Estimating rho and sigma of movestay manually

[Nobuya Fukugawa]

Dear Statalisters,
Here is the code I used.
---copy starts---
probit d x z
predict xb, xb
gen lambda = normalden(xb) / normal(xb) if treat== 1
replace lambda = -normalden(xb) / (1- normal(xb)) if treat== 0
gen lambda1 = lambda
gen lambda0 = -normalden(xb) / (1- normal(xb)) if treat== 0

regress y x i.id i.year lambda1 if treat== 1
regress y x i.id i.year lambda0 if treat== 0
---copy ends---

Notes
*lambda= inverse Mills ratio. A coefficient of lambda represents co-variance of the error term of selection equation, u, and the error term of either of the main equations, ε¹ or ε⁰.
** In movestay command, rho is correlation coefficient between u and ε¹ or ε⁰. Sigma is standard deviation of ε¹ or ε⁰. Variance of u is assumed to be one. Coefficient of lambda1=sigma1*rho1. Coefficient of lambda0=sigma0*rho0

Having estimated the coefficient of lambda, I need to know the followings.
Q1: Is rho of movestay is significant when estimated coefficient of lambda is significant?
Q2: How do I estimate rho and sigma of movestay separately and manually?

Any help would be appreciated.
I do recognize movestay is a user generated program, but I just would like to have a broad spectrum of opinion.
For reference, the paper introduced movestay is
https://www.stata-journal.com/sjpdf....iclenum=st0071

Nobuya Fukugawa

[Nick Cox]

Cross-posted at https://stats.stackexchange.com/ques...stata-manually

Please note our policy on cross-posting (FAQ Advice https://www.statalist.org/forums/help#crossposting), which is that you are asked to tell us about it.

[FernandoRios]

Hi Nobuya
The manual estimation of RHO from the movestay two step procedure is not a two lines of code (although a good programmer may actually debate this).
Nevertheless, couple of answer to your questions:
Q1: Is rho of movestay is significant when estimated coefficient of lambda is significant?
Mostly, yes. Lambda=rho * sigma. So lambda is significant when Rho is significant but also when sigma is significant. When interpreting selection models, however, we usually say if lambda is significant the correlation between the model selection process and the outcome regression is significant and different from zero.
Q2: How do I estimate rho and sigma of movestay separately and manually?
If you are using the MLE option of movestay, you cant estimate them manually, you need to do it within the ML function.
if what you are using is the two step option, you could:
1. estimate separate heckman regressions using two steps, and recover lambda sigma and rho from there, or:
2. Still do two step heckman, consult the manual formulas section of heckman tw step. It provides a very clear description on how rho and sigma are estimated.

HTH
Fernando