Definition
- Latent variable: \(f_i\)
- Observation: \(y_i\)
Suppose the observation is censored from above at \(y_U\) and below at \(y_L\) (type I Tobit model), i.e., \[ y_i= \begin{cases}y_i^* & \text { if } y_L<y_i^*<y_U, \\ y_L & \text { if } y_i^* \leq y_L, \\ y_U & \text { if } y_i^* \geq y_U. \end{cases} \] If \(y_i^\star \sim \mathcal{N}(f_i,\sigma_i^2)\), then the Tobit likelihood is, \[ p(y_i | f_i) = \begin{cases} \mathcal{N}(y_i|f_i, \sigma_i^2) & \text { if } y_L<y_i <y_U, \\ \Phi(y_L | f_i, \sigma_i^2) & \text { if } y_i = y_L, \\ 1 - \Phi(y_U | f_i, \sigma_i^2) & \text { if } y_i = y_U, \end{cases} \] where \(\Phi\) is the CDF of standard normal, \(\Phi(y_L | f_i, \sigma_i^2) \triangleq \Phi(\frac{y_L - f_i}{\sigma_i})\).
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Citation
BibTeX citation:
@online{li2023,
author = {Li, Chengkun},
title = {Tobit Likelihood},
date = {2023-06-16},
url = {https://pipme.github.io/posts/2023-06-16-Tobit-likelihood/index-gist.html},
langid = {en}
}
For attribution, please cite this work as:
Li, Chengkun. 2023. “Tobit Likelihood.” June 16, 2023. https://pipme.github.io/posts/2023-06-16-Tobit-likelihood/index-gist.html.