Consider the linear model

Y = b_{0} + b_{1}X_{1} + U and assume that there is another variable

X_{2} which has a direct effect on

Y and is also correlated with

X_{1}:

X_{2} = ρX_{1} + V, where E(V|X_{1}) = 0.

This variable lies in the error term:

U = b_{2}X_{2} + W, where E(W|X_{2}) = 0.

We can then rewrite the linear model as:

Y = b_{0} + b_{1}X_{1} + b_{2}X_{2} + W = b_{0} + b_{1}X_{1} + b_{2}(ρX_{1} + V) + W

= b_{0} + X_{1}(b_{1} + ρb_{2}) + (b_{2}V + W)

OLS will then not produce an estimate of b_{1} but of b_{1} + ρb_{2}. There is a bias.