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algorithm Model Results, Train:700 Test:200

The Metropolis-Hastings ARMA(1,1) model was implemented using the following assumptions for the prior distributions: the Autoregressive (phi) and Moving Average (theta) parameters are assumed to have a Uniformly distributed prior whose support is (-1,1), that is ,U ∼(−1, 1). The precision parameter is assumed to have a Gamma prior, that is, 1/σ2∼Gamma(2, 4).

Netcare Group Ltd: Estimated Metropolis-Hastings

-ARMA(1,1) model

Figure 5.74: Summary results of ARMA(1,1)- Metropolis Hastings model fitted on Netcare Group Limited returns series

Figure 5.74 shows summary results of the estimated Metropolis-Hastings ARMA(1,1) model fitted on Netcare Group Ltd returns series. The estimated Autoregressive, Moving Average, and precision parameters are−0.032,−0.069, and 85.843 respectively. The standard deviation (SD) for the precision pa- rameter is relatively large (about 4.591), this suggests that there is some de- gree of uncertainty in the estimated precision parameter. Since the Monte

Carlo (MC) errors for all the estimated parameters are small, this is a clear indication that there is a small amount of error in the estimates.

Figure 5.75: Netcare Group Ltd returns series predictions using ARMA(1,1)-MH model

Predicted Down Predicted Up

True Down 65 39

True Up 30 66

Accuracy=65.5%

Table 5.62: Confusion Matrix for ARMA(1,1)-MH model’s predictions of Netcare Group Ltd returns series

Figure 5.75 presents predictions of Netcare Group Ltd returns series ob- tained using the ARMA(1,1)-Metropolis-Hastings model. Yellow line in- dicates the predicted returns and blue line indicates the true returns. The structure of the predictions is similar to that in Figure 5.31 (they have small variance). The model did not adequately capture the variance in the Net- care series (although this does not necessarily imply that we will have poor forecasts). Accuracy results of the ARMA(1,1)-MH model’s predictions are

presented in Table 5.62. The model achieved an accuracy of 65%, hence the model is almost as good as a random walk model.

Santam Ltd: Estimated ARMA(1,1)-Metropolis Hastings

model

Figure 5.76: Summary results of the ARMA(1,1) Metropolis Hastings model fitted on Santam Ltd returns series

The estimated precision parameter and its standard deviation are close to that in Figure 5.74. In Bayesian statistics, standard deviations represents un- certainty in the estimated parameters. The degree of uncertainty in the Au- toregressive and Moving Average parameters is very low since their stan-

dard deviations are close to zero. All the estimated parameters lies within their 95% HPD interval.

Figure 5.77: Santam Ltd returns series predictions using ARMA(1,1)-MH model

Predicted Down Predicted Up

True Down 69 27

True Up 37 67

Accuracy=68%

Table 5.63: Confusion Matrix for ARMA(1,1)-MH predictions of Santam Ltd returns series

Figure 5.77 shows predictions from ARMA(1,1)-Metropolis Hastings model. In terms of the ability to predict the direction of Santam returns series, the ARMA(1,1)-MH model achieved an accuracy of 68%.

Sanlam Group Ltd: Estimated ARMA(1,1)-Metropolis

Hastings model

Figure 5.78: Summary results of ARMA(1,1) Metropolis Hastings model fitted on Santam Group Ltd returns series

The estimated precision parameter and its standard deviations are close to those in Figure 5.74, and 5.76. Since the MC errors for all the estimated model’s parameters are close to zero, this suggests that the amount of error in these estimates is small. Since all the estimated parameters lies within the 95% HPD interval, then we are sure that the true parameters also lies within this interval.

Figure 5.79: Sanlam Group Ltd returns series predictions using ARMA(1,1)-MH model

Predicted Down Predicted Up

True Down 66 35

True Up 32 67

Accuracy=66.5%

Table 5.64: Confusion Matrix for ARMA(1,1)-MH model’s predictions of Sanlam Group Ltd returns series

As seen in Figure 5.79, the variance of Sanlam returns series predictions seems to be less volatile and similar to that in Figure 5.75. The model achieved an accuracy of 66.5% and a misclassification rate of about 33.5%. This shows that the model has a “poor” ability to correctly predict the di- rection of movement of Sanlam returns series.

Nedbank Group Ltd: Estimated ARMA(1,1)-Metropolis

Hastings model

Figure 5.80: Summary results of ARMA(1,1) Metropolis Hastings model fitted on Nedbank Group Ltd returns series

A large standard deviation (SD) for the precision parameter (about 4.639) indicates that there is some degree of uncertainty in this parameter. On the other hand, the Monte Carlo error for this parameter is small, suggesting that the amount of error in this estimate is small. The Monte Carlo (MC) errors for the Autoregressive and Moving Average parameters are equal

(about 0.025), this value is extremely close to zero, this suggests that the amount of error in these estimates is very small.

Figure 5.81: Nedbank Group Ltd returns series predictions using ARMA(1,1)-MH model

Predicted Down Predicted Up

True Down 65 35

True Up 35 65

Accuracy=65%

Table 5.65: Confusion Matrix for ARMA(1,1)-MH model’s predictions of Nedbank Group Ltd returns series

Figure 5.81 shows forecasts for Nedbank returns series obtained using ARMA(1,1)- MH model. Blue indicates the true returns and yellow indicates the predic-

tions. The model has a “poor" ability to predict the direction of Nedbank returns series. It is as good as a random walk since its accuracy is only 65%.

Discussion:

Hastings model failed to achieve an accuracy of more than 68%. The poor performance could be due to the inability of the model to capture the dy- namics of the returns series. Stock returns are non-linear in nature, as a result a linear model such as the ARMA(1,1)-Metropolis Hastings model cannot accurately capture the complex structure of these returns series.

5.10

Artificial Neural Network (ANN) results,