When I run the linear regression using about 12 independent variables, I get insignificant F-test result overall. So I discarded variables to make the F-test significant while having no multicolinearity problem checking by VIF test.
Then I come up with a linear regression with significant F-test and no multicolinearity problem. However, I am left only with three independent variables and significant t-test results for these coefficients.
Shall I still use this result?