Tobit Model for Corner Solutions and Censored Data

Core Idea

Many outcomes are zero for a large share of people and positive and continuous for the rest, such as hours worked, charitable donations, or spending on a durable good. The Tobit model, proposed by James Tobin in 1958, treats the outcome as y = max(0, y*), where the latent variable y* = xβ + u has a normal error. Linear regression on such data is inconsistent, whether it uses all observations or only the positive ones. Tobit estimates β and σ by maximum likelihood, combining a probit-like term for the zeros with a normal-density term for the positive values. Its coefficients describe effects on the latent variable, so effects on the observed outcome must be computed separately.

How It's Best Learned

Plot simulated data where many observations pile up at zero, then fit OLS to all the data and to the positive data only, and show that both slopes are too flat. Next, write down the two pieces of the Tobit likelihood and connect the zero piece to the probit model students already know. Finish by computing a partial effect on E(y|x) and comparing it with the raw coefficient.

Common Misconceptions

• Dropping the zeros and running OLS on the positive observations fixes the problem. (Conditioning on y > 0 selects observations with high errors, so the error no longer has mean zero and OLS remains biased.)
• The Tobit coefficient β_j is the effect of x_j on the expected observed outcome. (β_j is the effect on the latent y*; the effect on E(y|x) is β_j times Φ(xβ/σ), which is smaller.)
• Tobit is valid whatever the error distribution. (Its consistency relies on the error being normal and homoskedastic; if these fail, the estimates are generally inconsistent.)

Explainer

You know the probit model handles a binary outcome by assuming a latent variable crosses a threshold. The Tobit model extends that idea to outcomes that are zero for many observations and positive and continuous for the rest. James Tobin introduced it in 1958 to study household spending on durable goods, where many households spend nothing in a given year.

The model starts with a latent variable, y* = xβ + u, with u normally distributed with mean zero and variance σ², independent of x. We observe y = y* when y* > 0, and y = 0 otherwise. For hours worked, you can think of y* as desired hours: some people would want negative hours at the going wage if that were possible, and they appear in the data as zero.

Why not use OLS? If you regress y on x using all observations, the pile of zeros pulls the fitted line toward a flatter slope, because the true relationship is curved: E(y|x) approaches zero for low xβ and approaches xβ for high xβ. If you keep only positive observations, you select people with large errors at each x, especially at low x, which again biases the slope. Both approaches are inconsistent.

Tobit uses maximum likelihood. Each zero contributes the probability that y* was at or below zero, 1 − Φ(xβ/σ). Each positive value contributes the normal density (1/σ)φ((y − xβ)/σ). Maximizing the product over the sample yields estimates of β and σ.

Interpretation needs care. The coefficient β_j measures the effect of x_j on y*. If the outcome is a real choice with many zeros, a corner solution, the effect of interest is on observed y. That effect is ∂E(y|x)/∂x_j = β_j Φ(xβ/σ). Since Φ lies between zero and one, the effect on the observed mean is smaller than β_j. With a coefficient of 80 hours per year of schooling and Φ = 0.6, the effect on expected hours is 48. When the data are censored for recording reasons, such as top-coded incomes, the latent variable is the target and β can be read directly.

The model's main weakness is its reliance on normal, homoskedastic errors, and on a single index governing both whether y is positive and how large it is. When participation and amount depend on different factors, two-part models or Heckman's selection model are often preferred.