# How to check the consistency of OLS estimator in macroeconomic models

Problem:
We have a model $$C_t = a + b Y_t + e_t$$ and $$Y_t = C_t + I_t$$ It's known that $$Cov(I, e)$$ is zero. A student estimates the following model: $$C_t = a + b Y_t + e_t$$ Are the estimators $$\hat a_{OLS} = \hat a$$ and $$\hat b_{OLS} = \hat b$$ consistent? If not, then recommend the consistent estimators.

My attempt:
I think that both of the estimators are inconsistent because the model is kind of a recursive one with two equations, while the student regress only one function.

Then I tried to prove it using formulas. I start with the slope estimator and there is a formula for the slope that reminds omitted variable bias:

$$\hat b = b + \frac{(1/n)\sum(y - \bar y)(u - \bar u)}{(1/n) \sum(y - \bar y)^2}$$

Then I take $$\text{plim}$$ of the numerator and denominator.

$$\text{plim}[ (1/n) \sum(y - \bar y)^2] = Var(y) \neq 0$$ $$\text{plim}[ (1/n)\sum(y - \bar y)(u - \bar u)] = Cov(y,u)$$

At this step, I don't know how to evaluate $$Cov(y,u)$$ is either zero or not zero.

I also assumed that this formula is wrong, and instead tried to use the formula for the OLS estimator of the slope. However, when I tried to substitute $$C_t$$ I received a recursive equation.