Included-variable bias

If you have two regression models $$E\left[Y|X\right]={\beta }_0+{\beta }_{1}X$$ and $$E[Y|X,Z]={\gamma }_0+{\gamma }_{1}X+{\gamma }_{2}Z$$ then typically ${\gamma }_1\ne {\beta }_1$, because they are different things¹

A common name for this phenomenon is omitted-variable bias. That’s an unfortunate name, because it implies a direction in a situation that’s completely symmetric. Yes, ${\hat{\beta }}_1$ is biased for ${\gamma }_1$, but ${\hat{\gamma }}_1$ is equally biased for ${\beta }_1$.

The idea that ${\gamma }_1$ is somehow natural and ${\beta }_1$ is wrong comes from the gold-standard² way of thinking about regression model choice: that there is a true model defined by having all its coefficients non-zero, and that your job is to find it. From this point of view, either ${\gamma }_2=0$, so ${\beta }_1$ is preferred but ${\beta }_1={\gamma }_1$, or ${\gamma }_2\ne 0$, so ${\gamma }_1$ is preferred.

If you want ${\beta }_1$ then ${\hat{\gamma }}_1$ has included-variable bias. If you want ${\gamma }_1$ then ${\hat{\beta }}_1$ has omitted-variable bias. Or you can stop trying to think of the $\beta$ and $\gamma$ as being estimates of the same things and just talk about which one you actually want to estimate.