# OLS, IV applied to basic macro model

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 $$Cov(I,u)=0$$

$$Cov(C,u)=Cov(\alpha_0+\alpha_1Y+u,u)=\alpha_1Cov(Y,u)+Var(u)$$

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?

$$Cov(Y,u)=Cov(\dfrac{\alpha_0+u+I}{1-\alpha_1},u)=\dfrac{1}{1-\alpha_1}Cov(u+I,u)=\dfrac{1}{1-\alpha_1}\sigma_u^2$$

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

$$Cov(Y,I)=Cov(\dfrac{\alpha_0+u+I}{1-\alpha_1},I)=\dfrac{1}{1-\alpha_1}Cov(u+I,I)=\dfrac{1}{1-\alpha_1}\sigma_I^2$$

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

$$Cov(C,I)=Cov(Y-I,I)=\dfrac{1}{1-\alpha_1}\sigma_I^2-\sigma_I^2=\sigma_I^2\left[\dfrac{\alpha_1}{1-\alpha_1}\right]$$

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

$$Var(Y)=Var(\dfrac{\alpha_0+u+I}{1-\alpha_1})=\left[\dfrac{1}{1-\alpha_1}\right]^2Var(u+I)=\left[\dfrac{1}{1-\alpha_1}\right]^2(\sigma_I^2+\sigma_u^2)$$

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

$$plim(\hat{\alpha_{1_{OLS}}})=\alpha_1+\dfrac{Cov(Y,u)}{Var(Y)}=\alpha_1+\dfrac{\dfrac{1}{1-\alpha_1}\sigma_u^2}{\left[\dfrac{1}{1-\alpha_1}\right]^2(\sigma_I^2+\sigma_u^2)}$$

$$=\alpha_1+\dfrac{(1-\alpha_1)\sigma_u^2}{(\sigma_I^2+\sigma_u^2)}$$

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

$$plim(\hat{\alpha_{1_{IV}}})=\alpha_1+\dfrac{Cov(I,u)}{Cov(Y,I)}=\alpha_1$$

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, $$Cov(Y,u)=Cov(u)/(1-\alpha_1)$$.