Pakistan's banking industry is vital for economic growth, and analyzing its performance, particularly efficiency, is crucial for stakeholders. While existing literature uses various measures, technical efficiency analysis, decomposable into pure technical and scale efficiencies, provides a holistic view. Previous studies in Pakistan have used non-parametric data envelopment analysis (DEA) and parametric stochastic frontier analysis (SFA), but these methods have limitations, such as biased efficiency estimates (DEA) or restrictive assumptions (SFA). This study addresses these limitations by employing a robust bootstrap DEA approach to analyze the technical efficiency of Pakistan's banking industry from 2006 to 2020. Furthermore, the study examines the impact of digitalization on banking efficiency, a topic with mixed findings in the international literature and lacking empirical evidence from Pakistan. Finally, the study explores absolute and conditional convergence of banking efficiency in Pakistan, aiming to determine if less efficient banks catch up to more efficient ones over time and how digitalization influences this process. This study uses data from 29 Pakistani banks, representing about 90% of the country's financial industry asset size.

The literature on the impact of digitalization on the banking sector highlights the significant implications of technology adoption. While several studies show a positive link between digital technology adoption and banking efficiency, others find a negative relationship, leaving the overall impact unresolved. There is a particular scarcity of research on this issue within the context of Pakistan's banking industry. Regarding banking efficiency convergence, existing studies reveal mixed results across different regions (EU, Latin America, China, Arab countries, and India). While some studies support the convergence hypothesis, demonstrating that less efficient banks catch up to more efficient ones over time, others show a lack of convergence or conditional convergence, where the convergence path depends on bank-specific factors. There is a significant gap in research on conditional banking efficiency convergence and the impact of digitalization on the convergence process, both in the international and Pakistani banking literature. This study seeks to address these gaps.

This study employs a two-stage approach. The first stage uses data envelopment analysis (DEA) to measure the technical efficiency of 29 Pakistani commercial banks from 2006 to 2020. Both traditional non-parametric DEA and a more robust bootstrap DEA approach are used to obtain bias-corrected efficiency estimates. The efficiency scores are decomposed into pure technical efficiency (PTE) and scale efficiency (SE) to identify the sources of inefficiency. The second stage uses econometric models to analyze the factors influencing banking efficiency and the convergence process. Specifically, Tobit regression and a two-step dynamic panel data system generalized method of moments (DPDSYS-GMM) are employed to assess the impact of digitalization (measured using ATM-based transactions, internet-based transactions, point-of-sale-based transactions, and a composite digitalization index) and other bank-specific and macroeconomic factors on banking efficiency. To analyze convergence, the study uses β-convergence and σ-convergence to examine absolute convergence and conditional β-convergence to assess the role of digitalization in the convergence process. The DPDSYS-GMM is chosen for its robustness in handling endogeneity, heteroskedasticity, and omitted variable bias in panel data settings. The data used includes bank-level data from the State Bank of Pakistan, macroeconomic data from the World Bank, and digitalization data from the International Telecommunication Union.

The first-stage DEA analysis reveals that the average bias-corrected overall technical efficiency (BSBC-OTE) of Pakistan's banking industry from 2006 to 2020 was 74%. The decomposition of efficiency shows that pure technical inefficiency, representing managerial inefficiency, accounts for the largest portion of overall inefficiency. The second-stage analysis, using both Tobit and DPDSYS-GMM models, demonstrates a significant and positive effect of digitalization on banking efficiency, supporting the hypothesis that increased digital transactions (ATM, internet, and point-of-sale) and a higher digitalization index lead to improved efficiency. Bank-specific factors, such as return on assets (ROA) and bank size, also positively influence efficiency. State-controlled banks are found to be significantly more efficient than private sector and special-purpose banks. Macroeconomic factors like interest rates and GDP growth rate also show a positive relationship with banking efficiency. The analysis of convergence reveals evidence of both absolute β-convergence and σ-convergence, indicating that less efficient banks caught up to more efficient ones, reducing cross-sectional efficiency dispersion. Conditional β-convergence analysis shows that digitalization significantly accelerates the convergence process, enabling less efficient banks to catch up faster. Other control variables, like loan growth ratio, loan loss provisions, and non-performing loan ratio, also positively influence the efficiency convergence process. The M2 money supply growth and GDP growth rate also positively affect efficiency convergence.

The findings of this study provide valuable insights into the dynamics of efficiency in Pakistan's banking industry and the role of digitalization in driving efficiency improvements and convergence. The positive impact of digitalization on efficiency aligns with the notion that technology adoption reduces costs, improves service delivery, and enhances competitiveness. The evidence of both absolute and conditional convergence suggests that the Pakistani banking industry is becoming more efficient and equitable, with digitalization acting as a catalyst for this convergence. The results highlight the importance of digitalization as a driver of efficiency gains and convergence in developing economies. The study's findings also underscore the importance of prudent lending practices and the influence of macroeconomic factors on both efficiency and convergence.

This research makes several significant contributions to the literature. First, it provides a comprehensive analysis of technical efficiency in Pakistan's banking sector using a robust bootstrap DEA methodology. Second, it offers empirical evidence on the positive relationship between digitalization and banking efficiency in Pakistan. Third, it demonstrates the existence of both absolute and conditional convergence in banking efficiency, highlighting the role of digitalization in accelerating this process. Future research could explore the long-term effects of digitalization, investigate other aspects of bank performance, and extend the analysis to other developing economies. Furthermore, exploring the impact of specific digital technologies (e.g., AI, blockchain) and the role of regulatory frameworks in shaping digital transformation and efficiency convergence would be beneficial.

The study has several limitations. First, the use of a specific DEA methodology may lead to variations in results compared to other approaches. Second, although the study used various digitalization indicators, incorporating additional factors such as investment in technology and R&D might offer a more comprehensive view. Third, the study's time frame (2006-2020) might not fully capture the long-term dynamics of convergence. A longer time series would be beneficial for more robust findings. Finally, the study is focused on Pakistan, limiting the generalizability of the findings to other contexts. Future research should consider these limitations and explore additional factors and contexts.