THE EM ALGORITHM FOR STANDARD STOCHASTIC FRONTIER MODELS

Andrade, Bernardo B. de; Souza, Geraldo S.

doi:10.1590/0101-7438.2019.039.03.0361

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Articles • Pesqui. Oper. 39 (3) • Sep-Dec 2019 • https://doi.org/10.1590/0101-7438.2019.039.03.0361 copy

THE EM ALGORITHM FOR STANDARD STOCHASTIC FRONTIER MODELS

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

The Expectation-Maximization (EM) algorithm is developed for the stochastic frontier models most used in practice with cross-section data. The resulting algorithms can be easily programmed into a computer and are shown to be worthy alternatives to general-purpose optimization routines currently used. The algorithms for the half normal and the exponential models have closed-form expressions whereas those for the truncated normal and gamma models will require the numerical solution of a nonlinear equation. Implementations of the EM algorithm either as a stand-alone routine or in accelerated form and also combined with Newton-like methods are discussed. We provide illustrations, along with R tools, for cost and production frontiers.

Keywords:
efficiency; EM acceleration; gamma; maximum likelihood

	${\hat{β}}_{0}$	${\hat{β}}_{1}$	${\hat{β}}_{2}$	${\hat{σ}}_{u}^{2}$	${\hat{σ}}_{v}^{2}$
MLE	1.18	0.73	0.24	0.15	0.22
Standard Errors
OPG (see eq. (14))	(.59)	(.03)	(.03)	(.10)	(.04)
Hybrid EM (see II.1)	(.60)	(.06)	(.05)	(.12)	(.05)
SQUAREM (see II.2)	(.56)	(.04)	(.03)	(.14)	(.05)
Stata14	(.60)	(.06)	(.05)	(.12)	(.05)

	${\hat{β}}_{0}$	${\hat{β}}_{1}$	${\hat{β}}_{2}$	${\hat{σ}}_{u}^{2}$ †	${\hat{σ}}_{v}^{2}$
MLE	1.59	0.70	0.26	0.07	0.20
Standard Errors
OPG (see eq. (14))	(.58)	(.04)	(.03)	(.02)	0.20	(.03)
Hybrid EM (see II.1)	(.60)	(.06)	(.05)	(.03)	(.03)
SQUAREM (see II.2)	(.58)	(.02)	(.03)	(.03)	(.03)
Stata14	(.60)	(.06)	(.05)	(.03)	(.03)

	${\hat{β}}_{0}$	${\hat{β}}_{1}$	${\hat{β}}_{2}$	${\hat{β}}_{3}$	${\hat{β}}_{4}$	${\hat{σ}}_{v}^{2}$	$\hat{α}$	$\hat{λ}$	$\hat{l}$
EM	-7.044	0.146	0.135	0.455	0.028	0.012	0.258	5.876	93.4
	(.22)	(.04)	(.04)	(.04)	(.002)	(.002)	(.15)	(.01)
SML	-7.034	0.145	0.138	0.445	0.028	0.011	1.082	9.577	91.6

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[1] *Corresponding author