Suppose that a quarterly economic time series is well represented by the model

∇???????? = 0.5 + (1 − 1.0???? + 0.5????2)????????

with ????2

???? = 0.04.

(a) Given ????48 = 130, ????47 = −0.3, ????48 = 0.2, calculate and plot the forecasts ????̂48(????)

for ???? = 1, 2, … , 12.

(b) Calculate and insert the 80% probability limits on the graph.

(c) Express the series and forecasts in integrated form.