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One-Step and Two-Step Estimation of the Effects of Exogenous Variables on Technical Efficiency Levels
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## TL;DR
This paper compares one-step and two-step estimation for stochastic frontier models when exogenous variables affect technical inefficiency. In Monte Carlo simulations, the one-step estimator performs well, while the two-step procedure shows serious bias in both the first and second stages.
## Key findings
- In the base case simulation (200 replications, sample size \(N=200\)), the one-step estimator is close to unbiased: \(\hat\beta=-0.0010\), \(\hat\delta=1.0010\), and average estimated efficiency \(E(\hat e^{-u})=0.5141\) versus the true \(0.5165\).
- Under the same base case, the two-step estimator is badly biased: \(\hat\beta=-0.3064\), estimated efficiency \(E(\hat e^{-u})=0.4482\) versus true \(0.52\) (table reports \(0.5165\) in the design and about \(0.52\) in results discussion), and the second-step estimate of \(\delta\) is \(0.3539\) versus the true value \(1\).
- The paper reports that the two-step bias persists even when \(x\) and \(z\) are independent: for \(\rho=0\), the second-step estimate of \(\delta\) is still \(0.4069\), and the estimated efficiencies remain underdispersed and downward biased.
- Increasing sample size from \(N=200\) to \(N=500\) and \(N=1000\) lowers standard deviations but does not remove the two-step biases; for example, the two-step \(\hat\delta\) stays around \(0.35\)–\(0.36\) while the true value is \(1\).
## Methods
- The paper studies a stochastic frontier model in which inefficiency depends on exogenous variables \(z\), emphasizing a “scaling property” specification where the one-sided inefficiency term is a function of \(z\) times a basic one-sided error.
- It uses Monte Carlo simulations with 2,000 replications per design point. The base case sets \(\beta=0\), \(\delta=1\), \(\rho=0.5\), \(\gamma=1\), \(\sigma_v=1\), \(\mu=0\), and \(N=200\).
- One-step estimation is maximum likelihood under the correctly specified model; two-step estimation first sets \(\delta=0\) in the frontier MLE, then regresses \(\ln(\hat u)\) on \(z\) to estimate \(\delta\).
## Limitations
- The paper’s own simulation evidence is limited to one simple model with the scaling property and a specific truncated-normal basic distribution; results may not generalize beyond that setup.
- The text makes evident, though it does not frame it as a formal limitation, that some parameters (\(\mu\) and \(\gamma\)) are “hard” to estimate and produced outliers severe enough that the authors truncated the most extreme 0.3% of replications for those parameters.
- Not stated in the paper.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Our simulations strongly confirm the relevance of this observation, since the two-step estimates of the effect of z on efficiency levels are seriously biased downward in all cases.”
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In the paper
“We first note that the one-step estimates look pretty good. In particular, there is no evidence of significant bias.”
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In the paper
“The parameters of the truncated normal base distribution ( µ and γ ) have rather large variances (their standard deviations are definitely not small relative to the parameter values themselves)”
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In the paper
“For the one-step estimator, the true value of δ is not terribly important.”
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In the paper
“But this does not seem to cause any problems for the parameters of main interest ( β , δ and the individual ri).”
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In the paper
“Second, the estimates of the ri (the technical efficiencies) are biased downward (mean = 0.45, compared to the mean of the actual ri of 0.52).”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Changing β has no substantive effects on our results.”
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In the paper
“First, the estimate of β is biased downward (mean = -0.31, compared to the true value = 0).”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Third, the second-step estimator of δ is very significantly biased downward (mean = 0.35, compared to the true value = 1).”
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In the paper
“Changing δ makes very little difference to the properties of the estimates of β , δ or σ v.”
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In the paper
“The bias of the estimate of β and of the two-step estimate of δ increases, and the estimates of the ri are also more biased, and less correlated with the true ri, compared to Case 1.”
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In the paper
“As expected, the bias of the first-step estimate of β is effectively zero when δ = 0, and it grows with δ .”
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In the paper
“All of these biases are in the expected direction, and the size of the biases is definitely not small.”
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In the paper
“They are less strongly correlated with the true ri and they are underdispersed, compared to the estimates from the one-step model.”
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In the paper
“The individual ri are seriously biased for δ = 1 but not for the smaller values of δ .”
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In the paper
“The two-step estimator of δ is biased for both δ = 1 and δ = 0.5, but the bias disappears as δ → 0.”
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“An interesting curiosity is the bias in the two-step estimator of δ based on the estimates of ui from the one-step model (mean estimate = 0.92, compared to the true value of δ = 1). This is not just finite-sample bias (it persists with larger sample sizes).”
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In the paper
“The mean of the estimate (both one-step and two-step) of β changes by the same amount that β is changed, so that the bias, standard deviation and MSE of the estimates of β are unchanged.”
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In the paper
“For this set of parameters, average technical efficiency is E(e -u ) = 0.5165.”
Strength:plain· not yet calibratedn = 200Tests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
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In the paper
“As δ → 0, our ability to estimate µ (and γ , to a lesser extent) deteriorates seriously.”
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In the paper
“The estimates of the individual ri become slightly more biased, and noticeably less strongly correlated with the true ri, when δ is small.”
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In the paper
“We find serious bias at all stages of this procedure. The size of the bias is very substantial and should argue strongly against two-step procedures.”
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“For the two-step estimates, surprisingly, imposing µ = 0 makes things worse (even though µ = 0 is a correct restriction).”
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In the paper
“As γ increases, the bias of the estimate of β increases, but the bias of the second-step estimate of δ decreases quite noticeably.”
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“Finally, the second-step estimate of δ is seriously biased even when ρ = 0, though its bias does grow with ρ .”
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In the paper
“The standard deviations of the estimators fall as N increases, but nothing else changes much.”
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In the paper
“For example, the mean of the two-step estimator of δ is 0.35 for σ v = 1, 0.12 for σ v = 3, and 0.05 for σ v = 5.”
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In the paper
“| 9 | µ = 1 | ˆ β | -0.0013 | 0.0963 | 0.0093 | | -0.3914 | 0.1191 | 0.1674 | |”
- Also stated in: § 4. CONCLUDING REMARKS · plain
“| 17 | N=500 | ˆ β | -0.0005 | 0.0572 | 0.0033 | | -0.3062 | 0.0661 | 0.0981 | |”
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- Also stated in: § 4. CONCLUDING REMARKS · plain
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“| 9 | | E ( ˆ e - u ) | 0.3850 | 0.0294 | 0.0285 | 0.8387 | 0.3499 | 0.0197 | 0.0513 | 0.6891 |”
- Also stated in: § 4. CONCLUDING REMARKS · plain
“| 17 | N=500 | E ( ˆ e - u ) | 0.5160 | 0.0220 | 0.0338 | 0.8145 | 0.4470 | 0.0146 | 0.0615 | 0.6591 |”
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- Also stated in: § 4. CONCLUDING REMARKS · plain
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In the paper
“For the two-step estimators, we see more differences as µ changes, but they are still not really important or striking.”
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In the paper
“The true value of δ is one, so these are large biases indeed.”
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In the paper
“| 9 | E ( e - u ) | ˆ µ | 1.0591 | 0.3951 | 0.1595 | | -37.2499 | 1.1759 | 1464.4405 | |”
- Also stated in: § 4. CONCLUDING REMARKS · plain
“| 13 | E ( e - u ) | ˆ µ | 0.0429 | 0.4471 | 0.2016 | -37.2787 | 0.1688 | 1389.7299 | |”
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- Also stated in: § 4. CONCLUDING REMARKS · plain
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“| 9 | | ˆ δ | 1.0002 | 0.0850 | 0.0072 | | - | - | - | |”
- Also stated in: § 4. CONCLUDING REMARKS · plain
“| 17 | N=500 | ˆ δ | 0.9995 | 0.0724 | 0.0052 | | - | - | - | |”
Strength:plain· not yet calibratedn = 500Tests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
- Also stated in: § 4. CONCLUDING REMARKS · plain
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In the paper
“The results are quite consistent with these expectations.”
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In the paper
“| 9 | | ˆ δ -2S | 0.9317 | 0.0888 | 0.0125 | 0.8025 | 0.4986 | 0.0566 | 0.2546 | 0.3469 |”
- Also stated in: § 4. CONCLUDING REMARKS · plain
“| 17 | N=500 | ˆ δ -2S | 0.9134 | 0.0736 | 0.0129 | 0.7911 | 0.3520 | 0.0336 | 0.4210 | 0.2375 |”
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- Also stated in: § 4. CONCLUDING REMARKS · plain
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“| 17 | N=500 | ˆ γ | 1.0208 | 0.2393 | 0.0577 | | 53.4005 | 4.3493 | 2764.7151 | |”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one. The quote only partly states this claim.
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In the paper
“For the one-step estimators, the value of ρ is not important. The results are essentially identical across all these cases.”
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In the paper
“For larger γ , we estimate β a little worse, but we estimate δ and the individual ri a little better.”
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
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“As µ moves from plus one to negative one, the estimates of β improve a little, and the estimates of the individual ri perhaps improve a little.”
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“As is often the case in simulations that involve numerical maximization, there were some problems with outliers, especially in estimation of γ and µ (the parameters of the truncated normal distribution).”
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In the paper
“This bias in the second-step regression, due to underdispersion in the estimates of u that do not take into account the effect of z on u, does not seem to be systematically discussed in the literature.”
What the sources say
- Not checked — no full text of the sourcecites Kumbhakar and Lovell (2000, p. 119) source
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In the paper
“That is, if the estimates of u are constructed ignoring the effect of z on σ u 2 , they will show less dependence on z than they should, and we should expect the second-step regressions to give downward biased estimates of the effect of z on u.”
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“Some but not all of the models in the literature have this property.”
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In the paper
“The one-step MLE will be based on the correctly specified model, and will therefore be consistent and asymptotically efficient.”
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In the paper
“For example, Caudill and Ford (1993) provide evidence on the bias of the estimated technological parameters, but not on the efficiency levels themselves or their relationship to the explanatory variables z.”
What the sources say
- Not checked — no full text of the sourcecites Caudill and Ford (1993) source
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In the paper
“For example, if h(z, δ ) = exp(z' δ ) and δ > 0, then large z will on average be associated with large u and also with large σ u 2”
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In the paper
“Presumably this is so because when δ is large, z correlates with u and is useful to help estimate u.”
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In the paper
“Saying the same thing, if we ignore the dependence of σ u 2 on z, we will shrink the observations with large u too much, and the observations with small u too little, and our estimates of u will be underdispersed.”
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“It is widely appreciated that the severity of the bias in estimation of the technological parameters (coefficients of x) depends on the magnitude of the correlation between x and z.”
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In the paper
“A second and less widely recognized problem is that the first-step technical efficiency measures are likely to be seriously underdispersed, so that the results of the second-step regression are likely to be biased downward. This is true regardless of whether x and z are correlated.”
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In the paper
“The first of these is that it captures the idea, which we find intuitively reasonable and appealing, that the shape of the distribution of u is the same for all firms.”
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In the paper
“That is, the second-step estimator of δ may be able to tell us whether or not z affects u (whether or not δ = 0) 3 but it cannot accurately estimate the effect of z on u when this effect exists.”
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“Models based on some other simple distributions, such as exponential, would also have the scaling property.”
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In the paper
“Our summary statistics are averages and are very sensitive to extreme outliers.”
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In the paper
“Importantly, this is true whether or not x and z are correlated.”
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In the paper
“One important implication of this viewpoint, which is sometimes missed, is that the variables z that determine inefficiency must not be functions of y.”
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In the paper
“This truncation of the results made very little difference for the parameters other than µ and γ .”
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In the paper
“Increasing σ v increases the amount of statistical noise in the model, and should increase the bias and/ or variance for each of the estimated parameters.”
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In the paper
“However, as long as δ is non-zero, this argument may be less than compelling, because the distribution of u depends on z whereas the distribution of v does not, and this is another way to distinguish u from v.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- Rests on an assumption: The distribution of u depends on z whereas the distribution of v does not.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Increasing γ while holding σ v constant has the effect of increasing the size of inefficiency relative to noise, and should tend to make it easier to estimate technical inefficiency precisely.”
Strength:hedged· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“For example, when µ is large and positive, the degree of truncation is very small, and the shape of the distribution (if it can be separated from that of v) contains a lot of information about the parameters µ and σ 2 .”
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In this case the first-step estimator of β is unbiased, and we then estimate the ri under the correct assumption that they do not depend on the zi, so we ought to do well in this case.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“As µ → ∞ , the distribution of u becomes normal and presumably becomes confounded with the normal distribution of v (statistical noise).”
Strength:hedged· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“When µ is large (in absolute value) but negative, on the other hand, we observe only the extreme right tail of the distribution, whose shape may not be very informative about µ and σ 2 , and the estimation problem may be harder just on that basis.”
Strength:hedged· not yet calibratedCausal:conditional· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“All in all, the two-step estimator performs well when the second step is not needed ( δ = 0) but is otherwise unreliable.”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“These biases are substantial enough that we would recommend against using two-step procedures in any circumstances that we can envision.”
Load-bearingStrength:hedged· not yet calibratedTests✓ Asserted: passes✗ Checkable: fails✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim. It may be too vague to check against evidence.
The authors state this as an assumption.
In the theoretical exploration of underdispersion, x and z are independent.
“To explore this point more precisely, suppose that x and z are independent, so that the first-step regression is unbiased.”
The paper relies on this without support.
The distribution of u depends on z whereas the distribution of v does not.
“because the distribution of u depends on z whereas the distribution of v does not”
The paper relies on this without support.
The biases identified above are expected to persist asymptotically.
“the biases we have identified above are expected to persist asymptotically”
The paper relies on this without support.
The one-step estimators are based on a correctly specified model and are asymptotically optimal.
“The one-step estimators are based on a correctly specified model and are asymptotically optimal.”
Citations
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Pitfalls of Normal-Gamma Stochastic Frontier Models
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Tannuri-Pianto, M., de Sousa, M. D. C. S., & Arcoverde, F. D. (2009) · Estudos Econômicos (São Paulo) · doi:10.1590/s0101-41612009000100008 · Cited by 10
Repkine, A. (2013) · South African Journal of Economics · doi:10.1111/saje.12019 · Cited by 10
In competency Aspects of Microfinance Industry: Via SFA approach
Riaz, M. (2015) · Journal of Economics and Behavioral Studies · doi:10.22610/jebs.v7i1(j).558 · Cited by 10
“As Wang and Schmidt (2002) showed that two step method renders biased coefficients, because the method undergo 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…”
Two dimensional efficiency measurements in vocational education
Fieger, P., Villano, R. A., Rice, J., & Cooksey, R. (2017) · International Journal of Productivity and Performance Management · doi:10.1108/ijppm-09-2015-0139 · Cited by 10
Stochastic Frontier Analysis Using SFAMB for Ox
Holtkamp, J., & Brümmer, B. (2017) · Journal of Statistical Software · doi:10.18637/jss.v081.i06 · Cited by 12
A Top-Down Economic Efficiency Analysis of U.S. HouseholdEnergy Consumption
Burnett, J. W., & Madariaga, J. (2018) · The Energy Journal · doi:10.5547/01956574.39.4.jbur · Cited by 10
“However, the estimation of stochastic frontier models in the presence of heteroskedasticity, if not accounted for, can yield biased parameter estimates (Caudill et al., 1995; Wang and Schmidt, 2002).”
BackgroundFerrera, J. M. C., Polo, C., & Salinas‐Jiménez, J. (2020) · Journal of Happiness Studies · doi:10.1007/s10902-020-00255-3 · Cited by 10
Tiffin, A. (2006) · IMF Working Paper · doi:10.5089/9781451864274.001 · Cited by 10
Thi, N. A. V., & Vencappa, D. V. (2007) · SSRN Electronic Journal · doi:10.2139/ssrn.968348 · Cited by 10
“(1991); Reichsneider and Stevenson (1991); Yuengert (1993); Simar, Lovell and Vanden Eeckaut (1994); Huang and Liu (1994); Caudill, Ford and Gropper (1995) and Battese and Coelli (1995), Wang and Schmidt (2002) and Wang (2003) amongst others.”
Ukraine: The Cost of Weak InstitutionsPreprint
Tiffin, A. (2006) · SSRN Electronic Journal · doi:10.2139/ssrn.920259 · Cited by 10
Too Small to Be Beautiful?: The Farm Size and Productivity Relationship in Bangladesh
Ahmed, M., & Gautam, M. (2018) · World Bank, Washington, DC eBooks · doi:10.1596/1813-9450-8387 · Cited by 10
Wang, T., Tian, J., & Yao, F. (2020) · Empirical Economics · doi:10.1007/s00181-020-01889-1 · Cited by 9
Profit efficiency in the provision of ecosystem services in the Cross Timbers forests
Susaeta, A. I., Sancewich, B., Klizentyte, K., et al. (2023) · Land Use Policy · doi:10.1016/j.landusepol.2023.106978 · Cited by 9
Parameswaran, M. (2004) · Indian Economic Review · Cited by 9
Minority owned banks and efficiency revisited
Kashian, R. D., McGregory, R., & Drago, R. (2017) · Journal of Productivity Analysis · doi:10.1007/s11123-017-0510-x · Cited by 9
Hampf, B. (2019) · Journal of Industrial Ecology · doi:10.1111/jiec.12832 · Cited by 9
“Note that in contrast to the original publication on the scaling property by Wang and Schmidt (2002), we do not account for the influence of exogenous variables (so-called z-variables).”
MethodResultEvaluation of Environmental Efficiency of Edible Canna Production in Vietnam
Hiền, V. T. T., Peng, K.-C., & Chung, R. H. (2019) · Agriculture · doi:10.3390/agriculture9110242 · Cited by 9
“Biased estimations of applying a two-step procedure in measuring inefficiency and assessing its determinants were also pointed out by Wang and Schmidt [31] and Kumbhakar et al.”
MethodRethinking Food Production: Nexus of Mobile Phones and Production Cost Minimization
Mwalupaso, G. E., Tian, X., & Geng, X. (2020) · International Journal of Environmental Research and Public Health · doi:10.3390/ijerph17072457 · Cited by 9
“Therefore, we employ the CD cost model and apply the approach by Wang and Schmidt [28] of estimating the parameters in both the stochastic frontier and the inefficiency model simultaneously using the maximum likelihood estimation (MLE) method.”
MethodMind the Gap: Measuring Academic Underachievement Using Stochastic Frontier Analysis
Mazrekaj, D., Witte, K. D., & Triebs, T. (2022) · Exceptional Children · doi:10.1177/00144029211073524 · Cited by 9
Uddin, M. M. M., Sharif, T., & Pillai, R. (2021) · Applied Economics Quarterly · doi:10.3790/aeq.67.1.71 · Cited by 10
Efecan, V., & Temiz, İ. (2023) · Utilities Policy · doi:10.1016/j.jup.2023.101494 · Cited by 10
Goh, K. H., & See, K. F. (2023) · Utilities Policy · doi:10.1016/j.jup.2022.101483 · Cited by 9
Machinery import, R&D spillover, and energy efficiency
Liu, F., Sim, J.-Y., Edziah, B. K., et al. (2023) · Journal of Environmental Planning and Management · doi:10.1080/09640568.2023.2166819 · Cited by 9
“Possibly this is because of the parametric nature of the SFA and the observations by Wang and Schmidt (2002) that the two-stage technique provides highly biased results.”
MethodThe Impact of Trade Openness on Technical Efficiency in U.S. Agriculture
Miljković, D., Shaik, S., Miljkovic, D., et al. (2010) · AgEcon Search (University of Minnesota, USA) · doi:10.22004/ag.econ.56334 · Cited by 9
Uddın, M. M., Brümmer, B., & Peters, K. J. (2014) · China Agricultural Economic Review · doi:10.1108/caer-06-2013-0088 · Cited by 9
“Wang and Schmidt (2002) suggest methods which incorporate these potentially explanatory variables in the variance parameter su.”
BackgroundBanking efficiency in emerging market economies
Matthews, K. (2010) · ORCA Online Research @Cardiff (Cardiff University) · Cited by 9
Stochastic Frontier Models in Sports Economics
Lee, Y. H. (2014) · International Journal of Sport Finance · doi:10.1177/155862351400900406 · Cited by 10
Pure and compensated efficiency of Swedish dairy farms
Manevska‐Tasevska, G., Rabinowicz, E., & Surry, Y. (2016) · Agricultural and Food Science · doi:10.23986/afsci.53182 · Cited by 9
Common Methodological Choices in Nonparametric and Parametric Analyses of Firms’ Performance
Orea, L., & Zofio, J. L. (2019) · Springer eBooks · doi:10.1007/978-3-030-23727-1_12 · Cited by 10
Comparison of dimension reduction methods for DEA under big data via Monte Carlo simulation
Chen, Z., & Han, S. (2021) · Journal of Management Science and Engineering · doi:10.1016/j.jmse.2021.09.008 · Cited by 9
Nonparametric estimation of production risk and risk preference functions
Kumbhakar, S. C., & Tsionas, E. G. (2009) · Advances in econometrics · doi:10.1108/s0731-9053(2009)0000025010 · Cited by 9
“This is basically a stochastic frontier model in which determinants of technical inefficiency are modeled through the scaling function, ( )ZXg , (see Wang and Schmidt (2002)).”
BackgroundJonghe, O. D., Disli, M., & Schoors, K. J. L. (2011) · SSRN Electronic Journal · doi:10.2139/ssrn.1966510 · Cited by 9
Valero-Carreras, D., Moragues, R., Aparicio, J., et al. (2023) · Computers & Operations Research · doi:10.1016/j.cor.2023.106485 · Cited by 8
Persistent and transient inefficiencies in the U.S. airline industry
Karanki, F., & Lien, G. (2024) · Transport Policy · doi:10.1016/j.tranpol.2024.10.018 · Cited by 8
Karanki, F., & Yu, C. (2025) · Transport Policy · doi:10.1016/j.tranpol.2025.01.036 · Cited by 8
Understanding the capacity of airport runway systems
Chen, K., Anupriya, Bansal, P., Anderson, R. J., Findlay, N. S., & Graham, D. J. (2025) · Transportation Research Part C Emerging Technologies · doi:10.1016/j.trc.2025.104998 · Cited by 10
Kehinde, A. D., Oyenpemi, L. O., & Oladimeji, O. (2025) · Social Sciences & Humanities Open · doi:10.1016/j.ssaho.2025.102068 · Cited by 8
Estimation of the impact of CAP subsidies as environmental variables on Romanian farms
Galluzzo, N. (2022) · ECONOMIA AGRO-ALIMENTARE · doi:10.3280/ecag2021oa12772 · Cited by 8
The role of hired labour on technical efficiency in an expanding dairy sector: The case of Ireland
García, L. F., Läpple, D., Dillon, E., & Thorne, F. S. (2024) · Australian Journal of Agricultural and Resource Economics · doi:10.1111/1467-8489.12553 · Cited by 8
Galli, F. (2024) · Spatial Economic Analysis · doi:10.1080/17421772.2024.2306953 · Cited by 8
Are Smart Tourism Destinations more productive efficient? The Spanish case
Baños-Pino, J. F., Sustacha, I., Boto‐García, D., & Tuero, E. A. D. V. (2025) · Tourism Economics · doi:10.1177/13548166241313413 · Cited by 8
Analyzing judicial courts' performance: inefficiency vs congestion
Espasa, M., & Esteller, M. (2015) · RECERCAT (Consorci de Serveis Universitaris de Catalunya) · Cited by 8
School externalities and scholastic performance: an efficiency analysis
Deutsch, J., Dumas, A., & Silber, J. (2019) · International Journal of Manpower · doi:10.1108/ijm-09-2017-0220 · Cited by 8
Calamunci, F. M. (2021) · Small Business Economics · doi:10.1007/s11187-020-00439-4 · Cited by 8
Adom, P. K. (2021) · Environmental Science and Pollution Research · doi:10.1007/s11356-021-14902-6 · Cited by 8
Investment Inefficiency and Corporate Social Responsibility
Engida, T. G., Parmeter, C. F., Rao, X., & Lansink, A. O. (2022) · Journal of Productivity Analysis · doi:10.1007/s11123-022-00641-4 · Cited by 8
Wen, F., Fang, X., Shan, A., et al. (2022) · The International Journal of Health Planning and Management · doi:10.1002/hpm.3534 · Cited by 8
Effects of free trade on export efficiency of COMESA member-states
Masunda, S., & Mhonyera, G. (2024) · Journal of Shipping and Trade · doi:10.1186/s41072-024-00164-1 · Cited by 8
“The two stage procedure is criticised for the lack of consistency in the assumption relating to the distribution of inefficiencies (Paul and Shankar 2018; Wang and Schimdt 2002).”
BackgroundGrain and sunflower on Russian farms in 2001: how efficient is crop production?
Grazhdaninova, M., & Brock, G. J. (2004) · Post-Communist Economies · doi:10.1080/1463137042000257537 · Cited by 8
Papaioannou, S., & Dimelis, S. P. (2017) · Information Economics and Policy · doi:10.1016/j.infoecopol.2017.05.001 · Cited by 10
Seismic soil response of scaled geotechnical test model on small shaking table
Çelebi, E., Göktepe, F., & Omid, A. J. (2019) · Arabian Journal of Geosciences · doi:10.1007/s12517-018-4197-8 · Cited by 8
Abegaz, M., & Lahiri, S. (2020) · Journal of International Development · doi:10.1002/jid.3517 · Cited by 8
Cost-Containment Policies and Hospital Efficiency: Evidence from a Panel of Italian HospitalsPreprint
Atella, V., Belotti, F., Daidone, S., et al. (2012) · SSRN Electronic Journal · doi:10.2139/ssrn.2038398 · Cited by 8
Efficiency and Contestability in the Colombian Banking System
Daude, C., & Pascal, J. (2015) · OECD Economics Department working papers · doi:10.1787/5js30twjgm6l-en · Cited by 8
“However, this approach suffers from potentially substantial biases (Wang and Schmidt, 2002).”
BackgroundInformation Technology and Value Creation in the Public Sector Organizations.
Pang, M. (2011) · Deep Blue (University of Michigan) · Cited by 9
Non-human and human service efficiency of long-term care facilities in China
Zhang, L., Han, Y., & Fang, Y. (2023) · Frontiers in Public Health · doi:10.3389/fpubh.2023.1066190 · Cited by 7
“In addition, the efficiency of LTCFs is affected by not only the five input factor variables but also other exogenous variables (72).”
BackgroundBank cost efficiency and credit market structure under a volatile exchange rate
Mamonov, M. E., Parmeter, C. F., & Prokhorov, A. B. (2024) · Journal of Banking & Finance · doi:10.1016/j.jbankfin.2024.107285 · Cited by 7
“To avoid estimation bias caused by the non-zero mean of u it (Wang and Schmidt, 2002), we simultaneously estimate Equations (2) and (3).”
BackgroundTesafa, F., Mulugeta, M., & Tsehay, S. (2024) · Heliyon · doi:10.1016/j.heliyon.2024.e41273 · Cited by 10
“Wang and Schmidst [69] also claimed that exogenous determinants of inefficiency (z i ) might affect its input choices (x i ), and hence efficiency might be dependent on explanatory variables.”
Nyirongo, B. B., & Khataza, R. (2025) · Discover Agriculture · doi:10.1007/s44279-025-00255-y · Cited by 7
Profit Efficiency: Insight into airline business models and strategic choices
Karanki, F., & Schaufele, R. (2025) · Journal of Air Transport Management · doi:10.1016/j.jairtraman.2025.102863 · Cited by 7
A stochastic semi‐non‐parametric analysis of regional efficiency in the European Union
Polo, C., Hernández, J. R., & Ricci‐Risquete, A. (2020) · Regional Science Policy & Practice · doi:10.1111/rsp3.12305 · Cited by 7
A parametric analysis of eco-efficiency and its determinants: Evidence from Norwegian dairy farms
Alem, H. (2023) · Agricultural Economics (Zemědělská ekonomika) · doi:10.17221/149/2023-agricecon · Cited by 7
Yane, S., & Berg, S. V. (2012) · Applied Economics · doi:10.1080/00036846.2012.663475 · Cited by 7
“…(1999) and Hadri et al. (2003) develop a half-normal doubly heteroscedastic frontier model; Greene (2004, 2005a, b) applies a truncated-normal heterogeneous mean model as well as true fixed or random model; and Wang and Schmidt (2002, 2009) and Alvarez et al. (2006) propose scaling-function models.”
Gronberg, T. J., Jansen, D. W., & Taylor, L. L. (2016) · Southern Economic Journal · doi:10.1002/soej.12173 · Cited by 7
Productive performance of organic crop farms in Finland 2010–2017
Kuosmanen, N., Yli‐Heikkilä, M., Väre, M., & Kuosmanen, T. (2021) · Organic Agriculture · doi:10.1007/s13165-020-00343-x · Cited by 7
Tax revenue inefficiency and political risk factors: The Hen or The egg?
Mebratu, A. A. (2023) · Cogent Business & Management · doi:10.1080/23311975.2023.2167546 · Cited by 7
“…Coelli (1995), who introduced a panel data model with time-varying inefficiency that reflects observable Page 8 of 18 heterogeneity, the parameters of the stochastic frontier and the inefficiency model are estimated simultaneously to avoid bias (Wang & Schmidt, 2002) using the maximum probability.”
MethodModeling issues in applied efficiency analysis: agriculture
Karagiannis, G. (2014) · Economics and Business Letters · doi:10.17811/ebl.3.1.2014.12-18 · Cited by 7
“…theoretically appropriate approach, the size variable included in the inefficiency effect function must not be measured in terms of the dependent variable (i.e., the observed output); it is better to measure size in terms of inputs, such as land or herd size, not outputs (Wang and Schmidt, 2002).”
BackgroundOwnership, Scale Economies and Efficiency in the Italian Water Sector
Cosmo, V. D. (2012) · Journal of Industry Competition and Trade · doi:10.1007/s10842-012-0131-z · Cited by 7
Kibirige, D. (2016) · Agriculture Forestry and Fisheries · doi:10.11648/j.aff.20160505.18 · Cited by 7
Estimation of the production profile and metafrontier technology gap: a quantile approach
Lai, H., Huang, C. J., & Fu, T. (2018) · Empirical Economics · doi:10.1007/s00181-018-1589-2 · Cited by 7
Family Ties, Social Capital and Small Businesses’ Efficiency. Evidence from the Italian Food Sector
Agostino, M., & Ruberto, S. (2022) · Journal of Family and Economic Issues · doi:10.1007/s10834-022-09882-9 · Cited by 7
“…of technical inefficiency, we adopt one-step SFA techniques (Battese & Coelli, 1995 and Greene, 2005), which allow a stochastic frontier function and a technical inefficiency model to be simultaneously estimated, avoiding the biases associated with the two-stage SFA approach (Schmidt & Wan, 2002).”
MethodSaastamoinen, A., & Kuosmanen, T. (2014) · Applied Economics · doi:10.1080/00036846.2014.914149 · Cited by 7
“Following Méon and Weill (2005), the macroeconomic data for the output, the capital stock and the labour input in the cross-section of 1990 are based on Easterly and Levine (2001). The data for these variables are obtained from the World Bank’s Global Development Network Growth database (World Bank, 2001).”
BackgroundMethodResultInfluentialPrevalence of sorghum anthracnose in different agroecologies of Uganda
Sserumaga, J. P., Biruma, M., Akwero, A., et al. (2013) · Uganda Journal of Agricultural Sciences · Cited by 7
Habiyaremye, N., Tabe‐Ojong, M. P. J., Ochieng, J., et al. (2019) · Scientific African · doi:10.1016/j.sciaf.2019.e00190 · Cited by 7
Profit Efficiency in Timber Supply: Marginalization and Other Relevant Influences
Baardsen, S., Lien, G., & Størdal, S. (2009) · Forest Science · doi:10.1093/forestscience/55.6.524 · Cited by 7
Efficiency and Foreign Ownership in Banking: An International ComparisonPreprint
Kasman, A., Kasman, S., & Carvallo, O. (2005) · RePEc: Research Papers in Economics · Cited by 7
Spillover and backward linkage effects of FDI: empirical evidence for the UK
HARRIS, R. I. D. (2009) · London School of Economics and Political Science Research Online (London School of Economics and Political Science) · Cited by 7
Paul, S., & Shankar, S. (2017) · SSRN Electronic Journal · doi:10.2139/ssrn.2913313 · Cited by 7
“Even if they are uncorrelated, the Monte Carlo simulations undertaken in Wang and Schmidt (2002) show that the second stage regression parameter estimates are likely to be biased downwards.”
BackgroundMarket Power and Efficiency in Banking: The Case of USA and CanadaPreprint
Din, S. U., Tripe, D., & Kabir, M. H. (2018) · SSRN Electronic Journal · doi:10.2139/ssrn.3118703 · Cited by 7
The comparison of stochastic frontier analysis with panel data models
Zhang, M. (2012) · Loughborough University Institutional Repository (Loughborough University) · Cited by 7
Cost efficiency of primary health care facilities in Ghana: stochastic frontier analysis
Arhin, K., Oteng-Abayie, E. F., & Novignon, J. (2023) · Discover Health Systems · doi:10.1007/s44250-023-00044-8 · Cited by 6
“5 is well suited [45].”
Do water cuts affect productivity? Case study of African manufacturing firms
Moyo, B. (2011) · Water SA · doi:10.4314/wsa.v37i3.68486 · Cited by 6
“Wang and Schmidt (2002) show that in the case of 2-step estimators of technical efficiency using a stochastic frontier production approach, simulations indicate that the bias due to the omitted variable problem is substantial.”
InfluentialOfori-Appiah, Y., Onumah, E. E., & Asem, F. E. (2021) · African Journal of Science Technology Innovation and Development · doi:10.1080/20421338.2020.1857544 · Cited by 6
The Unmet Medical Demand among China’s Urban Residents
Sheng, P., Yang, T., & Zhang, T. (2021) · International Journal of Environmental Research and Public Health · doi:10.3390/ijerph182111708 · Cited by 5
“However, the variables that affected the inefficient term shifted and scaled the frontier function and the distribution of the inefficient term; thus, the twostep approach always resulted in biased estimates [37,38].”
BackgroundSemi-parametric modelling of inefficiencies in stochastic frontier analysis
Forchini, G., & Theler, R. (2022) · Journal of Productivity Analysis · doi:10.1007/s11123-022-00656-x · Cited by 6
Irrigated Onion Production Efficiency in Humbo District, Southern Ethiopia
Tafesse, A., Dema, S., & Belay, A. M. (2023) · Cogent Economics & Finance · doi:10.1080/23322039.2023.2207928 · Cited by 6
“The two-step approach is strongly criticized by Wang and Schmidt (2002) for technical efficiency analysis in contrast to the one-step approach.”
MethodResultDoes Forced Solidarity Hamper Entrepreneurial Activity? Evidence from seven West-African Countries
Grimm, M., Gubert, F., Koriko, O., Lay, J., & Nordman, C. J. (2010) · HAL · Cited by 6
Zhang, J., Qu, B., Xu, Z., et al. (2012) · Asia-Pacific Journal of Accounting & Economics · doi:10.1080/16081625.2012.667380 · Cited by 6
“Following previous studies (e.g. Wang and Schmidt 2002, Yao et al. 2007, Jiang et al. 2009), this study adopts the estimation procedure proposed by Battese and Coelli (1995).”
MethodANALYSIS OF ECONOMIC EFFICIENCY IN SMALLHOLDER MAIZE PRODUCTION IN NORTHWESTERN KENYA
Mutoko, M. C. (2008) · doi:10.13140/rg.2.2.24926.77126 · Cited by 6
Fani, D. C. R., Ukpe, H., Ngo, N. V., et al. (2020) · GeoJournal · doi:10.1007/s10708-020-10163-x · Cited by 6
DETERMINANTS OF BANK COST EFFICIENCY: EMPIRICAL EVIDENCE FROM BANGLADESH
Sultana, I., & Rahman, M. M. (2020) · International Journal of Banking and Finance · doi:10.32890/ijbf2020.15.1.9931 · Cited by 6
The evolution of financial constraints
Christodoulou, D., Ho, S., & Prokhorov, A. B. (2021) · European Financial Management · doi:10.1111/eufm.12310 · Cited by 7
Morgan, C. M., Cecilia, N. R., Benhin, J., et al. (2015) · Journal of Development and Agricultural Economics · doi:10.5897/jdae2015.0633 · Cited by 6
“We applied a one-step maximum likelihood estimation procedure (Wang and Schmidt, 2002) to estimate each of the above equations simultaneously with those determinants of technical and allocative efficiency in maize production, specified in Equation 3. iiii iiiiii AEZMACEXTSFM…”
MethodCost (In)Efficiency and Institutional Pressures in Nursing Home Chains
Martin, C., & Jérôme, T. (2016) · European Accounting Review · doi:10.1080/09638180.2016.1169937 · Cited by 6
Technical Efficiency and Productivity Growth of Polish Crop Farms
Marzec, J., Pisulewski, A., & Prędki, A. (2019) · Gospodarka Narodowa · doi:10.33119/gn/108607 · Cited by 6
Wollni, M., & Wollni, M. (2007) · RePEc: Research Papers in Economics · doi:10.22004/ag.econ.9956 · Cited by 6
“Wang and Schmidt (2002) prove that the two-step approach leads to inconsistent results.”
ResultBargaining in Technology Markets: An empirical study of biotechnology alliancesPreprint
Kinukawa, S., & Motohashi, K. (2010) · RePEc: Research Papers in Economics · Cited by 6
Joint Variable Selection for Data Envelopement Analysis via Group SparsityPreprint
Qin, Z. T., & Song, I. (2014) · arXiv (Cornell University) · doi:10.48550/arxiv.1402.3740 · Cited by 6
“For those experiments concerned with correlated input variables, we adopt the following equation from Wang and Schmidt (2002) to establish the desired covariance structure of inputs. xi = ρi,jxj +w √ 1− ρ2i,j, i= 2,3,4, j =1,2,3, i 6= j. (30) Here, ρi,j is the correlation between xi and xj, and w…”
MethodThe influence of risk-taking on bank efficiency : evidence from Colombia
Sarmiento, M., & Galán, J. (2015) · doi:10.32468/be.894 · Cited by 6
Energy Efficiency in OECD and non-OECD Countries: Estimates and ConvergencePreprint
Liddle, B., & Sadorsky, P. (2020) · SSRN Electronic Journal · doi:10.2139/ssrn.3575110 · Cited by 6
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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