Heckman regression (Inverse mills ratio) significant or not?

Question

I am using a two-step Heckman regression model and I want to evaluate if probit looks okay, that the model converges, and that there are no "red" flags.

One of the estimators that I get is the inverse Mills ratio. Is this supposed to be statistically significant or not?

I am using an example from the book:

summary(heckit(lfp ~ age + I( age^2 ) + faminc + kids + educ,
               wage ~ exper + I( exper^2 ) + educ + city, Mroz87 ) )

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Tobit 2 model (sample selection model)
2-step Heckman / heckit estimation
753 observations (325 censored and 428 observed)
14 free parameters (df = 740)

Probit selection equation:
                Estimate   Std. Error t value  Pr(>|t|)    
(Intercept) -4.156806923  1.402085958  -2.965  0.003127 ** 
age          0.185395096  0.065966659   2.810  0.005078 ** 
I(age^2)    -0.002425897  0.000773540  -3.136  0.001780 ** 
faminc       0.000004580  0.000004206   1.089  0.276544    
kidsTRUE    -0.448986740  0.130911496  -3.430  0.000638 ***
educ         0.098182281  0.022984120   4.272 0.0000219 ***

Outcome equation:
              Estimate Std. Error t value  Pr(>|t|)    
(Intercept) -0.9712003  2.0593505  -0.472     0.637    
exper        0.0210610  0.0624646   0.337     0.736    
I(exper^2)   0.0001371  0.0018782   0.073     0.942    
educ         0.4170174  0.1002497   4.160 0.0000356 ***
city         0.4438379  0.3158984   1.405     0.160    

Multiple R-Squared:0.1264,  Adjusted R-Squared:0.116

Error terms:
              Estimate Std. Error t value Pr(>|t|)
invMillsRatio   -1.098      1.266  -0.867    0.386
sigma            3.200         NA      NA       NA
rho             -0.343         NA      NA       NA
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Answer 1

The Inverse Mills Ratio times its coefficient is supposed to pick up the expected value of the error in the wage equation conditional on working. This will reflect the idea that the women with large negative wage errors are not working, so the expected value of the wage error is no longer zero for some of the women who do work.

You can use the coefficient on the IMR to test for selection, since it represents the covariance between the errors in the wage and the participation equation under the assumptions of the model. The parameter $\rho$ is the correlation between these errors, since you can get a correlation by dividing the covariance by the product of the standard deviations. This is -1.098/(1*3.200) = -0.343 here, since the variance in the participation equation is normalized to one. The denominator is just two standard deviations (product of two positive numbers), so it is sufficient to test that the numerator is zero to learn about selection. In your model the t-statistics on the IMR is small and the p-value is large, so you cannot reject the null that the errors are uncorrelated. This does not mean that they are uncorrelated, only that the data is consistent with no selection.

Answer 2

an insignificant lambda may not indicate an absence of sample selection bias. If a sample is small and/or exclusion restrictions are weak, Heckman models are unlikely to produce significant lambdas—even in the presence of sample selection bias. for more information see https://media.terry.uga.edu/socrates/publications/2018/05/Heckman_Sample_Selection__2016_1.pdf