## 3 Materials and Methods
A key motivation for employing MMQR lies in the recognition that banking performance is not uniformly affected by macroeconomic and sectoral factors. Banks operating in different performance quantiles--low, median, and high profitability--face distinct constraints and opportunities, necessitating an analytical framework that accommodates this variation. By estimating the effects at multiple quantiles, MMQR allows for the identification of asymmetric relationships that would otherwise be obscured in standard regression models. For instance, economic growth or digital inclusion may have a stronger impact on highly profitable banks compared to those struggling with inefficiencies and regulatory burdens. Moreover, MMQR is robust to outliers and non-normal error distributions, making it particularly suited for financial datasets that often exhibit skewness and heavy-tailed distributions. Furthermore, the MMQR framework incorporates higher-order moments, namely Moment 1 (MOMENT1) and Moment 2 (MOMENT2), which enhance its ability to capture unobserved heterogeneity and improve the efficiency of estimations. These moments represent the first and second derivatives of the quantile regression process, respectively. * Moment 1 (MOMENT1) corresponds to the first moment of the distribution, capturing the linear component of the variation across quantiles. It reflects how the dispersion of profitability outcomes evolves as one moves from lower to higher quantiles. A significant MOMENT1 coefficient suggests that the impact of explanatory variables changes progressively along with the profitability distribution. * Moment 2 (MOMENT2) represents the second moment of the distribution and captures the curvature or non-linear variation in the quantile regression process. This higher-order moment provides insights into whether the relationship between predictors and banking profitability is concave or convex, offering a deeper understanding of the underlying structural dynamics. A significant MOMENT2 coefficient indicates that the marginal effect of an explanatory variable is not constant across quantiles but instead exhibits increasing or decreasing returns. The inclusion of these moments is significant for addressing potential issues related to heteroskedasticity and omitted variable bias, ensuring that the MMQR estimations remain robust and efficient. By leveraging these methodological advantages, this study provides a nuanced assessment of how macroeconomic and digitalization variables interact with banking profitability in the EU, offering policy-relevant insights that extend beyond conventional regression approaches. To enhance the robustness of our empirical framework and to address the concerns related to residual heteroskedasticity and potential model misspecification, we complemented the MMQR with Panel Estimated Generalized Least Squares (EGLS). The EGLS technique, as outlined by [PERSON] et al. [123] and [PERSON] [124], is specifically tailored for panel data and corrects for cross-sectional and period-specific heteroskedasticity, offering efficiency gains over standard OLS and Fixed Effects estimation in the presence of non-spherical error terms. In our model, we applied two EGLS variants:  
* Cross-section weighted EGLS, which adjusts for heteroskedasticity between countries (i.e., structural differences in variance across banking systems). * Period-weighted EGLS, which corrects for time-related volatility and ensures that shocks or structural changes during specific years (e.g., digital regulation peaks, pandemic effects) do not bias the estimates. The integration of instrumental variables, robust estimation techniques, and extensive diagnostic testing collectively demonstrates that the model successfully mitigates endogeneity concerns and delivers stable and reliable estimates of the effects of automation technologies on banking performance.