This [unedited] guest post is by a student in my PSTAT262MC class (background post). Please praise/critique/comment on its quality and importance to you.
Nathan Bennett says: Previoiusly I posted about the weekly gas prices in LA and weekly oil prices in the U.S. See my previous post for a detailed description of the data sets that I am studying. For this analysis I used a multivariate linear trend model with a seasonal component. The linear trend portion of the model was allowed to change with time. This portion of the model estimates the mean effect and slope for the data at each time point, and allows for a dependency in the evolution of the mean effect and slope. Since gas prices peak each year during the summer months, I used the first two harmonics of a Fourier seasonal model with a fixed period of 52 weeks. Moreover, I did not consider any evolution variance (or covariance) for this part of the model. This combined model does a good job modeling the data and gives reasonable predicted values. The model yields gas price predictions (95% Intervals) of \$3.09 (\$2.86, \$3.32) for January 11 and \$3.86 (\$1.64, \$6.08) for March 15; the model yields oil price predictions (95% Intervals) of \$75.32 (\$70.40, \$80.23) for January 11 and \$96.47 (\$40.18, \$152.77) for March 15. The figure below is a plot of the data and has these predicted values and bounds. While the model does give rather reasonable predictions for one and ten weeks into the future, the error bounds for these predicted values grow very rapidly. Hence, the one week estimate is a pretty good estimate of the prices, but the ten week prediction is probably not very accurate.