Yes, the LR test of rho=0 only appears in the maximum likelihood estimations, but not in the two-step process. Take a look at the following examples from the STATA manual:

use http://www.stata-press.com/data/r11/womenwk, clear
heckman wage educ age, select(married children educ age)
heckman wage educ age, select(married children educ age) twostep

(results omitted)

You need to understand what the LR test is for. according to the manual, the likelihood-ratio test reported at the bottom of the output is an equivalent test for rho = 0 and is computationally the comparison of the joint likelihood of an independent probit model for the selection equation and a regression model on the observed wage data against the Heckman model likelihood. Because chi2 = 61.20 in the above model (not the two-step one), this clearly justifies the Heckman selection equation with these data.

When do we choose the two-step version? well, maximum likelihood estimation of the parameters can be time consuming with large datasets, and the two-step estimates may provide a good alternative in such cases.

With more powerful computers today, however, MLE is not a problem in most cases, from my experience.

The Heckman selection model depends strongly on the model being correct, much more so than ordinary regression. There is a third option: the two-part model. according to the manual, running a separate probit or logit for sample inclusion followed by a regression, referred to in the literature as the two-part model is an especially attractive alternative if the regression part of the model arose because of taking a logarithm of zero values.

When to use which?

When the goal is to analyze an underlying regression model or to predict the value of the dependent variable that would be observed in the absence of selection, however, the Heckman model is more appropriate. When the goal is to predict an actual response, the two-part model is usually the better choice.

hope this long response helps! good luck.

解答太牛了！