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)}

Read more about this topic:  Tobit Model

Famous quotes containing the words likelihood and/or function:

    “What likelihood is there of corrupting a man who has no ambition?”
    —Samuel Richardson (1689–1761)

    “Nobody seriously questions the principle that it is the function of mass culture to maintain public morale, and certainly nobody in the mass audience objects to having his morale maintained.”
    —Robert Warshow (1917–1955)