spsur (version 1.0.0.4)

wald_deltas: Wald tests for spatial parameters coefficients.

Description

Function wald_deltas obtains Wald tests for linear restrictions on the spatial coefficients of a SUR model that has been estimated previously through the function spsurml. The restrictions can affect to coefficients of the same equation (i.e., \(\lambda_{g}=\rho_{g} forall g\)) or can involve coefficients from different equations (i.e., \(\lambda_{g}=\lambda_{h}\)). The function has great flexibility in this respect. Note that wald_deltas only works in a maximum-likelihood framework.

In order to work with wald_betas, the model on which the linear restrictions are to be tested needs to exists as an spsur object. Using the information contained in the object, wald_deltas obtains the corresponding Wald estatistic for the null hypotheses specified by the user through the R row vector and b column vector discussed, used also in spsurml. The function shows the resulting Wald test statistics and their corresponding p-values.

Usage

wald_deltas(results, R, b)

Arguments

results

: An object created with spsurml or spsur3sls. This argument serves the user to indicate the spatial SUR model, previously estimated by maximum-likelihood or 3sls, where the set of linear restrictions are to be tested.

R

: A row vector of order (1xGr) or (1x2Gr) showing the set of r linear constraints on the spatial parameters. The last case is reserved to "sarar" models where there appear G parameters \(\lambda_{g}\) and G parameters \(\rho_{g}\), 2G spatial parameters in total. The first restriction appears in the first G terms in R (2G for the "sarar" case), the second restriction in the next G terms (2G for the "sarar" case) and so on. Default = NULL.

b

: A column vector of order (rx1) with the values of the linear restrictions on the \(\beta\) parameters. Default = NULL.

Value

The output of the function is very simple and consists of two pieces of information, the value of the Wald statistic and the corresponding p-value, plus the degrees of freedom of the test.

Wald stat	The value of Wald test.
p_val	The p-value of Wald test.
q	Degrees of freedom of the corresponding \(\chi^{2}\) distribution.

See Also

spsurml, spsur3sls

Examples

# NOT RUN {
#################################################
######## CROSS SECTION DATA (G>1; Tm=1) ########
#################################################
rm(list = ls()) # Clean memory
data(spc)
Tformula <- WAGE83 | WAGE81 ~ UN83 + NMR83 + SMSA | UN80 + NMR80 + SMSA

#################################
## Estimate SUR-SLM model
spcsur.slm <-spsur3sls(Form = Tformula, data = spc, type = "slm", W= Wspc)
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(results = spcsur.slm, R = R1, b = b1)
# }
# NOT RUN {
#################################
## Estimate SUR-SEM model
#' ## It usually requires 1-2 minutes maximum
spcsur.sem <-spsurml(Form = Tformula, data = spc, type = "sem", W = Wspc)
summary(spcsur.sem)
## H_0: equality of the rho parameters of both equations.
R2 <- matrix(c(1,-1), nrow=1)
b2 <- matrix(0, ncol=1)
wald_deltas(results = spcsur.sem, R = R2, b = b2)

#################################
## Estimate SUR-SARAR model
## It usually requires 2-3 minutes maximum
spcsur.sarar <-spsurml(Form = Tformula, data = spc,
                       type = "sarar", W = Wspc)
summary(spcsur.sarar)
## H_0: equality of the lambda and rho parameters of both equations.
R3 <- matrix(c(1,-1,0,0,0,0,1,-1),nrow=2,ncol=4,byrow=TRUE)
b3 <- matrix(c(0,0), ncol=1)
wald_deltas(results = spcsur.sarar, R = R3, b = b3)

####################################
########  G=1; Tm>1         ########
####################################

#' #### Example 2: Homicides + Socio-Economics (1960-90)
#' # It could make an error out-of-memory in some computers
rm(list = ls()) # Clean memory
data(NCOVR)
Tformula <- HR80  | HR90 ~ PS80 + UE80 | PS90 + UE90

#################################
## A SUR-SLM model
NCOVRSUR.slm <-spsurml(Form = Tformula, data = NCOVR, type = "slm", W = W)
summary(NCOVRSUR.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(results = NCOVRSUR.slm, R = R1, b = b1)

#################################
## Estimate SUR-SEM model
NCOVRSUR.sem <-spsurml(Form = Tformula, data = NCOVR, type = "sem", W = W)
summary(NCOVRSUR.sem)
## H_0: equality of the rho parameters of both equations.
R2 <- matrix(c(1,-1), nrow=1)
b2 <- matrix(0, ncol=1)
wald_deltas(results = NCOVRSUR.sem, R = R2, b = b2)
# }