Inverse Mills ratio

In statistics, the inverse Mills ratio, named after John P. Mills, is the ratio of the probability density function to the cumulative distribution function of a distribution.

Use of the inverse Mills ratio is often motivated by the following property of the truncated normal distribution. If x is a random variable distributed normally with mean μ and variance σ², then

$\begin{align} & \operatorname{E}[\,x\,|\ x > \alpha \,] = \mu + \sigma^2 \frac {\phi\big(\tfrac{\alpha-\mu}{\sigma}\big)}{1-\Phi\big(\tfrac{\alpha-\mu}{\sigma}\big)}, \\ & \operatorname{E}[\,x\,|\ x < \alpha \,] = \mu + \sigma^2 \frac {-\phi\big(\tfrac{\alpha-\mu}{\sigma}\big)}{\Phi\big(\tfrac{\alpha-\mu}{\sigma}\big)}, \end{align}$

where α is a constant, ϕ denotes the standard normal density function, and Φ is the standard normal cumulative distribution function. The two fractions are the inverse Mills ratios.^[1]

A common application of the inverse Mills ratio (sometimes also called 'selection hazard') arises 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 squares estimation (OLS) will produce biased parameter estimates.^[2] 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 model. The inverse mills ratio must be generated from the estimation of a probit model, a logit cannot be used. The probit model assumes that the error term follows a standard normal distribution.^[3] The estimated parameters are used to calculate the inverse Mills ratio, which is then included as an additional explanatory variable in the OLS estimation.^[4] See Heckman correction for more details.

References

1. ^ Greene, W. (2003) Econometric Analysis, Prentice-Hall.
2. ^ Tobin, J. 1958. Estimation of relationships for limited dependent variables. Econometrica, 26(1): 24–36.
3. ^ Heckman, James. Sample Selection as a Specification Error. Econometrica, Vol. 47, No. 1 (Jan., 1979), pp. 153–161
4. ^ 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.