Built-in models:Tobit regression

Model

The Tobit model is a model proposed by James Tobin (Tobin, 1958) to describe the relationship between a limited dependent variable \(y_{i}\) and independent variables \(\mathbf{x}_{i}\). This model is widely used in econometrics and biometrics. To better understand the model, we can assume there is a latent (unobservable) variable \(y^{*}\) underlying the observed variable \(y\). The relationship between \(y^{*}\) and \(y\) is given by

\[ y_{i}=\begin{cases} y_{i}^{*} & \text{if }y_{i}^{*}>\tau \\ \tau & \text{otherwise} \end{cases} \]

In Tobin (1958), the threshold \(\tau=0\). The latent variable can be predicted using the independent variables as in

\[ y_{i}^{*}=\beta_{0}+\beta_{1}x_{1i}+\ldots+\beta_{q}x_{qi}+e_{i} \]

with \(e_{i}\sim N(0,\sigma^{2})\).

Code

model{
  for (i in 1:N){
     limit[i]<-M*(1-ind[i]) - 1000000*ind[i]
     mu[i]<-b[1]+b[2]*(x1[i]-mean(x1[]))+b[3]*(x2[i]-mean(x2[]))
     y[i]~dnorm(mu[i],pre.phi)I(limit[i],)
  }
  for (i in 1:3){
     b[i]~dnorm(0, 1.0E-6)
  }
  pre.phi~dnorm(.001,.001)
  phi<-1/pre.phi
}