Spatial-lag omission is bias, spatial-error omission is inefficiency

For the two-unit swap model \(y = \rho W y + u\) the lag regressor's covariance with the error is \(\rho/(1-\rho^2)\) (Research.P3.Spatial.lag_regressor_correlated); for a regressor x with error covariance Sigma the slope variances are \(x^T\Sigma x/(x^Tx)^2\) (OLS) and \(1/(x^T\Sigma^{-1}x)\) (GLS) (gls_more_precise); m uncorrected local tests at level alpha reject somewhere with probability \(1-(1-\alpha)^m\) (fwer_tendsto_one).

Usage

morie_spatial_omission(rho, x, Sigma, alpha = 0.05, m = 1L)

Arguments

rho

Spatial-lag coefficient with \(\rho^2 \ne 1\).

x

Regressor vector.

Sigma

Error covariance matrix.

alpha, m

Level and number of local tests.

Value

A list: lag_covariance, var_ols, var_gls, gls_more_precise, fwer.

References

Kikuchi G (2010). Neighborhood Structures and Crime. LFB Scholarly.

Examples

morie_spatial_omission(rho = 0.4, x = c(1, 2), Sigma = matrix(c(1, 0.5, 0.5, 1), 2), m = 50)
#> $lag_covariance
#> [1] 0.4761905
#> 
#> $var_ols
#> [1] 0.28
#> 
#> $var_gls
#> [1] 0.25
#> 
#> $gls_more_precise
#> [1] TRUE
#> 
#> $fwer
#> [1] 0.923055
#> 
#> attr(,"theorem")
#> [1] "Research.P3.Spatial.lag_regressor_correlated"