SEM One regressor depends on another regressor I have the following structural model, by which one of the regressors is partially explained by another.
$$
y_1= x_1+x_2+x_3+e \tag{1}
$$
$$
x_1= x_2 + u       \tag{2}
$$
The questions are:
a) Can I run equation (1) alone without getting the coefficients wrong?
b) If as I suspect I need to take into account the endogeneity of the regressors, which way would you suggest? SEM, SUR, ivreg?
 A: a) Yes, if $u$ and $e$ are independent, then you just work with equation (1). I recommend drawing this out as a directed graph to see it more clearly. According to Judea Pearl's approach, you get bias when there are either 1) omitted variables, or 2) unblocked "backdoor paths" between an explanatory variable and the response. If $u$ had was on the left side of $e$'s equation, then $u$ would be simultaneously causing both $x_2$ and $y$ without being taken into account by the model. To see this actually happen, first run the code below as is, and you'll see no bias. Then comment the line that links $u$ and $e$, and you'll see bias.
N <- 10000

x2 <- rnorm(N)
x3 <- rnorm(N)
u <- rnorm(N)
epsilon <- rnorm(N)
# epsilon <- .6 * u + rnorm(N)
x1 <- x2 + u
y <- x1 + x2 + x3 + epsilon

summary(lm(y ~ x1 + x2 + x3))

b) If you have a theoretical model for how $u$ and $e$ are related, you could use a likelihood-based method to estimate the parameters (which I suppose you could implement using SEM software). Having a model like that on hand is unlikely. Almost as unlikely is that you find a variable that predicts $x_2$ but not $e$ (which is not empirically testable). Then you can use the IV methods like 2SLS.
