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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
Battese, G. E., & Coelli, T. J. (1995) · Empirical Economics · doi:10.1007/bf01205442 · Cited by 5,978
Kumbhakar, S. C., & Lovell, C. A. K. (2000) · doi:10.1017/cbo9781139174411 · Cited by 4,358
Battese, G. E., & Coelli, T. (1988) · Journal of Econometrics · doi:10.1016/0304-4076(88)90053-x · Cited by 1,919
Kumbhakar, S. C., Ghosh, S., & McGuckin, J. T. (1991) · Journal of Business & Economic Statistics · doi:10.1080/07350015.1991.10509853 · Cited by 1,162
Systematic Departures from the Frontier: A Framework for the Analysis of Firm Inefficiency
Reifschneider, D. L., & Stevenson, R. (1991) · International Economic Review · doi:10.2307/2527115 · Cited by 686
Estimation of a non-neutral stochastic frontier production function
Huang, C. J., & Liu, J.-T. (1994) · Journal of Productivity Analysis · doi:10.1007/bf01073853 · Cited by 593
Frontier Estimation and Firm-Specific Inefficiency Measures in the Presence of Heteroscedasticity
Caudill, S. B., Ford, J. M., & Gropper, D. M. (1995) · Journal of Business and Economic Statistics · doi:10.1080/07350015.1995.10524583 · Cited by 390
Biases in frontier estimation due to heteroscedasticity
Caudill, S. B., & Ford, J. M. (1993) · Economics Letters · doi:10.1016/0165-1765(93)90104-k · Cited by 256
Pitfalls of Normal-Gamma Stochastic Frontier Models
Ritter, C., & Simar, L. (1997) · Journal of Productivity Analysis · doi:10.1023/a:1007751524050 · Cited by 174
Frontier Estimation and Firm-Specific Inefficiency Measures in the Presence of Heteroscedasticity
Caudill, S. B., Ford, J. M., & Gropper, D. M. (1995) · Journal of Business and Economic Statistics · doi:10.2307/1392525 · Cited by 116
Flagged first, then by citations
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Items 1,001–1,057 of 1,057
Hospital efficiency in Türkiye via spatial Stochastic Frontier Analysis
Eki̇nci̇, R., Kutlu, L., Yeşilyurt, F., et al. (2025) · Applied Economics · doi:10.1080/00036846.2025.2551935
Wang, T., Wang, Z., Yao, F., et al. (2025) · Journal of the Royal Statistical Society Series A (Statistics in Society) · doi:10.1093/jrsssa/qnaf134
Önen, E., & Efecan, V. (2025) · Health Policy · doi:10.1016/j.healthpol.2025.105433
Exposure to Real Estate in Bank Portfolios: Evidence from Japan’s Regional Banks
Harimaya, K., & Jinushi, T. (2025) · Asia-Pacific Financial Markets · doi:10.1007/s10690-025-09559-x
Does Pollution Prevention Enhance Firm Efficiency? Insights from U.S. Manufacturing FirmsPreprint
Memon, S. F., Kumbhakar, S. C., & Su, Y. (2025) · SSRN Electronic Journal · doi:10.2139/ssrn.5605887
PPPs Contribution to Economic Growth and Infrastructure Development: The Role of Governance
Sambe, A. M. (2025) · Springer proceedings in business and economics · doi:10.1007/978-3-032-04602-4_5
Market power in the EU banking sector in a time of unconventional monetary policy
Ortega, C. (2025) · Argumenta Oeconomica · doi:10.15611/aoe.2025.2.12
Rødseth, K. L., Kuosmanen, T., & Holmen, R. B. (2025) · European Journal of Operational Research · doi:10.1016/j.ejor.2025.11.011
Hussen, S., & Tenaye, A. (2026) · Discover Sustainability · doi:10.1007/s43621-026-03110-2
“This study employs the one-step maximum likelihood estimation approach, as recommended by [18], because of its methodological superiority over the traditional two-step procedure.”
Assessing the impossible trinity principle in BRICS grouping
Bonga‐Bonga, L. (2025) · BRICS Journal of Economics · doi:10.3897/brics-econ.6.e146580
“Unlike previous studies that did not account for this variability, it is based on the premise that pooled regression estimation without variation in the dependent variable can lead to inefficiencies (Wang & Schmidt, 2002).”
Gupta, R., & Singh, B. P. (2025) · Global Business Review · doi:10.1177/09721509251399744
Medina, D. M. C. (2015) · eDiss (Georg-August-Universität Göttingen) · doi:10.53846/goediss-4965
Liu, X., Ma, Z., & Li, H. (2025) · Journal of Service Science and Management · doi:10.4236/jssm.2025.186033
Pagliuca, L. G., Carrer, M. J., Damasceno, R., et al. (2025) · Precision Agriculture · doi:10.1007/s11119-025-10308-3
“In this study, the econometric approach with scaling property in inefficiency term, as developed by Wang and Schmidt ( 2002 ), Hadri et al. ( 2003 ), and Alvarez et al. (2006), was applied.”
Jang, H. (2025) · International Journal of Empirical Economics · doi:10.1142/s2810943025500222
Okwoche, P. U. (2026) · SSRN Electronic Journal · doi:10.2139/ssrn.6054988
Loussouarn, C. (2023) · Journal de gestion et d économie médicales · doi:10.54695/jges.225.0338
Delelegn, B. H., Entele, B. R., & Habtewold, T. M. (2026) · SN Business & Economics · doi:10.1007/s43546-025-01047-8
“Second, most employ two‑step stochastic frontier models that assume independence between inefficiency effects and explanatory variables, a simplification known to introduce endogeneity 1 3 bias (Wang and Schmidt 2002).”
InfluentialA Critical Assessment of Measures of Financing Constraints: An Application to Indian Manufacturing
Vincent, M. A., Bardhan, S., & Das, P. K. (2026) · Journal of Quantitative Economics · doi:10.1007/s40953-025-00493-3
Consumer food waste as household technical inefficiency across countries and time
Vieira, D., Barrera, E. L., & Ferreira, M. D. P. (2026) · American Journal of Agricultural Economics · doi:10.1002/ajae.70040
Operational Efficiency Post-Merger and Acquisition: Evidence from Indian Firms
Verma, S. K., & Kumar, S. (2025) · Journal of Transition Economics and Finance · doi:10.1142/s3082844926500016
The ABCs of Russia’s post-transition growth: what went wrong?
Ogloblin, C. G., & Brock, G. J. (2026) · Post-Communist Economies · doi:10.1080/14631377.2026.2621390
“Second, it models heteroscedasticity in the inefficiency term more accurately, which helps obtain more consistent estimates of the production function’s parameters and efficiency (Kumbhakar & Lovell, 2000; Wang & Schmidt, 2002).”
Farag, D., & Hassouba, T. A.-A. (2026) · SSRN Electronic Journal · doi:10.2139/ssrn.6184934
Panel stochastic frontier model with multi-factor error structure: an application on bank efficiency
Hsu, C.-C., Lin, C., & Yin, S.-Y. (2026) · Empirical Economics · doi:10.1007/s00181-025-02856-4
Owili, S. O., Otieno, D. J., Chimoita, E. L., & Baijukya, F. P. (2026) · Discover Agriculture · doi:10.1007/s44279-026-00521-7
“Since it is widely recognised that multi-step estimation procedures are generally associated with biased standard errors [91, 92], we apply a bootstrap procedure following Simar and Wilson [93] and Simar and Wilson [94] with 1000 replications to correct the standard errors.”
Czyżewski, B., Staniszewski, J., Kryszak, Ł., Grzelak, A., & Borychowski, M. (2026) · Economics and Environment · doi:10.34659/eis.2026.96.1.1038
Asmare, E., & Aragaw, Y. (2026) · Aquaculture Fish and Fisheries · doi:10.1002/aff2.70227
Bank competition in Africa : three essays
Léon, F. (2014) · doi:10.70675/6fc1d035zc8a0z4bbezb323z825e48dd81fe · Cited by 1
Menard, A.-R. (2014) · doi:10.70675/c6cbdfd5za342z4458z8a7cz9b4549dc3825
Efficiency and equality in water production and distribution
Cuevas, U. V. J. G. (2020) · doi:10.70675/6cd1e4dczc806z4a90z9d94z8d606d3165f0
Hossain, M. A., Hassan, S. M., Daula, A.-U., et al. (2026) · Discover Environment · doi:10.1007/s44274-025-00488-0
Profitability, efficiency and risk in banking and microfinance
Djekna, V. (2022) · theses.fr (ABES) · doi:10.70675/487ab79fze926z4604z9a99ze2b079a69021
smfa: Stochastic Metafrontier Analysis
Owili, S. O. (2026) · doi:10.32614/cran.package.smfa
Djekna, V., & Blavet, T. (2026) · Applied Economics · doi:10.1080/00036846.2026.2669353
Mazorodze, B. T. (2026) · Journal of Applied Social Science · doi:10.1177/19367244261450059
Evaluating the impact of waiting lists on hospital efficiency and costs
Rodrı́guez-Álvarez, A., & Roibás, D. (2026) · Journal of Productivity Analysis · doi:10.1007/s11123-026-00814-5
Atta, A. A. B., Shawabkeh, Y. T., Jundi, N. A. A., et al. (2026) · Investment Management and Financial Innovations · doi:10.21511/imfi.23(2).2026.35
The impact of industrial opt-out from utility sponsored energy efficiency programs
Ma, Y., Boyd, G. A., Doolin, M., & Weiss, J. (2026) · Energy Economics · doi:10.1016/j.eneco.2026.109491
Msukuma, M., Mkwanda, C., Khataza, R., Mathanda, H., Mgomezulu, W. R., & Makombe, G. (2026) · Sustainability · doi:10.3390/su18147198
Nguyen, D. T., Le, T. D., & Tran, L. H. T. (2026) · Eurasian economic review : · doi:10.1007/s40822-026-00385-8
“…) conditional expectation: Both the translog stability function and the stability inefficiency model are estimated using the one-step procedure of Battese and Coelli ( 1995), which is considered more efficient than the conventional two-step approach (Kumbhakar et al., 1991; Wang & Schmidt, 2002).”
Impact of ESG Performance on Cost Efficiency: Evidence from the US Banking Sector
Eki̇nci̇, R. (2026) · Sosyoekonomi · doi:10.17233/sosyoekonomi.2026.03.01
Mộng, T. Đ. T., Le, T., Ngo, T., et al. (2026) · Post-Communist Economies · doi:10.1080/14631377.2026.2712943
Profit efficiency of smallholder dairy production systems: Empirical evidence from Northern Uganda
Oyoo, P. O., Okwera, S., Okello, D. M., & Mugonola, B. (2026) · PLoS ONE · doi:10.1371/journal.pone.0333337
“This approach leads to skewed conclusions and diverging function distributions [42].”
Semiparametric inefficiency determinants and rank identification in stochastic frontier modelsPreprint
Liang, J., Maneesoonthorn, W. O., & Zhang, X. (2026) · SSRN Electronic Journal · doi:10.2139/ssrn.7369874
Technical Efficiency and Its Determinants Among Smallholder Soybean Farmers in Mozambique
Peace, A., & Popat, M. (2026) · Economies · doi:10.3390/economies14090368
“Estimating both equations simultaneously, rather than in two stages, avoids the bias that arises when the distributional assumptions of the inefficiency term are violated in a second-stage regression (H. Wang & Schmidt, 2002).”
Exploring Spatial Patterns in Inefficiency Using a Stationary Gaussian Random Field
Skevas, I. (2026) · Oxford Bulletin of Economics and Statistics · doi:10.1111/obes.70127
Energy Efficiency, Employment and Competitiveness: Firm Level Evidence from Ethiopian ManufacturingPreprint
Tesfaye, E. (2026) · SSRN Electronic Journal · doi:10.2139/ssrn.7412674
Capacity trilemma: robust capacity planning for demand, productive scale size and sustainability
Kang, W., Lee, C., & Fang, Y.-H. (2026) · Journal of the Operational Research Society · doi:10.1080/01605682.2026.2725868
Ogunsola, A. J., Osuma, G., Zwane, T. T., et al. (2026) · Frontiers in Environmental Science · doi:10.3389/fenvs.2026.1895193
Karanki, F., & Zhang, A. (2026) · Transportation Research Part A Policy and Practice · doi:10.1016/j.tra.2026.105261
Beza, D., Seyoum, A., & Walle, Y. (2026) · Research Square · doi:10.21203/rs.3.rs-10675488/v1
Prigoreanu, I., Galeş, D. C., & Ignat, G. (2026) · Agriculture · doi:10.3390/agriculture16192062
Pholphirul, P., & Rukumnuaykit, P. (2026) · Journal of Philanthropy · doi:10.1002/nvsm.70071
Alemán, A. S., & de León, O. P. (2026) · Transportation Research Part A Policy and Practice · doi:10.1016/j.tra.2026.105292
Ghosh, R., & Kathuria, V. K. (2014) · SSRN Electronic Journal · doi:10.2139/ssrn.2397296
The econometrics approach to the measurement of efficiency : a surveyPreprint
Rossi, M. A. (2015) · 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.
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