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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.”
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“If there are other variables, git, which do ‘explain’ uit, then they should have appeared in the model at the first stage, and estimates computed without them are biased in unknown directions (the ‘left out variable’ problem) (see Wang and Schmidt, 2002). There are two motivations for proceeding in this fashion noneth…”
BackgroundEvaluating Contextual Variables Affecting Productivity Using Data Envelopment Analysis
Banker, R. D., & Natarajan, R. (2007) · Operations Research · doi:10.1287/opre.1070.0460 · Cited by 789
“Wang and Schmidt (2002) explain that the problem is due to the estimator of technical efficiency u, derived from the first stage residual, having a “shrinkage” towards the mean and as a result showing less dependence on the contextual variables z than it should.”
BackgroundInfluentialOutreach and Efficiency of Microfinance Institutions
Hermes, N., Lensink, R., & Meesters, A. (2011) · World Development · doi:10.1016/j.worlddev.2009.10.018 · Cited by 724
“Wang and Schmidt (2002) show that a two-step approach suffers from the assumption that the efficiency term is independent and identically truncated, normally distributed in the first step, while in the second step the efficiency terms are assumed to be normally distributed and dependent on the…”
BackgroundAgriculture, Forestry and Fisheries
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“[See Wang and Schmidt (2002).] There are two important restrictions built into this model.”
BackgroundJournal of Development and Agricultural Economics
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Bank efficiency and foreign ownership: Do good institutions matter?
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Stochastic Frontier Analysis using Stata
Belotti, F., Daidone, S., Ilardi, G., & Atella, V. (2013) · The Stata Journal Promoting communications on statistics and Stata · doi:10.1177/1536867x1301300404 · Cited by 447
“Wang and Schmidt (2002) show that this approach leads to severely biased results; thus we shall focus only on model extensions based on simultaneous estimation.”
BackgroundInfluentialHeteroscedasticity and Non-Monotonic Efficiency Effects of a Stochastic Frontier Model
Wang, H. (2002) · Journal of Productivity Analysis · doi:10.1023/a:1020638827640 · Cited by 367
“The extensive Monte Carlo results presented by Schmidt and Wang (2002) give evidence in favor of the one-step approach.”
ResultSelf-citationEstimating fixed-effect panel stochastic frontier models by model transformation
Wang, H., & Ho, C.-W. (2010) · Journal of Econometrics · doi:10.1016/j.jeconom.2009.12.006 · Cited by 336
Self-citationWilliams, J. M., & Nguyen, N. T. (2005) · Journal of Banking & Finance · doi:10.1016/j.jbankfin.2005.03.011 · Cited by 301
Fractional regression models for second stage DEA efficiency analyses
Ramalho, E. A., Ramalho, J. J. S., & Henriques, P. D. D. S. (2010) · Journal of Productivity Analysis · doi:10.1007/s11123-010-0184-0 · Cited by 291
“See Simar and Wilson (2007) for an extensive list of references to the use of this approach, Coelli et al. (2005) and Daraio and Simar (2005) for other approaches to incorporating the influence of efficiency factors into DEA analysis, and Wang and Schmidt (2002) for an explanation of why, in the framework of stochasti…”
BackgroundDetermining management behaviour in European banking
Williams, J. M. (2004) · Journal of Banking & Finance · doi:10.1016/j.jbankfin.2003.09.010 · Cited by 266
Subsidies and Technical Efficiency in Agriculture: Evidence from European Dairy Farms
Latruffe, L., Bravo‐Ureta, B. E., Carpentier, A., Desjeux, Y., & Moreira, V. (2016) · American Journal of Agricultural Economics · doi:10.1093/ajae/aaw077 · Cited by 240
Market structure and the efficiency of European insurance companies: A stochastic frontier analysis
Fenn, P., Vencappa, D., Diacon, S. R., Klumpes, P. J. M., & O’Brien, C. (2007) · Journal of Banking & Finance · doi:10.1016/j.jbankfin.2007.09.005 · Cited by 227
Cost and profit efficiencies in transition banking: the case of new EU members
Kasman, A., & Yildirim, C. (2006) · Applied Economics · doi:10.1080/00036840600639022 · Cited by 226
“6 See Wang and Schmidt ( 2002 ) for a discussion of one-step and two-step methods.”
BackgroundUnderstanding Countries’ Tax Effort
Fenochietto, R., & Pessino, C. (2013) · IMF Working Paper · doi:10.2139/ssrn.2376661 · Cited by 239
Efficiency and productivity growth in the banking industry of Central and Eastern Europe
Koutsomanoli‐Filippaki, A., Margaritis, D., & Staikouras, C. (2008) · Journal of Banking & Finance · doi:10.1016/j.jbankfin.2008.09.009 · Cited by 216
An Empirical Analysis of State and Private-Sector Provision of Water Services in Africa
Kirkpatrick, C. H., Parker, D., & Zhang, Y. (2006) · The World Bank Economic Review · doi:10.1093/wber/lhj001 · Cited by 218
Efficiency Analysis: A Primer on Recent Advances
Parmeter, C. F., & Kumbhakar, S. C. (2014) · Foundations and Trends® in Econometrics · doi:10.1561/0800000023 · Cited by 218
Alvarez, A. M., Amsler, C. E., Orea, L., & Schmidt, P. (2006) · Journal of Productivity Analysis · doi:10.1007/s11123-006-7639-3 · Cited by 204
“4. As noted by Wang and Schmidt (2002) , the interpretation of δ does not depend on the”
BackgroundInfluentialSelf-citationShao, S., Yang, Z., Yang, L., & Ma, S. (2019) · The Energy Journal · doi:10.5547/01956574.40.4.ssha · Cited by 201
“…energy efficiency growth rate ( TFEEG ): Considering the corner-solution estimation problem in Battese and Coelli’s (1992, 1995) models (Wang, 2002; Wang and Schmidt, 2002) and the heterogeneity between different industrial sub-sectors, we adopt the fixed-effect SFA model (Greene, 2005) and…”
MethodChen, C., Delmas, M. A., & Lieberman, M. B. (2013) · Strategic Management Journal · doi:10.1002/smj.2199 · Cited by 194
“For example, Wang and Schmidt (2002) show that the SFA two-stage approach can generate substantially biased estimates in both stages.”
BackgroundAre small firms less efficient?
Yang, C., & Chen, K.-H. (2007) · Small Business Economics · doi:10.1007/s11187-007-9082-x · Cited by 179
“However, recent research has shown how inappropriate the two-step framework is and suggests that the single-stage estimation procedure is more appropriate ( Wang and Schmidt 2002; Greene 2005; Simar and Wilson 2007).”
BackgroundUrban land use efficiency and improvement potential in China: A stochastic frontier analysis
Liu, S., Xiao, W., Li, L., Ye, Y., & Song, X. (2020) · Land Use Policy · doi:10.1016/j.landusepol.2020.105046 · Cited by 183
Cuccia, T., Guccio, C., & Rizzo, I. (2015) · Economic Modelling · doi:10.1016/j.econmod.2015.10.049 · Cited by 167
Economic impacts of mobile versus fixed broadband
Thompson, H. G., & Garbacz, C. (2011) · Telecommunications Policy · doi:10.1016/j.telpol.2011.07.004 · Cited by 166
Jiang, C., Yao, S., & Zhang, Z. (2009) · China Economic Review · doi:10.1016/j.chieco.2009.05.005 · Cited by 165
Leverage and Corporate Performance: Does Institutional Environment Matter?
Weill, L. (2007) · Small Business Economics · doi:10.1007/s11187-006-9045-7 · Cited by 165
Institutional quality and its spatial spillover effects on energy efficiency
Sun, H., Edziah, B. K., Sun, C., & Kporsu, A. K. (2021) · Socio-Economic Planning Sciences · doi:10.1016/j.seps.2021.101023 · Cited by 166
Rao, E. J. O., Brümmer, B., & Qaim, M. (2012) · American Journal of Agricultural Economics · doi:10.1093/ajae/aas024 · Cited by 161
Potential Effects of Climate Change on the Productivity of U.S. Dairies
Key, N., & Sneeringer, S. (2014) · American Journal of Agricultural Economics · doi:10.1093/ajae/aau002 · Cited by 159
Estimating Food Waste as Household Production Inefficiency
Yu, Y., & Jaenicke, E. C. (2020) · American Journal of Agricultural Economics · doi:10.1002/ajae.12036 · Cited by 167
“Because of this concern, incorporating a comprehensive set of contextual variables has been a challenging task in productivity analysis research (Wang and Schmidt 2002; Alvarez et al. 2006; Simar andWilson 2007; Banker and Natarajan 2008).”
BackgroundGuidelines for using variable selection techniques in data envelopment analysis
Nataraja, N. R., & Johnson, A. L. (2011) · European Journal of Operational Research · doi:10.1016/j.ejor.2011.06.045 · Cited by 157
“ݕ = ݔ ଵ ݔ ଶ ݔ ଷ ݁ ି௨ (1) To establish the correlation between inputs we adopt the following equation from Wang and Schmidt (2002).”
Impact of land ownership on productivity and efficiency of rice farmers: The case of the Philippines
Koirala, K. H., Mishra, A., & Mohanty, S. (2015) · Land Use Policy · doi:10.1016/j.landusepol.2015.10.001 · Cited by 162
Mobile, fixed line and Internet service effects on global productive efficiency
Thompson, H. G., & Garbacz, C. (2007) · Information Economics and Policy · doi:10.1016/j.infoecopol.2007.03.002 · Cited by 151
Banker, R. D., Natarajan, R., & Zhang, D. (2018) · European Journal of Operational Research · doi:10.1016/j.ejor.2018.10.050 · Cited by 154
Stochastic Frontier Analysis of Hospital Inefficiency
Rosko, M. D., & Mutter, R. L. (2007) · Medical Care Research and Review · doi:10.1177/1077558707307580 · Cited by 151
“In contrast to the two-stage procedure, the one-stage procedure generates estimates of inefficiency (while imposing the assumed impact of the environmental pressure variables on inefficiency) and estimates the parameters of the inefficiency-effects variables in one step (Wang & Schmidt, 2002).”
BackgroundMethodOutreach and Efficiency of Microfinance InstitutionsPreprint
Hermes, N., Lensink, R., & Meesters, A. (2008) · SSRN Electronic Journal · doi:10.2139/ssrn.1143925 · Cited by 153
Yang, Z., Shao, S., Yang, L., & Miao, Z. (2018) · Energy Economics · doi:10.1016/j.eneco.2018.04.003 · Cited by 147
One-stage and two-stage DEA estimation of the effects of contextual variables
Johnson, A. L., & Kuosmanen, T. (2012) · European Journal of Operational Research · doi:10.1016/j.ejor.2012.01.023 · Cited by 149
“Following Wang and Schmidt (2002), we assume that the contextual variables are dependent on the input levels, and are generated according to equation”
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Mawejje, J., & Sebudde, R. K. (2019) · Economic Analysis and Policy · doi:10.1016/j.eap.2019.05.005 · Cited by 157
Precision agriculture technology adoption and technical efficiency
DeLay, N. D., Thompson, N. M., & Mintert, J. (2021) · Journal of Agricultural Economics · doi:10.1111/1477-9552.12440 · Cited by 140
“One, if ignored, heteroscedasticity can produce biased and inconsistent estimates of the SFA parameters, and two, by parameterising the variance of ui, we can model the determinants of technical efficiency (Kumbhakar & Lovell, 2000; Wang & Schmidt, 2002).”
BackgroundGuerrini, A., Carvalho, P. M. S., Romanò, G., et al. (2017) · Journal of Cleaner Production · doi:10.1016/j.jclepro.2017.01.079 · Cited by 140
McMillan, M., & Chan, W. H. (2006) · Education Economics · doi:10.1080/09645290500481857 · Cited by 139
“Wang and Schmidt ( 2002 ) compare one‐step and two‐step procedures for a stochastic model and found the two‐step procedure led to biased results, and therefore recommend against that approach.”
Kuosmanen, T., Saastamoinen, A., & Sipiläinen, T. (2013) · Energy Policy · doi:10.1016/j.enpol.2013.05.091 · Cited by 132
Consistent inference in fixed-effects stochastic frontier models
Belotti, F., & Ilardi, G. (2017) · Journal of Econometrics · doi:10.1016/j.jeconom.2017.09.005 · Cited by 131
“Furthermore, they are also important from the empirical perspective because they allow to test specific hypotheses of interest and policy implications, avoiding biased two-step procedures (Wang & Schmidt, 2002).”
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Economic reforms, efficiency and productivity in Chinese banking
Kumbhakar, S. C., & Wang, D. (2007) · Journal of Regulatory Economics · doi:10.1007/s11149-007-9028-x · Cited by 124
“2 Wang and Schmidt (2002) present extensive Monte Carlo results to show that the two-step estimation method suffers from serious biases.”
BackgroundRisk-adjusted efficiency and corporate governance: Evidence from Islamic and conventional banks
Safiullah, M., & Shamsuddin, A. F. M. (2018) · Journal of Corporate Finance · doi:10.1016/j.jcorpfin.2018.08.009 · Cited by 126
HARRIS, R. I. D., & Trainor, M. (2005) · Journal of Regional Science · doi:10.1111/j.0022-4146.2005.00364.x · Cited by 26
Modelling the joint impact of R&D and ICT on productivity: A frontier analysis approach
Pieri, F., Vecchi, M., & Venturini, F. (2018) · Research Policy · doi:10.1016/j.respol.2018.06.013 · Cited by 113
Estimate of transient and persistent energy efficiency in Africa: A stochastic frontier approach
Adom, P. K., Amakye, K., Abrokwa, K. K., et al. (2018) · Energy Conversion and Management · doi:10.1016/j.enconman.2018.04.038 · Cited by 120
Johnson, A. L., & Kuosmanen, T. (2011) · Journal of Productivity Analysis · doi:10.1007/s11123-011-0231-5 · Cited by 119
“Following Wang and Schmidt (2002), we assume that the contextual variables are dependent on the input levels, and are generated according to equation ( ) 21 3 1z x wρ ρ= − + − (9) where parameter ρ is some pre-specified correlation coefficient and w is an independent random number drawn from Uni[…”
BackgroundInfluentialMigration, local off-farm employment, and agricultural production efficiency: evidence from China
Yang, J., Wang, H., Jin, S., et al. (2015) · Journal of Productivity Analysis · doi:10.1007/s11123-015-0464-9 · Cited by 118
“However, this two-step approachwas criticized for the inconsistency between the independency assumption of uit in the first step and the dependency assumption in the second step (Wang and Schmidt 2002; Kumbhakar and Lovell 2000).”
Self-citationFilippini, M., Hrovatin, N., & Zorić, J. (2007) · Journal of Productivity Analysis · doi:10.1007/s11123-007-0069-z · Cited by 116
“This is quite often in the applied literature, although it contradicts the cost theory.9 8 This procedure has some serious econometric flaws which results in biased and inconsistent estimators (Wang and Schmidt, 2002).”
BackgroundProfit efficiency and its determinants in small and medium-sized enterprises in Spain
Pérez-Gómez, P., Arbelo-Pérez, M., & Arbelo, A. (2018) · BRQ Business Research Quarterly · doi:10.1016/j.brq.2018.08.003 · Cited by 143
Massimiliano Piacenza, B. (2006) · Journal of Productivity Analysis · doi:10.1007/s11123-006-7643-7 · Cited by 115
“suggested by Wang and Schmidt (2002) , we can call them Kumbhakar–Ghosh–McGuckin–Huang–Liu– Battese–Coelli model (KGMHLBC hereafter).”
Yang, Z., Shao, S., Li, C., et al. (2019) · Technological Forecasting and Social Change · doi:10.1016/j.techfore.2019.119878 · Cited by 114
Handling Endogeneity in Stochastic Frontier AnalysisPreprint
Karakaplan, M. U., & Kutlu, L. (2015) · SSRN Electronic Journal · doi:10.2139/ssrn.2607276 · Cited by 105
“8 The two-stage method suggested in here is different than the one that is criticized by Wang and Schmidt (2002) or the one implemented by Kutlu (2010), which requires bootstrapping.”
Wollni, M., & Brümmer, B. (2011) · Food Policy · doi:10.1016/j.foodpol.2011.11.004 · Cited by 113
“Wang and Schmidt (2002) illustrate that the two-step approach might lead to inconsistent results.”
BackgroundResultBank risk taking, efficiency, and law enforcement: Evidence from Chinese city commercial banks
Zhang, J., Wang, P., & Qu, B. (2011) · China Economic Review · doi:10.1016/j.chieco.2011.12.001 · Cited by 107
Moudud‐Ul‐Huq, S. (2020) · International Journal of Emerging Markets · doi:10.1108/ijoem-03-2019-0197 · Cited by 106
Feng, S. (2008) · NJAS - Wageningen Journal of Life Sciences · doi:10.1016/s1573-5214(08)80026-7 · Cited by 105
Stochastic Frontier Analysis Using StataPreprint
Belotti, F., Daidone, S., Ilardi, G., & Atella, V. (2012) · SSRN Electronic Journal · doi:10.2139/ssrn.2145803 · Cited by 111
Mechanization and efficiency in rice production in China
Shi, M., Paudel, K. P., & Chen, F. (2021) · Journal of Integrative Agriculture · doi:10.1016/s2095-3119(20)63439-6 · Cited by 104
Will land transfer always increase technical efficiency in China?—A land cost perspective
Liu, Y., Yan, B., Wang, Y., & Zhou, Y. (2018) · Land Use Policy · doi:10.1016/j.landusepol.2018.12.002 · Cited by 108
Measuring Excess Capital Capacity in Agricultural Production
Guan, Z., Kumbhakar, S. C., Myers, R. J., & Lansink, A. O. (2009) · American Journal of Agricultural Economics · doi:10.1111/j.1467-8276.2009.01271.x · Cited by 98
Stochastic Nonparametric Approach to Efficiency Analysis: A Unified Framework
Kuosmanen, T., Johnson, A. L., & Saastamoinen, A. (2015) · International series in management science/operations research/International series in operations research & management science · doi:10.1007/978-1-4899-7553-9_7 · Cited by 113
The influence of risk-taking on bank efficiency: Evidence from Colombia
Sarmiento, M., & Galán, J. (2017) · Emerging Markets Review · doi:10.1016/j.ememar.2017.05.007 · Cited by 98
Daidone, S., & D’Amico, F. (2009) · Journal of Productivity Analysis · doi:10.1007/s11123-009-0137-7 · Cited by 97
“Last but not least, Wang and Schmidt [15] have demonstrated that this model specification allows one-step estimation of the parameters, avoiding two-step procedures which give biased results if the model estimated at the first step is misspecified.”
DETERMINANTS OF INEFFICIENCY IN NORWEGIAN SALMON AQUACULTURE
Asche, F., & Roll, K. H. (2013) · Aquaculture Economics & Management · doi:10.1080/13657305.2013.812154 · Cited by 96
Chuang, H. H., Oliva, R., & Heim, G. R. (2019) · Production and Operations Management · doi:10.1111/poms.13055 · Cited by 101
“…the stochastic frontier model in one stage ensures consistent estimates and also avoids the conceptual inconsistencies of two-stage procedures, in which operational efficiency is estimated first without considering Z it and then efficiency estimates are regressed on Z it (Wang and Schmidt 2002).”
BackgroundHousehold energy consumption, energy efficiency, and household income–Evidence from China
Zheng, J., Dang, Y., & Assad, U. (2023) · Applied Energy · doi:10.1016/j.apenergy.2023.122074 · Cited by 103
Alleviating energy poverty in Europe: Front-runners and laggards
Rodrı́guez-Álvarez, A., Llorca, M., & Jamasb, T. (2021) · Energy Economics · doi:10.1016/j.eneco.2021.105575 · Cited by 92
“Moreover, we follow Wang and Schmidt (2002) who propose a modelling strategy in which the random variable u (representing inefficiency), has the following form: where h (.)”
Cost Efficiency in Regional Bus Companies: An Application of Alternative Stochastic Frontier Models
Farsi, M., Filippini, M., & Kuenzle, M. (2006) · Journal of Transport Economics and Policy · doi:10.3828/jtep.2006.40.1.95 · Cited by 93
“3 For the advantages of single stage models, see Wang and Schmidt (2002). single indicators.”
BackgroundToo small to be beautiful? The farm size and productivity relationship in Bangladesh
Gautam, M., & Ahmed, M. (2018) · Food Policy · doi:10.1016/j.foodpol.2018.03.013 · Cited by 92
“As noted, the sequential or two-step estimation of these equations may produce biased and inconsistent estimates of the production function (and hence the estimate of production efficiency) if the variables in the inefficiency functions are correlated with the inputs in the production function (Wang and Schmidt, 2002;…”
BackgroundClimate Change, Heat Stress, and U.S. Dairy ProductionPreprint
Key, N., Sneeringer, S., & Marquardt, D. (2014) · SSRN Electronic Journal · doi:10.2139/ssrn.2506668 · Cited by 92
“This two-step approach can, however, result in biased and inefficient estimates (Wang and Schmidt, 2002).”
Karagiannis, G., & Sarris, A. H. (2005) · Agricultural Economics · doi:10.1111/j.1574-0864.2005.00084.x · Cited by 87
Technical Efficiency of Smallholder Agriculture in Developing Countries: The Case of Ethiopia
Tenaye, A. (2020) · Economies · doi:10.3390/economies8020034 · Cited by 105
“Simultaneous estimation of both production technology and the covariates of technical inefficiency is preferred over the traditional two-step procedure (Wang and Schmidt 2002).”
MethodInfluentialBenedetti, I., Branca, G., & Zucaro, R. (2019) · Journal of Cleaner Production · doi:10.1016/j.jclepro.2019.117609 · Cited by 85
One-step and two-step estimation in SFA models
Schmidt, P. (2011) · Journal of Productivity Analysis · doi:10.1007/s11123-011-0228-0 · Cited by 89
“It relies heavily on Wang and Schmidt (2002) though there are some new insights.”
BackgroundInfluentialSelf-citationFinancial Development and the Efficiency of Microfinance InstitutionsPreprint
Hermes, N., Lensink, R., & Meesters, A. (2009) · SSRN Electronic Journal · doi:10.2139/ssrn.1396202 · Cited by 84
Are Women Better Bankers to the Poor? Evidence from Rural Microfinance Institutions
Hartarska, V., Nadolnyak, D. A., & Mersland, R. (2014) · American Journal of Agricultural Economics · doi:10.1093/ajae/aau061 · Cited by 82
Hu, J., Yang, C., & Chen, C. (2011) · Bulletin of Economic Research · doi:10.1111/j.1467-8586.2011.00417.x · Cited by 81
“C© 2011 Board of Trustees of the Bulletin of Economic Research and John Wiley & Sons Ltd be seriously biased, which suggests the single-stage SFA is more adequate (Wang and Schmidt, 2002).”
InfluentialResearch quality and efficiency
Groot, T. L. C. M., & Valderrama, T. G. (2006) · Research Policy · doi:10.1016/j.respol.2006.07.002 · Cited by 82
Self-sufficiency policy and irrigated rice productivity in the Senegal River Valley
Diagne, M., Demont, M., Seck, P. A., & Diaw, A. (2012) · Food Security · doi:10.1007/s12571-012-0229-5 · Cited by 82
“Determinants of technical efficiency Wang and Schmidt (2002) argue that the two-stage model is subject to significant biases due to omitted variables in the first stage and to underdispersion of estimates of the inefficiency function in the second stage.”
BackgroundNiaounakis, T. K., & Blank, J. L. T. (2017) · Local Government Studies · doi:10.1080/03003930.2017.1322958 · Cited by 78
Power infrastructure quality and manufacturing productivity in Africa: A firm level analysis
Moyo, B. (2013) · Energy Policy · doi:10.1016/j.enpol.2013.05.111 · Cited by 77
Wongnaa, C. A., Awunyo‐Vitor, D., Mensah, A., et al. (2019) · Scientific African · doi:10.1016/j.sciaf.2019.e00206 · Cited by 76
Suyanto, S., & Salim, R. A. (2011) · Applied Economics · doi:10.1080/00036846.2011.605554 · Cited by 75
“However, the stochastic frontier used in testing the spillover hypothesis of Kathuria’s study is a two-stage approach, which has been widely known having limitations that may lead to bias in estimations (Kumbhakar et al., 1991; Wang and Schmidt, 2002).”
BackgroundManlagñit, M. C. V. (2010) · Journal of Asian Economics · doi:10.1016/j.asieco.2010.10.001 · Cited by 74
Tauer, L. W., & Mishra, A. K. (2006) · Journal of Dairy Science · doi:10.3168/jds.s0022-0302(06)72544-9 · Cited by 74
Kažukauskas, A., Newman, C., & Sauer, J. (2013) · Agricultural Economics · doi:10.1111/agec.12068 · Cited by 74
Ownership structure, corporate governance and productive efficiency in China
Su, D., & He, X. (2011) · Journal of Productivity Analysis · doi:10.1007/s11123-011-0257-8 · Cited by 73
“Shleifer and Vishny (1997) argues that ownership concentration is, along with legal protection, one of two key determinants of corporate governance.”
BackgroundResultSpatial heterogeneity in composite indicator: A methodological proposal
Fusco, E., Vidoli, F., & Sahoo, B. K. (2017) · Omega · doi:10.1016/j.omega.2017.04.007 · Cited by 76
Profit efficiency and ownership of German hospitals
Herr, A., Schmitz, H., & Augurzky, B. (2010) · Health Economics · doi:10.1002/hec.1622 · Cited by 83
Abdulai, A. N., & Abdulai, A. (2016) · Environment and Development Economics · doi:10.1017/s1355770x16000309 · Cited by 78
“Wang and Schmidt (2002) demonstrate that in the stochastic frontier framework, a single-stage procedure is the appropriate approach to jointly estimate inefficiency and the determinants of inefficiency.”
BackgroundEducation production efficiency: Evidence from Brazilian universities
Zoghbi, A. C. P., Rocha, F., & Mattos, E. (2012) · Economic Modelling · doi:10.1016/j.econmod.2012.11.018 · Cited by 71
ESG investment and bank efficiency: Evidence from China
Cao, Q., Zhu, T. J., & Yu, W. (2024) · Energy Economics · doi:10.1016/j.eneco.2024.107516 · Cited by 69
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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