At a Mathematical Level, If the Two Conditions for Omitted $\quad E \left( u _ { i } \mid X _ { 1 i } , X _ { 2 i } , \ldots , X _ { k i } \right) \neq 0$

Short Answer

At a mathematical level, if the two conditions for omitted variable bias are satisfied, then a. $\quad E \left( u _ { i } \mid X _ { 1 i } , X _ { 2 i } , \ldots , X _ { k i } \right) \neq 0$
b. there is perfect multicollinearity.
c. large outliers are likely: $X _ { 1 i } , X _ { 2 i } , \ldots , X _ { k i }$ and $Y _ { i }$ have infinite fourth moments.
d. $\left( X _ { 1 i } , X _ { 2 i } , \ldots , X _ { k i } , Y _ { i } \right) , i = 1 , \ldots , n$ are not i.i.d. draws from their joint distribution.