# Bayesian estimation of the SEM probit model Bayesian estimation of the probit model with spatial errors (SEM probit model). ```r semprobit(formula, W, data, subset, ...) sem_probit_mcmc(y, X, W, ndraw = 1000, burn.in = 100, thinning = 1, prior=list(a1=1, a2=1, c=rep(0, ncol(X)), T=diag(ncol(X))*1e12, nu = 0, d0 = 0, lflag = 0), start = list(rho = 0.75, beta = rep(0, ncol(X)), sige = 1), m=10, showProgress=FALSE, univariateConditionals = TRUE) ``` ## Arguments - `y`: dependent variables. vector of zeros and ones - `X`: design matrix - `W`: spatial weight matrix - `ndraw`: number of MCMC iterations - `burn.in`: number of MCMC burn-in to be discarded - `thinning`: MCMC thinning factor, defaults to 1. - `prior`: A list of prior settings for $or rho ~ Beta(a1,a2)$ and $beta ~ N(c,T)$. Defaults to diffuse prior for beta. - `start`: list of start values - `m`: Number of burn-in samples in innermost Gibbs sampler. Defaults to 10. - `showProgress`: Flag if progress bar should be shown. Defaults to FALSE. - `univariateConditionals`: Switch whether to draw from univariate or multivariate truncated normals. See notes. Defaults to TRUE. - `formula`: an object of class "`formula`" (or one that can be coerced to that class): a symbolic description of the model to be fitted. - `data`: an optional data frame, list or environment (or object coercible by `as.data.frame` to a data frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which semprobit is called. - `subset`: an optional vector specifying a subset of observations to be used in the fitting process. - `...`: additional arguments to be passed ## Details Bayesian estimates of the probit model with spatial errors (SEM probit model) $$ z = X \beta + u, \\u = \rho W u + \epsilon, \epsilon \sim N(0, \sigma^2_{\epsilon} I_n)z = X*beta + u, \u = rho*W*u + e, e ~ N(0,sige * I_n) $$ which leads to the data-generating process $$ z = X \beta + (I_n - \rho W)^{-1} \epsilonz = X beta + (I_n - rho W)^{-1} e $$ where y is a binary 0,1 $(n x 1)$ vector of observations for $z < 0$ and $z >= 0$. $beta$ is a $(k x 1)$ vector of parameters associated with the $(n x k)$ data matrix X. The prior distributions are $beta ~ N(c,T)$, $\sigma^2_{\epsilon} \sim IG(a1, a2)$, and $rho ~ Uni(rmin,rmax)$ or $or rho ~ Beta(a1,a2)$. ## Returns Returns a structure of class `semprobit`: - **beta**: posterior mean of bhat based on draws - **rho**: posterior mean of rho based on draws - **bdraw**: beta draws (ndraw-nomit x nvar) - **pdraw**: rho draws (ndraw-nomit x 1) - **sdraw**: sige draws (ndraw-nomit x 1) - **total**: a matrix (ndraw,nvars-1) total x-impacts - **direct**: a matrix (ndraw,nvars-1) direct x-impacts - **indirect**: a matrix (ndraw,nvars-1) indirect x-impacts - **rdraw**: r draws (ndraw-nomit x 1) (if m,k input) - **nobs**: # of observations - **nvar**: # of variables in x-matrix - **ndraw**: # of draws - **nomit**: # of initial draws omitted - **nsteps**: # of samples used by Gibbs sampler for TMVN - **y**: y-vector from input (nobs x 1) - **zip**: # of zero y-values - **a1**: a1 parameter for beta prior on rho from input, or default value - **a2**: a2 parameter for beta prior on rho from input, or default value - **time**: total time taken - **rmax**: 1/max eigenvalue of W (or rmax if input) - **rmin**: 1/min eigenvalue of W (or rmin if input) - **tflag**: 'plevel' (default) for printing p-levels; 'tstat' for printing bogus t-statistics - **lflag**: lflag from input - **cflag**: 1 for intercept term, 0 for no intercept term - **lndet**: a matrix containing log-determinant information (for use in later function calls to save time) ## References LeSage, J. and Pace, R. K. (2009), **Introduction to Spatial Econometrics**, CRC Press, chapter 10 ## Author(s) adapted to and optimized for R by Stefan Wilhelm based on code from James P. LeSage ## See Also `sar_lndet` for computing log-determinants ## Examples ```r library(Matrix) # number of observations n <- 200 # true parameters beta <- c(0, 1, -1) sige <- 2 rho <- 0.75 # design matrix with two standard normal variates as "covariates" X <- cbind(intercept=1, x=rnorm(n), y=rnorm(n)) # sparse identity matrix I_n <- sparseMatrix(i=1:n, j=1:n, x=1) # number of nearest neighbors in spatial weight matrix W m <- 6 # spatial weight matrix with m=6 nearest neighbors # W must not have non-zeros in the main diagonal! i <- rep(1:n, each=m) j <- rep(NA, n * m) for (k in 1:n) { j[(((k-1)*m)+1):(k*m)] <- sample(x=(1:n)[-k], size=m, replace=FALSE) } W <- sparseMatrix(i, j, x=1/m, dims=c(n, n)) # innovations eps <- sqrt(sige)*rnorm(n=n, mean=0, sd=1) # generate data from model S <- I_n - rho * W z <- X %*% beta + solve(qr(S), eps) y <- as.double(z >= 0) # 0 or 1, FALSE or TRUE # estimate SEM probit model semprobit.fit1 <- semprobit(y ~ X - 1, W, ndraw=500, burn.in=100, thinning=1, prior=NULL) summary(semprobit.fit1) ```

semprobit function