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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.

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I think that your problem is that you considered the model as one equation. However, as you wrote yourself, there are two equation, not one. The problem you faced is called the simultaneous equations. Your equations are given in the structural form. Try to rewrite them in reduced form and then apply the logic to showed in your attempt.

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