For the climhyd data set, consider predicting the transformed flows It =

log it from transformed precipitation values Pt = √pt using a transfer function model of the form

(1 − B12)It = α(B)(1 − B12)Pt + nt, where we assume that seasonal differencing is a reasonable thing to do. You may think of it as fitting yt = α(B)xt + nt, where yt and xt are the seasonally differenced transformed flows and precipitations.

(a) Argue that xt can be fitted by a first-order seasonal moving average, and use the transformation obtained to prewhiten the series xt.

(b) Apply the transformation applied in

(a) to the series yt, and compute the cross-correlation function relating the prewhitened series to the transformed series. Argue for a transfer function of the form

α(B) = δ0 1 − ω1B .

(c) Write the overall model obtained in regression form to estimate δ0 and

ω1. Note that you will be minimizing the sums of squared residuals for the transformed noise series (1 − ωb1B)nt. Retain the residuals for further modeling involving the noise nt. The observed residual is ut = (1−ωb1B)nt.

(d) Fit the noise residuals obtained in

(c) with an ARMA model, and give the final form suggested by your analysis in the previous parts.

(e) Discuss the problem of forecasting yt+m using the infinite past of yt and the present and infinite past of xt. Determine the predicted value and the forecast variance.

Section 5.8