## 4 Results
The results of the _Kao Residual Cointegration Test_ (Table 4) confirm the presence of a long-run equilibrium relationship among the variables, rejecting the null hypothesis of no cointegration with a statistically significant ADF t-statistic \(-6.936554\), \(p=0.0000\)). This suggests that return on equity (ROE), research and development expenditure (RD), GDP per capita (GDP), E-banking adoption (EBANK), digital inclusion (DIGINCL), and digital economy indicators (DESI_IDT, DESI_HC), along with bank employment (BANKEMPL) and the number of ATMs (ATM), exhibit a stable long-term association. Such cointegration implies that these variables do not drift apart over time and are influenced by common structural factors shaping financial and digital economic performance in the European Union. Further validation through the Augmented Dickey-Fuller (ADF) test on the residuals strengthens the evidence for cointegration, as the negative and significant coefficient of RESID\((-1)\)\(-1.140605\), \(p=0.0000\)) indicates that deviations from equilibrium are corrected over time. The high adjusted R-squared (0.562447) suggests that a substantial proportion of variations in the dependent variable (changes in residuals) can be explained by past disequilibria, reinforcing the strength of the long-run relationships. The _Johansen Cointegration Tests_ (Table 5), employing both the Trace and Maximum Eigenvalue tests, provide further confirmation of these long-run relationships. The Trace test indicates two cointegrating equations, while the Maximum Eigenvalue test suggests three cointegrating equations, rejecting the null hypothesis at the 5% level. These results imply that banking performance, digital transformation, and economic development are jointly determined by multiple stable relationships. \begin{table}
\begin{tabular}{l c c} \hline \hline  & **t-Statistic** & **Prob.** \\ \hline ADF & \(-6.936554\) & 0.0000 \\ Residual variance & 0.120619 & \\ HAC variance & 0.073685 & \\ \hline ADF Test Equation & & \\ Dependent Variable: D(RESID) & Coefficient & Prob. \\ R-squared: 0.562447 & \(-1.140605\) & 0.0000 \\ Adjusted R-squared: 0.562447 & & \\ \hline \hline \end{tabular} Source: Author’s own calculation.  
\end{table} 
Table 4: Kao cointegration test. Turning to residual diagnostics (Table 6), the _likelihood ratio test for heteroskedasticity_ unequivocally rejects the null hypothesis of homoskedastic residuals across both cross-sections and time periods. This finding highlights structural differences across banking institutions and EU economies, suggesting that variability in banking profitability and digital transformation effects is contingent on country-specific factors. The significant log-likelihood differences between the restricted and unrestricted models confirm that failing to account for heteroskedasticity would lead to misleading inferences. Examining the _Panel Estimated Generalized Least Squares (EGLS) results_ (Table 7a-c), the statistically significant coefficients provide deeper insights into the determinants of banking profitability. The positive and significant impact of digital inclusion on return on equity suggests that broader accessibility to digital financial services fosters banking performance. However, the negative coefficient associated with digital integration (DESI_IDT) indicates that extensive reliance on digital technologies does not uniformly enhance profitability, possibly due to implementation costs or transitional inefficiencies. Banking employment exerts a negative influence on profitability, supporting the argument that labor-intensive banking models may be less efficient in the context of increasing digital adoption. The positive association between research and development and return on equity highlights the role of innovation in sustaining banking performance, reinforcing the idea that investment in technological progress is instrumental in fostering competitive advantages in the banking sector.