Why do we need to include precision variables in a regression model (i.e., a variable that is associated with the outcome but not the predictor of interest)?
Regression coefficient is often characterisized as a partial correlation coefficient which means it will show effect of particular variable X to the outcome variable Y after effects of other variables, Z, are controlled.
What happens when you omit Z and leave only X? Do coefficient for X change?
If variables Z and X are orthogonal, which rarely happens outside experimental data, these coefficients will not change but in other situations you cannot say that coefficient for X variable measures just effect of variation in X for Y.
In econometrics omission of important variables is called omitted variable bias and it states that marginal effect from the X variable to the Y will no longer be estimated without bias.
Goal of modeling is to find out all relevant variables and to check that residual variation behaves well.