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- 1. Dummy Variable Models
- 2. “ Using Dummy Variables in Wage Discrimination Cases” Multiple Regression Sandy: pages 603 - 613 Also read paper titled:
- 3. Are Male Nurses Discriminated Against? male nurses 0 female nurses Years of experience, X i W f _ 4 ^ W m _ 3 ^ ~ m W 3 ~ W f ~ 4 ~ ~ adjusted for experience not adjusted for experience o o o o o o o o o o o o + + + + + + + + + + + + + + + + + + + + + + + + + o o o ~
- 4. I. Dummy Variables - Adjusting the intercept . Adjusting the slope . Adjusting both intercept and slope .
- 5. Intercept Dummy Variables Dummy variables are binary (0,1) D t = 1 if red car, D t = 0 otherwise. y t = 1 + 2 X t + 3 D t + e t y t = speed of car in miles per hour X t = age of car in years Police: red cars travel faster . H 0 : 3 = 0 H 1 : 3 > 0
- 6. y t = 1 + 2 X t + 3 D t + e t red cars : y t = ( 1 + 3 ) + 2 X t + e t other cars : y t = 1 + 2 X t + e t y t X t miles per hour age in years 0 1 + 3 1 2 2 red cars other cars
- 7. Slope Dummy Variables y t = 1 + 2 X t + 3 D t X t + e t y t = 1 + ( 2 + 3 )X t + e t y t = 1 + 2 X t + e t y t X t value of porfolio years 0 2 + 3 1 2 stocks bonds Stock portfolio: D t = 1 Bond portfolio: D t = 0 1 = initial investment
- 8. Different Intercepts & Slopes y t = 1 + 2 X t + 3 D t + 4 D t X t + e t y t = ( 1 + 3 ) + ( 2 + 4 )X t + e t y t = 1 + 2 X t + e t y t X t harvest weight of corn rainfall 2 + 4 1 2 “ miracle” regular “ miracle” seed: D t = 1 regular seed: D t = 0 1 + 3
- 9. y t = 1 + 2 X t + 3 D t + e t 2 1 + 3 2 1 y t X t Men Women 0 y t = 1 + 2 X t + e t For men D t = 1. For women D t = 0. years of experience y t = ( 1 + 3 ) + 2 X t + e t wage rate . . Testing for discrimination in starting wage H 0 : 3 = 0 H 1 : 3 > 0
- 10. y t = 1 + 5 X t + 6 D t X t + e t 5 5 + 6 1 y t X t Men Women 0 y t = 1 + ( 5 + 6 )X t + e t y t = 1 + 5 X t + e t For men D t = 1. For women D t = 0. Men and women have the same starting wage, 1 , but their wage rates increase at different rates (diff.= 6 ). 6 > means that men’s wage rates are increasing faster than women's wage rates. years of experience wage rate
- 11. y t = 1 + 2 X t + 3 D t + 4 D t X t + e t 1 + 3 1 2 2 + 4 y t X t Men Women 0 y t = ( 1 + 3 ) + ( 2 + 4 ) X t + e t y t = 1 + 2 X t + e t Women are given a higher starting wage, 1 , while men get the lower starting wage, 1 + 3 , ( 3 < 0 ). But, men get a faster rate of increase in their wages, 2 + 4 , which is higher than the rate of increase for women, 2 , (since 4 > 0 ). years of experience An Ineffective Affirmative Action Plan women are started at a higher wage. Note : ( 3 < 0 ) wage rate
- 12. Testing Qualitative Effects <ul><li>1. Test for differences in intercept . </li></ul><ul><li>2. Test for differences in slope . </li></ul><ul><li>Test for differences in both </li></ul><ul><li>intercept and slope . </li></ul>
- 13. H 0 : vs 1 : H 0 : vs 1 : Y t 1 2 X t 3 D t 4 D t X t b 3 Est . Var b 3 ˜ t n 4 b 4 Est . Var b 4 ˜ t n 4 men: D t = 1 ; women: D t = 0 Testing for discrimination in starting wage. Testing for discrimination in wage increases. intercept slope e t
- 14. Why NOW wants one-sided test and Chauvinist Industries wants two-sided.
- 15. Are Two Regressions Equal? y t = 1 + 2 X t + 3 D t + 4 D t X t + e t variations of “The Chow Test” I. Assuming equal variances (pooling): men: D t = 1 ; women: D t = 0 H o : 3 = 4 = 0 vs. H 1 : otherwise y t = wage rate This model assumes equal wage rate variance. X t = years of experience
- 16. Testing H o : H 1 : otherwise and SSE R y t b 1 b 2 X t 2 t 1 T SSE U y t b 1 b X t b D t b D t X t 2 t 1 T SSE R SSE U 2 SSE U T 4 F T 4 intercept and slope
- 17. y t = 1 + 2 X t + e t II. Allowing for unequal variances: y tm = 1 + 2 X tm + e tm y tw = 1 + 2 X tw + e tw Everyone: Men only: Women only: SSE R Forcing men and women to have same 1 , 2 . Allowing men and women to be different. SSE m SSE w where SSE U = SSE m + SSE w F = (SSE R SSE U )/J SSE U /(T K) J = # restrictions K=unrestricted coefs. (running three regressions) J = 2 K = 4
- 18. Polynomial Terms y t = 1 + 2 X t + 3 X 2 t + 4 X 3 t + e t Linear in parameters but nonlinear in variables: y t = income; X t = age Polynomial Regression y t X t People retire at different ages or not at all. 90 20 30 40 50 60 80 70
- 19. y t = 1 + 2 X t + 3 X 2 t + 4 X 3 t + e t y t = income; X t = age Polynomial Regression Rate income is changing as we age : Slope changes as X t changes. y t X t = 2 + 2 3 X t + 3 4 X 2 t
- 20. Continuous Interaction y t = 1 + 2 Z t + 3 B t + 4 Z t B t + e t Exam grade = f(sleep: Z t , study time: B t ) Sleep and study time do not act independently. More study time will be more effective when combined with more sleep and less effective when combined with less sleep .
- 21. Your mind sorts things out while you sleep (when you have things to sort out.) y t = 1 + 2 Z t + 3 B t + 4 Z t B t + e t Exam grade = f(sleep: Z t , study time: B t ) Your studying is more effective with more sleep . continuous interaction y t B t = 2 + 4 Z t y t Z t = 2 + 4 B t
- 22. y t = 1 + 2 Z t + 3 B t + 4 Z t B t + e t Exam grade = f(sleep: Z t , study time: B t ) If Z t + B t = 24 hours, then B t = (24 Z t ) y t = 1 + 2 Z t + 3 (24 Z t ) + 4 Z t (24 Z t ) + e t y t = ( 1 + 24 3 ) + ( 2 3 + 24 4 ) Z t 4 Z 2 t + e t y t = 1 + 2 Z t + 3 Z 2 t + e t Sleep needed to maximize your exam grade : where 2 > 0 and 3 < 0 y t Z t = 2 + 2 3 Z t = 0 2 3 Z t =
- 23. Multicollinearity Correlation among the “ independent” variables. Note: They are independent of the error term, and not of one another.
- 24. Let yi represent the ith person's wage rate and Xi represent their months of work experience in the equation: yi = b1 + b2 Xi + ei (1) b1 = intercept (starting wage) b2 = increase in the person's wage for each additional month of work experience. ei = error term with mean zero and estimated variance s2.
- 25. yi = b1 + b2 Xi + b3 Mi + b4 Fi + ei (2) Fi = 1 if female Fi = 0 if male . Mi = 1 if male Mi = 0 if female .
- 26. yi = b1 + b2 Xi + b3 Mi + b4 Fi + ei (2) Unfortunately this equation contains an underidentified set of parameters (b1, b3, and b4) and cannot be estimated without some restriction on the coefficients.
- 27. To see this point, separate out the men's equation implied by equation (2) from the women's equation. For the men's equation Mi =1 and Fi =0. For men , equation (2) becomes: yi = (b1 + b3) + b2 Xi + ei (3) yi = b1 + b2 Xi + b3 Mi + b4 Fi + ei (2)
- 28. For women , Mi =0 and Fi =1. For women , equation (2) becomes: yi = (b1 + b4) + b2 Xi + ei (4)
- 29. Unfortunately, although we get estimates of the intercepts (b1 + b3) and (b1 + b4), the value of b1 cannot be separated from the values of b3 and b4. Some restriction is needed to achieve identification of b1, b3 and b4.
- 30. One such restriction is b1 = 0. We can drop the original intercept term, b1, since men and women already have their own intercept terms, b3 and b4 , respectively.
- 31. Underidentification of equation (2) can also be expressed in matrix terms. First, rewrite equation (2) putting the explanatory variables in a row vector multiplied by the corresponding column vector of their respective coefficients: y i 1 X i M i F i 2 3 4 i 5 1
- 32. This only represents the ith observation where i = 1, ..., n. To represent the entire set of n observations at once, we need to "pull the window shade down" as follows: y 1 y 2 M y n 1 X 1 M 1 F 1 1 X 2 M 2 F 2 M M M M 1 X n M n F n 1 2 3 4 1 2 M n (6)
- 33. Equation (6) presents us with an X matrix whose first column (the column of ones) is an exact linear combination of the last two columns (the M and F columns). Since Mi is always zero when Fi is equal to one and Mi is always one when Fi is equal to zero, then it always holds that Mi + Fi = 1. Therefore, the first column is equal to the sum of the last two columns.
- 34. Since Mi is always zero when Fi is equal to one and Mi is always one when Fi is equal to zero, then it always holds that Mi + Fi = 1. 1 1 M 1 M 1 M 2 M M n F 1 F 2 M F n ( 9 )
- 35. Equation (6) and, therefore,equation (2), represent a case of perfect multicollinearity . This means that a restriction must be introduced that drops one of these columns out of the regression. One such restriction is b1 = 0 , which means dropping the original intercept out of the regression model to provide the following reduced model: yi = b2 Xi + b3 Mi + b4 Fi + ei (10) Now men and women have separate intercepts and no common intercept is necessary.
- 36. yi = b2 Xi + b3 Mi + b4 Fi + ei b2 b3 b2 b4 yi Xi Male Female 0 yi = b3 + b2 Xi + ei yi = b4 + b2 Xi + ei For males Mi = 1 and Fi = 0. For females Mi = 0 and Fi = 1. Males and females have different starting salaries , b3 > b4 , but their salaries increase at the same rate, b2.
- 37. y i = b2 X i + b3 M i + b4 F i + e i b2 b3 b2 b4 y i X i Male Female 0 y i = b3 + b2 X i + e i y i = b4 + b2 X i + e i For males Mi = 1 and Fi = 0. For females Mi = 0 and Fi = 1. Males and females have different starting salaries , b3 > b4 , but their salaries increase at the same rate, b2. years of experience
- 38. y i = b1 + b5 M i X i + b6 F i X i + e i b6 b5 b1 y i X i Male Female 0 y i = b1 + b5 X i + e i y i = b1 + b6 X i + e i For males Mi = 1 and Fi = 0. For females Mi = 0 and Fi = 1. Males and Females have the same starting salary b1, but their salaries increase at different rates ( b5 vs. b6 ). b5 > b6 means that men salaries are increasing faster than women's salaries. years of experience
- 39. y i = b3 M i + b4 F i + b5 M i X i + b6 F i X i + e i b3 b4 For males Mi = 1 and Fi = 0. For females Mi = 0 and Fi = 1. b6 b5 y i X i Male Female 0 y i = b3 + b5 X i + e i y i = b4 + b6 X i + e i Females start with a higher starting salary, b4 , while men get the lower starting salary, b3 . But, men get a faster rate of increase in their salaries, b5 , which is higher than the rate of increase for females, b6 . ( b5 > b6 ). years of experience Chauvinist Industries Affirmative Action Plan
- 40. y i = b2 X i + b3 M i + b4 F i + e i b2 b3 b2 b4 y i X i Male Female 0 y i = b3 + b2 X i + e i y i = b4 + b2 X i + e i For males Mi = 1 and Fi = 0. For females Mi = 0 and Fi = 1. Males and females have different starting salaries , b3 > b4 , but their salaries increase at the same rate, b2. Back to our basic model: years of experience
- 41. Since under our null hypothesis the raw score test statistic: has a mean and a variance , we can standardize by subtracting the mean (zero) and dividing by the standard deviation (square root of the variance) to get the standardized test statistic: b 3 – b 4 Var ( b 3 – b 4 ) b 3 – b 4
- 42. To test the null hypothesis: Z ( b b ) 0 Var ( b b ) ~ ( 0 , 1 )
- 43. If the var iance of the y i , 2 , is unknown , then Var ( b 3 b 4 ) is also unknown and must be estimated from the exp ression : Est . Var ( b 3 b 4 ) Est . Var ( b 3 ) Est . Var ( b 4 ) 2 Est . Cov ( b 3 , b 4 )
- 44. Use the sample variance as an estimator of the population variance :
- 45. The values for the following expression are obtained in practice from the diagonal and off-diagonal elements of the estimated variance-covariance matrix : Est . Var ( b 3 b 4 ) Est . Var ( b 3 ) Est . Var ( b 4 ) 2 Est . Cov ( b 3 , b 4 )
- 46. y i = b1 + b2 X i + b3 M i b2 (b1 + b3) b2 b1 y i X i Male Female 0 y i = ( b1 + b3 ) + b2 X i y i = b1 + b2 X i Males and females have different starting salaries , b3 > 0 , but their salaries increase at the same rate, b2. years of experience Alternative : make women the default group ^ ^ ^
- 47. y i = b1 + b2 X i + b3 M i + b4 D i y i = (b1 + b3 + b4) + b2 X i y i = (b1 + b4) + b2 X i y i = (b1 + b3) + b2 X i y i = b1 + b2 X i characteristic dummy variables: male college grad: female college grad: male not a grad: female not a grad: ^ ^ ^ ^ ^
- 48. years of experience 0 X i M-D (male-degree) F-D (female-degree) M-N (male-no degree) F-N (female-no degree) y i wage rate very restrictive assumption y i = b1 + b2 X i + b3 M i + b4 D i b1 b1+b3 b1+b4 b1+b3+b4 very rigid !!! ^
- 49. Creating Composite Dummy Variables ( vs. characteristic dummy variables )
- 50. Job: Gender: Karnaugh map for gender vs. status of job : S I M 15 25 40 F 13 27 40 28 52 80 S = supervisor I = individual men : women :
- 51. Occupation vs. Job vs. Gender Gender: Occupation: Job: C T U S I S I S I M 2 4 3 5 10 16 40 F 1 6 0 7 12 14 40 3 10 3 12 22 30 80 C = Computer T = Other Technical U = Untechnical
- 52. Karnaugh Map for Occupation , Job Status, Gender , and Degree Status: Degree No Degree C T U S I S I S I D M 1 3 2 5 6 13 30 F 0 3 0 6 7 8 24 N M 1 1 1 0 4 3 10 F 1 3 0 1 5 6 16 3 10 3 12 22 30 80
- 53. composite dummy variables: This defines combined ( instead of separate ) general characteristics. y i = b1 + b2 X i + b3 MN i + b4 FD i + b5 MD i years of experience 0 X i M-D (male-degree) F-D (female-degree) M-N (male-no degree) F-N (female-no degree) y i wage rate b1 b1 + b3 b1 + b4 b1 + b5 ^
- 54. Multiple Regression Analysis value of residential property ( buying a home )
- 55. A i = bathrooms X i = sq. ft. living space H 0 : vs. H 1 : H 0 : vs. H 1 : ˆ Y i b 1 b 2 X i b 3 A i b 4 A i X i b 3 Est . Var b 3 ˜ t n 4 b 4 Est . Var b 4 ˜ t n 4
- 56. Testing Ho: H1 : otherwise and SSE R y i b 1 b 2 X i 2 i 1 n SSE U y i b 1 b X i b A i b A i X i 2 i 1 n
- 57. Sale of House with Bed and Bath Dummies 800 0 0 0 10.000 1000 0 0 1 20.000 1200 1 0 0 30.000 1500 1 0 0 40.000 1800 1 0 1 50.000 2000 1 0 1 60.000 2200 0 1 0 70.000 2500 0 1 0 80.000 3000 0 1 1 90.000 3500 0 1 1 100.000 PRICE = f ( SQFEET, D2BED, B3BED, A2BATH ) I. II. III. IV. PRICE (thousands) I. SQFEET = square feet of living space II. D2BED = dummy=1 if two-bedroom house III. D3BED = dummy=1 if three-bedroom house IV. A2BATH = dummy=1 if two-bathroom house
- 58. PRICE = f ( SQFEET, D2BED, B3BED, A2BATH ) Sale of House with Bed and Bath Dummies ANALYSIS OF VARIANCE SOURCE SUM-OF-SQUARES DF MEAN-SQ F-RATIO P REGRESSION 8191.943 4 2047.986 176.378 0.000 RESIDUAL 58.057 5 11.611 DURBIN-WATSON D STATISTIC: 2.216 FIRST ORDER AUTOCORRELATION COEFF: - 0.153 DEP VAR: PRICE N: 10 MULTIPLE R: 0.996 SQUARED MULTIPLE R: 0.993 ADJUSTED SQUARED MULTIPLE R: 0.987 STD ERROR OF ESTIMATE: 3.40
- 59. PRICE = f ( SQFEET, D2BED, B3BED, A2BATH ) Sale of House with Bed and Bath Dummies DEP VAR: PRICE N: 10 MULTIPLE R: 0.996 SQUARED MULTIPLE R: 0.993 ADJUSTED SQUARED MULTIPLE R: 0.987 STD ERROR OF ESTIMATE: 3.40 VARIABLE COEFF STD ERR T P(2-TAIL) INTERCEPT - 6.482 4.112 -1.576 0.176 SQFEET 0.021 0.005 3.958 0.011 D2BED 14.662 4.871 3.010 0.030 D3BED 29.803 10.575 2.818 0.037 A2BATH 4.883 3.953 1.235 0.272 ( for 1,000 square feet: 21 - 6.482 = 14.518 or $14,518 )
- 60. VARIABLE COEFF STD ERR T P(2-TAIL) INTERCEPT - 6.482 4.112 -1.576 0.176 SQFEET 0.021 0.005 3.958 0.011 D2BED 14.662 4.871 3.010 0.030 D3BED 29.803 10.575 2.818 0.037 A2BATH 4.883 3.953 1.235 0.272 for 1,000 square feet: 21 - 6.482 = 14.518 or $14,518 <ul><li>add a bathroom : </li></ul><ul><li>$14,518 </li></ul><ul><li>4,883 </li></ul><ul><ul><li>$19,401 </li></ul></ul><ul><li>add a bedroom : </li></ul><ul><li>$14,518 </li></ul><ul><li>14,662 </li></ul><ul><ul><li>$29,180 </li></ul></ul><ul><li>add 2 bedrooms : </li></ul><ul><li>$14,518 </li></ul><ul><li>29,803 </li></ul><ul><ul><li>$44,321 </li></ul></ul>add bath and 2 bedrooms: 14,518 + 4,883 + 29,803 = $49,204 Regression Analysis of Sale of Residential Property
- 61. Sales Value of Residential Property y = sales value of the property (dollars) X = square feet of living space D1 =dummy vble for one bedroom home D2 =dummy vble for two bedroom home D3 =dummy vble for three bedroom home A1 =dummy vble for one bathroom home A2 =dummy vble for two bathroom home For a one-bedroom, one-bathroom home, such that D2=0, D3=0, and A2=0, we have: y i b 1 b 2 X i b 3 D 2 i b 4 D 3 i b 5 A 2 i ^ y i b 1 b 2 X i 1 bedroom , 1 bathroom ^
- 62. Sales Value of Residential Property For a 2-bedroom, 1-bathroom home, we have D2=1, D3=0, and A2=0 ^ ^ y i b 1 b 2 X i b 3 D 2 i b 4 D 3 i b 5 A 2 i y i ( b 1 b 3 ) b 2 X i 2 bedroom , 1 bathroom
- 63. Sales Value of Residential Property For a 1-bedroom, 2-bathroom home, we have D2=0, D3=0, and A2=1 ^ ^ y i b 1 b 2 X i b 3 D 2 i b 4 D 3 i b 5 A 2 i y i ( b 1 b 5 ) b 2 X i 1 bedroom , 2 bathroom
- 64. Sales Value of Residential Property For a 2-bedroom, 2-bathroom home, we have D2=1, D3=0, and A2=1 y i b 1 b 2 X i b 3 D 2 i b 4 D 3 i b 5 A 2 i ^ y i ( b 1 b 3 b 5 ) b 2 X i 2 bedroom , 2 bathroom ^ y i ( b 1 b 4 b 5 ) b 2 X i 3 bedroom , 2 bathroom ^ y i ( b 1 b 4 ) b 2 X i 3 bedroom , 1 bathroom ^
- 65. square feet of living space 0 X i House Sales Model with Restricted Intercepts b b b D2-A2 (two bed, two bath) b b D2-A1 (two bed, one bath) b b D1-A2 (one bed, two bath) b D1-A1 (one bed,one bath) y i selling price b b b D3-A2 (three bed, two bath) b b D3-A1 (three bed, one bath) b y i b 1 b 2 X i b 3 D 2 i b 4 D 3 i b 5 A 2 i ^ ^ Rigid !!!
- 66. Creating Composite Dummy Variables ( vs. characteristic dummy variables )
- 67. Bath- rooms How do we create composite dummy variables ? Need to account for the interaction effect between bathrooms and bedrooms. <ul><ul><ul><ul><li>1 2 3 </li></ul></ul></ul></ul><ul><ul><ul><ul><li>1 6 8 26 40 </li></ul></ul></ul></ul><ul><ul><ul><ul><li>2 7 7 26 40 </li></ul></ul></ul></ul><ul><ul><ul><ul><li>13 15 52 80 </li></ul></ul></ul></ul>Bedrooms
- 68. Composite dummy variables are created for each nonempty cell. Create six composite dummy variables: D1A1=1 if one bed and one bath, or D1A1= 0 D1A2=1 if one bed and two bath, or D1A2= 0 D2A1=1 if two bed and one bath, or D2A1= 0 D2A2=1 if two bed and two bath, or D2A2= 0 D3A1=1 if three bed and one bath, or D3A1= 0 D3A2=1 if three bed and two bath, or D3A2= 0
- 69. Sales Value of Residential Property y = sales value of the property (dollars) X = square feet of living space D1 A1 = interaction one-bed & one-bath D1 A2 = interaction one-bed & two-bath D2 A1 = interaction two-bed & one-bath D2 A2 = interaction two-bed & two-bath D3 A1 = interaction three-bed & one-bath D3 A2 = interaction three-bed & two-bath y i b 1 b 2 X i b 3 D1A2 i b 4 D2A1 i b 5 D2A2 i ^ b 6 D3A1 i b 7 D3A2 i
- 70. This one equation with all these dummy variables actually is representing six equations . You must substitute in for each of the dummy variables to generate the six equations that are implied by this one dummy variable equation. For a one-bedroom, one-bathroom home, Since D1A1 = 1, while the others are zero: y i b 1 b 2 X i 1 bedroom , 1 bathroom ^ y i b 1 b 2 X i b 3 D1A2 i b 4 D2A1 i b 5 D2A2 i ^ b 6 D3A1 i b 7 D3A2 i
- 71. square feet of living space 0 X i House Sales Model with Unrestricted Intercepts D2-A2 (two bed, two bath) D2-A1 (two bed, one bath) D1-A2 (one bed, two bath) b D1-A1 (one bed,one bath) y i selling price D3-A2 (three bed, two bath) D3-A1 (three bed, one bath) b
- 72. one-bedroom , two-bathroom D1A2 =1, while the others are zero: now graph it ! =======> y i ( 1 b 3 ) b 2 X i 1 bedroom , 2 bathroom ^ y i b 1 b 2 X i b 3 D1A2 i b 4 D2A1 i b 5 D2A2 i ^ b 6 D3A1 i b 7 D3A2 i b
- 73. square feet of living space 0 X i House Sales Model with Unrestricted Intercepts D2-A2 (two bed, two bath) b b D2-A1 (two bed, one bath) D1-A2 (one bed, two bath) b D1-A1 (one bed,one bath) y i selling price D3-A2 (three bed, two bath) D3-A1 (three bed, one bath)
- 74. two-bedroom , one-bathroom now graph it ! =======> y i ( b 1 b 4 ) b 2 X i 2 bedroom , 1 bathroom ^ y i b 1 b 2 X i b 3 D1A2 i b 4 D2A1 i b 5 D2A2 i ^ b 6 D3A1 i b 7 D3A2 i D2A1 =1, while the others are zero:
- 75. square feet of living space 0 X i House Sales Model with Unrestricted Intercepts D2-A2 (two bed, two bath) b b D2-A1 (two bed, one bath) b b D1-A2 (one bed, two bath) b D1-A1 (one bed,one bath) y i selling price D3-A2 (three bed, two bath) D3-A1 (three bed, one bath)
- 76. two-bedroom , two-bathroom now graph it ! =======> y i ( b 1 b 5 ) b 2 X i 2 bedroom , 2 bathroom ^ y i b 1 b 2 X i b 3 D1A2 i b 4 D2A1 i b 5 D2A2 i ^ b 6 D3A1 i b 7 D3A2 i D2A2 =1, while the others are zero:
- 77. square feet of living space 0 X i House Sales Model with Unrestricted Intercepts b b D2-A2 (two bed, two bath) b b D2-A1 (two bed, one bath) b b D1-A2 (one bed, two bath) b 1 D1-A1 (one bed,one bath) y i selling price D3-A2 (three bed, two bath) D3-A1 (three bed, one bath)
- 78. square feet of living space 0 X i House Sales Model with Unrestricted Intercepts b b D2-A2 (two bed, two bath) b b D2-A1 (two bed, one bath) b b D1-A2 (one bed, two bath) b 1 D1-A1 (one bed,one bath) y i selling price b b D3-A2 (three bed, two bath) b b D3-A1 (three bed, one bath)
- 79. Creating Composite Dummy Variables ( vs. characteristic dummy variables )
- 80. Bath- rooms How do we create composite dummy variables ? Need to account for the interaction effect between bathrooms and bedrooms. <ul><ul><ul><ul><li>1 2 3 </li></ul></ul></ul></ul><ul><ul><ul><ul><li>1 6 8 26 40 </li></ul></ul></ul></ul><ul><ul><ul><ul><li>2 7 7 26 40 </li></ul></ul></ul></ul><ul><ul><ul><ul><li>13 15 52 80 </li></ul></ul></ul></ul>Bedrooms
- 81. Bedrooms vs. Baths vs. Garage Baths Bedrooms Cars in Garage: 1 2 3 1 2 1 2 1 2 1 2 4 3 5 10 16 40 2 1 6 0 7 12 14 40 3 10 3 12 22 30 80
- 82. Karnaugh Map for Bedrooms , Baths , Garage , and School : Adams Saint Joseph 1 2 3 1 2 1 2 1 2 A 1 1 3 2 5 6 13 30 2 0 3 0 6 7 8 24 J 1 1 1 1 0 4 3 10 2 1 3 0 1 5 6 16 3 10 3 12 22 30 80

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