Heteroskedasticity in Heckman two step model (Tobit II model)

[Rosamond Waston]

Hi,

I have a confusion regarding heckman two-step model. As it's well-know to us, the first stage of Heckman is a probit model, and we obtain Inverse Mills Ratio from the first stage and then use OLS regression and insert IMR as an extra explanatory variable. However, I would like to ask does the heteroscedasticity matter in the first stage probit model (because heterskedasticity in probit model itself is a severe problem)?

My code is presented below (x2 is endogeneous variable in the second stage, say wage equation, and its instrument is z3)

* Step 1. estimate the selection equation
probit LFP c1 c2 c3 x1 x2 x3 x5 x9 i.d i.x8 i.x7 
predict xb, xb
estimates store first



* Step 2. generate inverse mills ratio
gen mills = normalden(xb)/normprob(xb)
sort xb
scatter mills xb
drop xb

* Step 3. using 2sls regression
ivregress 2sls y x1 x3 x4 x5 x6 i.x7 i.x8 i.d mills (x2 = z3), robust 
estimates store myheck

The results from the second stage show that the IMR is insignificant (we have no selection bias).

Could you please let me know? Does the heteroskedasticity matter in the first stage (cuz it is a probit model)? Does it matter when lambda is siginificant? Shall we use

hetprob

to address for such heteroskedasticity issue?


Thank you in advance.

Regards,

[Rosamond Waston]

Estimation result presents below:



* Step 3. using 2sls regression
. ivregress 2sls y x1 x3 x4 x5 x6 i.x7 i.x8 i.d mills (x2 = z3), robust

Instrumental variables (2SLS) regression Number of obs = 261
Wald chi2(12) = 81.15
Prob > chi2 = 0.0000
R-squared = 0.2628
Root MSE = .67924

------------------------------------------------------------------------------
| Robust
y | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
x2 | .105599 .0327025 3.23 0.001 .0415033 .1696947
x1 | -.0620285 .1180802 -0.53 0.599 -.2934615 .1694044
x3 | .0303235 .0322925 0.94 0.348 -.0329686 .0936156
x4 | -9.99e-07 1.47e-06 -0.68 0.498 -3.89e-06 1.89e-06
x5 | .0006435 .0018409 0.35 0.727 -.0029647 .0042516
x6 | -.0003333 .0011244 -0.30 0.767 -.0025371 .0018705
1.x7 | .6336446 .325488 1.95 0.052 -.0043001 1.271589
|
x8 |
1 | -.1893487 .1673808 -1.13 0.258 -.5174089 .1387116
2 | -.1969385 .1218002 -1.62 0.106 -.4356624 .0417855
3 | .1915922 .1389663 1.38 0.168 -.0807768 .4639612
|
1.d | .1074228 .1072101 1.00 0.316 -.1027052 .3175508
mills | .4637238 .3498221 1.33 0.185 -.2219148 1.149362
_cons | 8.954886 2.033426 4.40 0.000 4.969444 12.94033
------------------------------------------------------------------------------
Instrumented: x2
Instruments: x1 x3 x4 x5 x6 1.x7 1.x8 2.x8 3.x8 1.d mills z3

[Joao Santos Silva]

Dear Rosamond Waston,

Others in this forum will be in a better position to comment but, yes, heteroskedasticity and normality matter. So, even using hetprob it is unlikely that the estimator will be valid (as you may guess, I am quite sceptical about the usefulness of this estimator). Not to mention that the estimator is not designed to work with endogenous explanatory variables, so I certainly would not do what you suggest above.

Best wishes,

Joao

[Rosamond Waston]

Dear Rosamond Waston,

Others in this forum will be in a better position to comment but, yes, heteroskedasticity and normality matter. So, even using hetprob it is unlikely that the estimator will be valid (as you may guess, I am quite sceptical about the usefulness of this estimator). Not to mention that the estimator is not designed to work with endogenous explanatory variables, so I certainly would not do what you suggest above.

Best wishes,

Joao

Dear Prof. Joao,

Many thanks for your replies.

However, I was wondering shall I just leave the heteroskedasticity in the first-stage probit model as it is and dont do on it? As I can see from many books, including Woodbridge, he said if the IMR is insignificant, the S.E does not matter too much. Also, they do not often talk about how to address the heteroskedasticity in Heckman? I am confused about how can I address the heteroskedasticity in the first step?

Many thanks

[Joao Santos Silva]

Dear Rosamond Waston,

There are methods to deal with these issues (non-normality and hetroskedasticity) but they are quite sophisticated and most of them will not be able to deal with endogeneity.

Best wishes,

Joao

[Rosamond Waston]

Dear Rosamond Waston,

There are methods to deal with these issues (non-normality and hetroskedasticity) but they are quite sophisticated and most of them will not be able to deal with endogeneity.

Best wishes,

Joao

Many thanks for your replies.

I think S.E in the first stage probit is not important, if we can obatin insignificant \lambda (IMR) at the second stage. Is this reasoning correct?



Regards,

[Joao Santos Silva]

I do not think you can trust the significance of the IMR term in your second step.