Stationary Points for Parametric Stochastic Frontier Models

Abstract

Stationary point results on the normal–half-normal stochastic frontier model are generalized using the theory of the Dirac delta, and distribution-free conditions are established to ensure a stationary point in the likelihood as the variance of the inefficiency distribution goes to zero. Stability of the stationary point and “wrong skew” results are derived or simulated for common parametric assumptions on the model. We discuss identification and extensions to more general stochastic frontier models.

Keywords:

• Dirac delta
• Generalized function
• Inefficiency estimation
• Ordinary least squares
• Singular distribution

Notes

1 This article is concerned with production function estimation, but the analysis can be applied to cost functions as well.

2 ALS also consider an exponential distributional assumption on inefficiency, leading to a normal–exponential model. Also, Badunenko, Henderson, and Kumbhakar (2012) emphasize that the signal to noise ratio is more important for identification than the pretruncated standard deviation of inefficiency.

3 Almanidis and Sickles (2012) examined the behavior of the likelihood for the N-DTN specification to understand how it produces an MLE that is asymptotically efficient relative to OLS regardless of the skew of the OLS residuals.

4 In this context, a stationary point is a solution to the maximum likelihood problem where the first-order conditions (FOC) equal zero. A stable stationary point is a solution where the Hessian is negative semidefinite.

5 Also, see Kobayashi (1991, 2009), Kobayashi and Shi (2005), Frieden (1983), and Arley and Buch (1950).

6 Intuitively, a Dirac delta can be thought of as a “function” on the real numbers that is everywhere zero, except at a single point in its domain (typically zero), where it is a spike of infinite height, yet its definite integral equals 1, in the same way that a probability density function integrates to one. Intuitively, a “delta sequence” is a sequence of probability density functions whose variance is shrinking toward zero about a single point (typically zero), resulting in a Dirac delta at that point. In other words, a Dirac delta is not a function per se, but it can be conceptualized as a delta sequence, solving the apparent contradiction that a spike of infinite height can possess a definite integral equal to one. In our context, the “sifting property” allows evaluation of the first- and second-order conditions of the likelihood function without having to worry that the likelihood is (in the limit) a function of an infinitely valued Dirac delta.

7 General results on the stability of the stationary point are not forthcoming (as we shall see), so we focus on the traditional parametric assumptions for our stability analysis. However, there may be scope for generalization based on Self and Liang (1987).

8 In the DTN case, when the pretruncated mean is nonpositive the skew of the distribution is positive. When the pretruncated mean is positive, there is scope for the distribution to have negative skew, however, the model is not identified as the variance of inefficiency goes to zero, therefore, we do not discuss this case.

9 See Horrace and Parmeter (2011) for a discussion of deconvolution in cross-sectional SFA.

10 We did not explore stability of the stationary point in these models. See Cho and Schmidt (2018) for results on stability in models with environmental variables.

11 Our results apply to the random effects, x-efficiency, selection, zero inefficiency, and spatial versions of the stochastic frontier model, as well.

12 For example, in ALS the distribution of u is known up to σ_u and can either be HN or exponential. The N-TN and N-DTN models have additional unknown parameters in the density of u and the likelihood function. Although the additional parameters make for a richer class of models, they make estimation more difficult in general. Moreover, these additional parameters are not identified when ${\sigma }_u\to 0$.

13 This is true for the N-TN and the N-DTN models (when the pretruncated mean is negative) and for the N-E model.

14 For probability densities that have Lebesgue measure, the dominated convergence theorem can be used to allow the interchanging of the limit and the integral. However, if the sequence of probability densities converge to a Dirac delta, then dominated convergence is not applicable.

15 Technically, it is a generalized function. See Frieden (1983) for a measure theoretical definition of the Dirac delta.

16 See Rothenberg (1971) for a discussion on the concept of observationally equivalence.

17 In particular, any derivatives of the likelihood with respect to the parameters of f_v will be sums and products of integrals, whose integrands contain the products of derivatives of f_v convoluted with delta sequence, $f_{u,m}$, so that all those integrals converge to exactly those same derivatives of f_v evaluated at u_i = 0.

18 The problem is not that the TN distribution does not satisfy Equation (6)(6) $$f_u(u,{\sigma }_u)=f_u^{\ast }(u/{\sigma }_u)/{\sigma }_u,$$(6) . The problem is our choice of notation for the density, which fails to capture the fact that the definite integral in the denominator must be scaled.

19 We did not explore stability of the stationary point in these models. See, Cho and Schmidt (2018) for results on stability in models with environmental variables.

20 For a complete discussion of the data see Erwidodo (1990)).

21 OLS estimates are used for the starting values for the parameters of the single-parameter families. We use zero for the starting values for the estimate of the pretruncated mean (μ) and the absolute value of the maximum value of OLS residuals for the starting value for the upper bound of inefficiency (B).

22 To save space, we only report results for continuous production inputs and do not report standard errors.

23 We also tried estimating the Li (1996) model where inefficiency is specified as a uniform distribution on the interval [0, b]. We computed the estimate of b using the method of moments, but it produced a complex valued estimate, so we stopped.

24 Constructing confidence interval using the inferential procedure for the N-TN and N-DTN would require conditioning on specific values for the pretruncated mean and the pretruncated mean and the upper bound, respectively. We are unsure how these values would be determined a priori.

25 In the presence of the wrong skew, the parameters are on the boundary of the parameter space and the likelihood becomes poorly behaved. We, therefore, appeal to a generalized LR test. The test has a nonstandard distribution, and the critical values are taken from Koddle and Palm (1986). Here our critical value is 2.706 at the 5% level.