day 2 review of regression ols1

Economics 20 - Prof. Anderson 1

M hnh hi qui n gin

y = 0 + 1x + u


D bo s dng m hnh chui thigian(Time Series Models for Forecasting)

n tp phng php hi quiReview of Regression

Nguyn Ngc AnhTrung tm Nghin cu Chnh sch v Pht trin

Nguyn Vit Cngi hc Kinh t Quc dn


Hi qui l g?

L mt cng c quan trng nht ca cc nhnghin cu kinh tHi qui l phng php m t v nh gi miquan h gia mt bin (gi l bin ph thuc, thng k hiu l y) vi mt hay nhiu bin khc(gi l bin c lp, x1, x2, ... , xk )


So snh hi qui v tng quan

Trong quan h tng quan, hai bin y v xl tng ng nhau. Trong m hnh hi qui, chng ta coi binc lp v bin ph thuc l hon ton khcnhau. Bin y c gi thit l c tnh ngunhin, cn bin x c gi thit l c nh(nhn gi tr c nh)


So snh hi qui v tng quan

M hnh hi qui cho php chng ta clng (estimate) v suy din thng k(inferences) cc tham s ca tng th.Trong kinh t lng, mc tiu ca chng tal c lng tc ng nhn qu ca vic X thay i mt n v i vi Y.


Nu so snh, th gic tng quan vic clng m hnh hi qui cng ging nh clng con s trung bnh. Trong m hnh hi qui, vic suy din thng k bao gm cc vic sauc lng (Estimation): Lm th no c lng

Kim nh gi thuyt (Hypothesis testing): Tham s c lng c c khc 0 hay khng?

Xy dng khong tin cy : Xy dng khong tin cy cho tham s c c

lng

M hnh hi qui n gin


M hnh hi qui n gin

M hnh ch bao gm mt bin c lp k=1. Trongm hnh ny bin y ch ph thuc vo mt bin xM hnh c th c nhiu bin x, nhng ta s xttrng hp ny sau. M hnh hi qui n gin c ths dng trong mt s trng hp : Lm pht v tht nghip Li nhun ca chng khon quan h th no vi

ri ro M phng quan h gia gi chng khon v c tc


M hnh hi qui n gin : V d Gi s ta c s liu nh :

Chng ta mun tm hiu mi quan h gia x v y

Year, t Excess return= rXXX,t rft

Excess return on market index= rmt - rft

1 17.8 13.72 39.0 23.23 12.8 6.94 24.2 16.85 17.2 12.3


Biu ri rc

0

5

10

15

20

25

30

35

40

45

0 5 10 15 20 25

Excess return on market portfolio

E

x

c

e

s

s

r

e

t

u

r

n

o

n

f

u

n

d

X

X

X


Tm ng ph hp nhtChng ta c th s dng phng trnh

y= + x c lng ng thng tt nht. l dc ca ng thngng thng ny cn gi l ng hi qui ca tng th (population regression line)Ta khng bit v , nn phi c lngng thng nh vy hon ton mang tnhxc nh (deterministic) c hp l khng?


Mt s k hiu v thut ng

Vit dng tng qut hn, vi m hnh hi qui tuyn tnh gin n, ta c y = + x+ u, y c gi l m hnh hi qui tuyn tnh catng thChng ta thng gi y l bin ph thuc v x lbin c lp/bin kim soat. l intercept, l slope ( dc)u l sai s ca ng hi qui tng th


Ti sao li c sai s u

- Chng ta c th b st nhng yu t c tc ngn yt- Vic o lng/ghi nhn s liu i vi bin s yt cth c sai- Nhng tc ng ngu nhin i vi bin s yt mchng ta khng th m hnh ha c


Biu din m hnh trn bng hnh nh


Mt s gi thit

Trung bnh ca cc sai s trong m hnhhi qui bng 0.

E(u) = 0 y khng phi l mt gi thit qu nngn, do chng tao lun c th dng chun ha trung bnh/k vng ton ca u, E(u) v khng.


Gi thit ca m hnh hi qui

Chng ta cn phi a ra gi thit v miquan h gia u v xChng ta mun gi thit rng, nhng thngtin m chng ta bit v x s khng chochng ta bit g v u, v nh vy, u v x lhon ton khng c quan h vi nhauE(u|x) = E(u) = 0, v iu ny dn tiE(y|x) = 0 + 1x


E(u|x) = E(u) = 0


Phng php bnh phng cc tiu

tng c bn ca vic hi qui l clng cc tham s ca tng th trn c smt mu s liuGi {(xi,yi): i=1, ,n} l mt mu ngunhin, c c l n m ta thu c t tng thVi mi quan st trong mu ny, ta s cyi = + xi + ui


.

..

.

y4

y1

y2y3

x1 x2 x3 x4

}

}

{

{

u1

u2

u3

u4

x

y E(y|x) = + x

ng hi qui ca tng th, im s liuv cc sai s


c lng vi phng php bnhphng cc tiu

c lng vi phng php bnh phng cctiu, chng ta cn thy rng, gi thit chnh cachng ta l E(u|x) = E(u) = 0, v iu ny cngha l

Cov(x,u) = E(xu) = 0

Ti sao? T l thuyt c bn v xc sut ta cCov(X,Y) = E(XY) E(X)E(Y)



Vi tng l tm ng ph hp nht, chng tac th xy dng bi ton cc tiuTc l chng ta mun tm cc tham s sao cho

biu thc di y t gi tr cc tiu :


.

==t

tt xyL 0)(2

==t

ttt xyxL 0)(2

( ) ( )==

+=n

iii

n

ii xyu

1

2

1

2 )(



Chng ta c th s dng o hm gii bi ton cc tiuny, chng ta nu ly o hm bc 1, theo va v gii ccphng trnh thu. Qua ta c th c lng c cc thams ca m hnh hi qui.

SXY = ng phng sai ca (X, Y)SX2 = phng sai ca (X)

2

12

1

)(

)()(X

XYN

i i

N

i ii

SS

XX

YYXX ==

=

=XY =


Tm tt v c lng tham s beta (slope estimate)

c lng v dc l ng phng saitnh trn mu gia y v x, chia cho phngsai mu ca x.Nu x v y c tng quan thun (dng) vinhau, th c lng c du dngNu x v y c tng quan nghch (m) vi

nhau, th c lng c du mChng ta ch cn x bin thin trong


OLS

V mt trc gic, OLS l vic c lng ngthng qua cc im s liu trong mu sao cho tngkhong cch bnh phng sai s l nh nht, nnc tn l bnh phng cc tiu. Sai s, , chnh l c lng cho sai s u v l ssai khc gia ng c lng (ng hi qui trn mu) v cc im s liu.


.

..

.

y4

y1

y2y3

x1 x2 x3 x4

}

}

{

{

1

2

3

4

x

y

xy 10 +=

ng hi qui mu, im s liuv cc sai s c lng


ng hi qui tng th l m hnh m chng ta chorng to ra s liu, v cc tham s thc l v .Hi qui tng thHi qui muv chng ta bit rng .

Chng ta s dng ng hi qui mu suy din vng m hnh ca tng th

Chng ta cng mun bit l cc c lng v c phi l cc c lng tt hay khng

tt xy +=ttt uxy ++=

ttt yyu =


Tnh cht ca OLS

Tng cc sai s (residual) OLS l bng 0Nh vy, trung bnh mu cc sai s OLS

cng bng 0ng phng sai mu gia cc bin c

lp v sai s OLS cng bng 0ng OLS s chy xuyn qua im trungbnh ca s liu


Biu din bng i s, ta c

xy

ux

n

uu

n

iii

n

iin

ii

10

1

1

1

0

0

thus,and 0

+==

==

=

==


Tnh cht ca c lng OLS

Tuyn tnh (linear)

Khng trch (unbiased)

Hiu qu nht (best)

Best Linear Unbiased Estimator

$ $

$$

$

$

$


S dng STATA c lng OLS

Thc hin hi qui trong STATA rt ginn. V c lng m hnh hi qui y theo x th ta ch cn nh lnhreg y x


c lng s dng STATA

regress testscr str, robust

Regression with robust standard errors Number of obs = 420 F( 1, 418) = 19.26 Prob > F = 0.0000 R-squared = 0.0512 Root MSE = 18.581 ------------------------------------------------------------------------- | Robust testscr | Coef. Std. Err. t P>|t| [95% Conf. Interval] --------+---------------------------------------------------------------- str | -2.279808 .5194892 -4.39 0.000 -3.300945 -1.258671 _cons | 698.933 10.36436 67.44 0.000 678.5602 719.3057 -------------------------------------------------------------------------


Mc ph hp ca m hnh(Goodness-of-Fit)

( )( )

SSR SSE SST co Ta(SSR)du con phuongbinh Phan tng

(SSE) thich giai duoc phuongbinh Phn tng

(SST)cach khoang phuongbinh Tng

:saunhu nghiainh somt co se taChng : thichgiai duoc khngphn thich vgiai duocPhn

phn 2 c gm lst quan mi coi thc taChng

2

2

2

+=

+=

i

i

i

iii

u

yy

yy

uyy


Chng minh rng SST = SSE + SSR

( ) ( ) ( )[ ]( )[ ]

( ) ( )( )

( )

=++=

++=+=

+=

0 rangbit tav

SSE 2 SSR

2

22

2

22

yyu

yyu

yyyyuu

yyu

yyyyyy

ii

ii

iiii

ii

iiii


Mc ph hp ca m hnh(Goodness-of-Fit)

Chng ta nh gi th no v ng hi qui m tac lng? C ph hp vi s liu hay khng?

C th tnh t l tng bnh phng khong cch(SST) c gii thch bi m hnh, v gi t lny l R-bnh phng ca m hnh hi qui.

R2 = SSE/SST = 1 SSR/SSTNm trong khong 0-1. Cng ln cng tt!!!!


Phn phi mu ca c lng OLS

c lng OLS c tnh ton da trn mt mu s liu, mt mu s liu khc s cho ta mt gi tr khc ca 1 . y c gi l tnh bt nh theo mu ca 1 . Chng ta mun nh gi mc bt nh ca 1 S dng 1 tin hnh kim nh gi thuyt nh 1 = 0 Xy dng khong tin cy cho 1

Tt c nhng iu ny i hi chng ta phi xem xt ti phn phi mu (sampling distribution) ca c lng OLS. lm c iu ny, ta phi xem xt Phn phi ca c lng OLS


Hm phn phi ca 1Cng ging nh trung bnh mu, Y , 1 cng c phn phi mu . Vy k vng ton ca E( 1 ) l bao nhiu

Nu nh E( 1 ) = 1, th c lng OLS l c lng khng trch Cn mun g hn?!

Phng sai ca 1 - var( 1 )? (cho chng ta bit c mc bt nh ca c lng)

Phn phi ca 1 trong cc mu nh l phn phi g ? Vn ny rt kh!!!!!

Phn phi ca 1 cc mu ln l phn phi g ? Vi cc mu ln, 1 c phn b l phn b chun

(normally distributed).


Tnh khng trch ca OLS (Unbiasedness)

Gi thit rng m hnh tng th l tuyntnh theo tham s c dng y = 0 + 1x + uGi thit rng chng ta s dng mt mu cqui m n, {(xi, yi): i=1, 2, , n}, c lyt m hnh tng th. Nh vy ta c th biudin m hnh mu l yi = 0 + 1xi + uiGi thit E(u|x) = 0 v nh vy E(ui|xi) = 0Gi thit rng xi c bin thin



xt tnh khng trch ca c lng, chng tavit li di dng tham s ca tng th. Vit mt cng thc ngin l

( )( )

=22

21 where,

xxs

syxx

ix

x

ii



( ) ( )( )( ) ( )( )( ) ( )( ) ii

iii

ii

iii

iiiii

uxx

xxxxx

uxx

xxxxx

uxxxyxx

++

=++

=++=

10

10

10



( )( ) ( )

( )( )

211

21

2

thusand ,asrewritten becan numerator the,so

,0

x

ii

iix

iii

i

suxx

uxxs

xxxxx

xx

+=+

==



( )

( ) ( ) 121121

1

then,1

thatso ,let

=

+=

+==

iix

iix

i

ii

uEdsE

uds

xxd



Cc c lng OLS ca tham s 1 v 0 lkhng trchVic chng minh tnh khng trch, da trn 04

gi thit. Nu mt gi thit m khng ng, thc lng OLS s khng phi l khng trchLu rng, tnh khng trch l tnh cht ca phpc lng (estimator) cn trong mt mu c th, th c lng thu c c th nhiu t khc vitham s thc t


Phng sai ca c lng OLS

Chng ta bit rng hm phn b (sampling distribution) ca c lng nm xungquanh tham s thcMun bit xem hm phn b ny c phn tn nh th noa thm mt gi thit na v phng saiGi thit l Var(u|x) = 2(Homoskedasticity)



Var(u|x) = E(u2|x)-[E(u|x)]2

E(u|x) = 0, so 2 = E(u2|x) = E(u2) = Var(u)Nh vy, 2 cng l phng sai khng iukin, v c gi phng sai ca sai s, c gi l sai s chun ca sai sC th ni rng : E(y|x)=0 + 1x vVar(y|x) = 2


..

x1 x2

Trng hp phng sai ng nht(Homoskedastic)

E(y|x) = 0 + 1x

y

f(y|x)


.xx1 x2

yf(y|x)

Phng sai khng ng nht(Heteroskedastic)

x3

. . E(y|x) = 0 + 1x



( )( ) ( )

( )12222222

2

2222

2

2

22

2

2

2

211

1

11

11

1

Varsss

dsds

uVardsudVars

udsVarVar

xx

x

ix

ix

iix

iix

iix

==

=

=

=

=

=

+=



Phng sai ca sai s, 2 cng ln, thphng sai ca c lng cng ln

xi bin thin cng nhiu, th phng saica c lng cng nhDo , mu ln s lm gim phng saica c lngVn l phng sai ca sai s chng ta

li khng bit


c lng phng sai ca sai s

Chng ta khng bit phng sai ca sai, 2, ca sai s l bao nhiu v chng ta khngquan st c sai s, ui

Chng ta ch quan st c , i

Chng ta c th s dng i c lngphng sai ca sai s



( )( ) ( )

( ) ( )2/21

is ofestimator unbiasedan Then,

22

21100

1010

10

==

=++=

=

nSSRun

u

xux

xyu

i

i

iii

iii



( )

( ) ( )( ) 21211

2

/se

, oferror standard the

have then wefor substitute weif

sd that recall

regression theoferror Standard

=

===

xx

s

i

x


Tm tt v phn phi mu ca 1Nu cc gi thit ca OLS l ng th Hm phn phi mu ca 1 c:

E( 1 ) = 1 (tc l, 1 l c lng khng trch) var( 1 ) = 4var[( ) ]1 i x i

X

X un

1

n.

Khi mu ln , 1 11

( )var( )

E

~ N(0,1) (CLT)

.


Kim nh gi thuyt v sai s chun ca 1

Mc tiu ca vic kim nh trong m hnh hi qui l s dng s liu kim nh mt gi thuyt v tng th nh 1 = 0, v a ra kt lun liu gi thuyt c ng hay khng

Gi thuyt trng v gi thuyt thay th hai pha H0: 1 = 1,0 vs. H1: 1 1,0

Trong 1,0 l mt gi tr gi thuyt

Gi thuyt trng v gi thuyt thay th mt pha : H0: 1 = 1,0 vs. H1: 1 < 1,0


Phng php kim nh: Xy dng thng k t hoc z, tnh p-value, hoc so snh vi gi tr ti hn ca hm phn phi N(0,1)) Ni chng ta c: t = (c lng gi tr mun kim nh)/sai s chun ca c

lng

Khi kim nh v trung bnh ca Y: ta c t = ,0/

Y

Y

Ys n

Khi kim nh 1, ta c t = 1 1,01

( )SE

,


Cng thc tnh SE( )1 1

2 =

2

2 2

1 estimator of (estimator of )

v

Xn =

2

12

2

1

1 1 2

1 ( )

n

ii

n

ii

vn

nX X

n

=

=

Trong iv = ( )i iX X u . OK. SE( 1 ) trng phc tp, nhng STATA tnh rt nhanh v

ta khng phi nh cc cng thc ny .


V dc lng ca m hnh hi qui: Test score = 698.9 2.28STR STATA cng cho ta c lng lch chun ca con s c lng l

SE( 0 ) = 10.4 SE( 1 ) = 0.52

Ta c th tnh cc kim nh thng k cho 1 vi gi thuyt Ho: 1,0 = 0 t-statistic testing 1,0 = 0 = 1 1,0

1

( )SE

= 2.28 0

0.52 = 4.38

mc ngha 1% gi tr l 2.58, nn ta c th bc b gi thuyt trng vi mc ngha 1%.

Ta cng c th tnh gi tr p-value . Nhng STATA lm h ht ri !


c lng s dng STATA

regress testscr str, robust

Regression with robust standard errors Number of obs = 420 F( 1, 418) = 19.26 Prob > F = 0.0000 R-squared = 0.0512 Root MSE = 18.581 ------------------------------------------------------------------------- | Robust testscr | Coef. Std. Err. t P>|t| [95% Conf. Interval] --------+---------------------------------------------------------------- str | -2.279808 .5194892 -4.39 0.000 -3.300945 -1.258671 _cons | 698.933 10.36436 67.44 0.000 678.5602 719.3057 -------------------------------------------------------------------------


Tm tt: kim nh H0: 1 = 1,0 v. H1: 1 1,0,

Tnh kim nh thng k t (t-statistic) t = 1 1,0

1

( )SE

=

1

1 1,0

2

Bc b gi thuyt trng vi mc ngha 5% nu |t| > 1.96 Bc b gi thuyt trng nu p


c kt qu STATA regress testscr str, robust

Regression with robust standard errors Number of obs = 420 F( 1, 418) = 19.26 Prob > F = 0.0000 R-squared = 0.0512 Root MSE = 18.581 ------------------------------------------------------------------------- | Robust testscr | Coef. Std. Err. t P>|t| [95% Conf. Interval] --------+---------------------------------------------------------------- str | -2.279808 .5194892 -4.38 0.000 -3.300945 -1.258671 _cons | 698.933 10.36436 67.44 0.000 678.5602 719.3057 ------------------------------------------------------------------------- Y = 698.9 2.28STR, , R2 = .05,

(10.4) (0.52) t (1 = 0) = 4.38, p-value = 0.000 (2-sided) Khong tin cy 95% ca 1 l (3.30, 1.26)

M hnh hi qui n ginD bo s dng m hnh chui thi gian (Time Series Models for Forecasting) n tp phng php hi qui Review of RegressioHi qui l g? So snh hi qui v tng quan So snh hi qui v tng quanM hnh hi qui n ginM hnh hi qui n gin : V d Biu ri rcTm ng ph hp nht Mt s k hiu v thut ngTi sao li c sai s u Biu din m hnh trn bng hnh nh Mt s gi thitGi thit ca m hnh hi qui E(u|x) = E(u) = 0Phng php bnh phng cc tiu c lng vi phng php bnh phng cc tiuc lng vi phng php bnh phng cc tiuc lng vi phng php bnh phng cc tiuTm tt v c lng tham s beta (slope estimate) OLSTnh cht ca OLS Biu din bng i s, ta cTnh cht ca c lng OLS S dng STATA c lng OLS c lng s dng STATA Mc ph hp ca m hnh (Goodness-of-Fit)Chng minh rng SST = SSE + SSRMc ph hp ca m hnh (Goodness-of-Fit)Phn phi mu ca c lng OLSHm phn phi ca Tnh khng trch ca OLS (Unbiasedness)Tnh khng trch ca OLS (Unbiasedness)Tnh khng trch ca OLS (Unbiasedness)Tnh khng trch ca OLS (Unbiasedness)Tnh khng trch ca OLS (Unbiasedness)Tnh khng trch ca OLS (Unbiasedness)Phng sai ca c lng OLS Phng sai ca c lng OLSPhng sai ca c lng OLSPhng sai ca c lng OLSc lng phng sai ca sai s c lng phng sai ca sai sc lng phng sai ca sai sTm tt v phn phi mu ca Kim nh gi thuyt v sai s chun ca Cng thc tnh SE( ) V d c lng s dng STATA Tm tt: kim nh H0: 1 = 1,0 v. H1: 1 1,0, c kt qu STATA

day 2 review of regression ols1

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