A linear statistical model is an equation between random variables:
Y = b0 + b1X1 + ... + bKXK + U,
Y, X1, ..., XK, and U are random variables
Y is the outcome, also called dependent variable.
X1, ..., XK are regressors, also called explanatory variables.
U is the error term.
The parameters b0, b1, ..., bK are real numbers.
This model writes the outcome of interest Y (say the wage) as a linear function of some explanatory variables (say age, gender, education) plus an error term U (potentially an omitted variable, say IQ).
U collects everything in Y that is not b0 + Σi=1nbkXk.
The b parameters can have a causal interpretation. The parameter bk (k≥1) is the marginal effect of Xk on Y keeping all other regressors and the error term constant:
bk = ∂Y |
Hence, keeping all Xl (l≠k) regressors and the error term U constant, we could look at how Y varies with Xk to know the causal effect of Xk on Y.
The exogeneity assumption is a restriction on the joint distribution of the error term and the regressors:
E(U|X) = 0
Differently put, for any values x1, ..., xK we have:
E(U| X1 = x1, ..., XK = xK) = 0.
The expected error for any value of the regressors is zero.
By the law of expected iterations, the exogeneity assumption implies: