Two-tailed or one-tailed test for testing statistical significance (multiple regression)?

Question

I want to establish if the effect of a variable x2 is statistically significant on a variable y. I want to do a hypothesis test using an F statistic for this but I am unsure if I need to make the test two tailed or one tailed?

Answer 1

You would solve the problem jointly, but since your only concern is over the status of the coefficient of $x_2$, you would do a two-sided t-test on $\beta_2$. An F-test on all variables would tell you if all variables are equal to zero, but if they were not, it would not tell you which one matters.

You are doing a two-sided test because if $\beta_2=0$, then $0\times{x_2}$ is always zero. If you reject that null, then $\beta_2$ matters (to some degree of statistical confidence).