Using the data in Exercise 7.1:

(a) Show (using least-squares) that the value ????̂0 of ????0 that minimizes ????∗(−0.5|0) is

????̂0 = (2)(0.50) + (4)(−0.25) + ⋯ + (−2)(0.0078)

12 + 0.52 + ⋯ + 0.00782 = − ∑????

????=0 ????????

????0

???? ∑????

????=0 ????2????

where ????0

???? = (????????|????, ????0 = 0) are the conditional values. Compare this expression for ????̂0 with that for the exact back-forecast [????0] in the MA(1) model, where the expression for [????0] is given preceding the equation (A7.3.9) in Appendix A7.3, and verify that the two expressions are identical.

(b) By first writing this model in the backward form ???????? = (1 − ????????)???????? and recursively computing the ????’s, show that the value of ????0 obtained in

(a) is the same as that obtained by the back-forecasting method.