2SLS regressions with data at different observation levels

[james xu]

Dear colleagues,
I am conducting a research study that examines how state-year level political uncertainty affects firm-state-year level cash flows. Worrying that the political uncertainty may be endogenous, I decide to use the state-year number of undocumented immigrants as my IV. However, I encountered some implementation difficulties and hope to get some suggestions.

At the very beginning, I used the old-fashion approach that I first regress political uncertainty on undocumented immigrants and then have the predicted value matched to my firm dataset and then regress firm cash flow on the fitted political uncertainty. The code I used follows:

reghdfe political_uncertainty undocumented state_GDP, a(state year) cluster(year)
predict pol_fitted, xb
reghdfe cash_flow pol_fitted firm_controls, a(firm state year) cluster(year)

The difficulty I encountered using this procedure was the calculation of the F-statistics of my instrument in the first stage. I could only estimate it using the fact that I only have one instrumental variable so the F-statistics of it would be the the square of the t-statistics. I also wanted to rely on the joint F-test score displayed on the upper right corner of the reghdfe command but I'm aware that it actually does not measure the same thing.
Thus, my first question would be how could I compute the F-statistics for testing weak instrument in my first stage?

Then, I also tried the ivreghdfe command directly after matching the undocumented immigrants data to my main data on state and year. The code I used follows:

ivreghdfe cash_flow (political_uncertainty=undocumented state_GDP) firm_controls, a(firm state year) cluster(year) first

Based on the first-stage estimation the first option gives me, it seems that all the undocumented, state_GDP, firm_controls are used for instrumenting for my endogenous variable, political_uncertainty, which is not what I wanted. Plus, for the first stage, I only wanted to run regressions on state-year level observations but now it automatically run on observations at firm-state-year level, which is the observation level of my main data.
Thus, my second question for this procedure would be how could I specify which variables are used as first stage controls while rest are used as my second stage controls.

I am aware that typically we require the same set of control variables in both stages (which is also how theory suggests), but my instrumental variable is actually at state-level. Basically, I'm regressing firm-level variables on state-level variables assuming that firms headquartered in the corresponding states have the exposure to the local political uncertainty.

It would be super-helpful that if someone could inform me how I could directly compute the F-statistics in the first stage for my first procedure.

Thank you in advance for any help you provide.

[Maxence Morlet]

In the first stage, the instrument is included. In the second stage, it is excluded. That is the exclusion restriction, i.e. the identifying assumption for all IVs. The instrument affects the outcome only through the endogenous variable, and thus may be omitted from the second stage. The rest of the model (e.g. controls and fixed effects) remains identical in both stages. The controls in the first stage do not count as instruments. Also, they do not have to be excluded from the second stage, which they are not.

It is however fundamental to obtain monotonicity correctness in your first stage. I suggest you read Blandhol et al. (2022) and follow recommendations therein.

"Plus, for the first stage, I only wanted to run regressions on state-year level observations but now it automatically run on observations at firm-state-year level, which is the observation level of my main data."

I don't think I really get this statement. You want data in your first stage to be aggregated, but then disaggregated in your second stage?

[Carlo Lazzaro]

James:
my guess is that Maxence is referring to When is TSLS Actually LATE? (nber.org)

[james xu]

In the first stage, the instrument is included. In the second stage, it is excluded. That is the exclusion restriction, i.e. the identifying assumption for all IVs. The instrument affects the outcome only through the endogenous variable, and thus may be omitted from the second stage. The rest of the model (e.g. controls and fixed effects) remains identical in both stages. The controls in the first stage do not count as instruments. Also, they do not have to be excluded from the second stage, which they are not.

It is however fundamental to obtain monotonicity correctness in your first stage. I suggest you read Blandhol et al. (2022) and follow recommendations therein.

"Plus, for the first stage, I only wanted to run regressions on state-year level observations but now it automatically run on observations at firm-state-year level, which is the observation level of my main data."

I don't think I really get this statement. You want data in your first stage to be aggregated, but then disaggregated in your second stage?

Thanks for your reply Maxence! To answer your last question, my data is kind of structured in the way that I am using state-level political uncertainty as a proxy to firm's exposure to state political uncertainty to estimate its effect on the firm-level outcomes, i.e., cash flows. And the instrumental variable I used in the first stage is also a state level variable. In the quoted statement, I'm actually saying that if I first match my instrument to my main data and use ivreghdfe or ivreg2, I will be forced to run my first stage at firm-level instead of state-level.

In response to your first part of the reply, I am aware that the same set of controls is needed in both stages as directed in the econometircs. However, my instrument is actually not at the same level as my dependent variable and even my endogenous variable is not at the same level as my dependent variable (state vs. firm level). Thus, I'm kind of constrained from using the command such as ivreghdfe, ivreg2, etc but instead estimate both stages separately.