The exogeneity condition and correlation

Question

My book says that the exogneity condition, the condition that should hold in order for $b$ to be consistent, is the following: $$\operatorname{plim}(\frac{1}{n}X'\epsilon)=0.$$

It then says that the $j$th component of this vector can be written as $\frac{1}{n} \sum_{i=1}^n x_{ji}\epsilon_i$, so that the condition is equivalent to that the explanatory variables should be asymptotically uncorrelated with the disturbances.

I don't however see why $\frac{1}{n} \sum_{i=1}^n x_{ji}\epsilon_i$ represents the correlation between the $x$ and \epsilon$ variables. Could anyone please explain me why this is so?

Answer 1

Note that $\epsilon$ is mean zero by assumption, then the covariance of $x_j$ and $\epsilon$ is simply $\mathbb{E}x_j\epsilon=:\sigma_{x_j\epsilon}$. Now, we also assume $\sigma_{x_j\epsilon}:=0$ for otherwise the OLS estimator is biased and inconsistent due to this endogeneity. The sample analogue of $\mathbb{E}x_j\epsilon$ is of course $$\frac{1}{n}\sum_i\epsilon_ix_{i,j},$$and so if this converges in probability to 0, this is the same as saying that you do not have any endogeneity problems for large enough samples. That this is also the large sample equivalence of $\text{Corr}(\epsilon,x_j)=0$ follows from the definition of correlation;$$\text{Corr}(\epsilon,x_j):=\frac{\sigma_{x_j\epsilon}}{\sigma_{\epsilon}\sigma_{x_j}},$$ which is zero iff $\sigma_{x_j\epsilon}=0.$