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| Suppose that data is unobserved if the dependent variable is less than zero. The expected value is then expressed as ''E[Y,,i,,|X,,i,,,Y,,i,,≥0] = bX,,i,, + E[U,,i,,|Y,,i,,≥0]''. | === 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,,''. 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^)'' * 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^)''. where ''φ,,i,,'' is shorthand for ''φ'' evaluated for a given ''γZ,,i,,/√σ,,2,2,,''; and ''ρ = σ,,1,2,,/√(σ,,1,1,,σ,,2,2,,)''. |
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 Yi = bXi + Ui.
Univariate
Suppose that the variable of interest is unobserved if it is less than zero. The expected value is then expressed as E[Yi|Xi,Yi≥0]. Substituting Yi with the model equation yields E[bXi + Ui|Xi,bXi + Ui≥0], and because the expectation is conditioned on a given Xi this simplifies to bXi + E[Ui|Xi,bXi + Ui≥0]. Algebraically this is rewritten as:
where σ is the standard deviation of the error term Ui. 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 bXi/σ is notated as λi.
Provided that the sample is censored (i.e., not truncated), it should be possible to estimate λi 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 Yi = bXi + σλi + Vi.
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:
Y1i = bXi + U1i
Y2i = γZi + U2i
Xi and Zi can be the same, but often the system is only solvable when Zi has more predictors.
Following the same procedures above, it can be demonstrated that:
where λi is specifically shorthand for λ evaluated for a given γZi/√σ2,2.
Adding these omitted variables leads to a model specified as:
Y1i = bXi + (σ1,2/√σ2,2)λi + V1i
Y2i = γZi + (σ2,2/√σ2,2)λi + V2i
E[V22] = σ2,2(1 + φiλi - λi2)
as φi goes to infinity (i.e., the chance of selection approaches 100%), this term approaches 0.
E[V1V2] = σ1,2(1 + φiλi - λi2)
as φi goes to infinity, this term approaches 0.
E[V12] = σ1,1[(1 - ρ2) + ρ2(1 + φiλi - λi2)]
as φi goes to infinity, this term approaches σ1,1(1 - ρ2).
where φi is shorthand for φ evaluated for a given γZi/√σ2,2; and ρ = σ1,2/√(σ1,1σ2,2).