Stochastic Frontier Estimation Capabilities: LIMDEP and Stata (Richard Hofler, University of Central Florida)

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1 Stochastic Frontier Estimation Capabilies: LIMDEP and (Richard Hofler, Uniersy of Central Florida) PREFACE Stochastic Frontier and Data Enelopment Analysis Software Only two of the large integrated econometrics programs crrently in general se proide programs and rotines for frontier and efficiency analysis, LIMDEP/NLOGIT and. The freeware program, FRONTIER 4.1 by Tim Coelli (find on the web) can also be sed for a small range of stochastic frontier models. FRONTIER is now rather old, howeer, and is not being pdated or maintained. This docment, prepared by Richard Hofler of the Uniersy of Central Florida, compares the capabilies of LIMDEP and for estimation of stochastic frontier models. The description is crrent as of 006. Some addional capabilies hae since been added to LIMDEP, notably the simlation based estimator for a sample selection corrected stochastic frontier model. Other featres and models are added to LIMDEP on an ongoing basis this will be one of the sbstantie pdates in the next ersion of LIMDEP (10.0) To my knowledge, no frther deelopment of the frontier capabilies beyond those listed here has been done or is ongoing in. Data Enelopment Analysis There are seeral packages that specialize in Data Enelopment Analysis (DEA) a search of the web for this topic will amply demonstrate the ariety of tools aailable for this mini-indstry. Some of them are qe extensie. LIMDEP also contains a program for data enelopment analysis. To my knowledge, LIMDEP is the only program (small or large) that proides both stochastic frontier and DEA capabilies. This rotine is, as of early 008, also ndergoing deelopment and will be extended in ersion 10.0 of LIMDEP. This docment does not describe featres of LIMDEP related to DEA. Tim Coelli has also deeloped another, separate program, DEAP for data enelopment analysis. Like FRONTIER, howeer, DEAP is rather old 1996, and is not crrent wh methodological deelopments of the last decade. Howeer, is freeware, and does proide the basic capabilies needed by the entry leel analyst. William Greene, New York, Febrary 1, 009

2 Cross Section Data Models Base model () y= β x+ where = U, U N(0, )and N(0, ) Cross Section Formlations Behaioral Assmptions maximizing (e.g., prodction) minimizing (e.g., cost) Distribtional Variations Normal exponential parameter = λ Normal gamma, parameters = λ,p Normal - trncated normal Assmptions abot mean of (inefficiency term) E[] = 0 () E[] = μ (N-trncated N wh constant mean) E[] = μ i = α z i (N-trncated N wh heterogeneos mean) = α z + w (w trncated N sch that 0 or 0) Variance of ar[] = (homoskedasticy) ar[] = i = exp( γ z i ) (heteroskedasticy) Scaling in Exponential (and Gamma) λ = exp(δ w) Variance of ar[] = (homoskedasticy) ar[] = = exp( δ w) (heteroskedasticy) Dobly Heteroskedastic ar[] = i = exp( γ z i ) and ar[] = = exp( δ w) Dobly Heteroscedastic and Nonzero Trncation E[] = μ i = α z i Other Reqires all data be in natral logarhms Confidence interals (note 1)

3 Panel Data Models Random Effects Base model () y = β x + i where i = N(0, ) Panel Data Formlations Random Effects Formlations ( - i : note time inariant) Behaioral Assmptions maximizing (e.g., prodction) minimizing (e.g., cost) Alarez-Amsler Scaling Model Distribtional Variations Normal - trncated normal Assmptions abot mean of (inefficiency term) E[] = 0 () E[] = μ (N-trncated N wh constant mean) E[] = μ i = α z i (N-trncated N wh heterogeneos mean) Other Battese-Coelli (199, 1995) model (note ) Reqires all data be in natral logarhms Confidence interals (note 1) Time-inariant inefficiency terms Time-arying inefficiency terms (note 3 & note 4)

4 Panel Data Models Fixed Effects Base model (Normal half-normal) (note 5) Unobsered heterogeney enters intercept α i (note i time inariant) y = αi + β x + i where = N(0, ) i Panel Data Formlations Fixed Effects Formlations (α i contains nobsered heterogeney; i : note time inariant) Behaioral Assmptions maximizing (e.g., prodction) minimizing (e.g., cost) Distribtional Variations Normal - trncated normal NOTE: LIMDEP fs seeral kinds of fixed effects models wh +/- AND a fixed effect in addion. Addional fixed effects may appear in the main eqation, in the mean of the trncated normal or in the ariance of the trncated normal (or any two of the three). Assmptions abot mean of (inefficiency term) E[] = 0 () E[] = μ (N-trncated N wh constant mean) E[] = μ i = α z i (N-trncated N wh heterogeneos mean)

5 3. Unobsered heterogeney enters mean of (inefficiency term) (note time arying) y = β x + where i = N( μi, ) Assmptions abot mean of (inefficiency term) E[] = 0 () μ i = α i + μ (N-trncated N wh constant mean) μ i = α i + α z i (N-trncated N wh dobly heterogeneos mean) Panel Data Models: Fixed Effects model (cont.) 4. Unobsered heterogeney enters inefficiency term ariance (note time arying) y = β x + where i = N(0, i) Assmptions abot mean of (inefficiency term) E[] = 0 () i = α i + μ (N-trncated N wh constant mean) (see aboe) i = α i + α z i (N-trncated N wh dobly heterogeneos mean) (see aboe) Variance of ar[] = = exp( αi + δ z )

6 Random Parameters (Random Effects) Formlation y = β x + i i where = N = (0, ) N μ (, ) μ = δ z i = exp( γ w ) i β (note: ) Behaioral Assmptions maximizing (e.g., prodction) minimizing (e.g., cost) Distribtional Variations Normal - trncated normal Assmptions abot mean of (inefficiency term) E[] = 0 () E[] = μ (N-trncated N wh constant mean) E[] = μ = δ ι z (N-trncated N wh heterogeneos mean) Random Parameters (Random Effects) Formlation (cont.) Variance of ar[] = (homoskedasticy) ar[] = i = exp( γ z i ) (firmwise heteroskedasticy) ar[] = t = exp( γ z t ) (timewise heteroskedasticy) ar[] = = exp( γ z ) (firmwise/ timewise heteroskedasticy)

7 Latent Class Formlation y j= β x + (note: β Jclasses) j j where = (0, j ) j N i = (0, j) j N Behaioral Assmptions maximizing (e.g., prodction) minimizing (e.g., cost) Distribtional Variations Normal - trncated normal Assmptions abot mean of (inefficiency term) E[] = 0 () E[] = μ (N-trncated N wh constant mean) E[] = μ = δ ι z (N-trncated N wh heterogeneos mean) Variance of ar[] = (homoskedasticy) ar[] = i = exp( γ ) (firmwise heteroskedasticy) ar[] = ar[] = t = = z i exp( γ ) (timewise heteroskedasticy) z t exp( γ ) (firmwise/ timewise heteroskedasticy) z Notes 1 Horrace & Schmidt (1996) contains formlas for calclating the correct CIs in the stochastic frontier model. I hae wrten LIMDEP commands to calclate those H & S CIs. The Battese-Coelli (1995) model concrrently estimates the parameters of the stochastic frontier model and the coefficients of a model that relates selected determinants of inefficiency ( z ariables) to the inefficiency estimates. 3 : It estimates the Battese-Coelli (199) model in which all firms series of inefficiency estimates follow the same time path. 4 LIMDEP: I hae wrten LIMDEP commands to allow each firm s series of inefficiency estimates to (potentially) follow s own niqe time path.