simulating south african equity index returns using garch models
TRANSCRIPT
Simulating South African equity Index returns using GARCH models
Niël Oberholzer* and Pierre J Venter
University of Johannesburg Department of Finance and Investment Management
*Corresponding author [email protected]
Abstract
The purpose of this paper is to simulate the GARCH processes obtained from fitting different
GARCH models to the FTSE/JSE All Share Index, FTSE/JSE Top 40 Index and the logarithmic
returns of the US-Dollar South African Rand exchange rate. The results obtained for the
FTSE/JSE All Share Index, FTSE/JSE Top 40 Index and the USD/ZAR exchange rate indicate
that log returns are mean reverting with signs of volatility clustering. The datasets of the
FTSE/JSE All Share Index, FTSE/JSE Top 40 Index are slightly negatively skewed with signs
of leptokurtosis. The results obtain from the volatility simulation process, via the fitting of the
GARCH (1,1) and GARCH (2,1) models, to the log return datasets, indicate no similarity in
the results, even though the model assumption for both GARCH models regarding errors being
independent, was satisfied.
Keywords: GARCH (1,1), GARCH (1,2), All Share, Top 40, USD/ZAR, volatility, simulation
1. Introduction
Volatility is a key variable in financial applications and modelling. The estimation, calculation
and modelling of volatility is a critical topic of research for academics, policy makers and
market practitioners alike. Volatility is a measure of risk as it measures the size of errors,
disbursements, made in the modelling of asset returns. The greater the volatility in financial
markets the greater the impact on the stability of financial markets and the global economy
(Yu, 2002, Engle et al., 2007).
As a theoretical concept volatility is of importance in the pricing of derivatives instruments,
the estimations of market and financial risk, risk management, derivative pricing and hedging,
market making, market timing, portfolio selection and many other financial activities. In each
of these the computation, modelling and predictability of volatility is a prerequisite (Minkah,
2007, Engle et al., 2007).
In finance the common model for logarithmic returns of asset is the normal distribution.
However, volatility clustering and leptokurtosis or fat tails, are seen as common violations of
the assumption of normality of data distribution of financial time series (Mandelbrot, 1963,
Minkah, 2007). The non-normality of financial data regarding assets returns are strongly
rejected by Mandelbrot (1963) which in turn inference that financial return processes behave
like non-Gaussian stable processes.
Resulting from better dealing and information systems financial time series data are often
available as high frequency data. According to Harris and Sollis (2003) this high frequency
data display properties of long-memory in other words the data reflects the presence of
statistically significant correlations between observations that are not close together.
Financial time series data displays certain important features and characteristics when used in
the calculation of volatility, (1) the leverage effect as descript by Black (1976), and (2)
heteroskedasticity, variance or changes in volatility, of the data. The structures of variance
cannot be explain by simple linear models like to random walk (RW) or the ordinary least
squares (OLS). The random walk model postulates that each price made takes a random step
away from the previous price or asset value and the each step is independently and identically
distributed in size iid(N~(0,σ)). The random walk model normally uses t-1 value to estimate
volatility at time t0. (Gokbulut and Pekkaya, 2014).
The ordinary least squares method extend the principal of the random walk model, under the
assumption a stationary mean, implying that volatility forecasts depends on the long-term
average of past volatility observations. The heteroskedasticity in equity returns is explained by
the examination of the behaviour of the conditioning information variables in relationship to
the ARCH effects (Bollerslev, 1986, Engle and Ng, 1993, Gokbulut and Pekkaya, 2014).
The GARCH model is a dependent model as it does not have any independent random
variables. This implies that in GARCH models variance changes over time as a computed
function of past squared deviations from the mean and past variances (Johnston and Scott,
2000).
The purpose of this paper is to simulate the GARCH processes obtained from fitting different
GARCH models (Bollerslev, 1986) to the (1) FTSE/JSE All Share Index, (2) FTSE/JSE Top
40 Index and (3) the US-Dollar South African Rand exchange rate logarithmic returns. The
simulated process will then be visually compared to the actual returns.
The remainder of the paper will be structured as follows, part 2 literature review, Part 2 the
date and part 3 for the methodology used in the research paper. Part 4 will be the description
of results and part 5 will be the conclusion of the study.
2. Data
This study uses daily data for the FTSE/JSE All Share Index, FTSE/JSE Top 40 Index and the
US-Dollar Rand (USD/ZAR) exchange rate for the period 4 January 2010 until 31 December
2014, 1248 observations were included in this study. All the datasets were obtained from
Thomson Reuters Eikon. The Indices and the exchange rate were converted to their logarithmic
returns.
3. Methodology
By employing a similar method to Dukich et al.(2010), a GARCH (1,1) and a GARCH (2,1)
model was fitted to each of the returns series. Thereafter residual diagnostics were performed
in order to determine which of the two models fitted mostly reflects the empirical nature of the
returns series. Finally, simulated GARCH (1,1) and GARCH (2,1) processes with the
respective coefficients were simulated and visually compared to the actual logarithmic returns
for each variable included in the study.
With regards to model specification, according to McNeil et al. (2006), the GARCH (1,1) and
GARCH(2,1) models are specified as follows:
2 2 2
1 1 1t t tu
2 2 2 2
1 1 2 2 1t t t tu u
4. Results
The results of the study will be divided in to three section. Each time series results will be
discussed in full before considering the results obtained from the next time series. The sequence
of result to be discusses will be the FTSE/JSE All Share Index then the FTSE/JSE Top 40 Index
and lastly the US-dollar South African Rand exchange rate results.
4.1 FTSE/JSE All Share Index
Figure 1 below display the time series and histogram for the FTSE/JSE All Share Index. The
log returns are mean reverting and show signs of volatility clustering. In addition, the histogram
shows that the log returns are slightly negatively skewed and leptokurtic (fat tails).
Figure 1: Log returns for FTSE/JSE All Share Index
Source: Researchers’ analysis
The auto correlation function (ACF) indicates the correlations for a time series between the
series xt and it’s successive. Thus, the ACF is a set of correlations between the series and lags
of itself over time. The ACF indicates the possible structure of the time series data. The optimal
results for the ACF is that there is not and significant correlations for any of the lags.
The partial autocorrelation function (PACF) measures the amount of correlation between a
variable and a log of itself that is not explained by correlations at all lower-order-lags.
The ACF plot as indicate in Figure 2 below, indicates that there appears to be serial correlation
among the log returns for the FTSE/JSE All Share Index returns. The result obtained for the
PACF, for the FTSE/JSE All Share Index returns, indicates that there is a long-term
dependence among the observations, as can be seen in Figure 2 below.
Figure 2: ACF and PACF results for the FTSE/JSE All Share Index
Source: Researchers’ analysis
In Figure 3 below, the ACF-plots of residuals and squared residuals for the two GARCH
models, GARCH (1,1) and GARCH (1,2), are shown. For the ACF plot of the residuals of
GARCH (1,1) and GARCH (1,2) only two observation at lag 2 and 18 are outside the two
standard deviation of the autocorrelation sample. As only two lag residuals are outside the two
standard deviations of the autocorrelation sample the supposition of independent errors are
accepted.
Figure 3: FTSE/JSE All Share Index ACF-plots of residuals and squared
residuals for the two GARCH models
Source: Researchers’ analysis
Figure 4 below indicated the normal probability plots and histogram results for the normality
assumption of the errors. As shown for both GARCH models the errors reflect normality, which
implies that the residuals will follow an approximate normal distribution.
Figure 4: Normal probability plots and histograms for the fitted GARCH models
for the FTSE/JSE All Share Index
Source: Researchers’ analysis
For further analysis, we examine the skewness and kurtosis of the residuals and finally perform
the Ljung-Box test in order to determine whether the residuals are independent. The histograms
and normal probability plots show that the residuals are approximately normal. The skewness
shows that the residuals are slightly negatively skewed with tails not too much heavier than the
normal distribution, as indicate in Table 1 below. Moreover, the Ljung-Box test indicates that
the residuals are independently distributed, as illustrated in Table 2 below.
Table 1: Skewness and Kurtosis
Skewness Kurtosis
JALSH -0.1108077 1.544576
GARCH(1,1) -0.2413688 0.6834049
GARCH(2,1) -0.2248144 0.7005966
Source: Researchers’ analysis
Table 2: Ljung-Box Test
X-squared p-value
GARCH(1,1) 1.1134 0.2913
GARCH(2,1) 0 0.9987
Source: Researchers’ analysis
Table 3 below lists the coefficients of the different fitted GARCH models. All three coefficients
of the GARCH (1,1) model are statistically significant at a five percent level of significance.
Furthermore, the sum of the alpha and beta coefficients is close to unity, this suggests that a
shock to the FTSE/JSE All Share Index will be persistent (Brooks 2014). Moreover, when the
GARCH (2,1) coefficients are considered, the intercept, alpha2 and beta coefficients are
statistically significant at a five percent level.
Table 3: GARCH coefficient(s) FTSE/JSE All Share
GARCH(1,1) Estimate t value
Omega 1.58E-06 2.815**
Alpha 0.0698 5.482***
Beta 0.9141 56.138***
GARCH(2,1) Estimate t value
Omega 1.80E-06 2.965**
Alpha1 0.0301 1.496
Alpha2 0.0454 2.046**
Beta 0.9059 53.061***
*(**) [***]: Statistically significant at a 10(5)[1] % level
Source: Researchers’ analysis
Figure 5 below indicate, for purpose of evaluation, the similitude data results for the logged
returns for the FTSE/JSE All Share Index and the two fitted GARCH models. The model
assumption for both GARCH models regarding errors being independent is satisfied. However,
the simulated volatility for both the GARCH (1,1) and GARCH(1,2) does not fit the log returns
series for the FTSE/JSE All Share Index.
Figure 5: Simulated GARH and log return plots for the FTSE/JSE All Share Index
Source: Researchers’ analysis
4.2 FTSE/JSE Top 40 Index
The log returns for the values of the FTSE/JSE Top 40 Index indicate large variability as
indicate in Figure 6 below. The large variability indicated the presence of volatility clustering
around the mean. The plot for the FTSE/JSE Top 40 Index is similar to the log return plot of
the FTSE/JSE All Share Index as can be seen in Figure 1 above. The histogram of the log
returns for the FTSE/JSE Top 40 Index is very slightly negatively skewed and leptokurtic (fat
tails).
Figure 6: Log returns for FTSE/JSE Top 40 Index
Source: Researchers’ analysis
The ACF plot for the FTSE/JSE Top 40 Index as illustrate in Figure 7 indicate serial correlation
among the log returns. The PACF plot, indicates that there is a long-term dependence among
the observations.
Figure 7: ACF and PACF results for the FTSE/JSE Top 40 Share Index
Source: Researchers’ analysis
The ACF-plot for the FTSE/JSE Top 40 Index residuals for the GARCH (1,1) and
GARCH(1,2) indicate that all values are within the two standard deviation autocorrelation
sample except for lag 2 and 18, as indicated in Figure 8 below. These result obtained is similar
to the residuals for the FTSE/JSE All Share Index as indicated in Figure 3 above. The ACF
squared for the GARCH (2,1) model indicate that lags 3, 7, 8, 16,18,21,27 and 28 are outside
of the two standard deviation autocorrelation sample. This implies that the squared residuals
show profound serial correlation.
Figure 8: FTSE/JSE Top 40 Index ACF-plots of residuals and squared residuals
for the two GARCH models
Source: Researchers’ analysis
The normal probability plot and histogram results for the normality assumption of the errors of
the GARCH (1,1) reflect normality. However, the plot and histogram for the GARCH (1,2)
reflects insignificant negative skewness and slightly more peakedness (leptokurtic) than the
histogram for the GARCH (1,1) model. These observations are shown in Figure 9 below.
Figure 9: Normal probability plots and histograms for the fitted GARCH models
for the FTSE/JSE Top 40 Index
Source: Researchers’ analysis
The histograms and normal probability plots show that the residuals are approximately normal
as indicated in Figure 9 above. Table 4 below indicates that the residuals are slightly negatively
skewed with kurtosis values that is close to normal. The results of the Ljung-Box test, Table
5, reflects that the residuals are independently distributed
Table 4: Skewness and Kurtosis
Skewness Kurtosis
JTOPI -0.060748 1.51019
GARCH(1,1) -0.195276 0.7175
GARCH(2,1) -0.108237 0.85735
Source: Researchers’ analysis
Table 5: Ljung-Box Test
X-squared p-value
GARCH(1,1) 1.5348 0.2154
GARCH(2,1) 1.8488 0.1739
Source: Researchers’ analysis
The GARCH coefficients of the different fitted GARCH models for the FTSE/JSE Top 40
Index is displayed in Table 6 below. The three coefficients for the GARCH (1,1) model are all
statistically significant at a five percent level of significance. The sum of the alpha and beta
coefficients is close to unity, this suggests that a shock to the FTSE/JSE All Share Index will
be persistent Brooks (2008).
Table 6: GARCH coefficient(s) FTSE/JSE Top 40 Index
GARCH(1,1) Estimate t value
Omega 1.87E-06 2.802***
Alpha 0.069 5.409***
Beta 0.915 56.224***
GARCH(2,1) Estimate t value
Omega 1.64E-05 4.952***
Alpha1 2.11E-09 0.0131
Alpha2 0.1770 5.394***
Beta 0.6740 13.501***
*(**) [***]: Statistically significant at a 10(5)[1] % level
Source: Researchers’ analysis
The results obtained for the GARCH (1,1) and GARCH (1,2) simulation for the FTSE/JSE Top
40 Index does not fit the log return series for the Index, as indicated in Figure 9.
Figure 9: Simulated GARH and log return plots for the FTSE/JSE All Share
Index FTSE/JSE Top 40 Index
Source: Researchers’ analysis
4.3 USD/ZAR
Figure 10 display the log returns of the USD/ZAR exchange rate. The log returns are mean
reverting with signs of volatilty clustering. The frequncy histogramme is slightle positively
skewed.
Figure 10: Log returns for the USD/ZAR
Source: Researchers’ analysis
In Figure 11 below the ACF-plot and PACF –plot results for the USD/ZAR exchange rate can
be seen. The plot results indicates that there appears to be serial correlation among the log
returns for the USD/ZAR exchange rate. The result obtained for the PACF, for the USD/ZAR
exchange rate, indicates that there is a long-term dependence among the observations.
Figure 11: ACF and PACF results for the USD/ZAR
Source: Researchers’ analysis
The results for the ACF-plots residuals and squared residuals for the USD/ZAR indicate that
no autocorrelation values are within the two standard deviations autocorrelation sample. The
hypothesis of independent errors are accepted.
Figure 14: The ACF-plots of residuals and squared residuals for the two
GARCH models
Source: Researchers’ analysis
Figure 12 below indicated the normal probability plots and histogram results for the normality
assumption of the errors for the USD/ZAR time series. The results obtained for both GARCH
models reflect normality, which implies that the residuals will follow an approximate normal
distribution
Figure 12: Normal probability plots and histograms for the fitted GARCH
models for the USD/ZAR
Source: Researchers’ analysis
As shown in Figure 12 above the histograms and normal probability plots show that the
residuals are approximately normal. The normal distribution of the USD/ZAR time series is
slightly positive skewed as reflected in Table 12. The tails of the distribution is not much
heavier than those of a normal distribution. The result for the Ljung-Box test reflects that the
residuals are independently distributed, as illustrated in Table 8 below.
Table 7: Skewness and Kurtosis
Skewness Kurtosis
ZAR 0.2026153 2.12348
GARCH(1,1) 0.2797318 0.7772856
GARCH(2,1) 0.2853443 0.8379899
Source: Researchers’ analysis
Table 8: Ljung-Box Test
X-squared p-value
GARCH(1,1) 1.1839 0.2766
GARCH(2,1) 0.0146 0.9039
Source: Researchers’ analysis
The coefficients of the fitted GARCH models for the USD/ZAR time series are indicated in
Table 8 below. All three coefficients of the GARCH (1,1) model are statistically significant at
a five percent level of significance. The result obtained by the summation of alpha and beta
indicate unity, implying that that a shock to the USD/ZAR exchange rate will be persistent.
Furthermore, when the GARCH (2,1) coefficients are considered, the intercept, alpha2 and beta
coefficients are statistically significant at a five percent level.
Table 8: GARCH coefficient(s) USD/ZAR
GARCH(1,1) Estimate t value
Omega 1.42E-06 2.841***
Alpha 0.063 6.26***
Beta 0.919 69.681***
GARCH(2,1) Estimate t value
Omega 1.94E-06 2.862***
Alpha1 1.93E-02 0.0168
Alpha2 0.0571 2.502***
Beta 0.8990 48.63***
*(**) [***]: Statistically significant at a 10(5)[1] % level
Source: Researchers’ analysis
Figure 13 below reflect the simulation results for the logged returns for the USD/ZAR
exchange rate and the two fitted GARCH models. The simulated volatility for both the GARCH
(1,1) and GARCH(1,2) does not fit the log returns series for the USD/ZAR exchange rate, even
though, the model assumption for both GARCH models regarding errors being independent is
satisfied.
Figure 13: Simulated GARH and log return plots for the USD/ZAR
Source: Researchers’ analysis
5. Conclusion
The purpose of this paper was to simulate the GARCH processes obtained from fitting different
GARCH models to the FTSE/JSE All Share Index, FTSE/JSE Top 40 Index and the US-Dollar
South African Rand exchange rate logarithmic returns. The results obtain for the FTSE/JSE
All Share Index, FTSE/JSE Top 40 Index and the USD/ZAR exchange rate indicate that log
returns are mean reverting with signs of volatility clustering. The datasets of the FTSE/JSE All
Share Index, FTSE/JSE Top 40 Index are slightly negatively skewed with signs of
leptokurtosis. This result is opposing the results of the USD/ZAR that is slightly positively
skewed.
The time series datasets for all the variables reflects serial correlation with a long-term
dependence amongst the variables. The ACF and PACF plots for the log returns for the
variables reflects the presents of auto correlation for certain of the lag periods. As certain of
the histogram values falls outside the two standard deviation parameter. The ACF plots of the
residuals for GARCH (1,1) and GARCH (2.1) for all the variables reflect normality under the
normality assumption of the errors.
The results obtain from the volatility simulation process, via the fitting of the GARCH (1,1)
and GARCH (2,1) models, to the log return datasets, indicate no similarity in the results. Even
though, the model assumption for both GARCH models regarding errors being independent
was satisfied.
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