Modelling of volatility in the South African mining sector : application of ARCH and GARCH models.

Where the work of others has been used, this is duly acknowledged in the text. The primary objectives of this study were to model volatility on the Johannesburg Stock Exchange (JSE) using data from two mining companies. Before using univariate GARCH models, the presence of an ARCH effect should be checked and this was found in both mining companies.

Autocorrelation in the mean was also present, which could then be removed by modeling the AR (Autoregressive) model and thus using the residual to run the GARCH models. The results show that there is a high persistence of volatility in both mining companies with higher volatility in Harmony Gold Limited mining company. The Dynamic Conditional Correlation (DCC) GARCH model was also used between the two daily data mining companies within the same period as that of the GARCH-type and stochastic volatility models.

The overall finding indicated that the correlation between the two mining companies varied over time. This thesis is dedicated to my brothers, Jean Claude Kubwimana and Gaspard Maniraho, for their guidance, support and love during the time we were together.

Introduction

Background
Literature Review
Objectives

Chapter Summary

Data Description
Data Exploration

Harmony Gold Mining Limited

Goudarzi & Ramanarayanan (2011) used GARCH, EGARCH and GJR-GARCH models to model asymmetric volatility in the Indian stock market. They found that volatility had a positive serial correlation in the market, as they expected. In the study by Mzamane (2013), univariate and multivariate GARCH models were proposed to extend the volatility model of the JSE index.

Mzamane planned to investigate market volatility using GARCH, GJR-GARCH, EGARCH and APARCH models. The findings show that the GJR-GARCH model was suitable for detecting volatility in the JSE index, with volatility in the residuals and leverage present in the JSE index returns. The primary objectives of this study are to assess the statistical properties of the GARCH models and to use these models to study volatility in the South African mining sector.

In this section, two datasets were used to model volatility in the JSE companies using ARCH, GARCH, multivariate and stochastic volatility models. The result clearly shows that the standard deviation is 3.6%, which implies a high degree of volatility in the market.

Table 2.1: Descriptive Statistic for Harmony Gold Mining Daily Returns

Close

Impala Platinum Holdings Limited

The data used for Impala Platinum Holdings Limited was a daily closing price from 3 January 1995 to 3 July 2014. There is also a wide gap between the low and high and this supports the high variability in price changes for the JSE mining sector of Impala Plat. - inum Holdings limited liability company. This is supported by highly significant J-B and Shapiro-Wilk statistics in Table 2.6, and also in Figure 2.10 reveals that there is an excess of kurtosis.

Thus, reject the null hypothesis, which states that there is no autocorrelation, and accept the alternative hypothesis, which states that there is autocorrelation in the return data. Finally, Table 2.8 shows higher test statistics compared to their corresponding critical values, and the p-values for all lags are less than the 0.005 significance level. The null hypothesis is rejected and thus concludes that there is heteroskedasticity in the return data.

Figure 2.11 shows that there is evidence of volatility clustering in the company Impala Platinum Holdings Limited. Furthermore, Figure 2.12 shows that there is some evidence of high volatility clustering, as well as some periods of medium and stable volatility clustering in the Impala Platinum Holdings Limited company.

Figure 2.10: Histogram of Daily Returns for Impala Platinum Holdings Limited

Chapter Summary

The ARCH model

The ARCH (1) model
Estimation of ARCH (1) model
Forecasting using ARCH (1) model
The ARCH (q) model
Estimation of ARCH (q) model
Forecasting using ARCH (q) model
The GARCH model
The GARCH (1,1) model
Estimation of GARCH (1,1) model
Forecasting using GARCH (1,1) model
The GARCH (p,q) model
Estimation of GARCH (p,q) model
Forecasting using GARCH (p,q) model
Extension of GARCH model
EGARCH model
Forecasting using EGARCH model
The GJR-GARCH model
The Asymmetric Power ARCH (APARCH) model
Model Selection Criteria
Testing for ARCH effect

Thus, the conditional mean of t equal to zero can be shown in the following equations. Normally the logarithm of the likelihood is used, while the maximization of the conditional likelihood function is equal to the maximization of the logarithm of the conditional likelihood. Note that maximizing Equation 3.8, with respect to τ, where τ = ( ˆα0,αˆ1) is a nonlinear optimization problem.

Under normality, the logarithm of the likelihood is used, and when the conditional likelihood is maximized, it is equivalent to maximizing the logarithm of the conditional likelihood. The ARCH model provides a way to describe the behavior of the conditional variance instead of showing us the causes of such behavior. This occurs because of the slow response of the ARCH model to the largely isolated shock in the return series (Tsay 2005).

The GARCH (1,1) model was introduced by Bollerslev (1986) after realizing that the ARCH model is simple and requires many parameters to fit the data (Tsay 2010). The estimation of the parameters of the GARCH (1,1) model can be solved in the same way as for the ARCH (1) model. However, the conditional variance of the GARCH (1,1) model depends on the past variance, and the initial value of the past conditional variance is required which is σ12.

To estimate the parameters of the GARCH (1,1) model, Bollerslev (1986) suggested that the unconditional variance for t should be used as an initial value for this variance, and that the equation can be written as. Consider equation 3.29 of the GARCH (1,1) model and assume that the predicted origin is k. The one-step-ahead volatility forecast is given by. where 2T and σ2T are both known at time index T. 3.37). The maximum likelihood estimator method can be used to estimate the parameters for the GARCH (p,q) model.

Maximizing the conditional probability function is equivalent to maximizing the logarithm of the conditional probability. The advantage of the EGARCH model is that even though the parameters are negative, σt2 is modeled (Su 2010). We assume that the parameters of the model are known and that the innovations are standard Gaussian (Tsay 2010).

Thus, the 1-step-ahead volatility forecast at origin k is given by ˆσk2(1) = σk+12, assuming that all the quantities on the right-hand side of the above equation are known. Note that model parameter estimation and forecast volatility are similar to ARCH and GARCH models.

SSR 1 m

Model checking
Chapter Summary
Introduction
Selection of the Best Model
Fitting the Model
Analysis of Harmony Gold Mining

Application for extensions of GARCH models

Analysis of Impala Platinum Holdings Limited

Application for extensions of GARCH models
Chapter Summary

Introduction

History of Multivariate GARCH model

Dynamic Conditional Correlation (DCC) model
Dynamic Conditional Correlation (DCC) model pa- rameter estimationrameter estimation
DCC model diagnostics
Application for DCC-GARCH model and Results

Parameter Estimation for DCC-GARCH (1,1) model
Diagnostic Checking for the DCC-GARCH (1,1) model

Chapter Summary
Introduction
State-Space Models
The Kalman Filter
The Kalman Smoother
The Lag One Covariance Smoother
Maximum Likelihood Estimation
The Expectation Maximization Algorithm (EM)
The Stochastic Volatility Model (SV)
Application of Stochastic Volatility (SV) Models

Introduction
The Stochastic Volatility Model for the Impala Platinum Hold- ings Limited dataings Limited data
The Stochastic Volatility Model for the Harmony Gold Mining Company dataCompany data

Chapter Summary
Result1
R Code for the symmetric GARCH Models
R Code for the Asymmetric GARCH Models
R Code for the DCC-GARCH (1,1) GARCH Mod- els
R Code for the Stochastic Volatility Models

The use of the ARCH and GARCH models was therefore applied since the ARCH effect was found in the data. The summary of the fitted model is shown in Table 4.2, and all the parameter estimates are significant at 5% level of significance, except µ which does not affect anything in the result. The parameter estimates of the GARCH (1,2) model with sstd are presented in Table 4.7 and the diagnosis of residuals under sstd is presented in Table 4.8.

This led to the use of the ARCH and GARCH models in Impala Platinum Holdings Limited data. After that, we can proceed to remove autocorrelation presented in the average, where the ACF and PACF of the squared returns represent an AR (4), therefore we model our ARMA-GARCH model, with several AR to AR (4) to the best equipped model. The parameter estimates of the GARCH (1,2) model with std are presented in Table 4.21 and diagnosis of residuals under std are presented in Table 4.22.

Extensions of the GARCH model, AR2+EGARCH (1,2) with std was found to be the best fitting model for the data. The DCC model is the extension of the CCC model developed by Bollerslev (1990) to model the conditional covariance matrix. In this section, the estimation of the parameters of the DCC model is determined using the likelihood, we follow the estimation procedure from Engle (2002), under the multivariate Gaussian distribution for the standardized error t. The likelihood function is given by.

Furthermore, the diagnostic test to check the adequacy of the model showed that the DCC model was good enough to estimate the volatility and correlation between the two mining companies. Therefore, the observation matrix Φt is the p×p transition matrix from the state equation, while ωt is assumed to be p×1 independent and identically distributed normal vectors with an average zero of the null vector and the covariance matrix Q. The observation equation is very important, since we cannot observe the state vector xt directly. Therefore, the SV model can be applied to the data using the logarithm of the squared modified residuals, which is given by

In addition, the SV model was applied to the residuals of the AR (2) model for the return series, as was done in Chapter 4. Therefore, the SV model was applied to the residuals of the AR (1) model for the return series , as shown in Chapter 4. In Chapter 5, the application of the DCC-GARCH model was also based on the daily returns from the univariate models.

The correlation parameters for the DCC-GARCH model ˆθ1 and ˆθ2 are significant at 5%, which implies that the correlations between the two mining companies are time-varying. Moreover, the diagnostic test to check the adequacy of the model showed that the DCC model was good enough to estimate the volatility and correlation of the two mining companies. 1990), 'Modeling the correlation in short-term nominal exchange rates: a multivariate generalized ARCH model', The Review of Economics and Statistics.

1983), 'Diagnostic Testing of ARMA Time Series Models Using Squared Residual Autocorrelations', Journal of Time Series Analysis Volatility Persistence, Long Memory, and the Unconditional Time-Varying Average: Evidence from 10 Equity Indices' , Quarterly Review of Economics and Finance.