Misspecification of Generalized Autoregressive Score Models: Monte Carlo Simulations and Applications

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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.
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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,
and,
 
 
2 2 2 2
1 1 1 1 1 1
1
t t t t
wz
 
 
 
,
which is finally written as,
(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+ 𝛼1𝜇𝑡1𝜎𝑡−1
2+ (𝛼1+𝛽1)𝑙𝑜𝑔𝜎𝑡−1
2(14)
AEGAS(1,1): 𝑙𝑜𝑔𝜎𝑡
2= 𝜔+ 𝛼1𝜇𝑡1𝜎𝑡−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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