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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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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
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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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Impact of Trade Liberalisation on Technical Efficiency of Vietnamese Manufacturing Firms
Chu, S. N., & KALIRAJAN, K. P. (2011) · Science Technology and Society · doi:10.1177/097172181101600302 · Cited by 24
ICT and R&D as inputs or efficiency determinants? Analysing Italian manufacturing firms (2007–2009)
Bonanno, G. (2015) · Eurasian economic review : · doi:10.1007/s40821-015-0035-z · Cited by 25
“As shown by Lensink and Meesters (2014) and Wang and Schmidt (2002), bias can arise in the two-step approach because, in estimating the production frontier, the inefficiency component is assumed to be identically and independently distributed, while it is determined by a number of explanatory…”
BackgroundLaureti, T. (2007) · International Advances in Economic Research · doi:10.1007/s11294-007-9132-9 · Cited by 24
Consistent Inference in Fixed-Effects Stochastic Frontier ModelsPreprint
Belotti, F., & Ilardi, G. (2017) · SSRN Electronic Journal · doi:10.2139/ssrn.3066766 · Cited by 25
Islam, A. M., & Hyland, M. (2019) · Ecological Economics · doi:10.1016/j.ecolecon.2019.04.024 · Cited by 23
Atkinson, S. E., Primont, D., & Tsionas, E. G. (2018) · Journal of Econometrics · doi:10.1016/j.jeconom.2017.12.009 · Cited by 23
Estimation of Profit Efficiency in the Hotel Industry Using a Bayesian Stochastic Frontier Model
Arbelo, A., Arbelo-Pérez, M., & Pérez-Gómez, P. (2018) · Cornell Hospitality Quarterly · doi:10.1177/1938965518762841 · Cited by 23
“This two-stage approach has been criticized by Wang and Schmidt (2002) and Rahman (2003), because this procedure is inconsistent with respect to the assumption that inefficiencies are identically distributed with one-sided error terms (Coelli, 1996).”
MethodLasting lending relationships and technical efficiency. Evidence on European SMEs
Agostino, M., Ruberto, S., & Trivieri, F. (2018) · Journal of Productivity Analysis · doi:10.1007/s11123-018-0532-z · Cited by 23
Kao, T., Su, H., & Chen, Y. (2019) · International Journal of Production Economics · doi:10.1016/j.ijpe.2019.02.020 · Cited by 23
Amos, S., & Zanhouo, D. A. K. (2019) · Borsa Istanbul Review · doi:10.1016/j.bir.2019.07.004 · Cited by 24
The Cost Efficiency and Clinical Quality of Institutional Long-Term Care for the Elderly
Laine, J., Linna, M., Noro, A., & Häkkinen, U. (2005) · Health Care Management Science · doi:10.1007/s10729-005-0397-3 · Cited by 26
“These arguments have met with wide acceptance, and have been further supported by Wang and Schmidt [ 31 ] and Battese and Coelli [32], who also concluded that the twostage approach can lead to serious bias.”
ResultRetail food environment and household food waste: An empirical study
Cuffey, J. M., Li, W., Yu, Y., et al. (2023) · Food Policy · doi:10.1016/j.foodpol.2023.102457 · Cited by 21
Trade and efficiency of manufacturing industries in South Africa
Mazorodze, B. T. (2019) · Journal of International Trade & Economic Development · doi:10.1080/09638199.2019.1640273 · Cited by 24
“We estimate the frontier equation and the technical inefficiency model using the maximum likelihood technique simultaneously to avoid the persistent bias observed in Wang and Schmidt (2002) that arises if estimation proceeds in two stages.”
BackgroundMethodHow does financial development reduce carbon emissions: evidence from BRI countries
Liu, C., Xu, J., & Zhao, J. (2022) · Environmental Science and Pollution Research · doi:10.1007/s11356-022-24105-2 · Cited by 22
“Unlike the traditional method, the two-tier SFA model can include multiple variables at one time and more reliably measure the unilateral influence and comprehensive net effect of the explained variables (Wang and Schmidt 2002).”
MethodAnalyzing Italian citrus sector by semi-nonparametric frontier efficiency models
Vidoli, F., & Ferrara, G. (2014) · Empirical Economics · doi:10.1007/s00181-014-0879-6 · Cited by 22
India’s Trade Potential and Free Trade Agreements: A Stochastic Frontier Gravity Approach
Kumar, S., & Prabhakar, P. (2017) · Global economy journal · doi:10.1515/gej-2016-0074 · Cited by 22
Huang, T., Shen, C.-H., Chen, K. C., et al. (2010) · Journal of Productivity Analysis · doi:10.1007/s11123-010-0181-3 · Cited by 22
Does modernization improve performance: evidence from Indian police
Kumar, S., Kumar, S., Kumar, S., et al. (2013) · European Journal of Law and Economics · doi:10.1007/s10657-013-9392-1 · Cited by 21
“Wang and Schmidt (2002) show that the estimates obtained using ‘two stages’ are sev rely biased, therefore we use ‘one stage’ stochastic frontier analysis to get the robust estimates of efficiency and its determinants.”
BackgroundMethodAlternative method to measure the VAT gap in the EU: Stochastic tax frontier model approach
Nerudová, D., & Dobranschi, M. (2019) · PLoS ONE · doi:10.1371/journal.pone.0211317 · Cited by 21
Specification of merger gains in the Norwegian electricity distribution industry
Saastamoinen, A., Bjørndal, E., & Bjørndal, M. (2016) · Energy Policy · doi:10.1016/j.enpol.2016.12.014 · Cited by 21
Efficiency of the Indian leather firms: some results obtained using the two conventional methods
Bhandari, A. K., & Maiti, P. (2011) · Journal of Productivity Analysis · doi:10.1007/s11123-011-0219-1 · Cited by 21
“function. This is in line with the practice followed by others (see Battese and Broca 1997; Bhandari and Maiti 2007; Huang and Liu 1994; Lundvall and Battese 2000; Wang and Schmidt 2002 ).”
ResultKebede, H., & Berhanu, W. (2012) · Journal of African Economies · doi:10.1093/jae/ejs012 · Cited by 21
Impacts of Off-Farm Employment on Irrigation Water Efficiency in North China
Yin, N., Huang, Q., Yang, Z., et al. (2016) · Water · doi:10.3390/w8100452 · Cited by 21
“The two-step approach produces biased estimates of output technical efficiencies because the relationship in the second step is not incorporated into the estimation in the first step [37,38].”
BackgroundLearning-by-Exporting? Firm-Level Evidence for UK Manufacturing and Services SectorsPreprint
HARRIS, R. I. D., & Li, Q. C. (2007) · RePEc: Research Papers in Economics · Cited by 21
Lin, J., Jin, S., & Guo, H. (2023) · Annals of Public and Cooperative Economics · doi:10.1111/apce.12435 · Cited by 20
Zhong, Z., Jiang, W., & Yang, S. (2025) · Journal of Rural Studies · doi:10.1016/j.jrurstud.2025.103650 · Cited by 20
Klein, N., Herwartz, H., & Kneib, T. (2019) · Journal of Econometrics · doi:10.1016/j.jeconom.2019.07.003 · Cited by 20
The impact of Brexit on bank efficiency: Evidence from UK and Ireland
Fernández, X. L., Saavedra, D. P., & Coto‐Millán, P. (2019) · Finance research letters · doi:10.1016/j.frl.2019.101338 · Cited by 20
Estimation of Technical Inefficiency in Production Frontier Models Using Cross-Sectional Data
Kumbhakar, S. C., & Wang, H. (2015) · Springer eBooks · doi:10.1007/978-81-322-2253-8_1 · Cited by 21
Self-citationAn Analysis of Technical Efficiency Variation in Indonesian Rice Farming
Haryanto, T., Talib, B. A., & Salleh, N. H. M. (2015) · Journal of Agricultural Science · doi:10.5539/jas.v7n9p144 · Cited by 20
“…this estimation procedure has been criticized can produce biased conclusion because it ignores the assumption independently and identically distributed (i.i.d) of error term in the second stage estimation (Coelli et al., 2005; Kumbakar et al., 1991; Wang & Schmidt, 2002; Kumbakar & Lovell, 2000).”
MethodBarros, C. P., & Williams, J. M. (2013) · International Review of Financial Analysis · doi:10.1016/j.irfa.2013.06.006 · Cited by 20
Orea, L., & Steinbuks, J. (2017) · Economic Inquiry · doi:10.1111/ecin.12539 · Cited by 20
“51 It is important to point out that because suppliers had the opportunity to sell their capacity in the CAISO ancillary services markets and the real-time energy market, the calculated Lerner indices are not the actual measure of the unilateral market power, unlike in Wolak (2003). Rather, we use this measure as a (m…”
MethodEffects of branch expansion on bank efficiency: evidence from Japanese regional banks
Harimaya, K., & Kondo, K. (2016) · Managerial Finance · doi:10.1108/mf-11-2014-0278 · Cited by 20
Pseudolikelihood estimation of the stochastic frontier model
Andor, M., & Parmeter, C. F. (2017) · Applied Economics · doi:10.1080/00036846.2017.1324611 · Cited by 20
“As the literature has shown, two-step procedures to model the determinants of inefficiency are generally biased (see, among others, Caudill and Ford 1993, Wang and Schmidt 2002 and the discussion in Parmeter and Kumbhakar 2014).”
MethodNon-parametric tests of productive efficiency with errors-in-variables
Kuosmanen, T., Post, T., & Scholtes, S. (2005) · Journal of Econometrics · doi:10.1016/j.jeconom.2005.08.003 · Cited by 20
Measuring and Explaining the Impact of Productive Efficiency on Economic Development
Jayasuriya, R. (2005) · The World Bank Economic Review · doi:10.1093/wber/lhi006 · Cited by 21
Measuring efficiency of Vietnamese banks
Ngo, T., & Tripe, D. (2017) · Pacific Accounting Review · doi:10.1108/par-06-2016-0064 · Cited by 20
Pedroso, R., Tran, D. H., Viet, T. Q., Le, A. V., Dang, K. T., & Le, K. P. (2018) · World Development Perspectives · doi:10.1016/j.wdp.2017.12.001 · Cited by 20
Gezahegn, T. W., Passel, S. V., Berhanu, T., D’Haese, M., & Maertens, M. (2019) · Agrekon · doi:10.1080/03031853.2019.1663223 · Cited by 19
“Note that h and w may overlap with each other and with the vector x, but must not be functions of q (Wang and Schmidt 2002); p, k are vectors of parameters to be estimated.”
BackgroundInfluentialGalluzzo, N. (2020) · European Countryside · doi:10.2478/euco-2020-0026 · Cited by 22
“In this research, the parameters in the model of technical efficiency have been estimated simultaneously in order to reduce the bias, which is the negative aspect of an assessment using a two-step approach (Wang & Schmidt, 2002; Manevska-Tasevska et al., 2016; Madau, 2007).”
MethodSubsidies vs. deposits and cost inefficiency in microfinance
Al‐Azzam, M., Mimouni, K., Smaoui, H., et al. (2022) · Journal of Economic Behavior & Organization · doi:10.1016/j.jebo.2022.04.002 · Cited by 19
Stochastic Frontier Analysis: Foundations and Advances I
Kumbhakar, S. C., Parmeter, C. F., & Zelenyuk, V. (2022) · Springer eBooks · doi:10.1007/978-981-10-3455-8_9 · Cited by 22
Do innovation and financial constraints affect the profit efficiency of European enterprises?
Bonanno, G., Ferrando, A., & Rossi, S. P. S. (2022) · Eurasian economic review : · doi:10.1007/s40821-022-00226-z · Cited by 19
“As shown by Lensink and Meesters (2014) and Wang and Schmidt (2002), the two-step approach suffers from the fact that the inefficiency is assumed to be identically and independently distributed in the main frontier equation, while it depends on other variables in the inefficiency equation.”
BackgroundDeterminants of profit efficiency: evidence from Korean savings banks
Han, Y., Kim, M. H., & Kim, W. J. (2012) · Applied Financial Economics · doi:10.1080/09603107.2011.636019 · Cited by 19
“This approach, so-called two-step approach, has been criticized by Wang and Schmidt ( 2002 ) and Alvarez et al .”
BackgroundMethodTechnical Efficiency of Rainfed and Irrigated Rice Production in Tanzania
Mkanthama, J., Makombe, G., Kihoro, J. M., et al. (2017) · Irrigation and Drainage · doi:10.1002/ird.2185 · Cited by 19
“It has been recognized that the two-step procedure gives biased estimates due to misspecification bias in the first step, a bias substantial enough to strongly argue against the two-step estimation procedure (Wang and Schmidt, 2002).”
BackgroundFDI, technical efficiency and spillovers: Evidence from Indian automobile industry
Sur, A., Nandy, A., & Zhang, X. (2018) · Cogent Economics & Finance · doi:10.1080/23322039.2018.1460026 · Cited by 19
“As pointed out by Wang and Schmidt (2002) the two-stage approach, is likely to yield biased estimates”
BackgroundMethodInfluentialThe impact of labour subsidies on total factor productivity and profit per employee
Mattsson, P. (2018) · Economic Analysis and Policy · doi:10.1016/j.eap.2018.09.007 · Cited by 24
Prakash, N., Singh, S., & Sharma, S. (2021) · Structural Change and Economic Dynamics · doi:10.1016/j.strueco.2021.07.007 · Cited by 19
von Cramon‐Taubadel, S., & Saldías, R. (2014) · Chilean journal of agricultural research · doi:10.4067/s0718-58392014000400006 · Cited by 23
Technology Import and Manufacturing Productivity in India: Firm Level Analysis
Rijesh, R. (2015) · Journal of Industry Competition and Trade · doi:10.1007/s10842-015-0193-9 · Cited by 19
Profit, Cost and Scale Efficiency for Latin American Banks: Concentration-Performance RelationshipPreprint
Tabak, B. M., Fazio, D. M., & Cajueiro, D. O. (2011) · SSRN Electronic Journal · doi:10.2139/ssrn.1864688 · Cited by 19
“This method avoids the bias of estimating the frontier and the correlates in two steps (Wang and Schmidt, 2002) by estimating both in only one-step.”
MethodOn the performance of creative industries: Evidence from Mexican metropolitan areas
Benita, F. (2018) · Papers of the Regional Science Association · doi:10.1111/pirs.12403 · Cited by 18
Oumer, A. M., Burton, M. P., Hailu, A., & Mugera, A. W. (2020) · Agricultural Economics · doi:10.1111/agec.12595 · Cited by 23
Institutions and performance of regulated firms: Evidence from electricity distribution in India
Jamasb, T., Llorca, M., Khetrapal, P., & Thakur, T. (2021) · Economic Analysis and Policy · doi:10.1016/j.eap.2021.02.002 · Cited by 18
Digital economy as a catalyst for low-carbon transformation in China: new analytical insights
Xu, M., & Tan, R. (2024) · Humanities and Social Sciences Communications · doi:10.1057/s41599-024-03330-9 · Cited by 23
Valid tests of whether technical inefficiency depends on firm characteristics
Kim, M., & Schmidt, P. (2008) · Journal of Econometrics · doi:10.1016/j.jeconom.2008.04.005 · Cited by 18
“This model can be estimated consistently by nonlinear least squares, as has been noted by Simar, Lovell, and van den Eeckaut (1994), Wang and Schmidt (2002)”
BackgroundMethodSelf-citationMensah, A., Brümmer, B., Mensah, A., et al. (2016) · RePEc: Research Papers in Economics · doi:10.22004/ag.econ.246269 · Cited by 18
Climate and landscape mediate patterns of low lentil productivity in Nepal
Paudel, G. P., Devkota, M., Keil, A., & McDonald, A. J. (2020) · PLoS ONE · doi:10.1371/journal.pone.0231377 · Cited by 18
“Furthermore, Wang and Schmidt [63] pointed out that the first stage is biased if dependent variables of the first and determinants of the second stage are correlated.”
BackgroundMethodEstimating German overqualification with stochastic earnings frontiers
Jensen, U., Gartner, H., & Rässler, S. (2010) · AStA Advances in Statistical Analysis · doi:10.1007/s10182-009-0121-6 · Cited by 18
Public external debt, informality and production efficiency in developing countries
Drine, I., & Nabi, M. S. (2009) · Economic Modelling · doi:10.1016/j.econmod.2009.10.014 · Cited by 18
Abedullah, M., Kouser, S., & Mushtaq, K. (2010) · The Pakistan Development Review · doi:10.30541/v49i1pp.57-72 · Cited by 18
“…efficiency in different crops and in different regions [Good, et al. (1993); Ahmed and Bravo-Ureta (1996); Wilson, et al. (1998); Wadud (1999); Wang and Schmidt (2002); Larson and Plessman (2002); Villano (2005); Abedullah, et al. (2007)] but little work has been done to estimate the…”
BackgroundTechnical Efficiency in Chilean Agribusiness Industry: A Metafrontier Approach
Lakner, S., Brenes‐Muñoz, T., & Brümmer, B. (2017) · Agribusiness · doi:10.1002/agr.21493 · Cited by 18
Effect of concentration in airline market on Spanish airport technical efficiency
Hidalgo‐Gallego, S., & Mantecón, I. M. (2019) · Journal of Air Transport Management · doi:10.1016/j.jairtraman.2019.02.003 · Cited by 19
Modeling the efficiency of Tunisian and Moroccan banks using the SFA approach
Hela, K., Hamad, S. B., & Triki, M. (2019) · International Journal of Productivity and Performance Management · doi:10.1108/ijppm-03-2018-0099 · Cited by 18
Tesfaw, Z., Zemedu, L., & Tegegn, B. (2021) · Cogent Food & Agriculture · doi:10.1080/23311932.2020.1865594 · Cited by 18
“According to Wang and Schmidt (2002) the two-step procedures have sources of bias.”
BackgroundVocational Training and Agricultural Productivity: Evidence from Rice Production in Vietnam
Ulimwengu, J. M., & Badiane, O. (2010) · The Journal of Agricultural Education and Extension · doi:10.1080/1389224x.2010.515062 · Cited by 18
“Since variables influencing agricultural efficiency ( ϵ h ) may also directly impact agricultural production ( q h ), we adopt the approach proposed by Wang and Schmidt ( 2002 ), and Liu and Myers ( 2009 ) where equation ( 3 ) is re-written as follows:where Z h include type of schooling, farmers’ health status, health…”
MethodSenyonga, L., & Bergland, O. (2018) · The Energy Journal · doi:10.5547/01956574.39.5.lsen · Cited by 15
Nguyen, H. T. M., Do, H., & Kompas, T. (2021) · World Development · doi:10.1016/j.worlddev.2021.105658 · Cited by 17
The impact of efficiency on asset quality in banking
Badunenko, O., Dadoukis, A., Fusi, G., et al. (2021) · European Journal of Finance · doi:10.1080/1351847x.2021.1946117 · Cited by 17
Haschka, R. E., & Herwartz, H. (2022) · Journal of Economics & Management Strategy · doi:10.1111/jems.12491 · Cited by 15
The role of local institutions in cooperative banks’ efficiency. The case of Italy
Agostino, M., Ruberto, S., & Trivieri, F. (2022) · International Review of Economics & Finance · doi:10.1016/j.iref.2022.11.001 · Cited by 17
Ali, N. S. Y., & See, K. F. (2023) · Journal of Cleaner Production · doi:10.1016/j.jclepro.2023.135982 · Cited by 17
Production analysis with asymmetric noise
Badunenko, O., & Henderson, D. J. (2023) · Journal of Productivity Analysis · doi:10.1007/s11123-023-00680-5 · Cited by 17
Corporate environmental performance and efficiency: Evidence from stochastic frontier analysis
da Silva, P. P. (2023) · Journal of Climate Finance · doi:10.1016/j.jclimf.2023.100022 · Cited by 17
Bardají, I., & Apezteguía, B. I. (2014) · Regional Environmental Change · doi:10.1007/s10113-014-0623-y · Cited by 17
“The choice of method used to consider the effects on efficiency is a single-stage estimation process (as recommended by Wang and Schmidt 2002 and Fried et al. 2008).”
MethodLand fragmentation with double dividends – the case of Tanzanian agriculture
Rao, X. (2018) · European Review of Agricultural Economics · doi:10.1093/erae/jby034 · Cited by 19
Model averaging estimators for the stochastic frontier model
Parmeter, C. F., Wan, A. T. K., & Zhang, X. (2019) · Journal of Productivity Analysis · doi:10.1007/s11123-019-00547-8 · Cited by 17
The efficiency of Japanese financial cooperatives: An application of parametric distance functions
Yamori, N., Harimaya, K., & Tomimura, K. (2017) · Journal of Economics and Business · doi:10.1016/j.jeconbus.2017.09.001 · Cited by 17
Tourism development and regional production efficiency: Evidence from southwestern China
Zhou, B., Xu, Y., & Lee, S. K. (2018) · Tourism Economics · doi:10.1177/1354816618810250 · Cited by 17
“Wang and Schmidt (2002) demonstrate that the one-stage approach improves the estimate of the frontier parameters, and is superior to the two-stage approach.”
BackgroundThe impact of crop rotation and land fragmentation on farm productivity in Albania
Pavel, C., Rajčániová, M., Guri, F., et al. (2018) · Studies in Agricultural Economics · doi:10.7896/j.1815 · Cited by 17
“This approach has often been criticised for generating biased results (Wang and Schmidt, 2002).”
ResultChoi, H., & Oh, D. (2019) · Journal of Productivity Analysis · doi:10.1007/s11123-019-00567-4 · Cited by 19
Martinez‐Cillero, M., & Reaños, M. A. T. (2022) · Journal of Agricultural Economics · doi:10.1111/1477-9552.12509 · Cited by 17
Kinda, T. (2011) · Applied Economics Letters · doi:10.1080/13504851.2011.587765 · Cited by 17
Stein, M., & Amanda, R. C. (2015) · Journal of Development and Agricultural Economics · doi:10.5897/jdae2014.0630 · Cited by 17
“…the levels of efficiency that are then regressed against another set of variables which are not firm specific is criticized on the potential of inducing a persistence bias that will be carried forward to the second stage thus affecting the estimates of efficiency (Wang and Schmidt, 2002).”
BackgroundDo creditor rights and information sharing affect the performance of foreign banks?
Kalyvas, A. N., & Mamatzakis, E. C. M. (2017) · Journal of International Financial Markets Institutions and Money · doi:10.1016/j.intfin.2017.07.001 · Cited by 17
“…set up, in which (in)efficiency estimates from the SFA model are regressed in a second-step to further variables, would provide inconsistent estimates (Wang and Schmidt, 2002) Therefore, employing the single-stage Battese and Coelli (1995) SFA model is justified in the context of this study.”
MethodGuliyev, O., Liu, A., Mwalupaso, G. E., & Niemi, J. K. (2019) · Sustainability · doi:10.3390/su11164332 · Cited by 17
“Stochastic frontier analysis was estimated using Maximum Likelihood Estimator (MLE) as guided by Wang and Schmidt [49] (Appendix A).”
MethodKumbhakar, S. C., & Tsionas, E. G. (2007) · Agricultural Economics · doi:10.1111/j.1574-0862.2007.00285.x · Cited by 18
The Influence of Risk-Taking on Bank Efficiency: Evidence from ColombiaPreprint
Sarmiento, M., & Galán, J. (2015) · SSRN Electronic Journal · doi:10.2139/ssrn.2627581 · Cited by 17
Miriti, P. K., Otieno, D. J., Chimoita, E. L., Bikketi, E., Njuguna, E. M., & Ojiewo, C. O. (2021) · Heliyon · doi:10.1016/j.heliyon.2020.e05845 · Cited by 16
“After estimation of MTEs, a two-limit-tobit model was applied to analyze determinants of TE because efficiency scores are bounded between 0 and 1 (Wang and Schmidt, 2002).”
MethodDo, H.-L., & Thuy, T. D. (2022) · Ecological Economics · doi:10.1016/j.ecolecon.2021.107326 · Cited by 16
Analysing SME investment, financing constraints and its determinants. A stochastic frontier approach
Martinez‐Cillero, M., Lawless, M., & O’Toole, C. M. (2023) · International Review of Economics & Finance · doi:10.1016/j.iref.2023.01.029 · Cited by 16
Female presence in local government and cost efficiency: the case of Italian municipalities
Bucci, V., Ferrara, G., & Resce, G. (2023) · Regional Studies · doi:10.1080/00343404.2023.2186386 · Cited by 16
Technologically related diversification: One size does not fit all European regions
Barbero, J., Diukanova, O., Gianelle, C., Salotti, S., & Santoalha, A. (2024) · Research Policy · doi:10.1016/j.respol.2024.104973 · Cited by 17
EFFICIENCY AND STOCK PERFORMANCE OF BANKS IN TRANSITION COUNTRIES: IS THERE A RELATIONSHIP?
Vardar, G. (2013) · International Journal of Economics and Financial Issues · Cited by 16
Bahta, S., Omore, A. O., Baker, D., Okike, I., Gebremedhin, B., & Wanyoike, F. N. (2020) · European Journal of Development Research · doi:10.1057/s41287-020-00279-8 · Cited by 17
Xu, Y., Zhang, Y., Lu, Y., & Chen, J. (2021) · Environmental Science and Pollution Research · doi:10.1007/s11356-021-15885-0 · Cited by 17
Mamiit, R. J., Yanagida, J. F., & Miura, T. (2021) · Frontiers in Sustainable Food Systems · doi:10.3389/fsufs.2021.727484 · Cited by 16
Pisulewski, A., & Marzec, J. (2022) · Journal of Agricultural and Applied Economics · doi:10.1017/aae.2022.26 · Cited by 16
“Due to the scaling property (Alvarez et al., 2006; Wang and Schmidt, 2002), the absolute values of the above-mentioned estimates may constitute a ranking of the impact of these factors.”
A selection of relevant issues in applied stochastic frontier analysis
Álvarez, A., & Arias, C. (2014) · Economics and Business Letters · doi:10.17811/ebl.3.1.2014.3-11 · Cited by 16
“…since excluding the explanatory variables of efficiency in the first stage will most likely yield biased estimates of the technological parameters Wang and Schmidt (2002) provide extensive Monte Carlo evidence that the size of this bias is important but also that the estimated efficiency levels…”
BackgroundDOES URBAN PROXIMITY ENHANCE TECHNICAL EFFICIENCY? EVIDENCE FROM CHINESE AGRICULTURE
Duvivier, C. (2013) · Journal of Regional Science · doi:10.1111/jors.12054 · Cited by 18
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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