Consider the Following Two Models Involving Binary Variables as Explanatory

question 43

Essay

Consider the following two models involving binary variables as explanatory
variables: $$\widehat { \text { Wage } } = \widehat { \beta } _ { 0 } + \widehat { \beta } _ { 1 } D \text { Femme and } \widehat { \text { Wage } } = \widehat { \phi } _ { 1 } D \text { Femme } + \widehat { \phi } _ { 2 } \text { Male }$$ where Wage is the hourly wage rate, DFemme is a binary variable that is equal to 1 if the person is a female, and 0 if the person is a male. Male $= 1 -$ DFemme. Even though you have not learned about regression functions with two explanatory variables (or regressions without an intercept), assume that you had estimated both models, i.e., you obtained the estimates for the regression coefficients.
What is the predicted wage for a male in the two models? What is the predicted wage for a female in the two models? What is the relationship between the $\beta \mathrm { s }$ and the $\phi s$ ? Why would you prefer one model over the other?

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Definitions:

Sodium

A soft, silvery-white, highly reactive metal and an essential element in animal nutrition, symbolized as Na.

Nitrogen

A colorless, odorless, tasteless gas that constitutes about 78% of the Earth's atmosphere; it is a critical element in the makeup of amino acids, proteins, and nucleic acids.

Ionic Compound

A substance made up of positively and negatively charged ions that are combined in a ratio to neutralize the charges.

Bromine

A reddish-brown liquid at room temperature, with the elemental symbol Br, and atomic number 35; it is a halogen and is used in various industrial applications.