## 3 Materials and Methods
Establishing cointegration confirms whether variables exhibit a shared trajectory over time, despite potential short-term fluctuations. This step is particularly relevant in the context of automation, where the economic effects of technological adoption may evolve gradually rather than manifest immediately. Recognizing these long-run relationships provides a more comprehensive understanding of the sustained impact of banking automation on banking performance. Furthermore, the study examines slope heterogeneity using the framework developed by [PERSON] and [PERSON] [121]. This methodological refinement determines whether the relationships between banking automation and banking performance vary significantly across different national contexts. The presence of heterogeneous slopes suggests that the impact of banking automation is shaped by country-specific characteristics such as economic performance, digital maturity, and integration of technology. By incorporating this aspect into the analysis, the study provides a differentiated understanding of how automation influences banking performance across diverse economies. To capture heterogeneity and distributional effects within the data, the study employs the Method of Moments Quantile Regression (MMQR), as introduced by [PERSON] and [PERSON]. Unlike conventional regression techniques that focus on mean effects, MMQR provides a nuanced analysis by examining the impact of banking automation across different quantiles of the dependent variable. This approach enables a deeper exploration of how banking automation affects banks with varying levels of economic performance, from low-productivity countries to highly advanced ones. Moreover, MMQR effectively addresses endogeneity concerns, ensuring that the estimated coefficients reflect causal relationships rather than spurious correlations. Figure 2: Econometric analysis process. Source: own processing. [PERSON] and [PERSON] [115] introduced an advanced quantile regression methodology that effectively captures the conditional distribution effects of robotic automation in EU countries while incorporating fixed effects, heterogeneity, and endogeneity within panel data structures. By extending beyond the conventional panel quantile regression approaches developed by [PERSON] and [PERSON]. [122], this method, known as the Method of Moments Quantile Regression with Fixed Effects (MMQR), provides a more comprehensive framework for analyzing the heterogeneous relationships between robotic automation, digitalization, and return on equity. Unlike earlier techniques, MMQR is specifically designed to accommodate panel data heterogeneity, allowing for a more precise estimation of the conditional covariance effects of robotic automation and other explanatory variables. An essential feature of MMQR lies in its ability to account for both fixed effects and endogeneity, making it particularly well-suited for panel data models where regressors exhibit heterogeneous influences. By addressing these complexities, the method ensures that the estimated quantile functions remain well-ordered, thereby resolving a fundamental limitation of earlier quantile regression approaches. Moreover, by incorporating non-crossing constraints, MMQR preserves the logical structure of the conditional distribution, ensuring robustness in the estimation process. The application of the Method of Moments Quantile Regression (MMQR) in this study is justified by its ability to capture the heterogeneous effects of economic and digitalization indicators on banking profitability across different levels of the return on equity (ROE) distribution. Unlike traditional Ordinary Least Squares (OLS) regression, which focuses on estimating the conditional mean relationship, MMQR provides a more comprehensive understanding of the distributional impacts of explanatory variables. This approach is particularly relevant in banking profitability analysis, where the determinants of ROE may exert varying influences depending on the financial health and competitive positioning of banks.