Causal effect of a variable with cross-sectional dimension

[Guest]

Turning to you as I am puzzled with something, and you might be familiar. Assume you have a model

Y observed for region r in country c and year t (in my data this is a cross-section of loans given to firms operating in several regions in the world from 2002 to 2018). X is observed for region r of country c and does not vary over time (culture in my data). Z is a characteristic of country c that causes exogenous variation of Y due to X and hence the interaction term. Z also does not vary over time (ancient folklore in my data). This is like a diff in diff but there is no pre-post treatment. It is like a heterogenous treatment effect, where another layer of the cross-sectional dimension causes exogenous variation of the treatment. Can this be a legitimate identification method? Any examples in the literature with the equivalent terminology?

Thank you kindly.

[Maxence Morlet]

What is index i? I am puzzled as to how Y varies over rct, whilst the error term over cit (?).

Is X continuous?

What you are doing here is that you are allowing culture to have a different effect on loans according to the level of ancient folklore.

Is Z caused by X? How correlated are the two? Beware of multicollinearity, which inflates standard errors... Also, given how you've described it, Z does not seem to be exogenous to your model...

You presumably have a repeated cross-sectional dataset, which allows you to include fixed-effect vectors, at the very least for time. Given that you have three sources of variation, you could probably interact fixed-effect vectors in order to shield against a larger number of confounding (unobserved) factors.

This is a just a start, it's quite difficult to answer your question given its vagueness...

[Guest]

The question is pretty straightforward and the only mistake was the dimension of the error term, which is the same as Y (obviously). X is continous and Z is correlated with X (actually determines X, so that it can also be used as an instrument of X). For various reasons that are irrelevant to this question, I prefer the specification above to the IV. Z is exogenous and predetermined because ancient folklore cannot directly determine current loan spreads. Now, if both X (the treatment) and Z were dummies, this would be like a 2x2 DID. I have the fixed effects you mention, but the question remains. Can this be like a DID without a pre-post interaction? Is there anything like that in the literature, to the best of your knowledge?

[Jared Greathouse]

The question is pretty straightforward

I disagree!!! All of this is super vague. Your question provides 0 data or code for us to work with, meaning it is on its face unclear. this is very bad, no good. We need code and data to help you, for 98% of queries on Statalist. See the FAQ for details.

Anyways.

This is like a diff in diff but there is no pre-post treatment

This isn't a thing. If I've understood you well (which I shouldn't have to understand you, because you should provide sample data and code!!), you're saying there's..... no pre-intervention period? Some units are always treated? This can't work, and doesn't make any sense. You NEED a pre-intervention period for DD to work, it literally, mathematically, cannot work without a pre-intervention period. This is why we ask for data and code, so we can see all the details.

For people to give meaningful feedback, I beseech you to use dataex to show us how your data look as well as the code you've tried. Otherwise, you'll get no help from pretty much anyone on this forum.

[Guest]

I see from your answers that you understand perfectly, which means you choose to be impolite. Let me repeat for the shake of an academic discussion. You have Y: cross section of loan spreads over 2002 to 2018. X is economic preferences for risk-taking varying with regions (within country) worldwide but not across time. Z is folklore varying with country and predetermines X. If X and Z where dummies based on their means, you could have the standard 2x2 matrix, similar to the DID. I understand this is not a DID. My Q is whether there is such an example in the lit. I guess not from your answer. If you prefer to reply in a difficult manner, better not to reply at all and delete the thread. Enough with impoliteness in this profession.

[Maxence Morlet]

That is in no way a DiD, just a regression with an interaction term. Yes, there are regressions with interaction terms in the literature. For a discussion of interaction terms in nonlinear models and their marginal effects, see Ai and Norton (2003).

And Guest, Jared was just re-iterating the FAQ, which you, as a user, are responsible for reading before posting. People here try to help you and want to help you, but cannot do so if you do not give them enough information.
In hindsight, neither Jared nor I were being rude. You'll notice we're the only ones who answered, and given your majestic eloquence and gratefulness (or complete lack thereof), you'll probaybly not get any other answers...

[Guest]

Thank you for your time.