Forbidden Regression Problem?

[Kyle Alex Smith]

I am quite confused about the forbidden regression problem, and unsure if I have one in this case.

Stage 1:

Probit Model: Prob(Vulnerability) = Set of regressors that predict vulnerability.

The predicted value "Predicted vulnerability" is a variable of interest in stage 2.

Stage 2:

Seemingly Unrelated Regression: DV = Predicted Vulnerability + Controls

Does having a probit first stage model with a SUR in the second stage a case of a forbidden regression? If so, what is the best solution for this?

It seems the discussion around forbidden regressions is centered around instrumental variable approaches. In my setting, the first stage model is mostly useful for just condensing multiple determinants of vulnerability into one predicted measure.

[George Ford]

It is FR.

Couple questions.
1. Why not just use DV from probit in second stage? Endogeneity?
2. Why using SUR?

If not SUR, then

etregress depvar [indepvars], treat(depvar_t = indepvars_t) [twostep]

[Kyle Alex Smith]

Hi George,

The construct I am interested in is like likelihood of a future distress. Vulnerability is a measure of a current distress (I'm realizing it's a bad variable name, I just pull it from prior lit).

I use SUR as I have two DV and my hypotheses compare the coefficients across the two SUR equations.

[George Ford]

Do the two DVs measure future distress? Or, is vulnerability a measure of lagged distress? Or, are you saying that certain observables are correlated with distress so you are predicting the propensity to be distressed in the first stage?

[Kyle Alex Smith]

The two DVs are donations and grants. I am seeking to see if they respond to predicted vulnerability (the likelihood of a future distress). Predicted Vulnerability and controls lagged one year.

Prob(Current Distress) = regressors shown to predict a future distress (lagged of course)

Derive predicted vulnerabiltity from this.

SUR:

Donations = a1Predicted Vulnerability + Controls (Predicted vulnerability and all controls lagged one year)

Grants = b1Predicted Vulnerability + Controls (Predicted vulnerability and all controls lagged one year)

Hypothesis: a1=b1

[George Ford]

I don't know what "distress" is measuring, but if its related to "happiness" then you've got a jointly determined system. If you think distress is not a function of donations/grants, then estimate stage 2 and forget about stage 1.
The first stage is not the likelihood of future distress, it is the likelihood of current distress. It is the prediction of a DV measured as current distress.

There is reason to suspect that they are jointly determined.

https://www.hbs.edu/ris/Publication%20Files/10-012_0350a55d-585b-419d-89e7-91833a612fb5.pdf
https://www.talkspace.com/blog/giving-back-mental-health-impact/

Is this two-stage thing done in the literature on this topic?

Might look at control function.

[Kyle Alex Smith]

The primary benefit of doing this in 2 stages is that the "predicted vulnerability" construct is a combination of around 6 measures (the regressors in the first stage). If I insert all 6 into stage 2 and justrun this as 1 stage, it becomes difficult to interpret my findings on this construct.

[Joro Kolev]

What you have written in #5 is not a Seemingly Unrelated Regression, but rather a Multivariate Regression, because the regressors in both your equations are the same.

When the regressors are the same in all equations, Seemingly Unrelated Regression reduces to a Multivariate Regression, which in turn is equivalent to OLS equation by equation.

What you have is more of a Generated Regressor problem, rather than a Forbidden Regression.

What you can do is the following:


1. Estimate your two equations of the system like George told you using -etregress-. Use full maximum likelihood, and not two stage or control function.

2. Use -suest- to carry out the across equations test you have in mind.

[Kyle Alex Smith]

Hi Joro,

Yes my understanding is that the SUR in my setting is equivalent to OLS equation by equation.

I will explore the usage of -etregress-.

Another question: I have two other variables of interest that are not generated from a first stage model. Will the coefficients on those variables be correct when using -etregress-, or should I run those tests separately and just include the 6 first stage variables as controls?

Thank you!

[Joro Kolev]

Everything will be correct if you use -etregress- because it correctly takes into account all interplay between the first and second stage.

If in the second stage you have a generated regressor, the standard errors and wrong for all variables, even for the ones which are not generated.

Hi Joro,

Yes my understanding is that the SUR in my setting is equivalent to OLS equation by equation.

I will explore the usage of -etregress-.

Another question: I have two other variables of interest that are not generated from a first stage model. Will the coefficients on those variables be correct when using -etregress-, or should I run those tests separately and just include the 6 first stage variables as controls?

Thank you!

[Kyle Alex Smith]

OK so I will try:

etregress donations a1VOI1 a2VOI2 Controls , treat(Distress = x1 x2 x3)

etregress grants b1VOI1 b2VOI2 Controls , treat(Distress = x1 x2 x3)


What would be the appropriate code to run a1=b1?

Previously with SUR I was using:

lincom _b[donations:programratio_tm1] - _b[grants:programratio_tm1]

[George Ford]

rough sketch, but I'd seek confirmation this is legit (Joro?)

sysuse auto, clear
etregress price headroom weight , treat(foreign = turn displacement)
eststo e1
etregress mpg headroom weight , treat(foreign = turn displacement)
eststo e2
suest e1 e2
test [e1_price]weight=[e2_mpg]weight

[Joro Kolev]

The code George displays implements what I have in mind.

And I have not seen this approach in textbooks or papers, so this is brand new super modern stuff. It should work because -etregress- is taking appropriate care of the relationship between first and second stage, and -suest- is taking appropriate care of the relationship between the two equations.

Most importantly, if this super modern stuff makes its way into a paper, Kyle Alex Smith must most definitely cite the two Georges, George Ford and Gueorgui Kolev as the proud inventors of the method.

rough sketch, but I'd seek confirmation this is legit (Joro?)

sysuse auto, clear
etregress price headroom weight , treat(foreign = turn displacement)
eststo e1
etregress mpg headroom weight , treat(foreign = turn displacement)
eststo e2
suest e1 e2
test [e1_price]weight=[e2_mpg]weight

[George Ford]

I'll get my 7 lines of code published in Econometrica ASAP!