# 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. ```r 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` ## Returns 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, tools:::Rd_expr_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](https://hdl.handle.net/2142/124329). ## Author(s) 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). ## See Also `lm`, `lm.RStests` ## Examples ```r columbus <- sf::st_read(system.file("shapes/columbus.gpkg", package="spData")[1]) 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")) res <- SD.RStests(lm_obj, col.listw, test="SDM") summary(res) all.equal(unname(res$SDM_Joint$statistic), unname(res$SDM_RSlag$statistic + res$SDM_adjRSWX$statistic)) all.equal(unname(res$SDM_Joint$statistic), unname(res$SDM_adjRSlag$statistic + res$SDM_RSWX$statistic)) res <- SD.RStests(lm_obj, col.listw, test="SDEM") summary(res) all.equal(unname(res$SDEM_Joint$statistic), unname(res$SDEM_RSerr$statistic + res$SDEM_RSWX$statistic)) summary(SD.RStests(lm_obj, nb2listw(col.gal.nb, style="C"), test="all")) summary(SD.RStests(lm_obj, col.listw, test="all", Durbin= ~ INC)) lm_obj0 <- lm(I(scale(CRIME)) ~ 0 + I(scale(INC)) + I(scale(HOVAL)), data=columbus) summary(SD.RStests(lm_obj0, col.listw, test="all")) columbusNA <- columbus columbusNA$HOVAL[15] <- NA lm_objNA <- lm(CRIME ~ INC + HOVAL, data=columbusNA) summary(SD.RStests(lm_objNA, col.listw, test="all")) ```

SD.RStests function