# sim_binomial_probit: Simulate the dependent variable of a SAR/SEM/SARAR model. In ProbitSpatial: Probit with Spatial Dependence, SAR and SEM Models

## Description

The function `sim_binomial_probit` is used to generate the dependent variable of a spatial binomial probit model, where all the data and parameters of the model can be modified by the user.

## Usage

 ```1 2``` ```sim_binomial_probit(W,X,beta,rho,model="SAR",M=NULL,lambda=NULL, sigma2=1,ord_iW=6,seed=123) ```

## Arguments

 `W` the spatial weight matrix (works for `"SAR"` and `"SEM"` models). `X` the matrix of covariates. `beta` the value of the covariates parameters. `rho` the value of the spatial dependence parameter (works for `"SAR"` and `"SEM"` models). `model` the type of model, between `"SAR"`, `"SEM"`, `"SARAR"` (Default is `"SAR"`). `M` the second spatial weight matrix (only if `model` is `"SARAR"`). `lambda` the value of the spatial dependence parameter (only if `model` is `"SARAR"`). `sigma2` the variance of the error term (Defaul is 1). `ord_iW` the order of approximation of the matrix (I_n-ρ W)^{-1}. `seed` to set the random generator seed of the error term.

## Details

The `sim_binomial_probit` generates a vector of dependent variables for a spatial probit model. It allows to simulate the following DGPs (Data Generating Process): SAR

z = (I_n-ρ W)^{-1}(Xβ+ε)

SEM

z = (Xβ+(I_n-ρ W)^{-1}ε)

SARAR

z = (I_n-ρ W)^{-1}(Xβ+(I_n-λ M)^{-1}ε)

where ε are independent and normally distributed with mean zero and variance `sigma2` (default is 1).

The matrix `X` of covariates, the corresponding parameters `beta`, the spatial weight matrix `W` and the corresponding spatial depndence parameter `rho` need to be passed by the user. The matrix (I_n-ρ W)^{-1} is computed using the `ApproxiW` function, that can either invert (I_n-ρ W) exactely, if `order_iW=0` (not suitable for `n` bigger than 1000), or using the Taylor approximation

(I_n-ρ W)^{-1}= I_n+ρ W+ρ^2 W^2+…

of order `order_iW` (default is approximation of order 6).

## Value

a vector of zeros and ones

`generate_W`, `SpatialProbitFit`.
 ``` 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) X <- cbind(1,rnorm(n,2,2),rnorm(n,0,1)) y <- sim_binomial_probit(W,X,beta,rho,model="SAR") #SAR model y <- sim_binomial_probit(W,X,beta,rho,model="SEM") #SEM model M <- generate_W(n,nneigh,seed=1) lambda <- -0.5 y <- sim_binomial_probit(W,X,beta,rho,model="SARAR",M=M,lambda=lambda) #SARAR ```