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 $X_2$?


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

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)

delta_0 + beta_2

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