Economics - Instrumental Variable: Does including past values of regressors into the IV cause exogeneity assumption to fail?

Setup: Annual panel of 125 countries. I am interested in the effect of x on y. As an instrument z I use an interaction of the variables gamma and delta. gamma fulfils the conditions of relevance and exogeneity. delta is the country’s propensity to receive x, which is an indicator variable based on how much x the country received in the past.
Question: Assuming that x in the past is correlated with today’s y – would then my instrument fail to comply with the exogeneity condition because through delta my instrument explains part of the variation in y (which is not through today’s x)?


Less is more. Stay pure. Stay poor.
I haven't used IV in this setting, but it sounds suspect. Can you draw a directed acyclic graph to depict your scenario? If it is correlated with future values not just through the path into the x variable, but through potentially a W variable then an assumption seems to be broken. If it is a week correlation, its affects may be less than not using the IV. Meaning it may be better than nothing, but biased.
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Thank you so much for your answer.
I use IV because reverse causality is likely (y effects x). The delta is based on x of past and future. So my question is if that means that the exogeneity condition (no correlation of IV and y exept throught x) still holds.



Less is more. Stay pure. Stay poor.
I apologize that this not my forte at all, but I find such topics interesting - so I will see if I can help. Do you only have a value for Y for one cross-section of time? Otherwise if you have historic Y as well marginal structural models can be use to control for time-varying confounding.

Or are you locked into just looking a certain cross-sectional value of Y. If there is a feedback from past Ys to past Xs, and you are using past Xs, a simple approach like your drawing seems suspect. Do you really need to control for the historic Xs? I may not be following your example, but I am thinking about it as say Y is your weight and X is an exercise routine. If historic Ys influence historic Xs, I just wonder if that creates a backdoor path when only using historic Xs. I am not seeing a confounding variable above linking X and Y that would justify using an IV?
I have to apologize for my impoliteness of answering so slow. The notifications keep ending up in my spam. I am sorry.

To put some flesh to the bone, the analysis is based on a panel data set containing of 125 countries and 35 years (so not only one cross-section). There is reason to believe that there is some kind of reverse causality. That’s the reason why IV is necessary.
To stick to your example: X = exercise, Y = weight, gamma = distance to the gym, delta = time invariant propensity score how likely it is for each person to go to gym (1= if the person went, 0= if never went, and everything in between). Now the question is if that propensity leads to a violation of the endogeneity condition...

I came up with a possible solution: The propensity score is only based on future values x, e.g. the last 15 years of the dataset and the analysis is only based on the first 20 years. In that way the propensity score cannot violate the exogeneity condition. Does that make sense?