For the series to be a random walk, the differences should

be

random, and there is

an upward

trend.

b$.$ Regardless of your answer in part a$,$ use a random walk model to forecast the next value $(2010)$ of the series. What is your forecast, and what is an approximate $95\%$ forecast interval, assuming normally distributed forecast errors? $($Round your answers to three decimal places.$)$

Forecast

Lower limit

Upper limit

c$.$ Forecast the series in three ways: $($i$)$ simple exponential smoothing $(\backslash$alpha $=0\mathrm{.}35),($ii$)$ Holt's method $(\backslash$alpha $=0\mathrm{.}5,\backslash$beta $=0\mathrm{.}1),$ and $($iii$)$ simple exponential smoothing $(\backslash$alpha $=0\mathrm{.}3)$ on trend$-$adjusted data, that is$,$ the residuals from regressing linearly versus time. $($These smoothing constants are close to optimal.$)$ For each of these, list the MAE, the RMSE, and the forecast for next year. $($Round your answers to three decimal places.$)$

Exponential smoothing method MAE RMSE Forecast

Simple forecast $(\backslash$alpha $=0\mathrm{.}35)$

Holt's forecast $(\backslash$alpha $=0\mathrm{.}5,\backslash$beta $=0\mathrm{.}1)$

Simple forecast on trend$-$adjusted data $(\backslash$alpha $=0\mathrm{.}3)$

Comment on any "problems" with forecast errors from any of these three approaches. Finally, compare the qualitative features of the three forecasting methods. For example, how do their short$-$run or longer$-$run forecasts differ? Is any one of the methods clearly superior to the others?

The

Holt's forecast

appears to be the most believable because it does the best job at projecting the

upward

trend we have seen in recent years.

d$.$ Do the data indicate convincingly that global warming has been occurring?

Yes