One-Step and Two-Step Estimation of the Effects of Exogenous Variables on Technical Efficiency Levels

It may state two separate claims as one.

In Case 1A, the bias of the estimate of β and of the two-step estimate of δ increases, and the estimates of ri are more biased and less correlated with true ri compared to Case 1.

“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.”

It may state two separate claims as one.

For the two-step estimator, the bias of the first-step estimate of β is effectively zero when δ = 0, and it grows with δ.

“As expected, the bias of the first-step estimate of β is effectively zero when δ = 0, and it grows with δ .”

It may state two separate claims as one.

All of these biases are in the expected direction, and the size of the biases is definitely not small.

“All of these biases are in the expected direction, and the size of the biases is definitely not small.”

It may state two separate claims as one.

In the base case, the two-step estimates of ri are less strongly correlated with true ri and are underdispersed compared to one-step estimates.

“They are less strongly correlated with the true ri and they are underdispersed, compared to the estimates from the one-step model.”

It may state two separate claims as one.

For the two-step estimator, the individual ri are seriously biased for δ = 1 but not for smaller values of δ.

“The individual ri are seriously biased for δ = 1 but not for the smaller values of δ .”

It may state two separate claims as one.

The two-step estimator of δ is biased for both δ = 1 and δ = 0.5, but the bias disappears as δ approaches 0.

“The two-step estimator of δ is biased for both δ = 1 and δ = 0.5, but the bias disappears as δ → 0.”

It may state two separate claims as one.

The two-step estimator of δ based on estimates of ui from the one-step model has a mean estimate of 0.92 compared to true value 1, and this is not just finite-sample bias (it persists with larger sample sizes).

“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).”

It may state two separate claims as one.

The mean of the estimate (both one-step and two-step) of β changes by the same amount that β is changed, so the bias, standard deviation and MSE of the estimates of β are unchanged.

“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.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

For the base case parameters, average technical efficiency is E(e-u)=0.5165.

“For this set of parameters, average technical efficiency is E(e -u ) = 0.5165.”

It may state two separate claims as one.

As δ approaches 0, the one-step estimator's ability to estimate μ, and to a lesser extent γ, deteriorates seriously.

“As δ → 0, our ability to estimate µ (and γ , to a lesser extent) deteriorates seriously.”

It may state two separate claims as one.

For the one-step estimator, the estimates of individual ri become slightly more biased and noticeably less strongly correlated with true ri when δ is small.

“The estimates of the individual ri become slightly more biased, and noticeably less strongly correlated with the true ri, when δ is small.”

It may state two separate claims as one.

The two-step procedure exhibits serious bias at all stages, and the size of the bias is very substantial.

“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.”

It may state two separate claims as one.

In Case 1A, imposing μ=0 makes the two-step estimates worse, even though μ=0 is a correct restriction.

“For the two-step estimates, surprisingly, imposing µ = 0 makes things worse (even though µ = 0 is a correct restriction).”

It may state two separate claims as one.

For the two-step estimators, as γ increases, the bias of the estimate of β increases, but the bias of the second-step estimate of δ decreases quite noticeably.

“As γ increases, the bias of the estimate of β increases, but the bias of the second-step estimate of δ decreases quite noticeably.”

It may state two separate claims as one.

The second-step estimate of δ is seriously biased even when ρ = 0, though its bias does grow with ρ.

“Finally, the second-step estimate of δ is seriously biased even when ρ = 0, though its bias does grow with ρ .”

It may state two separate claims as one.

The standard deviations of the estimators fall as N increases, but nothing else changes much.

“The standard deviations of the estimators fall as N increases, but nothing else changes much.”

It may state two separate claims as one.

The mean of the two-step estimator of δ is 0.35 for σv = 1, 0.12 for σv = 3, and 0.05 for σv = 5.

“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.”

It may state two separate claims as one. The quote only partly states this claim.

In the simulation, the two-step estimator of β is biased (mean around -0.3) while the one-step estimator of β is close to the true value (0).

“| 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 | |”

The quote only partly states this claim.

In the simulation, the one-step estimator of E(e-u) has a lower mean squared error than the two-step estimator across the parameter changes examined.

“| 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 |”

It may state two separate claims as one.

For the two-step estimators, there are more differences as μ changes, but they are still not really important or striking.

“For the two-step estimators, we see more differences as µ changes, but they are still not really important or striking.”

It may state two separate claims as one.

The true value of δ is one, so the two-step estimator of δ has large biases for σv = 1, 3, and 5.

“The true value of δ is one, so these are large biases indeed.”

It may state two separate claims as one. The quote only partly states this claim.

In the simulation, the two-step estimator of μ is severely biased (mean around -37) while the one-step estimator of μ is close to the true value.

“| 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 | |”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

In the simulation, the two-step estimator does not provide an estimate of δ (reported as '-').

“| 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 | | - | - | - | |”

It may state two separate claims as one.

The results are quite consistent with the expectations that standard deviations fall with N and biases persist.

“The results are quite consistent with these expectations.”

Probably not a substantial claim. The authors may not put it forward as true. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.

In the simulation, the one-step estimator of δ-2S has a higher reported correlation (corr*) than the two-step estimator.

“| 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 |”

It may state two separate claims as one. The quote only partly states this claim.

In the simulation, the two-step estimator of γ is severely biased (mean around 53 or higher) while the one-step estimator of γ is close to the true value.

“| 17 | N=500 | ˆ γ | 1.0208 | 0.2393 | 0.0577 | | 53.4005 | 4.3493 | 2764.7151 | |”

It may state two separate claims as one.

For the one-step estimators, the value of ρ is not important; results are essentially identical across all cases.

“For the one-step estimators, the value of ρ is not important. The results are essentially identical across all these cases.”

It may state two separate claims as one.

For the one-step estimators, for larger γ, β is estimated a little worse, but δ and individual ri are estimated a little better.

“For larger γ , we estimate β a little worse, but we estimate δ and the individual ri a little better.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

For the two-step estimators, 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.

“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.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

There were some problems with outliers in the simulations, especially in estimation of γ and μ.

“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).”

Probably not a substantial claim.

The bias in the second-step regression due to underdispersion in 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.

“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.”

• cites Kumbhakar and Lovell (2000, p. 119) sourceNot checked — no full text of the source

It may state two separate claims as one.

If estimates of u are constructed ignoring the effect of z on σu2, they will show less dependence on z than they should, and second-step regressions will give downward biased estimates of the effect of z on u.

“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.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.

Some but not all of the models in the literature have the scaling property.

“Some but not all of the models in the literature have this property.”

It may state two separate claims as one.

The one-step MLE is based on the correctly specified model and is therefore consistent and asymptotically efficient.

“The one-step MLE will be based on the correctly specified model, and will therefore be consistent and asymptotically efficient.”

It may state two separate claims as one.

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.

“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.”

• cites Caudill and Ford (1993) sourceNot checked — no full text of the source

It may state two separate claims as one.

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 σu2.

“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”

Probably not a substantial claim. It may state two separate claims as one. The quote only partly states this claim.

When δ is large, z correlates with u and is useful to help estimate u, which presumably explains why individual ri estimates are less biased and more correlated when δ is large.

“Presumably this is so because when δ is large, z correlates with u and is useful to help estimate u.”

It may state two separate claims as one.

If the dependence of σu2 on z is ignored, observations with large u will be shrunk too much and observations with small u too little, so estimates of u will be underdispersed.

“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.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.

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.

“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.”

It may state two separate claims as one.

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, regardless of whether x and z are correlated.

“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.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

The scaling property captures the idea that the shape of the distribution of u is the same for all firms.

“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.”

It may state two separate claims as one.

The second-step estimator of δ may be able to tell whether z affects u (whether δ = 0), but it cannot accurately estimate the effect of z on u when this effect exists.

“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.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Models based on some other simple distributions, such as exponential, would also have the scaling property.

“Models based on some other simple distributions, such as exponential, would also have the scaling property.”

Probably not a substantial claim. It may state two separate claims as one.

The summary statistics are averages and are very sensitive to extreme outliers.

“Our summary statistics are averages and are very sensitive to extreme outliers.”

It may state two separate claims as one.

The underdispersion of estimated firm-level efficiencies and the resulting downward bias of the second-step regression occur whether or not x and z are correlated.

“Importantly, this is true whether or not x and z are correlated.”

Probably not a substantial claim.

The variables z that determine inefficiency must not be functions of y.

“One important implication of this viewpoint, which is sometimes missed, is that the variables z that determine inefficiency must not be functions of y.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Truncating the results by discarding the 0.3% most extreme upper and lower tail estimates of μ and γ made very little difference for parameters other than μ and γ.

“This truncation of the results made very little difference for the parameters other than µ and γ .”

It may state two separate claims as one.

Increasing σv increases the amount of statistical noise in the model and should increase the bias and/or variance for each estimated parameter.

“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.”

It may state two separate claims as one.

As long as δ is non-zero, the argument that estimation is harder when μ is positive and large may be less compelling because the distribution of u depends on z whereas the distribution of v does not, providing another way to distinguish u from v.

“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.”

• AssumptionRests on an assumption: The distribution of u depends on z whereas the distribution of v does not.

It may state two separate claims as one.

Increasing γ while holding σv constant increases the size of inefficiency relative to noise and should tend to make it easier to estimate technical inefficiency precisely.

“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.”

It may state two separate claims as one.

When μ is large and positive, the degree of truncation is very small, and the shape of the distribution (if separable from v) contains a lot of information about μ and σ2.

“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 .”

It may state two separate claims as one.

When δ = 0, the first-step estimator of β is unbiased and the ri are estimated under the correct assumption that they do not depend on zi, so the two-step procedure ought to do well.

“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.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

As μ approaches infinity, the distribution of u becomes normal and presumably becomes confounded with the normal distribution of v (statistical noise).

“As µ → ∞ , the distribution of u becomes normal and presumably becomes confounded with the normal distribution of v (statistical noise).”

It may state two separate claims as one.

When μ is large in absolute value but negative, only the extreme right tail of the distribution is observed, whose shape may not be very informative about μ and σ2, and the estimation problem may be harder on that basis.

“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.”

It may state two separate claims as one.

All in all, the two-step estimator performs well when the second step is not needed (δ = 0) but is otherwise unreliable.

“All in all, the two-step estimator performs well when the second step is not needed ( δ = 0) but is otherwise unreliable.”

Probably not a substantial claim. It may be too vague to check against evidence.

The biases in two-step procedures are substantial enough to recommend against using two-step procedures in any circumstances the authors can envision.

“These biases are substantial enough that we would recommend against using two-step procedures in any circumstances that we can envision.”