Capacidad de diferenciación celular de las células de la membrana amniótica

In practice models are always imperfect and normally the data are noisy due to random (exogenous) or endogenous events. Ideally the analysis should include a stochastic element that can capture the effects of these random factors. Stochastic frontier estimators85 provide parametric estimates of efficiency and have been independently proposed by Aigner et al. (1977) and Meeusen and Van den Broeck (1977). SFA is an econometric technique for efficiency analysis based on regression analysis, that requires strong parametric assumptions for the functional form in terms of linking output and inputs and also distributional assumptions for noise and inefficiency, The main advantage of SFA is that it allows for noise in the data and makes possible stochastic inferences, while DEA basically assumes that data are noise-free, so without parameter estimates the method is deprived of providing inferences about elasticities or economies of scope (Thanassoulis et al., 2011).

In this case, the parameters of a model are first specified and then estimated using sample or simulated data (Salerno, 2003). SFA assumes, that the residual is separated into two components, one which illustrates the result of inefficiency and a second, that is considered as random. In practice, this involves assuming a specific distribution for each error component. Thus, the SFA production function can be written as, (Aigner et al., 1977): 𝑞_𝑘 = 𝑓(𝑥_𝑘1+ ⋯ + 𝑥_𝑘𝑁)𝑒𝐸𝑘 In log-forms: ln(𝑞_𝑘) = 𝑙𝑛[𝑓(𝑥_𝑘1+ ⋯ + 𝑥_𝑘𝑁) + 𝛦_𝑘] = 𝛽₀+ ∑ 𝛽_𝑖𝑙𝑛 𝑥_𝑘𝑖+ 𝛦_𝑘 𝑁 𝑖=1 Where 𝐸_𝑘= 𝑣_𝑘− 𝑢_𝑘and 𝑢_𝑘~𝑁(0, 𝜎_𝑢2_{); 𝑢}

𝑘 ≥ 0, 𝑣𝑘and 𝑢𝑘 are statistically independent86_{. The first component of the residuals 𝑣}

𝑘 is normal and is attributed to measurement error and random fluctuations, while the second component 𝑢_𝑘is one-

85 _{These models fall into the parametric stochastic model caste and most of these techniques are based on the ML principle} 86 _{The noise component 𝑣}

𝑘 has identical properties to the noise component of a linear regression model. The same properties are

valid for the inefficiency component except it has a non-zero mean 𝑢𝑘≥ 0. Both errors are uncorrelated to the explanatory

variables 𝑥𝑘𝑖. The main properties can be summarised into:

i. 𝐸(𝑣𝑘) = 0

ii. 𝐸(𝑣𝑘2) = 𝜎𝑣2 (homoscedastic)

iii. 𝐸(𝑣𝑘𝑣𝑗) = 0 ∀ 𝑘 ≠ 𝑗 (uncorrelated)

iv. 𝐸(𝑢𝑘2) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

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sided typically exponential or half-normal87 _{and is attributed to technical inefficiency}88_.

The parameters of the function can be estimated using MOLS (Førsund et al., 1980); Lovell, (1993) or MLE methods since it is a log linear operation and as such cannot be achieved using OLS. MLE estimators are asymptotically consistent and efficient estimators but the TE estimator may be inconsistent in some cases. In a stochastic frontier framework in the form introduced by Aigner et al. (1977), the maximum log- likelihood function considering a half normal distribution89 _𝑢

𝑘 ∼ 𝑁+(0, 𝜎𝑢) takes the form: ln L(q|𝛼, β, λ, 𝜎2) = 𝐾𝑙𝑛√2 √𝜋+ 𝐾𝑙𝑛 1 𝜎+ ∑ ln [1 − Φ(𝐸𝑘𝜆 1 𝜎)] − 1 2𝜎2∑ 𝐸𝑘 2 𝐾 𝑘=1 𝐾 𝑘=1 With 𝜆 =𝜎𝑢

𝜎𝑣 reflecting the asymmetry of the distribution of the error term. The larger

the value of 𝜆90_{, the more pronounced the asymmetry will be. If 𝜆 = 0 then the} symmetric error component 𝑣_𝑖𝑡 dominates the one-side error component 𝑢_𝑘 in the determination of 𝐸𝑘. Thus, the complete error term is determined solely by the random disturbance that is distributed normally (Mastromarco, 2008).

According to Greene (1980a, 1980b) the distribution of the composed error term is asymmetric since it incorporates the inefficiency term. Hence, Greene’s argument adopts an ML estimator that takes into consideration this information so more efficient estimates are produced, at least asymptotically. The Gamma distribution has been adopted to model the inefficiency error term due to its high flexibility but almost always the shapes of statistical noise and inefficiency are barely distinguishable.

Therefore, a stochastic approach produces efficiency measures that are separated from random shocks or measurement errors; however they are still potentially affected by misspecification errors. The imposition of a particular distributional form (e.g. half- normal or exponential) on that component of the residual that is attributed to technical inefficiency is an assumption that has no theoretical basis. Due to the allowance of stochastic errors and parameter estimation, parametric approaches, give a further insight into useful information such as, returns to scale and scope, and elasticities.

87 _{For a half normal distribution 𝑢}

𝑘~𝑖𝑖𝑑 𝑁+(𝜇, 𝜎2)

For an exponential distribution the variance of 𝑢𝑘 equals to 𝜎𝑢2 introduced by Meeusen and Van den Broeck (1977) and Aigner

et al., (1977).

Other distributions used in the literature are more flexible but more difficult to estimate: i. Truncated normal where 𝑢𝑘~𝑖𝑖𝑑 𝑁+(𝜇, 𝜎𝑢2) (Stevenson, 1980)

ii. Gamma 𝑢𝑘~𝑖𝑖𝑑 𝐺(𝜆, 𝑚) (gamma with mean λ and degrees of freedom 𝑚) Greene (1990)

iii. Exponential with mean 𝜆 , 𝑢𝑘~𝑖𝑖𝑑 𝐺(𝜆, 0)

88_{The stochastic element and the inefficiency element are independent from each other. Inefficiency is randomly distributed across}

DMUs similar to the deterministic frontier.

89 _{For an exponential or a truncated normal distribution see Mastromarco (2008).}

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The important features of the stochastic frontier model can be split into the deterministic part [exp (𝑓(𝑥𝑘1+ ⋯ + 𝑥𝑘𝑁)] , the stochastic element [exp(𝑣𝑘)] which is the symmetric part of the error and the TE component [exp(−𝑢_𝑘)] that represents the skewed part of the error term. The most common output-oriented measure of technical efficiency91 is the ratio of the observed output to the ideal output corresponding to the stochastic frontier output.

𝑇𝐸𝑘= 𝑞𝑘 exp [𝑓(𝑥𝑘1+ ⋯ + 𝑥𝑘𝑁) + 𝑣𝑘] =exp [𝑓(𝑥𝑘1+ ⋯ + 𝑥𝑘𝑁) + 𝑣𝑘− 𝑢𝑘] exp [𝑓(𝑥𝑘1+ ⋯ + 𝑥𝑘𝑁) + 𝑣𝑘] = exp (−𝑢𝑘)

The first step in predicting 𝑇𝐸_𝑘 is to estimate the parameters of the stochastic production frontier model. The measure of 𝑇𝐸_𝑘 varies between 0 < 𝑇𝐸_𝑘 < 1; this is an indicative measure of the output of the 𝑘-th DMU relative to the output produced by a fully efficient DMU being located to the frontier utilising the same mix of inputs. For a recent review see Kumbhakar and Lovell (2000) and in the context of panel data, stochastic models follow Schmidt and Sickles, (1984) and Cornwell et al., (1990). From the analysis a milestone step is the selection of the inefficiency distribution since it forms a fundamental assumption and not a decision based on the model’s characteristics. Generally, it is an a-priori decision and not testable. Nevertheless in most empirical work the various inefficiency estimates from different distributional assumptions are broadly more or less similar to each other.

By using MLE a direct estimate for the 𝑢_𝑘 is not feasible since the inefficiency parameter is unobservable. Therefore, the distribution of the inefficiency component provides sufficient information and can be deployed to get an estimate of the conditional mean of inefficiency 𝐸(𝑢_𝑘|𝐸𝑘). The main issue here is that there is no single way to generate the conditional mean, but there are two major estimators92 in the efficiency estimation literature (Kim and Schmidt 2000). The first is based on the early work of the Jondrow, Lovell, Materov, and Schmidt (1982), (JLMS) estimator. Calculating technical inefficiency can be based either on the mode of the distribution 𝑀(𝑢_𝑘|𝐸_𝑘) (the value of 𝑢 with the largest probability) or based on the mean of the distribution

𝐸(𝑢_𝑘|𝐸_𝑘) = [ 𝜎𝜆

1 + 𝜆2] [𝜇̃𝑘+

𝜙(𝜇̃_𝑘) Φ (𝜇̃_𝑘)]

Where 𝜇̃_𝑘 =−𝜆𝐸𝑘

𝜎 and 𝜙(. ) and Φ(. ) are the density and CDF of the standard normal distribution (Greene, 2008).

91 _{Exactly the same logic follows a cost frontier with the only difference being the sign of the inefficiency term:}

𝐶𝐸𝑘=

𝑓(𝑞𝑘; 𝑤𝑘)exp (𝑣𝑘)

𝐶𝑘

Where 0 < 𝐶𝐸𝑘≤ 1. So 𝑙𝑛𝐶𝑘= 𝑙𝑛𝑓(𝑞𝑘; 𝑤𝑘) + 𝑣𝑘− 𝑙𝑛𝐶𝐸𝑘= 𝑙𝑛𝑓(𝑞𝑘; 𝑤𝑘) + 𝑣𝑘+ 𝑢𝑘

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However, Battese and Coelli (1988) suggest an alternative estimator that calculates TE by the mean of the distribution of 𝐸(𝑇𝐸𝑘)|𝐸𝑘) = 𝐸(𝑒𝑥𝑝(−𝑢𝑘)|𝐸𝑘). For the truncated normal model (which includes the half-normal case), this is:

E(exp(−u_k)|E_k) = Φ [μk ∗ σ_∗− σ∗] Φ [μk∗ σ_∗] exp [−μ_k∗ +1 2σ∗ 2_] Where σ_∗2 ₌ σv2σu2 σ2 and μk∗ = μ̃k+ μ σu2

σ2 . The academic community has not yet settled on

which method to recommend since all methods produce estimates that are statistically inconsistent i.e. the estimate of u_k does not necessarily convert to the true value since the estimator is conditioned on a specific set of data.

SFA is an ingrained approach in economic theory and due to its statistical nature and various empirical applications, is quite a popular technique. However, there are a couple of constraints and limitations that should be stressed since, despite the computational facility of the simulation processes, the distributional assumption issue in every application is yet a major concern since the imposition of a particular distributional form remains an assumption that has no grounding in theory. Also, it requires large samples93 to ensure accurate results and any misspecification errors are incorporated into the measure of efficiency. Estimates are statistically inconsistent and this is reflected in the confidence intervals that attach to the inefficiency estimates that might be too wide for the method to gain credibility in practice (Johnes, 2004). However, in large samples by the central limit theorem we might expect the distribution of DMU efficiencies to be normal. Finally there are competing estimators to predict 𝑇𝐸, which do not always manage to converge in reality.