Linear regression with unusual error terms Suppose that $Y = a + b K + X$ with $0 < X$ and $0 < K$ where $X, Y$ and $K$ are random variables. What are then the expectations of the intercept and slope in the case of a linear regression of $Y$ on $K$?
 A: Just to be sure we share a common understanding of the question, I suppose you mean you have independent data $(X_i, K_i, y_i)$ where $(K_i,X_i)$ are $n$ independent and identically distributed random variables (you indicate they are positive, but this does not matter), the individual $(K_i, X_i)$ are independent (as you state in a comment), and you use ordinary least squares to estimate $a$ and $b$ in the model
$$y_i = a + b K_i + X_i.$$
Now, because you assumed $X \gt 0,$ necessarily $E[X_i] \gt 0$ (assuming these expectations are finite).  This is where your question differs from the usual regression setting.  The $X_i$ play the role of the "errors" in the model, but because they do not have a zero expectation, their expectation must be added to the constant $a.$
One way to see this is to write $$\varepsilon_i = X_i - E[X_i] = X_i - \mu$$  (where $\mu=E[X_i]$ is the common expectation).  These random variables are now legitimate errors because they have zero expectations.  Writing  for the common expectation, your model is the same as
$$y_i = a + b K_i + (\mu + \varepsilon_i) = (a + \mu) + b K_i + \varepsilon_i.$$
Thus, as always in least squares regression,

the expectation of the slope estimate is $b$ and the expectation of the intercept estimate is the true intercept, here seen equal to $a + \mu.$

Unless you know $\mu,$ you cannot determine $a$ from this model: you can only estimate the sum $a+\mu.$  (It sounds like your simulations used zero-mean variables $X_i,$ contrary to the assumptions stated in the question.  Indeed, the expectation of the estimate of $a$ equals $a$ if and only if $\mu=0.$)
