I have a problem which, in its general form, has been addressed on this forum: can you omit exogenous variables from the first stage regression in a 2SLS estimation? (see: here and here). The answer in both cases seems to be a clear "no", with the respondents suggesting that doing so will lead to an inconsistent estimator. I am not sure, however, how to apply this logic to two subsets of exogenous second stage variables: fixed effects, and lagged Y.
First: fixed effects (FEs). Since my data is in panel format, allowing me to observe multiple individuals i over time t, in the second stage I use individual i / year T fixed effects (note: t captures days, so this strategy leaves plenty of variation in the variable of interest that also varies by t). However, my best candidate instrument z only varies across i and years (T), but not within years. As a result, using i / T FEs in the first stage will suck out all the variation in z. However, if I only use i FEs, this will preserve variation in z that should allow me to instrument. My question is: just how bad would it be to use FEs that are less complicated in the first stage than in the second stage?
Second: lagged choice variable. There is good reason to believe there is inertia in the customer's decision when choosing the dependent variable y_t, and so I include a lagged variable y_t-1 as an explanatory variable. My question is: should I also include y_t-1 in the first stage, even though it makes no sense, intuitively, why y_t-1 would impact my endogenous variable w?
Thank you in advance for any help on this matter,
P.S. This is my first post on this forum, so I apologize if I failed to follow etiquette in any way