Consider a random walk (left{y_{t} ight}) as the partial sum of a white noise process (left{c_{t} ight}).

Consider a random walk \(\left\{y_{t}\right\}\) as the partial sum of a white noise process \(\left\{c_{t}\right\}\).

a. Show that the \(l\)-step forecast error is \(y_{T+l}-\widehat{y_{T+l}}=\sum_{j=1}^{l}\left(c_{T+j}-\bar{c}\right)\).

b. Show that the approximate variance of the \(l\)-step forecast error is \(l \sigma_{c}^{2}\).