I was wondering what the following OLS scenario would imply: a variable is endogenous (i.e. correlated with the error term) yet is statistically significant.
Alternatively, what if in, once again OLS, the variable is exogenous yet statistically insignificant (although the overall regression is indeed statistically significant - by say an F test).
Thanks!
In the first case, an endogenous variable may or may not be significant depending on the type of endogeneity. The coefficient of your endogenous variable may be over-estimated due to omitted variable bias which is related to confounding factors or because of simultaneity bias. In this case it is possible to find a significant effect since the coefficient of interest is too large relative to its standard error. This reasoning works the opposite way as well if you have a negative bias which under-estimates the coefficient of your endogenous variable. To have an idea about these biases it is often useful to apply the Frisch-Waugh theorem and reason about the sign of potential confounding or simultaneous relationships.
Another type of bias which typically leads to under-estimates and insignificant results (depending on the strength of the bias) is measurement error which introduces the so-called attenuation bias. If you still find a significant effect then either the effect is very strong or your attenuation bias was small (or both).
If you have an exogenous and insignificant variable but an overall significant F-test, then there must have been at least one other variable in the regression that had an effect. Remember that the F-test tests $$\beta_1 = \beta_2 = ... = \beta_k = 0$$ of all non-constant variables. If you only have one variable in your regression as in $$y_i = \alpha + \beta_1 x_i + u_i$$ then the regression F-test is the square of the t statistic for $\beta_1$. Hence there must have been another variable or there are two (or more) variables that are insignificant alone but jointly significant in explaining the outcome.
