Your task is to estimate the ice cream sales for a certain chain in New England

The company makes available to you quarterly ice cream sales (Y) and informs you that the price per gallon has approximately remained constant over the sample period. You gather information on average daily temperatures (X) during these quarters and regress Y on X, adding seasonal binary variables for spring, summer, and fall. These variables are constructed as follows: DSpring takes on a value of 1 during the spring and is zero otherwise, DSummer takes on a value of 1 during the summer, etc. Specify three regression functions where the following conditions hold: the relationship between Y and X is (i) forced to be the same for each quarter; (ii) allowed to have different intercepts each season; (iii) allowed to have varying slopes and intercepts each season. Sketch the difference between (i) and (ii). How would you test which model fits the data the best?
What will be an ideal response?

Answer: (i) Yi = β0 + β1Xi + ui;
(ii) Yi = β0 + β1Xi + β2DSpring + β3DSummer + β4DFall + ui;
(iii) Yi = β0 + β1Xi + β2DSpring + β3DSummer + β4DFall
+ β5(DSpring × Xi) + β6 (DSummer × Xi ) + β7 (DFall × Xi) + ui;
(iii) is the most general of the models, the others are nested. Hence you can use the F-test to see if certain restrictions hold. For example, (i) is a parsimonious representation of (iii) if all coefficients involving the seasonal binary variables are simultaneously equal to zero.