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How do you interpret the coefficient of inverse Mills ratio (lambda) in two step Heckman model?

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Let's say we have the following model: $$ y_{i}^{*}=x_{i}'\beta + \epsilon_{i} \quad \textrm{for} \quad i=1,\ldots,n $$

We can think about this in a few ways, but I think the typical procedure is to imagine us trying to estimate the effect of observed characteristics on the wage individual $i$ earns. Naturally, there are some people who choose not to work and potentially the decision to work can be modeled in the following way: $$ d_{i}^{*}=z_{i}'\gamma + v_{i} \quad \textrm{ for } \quad i =1,\ldots,n $$ If $d_{i}^{*}$ is greater than zero, we observe $y_{i} = y_{i}^{*}$ and if not we simply don't observe a wage for the person. I'm assuming that you know that OLS will lead to biased estimates as $E[\epsilon_{i}|z_{i},d_{i}=1]\neq 0$ in some circumstances. There are some conditions under which this might hold, which we can test via Heckman's Two-Step procedure. Otherwise, OLS is just misspecified.

Heckman tried to account for the endogeneity in this selection bias situation. So, to try to get rid of the endogeneity, Heckman suggested that we first estimate $\gamma$ via MLE probit, typically using an exclusion restriction. Afterward, we estimate an Inverse Mill's Ratio which essentially tells us the probability that an agent decides to work over the cumulative probability of an agent's decision, i.e.: $$\lambda_{i} = \frac{\phi(z_{i}'\gamma)}{\Phi(z_{i}'\gamma)}$$

Note: because we're using probit, we're actually estimating $ \gamma/\sigma_{v}$.

We'll call the estimated value above $\hat{\lambda}_{i}$. We use this as a means of controlling the endogeneity, i.e. the part of the error term for which the decision to work influences the wage earned. So, the second step is actually: $$y_{i} = x_{i}'\beta + \mu{\hat{\lambda_{i}}} + \xi_{i}$$

So, ultimately, your question is how to interpret $\mu$, correct?

The interpretation of the coefficient, $\mu$, is: $$ \frac{\sigma_{\epsilon{v}}}{\sigma_{v}^{2}} $$

What does this tell us? Well, this is the fraction of the covariance between the decision to work and the wage earned relative to the variation in decision to work. A test of selection bias is therefore a t-test on whether or not $\mu=0$ or ${\rm cov}(\epsilon,{v})=0$.

Hopefully that makes sense to you (and I didn't make any egregious errors).

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