The data to be used for this problem are contained in the S-Plus script file pprice.asc. Running this script will create the data frame PPRICE for which each row corresponds to a particular date.

• The first column contains the values of a variable named GasSpot which gives the spot price of natural gas on that date.

• The second column contains the values of a variable named SDTemp which gives the average temperature in San Diego over the 31 days preceding the date in question.

• The third column contains the values of a variable named PPower which gives the average over the 5 days preceding the date in question, of the spot price of firm on peak electric power at the Palo Verde station.

• Finally, the fourth column contains the values of a variable named FPower which gives the average over the two weeks following the date in question, of the spot price of firm on peak electric power at the Palo Verde station.

Form a data frame TRG with the first 250 rows of PPRICE. The entries of TRG correspond to the period from 2/4/1999 to 2/3/2000. You shall also need to form a data frame TST with the last 80 rows of PRICE. The entries of TST correspond to the period from 7/13/2001 to 11/9/2001. We avoid the period in between because of the extreme volatility of the natural gas and power prices. This does not mean that we are not interested in studying periods of high volatility, quite the contrary. It is merely because the economic fundamentals were not the only driving force during this crisis period.

The goal of the problem is to predict the values of the average price of (firm on peak) electric power over the next two weeks from past values of explanatory variables such as the weather (as quantified by the average temperature in San Diego), the price of natural gas, and possibly past values of the price of electricity at the same location. We use the data in TRG as a training sample to fit a regression model, and we compute the predictions given by such a model for the data in the testing sample TST.

Warning. The variable PPower should not be used in the first 4 questions. Moreover, for all the predictions considered below, the figure of merit should be the square root of the mean squared error.

1. Fit a least squares linear regression model for FPower against GasSpot and SDTemp using the data in TRG, use this model to predict the values of FPower in TST from the corresponding values of the explanatory variables, and compute the figure of merit.

2. Same question with least absolute deviations linear regression instead.

3. Same question using projection pursuit. Explain your work.

4. Same question using kernel regression. Again make sure that you explain all the steps you take, and justify your choice of kernel function and bandwidth.

5. Fit a least squares linear regression model for FPower against GasSpot, SDTemp and PPower using the data in TRG, as before, use the fitted model to predict the values of FPower in TST from the corresponding values of the three explanatory variables, and compute the figure of merit.

6. Same question with least absolute deviations linear regression instead.

7. Same question using projection pursuit.

8. Compare the numerical results obtained with the various methods, and explain why they could have been expected.