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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
Cites
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Biases in frontier estimation due to heteroscedasticity
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Pitfalls of Normal-Gamma Stochastic Frontier Models
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Frontier Estimation and Firm-Specific Inefficiency Measures in the Presence of Heteroscedasticity
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Personality traits and technical efficiency of Chinese rice farmers
Qian, C., Zhu, X., Antonides, G., & Heerink, N. (2024) · European Review of Agricultural Economics · doi:10.1093/erae/jbae019 · Cited by 6
The Effects of Non-Tariff Measures on Agricultural Trade Efficiency of South Africa Within the SADC
Mazorodze, B. T. (2025) · Journal of risk and financial management · doi:10.3390/jrfm18060286 · Cited by 5
“The simultaneous estimation of both equations acknowledges the bias that arises if the two-step equations are estimated in a stepwise fashion (Wang & Schmidt, 2002).”
Vinh, D. Q., & Phuong, L. H. (2022) · Asian Economic and Financial Review · doi:10.55493/5002.v12i8.4589 · Cited by 6
“However, Wang and Schmidt (2002) and Greene (2005) showed that there could be substantial bias if a two-step procedure is applied.”
BackgroundMethodMismeasurement and efficiency estimates: Evidence from smallholder survey data in Africa
Abay, K. A., Wossen, T., & Chamberlin, J. (2022) · Journal of Agricultural Economics · doi:10.1111/1477-9552.12514 · Cited by 5
Examining the effects of climate change adaptation on technical efficiency of rice production
Liu, Y., Ruiz‐Menjivar, J., Zavala, M., & Zhang, J. (2023) · Mitigation and Adaptation Strategies for Global Change · doi:10.1007/s11027-023-10092-3 · Cited by 5
“However, the estimation of the two-stage method may be biased and inconsistent (Kumbhakar et al. 1991; Wang and Schmidt 2002; Schmidt 2011).”
Muriu, A. N., Mugambi, P. J., & Mwito, M. M. (2024) · Research in Globalization · doi:10.1016/j.resglo.2024.100233 · Cited by 6
The use of the scaling property in a frontier analysis of a system of equations
Blank, J. L. T. (2020) · Applied Economics · doi:10.1080/00036846.2020.1763246 · Cited by 5
An Analysis of Technical Efficiency of Vegetables’ Household Production in Mongolia
Uuld, A., Magda, R., & Bilan, Y. V. (2021) · Agris on-line Papers in Economics and Informatics · doi:10.7160/aol.2021.130310 · Cited by 10
Bank size, competition, and efficiency: a post-GFC assessment of Australia and New Zealand
Din, S. U., Tripe, D., & Kabir, M. H. (2021) · New Zealand Economic Papers · doi:10.1080/00779954.2021.2020325 · Cited by 5
On the efficiency of subsidized bus services in rural areas: A stochastic metafrontier approach
Chung, Y.-S., & Chiou, Y. (2022) · Research in Transportation Business & Management · doi:10.1016/j.rtbm.2022.100811 · Cited by 5
Mesfin, H. M., Cecchi, F., Nillesen, E., & Tirivayi, N. (2022) · Economic Development and Cultural Change · doi:10.1086/719464 · Cited by 4
The inefficiency of exporting SMEs: Evidence from manufacturing industry
Forgione, A. F., & Migliardo, C. (2022) · Journal of International Trade & Economic Development · doi:10.1080/09638199.2022.2098527 · Cited by 5
“Moreover, our analysis applies the SFA one-step procedure, which eliminates the possible bias of the two-step technique (Wang and Schmidt 2002; Schmidt 2011).”
MethodLi, K., Liu, J., Xue, Y., et al. (2022) · Agriculture · doi:10.3390/agriculture12081078 · Cited by 5
“Then, wi is N(zδ, σ2 w) truncated on the left at zero [38].”
BackgroundBahta, S., Temoso, O., Ng’ombe, J. N., et al. (2023) · Frontiers in Sustainable Food Systems · doi:10.3389/fsufs.2023.1098642 · Cited by 5
Haschka, R. E., Herwartz, H., Coelho, C. C. D. A., et al. (2023) · Journal of Productivity Analysis · doi:10.1007/s11123-023-00694-z · Cited by 5
Estimation of endogenous firm productivity without instruments: an application to foreign investment
Fei, J., Huang, M., & Zhao, S. (2023) · Journal of Productivity Analysis · doi:10.1007/s11123-023-00709-9 · Cited by 5
Thapa, R. B., & Dhakal, S. C. (2024) · Cogent Food & Agriculture · doi:10.1080/23311932.2024.2353668 · Cited by 5
“The one-step estimators correctly specify the model as well as being asymptotically optimal if we further introduce the scaling property in the model proposed in (equation 4) by letting the ‘pre-truncation variance be proportional to the square of the pre-truncation mean’ (Wang & Schmidt, 2002).”
InfluentialMaking a virtue out of necessity: The effect of negative interest rates on bank cost efficiency
Avignone, G., Girardone, C., Pancaro, C., Pancotto, L., & Reghezza, A. (2025) · Journal of International Money and Finance · doi:10.1016/j.jimonfin.2025.103306 · Cited by 7
The Impact of Trade Openness and ICT on Technical Efficiency of Township Economies in South Africa
Mazorodze, B. T. (2025) · Economies · doi:10.3390/economies13050125 · Cited by 5
“It has long been demonstrated, however, that this traditional approach is biased (H. J. Wang & Schmidt, 2002).”
How output diversification affects bnak efficiency and risk: an intra-EU comparison study
Papanikolaou, N. I. (2009) · Figshare · Cited by 5
On the Distribution of Estimated Technical Efficiency in Stochastic Frontier Models: Revisited
Bhandari, A. K. (2011) · RePEc: Research Papers in Economics · Cited by 5
Charter versus traditional public schools: a panel study of the technical efficiency in Ohio
Palardy, J., Nesbit, T. M., & Adzima, K. A. (2012) · Education Economics · doi:10.1080/09645292.2012.748014 · Cited by 5
“In particular, we extend their model by incorporating a scaling property found by Wang and Schmidt (2002) and Alvarez et al. (2006).”
MethodOrea, L. (2011) · Oxford Economic Papers · doi:10.1093/oep/gpr028 · Cited by 5
Pitching DEA Against SFA in the Context of Chinese Domestic Versus Foreign Banks
Avkiran, N. K., & Zhu, Y. (2016) · International series in management science/operations research/International series in operations research & management science · doi:10.1007/978-1-4899-7705-2_5 · Cited by 5
Determinants of hospital costs and performance variation
Street, A. D., Scheller‐Kreinsen, D., Geißler, A., et al. (2010) · DepositOnce · doi:10.14279/depositonce-2470 · Cited by 5
“One stage approaches simultaneously estimate the stochastic frontier and the inefficiency model, which allows estimating the effect of hospital specific and environmental variables directly (Battese and Coelli 1995; Coelli et al. 2005; Hjalmarsson et al. 1996; Wang and Schmidt 2002).”
Econometric analysis of productivity: Theory and implementation in R
Sickles, R. C., Song, W., & Zelenyuk, V. (2019) · Handbook of statistics · doi:10.1016/bs.host.2018.11.007 · Cited by 5
Sun, K., & Salim, R. A. (2020) · Journal of Productivity Analysis · doi:10.1007/s11123-020-00589-3 · Cited by 5
“We then follow Yao et al. (2019) and assume that (1) vit iidNð0; σ2vÞ is a random shock, and (2) uit = uig(Zit; η), where ui iidNþð0; σ2uÞ is independent from Zit and is called base inefficiency level, gðZit; ηÞ ¼ expðη0ZitÞ is the scaling function (Wang and Schmidt 2002), and η is an Svector.”
An ex-ante DEA method for representing contextual uncertainties and stakeholder risk preferences
Topcu, T. G., & Triantis, K. P. (2021) · Annals of Operations Research · doi:10.1007/s10479-021-04271-1 · Cited by 5
Tirkaso, W. T., & Gren, I. (2021) · Journal of Environmental Planning and Management · doi:10.1080/09640568.2021.1987865 · Cited by 5
“The term u i denotes a non-negative deviation from the frontier cost function, i.e. minimum cost estimated for a given level of output, input prices and climate conditions, which can follow a half-normal, truncated, exponential, or gamma distributions (Wang and Schmidt 2002, 2).”
BackgroundMethodTrade and profit efficiency of manufacturing industries in South Africa
Mazorodze, B. T., Kaseeram, I., & Greyling, L. (2021) · Journal of International Trade & Economic Development · doi:10.1080/09638199.2021.1893374 · Cited by 5
“This approach is biased as the model estimated in the first stage is misspecified (Wang and Schmidt 2002).”
MethodImported inputs, firms' capabilities and productivity: evidence from emerging South Asian economies
Luqman, M., & Murtaza, G. (2023) · International Journal of Emerging Markets · doi:10.1108/ijoem-05-2022-0745 · Cited by 5
The threeIs of public schools: irrelevant inputs, insufficient resources and inefficiency
Henderson, D. J., Simar, L., & Wang, L. (2016) · Applied Economics · doi:10.1080/00036846.2016.1213363 · Cited by 5
Access Fees and Efficiency Frontiers with Selectivity and Latent Classes: Falkland Islands Fisheries
Mainardi, S. (2019) · Marine Resource Economics · doi:10.1086/702918 · Cited by 5
Wang, Y. (2019) · Applied Ecology and Environmental Research · doi:10.15666/aeer/1704_84178434 · Cited by 5
“Wang and Schmidt (2002) verified that one step method was superior to the two-steps method by Monte Carlo simulation.”
MethodInternational E-Banking: ICT Investments and the Basel Accord
Lin, H., & Lin, W. T. (2007) · Journal of Comparative International Management · Cited by 5
Jamasb, T. (2010) · RePEc: Research Papers in Economics · doi:10.17863/cam.5571 · Cited by 5
TECHNICAL EFFICIENCY IN RUSSIAN AGRICULTUREPreprint
Bruck, G., Grazhdaninova, M., Lerman, Z., et al. (2006) · RePEc: Research Papers in Economics · doi:10.22004/ag.econ.7182 · Cited by 5
Model Selection in Stochastic Frontier Analysis: Maize Production in KenyaPreprint
Liu, Y., & Liu, Y. (2006) · RePEc: Research Papers in Economics · doi:10.22004/ag.econ.21281 · Cited by 5
Lu, H., & Lu, H. (2006) · RePEc: Research Papers in Economics · doi:10.22004/ag.econ.25327 · Cited by 5
Buttinelli, R., & Zhu, X. (2025) · Journal of Cleaner Production · doi:10.1016/j.jclepro.2025.146070 · Cited by 4
Xin, M., Tan, S., Qiu, H., & Tang, J. (2025) · Journal of Agricultural Economics · doi:10.1111/1477-9552.12638 · Cited by 4
The Intra-Regional Economy Of Veracruz Province: What Can The Municipios Tell Us?
Brock, G. J. (2013) · The Journal of developing areas · doi:10.1353/jda.2013.0012 · Cited by 4
“The one step procedure is statistically more robust (Wang & Schmidt, 2002) than a two step procedure but does require the existence of a frontier with the underlying assumption of a truncated normal density function.”
The effect of farm size on economic efficiency: a case study of Vietnamese pangasius farms
Tho, H. H., Tuyet, N. L. H., & Tojo, N. (2021) · Fisheries Science · doi:10.1007/s12562-021-01549-y · Cited by 4
Gupta, R., & Singh, B. P. (2023) · Vision The Journal of Business Perspective · doi:10.1177/09722629231203934 · Cited by 4
Parajuli, S., & Thapa, R. B. (2024) · Cogent Food & Agriculture · doi:10.1080/23311932.2024.2360754 · Cited by 4
“Furthermore, Wang and Schmidt (2002) validate the biasedness with two conclusions after confirming the biasedness: 1.”
InfluentialExploring the corruption‐inefficiency nexus using an endogenous stochastic frontier analysis
Castro, M. F., & Guccio, C. (2024) · Southern Economic Journal · doi:10.1002/soej.12734 · Cited by 4
Varkey, L. M., Veettil, P. C., Singh, K. M., Borgohain, R., Bhattacharya, A., & Singh, S. (2024) · Gender Technology and Development · doi:10.1080/09718524.2024.2417472 · Cited by 5
“…in the classical linear model, where only the efficiency of the estimators is affected (Battese & colli, 1995; huang & liu, 1994; Kumbhakar, 1991; Wang & schmidt, 2002). in modeling technical efficiency, many studies that have adopted the production frontier model at least implicitly assume that…”
Khan, S. A. M. N., Baten, M. A., Nawawi, M. K. M., et al. (2016) · AIP conference proceedings · doi:10.1063/1.4966075 · Cited by 4
Khan, F., Salim, R. A., & Sun, K. (2018) · Applied Economics · doi:10.1080/00036846.2018.1470316 · Cited by 4
Buch, E., & Cabaleiro‐Casal, R. (2019) · Papers of the Regional Science Association · doi:10.1111/pirs.12491 · Cited by 6
SFA robustness to violated distributional assumptions: theory, simulations and empirical evidence
Bělín, M., & Hanousek, J. (2021) · Applied Economics · doi:10.1080/00036846.2021.1904124 · Cited by 4
“Pace Wang and Schmidt (2002), we recognize that the two-step procedure, in which regression residuals are correlated with explanatory variables not included in the first-step, might lead to biased results.”
BackgroundAjayi, V., & Weyman-Jones, T. G. (2021) · Energy Economics · doi:10.1016/j.eneco.2021.105650 · Cited by 4
“13 Our empirical analysis is programmed in Stata using the maximum likelihood code written by Wang (2002) 14 Coelli (1995) notes that under the null hypothesis of normal residual, the third moment of the OLS residual is asymptotically distributed as a normal random variable with mean 0 and variance…”
BackgroundMethodInfluentialEnvironmental policy and R&D productivity: A case study from the Korean Emissions Trading Scheme
Mo, J. Y. (2022) · Science and Public Policy · doi:10.1093/scipol/scac053 · Cited by 4
Sheng, P., & Liu, W. (2024) · Economic Analysis and Policy · doi:10.1016/j.eap.2024.02.008 · Cited by 6
Technical efficiency of public middle schools in New York City
Horrace, W. C., Rothbart, M. W., & Yang, Y. (2021) · Economics of Education Review · doi:10.1016/j.econedurev.2021.102216 · Cited by 4
Comparison of Stochastic Frontier “Effect” Models Using Monte Carlo Simulation
Lee, Y. H., & Shin, J. (2014) · Springer eBooks · doi:10.1007/978-1-4899-8008-3_8 · Cited by 6
Lin, H., Wen, M., & Yang, C. C. (2011) · North American Actuarial Journal · doi:10.1080/10920277.2011.10597634 · Cited by 4
“This paper adopts the stochastic frontier approach developed in Wang and Schmidt (2002) to consider how risk management decisions and firm characteristics affect efficiency and cost function within one step.”
BackgroundMeasuring Inefficiency of the Czech Labour Market
Němec, D. (2015) · Review of Economic Perspectives · doi:10.1515/revecp-2015-0017 · Cited by 4
A Eficiência do Consumo Residencial de Energia Elétrica no Brasil
Soares, T. C., Lopes, L. S., & da Cunha, D. A. (2017) · Economia Aplicada · doi:10.11606/1413-8050/ea117421 · Cited by 4
Urban water management reform : the case of China
Zhao, J. (2015) · doi:10.26481/dis.20150225jz · Cited by 4
Análisis de eficiencia del sector hospitalario: el caso colombiano
Castaño, A. (2013) · Cited by 4
Schettini, D. C. D. (2010) · Digital Library of Theses and Dissertations (Universidade de São Paulo) · doi:10.11606/t.12.2010.tde-10052010-141954 · Cited by 5
Lin, W. T., & Lin, H. (2006) · SSRN Electronic Journal · doi:10.2139/ssrn.929193 · Cited by 4
Globalisation and Technological Convergence in the EUPreprint
Mastromarco, C., Serlenga, L., & Shin, Y. (2012) · SSRN Electronic Journal · doi:10.2139/ssrn.2118483 · Cited by 4
(2013) · Palgrave Macmillan eBooks · doi:10.1057/9781137336583.1746 · Cited by 4
“Battese and Coelli (1995) propose an extended version of the model of Kumbhakar et al. (1991) to allow the use of panel data.33 Battese and Coelli (1995) specify inefficiency 32On this argument, see Wang and Schmidt (2002).”
BackgroundAn Investigation of Approaches to Performance Measurement: Applications to Long-term Care in Ontario
Hsu, A. T. (2015) · TSpace (University of Toronto) · Cited by 4
Does intercropping have a future in China? : insights from a case study in Gansu Province
Hong, Y. (2018) · doi:10.18174/450672 · Cited by 4
Islam, A., & Hyland, M. (2018) · World Bank, Washington, DC eBooks · doi:10.1596/1813-9450-8637 · Cited by 4
Estimation of Stochastic Frontier Panel Data Models with Spatial InefficiencyPreprint
Belotti, F., Ilardi, G., & Mortari, A. P. (2019) · SSRN Electronic Journal · doi:10.2139/ssrn.3396404 · Cited by 4
Lee, K., Ha, S., Gurudatt, N. G., et al. (2023) · SSRN Electronic Journal · doi:10.2139/ssrn.4535712 · Cited by 4
Negative rates and bank profit and cost efficiency: evidence for Eurozone
Siameti, P., Chronopoulos, D. K., & Dotsis, G. (2024) · European Journal of Finance · doi:10.1080/1351847x.2024.2366367 · Cited by 3
“In contrast to the two-step approach, our single-step analysis yields consistent estimates (see Battese and Coelli 1995; Wang and Schmidt 2002).”
Ayen, K., Kidane, T., & Wubet, A. (2025) · Frontiers in Sustainable Food Systems · doi:10.3389/fsufs.2024.1495820 · Cited by 3
Zhang, W. (2025) · Frontiers in Environmental Science · doi:10.3389/fenvs.2025.1573394 · Cited by 3
Bigerna, S., D’Errico, M. C., & Polinori, P. (2025) · Economic Analysis and Policy · doi:10.1016/j.eap.2025.06.011 · Cited by 3
Mantey, V., Missiame, A., Bosch, C., et al. (2025) · Environmental Development · doi:10.1016/j.envdev.2025.101373 · Cited by 3
Russo, S., Mirra, L., Caracciolo, F., & Giannoccaro, G. (2026) · Agricultural Water Management · doi:10.1016/j.agwat.2026.110148 · Cited by 4
Alem, H., Jumpah, E. T., & Bavorová, M. (2026) · Land Use Policy · doi:10.1016/j.landusepol.2026.108002 · Cited by 5
Eki̇nci̇, R. (2020) · Dokuz Eylül Üniversitesi Sosyal Bilimler Enstitüsü Dergisi · doi:10.16953/deusosbil.733488 · Cited by 3
“İkincisi, SSFA modeli ile yapılan etkinlik tahminleri iki-aşamalı SFA modeline dayalı olarak yapılan mekansal uygulamalarda görülen sapmalardan etkinlenmez (Wang & Schmidt, 2002).”
BackgroundImposing Monotonicity in Stochastic Frontier Models: An Iterative Nonlinear Least Squares Procedure
Belotti, F., & Ferrara, G. (2025) · Computational Economics · doi:10.1007/s10614-025-11086-w · Cited by 2
“Here, we assume that the inefficiency term depends on a set of exogenous covari - ates via the so called “scale” transformation (Wang & Schmidt, 2002) of some underlying process η i , such as where z i is a vector of contextual variables, γ the vector of unknown coefficients and η i an i.i.d.…”
Masunda, S., & Choga, I. (2021) · Problems and Perspectives in Management · doi:10.21511/ppm.19(4).2021.29 · Cited by 3
Zhao, S., & Parmeter, C. F. (2022) · Economics Letters · doi:10.1016/j.econlet.2022.110901 · Cited by 3
Determinants of Peruvian Export Efficiency: Poisson PML Estimation Approach
Soto, F. C. N., Morote, E., Macha-Huamán, R., et al. (2023) · Economies · doi:10.3390/economies11060169 · Cited by 3
Pathak, A., Leu, S., Robertson, M. L., & Siriwardana, M. (2024) · Applied Economics · doi:10.1080/00036846.2024.2310524 · Cited by 4
“Wang and Schmidt (2002) explain this bias in detail and Wang (2002) provides Monte Carlo evidence of the bias. also employ the interaction term tariffs*SOP to examine if tariff reductions help an industry gain efficiency based on its scale of operation.”
Rosario, J., & Shanmugam, K. R. (2024) · Journal of Economic Studies · doi:10.1108/jes-11-2023-0649 · Cited by 3
Parmeter, C. F. (2024) · Tourism Economics · doi:10.1177/13548166241266912 · Cited by 3
Do peers and agglomeration affect farm efficiency?
Min, S., & Kim, K. (2024) · Agricultural Economics (Zemědělská ekonomika) · doi:10.17221/99/2024-agricecon · Cited by 3
A spatial stochastic frontier model with fixed effects and endogenous environmental variables
Giannini, M. (2024) · Spatial Economic Analysis · doi:10.1080/17421772.2024.2414962 · Cited by 3
“…the fixed-effect parameters, to avoid the incidental parameters problem, and allowing exogenous determinants of inefficiency in the model (so-called environmental variables or ‘scaling effect’); Wang and Schmidt (2002) in the cross-sectional case, and Alvarez et al. (2006) in the panel case.”
Blank, J. L. T., & van Heezik, A. A. S. (2024) · International Transactions in Operational Research · doi:10.1111/itor.13575 · Cited by 3
Sun, K., Kumbhakar, S. C., & Lien, G. (2024) · European Review of Agricultural Economics · doi:10.1093/erae/jbae024 · Cited by 3
Mitigating simultaneity bias in seaport efficiency measurement
Rødseth, K. L., Kuosmanen, T., & Holmen, R. B. (2024) · Transportation Research Part A Policy and Practice · doi:10.1016/j.tra.2024.104333 · Cited by 4
Distance functions and the analysis of inefficiency
Esheba, M., & Serletis, A. (2025) · Macroeconomic Dynamics · doi:10.1017/s1365100525000148 · Cited by 4
Mazorodze, B. T. (2025) · Administrative Sciences · doi:10.3390/admsci15060229 · Cited by 5
Agbonlahor, M. (2010) · Journal of Humanities Social Science and Creative Arts · doi:10.51406/jhssca.v5i1.987 · Cited by 3
New Quantitative Approaches to Asset Selection and Portfolio Construction
Song, I. (2014) · doi:10.7916/d83n21jv · Cited by 3
“For those experiments concerned with correlated input variables, we adopt the following equation from Wang and Schmidt (2002) to establish the desired covariance structure of inputs.”
MethodThe paradox of energy regulation: The inefficiency of nuclear production
Coto‐Millán, P., de la Fuente, M., & Fernández, X. L. (2017) · Energy Sources Part B Economics Planning and Policy · doi:10.1080/15567249.2016.1209708 · Cited by 3
“2As demonstrated by Wang and Schmidt (2002), one stage reduces potential biases and contradictions in the calculation which are introduced in models with successive steps.”
BackgroundThe estimation of profit efficiency
Han, Y., & Kim, M. H. (2017) · Applied Economics Letters · doi:10.1080/13504851.2017.1397841 · Cited by 3
“Second, the two-step procedure leads to a potentially serious bias of profit inefficiency unless a scaling property is considered (Wang and Schmidt 2002).1 Although the bias may be somehow neglectable, identifying the source of profit inefficiency has not been available because of the formidable…”
BackgroundImpacto de las fusiones y adquisiciones en la eficiencia de la banca en Colombia
Sarmiento, M., Mutis, H., Cepeda, A., et al. (2018) · Revista de Economía Institucional · doi:10.18601/01245996.v20n38.07 · Cited by 3
Performance of agricultural cooperative banks in Japan
Harimaya, K., & Kagitani, K. (2019) · Agricultural Finance Review · doi:10.1108/afr-03-2019-0036 · Cited by 3
Banking market structure and efficiency: an assessment of the USA and Canada
Din, S. U., & Tripe, D. (2021) · Journal of Economic Studies · doi:10.1108/jes-03-2021-0157 · Cited by 3
Internal migration and technical efficiency: the case of Italy
Pipitone, V., Licari, F., Fodera, R., et al. (2021) · Applied Economics · doi:10.1080/00036846.2021.1980491 · Cited by 3
“On these grounds, Wang and Schmidt (2002) convincingly show that the two-steps approach leads to severely biased results.”
Background
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
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