The noise error component in stochastic frontier analysis

Abstract

With a little help from a handful of scholars, the noise component of the composed error in a production model created the stochastic frontier analysis field. But after that glorious moment, it was confined to obscurity. We review what little research has been done on it. We present two cases where it torments us from the shadows, by sabotaging identification, and by distorting the sample skewness. We examine the relation between predicted noise and predicted inefficiency. For the Normal-Half Normal and the Normal-Exponential error specification, we provide its conditional expectation as predictor and we examine its distribution in relation to the marginal law. We also derive the conditional distribution of the noise and we compute confidence intervals and the probability of over-predicting it. Finally, we present a model where the noise, as the carrier of uncertainty, induces directly inefficiency. We conclude by showcasing our theoretical results through an empirical illustration.

Appendix: Identification in the asymmetric Laplace-Exponential model

A way to write the Asymmetric Laplace-Exponential composed density is

$$\begin{aligned} g_{\varepsilon }(\varepsilon ) =\frac{\tau (1-\tau )}{\tau \sigma _u + \sigma _v}{\left\{ \begin{array}{ll} { {\frac{\sigma _u \exp \left\{ \varepsilon /\sigma _u\right\} \;-\;(\tau \sigma _u + \sigma _v)\exp \left\{ (1-\tau )\varepsilon /\sigma _v\right\} }{[(1-\tau )\sigma _u -\sigma _v]}}} , &{} \varepsilon \le 0 \\ &{} \\ \exp \left\{ -\frac{\tau }{\sigma _v}\varepsilon \right\} , &{} \varepsilon > 0. \end{array}\right. } \end{aligned}$$

Table 4 Two examples of identification failure in the Asymmetric Laplace-Exponential specification

The correspondence with the two scale parameters from the alternative Laplace representation as a difference of two Exponentials is

$$\begin{aligned} \tau = \frac{\sigma _2}{\sigma _1+\sigma _2},\qquad \sigma _v=\frac{\sigma _1\sigma _2}{\sigma _1+\sigma _2}. \end{aligned}$$

A reason to use the $(\tau ,\, \sigma _v)$ parametrization is because $\Pr (v\le 0) =\tau$, allowing for a likelihood-based estimation of a quantile probability. Using the above density, we present in Table 4 two cases of identification failure. In the left block, the true data generating process has a symmetric Laplace (case (iii) mentioned in the main text), while in the block to the right (which is the second example), the noise is Asymmetric Laplace (case (iv) mentioned in the main text). In each block, the first column with likelihood values is produced using the true $(\tau , \sigma _v,\sigma _u)$ parameter values, while the second corresponds to the parameter vector that characterize the alternative equivalent specification.

The $\sigma _1,\,\sigma _2,\,\sigma _3$ are the true values holding in the sample, and $\sigma _3$ relates to inefficiency. Based on these three values, for every value of $\varepsilon$, we see that the Asymmetric Laplace-Exponential density $g_{\varepsilon }(\varepsilon )$ takes the exact same value for two different combinations of values for the parameters $(\tau , \sigma _v, \sigma _u)$. So for any realized sample of values, the likelihood will have the same value under these two sets of estimates–but only one will be the true one (the left columns in each block of the Table). Moreover, these, identical as regards the value of the likelihood, parameter vectors represent totally different situations, e.g., in the case to the right, we may greatly overestimate the scale parameter (and so the mean) of the inefficiency (0.9 instead of the true 0.5).