y i OLS [0, 1] OLS x i = (1, x 1,i,, x k,i ) β = (β 0, β 1,, β k ) G ( x i β) 1 G i 1 π i π i P {y i = 1 x i } = G (

7 2 2008 7 10 1 2 2 1.1 2............................................. 2 1.2 2.......................................... 2 1.3 2........................................ 3 1.4................................................ 3 1.5......................................... 6 1.6............................................. 9 1.7 2........................................... 11 1.8 2.......................................... 14 2 17 2.1............................................. 17 2.2......................................... 18 2.3........................................ 18 2.4................................................ 18 2.5....................................... 19 2.6......................................... 19 2.7........................................ 22 1/25

1 2 1 2 1.1 2 y i 0 1 2 OLS [0, 1] 0 1 2 2 OLS x i = (1, x 1,i,, x k,i ) β = (β 0, β 1,, β k ) 0 1 0 G ( x i β) 1 G i 1 π i π i P {y i = 1 x i } = G ( x β ) (1) G 2 2 1.2 2 2 (1) G 2 π i = P {y i = 1 x i } = G ( x β ) = Λ ( x i β ) = exp { x i β} 1 + exp { x (2) i β} = exp {β 0 + β 1 x 1,i + + β k x k,i } 1 + exp {β 0 + β 1 x 1,i + + β k x k,i } Λ(z) = 1 π i = P {y i = 0 x i } = 1 P {y i = 1 x i } = 1 Λ ( x i β ) exp{z} 1+exp{z} ln L (β; y, x) = n { yi ln Λ ( x i β ) + (1 y i ) ln ( 1 Λ ( x i β ))} i=1 ˆβ 2/25

1 2 1.3 2 1.3 2 (1) G 2 π i = P {y i = 1 x i } = G ( x β ) = Φ ( x i β ) = x β 1 2π e t2 2 dt (3) Φ = z 1 2π e 1 2 t2 dt { n y i Φ ( x i ln L (β; y, x) = β) } Φ ( x i β) [ 1 Φ ( x i β)] ϕ ( x i β ) x i i=1 ϕ(z) ˆβ 1.4 1.4.1 1% micro data U.S. Census Bureau Census 2000 data 2000 1-Percent Public Use Microdata *1 1 25 34 13113 2 3 4 1.4.2 esrx 2 *1 http://www.census.gov/press-release/www/2003/pums.html *3 3/25

1 2 1.4 age paocf paocf1 paocf2 paocf3 citizen citizen0 msp msp1 msp2 msp3 msp4 msp5 white black asian engabil educ educ1 educ2 educ3 educ4 educ5 esr esrx sex 16 =0 6 =1 6 17 =2 6 6 17 =3 18 =4 6 =1 =0 6 17 =1 =0 6 6 17 =1 =0 =1 =2 =3 =4 =5 =1 =0 15 =0 =1 =2 =3 =4 =5 =6 =1 =0 =1 =0 =1 =0 =1 =0 =1 =0 =1 =0 =1 =0 =1 =0 =0 =1 =2 =3 =4 3 =0 =1 4th Grade =2 5th Grade 6th Grade=3 7th Grade 8th Grade=4 9th Grade=5 10th Grade=6 11th Grade=7 12th Grade =8 =9 1 =10 1 =11 Associate degree =12 Bachelor s degree =13 Master s degree =14 Professional degree =15 Doctorate degree =16 =1 =0 =1 =0 or =1 =0 =1 =0 or =1 =0 16 =0 =1 =2 =3 =4 =5 =6 =1 =0 =1 =2 1 * 3 4/25

1 2 1.4 Coefficient Std. Error z-statistic Prob. C 0.8894 0.3843 2.3141 0.0207 AGE 0.0011 0.0073 0.1431 0.8862 WHITE 0.4058 0.0678 5.9874 0.0000 BLACK 0.2446 0.0767 3.1892 0.0014 ASIAN 0.0991 0.0944 1.0502 0.2936 PAOCF1 0.7749 0.0566 13.6912 0.0000 PAOCF2 0.1726 0.0659 2.6201 0.0088 PAOCF3 0.7467 0.0613 12.1781 0.0000 MSP1 0.1638 0.0524 3.1271 0.0018 MSP2 0.3233 0.1027 3.1478 0.0016 MSP3 0.8998 0.2852 3.1554 0.0016 MSP4 0.1231 0.0915 1.3444 0.1788 MSP5 0.0990 0.0964 1.0268 0.3045 CITIZEN0 0.4990 0.0567 8.8018 0.0000 EDUC1 1.7001 0.3111 5.4640 0.0000 EDUC2 0.9772 0.3087 3.1661 0.0015 EDUC3 0.4127 0.3083 1.3386 0.1807 EDUC4 0.1611 0.3089 0.5216 0.6020 EDUC5 0.1677 0.3132 0.5354 0.5924 2 2 1 McFadden R-squared 0.10386 Mean dependent var 0.643255 S.D. dependent var 0.479057 S.E. of regression 0.447095 Akaike info criterion 1.170609 Sum squared resid 2617.409 Schwarz criterion 1.181449 Log likelihood 7656.097 Hannan-Quinn criter. 1.174229 Restr. log likelihood 8543.414 LR statistic 1774.634 Avg. log likelihood 0.583856 Prob(LR statistic) 0 3 2 1 0 1 0 1848 1202 3050 1 2830 7233 10063 4678 8435 13113 4 2 1 69.25% 5/25

1 2 1.5 1.5 1. Workfile File - New - Workfile 2. OK Workfile structure type Unstructured - Undated Observations 191433 3. 6/25

1 2 1.5 4. csv Workfile Proc - Import - Read Text-Lotus-Excel Text ASCII PC csv ny.csv 5. 21 21 Name for series or Number if named in file 21 * 4 Data order in Columns * 5 # of headers before data 1 * 6 delimiters Comma * 7 *4 21 *5 in Columns *6 1 1 *7 csv Comma Comma 7/25

1 2 1.5 6. 8/25

1 2 1.6 1.6 1. 2 Workfile Genr OK esr=1 esr=4 1 0 esrx esrx Enter equation esrx = esr = 1 or esr = 4 * 8 2. Workfile Genr *8 or *9 <> not equal *10 esrx = citizen0 = educ1 = { 1 if esr = 1 or 4 0 otherwise { 1 if citizen 5 0 otherwise { 1 if educ 8 0 otherwise *11 educ3 = { 1 if 10 educ 12 0 otherwise and *12 educ5 = { 1 if educ = 14 or 15 0 otherwise 9/25

1 2 1.6 Enter equation citizen0=citizen<>5 * 9 Enter equation paocf1=paocf=1 Enter equation paocf2=paocf=2 Enter equation paocf3=paocf=3 Enter equation msp1 =msp =1 Enter equation msp2 =msp =2 Enter equation msp3 =msp =3 Enter equation msp4 =msp =4 Enter equation msp5 =msp =5 Enter equation educ1 =educ <=8 * 10 Enter equation educ2 =educ =9 Enter equation educ3 =educ >=10 and educ <=12 * 11 Enter equation educ4 =educ =13 Enter equation educ5 =educ =14 or educ =15 * 12 10/25

1 2 1.7 2 1.7 2 1. esrx age white black asian paocf1 paocf2 paocf3 msp1 msp2 msp3 msp4 msp5 citizen0 educ1 educ2 educ3 educ4 educ5 2 P {esrx i = 1 age i, white i } = exp {c + β 1age i + β 2 white i + } 1 + exp {c + β 1 age i + β 2 white i + } 2. 2 25 34 Workfile Sample OK Sample range pairs @all * 13 IF condition sex = 2 and age >= 25 and age <= 34 3. Quick - Estimate Method BINARY 4. *13 11/25

1 2 1.7 2 Equation specification Binary estimation method esrx c age white black asian paocf1 paocf2 paocf3 msp1 msp2 msp3 msp4 msp5 citizen0 educ1 educ2 educ3 educ4 educ5 Logit 5. 12/25

1 2 1.7 2 6. View - Expectation-Prediction Evaluation Success of probability is greater than 0.5 7. 13/25

1 2 1.8 2 1.8 2 2 2 1. esrx age white black asian paocf1 paocf2 paocf3 msp1 msp2 msp3 msp4 msp5 citizen0 educ1 educ2 educ3 educ4 educ5 2 P {esrx i = 1 age i, white i } = {c+β1age i+β 2white i+ } 1 2π e t2 2 dt 2. 2 25 34 3. Quick - Estimate Method BINARY 4. Equation specification Binary estimation method esrx c age white black asian paocf1 paocf2 paocf3 msp1 msp2 msp3 msp4 msp5 citizen0 educ1 educ2 educ3 educ4 educ5 Probit 14/25

1 2 1.8 2 5. 15/25

1 2 1.8 2 6. View - Expectation-Prediction Evaluation Success of probability is greater than 0.5 7. 16/25

2 2 2.1 y y y i = x i β + u i, i = 1,, n x i = (1, x 1,i,, x k,i ) β = (β 0, β 1,, β k ) u i.i.d. 0 F y y 1, y 2,, y J y i y i threshold 1 κ 0 < yi κ 1 2 κ 1 < yi y i = κ 2.. J κ J 1 < yi < κ J J + 1 κ 0, κ 1,, κ J outcome y i J y i = 1 if and only if κ 0 < y i κ 1 κ 0 x i β < u i κ 1 + x i β y i = 2 if and only if κ 1 < y i κ 2 κ 1 x i β < u i κ 2 + x i β. y i = 3 if and only if κ J 1 < y i κ J κ J 1 x i β < u i κ J + x J β κ 0 = κ J = J 1 y i j π ij = P (y i = j x i ) = F (κ j x i β) F (κ j 1 x i β), j = 1, J (4) κ 0 = κ J = F ( ) = 0 F ( ) = 1 F 17/25

2 2.2 2.2 (4) F π ij = P (y i = j x i ) = Λ ( κ j x i β ) Λ ( κ j 1 x i β ) Λ(z) = exp(z) 1+exp(z) ln L (β, κ 1,, κ J 1 ; y, x) = n i=1 j=1 J d ij ln ( Λ ( κ j x i β ) Λ ( κ j 1 x i β )) ˆβ κ 1,, κ J 1 2.3 (4) F π ij = P (y i = j x i ) = Φ (κ j x i β) Φ (κ j 1 x i β) Φ(z) = z 1 2 exp { 1 2 t2} dt ln L (β, κ 1,, κ J 1 ; y, x) = n i=1 j=1 J d ij ln ( Φ ( κ j x i β ) Φ ( κ j 1 x i β )) ˆβ κ 1,, κ J 1 2.4 2.4.1 2 1% 30 micro data U.S. Census Bureau Census 2000 data 2000 1-Percent Public Use Microdata * 14 1 30 26510 *14 http://www.census.gov/press-release/www/2003/pums.html 18/25

2 2.5 2.5 2 2.6 1. engabil age white black asian earns educ1 educ2 educ3 educ4 educ5 citizen0 msp1 msp2 msp3 msp4 msp5 30 P {engbil i = j age i, white i } = exp {κ j (β 1 age i + β 2 white i + )} 1 + exp {κ j (β 1 age i + β 2 white i + )} exp {κ j 1 (β 1 age i + β 2 white i + )} 1 + exp {κ j 1 (β 1 age i + β 2 white i + )} 2. 30 Workfile Sample OK Sample range pairs @all * 15 IF condition engabil => 1 and age >= 30 3. Quick - Estimate Method ORDERED *15 19/25

2 2.6 4. Equation specification Error Distribution engabil age white black asian earns educ1 educ2 educ3 educ4 educ5 citizen0 msp1 msp2 msp3 msp4 msp5 * 16 Logit *16 20/25

2 2.6 5. 21/25

2 2.7 6. View - Prediction Evaluation 2.7 1. engabil age white black asian earns educ1 educ2 educ3 educ4 educ5 citizen0 msp1 msp2 msp3 msp4 msp5 30 P {engbil i = j age i, white i } = κj (β 1age i+β 2white i+ ) κj 1 (β 1age i+β 2white i+ ) 1 2π e t2 2 dt 2. 1 2π e t2 2 dt 3. Quick - Estimate Method ORDERED 22/25

2 2.7 4. Equation specification Error Distribution engabil age white black asian earns educ1 educ2 educ3 educ4 educ5 citizen0 msp1 msp2 msp3 msp4 msp5 * 17 Normal *17 23/25

2 2.7 5. 24/25

2 2.7 6. View - Prediction Evaluation 25/25