# Consistency under mild endogenity

Assume the usual linear model:

$$Y_i = X_i\beta + \varepsilon_i, \quad 1\leq i \leq n$$ whit $$E(\varepsilon_i)=0, Cov(\varepsilon_i, \varepsilon_j) = \sigma^2 \delta_{ij}$$ and $$Cov(X_i , \varepsilon_i )= 0$$.

Let $$\hat \beta$$ be a consisten estimator in the sense that $$\| (\hat \beta - \beta) X\| \overset{P}{\to} 0$$

Would this estimator be consistent in the same sense if the real model has a new (unobserved) term? $$Y_i = X_i\beta + W_i + \varepsilon_i, \quad 1\leq i \leq n$$

In this scenario $$(X_i, W_i, Y_i)$$ would be identically distributed for each $$1 \leq i \leq n$$ (this distribution changes with $$n$$ though), and $$Cov(W_i, X_i) \not = 0$$ but $$E(W_i) = 0$$ and $$\lim_{n\to \infty} Var(W_i) = 0$$.

I belive it would because endogenity is in a sense mild. But i am not sure how can I demonstrate something like this for any consistent estimator.

I did check this holds for $$\hat \beta_{ols}$$ \begin{align*} \hat \beta_{ols}&= \frac{ Cov(X,Y)}{Var(X)} + O_P(a_n) \\ &= \beta + \underbrace{ \frac{Cov(X, W) }{Var(X)}}_{\to 0} + O_P(a_n) \end{align*}