I want to regress a variable Y on another variable X (with appropriate control variables and fixed effects) in a panel data setting. Two approaches come to mind:
Use quantile regression;
Use OLS regression to regress Y on the quartiles of X by using interaction terms, that is, multiplying X by an indicator variable that takes value 1 if the observation belongs to a certain quartile. So basically we would have
y = intercept + D0.5*X + D0.75*X + D1.0*X + controls, where D0.5 is the indicator variable for the second quartile, D0.75 is the indicator variable for the third quartile, and so on.
What is the difference between the two approaches and in which cases would one be more appropriate than the other?
To answer the comments:
I am trying to see how X impacts Y for given quartiles of X. I expect that the impact of X on Y varies significantly across the quartiles of X. This is basically the hypothesis.
The observations are country-years. I expect X to only have an important impact on Y for high values of X (and for another variable, say X',I expect the opposite to hold). The idea is to check this hypothesis. What would you recommend?
Maybe it helps if I am more specific. X is an input factor in a country-year (hence panel data specification) and X' is another input factor. One theory suggests that X and X' should both have a statistically significant and positive impact on Y (dependent variable). Another suggests that X should have negative impact (becoming more negative for larger values of Y) and that X' should have a positive impact and larger as Y increases. The idea is to see how both these variables affect Y along the quantiles of Y and to test both theories. Both theories support that the direction of causality is from X to Y and not from Y to X.