Tobit and Censored Regression

Imagine trying to model how much retail investors want to short a stock, using data on actual short positions. Many investors would want a negative position if they could freely express one, but exchange or broker rules mean any desired negative amount just gets recorded as zero — the data is censored at zero, not truly zero. Running ordinary least squares on this data treats every recorded zero as a genuine "wants no position" observation, which biases the fitted relationship toward flatter, weaker effects than the true underlying desire actually has.

The Tobit model fixes this by assuming there's an unobserved "latent" variable $y_i^* = x_i\beta + \varepsilon_i$ (the true underlying desired amount, which can go negative) and the observed outcome is $y_i = \max(0, y_i^*)$ — anything below the censoring point gets clipped to zero before you ever see it. Because plain OLS on the observed $y_i$ effectively assumes those clipped values are the truth, it systematically underestimates $\beta$. Tobit instead fits $\beta$ by maximum likelihood, combining the probability mass at the censoring point (from a probit-style term) with the density of the uncensored observations above it, recovering an estimate of the effect on the true, uncensored quantity.

A close cousin, censored regression more generally, and truncated regression (where censored observations aren't merely clipped but are missing from the sample entirely — you never even see that an investor wanted a short) require a related but distinct correction, since truncation changes the sample composition rather than just the reported value.