IGARCH)

effect[18]. They also analysed the impact of misspecification on forecasted volatilities, showing that innovations

with light tails can lead to a remarkable over-estimation of volatilities. The real size and power of the likelihood

ratio and the Lagrange Multiplier (LM) misspecification tests when periodic long memory GARCH models are

involved were analysed in[19]. The performance of these tests was studied by means of Monte Carlo

simulations with respect to the class of generalized long memory GARCH models, and by means of a Monte

Carlo analysis the real size and power of these tests were derived, evidencing their reliability apart from some

special and limited cases. The test performances were however influenced by the sample length with about a

thousand observations needed to obtain reliable conclusions.

On GAS modelling and its variants specification, the dynamic properties of Generalized Autoregressive Score

(GAS) models were characterized by identifying the regions of the parameter space that implied stationarity and

ergodicity of the corresponding nonlinear time series process[20]. They showed how these regions are affected

by the choice of parameterization and scaling, which are key factors for the class of GAS models compared to

other observation driven models.

As a follow-up by[11], the observation driven time series models used the scaled score of the likelihood

function as the mechanism for updating the parameters over time. This approach provides a unified and

consistent framework for introducing time varying parameters in a wide class of non-linear models. They

developed a framework for time varying parameters which is based on the score function of the predictive

model density at time t and concluded that by scaling the score function appropriately, standard observation

driven models such as Generalized Autoregressive Conditional Heteroscedasticity (GARCH), Autoregressive

Conditional Duration (ACD) and Autoregressive Conditional Intensity (ACI) models can be recovered.

A novel GAS model for predicting volume of shares (relative to the daily total), inspired by empirical

regularities of the observed series (intra-daily periodicity pattern, residual serial dependence) was proposed

in[21]. An application of the proposed GAS model to New York Stock Exchange (NYSE) ticketers confirmed

the suitability of the proposed model in capturing the features of intra-daily dynamics of volume shares.

A new observation-driven time-varying parameter framework to model the financial return and realized variance

jointly was proposed in[22]. The latent dynamic factor was updated by the scaled local density score as a

function of past daily return and realized variance. The proposed GAS variant adapted quickly to drastic

volatility changes by incorporating realized measures of volatility based on high frequency data and they

demonstrated the promising performance of the proposed model by applying it to a number of equity returns,

even during the 2008 financial crisis.

The consistency and asymptotic normality of the Maximum Likelihood Estimators (MLE) for a class of time

series models driven by score function of the predictive likelihood was studied in[23]. They formulated

primitive conditions, and asymptotic normality under correct specification and under misspecification of the

GAS models.

The theoretic optimality properties of the score function of the predictive likelihood as a device to update

parameters in GAS models was investigated in[24]. Their results provided a new theoretical justification for the

class of GAS models, which covers the GARCH model as a special case. Their main contribution was to show

that only parameter updates based on the score always reduce the local Kullback-Leibler divergence between the

true conditional density and the model implied conditional density and they found out that it holds irrespective

International Journal of Mathematics Trends and Technology (IJMTT) - Volume 65 Issue 3 - March 2019

ISSN: 2231 – 5373 http://www.ijmttjournal.org Page 74

of the severity of the model misspecification. They concluded that updates based on the score function

minimized the local Kullback-Leibler divergence between the true conditional data density and the model

implied conditional density.

A new class of flexible Copula models where the evolution of the dependence parameters follows a Markov-

Switching Generalized Autoregressive Score (SGASC) dynamics was developed in[25]. Maximum Likelihood

Estimation is consistently performed using the Inference Function for Margins (IFM) approach and a version of

the Expectation-Maximisation (EM) algorithm specifically tailored to this class of models. They used their

developed SGASC model to estimate the Conditional Value-at-Risk (CoVaR), which is defined as the VaR of a

given asset conditional on another asset (or portfolio) being in financial distress, and the Conditional Expected

Shortfall (CoES). Their empirical investigation shows that the proposed SGASC models are able to explain and

predict the systemic risk contribution of several European countries. Also, they found out that the SGASC

models outperformed competitors using several CoVaR back testing procedures.

III. THE GAS MODELS AND THEIR VARIANTS

The GAS model specification was derived from the classical GARCH model of [1]which is given as,

t t t t

yz

(1)

2 2 2

1 1 1 1t t t

w

(2)

where

t

y

is the returns time series decomposed as in (1),

w

,

1

and

1

are the parameters defined with the

conditions

0w

,

10

,

10

and

11

1

to ensure covariance stationarity of the model in (2).

The jump volatility model as proposed in[2], [3]is given by re-writing GARCH (1,1) as,

2 2 2 2

1 1 1 1 1t t t t

wz

and,

2 2 2 2

1 1 1 1 1 1

1

t t t t

wz

,

which is finally written as,

2 2 2

1 1 1 1 1t t t t

wu

(3)

where

1 1 1

and

2

11

tt

uz

is proportional to the score of the conditional distribution of

t

with

respect to

2

1t

. This is Beta-GARCH model because and

11

t

uv

has a Beta distribution, and the

innovations

t

u

are given as,

21

tt

uz

,

0,1

t

uN

; (4)

2

2

11

2

t

t

t

vz

uvz

,

0,1,

t

z T v

; (5)

and

*

11

2t

tt

tI

t

v z z

uvg

0,1, , ;

t

z skT v

(6)

where

*

tt

z s z m

,

* * *

0 0 ,

t t t t

I sig n z I z I z

*2

2

12t

t

tI

z

gv

,

121

2

2

vv

mv

and

22

2

11sm

Now, combining (3) with (4) gives the GAS-Normal (GAS-N) model; combining (3) with (5) gives the GAS-

Student-t (GAS-T) model and combining (3) with (6) gives the GAS-Skewed-Student-t (GAS-SKT) model.

International Journal of Mathematics Trends and Technology (IJMTT) - Volume 65 Issue 3 - March 2019

ISSN: 2231 – 5373 http://www.ijmttjournal.org Page 75

The Exponential GARCH (EGARCH) and Asymmetric Exponential GARCH (AEGARCH) types of the GAS

model were also considered in[3], each with the three distributional assumptions applied. The EGAS model is

given as,

22

1 1 1 1

lo g lo g

t t t

wu

(7)

specified without the leverage effect.

1

Now, combining (7) with (4) gives the EGAS-Normal (EGAS-N) model;

combining (7) with (5) gives the EGAS-Student-t (EGAS-T) model and combining (7) with (6) gives the

EGAS-Skewed-Student-t (EGAS-SKT) model.

Introducing the leverage effect into (7), we have the AEGAS model,

22

1 1 1 1 1 1

lo g lo g

t t t t

w u l

(8)

where

11

t t t

l sign z u

when Normal and Student-t distributions are considered, and

*

11

t t t

l sign z u

for the Skewed Student-t distribution.

2

IV. MODEL MISSPECIFICATION TESTS

Each model under the distributional assumption is evaluated using[26] Information Criterion (AIC),

11

ˆ

2 ; 2

N

Nt

A IC N L y N

; (10)

where

.

N

L

is the maximized log-likelihood function, simplified using numerical derivatives,

ˆN

is the

ML estimator of the parameter vector

based on a sample of size N, and

gives the dimension of

. The

excess kurtosis is then computed based on the formula,

4

2

23

t

t

E

k

E

(11)

where

2

11

1

t

w

E

is the estimate of unconditional variance, and the fourth moment about the mean,

2

11

4

2

2

1 1 1 1 1

13

1 1 2

z

t

z

wk

Ek

and

z

k

is the excess kurtosis from the assumed

GARCH distribution pro cess

t

z

.

V. MONTE CARLO EXPERIMENT AND RESULT DISCUSSION

Though the structural and distributional properties of classical GARCH model have been investigated

theoretically and by simulations but the properties of GAS model and its variants are yet to be established. The

Monte Carlo (MC) simulations experiment carried out in this work investigated both the fitness performance of

the models as well as the measure of tail effect of the model residuals. Four Data Generating Processes (DGPs)

considered are: GARCH(1,1) : 𝜎𝑡

2= 𝜔+ 𝛼1𝜀𝑡−1

2+𝛽1 𝜎𝑡−1

2(12)

GAS(1,1) : 𝜎𝑡

2= 𝜔+ 𝛼1𝜇𝑡 −1𝜎𝑡−1

2+ (𝛼1+𝛽1)𝜎𝑡−1

2 (13)

EGAS(1,1): 𝑙𝑜𝑔𝜎𝑡

2= 𝜔+ 𝛼1𝜇𝑡−1𝜎𝑡−1

2+ (𝛼1+𝛽1)𝑙𝑜𝑔𝜎𝑡−1

2(14)

AEGAS(1,1): 𝑙𝑜𝑔𝜎𝑡

2= 𝜔+ 𝛼1𝜇𝑡−1𝜎𝑡−1

2+𝛾1𝜏𝑡−1 + (𝛼1+𝛽1)𝑙𝑜𝑔𝜎𝑡−1

2(15)

1

The EGAS specification has no asymmetric parameter, unlike the classical EGARCH model of Nelson [27].

2

Note,

2

2

1

1

t

El

in the three symmetric distributions, while

0

t

El

for the Skewed Student-t

distribution.

International Journal of Mathematics Trends and Technology (IJMTT) - Volume 65 Issue 3 - March 2019

ISSN: 2231 – 5373 http://www.ijmttjournal.org Page 76

where 𝜏𝑡−1 =𝑠𝑖𝑔𝑛 (−𝑧𝑡 )(𝜇𝑡 + 1). For each of the DGP in (12)-(15), a sample of 1000 time series was

generated after making control for the initialization error, and each generated following Normal, Student-t and

Skewed Student-t distributions. The sum 𝛼+𝛽 is referred to as the persistence of the conditional variance

process. For financial return series, estimates of 𝛼 and 𝛽 are often in the ranges [0.02, 0.25] and [0.75,0.98],

respectively with 𝛼 often in the lower part of the interval and 𝛽 in the upper part of the interval, such that the

persistence is close but rarely exceeding 128. We can then make classification into low, medium and high

persistence. The parameters of the models were varied and classified in14as low, medium and high volatility

persistence realizations as given below:

Low Persistence:

1 1 1 1

, , , , 0.04,0.05,0 .6 5,0.7,0.01

Medium Persistence:

1 1 1 1

, , , , 0.04,0.1,0.8,0 .9,0.01

High Persistence:

1 1 1 1

, , , , 0.04,0.09,0.9,0.99,0.01

where the values of the intercept

and

asymmetric parameter

1

remained constant throughout and these do not affect volatility persistence. The

value of

1 1 1

for the case of GAS(1,1), EGAS(1,1) and AEGAS(1,1) models.

The estimates of Akaike Information Criteria (AIC) and Excess Kurtosis from the Monte Carlo Experiments are

given in table 1-3. The AIC of the DGP is denoted with single asterisk whereas the AIC of the best performed

misspecified model is denoted with double asterisks. The results presented in table 1 showed that when the DGP

is GAS-N, at low persistence, the misspecified model is EGAS-N while at both medium and high persistence;

the misspecified model is GAS-SKT. When the DGP is both GAS-T and GAS-SKT at all persistence levels, the

misspecified model is EGAS-SKT. Table 2 showed that when the DGP is EGAS-N at both low and medium

persistence, the misspecified model is GAS-N while at high persistence; the misspecified model is EGAS-SKT.

When the DGP is EGAS-T at all persistence levels, the misspecified model is EGAS-SKT whereas when the

DGP is EGAS-SKT at all persistence levels, the misspecified model is GAS-T. In table 3, the results showed

that when the DGP is AEGAS-N at low persistence, the misspecified model is EGAS-N while at both medium

and high persistence; the misspecified model is AEGAS-SKT. When the DGP is both AEGAS-T and AEGAS-

SKT at all persistence levels, the misspecified model is EGAS-SKT.

The results of this paper also showed that when the probability distribution of the residuals of the DGPs is

normal, the probability distribution of the misspecified model will be normal since all the excess kurtosis

observed under the three DGPs, at low, medium and high persistence were either negatively low or positively

low and close to zero whereas when the probability distribution of the residuals of the DGPs is non-normal

(skewed), the probability distribution of the residuals will be non-normal(Skewed) since the excess kurtosis

observed under the three DGPs at low, medium and high persistence were positive and greater than zero.

VI. TABLE1: ESTIMATES OF AIC AND EXCESS KURTOSIS WHEN THE DGP IS GAS

Persistence

Assumed

Distribution

GAS (1,1)

EGAS (1,1)

AEGAS (1,1)

AIC

Ex. Kurt

AIC

Ex. Kurt

AIC

Ex. Kurt

When the DGP is GAS-N

Low

Normal

0.9188*

-0.1037

0.9188**

-0.1050

0.9196

-0.1223

Student-t

0.9209

-0.1037

0.9209

-0.1050

0.9217

-0.1224

Skewed-t

0.9196

-0.1039

0.91967

-0.1051

0.9204

0.1225

Medium

Normal

2.0429*

-0.1130

2.0442

-0.0986

2.0445

-0.1222

Student-t

2.0450

-0.1127

2.0463

-0.0985

2.0466

-0.1220

Skewed-t

2.0435**

-0.1117

2.0447

-0.0941

2.0448

-0.1193

High

Normal

4.6419*

-0.0823

4.6422

-0.0842

4.6436

-0.0965

Student-t

4.6440

-0.0822

4.6443

-0.0842

4.6457

-0.0965

Skewed-t

4.6419**

-0.0846

4.6427

-0.0840

4.6440

-0.0982

When the DGP is GAS-T

Low

Normal

0.8556

1.8835

0.8770

2.8232

0.8579

2.0572

Student-t

0.8162*

2.8569

0.8162

2.8616

0.8183

2.8441

Skewed-t

0.8138**

2.8423

0.8138**

2.8451

0.8158

2.8502

Medium

Normal

1.9687

2.6440

1.9712

2.5112

1.9713

2.6193

Student-t

1.9059*

2.9442

1.9051

3.0642

1.9070

3.0378

Skewed-t

1.9034

2.9299

1.9027**

3.0443

1.9046

3.0223

High

Normal

4.2293

2.8318

4.2350

2.7036

4.2359

2.7969

Student-t

4.1567*

2.9177

4.1563

2.9413

4.1583

2.9316

International Journal of Mathematics Trends and Technology (IJMTT) - Volume 65 Issue 3 - March 2019

ISSN: 2231 – 5373 http://www.ijmttjournal.org Page 77

Skewed-t

4.1544

2.8900

4.1540**

2.9077

4.1560

2.9048

When the DGP is GAS-SKT

Low

Normal

0.8549

2.1361

0.8757

2.8379

0.8563

2.0975

Student-t

0.8152

2.8729

0.8152

2.8779

0.8172

2.8602

Skewed-t

0.8136*

2.8587

0.8136**

2.8619

0.8156

2.8648

Medium

Normal

1.9671

2.6423

1.9695

2.3060

1.9697

2.6130

Student-t

1.9045

2.9597

1.9038

3.0833

1.9055

3.0518

Skewed-t

1.9029*

2.9296

1.9022**

3.0684

1.9040

3.0406

High

Normal

4.2243

2.8137

4.2299

2.6793

4.2309

2.7637

Student-t

4.1522

2.9233

4.1518

2.9502

4.1538

2.9349

Skewed-t

4.1508*

2.9017

4.1503**

2.9230

4.1523

2.9129

VII. TABLE 2: ESTIMATES OF AIC AND EXCESS KURTOSIS WHEN THE DGP IS EGAS

Persistence

Assumed

Distribution

GAS (1,1)

EGAS (1,1)

AEGAS (1,1)

AIC

Ex. Kurt

AIC

Ex. Kurt

AIC

Ex. Kurt

When the DGP is EGAS-N

Low

Normal

3.0716**

-0.1031

3.0715*

-0.1060

3.0724

-0.1224

Student-t

3.0737

-0.1030

3.0736

-0.1060

3.0745

-0.1224

Skewed-t

3.0723

-0.1031

3.0724

-0.1060

3.0732

-0.1235

Medium

Normal

3.4071**

-0.1096

3.4073*

-0.1039

3.4079

-0.1225

Student-t

3.4092

-0.1092

3.4094

-0.1036

3.4101

-0.1222

Skewed-t

3.4079

-0.1081

3.4079

-0.0992

3.4085

-0.1193

High

Normal

7.5633

-0.0312

7.5604*

-0.0656

10.6596

Student-t

7.5653

-0.0309

7.5626

-0.0650

7.9513

Skewed-t

7.5632

-0.0326

7.5609**

-0.0631

7.5621

-0.0815

When the DGP is EGAS-T

Low

Normal

3.0064

2.0375

3.0290

2.8148

3.0307

2.8140

Student-t

2.9683

2.8509

2.9683*

2.8567

2.9703

2.8565

Skewed-t

2.9660

2.8720

2.9659**

2.8397

2.9679

2.8396

Medium

Normal

3.3255

2.5689

3.3264

2.5350

3.3284

2.5430

Student-t

3.2632

2.8334

3.2620*

2.9804

3.2639

2.9615

Skewed-t

3.2606

2.8154

3.2595**

2.9545

3.2615

2.9402

High

Normal

7.1818

2.9987

7.1894

2.8448

7.1901

2.9567

Student-t

7.1077

3.0177

7.1071*

3.0287

7.1091

3.0260

Skewed-t

7.1053

2.9927

7.1046**

2.9956

7.1066

3.0010

When the DGP is EGAS-SKT

Low

Normal

3.0065

2.1177

3.0277

2.8290

3.0099

2.0210

Student-t

2.9673

2.8666

2.9673

2.8728

2.9692

2.8731

Skewed-t

2.9659**

2.8878

2.9657*

2.8562

2.9677

2.8563

Medium

Normal

3.3237

2.5641

3.3246

2.5281

3.3266

2.5330

Student-t

3.2618

2.8451

3.2606

2.9962

3.2624

2.9722

Skewed-t

3.2601**

2.8315

3.2590*

2.9758

3.2609

2.9558

High

Normal

7.1765

2.9816

7.1840

2.8182

7.1849

2.9215

Student-t

7.1029

3.0230

7.1023

3.0373

7.1043

3.0278

Skewed-t

7.1014**

3.0043

7.1007*

3.0109

7.1027

3.0076

VIII. TABLE 3: ESTIMATES OF AIC AND EXCESS KURTOSIS WHEN THE DGP IS

AEGAS

Persistence

Assumed

Distribution

GAS (1,1)

EGAS (1,1)

AEGAS (1,1)

AIC

Ex. Kurt

AIC

Ex. Kurt

AIC

Ex. Kurt

When the DGP is AEGAS-N

Low

Normal

3.0742

-0.0887

3.0741**

-0.0923

3.0735*

-0.1219

Student-t

3.0763

-0.0885

3.0762

-0.0922

3.0755

-0.1219

Skewed-t

3.0749

-0.0883

3.0750

-0.0917

3.0742

-0.1227

Medium

Normal

3.4122

-0.0908

3.4123

-0.0858

3.4111*

-0.1207

Student-t

3.4143

-0.0902

3.4144

-0.0854

3.4131

-0.1206

Skewed-t

3.4131

-0.0887

3.4131

-0.0802

3.4116**

-0.1173

High

Normal

7.5943

-0.0026

7.5913

-0.0403

7.5905*

-0.0796

Student-t

7.5964

-0.0020

7.5934

-0.0398

7.5926

-0.0792

Skewed-t

7.5944

-0.0027

7.5921

-0.0359

7.5907**

-0.0783

When the DGP is AEGAS-T

Low

Normal

3.0084

1.9840

3.0313

2.7744

3.0328

2.7737

Student-t

2.9712

2.8144

2.9712

2.8210

2.9731*

2.8206

International Journal of Mathematics Trends and Technology (IJMTT) - Volume 65 Issue 3 - March 2019

ISSN: 2231 – 5373 http://www.ijmttjournal.org Page 78

Skewed-t

2.9690

2.8388

2.9688**

2.8038

2.9707

2.8046

Medium

Normal

3.3335

2.5298

3.3344

2.4956

3.3360

2.5245

Student-t

3.2718

2.7969

3.2706

2.9464

3.2726*

2.9556

Skewed-t

3.2693

2.7784

3.2682**

2.9205

3.2701

2.9350

High

Normal

7.2642

2.9172

7.2728

2.9724

7.2712

2.8943

Student-t

7.1909

2.9826

7.1904

2.9847

7.1920*

3.0194

Skewed-t

7.1888

2.9548

7.1881**

2.9488

7.1896

2.9926

When the DGP is AEGAS-SKT

Low

Normal

3.0051

1.8743

3.0298

2.7902

3.0313

2.7880

Student-t

2.9699

2.8313

2.9698

2.8382

2.9718

2.8363

Skewed-t

2.9685

2.8564

2.9683**

2.8216

2.9703*

2.8206

Medium

Normal

3.3310

2.5266

3.3320

2.4903

3.3336

2.5154

Student-t

3.2698

2.8101

3.2685

2.9639

3.2705

2.9673

Skewed-t

3.2681

2.7963

3.2670**

2.9437

3.2689*

2.9510

High

Normal

7.2529

2.8990

7.2614

2.6980

7.2599

2.8580

Student-t

7.1804

2.9903

7.1799

2.9965

7.1816

3.0210

Skewed-t

7.1791

2.9700

7.1784**

2.9685

7.1800*

2.9990

* DGP ** Best performed misspecified model

IX. RESULTS OF CRUDE OIL AND GAS PRICES

We apply both daily crude oil and Gas prices to test the effect of misspecification of volatility models. The

crude oil prices are the European Brent prices (US dollars/barrel) while the gas prices are the Henry Hub

Natural gas spot prices (US Dollars per Million Btu), both obtained from the website of US Energy Information

Administrations (http://www.eia.gov/). The oil prices span between 20 May 1987 and 29 September 2014 while

the natural gas series span between 07 January 1997 and 09 March 2015.

The plot of the crude oil prices is given in Figure 5.1. We observe stability in the prices of crude oil from 1987

to 1999 with a major spike in 1990. We observe a gradual increase in the prices of crude oil from 2000 to 2008

with the prices of crude oil getting to its peak in 2008. We also observe a fall in 2008 and a gradual increase in

the prices of crude oil from 2008 to 2011 and the prices were stable from 2011 to 2015.

X. FIGURE 5.1: TIME PLOT OF CRUDE OIL PRICES (US DOLLAR/BARREL)

The plot of the natural gas prices is given in Figure 5.2. We observe major spike in the prices of natural gas

in 2001, 2003, 2005, 2008, 2010 and 2014. We observe fall in prices of natural gas after each spike and stability

of prices of natural gas before the next spike.

0.9

1.1

1.3

1.5

1.7

1.9

2.1

1

214

427

640

853

1066

1279

1492

1705

1918

2131

2344

2557

2770

2983

3196

3409

3622

3835

4048

4261

4474

4687

4900

5113

5326

5539

5752

5965

6178

6391

6604

6817

20 May 1987 29 September 2014

International Journal of Mathematics Trends and Technology (IJMTT) - Volume 65 Issue 3 - March 2019

ISSN: 2231 – 5373 http://www.ijmttjournal.org Page 79

XI. FIGURE 5.2: TIME PLOT OF GAS PRICES (US DOLLAR/BTU)

The estimates of Akaike Information Criteria (AIC) and Excess Kurtosis from the model estimation

using crude oil and natural gas prices are presented in table 4. The results showed that the specified model for

the crude oil prices is AEGAS-SKT while the misspecified model is EGAS-SKT. The specified model for the

natural gas prices is EGAS-T while the misspecified model is EGAS-SKT.

We observed positive estimates of excess kurtosis throughout Table 4 which are all greater than zero. This

implies that the estimated residuals for the specified models deviate from normal distribution and they have

fatter tails than the normal distribution.

XII. TABLE 4: MISSPECIFICATION TESTS FOR MODELS FOR CRUDE OIL AND NATURAL GAS

PRICES

Estimated Model

Distribution

Assumed

Crude Oil Prices

Natural Gas Prices

AIC

Ex. Kurt

AIC

Ex. Kurt

GAS

Normal

-6.6929

1.9429

-5.4666

8.7822

T

-6.7451

2.4417

-5.5458

8.6593

Skewed-t

-6.7463

2.4535

-5.5455

8.6125

EGAS

Normal

-6.6916

1.9525

-5.4147

10.506

T

-6.7465

2.3753

-5.5516*

8.6237

Skewed-t

-6.7476**

2.3888

-5.5513**

8.5867

AEGAS

Normal

-6.6918

1.9381

-5.4229

11.931

T

-6.7471

2.4110

-5.5512

8.3575

Skewed-t

-6.7482*

2.4264

-5.5509

8.3358

* Specified model ** Best performed misspecified model

XIII. CONCLUSION

This paper has investigated the misspecification of GAS models and its variants using Monte Carlo

simulation approach. The work was extended to real life situation by using the daily prices of crude oil and

natural gas prices. The estimation involved investigating the misspecification of GAS models and their variants

assuming normal, student-t and Skewed Student-t probability distributions for the GARCH variants. Model

selection performance was then investigated using information criteria and tail coefficient (kurtosis). We

therefore present the results for studying the misspecification of the GAS variantsand residual tail behaviour as

summarized in table 5 and table 6 respectively.

Xiv. Table 5: summary of fitness performance of the misspecified models

DGP

Best Performed Misspecified Model

Low Persistence

Medium Persistence

High Persistence

GAS-N

EGAS-N

GAS-SKT

GAS-SKT

GAS-T

GAS-SKT & EGAS-SKT

EGAS-SKT

EGAS-SKT

GAS-SKT

EGAS-SKT

EGAS-SKT

EGAS-SKT

EGAS-N

GAS-N

GAS-N

EGAS-SKT

EGAS-T

EGAS-SKT

EGAS-SKT

EGAS-SKT

EGAS-SKT

GAS-SKT

GAS-SKT

GAS-SKT

0

2

4

6

8

10

12

14

16

18

20

1

139

277

415

553

691

829

967

1105

1243

1381

1519

1657

1795

1933

2071

2209

2347

2485

2623

2761

2899

3037

3175

3313

3451

3589

3727

3865

4003

4141

4279

4417

07 January 1997 09 March 2015

International Journal of Mathematics Trends and Technology (IJMTT) - Volume 65 Issue 3 - March 2019

ISSN: 2231 – 5373 http://www.ijmttjournal.org Page 80

AEGAS-N

EGAS-N

AEGAS-SKT

AEGAS-SKT

AEGAS-T

EGAS-SKT

EGAS-SKT

EGAS-SKT

AEGAS-SKT

EGAS-SKT

EGAS-SKT

EGAS-SKT

XV. TABLE 6: Summary Of The Probability Distribution Of The Residuals (Tail Behavior)

DGP

Tail Behavior

GAS-N

Normal

GAS-T

Skewed

GAS-SKT

Skewed

EGAS-N

Normal

EGAS-T

Skewed

EGAS-SKT

Skewed

AEGAS-N

Normal

AEGAS-T

Skewed

AEGAS-SKT

Skewed

The crude oil and gas prices were used to confirm the results of the Monte Carlo experiment. The specified

models for the crude oil and gas prices are AEGAS-SKT and EGAS T respectively while the misspecified

model for both the crude oil and gas prices is EGAS-SKT. This result agrees with the outcome of the Monte

Carlo experiment as noted in table 5, that is, the misspecification model for both AEGAS-SKT and EGAS-T is

EGAS-SKT.

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