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An efficient and well behaved capital market can be regarded as a prerequisite for the sustainable financial development for an economy. For making the stock market efficient and reducing uncertainty, volatility measure is necessary for the policy makers. The main objective of this paper is to examine relative ability of various models to forecast future volatility and to devise appropriate volatility model for capturing variability in stock returns of Dhaka Stock Exchange (DSE). By exploiting daily data spanning from 27
^{th}
November, 2001 to 31
^{st}
July, 2013, it was found that, from volatility persistency perspective MA(2)-GARCH(2, 1) is better due to both in sample and out of sample accuracy. In contrast, from capturing asymmetric effect perspective MA(2)-EGARCH(1, 3) is better. Thus, there was no clear winner and hence the decision should depend on the purpose of the concerned people.

Stock Market volatility is the variability in stock prices during a period which is perceived as a measure of risk by investors. It may affect business investments, financial market performance and economic performance directly [

Like other developing countries in Bangladesh stock market is an emerging market but has been experiencing inefficiency from its inception. To make the market efficient and reduce uncertainty, volatility measure is necessary for the policy makers. The main objective of this paper is to examine relative ability of various models to forecast future volatility and to devise appropriate volatility model for capturing variability in stock returns of Dhaka Stock Exchange (DSE).

On this background already an enormous amount of effort has been made from the researchers to model the variance dynamics of stock market return and its different characteristics in Bangladesh [

The rest of the article is organized as follows: Section 2 provides an overview of existing literature, Section 3 discusses about data, variable construction and model specification and Section 4 contains the estimation results and findings. Finally, Section 5 concludes.

Several researchers have examined the volatility of stock returns of Dhaka Stock Exchange (DSE). They considered different models for different time period and sample size and found contradictory results in some cases. Rayhan et al. [

Earlier Huq et al. [

With a view to capture the asymmetric effect during 90s Chowdhury [

It is widely accepted that while modeling the variance the mean equation is of vital importance. As failure of appropriate model detection for mean might result in less efficiency of parameters used in variance model due to potential presence of autocorrelation. This issue has largely been ignored in the stock market volatility literature in Bangladesh. Thus, the current study would at first optimize the mean equation while quest for modeling the variance dynamics in stock return. The main contribution here would be to set up appropriate volatility model for efficiently capturing variance dynamics in stock returns of Dhaka Stock Exchange (DSE).

This paper has exploited the daily data on general stock price index of DSE spanning from 27^{th} November, 2001 to 31^{st} July, 2013^{1}. Although there are regular fluctuations in the movement of stock price index, generally they do contain the unit root property and hence can be characterized as nonstationary in nature. Thus, we have used logarithmic transformation to convert the data into stock market return which would have greater possibility to be stationary and appropriate for analysis. We have used the following formula to measure the return:

r t = ln ( P ) − ln ( P t − 1 )

Here, r t stands for stock market return at day t, P t and P t − 1 stands for general stock price index at day t and day before t. EViews 9 has been used as the statistical software for performing quantitative exercise.

^{1}The data has been limited to this period because of the change in Index definition during 2013.

The perfect modeling of conditional mean can be considered as a prerequisite of correct model specification of conditional variance of stock market return. Along with independent variables researchers of volatility modeling usually augment the conditional mean model either with Autoregressive (AR) or with Moving Average (MA) or even with a mixture of these two (Autoregressive Moving Average, ARMA) process [

r t = μ + ∑ i = 1 p θ i r t − i + ε t

r t = μ + ∑ i = 1 q ϕ i ε t − i + ε t

where, μ is the constant term, θ 1 , θ 2 , ⋯ , θ p and ϕ 1 , ϕ 2 , ⋯ , ϕ q are the lagged coefficients and ε t is a white noise process. For modeling the volatility in stock market return using the above two different types of mean equation, the variance equation is specified following family of GARCH models namely, Standard GARCH, APARCH, EGARCH and IGRACH models. GARCH models developed by Bollerslev [

ε t = h t v t

Here, v t ∼ i i d ( 0 , 1 ) , and specification of h t will determine different varieties of GARCH family models while each of which will serve specific purpose. For analyzing the different feature of stock market return by modeling its volatility we have estimated the following models:

GARCH ( p , q ) : h t = η + ∑ i = 1 q α i ε t − i 2 + ∑ i = 1 p β j h t − j

APARCH ( p , q ) : h t = η + ∑ i = 1 q α i ( | ε t − i | − γ i ε t − i ) δ + ∑ j = 1 p β j ( h t − j ) δ

EGARCH ( p , q ) : ln h t = η + ∑ i = 1 q ( α i ε t − i h t − i + λ i | ε t − i h t − i | ) + ∑ j = 1 p β j ln ( h t − j )

IGARCH ( p , q ) : h t = ∑ i = 1 q α i ε t − i 2 + ∑ j = 1 p β j h t − j , where, ∑ i = 1 q α i + ∑ j = 1 p β j = 1

Here, in GARCH(p, q) model, p and q denotes the lag order of GARCH and ARCH terms respectively. In order to have well behaved variance restrictions on the parameters needed to be imposed, for instance, η > 0 , α i ≥ 0 and β j ≥ 0 . The volatility persistence will be measured by summing up the ARCH and GARCH coefficients ( ∑ i = 1 q α i + ∑ i = 1 p β j ) which is expected to be less than unity to have a stationary residual and nonnegative variance.

APARCH model developed by Ding, Granger & Engel [

Another volatility model that captures the asymmetric effect is EGARCH developed by Nelson [

Engle & Bollerslev [

To choose among the conditional heteroscedasticity models we have compared their out of sample volatility forecasting performance. In particular different models with several specifications have been compared in terms of Root Mean Square Error (RMSE), Mean Absolute Error (MAE), Mean Absolute Percent Error (MAPE) and Theil Inequality (TI) for identifying the most appropriate one with the current data.

As discussed in Section 3 it is imperative to identify the conditional mean model appropriately for modeling the volatility in concerned variable. Therefore, selection of estimation method of conditional mean model for stock return series is of vital importance.

The conditional mean model for stock return has been developed with two specifications one with AR terms and the other with MA terms.

Since ARCH effect was present we have tried to model the volatility clustering using GARCH family models and explain the characteristics of the market while capturing its variance dynamics. To begin with we have estimated standard GARCH model both with AR(2) and MA(2) conditional mean specification.

Variables | Coefficients | |
---|---|---|

Dependent Variable, r_{t} | ||

(AR) | (MA) | |

μ | 0.0005** | 0.0005** |

(0.0002) | (0.0002) | |

r t − 1 | 0.0401** | |

(0.0184) | ||

r t − 2 | −0.0430** | |

(0.0184) | ||

ε t − 1 | 0.0402** | |

(0.0184) | ||

ε t − 2 | −0.0397** | |

(0.0184) | ||

H_{O}: No ARCH Effect | ||

F-Statistic | 134.9106 | 134.8517 |

Probability | 0.0000 | 0.0000 |

Note: Standard Errors are in Parenthesis. *** indicates significant at 10 per cent level, ** significant at 5 per cent level and * indicates that at 1 per cent level.

Coefficients | GARCH | ||||
---|---|---|---|---|---|

(1, 1) | (1, 2) | (1, 3) | (1, 4) | (2, 1) | |

μ | 0.0014** | 0.0034* | 0.0033* | 0.0013** | 0.0013** |

(0.0006) | (0.0003) | (0.0003) | (0.0006) | (0.0005) | |

ϕ 1 | 0.0862*** | 0.0629** | 0.0657*** | 0.0987*** | 0.1096** |

(0.0469) | (0.0312) | (0.0363) | (0.0523) | (0.0461) | |

ϕ 2 | −0.0384 | −0.0435** | −0.0197 | −0.0322 | −0.0389 |

(0.0323) | (0.0215) | (0.0213) | (0.0422) | (0.0366) | |

η | 1.25E−05 | 6.11E−05* | 3.49E−05* | 5.38E−06* | 1.29E−05** |

(7.73E−06) | (2.58E−06) | (2.88E−06) | (1.86E−06) | (5.91E−06) | |

α 1 | 0.3465* | 0.3195* | 0.3365* | 0.2770* | 0.1839** |

(0.1104) | (0.0490) | (0.0332) | (0.0911) | (0.0850) | |

α 2 | 0.4115 | ||||

(0.3785) | |||||

β 1 | 0.6684* | 0.7351* | 0.9718* | 1.4214* | 0.5159* |

(0.0789) | (0.1755) | (0.1252) | (0.0963) | (0.1580) | |

β 2 | −0.2042*** | −0.4266* | −1.1294* | ||

(0.1121) | (0.0965) | (0.1612) | |||

β 3 | 0.0768* | 0.5980* | |||

(0.0199) | (0.1286) | ||||

β 4 | −0.1167** | ||||

(0.0496) | |||||

Q1 (5) | 33.554* | 31.074* | 32.508* | 36.777* | 32.264* |

Q1 (10) | 39.896* | 37.997* | 38.392* | 44.698* | 40.258* |

Q2 (5) | 0.1161 | 0.1643 | 0.0911 | 0.1289 | 0.2274 |

Q2 (10) | 0.3068 | 1.1046 | 0.1850 | 0.4465 | 0.5983 |

Log Likelihood | 8700.438 | 8530.647 | 8600.336 | 8740.959 | 8746.996 |

F Stat. | 0.0512 | 6.42E−05 | 0.0252 | 0.0528 | 0.0132 |

Prob. | 0.8208 | 0.9936 | 0.8739 | 0.8093 | 0.9084 |

Note: Robust Standard Errors are in Parenthesis. *** indicates significant at 10 per cent level, ** indicates significant at 5 per cent level and * indicates that at 1 per cent level.

the constant, η and the ARCH coefficient, α is positive and significant. However, though the GARCH coefficients denoted by β has been found to be significant in all specifications the sign for many of them was not conventional. Also as the results show in most of the cases the sum of ARCH and GARCH coefficients turn out to be greater than or close to unity implying that for those, variance won’t remain well behaved. The one exception can be found in GARCH(2, 1) where the first ARCH coefficient along with the GARCH coefficient was found to be positive significant while the second ARCH coefficient was insignificant. The sum of ARCH and GARCH coefficients remained less than unity with a positive significant constant. Thus it satisfies all the restrictions and might be a potential model.

The residuals of the GARCH models are needed to be white noise. Thus, a diagnostic test in the form of Ljung-Box Q test has been performed under the null hypothesis, (H_{0}: No Serial Correlation in the Error Term). We calculate Q-Statistics for the standardized residuals (Q1) and for their squared values (Q2). It can be seen that all Q1-Statistics are significant at 1 per cent level while the Q2-Statistics were not. Therefore autocorrelation was found when we test based on level residuals and that was absent when test based on squared residuals. Nevertheless as F-statistic turn out to be insignificant for all models, it can be argued that none of them have ARCH effect. As the post estimation diagnostic results was found to be same for all but AR(2) − GARCH(2, 1) is the one which satisfies the restrictions, it can be treated as the appropriate one. Here the models were not augmented further as the coefficients were not found to be significant.

While the variance modeling was performed with MA mean specification the findings turn out to be almost same. However, MA(2) − GARCH(2, 1) specification has been observed to follow all the required restrictions with the similar post diagnostic properties as AR(2) − GARCH(2, 1). It is evident from the findings that among these two the former one have higher (8746.99) log likelihood than the later (8739.313), also the information criteria for the former one (SIC = −5.9597) is found to be lower than the later (SIC = −5.9585). Therefore, modeling volatility of stock return with MA mean specification has been revealed to be more appropriate than its AR counterpart.

For introducing nonlinearity in the variance equation and analyzing the asymmetric feature of volatility in the stock market return we have estimated APACRH model. As MA specification was found to provide more appropriate result, for mean equation MA(2) specification was used.

As APARCH model failed to capture the asymmetric volatility effect which could potentially be present in the stock market, we have given effort to examine this feature using another volatility modeling approach called EGARCH. Unlike the earlier one it doesn’t need to impose any non negativity restrictions while capturing the asymmetric volatility effect. Also, along with “asymmetry parameter” it estimates another parameter called “size parameter” measuring the size of shocks. Thus in contrast to APARCH, EGARCH can measure the existence of leverage effect as well as the magnitude of shock.

The results on diagnostic indicators shows that MA(2) − EGARCH(1, 2) specification have autocorrelation problem as well as the model still contains ARCH effect (as F-Statistic is significant). Nonetheless, MA(2) − EGARCH(1, 1) and MA(2) − EGARCH(1, 3) have no autocorrelation when we considers the squared residuals. Also, they do not have any ARCH effect reveled by insignificant F-Statistic. So, these two specifications are better than the earlier one. Among them log likelihood is maximum (8815.42) and information criteria is minimum (SIC = −6.0010) for MA(2) − EGARCH(1, 3). Thus, it could be the potentially appropriate model for capturing the asymmetric effect in stock market return.

Finally we have given effort to model the volatility clustering in stock return addressing the restriction saying that “persistence parameters sum up to unit”. The rationale for this restriction is that earlier in some of the GARCH models it was found that the sum of the coefficients were close to or more than unit implying that the variance of stock return might be nonstationary. Imposing this restriction in standard GARCH models leads to IGARCH specification.

With a view to choose among the models we have compared the performance among them in terms of accuracy of out of sample volatility forecasting. The

Variables | EGARCH with Normal Distribution | ||
---|---|---|---|

(1, 1) | (1, 2) | (1, 3) | |

μ | 0.0001 | 0.0001 | 0.0005** |

(0.0002) | (0.0003) | (0.0002) | |

ϕ 1 | 0.1489* | 0.1411* | 0.1542* |

(0.0220) | (0.0226) | (0.0222) | |

ϕ 2 | 0.0135 | 0.0482*** | 0.0325 |

(0.0556) | (0.0260) | (0.0580) | |

η | −1.2659* | −0.2956* | −0.5718* |

(0.4370) | (0.0737) | (0.1038) | |

α | 0.4865* | 0.1611* | 0.3038* |

(0.1149) | (0.0201) | (0.0432) | |

λ | −0.1527*** | 0.0147 | −0.0988** |

(0.0914) | (0.0090) | (0.0526) | |

β 1 | 0.8945* | 0.0200* | 1.7725* |

(0.0450) | (0.0042) | (0.0846) | |

β 2 | 0.9594* | −1.4774* | |

(0.0045) | (0.1270) | ||

β 3 | 0.6649* | ||

(0.0408) | |||

Q1 (5) | 21.938* | 21.402* | 27.451* |

Q1 (10) | 28.350* | 29.040* | 33.939* |

Q2 (5) | 0.0892 | 25.218* | 0.2585 |

Q2 (10) | 0.2739 | 27.467* | 1.0130 |

Log Likelihood | 8744.181 | 8715.984 | 8815.427 |

F Statistic | 0.0467 | 13.547* | 0.0102 |

Probability | 0.8288 | 0.000 | 0.9193 |

Note: Robust Standard Errors are in Parenthesis. *** indicates significant at 10 per cent level, ** indicates significant at 5 per cent level and * indicates that at 1 per cent level.

pseudo sample has been created using the observations for the period 27^{th} November, 2001 to 30^{th} December 2010. The forecasting accuracy of different models was compared in terms of RMSE, MAE, MAPE and TI for the period 2^{nd} January 2011 to 31^{st} July, 2013.

MA(2) − GARCH | ||||||||
---|---|---|---|---|---|---|---|---|

(1, 1) | (1, 2) | (1, 3) | (1, 4) | (2, 1) | ||||

RMSE | 0.023 | 0.023 | 0.023 | 0.023 | 0.023 | |||

MAE | 0.016 | 0.016 | 0.016 | 0.016 | 0.016 | |||

MAPE | 196.103 | 199.519 | 201.053 | 201.076 | 180.713 | |||

TI | 0.880 | 0.876 | 0.879 | 0.872 | 0.875 | |||

MA(2) − EGARCH | MA(2) − APARCH | |||||||

(1, 1) | (1, 3) | (1, 1) | ||||||

RMSE | 0.024 | 0.024 | 0.023 | |||||

MAE | 0.016 | 0.016 | 0.016 | |||||

MAPE | 185.595 | 203.204 | 179.875 | |||||

TI | 0.858 | 0.850 | 0.781 | |||||

satisfies all the required restrictions and revealed the volatility clustering feature of stock return appropriately. Thus if the purpose is only modeling the volatility clustering and forecasting the future volatility then MA(2) − GARCH(2, 1) will perform better.

When we have tried to forecast the volatility in stock return while addressing the asymmetric affect it was found again that in terms of RMSE and MAE, MA(2) − EGARCH(1, 1), MA(2) − EGARCH(1, 3) and MA(2) − APARCH(1, 1) all are same. In terms of TI, MA(2) − EGARCH(1, 3) is better than MA(2) − EGARCH(1, 1) while its other way around in terms of MAPE. However, among EGARCH and APARCH, MA(2) − APARCH(1, 1) has the lowest value both for TI and MAPE. Nevertheless, earlier it was observed that APARCH model failed to capture the asymmetric volatility effect. Thus, if the purpose is to capture the asymmetric volatility effect along with forecasting then the appropriate model would be MA(2) − EGARCH(1, 3) as it was able to capture the asymmetric volatility effect appropriately. It also had maximum likelihood and minimum information criteria when full sample was used and have a lower TI value when out of sample forecasting is considered.

An efficient and well behaved capital market can be regarded as a prerequisite for the sustainable financial development for an economy. The importance appeared to be even more crucial when the country is in early stage of development as characteristics and behavior of this market is and usually does maintain a close relation to the other macroeconomic indicators. Therefore modeling and forecasting the variance dynamics of stock market return in Bangladesh has gained a greater attention from the academicians and researchers, since the country has been trying to develop an efficient capital market from a long ago. Modeling of conditional variance of stock return also has greater relevance for taking decision by the investors and policy makers regarding portfolio optimization, asset pricing and finally risk management. An appropriate variance model will help concerned people to have a better forecast of volatility which in turn would be significant for having efficient portfolio distribution, better risk management capacity and more specific derivative prices for particular financial instrument.

Identifying appropriate volatility model for capturing fluctuations in stock returns is of significant policy relevance to the policy makers as well. The importance lies on the fact that unregulated fluctuation in asset return could influence investment decision that can manifest in the real sector with adverse consequences for economic growth and development. Undue fluctuation of stock return could also impose challenges to monetary policy formulation as increase in stock prices stimulates interest rate which eventually could generate inflationary spree in the economy [

In the above context this paper goes with challenge of exploring comparative ability of capturing in-sample and out-of-sample volatility of different conditionally heteroscedastic econometric models regarding stock market return in Bangladesh. More specifically, it explored daily data for the period 27^{th} November, 2001 to 31^{st} July, 2013 from DSE and has used different GARCH class models to compare their performance form in-sample estimation accuracy and out-of-sample forecasting accuracy perspective to come up with the best performing one. By developing appropriate mean equation for addressing the autocorrelation problem the paper compared different order of variance models namely, GARCH, APARCH, EGARCH and IGRACH. While concerning in sample estimation accuracy it was found that MA(2) − GARCH(2, 1) out performs AR(2) − GARCH(2, 1) following post estimation diagnostics. When nonlinearity was allowed in variance equation to capture the asymmetric effect MA(2) − EGARCH(1, 3) was found to be successful over MA(2) − APARCH(1, 1). By assuming that variance could potentially be remained nonstationary it was observed that MA(2) − IGARCH(1, 2) has more in sample accuracy than its other counterparts. As far as out of sample volatility forecasting is concerned it was found that for only modeling the volatility clustering MA(2) − GARCH(2, 1) out performs the other. However, for modeling the volatility clustering addressing the asymmetric effect MA(2) − EGARCH(1, 3) provides more out of sample accurate result among its competing ones. Therefore, it can be argued with evidence that there is no clear winner. The decision mainly depends on the purpose of the concerned people. From volatility persistency perspective MA(2) − GARCH(2, 1) is better due to both in sample and out of sample accuracy. In contrast, from capturing asymmetric effect perspective MA(2) − EGARCH(1, 3) is better.

One potential shortcoming of the current effort could be that it did not incorporate the issue of existence of structural break in the variance dynamics while modeling the conditional heteroscedasticity. Nevertheless, addressing the structural break in variance dynamics could remain as a further area of research.

The authors declare no conflicts of interest regarding the publication of this paper.

Abdullah, S.M., Kabir, M.A., Jahan, K. and Siddiqua, S. (2018) Which Model Performs Better While Forecasting Stock Market Volatility? Answer for Dhaka Stock Exchange (DSE). Theoretical Economics Letters, 8, 3203-3222. https://doi.org/10.4236/tel.2018.814199

Variable | Mean | Median | Std. Deviation | Skewness | Kurtosis |
---|---|---|---|---|---|

DSE General Index | 2810.734 | 2074.550 | 1935.526 | 0.897 | 2.847 |

Augmented Dickey Fuller (ADF) Test | Kwiatkowski ?Philips?Schmidt?Shin (KPSS) Test | ||||||
---|---|---|---|---|---|---|---|

H_{0}: Stock Return has a Unit Root | H_{0}: Stock Return is Stationary | ||||||

Intercept | Trend and Intercept | Intercept | Trend and Intercept | ||||

Test Statistic | Probability | Test Statistic | Probability | Test Statistic | 1% Critical Value | Test Statistic | 1% Critical Value |

−52.0260* | 0.0000 | −52.0246 | 0.0000 | 0.174 | 0.739 | 0.133 | 0.216 |

Optimum Order of Lag is Zero | Optimum Order of Lag is Zero | Optimum Bandwidth is Nine | Optimum Bandwidth is Eight |

Note: *indicates significant at 1% level. In case of selecting optimal lag length for ADF test, the SIC has been minimized. The optimum bandwidth for KPSS test has been selected using Newey-West method and following Bartlett Kernel for spectral estimation.

Coefficients | GARCH | |||||
---|---|---|---|---|---|---|

(1, 1) | (1, 2) | (1, 3) | (1, 4) | (1, 5) | (2, 1) | |

μ | 0.0014** | 0.0014** | 0.0031* | 0.0013** | 0.0012*** | 0.0012** |

(0.0006) | (0.0006) | (0.0003) | (0.0006) | (0.0006) | (0.0005) | |

θ 1 | 0.0812*** | 0.0936*** | 0.0313 | 0.0950*** | 0.1005** | 0.1039** |

(0.0483) | (0.0507) | (0.0388) | (0.0531) | (0.0512) | (0.0477) | |

θ 2 | −0.0310 | −0.0391 | −0.0319 | −0.0290 | −0.0352 | −0.0360 |

(0.0402) | (0.0460) | (0.0236) | (0.0548) | (0.0543) | (0.0492) | |

η | 1.25E−05 | 9.83E−06* | 3.53E−05* | 5.29E−06* | 3.50E−06* | 1.27E−05** |

(7.68E−06) | (3.08E−06) | (2.84E−06) | (1.79E−06) | (6.07E−07) | (5.87E−06) | |

α 1 | 0.3458* | 0.3187* | 0.3498* | 0.2738* | 0.2282** | 0.1829** |

(0.1111) | (0.0703) | (0.0341) | (0.0878) | (0.0911) | (0.0841) | |

α 2 | 0.4092 | |||||

(0.3770) | ||||||

β 1 | 0.6695* | 0.9743* | 0.9816* | 1.4296* | 1.7836* | 0.5187* |

(0.0791) | (0.0821) | (0.1204) | (0.1068) | (0.0961) | (0.1580) | |

β 2 | −0.2596* | −0.4264* | −1.1306* | −1.9036* | ||

(0.0812) | (0.0910) | (0.1831) | (0.1880) | |||

β 3 | 0.0732* | 0.5869* | 1.5081* | |||

(0.0160) | (0.1394) | (0.2141) | ||||

β 4 | −0.1099** | −0.8152* | ||||

(0.0548) | (0.1469) | |||||

β 5 | 0.2462* | |||||

(0.0517) | ||||||

Q1 (5) | 35.381* | 34.211* | 48.953* | 37.973* | 35.341* | 34.203* |

Q1 (10) | 41.864* | 41.195* | 56.104* | 45.295* | 45.573* | 42.249* |

Q2 (5) | 0.1167 | 0.1709 | 0.1029 | 0.1291 | 0.141 | 0.227 |

Q2 (10) | 0.3078 | 0.3840 | 0.3891 | 0.449 | 0.423 | 0.601 |

Log Likelihood | 8693.102 | 8710.725 | 8584.991 | 8733.666 | 8743.978 | 8739.313 |

F Stat. | 0.0518 | 0.0564 | 0.024 | 0.058 | 0.040 | 0.013 |

Prob. | 0.8199 | 0.8122 | 0.874 | 0.809 | 0.841 | 0.906 |

Note: Robust Standard Errors are in Parenthesis. *** indicates significant at 10 per cent level, ** indicates significant at 5 per cent level and * indicates that at 1 per cent level.

Coefficients | APARCH(1, 1) |
---|---|

μ | 0.0004** |

(0.0002) | |

ϕ 1 | 0.1231* |

(0.0224) | |

ϕ 2 | −0.0088 |

(0.0460) | |

η | 0.0002 |

(0.0001) | |

α | 0.2867* |

(0.0763) | |

γ | 0.2901 |

(0.1907) | |

δ | 1.3731* |

(0.2603) | |

β | 0.7070* |

(0.0812) | |

Q1 (5) | 26.810* |

Q1 (10) | 33.524* |

Q2 (5) | 0.0719 |

Q2 (10) | 0.2538 |

Log Likelihood | 8740.570 |

F Stat. | 0.0357 |

Prob. | 0.8501 |

Variables | IGARCH with Normal Distribution | ||
---|---|---|---|

(1, 1) | (1, 2) | (1, 3) | |

μ | 0.0003 | 0.0001 | 0.0003 |

(0.0002) | (0.0002) | (0.0002) | |

ϕ 1 | 0.0908* | 0.1123* | 0.0879* |

(0.0335) | (0.0305) | (0.0326) | |

ϕ 2 | −0.0390 | −0.0465 | −0.0468 |

(0.0280) | (0.0298) | (0.0287) | |

α | 0.0109 | 0.0292 | 0.0140** |

(0.0089) | (0.0188) | (0.0069) | |

β 1 | 0.9890* | −0.0015 | 1.4322* |

(0.0089) | (0.0040) | (0.0109) | |

β 2 | 0.9722* | −1.4204* | |

(0.0225) | (0.0233) | ||

β 3 | 0.9741* | ||

(0.0194) | |||

Q1 (5) | 16.274* | 15.658* | 20.861* |

Q1 (10) | 25.969* | 25.727* | 32.325* |

Q2 (5) | 4.963 | 3.336 | 9.628** |

Q2 (10) | 5.545 | 3.890 | 11.767 |

Log Likelihood | 8529.451 | 8582.327 | 8573.475 |

F Statistic | 2.820*** | 1.578 | 4.475** |

Probability | 0.093 | 0.209 | 0.034 |