Efficiency and Productivity Measurement: Bootstrapping DEA Scores. D.S. Prasada Rao School of Economics The University of Queensland, Australia. Measures of Reliability for DEA Scores.
D.S. Prasada Rao
School of Economics
The University of Queensland, Australia
As DEA is a non-parametric and non-stochastic approach, efficiency scores from DEA have been treated as non-stochastic.
However, there are attempts to see how DEA scores are affected by changes in data – mainly to see the effect of outliers.
Simar and Wilson have been working on the problem of generating standard errors for DEA scores using “bootstrap” technique.
An alternative to the bootstrap technique is the technique of “jackknife” which is a simpler technique.
Monte Carlo simulation experiments are often used to estimate
the sampling distributions of econometric estimators. Such
experiments typically involve several steps:
Specify a data generating process (DGP)
The distribution of the estimates obtained in step 3 approximates the sampling distribution of the estimator. The bootstrap is a form of Monte Carlo experiment where the DGP is unknown.
Methods for conducting a DEA bootstrap have been suggested by
We only discuss the Lothgren-Tambour (LT) method because
Let us consider input-oriented DEA models where the output vectors q1, …, qI are treated as fixed. We need to specify a DGP that will allow us to generate data on x1, …, xI.
Let Then is a technically-efficient input combination capable of producing qi. Suppose the process generating the distances for all firms is Then a DGP for x1, …, xI is completely characterised by
q1, …, qI and F.
x2 = ρ2x2 = (2, 4)
x2 = (1, 2)
q = 1
1 2 3 4 5
Let denote the DEA estimate of ρi (computed as the inverse of the optimised value of the DEA objective function). We estimate by projecting xi onto the estimated frontier:
i = 1, …, I,
We estimate F using the empirical distribution function (EDF) of the
x2 =(2, 4)
q = 1
1 2 3 4 5
To obtain B bootstrap samples:
Use the observed data to estimate the input-oriented DEA model, and project the observed data points onto the frontier using Set b = 1.
These B bootstrap samples can be used to construct confidence
In the hospital example and
To illustrate generation of the first bootstrap sample, suppose 4 drawings from the U(0,1) distribution happen to be 0.46, 0.76, 0.18 and 0.92. This implies and
We then solve the DEA problem using this data.
Let be the computed DEA score for firm i in the sample.
Suppose be the scores generated from the bootstrapped sampling procedure which is conducted B times. Then we can compute bias and SE as: