A dynamic stochastic frontier model with threshold effects: U.S. bank size and efficiency


Common/Single frontier methodologies that are used to analyze bank efficiency and performance can be misleading because of the homogeneous technology assumption. Using the U.S. banking data over 1984-2010, our dynamic methodology identifies a few data-driven thresholds and distinct size groups. Under common frontier assumption, the largest banks appear to be 22% less efficient on average than how they are in our model. Also, in the common frontier model, smaller banks seem to be relatively more efficient compared to their larger counterparts. Hence, common policies or regulations may not be well-balanced about controlling the banks of different sizes on the spectrum.

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  1. 1.

    Throughout this paper, whenever we refer to “technology”, we mean the production possibilities frontier (or simply the frontier) corresponding to that particular technology. Hence, at a given time period, if two groups of banks have different frontiers, we say that these groups of banks have different production technologies. See Färe and Primont (1990) for relevant discussions.

  2. 2.

    For other banking studies that examine efficiencies include Berger and Humphrey (1997), Vivas (1997), Muñiz (2002), Brissimis et al. (2010), Paradi et al. (2012), Galán et al. (2015), Dong et al. (2016), Tsionas (2017), and Delis et al. (2017).

  3. 3.

    See Tsionas (2002), El‐Gamal and Inanoglu (2005), Greene (2005), and Almanidis (2013) for a few examples.

  4. 4.

    This means that the econometric issue in estimating the best-practice technologies may not necessarily be solved by simply increasing the flexibility of the functional forms or using non-parametric methods that ignore state dependence.

  5. 5.

    Alternative approaches for dealing with heterogeneous technologies in the stochastic frontier models were considered by Tsionas (2002), Huang (2004), and Greene (2005) in a random parameters model framework, as well as the latent class model employed by Orea and Kumbhakar (2004), El‐Gamal and Inanoglu (2005), and Greene (2005). These methodologies require the parameters to be re-estimated with new information and this update may become computationally expensive, especially when the sample size is relatively large. In contrast, the threshold effect model’s parsimonious setup would allow the individual units to switch groups, as a result of change in certain characteristics such as size, without requiring a re-estimation of the parameters.

  6. 6.

    There are many different size groups defined and used by banking researchers and authorities. The cutoff points in almost all cases are arbitrarily selected.

  7. 7.

    In this paper, whenever we refer to a bank as “small” or “large”, this reference would be vaguely saying that the bank belongs to one of the bank groups in our sample that has relatively smaller or larger asset sizes compared to the other groups in our sample. Hence, in this context, even though a bank that has $1 billion asset size may be considered as large compared to smaller community banks in the sector, we consider this bank as a relatively small bank compared to the other banks in our sample with much larger asset sizes.

  8. 8.

    Durbin and Koopman (2012) provide details on the Kalman filter estimation techniques.

  9. 9.

    Other examples to the models where the one-sided error term is dynamic include Hultberg et al. (2004) and Ahn and Sickles (2000).

  10. 10.

    The Call Reports contain detailed data on a bank’s on-balance and off-balance sheet assets and liabilities, capital structure, income from earning assets, expenses, and other bank-specific structural and geographical characteristics.

  11. 11.


  12. 12.

    It is worth mentioning that, the number of unique community banks over the sample period exceeds 11,000 and their total number of observations is more than 750,000.

  13. 13.

    See DeYoung (2013) for more discussion.

  14. 14.

    See Kaparakis et al. (1994), Wheelock and Wilson (1995, 2001), Sickles (2005), Greene (2005), Kutlu (2012), and Almanidis (2013) among others.

  15. 15.

    See Baltensperger (1980) and Berger and Humphrey (1992) for the production and value-added approaches discussions.

  16. 16.

    Note that, although the total assets variable is a commonly employed threshold variable in the banking literature and in practice to assign banks in group sizes, there are other variables/factors that can also be employed separately or in combination to segment the banking industry. Such variables/factors may include a strategy mix, marketing, risk-taking, etc. A proper quantification of these factors in a multi-threshold model framework would potentially complete another piece of the banking technology/strategy puzzle.

  17. 17.

    In our view, these output and input variables cover the majority of outputs and inputs produced by an average bank operating in the U.S. commercial banking industry.

  18. 18.

    A translog function provides the second-order Taylor series approximation to any arbitrary function at a single point. In addition, the translog output distance function does not restrict the returns to scale measures and factor demand elasticities to be constant, as is required in the Cobb-Douglas output distance function case.

  19. 19.

    See Färe and Primont (2012) for more discussion on the scale elasticity measures for distance functions.

  20. 20.

    Appendix A explains the CSSW and LCSS models in detail.

  21. 21.

    For example, the fixed effects estimator of Schmidt and Sickles (1984), CSS estimators, and the Kalman filter estimator of DKS.

  22. 22.

    See Berger (1993), Berger and Hannan (1998), and Kutlu (2012) for more details.

  23. 23.

    To preserve the space, we do not report the estimates of the CSSW model, which are qualitatively similar to the estimates obtained under the LCSS model. The CSSW model estimates for the six groups and the single group are available upon request.

  24. 24.

    The details are provided in the website of the bank at www.westamerica.com

  25. 25.

    This is based on the empirical distribution of the returns to scale values around their median values.

  26. 26.

    See for example McAllister and McManus (1993) and Wheelock and Wilson (2001).


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Correspondence to Levent Kutlu.

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Almanidis, P., Karakaplan, M.U. & Kutlu, L. A dynamic stochastic frontier model with threshold effects: U.S. bank size and efficiency. J Prod Anal 52, 69–84 (2019). https://doi.org/10.1007/s11123-019-00565-6

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  • Dynamic Stochastic Frontier
  • Bank Efficiency
  • Bank Heterogeneity

JEL classification

  • C13
  • C23
  • D24
  • G21
  • G28