spsurml: Maximum likelihood estimation of spatial SUR model.

Description

This function estimates spatial SUR models using maximum-likelihood methods. The number of equations, time periods and cross-sectional units is not restricted. The user can choose between different spatial specifications as described below. The estimation procedure allows for the introduction of linear restrictions on the $\beta$ parameters associated to the regressors.

Usage

spsurml(
  Form = NULL,
  data = NULL,
  R = NULL,
  b = NULL,
  W = NULL,
  X = NULL,
  Y = NULL,
  G = NULL,
  N = NULL,
  Tm = NULL,
  p = NULL,
  demean = FALSE,
  type = "sim",
  cov = TRUE,
  control = list(tol = 0.05, maxit = 200, trace = TRUE)
)

Value

Output of the maximum-likelihood estimation of the specified spatial SUR model. A list with:

`call`	Matched call.
`type`	Type of model specified.
`betas`	Estimated coefficients for the regressors.
`deltas`	Estimated spatial coefficients.
`se_betas`	Estimated standard errors for the estimates of beta.
`se_deltas`	Estimated standard errors for the estimates of the spatial coefficients.
`cov`	Estimated covariance matrix for the estimates of beta's and spatial coefficients.
`llsur`	Value of the likelihood function at the maximum-likelihood estimates.
`R2`	Coefficient of determination for each equation, obtained as the squared of the correlation coefficient between the corresponding explained variable and its estimate. spsurml also shows a global coefficient of determination obtained, in the same manner, for the set of G equations.
`Sigma`	Estimated covariance matrix for the residuals of the G equations.
`Sigma_corr`	Estimated correlation matrix for the residuals of the G equations.
`Sigma_inv`	Inverse of `Sigma`, the (GxG) covariance matrix of the residuals of the SUR model.
`residuals`	Residuals of the model.
`df.residuals`	Degrees of freedom for the residuals.
`fitted.values`	Estimated values for the dependent variables.
`BP`	Value of the Breusch-Pagan statistic to test the null hypothesis of diagonality among the errors of the G equations.
`LMM`	Marginal Lagrange Multipliers, LM($\rho$\|$\lambda$) and LM($\lambda$\|$\rho$), to test for omitted spatial effects in the specification.
`G`	Number of equations.
`N`	Number of cross-sections or spatial units.
`Tm`	Number of time periods.
`p`	Number of regressors by equation (including intercepts).
`demean`	Logical value used for demeaning.
`Y`	Vector Y of the explained variables of the SUR model.
`X`	Matrix X of the regressors of the SUR model.

Control arguments

`tol`	Numerical value for the tolerance for the estimation algorithm until convergence. Default = 1e-3.
`maxit`	Maximum number of iterations until convergence; it must be an integer value. Default = 200.
`trace`	A logical value to show intermediate results during the estimation process. Default = TRUE.

Details

The list of (spatial) models that can be estimated with the spsurml function are:

"sim": SUR model with no spatial effects $$ y_{tg} = X_{tg} \beta_{g} + \epsilon_{tg} $$
"slx": SUR model with spatial lags of the regressors $$ y_{tg} = X_{tg} \beta_{g} + WX_{tg} \theta_{g} + \epsilon_{tg} $$
"slm": SUR model with spatial lags of the explained variables $$y_{tg} = \lambda_{g} Wy_{tg} + X_{tg} \beta_{g} + \epsilon_{tg} $$
"sem": SUR model with spatial errors $$ y_{tg} = X_{tg} \beta_{g} + u_{tg} $$ $$ u_{tg} = \rho_{g} Wu_{tg} + \epsilon_{tg} $$
"sdm": SUR model of the Spatial Durbin type $$ y_{tg} = \lambda_{g} Wy_{tg} + X_{tt} \beta_{g} + WX_{tg} \theta_{g} + \epsilon_{tg} $$
"sdem": SUR model with spatial lags of the regressors and spatial errors $$ y_{tg} = X_{tg} \beta_{g} + WX_{tg} \theta_{g} + u_{tg} $$ $$ u_{tg} = \rho_{g} W u_{tg} + \epsilon_{tg} $$
"sarar": SUR model with spatial lags of the explained variables and spatial errors $$ y_{tg} = \lambda_{g} Wy_{tg} + X_{tg} \beta_{g} + u_{tg} $$ $$ u_{tg} = \rho_{g} W u_{tg} + \epsilon_{tg} $$

References

Mur, J., L<U+00F3>pez, F., and Herrera, M. (2010). Testing for spatial effects in seemingly unrelated regressions. Spatial Economic Analysis, 5(4), 399-440.
L<U+00F3>pez, F.A., Mur, J., and Angulo, A. (2014). Spatial model selection strategies in a SUR framework. The case of regional productivity in EU. Annals of Regional Science, 53(1), 197-220.
Breusch T, Pagan A (1980) The Lagrange multiplier test and its applications to model specification in econometrics. Rev Econ Stud 47: 239-254

Examples

Run this code

# NOT RUN {
#################################################
######## CROSS SECTION DATA (G>1; Tm=1) ########
#################################################

#### Example 1: Spatial Phillips-Curve. Anselin (1988, p. 203)
## It usually requires 2-3 minutes maximum...
rm(list = ls()) # Clean memory
data(spc)
Tformula <- WAGE83 | WAGE81 ~ UN83 + NMR83 + SMSA | UN80 + NMR80 + SMSA
spcsur.sim <-spsurml(Form = Tformula, data = spc, type = "sim", W = Wspc)
summary(spcsur.sim)

## A SUR-SLX model
spcsur.slx <-spsurml(Form = Tformula, data = spc, type = "slx", W = Wspc)
summary(spcsur.slx)
# }
# NOT RUN {
## A SUR-SLM model
spcsur.slm <-spsurml(Form = Tformula, data = spc, type = "slm", W = Wspc)
summary(spcsur.slm)
rm(spcsur.slm) # remove

## A SUR-SEM model
spcsur.sem <-spsurml(Form = Tformula, data = spc, type = "sem", W = Wspc)
summary(spcsur.sem)
rm(spcsur.sem) # remove

## A SUR-SDM model
spcsur.sdm <-spsurml(Form = Tformula, data = spc, type = "sdm", W = Wspc)
summary(spcsur.sdm)
rm(spcsur.sdm) # remove

## A SUR-SDEM model
spcsur.sdem <-spsurml(Form = Tformula, data = spc, type = "sdem", W = Wspc)
summary(spcsur.sdem)
rm(spcsur.sdem) # remove

## A SUR-SARAR model
spcsur.sarar <-spsurml(Form = Tformula, data = spc, type = "sarar", W = Wspc)
summary(spcsur.sarar)
rm(spcsur.sarar) # remove
# }
# NOT RUN {
#################################################
########  G=1; Tm>1         ########
#################################################

#### Example 2: Homicides + Socio-Economics (1960-90)
# Homicides and selected socio-economic characteristics for continental
# U.S. counties.
# Data for four decennial census years: 1960, 1970, 1980 and 1990.
# \url{https://geodacenter.github.io/data-and-lab/ncovr/}
# }
# NOT RUN {
## It usually requires 1-2 minutes maximum...
rm(list = ls()) # Clean memory
data(NCOVR)
Tformula <- HR80  | HR90 ~ PS80 + UE80 | PS90 + UE90
## A SUR-SIM model
NCOVRSUR.sim <-spsurml(Form = Tformula, data = NCOVR, type = "sim", W = W)
summary(NCOVRSUR.sim)
rm(NCOVRSUR.sim)

## A SUR-SLX model
NCOVRSUR.slx <-spsurml(Form = Tformula, data = NCOVR, type = "slx", W = W)
summary(NCOVRSUR.slx)
rm(NCOVRSUR.slx)
# }
# NOT RUN {
## It usually requires 1-2 minutes maximum...
## A SUR-SLM model
NCOVRSUR.slm <-spsurml(Form = Tformula, data = NCOVR, type = "slm", W = W)
summary(NCOVRSUR.slm)
rm(NCOVRSUR.slm)

## A SUR-SDM model
NCOVRSUR.sdm <-spsurml(Form = Tformula, data = NCOVR, type = "sdm", W = W)
summary(NCOVRSUR.sdm)
rm(NCOVRSUR.sdm)

## A SUR-SEM model
NCOVRSUR.sem <-spsurml(Form = Tformula, data = NCOVR, type = "sem", W = W)
summary(NCOVRSUR.sem)
## A SUR-SDEM model
NCOVRSUR.sdem <-spsurml(Form = Tformula, data = NCOVR, type = "sdem",W = W)
summary(NCOVRSUR.sdem)

## A SUR-SARAR model
NCOVRSUR.sarar <-spsurml(Form = Tformula, data = NCOVR,
                          type = "sarar", W = W)
 summary(NCOVRSUR.sarar)
# }
# NOT RUN {
##############################################
######## SUR with G>1; Tm>1  ###
##############################################
# }
# NOT RUN {
## It usually requires 2-3 minutes maximum...
rm(list = ls())  # Clean memory
#### Reshape NCOVR in panel format
data(NCOVR,package="spsur")
N <- nrow(NCOVR)
Tm <- 4
index_time <- rep(1:Tm, each = N)
index_indiv <- rep(1:N, Tm)
pHR <- c(NCOVR$HR60, NCOVR$HR70, NCOVR$HR80, NCOVR$HR90)
pPS <- c(NCOVR$PS60, NCOVR$PS70, NCOVR$PS80, NCOVR$PS90)
pUE <- c(NCOVR$UE60, NCOVR$UE70, NCOVR$UE80, NCOVR$UE90)
pDV <- c(NCOVR $DV60, NCOVR$DV70, NCOVR$DV80, NCOVR$DV90)
pFP <- c(NCOVR$FP59, NCOVR$FP70, NCOVR$FP80, NCOVR$FP90)
pSOUTH <- rep(NCOVR$SOUTH, Tm)
pNCOVR <- data.frame(indiv = index_indiv, time = index_time,
                     HR = pHR, PS = pPS, UE = pUE, DV = pDV,
                     FP = pFP, SOUTH = pSOUTH)
rm(NCOVR) # Free memory...
pform <- HR | DV | FP ~ PS + UE | PS + UE + SOUTH | PS

## SIM (easy to compute...)

psur_sim <- spsurml(Form = pform, data = pNCOVR, W = W, type = "sim")
summary(psur_sim)

## SLM (cov = FALSE to prevent overflows of memory)
psur_slm <- spsurml(Form = pform, data = pNCOVR, W = W,
                        type = "slm", cov = FALSE)
psur_slm$deltas
psur_slm$betas
psur_slm$Sigma_corr
rm(psur_slm)
## SEM (cov = FALSE to prevent overflows of memory)
### Only execute if you have enough memory...
psur_sem <- spsurml(Form = pform, data = pNCOVR, W = W,
                        type = "sem", cov = FALSE)
psur_sem$deltas
psur_sem$betas
psur_sem$Sigma_corr
psur_sem
# }
# NOT RUN {
##############################################
######## Demeaning Examples with G>1; Tm>>1  ###
##############################################
# }
# NOT RUN {
#' rm(list = ls())  # Clean memory
set.seed(123456)
Tm <- 10 # Number of time periods
G <- 3 # Number of equations
N <- 100 # Number of spatial elements
p <- 3 # Number of independent variables
Sigma <- matrix(0.5, ncol = G, nrow = G)
diag(Sigma) <- 1
Betas <- rep(1:3, G)
lambda <- 0.5
rho <- 0.0 # spatial autocorrelation error term = 0
#  random coordinates
co <- cbind(runif(N,0,1),runif(N,0,1))
W <- spdep::nb2mat(spdep::knn2nb(spdep::knearneigh(co, k = 5, longlat = FALSE)))
DGPsim <- dgp_spsur(Sigma = Sigma, Betas = Betas, rho = rho, lambda = lambda,
                 Tm = Tm, G = G, N = N, p = p, W = W)

## SLM without demeaning

SUR_slm  <-spsurml(Y= DGPsim$Y, X = DGPsim$X, G = G, N = N, Tm = Tm,
                  p = p, W = W, type = "slm")
summary(SUR_slm)

# SLM with demeaning

SUR_slm_dem  <-spsurml(Y= DGPsim$Y, X = DGPsim$X, G = G, N = N, Tm = Tm,
                  p = p, W = W, type = "slm", demean = TRUE)
summary(SUR_slm_dem)
# }

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