Stochastic frontier analysis

   Stochastic Frontier Analysis
   
   Stochastic Frontier Analysis 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).
   A production frontier model can be written as:

y_{i}=f(x_{i};β)⋅TE_{i}

       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 β is a vector of technology parameters to be estimated.
   TEi denotes the technical efficiency defined as the ratio of observed output to maximum feasible output
   TEi=1 shows that the i-th firm obtains the maximum feasible output, while TEi<1 provides a measure of the shortfall of the observed output from maximum feasible output.
   We include now a stochastic component that describes random shocks that affect the production process and are not directly attributable to the producer or the underlying technology. These shocks may come from weather changes, economic adversities or plain luck. Let's denote these producer specific effects with exp{v_{i}}. Later we will see why we prefer the exponential form.
   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. Let's denote these producer specific effects with exp{v_{i}}.
   The stochastic production frontier will become:

y_{i}=f(x_{i};β)⋅TE_{i}⋅exp{v_{i}}

   We assume that TEi is also a stochastic variable, with a specific distribution function, common to all producers.
   We can also write it as an exponential TEi=exp{-u_{i}}, where ui≥0, since we required TEi≤1.
   Now, if we also assume that f(x_{i},β) takes the log-linear Cobb-Douglas form, we can write our model as:
   

ln y_{i}=β₀+∑β_{n}ln x_{ni}+v_{i}-u_{i}

   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, with a specific distribution to be determined, hence the name of "composed error model" as is often referred. 
   This model was simultaneously proposed by Aigner, Lovell and Schmidt (1977) and Meeusen and Van den Broeck (1977).