Limited Dependent Variable Models

1. Core Model Classification

2. Key Conceptual Distinction

• Censoring (Tobit): You observe all individuals, but some values are cut or capped.

• Truncation: Some individuals are not in the dataset at all.

• Heckman: Some individuals are missing because of a decision process, and this creates selection bias.

3. Correct Classification of Heckman

• Heckman is not censoring.

• Heckman is not simple truncation.

• Heckman is best understood as non-random truncation (selection bias).

4. Core Decision Rule

• Cut values → Tobit

• Missing people due to rule → Truncated

• Missing people due to decision → Heckman

5. Common Mistake Patterns (From Practice)

Mistake 1: Overusing Tobit

Incorrect thinking:

• “0 + positive values → Tobit”

Correction:

• Always ask: Why is the value 0?

  • If due to threshold/censoring → Tobit

  • If due to behavioral decision → Heckman

Example:

• Loan default problem:

  • Default (0) is a decision

  • Repayment amount is an outcome

  • Two processes → Heckman

Mistake 2: Confusion about “random vs non-random”

Clarification:

• “Random” does not mean literally random.

• It means: inclusion is not driven by unobserved factors affecting Y.

Examples:

• Income cutoff (>100K) → Truncated

• Survey response (choice to respond) → Heckman

Correction:

• High-income individuals are different, but they did not choose inclusion → Truncated

Mistake 3: Confusion between Tobit and Heckman overlap

• Tobit can be used when there are zeros and a single process is assumed.

• Heckman is more appropriate when there are two processes (decision + outcome).

Exam Strategy:

• If behavioral decision is clearly present → choose Heckman

6. Model Intuition

Tobit (Single Process Model)

• Assumes the same factors determine:

  • whether (Y > 0)

  • how large (Y) is

• Limitation: unrealistic when participation and outcome differ

Heckman (Two-Process Model)

Step 1: Selection (Probit model)

• Determines whether observation is included

Step 2: Outcome (Regression model)

• Determines value of Y given selection

Key Feature:

• Allows different factors to influence participation and outcome

Truncated Regression

• Only part of the population is observed

• Missing observations are due to a rule, not behavior

• Corrects for truncation bias, but not selection bias

7. Final Decision Framework

Step 1: Are some individuals missing?

• No → Tobit

• Yes → go to Step 2

Step 2: Why are they missing?

• Data rule or cutoff → Truncated

• Behavioral decision → Heckman

Step 3: Are zeros present?

• Yes → Tobit or Heckman

• No → Truncated

8. Exam Flow

Ask the following:

1. Do I observe zeros?

   • Yes → Tobit or Heckman

2. Why do zeros exist?

   • Threshold/censoring → Tobit

   • Behavioral decision → Heckman

3. Are observations missing?

   • Yes → Truncated or Heckman

9. High-Value Exam Traps

• “0 present → always Tobit” → Incorrect

• “Missing data → always Heckman” → Incorrect

• “Different people → selection bias” → Incorrect

Correct:

• Only behavioral selection implies Heckman

10. Final Summary

• Tobit: censored values (0 + positive values, all observed)

• Truncated: missing observations due to rule/filter

• Heckman: missing observations due to behavioral decision (selection bias)

11. Key Insight for Difficult Questions

• If the problem involves:

  • A decision (participation/default/apply)

  • Followed by an outcome (amount/wage/spending)

→ Treat as two-step process → Heckman is most appropriate