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One-Step and Two-Step Estimation of the Effects of Exogenous Variables on Technical Efficiency Levels
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## TL;DR
This paper compares one-step and two-step estimation for stochastic frontier models when exogenous variables affect technical inefficiency. In Monte Carlo simulations, the one-step estimator performs well, while the two-step procedure shows serious bias in both the first and second stages.
## Key findings
- In the base case simulation (200 replications, sample size \(N=200\)), the one-step estimator is close to unbiased: \(\hat\beta=-0.0010\), \(\hat\delta=1.0010\), and average estimated efficiency \(E(\hat e^{-u})=0.5141\) versus the true \(0.5165\).
- Under the same base case, the two-step estimator is badly biased: \(\hat\beta=-0.3064\), estimated efficiency \(E(\hat e^{-u})=0.4482\) versus true \(0.52\) (table reports \(0.5165\) in the design and about \(0.52\) in results discussion), and the second-step estimate of \(\delta\) is \(0.3539\) versus the true value \(1\).
- The paper reports that the two-step bias persists even when \(x\) and \(z\) are independent: for \(\rho=0\), the second-step estimate of \(\delta\) is still \(0.4069\), and the estimated efficiencies remain underdispersed and downward biased.
- Increasing sample size from \(N=200\) to \(N=500\) and \(N=1000\) lowers standard deviations but does not remove the two-step biases; for example, the two-step \(\hat\delta\) stays around \(0.35\)–\(0.36\) while the true value is \(1\).
## Methods
- The paper studies a stochastic frontier model in which inefficiency depends on exogenous variables \(z\), emphasizing a “scaling property” specification where the one-sided inefficiency term is a function of \(z\) times a basic one-sided error.
- It uses Monte Carlo simulations with 2,000 replications per design point. The base case sets \(\beta=0\), \(\delta=1\), \(\rho=0.5\), \(\gamma=1\), \(\sigma_v=1\), \(\mu=0\), and \(N=200\).
- One-step estimation is maximum likelihood under the correctly specified model; two-step estimation first sets \(\delta=0\) in the frontier MLE, then regresses \(\ln(\hat u)\) on \(z\) to estimate \(\delta\).
## Limitations
- The paper’s own simulation evidence is limited to one simple model with the scaling property and a specific truncated-normal basic distribution; results may not generalize beyond that setup.
- The text makes evident, though it does not frame it as a formal limitation, that some parameters (\(\mu\) and \(\gamma\)) are “hard” to estimate and produced outliers severe enough that the authors truncated the most extreme 0.3% of replications for those parameters.
- Not stated in the paper.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Our simulations strongly confirm the relevance of this observation, since the two-step estimates of the effect of z on efficiency levels are seriously biased downward in all cases.”
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In the paper
“We first note that the one-step estimates look pretty good. In particular, there is no evidence of significant bias.”
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In the paper
“The parameters of the truncated normal base distribution ( µ and γ ) have rather large variances (their standard deviations are definitely not small relative to the parameter values themselves)”
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In the paper
“For the one-step estimator, the true value of δ is not terribly important.”
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In the paper
“But this does not seem to cause any problems for the parameters of main interest ( β , δ and the individual ri).”
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In the paper
“Second, the estimates of the ri (the technical efficiencies) are biased downward (mean = 0.45, compared to the mean of the actual ri of 0.52).”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Changing β has no substantive effects on our results.”
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In the paper
“First, the estimate of β is biased downward (mean = -0.31, compared to the true value = 0).”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Third, the second-step estimator of δ is very significantly biased downward (mean = 0.35, compared to the true value = 1).”
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In the paper
“Changing δ makes very little difference to the properties of the estimates of β , δ or σ v.”
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In the paper
“The bias of the estimate of β and of the two-step estimate of δ increases, and the estimates of the ri are also more biased, and less correlated with the true ri, compared to Case 1.”
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In the paper
“As expected, the bias of the first-step estimate of β is effectively zero when δ = 0, and it grows with δ .”
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In the paper
“All of these biases are in the expected direction, and the size of the biases is definitely not small.”
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In the paper
“They are less strongly correlated with the true ri and they are underdispersed, compared to the estimates from the one-step model.”
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In the paper
“The individual ri are seriously biased for δ = 1 but not for the smaller values of δ .”
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In the paper
“The two-step estimator of δ is biased for both δ = 1 and δ = 0.5, but the bias disappears as δ → 0.”
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“An interesting curiosity is the bias in the two-step estimator of δ based on the estimates of ui from the one-step model (mean estimate = 0.92, compared to the true value of δ = 1). This is not just finite-sample bias (it persists with larger sample sizes).”
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In the paper
“The mean of the estimate (both one-step and two-step) of β changes by the same amount that β is changed, so that the bias, standard deviation and MSE of the estimates of β are unchanged.”
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In the paper
“For this set of parameters, average technical efficiency is E(e -u ) = 0.5165.”
Strength:plain· not yet calibratedn = 200Tests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
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In the paper
“As δ → 0, our ability to estimate µ (and γ , to a lesser extent) deteriorates seriously.”
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In the paper
“The estimates of the individual ri become slightly more biased, and noticeably less strongly correlated with the true ri, when δ is small.”
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In the paper
“We find serious bias at all stages of this procedure. The size of the bias is very substantial and should argue strongly against two-step procedures.”
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“For the two-step estimates, surprisingly, imposing µ = 0 makes things worse (even though µ = 0 is a correct restriction).”
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In the paper
“As γ increases, the bias of the estimate of β increases, but the bias of the second-step estimate of δ decreases quite noticeably.”
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“Finally, the second-step estimate of δ is seriously biased even when ρ = 0, though its bias does grow with ρ .”
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In the paper
“The standard deviations of the estimators fall as N increases, but nothing else changes much.”
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In the paper
“For example, the mean of the two-step estimator of δ is 0.35 for σ v = 1, 0.12 for σ v = 3, and 0.05 for σ v = 5.”
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In the paper
“| 9 | µ = 1 | ˆ β | -0.0013 | 0.0963 | 0.0093 | | -0.3914 | 0.1191 | 0.1674 | |”
- Also stated in: § 4. CONCLUDING REMARKS · plain
“| 17 | N=500 | ˆ β | -0.0005 | 0.0572 | 0.0033 | | -0.3062 | 0.0661 | 0.0981 | |”
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- Also stated in: § 4. CONCLUDING REMARKS · plain
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“| 9 | | E ( ˆ e - u ) | 0.3850 | 0.0294 | 0.0285 | 0.8387 | 0.3499 | 0.0197 | 0.0513 | 0.6891 |”
- Also stated in: § 4. CONCLUDING REMARKS · plain
“| 17 | N=500 | E ( ˆ e - u ) | 0.5160 | 0.0220 | 0.0338 | 0.8145 | 0.4470 | 0.0146 | 0.0615 | 0.6591 |”
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- Also stated in: § 4. CONCLUDING REMARKS · plain
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In the paper
“For the two-step estimators, we see more differences as µ changes, but they are still not really important or striking.”
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In the paper
“The true value of δ is one, so these are large biases indeed.”
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In the paper
“| 9 | E ( e - u ) | ˆ µ | 1.0591 | 0.3951 | 0.1595 | | -37.2499 | 1.1759 | 1464.4405 | |”
- Also stated in: § 4. CONCLUDING REMARKS · plain
“| 13 | E ( e - u ) | ˆ µ | 0.0429 | 0.4471 | 0.2016 | -37.2787 | 0.1688 | 1389.7299 | |”
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- Also stated in: § 4. CONCLUDING REMARKS · plain
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“| 9 | | ˆ δ | 1.0002 | 0.0850 | 0.0072 | | - | - | - | |”
- Also stated in: § 4. CONCLUDING REMARKS · plain
“| 17 | N=500 | ˆ δ | 0.9995 | 0.0724 | 0.0052 | | - | - | - | |”
Strength:plain· not yet calibratedn = 500Tests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
- Also stated in: § 4. CONCLUDING REMARKS · plain
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In the paper
“The results are quite consistent with these expectations.”
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In the paper
“| 9 | | ˆ δ -2S | 0.9317 | 0.0888 | 0.0125 | 0.8025 | 0.4986 | 0.0566 | 0.2546 | 0.3469 |”
- Also stated in: § 4. CONCLUDING REMARKS · plain
“| 17 | N=500 | ˆ δ -2S | 0.9134 | 0.0736 | 0.0129 | 0.7911 | 0.3520 | 0.0336 | 0.4210 | 0.2375 |”
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- Also stated in: § 4. CONCLUDING REMARKS · plain
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“| 17 | N=500 | ˆ γ | 1.0208 | 0.2393 | 0.0577 | | 53.4005 | 4.3493 | 2764.7151 | |”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one. The quote only partly states this claim.
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In the paper
“For the one-step estimators, the value of ρ is not important. The results are essentially identical across all these cases.”
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In the paper
“For larger γ , we estimate β a little worse, but we estimate δ and the individual ri a little better.”
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
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“As µ moves from plus one to negative one, the estimates of β improve a little, and the estimates of the individual ri perhaps improve a little.”
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“As is often the case in simulations that involve numerical maximization, there were some problems with outliers, especially in estimation of γ and µ (the parameters of the truncated normal distribution).”
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In the paper
“This bias in the second-step regression, due to underdispersion in the estimates of u that do not take into account the effect of z on u, does not seem to be systematically discussed in the literature.”
What the sources say
- Not checked — no full text of the sourcecites Kumbhakar and Lovell (2000, p. 119) source
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In the paper
“That is, if the estimates of u are constructed ignoring the effect of z on σ u 2 , they will show less dependence on z than they should, and we should expect the second-step regressions to give downward biased estimates of the effect of z on u.”
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“Some but not all of the models in the literature have this property.”
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In the paper
“The one-step MLE will be based on the correctly specified model, and will therefore be consistent and asymptotically efficient.”
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In the paper
“For example, Caudill and Ford (1993) provide evidence on the bias of the estimated technological parameters, but not on the efficiency levels themselves or their relationship to the explanatory variables z.”
What the sources say
- Not checked — no full text of the sourcecites Caudill and Ford (1993) source
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In the paper
“For example, if h(z, δ ) = exp(z' δ ) and δ > 0, then large z will on average be associated with large u and also with large σ u 2”
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In the paper
“Presumably this is so because when δ is large, z correlates with u and is useful to help estimate u.”
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In the paper
“Saying the same thing, if we ignore the dependence of σ u 2 on z, we will shrink the observations with large u too much, and the observations with small u too little, and our estimates of u will be underdispersed.”
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“It is widely appreciated that the severity of the bias in estimation of the technological parameters (coefficients of x) depends on the magnitude of the correlation between x and z.”
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In the paper
“A second and less widely recognized problem is that the first-step technical efficiency measures are likely to be seriously underdispersed, so that the results of the second-step regression are likely to be biased downward. This is true regardless of whether x and z are correlated.”
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In the paper
“The first of these is that it captures the idea, which we find intuitively reasonable and appealing, that the shape of the distribution of u is the same for all firms.”
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In the paper
“That is, the second-step estimator of δ may be able to tell us whether or not z affects u (whether or not δ = 0) 3 but it cannot accurately estimate the effect of z on u when this effect exists.”
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“Models based on some other simple distributions, such as exponential, would also have the scaling property.”
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In the paper
“Our summary statistics are averages and are very sensitive to extreme outliers.”
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In the paper
“Importantly, this is true whether or not x and z are correlated.”
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In the paper
“One important implication of this viewpoint, which is sometimes missed, is that the variables z that determine inefficiency must not be functions of y.”
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In the paper
“This truncation of the results made very little difference for the parameters other than µ and γ .”
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In the paper
“Increasing σ v increases the amount of statistical noise in the model, and should increase the bias and/ or variance for each of the estimated parameters.”
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In the paper
“However, as long as δ is non-zero, this argument may be less than compelling, because the distribution of u depends on z whereas the distribution of v does not, and this is another way to distinguish u from v.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- Rests on an assumption: The distribution of u depends on z whereas the distribution of v does not.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Increasing γ while holding σ v constant has the effect of increasing the size of inefficiency relative to noise, and should tend to make it easier to estimate technical inefficiency precisely.”
Strength:hedged· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“For example, when µ is large and positive, the degree of truncation is very small, and the shape of the distribution (if it can be separated from that of v) contains a lot of information about the parameters µ and σ 2 .”
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In this case the first-step estimator of β is unbiased, and we then estimate the ri under the correct assumption that they do not depend on the zi, so we ought to do well in this case.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“As µ → ∞ , the distribution of u becomes normal and presumably becomes confounded with the normal distribution of v (statistical noise).”
Strength:hedged· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“When µ is large (in absolute value) but negative, on the other hand, we observe only the extreme right tail of the distribution, whose shape may not be very informative about µ and σ 2 , and the estimation problem may be harder just on that basis.”
Strength:hedged· not yet calibratedCausal:conditional· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“All in all, the two-step estimator performs well when the second step is not needed ( δ = 0) but is otherwise unreliable.”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“These biases are substantial enough that we would recommend against using two-step procedures in any circumstances that we can envision.”
Load-bearingStrength:hedged· not yet calibratedTests✓ Asserted: passes✗ Checkable: fails✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim. It may be too vague to check against evidence.
The authors state this as an assumption.
In the theoretical exploration of underdispersion, x and z are independent.
“To explore this point more precisely, suppose that x and z are independent, so that the first-step regression is unbiased.”
The paper relies on this without support.
The distribution of u depends on z whereas the distribution of v does not.
“because the distribution of u depends on z whereas the distribution of v does not”
The paper relies on this without support.
The biases identified above are expected to persist asymptotically.
“the biases we have identified above are expected to persist asymptotically”
The paper relies on this without support.
The one-step estimators are based on a correctly specified model and are asymptotically optimal.
“The one-step estimators are based on a correctly specified model and are asymptotically optimal.”
Citations
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Efficiency Analysis of Knowledge Firms as Knowledge Producers Using Stochastic Frontier Model
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An Assessment of the Efficiency of Decentralization in the Execution of Public Works
Guccio, C., Pignataro, G., & Rizzo, I. (2019) · New frontiers in regional science: Asian perspectives · doi:10.1007/978-981-13-3107-7_10 · Cited by 1
Technical Efficiency of Rice Farmers using a Stochastic Frontier Analysis
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Zelenyuk, N., & Zelenyuk, V. (2021) · Edward Elgar Publishing eBooks · doi:10.4337/9781788976480.00016 · Cited by 1
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Afiatno, B. E., & Joyoutomo, K. D. (2022) · Journal of Developing Economies · doi:10.20473/jde.v7i1.34928 · Cited by 1
Recent Advances in the Construction of Nonparametric Stochastic Frontier Models
Parmeter, C. F., & Kumbhakar, S. C. (2023) · Lecture notes in economics and mathematical systems · doi:10.1007/978-3-031-29583-6_10 · Cited by 1
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Sources of Profit Inefficiency for Faba Bean Producers in Ethiopia
Muleta, G., & Mebratu, T. (2024) · American Journal of Agricultural Science Engineering and Technology · doi:10.54536/ajaset.v8i2.2610 · Cited by 1
“This will overcome the estimation bias of the widely used two-step approach (Battese & Coelli, 1995; Kumbhakar et al. , 1991; Wang & Schmidt, 2002).”
MethodFeed efficiency estimation from stochastic feed requirement frontier
Atsbeha, D. M. (2024) · Journal of Dairy Science · doi:10.3168/jds.2023-24301 · Cited by 1
“This 2-stage approach is however heavily criticized in the production economics literature (see Battese and Coelli 1995; Wang and Schmidt 2002).”
BackgroundRural institutions and the technical efficiency of teff production in Ethiopia
Hailu, G., Weersink, A. J., & Minten, B. (2025) · Journal of Co-operative Organization and Management · doi:10.1016/j.jcom.2024.100259 · Cited by 1
Tax Potential and Tax Efforts in Indian States: An Evidence from Panel Stochastic Frontier Approach
Kaur, B., & Kumar, S. (2025) · The Indian Economic Journal · doi:10.1177/00194662251372329 · Cited by 1
An research on the influence factors of China's high-tech industry's R&D activities efficiency
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Assessing Two Airline Models: Legacy vs. Low Cost Carriers
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YIELD UNCERTAINTY AND MILK SUPPLY RESPONSE IN TWO-TIER PRICE SYSTEMS
Atsbeha, D. M. (2017) · Journal of Agricultural and Applied Economics · doi:10.1017/aae.2016.40 · Cited by 1
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“In addition, the stochastic frontier parameters and the technical inefficiency are estimated by maximum likelihood (Wang & Schmidt, 2002).”
MethodL'efficienza rispetto alla frontiera delle possibilità produttive
Laureti, T. (2007) · Studi e saggi · doi:10.36253/88-8453-426-7 · Cited by 1
The Measurement of Firms’ Efficiency Using Parametric Techniques
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CLEMENTI, L. D. D. S. E. U. D. R. –. L. S. E. –. U. L. D. B., & Maruotti, A. (2013) · Statistica · doi:10.6092/issn.1973-2201/3509 · Cited by 1
“Furthermore, Wang and Schmidt (2002) found that the two-stage approach leads to biased estimates in an SFA study of technical efficiency; hence we will use one single step estimators in our empirical analysis.”
MethodApplication of Production Economics in the Electricity Distribution Sector
Mydland, Ø., & Lien, G. (2021) · Springer eBooks · doi:10.1007/978-981-10-3450-3_37-1 · Cited by 1
Technical Efficiency of Tef Producers: Evidence from West Shewa, Ethiopia
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The importance of efficiency for life insurer profit regarding Canadian life insurers
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A Non-Linear Stochastic Frontier ModelPreprint
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Performance of the Fruit Crop Industry in Ghana: Empirical Results and Policy Implications
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A Longitudinal Parametric Approach to Estimate Local Government E¢ ciencyPreprint
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Cost efficiency of Italian Commercial Banks: \na Stochastic Frontier analysis
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Banks in China: An Analysis of Cost and Profit EfficienciesPreprint
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Efficiency of U.S. Grass-Fed Beef FarmsPreprint
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Technical and environmental efficiency of smallholder palm oil and rubber production
Holtkamp, A. M. (2017) · doi:10.53846/goediss-6422 · Cited by 1
“(1977) to distributions dependent on observation characteristic variables (Wang and Schmidt, 2002).”
BackgroundMicrofoundations for Performance, Competition and Econometric ImplicationsPreprint
Kutlu, L., Mamatzakis, E. C. M., & Tsionas, E. G. (2017) · SSRN Electronic Journal · doi:10.2139/ssrn.3073430 · Cited by 1
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Three Essays on Applied Semiparametric Methods
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“Let uit = uig(zit; η) which exhibits “scaling” property (Wang and Schmidt (2002)). ui ∼ Chapter 3 45 IID|N(0, σ2u)| and the scaling function g(zit; η) = eηzit is known up to parameter η. Define the conditional density of it to be h( it|zit, θ0), where the true parameter θ0 = (σ2u, σ2v , η)′.”
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Essays on Distance Functions and Inefficiency Measurement
Esheba, M. (2018) · Libraries and Cultural Resources (University of Calgary) · doi:10.11575/prism/32405 · Cited by 1
Three Essays on Nonparametric Hypothesis Testing and Stochastic Frontier Analysis
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Janoudi, S. (2014) · Figshare · Cited by 1
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Ding, C., Xi, Y., & Chen, Z. (2012) · Economic Research-Ekonomska Istraživanja · doi:10.1080/1331677x.2012.11517525
Essays on Agricultural Production, Risk, and Productivity
Rao, X. (2015) · University of Minnesota Digital Conservancy (University of Minnesota)
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Bazzana, F. (2007) · RePEc: Research Papers in Economics
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Duvivier, C. (2011) · RePEc: Research Papers in Economics
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Meesters, A. (2010) · SSRN Electronic Journal · doi:10.2139/ssrn.1625815
U.S. Dairy Farm Cost EfficiencyPreprint
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Herr, A., Schmitz, H., & Augurzky, B. (2009) · SSRN Electronic Journal · doi:10.2139/ssrn.1495602 · Cited by 1
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Dormont, B., & Martin, C. (2011) · RePEc: Research Papers in Economics
Technical efficiency in the mediterranean countries' agricultural sector
HassineBelghith, N. B. (2007)
Ayuso, I. Á., & Rodríguez, D. (2006) · RePEc: Research Papers in Economics
Dhehibi, B. (2009) · doi:10.17265/2161-6264/2012.05b.007
“these techniques, include the researches [5-26].”
Cost Efficiency of Critical Access HospitalsPreprint
Nedelea, I. C., Fannin, J. M., Barnes, J. N., et al. (2010) · RePEc: Research Papers in Economics · doi:10.22004/ag.econ.56228
Lang, G. (2004) · RePEc: Research Papers in Economics
Salim, R. A., & Suyanto, S. (2010) · eSpace (Curtin University)
Maiorano, F. (2009) · The Journal of Economic Asymmetries · doi:10.1016/j.jeca.2009.02.009
Kim, M., & McPherson, M. A. (2012) · SSRN Electronic Journal · doi:10.2139/ssrn.2016566
“However, it is a simple function of time and allows a multiplicative decomposition of u it which has the so-called " scaling property " (Wang and Schmidt 2002). u is a product of a function of exogenous variables, z and the part of u that does not depend on z, u*, so that u = f(z; δ)u*. Battese and…”
Sugathan, A., Malgan, D., Chandrashekar, S., et al. (2013) · SSRN Electronic Journal · doi:10.2139/ssrn.2210741
“Here MLE tractability is achieved by the use of ‘scaling-property’ (Alvarez et al., 2006; Wang and Schmidt, 2002) to represent the inefficiency component uit as a product of a positive non-stochastic time-varying function and a stochastic time-invariant inefficiency term as uit = hit.u∗i , (3) hit…”
BackgroundMethodFiscal Federalism and Spatial Interactions among Governments
Chen, L. (2012) · UKnowledge (University of Kentucky)
Roh, J.-W. (2011) · The Journal of Society for e-Business Studies · doi:10.7838/jsebs.2011.16.2.111
“그러나 Wang and Schmidt[24]가 2단계”
HEALTHCARE ACCESSIBILITY SHORTFALLS AND HOSPITAL INFRASTRUCTURES IN CHILE
Mainardi, S. (2009) · Revista de análisis económico · doi:10.4067/s0718-88702009000200002
“…which supposedly cause systematic deviations from the frontier can produce biased parameters and spuriously under-dispersed inefficiency estimates (Wang and schmidt, 2002). an additional source of bias can arise from correlation of unobserved individualspecific effects with one or more…”
Cost Efficiency of Italian Commercial Banks:A Stochastic Frontier Analysis
Fontani, A., & Vitali, L. (2014) · Universal Journal of Industrial and Business Management · doi:10.13189/ujibm.2014.020304
“The two-step procedure has been much criticized (see Wang and Schmidt 2002).”
MethodChen, Y.-H., & Kao, M. (2015) · International Business Research · doi:10.5539/ibr.v8n2p71
“As pointed out by Wang and Schmidt (2002), the one-step estimators generally performed quite well in finite sample, whereas both estimators of the two-step procedure were seriously biased.”
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Weaver, R. D., Curtiss, J., Brümmer, B., et al. (2005) · RePEc: Research Papers in Economics · doi:10.22004/ag.econ.24528
Anàlisi de l'eficiència de l'empresa aeroportuària en l'actual marc desregulador.
Bierge, E. T. I. (2006) · LA Referencia (Red Federada de Repositorios Institucionales de Publicaciones Científicas)
Modebadze, G. (2011)
Public External Indebtedness and Productive Efficiency in Developing Countries
Drine, I., & Nabi, M. S. (2009)
Namonje, T. (2015) · Michigan State University Libraries · doi:10.25335/m5pb0q
Technology Invention and Diffusion in Residential Energy Consumption. A Stochastic Frontier ApproachPreprint
Marin, G., & Palma, A. (2016) · SSRN Electronic Journal · doi:10.2139/ssrn.2718094
“Although relatively easy to be implemented, this approach can produce biased results both in the case of heteroscedasticity, i.e. when the vector of inputs and the vector of auxiliary variables are correlated, as well as when is correlated with the idiosyncratic term (Wang and Schmidt, 2002).”
BackgroundInfluentialA Production Model to Measure Technical Efficiency in the Refrigerated Warehouse Industry
Farnsworth, B. E. (2015) · OakTrust (Texas A&M University Libraries)
Essays on banking sector performance in the CISs
Sharipova, A. (2016) · Middlesex University Research Repository (Middlesex University Of London)
Entry efficiency and barriers to entry in the Spanish retail marketPreprint
Orea, L. (2009) · RePEc: Research Papers in Economics
Metadata
- Authors
- Wang, H.-J., & Schmidt, P.
- Affiliations
- Institute of Economics, Academia Sinica (TW)
- Michigan State University (US)
- Venue
- Journal of Productivity Analysis
- Published
- 2002-09
- Volume
- 18
- Issue
- 2
- Pages
- 129-144
- ISSN
- 0895-562X (ISSN-L)
- 1573-0441
- OpenAlex
- W1524136776
- Semantic Scholar
- 0ba29f8abcbf8683e8286f2af42fa42ac20b1c6f
- Cited by
- 1,045
- References
- 10
- Citations by year
- 2026: 42 · 2025: 56 · 2024: 63 · 2023: 65 · 2022: 61 · 2021: 74 · 2020: 63 · 2019: 64 · 2018: 70 · 2017: 51 · 2016: 48 · 2015: 55 · 2014: 49 · 2013: 39 · 2012: 34
- FWCI
- 17.56
- Type
- journal article
- Language
- en
- Open access
- Licence
- Locations
- Journal of Productivity Analysis (publishedVersion)
- Munich Personal RePEc Archive (Ludwig Maximilian University of Munich) (submittedVersion)
- RePEc: Research Papers in Economics (submittedVersion)
- Topics
- Social Sciences › Decision Sciences › Management Science and Operations Research › Efficiency Analysis Using DEA
- Social Sciences › Economics, Econometrics and Finance › Economics and Econometrics › Economic Growth and Productivity
- Social Sciences › Economics, Econometrics and Finance › Economics and Econometrics › Spatial and Panel Data Analysis
- Keywords
- stochastic frontier analysis, technical efficiency, firm characteristics, one-step estimation, maximum likelihood estimation, Monte Carlo simulation
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
A language model drew the claims from the paper on 2026-10-11, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-10-11 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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