TERM
Date. ☐
Weeh I: Simple Linear Megression Mode!
= Detinstion of Simple lincar Rigression (SUR) Modas
fromework used to study how a dependint varcable ly) vailes
with changes in on indipen dint variable lod within speritic popula tios
model is detined by follouing population equatien
y = Bo 1 Brz 1u
.y= dependint varcabи
t = indepen dint variable, esplenctog vrinble used to study veriatrenan
Bo intirapt paramiler, oles ot y when x=0
Bi= slops paramitue; repisseat the citeris paribus lallelre gve1)
eltect ot o on y. It indicots espentid Dy for one und thenge
u= erro, tum; uccounts tor ell unobserred tactors otherthen x
thet ntlmna y
SUR modl allows other tactors contained in u to uffect y
it addresses functienel form irsue : y ussumed to be lwaly
☑
Hluledto 7
☑
it also addresses ateris paribes 1suки
05 B. ox t sи -72 adn
y:
=
sourcas of rorcetin in
Holding all othe factors fired Cou=0(
ateris poribus eltect
-Erumph (wage & Educntion)
. stady hoow education intlwna wage
wage= Bo t B. educt u
whert it contuins unobserred tactors eig. ubility, expericna, wrent job
•aleris parikus effact ot educ on wages
Swoge Pi seduc whin su=a
However, taactiunal torm assumptios implies that another year
of edacotion is worth the semm dilloi emount no mether how
mach iducation starts with. Clinearity issue)
•In reality, this may net be rralistic as relucas to edacatiea
otlea vary leig. finishing digree extra yea of pumay schoel
,Date. 1
=) SUR AS model
conditienal distribution of y given x
tor
- onalysing how voles of dependint voriable ly)pe spreed out tor
specitrc fixed volws ot indipindint variobu (2)
y
ナフス
えて 23
- Stwight line represent fot B; condekenal diskibution oty
ot 3 dilt vols ot & aresuperimposed
• it we cordition on x os, me looking ot the shie of the erpatation with e
-distribution in y for cach slica is de to varcetion inu for
this particular sha
- For ony specitic sliu of populrtion whire x is constent,the vaicetion
we see in y entirely caused by unebserved factors contained willeen
is u
tor that paticula slice
disabutun of u doesn't choree
- Exomple (uond z indipendint) regrdless of rolur x
・under indipendinc, probekility of y =yo1x=s is girinkg
Ply=yo1x=S) = P(u=yo- Bo-kis) hdmis
•suppose P(u=1) = Plu= 1)=AS,then E discee distibntion
PC y = Bo t BiSt1 1x=5) =PLu=1) = 0.5
Ply= BotfiS-1 1x= 5) = P(u=-1)= 0.5
• it u follows a normal distibution unNC0, 6²) ,then
Ly cendilional distribution of y is also mormal
ylx =5 ~N (BolBIS, 6)
Chth,5t
o this mmons y is uabred exuctly on the pupulation reguession line
with a spiead detined by the varcam of unokserved (62)
, Date./
= Restricting dipendina between a ondn
- To identify B., wenied to restict dpendina bepnr поирк
Without resticticas, we connct distinguish eftect ot a trom ticts
ot unokservad facters
Potential Assumptioas
Assumptioл Descriptren Pros Ilins
Full Indpendna u ond a completly independint ench otler overly strong and unndide
Uncoralated Co Ca,u)=0 Work for mony bul
Ilpatetica bostats pualyss
Mean- independna Elu12) =o tor all valus ofe mean stay constont buat
wand alloned to 4e
Le ElulxJ = Elu) =7 Elu)=0 dy to eutenn
ot Bo intrор
57Elulc0)=0 mutually depeadint
- Mean - 1ndipendiına
•the assumption requiras that the average of nobrened tactors Lu)
sleys cunstont lot wo) tor cach slice ot popalaticn doted byz
.while the mmon must be trd ,u and x can still be mutuelly depeadent
inother ways eg. Vor Culx) is permitled to chonge as a changes
eo conditinal meon assumption CELul2)=0) allew us to dotice
Perntation Legussion Function (PEFJ end idintity tu k.
-Erampl lwage and educ)
•Suppse u= abilily 7c= educ, El u l z= 8J = E (u / x 12) = (ulz=16) =
so thu ary obility is same in dilt portrens of pepulatien with 8, 7,16 yea
ot eduwt1on
•Becwuse peopl thouse edac leit! partly baed on abihty,this assumptin olmst false
Suppuse inrtiad that o distutr pandnly assign years of educ regordlus ot
ability, then ablity ond educ will be mutually Incpendrat.
I meon- incpendena is satistied
=7 Populatnn Rgrersion Fonctica CPRE ええんさ
- PRF tells the arg my for all n whe achikut a specitie 2.
- Beconse expected rale is linear opecator, Elulx)=O implles
Ely lx)= E CBotB.ctulx) = fot Bix +ECulx)e Bo thiz
- AElylas = B,Dx Bi intirpritid as the epeetd raspond ngto Oinz
, Date.
Deriving Ordinorg last Square Estimats
- to estimete popalation paometirs Po cAd B.,we need sompl of r
lit langg.):is 1,2.. m} be a somple sien It obseruptio0) ler1
Think as rondom samy
- Plug ony obs to pep egn: yi = B.t F,xitui is1,2
We obsernt x. ond yi but not ui Chowever; we loow vi is there
Similarly, values of B. ond k. unknoun to us, but exs!
Moment conditicas in popalation (population assumptiaas)
•Zero conditicnel mean assumption ECulx)co implud.
① ECu) =o: this condilicn ditiae thiintrupt Bo. On ang, unrbsened facde coaas
ECux)=0: mems x ond u uncoulaud.Hensurds enplanatiy varia bи
Clalam) =Eluz)
not relaled to unobsevnd facto!
u= y-6o-Bix inlo egns
Elyß・ Bia) -。 ECてしュープェー月スノ二〇
Thise 2 conditions in pop deleine Boand Bo, bul pop duslrikatıon nkmou
Üse sarpu analogs which is a muthad of moments appruach to slind
亢意新y-声。-食え1二0 属二产房庄和意工八如去意如
These are called ordinong lart sguae eslimo tes.
Daly valid wher Eizi (ri-)²70, whin zizz rales out
15-153で-127:2
房二ら一层え
3 (2-2/2
OR
Pay corkloon 655.d
Why isit rulled Ous islimaus:
DB.eB, diline fitted volar for y when 2=zi as gi= Bitkx
Residaal for
or any i ú; yi-yi = Ai -fo -fizi and h ae nresidas
Mirimise sum ot sgueered pisduals Diaa lgi-Fr2112
Elu)=0 Elux)= aM the first order cunditions ok ocs
EL.087700 €(alg-thu ha))= men independen osney tion strate obsestioneer
Date. ☐
Weeh I: Simple Linear Megression Mode!
= Detinstion of Simple lincar Rigression (SUR) Modas
fromework used to study how a dependint varcable ly) vailes
with changes in on indipen dint variable lod within speritic popula tios
model is detined by follouing population equatien
y = Bo 1 Brz 1u
.y= dependint varcabи
t = indepen dint variable, esplenctog vrinble used to study veriatrenan
Bo intirapt paramiler, oles ot y when x=0
Bi= slops paramitue; repisseat the citeris paribus lallelre gve1)
eltect ot o on y. It indicots espentid Dy for one und thenge
u= erro, tum; uccounts tor ell unobserred tactors otherthen x
thet ntlmna y
SUR modl allows other tactors contained in u to uffect y
it addresses functienel form irsue : y ussumed to be lwaly
☑
Hluledto 7
☑
it also addresses ateris paribes 1suки
05 B. ox t sи -72 adn
y:
=
sourcas of rorcetin in
Holding all othe factors fired Cou=0(
ateris poribus eltect
-Erumph (wage & Educntion)
. stady hoow education intlwna wage
wage= Bo t B. educt u
whert it contuins unobserred tactors eig. ubility, expericna, wrent job
•aleris parikus effact ot educ on wages
Swoge Pi seduc whin su=a
However, taactiunal torm assumptios implies that another year
of edacotion is worth the semm dilloi emount no mether how
mach iducation starts with. Clinearity issue)
•In reality, this may net be rralistic as relucas to edacatiea
otlea vary leig. finishing digree extra yea of pumay schoel
,Date. 1
=) SUR AS model
conditienal distribution of y given x
tor
- onalysing how voles of dependint voriable ly)pe spreed out tor
specitrc fixed volws ot indipindint variobu (2)
y
ナフス
えて 23
- Stwight line represent fot B; condekenal diskibution oty
ot 3 dilt vols ot & aresuperimposed
• it we cordition on x os, me looking ot the shie of the erpatation with e
-distribution in y for cach slica is de to varcetion inu for
this particular sha
- For ony specitic sliu of populrtion whire x is constent,the vaicetion
we see in y entirely caused by unebserved factors contained willeen
is u
tor that paticula slice
disabutun of u doesn't choree
- Exomple (uond z indipendint) regrdless of rolur x
・under indipendinc, probekility of y =yo1x=s is girinkg
Ply=yo1x=S) = P(u=yo- Bo-kis) hdmis
•suppose P(u=1) = Plu= 1)=AS,then E discee distibntion
PC y = Bo t BiSt1 1x=5) =PLu=1) = 0.5
Ply= BotfiS-1 1x= 5) = P(u=-1)= 0.5
• it u follows a normal distibution unNC0, 6²) ,then
Ly cendilional distribution of y is also mormal
ylx =5 ~N (BolBIS, 6)
Chth,5t
o this mmons y is uabred exuctly on the pupulation reguession line
with a spiead detined by the varcam of unokserved (62)
, Date./
= Restricting dipendina between a ondn
- To identify B., wenied to restict dpendina bepnr поирк
Without resticticas, we connct distinguish eftect ot a trom ticts
ot unokservad facters
Potential Assumptioas
Assumptioл Descriptren Pros Ilins
Full Indpendna u ond a completly independint ench otler overly strong and unndide
Uncoralated Co Ca,u)=0 Work for mony bul
Ilpatetica bostats pualyss
Mean- independna Elu12) =o tor all valus ofe mean stay constont buat
wand alloned to 4e
Le ElulxJ = Elu) =7 Elu)=0 dy to eutenn
ot Bo intrор
57Elulc0)=0 mutually depeadint
- Mean - 1ndipendiına
•the assumption requiras that the average of nobrened tactors Lu)
sleys cunstont lot wo) tor cach slice ot popalaticn doted byz
.while the mmon must be trd ,u and x can still be mutuelly depeadent
inother ways eg. Vor Culx) is permitled to chonge as a changes
eo conditinal meon assumption CELul2)=0) allew us to dotice
Perntation Legussion Function (PEFJ end idintity tu k.
-Erampl lwage and educ)
•Suppse u= abilily 7c= educ, El u l z= 8J = E (u / x 12) = (ulz=16) =
so thu ary obility is same in dilt portrens of pepulatien with 8, 7,16 yea
ot eduwt1on
•Becwuse peopl thouse edac leit! partly baed on abihty,this assumptin olmst false
Suppuse inrtiad that o distutr pandnly assign years of educ regordlus ot
ability, then ablity ond educ will be mutually Incpendrat.
I meon- incpendena is satistied
=7 Populatnn Rgrersion Fonctica CPRE ええんさ
- PRF tells the arg my for all n whe achikut a specitie 2.
- Beconse expected rale is linear opecator, Elulx)=O implles
Ely lx)= E CBotB.ctulx) = fot Bix +ECulx)e Bo thiz
- AElylas = B,Dx Bi intirpritid as the epeetd raspond ngto Oinz
, Date.
Deriving Ordinorg last Square Estimats
- to estimete popalation paometirs Po cAd B.,we need sompl of r
lit langg.):is 1,2.. m} be a somple sien It obseruptio0) ler1
Think as rondom samy
- Plug ony obs to pep egn: yi = B.t F,xitui is1,2
We obsernt x. ond yi but not ui Chowever; we loow vi is there
Similarly, values of B. ond k. unknoun to us, but exs!
Moment conditicas in popalation (population assumptiaas)
•Zero conditicnel mean assumption ECulx)co implud.
① ECu) =o: this condilicn ditiae thiintrupt Bo. On ang, unrbsened facde coaas
ECux)=0: mems x ond u uncoulaud.Hensurds enplanatiy varia bи
Clalam) =Eluz)
not relaled to unobsevnd facto!
u= y-6o-Bix inlo egns
Elyß・ Bia) -。 ECてしュープェー月スノ二〇
Thise 2 conditions in pop deleine Boand Bo, bul pop duslrikatıon nkmou
Üse sarpu analogs which is a muthad of moments appruach to slind
亢意新y-声。-食え1二0 属二产房庄和意工八如去意如
These are called ordinong lart sguae eslimo tes.
Daly valid wher Eizi (ri-)²70, whin zizz rales out
15-153で-127:2
房二ら一层え
3 (2-2/2
OR
Pay corkloon 655.d
Why isit rulled Ous islimaus:
DB.eB, diline fitted volar for y when 2=zi as gi= Bitkx
Residaal for
or any i ú; yi-yi = Ai -fo -fizi and h ae nresidas
Mirimise sum ot sgueered pisduals Diaa lgi-Fr2112
Elu)=0 Elux)= aM the first order cunditions ok ocs
EL.087700 €(alg-thu ha))= men independen osney tion strate obsestioneer
