13 2018 7 27 1 / 31
1. 2. 2 / 31
y i = β 0 + β X x i + β Z z i + β XZ x i z i + u i, E(u i x i, z i ) = 0, E(u i u j x i, z i ) = 0 (i j), V(u i x i, z i ) = σ 2, i = 1, 2,, n x i z i 1 3 / 31
y i = β 0 + β X x i + β Z z i + β XZ x i z i + u i x i y i x i = β X + β XZ z i. x i x i 1 y i z i y i = β 0 + β X x i + u i y i = β 1 x i x i z i 4 / 31
β X β XZ 0 z i 0 β X > 0, β XZ > 0 z i = 0 z i β X > 0, β XZ < 0 zi = 0 z i β X < 0, β XZ > 0 zi = 0 z i β X < 0, β XZ < 0 zi = 0 x i 5 / 31
β X β XZ 0 x i z i y i 6 / 31
Stata Stata generate c. #c. c. # 2 3 generate 7 / 31
c i = β 0 + β Y y i + β M m i + β D d i + β YM y i m i + u i (1) c i : y i : m i : d i : 0 8 / 31
1 1. Stata 2. File Log Begin... 2018microdata1 lecture20180727.smcl 3. File Open 4. consumption2009.dta 9 / 31
5. Statistics Linear models and related Linear regression 6. Model Dependent variable: expenditure_th 7. Independent variables: income_th, deposit_th, female 8. Model Independent variable: c.income_th#c.deposit_th c. # 10 / 31
9. SE/Robust Standard error type: Robust White 10. Reporting Set table formats 11. Coef/SE/CI Decimal format 4 decimals 4 12. p-value Decimal format 4 decimals p 4 13. Test statistic Decimal format 2 decimals t 2 11 / 31
14. Format settings for coefficient tables OK 15. regress - Linear regression OK 12 / 31
1% H 0 H0 1 H0 H0 13 / 31
ln y i = β 0 + β 1 ln x i + u i, E(u i x i ) = 0, E(u i u j x i ) = 0 (i j), V(u i x i ) = σ 2, i = 1, 2,, n ln y i ln x i ln 1 14 / 31
β 1 ln y i ln x i β 1 = d ln y i d ln x i. ln xi ln x i 1 15 / 31
d ln y i d ln x i = d ln yi d ln x i = d ln y i dy i = d ln y i dy i dy i y i. dx i x i dy i dx i dy i dx i dx i d ln x i 1. d ln x i dx i d ln y i dy i = 1 y i, d ln x i dx i = 1 x i 16 / 31
d ln y i dy i dy i dx i 1 d ln x i dx i = 1 dy i 1 y i dx 1 i x i = 1 dy i y dx i i x i 1 = dy i y i = dy i y i. dx i x i dx i x i d ln y i d ln x i = dy i y i dx i x i 17 / 31
β 1 = d ln y i d ln x i = dy i y i. dx i x i dx i : x i x i dy i : y i y i dx i x i : x i x i dy i y i : y i y i 18 / 31
β 1 = dy i y i dx i x i β 1, = y i x i. dy i y i dx i x i x i 1% y i % y i x i elasticity y i x i e.g., β 1 x i 1% y i β 1 % 19 / 31
ln y i = β 0 + β 1 ln x i + u i, i = 1, 2,..., n ln x i β 1 y i x i β 1 y i x i e.g., β 1 OLS ˆβ 1 y i x i H 0 : β 1 = 0 vs H 1 : β 1 0 0 y i x i 20 / 31
level Level-Level Model: y i = β 0 + β 1 x i + u i β1 x i 1 y i β 1 Log-Log Model: ln y i = β 0 + β 1 ln x i + u i β 1 x i 1% y i β 1 % 100 β 1 % 21 / 31
Log-Level Model: ln y i = β 0 + β 1 x i + u i β1 x i 1 y i 100 β 1 % β 1 y i x i semi-elasticity Level-Log Model: y i = β 0 + β 1 ln x i + u i β1 x i 1% y i 1 100 β 1 22 / 31
Log-Log Model Log-Level Model Log-Log Model Level-Level Model R 2 R 2 23 / 31
Stata generate Command generate ( )=ln(( )) generate ln ( 0 0 0 0 24 / 31
ln c i = β 0 + β Y ln y i + β M ln m i + β D d i + u i (2) % d i 0 25 / 31
2 1. Command generate lnexpenditure=ln(expenditure) Enter 2. Command generate lnincome=ln(income) Enter 3. Command generate lndeposit=ln(deposit) Enter 4. 26 / 31
5. Command list prefecture male female expenditure lnexpenditure income lnincome deposit lndeposit Enter list more 6 2 27 / 31
6. Statistics Linear models and related Linear regression 7. Model Dependent variable: lnexpenditure 8. Independent variables: Independent variables: lnincome, lndeposit, female 9. OK 28 / 31
0.7197 1% H0 0.7197 1% 0.7197% 1 0.039 5% H0 0.7197 1% 0.039% 29 / 31
0.1246 1% H0 12.46% 30 / 31
3 1. File Log Close 2. Stata lecture20180727.smcl 2018microdata1 3. lecture20180727.smcl 31 / 31