stochastic frontier analysis

Stochastic frontier analysis (SFA) is a method of economic modeling. It has its starting point in the stochastic production frontier models simultaneously introduced by Aigner, Lovell and Schmidt (1977) and Meeusen and Van den Broeck (1977).

The ''production frontier model'' without random component can be written as:

$y_i = f(x_i ;\beta ) \cdot TE_i$
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where ''y_i'' is the observed scalar output of the producer ''i'', ''i=1,..I, x_i'' is a vector of ''N'' inputs used by the producer ''i'', ''f(x_i, β)'' is the production frontier, and $\beta$ is a vector of technology parameters to be estimated.

''TE_i'' denotes the technical efficiency defined as the ratio of observed output to maximum feasible output.
''TE_i = 1'' shows that the ''i-th'' firm obtains the maximum feasible output, while ''TE_i < 1'' provides a measure of the shortfall of the observed output from maximum feasible output.

A stochastic component that describes random shocks affecting the production process is added. These shocks are not directly attributable to the producer or the underlying technology. These shocks may come from weather changes, economic adversities or plain luck. We denote these effects with $\exp \left\$. Each producer is facing a different shock, but we assume the shocks are random and they are described by a common distribution.

The stochastic production frontier will become:

$y_i = f(x_i ;\beta ) \cdot TE_i \cdot \exp \left\$

We assume that ''TE_i'' is also a stochastic variable, with a specific distribution function, common to all producers.

We can also write it as an exponential $TE_i = \exp \left\$, where ''u_i ≥ 0'', since we required ''TE_i ≤ 1''. Thus, we obtain the following equation:

$y_i = f(x_i ;\beta ) \cdot \exp \left\ \cdot \exp \left\$

Now, if we also assume that ''f(x_i, β)'' takes the log-linear Cobb–Douglas form, the model can be written as:

$\ln y_i = \beta _0 + \sum\limits_n$

where ''v_i'' is the “noise” component, which we will almost always consider as a two-sided normally distributed variable, and ''u_i'' is the non-negative technical inefficiency component. Together they constitute a compound error term