I do not fully understand how to interpret the difference between two statistical models where they only differ based on whether a certain variable is included on the right hand side.
If the results do not change much, then do we say that that this omitted variable has little effect on the result?
If the coefficient on the variable of interest changes significantly in magnitude and is no longer significant, we say that the variable of interest on the right hand side is not robust to that omitted variable. That omitted variable effects the relationship between the variable of interest on the right hand side and the dependent variable.
What is the estimate becomes more statistically significant when this variable is included? It appears that including these variables makes the results more precisely estimated. How can this be? What does that mean about the relationship between the dependent variable and the main variable of interest on the right hand side if the relationship is more precisely estimated only with the additional control variable?