1 Adjusted Parameters

Transcription

1 1 Adjued Parameer Here, we li he exac calculaion we made o arrive a our adjued parameer Thee adjumen are made for each ieraion of he Gibb ampler, for each chain of he MCMC The regional memberhip of each ae are aken from he US Cenu webie, and are lied uing e noaion: R 1 = (7, 19, 21, 29, 30, 32, 38, 39, 45), R 2 = (1, 4, 8, 9, 10, 17, 18, 20, 24, 33, 36, 40, 42, 43, 46, 48, 51), R 3 = (13, 14, 15, 16, 22, 23, 25, 27, 34, 35, 41, 49), R 4 = (2, 3, 5, 6, 11, 12, 26, 28, 31, 37, 44, 47, 50), (Norh) (Souh) (Midwe) (We) where hee number refer o he indice of he ae a hey are lied alphabeically by heir full name, wih he Diric of Columbia lied la The number of ae in each region i 9, 17, 12, and 13, repecively We alo wrie he mapping of ae o region uing he vecor W = (2, 4, 4, 2, 4, 4, 1,, 3, 4, 2) There are wo ype of noaion ued here The fir i he convenional ue of a do a a ubcrip o denoe an average over one dimenion of an array For example, uppoe x i an N M array Then x j = 1 N x i = 1 M x = N x ij, 1 NM M x ij, j=1 We define anoher ype of noaion o denoe an average over a ube of indice in an array In hi cae, we ue parenhee around one index in a ubcrip inead of a do, and for hi paper, he number inide he parenhee i alway a 1, 2, 3, or 4, o denoe an average of he ube of ae in one of he four region So, for example, he average of he ae-year ineracion effec in year for ae in region 1 (Norh) i N M j=1 x ij α ae-year (1) = 1 R 1 α ae-year R R 1 1i, (1) 1

2 which i an average of R 1 = 9 value Thi noaion i ued wherever here i an average over a ube of ae, which i ofen, becaue of he many inance in which ae-indexed parameer are cenered around heir regional mean A key ep in compuing adjued parameer i compuing he finie-populaion lope for ae Fir, define he cenered ae-year effec α ae-year = α ae-year α ae-year α ae-year + α ae-year Nex, define he finie-populaion ae lope a he eimaed lope from regreing a given ae cenered ae-year effec on X year : ˆδ ae = =1 Xyear α ae-year =1 (Xyear ) 2 Now, he adjued parameer are defined a follow, where he adjued parameer are on he lef had ide of hee equaion, and are denoed wih a prime ymbol ( ): µ = α ae-year + α age-ae + α degree-ae, α ae-year = α ae-year α ae-year α ae-year + α ae-year α ae = α ae-year α ae-year () + α age-ae α age-ae () +α degree-ae α degree-ae () (βx ae βx ae ()), αr region = α ae-year (r) α ae-year + α age-ae α age-ae +α degree-ae α degree-ae βx ae (r), α year = α ae-year α ae-year (µ δ δ age-ae )X year, ae ˆδ X year, α degree d = α degree-ae d α degree-ae α degree-ae d = α degree-ae d α degree-ae d() α degree-ae + α degree-ae () α degree-region dr = α degree-ae d(r) α degree-ae d α degree-ae + α degree-ae αa age = αa age-ae α age-ae αa age-ae = αa age-ae α age-ae a() α age-ae + α age-ae () αar age-region = α age-ae a(r) αa age-ae α age-ae + α age-ae 2

3 β black = β black-ae β black-ae = β black-ae β black-ae (), βr black-region = β(r) black-ae β black-ae, β female = β female-ae β female-ae = β female-ae β female-ae (), βr female-region = β(r) female-ae β female-ae, β black-female = β black-female-ae β black-female-ae = β black-female-ae β black-female-ae (), βr black-female-region = β(r) black-female-ae β black-female-ae, µ δ = δ ae = δ region r = =1 Xyear (α ae-year ae ˆδ ae ˆδ (r) α ae-year + δ age-ae =1 (Xyear ) 2, ae ˆδ () (γz ae γz ae ()) age-ae δ age-ae γz ae (r) + δ age-ae ) δ age-ae δ age a = δ age-ae a δ age-ae δa age-ae = δa age-ae δ age-ae a() δ age-ae + δ age-ae () δar age-region = δ age-ae a(r) δa age-ae δ age-ae + δ age-ae 3

4 δ black = δ black-ae δ black-ae = δ black-ae δ black-ae (), δr black-region = δ(r) black-ae δ black-ae, δ female = δ female-ae δ female-ae = δ female-ae δ female-ae (), δr female-region = δ(r) female-ae δ female-ae, δ black-female = δ black-female-ae δ black-female-ae = δ black-female-ae δ black-female-ae (), δr black-female-region = δ(r) black-female-ae δ black-female-ae, 4

5 Wih adjued parameer defined hi way, we have decompoed he mean of he log odd of deah penaly uppor by individual i ino i componen par, uch ha he righ hand ide of he likelihood equaion in Equaion 1 i equal o: µ + α year + α region + α ae + βx ae + α ae-year + α degree d + α age a + (δ age a + X year µ δ (Naional average yearly effec) + δ region X year ae X year + γz ae X year + α degree-region d + α age-region a + δ age-region a + (β black + β black-region + (β female + β female-region + α degree-ae d + α age-ae a + δa age-ae )X year + (β black-female + β black-female-region + (δ black + δ black-region + (δ female + δ female-region + (δ black-female + δ black-female-region (Regional rend) (Sae rend) (Sae-level variable rend) (Sae-year ineracion effec) (Degree effec) (Age effec) (Age lope) + β black-ae )X black (Black inercep) + β female-ae )X female (Female inercep) + β black-female-ae )X black-female (Black-female ineracion inercep) black-ae )X black X year female-ae )X female X year (Black lope) (Female lope) black-female-ae )X black-female X year (Black-female ineracion lope) 5