Consider the simple white noise process, $Z_t = a_t$. Discuss the consequences of over-differencing by examining the ACF and AR representation of the differenced series, $W_t = Z_t - Z_{t-1}$.
Answer: The ACF is easily found to be: $$\rho_k = \left\{\begin{array}{ccc} \frac{-1}{2} & , & k = 1 \\ 0 & , & k > 0\end{array}\right.$$
The MA representation of $W_t$ is: $$W_t = Z_1 - Z_{t - 1} = a_t - a_{t - 1} = (1 - B)a_t$$
To find out if the process is invertible, we let $1 - B = 0$ which means $B = 1$. However, this value is not outside the unit circle, implying that the process is not invertible, hence no AR representation exists.
So I am confused about why the question asks to "examine using the AR representation" if one does not exist!
