[Nursena Sagir]

Dear Sebastian,

Is there any reference that I can understand this better and explain why I can treat X_it as predetermined? I had difficulties to explain it in the method section of my paper. Or can you elaborate more on your reasoning?

Best regards,
Nursena

[Sebastian Kripfganz]

As long as you do not change the second equation, my earlier statement about X_it being predetermined still stands.

Once X_it becomes a direct function of Y_it, as in your amended second equation, X_it becomes endogenous.

[Nursena Sagir]

Dear Sebastian,

I have a follow-up question.

I think there are at least two things that are potentially confusing here.


In your two-equations example. X_it is predetermined because it is a function of Y_it-1. But X_it-1 is uncorrelated with the error term in ε_it (and any future error term) in equation 1, which is all that is needed for it to be a valid instrument (in the FOD-transformed model).

You have stated that for the two-equations example below:

1. Yit = Yit-1 + Yit-2 + Xit-1 + Xit-2 + εit
2. Xit = Xit-1 + Xit-2 + Yit-1 + Yit-2 + εit

If I change the first equation to:

1. Yit = Yit-1 + Yit-2 + Xit + Xit-1 + εit

which includes contemporaneous X, can I still assume that Xit is predetermined? When I look at the incJ test, it does not reject the null hypothesis that the additional overidentifying restriction for predetermined X is valid (p-value =0.95).

Would your response change if I have second equation as:

2. Xit = Xit-1 + Xit-2 + Yit + Yit-1 + εit

Thank you in advance!

Best regards,
Nursena

[Sebastian Kripfganz]

Yes, some of those lagged interaction terms will be equal to each other. As far as I can tell, this should be fine.

[Nursena Sagir]

Dear Sebastian,

Thank you for your explanation. Specifying them explicitly added relevant instruments for interaction with the first lags but not with the second lags. Is this because L1.(0.gender#cL.depression) = (0.gender#cL2.depression), L2.(0.gender#cL.depression)=L1.(0.gender#cL2.depre ssion) so they are dropped in model 4 below? Can I still rely on the produced output with these dropped instruments?

Instruments corresponding to the linear moment conditions:
 1, model(fodev):
   L1.depression L2.depression L3.depression L4.depression
 2, model(fodev):
   L.income L1.L.income L2.L.income L3.L.income L4.L.income
 3, model(fodev):
   L1.(0.gender#cL.depression) L2.(0.gender#cL.depression)
   L3.(0.gender#cL.depression) L4.(0.gender#cL.depression)
 4, model(fodev):
   L4.(0.gender#cL2.depression)
 5, model(fodev):
   0.gender#cL.income L1.(0.gender#cL.income) L3.(0.gender#cL.income)
   L4.(0.gender#cL.income)
 6, model(fodev):
   L1.(0.gender#cL2.income) L4.(0.gender#cL2.income)
 7, model(level):
   3bn.wave 4.wave 5.wave 6.wave 7.wave 8.wave 9.wave 10.wave 11.wave 12.wave
   13.wave 14.wave 15.wave 16.wave 17.wave

Best regards,
Nursena

[Sebastian Kripfganz]

In the specification of the instruments, the base category for the gender dummy is ignored. The command creates interaction terms with both 0.gender and 1.gender. This leads to perfect collinearity with some of the non-interacted instruments, which is why some of them are dropped. You could explicitly specify interaction terms as c.L1.depression#0.gender etc to ensure that only this particular group is interacted.

[Nursena Sagir]

Dear Sebastian,

Thank you for your reply. I use the following model as main estimation.

 
 xtdpdgmm L(0/2).(depression) L(1/2).income, model(fodev) collapse gmm(depression, l(1 4))  gmm(L.income, l(0 4)) teffects two vce(r) nocons overid

And the produced instrument list is below as expected.

 
Instruments corresponding to the linear moment conditions:
 1, model(fodev):
   L1.depression L2.depression L3.depression L4.depression
 2, model(fodev):
   L.income L1.L.income L2.L.income L3.L.income L4.L.income
 3, model(level):
   3bn.wave 4.wave 5.wave 6.wave 7.wave 8.wave 9.wave 10.wave 11.wave 12.wave
   13.wave 14.wave 15.wave 16.wave 17.wave

However, when I add interaction terms for gender for all regressors, instrument set looks wrong.

 
 xtdpdgmm L(0/2).(depression) L(1/2).income c.L1.depression#ib1.gender c.L2.depression#ib1.gender c.L1.income#ib1.gender c.L2.income#ib1.gender, model(fodev) collapse gmm(depression, l(1 4))  gmm(L.income, l(0 4)) gmm(c.L1.depression#ib1.gender, l(1 4)) gmm(c.L2.depression#ib1.gender, l(1 4))  gmm(c.L1.income#ib1.gender, l(0 4))  gmm(c.L2.income#ib1.gender, l(0 4)) teffects two vce(r) nocons overid

As you can see below, it drops L3.depression from row 1 and L.income from row 2 as instrument. Then, shows arbitrary matches for gender categories and lags.

Instruments corresponding to the linear moment conditions:
 1, model(fodev):
   L1.depression L2.depression L4.depression
 2, model(fodev):
   L1.L.income L2.L.income L3.L.income L4.L.income
 3, model(fodev):
   L2.(0.gender#cL.depression) L4.(0.gender#cL.depression)
   L1.(1b.gender#cL.depression) L2.(1b.gender#cL.depression)
   L4.(1b.gender#cL.depression)
 4, model(fodev):
   L2.(0.gender#cL2.depression) L4.(0.gender#cL2.depression)
   L4.(1b.gender#cL2.depression)
 5, model(fodev):
   0.gender#cL.income L3.(0.gender#cL.income) 1b.gender#cL.income
 6, model(fodev):
   0.gender#cL2.income L1.(0.gender#cL2.income) L4.(0.gender#cL2.income)
   L3.(1b.gender#cL2.income) L4.(1b.gender#cL2.income)
 7, model(level):
   3bn.wave 4.wave 5.wave 6.wave 7.wave 8.wave 9.wave 10.wave 11.wave 12.wave
   13.wave 14.wave 15.wave 16.wave 17.wave

Do you have any recommendation for me to correct the estimation code?

Thank you in advance.

Best regards,
Nursena

[Sebastian Kripfganz]

In a nutshell: Yes.

You should still think about what your model assumptions imply about the error term, because this will ultimately decide the instrument validity. If X_it is correlated with ε_it (because of simultaneity), then X_i,t-1 is predetermined with respect to ε_it (i.e., X_i,t-1 is uncorrelated with ε_it but correlated with ε_i,t-1). In the first-differenced model, the error term is ε_it-ε_i,t-1. Thus, X_i,t-1 would be correlated with this first-differenced error term but X_i,t-2 is not (assuming that the errors are not serially correlated). With the FOD transformation, the transformed errors are only a function of current (period t) and future errors. Thus, X_i,t-1 remains a valid instrument.

[Nursena Sagir]

Dear Sebastian,

Thank you for your reply.

In your two-equations example. X_it is predetermined because it is a function of Y_it-1. But X_it-1 is uncorrelated with the error term in ε_it (and any future error term) in equation 1, which is all that is needed for it to be a valid instrument (in the FOD-transformed model).

I'd like to clarify my understanding of instrument validity under different transformations.

For my model: Yit = Yit-1 + Yit-2 + Xit-1 + Xit-2 + εit

Under FOD transformation:

• I can use L.X_it-1 as an instrument for X_it-1
• And similarly L.X_it-2 as an instrument for X_it-2
• While having Yit-1 and Yit-2 as regressors

Under first-difference transformation:

• I need to start from L2.X_it-1 as instrument bcs. X_it-1 appears in the differenced equation as (X_it-1 - X_it-2)

Is my understanding of these different requirements under FOD versus first-differences correct?

Thank you for your help in clarifying this.

Best regards,
Nursena

[Sebastian Kripfganz]

I think there are at least two things that are potentially confusing here.

First, in my quoted post, lag 0 of L.n refers to L.n itself, which is lag 1 of n.

Second, the FOD transformation and the first-difference transformation are not the same. (X_i,t-1 - X_i,t-2) is the first-difference transformation of X_i,t-1. The FOD transformation would subtract the average of X_i,t-1 X_it X_i,t+1 ... X_i,T from X_i,t-1. This average does not contain X_i,t-2.

In your two-equations example. X_it is predetermined because it is a function of Y_it-1. But X_it-1 is uncorrelated with the error term in ε_it (and any future error term) in equation 1, which is all that is needed for it to be a valid instrument (in the FOD-transformed model).

[Nursena Sagir]

Dear Sebastian,

I have a question regarding to your quote below:

Please have a look at my 2019 London Stata Conference presentation. Slide 67 tells you the admissible lags under forward-orthogonal deviations. For strictly exogenous regressors (which I implicitly assumed here for w and k), any lag can be used. For predetermined regressors (L.n), lag 0 is the first admissible lag. In my specification, I assumed indeed that there are no endogenous variables (with respect to the idiosyncratic error term) in the model.

For the level model, slide 31 tells you the additionally available instruments. For strictly exogenous and predetermined regressors, lag 0 of the first-differenced instruments is usually used (because it has the strongest correlation with the instrumented variables compared to any other lag). It is common practice not to use multiple lags as instruments for the level model (based on the idea that further lags would be redundant if all available lags were used for the transformed model).

I did not understand how lag 0 is the first admissible lag if L.n is a regressor in the estimation equation. I will give you a example.

Assume that you would like to run a model with reverse causality:

1. Yit = Yit-1 + Yit-2 + Xit-1 + Xit-2 + εit
2. Xit = Xit-1 + Xit-2 + Yit-1 + Yit-2 + εit

But, when differenced in FOD estimation, Xi,t-1 appears directly in (Xi,t-1 - Xi,t-2) in equation 1. I do not understand how can I use Xi,t-1 as an instrument - not because of correlation with the error terms, but because it's a regressor in the differenced equation. But when I define Xit-1 as predetermined regressor it means lag 0 (Xit-1) is the first admissible lag. Or if I define it as endogenous, then lag 1 (Xit-2) can be used but it is also included as regressor. It this something that I should be concerned? Should I start from lag 3 and use gmm(Xi,t-1, lag(3 .))?

Or should I just think that the sequential exogeneity condition (allowing εit to correlate with future x values) accounts for reverse causality regardless of how I specify the lag structure in the model. So, I can use regressors as theory points and classify the instruments based on coherence tests?

Thank you for your reply.

Best regards,
Nursena

[Tullio Gregori]

Thank you Sebastian
you are right and maybe in this setting we should go for the Sargan's difference test as suggested by Sarafidis, Yamagata and Robertson

https://www.sciencedirect.com/scienc...608001826#sec3

thanks again

[Sebastian Kripfganz]

The CD test is not designed for use after GMM estimation. I do not have enough experience with this test to provide a more solid judgement.

[Tullio Gregori]

Dear Sebastian
in your post 367
https://www.statalist.org/forums/for...40#post1652040
you suggest an incremental Hansen test for the validity of the moment conditions for the lagged dependent variable to check for cross-sectional dependence
I wonder if a standard cd test on residuals may suffice.
Thanks for your reply

[Sarah Magd]

Thank you very much Prof. Sebastian Kripfganz

What we do in the first step, before using GMM, is estimate our models with FE and show different specifications. We also report the R² and adjusted R². As I understand from your answer, I can rely on the R² values to compare my specifications, since R² is not affected by the endogeneity problem caused by the simultaneity issue. Is that correct?