Ice cream consumption was measured over 30 four-week periods from March 18, 1951 to July 11, 1953. The purpose of the study was to determine if ice cream consumption depends on the variables price, income, or temperature. The variables Lag-temp and Year have been added to the original data.
Figure 1 shows the variable IC plotted against Date. The colors represent the different years of the study (blue = 1951, yellow = 1952, pink = 1953). This plot shows that ice cream consumption has a seasonal cycle. There also appears to be a gradually increasing trend in ice cream consumption from year to year. This is time series data, and the assumption that the errors of any ordinary regression model are independent will probably be violated.
The original study concluded that of the three possible predictor variables, Price, Income, and Temp, only Temp had a significant effect on the dependent variable IC. However, this study did not include the variable Year.
Velleman (1995) reanalyzed these data and determined that the model including the predictors Lag-temp (Temp lagged by one time period) and Year were significant predictors of ice cream consumption. Figure 2 shows a plot of IC vs. Lag-temp colored by Year with individual regression lines superimposed for each year. Since the variable Lag-temp varies with the season, including it in the model corrects most of the problems of autocorrelation among the errors. Figure 3 shows the results of the linear model predicting IC by Lag-temp and Year. Note the positive effect of Year and Lag-temp and the high adjusted R-square of 85.1%.