Tobit Model - The Likelihood Function

The Likelihood Function

Below are the likelihood and log likelihood functions for a type I Tobit. This is a Tobit that is censored from below at when the latent variable . In writing out the likelihood function, we first define an indicator function where:

$I(y_j) = \begin{cases} 0 & \textrm{if} \; y_j = y_L \\ 1 & \textrm{if} \; y_j \neq y_L. \end{cases}$

Next, we mean to be the standard normal cumulative distribution function and to be the standard normal probability density function. For a data set with N observations the likelihood function for a type I Tobit is

$\prod _{j=1}^N \left(\frac{1}{\sigma}\phi \left(\frac{Y_j-X_j\beta }{\sigma }\right)\right)^{I\left(y_j\right)} \left(1-\Phi \left(\frac{X_j\beta-y_L}{\sigma}\right)\right)^{1-I\left(y_j\right)}$

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