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*}


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