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The preceding preliminary analysis suggests that the proposed multifactor model of the return generating process provides a reasonable description of the process driving South African stock market returns. The model explains almost 60 percent of the variation in returns with the domestic risk factors explaining at least 16.1 percent of the variation in returns in addition to UFTW_t. All factors are statistically significant in the unrestricted model, the model is generally robust to outliers and parameter instability does not appear to be inherent to the model. However, regression diagnostics indicate that ARCH effects are present in both versions of the unrestricted model (Table 8.2, Panel B & Table 8.3, Panel A). This suggests that the ARCH/GARCH framework is more appropriate for modelling returns on the JSE All-Share Index (see section 6.4.3 & 6.4.4). Furthermore, the framework also provides insight into the conditional variance of South African stock returns. Table 8.4 reports the results of the unrestricted and restricted versions of the model. Selection of the ARCH/GARCH model (ARCH (section 6.3.1), GARCH (section 6.3.2), IGARCH (section 6.3.3) or EGARCH (section 6.3.5)), the number of ARCH and GARCH terms and the conditional error distribution is based upon the unrestricted model (including outliers) using AIC. The ARCH/GARCH specification, the number of ARCH and GARCH parameters and the conditional error distribution applied in the estimation of the unrestricted model, are extended to the restricted versions of the model.

The results in Panel A of Table 8.4 are those of the unrestricted model whereas the results in Panel B are those of the unrestricted model after controlling for outliers. The model in Panel C excludes UFTW_t and the model in Panel D is a single-factor model incorporating only UFTWt. The results in Panel A of Table 8.4 are comparable to those of Panel B in Table 8.2.

All factors are individually and jointly statistically significant. As before, returns on the JSE All-Share Index are positively related to UFTW , _t UBP_t−1,UM3_t−1, UOIL _t UZARUS and _t UCI and negatively related to t UCPI_t−1 and URBAS . Whereas most coefficients remain _t similar in magnitude, the coefficient on URBAS decreases from an absolute value of 1.318 in _t Panel B of Table 8.2 to 0.800 in Panel A of Table 8.4. Together, these factors explain 56.2

percent of the variation in returns and the AIC statistic of -5.322 is lower than the AIC statistic in Panel B of Table 8.2 suggesting that the ARCH/GARCH framework yields a better description of returns on the JSE All-Share Index relative to the LS model. Notably, regression diagnostics indicate that the GARCH(1,1) model with normally distributed errors provides an adequate description of the return generating process. Residuals and squared residuals are white noise and the ARCH LM test suggests that ARCH effects are no longer present (see Sadorsky & Henriques, 2001; Gujarati, 2003). This strengthens the argument that the ARCH/GARCH framework is a more appropriate econometric framework for the modelling of returns.

Table 8.4: ARCH/GARCH model of JSE All-Share Index returns

Panel A Panel B Panel C Panel D

ARCH/GARCH GARCH(1,1) GARCH(1,1) GARCH(1,1) GARCH(1,1)

Distribution Normal Normal Normal Normal

Notes:

1. *** Indicates statistical significance at the 1 percent level of significance. ** Indicates statistical significance at the 5 percent level of significance. * Indicates statistical significance at the 10 percent level of significance.

2. F-statistics are reported for Wald’s test of linear restrictions testing the null hypothesis of coefficients jointly equalling zero (Nelson, 1991; McMillan & Ruiz, 2009).

3.Q(1) and Q(5) are Ljung-Box test statistics for residual serial correlation at the 1^st and 5^th orders.

4.Q²(1) and Q²(5)are Ljung-Box test statistics for squared residual serial correlation at the 1^st and 5^th orders.

5. ARCH(1) and ARCH(5) are LM test statistics for residual ARCH effects at the 1^st and 5^th orders.

Source: Compiled by author

The results in Panel B indicate that there are no substantial changes in the magnitudes and signs of the estimated coefficients after controlling for outliers. However, the coefficient on

1

UCPIt₋ is now marginally statistically insignificant (p-value of 0.131). Notably, the decrease in the absolute size of the coefficient on UCPI_t−1 in Table 8.4 is less pronounced than the decrease under the LS methodology after controlling for outliers. The question that arises is whether the unrestricted model in Panel A or the unrestricted model which excludes outliers in Panel B should be chosen as the multifactor model of the return generating process of South African stock returns. The R² indicates that after controlling for outliers in Panel B, the model explains 62.6 percent of the variation in returns – an improvement over the unrestricted model in Panel A. Furthermore, the AIC statistic is now -5.393 suggesting that removing outliers improves the fit of the model. Regression diagnostics indicate that the unrestricted model after controlling for outliers is also appropriate. However, although this version of the unrestricted model explains a greater amount of the variation in returns and the AIC statistic is lower, the coefficient on the dummy factor is statistically insignificant suggesting that the ARCH/GARCH framework, unlike the LS framework, is robust to outliers. Furthermore, because the coefficient on the dummy factor is statistically insignificant, it can be argued that it is unnecessary to control for outliers under the ARCH/GARCH framework. Following from this argument, it can be further postulated that the strongest argument for retaining outliers is that the removal of outliers will artificially improve the characteristics of the model (Brooks, 2008). In light of these arguments, the unrestricted model in Panel A is accepted as the most appropriate multifactor model of the return generating process.

The results of the restricted models are reported in Panel C and Panel D. The purpose of these models in the present context is to act as benchmarks and these models can be considered as

“naive models” representative of simpler models. This is motivated by Reinganum’s (1981) argument that there is no justification for accepting a more complex model that does not convey more information relative to a simpler model . In a similar vein, Elton et al. (1995) suggest that a more complex model can only be considered for further use if it outperforms a simpler model. One approach to determining whether a multifactor model is more appropriate relative to a single-factor model or a simpler multifactor model, is to determine whether the unrestricted model encompasses the two models (Brooks, 2008). In the context of this study, this suggests that an unrestricted model should be accepted if and only if it provides more insight into the return generating process of South African stock returns by explaining a

greater amount of variation relative to simpler specifications and results in a more adequate fit (see section 2.2: 15 & 2.2.4).

The results in Panel C - those of the restricted model incorporating the domestic risk factors - indicate that these factors explain 21.7 percent of the variation in returns on the JSE All-Share Index. This suggests that there is value in incorporating these factors into a multifactor model. The results in Panel D indicate that UFTW explains 41.6 percent of the variation in _t returns. The AIC statistic of -5.039 is lower than the AIC statistic of -4.653 of the restricted model in Panel C suggesting that the restricted single-factor model provides a better fit relative to the seven-factor model. However, the unrestricted model in Panel A performs better relative to both restricted models in Panel C and Panel D. The R suggests that the ² unrestricted model explains a higher percentage of variation in returns than either of the restricted models in Panel C and Panel D. The AIC statistic is lower than that of the two restricted models suggesting that this model is more suitable relative to these simpler specifications. Furthermore, if the unrestricted model is considered to be a direct extension of the single-factor model in Panel D, then the seven domestic risk factors explain at least an additional 14.6 percent of the variation in returns on the JSE All-Share Index under the ARCH/GARCH framework. This represents an increase of 35.096 percent in R .^{2 130} These findings suggest that the unrestricted model in Panel A encompasses the restricted models in Panel C and Panel D in terms of explanatory power and adequacy of fit (addressed further in section 8.4.5 & 8.5.4). Therefore, the unrestricted multifactor model is a more appropriate model of the return generating process (Reinganum, 1981; Brooks, 2008).

130 Estimated as a percentage change between the R of the unrestricted model and that of the single-factor ² model.