Rao's score (a.k.a Lagrange Multiplier) diagnostics for spatial dependence in linear models

The function reports the estimates of tests chosen among five statistics for testing for spatial dependence in linear models. The statistics are the simple RS test for error dependence (“RSerr”), the simple RS test for a missing spatially lagged dependent variable (“RSlag”), variants of these adjusted for the presence of the other (“adjRSerr” tests for error dependence in the possible presence of a missing lagged dependent variable, “adjRSlag” the other way round), and a portmanteau test (“SARMA”, in fact “RSerr” + “adjRSlag”). Note: from spdep 1.3-2, the tests are re-named “RS” - Rao's score tests, rather than “LM” - Lagrange multiplier tests to match the naming of tests from the same family in SDM.RStests.

Usage

lm.RStests(model, listw, zero.policy=attr(listw, "zero.policy"), test="RSerr",
 spChk=NULL, naSubset=TRUE)
lm.LMtests(model, listw, zero.policy=attr(listw, "zero.policy"), test="LMerr",
 spChk=NULL, naSubset=TRUE)
# S3 method for class 'RStestlist'
print(x, ...)
# S3 method for class 'RStestlist'
summary(object, p.adjust.method="none", ...)
# S3 method for class 'RStestlist.summary'
print(x, digits=max(3, getOption("digits") - 2), ...)
<!-- %tracew(listw) -->

Arguments

model: an object of class lm returned by lm, or optionally a vector of externally calculated residuals (run though na.omit if any NAs present) for use when only "RSerr" is chosen; weights and offsets should not be used in the lm object
listw: a listw object created for example by nb2listw, expected to be row-standardised (W-style)
zero.policy: default attr(listw, "zero.policy") as set when listw was created, if attribute not set, use global option value; if TRUE assign zero to the lagged value of zones without neighbours, if FALSE assign NA
test: a character vector of tests requested chosen from RSerr, RSlag, adjRSerr, adjRSlag, SARMA; test="all" computes all the tests.
spChk: should the data vector names be checked against the spatial objects for identity integrity, TRUE, or FALSE, default NULL to use get.spChkOption()
naSubset: default TRUE to subset listw object for omitted observations in model object (this is a change from earlier behaviour, when the model$na.action component was ignored, and the listw object had to be subsetted by hand)
x, object: object to be printed
p.adjust.method: a character string specifying the probability value adjustment (see p.adjust) for multiple tests, default "none"
digits: minimum number of significant digits to be used for most numbers
...: printing arguments to be passed through

Details

The two types of dependence are for spatial lag $\rho$ and spatial error $\lambda$:

$$ \mathbf{y} = \mathbf{X \beta} + \rho \mathbf{W_{(1)} y} + \mathbf{u}, $$ $$ \mathbf{u} = \lambda \mathbf{W_{(2)} u} + \mathbf{e} $$

where $\mathbf{e}$ is a well-behaved, uncorrelated error term. Tests for a missing spatially lagged dependent variable test that $\rho = 0$, tests for spatial autocorrelation of the error $\mathbf{u}$ test whether $\lambda = 0$. $\mathbf{W}$ is a spatial weights matrix; for the tests used here they are identical.

Value

A list of class RStestlist of htest objects, each with:

statistic: the value of the Rao's score (a.k.a Lagrange multiplier) test.
parameter: number of degrees of freedom
p.value: the p-value of the test.
method: a character string giving the method used.
data.name: a character string giving the name(s) of the data.

References

Anselin, L. 1988 Spatial econometrics: methods and models. (Dordrecht: Kluwer); Anselin, L., Bera, A. K., Florax, R. and Yoon, M. J. 1996 Simple diagnostic tests for spatial dependence. Regional Science and Urban Economics, 26, 77–104 doi:10.1016/0166-0462(95)02111-6 ; Malabika Koley (2024) Specification Testing under General Nesting Spatial Model, https://sites.google.com/view/malabikakoley/research.

Author

Roger Bivand Roger.Bivand@nhh.no and Andrew Bernat

Examples

data(oldcol)
oldcrime.lm <- lm(CRIME ~ HOVAL + INC, data = COL.OLD)
summary(oldcrime.lm)
#> 
#> Call:
#> lm(formula = CRIME ~ HOVAL + INC, data = COL.OLD)
#> 
#> Residuals:
#>     Min      1Q  Median      3Q     Max 
#> -34.418  -6.388  -1.580   9.052  28.649 
#> 
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)  68.6190     4.7355  14.490  < 2e-16 ***
#> HOVAL        -0.2739     0.1032  -2.654   0.0109 *  
#> INC          -1.5973     0.3341  -4.780 1.83e-05 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 11.43 on 46 degrees of freedom
#> Multiple R-squared:  0.5524,	Adjusted R-squared:  0.5329 
#> F-statistic: 28.39 on 2 and 46 DF,  p-value: 9.341e-09
#> 
lw <- nb2listw(COL.nb)
res <- lm.RStests(oldcrime.lm, listw=lw, test="all")
summary(res)
#> 	Rao's score (a.k.a Lagrange multiplier) diagnostics for spatial
#> 	dependence
#> data:  
#> model: lm(formula = CRIME ~ HOVAL + INC, data = COL.OLD)
#> test weights: lw
#>  
#>          statistic parameter  p.value   
#> RSerr     5.723131         1 0.016743 * 
#> RSlag     9.363684         1 0.002213 **
#> adjRSerr  0.079495         1 0.777983   
#> adjRSlag  3.720048         1 0.053763 . 
#> SARMA     9.443178         2 0.008901 **
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
if (require("spatialreg", quietly=TRUE)) {
  oldcrime.slx <- lmSLX(CRIME ~ HOVAL + INC, data = COL.OLD, listw=lw)
  summary(lm.RStests(oldcrime.slx, listw=lw, test=c("adjRSerr", "adjRSlag")))
}
#> 	Rao's score (a.k.a Lagrange multiplier) diagnostics for spatial
#> 	dependence
#> data:  
#> model: lm(CRIME ~ HOVAL + INC + lag.HOVAL + lag.INC, data = COL.OLD,
#> listw = lw)
#> test weights: lw
#>  
#>              statistic parameter p.value
#> GNM_adjRSerr   0.10933         1  0.7409
#> GNM_adjRSlag   0.61580         1  0.4326