# 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 $$X_2$$?

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