5.1 The data set labeled fracdiff.dat is n = 1000 simulated observations from a fractionally differenced ARIMA(1, 1, 0) model with φ = .75 and d = .4.

(a) Plot of the data and comment.

(b) Plot the ACF and PACF of the data and comment.

(c) Estimate the parameters and test for the significance of the estimates

φ and  d.

(d) Explain why, using the results of part

(a) and (b), it would seem reasonable to difference the data prior to the analysis. That is, if xt represents the data, explain why we might choose to fit an ARMA model to ∇xt.

(e) Plot the ACF and PACF of ∇xt and comment.

(f) Fit an ARMA model to ∇xt and comment.