## 3 Materials and Methods
The benchmark econometric model is expressed following the approach of [PERSON] and [PERSON] [115]:  
\[ \begin{split} ROE_{i,t}=\beta_{0}+\beta_{1}ATM_{i,t}+\beta_{2}DESI \_IDT_{i,t}+\beta_{3}DESI\_HC_{i,t}+\beta_{4}EBANK_{i,t}+\\ \beta_{5}DIGINCL_{i,t}+\beta_{6}RD_{i,t}+\beta_{7}GDP_{i,t}+\beta_{8 }BANKMPL_{i,t}+\epsilon_{i,t}\end{split}  \tag{1}\]  
where \(ROE_{i,t}\) is the return on equity for the country \(i\) at time \(t\), and \(\epsilon_{i,t}\) is the error term. This equation represents the baseline panel regression model estimating the effects of automation and digitalization variables on return on equity (ROE), which is the primary performance indicator for banking institutions. Subscripts \(i\) and \(t\) denote country and time (year), respectively. On the left-hand side, \(ROE_{i,t}\) is the dependent variable. The right-hand side includes the following independent variables:  
* \(ATM_{i,t}\): density of automated teller machines (a mechatronic proxy);
* \(DESI\_IDT_{i,t}\) and \(DESI\_HC_{i,t}\) indicators of digital integration and human capital;
* \(EBANK_{i,t}\) and \(DIGINCL_{i,t}\): digital financial service usage and inclusion;
* \(RD_{i,t}\): R&D expenditures;
* \(GDP_{i,t}\): economic growth;
* \(BANK\)EMPL\({}_{i,t}\): number of employees in the banking sector;
* \(\epsilon_{i,t}\) captures the unobserved error term. Instead of estimating this model using the least squares (OLS) method, MMQR seeks to estimate conditional quantile functions for different values of \(\tau\), where \(\tau\) (0,1) defines various points in the conditional distribution of ROE. Thus, the model becomes the following:  
\[Q_{ROE}(\tau|X)=\mu(X)+\sigma(X)\cdot q_{U}(\tau) \tag{2}\]  
where \(Q_{ROE}(\tau|X)\) is the quantile \(\tau-a\) of the conditional distribution of bank ROE, \(\mu(X)\) and \(\sigma(X)\) being functions of the independent variables that control the location and dispersion of ROE. The error distribution is modeled to respect the conditions of the moments specific to quantile regression, following the approach of [PERSON] and [PERSON] [115]. It expresses the conditional quantile function of ROE given the covariates X. * \(Q_{ROE}(\tau|X)\) is the \(\tau\)-th quantile (e.g., 25 th, 50 th, 75 th, or 90 th percentile) of the conditional distribution of ROE;
* \(\mu(X)\) denotes the location function, reflecting how the center of the conditional distribution shifts with changes in X;
* \(\sigma(X)\) represents the scale function, capturing heteroskedasticity or how dispersion in ROE changes with covariates;
* \(q_{U}(\tau)\) is the \(\tau\)-th quantile of a standard uniform error term U, which introduces non-linearity into the model. This equation enables us to model distributional heterogeneity, revealing how the effects of covariates vary across different performance levels of banks. Thus, the application of the MMQR method for analyzing the impact of robotic automation and AI solutions in the banking sector involves several essential methodological steps. First, it is necessary to prepare the dataset, which includes the collection of information on the number of ATMs, DESI indicators, proportion of internet banking users, R&D expenditure, GDP per capita, and number of employees in the banking system for each EU country in the period of 2017-2022. The variables are checked for outliers, missingness, and multicollinearity, and logarithmic transformations are applied to normalize the distributions where necessary. Second, the MMQR model is estimated to use relevant quantiles such as \(\tau=0.25\), \(0.50\), \(0.75\), and \(0.90\) to capture the effects on low, medium, and high profitability banks. For each quantile, we solve the system of equations based on the moment conditions:  
\[E[\psi(Y,X;\theta)]=0,\ E[\phi(Y,X;\theta)]=0 \tag{3}\]  
where \(\theta\) is the vector of the model's estimator coefficients, and \(\psi\) and \(\phi\) are functions defined based on the distribution of model errors. Equation (3) represents the moment conditions central to the estimation of MMQR.