When the key regressor is highly correlated with time, is it reasonable to use time trends instead of year fixed effects?

External question — Cross Validated Stack Exchange Author: user21213679 Original post: https://stats.stackexchange.com/questions/675106 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. I am estimating a panel regression with year fixed effects. However, the key variable x increases steadily over time and is highly correlated with calendar year. When I include year fixed effects, most of the variation in x seems to be absorbed by the year dummies, and the coefficient on x becomes very imprecise. An alternative specification I am considering is a linear or quadratic time trend instead of year fixed effects. In other words, instead of including a dummy for each year, I include variables such as year and year squared. My questions are: Is it methodologically reasonable to use parametric time trends instead of year fixed effects when the key regressor is strongly trending over time? Are there textbook or journal references that discuss this trade-off between year fixed effects and time trends? Are there recommended diagnostics or robustness checks researchers use to justify this choice?