Questions: Tobit Model for Corner Solutions and Censored Data

Question 1

In a Tobit model of annual hours worked, the estimated coefficient on years of education is 80, and at the average x the probability of working, Φ(xβ/σ), is 0.6. What is the approximate partial effect of one more year of education on expected hours, E(y|x)?

• A. 80 hours
• B. 48 hours
• C. 133 hours
• D. 0.6 hours

Answer: (B)

For the Tobit model, ∂E(y|x)/∂x_j = β_j Φ(xβ/σ) = 80 × 0.6 = 48. The raw coefficient of 80 is the effect on latent desired hours, which overstates the effect on observed hours because some people remain at zero.

Question 2

OLS estimated only on observations with y > 0 gives consistent estimates of β in a Tobit setting.

• T. True
• F. False

Answer: False

Among people with a given x, those with y > 0 tend to have larger errors u. So E(u | x, y > 0) is positive and depends on x, which makes OLS on the truncated sample biased and inconsistent.

Question 3

For an observation with y = 0, what is its contribution to the Tobit likelihood?

• A. The normal density at zero
• B. The probability that y* ≤ 0, which is 1 − Φ(xβ/σ)
• C. Zero, because such observations are dropped
• D. The logistic CDF of xβ

Answer: (B)

A zero tells us only that the latent variable was at or below zero. Its probability is P(u ≤ −xβ) = Φ(−xβ/σ) = 1 − Φ(xβ/σ). Positive observations contribute the normal density (1/σ)φ((y − xβ)/σ).

Question 4

Explain the difference between a corner-solution outcome and a censored outcome, and why the distinction affects interpretation.

Model answer: A corner solution is a choice variable that truly equals zero for many people, such as hours worked, so the observed y is the quantity of interest. Censoring is a data problem, such as incomes top-coded above a threshold, where the true value exists but is not recorded, so the latent y* is the quantity of interest. A likelihood of the same form can be used, with the censoring point and direction adjusted, but with a corner solution you report effects on E(y|x), and with censoring you interpret β directly.

Wooldridge's introductory text stresses this distinction because the same Tobit output answers different questions in the two cases.