Construction of new homes is a popular measure of the health of the economy. A slowdown in the construction industry means more unemployment, fewer sales for home stores, and a general economic malaise. These data from the Census Bureau track (seasonally adjusted) sales of new homes and the number of permits issued for constructing new homes, monthly from January 1990 to April 2008. Because builders must apply for permission from the local government to build a house, the number of permits issued today is a possible leading indicator for the number of homes that will be on the market in future months. Let yt denote the number of single-family homes completed and xt denote the number of permits issued for single-family homes. (Both counts are in thousands.)
(a) Compare the timeplot of homes completed to the timeplot of permits issued. Do the two time series line up, or does one seem shifted from the other?
(b) Compare the association between yt and xt to the association between yt and xt -6. Which variable, number of current permits or number of permits issued six months earlier, appears to be the better explanatory variable (taken by itself)?
(c) Build a multiple regression designed to predict the number of single-family homes to be completed in July 2008 (three months past the end of this time series). Justify your choice of this model and verify that it meets the conditions required for a multiple regression model.
(d) Use your model to predict housing completions in July 2008 and include a 95% prediction interval. Note any caveats that you would offer with the forecast.