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I am estimating a model with two survey questions, which I expect to be endogenous by 2SLS:

$$ Outcome_i = B_0 + B_1SurveyQuestionA_i + B_2SurveyQuestionB_i + B_3Control + u$$

I have an instrumental variable (IV) for $SurveyQuestionA_i$ and an IV for $SurveyQuestionB_i$.

EDIT: The survey questions deal with the perception of the issue related to A and the perception of the issue related to B.

Ben Lambert in this video shows how to do calculate both first stages (although he apparently makes an error when discussing the conditions at the end, see the comments), which is simply including both instruments in both stages;

$$ SurveyQuestionA_i = C_0 + C_1IV_A + C_2IV_B + C_3Controls + v $$ $$ SurveyQuestionB_i = D_0 + D_1IV_A + D_2IV_B + D_3Controls + v $$

In my example, I thought that was kind of odd. Because in my scenario, the IV for $SurveyQuestionA_i$ makes very little sense for $SurveyQuestionB_i$ and vice versa.

Is it not allowed to simply use $IV_A$ for $SurveyQuestionA_i$ and $IV_B$ for $SurveyQuestionB_i$? If not, why not?

NOTE: In case it matters for the question at hand; I am actually utilising a Control Function/ Two Stage Residual Inclusion for the estimation. I did however not want to unnecessarily over-complicate the example.

EDIT: After Adrian's comment I decided to add the output that I have.

First Stages

            SurveyQuestionA |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
---------------------------+----------------------------------------------------------------
                   IV_B     |   .1600725   .0270538     5.92   0.000     .1070479     .213097
             IV_A           |   .0009261   .0002869     3.23   0.001     .0003636    .0014885


--------------------------------------------------------------------------------------------
          SurveyQuestion_B  |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
---------------------------+----------------------------------------------------------------
                   IV_B    |   .4611017   .0273291    16.87   0.000     .4075377    .5146657
             IV_A          |   .0002393   .0002889     0.83   0.407    -.0003268    .0008055

Per Adrian's request, the causal diagram is quite simple (where the arrows are the causal directions). The endogenous variables are correlated.

enter image description here

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  • $\begingroup$ Do you have a causal diagram for your variables? $\endgroup$ May 24 at 17:15
  • $\begingroup$ @AdrianKeister Dear Adrian, I have added the assumed causal diagram. If there it is not completely clear, please let me know. $\endgroup$
    – Tom
    May 25 at 7:16
  • $\begingroup$ Well, there's nothing unclear about your diagram, except this: your diagram is not a normal setup for an instrumental variable. Instrumental variables are used to adjust for confounding variables in certain settings. Like this: stats.stackexchange.com/questions/563/… So I want to ask: is there some correlation among your endogenous variables? $\endgroup$ May 25 at 12:39
  • $\begingroup$ @AdrianKeister Yes, there is definitely correlation among the endogenous variables. They are quite subjective survey questions, dealing with what constitutes an issue for a firm, which are simultaneously measured in the survey. The IV's however are more "objective measures" which temporally precede the survey questions. $\endgroup$
    – Tom
    May 25 at 12:51
  • $\begingroup$ Which variables are endogenous? If they are correlated, do you have arrows between them? Bi-directional arrows? Which variables are you considering as the instrumental variables? $\endgroup$ May 25 at 18:55

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