Stochastic frontier models

A. Fixed Effects model

The simplest panel data model in an efficiency measurement context is the fixed

effects model, such as that proposed by Schmidt and Sickles, and applied by Evans 25

et al. The time invariant fixed effects model is simple because it requires no 26

assumption with respect to the distribution of 𝒖i nor does it require the assumption

That is, the first order derivatives 𝜕y/𝜕𝐱1 and 𝜕y/𝜕𝐱2are positive, and the second order derivatives 24

𝜕2_{y/𝜕𝐱12 and 𝜕}2_{y/𝜕𝐱22 = β11 and β12 are negative.} Schmidt, P., and R. Sickles. Supra n. 17 25

Evans D., A. Tandon, C. Murray, and J. Lauer. Supra n. 15 26

that the 𝒖i are uncorrelated with the 𝒗it and the other regressors. The 𝒖i values are treated as state-specific constants so that the basic model in (1) becomes

(2)

where α represents the frontier intercept and the 𝒖i capture the degree of non-

compliance. The fixed effects model can be estimated using the ‘within’ estimator, which corresponds to the Corrected Ordinary Least Squares (COLS) estimator where the parameters in (2) are estimated by OLS and the intercept is shifted up to the observation of the best performing state so that all other residuals are non- positive. Here, the state with the maximum αi is assumed to be fully compliant (when

𝒖i = 0) and every other state-specific αi is measured as a deviation from this

benchmark. The differences in 𝒖i amongst states should, therefore, be interpreted as

relative rather than actual non-compliance.

(3)

There is therefore an implicit overestimation of compliance in the fixed effects model; it is unlikely that the best performing state in the sample is fully compliant. This would not be problematic if all the analysis was seeking to do was to provide a ranking of all states with respect to compliance. But this is not all that is required. What is additionally sought is a more precise estimate of compliance. Since there is no way of testing the extent of compliance-overestimation, the revealed fixed effects estimates must be interpreted with this limitation in mind.

Nevertheless, since there is no requirement to make any assumptions with respect to

𝒖i, estimates of the 𝒖i and the parameters in the fixed effects model are consistent as yit =(α−ui)+X / itβ+vit yit =αi +X / itβ+vit

N → ∞ and/or T → ∞. At the same time however, because the fixed effects model

creates no space for other time invariant variables to enter it, whatever the model may gain in robustness it may well lack in precision. Specifically, the fixed effects, that

is the αi and ultimately the 𝒖i, will be capturing not only the variation in time

invariant non-compliance across states but will also be capturing any time invariant

heterogeneity. A problem emphasised by Greene and others. The degree to which 27

this problem is indeed problematic is dependent on the nature of the data under investigation: the degree of heterogeneity. In this analysis the data cover the whole world, encompassing quite different disease and population environments. Time invariant heterogeneity is, therefore, likely to play an influential role in a state’s ability to ‘produce’ health and ignoring it in the modelling process will likely result in

erroneous estimates of 𝒖i. As a result, the fixed effects model can be treated as a

base model against which others, that can accommodate time invariant heterogeneity, may be compared.

B. Heterogeneous Random Effects model

In the foregoing fixed effects model it was assumed that the 𝒖i were fixed but were

allowed to be correlated with the regressors and with the 𝒗it. In the random effects

model the alternate assumption is made: the 𝒖i are randomly distributed with a

constant mean and variance, but are assumed to be uncorrelated with the regressors

and with the 𝒗it. The 𝒗it are assumed to be symmetric, have zero mean and constant

variance. Pitt and Lee first applied the time invariant random effects model to a panel

data version of stochastic frontier analysis. By assuming that the 28 𝒖i are random,

Greene, W.H. “Reconsidering Heterogeneity in Panel Data Estimators of the Stochastic Frontier 27

Model.” (2005); Greene, W.H. Supra n. 21; Kumbhakar, S.C., and K. Lovell. Stochastic Frontier Analysis. 2000

Pitt, M., and L. Lee. “The Measurement and Sources of Technical Inefficiency in the Indonesian 28

rather than fixed, the random effects model provides an entrance for time invariant covariates.

As noted above, with respect to measuring compliance across states worldwide, there

are important variables, (𝒛’s) besides inputs, (𝐱’s) that could influence the position or

the shape of the frontier and/or the efficiency distribution. There are, however, a

number of ways in which this heterogeneity (the 𝒛’s) can enter the model: i) as

additional shift parameters in the production function; ii) in the conditional mean of the 𝒖i; and/or iii) in the variance of either or both parts of the combined error term,

σ2_{𝒗 and/or σ}2_{𝒖. Fortunately, this choice need not be dilemmatic. The previous}

chapters have provided a theoretical basis upon which to make the decision. In Chapter 3, it was argued that a state’s ability to fulfil the right to health is not solely determined by the inputs it has at its disposal, those being financial and human resources. In addition, other environmental factors over which the state has little to

no control may also facilitate or impede a state in its efforts to do so. These 𝒛’s have

already been identified as mortality density and population density and, since they are additional determinants of ability, they should appear as shift parameters in the production function. As such, the standard stochastic frontier model in (1) becomes

(4)

where 𝒛’i is the vector of time invariant environmental variables. These 𝒛’s are state-

specific and can shift the frontier by changing α or can change the shape of the

frontier by influencing both α andβ.

y_it =α +X/

itβ+z

/

The random effects model can be estimated with estimators based on OLS, which

require no assumption on the distribution of the 𝒖i, (although the 𝒖i are still required

to be non-negative). However, if an assumption on the distribution is tenable, maximum likelihood estimation (MLE) is feasible, which makes estimation more precise since it can exploit distributional information which OLS estimators cannot.

In this case, the distribution of the error components in (4) remains to be

determined. Given the non-negativity of 𝒖i the choices are generally limited to half-

normal, truncated-normal, exponential, or gamma. For simplicity, here the normal- half-normal distribution is applied to the model so that

1. 𝒗it ~ iid N(0, σ2_{𝒗) the random error term is assumed to be normally distributed}

2. 𝒖i ~ iid N+(0, σ2𝒖) the non-compliance term is non-negative and half-

normal. 29

The next step is to obtain estimates of state-specific non-compliance. Because the process for estimating the parameters only produces an estimate of the combined

error term, 𝓔i, Jondrow et al.’s conditional mean estimator (JLMS) is applied to 30

separate non-compliance from the combined error term.

Whilst some authors have noted that the assumption in ii. (that the mean value of 𝒖i is zero) is a 29

significant restriction to the stochastic frontier analysis, it is used and treated here as a first-step upon which more sophisticated extensions can be built.

Jondrow, J., K. Lovell, I. Materov, and P. Schmidt. “On the Estimation of Technical Inefficiency in 30