Indicators of ill-conditioned data sets and model misspecification in Data Envelopment Analysis: An extended facet approach

Data Envelopment Analysis employs mathematical programming to measure the relative efficiency of Decision Making Units (DMUs). This paper is concerned with development of indicators to determine whether or not the specification of the input and output space is supported by data in the sense that the variation in data is sufficient for estimation of a frontier of the same dimension as the input output space. Insufficient variation in data implies that some inputs/outputs can be substituted along the efficient frontier but only in fixed proportions. Data thus locally supports variation in a subspace of a lower dimension rather than in the input output space of full dimension. Each segment of the efficient frontier is in this sense subject to local collinearity. Insufficient variation in data provides a bound on admissible disaggregations in cases where substitution in fixed proportions is incompatible with a priori information concerning the production process. A data set incapable of estimating a frontier of full dimension will in this case be denoted ill-conditioned. It is shown that the existence of well-defined marginal rates of substitution along the estimated strongly efficient frontier segments requires the existence of Full Dimensional Efficient Facets (FDEFs). A test for the existence of FDEFs is developed, and an operational two-stage procedure for efficiency evaluation relative to an overall non-fixed technology is developed; the two-stage procedure provides a lower and an upper bound on the efficiency index for each DMU.