An integrated approach of data envelopment analysis and boosted generalized linear mixed models for efficiency assessment

Abstract

Performance evaluation is an important part in the management of any decision-making unit (DMU) as it identifies sources of managerial inefficiencies and provides a policy for inefficient DMUs to improve their efficiency. The latter is generally affected by environmental variables that are beyond managerial control. Modeling the impact of these environmental variables is a critical issue for both researchers and practitioners. Researchers developed and proposed several methods to deal with this issue in general and in the data envelopment analysis (DEA) literature in particular. However, the available two-stage DEA methods do not account for interdependence between observations and they are of limited use when the number of variables is fairly large. This paper proposes an integrated framework combining DEA, and boosted generalized linear mixed models (GLMMs) that accounts for the interdependence problem when studying the impact of environmental variables on performance. Additionally, the framework carries out variable selection. The framework is illustrated with a sample of 151 commercial banks from Middle East and North African countries.

Appendix

See Table 6.

Table 6 Environmental factors used in previous studies on bank performance

bGLMM algorithm

The bGLMM algorithm (Boosted GLMM) of Tutz and Groll (2010) is now briefly summarized (see the original article for all the computational details).

• 1. Initialization Choose a number of iteration $l_{max}$. Start with the model without any fixed effects (only an intercept). Compute initial values for the parameters $\beta _0 ,Q$ and for the random effects b. For example with the PQL method. Initialize the fixed effects parameters $\beta _1 ,\ldots ,\beta _p$ to 0.

• 2. Iteration For $l=1,\ldots ,l_{max}$ Refitting of residuals

  • (i) Computation of parameters. Based on the current model, for each fixed effect covariate $r=1,\ldots ,p$, compute an update of the estimated parameter. This is done separately (component-wise) for each r. The update is performed based on a one-step Fisher scoring.

  • (ii) Selection step. Let r be the component in $\left\{ {1,\ldots ,p} \right\}$ leading to the smallest value of the AIC (or BIC) and let $\hat{\beta }_0^*,\hat{\beta } _r^{*} ,\hat{b}^{*}$ be the corresponding estimates.

  • (iii) Update the parameters. Set $\hat{\beta } _0^{\left( l \right) } =\hat{\beta } _0^{\left( {l-1} \right) } +\hat{\beta } _0^*$, $\hat{\beta } _r^{\left( l \right) } =\hat{\beta } _r^{\left( {l-1} \right) } +\hat{\beta } _r^*, \quad \hat{b} ^{\left( l \right) }=\hat{b} ^{\left( {l-1} \right) }+\hat{b}^{*}$. Based on these updated parameters, compute the update $\hat{Q} ^{\left( l \right) }$ (for instance as an approximate REML-type estimate).

• 3. Final selection Select the final solution among all the solutions obtained for $l=1,\ldots ,l_{max}$ as the one with the smallest AIC (or BIC).