karerbirás´m¨artiselikmrurwerh:ssüúglieneg pdf... · 2013. 4. 9. ·...

saklviTüal&yPUminÞPñMeBj ROYAL UNIVERSITY OF PHNOM PENH

edá̈tWm¨ǵKNitviTüa 262 MATHEMATICS DEPARTMENT

CMBUken¼bgHajBIKMrUrWERh:ssüúglIenEG‘rkøasik EdlCab´TaḱTgnwg k Gefr (Y nig X2, X3, ..., Xk)

CaTMrg´mäRTIs . KMrU k Gefr CakarBRgIkd¾smrmüBIKMrUBIrGefr EdlánBnül´rYcmkehIy . dUecñ¼ CMBUken¼bgHajBI

bBaØtifµImYycMnYnCaTMrg´´ma¨RTIs .

BiCKNitmäRTIspþl´viFIgaysMrab´KMrUrWERh:ssüúg EdlCab́Tak´TgnwgGefreRcIn . enAeBleyIgman k

Gefr eKGaceda¼Rsayedaybeg;ItrUbmnþCaTMrg´m̈aRTIs .

9¿1 KMrUrWERh:ssüúglIenEG‘r k Gefr (The k Variable Linear Regression Model) RbsinebIeyIgeFVITUeTAnIykmµKMrUrWERh:ssüúgBIr nig 3 Gefr , PRF EdlmanGefrTaḱTg Y nig GefrKit

bBa©Úl k - 1 , X2 , X3 , ... , Xk GacsresrCa £

PRF : Yi = 1 + 2X2i + 3X3i + ... + kXki + ui , i = 1, 2,3, ... , n (9.1.1)

Edl1 = cMnuckat´G&kß , 2 , ... , k = emKuNRáb́TisedayEpñk, u = tYclkr{kas nig i = tMélseg;tTI i,

n CaTMhMbÜBuyLasüúg. PRF (9.1.1) RtUvbkRsayCa£ vapþléGaymFüm rWsgÇwmKNit E(Y/X2, X3 , ... ,Xki)

én Y eRkamlkçxNÐtMélefr (kñúgkareFVIKMrUtagRcMEdl) én X2, X3 , ... , Xk , .

smIkar (9.1.1) GacsresrCasMnMuén n smIkar £

Y1 = 1 + 2X21 + 3X31 + ... + kXk1 + u1

Yi = 1 + 2X22 + 3X32 + ... + kXk2 + u2 (9.1.2)

Yn = 1 + 2X2n + 3X3n+ ... + kXkn + un

eyIgGacsresrRbB&n§smIkar (9.1.2) eTACaTMrg´m̈aRTIsdUcxageRkam £

Y1 1 X21 X31 ... Xk1 1 u1

Y2 1 X22 X32 ... Xk2 2 u2 = + (9.1.3)

Yn 1 X2n X3n ... Xkn k un

y = X u

n 1 n k k 1 n 1

y = vicT&rCYrQr n 1 éntMélseg;telIGefrTaḱTg Y

X = mäRTIs n k Edlman n tMélseg;telI k – 1 Gefr X2, ... , Xk , CYrQrTI 1 EdlmanFatu 1

tageGaytYcMnuckat´G&kß (mäRTIsen¼GacehAfa ma¨RTIsTinñn&y (Data Matrix) ) .

CMBUk 9

kareRbIRás´mäRTIselIKMrUrWERh:ssüúglIenEG ‘r THE MATRIX APPROACH TO LINEAR REGRESSION MODEL



= viT&rCYrQr k 1 éntMélá¨räEm¨tmins:al´ 1 , 2 ,..., k

u = viT&rCYrQr n1 én n clkr ui

eRbIc,ab́plKuN nigplbUkmäRTIs eKGacepÞógpÞat́fa RbB&n§smIkar (9.1.2) nig (9.1.3) smmUlKña .

RbB&n§smIkar (9.13) ehAfamäRTIstMNag (Matrix Representation) énKMrUrWERh:ssüúglIenEG‘r ( k Gefr) .

eKGacsresredayxøICa £

y = X + u (9.1.4)

n 1 n k k 1 n 1

RbsinebIminRclMelIvimaRt rWlMdab́énmäRTIs X nig y , nig u, smIkar (9.1.4) GacsresredaygayCa £

y = X + u (9.1.5)

CakarcgðúlbgHajénmäRTIstMNag eyIgBinitüemIlKMrUkareRbIRás´-cMnUlBIrGefr EdlánBnül´kñúgCMBUk 3, Yi =

1 + 2Xi + ui Edl Y CacMNayeRbIRás´ nig X CacMnUl . edayeRbITinñn&ypþléGaykñúgtarag 3.2, eyIg

GacsresrCaTMrg´mäRTIs £

70 1 80 u1

65 1 100 u2

90 1 120 u3

95 1 140 1 u4 110 = 1 160 + u5 (9.1.6)

115 1 180 2 u6 120 1 200 u7

140 1 220 u8

155 1 240 u9

150 1 260 u10

y = X + u

10 1 10 2 2 1 10 1

dUckñúgkrNIBIrGefrnig 3 Gefr eKalbMNgrbśeyIgKWRtUvá¨n´RbmaNtMélá¨räEm¨ténBhurWERh:ssüúg

(9.1.1) nig TaysnñidæanGMBItMélTaMgena¼ BITinñn&yEdlman . CaTMrg´m̈aRTIs krNIen¼Cakará¨n´RbmaN

nig TajsnñidæanGMBItMélen¼ () . sMrabéKalbMNgénkará¨n´RbmaN eyIgGaceRbIviFI OLS rWviFI ML . büEnþ

dUcánkt́sMKal´mun viFITaMgBIren¼pþléGaytMélá¨n´RbmaNesµIKñaénemKuNrWERh:ssüúg . dUecñ¼ eyIgGackMnt´

ykviFI OLS .

9¿2 karsnµténKMrUrWERh:ssüúglIenEG‘rkøasikCaTMrǵmäRTIs (Assumptions of the Classical Linear Regression Model in Matrix Notation)

karsnµt EdlCaRKw¼énKMUrrWRh:ssüúglIenEG‘rkøasik RtUvpþléGaykñúgtarag 9.1 . karsnµtTaMgen¼

RtUvbgHajCaTMrg´s;aEl nigTMrg´mäRTIs . karsnµt (1) pþléGaykñúg (9.2.1) mann&yfa sgÇwmKNiténvicT&r u

esµIsUnü KWfa E(u) = 0 £

u1 E(u1) 0

u2 E(u2) 0

E = = (9.2.1)

un E(un) 0



tarag 9.1 £ karsnµténKMrUrWERh:ssüúglIenEG‘rkøasik

TMrg´s;aEl TMrg´m̈aRTIs

1. E(ui) = 0, i (3.2.1) 1. E(u) = 0

u nig 0 CavicT&rCYrQr n 1 (0 CavicT&rsUnü)

2. E(uiuj) = 0, i j (3.2.5) 2. E(uu') = 2 I

I CamäRTIsÉkta n n

3. X2, X3, ... , Xk CatMél

minEmn{kas (tMélefr)

3. m̈aRTIs (n k) X minEmn{kas KWfa vamansMnMutMélefr

4. KµanTMnak´TMnglIenEG‘r (7.1.7)

Cak´lak´rvagGefrX KWfaKµanBhukUlIenEG‘r

4. r¨g´én X KW (X) = k

k CacMnYnCYrQrkñúg X nig k < n (cMnYntMélseg;t)

5.sMrab́karBinitüemIlsmµtikmµ (4.2.4)

ui N(0, 2)

5. vicT&r u manráyn&rm̈aléRcInGefr KWfa

u N (0, 2I)

karsnµt 2 [smIkar (9.2.2) ] CaviFIxøIénkarsresrkarsnµtTaMgBIrEdlpþléGaykñúg(3.2.5)nig (3.2.2)

CaTMrg´s;aEl . edIm,Iyl´krNIen¼ eyIgGacsresr £

u1

u2

E(uu') = E = [ u1 u2 ... un]

un

u' Ca Transpose énviT&rCYrQr u . edayeFVIRbmaNviFIKuN eyIgán £

2

21

22212

12121

...

...

...

)'(

nnn

n

n

uuuuu

uuuuu

uuuuu

EuuE

edayeRbIRbmaNviFIsgÇwmKNit E elIFatunImYy@énmäRTIxagelI eyIgán £

)()...()(

)()...()(

)()...()(

)'(2

21

22212

212121

nnn

n

uEuuEuuE

uuEuEuuE

uuEuuEuE

uuE (9.2.2)

BIeRBa¼karsnµténGUmÜs:IdasÞIsIuFI nig lkçN£KµankUrwLasüúges‘rI / mäRTIs (9.2.2) GacsresrCa £

2

2

2

...000

0...00

0...00

)'(

uuE

1...000

0...010

0...001

2



= 2 I (9.2.3)

I Cam̈aRTIsÉkta N N .

mäRTIs(9.2.2) [nigm̈aRTIstMNag pþléGaykñúg (9.2.3)] ehAfa mäRTIsvärü¨g´-kUvärü̈g´énclkr ui .

FatuelIGg;t́RTUgénmäRTIsen¼ (rt´BIxageqVgEpñkelI rhUtdl´xagsþaMEpñkeRkam) pþléGayvärü¨g´ nigFatuminenA

elIGg;t́RTUgpþl´kUvärü̈g´ . kMnt´sMKal´fa mäRTIsvärü̈g´-kUvärü̈g´manlkçN£qøú¼ (FatuxagelI nigxageRkamGg;t´RTUg

Føú¼Kña ) .

karsnµt 3 bgHajfa mäRTIs n k X minEmn{kas KWfa vamantMélefr . dUcánkt´sMKal´mun

viPaKrWERh:ssüúgrbśeyIg CaviPaKrWERh:ssüúgmanlkçxNÐelItMélefrénGefr X .

karsnµt 4 bgHajfa mäRTIs X manr¨gĆYrQrTaMgGś esµInwg k (cMnYnCYrQrkñúgmäRTIs) . krNIen¼man

n&yfa CYrQrénmäRTIs X minGaRs&ylIenEG‘r KWfa KµanTMnak´TMnglIenEG‘rBitRàkdkñúgcMeNamGefr X . mü̈ag

eTot KµanBhukUlIenEG‘r . CaTMrg´s;aEl eKGacniyayfa KµansMnMutMél 1, 2, ... , n (minsUnüTaMgGś) Edl£

1X1i + 2X2i + ... + kXki = 0 (9.2.4)

X1i = 1, i ( eFVIeGaymanma¨RTIsCYrQrmanFatu 1 kñúgm̈aRTIs X ) . CaTMrg´m̈aRTIs , (9.2.4) GacsresrCa £

' x = 0 (9.2.5)

' CavicT&rCYredk 1 k nig x CavicT&rCYrQr k 1 .

RbsinebImanTMnaḱTMnglIenEG‘rBitRàkddUcCa (9.2.4), eKfa Gefr CaGefrkUlIenEG‘r . mü̈ageTot

RbsinebI (9.2.4) Bit ( 1 = 2 = 3 = ... = 0) ena¼Gefr X ehAfa GefrminGaRs&ylIenEG‘r . ehtupl

smrmüsMrab´karsnµténlkçN£KµanBhukUlIenEG‘r RtUvànpþléGaykñúgCMBUk 7 nig eyIgnwgCYbkarsnµten¼kñúg

CMBUk 10 .

9¿3 kará¨ńRbmaN OLS (OLS Estimation) edIm,IrktMélá¨n´RbmaN OLS én CatMbUgeyIgsresrrWERh:ssüúgKMrUtag k Gefr (SRF) Ca £

Yi = ikikii uXXX ˆ...ˆˆˆ 33221 (9.3.1)

EdleKGacsresredayxøICaTMrg´mäRTIs £

y = X ̂ + û (9.3.2)

nigCaTMrg´m̈aRTIs £

nkknnn

k

k

n u

u

u

XXX

XXX

XXX

Y

Y

Y

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

...1

...1

...1

2

1

2

1

32

23222

13121

2

1

(9.3.3)

y X ̂ + û

n 1 n k k 1 n 1

̂ CavicT&rCYrQrmanFatu k énsnÞsßn_á¨n´RbmaN OLS énemKuNrWERh:ssüúg nig û CavicT&rCYrQr

n 1 én n sMnl´ .



dUckñúgKMrUBIr nig3 Gefr, kñúgkrNI k Gefr snÞsßn_á¨n´RbmaN OLS RtUvKNnaedayrktMélGb,brmaén

2îu = (Yi - 2

221 )ˆ...ˆˆ kiki XX (9.3.4)

2îu CaplbUksMnl´kaer (RSS) . CaTMrg´mäRTIs krNIen¼ CakarrktMélGb,brmaén 'û û edayehtufa £

'û û =

n

n

u

u

u

uuu

ˆ

ˆ

ˆ

ˆ...ˆˆ2

1

21 = 222

221 ˆˆ...ˆˆ in uuuu (9.3.5)

}LÚven¼ BI (9.3.2) eyIgán £

û = y - X ̂ (9.3.6)

dUecñ¼

'û û = ( y - X ̂ )' (y -X ̂ )

= y'y - 2' Xý + '̂ X'X ̂ (9.3.7)

edayáneRbIlkçN£ Tranpose énmäRTIs:

(X ̂ )' = '̂ X'

nigedayehtufa '̂ Xý CatMéls;aEl (cMnYnBit) tMélen¼esµInwg Transpose rbśva y'X ̂ .

smIkar (9.3.7) Cam̈aRTIstMNagén(9.3.4) . CaTMrg´s;aEl viFI OLS mantMélá¨n´RbmaN 1, 2,...

,k EdleFVIeGay 2îu mantMélGb,brma . eKeFVIDIepr¨gÉsül (9.3.4) edayEpñkeFobnwg 1̂ , 2̂ , ... , k̂

nigkMnt́eGayesµIsUnü . dMeNIren¼pþléGaysmIkarelI k GBaØti . dUcánbgHajkñúgesckþIbEnSm 9A.1 smIkar

TaMgen¼mandUcxageRkam £

n ikikii YXXX ˆ...ˆˆˆ 33221

iikiikiiii YXXXXXXX 2232322221

ˆ...ˆˆˆ

iikiikiiii YXXXXXXX 3323323231

ˆ...ˆˆˆ

(9.3.8)

ikikikikiikiki YXXXXXXX 2

33221ˆ...ˆˆˆ

CaTMrg´mäRTIs smIkar (9.3.8) GacsresrCa £

nknkk

n

n

kkiikiikiki

kiiiiii

kiiiiii

kiii

Y

Y

Y

Y

XXX

XXX

XXX

XXXXXX

XXXXXX

XXXXXX

XXXn

3

2

1

21

33231

22221

3

2

1

232

323233

232222

32

...

...

...

1...11

ˆ

ˆ

ˆ

ˆ

...

...

...

...

...

(X'X) ̂ X' y

(9.3.9)

rWeKGacsresredayxøI £

(X'X) ̂ = Xý (9.3.9)



kMnt´sMKal´lkçN£énm̈aRTIs (X'X) : (1): vapþléGayplbUkkaer nigplKuNExVgénGefr X Edl

mantYmYyCatYcMnuckat́G&kß EdlmantMél 1 sMrab́RKb´tMélseg;t . FatuelIGg;t́RTUgpþléGayplbUkkaeredIm

nigFatuenAxageRkAGg;t́RTUgpþléGayplbUkénplKuNExVg (Bakü ²edIm³ mann&yfa ÉktargVasédIm ) . (2):

vaqøú¼Kña edayehtufaplKuNExVgrvag X2i nig X3i dUcKñanwgrvag X3i nig X2i . (3): vamanlMdab´ (kk) KWfa k

CYredk nig k CYrQr .

kñúg (9.3.10) tMéls:al´ KW (X´X) nig (X'y) ( plKuNExVgrvagGefr X nig y ) nig GBaØti KW ̂ .

}LÚven¼ edayeRbImäRTIs RbsinebIcMraś (X'X)-1

én (X'X) man ena¼edayKuNGg:TaMgBIrén (9.3.10) nwg

cMraś eyIgán £

(X'X)-1

(X'X) ̂ = (X'X)-1X'y

büEnþedayehtufa (X'X)-1

(X´X) = I (mäRTIsÉktalMdab´ k k )/ eyIgán £

I ̂ = (X'X)-1X'y

rW ̂ = (X'X)-1 X' y (9.3.11)

k 1 k k (k n) (n 1)

smIkar (9.3.11)CalT§plRKw¼énRTwsþI OLS CaTMrg´m̈aRTIs . lT§plen¼bgHajfa etIvicT&r ̂ GacRtUv

á¨n´RbmaNBITinñn&ypþléGayrWeT . eTa¼Ca (9.3.11) RtUvánKNnaBI (9.3.9) k¾eday eKGacrkvaedaypÞal´BI

(9.3.7)edayKNnaDIepr¨gÉsülén uu ˆ'ˆ eFobnwg ̂ . sMraybBa¢ak´pþléGaykñúgesckþIbEnSm 9A Epñk 9A.2 .

bMNkRsay

CakarcgðúlbgHajénviFImäRTIs EdlánBnül´ eyIgeda¼Rsay«TahrN_kareRbIRás´-cMnUlénCMBUk 3 eLIgvij EdlTinñn&yRtUvbeg;IteLIgvijtam (9.1.6) . sMrab´krNIBIrGefr eyIgman £

̂ =

2

1

ˆ

ˆ

(X'X) =

23

2

1

321

1

1

1

1

1...111

ii

i

N

n XX

Xn

X

X

X

X

XXXX

nig X'y =

ii

i

n

n YX

Y

Y

Y

Y

Y

XXXX3

2

1

321

1...111

edayeRbITinñn&ypþléGaykñúg (9.1.6), eyIgán £

X'X =

3220001700

170010



nig X'y =

205500

1110

edayeRbIc,abćMras´mäRTIspþléGaykñúgesckþIbEnSm B / eyIgeXIjfa cMraśénmäRTIsxagelI (X'X) KW £

(X'X)-1

=

0000303,0005152,0

005152,097576,0

dUecñ¼ £

̂ =

205500

1110

0000303,0005152,0

005152,097576,0

ˆ

ˆ

2

1

=

5079,0

4545,24

BImuneyIgánKNna 1̂ = 24,4545 nig 2̂ = 0,5091 edayeRbIkmµviFIkMuBüÚT&r . PaBxusKñarvag

tMélá¨n´RbmaNTaMgBIrGaRs&yedaylMeGogkarkat́xÞg´ . kt´sMKal´fa kñúgkareRbIm̈asIunKitelx vaCakarcaMác´

edIm,IrklT§plkMnt́eGayáneRcInxÞgédIm,IeFVIeGaykMritlMeGogtUc .

mäRTIsv̈arü¨g-́kUv̈arüg̈´én ̂

viFIm̈aRTIsGaceFVIeGayeyIgrkrUbmnþ minRKańEtsMrab´värü¨g´én î (tMélNamYyén ̂ ) büeNÑa¼eT büEnþEfm

TaMgsMrab´kUvärü¨g´rvagFatuBIrén ̂ ( î nig ĵ ) . eyIgRtUvkarvärü¨g´ nigkUvärü̈g´sMrab́eKalbMNgsnñidæansSiti .

tamniymn&y ma¨RTIsvärü̈g-kUvärü̈g´én ̂ KW [RtUvKñanwg (9.2.2)] :

var-cov( ̂ ) = E{[ ̂ - E( ̂ )][ ̂ -E( ̂ )]'}

EdlGacsresrCa £

var-cov( ̂ ) =

)ˆvar(...)ˆ,ˆcov()ˆ,ˆcov(

)ˆ,ˆcov(...)ˆvar()ˆ,ˆcov(

)ˆ,ˆcov(...)ˆ,ˆcov()ˆvar(

21

2212

1211

kkk

k

k

(9.3.12)

eKbgHajkñúgesckþIbEnSm 9A Epñk 9A.3 fa ma¨RTIsvärü̈g´-kUvärü̈g´xagelIGacRtUvKNnaBIrUbmnþxageRkam £

var-cov( ̂ ) = 2(X'X)-1 (9.3.13)

Edl2Cavärü¨g´GUmÜs:IdasÞIsIuFI én ui nig (X´X)-1

Cam̈aRTIsRcas EdlánmkBI(9.3.11) EdlpþléGaysnÞsßn_

á¨n´RbmaN OLS ̂ .

kñúgKMrUrWERh:ssüúgBIr nig 3 Gefr snÞsßn_á¨n´RbmaNminlMeGogén 2 RtUvpþléGayeday 2̂ =

)2/(ˆ2 nui nig 2̂ = )3(ˆ2 nui erogKña . kñúgkrNI k Gefr rUbmnþRtUvKñaKW £

2̂ =kn

ui

2ˆ

= kn

uu

ˆ'ˆ (9.3.14)

Edlman df = n – k (ehtuGVI?) .



eTa¼Ca tameKalkarN_ uu ˆ'ˆ GacRtUvKNnaBIsMnl´á¨n´RbmaNk¾eday tamkarGnuvtþ eKGacKNnatMél

en¼edaypÞal´dUcxageRkam . rMlwkfa 2îu (=RSS) = TSS – ESS kñúgkrNI 2 Gefr eyIgGacsresr £

2îu = 2iy -

22̂

2ix (3.3.6)

nigkñúgkrNI 3 Gefr £

2îu = 2iy - 2̂ yix2i - 3̂ yix3i (7.4.19)

edayBRgIkeKalkarN_en¼ eKeXIjfa sMrab´KMrU k Gefr £

2îu = 2iy - 2̂ yi x2i - ... - k̂ yi xki (9.3.15)

CaTMrg´mäRTIs £

TSS : 2iy = y 'y – n 2Y (9.3.16)

ESS : 222 ''ˆˆ...ˆ YnyXxyxy kiikii (9.3.17)

Edl n 2Y ehAfatYkMEntMrUvsMrab́mFüm . dUecñ¼ £

uu ˆ'ˆ = y'y - yX ''̂ (9.3.18)

enAeBl uu ˆ'ˆ RtUvánKNna/ 2 RtUvKNnaedayRsYlBI (9.3.14) EdleFVIeGayeyIgGacá¨n´RbmaNmäRTIs

värü¨g´- kUvärü̈g´ (9.3.13) .

sMrab́«TarN_cgðúlbgHajrbśeyIg £

uu ˆ'ˆ = 132100 – [24,4545 0,5091]

205500

1110 = 337,373

dUecñ¼ 2̂ =337,273/8 = 42,1591 EdlCatMélRbhak´RbEhlnwgtMélKNnamunkñúgCMBUk 3 .

lkçN£énvicT&r OLS ̂ (Property of OLS Vector ̂ ) kñúgkrNIBIrGefr nig 3 Gefr eyIgdwgfa snÞsßn_á¨n´RbmaN OLS KWlIenEG‘r nigminlMeGog nigkñúgfñak´

énsnÞsßn_à¨n´RbmaNlIenEG‘rminlMeGogTaMgGś vamanvärü̈g´´Gb,brma (lkçN£ Gauss-Markov) . Casegçb

snÞsßn_á¨n´RbmaN OLS CasnÞsßn_à¨n´RbmaNlIenEG‘r nigminlMeGoglðbMput (BLUE). lkçN£en¼Gac

RtUvBRgIkeTAvicT&r ̂ KWfa ̂ lIenEG‘r [tMélnImYy@rbśvaCaGnuKmn_lIenEG‘rénY ( GefrTak´Tg) ] . E( ̂ ) =

̂ KWfasgÇwmKNiténFatunImYy@én ̂ esµInwgFatuRtUvKñaéntMélBit nigkñúgfñaḱénsnÞsßn_á¨n´RbmaNlIenEG‘r

minlMeGogTaMgGśén , snÞsßn_á¨n´RbmaN OLS ̂ manvärü¨g´Gb,brma .

sMraybBa¢aḱRtUvpþléGaykñúgesckþIbEnSm 9A Epñk 9A.4 . dUcánbgHajkñúgkarBnül´dMbUg krNI k

Gefr PaKeRcInCakarBRgIkénkrNIBIr rW 3 Gefr .

9¿4 emKuNkMnt ́R2 CaTMrg´m̈aRTIs (The Coefficient of Determination R2 in Matrix Notation) emKuNkMnt́ R

2 RtUvánkMnt́Ca £

R2 =

TSS

ESS

kñúgkrNIBIrGefr £



R2=

2

222ˆ

i

i

y

x

(3.5.6)

nigkñúgkrNI 3 Gefr £

R2 =

2

3322ˆˆ

i

iiii

y

xyxy

(7.5.5)

edayeFVITUeTAnIykmµ eyIgán sMrab́krNI k Gefr £

R2 =

2

3322ˆˆˆ

i

kiikiiii

y

xyxyxy

(9.4.1)

edayeRbI (9.3.16) nig (9.3.17) smIkar (9.4.1) GacsresrCa £

R2 =

2

2

'

'ˆ

Ynyy

YnyX

(9.4.2)

EdlpþléGaym̈aRTIstMNagén R2 .

sMrab́«TahrN_cgðúlbgHajrbséyIg £

205500

11105079,03571,24''ˆ yX = 131409,831

y'y = 132100

nig n 2Y = 123210

bBa©ÚltMélTaMgen¼kñúg (9.4.2) eyIgeXIjfa R2 = 0,9224 EdlmantMélRbhak´RbEhlKñanwgtMél

KNnamun (manlMeGogkat́xÞg´tUc) .

9¿5 mäRTIskUrwTLasüúg (The Correlation Matrix) kñúgCMBUkmun eyIgáns:al´kUrwLasüúglMdab́sUnü (rWkUrWLasüúggay) kUrWLasüúg r12, r13, r23 nig

kUrWLasüúglMdab´ 1 (rWkUrWLasüúgedayEpñk) r12.3, r13.2, r23.1 nigGnþrTMnak´TMngrbs´va . kñúgkrNI k Gefr eyIg

nwgman k(k – 1)/2 emKuNkUrWLasüúglMdab́sUnüTaMgGś (ehtuGVI ?) . kUrWLasüúg k(k – 1)/2 TaMgen¼ GacRtUv

daḱeTACamäRTIs EdlehAfa mäRTIskUrWLasüúg (Correlation Matrix) R dUcxageRkam £

kkkkk

k

k

rrrr

rrrr

rrrr

R

...

...

...

321

2232221

1131211

1...

...1

...1

321

22321

11312

kkk

k

k

rrr

rrr

rrr

(9.5.1)

EdlsnÞsßn_ 1 (dUcmun) kMntéGayGefrTaḱTg Y ( r12 £ emKuNkUrWLasüúgrvag Y nig X2¿¿¿¿) nig edayeRbI

lkçN£emKuNkUrWLasüúgénGefrmYyeFobnwgGefrxøÜnva Canic©kalesµI 1 (r11 = r22 = r33 = ... = rkk = 1) .



BIm̈aRTIskUrWLasüúg R eKGacrkemKuNkUrWLasüúglMdab´ 1 (emIlCMBUk 7) niglMdab´x



edaysnµtfa clkr ui Caráyn&rmäl´ nigsmµtikmµsUnüKW 2 = 3 = ... = k = 0 nig tamCMBUk 8

eyIgGacbgHajfa £

F = )/()''ˆ'(

)1/()''ˆ( 2

knyXyy

kYnyX

(9.7.1)

eKarBtamráy F edayman df = k – 1 nig df = n – k .

tarag 9.2 : TMrg´m̈aRTIséntarag ANOVA sMrab́KMrUrWERh:ssüúglIenEG‘r k Gefr

RbPBbMErbMrYl SS df MSS

GaRs&yedayrWERh:ssüúg

( GaRs&yedayX2, X3, ..., Xk)

2''ˆ YnyX k – 1

1

''ˆ 2

k

YnyX

GaRs&yedaysMnl´ yXyy ''ˆ' n – k

kn

yXyy

''ˆ'

srub y'y – n 2Y n - 1

kñúgCMBUk 8 eyIgeXIjfa eRkamkarsnµt EdlánbgHajmun manTMnak´TMngmYyrvag F nig R2 KW £

F = )/()1(

)1/(2

2

knR

kR

(8.5.11)

dUecñ¼tarag ANOVA 9.2 GacRtUvsresrCa tarag 9.3 . RbeyaCn_éntarag 9.3 eFobnwgtarag 9.2 KWfa viPaK

TaMgmUlGacRtUveFVIántam R2 . eKmincaMác´BinitüemIltY (y'y – n 2Y ) BIeRBa¼vaminmankñúgpleFob F .

tarag 9.3 : tarag ANOVA k Gefr CaTMrg´m̈aRTIselItMél R2


GaRs&yedayrWERh:ssüúg

( GaRs&yedayX2, X3, ..., Xk)

)'( 22 YnyyR k – 1

1

)'( 22

k

YnyyR

GaRs&yedaysMnl´ )')(1( 22 YnyyR n – k

kn

YnyyR

)')(1( 22

srub y'y – n 2Y n - 1

9¿8 karBinitüemIllkçxNÐlIenEG‘r £ karBinitüemIlTUeTA F edayeRbITMrg´m̈aRTIs (Testing Linear Restrictions: General F Testing Using Matrix Notation) kñúgEpñk 8.7 eyIgánbgHajkarBinitüemIlTUeTA F edIm,IBinitüemIlsuBlPaBénlkçxNÐlIenEG‘r Edl

manelItMélà¨räEm̈tmYy rWeRcInénKMrUrWERh:ssüúglIenEG‘r k Gefr . karBinitüemIlsmrmüRtUvánpþléGaykñúg

(8.7.9) [ rWsmIkarsmmUlrbs´va (8.7.10)] . TMrg´mäRTIsén (8.7.9) GacRtUvTajánedaygay .

tag £

Rû = vicT&rsMNl´BIrWERh:ssüúgkaertUcbMputmanlkçxNÐ



URû = vicT&rsMnl´BIrWERh:ssüúgkaertUcbMputKµanlkçxNÐ

dUecñ¼ £ RR uu ˆ'ˆ = 2ˆRu = RSS BIrWERh:ssüúgmanlkçxNÐ

URUR uu ˆ'ˆ = 2ÛRu = RSS BIrWERh:ssüúgKµanlkçxNÐ

m = cMnYnlkçxNÐlIenEG‘r

k = cMnYná¨räEm¨t ( rYmmancMnuckat´G&kß) kñúgrWERh:ssüúgKµanlkçxNÐ

n = cMnYntMélseg;t

dUecñ¼ TMrg´mäRTIsén (8.7.9) KW £

F = )/()ˆ'ˆ(

/)ˆ'ˆˆ'ˆ(

knuu

muuuu

URUR

URURRR

(9.8.1)

EdleKarBtamráy F edayman df = (m, n – k) . tamFmµta RbsinebItMélKNna F BI (9.8.1) FMCagtMél

RKITik F, eyIgGacbdiesFrWERh:ssüúgmanlkçxNÐ eRkABIen¼eyIgminbdiesF .

9¿9 karTsßn_TaytMéledayeRbIBhurWERh:ssüúg £ TMrǵmäRTIs

(Prediction Using Multiple Regression: Matrix Formulation) kñúgEpñk 8.10 eyIgánBnül´ (edayeRbITMrg´s;aEl) viFIEdlBhurWERh:ssüúgGacRtUveRbIsMrab́Tsßn_Tay

(1): tMélmFümnig (2): tMélÉktþ£én Y eBlmantMélrWERh:ssüúg X . kñúgEpñken¼ eyIgbgHajviFITsßn_Tay

tMélen¼CaTMrg´mäRTIs . eyIgk¾bgHajBIrUbmnþedIm,Iá¨n´RbmaNvärü¨g´ niglMeGogKMrUéntMélTsßn_Tay . kñúgCMBUk 8

eyIgánkt´sMKal´fa rUbmnþTaMgen¼RtUváneRbICaTMrg´m̈aRTIsgayRsYlCag eRBa¼kenßams;aEl rWkenßamBiCKNit

énrUbmnþTaMgen¼mankarsµúKsµaj .

karTsßn_TaytMélmFüm (Mean Prediction)

tag

kX

X

X

x

0

03

02

0

1

(9.9.1)

CavicT&réntMélGefr X EdleyIgcg´Tsßn_TaytMél 0Ŷ (karTsßn_TaytMélmFümén Y ) .

}LÚven¼ BhurWERh:ssüúgá¨n´RbmaNCaTMrg´s;aElKW £

iŶ = kikii XXX ˆ...ˆˆˆ 33221 (9.9.2)

EdlCaTMrg´m̈aRTIs GacsresredayxøICa £

iŶ = ̂'ix (9.9.3)

Edl ix' = [ 1 X2i X3i ... Xki ] nig

k

ˆ

ˆ

ˆ

ˆ 2

1



smIkar (9.9.2) rW (9.9.3) CakarTsßn_TaytMél énYi RtUvKñanwg ix' .

RbsinebI ix' dUcánpþléGaykñúg (9.9.1) nig (9.9.3) GacsresrCa £

̂')'/ˆ( 00 xxYi (9.9.4)

EdltMélén x0 RtUvánkMnt́ . kt́sMKal´fa (9.9.4) pþléGaykarTsßn_TayminlMeGogén E(Yi /x'0) eday

ehtufa E(x'0 ̂ ) = x'0 (ehtuGVI?) .

karTsßn_TaytMélÉktþ£ (Individual Prediction)

dUceyIgándwgBICMBUk 5 nig 8 karTsßn_TaytMélÉktþ£én Y ( Y0) CaTUeTApþléGayeday (9.9.3) rW

(9.9.4) . KWfa £

(Y0 / x'0) = x'0 ̂ (9.9.5)

dUecñ¼ sMrab́«TahrN_cgðúlbgHajénEpñk 8.11 TMrg´m̈aRTIsénkarTsßn_TaytMélmFüm nigkarTsßn_TaytMél

Éktþ£ KW £

x0 = x1971 =

16

567

1

nig

7363,2

7266,0

1603,53

̂

dUecñ¼

7363,2

7266,0

1603,53

165671)'/ˆ( 19711971 xY = 508,9297 (9.9.6)

= (8.10.2)

nig 9297,508)'/( 19711971 xY (ehtuGVI?) (9.9.7)

värü¨g´énkarTsßn_TaytMélmFüm (Variance of Mean Prediction) rUbmnþedIm,Iá¨n´RbmaNvärü¨g´én ( )'/ˆ 00 xY KW£

var( )'/ˆ 00 xY = 2x'0 (X'X)

-1x0 (9.9.8)

2 Cav̈arü¨g´én ui , x'0 CatMéls:al´énGefr X EdleyIgcg´Tsßn_Tay nig (X'X) Cam̈aRTIspþléGaykñúg

(9.3.9) KWfaCamäRTIsEdleRbIedIm,Iá¨n´RbmaNBhurWERh:ssüúg . CMnYs 2 edaysnÞsßn_á¨n´RbmanminlMeGog

rbśva 2̂ eyIgGacsresrrUbmnþ (9.9.8) Ca £

var( )'/ˆ 00 xY = 2̂ x'0 (X'X)

-1x0 (9.9.9)

sMrab́«TahrN_cgðúlbgHajénEpñk 8.10 eyIgmantMélxageRkam £

2̂ =6,4308 X'X-1 =

1120,00063,06532,1

0063,00004,00982,0

6532,10982,03858,26



edayeRbITinñn&yTaMgen¼ BI (9.9.9) eyIgán £

var(

16

567

1

)'(1656714308,6)'/ˆ 119711971 XXxY =6,4308 (0,5688)

= 3,6580 (9.9.10)

= (8.10.3)

nig se( 1971Ŷ /x1971) = 9126,16580,3 (9.9.11)

=(8.10.3)

dUecñ¼ BIkarBnül´kñúgCMBUk 5 nig 8 eyIgeXIjfa cenøa¼TMnukcitþ 100(1-) % elItMélmFümRtg´ x0 KW £

01

02

2/0 )'(ˆˆ xXXxtY E(Y/x0) 0

10

22/0 )'(ˆ

ˆ xXXxtY (9.9.12)

sMrab́«TahrN_rbśeyIg cenøa¼TMnukcitþ 95% sMrab́tMélmFüm KWdUcbgHajkñúg (8.10.6) :

504,7518 E(Y1971) 513,0868

värü¨g´énkarTsßn_TaytMélÉktþ£ (Variance of Individual Prediction) rUbmnþsMrab´värü̈g´énkarTsßn_TaytMélÉktþ£ KW £

var(Y0 / x0) = 2̂ [1+ x'0(X'X)

-1x0] (9.9.13)

Edl var(Y0/x0) Ca E[(Y0 - 2

0 )/ˆ XY ] [GñkGanGacemIlpgEdr (5.10.6)] .

dUecñ¼edayeRbITinñn&yrbśeyIg eyIgán £

var(Y1971 / X1971) = 6,4308 (1 + 0,5688) = 10,0887 (9.9.14)

= (8.10.4)

nig se(Y1971 / x1971) = 0887,10 = 3,1763 (9.9.15)

= (8.10.4)

RbsinebIeyIgcg´beg;Itcenøa¼TMnukcitþ 100(1-) %sMrab́karTsßn_TaytMélÉktþ£ eyIgeFVIdUckñúg (9.9.12)elIk

ElgEtlMeGogKMrUénkarTsßn_TaytMélRtUveKKNnaBI (9.9.13) . lMeGogKMrUénkarTsßn_TaysMrab´karTsßn_Tay

tMélÉktþ£RtUveKrMBwgfa mantMélFMCagtMéllMeGogKMrUénkarTsßn_TaytMélmFüm [emIl (8.10.7) ] .

9¿10 esckþIsegçbénviFImäRTIs £ «TahrN_cgðúlbgHaj

(Summary of the Matrix Approach: An Illustrative Example)

CakarsegçbviFIm̈aRTIskñúgviPaKrwERh:ssüúg eyIgbgHaj«TahrN_CaelxEdlCab´Tak´TgnwgbIGefr .

rMlwk«TahrN_cgðúlbgHajénCMBUk 8 EdlCab´Tak´TgnwgrWERh:ssüúgéncMNaykareRbIRás´pÞal´xøÜnRbmUlpþúMelI

cMnUlpÞal´xøÜnRbmUlpþúMbnÞab́BIbg´Bn§ nigFanaräb́́rg nigGefreBlsMrab´GMlugeBl 1956-1970 . eKánbgHajfa

Gefrninñakar t GactageGay (kñúgcMeNamGefrepßgeTot) bÜBuyLasüúgsrub rWRbmUlpþúM £ cMNayeRbIRás´RbmUlpþúM

RtUveKrMBwgfa ekIneLIg enAeBlbÜBuyLasüúgfycu¼ . viFImYyénkarpþac´}T§iBlénb̈UBuyLasüúg KWRtUvbþÚrcMNay

kareRbIRás´RbmUlpþúM nigtYelxcMnUlRbmUlpþúMeTACamUldæankñúgmnusßmñak´ rWmYymuxedayEcknwgcMnYnb̈UBuyLasüúg

srub . rWERh:ssüúgéncMnayeRbIRás´RbmUlpþúMkñúgmnusßmñak´ elIcMnUlkñúgmnusßmñak´@ nwgpþléGayTMnaḱTMngrvag

cMNayeRbIRás´ nigcMnUleFobnwgbMErbMrYlb̈UBuyLasüúg (rW}T§iBlkMrit) . GefrninñakarGacenAEtrkßaTukkñúgKMrUCa



}T§ilRbmUlpþúMsMrab}́T§iBlepßgeTot Edlman}T§iBlelIcMNaykareRbIRás´ («/ bec©kviTüa) . sMrab́eKalbMNg

BiesaFn_ KMrUrwRh:ssüúgrbśeyIgKW £

Yi = iii uXX ˆˆˆˆ

33221 (9.10.1)

Y = cMNayeRbIRás´kñúgmnusßmñak´@/ X2 = cMnUlpÞal´xøÜnkñúgmnusßmñak´@bnÞab´BIbg´Bn§nigFanaräb´rg nig X3 =

eBl . Tinñn&yRtUvkaredIm,IeFVIrWERh:ssüúg (9.10.1) pþléGaykñúgtarag 9.4 .

CaTMrg´mäRTIs dMeNa¼RsayrbséyIgGacRtUvbgHajxageRkam £

15ˆ14ˆ13ˆ12ˆ11ˆ10ˆ

9ˆ8ˆ7ˆ6ˆ5ˆ4ˆ3ˆ2ˆ1ˆ

3ˆ2ˆ1ˆ

1525951

1425351

1324871

1224041

1123361

1022391

921261

820161

719691

619101

518831

418811

318311

218441

118391

2324

2316

2257

2165

2128

2048

1948

1867

1815

1756

1749

1735

1666

1688

1673

u

u

u

u

u

u

u

u

u

u

u

u

u

u

u

(9.10.2)

y = X ̂ + û

(15 1) (15 3) (3 1) (15 1)

tarag 9.4 £ cMNayeRbIRás´pÞal´xøÜnkñúgmnusßmñaḱ@ (PPCE)

nigcMnUlpÞal´xøÜnkñúgmnusßmñaḱbnÞab́BIbg´Bn§ nigFanaräb́rgenAGaemrik (1956-1970) CaduløaqñaM 1958

PPCE (Y) PPDI (X2) eBl (X3)

1673 1839 1 ( = 1956) 1688 1844 2 1666 1831 3 1735 1881 4 1749 1883 5 1756 1910 6 1815 1969 7 1867 2016 8 1948 2126 9 2048 2239 10 2128 2336 11 2165 2404 12 2257 2487 13 2316 2535 14 2324 2595 15 (=1970)

RbPB £ Economic Report of the President, January 1972, Table B-16

BITinñn&yxagelI eyIgán £

Y = 1942,333 2X =2126,333 3X =8,0

(Yi - Y )2 = 830121,333

(X2i - 2X )2 = 1103111,333 (X3i - 3X )

2 = 280,0



X'X =

nn

n

n

XX

XX

XX

XX

XXXX

XXXX

32

3323

3222

3121

3333231

2232221

1

1

1

1

...

...

1...111

=

23323

32222

32

iiii

iiii

ii

XXXX

XXXX

XXn

=

1240272144120

272144513,6892231895

1203189515

(9.10.3)

X'y =

247934

62905821

29135

(9.10.4)

edayeRbIc,ab´éncMraśmäRTIspþléGaykñúgesckþIbEnSm B eKeXIjfa £

(X'X)-1

=

054034,00008319,0336707,1

0008319,00000137,00225082,0

336707,10225082,0232491,37

(9.10.5)

dUecñ¼ £

̂ = (X'X)-1 Xý =

04356,8

74198,0

28625,300

(9.10.6)

plbUksMnl´kaerGacRtUvKNna £

2îu = uu ˆ'ˆ = y'y - yX ''̂

= 57420003 – [300,28625 0,74198 8,04356]

247934

62905821

29135

(9.10.7)

= 1976,85574

dUecñ¼eyIgán £

73797,16412

ˆ'ˆˆ 2

uu (9.10.8)

mäRTIsvärü¨g´-kUvärü̈g´sMrab́ ̂ GacRtUvsresrCa £

var-cov( ̂ ) = 2̂ (X'X)-1 =

90155,813705,020634,220

13705,000226,070794,3

20634,22070794,3650,6133

(9.10.9)



FatuGg;t́RTUgénm̈aRTIsen¼pþléGayvärü¨g´én 1̂ , 2̂ nig 3̂ erogKña nigrWskaerviC¢manrbs´vapþ;léGaylMeGog

KMrURtUvKña .

BITinñn&yxagelI eKGacepÞógpÞat´fa £

ESS : ̂ X'y – n 2Y = 828144,47786 (9.10.10)

TSS : y'y – n 2Y = 830121,333 (9.10.11)

dUecñ¼ £

R2 =

2

2

'

''ˆ

Ynyy

YnyX

= 99761,0

333,830121

47786,828144 (9.10.12)

edayeRbI (7.8.4) emKuNkMnt´kMEntMrUv (Adjusted Coefficient of Determination) GacRtUvKNna £

2R =0,99722 (9.10.12) edayRbmUlpþúMlT§pl eyIgán£

iŶ = 300,28625 + 0,74198 X2i + 8,04356X3i

(78,31763) (0,04753) (2,98354) (9.10.14)

t = (3,83421) (15,61077) (2,69598)

R2 = 0,99761 2R = 0,99722 df = 12

bMNkRsayén (9.10.14) KW £ RbsinebI X2 nig X3 yktMélefrRtg´ 0 ena¼tMélmFüméncMnayeRbIRás´pÞal´

xøÜnkñúgmnusßmñaḱ@ RtUvá¨n´RbmaN esµInwg $ 300 . tamFmµta bMNkRsayéncMnuckat´G&kßen¼minc,as´las´ .

emKuNrWERh:ssüúgedayEpñk 0,74198 mann&yfa edayrkßatMélGefrepßgeTotefr kMenIncMnUlpÞal´xøÜnkñúg

mnusßmñak´@ $1 eFVIeGaymankMenIncMnayeRbIRás´pÞal´xøÜnCamFümkñúgmnusßmñak´@ $ 0,74 . Casegçb MPC

RtUvá¨n´RbmaNesµInwg 0,74 rW 74% . dUcKñaen¼Edr edayrkßaGefrTaMgGsépßgeTotefr cMNayeRbIRás´pÞaĺ

xøÜnCamFümkñúgmnusßmñak´@ ekIneLIg edaykMrit $ 8 kñúgmYyqñaM kñúgGMlugqñaM 1956-1970 . R2 = 0,9976

bgHajfa GefrKitbBa©ÚlBIrRtUvánbBa©Úl 99 % énbMErbMrYlelIcMNayeRbIRáśkñúgmnusßmñakénAGaemrikkñúgGMlug

qñaM 1956-1970 . eTa¼Ca 2R Føaḱcu¼ bnþick¾eday vaenAEtmantMélx



GviC¢man (kUvärü¨g´rvagsnÞsßn_á¨n´RbmaNTaMgBIr KW – 0,13705 ) . dUecñ¼ eyIgminGaceRbIkarBinitüemIl t edIm,I

BinitüemIlsmµtikmµsUnü 2=3 = 0 .

büEnþrMlwkfa smµtikmµsUnü 2 = 3 = 0 RBmKña GacRtUvBinitüemIledayviFIviPaKvärü¨g´ nigkarBinitüemIl

F EdlánBnül´kñúgCMBUk 8 . sMrab́dMeNa¼RsayrbséyIg taragviPaKvärü¨g´ KWCatarag 9.5 . eRkamkarsnµt

Fmµta eyIgán £

F = 52,251373797,164

3893,414072 (9.10.15)

tarag 9.5 : tarag ANOVA sMrab́Tinñn&ytarag 9.4


GaRs&yedayX2, X3, 828144,47786 2 414072,3893

GaRs&yedaysMnl´ 1976,85574 12 164,73797

srub 830121,33360 14

EdlCaráy F edayman df = 2 nig df = 12 . tMélKNna F mansar£sMxan´sSitix



eyIgTukeGayGñkGanepÞógpÞat´ edayeRbI (9.9.13) fa £

var(Y1971 / x'1971 ) = 213,3806 (9.10.19)

nig se(Y1971 / x'1971 ) = 14,6076

kMnt´sMKal´ £ var(Y1971 / x'1971 ) = E[Y1971 - 1971Ŷ / x'1971]2 .

kñúgEpñk 9.5 eyIgánbgHajmäRTIskUrWLasüúg R . sMrab´Tinñn&yrbséyIg mäRTIskUrWLasüúgKW £

R =

19664,09743,0

9664,019980,0

9743,09980,01

(9.10.20)

kt́sMKal´fa kñúg(9.10.20) eyIgánkMnt́RBMEdnm̈aRTIskUrWLasüúgtamGefrénKMrU edIm,IeGayeyIgGackMnt́fa

etIGefrmYyNaCab´TaḱTgkñúgkarKNnaénemKuNkUrWLasüúg . dUecñ¼ emKuN 0,9980 kñúgCYrTI 1 énmäRTIs

(9.10.20) RábéyIgeGaydwgfa vaCaemKuNkUrWLasüúgrvag Y nig X2 ( r12) . BIkUrWLasüúglMdab´sUnü Edlpþl´

eGaykñúgmäRTIs (9.10.20) eKGacrkemKuNkUrWLasüúglMdab´ 1 edaygay (emIllMhat´ 9.7) .

9¿11 esckþIsegçb nigkarsnñidæan

eKalbMNgd¾FMénCMBUken¼ KWRtUvbgHajviFIm̈aRTIselIKMrUrWERh:ssüúglIenEG‘rkøasik . eTa¼CamanbBaØt imYy cMnYntUcénviPaKrWERh:ssüúgRtUvànBnül´k¾eday TMrg´m̈aRTIspþléGayviFIxøIéndMeNa¼RsayKMrUrwERh:ssüúglIenEG‘r

EdlCab´TaḱTgnwgGefrmYycMnYn .

kñúgkarbBa©bĆMBUken¼ kt́sMKal´fa RbsinebI Gefr Y nig X RtUvvas´TMrg´KMlat (KWfaKMlatBImFümKMrU

tagrbs´va) mankarERbRbYlmYycMnYntUcelIrUbmnþ EdlánBnül´mun . karERbRbYlen¼RtUváncu¼kñúgtarag 9.6 .

dUctaragen¼ánbgHaj CaTMrg´KMlat kMEntMrUvsMrab´mFüm n 2Y RtUvykecjBI TSS nig ESS (ehtuGVI ?) .

karát´tYtMélen¼eFVIeGaymankarpøas´bþÚrkñúgrUbmnþsMrab´ R2 . eRkABIen¼ rUbmnþPaKeRcIn EdlánbkRsaytam

ÉktargVas´dIm enAEtBitsMrab´TMrg´KMlat .

tarag 9.6 : KMrUrWERh:ssüúg k GefrtamÉktargVasédIm nigCaTMrg´KMlat * (Deviation Form)

Y X2 X3

Y

X2

X3



ÉktargVas´edIm TMrg´KMlat

y =X ̂ + û (9.3.2) y =X ̂ + û

CYrQrmanFatu 1 kñúgm̈aRTIs X RtUvát́

(ehtuGVI?)

̂ = (X'X)-1X'y (9.3.11) dUcKña

var-cov( ̂ ) =2(X'X)-1 (9.3.13) dUcKña

uu ˆ'ˆ = y'y - yX ''̂ (9.3.18) dUcKña

2iy =y'y – n 2Y (9.3.16) 2iy = y'y (9.11.1)

ESS = yX ''̂ - n 2Y (9.3.17) ESS = yX ''̂ (9.11.2)

R2 =

2

2

'

''ˆ

Ynyy

YnyX

(9.4.2) R

2 =

yy

yX

'

''̂ (9.11.3)

* : kt́sMKal´fa eTa¼CakñúgkrNITaMgBIr nimitþisBaØasMrab́m̈aRTIs nigvicT&r dUcKñak¾eday kñúgTMrg´KMlat Fatuénm̈aRTIs

nigvicT&rRtUvsnµtfa CaKMlat CaCagTinñn&yedIm . kt́sMKal´pgEdrfa kñúgTMrg´KMlat ̂ manlMdab́ k – 1 nig

var-cov ( ̂ ) manlMdab´ (k – 1)(k – 1) .

karerbirás´m¨artiselikmrurwerh:ssüúglieneg pdf... · 2013. 4. 9. ·...

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