Outline

1. Outline • When X’s are Dummy variables • EXAMPLE 1: USED CARS • EXAMPLE 2: RESTAURANT LOCATION • Modeling a quadratic relationship • Restaurant Example

2. Qualitative Independent Variables • In many real-life situations one or more independent variables are qualitative. • Including qualitative variables in a regression analysis model is done via indicator variables. • An indicator variable (I) can assume one out of two values, “zero” or “one”. 1 if a degree earned is in Finance 0 if a degree earned is not in Finance 1 if the temperature was below 50o 0 if the temperature was 50o or more 1 if a first condition out of two is met 0 if a second condition out of two is met 1 if data were collected before 1980 0 if data were collected after 1980 I=

3. 1 if the color is white 0 if the color is not white I1 = 1 if the color is silver 0 if the color is not silver I2 = Example 1 • The dealer believes that color is a variable that affects a car’s price. • Three color categories are considered: • White • Silver • Other colors • Note: Color is a qualitative variable. And what about “Other colors”? Set I1 = 0 and I2 = 0

4. To represent a qualitative variable that has m possible categories (levels), we must create m-1 indicator variables. • Solution • the proposed model is y = b0 + b1(Odometer) + b2I1 + b3I2 + e • The data White car Other color Silver color

5. There is insufficient evidence to infer that a white color car and a car of “Other color” sell for a different auction price. There is sufficient evidence to infer that a silver color car sells for a larger price than a car of the “Other color” category.

6. Price 6498 - .0278(Odometer) 6395.2 - .0278(Odometer) 6350 - .0278(Odometer) Odometer From Excel we get the regression equation PRICE = 6350-.0278(ODOMETER)+45.2I1+148I2 For one additional mile the auction price decreases by 2.78 cents. A white car sells, on the average, for $45.2 more than a car of the “Other color” category A silver color car sells, on the average, for $148 more than a car of the “Other color” category The equation for a car of silver color Price = 6350 - .0278(Odometer) + 45.2(0) + 148(1) The equation for a car of white color The equation for a car of the “Other color” category. Price = 6350 - .0278(Odometer) + 45.2(1) + 148(0) Price = 6350 - .0278(Odometer) + 45.2(0) + 148(0)

7. Example 2 Location for a new restaurant • A fast food restaurant chain tries to identify new locations that are likely to be profitable. • The primary market for such restaurants is middle-income adults and their children (between the age 5 and 12). • Which regression model should be proposed to predict the profitability of new locations?

8. Revenue Revenue Income age Low Middle High Low Middle High • Solution • The dependent variable will be Gross Revenue • There are quadratic relationships between Revenue and each predictor variable. Why? • Members of middle-class families are more likely to visit a fast food family than members of poor or wealthy families. Revenue = b0 + b1Income + b2Age + b3Income2 +b4Age2 + b5(Income)(Age) +e • Families with very young or older kids will not visit the restaurant as frequent as families with mid-range ages of kids.

9. Example 2 • To verify the validity of the model proposed in example 19.1, 25 areas with fast food restaurants were randomly selected. • Data collected included (see Xm19-02.xls): • Previous year’s annual gross sales. • Mean annual household income. • Mean age of children

11. The model provides a good fit

12. The model can be used to make predictions. However, do not interpret the coefficients or test them. Multicollinearity is a problem!! In excel: Tools > Data Analysis > Correlation

13. Regression results of the modified model Multicolinearity is not a problem anymore