I am fitting the usual linear regression model
$$y_j = x_j^T\beta + e_j,$$ where the errors $e_j$ are iid normal with unknown variance. If the vector of covariates $x_j$ contain spurious variables, will they always vanish with large samples? This is, the corresponding estimators converge to zero when the sample grows? What if my model is incorrect (like the true model is non linear)?