ISSN 1450-2267 Vol. 57 No 2 November, 2018, pp.167-178 http://www.europeanjournalofsocialsciences.com/
Using GARCH Models for Modelling and Forecasting Volatility an Empirical Study of the Egyptian Stock Market
Mohamed E. M. Abdelhafez
Corresponding Author, College of Business Administration Kuwait University
E-mail: [email protected]
Abstract
In this paper, we consider the problem of modeling and forecasting volatility of daily time series of Egyptian Stock market using GARCH models. These models are considered as generalization of the ARCH models that take into account the non-constant variance in the time series. Different Symmetric and Asymmetric GARCH models are compared using Akaike Information Criterion (AIC) and Schwarz Information Criterion (SIC) to determine the appropriate models to analyze the data. The results indicate that the models GARCH (1, 2), GARCH (2, 1), TARCH (1, 1) and EGARCH (1, 1) are appropriate to analyze data from the Egyptian stock market. We also study the problem of forecasting conditional variations using these models. It has been shown that TARCH (1, 1) is the best-fitted model to capture the leverage effect. Furthermore, the forecasting evaluation shows that TARCH (1, 1) model outperforms the other GARCH models.
Keywords: GARCH, Volatility, Egyptian Stock market, Forecasting, Leverage.
1. Introduction
Financial markets data are usually volatile due to rapid changes in asset prices and returns on these assets. This leads to the instability of the variation of these prices and returns over time. The phenomena of changing the variance over time is called Heteroskedasticity. Suppose that one day there was a significant decline in the market in terms of returns and asset prices, this leads to a negative feeling in the market resulting in an attempt to get rid of stocks quickly, leading to further decline in prices. In other words, the increase in the variance resulting from lower prices initially leads to more volatility in falling prices and returns. This means that the increase in variance is associated with another increase in variation in the period of price declines. Therefore, we say that the period of increasing the variance is conditional on the initial sales and this phenomenon is usually called Conditional Heterocedasticity (CH).
The conditional variation over time can be expressed linearly as an Autoregressive Model (AR) in the previous squared residuals. This model is called the Autoregressive Conditionally Heterocedastic Model (ARCH) model. Thus, we can say that the ARCH model is appropriate when random error variation follows the autoregression model AR, Engle [9].
The ARCH (q) model takes the following form:
( )
0.
0
,
1 0 ,
1 2
≥
>
+
=
=
=
−
i q
i
i t i t
t t t t
and h
, N e e h
α α
ε α α
ε
ο ο
(1) By examining the conditional heterocedastic series, it may be appropriate to assume that the conditional variation in time t depends on the conditional variation in previous periods. In this case, it is preferable to use the ARMA (p, q) model instead of the AR (q) model to express these conditional heterocedasticities. Hence, the general model of this time series is called the Generalized Autoregressive Conditionally Heterocedastic Model (GARCH), Bollerslev, T. [5].
The GARCH (p, q) model is a time series model that expresses the conditional heterocedastic in time t as a linear function of the previous squared residuals ε , and previous conditional variances, In 2 GARCH (p, q) models, p is the number of previous conditional variance terms (GARCH terms) and q is the number of previous random square error terms, (ARCH terms).
ARCH and GARCH models have been widely discussed in many researches and books to study their characteristics and to evaluate their features in the various applications of financial markets, Bera and Higgins [4], Bekaert [3] and Bollerslev et al. [6].
There have been some modifications to the GARCH model to overcome the deficiencies in this model. One of these deficiencies is that the model is symmetric in terms of responding to changes that occur in random error fluctuations because the conditional heterocedastic in time t is a function of conditional heterocedastic in previous periods as well as a function in random square errors in previous periods. That is, the model expresses the value only to the magnitude of the change and does not reflect the signal of this change (negative or positive). This deficiency is appeared if the correlation between the returns of equity and changes in variance is negative. That is, the variation tends to rise in response to bad news and tends to decline in response to good news. Another limitation in the GARCH model is the constraints on the parameters of this model are not negative in order to ensure that the conditional heterocedastic is positive. Examples of models that overcome the deficiencies in the GARCH model are TARCH and EGARCH models.
GARCH models have been applied to capital markets in many countries to select the appropriate model as well as to use it in forecasting. See for example, Balaban et al. [2], Chukwugor- Ndu, C. [7], Ebeid et al. [8], Koutmos, GF [17], Lee et al. [18], McMillan et al. [19] and Mishra, PK [21].
In addition, Haroutounian and Price [13] studied variability in the Czech Republic, Greece, Poland, and Slovakia using different GARCH models either univariate or multivariate. The study concluded that there were no asymmetric effects in these markets despite strong evidence that GARCH models should be used in the analysis. Erginbay et al., [11] studied the volatility of stock markets in five European countries: Bulgaria, Czech Republic, Poland, Hungary, and Turkey using the GARCH, GJR-GARCH, and EGARCH models for daily data for the period from 2001 to 2012. In the African countries, different GARCH models have been used to analyze time series related to Capital markets and also to apply them to forecasting; e.g., Kalu, OE, [16], Samouilhan, and Shannon, G. [25], Mecagni et al. [20] andTooma, EA [27].
The following is a general overview of some GARCH models used in this paper. We consider two symmetric GARCH models, and two asymmetric GARCH Models.
2. Symmetric GARCH Models
2.1. GARCH (p, q) Model: Bollerslev, T. [5]
The GARCH (p, q) model is a time series model that expresses the variation at time t as a linear function in the previous squared residuals and the previous conditional variances, which have occurred
in the past. In the GARCH (p, q) model, p is the number of the previous conditional variance ht−j (GARCH terms) and q represent the number of previous random squared error termsεt−2i (ARCH terms).
Therefore, the GARCH (p, q) model takes the following form:
( )
0.
0 , 0
1
0 ,
1 1
2
≥
≥
>
+ +
=
= +
=
=
−
=
−
j i
p
j
j t j q
i
i t i t
t t t t
t t
and
h h
, N e e h r
β α
α
β ε
α α
ε
ε µ
ο ο
(2) In order to ensure the stationarity of this model, it must meet the following requirement: the sum of all parameters must be less than one. i.e. max( ,
(
))
11
<
+= q p
i
i
i β
α . The ARCH (q) model is a special case of the GARCH model (p, q) whenp=0.
2.2. GARCH-In-MEAN Model: Engle et al. [10]
This model allows that the conditional mean of a time series to depend on the conditional variance (or standard deviation). That is, it assumes that the total change in return due to the purchase of a security or portfolio is the average risk plus sudden changes (increase or decrease) in the capital gains provided that the retention of this security or portfolio is for long period of time. This is called long-term investment. The GARCH-In-Mean model is used to examine the risk of long-term investment on return. The GARCH-In-Mean (1, 1) model is as follows:
2 1 1 2
1 1 2
1
−
−
−
+ +
=
+ +
=
t t
t
t t t
h h
h r
β ε α α
ε δ
µ
ο (3)
Note that the conditional variance equation in (3) is quite similar to the conditional variance equation of the GARCH (1, 1) model. The difference between them is in the conditional mean equation. The parameterδ in equation (3) is called the risk premium parameter. It reflects the extent of the risk as the positive value of δ indicates a positive relationship between the return and variance or changes in return, in the sense that there may be an increase in the average return due to the increase in conditional variance. The greater the conditional variation, the greater the risk. Takingδ =0 in the previous conditional mean equation (3), we arrive at the conditional mean equation in the GARCH (1, 1) model.
3. Asymmetric GARCH Models
Symmetric GARCH models are unable to capture the asymmetry or leverage effects of the time varying variance. This is because that the conditional variances depend only on the squared residuals,
2
εt . Thus, negative and positive values ofεt that are equal in absolute value have the same effect on conditional variation, whereas negative values and positive values for εt can have different impacts on the conditional variance of the following returns. A feature of stock returns is that the fast volatile periods and the variance often begin with large negative values forεt . The symmetric GARCH models cannot express the asymmetric effects of negative and positive values forεt that can have great impact on market returns. For example, a negative signal can have a greater impact than a positive signal because this leads to a lower stock price and thus increases the risk to investors holding these shares.
Two of the models that describe asymmetry are Threshold GARCH (TARCH or GJR) Model and EGARCH model.
3.1. Threshold GARCH (TARCH orGJR) Model Glosten, et al [12] and Zakoian, [28]
This model allows the conditional variance to have a different response to the past negative and positive shocks.
( )
0.
0 , 0
, 1
0 ,
1 1
2 1
2
≥
≥
>
+ +
+
=
= +
=
= =
−
−
−
=
−
j i
p
j
q
i
i t i t i j
t j q
i
i t i t
t t t t
t t
and
d h
h
, N e e h r
β α
α
ε γ β
ε α α
ε
ε µ
ο ο
(4) Whereγ is the asymmetric response parameter or leverage parameter and dt−i is defined as:
≥
= <
−
−
− 0 0
0
1
i t
i t i
t if
d if
ε ε
The TARCH (p, q) model reduces to the GARCH (p, q) model whenγj =0. In the TARCH model, the impact of good news is expressed by αand the impact of bad news is expressed by α+γ . In addition, if γ ≠0the news impact is asymmetric.
3.2. Exponential GARCH (EGARCH) Model:Nelson [22]
In general, this model takes the form:
log
1 1
1
= −
−
= −
−
=
− + +
+
= +
=
q
j t j
j t j q
j t j
j t j p
i
i t i t
t t
h h
h h
Log r
γ ε α ε
β α
ε µ
ο
(5) Whereγ j is the asymmetric or leverage effect parameter. The conditional variance value will be positive even if the parameters of the model are negative because the model expresses the conditional variance logarithm. Therefore, there are no restrictions on the parameters in the model to ensure that conditional variance is positive. For the parameterγ j, it will be negative if the relationship is negative between the current return and the variance in the future. This indicates that an asymmetric effect exists and the impact of bad news is increasing volatility.
4. Research Data
4.1. Brief on the Egyptian Stock Exchange
The Egyptian Stock Exchange is one of the oldest stock exchanges in the world. There were attempts to establish an organized market for the sale and purchase of stocks during the nineteenth century, the last of which was in 1898 and was in a hotel, but the first official stock exchange was established in Cairo in 1903 and then followed by the Alexandria Stock Exchange. The Egyptian stock exchange has been operating effectively since that date, but there has been a period of recession following the nationalization decisions of 1961 and then it resumed its activity again from 1974, when a number of laws and declarations aimed at the development and growth of the capital market were issued in Egypt.
This led to a significant increase in the number of companies listed on the stock exchange.
4.2. Data and Statistical Analysis
In this section, we analyze the Egyptian Stock Exchange data. Data on the Egyptian public index has been obtained for 2300 daily readings (from Sunday to Thursday of each week) for nine consecutive years. These data are available on the Capital Market Authority.. Returns on the indices in a stock market for a single period are defined as the increase or decrease in price since the last time point in the time series. The data has been transformed so that we can get the Egyptian general index returns by using the following equation:
1
1
−
=
− t
t
t y
r y
Where,
t =
r the rate of return at time t
t =
y the stock index at time t
−1 =
yt the stock index just before time t
The statistical analysis of data includes the following GARCH models:
1. Symmetric GARCH models: GARCH (1, 1), GARCH (1, 2), GARCH (2, 1), GARCH (2, 2) and ARCH-in-Mean (1, 1).
2. Asymmetric GARCH models: TARCH model (1, 1) and EGARCH model (1, 1) The statistical analysis procedures of the data will consider the following points:
First: Estimating the parameters of the model using Maximum Likelihood Method Second: Forecasting
Third: Model tests
Fourth: Choosing the best fitting model.
5. Parameter Estimation of Symmetric GARCH Models
5.1. GARCH (1, 1) ModelPrevious statistical studies on financial time series analysis using Symmetric-GARCH models have indicated the efficient use of the GARCH (1, 1) model in the analysis of financial market data. This model takes the following form:
0 , 0 , 0
,
1 1
1 1 2
1 1
≥
>
>
+ +
= +
=
−
−
β α
α
β ε α α
ε µ
ο
ο t t
t
t t
h h
r
Wherep=1,q=1,αο is a constant, α1is the parameter of the ARCH term in the model, and β1 is the parameter for the GARCH term in the model.
Using the Maximum Likelihood Estimation (MLE) method, the estimated model is as follows:
(
2.38 07) (
0.0492)
(
0.0112)
372739 .
0 220016
. 1 06 53 . ˆ 4
1 2
1
−
+ +
−
= − −
E
h E
ht εt t
From table (1) below, it is clear that the parameters of the model are significant. However, we observe thatα1+β1 >1 and this means that the random error series is nonstationary and the model does not hold for the returns.
5.2. GARCH (1, 2) Model
The model takes the following form:
0 , 0 ,
0 , 0
,
1 2
1
1 1 2
2 2 2
1 1
≥
>
>
>
+ +
+
= +
=
−
−
−
β α
α α
β ε α ε α α
ε µ
ο
ο t t t
t
t t
h h
r
Wherep=1,q=2, and the estimated variance equation will be as follows:
(
2.34 06) (
0.0236112) (
0.010444)
(
0.044422)
436182 .
0 093602
. 0 210266
. 0 05 18 . ˆ 3
1 2
2 2
1
−
+ +
+
−
= − − −
E
h E
ht εt εt t
Table (1) below shows that the parameters of the model are significant. This indicates that the model is appropriate to analyze the stock market data in Egypt. We also note that the sum of the parameters of the model α1+α2 +β1 <1 and this indicates the stationarity of random error series while this condition was not achieved with the GARCH (1, 1) model.
5.3. GARCH (2, 1) Model
The model takes the following form:
0 , 0 , 0 , 0
,
2 1
1
2 2 1 1 2
1 1
≥
≥
>
>
+ +
+
= +
=
−
−
−
β β
α α
β β
ε α α
ε µ
ο
ο t t t
t
t t
h h
h r
Wherep=2,q=1, and the estimated variance equation will be as follows:
(
1.11 06) (
0.036166)
(
0.036154)
(
0.011629)
087007 .
0 493001
. 0 507809
. 0 05 42 . ˆ 1
2 1
2 1
−
− +
+
−
= − − −
E
h h
E
ht εt t t
The results in table (1) below show that the parameters of the model are significant. This indicates the appropriateness of the model to analyze the stock market data in Egypt. We also note that the sum of the parameters of the model α1 +β1 +β2 <1 and this indicates the random error series is stationary.
5.4. GARCH (2, 2) Model
The model takes the following form:
0 , 0 , 0 , 0 , 0
,
2 1
2 1
2 2 1 1 2
2 2 2
1 1
≥
≥
>
>
>
+ +
+ +
= +
=
−
−
−
−
β β
α α
α
β β
ε α ε α α
ε µ
ο
ο t t t t
t
t t
h h
h r
Wherep=2,q=2.
From Table (1) it is clear that most of the model parameters are not significant and therefore we will exclude this model. In addition, the higher order models of p, q will be also excluded when analyzing the stock market data in Egypt.
5.5. GARCH-in-MEAN (1, 1) Model
In general, the used model takes the following form:
2 1 1 2
1 1 2
1
−
−
−
+ +
=
+ +
=
t t
t
t t t
h h
h r
β ε α α
ε δ
µ
ο
Thus, the estimated model using the MLE method is as follows:
(
2.79E-07) (
0.062333)
(
0.008859)
264413 .
0 615949
. 1 06 99 . 6
061065 .
0 000557 .
ˆ 0
2 1 2
1 2
1
−
−
−
+ +
−
=
−
−
=
t t
t
t t
h E
h
h y
ε
From Table (1) below, it is clear that the parameters of the model are significant. However, the parameter δ of conditional variance in the mean equation has a negative sign and is not significant.
This means that the greater the holding of the stock, the greater the risk and the lower the return. Here we advise the investor to exit quickly from this financial portfolio (or the stock). The results makes us prefer to exclude the GARCH-in-Mean (1, 1) model.
Table 1: Results of Estimates for Symmetric Models of Symmetric GARCH
Models Coefficients AIC SIC Log Likelihood
GARCH(1, 1)
* 1
* 1
*
*
372739 .
0
220016 .
1
06 53 . 4
000339 .
0
=
=
−
=
−
=
β α α µ
ο E
-7.358917 -7.348933 8466.754
GARCH(1, 2)
* 1
* 2
* 1
*
*
436182 .
0
09360 . 0 210266 .
0
05 18 . 3
000384 .
0
=
−
=
=
−
=
−
=
β α α α µ
ο E
-7.130567 -7.118087 8205.152
GARCH(2, 1)
* 2
* 1
* 1
*
*
087007 .
0 493001 .
0
220016 .
1
05 42 . 1
000606 .
0
−
=
=
=
−
=
−
=
β β α α µ
ο E
-7.289127 -7.276647 8387.496
GARCH(2, 2)
061132 .
0 421614 .
0
074892 .
0 317683 .
0
05 15 . 3
05 64 . 4
2 1 1
* 1
−
=
=
−
=
=
−
=
−
−
=
β β α α α µ
ο E
E
-7.167614 -7.152639 8248.757
GARCH-in-Mean
* 1
* 1
*
*
264413 .
0
061065 .
0
615949 .
1
06 99 . 6
000557 .
0
=
−
=
=
−
=
−
=
β δ α α µ
ο E
-7.335582 -7.323102 8440.919
(*) means that the estimated parameters of the model are statistically significant
6. Parameter Estimation of Asymmetric GARCH Models
One of the main constraints of symmetric GARCH models is symmetry in the model's response to fluctuations in stock returns, whether rising or falling. In this case, it is preferable to use asymmetric GARCH models to express data in a better way.
6.1. TARCH (1, 1) or GJR Model
In general, the used model will be of the form:
1 1 1 2
1 1 2
1
1 − + − − + −
+
= +
=
t t
t t
t
t t
h d
h r
β ε
γ ε α α
ε µ
ο
wherep =1,q =1,γ is the asymmetry parameter and dt−i is defined as:
≥
= <
−
−
− 0 0
0
1
1 1 1
t t
t if
d if
ε ε
It should be noted that the GARCH model is a special case of the TARCH model. If we put
1 =0
γ in the TARCH model, which means no asymmetric effects exist, we obtain the GARCH model.
According to the data, the estimated model is as follows:
(
2.64E-07) (
0.070650)
(
0.096818)
(
0.014738)
364511 . 0 638472 .
0 518907 . 1 06 80 . ˆ 4
1 2
1 2
1 − −
− − +
+
−
= t t t
t E h
h ε ε
From table (2) below, it is clear that the parameters of the model are significant and this indicates that the model is appropriate. In addition, the value of the asymmetry parameterγ is -0.63847 and it is significant, indicating the asymmetry of the volatility. The results of the estimates show that negative effects (such as information on price declines) lead in the coming period to greater conditional variance than positive effects, leading to further price declines.
6.2. EGARCH (1, 1) Model
In general, the used model will be of the form:
1 1 1
1 1 1 1log
−
−
−
−
− + +
+
= +
=
t t t
t t
t t t
h h
h h
Log r
γ ε α ε
β α ε µ
ο
The estimated model is:
(
1.014395) (
0.101314)
(
0.013955)
(
0.013836)
054887 . 0 163030
. 0 233799 . 0 37168 . ˆ 12
1 1 1
1 1 1
−
−
−
−
− + −
−
−
=
t t t
t t
t h h
h h
Log ε ε
From Table (2) below, it is clear that the parameters of the model are significant and this indicates the model is appropriate to analyze the stock market data in Egypt. We also note that the asymmetry parameterγ =−0.054887 is significant and this means that the volatility is asymmetry and it is negative indicating that negative shocks imply higher conditional variance than positive shocks.
Table 2: Results of Estimates for Asymmetric Models of Asymmetric GARCH
Models Coefficients AIC SIC Log Likelihood
TARCH(1, 1)
* 1
* 1
* 1
*
*
364511 .
0
638472 .
0 518907 .
1
06 80 . 4
000216 .
0
=
−
=
=
−
=
−
=
β γ α α µ
ο E
-7.364258 -7.096233 8473.897
EGARCH(1, 1)
* 1
* 1
*
*
233799 .
0 05488 . 0
37168 . 12
000658 .
0
−
=
−
=
=
−
=
β α α µ
ο -7.096233 -7.083753 8165.668
(*) means that the estimated parameters of the model are statistically significant
7. Forecasting Using GARCH Models
In order to evaluate the volatility forecasting performance of the symmetric and asymmetric GARCH models, the following statistical measures are used.
1. Mean Absolute Percentage Error (MAPE):
t t t
h h h
MAPE N −
= 1 ˆ
2. Mean Absolute Error (MAE):
=
−
=
n
t
t h
N h MAE
1 t
1 ˆ
3. Mean Square Error (MSE):
(
ˆ)
1 2
1
=
−
=
n
t
t
t h
N h MSE
Table 3: Results of Prediction Using GARCH Models
Model Forecast sample µ MSE MAE MAPE
GARCH (1, 1) 100 -0.000339 0.006538 0.002737 135.5229
GARCH (1, 2) 100 -0.000606 0.006457 0.002426 120.3176
GARCH (2, 1) 100 0.000384 0.006588 0.002937 187.3767
TARCH (1, 1) 100 -0.000216 0.006419 0.002360 116.2573
EGARCH (1, 1) 100 0.000658 0.006447 0.002396 157.2470
Looking at Table (3) above, we find some important remarks: Forecasting is in-sample volatility forecasts (100 observations). In addition, the forecasting for the conditional mean is constant.
Finally, the volatility forecasting results indicate that the best model is the TARCH (1, 1) model where it has the lowest values for the three error measures: MSE, MAE and MAPE.
8. Models Tests
8.1. Durbin-Watson TestDurbin-Watson Test is one of the tests that reveals the existence of a serial correlation of the first degree (for one period). The test statistic is:
( )
=
=
− −
= T
t t T
t
t t
d
1 2 2
2 1
ˆ ˆ ˆ
ε ε ε
The closer the statistical value of 2 indicates the lack of serial correlation of random errors. The following table (4) illustrates the results of this test with the used GARCH models.
Table 4: Durbin-Watson test results
Durbin-Watson Model
1.794426 GARCH (1, 1)
1.774824 GARCH (1, 2)
1.827484 GARCH (2, 1)
1.802214 TARCH (1, 1)
1.831774 EGARCH (1, 1)
Note that the value of the test statistic is close to 2 in all models and this indicates the lack of serial correlation of random errors.
8.2. Lagrange Multiplier Test
After estimating the different GARCH models, we perform this test to ensure that there are no additional effects of additional ARCH parameters in the respective models. This means that the models used were all correct models and no need to use models that are more complicated. The test is as follows:
( )
(
ARCH)
ne at least o H
ARCH No
H
i p
0 :
0 :
1
2 1
≠
=
=
=
=
α α α
ο α ⋯
The test statistic is TR2which followχp2 and can be approximated by F- distribution,F
(
p,T − p−1)
, where T is the number of observations and R2 is the coefficient of determination resulting from the autoregressive equation:2 2
2 2 2
1 1 2
p t p t
t
t =α +α ε − +α ε − + +α ε −
ε ο ⋯
Using this test, we obtained the following results:
Table 5: Results of the tests of ARCH terms In different models after estimation
Model F - Value p-value
GARCH (1, 1) 0.010834 0.917
GARCH (1, 2) 0.513984 0.473
GARCH (2, 1) 0.010926 0.917
TARCH (1, 1) 0.015588 0.901
EGARCH (1, 1) 0.325421 0.568
The results in the above table show that the parameters of the ARCH model are not significant, indicating the quality of these models.
8.3. Choosing the Best Fitting Model
To evaluate the performance of the GARCH models used to analyze the Egyptian stock market data, we use the following selection criteria:
1. Akaike Information Criterion (AIC) 2. Schwarz Information Criterion (SIC) 3. MLE values
Table 6: Results of different GARCH models tests
Model MLE AIC SIC
GARCH (1, 1) 8466.75 -7.3589 -7.3489
GARCH (1, 2) 8205.15 -7.1305 -7.1180
GARCH (2, 1) 8387.49 -7.2891 -7.2766
TARCH (1, 1) 8473.89 -7.3642 -7.3517
EGARCH (1, 1) 8165.66 -7.0960 -7.0830
From Table (6) above, it is clear that the TARCH (1, 1) model has the highest MLE value (8473.89) and the lowest AIC and SIC values (-7.3642 and -7.3517, respectively). Thus, we can conclude that model TARCH (1, 1) is the best model for the Egyptian stock market.
9. Conclusion
Symmetric and Asymmetric GARCH models are used to analyze the financial market data in Egypt.
The symmetric models used in the analysis are GARCH (1, 1), GARCH (1, 2), GARCH (2, 1), GARCH (2, 2) and GARCH-IN-MEAN (1, 1). The asymmetric models that take into account whether changes in the market are negative or positive and respond differently in each case. The asymmetric models used in the statistical analysis are TARCH (1, 1) and EGARCH (1, 1). The results indicated the exclusion of three symmetric models: GARCH (1, 1), GARCH (2, 2) and GARCH-in-MEAN (1, 1) and the appropriateness of the other models for analyzing the financial market data in Egypt. The best model in expressing volatility in the Egyptian capital market as well as in forecasting the conditional variance is the TARCH (1, 1) model.
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