Censored and Truncated Regression Models

A censored regression model is appropriate when the dependent variable is unavailable is above or below some threshold.

A truncated regression model is appropriate when cases are systemically not collected/unreported when the dependent variable is above or below some threshold.

The Tobit model, named for Tobin (1958), is a special case of a censored regression model.

Description

This is a modification of the OLS model, where the dependent variable Y is related to the independent variable(s) X as Y_i = bX_i + U_i.

Univariate

Suppose that the variable of interest is unobserved if it is less than zero. The expected value is then expressed as E[Y_i|X_i,Y_i≥0]. Substituting Y_i with the model equation yields E[bX_i + U_i|X_i,bX_i + U_i≥0], and because the expectation is conditioned on a given X_i this simplifies to bX_i + E[U_i|X_i,bX_i + U_i≥0]. Algebraically this is rewritten as:

where σ is the standard deviation of the error term U_i. The insertion of that standard deviation term transforms this into a formula that is easily decomposed into terms of the p.d.f. and c.d.f. of the standard normal distribution. Altogether, the expected value is:

The hazard ratio or inverse Mills' ratio (IMR) is notated as λ here. Sometimes λ evaluated for a given bX_i/σ is notated as λ_i.

Provided that the sample is censored (i.e., not truncated), it should be possible to estimate λ_i using a probit model. This reveals that selection bias seen in the initial model can be treated as omitted variable bias, and can be corrected by using the model Y_i = bX_i + σλ_i + V_i.

Bivariate

Suppose the variable of interest is unobserved if a second variable is less than zero, and suppose that these are drawn from a joint normal distribution. In other words, the model is specified as:

• Y_1i = bX_i + U_1i

• Y_2i = γZ_i + U_2i

  • X_i and Z_i can be the same, but often the system is only solvable when Z_i has more predictors.

Following the same procedures above, it can be demonstrated that:

where λ_i is specifically shorthand for λ evaluated for a given γZ_i/√σ_2,2.

The first equation is also sometimes rewritten in terms of the error correlation ρ = σ_1,2/√(σ_1,1σ_2,2): bX_i + ρ(√σ_1,1)λ_i.

Adding these omitted variables leads to a model specified as:

• Y_1i = bX_i + (σ_1,2/√σ_2,2)λ_i + V_1i

• Y_2i = γZ_i + (σ_2,2/√σ_2,2)λ_i + V_2i

• E[V₂²] = σ_2,2(1 + φ_iλ_i - λ_i²)

  • where φ_i is shorthand for φ evaluated for a given γZ_i/√σ_2,2.

  • as φ_i goes to infinity (i.e., the chance of selection approaches 100%), this term approaches 0.

• E[V₁V₂] = σ_1,2(1 + φ_iλ_i - λ_i²)

  • as φ_i goes to infinity, this term approaches 0.

• E[V₁²] = σ_1,1[(1 - ρ²) + ρ²(1 + φ_iλ_i - λ_i²)]

  • as φ_i goes to infinity, this term approaches σ_1,1(1 - ρ²).

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