Xbeta is the structural part on the right-hand side of the model. Predicted value is the predicted dependent variable value. For censored variables, if the predicted value is outside the boundaries, it is reported as the closest boundary. For discrete variables, it is the level whose boundaries Xbeta falls between. Residual is defined only for continuous variables and is defined as
Error standard deviation is in the model. It varies only when the HETERO statement is used.
Marginal effect is defined as a contribution of one control variable to the response variable. For the binary choice model with two response categories, , , ; and ordinal response model with response categories, , define
The probability that the unobserved dependent variable is contained in the jth category can be written as
The marginal effect of changes in the regressors on the probability of is then
where . In particular,
The marginal effects in the Box-Cox regression model are
The marginal effects in the truncated regression model are
where and .
The marginal effects in the censored regression model are
Expected and conditionally expected values are computed only for continuous variables. The inverse Mills ratio is computed for censored or truncated continuous, binary discrete, and selection endogenous variables.
Let and be the lower boundary and upper boundary, respectively, for the . Define and . Then the inverse Mills ratio is defined as
for a continuous variable and defined as
for a binary discrete variable.
The expected value is the unconditional expectation of the dependent variable. For a censored variable, it is
For a left-censored variable (), this formula is
where .
For a right-censored variable (), this formula is
where .
For a noncensored variable, this formula is
The conditional expected value is the expectation given that the variable is inside the boundaries:
Probability applies only to discrete responses. It is the marginal probability that the discrete response is taking the value of the observation. If the PROBALL option is specified, then the probability for all of the possible responses of the discrete variables is computed.
Technical efficiency for each producer is computed only for stochastic frontier models.
In general, the stochastic production frontier can be written as
where denotes producer i’s actual output, is the deterministic part of production frontier, is a producer-specific error term, and is the technical efficiency coefficient, which can be written as
In the case of a Cobb-Douglas production function, . See the section Stochastic Frontier Production and Cost Models.
Cost frontier can be written in general as
where denotes producer i’s input prices, is the deterministic part of cost frontier, is a producer-specific error term, and is the cost efficiency coefficient, which can be written as
In the case of a Cobb-Douglas cost function, . See the section Stochastic Frontier Production and Cost Models. Hence, both technical and cost efficiency coefficients are the same. The estimates of technical efficiency are provided in the following subsections.
Normal-Half Normal Model
Define and . Then, as it is shown by Jondrow et al. (1982), conditional density is as follows:
Hence, is the density for .
Using this result, it follows that the estimate of technical efficiency (Battese and Coelli, 1988) is
The second version of the estimate (Jondrow et al., 1982) is
where
Normal-Exponential Model
Define and . Then, as it is shown by Kumbhakar and Lovell (2000), conditional density is as follows:
Hence, is the density for .
Using this result, it follows that the estimate of technical efficiency is
The second version of the estimate is
where
Normal-Truncated Normal Model
Define and . Then, as it is shown by Kumbhakar and Lovell (2000), conditional density is as follows:
Hence, is the density for .
Using this result, it follows that the estimate of technical efficiency is
The second version of the estimate is
where