(a) Problems were experienced in obtaining a satisfactory fit to a series, the last 16 values of which were recorded as follows:

129, 135, 130, 130, 127, 126, 131, 152, 123, 124, 131, 132, 129, 127, 126, 124 Plot the series and suggest where the difficulty might lie.

(b) In fitting a model of the form (1 − ????1???? − ????2????2)???????? = (1 − ????????)???????? to a set of data, convergence was slow and the coefficient estimates in successive iterations oscillated wildly. Final estimates having large standard errors were obtained as follows: ????̂
1 = 1.19, ????̂
2 = −0.34, ????̂ = 0.52.

Can you suggest an explanation for the unstable behavior of the model? Why should preliminary identification have eliminated the problem?

(c) In fitting the model ∇2???????? = (1 − ????1???? − ????2????2)???????? convergence was not obtained.

The last iteration yielded the values ????̂

1 = 1.81, ????̂

2 = 0.52.

Can you explain the difficulty?