Abstract
Credit risk is crucial to understanding banks’ production technology and should be explicitly accounted for when modeling the latter. The banking literature has largely accounted for risk using ex-post realizations of banks’ uncertain outputs and the variables intended to capture risk. This is equivalent to estimating an ex-post realization of bank’s production technology which, however, may not reflect optimality conditions that banks seek to satisfy under uncertainty. The ex-post estimates of technology are likely to be biased and inconsistent, and one thus may call into question the reliability of the results regarding banks’ technological characteristics broadly reported in the literature. However, the extent to which these concerns are relevant for policy analysis is an empirical question. In this paper, we offer an alternative methodology to estimate banks’ production technology based on the ex-ante cost function. We model credit uncertainty explicitly by recognizing that bank managers minimize costs subject to given expected outputs and credit risk. We estimate unobservable expected outputs and associated credit risk levels from banks’ supply functions via nonparametric kernel methods. We apply this framework to estimate production technology of U.S. commercial banks during the period from 2001 to 2010 and contrast the new estimates with those based on the ex-post models widely employed in the literature.
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Notes
Some studies investigating banks’ profitability account for liquidity risk using ex-post liquidity ratios (e.g., long-term loans to liquid liabilities, liquid assets to total assets, or liquid assets to deposits). See Shen et al. (2009) and references therein.
Clearly, the fundamental principle of the production process in banking itself is quite certain, i.e., to borrow funds from one group of customers in the form of deposits and lend these funds to another group in the form of loans. However, the amount of loans that will ultimately generate income for a bank is uncertain because not all issued loans are paid back duly. Since the fraction of the nonperforming loans is unknown to banks in advance, the latter makes the production of performing (earnings) loans uncertain.
The terms “ex-ante cost function” and “ex-post cost function” were first coined by Pope and Chavas (1994).
Restrepo-Tobón et al. (2013) similarly document that one is more likely to find the evidence of non-increasing returns to scale among commercial banks if unobserved effects are ignored.
This narrowed definition is consistent with a more realistic banks’ objective, i.e., to maximize expected profits as opposed to maximize the potential for profits. It is also consistent with the proposition that bank managers (or banks themselves) maximize expected utility drawn from actual profits, not from the potential for profits.
An alternative approach is to estimate the dual profit function under the premise of profit maximization. This is mostly popular amid the studies of inefficiency in the banking industry in the stochastic frontier framework (e.g., Berger and Humphrey 1997).
Here we use the narrow definition of earning assets as discussed above: performing loans (\(y^+\)), rather than total issued loans (\(y\)), are treated as the output.
Here we abstract from other potential sources of biases across both the ex-post and ex-ante cost functions, such as biases due to the misspecification of the model, etc.
These earning outputs are computed by subtracting the value of nonperforming loans and securities from the corresponding reported total values, i.e., \(y^+_m=y_m-y^-_m\quad \forall \quad m=1,\dots ,4\).
As previously discussed, note that the total variable cost (\(C\)) includes expenses associated with total issued loans and securities (both those which turn out being performing and nonperforming) because the bank managers allocate inputs ex ante, i.e., before they know which loans would eventually become nonperforming.
Note that one does not need to regress outputs on inputs in order to obtain expected output levels per se: they can rather be recovered indirectly inside the numerical optimization algorithm (see Pope and Just 1996 for details). This approach, however, is still subject to Moschini’s (2001) criticism.
We opt for the local-constant estimator as opposed to the local-polynomial estimator (which has the same asymptotic variance but smaller bias) because the selection of optimal bandwidths for the former is less computationally demanding than it is for the alternative. The latter is a non-negligible issue given a large sample size of the dataset we use, as well as the number of estimations we need to perform.
In order to control for year and fixed effects, we also include the time trend, as well as an unordered bank-index variable. This is similar in nature to the least squares dummy variable approach in parametric panel data models with fixed effects.
We use a second-order Gaussian kernel for continuous covariates and Racine and Li’s (2004) kernels for ordered and unordered covariates (i.e., the time trend and the bank index, respectively).
Given that the cross-validation (CV) function is often not smooth in practice, we use multiple starting values for bandwidths when optimizing the function in order to ensure a successful convergence. Also, although we use constant bandwidths in our analysis, we acknowledge that one may instead prefer the use of adaptive bandwidths which adjust to the local sparseness of the data (if there is any). The selection of the adaptive bandwidths would, however, be more computationally demanding, especially given a relatively large number of dimensions in which the CV function needs to be optimized.
McAllister and McManus (1993) use a similar nonparametric procedure to estimate the expected rate of return and associated expected risk for U.S. banks.
To conserve space, we do not report the detailed results from the first stage (they are available upon request) and directly proceed to the discussion of the main results from the second stage.
Note that, like in the first stage, the ex-ante cost function could have been alternatively estimated via nonparametric kernel methods. In this paper, we however, opt for (admittedly more restrictive) translog specification. We acknowledge that several papers have documented that the translog form may sometimes be a poor approximation of banks’ cost function (e.g., Wheelock and Wilson 2001, 2012). We nevertheless opt for this parametric specification in order to facilitate the comparison of our findings with the results in the existing banking literature that overwhelmingly favors the translog specification.
This result may also stem from the fact that financial regulations require equity (financial) capital to expand in proportion to loans.
When computing these summary statistics, we omit the first and the last percentiles of the distribution of the returns to scale estimates, in order to minimize the influence of outliers. However, the omitted estimates correspond to the same observations across all five models, in order to keep the results comparable. We, therefore, can still cross-reference results from different models at the bank level.
Standard errors are constructed using the delta method.
We note that one ought to be careful here when comparing rankings using the Spearman’s rank correlation coefficient because the latter does not account for the estimation error associated with the estimation of scale economies.
To conserve space, we do not report detailed results from these models; they are available upon request.
Here, the “dot” designates the growth rate and \(s_j\) is the cost share of the \(j\)th input.
Since we have four outputs, we follow the literature and use the revenue-shared weighted output growth when computing the TFP growth, i.e., \(\dot{TFP}=\sum ^4_{m=1}r_m\dot{y}_m-\sum ^5_{j=1}s_j\dot{x}_j\), where \(r_m\) is the revenue share of the \(m\)th output.
Since the Divisia index is bank specific, in order to construct the plot, we use the asset-weighted average annual TFP growth rates.
For more on the decomposition of the TFP growth, see Kumbhakar and Lovell (2000).
In the case of the ex-post Model V, it is easy to show that the risk component of the TFP growth is defined as the negative of \(\frac{\partial {\ln C(\cdot )}}{\partial {\ln npl}}\dot{npl}\), where \(\dot{npl}\) is the growth rate of the ratio of total nonperforming loans to issued loans (an ex-post measure of credit risk).
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Acknowledgments
Restrepo acknowledges financial support from the Colombian Fulbright Commission, the Colombian Administrative Department of Science, Technology and Innovation (Colciencias) and EAFIT University.
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Malikov, E., Restrepo-Tobón, D. & Kumbhakar, S.C. Estimation of banking technology under credit uncertainty. Empir Econ 49, 185–211 (2015). https://doi.org/10.1007/s00181-014-0849-z
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DOI: https://doi.org/10.1007/s00181-014-0849-z