Cross-Efficiency Evaluation and Decomposition with Directional Distance Function in Parallel Two-Stage Systems : An Application to Mutual Funds

The existing cross-efficiency evaluation methods under network structures are all limited to two basic network systems: series and parallel. Based on the directional distance function (DDF), we firstly propose the cross-efficiency evaluation method for a parallel two-stage system, which consists of several independent parallel processes and each of them is a sub-system with a general two-stage series structure. The proposed method comprehensively measures the cross-efficiencies of decision making units (DMUs) by simultaneously reducing inputs and increasing outputs. Due to the complex nested network structure of the system, we design weight sets to aggregate cross-efficiencies and decompose cross-efficiency scores, respectively. For each DMU, the resulting system cross-efficiency score can be decomposed into a weighted average of all the process cross-efficiency scores, a weighted average of the stage cross-efficiency scores in all the processes, and a weighted average of the cross-efficiency scores of overall stages, respectively. In each process, the process cross-efficiency score can be decomposed into a weighted average of the two stage cross-efficiency scores. To avoid unboundedness and meanwhile ensure a unique cross-efficiency for each DMU, a model for avoiding zero multipliers and a neutral model are provided for the parallel two-stage system. Finally, an empirical study on 40 open-ended equity funds in China from 2018 to 2020 verifies the rationality and practicability of the newly proposed method