Amemiya (1984) Tobit Models: A Survey

Summary

Amemiya's survey unifies the sprawling literature on Tobit models — regression models in which the range of the dependent variable is constrained (censored or truncated), named after Tobin's (1958) limited-dependent-variable model. Its organizing device is a classification of roughly 95% of econometric applications into five basic types, distinguished by the form of the likelihood function, on the principle that a similar likelihood implies similar estimation and computation methods. Part I develops the estimation theory for the Standard (Type 1) Tobit model — probit ML, least squares, Heckman's two-step, nonlinear least squares, the Tobit MLE, and the EM algorithm — and examines the estimator's properties under non-standard assumptions (heteroskedasticity, serial correlation, non-normality). Part II defines Types 2–5 and surveys their applications.

Key Claims

• Classification by likelihood form. Tobit models are grouped into five types by how the likelihood factors over the observed/latent regions. Two models that look superficially different can belong to the same type; the classification is "statistically most useful" because it groups models by the estimation machinery they require.
  • Type 1 (Standard Tobit): latent $y^* = \mathbf{x}'\boldsymbol\beta + u$, observe $y=\max(0,y^*)$. A single index $\mathbf{x}'\boldsymbol\beta$ governs both the probability of the limit outcome $P(y=0)$ and the conditional mean $E(y\mid y>0)$.
  • Type 2: a first index determines whether a second, distinct variable $y_2$ is observed (sample-selection / selection structure; Gronau 1973, Heckman 1974/1976a/1979).
  • Type 3: as Type 2 but the selection variable $y_1$ is itself continuously observed when positive (Heckman 1974 shadow-price labor supply).
  • Type 4: two censored variables with a selection mechanism (Kenny–Lee–Maddala–Trost 1979; Nelson–Olson 1978).
  • Type 5: bivariate sign-selection — the sign of $y_1$ selects which of two regimes generates $y_2$ (Lee 1978, Lee–Trost 1978; Heckman 1978 dummy endogenous variable).
• Least squares is inconsistent for Tobit ("Tobit bias"): regressing $y$ on $\mathbf{x}$ over the full sample, or over positive observations only, yields biased and inconsistent estimates because $E(y\mid \mathbf{x})$ is nonlinear (Greene 1981, 1983).
• The Tobit MLE has a globally concave log-likelihood (in a reparameterization; Olsen 1978a), so the maximum-likelihood estimator is unique — an unusually clean computational guarantee for a nonlinear model.
• Heckman's two-step estimator offers a simple consistent alternative: estimate the selection/probit equation first, then correct the regression for the inverse-Mills-ratio (selectivity) term.
• The EM algorithm (Dempster–Laird–Rubin 1977) treats the censored observations as missing data, alternating an E-step (imputing the latent $y^*$ moments) and an M-step (a least-squares update) — a natural fit for the Tobit likelihood.
• Fragility to distributional assumptions is the central caveat. Unlike OLS in the normal linear model, the Tobit MLE is inconsistent under heteroskedasticity (Hurd 1979; Arabmazar–Schmidt 1981) and under non-normality (Arabmazar–Schmidt 1982; Goldberger 1980). This motivates tests for normality/heteroskedasticity and semiparametric alternatives.
• Powell's least-absolute-deviations (LAD/CLAD) estimator for censored and truncated regression is consistent and asymptotically normal under weak conditions, without assuming normality or homoskedasticity — an early semiparametric robustification of the Tobit model.

Concepts Introduced or Extended

• Tobit Model
• Selection Model
• Binary Probit
• EM Algorithm
• Data Augmentation

Entities Mentioned

• Takeshi Amemiya

Quotes

"Tobit models refer to regression models in which the range of the dependent variable is constrained in some way."

"I will try to accomplish this objective by means of classifying the diverse Tobit models into five basic types … according to the form of the likelihood function."

My Take

The five-type taxonomy is the reason this survey endured: it turned a chaotic applied literature into a small, teachable set of likelihood structures, and it remains the vocabulary ("Type 2 Tobit," "Type 3 Tobit") econometricians still use. From the vantage of this wiki, the most consequential message is the misspecification fragility: the Tobit MLE buys efficiency by fully committing to normality and homoskedasticity, and loses consistency when either fails — a sharp contrast with OLS's robustness in the uncensored normal model. That fragility is exactly what later work attacks, from Powell's LAD/CLAD (semiparametric) to the Bayesian data-augmentation treatment (Chib and others), where the latent $y^*$ that Amemiya handles via the EM E-step becomes an augmented variable sampled inside a Gibbs loop — the same missing-data insight, recast for MCMC.