In each of the following three sets of graphs is shown a realisation of the process Xt = Xt1 + ut + ut1, ut WN(0, 1) together with its estimated autocorrelation and partial autocorrelation functions and their 99% confidence intervals. Give values for and in all cases and state if the process is stationary and invertible.

(i) Xt Autocorrelation Partial autocorrelation 1.0 1.0 0.9 0.8 0.7 0.8 0.9 0.7 0.6 ACF 0.4 0.0 0.1 0.2 0.3 Partial ACF 0.0 0.1 0.2 0.3 0.4 0.5 0.6 1 1 0 100 200 300 400 500 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 10 Time Lag Lag (ii) Xt Autocorrelation Partial autocorrelation 0.8 0.9 1.0 0.9 1.0 0.8 0.7 ACF M 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Partial ACF 0.0 0.1 0.2 0.3 0.4 0.5 0.6 N Y T 1 1 0 100 200 300 400 500 0 2 3 4 5 6 8 9 1 2 3 9 7 8 9 10 Time Lag Lag (iii) X Autocorrelation Partial autocorrelation 1.0 3 N 0.7 0.9 1.0 0.7 0.8 LLLLL ACF 0.2 0.3 0.4 0.5 0.6 0.4 0.5 0.6 0.7 0.8 0.9 Partial ACF 0.2 0.3 11 -2 0.0 0.1 0.0 0.1 3 - 4 8 ch 7 1 1 10 2 6 2 3 6 9 300 200 400 0 3 9 10 5 500 4 0 100 8 7 Time Lag Lag