R: Maximum likelihood estimation of spatial SUR model.

spsurml {spsur}

R Documentation

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(formula = NULL, data = NULL, na.action,
               listw = NULL, type = "sim", Durbin = NULL,
               method = "eigen", zero.policy = NULL, interval = NULL,
               trs = NULL, R = NULL, b = NULL, X = NULL, Y = NULL, 
               G = NULL, N = NULL, Tm = NULL,p = NULL, 
               control = list() )

Arguments

`formula`	An object type `Formula` similar to objects created with the package Formula describing the equations to be estimated in the model. This model may contain several responses (explained variables) and a varying number of regressors in each equation.
`data`	An object of class data.frame or a matrix.
`na.action`	A function (default `options("na.action")`), can also be `na.omit` or `na.exclude` with consequences for residuals and fitted values. It may be necessary to set `zero.policy` to `TRUE` because this subsetting may create no-neighbour observations.
`listw`	A `listw` object created for example by `nb2listw` from spatialreg package; if `nb2listw` not given, set to the same spatial weights as the `listw` argument. It can also be a spatial weighting matrix of order (NxN) instead of a `listw` object. Default = `NULL`.
`type`	Type of spatial model specification: "sim", "slx", "slm", "sem", "sdm", "sdem", "sarar" or "gnm". Default = "sim".
`Durbin`	If a formula object and model is type "sdm", "sdem" or "slx" the subset of explanatory variables to lag for each equation.
`method`	Similar to the corresponding parameter of `lagsarlm` function in spatialreg package. "eigen" (default) - the Jacobian is computed as the product of (1 - rho*eigenvalue) using `eigenw`, and "spam" or "Matrix_J" for strictly symmetric weights lists of styles "B" and "C", or made symmetric by similarity (Ord, 1975, Appendix C) if possible for styles "W" and "S", using code from the spam or Matrix packages to calculate the determinant; "Matrix" and "spam_update" provide updating Cholesky decomposition methods; "LU" provides an alternative sparse matrix decomposition approach. In addition, there are "Chebyshev" and Monte Carlo "MC" approximate log-determinant methods; the Smirnov/Anselin (2009) trace approximation is available as "moments". Three methods: "SE_classic", "SE_whichMin", and "SE_interp" are provided experimentally, the first to attempt to emulate the behaviour of Spatial Econometrics toolbox ML fitting functions. All use grids of log determinant values, and the latter two attempt to ameliorate some features of "SE_classic".
`zero.policy`	Similar to the corresponding parameter of `lagsarlm` function in spatialreg package. If `TRUE` assign zero to the lagged value of zones without neighbours, if `FALSE` assign `NA` - causing `spsurml()` to terminate with an error. Default = `NULL`.
`interval`	Search interval for autoregressive parameter. Default = `NULL`.
`trs`	Similar to the corresponding parameter of `lagsarlm` function in spatialreg package. Default `NULL`, if given, a vector of powered spatial weights matrix traces output by `trW`.
`R`	A row vector of order (1xpr) with the set of r linear constraints on the beta parameters. The first restriction appears in the first p terms, the second restriction in the next p terms 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`.
`X`	A data matrix of order (NTmGxp) with the observations of the regressors. The number of covariates in the SUR model is p = `sum(p_{g})` where `p_{g}` is the number of regressors (including the intercept) in the g-th equation, g = 1,...,G). The specification of "X" is only necessary if not available a `Formula` and a data frame. Default = `NULL`.
`Y`	A column vector of order (NTmGx1), with the observations of the explained variables. The ordering of the data must be (first) equation, (second) time dimension and (third) cross-sectional/spatial units. The specification of Y is only necessary if not available a `Formula` and a data frame. Default = `NULL`.
`G`	Number of equations.
`N`	Number of cross-section or spatial units
`Tm`	Number of time periods.
`p`	Number of regressors by equation, including the intercept. p can be a row vector of order (1xG), if the number of regressors is not the same for all the equations, or a scalar, if the G equations have the same number of regressors. The specification of p is only necessary if not available a `Formula` and a data frame.
`control`	list of additional arguments.

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} = \rho_{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} = \lambda_{g} Wu_{tg} + \epsilon_{tg}
"sdm": SUR model of the Spatial Durbin type

y_{tg} = \rho_{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} = \lambda_{g} W u_{tg} + \epsilon_{tg}
"sarar": SUR model with spatial lags of the explained variables and spatial errors

y_{tg} = \rho_{g} Wy_{tg} + X_{tg} \beta_{g} + u_{tg}

u_{tg} = \lambda_{g} W u_{tg} + \epsilon_{tg}
"gnm": SUR model with spatial lags of the explained variables, regressors and spatial errors

y_{tg} = \rho_{g} Wy_{tg} + X_{tg} \beta_{g} + WX_{tg} \theta_{g} + u_{tg}

u_{tg} = \lambda_{g} W u_{tg} + \epsilon_{tg}

Value

Object of spsur class with the output of the maximum-likelihood estimation of the specified spatial SUR model. A list with:

`call`	Matched call.
`type`	Type of model specified.
`method`	Value of `method` argument to compute the Jacobian
`Durbin`	Value of `Durbin` argument.
`coefficients`	Estimated coefficients for the regressors.
`deltas`	Estimated spatial coefficients.
`rest.se`	Estimated standard errors for the estimates of beta.
`deltas.se`	Estimated standard errors for the estimates of the spatial coefficients (`deltas`).
`resvar`	Estimated covariance matrix for the estimates of beta's and spatial coefficients (`deltas`).
`LL`	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 fitted values.
`R2 pooled`	Global coefficient of determination obtained for the set of the G equations. It is computed in the same way than uniequational `R2` but joining the dependent variable and fitted values in single vectors instead of one vector for each equation.
`Sigma`	Estimated covariance matrix for the residuals of the G equations.
`fdHess`	Logical value of `fdHess` argument when computing numerical covariances.
`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).
`Y`	If data is NULL, vector Y of the explained variables of the SUR model.
`X`	If data is NULL, matrix X of the regressors of the SUR model.
`W`	Spatial weighting matrix.
`zero.policy`	Logical value of `zero.policy` .
`interval`	Search interval for spatial parameter.
`listw_style`	Style of neighborhood matrix `W`.
`trs`	Either `NULL` or vector of powered spatial weights matrix traces output by `trW`.
`insert`	Logical value to check if `is.null(trs)`.

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`.
`fdHess`	Compute variance-covariance matrix using the numerical hessian. Suited for large samples. Default = `FALSE`
`Imult`	default 2; used for preparing the Cholesky decompositions for updating in the Jacobian function
`super`	if `NULL` (default), set to `FALSE` to use a simplicial decomposition for the sparse Cholesky decomposition and method "Matrix_J", set to as.logical(NA) for method "Matrix", if `TRUE`, use a supernodal decomposition
`cheb_q`	default 5; highest power of the approximating polynomial for the Chebyshev approximation
`MC_p`	default 16; number of random variates
`MC_m`	default 30; number of products of random variates matrix and spatial weights matrix
`spamPivot`	default "MMD", alternative "RCM"
`in_coef`	default 0.1, coefficient value for initial Cholesky decomposition in "spam_update"
`type`	default "MC", used with method "moments"; alternatives "mult" and "moments", for use if trs is missing
`correct`	default `TRUE`, used with method "moments" to compute the Smirnov/Anselin correction term
`trunc`	default `TRUE`, used with method "moments" to truncate the Smirnov/Anselin correction term
`SE_method`	default "LU", may be "MC"
`nrho`	default 200, as in SE toolbox; the size of the first stage lndet grid; it may be reduced to for example 40
`interpn`	default 2000, as in SE toolbox; the size of the second stage lndet grid
`SElndet`	default `NULL`, may be used to pass a pre-computed SE toolbox style matrix of coefficients and their lndet values to the "SE_classic" and "SE_whichMin" methods
`LU_order`	default `FALSE`; used in "LU_prepermutate", note warnings given for lu method
`pre_eig`	default `NULL`; may be used to pass a pre-computed vector of eigenvalues

Author(s)

Fernando Lopez	fernando.lopez@upct.es
Roman Minguez	roman.minguez@uclm.es
Jesus Mur	jmur@unizar.es

References

Anselin, L. (1988). Spatial econometrics: methods and models. Dordrecht: Kluwer
Bivand, R.S. and Piras G. (2015). Comparing Implementations of Estimation Methods for Spatial Econometrics. Journal of Statistical Software, 63(18), 1-36. <doi: 10.18637/jss.v063.i18>
Bivand, R. S., Hauke, J., and Kossowski, T. (2013). Computing the Jacobian in Gaussian spatial autoregressive models: An illustrated comparison of available methods. Geographical Analysis, 45(2), 150-179. <doi:10.1111/gean.12008>
Breusch T., Pagan A. (1980). The Lagrange multiplier test and its applications to model specification in econometrics. Rev Econ Stud 47: 239-254
Cliff, A.D. and Ord, J.K. (1981). Spatial processes: Models and applications, Pion.
LeSage J and Pace, R.K. (2009). Introduction to Spatial Econometrics. CRC Press, Boca Raton.
Lopez, 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. <doi:10.1007/s00168-014-0624-2>
Minguez, R., Lopez, F.A. and Mur, J. (2022). spsur: An R Package for Dealing with Spatial Seemingly Unrelated Regression Models. Journal of Statistical Software, 104(11), 1–43. <doi:10.18637/jss.v104.i11>
Mur, J., Lopez, F., and Herrera, M. (2010). Testing for spatial effects in seemingly unrelated regressions. Spatial Economic Analysis, 5(4), 399-440. <doi:10.1080/17421772.2010.516443>
Ord, J.K. (1975). Estimation methods for models of spatial interaction, Journal of the American Statistical Association, 70, 120-126.

Examples


#################################################
######## CROSS SECTION DATA (G>1; Tm=1) ########
#################################################

#### Example 1: Spatial Phillips-Curve. Anselin (1988, p. 203)
rm(list = ls()) # Clean memory
data(spc)
Tformula <- WAGE83 | WAGE81 ~ UN83 + NMR83 + SMSA | UN80 + NMR80 + SMSA
spcsur.sim <- spsurml(formula = Tformula, data = spc, type = "sim")
summary(spcsur.sim)
# All the coefficients in a single table.
print(spcsur.sim)
# Plot of the coefficients of each equation in different graphs
plot(spcsur.sim) 

## A SUR-SLX model 
## (listw argument can be either a matrix or a listw object )
spcsur.slx <- spsurml(formula = Tformula, data = spc, type = "slx", 
  listw = Wspc, Durbin = TRUE)
summary(spcsur.slx)
# All the coefficients in a single table.
print(spcsur.slx)
# Plot of the coefficients in a single graph
if (require(gridExtra)) {
  pl <- plot(spcsur.slx, viewplot = FALSE)
  grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]], 
                        nrow = 2)
} 

## VIP: The output of the whole set of the examples can be examined 
## by executing demo(demo_spsurml, package="spsur")
  

### A SUR-SLM model
spcsur.slm <- spsurml(formula = Tformula, data = spc, type = "slm", 
                     listw = Wspc)
summary(spcsur.slm)
if (require(gridExtra)) {
  pl <- plot(spcsur.slm, viewplot = FALSE) 
  grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]], 
               pl$pldeltas, nrow = 3)
}


### A SUR-SDM model
spcsur.sdm <- spsurml(formula = Tformula, data = spc, type = "sdm", 
                     listw = Wspc)
summary(spcsur.sdm)
print(spcsur.sdm)
if (require(gridExtra)) {
  pl <- plot(spcsur.sdm, viewplot = FALSE) 
  grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]], 
               pl$pldeltas, nrow = 3)
}

## A SUR-SDM model with different spatial lags in each equation
TformulaD <- ~ UN83 + NMR83 + SMSA | UN80
spcsur.sdm2 <- spsurml(formula = Tformula, data = spc, type = "sdm", 
                      listw = Wspc, Durbin = TformulaD)
summary(spcsur.sdm2)
if (require(gridExtra)) {
  pl <- plot(spcsur.sdm2, viewplot = FALSE) 
  grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]], 
               pl$pldeltas, nrow = 3)
}


##################################################
#########  CLASSIC PANEL DATA 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/}


### It usually requires 1-2 minutes maximum...
rm(list = ls()) # Clean memory
### Read NCOVR.sf object
data(NCOVR, package = "spsur")
nbncovr <- spdep::poly2nb(NCOVR.sf, queen = TRUE)
### Some regions with no links...
lwncovr <- spdep::nb2listw(nbncovr, style = "W", zero.policy = TRUE)
Tformula <- HR80  | HR90 ~ PS80 + UE80 | PS90 + UE90
### A SUR-SIM model
NCOVRSUR.sim <- spsurml(formula = Tformula, data = NCOVR.sf, type = "sim")
summary(NCOVRSUR.sim)
if (require(gridExtra)) {
  pl <- plot(NCOVRSUR.sim, viewplot = FALSE) 
  grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]], nrow = 3)
}
### A SUR-SLX model
NCOVRSUR.slx <- spsurml(formula = Tformula, data = NCOVR.sf, type = "slx", 
                       listw = lwncovr, zero.policy = TRUE)
print(NCOVRSUR.slx)
if (require(gridExtra)) {
  pl <- plot(NCOVRSUR.slx, viewplot = FALSE) 
  grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]], nrow = 2)
}

### A SUR-SLM model
### method = "Matrix" (Cholesky) instead of "eigen"
### (fdHess = TRUE to compute numerical covariances )
NCOVRSUR.slm <- spsurml(formula = Tformula, data = NCOVR.sf, 
                       type = "slm", listw = lwncovr, method = "Matrix", 
                       zero.policy = TRUE, control = list(fdHess = TRUE))
summary(NCOVRSUR.slm)

if (require(gridExtra)) {
  pl <- plot(NCOVRSUR.slm, viewplot = FALSE) 
  grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]], 
               pl$pldeltas, nrow = 3)
}
# LR test for nested models
anova(NCOVRSUR.sim, NCOVRSUR.slm)

[Package spsur version 1.0.2.5 Index]