A Indirect effect and spuriousity

[monsmatglad]

Say one has a regression result (ols) with significant coefficients for all independent variables. Then a new variable (Z) is added. This new variable is either something that reveals a spurious relationship among one of the initially included variables (x) and the dependent variable (y), or represents an effect that is "between" the independent variable (z) and the dependent (z) - an indirect effect. Will both these situations cause the size (and significance) of the independent variable (x) to decline? I have a statistics course, and from what I remember, my lecturer told me that adding an additional independent variable which results in one of the original variables to have a smaller coefficient, is a sign of either an indirect effect or a spurious relation. Is this correct?

Mons

 

[FactChecker]

I wouldn't say that any situation forces one to conclude that one of the variable effects is indirect or spurious. Both X and Z variables may have a component showing an effect on Y that is not reflected in the other variable at all.

If the new variable, Z, is correlated with a variable, X, already in the model, then if it is added, the coefficient of X will change.
If the correlation between Z and X is positive, and Z is added with a coefficient of the same sign as the X coefficient, then the magnitude of the coefficient of X will decrease. Likewise if the ZX correlation is negative and have the same sign of coefficient. The statistical significance of X may be decreased to the point where it should be removed from the model.
Other combinations would make the coefficient of X increase in magnitude.