Question

2. Describe how you would construct a purely seasonal model for the following monthly series. In partic- ular, what dummy variable() would you use to capture the relevant effects? (a) A sporting goods store finds that detrended monthly sales are roughly the same for each month in a given 3-month season. For example, sales are similar in the winter months of January, February, and March, in the spring months of April, May, and June, and so on. (b) A campus bookstore finds that detrended sales are roughly the same for all first, all second, all third, and all fourth months of each trimester (C) A Christmas ornament store is only open in November and December, so sales are zero in all other months.

Answer 1

a)

Let us assign dummy variables such that:

X_1i =1 if month = January, February, March; 0 otherwise

X_2i = 1 if month = April, May, June; 0 otherwise

X_3i = 1 if month = July, August, September; 0 otherwise

Y_i = Monthly Sales

u_i = Residual term

Hence we construct the model:

Y_i = a₀ + a₁X_1i + a₂X_2i + a₃X_3i + u_i

Where a_j = Incremental effect of that particular 3-month season on Monthly Sales ; i = 1,2,3

a₀ = Mean effect of October, November, December season on Monthly Sales

b)

Let us assign dummy variables such that:

X_1i = 1 if month = 1^st trimester; 0 otherwise

X_2i = 1 if month = 2^nd trimester; 0 otherwise

X_3i =1 if month = 3^rd trimester; 0 otherwise

Y_i = Monthly Sales

u_i = Residual term

Hence we construct the model:

Y_i = a₀ + a₁X_1i + a₂X_2i + a₃X_3i + u_i

Where a_j = Incremental effect on Monthly Sales of that particular trimester ; i = 1,2,3

a₀ = Mean effect of 4^th trimester on Monthly Sales

c) Let us assign the dummy variables as follows:

X_i = 1 if month = November, December; 0 otherwise

Y_i = Monthly Sales

u_i = Residual term

Hence we construct the model:

Y_i = aX_i + u_i

where a = Incremental effect on sales due to the month being November/December