Table 1.

Results of a simulation study from 500 replications under the CAR model. A regular 20 × 20 grid design and an adjacent neighborhood covariance structure with γ = 0.95 were used. The true parameters were β₀ = 0, β_x = 0.5, $\theta =1{\sigma }_{\epsilon }^2=0.3,{\sigma }_{\mathrm{\sum }}^2=1.2,{\sigma }_e^2=0.5$ and α₀ = 1.4. The measurement error variance was ${\sigma }_u^2=0.5$. Inside the brackets are estimated standard errors

Sample Size	Parameter	Mean of naive est	model based Var. of naive est	MSE	Mean of MLE	model based Var. of MLE	MSE
49	β0	0.221	0.095 (0.161)	0.200	−0.045	0.150 (0.210)	0.210
	βx	0.326	0.011 (0.012)	0.040	0.513	0.030 (0.043)	0.043
	θ	1.141	0.355 (0.383)	0.401	1.041	0.283 (0.367)	0.368
	${\sigma }_{\epsilon }^2$	0.461	0.054(0.063)	0.090	0.374	0.045 (0.056)	0.061
100	β0	0.261	0.062(0.072)	0.14	−0.042	0.079 (0.090)	0.09
	βx	0.323	0.005 (0.005)	0.036	0.507	0.0157 (0.0157)	0.016
	θ	1.066	0.222 (0.265)	0.269	0.957	0.209 (0.239)	0.240
	${\sigma }_{\epsilon }^2$	0.399	0.0268 (0.033)	0.043	0.326	0.0312 (0.029)	0.030
400	β0	0.247	0.0160 (0.0154)	0.076	0.0032	0.020 (0.022)	0.022
	βx	0.318	0.0012 (0.0012)	0.034	0.503	0.0035(0.0035)	0.0035
	θ	1.015	0.067 (0.073)	0.073	0.989	0.062 (0.072)	0.072
	${\sigma }_{\epsilon }^2$	0.376	0.0068 (0.0069)	0.012	0.304	0.0068 (0.0072)	0.0072