## 3 Materials and Methods
These expectations imply that the estimators are derived from a system of equations where the weighted residuals of the model have a mean of zero, satisfying the generalized method of moments (GMM) framework. \(\psi(Y,X;\theta)\) and \(\phi(Y,X;\theta)\) are moment functions that depend on the observed outcome \(Y\), covariates \(X\), and parameters \(\theta\). The moment conditions ensure identification and consistency of the estimators by leveraging distributional information beyond the conditional mean, accommodating fixed effects, endogeneity, and heteroskedasticity in the data. These moment conditions underpin the MMQR approach, allowing for robust quantile-specific inference in a heterogeneous panel setting. Next, it is essential to check the validity of the model by a series of statistical tests. Heteroscedasticity testing is performed using the Breusch-Pagan test, to confirm that the dispersion of errors varies across quantiles. The Wald test is also performed to check the significance of the coefficients at different quantiles. Ultimately, the results are analyzed comparatively across quantiles to identify whether the impact of bank automation and AI-driven solutions in banking differs across banks with different performance levels. The interpretation of the MMQR coefficients allows us to understand how automation and AI-driven solutions influence the financial stability of banks and whether these technologies contribute to improving profitability in a uniform or differentiated way depending on the performance of the analyzed institutions. The adoption of the Method of Moments Quantile Regression method in this context provides an advanced perspective on the impact of automation and AI on the European banking sector, overcoming the limitations of classical methods through a distributed analysis of the effects across different performance levels. As the integration of advanced technologies may have diverging effects on banking institutions with low profitability compared to more profitable ones, this methodology allows a more nuanced understanding of the phenomenon and supports the formulation of effective policies to adapt the banking system to the emerging technological transformations. The analysis begins (Figure 2) with an examination of descriptive statistics, which establishes the foundational understanding of the dataset by summarizing central tendencies, dispersion, and distribution patterns. This preliminary investigation is essential for detecting trends, potential outliers, and anomalies that could influence subsequent econometric modelling. By providing insights into variability and skewness, this stage informs the selection of appropriate econometric techniques, thereby enhancing the robustness of the study. Given the panel structure of the dataset, the next stage involved testing for cross-sectional dependency, following the methodologies proposed by [PERSON] [116] and [PERSON] et al. [117]. Since banks are often influenced by common external shocks, regional dynamics, or global technological advancements, neglecting such dependencies could result in biased and inconsistent estimates. Addressing this issue ensures that the estimation process accounts for the interconnected nature of the observations, thereby increasing the validity of the empirical findings. Ensuring the reliability of time-series data requires an evaluation of stationarity properties, which is conducted using the unit root tests proposed by [PERSON], [PERSON], and [PERSON] [118]. Establishing whether variables exhibit mean-reverting behavior or stochastic trends is significant for preventing spurious regressions, where observed correlations between variables might simply reflect common trends rather than genuine economic relationships. Addressing stationarity issues strengthens the credibility of the subsequent econometric inferences. To ascertain the presence of stable long-term relationships among the variables, the study applies cointegration tests based on the methodologies of [PERSON] [119] and [PERSON] [120].