1 Introduction
A stochastic frontier analysis (SFA) is one of the most popular tool in Econometrics for analysis of the technical efficiency (Aigner et al (1977); Meeusen and van Den Broeck (1977)
). A technical efficiency is a measure of the ratio of actual output to the maximum possible output. Stochastic frontier model is a production function model with two error terms — symmetrical statistical noise and nonnegative error term representing technical inefficiency. A large number of SFA models implies heteroscedasticity of the inefficiency in order to obtain robust estimates of the frontier function’s parameters as well as the estimates of the technical efficiency (
Kumbhakar and Lovell (2000)). Under such assumption estimates of technical efficiency depend on inputs of the model and the uncertainty of an inefficiency error, which in turn depends on heteroscedasticity factors.The signs of dependency of inputs on the technical efficiency are evident from estimates of coefficients. An effect of the uncertainty of the inefficiency error on the technical efficiency is unclear, as the relationship in the model is more complex. While, this sign can provide more inside on what’s going on in SFA models as well as provide more evidence on quality of those models.
Similar problems were considered before. In (Wang, 2002; Ray et al, 2015) authors derived an expression for marginal effects of the heteroscedasticity factors on the expected value of inefficiency and shown that the sign of the coefficient for heteroscedasticity factor equals to the sign for marginal effect. In Bera and Sharma (1999) authors evaluated monotonicity of the dependence of the technical efficiency on the overall error of estimate of the technical efficiency. In Kumbhakar and Lovell (2000) authors estimated marginal effects of the technical efficiency with respect to heteroscedasticity factors. However, according to our knowledge the dependence of the technical efficiency defined in a usual way on the uncertainty of the inefficiency error has not been yet identified. Moreover, direct application of the techniques from the articles above doesn’t allow identification of the desired dependence.
This work considers the dependence of the technical efficiency on the uncertainty of the inefficiency term. We infer the signs of this dependence for different distribution assumptions of the inefficiency term. In additional we investigate the signs and values for dependence of the technical efficiency on the uncertainty of the inefficiency for manufacturing industry of Russia taken from Krasnopeeva et al (2016).
This study is organized as follows:
2 Marginal effects for the technical efficiency with respect to the uncertainty of the inefficiency
Stochastic frontier analysis was proposed in (Aigner et al (1977); Meeusen and van Den Broeck (1977)). Further improvements of the methods as well as for crosssectional models as for panel data models are presented e.g. in Battese and Coelli (1988); Cornwell et al (1990); Heshmati et al (1995); Kumbhakar and Lovell (2000); Greene (2005); Kumbhakar et al (2014).
The basic SFA model is a parametric production function with a random error, consisting of two components — stochastic noise and inefficiency term (Battese and Coelli (1988); Kumbhakar and Lovell (2000)). For crosssectional data the basic SFA model is the following:
(1) 
wherein is output, are inputs, , , a production function usually takes the loglinear CobbDouglas form or the transcendental logarithmic specification (translog). Errors и are independent and unobservable components of . The is a statistical noise term. The is nonnegative distributed inefficiency term. Here we consider specification of the model (1) with and exponential distribution for .
In accordance with (Kumbhakar and Lovell, 2000; Wang and Schmidt, 2002) ignoring the heteroscedasticity of the inefficiency term causes biased estimates of the frontier function’s parameters as well as the estimates of the technical efficiency.
Exponential distribution of inefficiency
The exponential distribution of
has probability density function:
(2) 
Here and below we omit th index for short.
The simplest definitions are observation nonspecific measure of technical efficiency, , and the observation nonspecific measure of inefficiency, :
(3)  
(4) 
Those measures have a number of limitations due to the Independence of observations and therefore are not widely used in the literature.
The most common definitions are observation specific measure of technical efficiency, , and the observationspecific measure of inefficiency, , (see Battese and Coelli (1988), Kumbhakar and Lovell (2000)):
(5)  
(6)  
(7) 
where is the probability density function and is the cumulative density function of the standard normal distribution.
Proposed by (Caudill and Ford, 1993; Caudill et al, 1995; Hadri, 1999; Wang, 2003)
heteroscedasticity can be parameterized by a vector of observable variables and the associated parameters.
(8) 
where are heteroscedasticity factors (observable variables), and are parameters.
It is easy to calculate marginal effects of the on the mean of from (3):
Marginal effects of the kth factor on the mean of are:
The same way we can calculate marginal effects of the on the from (4):
Marginal effects of the kth factor on the are:
One can see that the sign of the marginal effect of the kth factor coincides with sign of parameter with the corresponding factor . So, inefficiency increases with increased value of heteroscedasticity factor (and uncertainty of correspondingly) while technical efficiency decreases.
Obviously, such conclusions seems to be fair also for observationspecific definitions of inefficiency and technical efficiency in (5)(6). But we could not find evidence of this statement in the literature.
That’s why we decide to obtain formula for the marginal effect for the inefficiency (5) and the technical efficiency (6) with respect to the uncertainty of the inefficiency in statements 12 of the model (1)(2) and to prove the statement that inefficiency increases and technical efficiency decreases with increasing the uncertainty of the inefficiency for exponential specification in Theorems 2.12.2.
Nevertheless, does obtained results are valid for any nonnegative distribution of inefficiency term ? We shown that in general it’s not true. We shown a counterexample of the nonnegative distribution of inefficiency term , for which marginal effect for the technical efficiency with respect to the uncertainty of the inefficiency will be positive in Theorem 2.3.
Now we are ready to obtain the marginal effects for the inefficiency and technical efficiency with respect to the uncertainty of the error of inefficiency . The marginal effect is the partial derivative .
Statement 1
In case the marginal effect of the efficiency (5) with respect to equals:
(9) 
Statement 2
In case the marginal effect of the technical efficiency (6) with respect to equals:
(10) 
Assuming heteroscedasticity of the error we get the uncertainty , , where are heteroscedasticities.
Now the marginal effect for the technical efficiency with respect to heteroscedasticity factor taking into account (17) is:
(11) 
At Figure 1 there is a dependence of the technical efficiency on the uncertainty of the error . It was calculated using (6) for different and fixed . For these values the technical efficiency decreases monotonically as the uncertainty of the error of inefficiency increases. Let us provide a formal statement on this dependence in Theorem 2.2 and for all and .
Theorem 2.1
For the model (1) with the inefficiency error if the uncertainty increases, the inefficiency increases:
(12) 
Theorem 2.2
For the model (1) with the inefficiency error if the uncertainty increases, the technical efficiency decreases:
(13) 
Theorem 2.3
There exists a parametric family of distributions of bounded on such that the marginal effect for the technical efficiency with respect to the uncertainty of the error of the inefficiency for the model (1) has different sign for different values of parameters.
The proof provides an example of such parametric family of distributions.
3 Empirical results
In this section we investigate the dependence of the technical efficiency on the uncertainty of the inefficiency error using empirical estimation. The used dataset consists of 2004–2013 data for Russian manufacturing industry companies.
As a dependent variable of estimated model we use turnover (variable ) of a firm. As inputs we use: fixed assets (); assets (); other assets (assets subtracting fixed assets, ); labor (). Other assets allow to compare the technical efficiency of companies with different production models, based on ownership of equipment and its lease as in Ipatova and Peresetsky (2013). As a heteroscedasticity factor we use export activity variable (), which equals if there exists an year in range when this company is an exporter and otherwise. The data contains information about . Some statistics regarding the inputs is given in Table 1.
To get the final dataset we use procedure similar to one in article Bessonova et al (2003); Fiorentino et al (2006) to get rid of bankrupt and shadow companies.
The obtained dataset consists of pairs companyyear in . The final panel is unbalanced, and the number of companies varies at each period.
Input  Mean  St.deviation  Min  Max 

Turnover  112819.3  463140.7  100.10  28705324 
Fixed assets  33236.9  232354.1  1.10  10696317 
Labor  132.6  294.6  10.00  8564 
Assets  83899.5  452009.9  3.41  22451868 
Other assets  50662.6  245962.9  1.10  11773889 
Export activity  0.3  0.46  0.00  1.00 
The model (1) is used for panel data in the following specification:
(14) 
where has a parametric translog or Cobb–Douglas specification. Index corresponds to companies, and index corresponds to year of observation. — turnover, — inputs,
. Taking into account differences between branches of industry and changes in macroeconomic situation we use auxiliary dummy variables
, which corresponds to subindustries and time effects. The and has exponential distribution.We use indicators for different industries to be able to compare estimates of the stochastic efficiency scaled to the interval , with corresponds to the maximum efficiency similar to Kumbhakar and Lovell (2000).
Due to the significant sample heterogeneity we assume heteroscedasticity of the inefficiency in order to obtain robust estimates of SFA parameters and technical efficiency Kumbhakar and Lovell (2000).
We assume that uncertainty of the noise depends on the firm’s size () and uncertainty of the inefficiency depends on the firm’s size () and export activity:
(15) 
Input  Cobb  translog 

Douglas  
0.717  1.914  
(0.003)  (0.015)  
0.034  0.103  
(0.001)  (0.006)  
0.316  0.035  
(0.002)  (0.008)  
  0.010  
(0.001)  
  0.223  
(0.004)  
  0.045  
(0.001)  
  0.030  
(0.001)  
  0.030  
(0.001)  
  0.049  
(0.002)  
constant  5.025  3.715 
(0.018)  (0.038)  
Economic sector  In model  
Year  In model  
0.198  0.184  
(0.006)  (0.006)  
0.261  0.246  
(0.017)  (0.016)  
0.126  0.144  
(0.004)  (0.005)  
LogLikelihood  192657.7  187400.1 
Sample size  157185 
The estimates of parameters and their standard deviations in brackets for Cobb–Douglas and translog specifications. All coefficients are nonzero with at least 1% significance
From results presented in Table 2 we can see that for both CobbDouglas and translog specifications the uncertainties of inefficiency errors are significant, and the signs of the corresponding effects are negative (and positive for random error). Using Theorem 2.2 obtained results shows that exporting companies are more technical efficient then nonexporting companies as well as increasing in assets relates with increasing in technical efficiency of companies.
4 Conclusions and discussions
We considered stochastic frontier model for exponential distribution of the inefficiency error. For this specification the marginal effect for the technical efficiency with respect to the uncertainty of the error of the inefficiency is negative. However, there exist quite natural distributions with the support at and with the positive marginal effect. The real data experiments confirm that for exponential distribution there is a significant dependence of the uncertainty of the inefficiency error on the technical efficiency.
References
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Appendix A Proofs of the statements
Let us prove Statement 1.
Proof
Let us prove Statement 2.
Proof
We can get from (6) the marginal effect for the technical efficiency with respect to the uncertainty of the inefficiency error:
(17) 
∎
Let now us prove Theorem 2.1
Proof
From (16) we can see that the denominator of the expression is positive. So we should proof positive sign of the expression’s numerator.
Lemma 1
Expression monotonically increases from 0 to 1 in the interval .
Proof
Obviously that .
Therefore
(18)  
(19) 
Now we can calculate the derivative of and proof that it is positive:
(20) 
Since is positive, we should proof now that g(z) is positive too. Let proof first auxiliary lemma 2:
Lemma 2
for .
Proof
If then and .
If then .
Therefore . ∎
The derivative of is .
Lemma 3
.
Proof
Obviously, . Also we can find that .
Therefore . ∎
Let us prove Theorem 2.2 now.
Proof
From (6) replacing by we get:
(21) 
If increases, then increases too. So, it is sufficient to prove the following inequality:
(22) 
In a way similar to (17) we obtain the partial derivative of the technical efficiency with respect to . From (21) we get:
(23) 
where is the cumulative density function of the standard normal distribution, and is the probability density function of the standard normal distribution.
As the first term in the right side of (23) is positive, then inequality (22) hold, if :
(24) 
where .
Now we calculate the derivative of on from (24):
(25) 
The main steps of the proof are the following:

increases monotonically from for to for

As has different behavior for different . we consider intervals: ; ; ; ; .
For some it holds that:
So, for it holds that:
(26)  
If , then:
(27)  
To prove inequality (24) we consider the following intervals: ; ; ; ; . Similar arguments are used in Zaitsev et al (2013).
1. Let .
We prove that . Then if , и , then for , and, consequently, (22) holds: .
For to prove (25) it is sufficient to prove:
(28) 
As , , then and consequently . Then we get and in the end we get that the initial inequality holds.