Controlling for both time fixed effects and "entity invariant" variables I have a question about a panel data regression model in general. I have firm-level data in a single country. Is it okay to control for both time fixed effects and entity-invariant variables, such as GDP growth and interest rate (which are the same across firms but vary across years)? Ideally, I want my model to be as parsimonious as possible, i.e., without controlling for the entity-invariant variables.
However, my regression result without including them is quite bad (none are statistically significant), while the results are significant by including them. I am not sure whether the latter result is valid.
 A: 
Is it okay to control for both time fixed effect and entity-invariant variables, such as GDP growth and interest rate (which are the same across firms but vary across years)?

No.
But let's get our terminology straight. The term "entity-invariant" means: common across entities but changes over time. In short, the year fixed effects will adjust for the unobserved factors that evolve over the years but are constant across entities.
In practice, including a full set of year dummies adjusts for both observed and unobserved factors, which means most software packages would drop your entity-invariant covariates. It appears you're sampling firms/industries within the same country, so annual GDP growth for the units within those countries should be the same. In short, all units (i.e., firms, industries, etc.) experience the same rate of change over time, assuming they're all nested within the same country. Remember, GDP is a core measure of economic prosperity—for the whole country.

However, my regression result without including them is quite bad (none are statistically significant), while the results are significant by including them.

I wouldn't assume it's a "bad" result simply because you're missing a few asterisks next to the coefficients. Maybe a statistically insignificant result has a meaningful interpretation. It's hard to offer further guidance without actually seeing your results, but I would recommend either dropping the entity-invariant variables, or modeling "time" in a different way.
