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 |
data |
the data set containing the variables of the model. |
W |
the spatial weight matrix of class |
DGP |
the data generating process of |
method |
the optimisation method: |
varcov |
the likelihood function is computed using the
variance-covariance matrix ( |
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
:
the estimated parameters for the covariates
the estimated spatial dependence parameter
all estimated parameters
the log-likelihood associated to the estimated model
same as formula
number of observations
number of covariates or explanatory variables
the vector of the dependent variable
the matrix of covariates or explanatory variables
estimation time
the chosen DGP (SAR or SEM)
the estimation method (see Details)
the matrix used in the approximation (see Details)
the spatial weight matrix
the order of approximation used in the conditional method
the order of approximation used in the likelihood
function for the full-lik
method
the order of approximation used in the gradient functions
for the full-lik
method
the relative convergence tolerance
the pruning used in the gradient functions
an environment
containing information for use in later
function calls to save time
a integer giving any additional information or NULL.
1 2 3 4 5 6 7 8 9 10 11 | 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")
|
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