Omission of a variable in the K-variable regression model. Refer to Eq. (13.3.3), which shows the bias in omitting the variable X3 from the model Y_i = β₁+β₂X_2i + β₃X_3i + u_i . This can be generalized as follows: In the k-variable model Y_i = β₁ + β₂X_2i + · · ·+β_k X_ki + u_i , suppose we omit the variable X_k . Then it can be shown that the omitted variable bias of the slope coefficient of included variable X_j is:

E(β̂_j ) = β_j + β_kb_kj = 2, 3, . . . , (k − 1)

where b_kj is the (partial) slope coefficient of X_j in the auxiliary regression of the excluded variable X_k on all the explanatory variables included in the model.*