A very basic question on ENDOGENEITY In the regression model
$Y$ = $\beta_0$ +  $\beta_1X_1$ + $\beta_2X_2$ +.......+ $\beta_kX_k$ + $\epsilon$
where 
$\epsilon$ = $\delta_0X_2$ + $\lambda$
Will this also be the case of endogeneity since E($\epsilon|X$) is not independent of $X_2$? 
 A: If assume the true model is
$$Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \epsilon$$
and that $\epsilon = \delta_0 X_2 + \lambda$ it follows that
$$Y = \beta_0 + \beta_1 X_1 + (\beta_2+\delta_0) X_2 + \lambda$$
and estimating this model using the OLS estimator 
$$\hat \theta_{OLS} = \left( \frac{1}{N} \sum_i \mathbf x_i \mathbf x_i^\top\right)^{-1} \left( \frac{1}{N} \sum_i \mathbf x_i y_i\right)$$
where $\mathbf x_i = (1,x_{i1},x_{i2})^\top$ will result in consistent estimator $\hat \theta_1$ as an estimator of $\beta_0$ and $\hat \theta_2$ is estimator of $\beta_1$ and $\hat \theta_3$ as an estimate of $(\beta_2 + \delta_0)$ if 
$$\mathbf E[\lambda \lvert \mathbf x] = 0$$
however the fact that $\epsilon = \delta_0 X_2 + \lambda$ means that you cannot identiy $\beta_2$ nor $\delta_0$ but only the combined effect $\beta_2 + \delta_0$.
Simulation in R
N<-10000
x1 <- rnorm(N)
x2 <- rnorm(N)


beta_0 <- 2
beta_1 <- 1
beta_2 <- -1


delta_0 <- 2
lambda <- rnorm(N)
epsilon <- delta_0 * x2 + lambda

y <- beta_0 + beta_1*x1 + beta_2 * x2 + epsilon

model<-lm(y~x1 + x2)
summary(model)


beta_0
beta_1
delta_0 + beta_2

