Have inputs been substituted for each other? Nonparametric tests for factor substitution in banking data envelopment analysis

ABSTRACT Standard Data Envelopment Analysis radial models such as CCR and BCC assume that inputs are substituted for each other along the efficiency frontier. However, this assumption is rarely verified empirically, creating the risk of specification bias if factors are not substituted. This paper introduces a nonparametric testing protocol to determine whether inputs have, in fact, been substituted for each other along the efficiency frontier. It employs Local Weighted Quantile Regression to estimate the partial derivatives of the input frontier directly from the data, classifying input pairs as substitutes or non-substitutes based on the sign and statistical significance of the estimated frontier slopes. The methodology is illustrated using Monte Carlo simulations of known substitute and non-substitute technologies. When applied to 2,534 U.S. commercial banks, the protocol finds no evidence of substitution for any input pair: 13 of 15 specifications classify the inputs as non-substitutes that increase and decrease together, and the remaining two yield positive but imprecisely estimated slopes. The paper then shows that applying the standard CCR model to the bank data results in substantial biases in bank efficiency scores. Finally, the paper identifies models that have been used to account for fixed proportion technologies.