). 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 non-negative 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 cross-sectional 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 cross-sectional data the basic SFA model is the following:
wherein is output, are inputs, , , a production function usually takes the log-linear Cobb-Douglas form or the transcendental logarithmic specification (translog). Errors и are independent and unobservable components of . The is a statistical noise term. The is non-negative 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
Here and below we omit -th index for short.
The simplest definitions are observation non-specific measure of technical efficiency, , and the observation non-specific measure of inefficiency, :
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 observation-specific measure of inefficiency, , (see Battese and Coelli (1988), Kumbhakar and Lovell (2000)):
where is the probability density function and is the cumulative density function of the standard normal distribution.
heteroscedasticity can be parameterized by a vector of observable variables and the associated parameters.
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 k-th factor on the mean of are:
The same way we can calculate marginal effects of the on the from (4):
Marginal effects of the k-th factor on the are:
One can see that the sign of the marginal effect of the k-th 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 observation-specific 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 2.1-2.2.
Nevertheless, does obtained results are valid for any non-negative distribution of inefficiency term ? We shown that in general it’s not true. We shown a counterexample of the non-negative 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 .
In case the marginal effect of the efficiency (5) with respect to equals:
In case the marginal effect of the technical efficiency (6) with respect to equals:
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:
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 .
For the model (1) with the inefficiency error if the uncertainty increases, the inefficiency increases:
For the model (1) with the inefficiency error if the uncertainty increases, the technical efficiency decreases:
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.
The obtained dataset consists of pairs company-year in . The final panel is unbalanced, and the number of companies varies at each period.
The model (1) is used for panel data in the following specification:
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:
|Economic sector||In model|
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 Cobb-Douglas 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 non-exporting 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.
- Aigner et al (1977) Aigner D, Lovell CK, Schmidt P (1977) Formulation and estimation of stochastic frontier production function models. Journal of econometrics 6(1):21–37
- Battese and Coelli (1988) Battese GE, Coelli TJ (1988) Prediction of firm-level technical efficiencies with a generalized frontier production function and panel data. Journal of econometrics 38(3):387–399
- Bera and Sharma (1999) Bera AK, Sharma SC (1999) Estimating production uncertainty in stochastic frontier production function models. Journal of Productivity Analysis 12(3):187–210
- Bessonova et al (2003) Bessonova E, Kozlov K, Yudaeva K (2003) Trade liberalization, foreign direct investment, and productivity of russian firms. The Centre for Economic and Financial Research
- Caudill and Ford (1993) Caudill S, Ford J (1993) Biases in frontier estimation due to heteroscedasticity. Economics Letters 41:17–20
- Caudill et al (1995) Caudill S, Ford J, Gropper D (1995) Frontier estimation and firm-specific inefficiency measures in the presence of heteroscedasticity. Journal of Business & Economic Statistics 13:105–111
- Cornwell et al (1990) Cornwell C, Schmidt P, Sickles RC (1990) Production frontiers with cross-sectional and time-series variation in efficiency levels. Journal of econometrics 46(1-2):185–200
- Fiorentino et al (2006) Fiorentino E, Karmann A, Koetter M (2006) The cost efficiency of german banks: a comparison of sfa and dea. Discussion Paper, Series 2: Banking and Financial Supervision 2006, 10
- Greene (2005) Greene W (2005) Reconsidering heterogeneity in panel data estimators of the stochastic frontier model. Journal of econometrics 126(2):269–303
- Hadri (1999) Hadri K (1999) Estimation of a doubly heteroscedastic stochastic frontier cost function. Journal of Business & Economic Statistics 17:359–363
- Heshmati et al (1995) Heshmati A, Kumbhakar SC, Hjalmarsson L (1995) Efficiency of the swedish pork industry: A farm level study using rotating panel data 1976–1988. European Journal of Operational Research 80(3):519–533
- Ipatova and Peresetsky (2013) Ipatova I, Peresetsky A (2013) Technical efficiency of russian plastic and rubber production firms. Applied Econometrics 32(4):71–92
- Krasnopeeva et al (2016) Krasnopeeva N, Nazrullaeva E, Peresetsky A, Shchetinin E (2016) To export or not to export? the link between the exporter status of a firm and its technical efficiency in russia’s manufacturing sector. Voprosy Economiki 7
- Kumbhakar and Lovell (2000) Kumbhakar SC, Lovell CK (2000) Stochastic frontier analysis. Cambridge university press
- Kumbhakar et al (2014) Kumbhakar SC, Lien G, Hardaker JB (2014) Technical efficiency in competing panel data models: a study of norwegian grain farming. Journal of Productivity Analysis 41(2):321–337
- Meeusen and van Den Broeck (1977) Meeusen W, van Den Broeck J (1977) Efficiency estimation from cobb-douglas production functions with composed error. International economic review pp 435–444
- Ray et al (2015) Ray SC, Kumbhakar SC, Dua P (2015) Benchmarking for Performance Evaluation. Springer
- Wang (2002) Wang H (2002) Heteroscedasticity and non-monotonic efficiency effects of a stochastic frontier model. Journal of Productivity Analysis 18:241–253
- Wang (2003) Wang H (2003) A stochastic frontier analysis of financing constraints on investment: the case of financial liberalization in taiwan. Journal of Business & Economic Statistics 21:406–419
- Wang and Schmidt (2002) Wang H, Schmidt P (2002) One-step and two-step estimation of the effects of exogenous variables on technical efficiency levels. Journal of Productivity Analysis 18:129–144
- Zaitsev et al (2013) Zaitsev A, Burnaev E, Spokoiny V (2013) Properties of the posterior distribution of a regression model based on Gaussian random fields. Automation and Remote Control 74(10):1645–1655
Appendix A Proofs of the statements
Let us prove Statement 1.
Let us prove Statement 2.
We can get from (6) the marginal effect for the technical efficiency with respect to the uncertainty of the inefficiency error:
Let now us prove Theorem 2.1
From (16) we can see that the denominator of the expression is positive. So we should proof positive sign of the expression’s numerator.
Expression monotonically increases from 0 to 1 in the interval .
Obviously that .
Now we can calculate the derivative of and proof that it is positive:
Since is positive, we should proof now that g(z) is positive too. Let proof first auxiliary lemma 2:
If then and .
If then .
Therefore . ∎
The derivative of is .
Obviously, . Also we can find that .
Therefore . ∎
Let us prove Theorem 2.2 now.
From (6) replacing by we get:
If increases, then increases too. So, it is sufficient to prove the following inequality:
where is the cumulative density function of the standard normal distribution, and is the probability density function of the standard normal distribution.
Now we calculate the derivative of on from (24):
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:
If , then:
1. Let .
We prove that . Then if , и , then for , and, consequently, (22) holds: .
For to prove (25) it is sufficient to prove:
As , , then and consequently . Then we get and in the end we get that the initial inequality holds.
Now we’ll show that for . From (24) we get: