Netcare Group Limited (Ltd): Estimated State Space
ARMA(1,1) model
Figure 5.66:Summary results of the State Space ARMA(1,1) fitted on Net- care Group Ltd returns series
Figure 5.66 shows summary results of the estimated State Space ARMA(1,1) model fitted on Netcare Group Ltd returns series. Both the estimated Au- toregressive (phi) and Moving Average (theta) parameters lies within the computed intervals, this suggests that the true parameters also lie within this intervals. The standard errors are relatively small, this is a clear indi- cation that the estimated parameters are close to the true parameters. The Moving Average (theta) and the Variance (sigma2) parameters are statisti- cally significant since their Z-statistics are greater than 1.96 (critical value). On the other hand, the Autoregressive parameter is statistically insignificant since its Z-statistic is “slightly” less than the 1.96 critical value. The p-value for the Ljung Box test statistic is 0.59, this value is extremely large, sug- gesting that we have weak evidence against the null hypothesis of indepen- dence of the residuals. The residuals from the fitted State Space ARMA(1,1) model are not normally distributed since the p-value for the Jarque Bera
test is less than 0.05 significance level. The p-value for the Heteroscedastic- ity test is 0.02, this value is less than the 0.05 level of significance, suggesting that we reject the null hypothesis that the residuals have a constant variance. The distribution of the residuals has “heavier” tails and is “slightly" skewed to the left since it has a kurtosis value of 4.65 and a negative skewness value. Since the normality and constant variance assumptions are violated, this suggests that an ARMA model which include a conditional variance process such as GARCH-Student-t process would be more appropriate for modelling the Netcare returns series. The estimated State Space ARMA(1,1) model is
rt =0.3854rt−1+ (−0.5182)et−1+et, (5.33)
whereet ∼N(µ,σ2=0.0003).
Figure 5.67: Netcare Group Ltd returns series predictions using the State Space ARMA(1,1) model
Predicted Down Predicted Up
True Down 78 26
True Up 25 71
Accuracy=74.5%
Table 5.58: Confusion Matrix for State Space ARMA(1,1) model’s predic- tions of Netcare returns series
Figure 5.67 shows the actual (true) and predicted returns obtained using State Space ARMA(1,1) model. Confusion matrix in Table 5.58 shows pre- diction accuracy of the State Space ARMA(1,1) model . The model achieved an accuracy of 74.5%, so it has a “strong” ability to predict the direction of movement of Netcare Group returns prices.
Santam Ltd: Estimated State Space ARMA(1,1) model
Figure 5.68: Summary results of the State space ARMA(1,1) model fitted on Santam Ltd returns series
The standard errors for the estimated parameters are fairly small, this sug- gests that these parameters are close to the true values. The estimated pa- rameters are all significantly different from zero since their Z-statistics are greater than the 1.96 critical value. The null hypothesis that residuals are independent at 5% significance level is accepted since the p-value for the
Ljung Box test statistic (about 0.97) is greater than 0.05. The Heteroscedas- ticity test is inconclusive since the p-value of the test is equal to the sig- nificance level (0.05). The Jarque Bera test statistic value is extremely large (about 378), this indicates that we cannot accept the null hypothesis that the residuals are normally distributed. A skewness value of 0.00 suggests that the distribution of residuals is symmetric. The distribution of the resid- uals has “heavier” tails than that of a normal distribution since its kurto- sis value is larger than 3. Due to violations of some of the assumptions of the ARMA modelling process, an ARMA model which incorporate a con- ditional GARCH process would be more appropriate for modelling Santam returns series as the model would be able to capture the statistical dynamics of the series. The estimated State Space ARMA(1,1) model for Santam Ltd returns series is
rt =0.5368rt−1+ (−0.6020)et−1+et, (5.34)
where the error termet ∼N(µ,σ2=0.0002)
Figure 5.69: Santam Ltd returns series predictions using State Space ARMA(1,1) model
Predicted Down Predicted Up
True Down 72 24
True Up 28 76
Accuracy=74%
Table 5.59: Confusion Matrix for the State Space ARMA(1,1) model’s pre- dictions of Santam returns series
As seen in Figure 5.69, the variance of the predictions made by the State Space ARMA(1,1) model is very small. (Hansson, 2017) also observed that when predicting the log returns series, Box and Jenkins ARMA-type models operates around the mean. In terms of predicting the direction of movement of Santam returns prices, the model achieved an accuracy of 74%.
Sanlam Group Ltd: Estimated State Space ARMA(1,1) model
Figure 5.70: Summary results of State Space ARMA(1,1) model fit on San- lam Group Ltd returns series
Figure 5.70 shows summary results of the estimated State Space ARMA(1,1) model fitted on Sanlam Group Ltd returns series. Small standard error are an indication that the estimated parameters are close to the true parame- ters. The Z-statistics for the Moving Average (theta) and Variance (sigma2) parameters are greater than the 1.96 critical value, this suggests that these
parameters are significantly different from zero. On the other hand, the Au- toregressive parameter (phi) is statistically insignificant since its Z-statistic value is less than the 1.96 (critical value). The null hypothesis that the resid- uals are independent at 5% significance level is accepted since the p-value for the Ljung Box test statistic is greater than 0.05 (level of significance). The residuals of the fitted State Space ARMA(1,1) have a constant variance since the p-value for the Heteroscedasticity test is greater than 0.05 significance level. Results of the Jarque-Bera test suggests that the residuals are not nor- mally distributed since the p-value of the test statistic is less than 0.05 (level of significance). The estimated State Space ARMA(1,1) model for Sanlam Group Ltd returns series is
rt =0.5164rt−1+ (−0.5832)et−1+et, (5.35)
where et is a normal distributed process with mean µand varianceσ2, that
is,et ∼N(µ,σ2).
Figure 5.71: Sanlam Group Ltd returns series predictions using State Space ARMA(1,1) model
Predicted Down Predicted Up
True Down 62 39
True Up 36 63
Accuracy=62.5%
Table 5.60: Confusion Matrix for State Space ARMA(1,1) model’s predic- tions of Sanlam returns series
Plot in Figure 5.71 shows that the predictions from the State Space ARMA(1,1) model are extremely close to the mean and they have low or small variance. The predictions looks similar to those in Figure 5.69. The accuracy achieved by the model is very poor (about 62.5%). The model is as good as a random walk.
Nedbank Group Ltd: Estimated State Space ARMA(1,1)
model
Figure 5.72: Summary results of State Space ARMA(1,1) model fitted on Nedbank Group Ltd returns series
The estimated coefficients for the Autoregressive, Moving Average, and the Variance parameters are 0.4628, −0.6247, and 0.0002 respectively. All these estimated coefficients lies within the confidence intervals, suggesting that the true parameters are also within these intervals. All the estimated State Space ARMA(1,1) model’s parameters are statistically significant since their
Z-statistics are greater than the 1.96 critical value. The Ljung Box test for independence of residuals is inconclusive since the p-value of the test is equal to the 0.05 level of significance. The Jarque-Bera test rejects the null hypothesis that the residuals are normally distributed at 5% significance level. Residuals from the fitted State Space ARMA(1,1) model do not have a constant variance since the p-value for the Heteroscedasticity test is less than the 0.05 level of significance. Consequently, since the normality and the heteroscedasticity test are violated, this suggests that an ARMA model with a conditional variance process (such as the GARCH or EGARCH) should be used, so that the dynamics of the returns series can be better captured. The estimated State Space ARMA(1,1) model for the Nedbank Group Ltd returns series is
rt =0.4628rt−1+ (−0.6247)et−1+et, (5.36)
whereet ∼N(µ,σ2=0.0002).
Figure 5.73: Nedbank Group Ltd returns series predictions using State Space ARMA(1,1) model
Predicted Down Predicted Up
True Down 72 28
True Up 33 67
Accuracy=69.5%
Table 5.61: Confusion Matrix for State Space ARMA(1,1) model’s predic- tions of Nedbank returns series
The predictions made by the State Space ARMA(1,1) model have higher variance than those in Figure 5.69, and 5.71. The model achieved an accu- racy of 69.5% and a misclassification rate (percentage of incorrect predic- tions) of 30.5%.
Discussion
In some cases, the assumptions of the ARMA modelling process were not satisfied, this includes the Normality and Heteroscedasticty assumptions. Consequently, an ARMA model with a conditional variance process could be used to accurately capture the dynamics of the returns series. The ad- vantage of including a conditional variance process in the ARMA process is that we can assume other statistical distribution such as Skewed Student- t, Student-t, and Generalised Error Distribution (GED) for the returns se- ries. The highest accuracy achieved by the State Space ARMA(1,1) model is 74.5% for the Netcare returns series and the lowest is 62.5% for the Nedbank returns series (suggesting that this model is as good as a random walk for this particular series).