Rao's score and adjusted Rao's score tests of linear hypotheses for spatial Durbin and spatial Durbin error models

Rao's score and adjusted Rao's score tests of linear hypotheses applied to a fitted linear model to examine whether either the spatially lagged dependent variable lag or the spatially lagged independent variable(s) WX should be included in the model, or both (SDM). Adjusted tests are provided for lag and WX adapting to the presence of the other, and a joint test for both. The joint test is equal to the unadjusted of one plus the adjusted of the other. In addition, draft tests are added from Koley (2024, section 6) for spatial Durbin error models to examine whether either the spatially lagged error err or the spatially lagged independent variable(s) WX should be included in the model, or both (SDEM); because of orthogonality, no adjusted tests are required.

Because the spatial lags of categorical variables are poorly understood, from version 1.3-11 warnings are given when categorical variables are included in the Durbin term.

Usage

SD.RStests(model, listw, zero.policy = attr(listw, "zero.policy"), test = "SDM",
 Durbin = TRUE)
have_factor_preds_mf(mf)
warn_factor_preds(x)

Arguments

model: an object of class lm returned by lm
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: test=“SDM” computes the SDM tests, a character vector of tests requested chosen from SDM_RSlag, SDM_adjRSlag, SDM_RSWX, SDM_adjRSWX, SDM_Joint, test=“SDEM” computes the SDEM tests, a character vector of tests requested chosen from SDEM_RSerr, SDEM_RSWX, SDEM_Joint; test=“all” computes all the tests
Durbin: default TRUE for Durbin models including WX; if TRUE, full spatial Durbin model; if a formula object, the subset of explanatory variables to lag
mf: model.frame object
x: object created by have_factor_preds_mf

Value

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

statistic: the value of the 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

Malabika Koley and Anil K. Bera (2024) To use, or not to use the spatial Durbin model? – that is the question, Spatial Economic Analysis, 19:1, 30-56, doi:10.1080/17421772.2023.2256810 ; Malabika Koley (2024) Specification testing of spatial econometric models, Ph.D. dissertation, University of Illinois at Urbana-Champaign, pp. 78-118, 148-156; https://hdl.handle.net/2142/124329.

Author

Roger Bivand Roger.Bivand@nhh.no, Malabika Koley and Anil K. Bera

Note

The results in the example below agree with those in Table 3, p. 22 in Koley and Bera (2024).

Examples

columbus <- sf::st_read(system.file("shapes/columbus.gpkg", package="spData")[1])
#> Reading layer `columbus' from data source 
#>   `/home/rsb/lib/r_libs/spData/shapes/columbus.gpkg' using driver `GPKG'
#> Simple feature collection with 49 features and 20 fields
#> Geometry type: POLYGON
#> Dimension:     XY
#> Bounding box:  xmin: 5.874907 ymin: 10.78863 xmax: 11.28742 ymax: 14.74245
#> Projected CRS: Undefined Cartesian SRS with unknown unit
col.gal.nb <- read.gal(system.file("weights/columbus.gal", package="spData")[1])
col.listw <- nb2listw(col.gal.nb, style="W")
columbus$fEW <- factor(columbus$EW)
columbus$fDISCBD <- ordered(cut(columbus$DISCBD, c(0, 1.5, 3, 4.5, 6)))
f <- formula(CRIME ~ INC + HOVAL + fDISCBD*fEW)
lm_obj <- lm(f, data=columbus)
summary(lm.RStests(lm_obj, col.listw, test="all"))
#> 	Rao's score (a.k.a Lagrange multiplier) diagnostics for spatial
#> 	dependence
#> data:  
#> model: lm(formula = f, data = columbus)
#> test weights: col.listw
#>  
#>          statistic parameter p.value
#> RSerr     0.163267         1  0.6862
#> RSlag     0.021092         1  0.8845
#> adjRSerr  0.831452         1  0.3619
#> adjRSlag  0.689277         1  0.4064
#> SARMA     0.852544         2  0.6529
res <- SD.RStests(lm_obj, col.listw, test="SDM")
#> Warning: use of spatially lagged factors (categorical variables)
#> fDISCBD, fEW
#> is not well-understood
#> Warning: In addition variable:
#> fDISCBD
#> is ordered (ordinal) with contrasts:
#> contr.poly
summary(res)
#> 	Rao's score test spatial Durbin diagnostics
#> data:  
#> model: lm(formula = f, data = columbus)
#> weights: col.listw
#>  
#>              statistic parameter p.value
#> SDM_RSlag     0.021092         1  0.8845
#> SDM_adjRSlag  0.163267         1  0.6862
#> SDM_RSWX     11.804388         9  0.2246
#> SDM_adjRSWX  11.946563         9  0.2163
#> SDM_Joint    11.967655        10  0.2872
all.equal(unname(res$SDM_Joint$statistic),
 unname(res$SDM_RSlag$statistic + res$SDM_adjRSWX$statistic))
#> [1] TRUE
all.equal(unname(res$SDM_Joint$statistic),
 unname(res$SDM_adjRSlag$statistic + res$SDM_RSWX$statistic))
#> [1] TRUE
res <- SD.RStests(lm_obj, col.listw, test="SDEM")
#> Warning: use of spatially lagged factors (categorical variables)
#> fDISCBD, fEW
#> is not well-understood
#> Warning: In addition variable:
#> fDISCBD
#> is ordered (ordinal) with contrasts:
#> contr.poly
summary(res)
#> 	Rao's score test spatial Durbin diagnostics
#> data:  
#> model: lm(formula = f, data = columbus)
#> weights: col.listw
#>  
#>            statistic parameter p.value
#> SDEM_RSerr   0.16327         1  0.6862
#> SDEM_RSWX   11.80439         9  0.2246
#> SDEM_Joint  11.96765        10  0.2872
all.equal(unname(res$SDEM_Joint$statistic),
 unname(res$SDEM_RSerr$statistic + res$SDEM_RSWX$statistic))
#> [1] TRUE
summary(SD.RStests(lm_obj, nb2listw(col.gal.nb, style="C"), test="all"))
#> Warning: Spatial weights matrix not row standardized
#> Warning: use of spatially lagged factors (categorical variables)
#> fDISCBD, fEW
#> is not well-understood
#> Warning: In addition variable:
#> fDISCBD
#> is ordered (ordinal) with contrasts:
#> contr.poly
#> 	Rao's score test spatial Durbin diagnostics
#> data:  
#> model: lm(formula = f, data = columbus)
#> weights: nb2listw(col.gal.nb, style = "C")
#>  
#>              statistic parameter p.value
#> SDM_RSlag      0.73504         1  0.3913
#> SDM_adjRSlag   0.19451         1  0.6592
#> SDM_RSWX      11.11186        10  0.3489
#> SDM_adjRSWX   10.57132        10  0.3919
#> SDM_Joint     11.30636        11  0.4180
#> SDEM_RSerr     0.19451         1  0.6592
#> SDEM_RSWX     11.11186        10  0.3489
#> SDEM_Joint    11.30636        11  0.4180
summary(SD.RStests(lm_obj, col.listw, test="all", Durbin= ~ INC))
#> 	Rao's score test spatial Durbin diagnostics
#> data:  
#> model: lm(formula = f, data = columbus)
#> weights: col.listw
#> Durbin: ~ INC
#>  
#>              statistic parameter p.value
#> SDM_RSlag    0.0210916         1  0.8845
#> SDM_adjRSlag 0.0046891         1  0.9454
#> SDM_RSWX     0.0344804         1  0.8527
#> SDM_adjRSWX  0.0180779         1  0.8930
#> SDM_Joint    0.0391696         2  0.9806
#> SDEM_RSerr   0.1632670         1  0.6862
#> SDEM_RSWX    0.0344804         1  0.8527
#> SDEM_Joint   0.1977475         2  0.9059
lm_obj0 <- lm(I(scale(CRIME)) ~ 0 + I(scale(INC)) + I(scale(HOVAL)),
 data=columbus)
summary(SD.RStests(lm_obj0, col.listw, test="all"))
#> 	Rao's score test spatial Durbin diagnostics
#> data:  
#> model: lm(formula = I(scale(CRIME)) ~ 0 + I(scale(INC)) +
#> I(scale(HOVAL)), data = columbus)
#> weights: col.listw
#>  
#>              statistic parameter  p.value   
#> SDM_RSlag       7.8250         1 0.005153 **
#> SDM_adjRSlag    4.6111         1 0.031765 * 
#> SDM_RSWX        6.0609         2 0.048295 * 
#> SDM_adjRSWX     2.8470         2 0.240873   
#> SDM_Joint      10.6720         3 0.013638 * 
#> SDEM_RSerr      4.6111         1 0.031765 * 
#> SDEM_RSWX       6.0609         2 0.048295 * 
#> SDEM_Joint     10.6720         3 0.013638 * 
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
columbusNA <- columbus
columbusNA$HOVAL[15] <- NA
lm_objNA <- lm(CRIME ~ INC + HOVAL, data=columbusNA)
summary(SD.RStests(lm_objNA, col.listw, test="all"))
#> 	Rao's score test spatial Durbin diagnostics
#> data:  
#> model: lm(formula = CRIME ~ INC + HOVAL, data = columbusNA)
#> weights: col.listw
#>  
#>              statistic parameter  p.value   
#> SDM_RSlag       7.6010         1 0.005834 **
#> SDM_adjRSlag    4.4716         1 0.034463 * 
#> SDM_RSWX        6.0812         2 0.047807 * 
#> SDM_adjRSWX     2.9518         2 0.228576   
#> SDM_Joint      10.5527         3 0.014407 * 
#> SDEM_RSerr      4.4716         1 0.034463 * 
#> SDEM_RSWX       6.0812         2 0.047807 * 
#> SDEM_Joint     10.5527         3 0.014407 * 
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1