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#' @name wald_deltas
#' @rdname wald_deltas
#'
#' @title Wald tests for spatial parameters coefficients.
#'
#' @description
#' Function \code{\link{wald_deltas}} obtains Wald tests for linear
#' restrictions on the spatial coefficients of a SUR model that has been
#' estimated previously through the function \code{\link{spsurml}}. The
#' restrictions can affect to coefficients of the same equation
#' (i.e., \eqn{\lambda_{g}=\rho_{g} forall g}) or can involve coefficients
#' from different equations (i.e., \eqn{\lambda_{g}=\lambda_{h}}). The
#' function has great flexibility in this respect. Note that
#' \code{\link{wald_deltas}} only works in a maximum-likelihood framework.
#'
#' In order to work with \code{\link{wald_betas}}, the model on which the
#' linear restrictions are to be tested needs to exists as an \emph{spsur}
#' object. Using the information contained in the object,
#' \code{\link{wald_deltas}} obtains the corresponding Wald statistic for
#' the null hypotheses specified by the user through the \emph{R} row vector
#' and \emph{b} column vector discussed, used also in \code{\link{spsurml}}.
#' The function shows the resulting Wald test statistics and their
#' corresponding p-values.
#'
#' @usage wald_deltas (obj , R , b)
#'
#' @param obj An \code{spsur} object created by \code{\link{spsurml}},
#' \code{\link{spsur3sls}} or \code{\link{spsurtime}}.
#' @param R A row vector of order \emph{(1xGr)} or \emph{(1x2Gr)} showing
#' the set of \emph{r} linear constraints on the spatial parameters. The
#' last case is reserved to "sarar" models where there appear
#' \emph{G} parameters \eqn{\lambda_{g}} and \emph{G} parameters
#' \eqn{\rho_{g}}, \emph{2G} spatial in total. The \emph{first}
#' restriction appears in the first \emph{G} terms in \emph{R}
#' (\emph{2G} for the "sarar" case), the second restriction
#' in the next \emph{G} terms (\emph{2G} for the
#' "sarar" case) and so on.
#' @param b A column vector of order \emph{(rx1)} with the values
#' of the linear restrictions on the \eqn{\beta} parameters.
#'
#' @return Object of \code{htest} including the Wald
#' statistic, the corresponding p-value, the degrees of
#' freedom and the values of the sample estimates.
#'
#' @author
#' \tabular{ll}{
#' Fernando Lopez \tab \email{fernando.lopez@@upct.es} \cr
#' Roman Minguez \tab \email{roman.minguez@@uclm.es} \cr
#' Jesus Mur \tab \email{jmur@@unizar.es} \cr
#' }
#'
#' @seealso
#' \code{\link{spsurml}}, \code{\link{spsur3sls}}
#' @examples
#'
#' #################################################
#' ######## CROSS SECTION DATA (G>1; Tm=1) ########
#' #################################################
#' rm(list = ls()) # Clean memory
#' data(spc, package = "spsur")
#' lwspc <- spdep::mat2listw(Wspc, style = "W")
#' Tformula <- WAGE83 | WAGE81 ~ UN83 + NMR83 + SMSA | UN80 + NMR80 + SMSA
#'
#' #################################
#' ## Estimate SUR-SLM model
#' spcsur.slm <-spsurml(formula = Tformula, data = spc,
#' type = "slm", listw = lwspc)
#' summary(spcsur.slm)
#' ## H_0: equality of the lambda parameters of both equations.
#' R1 <- matrix(c(1,-1), nrow=1)
#' b1 <- matrix(0, ncol=1)
#' wald_deltas(spcsur.slm, R = R1, b = b1)
#' @export
wald_deltas <- function(obj , R , b){
z <- obj
deltas <- Matrix::Matrix(matrix(z$deltas, ncol = 1))
rownames(deltas) <- names(z$deltas)
cov_deltas <- Matrix::Matrix(z$resvar[rownames(deltas),
rownames(deltas)])
R <- Matrix::Matrix(R)
b <- Matrix::Matrix(matrix(b,ncol=1))
holg <- (R %*% deltas) - b
parameter <- nrow(as.matrix(R))
attr(parameter, "names") <- "df"
statistic <- as.numeric(Matrix::t(holg) %*%
Matrix::solve(R %*% cov_deltas %*%
Matrix::t(R),holg) )
attr(statistic, "names") <- "Wald test"
method <- paste("Wald test on spatial delta parameters")
p.value <- pchisq(statistic, df = parameter,
lower.tail = FALSE)
estimate <- as.numeric(deltas)
names(estimate) <- rownames(deltas)
data.name <- z$call[[3]]
res <- list(statistic = statistic, parameter = parameter,
p.value = p.value, # estimate = estimate,
method = method, data.name = data.name)
class(res) <- "htest"
res
}
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