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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)?

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What type of model are we talking about here? Linear regression?! –  Patrick Coulombe Mar 11 at 3:33
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Any type of regression. –  guestom Mar 11 at 4:21
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You seem to be asking about what are more usually called covariates. –  Glen_b Mar 11 at 7:09
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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.

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