**3. Results**

One of the first thing to notice after the initial data preparation performed in step one of our methodology described above was that the overall performance (return) for the entire period (1 January 2017—12 March 2020) for the gender equality indices was mixed and inconclusive in comparison with the MSCI World indices. Still, when studying the first two moments of the distribution of daily returns, we observe that the mean values of the distributions are similar and not statistically different from zero (see Appendix A, Table A2). Also, the standard deviation is larger in comparison with the mean, and all the series present significant negative skewness and excess kurtosis. Thus, from this point of view, none of them deviates from the general characteristics of high-frequency equity returns as they were described by Cont [44]. In addition, we observe that the values of the fifth percentile of the distributions of daily returns (equivalent to 95% confidence level historical Value-at-Risk) are quite similar.

When comparing the distributions of daily conditional volatilities estimated using the EGARCH (1,1) model described in the *second stage* of our methodology, we observe, however, some small but relevant differences (see Figure 1):


**Figure 1.** *Cont.*

**Figure 1.** Comparison of the distributions of daily conditional volatilities: (**a**) MWXO; (**b**) BGEI; (**c**) MXWO0FN; (**d**) BGEIF. Source: Authors' work. Source: Authors' calculations.

Furthermore, we observe that in general the evolution of daily conditional volatility is highly synchronized between the gender equality indices and their correspondent overall MSCI indices, which already hints a high level of correlation between both the returns series and the volatility regimes, bearing in mind that linear correlation does not necessarily mean causality and spill-over (as we will argue bellow, based on our results from the VAR model). Examining Figure 2 bellow we can confirm the observation that gender equality indices in general exhibit higher daily conditional volatility (as mentioned above while interpreting the results from Table 1), and we can also observe the synchronized reaction of the indices to risk events (including the beginning of the Covid-19 pandemic in the right hand part of the charts).


**Table 1.** Relevant distributional characteristics of daily conditional volatility.

Source: Authors' calculations.

The high level of synchronization of the daily volatility regimes is confirmed by Figure 3 below, where we present for each observation in our sample whether the volatility regimes of the two biomes of variables (BGEI vs. MXWO and BGEIF vs. MXWO0FN, respectively) were at the same level (both high or both low) or were decoupled. Going into detail, we observe that for only 48 out of 834 observations (that is, 5.76% percent of the time), the volatility regimes of the gender equality indices were not in sync with the overall MSCI indices. As previous studies concluded, the volatility regimes of the cross-sectoral indices were not necessary aligned with the ones of the financial sector indices (confirming the different behaviour of financial sector equity indices, which are more volatile in comparison with cross-sectoral diversified ones).

**Figure 2.** Comparable evolution of daily conditional volatility: (**a**) BGEI vs. MXWO; (**b**) BGEIF vs. MXWO0FN. Source: Authors' calculations.

**Figure 3.** Comparable evolution of daily volatility regimes: (**a**) BGEI vs. MXWO volatility regimes; (**b**) BGEIF vs. MXWO0FN volatility regimes. Source: Authors' calculations.

Our results show not only that volatility and volatility regimes of gender equality indices are corelated with overall MSCI indices, but also with the returns themselves. The value of the Pearson linear correlation coefficient for the entire sample of daily returns is 0.942 for the pairing of MXWO and BGEI and 0.953 for the pair made by MXMO0FN and BGEIF, respectively (see Appendix A, Table A3). Furthermore, the study of the daily conditional correlations computed using a DCC MV GARCH (1,1) model as described in *stage three* of our methodology confirms that, during the entire period investigated by us, the conditional correlations between the gender equality indices and the overall MSCI indices were very high, indifferent of the volatility regime (see Figure 4 below). While the correlations among the financial indices appear to be more stable over time in comparison with the correlations among cross-sectoral indices, the conclusion remains that for the entire period the daily returns of the gender equality indices show a high level of linear correlation with the overall MSCI indices.

**Figure 4.** Evolution of dynamic conditional correlations of daily (logarithmic) returns. Source: Authors' calculations.

Going further with our analysis, we were interested to test the above findings using a different method. As described in *stage five* of our methodology, we have calibrated simple linear quantile regressions among the equity indices included in our sample, and our results presented in Appendix A, Table A4 show that all the slope coefficients (for all quantiles tested) are statistically significant. Regarding the regressions between the gender equality indices and the overall MSCI indices, the values of the slope coefficients are close to 1 and relatively stable in relation with the value of the quantile, as we can observe from Figure 5 below.

**Figure 5.** Slope coefficients resulted from quantile liner regression models for daily returns: (**a**) BGEI ~ MXWO + c + ε; (**b**) BGEIF ~ MXWO0FN + c + ε. Source: Authors' work.

A more in-depth view of the results presented in Figure 5 and Appendix A, Table A4 confirms frequent findings in the financial literature that daily returns for financial sector assets are often more volatile and exhibit more skewness and fatter tails. Specifically, our results show that slope coefficients for simple linear regressions between the cross-sectoral indices are more stable in relation with the value of the quantile, while the slope coefficients for the linear regressions between the financial indices tend to be higher for the left tail quantiles, and tend to decrease for the right tail quantile. This is consistent with studies showing that, especially for financial sector assets, correlations tend to increase during bad times and decrease during good times.

For the last phase of our research, we were interested to see whether the strong link between the gender equality indices and the overall MSCI indices could be (in part) explained by causality or spill-over effects. In order to investigate this, we chose to calibrate vector autoregressive models, as described in *stage five* of our methodology, for the cross-sectoral indices and for the financial sector indices separately. The most relevant results returned by the model are presented in Figure 6.

**Figure 6.** Response to Cholesky one s.d. (d.f. adjusted) innovations ± 2 s.e. bands derived from VAR (2) model of daily (logarithmic) returns: (**a**) response of BGEI to MXWO; (**b**) response of BGEIF to MXWO0FN. Source: Authors' work.

Based on the results (presented in detail in Appendix A, Tables A5 and A6), we argue that there is only very little evidence to support any statistically significant causality or spill-over effects from the overall market to the gender equality indices. In our view, this could mean that the high degree of correlation observed from the results of all the different models employed and presented in our study are probably mainly explained by the contemporaneous co-movement of the prices, which supports the hypothesis of the similar behaviour of investors in relation with the gender equality assets vs. the general market (entire universe of assets).
