Q1

Following the hint, we plug the 1st equation into the 2nd equation to substitute the \(gGDP_{t-1}\) and get: \[int_t = \gamma_0 + \gamma_1(\alpha_0 + \delta_0 int_{t-1} + \delta_1 int_{t-2} + u_{t-1} - 3) + v_t\] \[\implies int_t = (\gamma_0 +\gamma_1\alpha_0-3\gamma_1) + \gamma_1\delta_0 int_{t-1} + \gamma_1\delta_1 int_{t-2} + \gamma_1u_{t-1} + v_t\]

Then multiply both sides by \(u_{t-1}\) and take expectations, we get: \[E(int_t u_{t-1}) = E[(\gamma_0 +\gamma_1\alpha_0-3\gamma_1)u_{t-1} + \gamma_1\delta_0 int_{t-1}u_{t-1} + \gamma_1\delta_1 int_{t-2}u_{t-1} + v_tu_{t-1}+ \gamma_1u_{t-1}^2].\] Note that \(u_{t-1}\) has zero mean and uncorrelated with \(int_{t-1}\), \(int_{t-2}\) and \(v_t\), the first 4 terms of the RHS will be rounded out to zero, so we have: \[E(int_t u_{t-1}) = Cov(int_t,u_{t-1}) = E[\gamma_1u_{t-1}^2]\]

Also note that \(E(u_{t}^2) = \sigma_u^2\) for all t and \(\gamma_1 > 0\), therefore we have, \[Cov(int_t,u_{t-1}) = \gamma_1\sigma_u^2>0,\]

and this violates the strict exogeneity of Gaussian Markov Assumptions.

Q2

The code is shown below:

library(dplyr)
library(tidyverse)
library(broom)
library(wooldridge)
yr79 <- ifelse(intdef$year > 1979, 1, 0)
data2 <- cbind(intdef,yr79)
doReg1 = data2 %>% do(reg1=lm(i3~inf+def,data=.))
doReg = data2 %>% do(reg=lm(i3~inf+def+yr79,data=.))
tidy(doReg1,reg1)
tidy(doReg,reg)

As we see, the coefficient of the dummy year after 1979 is roughly 1.56 and it is a statistical significant value, which implies that the there is some special factors after 1979 that may affect the \(i3\). Also, we observe a decrease in \(def\) when we include the \(yr79\).

Q3

doReg2 <- fertil3%>%do(reg2 = lm(gfr ~ pe + I(pe_1 - pe) + I(pe_2 - pe_1) + ww2 + pill, data = .))
tidy(doReg2,reg2)

The standard error of \(pe_t\) is 0.02980, roughly 0.030.