Estimation of Stochastic Frontier Models with Fixed Effects through Monte Carlo Maximum Likelihood

Table 1

Simulation results for the stochastic frontier model.

		𝑁=500, 𝑇=4			𝑁=250, 𝑇=8			𝑁=125, 𝑇=16
Estimator	Parameter	Mean	MSE	Bias	Mean	MSE	Bias	Mean	MSE	Bias
Linear fixed effects	𝛽1=0.7	0.7008	0.0005	0.1%	0.7001	0.0004	0.0%	0.7007	0.0004	0.1%
	𝛽2=0.4	0.3997	0.0005	−0.1%	0.4004	0.0004	0.1%	0.3999	0.0004	0.0%
	𝜎=(𝜎^2𝑣+𝜆^2)	0.4242	0.0001	0.0%	0.4243	0.0001	0.0%	0.4243	0.0001	0.0%
Monte Carlo maximum likelihood	𝛽1=0.7	0.7002	0.0004	0.0%	0.6992	0.0004	−0.1%	0.7001	0.0004	0.0%
	𝛽2=0.4	0.3993	0.0004	−0.2%	0.3997	0.0004	−0.1%	0.3997	0.0004	−0.1%
	log𝜎^2𝑣	−2.4094	0.0100	−0.1%	−2.4123	0.0065	−0.2%	−2.4130	0.0054	−0.2%
	log𝜆^2	−2.4225	0.0196	−0.6%	−2.4134	0.0144	−0.2%	−2.4112	0.0127	−0.1%
	𝜎𝑣=0.3	0.3002	0.0002	0.1%	0.2996	0.0001	−0.1%	0.2995	0.0001	−0.2%
	𝜆=0.3	0.2985	0.0004	−0.5%	0.2997	0.0003	−0.1%	0.3000	0.0003	0.0%
Simple random effects	𝛽1=0.7	0.8518	0.0236	21.7%	0.7845	0.0077	12.1%	0.7419	0.0022	6.0%
	𝛽2=0.4	0.5513	0.0234	37.8%	0.4851	0.0078	21.3%	0.4417	0.0022	10.4%
	log𝜎^2𝑣	−2.2901	0.0237	4.9%	−2.3711	0.0083	1.5%	−2.4030	0.0055	0.2%
	log𝜆^2	−2.4197	0.0210	−0.5%	−2.4152	0.0153	−0.3%	−2.4114	0.0128	−0.1%
	𝜎𝑣=0.3	0.3186	0.0006	6.2%	0.3058	0.0002	1.9%	0.3010	0.0001	0.3%
	𝜆=0.3	0.2990	0.0005	−0.3%	0.2995	0.0003	−0.2%	0.3000	0.0003	0.0%
Correlated random effects	𝛽1=0.7	0.7006	0.0004	0.1%	0.6997	0.0004	0.0%	0.7006	0.0003	0.1%
	𝛽2=0.4	0.3997	0.0004	−0.1%	0.4002	0.0004	0.1%	0.4003	0.0003	0.1%
	log𝜎^2𝑣	−2.4129	0.0097	−0.2%	−2.4153	0.0066	−0.3%	−2.4176	0.0056	−0.4%
	log𝜆^2	−2.4150	0.0187	−0.3%	−2.4119	0.0143	−0.2%	−2.4106	0.0126	−0.1%
	𝜎𝑣=0.3	0.2996	0.0002	−0.1%	0.2991	0.0002	−0.3%	0.2988	0.0001	−0.4%
	𝜆=0.3	0.2996	0.0004	−0.1%	0.2999	0.0003	0.0%	0.3001	0.0003	0.0%