# Tobit or Censored Regression Model
# Three methods : Heckman 2 step procedure, MLE, Powell's median (quantile) estimator
# For the MLE, need "survival" package
# For the Powell estimator, need "quantreg" package


# 0. Data generation

n  <- 500
x1 <- rnorm(n,sd=3)
x2 <- runif(n,min=-2,max=2)
ys <- 1 + x1 - x2 + rt(n,df=8)
c  <- (ys > 0)
y  <- pmax(ys,0)


# 1. Heckman 2 step estimator

probit <- glm(c~x1+x2,family=binomial(link="probit"))

lambda <- dnorm(cbind(1,x1,x2)%*%(probit$coef))/pnorm(cbind(1,x1,x2)%*%(probit$coef))

hfit <- lm(y ~ x1 + x2 + lambda)


# 2. MLE

tfit <- survreg(Surv(y, c, type='left') ~ x1 + x2, dist='gaussian')

predict(tfit,type="response")


# 3. Powell estimator

require(quantreg)

pfit <- rq.fit.fcen(cbind(1,x1,x2),y,1-c,left=TRUE)

pfit$coef