# Can I add a variable that varies only with time in Least Squares Regression model with a time-fixed effects term?

I'm estimating this equation for the trade flows between various countries at time t: ($$i$$ and $$j$$ are countries)

The last 3 terms are control variables, one of which is oil prices. Can I add oil prices as a control variable if I already have a time fixed effects term, $$\beta_3$$. Further, can I have country pair fixed effects, $$\beta_6$$, when I'm also controlling for variables that are fixed for two countries across time (for example, if the two countries were in a colonial relationship in the past, this variable wouldn't change with time and would stay fixed for the two countries for the entire period).

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– 1muflon1
Feb 27 at 10:13

Your model has $$\beta_3 * t$$, which is a linear time trend, not time dummies. If that's correct, you are controlling for only a linear trend. Because oil prices do not have a perfect linear trend, you can include them.
But I am not sure you really want the linear trend specification instead of time dummies (say, $$\beta_{3t}$$). For a model with common time effects (year dummies), time series variables are not allowed due to perfect collinearity. Thus, if the time series variables are the key variables, you have a trouble. But if you only want to control for them, you can just omit them because any common time effects will be eliminated by the time dummies.
One possible solution is to take the correlated random effects approach, but there is the issue of RE $$\ne$$ FE. You will have to defend your RE model. That's a different issue.