Dependence modeling in stochastic frontier analysis

2.1 Canonical independence framework

The main difference between a standard production function setup and the SFM is the presence of two distinct error terms in the model. The u i term captures inefficiency, shortfall from maximal output dictated by the production technology, while the v i term captures stochastic shocks. The standard neoclassical production function model assumes full efficiency – so SFA embraces it as a special case, when u i = 0 , ∀ i , and allows the researcher to test this statistically.^[2] It is commonly assumed that inputs are exogenous, in the sense that x is independent of u and v , x ⊥ ( u , v ) , and the two components of the error term are independent, u ⊥ v .

Many estimation methods require distributional assumptions for both u and v (beyond the assumption of independence). For an assumed distributional pair, one can obtain the implied distribution for ε i and then estimate all of the parameters of the SFM with the maximum likelihood estimator (MLE). The most common assumption is that v i ∼ N ( 0 , σ v 2 ) and u i is from a Half Normal distribution, N + ( 0 , σ u 2 ) , or u i is from an Exponential distribution with parameter σ u .

The most popular case for the density of the composed error ε is obtained for the Normal Half Normal specification under independence u ⊥ v . According to Aigner et al. [3], the distribution function of a sum of a normal and Truncated Normal was first derived by Weinstein [94]. Let f v and f u denote the density of v and u , respectively. For the Normal Half Normal case, f v ( v ) = ϕ v σ v and f u ( u ) = 2 σ u ϕ u σ u , where ϕ ( ⋅ ) is the standard Normal probability density function (pdf). The closed form expression for the pdf can be obtained by convolution as follows:

where Φ ( ⋅ ) is the standard normal cumulative distribution function (cdf), with the parameterization σ = σ u 2 + σ v 2 and λ = σ u / σ v . λ is commonly interpreted as the proportion of variation in ε due to inefficiency. The density of ε in (2.2) can be characterized as that of a Skew Normal random variable with location parameter 0, scale parameter σ , and skew parameter − λ .^[3] This connection has only recently appeared in the efficiency and productivity literature [24].

It is worth noting that the closed form expression in (2.2) is equivalent to

where expectations are taken with respect to the relevant distribution. This suggests an alternative, simulation-based, way to construct the density by sampling from the distribution of u or v and evaluating the corresponding sample averages. Among the two sampling options (from the distribution of u or from the distribution of v ), sampling the u ’s is more practical as it avoids to need to ensure that v − ε > 0 . Sampling u can be easily done by sampling from the standard normal distribution and taking the absolute values σ u ∣ N ( 0 , 1 ) ∣ (in the case of the Half Normal distribution).^[4]

Our mathematical formulation will focus on a production frontier as it is the most popular object of study. The framework for dual characterizations (e.g., cost, revenue, profit) or other frontiers is similar and follows with only minor changes in notation. For example, the cost function formulation is obtained by changing the sign in front of u to a “ + ,” which will represent excess, rather than shortfall, of cost above the minimum level.

2.2 Modeling dependence

Smith [81] relaxed the assumption of independence between u and v by introducing a copula function to model their joint distribution. This is one of the first relaxations of the independence assumptions available for SFA and it allowed testing the adequacy of this assumption. If the marginal distributions of u and v are linked by a copula density c ( ⋅ , ⋅ ) , then their joint density can be expressed as follows:

where F u and F v denote the respective cdfs. It then follows by a similar construction to (2.2) that the density of ε can be written as

For commonly used copula families, this density does not have a close form expression similar to (2.2), even in the Normal Half Normal case, so a simulation-based approach would often need to be used, where we simulate many draws of u and evaluate the sample analogue of the following expectation with respect to the distribution of u :

Smith [81] found that ignoring the dependence can lead to biased estimates and discussed how one can test whether the independence assumption nested within this model is adequate. It is easy to see that the model in (2.2) is a special case of (2.3) when c ( ⋅ , ⋅ ) is the independence (or product) copula.

From f ( ε ) , along with the assumption of independence over i , the log-likelihood function can be written as follows:

where ε i = y i − m ( x i ; β ) . The SFM can be estimated using the traditional MLE, if an analytic expression for the integrals is available, or the maximum simulated likelihood estimator (MSLE), if we need to use a simulation-based approach to evaluate the integrals of the density [60].^[5] The benefit of MLE/MSLE is that under the assumption of correct distributional specification of ε , the MLE is asymptotically efficient (i.e., consistent, asymptotically Normal, and its asymptotic variance reaches the Cramer–Rao lower bound). A further benefit is that a range of testing options are available. For instance, tests related to β can easily be undertaken using any of the classic trilogy of tests: Wald, Lagrange multiplier, or likelihood ratio. The ability to readily and directly conduct asymptotic inference is one of the major benefits of SFA over data envelopment analysis (DEA).^[6]

Two main issues that practitioners face when confronting dependence are the choice of copula model and the assumed error distributions that best fit the data. As Wiboonpongse et al. ([95], p. 34) note “The impact of the independence assumption on technical efficiency estimation has long remained an open issue.” Analytical criteria such as AIC or BIC can be used for these purposes, see both [9] and [95] for detailed reviews.

More specifically, Wiboonpongse et al. [95] use MSLE and systematically consider several copula families including the Student-t, Clayton, Gumbel and Joe families as well as their relevant rotated versions. Their data is a cross section from coffee production in Thailand and they use AIC and BIC to determine which copula model is most appropriate. Wiboonpongse et al. [95] also assume the marginals of the two error components are Normal and Half Normal, then apply a range of copulas to inject dependence. In their empirical application they have a total of 111 observations. The Clayton copula is found to be the best and plots of technical efficiencies across 111 farmers for independence and the best fitting copula model found near uniformly lower TE scores (though not much different). Finally, it also appears that the ranks are preserved (see their Figure 3).

An unintended benefit of modelling dependence is that it may alleviate the “wrong skewness” issue that is common in the canonical Normal Half Normal SFM [77,89]. The wrong skewness issue arises when the OLS residuals display skewness that is of the wrong sign compared to that stemming from the frontier model (so positive when estimating a production frontier). For specific distributional pairs the model cannot separately identify the variance parameters for both v and u . It was noted in [19] that the third central moment of the composed error is

It is clear that the skewness of ε only depends on the skewness of u when v is assumed to be symmetric and independent of u . Once u and v are allowed to be dependent and/or v is allowed to be asymmetric, then the skewness of the composed error does not have to align with the skewness of inefficiency. Thus, modelling dependence is one way in which some of the empirical vagaries of the SFM can be overcome [93].

4.1 A benchmark specification

The benchmark panel SFM can be written as follows:

This model differs from (2.1) in many ways. All observed variables and error terms inherited from (2.1) now have a double-index for both firms, i , and time, t = 1 , … , T . In addition, the model contains the so-called firm-specific heterogeneity, c i , and the time-invariant component of inefficiency η i . Compared with (2.1), c i encapsulates any unobserved factors that affect output (other than inputs) without changing over time such as unmeasured management or operational specifics of the firm. If such factors are present, the dependence between c i and x i causes omitted variable biases and invalidates inference based on cross-sectional SFM. In panel models, when ignored, such factors can serve as common sources of dependence in the error term ε i t which also leads to invalid inference.

Another distinguishing feature of (4.1) is the presence of η i , a component of inefficiency which is time-invariant. This means that inefficiency is composed of both time-invariant and time variant components, which are sometimes interpreted as long-run and short-run inefficiency. Since both c i and η i are unobserved, it will generally be difficult to decompose α i into its subsequent firm-specific and time-invariant inefficiency components.

Classical panel methods (i.e., methods that assume that η i and u i t do not exist) allow for various forms of dependence between c i and x i t . For example, estimation under the fixed effects (FE) framework allows x i t to be correlated with c i and uses a within transformation to obtain a consistent estimator of β . Alternatively, estimation in the random effects (RE) framework assumes that x i t and c i are independent and uses OLS or GLS. The difference between OLS and GLS arises due to the fact that the variance-covariance matrix of the composed error term c i + v i t is no longer diagonal, and so, feasible GLS is asymptotically efficient.

The early work on panel SFM assumed inefficiency to be time-invariant. This allowed handling dependence within panels using classical panel methods such as FE and RE estimation [76]. The standard time-invariant SFM is

where c i ≡ β 0 − η i . Under the FE framework, the serially independent inefficiency η i , i = 1 , … , n , is allowed to have arbitrary dependence with x i t . In cases in which there are time-invariant variables of interest in the production model, one can use the RE framework, which also requires no distributional assumptions on v and η and can be estimated with OLS or GLS. Alternatively, in such cases, one can rely on distributional assumptions as in [69], where v i t is assumed to follow a Normal distribution and η i Half Normal.

Table 1 contains a summary of the classical SFMs allowing for specific forms of serial dependence in u i t . It also lists any additional dependence structures permitted in these different models such as dependence between c i and x i t . See [67] and [50] for a detailed discussion of these methods.

4.2 Quasi MLE

If there is no ( c i , η i ) or if ( c i , η i ) is assumed to be part of u i t or v i t , which are independent of x , then we can view the panel model as a special case of the cross-sectional model (2.1), only with the double-index i t . The MLE method described in Section 2 applies in this case but it uses the sample likelihood obtained leveraging the assumption of independence over both i and t , not just i . Because independence over t is questionable in panels, this version of MLE is often referred to as quasi-MLE (QMLE).

Let θ = ( β ′ , σ 2 , λ ) ′ and let f i t denote the density of the composed error term evaluated at ε i t = v i t − u i t . Then,

and the QMLE of θ can be written as

The QMLE is known to be consistent even if there is no independence over t but to obtain the correct standard errors one needs to use the so-called “sandwich,” or misspecification-robust, estimator of the QMLE asymptotic variance matrix. QMLE is known to be dominated in terms of precision by several other estimators that use the dependence information explicitly (and correctly). However, an appeal of the QMLE in this setting is that assuming independence is more innocuous in the sense that it does not lead to estimation bias, only to a lack of precision, when compared to a misspecification of the type of dependence that can lead to distinct biases.

Amsler et al. [5] proposed several estimators that model time dependence in panels. One such estimator can be obtained in the Generalized Method of Moments (GMM) framework. Let s i t ( θ ) denote the score of the density function f i t ( θ ) , i.e.,

where ∇ θ denotes the gradient with respect to θ . Then, the QMLE of θ solves

and is identical to the GMM estimator based on the moment condition

where expectation is with respect to the distribution of ε . However, under time dependence, summation (over t ) of the scores in (4.6) is not the optimal weighting. The theory of optimal GMM suggests using correlation of s i t over t by applying the GMM machinery to the T score functions written as follows:

The optimal GMM estimator based on these moment conditions has the smallest asymptotic variance than that of any other estimator using these moment conditions. In a classical (non-SFA) panel data setting, Prokhorov and Schmidt [70] call this estimator Improved QMLE (IQMLE).

4.3 Using a Copula

Alternative estimators that allow explicit modelling of dependence between cross-sectional errors over t have to construct a joint distribution of those errors. Amsler et al. [5] offered two ways of doing so. One is to apply a copula to form f ε , the joint (over t ) density of the composed errors ε i = ( ε i 1 , … , ε i T ) . The other is to use a copula to form f u , the joint distribution of u i = ( u i 1 , … , u i T ) .

Given the Normal/Half Normal marginals of ε ’s in (2.2) and a copula density c ( ⋅ , … , ⋅ ) , the joint density f ε can be written as follows:

where, as before, f i t ( θ ) ≡ f ( ε i t ) = f ( y i t − x i t ′ β ) is the pdf of the composed error term evaluated at ε i t and F i t ( θ ) ≡ ∫ − ∞ ε i t f ( s ) d s is the corresponding cdf. Once the joint density is obtained we can construct a log-likelihood and run MLE. If we let the copula density have a scalar parameter ρ , then the sample log-likelihood can be written as follows:

The first term in the summation is what distinguishes this likelihood from QMLE – an explicit modelling of dependence between the composed errors at different t .

In a GMM framework, the MLE that maximizes (4.7) is identical to the GMM estimator based on the moment conditions

where c i ( θ , ρ ) = c ( F i 1 ( θ ) , … , F i T ( θ ) ; ρ ) . Again, efficiency improvement is, in some circumstances, possible if we instead use the optimal GMM machinery on the moment conditions

However, this improvement may now come at the price of a bias as the copula-based moment conditions may be misspecified causing inconsistency of GMM and offsetting any benefit of higher precision. So assuming a wrong kind of time dependence may be worse than assuming independence (over time). Prokhorov and Schmidt [70] explored these circumstances.

The alternative copula-based specification is to form the joint distribution of ( u i 1 , … , u i T ) rather than that of ( ε i 1 , … , ε i T ) . A challenge of this specification is that a T -dimensional integration will be needed to form the likelihood in this case. Let f u ( u ; θ , ρ ) denote the copula-based joint density of the one-sided error vector and let f u ( u ; θ ) denote the marginal density of an individual one-sided (Half Normal) error term. Then,

where F u ( u ; θ ) ≡ ∫ 0 u f u ( s ; θ ) d s is the cdf of the Half Normal error term.

To form the sample likelihood we need the joint density of the composed error vector ε . Given the density of u and assuming, as before, that v ⊥ u , this density can be obtained as follows:

where E u ( θ , ρ ) denotes the expectation with respect to f u ( u ; θ , ρ ) and f v ( v ; θ ) = ϕ ( v ) is the multivariate Normal pdf of v , where all v ’s are independent and have equal variance σ v 2 .

Similar to the previous section, this integral has no analytical form. Additionally, this is a T -dimensional integral, which is computationally strenuous to evaluate using numerical methods. However, it has the form of an expectation over a distribution we can sample from and this, as before, permits application of MSLE, where we simulate the u ’s and estimate f ε ( ε ) by averaging ϕ ( ε + u ) over the draws. To be precise, let S denote the number of simulations. The direct simulator of f ε ( ε ) can be written as follows:

where u s ( θ , ρ ) is a draw from f u ( u ; θ , ρ ) constructed in (4.9). Then, a simulated log-likelihood can be obtained as follows:

where, as before, ε i = ( ε i 1 , … , ε i T ) and ε i t = y i t − x i t ′ β .

This method is a multivariate extension to the simulation-based estimation of univariate densities discussed earlier. An important additional requirement is the ability to sample from the copula; see ([64], Ch. 2) for a discussion of how to sample from the copula to allow dependence. Other than that, similar asymptotic arguments suggest that MSLE is asymptotically equivalent to MLE [29].

5.1 A likelihood framework

Likelihood-based alternatives allow for explicit modelling and estimation of the dependence structure that underlies endogeneity. This has recently been studied by Kutlu [54], Karakaplan and Kutlu [44], Tran and Tsionas [87,88], and Amsler et al. [6]. Our discussion here follows [6] as their derivation of the likelihood relies on a simple conditioning argument as opposed to the earlier work relying on the Cholesky decomposition or alternative approaches. While all approaches lead to a likelihood function, the conditioning idea of Amsler et al. [6] is simpler and more intuitive.

Consider the stochastic frontier system:

where x i = ( x 1 i , x 2 i ) , β = ( β 1 , β 2 ) , w i = ( x 1 i , q i ) is the vector of instruments, η i (different from η i in Sections 3.1–3.2) is uncorrelated with w i and endogeneity of x 2 i arises through c o v ( ε i , η i ) ≠ 0 . Here simultaneity bias (and the resulting inconsistency) exists because η i is correlated with either v i , u i , or both.

We start with the case of dependence between η i and v i while u i is independent of ( η i , v i ) . Assume that, conditional on w i , ψ i = ( v i , η i ) ′ ∼ N ( 0 , Ω ) , where

To derive the likelihood function, [6] condition on the instruments, w . Doing this yields f ( y , x 2 ∣ w ) = f ( y ∣ x 2 , w ) ⋅ f ( x 2 ∣ w ) . With the density in this form, the log-likelihood follows suite: ln ℒ = ln ℒ 1 + ln ℒ 2 , where ln ℒ 1 corresponds to f ( y ∣ x 2 , w ) and ln ℒ 2 corresponds to f ( x 2 ∣ w ) . These two components can be written as

where ε ˜ i = y i − β 0 − x i β − μ c i , μ c i = Σ v η Σ η η − 1 η i , σ 2 = σ v 2 + σ u 2 , λ c = σ u / σ c , and σ c 2 = σ v 2 − Σ v η Σ η η − 1 Σ η v . The subtraction of μ c i in ln ℒ 1 is an endogeneity correction while it should be noted that ln ℒ 2 is nothing more than the standard likelihood function of a multivariate normal regression model (as in (5.2)). Estimates of the model parameters ( β , σ v 2 , σ u 2 , Γ , Σ v η ) and Σ η η can be obtained by maximizing ln ℒ .

While direct estimation of the likelihood function is possible, a two-step approach is also available [54]. However, as pointed out by both Kutlu [54] and Amsler et al. [6], this two-step approach will have incorrect standard errors. Even though the two-step approach might be computationally simpler, it is, in general, different from full optimization of the likelihood function of Amsler et al. [6]. This is due to the fact that the two-step approach ignores the information provided by Γ and Σ η η in ln ℒ 1 . In general, full optimization of the likelihood function is recommended as the standard errors (obtained in a usual manner from the inverse of the Fisher information matrix) are valid.^[7]

5.2 A GMM framework

An insightful avenue to model dependence due to endogeneity in the SFM that differs from the traditional corrected methods or maximum likelihood stems from the GMM framework as proposed by Amsler et al. [6], who used the insights of Hansen et al. [35]. Similar to our discussion on the use of GMM in panel estimation, the idea is to use the first-order conditions for maximization of the likelihood function under exogeneity as a GMM problem:

where ϕ i = ϕ ( λ ε i σ ) and Φ i = Φ ( λ ε i σ ) . Note that these expectations are taken over x i and y i (and by default, ε i ) and solved for the parameters of the SFM.

The key here is that these first-order conditions (one for σ 2 , one for λ , and the vector for β ) are valid under exogeneity and this implies that the MLE is equivalent to the GMM estimator. Under endogeneity however, this relationship does not hold directly. But the seminal idea of Amsler et al. [6] is that the first-order conditions (5.3) and (5.4) are based on the distributional assumptions on v and u , not on the relationship of x with v and/or u . Thus, these moment conditions are valid whether x contains endogenous components or not. The only moment condition that needs to be adjusted is (5.5). In this case, the first-order condition needs to be taken with respect to w , the exogenous variable, not x . Doing so results in the following amended first-order condition:

where ϕ i and Φ i are identical to those in (5.5). It is important to acknowledge that this moment condition is valid when ε i and w i are independent. This is a more stringent requirement than the typical regression setup with E [ ε i ∣ w i ] = 0 . As with the C2SLS approach, the source of endogeneity for x 2 does not need to be specified (through v and/or u ).

5.3 An economic model of dependence

From an economic theory perspective, there are grounds to model dependence between u i and η i , not only between v i and η i . A system similar to (5.1)–(5.2) arises as a result of appending the SFM with the first-order conditions of cost minimization^[8] ([52], Chapter 8):

for input prices P , the first-order conditions in this case are

where m j ( x ; β ) is the partial derivative of m ( x ; β ) with respect to x j . These first-order conditions are exact, which usually does not arise in practice, rather, a stochastic term is added which is designed to capture allocative inefficiency. That is, our empirical first-order conditions are m j ( x ; β ) m 1 ( x ; β ) = P j P 1 e η j for j = 2 , … , J , where η j captures allocative inefficiency for the jth input relative to input one (the choice of input to compare is without loss of generality).

The idea behind allocative inefficiency is that firms could be fully technically efficient, and still have room for improvement due to over or under use of inputs, relative to another input, given the price ratio. On the other hand, firms can be technically inefficiency because of allocative inefficiency and vice versa so independence between u and η is hard to justify. Additionally, if firms are cost minimizers and one estimates a production function, the inputs will be endogenous as these are choice variables to the firm. In this case, input prices can serve as instruments.

Combining the SFM, under the Cobb–Douglas production function, with the information in the J − 1 conditions in (5.8) with allocative inefficiency built in, results in the following system:

where x i j is the log of input j of firm i , p j is the log of input j price, β j is the coefficient on input j in (5.9), and η i = ( η i 2 , … , η i J ) are the allocative inefficiencies for J − 1 inputs with respect to input one. See Schmidt and Lovell [74,75] for details.

5.4 A copula-based approach

Amsler et al. [6] used copulas to obtain a joint distribution for u , v , and η , whereas Amsler et al. [8] developed a new copula family for u and η with properties that reflect the nature of allocative (symmetric) and technical (one-sided) inefficiencies. Here we provide the derivation of a copula-based likelihood for the most general case that allows us to model dependence between all the components of ( u , v , η ) .

We keep the Half Normal marginal for u , Normal marginals for the elements of ψ = ( v , η ′ ) ′ as before, and assume a copula density c ( ⋅ , … , ⋅ ) . Amsler et al. [6] used the Gaussian copula, which implies that the joint distribution of ψ is Normal but this is largely done for convenience. This gives the joint density of ( u , v , η ) :

However, we need the joint density of ε and η in order to form a sample log-likelihood. This density can be obtained by integrating u out of f u , v , η ( u , v , η ) as follows:

Again, we can use simulation techniques to evaluate this density. Specifically, given S draws of u s , s = 1 , … , S , the direct simulator can be written as

and this leads to MSLE using the log-likelihood

where ε i = y i − x i ′ β and η i = x 2 i − w i Γ as follows from the system in (5.1)–(5.2).

The MSLE will produce estimates of all the parameters of the model, that is, β , Γ , σ u 2 , σ v 2 , variances of η j and whatever copula parameters appear in c . This permits modelling and testing the validity of independence assumptions between all error terms in the system including the assumption of exogeneity.

5.5 Dependence on determinants of inefficiency

To conclude this section, we consider the extension to a setting when inefficiency depends on covariates and some of these determinants of inefficiency may be endogenous [7,55]. These models can be estimated using traditional instrumental variable methods. However, given that the determinants of inefficiency enter the model nonlinearly, nonlinear methods are required.

Amsler et al. [7] considered the model

where u ∗ is the baseline inefficiency and u has the property that the scale of its distribution (relative to the distribution of u ∗ ) changes depending on the determinants z (the so-called scaling property). The covariates x i and z i are partitioned as

where x 1 i and z 1 i are exogenous and x 2 i and z 2 i are endogenous. The set of instruments used to combat endogeneity are defined as

where q i are the traditional outside instruments. Identification of all the parameters requires that the dimension of q be at least as large as the dimension of x 2 plus the dimension of z 2 (the rank condition).

In the model of Amsler et al. [7], endogeneity arises through dependence between a variable in the model ( x 2 and/or z 2 ) and noise, v . That is, both x and z are assumed to be independent of baseline inefficiency u ∗ . Given that E [ u i ] is not constant, the COLS approach to deal with endogeneity proposed by Amsler et al. [6] cannot be used here. To develop an appropriate estimator, add and subtract the mean of inefficiency to produce a composed error term that has mean 0,

Proper estimation through instrumental variables requires that the following moment condition holds

The nonlinearity of these moment conditions would necessitate use of nonlinear two-stage least squares (NL2SLS) [4].

Latruffe et al. [55] have a similar setup to Amsler et al. [7], using the model in (5.13), but develop a four-step estimator for the parameters; additionally, only x 2 is treated as endogenous. Latruffe et al.’s [55] approach is based on [23] using the construction of efficient moment conditions. The vector of instruments proposed in [55] is defined as

where q i ′ γ captures the linear projection of x 2 on the external instruments q . The four-stage estimator is defined as

1. Regress x 2 on q to estimate γ . Denote the OLS estimator of γ as γ ^ .

2. Use NLS to estimate the SFM in (5.13). Denote the NLS estimates of ( β , δ ) as ( β ¨ , δ ¨ ) . Use the NLS estimate of δ and the OLS estimate of γ in Step 1 to construct the instruments w i ( γ ^ , δ ¨ ) .

3. Using the estimated instrument vector w i ( γ ^ , δ ¨ ) , calculate the NL2SLS estimator of ( β , δ ) as ( β ˜ , δ ˜ ) . Use the NL2SLS estimate of δ and the OLS estimate of γ in Step 1 to construct the instruments w i ( γ ^ , δ ˜ ) .

4. Using the estimated instrument vector w i ( γ ^ , δ ˜ ) , calculate the NL2SLS estimator of ( β , δ ) as ( β ^ , δ ^ ) .

This multi-step estimator is necessary in the context of efficient moments because the actual set of instruments is not used directly, rather w i ( γ , δ ) is used, and this instrument vector requires estimates of γ and δ . The first two steps of the algorithm are designed to construct estimates of these two unknown parameter vectors. The third step then is designed to construct a consistent estimator of w i ( γ , δ ) , which is not done in Step 2 given that the endogeneity of x 2 is ignored (note that NLS is used as opposed to NL2SLS). The iteration from Step 2 to Step 3 does produce a consistent estimator of w i ( γ , δ ) , and as such, Step 4 produces consistent estimators for β and δ . While Latruffe et al. [55] proposed a set of efficient moment conditions to handle endogeneity, the model of Amsler et al. [7] is more general because it can handle endogeneity in the determinants of inefficiency as well. Finally, the presence of z is attractive since this allows the researcher to dispense with distributional assumptions on v and u .