$($Q$2)$ Suppose that the following environment holds: $r_t=+e_t$ with $[I_{t-1}|]=0,E[e_t^2]={}_e^2$ and where $r_{t+1}=ln(\frac{P_{t+1}}{P_t})-=p_{t+1}-p_t.$ We also let $r_{t,t+2}=ln(\frac{P_{t+2}}{P_t})-=p_{t+2}-p_t$ denote the $2-$period continuously compounded returns.

$($i$)$ Are the $r_t\text{'}$s $($single period returns$)$ serially correlated? Why or why not? Justify your answer.

$($ii$)$ Are the $r_{t,t+2}\text{'}$s serially correlated? Why or why not? Justify your answer.

$($iii$)$ Using your answers to $($i$)$ and $($ii$)$ briefly discuss the dangers of using long$-$horizon returns for assessing the weak form of the efficient markets hypothesis.