I am working with censored data and would like to employ a selection model. As far as I can see, the most frequently applied selection model is the Heckman selection model that assumes a two stage process.

First we have a partly latent variable $Y\in \mathbb{R}$, where we only observe $\Delta Y \in \mathbb{R_+}$, with $\Delta=I(Y>0)$ beeing an indicator function that is one if $Y$ is greater than zero which is seen as equivalent to $Y$ beeing observed. Furthermore we assume that the latent variable $Y$ can be modelled using a linear model, using some covariates $X$: $Y=X\beta + \epsilon$ and that we can use covariates $Z$ in order to employ a probit model for the probability of response, i.e. $P(\Delta=1|Z)=\Phi(Z\gamma)$, with $\Phi(\cdot)$ denoting a standard normal CDF.

Using this, plus the assumption that the errors from both stages are jointly normal, as well as $Z\subset X$, we can arrive at $$E(\Delta Y|X,\Delta=1)=E(Y|X,\Delta=1)=X\beta + E(\epsilon|X,\Delta=1)=X\beta + \rho \sigma_\epsilon\lambda(Z\gamma) $$ with $\lambda(\cdot)$ representing the inverse Mill Ratio evaluated at $Z\gamma$. So far so good. But for some theoretical reasons I cannot assume in my application, that $P(\Delta=1|Z)$ is modelled correctly by a standard normal, but rather $P(\Delta=1|Z)=\Lambda(Z\gamma)$ with $\Lambda(\cdot)$ beeing the logistic function. Now the crucial assumption of joint normality of the errors breaks down as the errors from the logistic regression come from the type I generalized extreme value distribution.

Is there a theoretic possibility to fix this problem or do I have to rely on some numerical approximations for $E(\epsilon|X,\Delta=1)$


2 Answers 2


The inverse Millls ratio is just the trick used in the Heckman's two-stage estimation procedure, as a result of the bivariate normal assumption as you mention

It is inconsistent and in the case of different marginal distributions the only way to go is to specify a joint likelihood

I would look into copulas as they give a very easy way of specifying nonlinear joint distributions of different marginal distributions, for an application in the sample selection context see this paper where they use a logistic for the selection equation and a student-t for the outcome equation

  • 1
    $\begingroup$ Please provide the title of the mentioned paper as the provided link is not working. Also mention software implementation of this technique. $\endgroup$ Aug 9, 2017 at 15:42
  • $\begingroup$ The link worked for me; the name of the paper is "Applying the Copula Approach to Sample Selection Modelling" $\endgroup$ Jun 12, 2023 at 15:37

Lee (1983) shows that this is possible in principle with any distribution and provides an example with a logit first stage. I am not aware of any off-the-rack implementation, though.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.