formula_2 is a vector. formula_3 is a matrix, usually with a column of ones and perhaps with additional columns for other covariates. Consider how an instrument allows formula_4 to be recovered. Recall that OLS solves for formula_5 such that formula_6 (when we minimize the sum of squared errors, formula_7, the first-order condition is exactly formula_8.) If the true model is believed to have formula_9 due to any of the reasons listed above—for example, if there is an omitted variable which affects both formula_3 and formula_2 separately—then this OLS procedure will "not" yield the causal impact of formula_3 on formula_2. OLS will simply pick the parameter that makes the resulting errors appear uncorrelated with formula_3.