Editorial Manager(tm) for Journal of Productivity Analysis Manuscript Draft

Manuscript Number:

Title: A Stochastic Frontier Model with Correction for Sample Selection

Article Type: Original Research

Keywords: Stochastic Frontier; Sample Selection; Simulation; Efficiency

Corresponding Author: Dr William Greene, PhD

Corresponding Author's Institution: New York University, Stern School of Business

First Author: William Greene, PhD

Order of Authors: William Greene, PhD

Manuscript Click here to download Manuscript: StochasticFrontier-Selection-JPA-Revised.doc Click here to view linked References 1 2 3 4 A Stochastic Frontier Model with Correction for Sample Selection 5 * 6 William Greene 7 Department of Economics, Stern School of Business, 8 New York University, 9 March, 2008 10 Revised April, 2009 11 12 ______13 14 Abstract 15 16 Heckman’s (1976, 1979) sample selection model has been employed in three decades of 17 18 applications of linear regression studies. This paper builds on this framework to obtain a 19 20 sample selection correction for the stochastic frontier model. We first show a 21 22 surprisingly simple way to estimate the familiar normal-half normal stochastic frontier 23 24 model using maximum simulated likelihood. We then extend the technique to a 25 stochastic frontier model with sample selection. In an application that seems 26 27 superficially obvious, the method is used to revisit the World Health Organization data 28 29 [WHO (2000), Tandon et al. (2000)] where the sample partitioning is based on OECD 30 31 membership. The original study pooled all 191 countries. The OECD members appear 32 33 to be discretely different from the rest of the sample. We examine the difference in a 34 35 sample selection framework. 36 37 38 39 JEL classification: C13; C15; C21 40 41 Keywords: Stochastic Frontier, Sample Selection, Simulation, Efficiency 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 * 44 West 4 th St., Rm. 7-78, New York, NY 10012, USA, Telephone: 001-212-998-0876; e-mail: 60 [email protected] , URL pages.stern.nyu.edu/~wgreene . 61 62 63 1 64 65 1 A Stochastic Frontier Model with Correction for Sample Selection 2 3 4 5 1 Introduction 6 7 Heckman’s (1976, 1979) sample selection model has been employed in three decades of 8 9 applications of linear regression studies. Numerous applications have extended 10 Heckman’s approach to nonlinear settings such as the binary probit and Poisson 11 12 regression models. The first is Wynand and van Praag’s (1981) development of a probit 13 14 model for insurance purchase. Among a number of other recent applications, Bradford et 15 16 al. (2001) extended Heckman’s method to a stochastic frontier model for hospital costs. 17 18 The familiar approach in which a sample selection correction term is simply added to the 19 20 model of interest (see (7) and (8)) is not appropriate for nonlinear models such as the 21 stochastic frontier. In this study, we build on the maximum likelihood estimator of 22 23 Heckman’s sample selection corrected linear model and the extension to nonlinear 24 25 models by Terza (1996, 2009) to obtain a counterpart for the stochastic frontier model. 26 27 We first show a surprisingly simple way to estimate the familiar normal-half normal 28 29 stochastic frontier model using maximum simulated likelihood. The next step is to 30 31 extend the technique to a stochastic frontier model in the presence of sample selection. 32 The method is used to revisit the World Health Organization (2000) data [see also 33 34 Tandon et al. (2000)] where the sample partitioning is based on OECD membership. The 35 36 original study pooled all 191 countries (in a panel, albeit one with negligible within 37 38 groups variation). The OECD members appear to be discretely different from the rest of 39 40 the sample. We examine the difference in a sample selection framework. 41 42 43 2. A Selection Corrected Stochastic Frontier Model 44 45 The stochastic frontier model of Aigner, Lovell and Schmidt (1977) (ALS) is 46 47 specified with 48 49 β 50 yi = ββ′xi + vi - ui 51 where ui = | σuUi| = σu |Ui|, Ui ~ N[0,1], (1) 52 vi = σvVi , Vi ~ N[0,1]. 53 54 55 A vast literature has explored variations in the specification to accommodate, e.g., 56 heteroscedasticity, panel data formulations, etc. 1 It will suffice for present purposes to 57 58 59 60 1 See Greene (2008a) for further development of the model and a survey of extensions and applications. 61 62 63 2 64 65 1 A Stochastic Frontier Model with Correction for Sample Selection 2 3 4 5 work with the simplest form. Extensions will be considered later. The model can be 6 estimated by modifications of ordinary least squares [e.g., Greene (2008a)], the 7 8 generalized method of moments [Kopp and Mullahy (1990)] or, as is conventional in the 9 10 recent literature, by maximum likelihood (ALS). [A spate of Bayesian applications has 11 12 also appeared in the recent literature, e.g., Koop and Steel (2001).] In this study, we will 13 14 suggest, a fourth estimator, maximum simulated likelihood (MSL). The simulation based 15 16 estimator merely replicates the conventional estimator for the base case, in which the 17 closed form is already available. The log likelihood function for the sample selection 18 19 model does not exist in closed form, so some approximation method, such as MSL is 20 21 necessary. 22 23 24 2.1 Maximum Likelihood Estimation of the Stochastic Frontier Model 25 26 The log likelihood for the normal-half normal model for a sample of N 27 28 observations is 29 30 βββ σ λ N  1( 2) − σ− 1 ε σ2 + Φ−γε σ  log L( , , ) = ∑ logπ log (/)i log( i /) (2) 31 i=1  2 2  32 where εi = yi - βββ′xi = vi – ui, 33 γ = σ /σ , 34 u v σ σ2 + σ 2 35 = v u 36 37 Φ 38 and (.) denotes the standard normal cdf. The density satisfies the standard regularity 39 40 conditions, and maximum likelihood estimation of the model is a conventional problem 41 handled with familiar methods. Estimation is straightforward and has been installed in 42 43 the menu of supported techniques in a variety of programs including LIMDEP , Stata and 44 45 TSP .2 46 47 Conditioned on ui, the central equation of the model in (2.1) would be a classical 48 49 linear regression model with normally distributed disturbances. Thus, 50 51 −1 −βββ′ +σ2 σ 2 52 exp[2 (yix iui |U |) / v ] f(yi|xi,|Ui|) = . (3) 53 σ2 π 54 v 55 56 57 58 59 2 Details on maximum likelihood estimation of the model can be found in ALS and elsewhere, e.g., Greene 60 (2008b, Ch. 16). 61 62 63 3 64 65 1 A Stochastic Frontier Model with Correction for Sample Selection 2 3 4 5 The unconditional log likelihood for the model is obtained by integrating the unobserved 6 random variable, |U |, out of the conditional density. Thus, 7 i 8 9 −1 −βββ′ +σ2 σ 2 10 exp[2 (yix iui |U |) / v ] f(yi|xi) = ∫ p(| Ui |) dU | i | , 11 |Ui | σ π v 2 12 φ 13 (|Ui |) 1 2 2 where p(| Ui|) = =exp[ − |U |] , | Ui| > 0, (4) 14 Φ(0) 2 i π 15 N then log L(βββ,σ ,σ ) = logf ( y |x ), 16 u v ∑i=1 i i 17 18 φ Φ 19 where is the standard normal density and is the standard normal cdf . The closed 20 form of the integral appears in (2). 3 Consider using simulation to approximate the 21 22 integrals; 23 24 −1 −βββ′ +σ2 σ 2 1 R exp[2 (yix i uir |U |) / v ] 25 f(yi|xi) ≈ ∑ , (5) r=1 σ π 26 R v 2 27 28 where Uir is R random draws from the standard normal population. (There is no closed 29 30 form for the extension of the model that appears below.) The simulated log likelihood is 31 32 33 1 2 2 34 N 1 R exp[− (y −βββ′x +σ |U |) / σ ]  βββ σ σ 2 i i uir v 35 logLS (, u , v ) = ∑ log  ∑ =  . (6) i=1 R r 1 σ2 π  36  v  37 38 39 The maximum simulated likelihood estimators of the model parameters are obtained by 40 4 41 maximizing this function with respect to the unknown parameters. 42 43 44 2.2 Sample Selection in the Linear Model 45 46 Heckman’s (1979) sample selection model for the linear regression case is 47 48 specified as 49 50 51 di = 1[ ααα′zi + wi > 0], wi ~ N[0,1] 52 2 yi = βββ′xi + εi, εi ~ N[0, σε ] 53 ε ρσ σ 2 54 (wi, i) ~ N 2[(0,1), (1, ε, ε )] (7) 55 (yi,xi) observed only when di = 1. 56 57 58 3 See Weinstein (1964). 59 4 See Gourieroux and Monfort (1996), Train (2003), Econometric Software (2007), Greene (2008b) and 60 Greene and Misra (2004). 61 62 63 4 64 65 1 A Stochastic Frontier Model with Correction for Sample Selection 2 3 4 5 Two familiar methods have been developed for estimation of the model parameters. 6 Heckman’s (1979) two step, limited information method builds on the result 7 8 9 E[yi|xi,di=1] = βββ′xi + E[ εi|di=1] 10 β ρσ φ α Φ α 11 = ββ′xi + ε (αα′zi)/ (αα′zi) (8) 12 = βββ′xi + θλi. 13 14 In the first step, ααα in the probit equation is estimated by unconstrained single equation 15 16 maximum likelihood and the inverse Mills ratio (IMR), λ=φˆ (αˆ′z )/( Φ α ˆ ′ z ) is computed 17 i i i 18 for each observation. The second step in Heckman’s procedure involves linear regression 19 20 of y on the augmented regressor vector, x * = (x , λˆ ), using the observed subsample, with 21 i i i i 22 ααα 23 a correction of the OLS standard errors to account for the fact that an estimate of is 24 used in the constructed regressor. 25 26 The full information maximum likelihood estimator for the model is developed in 27 28 Heckman (1976) and Maddala (1983). The log likelihood function for the sample 29 30 selection model is 31 32 33 1 ′ 2 2  exp(− ((y −βββ x ) / σ ε ) )  34 2 i i ×   σ π 35 ε 2   N d   +  36 β σ αρ = i log(,L ββ ε ,,) log ρ−β′ σ+ α ′   37 ∑ i=1 ((y x )/)ε z  Φi i i   38 2   1− ρ    (9) 39   − Φ − ααα′ 40 (1di ) (z i )  41     42 N 1 ε  ( ρε / σ + ααα′z = φi Φ  iε i  + − Φ− ααα′  43 ∑ = log di   (1di ) (z i ) . i 1  σ σ − ρ 2   44  ε ε  1   45 46 This has become a conventional, if relatively less frequently used estimator that is built 47 48 into most contemporary software. 49 50 51 2.3 Estimating a Stochastic Frontier Model with Sample Selection. 52 53 54 The received literature contains many studies in which authors, have extended 55 Heckman’s selectivity model to nonlinear settings, such as count data (e.g., Poisson 56 57 regression – Greene (1994)), nonlinear regression, and binary choice models. The first 58 59 application of the sample selection treatment in a nonlinear setting was Wynand and van 60 61 62 63 5 64 65 1 A Stochastic Frontier Model with Correction for Sample Selection 2 3 4 5 Praag’s (1981) development of a probit model for binary choice. The typical approach 6 taken to control for selection bias , motivated by (8), is to fit the probit model in (7), as in 7 8 the first step of Heckman’s two step estimator, then append λˆ (from (8)) to the linear 9 i 10 index part of the nonlinear model wherever it happens to appear. The approach is 11 12 inappropriate. The term λˆ in (8) arises as E[ ε |d =1] in a linear model. The expectation 13 i i i 14 βββ ε 15 of some nonlinear g( ′xi + i) subject to selection will generally not produce the form 16 βββ ε βββ θλ 17 E[ g( ′xi + i)| di=1] = g( ′xi + i) which can then be carried back into the otherwise 18 unchanged nonlinear model. See, e.g., Terza (1994, 1996, 1998) who develops the result 19 20 in detail for nonlinear regressions such as the exponential conditional mean case. 21 22 Indeed, in some cases, such as the probit and count data models, the εi for which the 23 24 expectation given di = 1 is taken does not even appear in the original model; it is unclear 25 26 as such what the correction is correcting. 27 28 The distribution of the observed random variable conditioned on the selection will 29 generally not be what it was without the selection (with or without the addition of the 30 31 inverse Mills ratio, λ to the index function). Thus, the addition of λ to the original 32 i i 33 likelihood function generally does not produce the appropriate log likelihood in the 34 35 presence of the sample selection. This can be seen even for the linear case in (9). The 36 37 least squares estimator of βββ (with λi added to the equation) is not the MLE in (9); it is 38 39 merely a feasible consistent estimator. Two well worked out specific cases do appear in 40 41 the literature. Maddala (1983) and Boyes, Hoffman and Lowe (1989) obtained the 42 appropriate closed form log likelihood for a probit model subject to sample selection. 43 44 The resulting formulation is a type of bivariate probit model, not a univariate probit 45 46 model based on ( xi,λi). Another well known example is the open form result for the 47 48 Poisson regression model obtained by Terza (1996,1998). 5 49 50 The combination of efficiency estimation and sample selection appears in several 51 52 studies. Bradford, et al. (2001) studied patient specific costs for cardiac revascularization 53 54 in a large hospital. They state “... the patients in this sample were not randomly assigned 55 to each treatment group. Statistically, this implies that the data are subject to sample 56 57 selection bias. Therefore, we utilize a standard Heckman two-stage sample-selection 58 59 60 5 See, also, Winkelman (1998). 61 62 63 6 64 65 1 A Stochastic Frontier Model with Correction for Sample Selection 2 3 4 5 process, creating an IMR from a first-stage probit estimator of the likelihood of CABG or 6 PTCA. This correction variable is included in the frontier estimate....” (page 306). 6 7 8 Sipiläinen and Oude Lansink (2005) have utilized a stochastic frontier, translog 9 10 model to analyze technical efficiency for organic and conventional farms. They state 11 12 “Possible selection bias between organic and conventional production can be taken into 13 14 account [by] applying Heckman’s (1979) two step procedure.” (Page 169.) In this case, 15 16 the inefficiency component in the stochastic frontier translog distance function is 17 distributed as the truncation at zero of a U with a heterogeneous mean. 7 The IMR is 18 i 19 added to the deterministic (production function) part of the frontier function. 20 21 Other authors have acknowledged the sample selection issue in stochastic frontier 22 23 studies. Kaparakis, Miller and Noulas (1994) in an analysis of commercial banks and 24 25 Collins and Harris (2005) in their study of UK chemical plants both suggested that 26 27 “sample selection” was a potential issue in their analysis. Neither of these formally 28 modified their stochastic frontier models to accommodate the result, however. 29 30 If, to motivate the sample selection treatment, we specify that the unobservables 31 32 in the selection model are correlated with the noise in the stochastic frontier model, then 33 34 the stochastic frontier model with sample selection can be cast as an extension of 35 36 Heckman’s specification for the linear regression model. The combination of the models 37 in (1) and (7) is 38 39 40 ααα 41 di = 1[ ′zi + wi > 0], wi ~ N[0,1] 2 42 yi = βββ′xi + εi, εi ~ N[0, σε ] 43 (yi,xi) observed only when di = 1. 44 εi = vi - ui 45 σ σ 46 ui = | uUi| = u |Ui| where Ui ~ N[0,1] (10) 47 vi = σvVi where Vi ~ N[0,1]. 48 2 (wi,v i) ~ N2[(0,1), (1, ρσv, σv )] 49 50 51 The conditional density for an observation in this specification is 52 53 6 The authors opt for a GMM estimator based on Kopp and Mullahy’s (1990) (KM) relaxation of the 54 distributional assumptions in the standard frontier model. It is suggested, that KM “find that the traditional 55 maximum likelihood estimators tend to overestimate the average inefficiency.” (Page 304.) KM did not, in 56 fact, make the latter argument, and we can find no evidence to support it in the since received literature. 57 KM’s support for the GMM estimator is based on its more general, distribution free specification. We do 58 note Newhouse (1994), whom Bradford et al cite, has stridently argued against the stochastic frontier model 59 as well, but not based on the properties of the MLE. 60 7 See Battese and Coelli (1995). 61 62 63 7 64 65 1 A Stochastic Frontier Model with Correction for Sample Selection 2 3 4

5  −1 −βββ′ +σ2 σ 2   6 exp( 2 (yix iui | U |) / v ) ) 7  ×    σ π  8 v 2   + 9 di   f (yi|xi,|Ui|, zi,di,) =  ρ−(yβ′x +σ | U |) / σ+ α ′ z    (11) 10  Φi iuiε i   2   11  1− ρ    12   − Φ − ααα′ 13  (1di ) (z i )  14 15 σ 16 Save for the appearance of the unobserved inefficiency term, u|Ui|, (11) is the same as 17 (9). Terza (1996, 2009) develops the log likelihood function for a generic extension of 18 19 Heckman’s result in (9) to nonlinear models. The result in (11) shows an application to 20 21 the stochastic frontier case – see (34:SS) in Terza (2009). 22 23 Sample selection arises as a consequence of the correlation of the unobservables 24 25 in the production or cost equation, vi, with those in the sample selection equation, wi. 26 27 Two other applications of this general approach to modeling sample selection or 28 endogenous switching in the stochastic frontier model have appeared in the recent 29 30 literature. In Kumbhakar, Tsionas and Similainen (2009), the model framework is very 31 32 similar to that in (10), but the selection mechanism is assumed to operate through ui 33 34 rather than vi. In particular, the disturbance in their counterpart to the equation for di is 35 36 wi + δui; in essence, the inefficiency in the production process produces an “inclination” 37 38 towards, in their case, organic farming. In Lai, Polachek and Wang’s (2009) application 39 40 to a wage equation, the wi in the selection mechanism is correlated (through a copula 41 function) with ε , not specifically with v or u . In both of these cases, the log likelihood is 42 i i i 43 substantially more complicated than the one used here. More importantly, the difference 44 45 in the assumption of the impact of the selection effect is substantive. 46 47 The log likelihood for the model in (10) is formed by integrating out the 48 49 unobserved | Ui| then maximizing with respect to the unknown parameters. Thus, as in (4) 50 51 and (5), 52 53 N 54 log L(βββ,σu,σv,ααα,ρ) = logfy ( |x ,, z dUpUdU ,| |)(| |)| | . (12) ∑i=1 ∫ |U | iiiii i i 55 i 56 57 The integral in (12) is not known; it must be approximated. The simulated log likelihood 58 59 function is 60 61 62 63 8 64 65 1 A Stochastic Frontier Model with Correction for Sample Selection 2 3 4 5  exp(−1 (y −βββ′x +σ | U |)/)2 σ 2 )    2 i i uir v × 6 di   σ2 π  7   v  8 N1 R  ρ−β′ +σ σ+ α ′   9 βββ σ σ ααα ρ (yβx ||)/ U ε α z log LS( , u, v, , )= ∑=log ∑ =  Φi i u ir i   . (13) i1R r 1 10  1− ρ 2   11     12   13   +(1 −d ) Φ− (ααα′z ) 14  i i  15 16 To simplify the estimation, we will use a two step approach. The single equation 17 18 MLE of ααα in the probit equation in (7) is consistent, albeit inefficient. For purposes of 19 20 estimation of the parameters of the stochastic frontier model, however, ααα need not be 21 22 reestimated. We take the estimates of ααα as given in the simulated log likelihood in (13), 23 24 then use the Murphy and Topel (2002) correction to adjust the standard errors in 25 26 essentially the same fashion as Heckman’s correction of the canonical selection model in 27 28 (8). Thus, the conditional simulated log likelihood function is 29 30  −1 −βββ′ +σ2 σ 2   31 exp( 2 (yix i uir | U |) / v ) ) 32  ×    σ π   33 v 2  N1 R d    34 log L (βββ,σ ,σ , ρ)= log i   . (14) S,C u v ∑i=1 ∑ r = 1  ρ−(yβββ′x +σ | U |) / σ+ a   35 R  Φi iuirε i   2   36  1− ρ    37  + − Φ−  38  (1di ) ( a i )  39 40 where a = αααˆ ′z . With this simplification, the nonselected observations (those with d = 0) 41 i i i 42 do not contribute information about the parameters to the simulated log likelihood. Thus, 43 44 the function we maximize becomes 45 46 47 exp(−1 (y −βββ′x +σ | U |)2 / σ 2 ) )  48 2 i i uir v ×  49 σ π 1 R v 2  50 βββ σ σ ρ log LS,C ( , u, v, ) = ∑=log ∑ =   . (15) 51 di 1R r 1 ρ−βββ′ +σ σ+   (yix i u ||)/ U irε a i 52 Φ   − ρ 2  53 1   54 55 The parameters of the model are estimated using a conventional gradient based approach, 56 57 the BFGS method. We use the BHHH estimator to estimate the asymptotic standard 58 59 errors for the parameter estimators. When ρ equals zero, the maximand reduces to that of 60 61 62 63 9 64 65 1 A Stochastic Frontier Model with Correction for Sample Selection 2 3 4 5 the maximum simulated likelihood estimator of the basic frontier model shown earlier. 6 This provides us with a method of testing the specification of the selectivity model 7 8 against the simpler model using a (simulated) likelihood ratio test. 9 10 11 2.4 Estimating Observation Specific Inefficiency 12 13 14 The end objective of the estimation process is to characterize the inefficiency in 15 the sample, u or the efficiency , exp(-u ). Aggregate summary measures, such as the 16 i i 17 sample mean and variance are often provided (e.g., Bradford, et al. (2001) for hospital 18 19 costs). Researchers also compute individual specific estimates of the conditional means 20 21 based on the Jondrow et al. (1982) (JLMS) result, 22 23 σλ φµ  −λε 24 (i ) i E[|] u ε= µ+  , µ= , εi = yi - βββ′xi. (16) 25 ii1+λ2 i Φµ ( ) i σ 26 i  27 28 The standard approach computes this function after estimation based on the maximum 29 30 likelihood estimates. In principle, we could repeat this computation with the maximum 31 32 simulated likelihood estimates. An alternative approach takes advantage of the 33 34 simulation of the values of ui during estimation. Using Bayes theorem, we can write 35 36 pu(,)ε p (|)() ε upu 37 p(|) u ε = ii = ii i . (17) i i p(ε ) ε 38 i ∫ p(ii | u )() pudu i i 39 ui 40 41 Recall ui = σu|Ui|. Thus, equivalently, 42 43 pU[(σε | |),] p [|( εσ | UpU |)]( σ | |) 44 p[(σ | U |) |] ε = uii = iui ui . (18) u i i ε εσ σ σ 45 p(i ) p[ |( | UpUd |)]( | |)( | U |) ∫ u iui ui ui 46 i 47 48 The desired expectation is, then 49 50 51 σ εσ σ σ (ui |Up |)[ iui |( | Up |)]( ui | Ud |)( ui | U |) 52 ∫ σ |U | σ ε = u i E[(u | U i |) | i ] . (19) 53 p([εσ |( | UpUdU |)]( σ | |)( σ | |) ∫ σ iui ui ui 54 u |U i | 55 56 These are the terms that enter the simulated log likelihood for each observation. The 57 58 simulated denominator would be 59 60 61 62 63 10 64 65 1 A Stochastic Frontier Model with Correction for Sample Selection 2 3 4 −1 −βββˆ′ +σ2 σ 2  5 exp( 2 (yix iˆ uir | U |)/) ˆ v ) 6 ×  σ π  7 1 R ˆ v 2 1 R Bˆ =   = fˆ (20) 8 i ∑r=1 ∑ r = 1 ir R ρ−ˆ(yβββˆ′x +σˆ ||)/ U σ+ ˆ a   R 9 Φi i u irε i   2  10 1− ρˆ   11 12 13 ˆ =1 R σ ˆ ε while the numerator is simulated with Ai∑ (ˆ u | U ir |) f ir . The estimate of E[ ui| i] is 14 R r=1 15 then 16 17 R fˆ ˆ ˆ =σ =Replication of the Pitt and Lee (1981) random effects form of the model, again 40 41 with any distribution from which draws can be simulated, is simple. The term Bi defined 42 43 in (20) that enters the log likelihood becomes 44 45 46 exp(−1 (y −βββ′x +σ | U |)/)2 σ 2 )  47 2 it it u ir v ×  48 σ π 1 R T v 2  1 R T = i = i ˆ 49 Bi ∑∏  ∑ ∏ f irt (22) R r=1 t=1 ′  R r = 1 t = 1 50 ρ(y −βββ x +σ | U |) / σ+ε a  Φit it uir i   51 − ρ 2  52 1   53 54 Further refinements, such as a counterpart to Battese and Coelli (1992, 1995) and 55 56 Stevenson’s (1980) truncation model may be possible as well. This remains to be 57 58 investigated. 59 60 61 62 63 11 64 65 1 A Stochastic Frontier Model with Correction for Sample Selection 2 3 4 5 3. Applications 6 7 In 2000, the World Health Organization published its millennium edition of the 8 9 World Health Report (WHR) [WHO (2000).] The report contained Tandon et al.’s 10 11 (2000) (TMLE) frontier analysis of the efficiency of health care delivery for 191 12 countries. The frontier analysis attracted a surprising amount of attention in the popular 13 14 press (given its small page length, minor role in the report and highly technical nature), 15 16 notably for its assignment of a rank of 37 to the United States’ health care system. 17 18 [Seven years after its publication, the report still commanded attention, e.g., New York 19 20 Times (2007).] The authors provided their data and methodology to numerous 21 22 researchers who have subsequently analyzed, criticized, and extended the WHO study. 23 [E.g., Gravelle et al. (2002a,b), Hollingsworth and Wildman (2002) and Greene (2004).] 24 25 TMLE based their analysis on COMP , a new measure of health care attainment 26 27 that they created. (The standard measure at the time was disability adjusted life 28 29 expectancy, DALE .) “In order to assess overall efficiency, the first step was to combine 30 31 the individual attainments on all five goals of the health system into a single number, 32 which we call the composite index. The composite index is a weighted average of the five 33 34 component goals specified above. First, country attainment on all five indicators (i.e., 35 36 health, health inequality, responsiveness-level, responsiveness-distribution, and fair- 37 38 financing) were rescaled restricting them to the [0,1] interval. Then the following weights 39 40 were used to construct the overall composite measure: 25% for health (DALE), 25% for 41 42 health inequality, 12.5% for the level of responsiveness, 12.5% for the distribution of 43 responsiveness, and 25% for fairness in financing. These weights are based on a survey 44 45 carried out by WHO to elicit stated preferences of individuals in their relative valuations 46 47 of the goals of the health system.” (TMLE, page 4.) (It is intriguing that in the public 48 49 outcry over the results, it was never reported that the WHO study did not, in fact, rank 50 51 countries by health care attainment, COMP , but rather by the efficiency with which 52 53 countries attained their COMP . That is, countries were ranked by the difference between 54 their COMP and a constructed country specific optimal COMP *.) In terms of COMP , 55 56 itself, the U.S. ranked 15 th in the study, not 37 th , and France did not rank first as widely 57 58 reported, Japan did. The full set of results needed to reach these conclusions are 59 60 contained in TMLE (2000). 61 62 63 12 64 65 1 A Stochastic Frontier Model with Correction for Sample Selection 2 3 4 5 The data set used by TMLE contained five years (1993-1997) of observations on 6 the time varying variables COMP , per capita health care expenditure and average 7 8 educational attainment, and time invariant, 1997 observations on the set of variables 9 10 listed in Table 1. TMLE used a linear fixed effects translog production model, 11 12 13 log COMP it = β1 + β2log HExp it + β3log Educ it 14 + β log 2Educ + β log 2HExp (23) 15 4 it 5 it 16 + β6 log HExp it ×logEduc it - ui + vit 17 18 in which health expenditure and education enter loglinearly and quadratically. (They 19 20 ultimately dropped the last two terms in their specification.) Their estimates of ui were 21 22 computed from the estimated constant terms in the linear fixed effects regression. Since 23 24 their analysis was based on the fixed effects regression, they did not use the time 25 26 invariant variables in their regressions or subsequent analysis. [See Greene (2004) for 27 discussion.] Their overall efficiency indexes for the 191 WHO member countries are 28 29 published in the report (Table 1, pages 18-21) and used in the analysis below. 30 31 Table 1 lists descriptive statistics for the TMLE efficiencies and for the variables 32 33 present in the WHO data base. The COMP , education and health expenditure are 34 35 described for the 1997 observation. Although these variables are time varying, the 36 amount of within group variation ranges from very small to trivial. [See Gravelle et al. 37 38 (2002a) for discussion.] The time invariant variables were not used in their analysis. The 39 40 data in Table 1 are segmented by OECD membership. The OECD members are 41 42 primarily 30 of the wealthiest countries (thought not specifically the 30 wealthiest 43 44 countries). The difference between OECD countries and the rest of the world is evident. 45 46 Figure 1 plots the TMLE efficiency estimates versus per capita GDP, segmented by 47 OECD membership. The figure is consistent with the values in Table 1. This suggests 48 49 (but, of course, does not establish) that OECD membership may be a substantive 50 51 selection mechanism. OECD membership is based on more than simply per capita GDP. 52 53 The selectivity issue is whether other factors related to OECD membership are correlated 54 55 with the stochastic element in the production function. 56 57 Figure 1 plots TMLE’s estimated efficiency scores against per capita GDP for the 58 191 countries stratified by OECD membership. The difference is stark. The layer of 59 60 points at the top of the figure for the OECD countries suggests that wealth produces 61 62 63 13 64 65 1 A Stochastic Frontier Model with Correction for Sample Selection 2 3 4 5 efficiency in the outcome. The question for present purposes is whether the selection 6 based on the observed GDP value is a complete explanation of the difference, or whether 7 8 there are latent factors related to OECD membership that also impact the placement of 9 10 the frontier function. We will use the sample selection model developed earlier to 11 12 examine the issue. We note, it is not our intent here to replace the results of the WHO 13 14 study. Rather, this provides a setting for demonstrating the selection model. Since we 15 16 will be using a stochastic frontier model while they used a fixed effects linear regression, 17 it will be difficult to make a direct comparison of the results. [The issue is examined in 18 19 detail in Greene (2004).] TMLE also used an elaborate normalization based on a turn of 20 21 the last century benchmark to anchor their efficiency estimates to a “minimal” level of 22 23 health care. And, of course, they used a panel data (fixed effects) estimator whereas we 24 25 have used a cross section. As such, it seems unlikely that the specific estimates of 26 27 inefficiency would be very similar. We can, however, see whether general conclusions 28 do hold up in the two settings. For example, if both approaches are addressing the same 29 30 broad concept of efficiency relative to the production function in (23), then the rankings 31 32 of countries might well be broadly similar. It is interesting to compare the rankings of 33 34 countries produced by the two methodologies, though we will do so without naming 35 36 names. 37 We have estimated the stochastic frontier models for the logCOMP measure using 38 39 TMLE’s truncated specification of the translog model. Since the time invariant data are 40 41 only observed for 1997, we have used the country means of the logs of the variables 42 43 COMP , HExp and Educ in our estimation. Table 2 presents the maximum likelihood and 44 45 maximum simulated likelihood estimates of the parameters of the frontier models. The 46 47 MSL estimates are computed using 200 Halton draws for each observation for the 48 simulation. [See Greene (2008b) or Train (2003) for discussion of Halton sequences.] 49 50 By using Halton draws rather than pseudorandom numbers, we can achieve replicability 51 52 of the estimates. To test the specification of the selection model, we have fit the sample 53 54 selection model while constraining ρ to equal zero. The log likelihood functions can then 55 56 be compared using the usual chi squared statistic. The results provide two statistics for 57 58 the test, then, the Wald statistic (t ratio) associated with the estimate of ρ and the 59 60 likelihood ratio statistic. Both Wald statistics fail to reject the null hypothesis of no 61 62 63 14 64 65 1 A Stochastic Frontier Model with Correction for Sample Selection 2 3 4 5 selection. For the LR statistics (with one degree of freedom) we do not reject the base 6 model for the non-OECD countries, but we do for the OECD countries, in conflict with 7 8 the t test. Since the sample is only 30 observations, the standard normal and chi squared 9 10 limiting distribution used for the test statistic may be suspect. We would conclude that 11 12 the evidence does not strongly support the selection model. It would seem that the 13 14 selection is dominated by the observables, presumably primarily by per capita income. 15 16 Figure 2 plots the estimated efficiency scores from the stochastic frontier model 17 versus those in the WHO report. (We did not reestimate the TMLE values; those shown 18 19 in the figure appear in the tables in the WHO report.) As anticipated in Greene (2004), 20 21 the impact of the fixed effects regression is to attribute to inefficiency effects that might 22 23 be better explained by cross country heterogeneity. These effects would be picked up by 24 25 the noise term in the frontier model. The heavy diagonal line in the figure shows the 26 ε 27 effect; save for the very largest values, the MSL estimates of E[ui| i] are well below their 28 counterparts computed using the TMLE fixed effects estimator. 29 30 Figure 3 shows a plot of the two estimators of the inefficiency scores in the 31 32 selectivity corrected frontier model, the JLMS estimator and the simulated values of 33 34 E[ u|ε] computed during the estimation. These are based on the parameters of the 35 36 selectivity model in (11) As noted earlier, they are strikingly similar. 37 38 Finally, Figure 4 shows a plot of the country ranks based on the stochastic frontier 39 40 model versus the country ranks implicit in the WHO estimates for the non-OECD 41 countries. The Spearman rank correlation of the two series is 0.66, which seems higher 42 43 than the figure would suggest. The (visually) quite weak correlation in the two sets of 44 45 results conflicts with our earlier suggestion. In sum, there are a long list of substantive 46 47 differences between the approach taken here and the one in TMLE. There are at least 48 49 three sources of difference. First, TMLE used a fixed effects linear regression whereas 50 we have used a stochastic frontier model. Second, we have used the time invariant 51 52 variables in Table 1 to control for cross country heterogeneity whereas ETML did not 53 54 make use of these. Third, we have accounted for the nonrandom sample selection in the 55 56 OECD and NonOECD subsamples. None of these, alone or together should necessarily 57 58 produce a change in the rankings of observations. The impacts of each source of variation 59 60 might be the subject of some fruitful further analysis. The TMLE study was ultimately 61 62 63 15 64 65 1 A Stochastic Frontier Model with Correction for Sample Selection 2 3 4 5 focused on the ranks of the counties, not on the inefficiency levels themselves. The 6 disparity in the ranks produced by the methods considered here should be of significant 7 8 concern. 9 10 The analysis described here is essentially microeconomic, behavioral in nature. 11 12 One might question the theoretical underpinnings of a behavioral model of optimization 13 14 and efficiency applied to macroeconomic data such as these. Another recent study, 15 16 Rahman, Wiboonpongse, Sriboonchitta and Chaovanapoonphol (2009) used the methods 17 described in this paper to analyze production efficiency of rice producers in Thailand. In 18 19 this study, the authors analyzed the switch by Thai farmers from lower quality rice 20 21 varieties to a higher quality, Jasmine variety. Their sample included 207 farmers in the 22 23 former group and 141 in the latter. They were able to examine the production process in 24 25 much greater detail than we have here. In their results, the “correction” for selection into 26 27 the high quality market produced quite marked differences in the estimated production 28 frontier and a highly significant “selection effect.” 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 16 64 65 1 A Stochastic Frontier Model with Correction for Sample Selection 2 3 4 5 4. Conclusions 6 7 We have proposed a maximum simulated likelihood estimator for ALS’s normal – 8 9 half normal stochastic frontier model. The normal–exponential model, a normal–t model, 10 11 or normal–anything else model would all be trivial modifications. The manner in which 12 the values of u are simulated is all that changes from one to the next. The identical 13 i 14 simulation based estimator of the inefficiencies is used as well. We note that in a few 15 16 other cases, such as the t distribution, simulation (or MCMC) is the only feasible method 17 18 of proceeding. [See Tsionas, Kumbhakar and Greene (2008).] The model is then 19 20 extended using Heckman’s (1976) formulation for the linear model and Terza’s 21 22 (1986,2009) extension to nonlinear models to produce a sample selection correction for 23 the stochastic frontier model. 24 25 The assumption that the unobservables in the selection equation are correlated 26 27 with the heterogeneity in the production function but uncorrelated with the inefficiency is 28 29 an important feature of the model. It seems natural and appropriate in this setting – one 30 31 might expect that observations are not selected into the sample based on their being 32 inefficient to begin with. Nonetheless, that, as well, is an issue that might be further 33 34 considered. (Note, again, the alternative approaches by Kumbhakar, Tsionas and 35 36 Sipilainen (2009) and by Lai, Polachek and Wang (2009).) A related question is whether 37 38 it is reasonable to assume that the heterogeneity and the inefficiency in the production 39 40 model should be assumed to be uncorrelated. Some progress has been made in this 41 42 regard, e.g., in Smith (2003), and, by implication, Lai et al. (2009), but the analysis is 43 tangential to the model considered here. 44 45 We have revisited the WHO (2000) study, and found that the results vary greatly 46 47 depending on the specification. However, it does appear that our expectation that 48 49 ‘selection’ on OECD membership is an important element of the measured inefficiency in 50 51 the data was not supported statistically. The results suggest that the obvious pattern in 52 53 Figure 1 that separates OECD from nonOECD members is explained by observables 54 (such as per capita GDP) and not unobservables as would be implied by the sample 55 56 selection model. 57 58 59 60 61 62 63 17 64 65 1 A Stochastic Frontier Model with Correction for Sample Selection 2 3 4

5 Table 1 Descriptive Statistics for WHO Variables, 1997 Observations* 6 7 Non -OECD OECD All 8 Mean Std. Dev. Mean Std. Dev. Mean Std. Dev 9 COMP 70.30 10.96 89.42 3.97 73.30 12.34 10 HEXP 249.17 315.11 1498.27 762.01 445.37 616.36 11 EDUC 5.44 2.38 9.04 1.53 6.00 2.62 12 GINI 0.399 0.0777 0.299 0.0636 0.383 0.0836 13 VOICE -0.195 0.794 1.259 0.534 0.0331 0.926 14 GEFF -0.312 0.643 1.166 0.625 -0.0799 0.835 15 TROPICS 0.596 0.492 0.0333 0.183 0.508 0.501 16 POPDEN 757.9 2816.3 454.56 1006.7 710.2 2616.5 17 PUBFIN 56.89 21.14 72.89 14.10 59.40 20.99 18 GDPC 4449.8 4717.7 18199.07 6978.0 6609.4 7614.8 19 Efficiency 0.5904 0.2012 0.8831 0.0783 0.6364 0.2155 20 161 30 191 21 Sample 22 * Variables in the data set are as follows: 23 COMP = WHO health care attainment measure. 24 HEXP = Per capita health expenditure in PPP units. 25 EDUC = Average years of formal education. 26 GINI = World bank measure of income inequality. 27 VOICE = World bank measure of democratization. 28 GEFF = World bank measure of government effectiveness. 29 TROPICS = Dummy variable for tropical location. 30 POPDEN = Population density in persons per square kilometer. 31 PUBFIN = Proportion of health expenditure paid by government. 32 GDPC = Per capita GDP in PPP units. 33 Efficiency = TMLE estimated efficiency from fixed effects model. 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 18 64 65 1 A Stochastic Frontier Model with Correction for Sample Selection 2 3 4 Table 2 Estimated Stochastic Frontier Models a (Estimated standard errors in parentheses) 5 Non -OECD Co untries OECD Countries 6 7 Stochastic Sample Selection Stochastic Sample Selection Frontier Frontier 8 3.76162 3.74915 3.10994 3.38244 9 Constant (0.05429) (0.05213) (1.15519) (1.42161) 10 0.08388 0.08842 0.04765 0.04340 11 LogHexp (0.01023) (0.010228) (0.006426) (0.008805) 12 0.09096 0.09053 1.00667 0.77422 13 LogEduc (0.075150) (0.073367) (1.06222) (1.2535) 14 0.00649 0.00564 -0.23710 -0.18202 15 2 Log Educ (0.02834) (0.02776) (0.24441) (0.28421) 16 0.12300 0.12859 0.02649 0.01509 17 σσσu 0.05075 0.04735 0.00547 0.01354 18 σσσv 19 λλλ 2.42388 2.71549 4.84042 1.11413 20 σσσ 0.13306 0.13703 0.02705 0.02027 21 0.63967 -0.73001 22 0.0000 0.0000 ρρρ (1.4626) (0.56945) 23 24 logL 160.2753 161.0141 62.96128 65.44358 25 LR test 1.4776 4.9646 26 N 161 30 27 aThe estimated probit model for OECD membership (with estimated standard errors in parentheses) is 28 OECD = -8.2404 (3.369) + 0.7388LogPerCapitaGDP (0.3820) 29 + 0.6098GovernmentEffectiveness (0.4388) 30 + 0.7291Voice (0.3171) 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 19 64 65 1 A Stochastic Frontier Model with Correction for Sample Selection 2 3 4

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 Figure 1 Efficiency Scores Related to Per Capita GDP. 27 Larger points indicate OECD members 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 20 64 65 1 A Stochastic Frontier Model with Correction for Sample Selection 2 3 4

5 6 7 8 Efficiencies from Selection SF Model vs. WHO Estimates 9 1.00 10 11 12 13 .80 14 15 16 17 18 .60 19 20 21 WHOEFF WHOEFF 22 .40 23 24 25 26 .20 27 28 29 30 .00 31 .650 .700 .750 .800 .850 .900 .950 1.000 32 33 EFFSEL 34 35 36 37 Figure 2 Estimated Efficiency Scores 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 21 64 65 1 A Stochastic Frontier Model with Correction for Sample Selection 2 3 4 5 6 Simulation vs. Plug-in Efficiency Estimates 7 1.000 8 9 .950 10 11 .900 12 13 .850 14 15 .800 16 17 .750 EFFJLMS EFFJLMS 18 19 .700 20 21 .650 22 23 .600 24 25 .550 26 .650 .700 .750 .800 .850 .900 .950 1.000 27 EFFSIM 28 29 30 31 Figure 3 Alternative Estimators of Efficiency Scores 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 22 64 65 1 A Stochastic Frontier Model with Correction for Sample Selection 2 3 4

5 6 7 8 9 175 10 11 12 13 140 14 15 16 105 17 18 19 RANKW RANKW 20 70 21 22 23 24 35 25 26 27 28 0 29 0 35 70 105 140 175 30 RANKS 31 32 33 34 Figure 4 Ranks of Countries Based on WHO and 35 Simulation Efficiency Estimates 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 23 64 65 1 A Stochastic Frontier Model with Correction for Sample Selection 2 3 4 5 References 6 7 Aigner, D., K. Lovell, and P. Schmidt, 1977, “Formulation and Estimation of Stochastic 8 Frontier Production Function Models,” Journal of Econometrics , 6, pp. 21-37. 9 Battese, G. and T. 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1 2 3 4 A Stochastic Frontier Model with Correction for Sample Selection 5 * 6 William Greene 7 Department of Economics, Stern School of Business, 8 New York University, 9 March, 2008 10 ______11 12 13 Abstract 14 15 Heckman’s (1979) sample selection model has been employed in three decades of 16 17 applications of linear regression studies. The formal extension of the method to nonlinear 18 19 models, however, is of more recent vintage. A generic solution for nonlinear models is 20 21 proposed in Terza (1998). We have developed simulation based approach in Greene 22 (2006). This paper builds on this framework to obtain a sample selection correction for 23 24 the stochastic frontier model. We first show a surprisingly simple way to estimate the 25 26 familiar normal-half normal stochastic frontier model (which has a closed form log 27 28 likelihood) using maximum simulated likelihood. The next step is to extend the 29 30 technique to a stochastic frontier model with sample selection. Here, the log likelihood 31 does not exist in closed form, and has not previously been analyzed. We develop a 32 33 simulation based estimation method for the stochastic frontier model. In an application 34 35 that seems superficially obvious, the method is used to revisit the World Health 36 37 Organization data [WHO (2000), Tandon et al. (2000)] where the sample partitioning is 38 39 based on OECD membership. The original study pooled all 191 countries. The OECD 40 41 members appear to be discretely different from the rest of the sample. We examine the 42 difference in a sample selection framework. 43 44 JEL classification: C13; C15; C21 45 46 47 Keywords: Stochastic Frontier, Sample Selection, Simulation, Efficiency 48 49 50 51 52 53 54 55 56 57 58 59 * 44 West 4th St., Rm. 7-78, New York, NY 10012, USA, Telephone: 001-212-998-0876; e-mail: 60 [email protected], URL www.stern.nyu.edu/~wgreene. 61 62 63 1 64 65