Short version of the question:
Is it possible to create a dependent variable in the first step of the Heckman Selection model such that it is possible to obtain the values for the calculation of the Inverse Mills Ratio for the second step by OLS? If so, are there any pitfalls regarding it?
Long version:
A two-step selection model is used, e.g., in the case that due to a selection process a significant part of the outcome variable $y$ is zero. To estimate unbiased coefficients of the independent variables, Heckman (1979) proposed a two-step estimator, where in the first step a Probit regression of a dummy indicating that $y>0$ on explanatory variables. Then the inverse mills ratio of every observation is calculated and included in a second-step OLS regression of the observations with $y>0$ of $y$ on explanatory variables.
The inverse Mills ratio is the ratio of the probability density function and the cumulative density function of the normal distribution evaluated at the predicted outcomes $x*b_2$ devided by the standard error of the Probit estimation (normally the standard error is assumed to be one).
The Selection model is the following (based on Wooldridge): $$ \begin{aligned} \text{Equation 1:}~~~~~~~~~~~~~~ y1 &= x_1b_1+u_1 \\ \text{Equation 2:}~~~~~~~~~~~~~~ y2 &= 1[xb_2+v_2>0] \end{aligned} $$ where $x_1$ is contained in $x$. It is assumed that $E(u_1|v_2)=c_1*v_2$, i.e. we assume linearity. Moreover, $v_2\sim\mathcal N(0,1)$ and $(u_1,v_2)$ are independent of $x$ with zero mean. $(x,y_2)$ are always observed, $y_1>0$ only if $y_2=1$.
$y_1>0$ only for $y_2=1$
Running OLS on the sample, where $y_1>0$ and thus $y_2=1$ yields biased estimates because of the correlation of the error terms. One has to include a correction for this: $$ E(y_1|x_1,y_2)=x_1*b_1+c_1*E(v_2|x,y_2)=x_1*b_1+c_1*h(x,y_2) $$ where $h(x,y_2)=E(v_2|x,y_2)$. In the censored sample where $y_2=1$, so we have: $$ E(y_1|x_1,1)=x_1*b_1+c_1*E(v_2|x,y_2=1)=x1*b_1+c_1*h(x,1) $$ As $E(v_2|x,y_2=1)=E(v_2|v_2>-b_2x_2)$=inverse Mills ratio (IMR) of $(b_2*x_2)$, we have: $$ E(y_1|x_1,1)=x_1*b_1+c_1*IMR(b_2*x_2) $$ as a corrected regression of the censored sample.
Estimates for $b_2$ can be obtained by running a Probit regression of a dummy indicating that $y_1>0$ on independent variables $x_2$. As the Probit estimates are the inverse of the standard normal cumulative, $x_1*b_2$ of the Probit estimates have the desired feature that e.g. for the predicted probability 0.5 of $y_2=1$ half of $(xb_2+v_2)$ are below and half are above zero and thus the error term is positive for half of these observations.
Why doesn't one just create a variable $y_2$ that is above zero for $y_1>0$ and below zero otherwise and runs an OLS regression on this variable as the first step regression and construct the Inverse Mills Ratio using the obtained estimates? If it is possible, what features does this variable need to have?
An advantage of this procedure would be the possibility to estimate a fixed effects model in the first step with panel data by time-demeaning all observations.
Literature:
- Heckman, James (1979): "Sample Selection as a Specification Error". Econometrica, 47 (1), 153–161
- Wooldridge: Econometric Analysis of Cross-Section and Panel Data
