Inverse Mills ratio

The inverse Mills' ratio is a concept in statistics. It is the ratio of the probability density function over the cumulative distribution function of a distribution.

The inverse Mills' ratio (sometimes also called 'selection hazard') is used in regression analysis to take account of a possible selection bias. If a dependent variable is censored, i.e. not for all observations a positive outcome is observed, it causes a concentration of observations at zero values. This problem was first acknowledged by Tobin (1958), who showed that if this is not taken into consideration in the estimation procedure, an ordinary least square estimation (OLS) will produce biased parameter estimates. With censored dependent variables there is a violation of the Gauss-Markov assumption of zero correlation between independent variables and the error term. Heckman (1976) proposed a two-stage estimation procedure using the inverse Mills' ratio to take account of the selection bias. In a first step, a regression for observing a positive outcome of the dependent variable is modeled with a probit or logit model. The estimated parameters are used to calculate the inverse Mills' ratio, which is then included as an additional explanatory variable in the OLS estimation.

Most standard statistical packages have the procedure programmed already.


 * Tobin, J. 1958. Estimation of relationships for limited dependent variables. Econometrica, 26(1): 24-36.


 * Heckman, J. J. 1976. The common structure of statistical models of truncation, sample selection and limited dependent variables and a simple estimator for such models. Annals of Economic and Social Measurement, 5(4): 475-492.