An economist wants to estimate a regression equation relating demand for a product (Y) to its price (X1) and income (X2). It is to be based on 12 years of quarterly data. However, it is known that demand for this product is seasonal; that is, it is higher at certain times of the year than others.
a. One possibility for accounting for seasonality is to estimate the model
y = β0 + β1x1 + β2x2 + β3x3 + β4x4 + β5x5 + β6x6 + ε
where x3, x4, x5, and x6 are dummy variable values, with
x3 = 1 in first quarter of each year, 0 otherwise
x4 = 1 in second quarter of each year, 0 otherwise
x5 = 1 in third quarter of each year, 0 otherwise
x6 = 1 in fourth quarter of each year, 0 otherwise
Explain why this model cannot be estimated by least squares.
b. For a model that can be estimated is as follows:
y = β0 + β1x1 + β2x2 + β3x3 + β4x4 + β5x5 + β6x6 + ε
interpret the coefficients on the dummy variables in the model.