The monthly and annual values of the average surface air temperature of the earth (in degrees Celsius). (Actually, the data are indexes, relative to the period 1951−1980 where the average temperature was about 14 degrees Celsius. So if you want the actual temperatures, you can add 14 to all values.) A look at the time series shows a gradual upward trend, starting with negative values and ending with (mostly) positive values. This might be used to support the claim of global warming. For this problem, use only the annual averages in column N. 

a. Is this series a random walk? Explain.

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?

c. Forecast the series in three ways: (i) simple exponential smoothing (α = 0.35), (ii) Holt’s method (α = 0.5, β = 0.1), and (iii) simple exponential smoothing (α = 0.3) 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 MAPE, the RMSE, and the forecast for next year. Also, 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? 

d. Does your analysis predict convincingly that global warming has been occurring? Explain.