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Technical efficiency and inefficiency: SFA misspecification

The effect of external factors $z$ on technical inefficiency ($TI$) in stochastic frontier (SF) production models is often specified through the variance of inefficiency term $u$. In this setup the signs of marginal effects of $z$ on $TI$ and technical efficiency $TE$ identify how one should control $z$ to increase $TI$ or decrease $TE$. We prove that these signs for $TI$ and $TE$ are opposite for typical setups with normally distributed random error $v$ and exponentially or half-normally distributed $u$ for both conditional and unconditional cases. On the other hand, we give an example to show that signs of the marginal effects of $z$ on $TI$ and $TE$ may coincide, at least for some ranges of $z$. In our example, the distribution of $u$ is a mixture of two distributions, and the proportion of the mixture is a function of $z$. Thus if the real data comes from this mixture distribution, and we estimate model parameters with an exponential or half-normal distribution for $u$, the estimated efficiency and the marginal effect of $z$ on $TE$ would be wrong. Moreover, for a misspecified model, the rank correlations between the true and the estimated values of TE could be small and even negative for some subsamples of data. These results are demonstrated by simulations.