Essential Statistical Concepts, Definitions, and Terminology - Sampling Spatial Units for Agricultural Surveys - page 26

> SEM < - errorsarlm((YIELD~N+N2), data¼ LasRosas@data, LasRosas_listw)
> summary(SEM)

Call:errorsarlm(formula ¼ (YIELD ~ N + N2), data ¼ LasRosas@data,
listw ¼ LasRosas_listw, tol.solve ¼ 1e-18)

Residuals:
Min 1Q Median 3Q Max

-21.1655173 -2.2527988 0.0080839 2.1378768 26.9502085

Type: error
Coefficients: (asymptotic standard errors)
Estimate

Std. Error z value Pr( > |z|)

(Intercept) 59.003675276 0.884378583 66.7177 < 2.2e-16

< 2.2e-16

N 0.108269927 0.006378146 16.9751

N2 -0.000220194 0.000045639 -4.8247 0.000001402

Lambda: 0.89754, LR test value: 2459.7, p-value: < 2.22e-16
Asymptotic standard error: 0.012503
z-value: 71.787, p-value: < 2.22e-16
Wald statistic: 5153.4, p-value: < 2.22e-16

Log likelihood: -4909.478 for error model
ML residual variance (sigma squared): 13.838, (sigma: 3.7199)
Number of observations: 1738
Number of parameters estimated: 5
AIC: 9829, (AIC for lm: 12287)

Note that the parameter of Eq. ( 1.52 ) is replaced with Lambda in the R output.

Considering Eq. ( 1.52 ) and using a little algebra

ˁ

ð I ˁ W Þ y ¼ I ˁ ð W Þ X ʲ þ ε;

and hence

y ¼ ˁ Wy þ X ʲ ˁ WX ʲ þ ε;

or (Anselin 1988 )

:

2 I

y ¼ ˁ Wy þ X ʲ þ WX ˈ þ ε ε / N 0, ˃ ð 1 : 53 Þ

The model in Eq. ( 1.53 ) is known in the spatial econometrics literature as the
spatial Durbin model (SDM, LeSage and Pace 2009 ).
The terms Wy and WX are called spatially lagged variables. The more complex
SDM model in Eq. ( 1.53 ) can be reduced to the simpler SEM model in Eq. ( 1.52 ), if
some constraints on the coefficients of SDM are satisfied. For more details about