3. Methodology
3.3 Four-stage DEA approach
Fried et al. (1999) used a slack-based measure, the four-stage DEA procedure, to estimate the influence by the environment-adjusted variables of American hospital-affiliated nursing homes in 1993. They believed that the characteristics of the external environment could influence the ability of management to transform input to output. This procedure for incorporating the operational environment into a measure of technical efficiency can obtain a separate measure of managerial inefficiency.
As introduced by Fried et al. (1999), the first stage is to compute a DEA frontier by using the traditional inputs and outputs according to DEA model theory. The external variables are excluded. The efficiency scores as well as input slacks and output surpluses are computed for each observation.
In the second stage, a system of equations is specified in which the dependent variable for each equation is the sum of radial and non-radial input slack for an input-oriented model or radial plus non-radial output surplus for an output-oriented model. The independent variables are used to measure the features of the external operational environment. This equation system identifies the variation in total measures of inefficiency attributable to factors outside the control of management.
The third stage is to use the parameter estimates from the second stage to predict the total input slack or output surplus, depending upon model orientation. These predicted values represent the ‘allowable’ slack or surplus, due to the operational environment, and are used to compute adjusted values for the primary inputs or outputs.
The fourth stage is to re-run the DEA model under the initial output specification by using the adjusted input data set. The new radial efficiency measures incorporate the influences of the external variables into the production process, and isolate the managerial component of inefficiency. Figure 3 illustrates the four-stage DEA procedure.
z CRS efficiency scores z Target value of inputs z Slacks of inputs CRS DEA
Model Inputs & Outputs
Figure 3. Illustration of four-stage procedure
z Slacks of Adjust the input data set Significant factors ofThe detail of the procedure is disclosed as below.
The first stage begins with a specification of production technology. The efficiency
scores, the value of θ in equation (1), and the target value of the input variables during the sample period for each firm are computed by using the CRS DEA model. The definition of input target value is an input level which is utilized by a firm to be efficient.
The second stage is to estimate the N input equations by using Tobit regression because
of the estimative range from zero to infinity. The dependent variables are radial plus non-radial input slack equal to the absolute value of input actual value minus input target value. The independent variables are measures of external conditions applicable to the particular input. The objective is to quantify the effect of external conditions on the excessive use of inputs. The N equations are specified as:
( , , ) (2)
t t t t
i j j i j j j
IS = f Z β ε
1 1 1
i = ,...,K ; j = ,...,N ; t = ,...,T ;
where
IS is the total radial plus non-radial slack of input j for firm i in time t based on the
i jt DEA results from stage 1,Z is a vector of variables characterizing the operational
i jt environment for firm i that may affect the utilization of input j ,β
tj is a vector of coefficients to be estimated; andε
tj is a disturbance term. These equations explain thevariation in total by-variable measures of inefficiency. Note that the explanatory variables characterizing the operational environment in equation (2) are not restricted to be the same
across equations, needing not have a linear relationship with the dependent variables and can be a mixture of continuous and categorical variables.
The output surplus in this stage for an input oriented model is omitted. As Fried et al.
(1999) mentioned, “An input oriented model takes output as given and measures inefficiency by the potential reduction in inputs. Output surplus exists in empirical applications because the data set is sparse for some output vectors. Where it does exist, it is likely to be composed mostly of zeros and have insufficient variation to be useful in the estimation.”
The third stage is to use the estimated coefficients from the regression to predict total
input slack for each input and for each unit based on its external variables:1
As proposed by Fried et al. (1999), the predictions are used to adjust the primary input data for each unit. The each adjusted input data of each input for each sample firm is the original input plus the difference between maximum predicted slack and predicted slack as equation (4) below:
where {
t
}
Max IS
∧ j means the maximum predicted slack of the firms for input j in the same sample year t. The purpose of adjusting the primary input data by the difference between maximum predicted slack and predicted slack is to establish a base equal to the least favorable set of external conditions. A firm with the maximum predicted slack means it would get the most severe penalty by the operational environment. The environment where a firm with the maximum predicted slack operates in is the least favorable external environment. According to equation (4), this firm would not have to adjust its inputs at all. A firm with external variables generating a lower level of predicted slack would have its input vector adjusted to put it on the same basis as the firm with the least favorable external environment. The reason why choosing to use the least favorable operational environment as the base is to provide a performance target that managers can attain regardless of their operational environment. Managers will have no excuse of operational environment for failing to achieve the performance target.A special attention must be paid that the input adjustment takes the form of an increase in the original input. Predicted slack below the maximum predicted slack is attributable to external conditions more favorable than the least favorable conditions prevailing in the sample for that input. The purpose of the adjustment is to penalize the firm for the fewer inputs required to operate under favorable external conditions. Besides, another advantage is to avoid the possibility of a negative value for an adjusted input from the estimation of the Tobit regression, rendering the DEA problem for that unit without a solution. By increasing the input vector and leaving the output vector unchanged, the firm's performance is purged of the external advantage. This makes it possible to isolate managerial inefficiency by re-running the DEA model on the adjusted data set.
The final stage is to use the adjusted data set to re-run the DEA model under the initial
input-output specification and generate new radial measures of inefficiency. These radial scores measure the inefficiency that is attributable to management.
S
S C B
D
x
1/ y x
2/ y
0
S’
C’
B’ D’
S’
A
Figure 4. Shift in frontier after the slack-based adjustment in the CRS DEA model
Figure 4 illustrates the shifting course of frontier after conducting the slack-based adjustment in the CRS DEA model. Firm A with the maximum predicted slacks would get the most severe penalty by the operational environment. After conducting the slack-based adjustment in stage two and three, firm A would have the minimum adjustment (zero) while firm B would be added a punished input vector matrix from B to B’, and so do C and D to C’
and D’ respectively. The benefit from the operational environment of each firm would be eliminated. All firms would be brought into the same operational environment, the least favorable environment, to be evaluated. After re-running the DEA model in stage four, the