# Heckman regression (Inverse mills ratio) significant or not?

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 ) )

--------------------------------------------
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

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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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.