SpatialProbitFit: Fit a spatial probit model.
In ProbitSpatial: Probit with Spatial Dependence, SAR and SEM Models

Description Usage Arguments Details Value Examples

Approximate likelihood estimation of the spatial autoregressive probit model (SAR) or spatial error probit model (SEM).

1 2	SpatialProbitFit(formula,data,W, DGP='SAR',method="conditional",varcov="varcov",control=list())

`formula`	an object of class `formula`: a symbolic description of the model to be fitted.
`data`	the data set containing the variables of the model.
`W`	the spatial weight matrix of class `"dgCMatrix"`.
`DGP`	the data generating process of `data`: SAR or SEM (Default is SAR).
`method`	the optimisation method: `"conditional"` or `"full-lik"` (Defaul is `"conditional"`, see Details).
`varcov`	the likelihood function is computed using the variance-covariance matrix (`"varcov"`) or the precision matrix (`"precision"`)? Default is `"varcov"`.
`control`	a list of control parameters. See Details.

The estimation is based on the approximate value of the true likelihood of spatial autoregressive (SAR) or spatial error (SEM) probit models. The DGP of the spatial autoregressive model (SAR) model is the following

y = (I_n-ρ W)^{-1}(Xβ + ε),

where the disturbances ε are iid standard normally distributed, W is a sparse spatial weight matrix and ρ is the spatial lag parameter. The variance of the error term is equal to Σ=σ^2((I_n-ρ W)^{-1}((I_n-ρ W)^{-1})^{t}). The DGP of the spatial error model (SEM) is as follows

y = Xβ+(I_n-ρ W)^{-1}ε,

where the disturbances ε are iid standard normally distributed, W is a sparse spatial weight matrix and ρ is the spatial error parameter. The variance of the error term is equal to Σ=σ^2((I_n-ρ W)^{-1}((I_n-ρ W )^{-1})^{t}).

The approximation is inspired by the Mendell-Elston approximation of the multivariante normal probabilities (see References). It makes use of the Cholesky decomposition of the variance-covariance matrix Σ.

The SpatialProbitFit command estimates the model by maximising the approximate log-likelihood. We propose two optimisation method:

"conditional":: it relies on a standard probit estimation (we use speedglm) which applies to the model estimated conditional on ρ.
"full-lik":: it minimises the full-log-likelihood using the analytical gradient functions. The optimisation is performed by means of the optim function with method = "BFGS".

In both cases a "conditional" estimation is performed. If method="conditional", then SpatialProbitFit returns the results of this first estimation. In case method="full-lik", the function tries to improve the log-likelihood by means of a further exploration around the value of the parameters found by the conditional step. The conditional step is usually very accurate and particularly fast. The second step is more time consuming and does not always improve the results of the first step. We dissuade the user from using the full-likelihood method for sample sizes bigger than ten thousands, since the computation of the gradients is quite slow. Simulation studies reported in Martinetti and Geniaux (2015) prove that the conditional estimation is highly reliable, even if compared to the full-likelihood ones.

In order to reduce the computation time of the function SpatialProbitFit, we propose a variant of the likelihood-function estimation that uses the inverse of the variance-covariance matrix (a.k.a. precision matrix). This variant applies to both the "conditional" and the "full-lik" methods and can be invoked by setting varcov="precision". Simulation studies reported in Martinetti and Geniaux (2015) suggest that the accuracy of the results with the precision matrix are sometimes worst than the one with the true variance-covariance matrix, but the estimation time is considerably reduced.

The control argument is a list that can supply any of the following components:

iW_CL: the order of approximation of (I_n-ρ W)^{-1} used in the "conditional" method. Default is 6, while 0 means no approximation (it uses exact inversion of matrixes, not suitable for big sample sizes). See Martinetti and Geniaux (2015) for further references.
iW_FL: the order of approximation of (I_n-ρ W)^{-1} used in the computation of the likelihood function for the "full-lik" method. Default is 0, meaning no approximation.
iW_FG: the order of approximation of (I_n-ρ W)^{-1} used in the computation of the gradient functions for the "full-lik" method. Default is 0, meaning no approximation.
reltol: relative convergence tolerance. It represents tol in optimize function for method="conditional" and reltol in optim function for method="full-lik". Default is 1e-5.
prune: the pruning value used in the gradients. Default is 0, meaning no pruning. Typacl values are around 1e-3 and 1e-6. They help reducing the estimation time of the gradient functions.

Return a structure of class SpatialProbit:

beta: the estimated parameters for the covariates
rho: the estimated spatial dependence parameter
coeff: all estimated parameters
loglik: the log-likelihood associated to the estimated model
formula: same as formula
nobs: number of observations
nvar: number of covariates or explanatory variables
y: the vector of the dependent variable
X: the matrix of covariates or explanatory variables
time: estimation time
DGP: the chosen DGP (SAR or SEM)
method: the estimation method (see Details)
varcov: the matrix used in the approximation (see Details)
W: the spatial weight matrix
iW_CL: the order of approximation used in the conditional method
iW_FL: the order of approximation used in the likelihood function for the full-lik method
iW_FG: the order of approximation used in the gradient functions for the full-lik method
reltol: the relative convergence tolerance
prune: the pruning used in the gradient functions
env: an environment containing information for use in later function calls to save time
message: a integer giving any additional information or NULL.

n <- 1000
nneigh <- 3
rho <- 0.5
beta <- c(4,-2,1)
W <- generate_W(n,nneigh,seed=123)
X <- cbind(1,rnorm(n,2,2),rnorm(n,0,1))
colnames(X) <- c("intercept","X1","X2")
y <- sim_binomial_probit(W=W,X=X,beta=beta,rho=rho,model="SAR")
d <- as.data.frame(cbind(y,X))
mod <- SpatialProbitFit(y~X1+X2,d,W,
       DGP='SAR',method="conditional",varcov="varcov")