"Let ct be the cardiovascular mortality series (cmort) discussed in Example 3.5 of the textbook, and

xt=ct

be the differenced data.

(a) Plot xt and compare it to the actual data plotted in Figure 3.2 in the textbook. Why does differencing seem reasonable in this case?

(b) Calculate and plot the sample ACF and PACF of xt and using Table 4.1 in the textbook argue that an AR(1) is appropriate for xt.

(c) Fit an AR(1) to xt using maximum likelihood (basically unconditional least squares) as in Section 4.3 in the textbook. The easiest way to do this is to usesarima fromastsa. Comment on the significance of the regression parameter estimates of the model. What is the estimate of the white noise variance?

(d) Examine the residuals and comment on whether or not you think the residuals are white.

(e) Assuming the fitted model is the true model, find the forecasts over a four-week horizon, xn+m for m = 1,2,3,4, and the corresponding 95% prediction intervals; n = 508 here. The easiest way to do this is to use sarima.for from astsa.

(f) Show how the values obtained in part (e) were calculated.

(g) What is the one-step-ahead forecast of the actual value of cardiovascular mortality; i.e., what is cn+1?"