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A NOVEL HYBRID DISTRIBUTED INNOVATION EGARCH MODEL FOR INVESTIGATING THE VOLATILITY OF THE STOCK MARKET
1Mubarak M.T., 2Adubisi O.D., and 3Abbas U.F.
13School of Science and Technology, Gombe State Polytechnic, Bajoga, Gombe State, Nigeria. 2Department of Mathematics and Statistics, Federal University Wukari, Nigeria. [email protected], [email protected], [email protected]
Abstract
When calculating risk and making decisions, investors and financial institutions heavily rely on the modeling of asset return volatility. For the exponentiated generalized autoregressive conditional heteroscedasticity (EGARCH) model, we created a unique innovation distribution in this study called the type-II-Topp-Leone-exponentiated-Gumbel (TIITLegu) distribution. The key mathematical characteristics of the distribution were determined, and Monte Carlo experiments were used to estimate the parameters of the novel distribution using maximum likelihood estimation (MLE) procedure. The performance of the EGARCH (1,1) model with TIITLegu distributed innovation density in relation to other innovation densities in terms of volatility modeling is examined through applications using two Nigerian shock returns. The results of the diagnostic tests indicated that, with the exception of the EGARCH (1,1)-Johnson (SU) reparametrized (JSU) innovation density, the fitted models have been sufficiently specified. The parameters for the EGARCH (1,1) model with different innovation densities are significant at various levels. Furthermore, in out-of-sample prediction, the fitted EGARCH (1,1)-TIITLegu innovation density performed better than the EGARCH (1,1)-existing innovation densities. As a result, it is decided that the EGARCH-TIITLegu model is the most effective for analyzing Nigerian stock market volatility.
Keywords: EGARCH, Innovations density, Maximum likelihood estimation, Simulation.
I. Introduction
The Nigerian stock market has grown in terms of the number of stock exchanges and other financial intermediaries, the number of listed stocks, trading volumes, market capitalization, investor population, stock exchange turnover, and stock price indexes over time. The stock market's performance is a key measure of a country's progress and development. Because it reflects the potential viability and financial strength of corporate entities registered on the stock exchange, the stock market is one of the yardsticks for assessing an economy's growth and development. However, the state of investor confidence in different economic sectors is reflected in the stock market. It shows hopes for the stability of the financial system and reflects the strength of the producing sector [1-2]. A survey of pertinent literature reveals that while predicting stock market volatility, academics have neglected to account for the contributions of alternate innovation distributions. Since financial time series have leptokurtic and autocorrelation characteristics, mis-specification may result from applying the incorrect innovation density in the EGARCH volatility model. Additionally, an
erroneous innovation density specification can result in a significant loss of efficiency for the relevant estimators, an invalid risk assessment, incorrectly priced options, and an improper valuation of Value-at-Risk (VaR).
In making financial decisions such as portfolio selection, risk management, and option pricing, financial return volatility is an important metric to consider. Therefore, in order to model and anticipate the volatility of asset returns, it is imperative to create a model with strongly driven conditional innovation density. According to [3], a suitable volatility model is one that both accurately represents the disturbance term's heteroscedasticity and reflects the stylized facts that are present in stock return series. Financial institutions usually employ generalized autoregressive conditional heteroskedasticity (GARCH) models to predict return volatility for stocks, bonds, and market indexes, while they can also be used to analyze other types of financial data, such as macroeconomic data. In order to aid in their decisions about asset allocation, hedging, risk management, and portfolio optimization, they use the information that is produced to estimate the returns of current investments, help decide pricing, and assess which assets may yield larger returns [4]. However, these returns exhibit leverage effects, heavy tail, volatility clustering, significant skewness and excess leptokurtic behaviours which the symmetric GARCH model with normal distributed innovations in most cases fail to capture [5-7]. Despite the skewed form and appealing characteristics of the student-t in GARCH models, the tail behaviour is still too short to adequately describe skewness and fat tails in asset returns [8-9]. According to [10], volatility models that do not allow for conditional variance asymmetry typically result in inaccurate volatility estimations and projections. To address this flaw in GARCH's treatment of financial time series, the asymmetric exponential GARCH model was developed using the generalized error distributed innovation. Specifically, to accommodate asymmetric impacts between asset returns that are positive and negative [10-11]. Few studies had been done on returns volatility modeling utilizing the EGARCH model with common innovation densities, which raises questions about the choice of innovation densities [11-17].
Quite a few academics have focused on establishing novel distributions for volatility model innovations, and have conducted various studies in this area of altering the innovation density of the EGARCH volatility model. [18] advocated the use of the beta-student-t distribution for the EGARCH model in the estimation of volatility. [19] discovered that more flexible GARCH-type models are sufficiently acceptable in predicting volatility for all density assumptions. Using simulated and actual data, the Bayesian analysis of a stochastic model with generalized hyperbolic skew Student-t distribution. [20] proposed the EGARCH model with the beta-skew Student-t density for predicting daily volatility. [21] discovered that using leptokurtic distributions in GARCH-type models helped them produce more accurate volatility projections. [22-23] evaluated the daily volatility of stock index returns using a new generalization of the skew Student-t distribution, and demonstrated that it performed better than some innovation densities. [2] and [24] proposed the exponentiated half-logistic skew-t and generalized odd generalized skew-t densities for evaluating the daily volatility of bitcoin, Nigeria inflation and first bank Nigeria stock returns, and found that it performed better than other innovation densities in the GARCH-type models. [7] proposed the GARCH model with the exponentiated Gumbel density for predicting daily return volatility of the S&P 500 index. [9] proposed the odd generalized exponential Laplace density for evaluating the daily volatility of the Nigeria stock exchange, and found that it performed better than other innovation densities in the GARCH-type models. These models informed entrepreneur and investors on volatile nature of stock prices and bitcoin rates. However, developing robust distributions remains critical in improving the accuracy of the monetary risk system. Therefore, this research set out to propose a new innovation density for the asymmetric EGARCH model by introducing a novel distribution. This research is focused on the distinctiveness of the structural properties of the novel distribution and modification of the distributional assumption of the
Mubarak,M.T., Adubisi,O.D., Abbas,U.F. RT&A, No 3 (79) A NOVEL HYBRID DISTRIBUTED INNOVATION EGARCH MODEL_Volume I9, September 2024
innovations of the EGARCH volatility model.
The article is arranged as follows: Section 2 presents the developed novel distribution and its
standardized form. Section 3 presents the properties of the novel distribution. Section 4 presents the
maximum likelihood estimation procedure and Monte-Carlo simulation process. Section 5 presents
the EGARCH model with the novel innovation density function including methods for selecting
models and appraising predictions. Section 6 reports the empirical results of both estimation and
forecast assessment, and conclusion in Section 7.
2. Distribution Genesis
The cumulative distribution function (cdf) of the Type II Topp Leone (TIITL-G) family of distributions developed by [25] is specified as
F(x) = 1-[1-G2(x)]9 (1)
and the corresponding probability density function (pdf) is
f(x) = 28g(x)G(x)[1-G2(x)]e-1 (2)
where 8 > 0 is the shape parameter, G(x) and g(x) are the baseline cdf and pdf, respectively. The cdf of the baseline distribution titled the exponentiated Gumbel (EGU) distribution introduced by [26] is specified as
G(x) = 1-\l-el J}, (3)
and the pdf is given as
g(x)=-\l-e[ei' * )]) ei^ "^e^-*--), (4)
where a > 0, a > 0, ^ E K are shape, scale and location parameters, and x E K.
The cdf and pdf of the type II Topp Leone exponentiated Gumbel (TIITLEGU) model derived by
inserting Equations (3) and (4) into Equations (1) and (2), respectively, are specified as
F(x\a,e,y.,a) = 1-{1-[®(x)]2}e, (5)
f(x;a,9,ii,a) = 28<&(x){1 - [0(x)]2}e-1, (6)
Hither, the pdf and cdf of the EGU are represented with O(x) and 0(x). a > 0,8 > 0,a > 0 are the shape and scale parameters, and ^ E K is the location parameter. More so, the survival and hazard rate functions are specified as follows
S(x;a,8,v,a) = {1-[Q(x)]2}9 (7)
h(x; a, e,\i, a) = 28®(x)Q(x){1 - [0(x)]2}-1, (8)
Figures 1 depicts that the TIITLEGU density function can be very useful in describing symmetric, heavy-tailed, unimodality, leptokurtic and skew patterns of most data sets. Hence, a viable alternative innovation density for increasing the accuracy of the EGARCH volatility model prediction.
2.1 Standardized TIITLegu Model
The standardized TIITLegu model is obtained via the transformation et = z—^, where E(zt) = 0 and var(zt) = 1. The random variable zt can be expressed as zt = = -iL, then — = -=. Therefore,
Jh2 Jh2 dEt Jh2
the standardized TIITLEGU density function is
f(et;a,e,ht) =^TT}a-1(1-T})e(->i!)(1-va)[1 - (1 - va)2]6-1 ^ (9)
(htf (h2)2
where ^ = 1- e e t , ^ and —h denote the mean and standard deviation. The standardized
a
TIITLegu is the novel hybrid distributed innovation function for the EGARCH volatility model.
Figure 1: The TIITLegu density function (pdf) plots with selected parameter values.
3. Properties of the TIITLegu Model 3.1 Quantile and Median Functions
The quantile function (qf) of the TIITLegu model derived by inverting Ec
Q(u) = —a log
— log ( 1 — {l — [l — (1 — u)ëj2'
(5) is specified as
where u ~ Uniform(0,1). The median (M) of the TIITLegu model is specified as
+ ß
(10)
(11)
M = -a log -log(1 - {1 - [l - (0.5)e]2 By means of the quantile function in Eq (10), various quantile measures can be estimated. 3.2 Raw-Moment and Moment-Generating-Function
The raw-moment (rm) of the TIITLegu model is derived using the expansion series approach. Let X ~ TIITLecu(x; a, 8,^, a), the rth raw-moment of X is specified as
= J+œ xr f (x; a, 8,^, a)dx (12)
The expanded form of the TIITLegu pdf using the series expansion is
/(x) = £m=0^me
-(m+1)-
jm=^mt
where = ^^o-U A } " "
By inserting the expanded form in Eq. (13) into Eq. (12), we have
(13)
Let,
1
1
1
1
G
z = (m + 1) —- ^ x = ■
G
dx
dx = •
1
dz,
dz -£(m+1) -%(m+1)
Hence, the rm of the TIITLEGU is specified as
Vr = Zm=0 Vm fo (-t(m+1))
Simplifying Eq. (15) leads to
- r-1
¿■=ZX=oam(-;(rn + 1)) f0+m(z)
e-z-
e Zdz,
dz,
(15)
(16)
Using the gamma integral representation r(a + 1) = f (y)a e ydy. Therefore, the rm of the TIITLEGU is specified as
r(r+1)
2a9 ,
(->+1))
(-i)i+j+k+m(k + i)m(9-^)(2i-1)(a(j + l)-1)
(17)
where Vm = — Ymj,k=o
Table 1 reports the first four raw-moments, standard-deviation (SD), dispersion index (DI) and coefficient of variation (CV) for the TIITLegu model utilizing Eq. (17) with fixed V = 0, a = 1.
Table 1: Summary statistics for the TIITLegu model
ßr a = 1.5,6 = 1.5 a = 2.5,6 = 2.0 a = 3.3,6 = 3.5 a = 3.7,6 = 37
ß1 0.4810 0.0969 0.0057 0.0022
ß2 0.5850 0.0577 0.0016 0.0006
ß3 0.9590 0.0472 0.0006 0.0002
ß4 1.9600 0.0483 0.0003 0.0000
SD 0.5950 0.2200 0.0396 0.0244
DI 0.7350 0.4990 0.2750 0.2710
CV 1.2400 2.2700 6.9500 11.1000
Additionally, the moment generating function (mgf) of the TIITLEGU is specified as
Mr(t) = ^mm=o'Zm=0Vm—T{r+l1t-
2a0 ,
where ßm = —Y.™j,k=o 3.3 Order Statistics
(-%(m+1)) r!
(-1)i+j+k+m(k + 1)rn(e-1)(2i-1)(a(j+1)-1)
(18)
Let x1,x2, ...,xn denote a random sample from the TIITLegu and x1:n < x2.n < statistics obtained from the sample. The pth order statistics (os) is specified as
fp.n (xX) = '
f(x)
Jp:n\--j B(p,n-p + 1)
By inserting Eqs. (5) and (6) into Eq. (19), we have
[F(x)]p-1[1 - F(x)]n-p
fp.n(x =
28<b(x){1-[Q(x)]2
[1-{1- [0(x)]2}°]P-1[1 -(1-{1- [0(x)]2}o)]
< xn.n be the order
Jp:n\--j B(p,n-p + 1)
Simplifying Eq. (20), the os of the TIITLegu is specified as
2->8Mn-p
fp.n(x)=2S0(x){1-!0(x)]2}e(;-p+1)-1[i
p. n
-{1-[Q(x)]2} o]
6~\P—1
(19)
(20)
B(p,n-p+1)
The minimum and maximum order statistics is derived by inserting p = 1 and p = n into Eq. (21)
(21)
4. Estimation and Simulation Study 4.1 Maximum Likelihood Estimation
Let x1,x2,...,xn denote the observed random-values from the TIITLEGU(x; a, 8,v, a). Assuming that
z
1
m!
0-1
Mubarak,M.T., Adubisi,O.D., Abbas,U.F. RT&A, No 3 (79) A NOVEL HYBRID DISTRIBUTED INNOVATION EGARCH MODEL_Volume I9, September 2024
^ = 0, a = 1, without loss of generality, the log-likelihood-function (LL) is specified as LL (в, a ) = n log2 + n log в + n log a + (a - 1) £?=i logfo} + log{1 -
+ Z?=i X + Z|=i log(1 - T|f) + (в -1) 2?=i log[1 - (1- |f )2] (22)
where = 1 - e[-e( *)]. Differentiating Eq. (22) with respect to в and a gives the follows:
^ = ^ + £r=iiog[1-(1-if)2] ^=i+^r=i iog{ii} - ^r^i +(в -1) z?=i 2(i-!^;ff{,?i}
In this study, R-programming (optim function) is used in finding the maximum likelihood (ML) estimates (в, a) of the TIITLegu parameters.
4.2 Simulation-Study
The simulation process for the ML is obtainable as follows: N = 10,000 samples (replicates) are generated from the TIITLegu model with sizes n' = 20,50, 150, 250, 500 and 1000 using R-programming. The precision of the ML estimates is evaluated via the mean estimates (MEs), absolute bias (Absbias), mean square errors (MSEs) and root mean square roots (RMSEs). The MLE is a suitable technique for estimating the TIITLegu parameters Based on the simulation study. The results reported in Table 2 indicate that the parameter estimates are quite stable and very close to the true parameter values for the various sample sizes. The ME tend to be closer to the true values of the parameter with minimum MSEs, and RMSEs values as the sample size increases.
Table 2: Numerical values of MEs, Absbias, MSEs and RMSEs
a = 1.5, в = 1.5 a = 3.5,6 = 3.3
n' Par. ME Absbias MSE RMSE n' Par. ME Absbias MSE RMSE
20 a 1.7238 0.2238 2.1842 1.4779 20 a 3.7210 0.4210 8.8988 2.9831
в 3.1475 1.6475 10.9000 3.3015 в 7.2017 3.7017 51.1596 7.1526
50 a 1.6239 0.1239 1.0649 1.0320 50 a 3.5275 0.2275 4.4887 2.1187
в 2.4156 0.9156 4.5588 2.1351 в 5.7094 2.2094 23.6443 4.8625
150 a 1.5312 0.0312 0.3634 0.6029 150 a 3.3703 0.0703 1.8493 1.3599
в 1.8985 0.3985 1.3260 1.1515 в 4.6138 1.1138 8.9932 2.9989
250 a 1.5156 0.0156 0.2146 0.4633 250 a 3.3349 0.0349 1.1736 1.0833
в 1.7484 0.2484 0.7015 0.8375 в 4.2532 0.7532 5.2725 2.2962
500 a 1.5089 0.0089 0.1105 0.3323 500 a 3.3155 0.0155 0.6437 0.8023
в 1.6267 0.1267 0.3019 0.5495 в 3.9323 0.4323 2.6383 1.6243
1000 a 1.5033 0.0033 0.0561 0.2368 1000 a 3.3028 0.0028 0.3303 0.5747
в 1.5653 0.0653 0.1336 0.3655 в 3.7293 0.2293 1.1773 1.0850
a = 2.0,0= 2.5 a = 4.1, в = 4.5
n' Par. ME Absbias MSE RMSE n' Par. ME Absbias MSE RMSE
20 a 2.8662 0.3662 6.1123 2.4723 20 a 5.0225 0.5225 15.9168 3.9896
в 4.2505 2.2505 19.2609 4.3887 в 8.4532 4.3532 69.0055 8.3070
50 a 2.6997 0.1997 2.9340 1.7129 50 a 4.7706 0.2706 8.0638 2.8397
в 3.2939 3.2939 8.5372 2.9218 в 6.8341 2.7341 35.0702 5.9220
150 a 2.5439 0.0439 1.0106 1.0053 150 a 4.5916 0.0916 3.5820 1.8926
в 2.5848 0.5848 2.7012 1.6435 в 5.5217 1.4217 13.8556 3.7223
250 a 2.5253 0.0253 0.6185 0.7865 250 a 4.5407 0.0407 2.2462 1.4987
в 2.3669 0.3669 1.4303 1.1959 в 5.0808 0.9808 8.5140 2.9179
500 a 2.5110 0.0110 0.3252 0.5702 500 a 4.5209 0.0209 1.2639 1.1243
в 2.1968 0.1968 0.6399 0.7999 в 4.6527 0.5527 4.0301 2.0075
1000 a 2.5028 0.0028 0.1635 0.4043 1000 a 4.5031 0.0031 0.6643 0.8150
в 2.1011 0.1011 0.2767 0.5260 в 4.4001 0.3001 1.8519 1.3608
Mubarak,M.T., Adubisi,O.D., Abbas,U.F. RT&A, No 3 (79) A NOVEL HYBRID DISTRIBUTED INNOVATION EGARCH MODEL_Volume I9, September 2024
5. EGARCH-Model
The asymmetric EGARCH model introduced by [10] is considered as a variate of the ARCH/GARCH model introduced by [3] and [27] for modeling time-varying volatility. The EGARCH differ from the symmetric GARCH variance structure given that the natural log variance is used in other for the parameters to be unrestricted and can take negative values while guaranteeing a positive conditional variance. Moreso, the EGARCH model includes the asymmetric impact of positive and negative shocks on volatility. The return of daily prices of assets is represented by rt and the EGARCH (1,1) model is defined as:
rt = M + et = ZtVft2, Zt ~ I ¿. d,
(23)
£t-i 1 ,, £t-i
10 + pi wg'it-l +
log = ¿0 + £1 log ftt-1 + ¿1 "T-^ + Ki
where A0 >0, A1 > 0, p1 > 0 are the model parameters, zt is the conditional innovation density with £(zt) = 0 and var(zt) = 1, is the conditional mean, y1 is the leverage parameter, logft^ is conditional log variance at present day t, et-1 and logft2-1 are the error and conditional log variance at preceding day t — 1, respectively. The commonly utilized conditional innovation densities are well described in the literature.
5.1 The Novel Conditional Innovation Density
The log-likelihood (LL) function of the standardized TIITLegu model presented in Eq. (9), is specified as
Lifl9) = nlog2 + nlogS + nlog« — ^logft? + (« — 1)I?=ilog{77t) + Z?=ilog{1 — +
E=ilog
+ Zl=ilog(1 - ^f) + (0 -1) Z?=ilog[1 -(1 - ^)2] - ^logft2 (24)
where $ = (а, в, a, 0 are the shape parameters and is the EGARCH volatility model with
vector parameters and = 1 — e
5.2 Model Selection Criteria
The modified Akaike information criteria (AIC) and Bayesian information criteria (BIC) proposed by [28] are utilized in selecting the best model under the conditional innovation densities. The modified AIC and BIC criteria are given by
ЖС = — — — (25)
w w 4 '
B/c = fcb^(W)— 2LL (26)
w w 4 '
where k is the total number of estimated parameters, the estimated log-likelihood value and sample size are denoted by LL and N, respectively. The EGARCH model with the least AIC and BIC values is regarded as the most suitable model under the specified innovation density.
5.3 Forecasts Performance
The forecasts performance of the EGARCH models is appraised using the mean square error (MSE), root mean square root (RMSE), and mean absolute error (MAE). The performance measures for the volatility forecasts are given by
eV"t
mse = 1sf=1(/Tt - ht)
RMSE = JlX?=1(h~t - ht)2 MSE = lYH>=1\ht-ht\
(27)
(28)
„ (29)
Where ht and ht represent the volatility forecast and realized volatility, and N is the sample size. The model with the least performance measures is regarded as the most appropriate for predicting the volatility of the daily log-returns.
6. Empirical Results
6.1 Data Report
To appraise the performance of the novel distributed innovation density in the EGARCH model, the United bank of Africa (UBA) and Total energies Nigeria (TEN) stock prices log-returns are utilized. The UBA dataset consists 5468 daily log-returns from 1/2/2000 to 5/1/2024 and TEN datasets consist 5510 daily returns from 2/1/2001 to 5/1/2024. The estimation process is executed using 5218 daily log-returns from 1/2/2000 to 30/12/2022 for the UBA while 5268 daily returns from 2/1/2001 to 30/12/2022 for the TEN. The forecast evaluation of the models is carried-out with 250 daily returns from 3/1/2023 to 5/1/2024 for both UBA and TEN. The summary statistics of the daily returns for the estimation processes are reported in Table 2. More so, the graphical plots of the daily returns, squared-returns and absolute-returns with their respective sample autocorrelation function (ACF) for both UBA and TEN are depicted in Figures 3 and 4.
Tables 2 reports positive skewness and high excess kurtosis, leading to large Jarque-Bera (JB) statistic (p < 0.001) signifying that the daily returns for the estimation process have non-normality characteristics. Figure 2 displays the density function of a normal distribution that has the same mean and standard deviation as those of the UBA and TEN return series. The plots provide a visual check of the normality assumption for the daily returns. The deviation between the solid (return series) and dashed line (Normal distribution) indicates that the daily returns are not normally distributed.
UBA return TEH return
Figure 2: Empirical density function of the UBA and TEN daily returns.
Further, the ARCH Lagrange-multiplier (LM) and Ljung Box-Q tests at lag 10, indicates the incidence of conditional heteroscedasticity and autocorrelation in the returns while the Augmented Dickey-Fuller (ADF) test with its p-value indicates that the returns for the UBA and TEN are stationary.
Table 2: Summary statistics for the daily returns
UBA returns (Estimation process)
Number of Mean Median Minimum Maximum Std Dev.
observations
5218 -0.004 0.000 -204.521 206.978 5.005
Skewness Kurtosis Jarque-Bera ARCH (10) Q (10) ADF test
0.087 1096.432 26157 (< 0.0001) 894.89 (< 0.0001) 901.99 (< 0.0001) -24.7 (<0.01)
TEN returns (Estimation process)
Number of Mean Median Minimum Maximum Std Dev.
observations
5268 0.022 0.000 -231.941 235.569 6.790
Skewness Kurtosis Jarque-Bera ARCH (10) Q (10) ADF test
0.999 1027.144 23175 (< 0.0001) 107.55 (< 0.0001) 117.84 (< 0.0001) -27.5 (<0.01)
Figure 3: UBA daily returns, squared returns, absolute returns and sample autocorrelations.
6.2 Parameters Estimation of the EGARCH (1,1) Model
The EGARCH (1,1) model specified in Eq. (22) is estimated under ten different innovation densities: normal (NORM), student-t (ST), generalized error (GE), skew normal (SNORM), skew student-t (SST), skew generalized error (SGE), generalized hyperbolic (GHYP), Johnson (SU) reparametrized (JSU), Normal inverse Gaussian (NIG) and the novel type II Topp Leone exponentiated Gumbel (TIITLegu). Tables 3 and 4 reports the estimated parameters of the EGARCH (1,1) models. The rugarch package in R-programming is used in estimating the parameters of the EGARCH-NORM, EGARCH-ST, EGARCH-GE, EGARCH-SNORM, EGARCH-SST, EGARCH-SGE, EGARCH-GHYP, EGARCH-JSU and EGARCH-NIG while the Optim function in R-programming is utilized to maximize the log-likelihood function of EGARCH-TIITLegu. As reported in Table 5, the EGARCH-TIITLegu model has the highest log-likelihood (LL) value and exhibits superior fit to the standardized residuals compare to others for both return series.
Figure 4: TEN daily returns, squared returns, absolute returns and sample autocorrelations. Table 3: EGARCH Model Parameter Estimates with Innovation Densities (UBA returns).
Model Cond. Distr. Ao A! ßi Vi Ghlambda Skew Shape1 Shape2
EGARCH (1,1) NORM 0.5707***' -0.0581 ***' 0.7492 ***' 0.4881 ***' - - -
STD 0.4320***' 0.0960 ' 0.888 ***' 2.1239 ***' - 2.1000' ***' -
GED 8.5262'***' 1.2433 0.1171 6.5026 ***' - 0.1019' ***' -
SNORM 0.5680' ***' -0.0488 ***' 0.7488 ***' 0.4867 ***' 0.9781 ***' - -
SSTD 0.6725' ***' 0.2855 ' 0.8914 ***' 6.3057' ***' 1.0003' ***' 2.0100' ***' -
SGED 1.6151 ***' 0.7086 ***' 0.3730 ***' 3.6658' ***' 1.0005' ***' 0.3956' ***' -
GHYP 0.0240' ***' 0.0524 ***' 0.9858 ***' 0.2100' ***' -0.9677 ***' -0.0547 0.2500' ***' -
JSU 0.2856 ' 0.0571 ***' 0.8828 ***' 1.1730' ***' 0.0001 0.8269' ***' -
NIG 0.7949' ***' 0.3175 ***' 0.9532 ***' 4.3156' ***' 0.0172 0.0100' ***' -
TIITLegu 2.327e-10'' 3.929e-10'' 0.9828 ***' 0.7799' ***' - 1.9433' ***' 3.5128' ***'
Significance levels: 0 o ooi '**■' 0.01 '*', 0.05 .', 0.1' 1
Table 4: EGARCH Model Parameter Estimates with Innovation Densities (TEN returns).
Model Cond. Distr. A0 A! ßi Vi Ghlambda Skew Shape1 Shape2
EGARCH (1,1) NORM 0.4283'***' -0.0869'***' 0.7781 ***' 0.2932' ***' - - -
STD -0.0587 0.2583'***' 0.8234 ***' 0.6745' ***' - 2.1000' ***' -
GED 1.3745'***' 0.0974 -0.1137 ***' 1.1712' ***' - - 0.1024' ***' -
SNORM 0.4296'***' -0.0858' ***' 0.7765 ***' 0.3024' ***' 0.9552' ***' - -
SSTD 0.2775'***' 0.7821 ***' 0.8283 ***' 1.8532' ***' 1.0011 ***' 2.0100' ***' -
SGED 0.6894'***' 0.0739' ***' 0.2451 ***' 0.0659' ***' 1.1175' ***' 0.3868' ***' -
GHYP -0.0226'***' -0.0294 ***' 0.9880 ***' 0.0905' ***' -02818'***' 0.0025 0.2500' ***' -
JSU 0.4089'***' -0.0000 0.3658 ***' -0.0000 0.2953' ***' 0.2616' ***' -
NIG -0.0434' ***' 0.0071 ***' 1.0000 ***' 0.0177' ***' -0.0071 0.0100' ***' -
TIITLegu 0.0552' ***' 0.0579' ***' 0.5195' ***' 1.5234' ***' - 5.2086' ***' 9.8771'***'
Significance levels: 0 '***', 0.001 '**', 0.01 '*', 0.05 '.', 0.1'', 1
Tables 3 and 4 reports that the parameter estimates of the EGARCH conditional variance specifications are highly statistically significant and y is highly significant which shows that the
daily log-returns have leverage effect. Therefore, the impact of the shocks is asymmetric in nature that is, the impact of positive shocks on volatility are higher than negative shocks of the similar size. The AIC and BIC values also reported in Table 5 suggest that the EGARCH-TIITLegu model is best for investigating the volatility of the Nigerian stock market.
Table 5: Comparison of the Innovation Densities for estimated models.
Model Innovation Dist. Log- Likelihood AIC BIC Log- Likelihood AIC BIC
(UBA Returns). (TEN Returns).
EGARCH (1,1) NORM -12581.400 4.8242 4.8305 -11134.030 4.4296 4.461
STD -11070.810 4.2456 4.2532 -8291.421 3.3071 3.3149
GED -5747.029 2.2051 2.2126 19540.740 -7.7859 -7.7781
SNORM -12578.510 4.8235 4.8310 -11124.140 4.4361 4.4439
SSTD -11046.030 4.2365 4.2453 -8173.160 3.2603 3.2694
SGED -10738.730 4.1187 4.1275 -4397.584 1.7555 1.7646
GHYP -10923.350 4.1899 4.1999 -8299.449 3.3111 3.3215
JSU -11066.350 4.2443 4.2531 9042.654 -3.6013 -3.5922
NIG -10092 3.8709 3.8797 -4979.278 1.9874 1.9965
TIITLegu 13546.100 -5.1894 -5.1806 25184.930 -9.5588 -9.5501
Note(s): Bolded values indicate the highest log-likelihood value, and the least AIC and BIC values
Tables 6 and 7 reports the diagnostic tests results for the EGARCH (1,1) model under the various innovation densities. From Table 6, the Ljung Box-Q statistic (p > 0.05) specifies that the squared standardized residuals from the EGARCH-TIITLEGU model exhibit no sign of serial-correlation. Likewise, the ARCH-LM statistic (p > 0.05) indicates that the standardized residuals from the EGARCH-TIITLEGU model exhibit no additional conditional heteroscedasticity, that is, the conditional variance equation are correctly specified. Therefore, the results disclose that standardized TIITLegu density is an improved distributed innovation function for the EGARCH (1,1) model.
Table 6: Estimated EGARCH (1,1) models diagnostic tests (UBA returns).
Model Innovation Dist. Ljung-Box Q-Statistic p-value ARCH-LM Statistic p-value
EGARCH (1,1) NORM 0.007 0.999 0.007 0.999
STD 0.008 0.999 0.008 0.999
GED 0.010 0.999 0.010 0.999
SNORM 0.007 0.999 0.007 0.999
SSTD 0.008 0.999 0.008 0.999
SGED 0.007 0.999 0.007 0.999
GHYP 0.066 0.999 0.067 0.999
JSU 0.008 0.999 0.008 0.999
NIG 0.021 0.999 0.021 0.999
TIITLegu 0.013 0.999 0.013 0.999
Note(s): Significance level: a = 0.05 .
Table 7: Estimated EGARCH (1,1) models diagnostic tests (TEN returns).
Model Innovation Dist. Ljung-Box Q-Statistic p-value ARCH-LM Statistic p-value
EGARCH (1,1) NORM 0.021 0.999 0.021 0.999
STD 0.091 0.999 0.092 0.999
GED 5.936 0.981 5.923 0.981
SNORM 0.021 0.999 0.020 0.999
SSTD 0.088 0.999 0.088 0.999
SGED 0.010 0.999 0.010 0.999
GHYP 0.045 0.999 0.046 0.999
JSU 1251.5 2.2E-16 2317.7 2.2E-16
NIG 0.477 0.999 0.474 0.999
TIITLegu 0.039 0.999 0.039 0.999
Note(s): Significance level: a = 0.05 .
6.3 Forecasts Evaluation of the EGARCH Models
The forecast evaluation metrics for the out-of-sample are reported in Table 8, and the least MSE, RMSE and MAE values belong to the EGARCH-TIITLEGU model. Therefore, the EGARCH-TIITLEGU model is statistically efficient and displays superior capability in forecasting the volatility of the Nigerian stock market relative to other models.
Table 8: Forecasts evaluation metrics of the estimated EGARCH (1,1) models.
Model Innovatio n Dist. MSE RMSE MAE MSE RMSE MAE
(UBA returns) (TEN returns)
EGARCH (1,1) NORM 8.155 2.856 1.752 1.731 1.316 0.213
STD 8.151 2.855 1.751 1.733 1.316 0.197
GED 8.151 2.855 1.751 1.733 1.316 0.197
SNORM 8.121 2.849 1.752 1.730 1.315 0.267
SSTD 8.151 2.855 1.751 1.733 1.316 0.197
SGED 8.150 2.855 1.751 1.777 1.332 0.354
GHYP 8.159 2.856 1.753 1.733 1.316 0.197
JSU 8.151 2.855 1.751 1.733 1.316 0.198
NIG 8.151 2.855 1.751 1.733 1.316 0.197
TIITLegu 8.050 2.837 1.736 1.547 1.244 0.182
Note(s): Bolded values indicate the conditional distribution with the least MSE, RMSE and MAE.
7. Conclusion
The estimation of the TIITLegu model parameters using the MLE procedure, and introduction as a novel distributed innovation function for the EGARCH-volatility model is considered. The density and cumulative functions, failure rate function, quantile function, standardized density function and other mathematical properties are derived. Monte-Carlo experiments are carried out to study the performance of the MLE procedure. The experiments results indicate that the MLE is asymptotically unbiased and consistent given that the ME tend to be closer to the true values of the parameter with minimum MSEs, and RMSEs as the sample size increases.
Additionally, the standardized TIITLEGU density is presented as a novel distributed innovation function for the EGARCH volatility model for investigating the volatility of the Nigerian
stock market via the UBA and TEN returns. The empirical findings showed that the EGARCH-TIITLegu model has the highest log-likelihood, and least AIC and BIC values. Equally, the EGARCH-TIITLEGU model has the least forecast evaluation metrics among other models. In conclusion, the EGARCH-TIITLEGU model is best for investigating the volatility of the Nigerian stock market.
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