I am preparing for my final in Econometrics but I am confused over a new problem I encountered. I think I have solved it but I am unsure whether I am not making any gross mistakes.

This is the study of a simple macroeconomic model that is defined as:

$$C_t=\alpha_0+\alpha_1Y_t+u_t$$ and $$Y_t=C_T+I_t$$ We further assume that $Cov(I,u)=0$ and denote $\sigma_I^2$ and $\sigma_u^2$ the variance of $I$ and $u$.

Questions are as follows:

a) Express $Y_t$ in terms of $I_t$, $u_t$, and $\alpha's$

$Y_t=\dfrac{\alpha_0+u_t+I_t}{1-\alpha_1} $

b) Express $Cov(Y,u)$ in terms of $\sigma_I^2$, $\sigma_u^2$, and $\alpha's$

I attempt the following:

$Cov(Y,u)=Cov(C+I,u)=Cov(C,u)+Cov(I,u)=Cov(C,u)$ as we know $Cov(I,u)=0$


And this is where I get confused as I loop back to the beginning. Should I approach it using $Y_t$ derived in question 1?


c) Express $Cov(Y,I)$ in terms of $\sigma_I^2$, $\sigma_u^2$, and $\alpha's$


d) Express $Cov(C,I)$ in terms of $\sigma_I^2$, $\sigma_u^2$, and $\alpha's$


e) Express $Var(Y)$ in terms of $\sigma_I^2$, $\sigma_u^2$, and $\alpha's$


f) What is $plim(\hat{\alpha_{1_{OLS}}})$



f) What is $plim(\hat{\alpha_{1_{IV}}})$ using $I$ as an instrument for GNP (Instrumental Variable approach)



Yes. Given that you have solved (a), it is easiest to solve (b) by using the result in (a). It gives you $Cov(Y,u)(1-\alpha_1)=Cov(\alpha_0,u)+Cov(u,u)+Cov(I,u)$. Since $Cov(\alpha_0,u)=Cov(I,u)=0$ and $Cov(u,u)=Var(u)$, $Cov(Y,u)(1-\alpha_1)=Var(u)$. Hence, $Cov(Y,u)=Cov(u)/(1-\alpha_1)$.


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