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I have been reading several papers that use 2SLS. These papers usually also show the OLS regression results. I noticed that when authors report results for 2SLS, they also include control variables that were used in the OLS regression.

If 2SLS is supposed to solve the omitted variable bias, couldn't we omit the control variables that are included in OLS because we do not have to worry about omitted variable bias? For what reason would authors include those variables?

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    $\begingroup$ I'm voting to close this question as off-topic because it belongs to stats.stackexchange.com $\endgroup$ – Art Nov 11 '19 at 2:33
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    $\begingroup$ Another thing to note before this is migrated, 2SLS is not meant to control for omitted variable bias per se $\endgroup$ – Brennan Nov 11 '19 at 5:13
  • $\begingroup$ @Art Seems like if this is off-topic then the Econometrics tag is nearly useless and all we have are applied econometrics. Perhaps you could post a question on meta to clarify this. $\endgroup$ – Giskard Nov 11 '19 at 7:20
  • $\begingroup$ Fair point. I guess the line for me is if it has some economic application then it's ok here. Seems to me that this question is general enough to be in Cross Validated. Edit: On the "Welcome" thread, it says, "Econometrics: Totally on topic, but especially for theoretical questions, you may subsequently try the "All of Statistics" .SE site Cross-Validated." I guess this to me is on the theoretical side. $\endgroup$ – Art Nov 11 '19 at 7:24
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    $\begingroup$ @Art If you do not tag me I do not get a notification of the message. Quoting from your quote, emphasis by me: "on topic [...] you may subsequently try" $\endgroup$ – Giskard Nov 11 '19 at 7:58
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The reason for that is that 2SLS does not solve omitted variable bias without its cost.

First, for every omitted variable you would have to find suitable instrument. Finding instruments is incredibly hard so why would you purposely create more omitted variable bias? Unless you are trying to challenge yourself for fun or doing it as a practice exercise for students it does not make sense.

Second, actually 2SLS still has bias - the advantage of 2SLS is that bias decreases as strength of the instrument increases, so with good instruments 2SLS is better than OLS in terms of bias. But in practical applications you rarely have perfect instruments- so even in case you could feasibility instrument the other variables you omit on purpose you will be just adding to the bias of your model.

Hence you should always include all relevant controls that you can include

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