= 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 [[EstimationOfRelationshipsForLimitedDependentVariables|Tobin (1958)]], is a special case of a censored regression model.

<<TableOfContents>>

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== Description ==

This is a modification of the [[Statistics/OrdinaryLeastSquares|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:

{{attachment:expectation1.svg}}

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 [[Statistics/NormalDistribution|p.d.f. and c.d.f. of the standard normal distribution]]. Altogether, the expected value is:

{{attachment:expectation2.svg}}

The '''hazard ratio''' or '''inverse [[Statistics/MillsRatio|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 [[Statistics/ProbitModel|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:

{{attachment:expectation3.svg}}

{{attachment:expectation4.svg}}

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,,1,1,,σ,,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[V,,2,,^2^] = σ,,2,2,,(1 + φ,,i,,λ,,i,, - λ,,i,,^2^)''
   * 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[V,,1,,V,,2,,] = σ,,1,2,,(1 + φ,,i,,λ,,i,, - λ,,i,,^2^)''
   * as ''φ,,i,,'' goes to infinity, this term approaches 0.
 * ''E[V,,1,,^2^] = σ,,1,1,,[(1 - ρ^2^) + ρ^2^(1 + φ,,i,,λ,,i,, - λ,,i,,^2^)]''
   * as ''φ,,i,,'' goes to infinity, this term approaches ''σ,,1,1,,(1 - ρ^2^)''.



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CategoryRicottone