A time series representing a global mean land--ocean temperature index from 1880 to 2009 is available in a file called ‘‘gtemp’’ in the astsa package of R. The data are temperature deviations, measured in degree centigrades, from the 1951--1980 average temperature, as described by Shumway and Stoffer (2011, p. 5). Assume that a thirdorder autoregressive model is appropriate for the first differences ???????? = (1 − ????)???????? of this series.

(a) Plot the time series ???????? and the differenced series ???????? using R.

(b) Generate forecasts and associated probability limits for up to 20 time periods ahead for this series using the function sarima.for() without including a constant term in the model.

(c) Generate the same forecasts and probability limits as in part

(b) but with a constant term now added to the model. Discuss your findings and comment on the implications of including a constant in this case.