Tobit Model

Definition

A Tobit model (limited-dependent-variable model) is a regression model in which the observed range of the dependent variable is constrained — censored (values outside a range are piled at a limit, but the covariates are still observed) or truncated (observations outside the range are lost entirely). The canonical case, due to Tobin (1958), posits a latent variable $y^* = \mathbf{x}'\boldsymbol\beta + u$, $u\sim N(0,\sigma^2)$, of which one observes only $y=\max(0, y^*)$. The name (Goldberger 1964) puns on "Tobin" + "probit."

Key Ideas

• Latent-variable formulation. A continuous latent $y^*$ is generated by a linear model, but the observation rule (a max/min, an indicator, or a selection equation) maps it to the constrained observed variable.
• Censoring vs. truncation. Censored: the covariates $\mathbf{x}$ are observed even when $y$ is at the limit. Truncated: nothing is observed outside the range, so the likelihood conditions on being in-range.
• One index, two margins (Type 1). In the standard Tobit, the same linear index $\mathbf{x}'\boldsymbol\beta$ controls both the probability of the limit outcome $P(y=0)=\Phi(-\mathbf{x}'\boldsymbol\beta/\sigma)$ and the conditional mean of the non-limit outcomes — a restriction that Cragg-type and Type 2–5 generalizations relax.
• Amemiya's five-type taxonomy. Tobit models are classified by the form of the likelihood function into five types (Standard; sample-selection; observed-selection; double-censored; sign-selection), because a shared likelihood shape implies shared estimation methods.

How It Works

The standard-Tobit log-likelihood combines a discrete (probit) part for limit observations and a continuous (normal density) part for non-limit observations: $$\log L = \sum_{y_i=0}\log\Phi\!\left(-\tfrac{\mathbf{x}_i'\boldsymbol\beta}{\sigma}\right) + \sum_{y_i>0}\log\left[\tfrac{1}{\sigma}\phi\!\left(\tfrac{y_i-\mathbf{x}_i'\boldsymbol\beta}{\sigma}\right)\right].$$ Estimation options (all surveyed by Amemiya):

• Tobit MLE — globally concave in the reparameterization $(\boldsymbol\alpha,h)=(\boldsymbol\beta/\sigma,1/\sigma)$ (Olsen 1978a), so the maximizer is unique.
• EM algorithm — treat censored $y^*$ as missing data; E-step imputes the truncated-normal moments, M-step is a least-squares update.
• Heckman two-step — for selection-type (Type 2/3) models: fit the selection probit, then add the inverse-Mills-ratio as a regressor to purge selectivity bias.
• OLS is inconsistent — because $E(y\mid\mathbf{x})$ is nonlinear; naive least squares on all data or on positives-only is biased (Greene 1981).

Why It Matters

• Tobit and its selection-model descendants are the standard tools whenever an economic variable is bounded, corner-solution, or observed only for a self-selected subsample (expenditure, hours worked, wages of the employed, insurance claims).
• The framework connects censored regression, sample selection, and disequilibrium models under one likelihood-based umbrella.
• Its latent-variable structure is the classical antecedent of the Bayesian data-augmentation treatment (Chib), where the latent $y^*$ is sampled inside a Gibbs loop rather than integrated out.

Open Questions

• Distributional fragility. The Tobit MLE is inconsistent under heteroskedasticity or non-normality (unlike OLS in the uncensored normal model). How much robustness is worth how much efficiency?
• Semiparametric alternatives. Powell's LAD/CLAD estimator restores consistency without normality/homoskedasticity; how do such semiparametric estimators trade off efficiency and finite-sample behavior?
• Exclusion restrictions in selection models. Identification of Type 2–5 models often leans on exclusion restrictions whose credibility is frequently debated.

Related

• Selection Model
• Binary Probit
• EM Algorithm
• Data Augmentation
• Method of Simulated Scores
• Amemiya (1984)
• Takeshi Amemiya