Sales of a product are influenced by the salesperson’s level of education and gender, as well as consumer income, ethnicity, and wealth.
a. Formulate this statement as a multiple regression model (form only, without parameter estimation).
b. Specify dummy variables.
c. If the effects of consumer income and wealth are not additive alone, and an interaction is expected, specify a new variable to test for the interaction.
Let us adopt the following symbols for the variables:
· Sales: S ($)
· Salesperson's level of education: E (Years of education).
· Gender (M): We use a single dummy variable for two categories. If the observation is male, then the variable M=1, if female then M =0.
· Consumer income: I ($)
· Wealth: W ($)
· Ethnicity: Let us consider three categories Caucasian, Hispanic and Asian. For the three categories we need two dummy variables.
o Asian (A), which takes the value 1, if it is an Asian and 0 otherwise.
o Hispanic (H), which takes the value 1, if it is a Hispanic and 0 otherwise.
b0 is a constant; b1, b2, b3 etc. are coefficients, e is the residual error term.
The general model will be:
S = b0 + b1E + b2M + b3I + b4W + b5A + b6H + e
a. The multiple regression model can be formulated as follows:
Sales = β0 + β1(education) + β2(gender) + β3(income) + β4(ethnicity) + β5(wealth)
Here, Sales represents the dependent variable, and education, gender, income, ethnicity, and wealth are the independent variables. β0, β1, β2, β3, β4, and β5 represent the regression coefficients which need to be estimated.
b. Dummy variables need to be specified for variables like gender and ethnicity, which have categorical values. To include gender in the model, we can create a binary dummy variable as follows:
gender_dummy = 1 if the salesperson is male
0 if the salesperson is female
Similarly, to include ethnicity, we can create multiple dummy variables based on the number of categories. For example, if there are three ethnicities (A, B, C), we can create two dummy variables as follows:
ethnicity_dummy1 = 1 if the consumer belongs to ethnicity A
0 otherwise
ethnicity_dummy2 = 1 if the consumer belongs to ethnicity B
0 otherwise
c. If an interaction is expected between consumer income and wealth, we can create a new variable to test for the interaction effect. Let's call this variable "income_wealth_interaction". This new variable can be created by multiplying the income and wealth variables:
income_wealth_interaction = income * wealth
Now, we can add this interaction term to the original multiple regression model:
Sales = β0 + β1(education) + β2(gender) + β3(income) + β4(ethnicity) + β5(wealth) + β6(income_wealth_interaction)
The term β6 represents the coefficient for the interaction effect between income and wealth.