saklviTüal&yPUminÞPñMeBj ROYAL UNIVERSITY OF PHNOM PENH

edátWm¨gKNitviTüa 145 MATHEMATICS DEPARTMENT

TidæPaBxø¼@ènviPaKrWERh:ssüúglIenEG‘r GacRtUveKBnül´edaygaykñúgkarsikßaKMrUrWERh:ssüúglIenEG‘rBIr

Gefr EdleyIgánBiPakßamkdl´eBlen¼ . CadMbUg eyIgsikßakrNIrWERh:ssüúgkat´tamKl´G&kß KWfakñúgsSan-

PaB EdltYcMnuckat´G&kß 1 minmankñúgKMrU . bnÞab´mkeyIgseg;telIÉktargVas (Unit of Measurement)

ena¼KW£ etIeKvasGefr X nig Y y¨agdUcemþc nigfa etIbMErbMrYl 1 ÉktargVas´man}T§iBlelIlT§pl rWERh:ssüúg

EdrrWeT . CacugbBa©b eyIgnwgseg;temIlbBaHaénTMrg´GnuKmn_ (Functional Form) ènKMrUrWERh:ssüúglIenEG‘r .

mkdl´eBlen¼ eyIgànBinitüemIlKMrUEdl lIenEG‘relItMèlàraEmt nigelIGefr . buEnþKYrrMlwkfa RTwsþIrWERh:s

süúgEdlànbkRsaykñúgCMBUkmuntMrUveGaymanEtlkçN£lIenEG‘relItMèlàraEmtbueNÑa¼ . lIenEG‘relIGefrCa

krNImincaMàc . edaykarseg;temIlKMrU EdllIenEG‘relItMèlàraEmt EtmincaMàc´lIenEG‘relIGefr eyIgnwg

bgHajkñúgCMBUken¼ faetIeKGaceRbIKMrUBIrGefredIm,Ieda¼RsaybBaHaCakEsþgy¨agNaxø¼ ?

enAeBleKylKMnitdMbUgkñúgCMBUken¼ eKGacBRgIkvaeTAkñúgKMrUBhurWERh:ssüúgedaygaydUceyIgnwgsikßa

kñúgCMBUk 7 nig 8 .

6¿1 rWERh:ssüúgkat´tamKl´G&kß (Regression through the Origin)

CYnkal PRFBIrGefr manTMrg´ £

Yi = 2Xi + ui (6.1.1)

kñúgKMrUen¼ KµantYcMnuckat´G&kß rW vamantMélesµIsUnü ehtudUecñ¼eKehAKMrUen¼fa rWERh:ssüúgkat´Kl´G&kß .

CakarcgðúlbgHaj seg;temIlKMrUkMntéføRTBüskmµedImTun [Capital Asset Pricing Model

(CAPM)] ènRTwsþIb&NÑPaKh‘unTMenIb EdlCaTMrg´cMnUl-eRKa¼fñak(Risk-Premium Form) GacsresrCa £

(ERi - rf )= i (ERm – rf) (6.1.2)

Edl ERi = kMritcMnUlBIvinieyaKsgÇwmelIb&NÑPaKh‘un i

ERm = kMritcMnUlBIvinieyaKsgÇwmelIb&NÑPaKh‘unTIpßar EdlmantMèlCasnÞsßn_b&NÑPaKh‘unbNþak´ S&P 500

rf = kMritRák´cMnUlBIvinieyaKKµaneRKa¼fñak´ «¿ cMnUlelIvik&yb&RtrtnaKarry£eBl 90 éf¶

i = emKuN EdlCargVas´éneRKa¼fñak´tamRbB&n§ «¿ eRKa¼fñak´EdlminGacRtUveCosvagán . müageTotrgVas´

elIkMritcMnUlBIvinieyaKb&NÑPaKh‘un i ERbRbYlGaRs&yelITIpßar . i > 1 Cab&NÑPaKh‘unERbRbYl nig i < 1

Cab&NÑPaKh‘unTb´Tl´ (Defensive Security) (kt´sMKal´ £ eKminRtUvyl´RclM i kñúgkrNIen¼CamYyemKuN

Ráb´TisénrWERh:ssüúgBIrGefr 2 ) .

CMBUkTI 6

BRgIkbEnSmelIKMrUrWERh:ssüúglIenEG‘rBIrGefr

EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODEL

saklviTüal&yPUminÞPñMeBj ROYAL UNIVERSITY OF PHNOM PENH

edátWm¨gKNitviTüa 146 MATHEMATICS DEPARTMENT

RbsinebITIpßarTundMeNIrkarlð ena¼ CAPM bgHajfa cMnUleRKa¼fñaksgÇwmrbs´´b&NÑPaKh‘un i (=ERi

– rf ) esµInwgplKuNénemKuN nwgcMnUleRKa¼fñakTIpßarsgÇwm (=ERm – rf) . RbsinebI CAPM GaceRbIán

eyIgmansSanPaBdUckñúgrUb 6.1 . bnÞat´EdlbgHajkñúgrUb ehAfabnÞat´TIpßarb&NÑPaKh‘un [Security Market

Line (SML)] .

rUb 6.1 : eRKa¼fñak´tamRbB&n§

sMrabeKalbMNgBiesaFn_ (6.1.2) GacsresrCa £

Ri – rf =i (Rm – rf ) + ui (6.1.3)

rW Ri – rf =i + i (Rm – rf ) + ui (6.1.4)

KMrUen¼ehAfa KMrUTIpßar (Market Model) . RbsinebI CAMP GaceRbIán ena¼ i RtUveKrMBwgfa mantMél esµI

sUnü (emIlrUb 6.2) .

rUb 6.2 : KMrUTIpßarènRTwsþIb&NÑPaKh‘un (snµtyk i = 0 )

ktsMKal´fakñúg (6.1.4) GefrTak´Tg Y KW (Ri – rf ) nig GefrKitbBa©Úl X KW i (emKuNERbRbYl) buEnþ

minEmnCa (Rm – rf ) GefrKitbBa©Úl . dUecñ¼ edIm,IeFVIrWERh:ssüúg (6.1.4) eKRtUvEtánRbmaN i Camun Edl

CaFmµtaRtUveKTajmkBIbnÞat´lkçN£ dUcbgHajkñúglMhat´ 5.5 (esckþIlMGitemIlkñúglMhat´ 8.34) .

0

ERi –rf

ERi –rf

1

bnÞat´TIpßarb&NÑPaKh‘un

i

Ri – rf

i

eRKa¼fñak´tamRb&Bn§

cMnUl

eRKa¼

fñak

´b&N

ÑPaK

h‘un

saklviTüal&yPUminÞPñMeBj ROYAL UNIVERSITY OF PHNOM PENH

edátWm¨gKNitviTüa 147 MATHEMATICS DEPARTMENT

dUc«TahrN_bgHaj CYnkalRTwsþICaRKw¼tMrUveGayKµantYcMnuckat´G&kßkñúgKMrU . krNIepßgeTot EdlKMrU

mancMnuckat´sUnüsmRsb KWCasmµtikmµcMnUlGciéRnþrbs´ Milton Friedman EdlG¼Gagfa kareRbIRásGciéRnþ

smamaRtnwgcMnUlGciéRnþ ; RTwsþIviPaKcMNayEdlRtUveKdakCasMeNIfacMNayGefrénplitkmµsmamaRteTAnwg

Tinñpl ; nigbMNkRsayénRTwsþIGñkrUbiyvtSúniymxø¼ EdlG¼Gagfa kMritbMErbMrYléfø (KWkMritGtiprNa) smamaRt

eTAnwgkMritbMErbMrYlénkarp:tp:g´Rák´ .

etIeKánRbmaNKMrUdUcCa (6.1.1) y¨agdUcemþc ? etIKMrUkMntnUvbBaHaGVIxø¼? edIm,IeqøIynwgsMnYrTaMgen¼ CadMbUg

eyIgsresr SRF én (6.1.1) Ca £

Yi = 2 Xi + iu (6.1.5)

}LÚven¼edayeRbIviFI OLS elI (6.1.5) eyIgTTYlánrUbmnþxageRkamsMrab´ 2 nigv¨arü¨g´rbs´va (sMray

bBa¢ak´pþl´eGaykñúgesckþIbEnSm 6A Epñk 6A.1):

22

ˆ

i

ii

X

YX

(6.1.6)

2

2

2 )ˆvar(

iX

(6.1.7)

Edl 2 RtUvánRbmaNeday £

1

ˆˆ

22

n

ui (6.1.8)

eKGaceRbobeFobrUbmnþTMagen¼CamYyrUbmnþ EdlTTYlánenAeBltYcMnuckat´G&kßRtUvbBa©ÚlkñúgKMrU £

22

ˆ

i

ii

x

yx

(3.1.6)

2

2

2 )ˆvar(

ix

(3.3.1)

2

ˆˆ

22

n

ui (3.3.5)

PaBxusKñarvagsMnMurUbmnþTMagBIrCakarc,as´lasNas´ £ kñúgKMrUEdlminmantYcMnuckat´G&kß eyIgeRbIplbUkeday

kaernigplKuNedIm buEnþkñúgKMrUmancMnuckat´G&kß eyIgeRbIplbUkkaer nigplKuNEktMrUv (BImFüm). TI 2 df sMrab´

tMélKNna 2 esµInwg n –1 kñúgkrNITI 1 nig n –2 kñúgkrNITI 2 (ehtuGVI?) .

eTa¼CaKMrUKµancMnuckat´G&kß rWcMnuckat´G&kßesµIsUnü GacCakarsmrmütamkareRbIRás´k¾eday KMrUen¼man

lkçN£Biessxø¼ EdlcaMácRtUvktsMKal´ . CadMbUg iu EdlCanic©kalesµIsUnüsMrabKMrU EdlmantYcMnuckat´

G&kß (KMrUtamTMlab´) mincaMácesµIsUnü enAeBltYena¼minman . Casegçb iu mincaMác´esµIsUnüsMrab´KMrUkat´

tamKl´G&kß . TI 2 emKuNkMnt r2 Bnül´kñúgCMBUk 3 EdlCanic©kalCacMnYnviC¢mansMrab´KMrUtamTMlab´ GacCYn

kalCacMnYnGviC¢mansMrab´KMrUminmancMnuckatG&kß . lT§pldUcKñaen¼ekIteLIg BIeRBa¼ r2Bnül´kñúgCMBUk 3 snµt

saklviTüal&yPUminÞPñMeBj ROYAL UNIVERSITY OF PHNOM PENH

edátWm¨gKNitviTüa 148 MATHEMATICS DEPARTMENT

yagc,asfa cMnuckat´G&kßRtUvbBa©ÚlkñúgKMrU . dUecñ¼ r2 EdlKNnatamTMlab´ GacminsmRsbsMrabKMrUkattam

Kl´G&kß .

r2 sMrabKMrUkat´tamKl´G&kß (r2

for Regression-through-Origin Model)

dUcánktsMKal´ nigkarBiPakßakñúgesckþIbEnSm 6A Epñk 6A.1 r2 kñúgCMBUk 3 minsmrmüsMrabrWERh:s

süúg EdlminmancMnuckat´G&kß . buEnþeKGacKNna r2 edIm sMrab´KMrUEbben¼ EdlkMnteday £

r2 (edIm)=

22

2)(

ii

ii

YX

YX

(6.1.9)

ktsMKal´£ manplbUkkaer nigplKuNedIm (KWfa minEktMrUvtammFüm) .

eTa¼Ca r2 (edIm) eKarBtamlkçxNÐénTMnak´TMng 0 < r < 1 k¾eday eKminGaceRbobeFobedaypÞal

nigtMél r2 tamTMlab´. edayehtuen¼ GñkniBn§xø¼minbgHajtMél r

2 sMrab´KMrUrWERh:ssüúgcMnuckat´G&kßesµIsUnü .

BIeRBa¼EtlkçN£BiessénKMrUen¼ eKcaMácRtUvEtRby&tñRbEygkñúgkareRbIKMrUcMnuckat´G&kßsUnü . RbsinebI

minmankarsgÇwmsmehtupl eKRtUveRbIKMrUmancMnuckat´G&kßtamTMlab´ . krNIen¼manRbeyaCn_BIr £TI 1

RbsinebItYcMnuckat´G&kßRtUvbBa©ÚlkñúgKMrU b¨uEnþvakøayCaminmansarsMxan´sSiti (minesµIsUnü)/ sMrab´eKalbMNgGnuvtþ

eyIgeFVIrWERh:ssüúgkattamKl´G&kß . TI 2 (sMxan´CagTI 1) RbsinebItamBitmancMnuckat´G&kßkñúgKMrUbuEnþ eyIg

enAEteFVIkartMrUvrWERh:ssüúgkattamKl´G&kß/ eyIgnwgbeg;ItkMhusbBa¢ak´ (Specification Error) EdlmineKarB

tamkarsnµt 9 ènKMrUrWERh:ssüúglIenEG‘rkøasik .

«TahrN_cgðúlbgHaj £ bnÞatlkçN£énRTwsþIb&NÑPaKh‘un (An Illustrative Example: The

Characteristic Line of Portfolio Theory)

tarag 6.1 pþl´eGayTinñn&yelIkMritcMnUlBIvinieyaKRbcaMqñaM (%) elI Afuture Fund (CasgHhFnEdl

eKalbMNgvinieyaKrbs´xøÜnCasMxan´ KWcMenjTunGtibrima) nig elIb&NÑPaKh‘unTIpßar (RtUvvas´edaysnÞsßn_

Fisher Index sMrab´kMlugqñaM 1970-1980) .

kñúglMhat´ 5.5 eyIgánBnül´bnÞat´lkçN£énviPaKvinieyaK EdlGacsresrCa £

Yi = i +iXi + ui (6.1.10)

Edl Yi = kMritcMnUlBIvinieyaKRbcaMqñaM (%) elI Afuture Fund

Xi = kMritcMnUlBIvinieyaKRbcaMqñaM (%) elIb&NÑPaKh‘unTIpßar

i = emKuNRábTis (ehAfa emKuN kñúgRTwsþIb&NÑPaKh‘un)

i = cMnuckat´G&kß

kñúgGtSbTniBn§ minmanmtiCaTUeTAGMBItMél i mun . lT§plBiesaFn_xø¼ánbgHajfa vamantMélviC¢man nig

mansar£sMxan´sSiti nigxø¼bgHajfa vaminxusBIsUnüCasar£sMxan´sSiti . kñúgkrNITI2en¼ eyIgGacsresrKMrUCa £

Yi = i Xi + ui (6.1.11)

EdlCarwERh:ssüúgkattamKl´G&kß .

Formatted

saklviTüal&yPUminÞPñMeBj ROYAL UNIVERSITY OF PHNOM PENH

edátWm¨gKNitviTüa 149 MATHEMATICS DEPARTMENT

RbsinebIeyIgsMerccitþeRbIKMrU (6.1.1) eyIgTTYlánlT§plrWERh:ssüúgxageRkam (emIllT§plkmµviFI

kMuBüÚT&r SAS kñúgesckþIbEnSm 6A Epñk 6A.2) :

Yi = 1,0899Xi

(0,1916) r2 (edIm) = 0,7825 (6.1.12)

t = (5.6884)

EdlbgHajfa i FMCagsUnüCasar£sMxan . bMNkRsayen¼KWfa kMenIn 1% elIkMritcMnUlTIpßarBIvinieyaK naM

eGayman CamFüm kMenIn 1,09 % énkMritcMnUlBIvinieyaKelI Afuture Fund .

etIeyIgGacdwgc,as´fa KMrU (6.1.11) minEmn KMrU (6.1.10) RtwmRtUv CaBiesselIsMenIEdlfa

minmankaryl´eXIjsmehtuplelIsmµtikmµEdl i BitCaesµIsUnü rWeT ? eKGacepÞógpÞat´edaykareFVI rWERh:s-

süúg (6.1.10) . edayeRbITinñn&ypþl´eGaykñúgtarag 6.1 eyIgTTYlánlT§plxageRkam £

iY =1,2797 + 1,0691Xi

(7,6886) (0,2383) (6.1.13)

t = (0,1664) (4,4860) r2 = 0,7155

ktsMKal´ £ tMél r2én (6.1.12) nig (6.1.13) minGacRtUveRbobeFobedaypÞal´ . BIlT§plTaMgen¼ eKmin

Gac bdiesFsmµtikmµEdlcMnuckat´G&kßBit esµIsUnü edaykarbBa¢ak´kareRbIRás´ (6.1.1) EdlCaKMrUrWERh:ssüúg

kattamKl´G&kß .

ktsMKal´fa minmanPaBxusKñaeRcInelIlT§plén (6.1.12) nig (6.1.13) eTa¼CalMeGogKMrUán´

RbmaN én TabCagbnþicsMrabKMrUrWERh:ssüúgkatKl´G&kß k¾eday . krNIen¼ RtUvKñanwgGMn¼GMnagrbs´ Theil

Edlfa RbsinebI i tamBitesµIsUnü ena¼emKuNRábTisGacRtUvvas´edaykMritCak´lakeRcInCag £ edayeRbI

Tinñn&ykñúgtarag 6.1 niglT§plrWERh:ssüúg GñkGanGacepÞógpÞat´ánfa cenøa¼TMnukcitþ 95% sMrab´emKuNRáb´

Tis énKMrUrWERh:ssüúgkattamKl´G&kß KW (0,6566, 1,5232) rIÉsMrabKMrU (6.1.13) KW (0,5195, 1,6186) KWfa

cenøa¼TMnukcitþmuntUcCagcenøa¼TMnukcitþTI 2 .

tarag 6.1: kMritcMnUlRbcaMqñaMBIvinieyaKelI Afuture Fund nig

elI Fisher Index (b&NÑPaKh‘unTIpßar) qñaM 1971-1980

qñaM

cMnUlBIvinieyaKelI

Afuture Fund (%)

Y

cMnUlBIvinieyaKelI

Fisher Index (%)

X 1971 67,5 19,5

1972 19,2 8,5

1973 -35,2 -29,3

1974 -42,0 -26,5

1975 63,7 61,9

1976 19,3 45,5

1977 3,6 9,5

1978 20,0 14,0

1979 40,3 35,3

1980 37,5 31,0

saklviTüal&yPUminÞPñMeBj ROYAL UNIVERSITY OF PHNOM PENH

edátWm¨gKNitviTüa 150 MATHEMATICS DEPARTMENT

RbPB £ Haim Levy and Marshell Sarnat, Portfolio and Investment Selection: Theory and

Practice, Prentice Hall International, Englewood Cliffs, N.J, 1984, pp.730 and 738.

Tinñn&yen¼TTYlánedayGñkniBn§ BI Weignberg Investment Service, Investment Companies, 1981

edition.

6¿2 karkMnt´kMrit nigÉktþaénrgVas´(Scaling and Units of Measurement)

edIm,Iyl´KMnitEdlbkRsaykñúgEpñken¼ cUrseg;temIlTinñn&ypþleGaykñúgtarag 6.2. Tinñn&ykñúgtarag

en¼sMedAelIkarvinieyaKkñúgRsukÉkCnsrubrbs US (GPDI) nig NGP CaduløakñúgqñaM 1972 sMrab´kMlugeBl

1974-1983 . CYrQrTI 1 nigCYrTI 2 pþl´eGayTinñn&yelI GPDI CaxñatBan´landuløa nig landuløaerogKña

rIÉ CYrTI 3 nig TI 4 pþl´eGayTinñn&yelI GNP CaxñatBan´landuløa niglanduløa erogKña .

tarag 6.2: karvinieyaKkñúgRsukÉkCnsrub (GPDI) nig GNP

CaduløaqñaM 1972 kñúg US kMlugqñaM 1974-1983

qñaM GPDI

(Ban´landuløa qñaM 1972)

(1)

GPDI

(landuløa qñaM 1972)

(2)

GNP

(Banlanduløa qñaM1972)

(3)

GNP

(landuløaqñaM 1972)

(4)

1974 195,5 195500 1246,3 1246300

1975 154,8 154800 1231,6 1231600

1976 184,5 184500 1298,2 1298200

1977 214,2 214200 1369,7 1369700

1978 236,7 236700 1438,6 1438600

1979 236,3 236300 1479,4 1479400

1980 208,5 208500 1475,0 1475000

1981 230,9 230900 1512,2 1512200

1982 194,3 194300 1480,0 1480000

1983 221,0 22100 1534,7 1534700

RbPB £ Economic Report of the President, 1985, p.234 (sMrab´Tinñn&yCaxñatBan´landuløa)

«bmafa kñúgrWERh:ssüúgén GPDI elI GNP GñkRsavRCaveRbITinñn&yvas´CaxñatBan´landuløa buEnþmñak´

eToteRbITinñn&yelIGefrTaMgen¼vas´Caxñatlanduløa . etIlT§plrWERh:ssüúgnwgdUcKñakñúgkrNITaMgBIr? RbsinebI

mindUcKña etIeKKYreRbIxñatmYyNa ? Casegçb etIÉktaEdl Gefr Y nig X EdleKkMnt eFVIeGaymanlT§pl

rWERh:ssüúgxusKñarWeT? RbsinebIxus etIeKalkarN_smehtuflmYyNa KYrEtRtUveKeRbIsMrab´viPaK rWERh:ssüúg?

edIm,IeqøIynwgsMnYrTaMgen¼ eyIgbnþdMeNIrCaRbB&n§ £

Yi = ii uX ˆˆˆ21 (6.2.1)

Edl Y = GPDI nig X = GNP . eyIgkMnt £

ii YwY 1* (6.2.2)

ii XwX 2* (6.2.3)

Edl w1 nig w2 CacMnYnefr nigehAfa ktþakMrit (Scale Factors) . w1 GacesµI rWxusBI w2 .

saklviTüal&yPUminÞPñMeBj ROYAL UNIVERSITY OF PHNOM PENH

edátWm¨gKNitviTüa 151 MATHEMATICS DEPARTMENT

BI (6.2.2) nig (6.2.3) Cakarc,asfa *iY nig *

iX RtUveKdak´kMritCa Yi nig Xi . dUecñ¼ RbsinebI Yi

nig Xi RtUvvas´CaxñatBan´landuløa nig eKcg´vas´vaCaxñatlanduløa eyIgnwgman *iY =1000Yi nig *

iX =

1000Xi . dUecñ¼ eKman w1 =w2 = 1000 .

}LÚven¼cUrseg;temIlrWERh:ssüúgedayeRbIGefr *iY nig *

iX £

*iY = *

1 + *2 Xi + *ˆiu (6.2.4)

Edl *iY = w1Yi ;

*iX =w2Xi ; nig *ˆiu =w1 iu (ehtuGVI?)

eyIgcgrkTMnak´TMngrvagKUtMél £

1. 1 nig *1

2. 2 nig *2

3. var( 1 )nig var( *1 )

4. var( 2 ) nig var ( *2 )

5. 2 nig2*

6. 2xyr nig

2**yx

r

BIRTwsþIkaertUcbMput eyIgdwg (emIlCMBUk 3) fa £

1 =Y - 2 X (6.2.5)

2 = 2i

ii

x

yx

(6.2.6)

var( 1 ) = 2

2

2

i

i

xn

X

(6.2.7)

var( 2 ) = 2

2

ix

(6.2.8)

2 = 2

2

n

ui (6.2.9)

edayeRbIviFI OLS elI (6.2.4) eyIgTTYlán £

*1 = *Y - *

2*X (6.2.106)

*2 =

2*

**

i

ii

x

yx

(6.2.11)

var( *1 ) =

2*

2*

2*

.

i

i

xn

X

(6.2.12)

var ( *2 )=

2*

2*

ix

(6.2.13)

Formatted

Formatted

Formatted

saklviTüal&yPUminÞPñMeBj ROYAL UNIVERSITY OF PHNOM PENH

edátWm¨gKNitviTüa 152 MATHEMATICS DEPARTMENT

2* = 2

ˆ2*

n

ui (6.2.14)

BIlT§plTaMgen¼ eKgaynwgbeg;ItTMnak´TMngrvagsMnMuTaMgBIréntMélánRbmaNáraEmt . GVIEdleKRtUveFVIKW kMnt´

TMnak´TMngTaMgen¼eLIgvij £ *iY =w1Yi ( rW *

iy =w1yi); *iX =w2Xi (rW *

ix =w2xi) ; *ˆiu =w1 iu ;

*Y =w1Y nig *X =w2 X . edayeRbIkarkMntEbben¼ GñkGanGacepÞógpÞatánedaygayfa £

*2 =

2

1

w

w2 (6.2.15)

*1 = 1w

1 (6.2.16)

2* = 2

1w 2 (6.2.17)

var( *1 ) = 2

1w var(1 ) (6.2.18)

var( *2 ) =

2

2

1

w

wvar( 2 ) (6.2.19)

2xyr = 2

**yxr (6.2.20)

BIlT§plxagelI eKKYrEtc,as´fa ebImanlT§plrWERh:ssüúg EpðkelIkMritrgVas´mYy eKGacTajyk

lT§pl EpðkelIkMritrgVas´mYyeTot enAeBlktþakMrit w s:al´ . kñúgkarGnuvtþ eKKYrEteRCIserIsÉktargVas´

smrmü . manRbeyaCn_tictYcNas´kñúgkareRbIelx 0 kñúgkarsresrCaxñatlan rWxñatBanlan .

BIlT§plpþl´eGaykñúg (6.2.15) rhUtdl´(6.2.20) eKGacTajykkrNIBiessmYycMnYnánedaygay .

«TahrN_ RbsinebI w1= w2 (ktþakMritesµIKña) ena¼emKuNRáb´Tis niglMeGogKMrUrbs´vaminrgnUv}T§iBlBIbþÚrBI

(Yi,Xi) eTA kMrit ( *iY , *

iX ) . eTa¼Cay¨agNak¾edaycMnuckat´G&kß niglMeGogKMrUrbs´va RtUvKuNnwgRtUvKuNnwg

w1 . buEnþRbsinebIkMrit X minERbRbYl (w1 = 1) nig kMrit Y ERbRbYltamktþa w1 ena¼emKuNRábTisnig

emKuNcMnuckat´G&kß niglMeGogKMrURtUvKñarbs´va RtUvKuNnwgktþa w1 EtmYy . CacugeRkayen¼ RbsinebIkMrit Y min

ERbRbYl (w1=1) buEnþ kMrit X ERbRbYltamktþa w2 ena¼emKuNRábTisniglMeGogKMrURtUvKñarbs´vaRtUvKuNnwgktþa

(1/w2) buEnþemKuNcMnuckat´G&kß niglMeGogKMrUrbs´vaminERbRbYl .

eTa¼Cay¨agNak¾eday eKKYrktsMKal´fa bMElgBI (Y,X) eTAkMrit (Y*,X

*) mineFVIeGaymankarERbRbYl

lkçN£énsnÞsßn_ánRbmaN OLS EdlánBiiPakßakñúgCMBUkmun .

«TahrN_Caelx £ TMnak´TMngrvag GPDI nig GNP enA US (1974-1983)

edIm,IbgHajPsþútagGMBIlT§plRTwsþIxagelI eyIgRtlbeTA«TahrN_éntarag6.2 nigseg;temIllT§pl

rWERh:ssüúgxageRkam . (tYelxkñúgrgVg´RkckCalMeGogKMrUánRbmaN) .

GPDI nig GNP CaxñatBan´landuløa £

GPDIt = -37,0015205 + 0,17395.GNPt

Formatted

Formatted

Formatted

Formatted

Formatted

saklviTüal&yPUminÞPñMeBj ROYAL UNIVERSITY OF PHNOM PENH

edátWm¨gKNitviTüa 153 MATHEMATICS DEPARTMENT

(76,2611278) (0,055406) (6.2.21)

r2 = 0,5641

GPDI nig GNP Caxñatlanduløa £

GPDIt = -37001,5205 + 0,17395 GNPt

(76261,1278) (0,05406) (6.2.22)

r2 =0,5641

ktsMKal´fa cMnuckat´G&kß niglMeGogKMrUrrbs´va KW 1000 (w1 = 1000 kñúgkarbMElgBIxñatBan´lan eTACaxñat

landuløa) KuNnwgtMélRtUvKñakñúgrWERh:ssüúg(6.2.21) buEnþemKuNRábTisniglMeGogKMrUrbs´vaminERbRbYlGaRs&y

edayRTwsþI .

GPDI xñatBan´landuløa nig GNP xñatlanduløa £

GPDIt =-37,0015205 + 0,00017395 GNPt

(76,2611278) (0,00005406) (6.2.23)

r2 = 0,5641

dUcánrMBwgTuk emKuNRábTisnigKMlatKMrUrbs´va KWmantMél (1/1000) éntMélrbs´vakñúg (6.2.21) edayehtu

fa manEtkMrit X rW GNP ERbRbYl .

GPDI xñatlanduløa nig GNP xñatBan´landuløa £

GPDIt = -37001,5205 + 173,95 GNPt

(76261,1278) (54,06) (6.2.24)

r2 = 0,5641

ktsMKal´facMnuckat´G&kß nigemKuNRábTis rYmTaMglMeGogKMrUerogKñarbs´va mantMél 1000 dgeFobnwgtMél

rbs´vakñúg (6.2.21) GaRs&yedaylT§plRTwsþIrbs´eyIg .

kMnt´sMKal´elIbMNkRsay

edayehtufa emKuNRábTis 2 RKan´EtCakMritbMErbMrYl ena¼vaRtUvvas´CaÉktaénpleFob £

dUecñ¼kñúgrWERh:ssüúg (6.2.21) bMnkRsayemKuNRábTis 0,17395 KWfa RbsinebI GNP ERbRbYl 1Ékta

(mYyBan´landuløa) ena¼ GDPI CamFümnwgERbRbYledaytMél 0,17395 Ban´landuløa . kñúgrWERh:ssüúg

(6.2.23) bMErbMrYl 1Éktaén GNP (1 landuløa) naMeGaymanbMErbMrYlCamFüm 0,00017395 Banlanduløa

elI GPDI . lT§plTaMgBIren¼rg}T§iBldUcKñaBI}T§iBlén GNP elI GPDI KWfa vaRKan´EtsresrCaÉkta

rgVas´xusKñaEtbueNÑa¼ .

6¿3 TMrg´GnuKmn_énKMrUrWERh:ssüúg (Functional Forms of Regression Models)

dUcánktsMKal´kñúgCMBUk 3 GtSbTen¼Cab´TakTgCasMxan´nwgKMrUEdllIenEG‘relItMèlàraEmt buEnþ

vaGaclIenEG‘r rWminEmnlIenEG‘relIGefr . kñúgEpñkxageRkam ey Ignwgseg;temIlKMrUrWERh:ssüúgEdleRbIRás´

ÉktaénGefrTak´Tg Y

ÉktaénGefrKitbBa©Úl X

saklviTüal&yPUminÞPñMeBj ROYAL UNIVERSITY OF PHNOM PENH

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CaTUeTA EdlGacminEmnlIenEG‘relIGefr buEnþvalIenEG‘relItMèlàraEmt rWEdlGacRtUveKbeg;IteGayman

lkçN£lIenEG‘rEbben¼ edaybMElgsmrmüènGefr. kñúgkrNIBiess eyIgBiPakßaelIKMrUrWERh:ssüúgxageRkam £

1. KMrUlIenEG‘relakarIt (The Log-Linear Model)

2. KMrUBak´kNþalelakarIt (Semilog Model)

3. KMrURcas (Reciprocal Models)

eyIgBiPakßaGMBIlkçN£BiessénKMrUnImYy@ Edlsmrmü nigGMBIviFIRtUveKánRbmaN . kñúgKMrUnImYy@RtUv

eKBnül´edayman«TahrN_smRsb . 6¿4 viFIvas´bMlas´bþÚr £ KMrUlIenEG‘relakar It (How to Measure Elasticity: The Log-Linear Model)

seg;temIlKMrUxageRkam EdleKehAfa KMrUrWERh:ssüúg Giucs,¨ÚNgEsül (Exponential Regression

Model) :

Yi = 12

iX iue (6.4.1)

EdlGacsresrCa £

lnYi = ln1 + 2lnXi + ui (6.4.2)

Edl ln = elakarItFmµCati (elakarIteKal e =2,718) .

RbsinebIeyIgsresr (6.4.2) Ca £

lnYi = + 2lnXi + ui (6.4.3)

Edl = ln 1 , KMrUen¼lIenEG‘relItMèlàraEmt nig 2 KWvalIenEG‘relIelakarItènGefr Y nig X nig

GacRtUvánRbmaNedayrWERh:ssüúg OLS . BIeRBa¼EtlkçN£lIenEG‘ren¼ KMrUEbben¼ehAfa KMrUlIenEG‘r

elakarIt-elakarIt/ elakarItDub rWelakarIt (log-log, double-log or log-linear models) .

RbsinebIkarsnµtènKMrUrWERh:ssüúglIenEG‘rkøasikRtUvbMeBj tMèlàraEmtèn (6.4.3) GacRtUvánRbmaN

edayviFI OLS edaykartag £

*iY = +2

*iX + ui (6.4.4)

Edl *iY =lnYi nig *

iX =lnXi . snÞsßn_ánRbmaN OLS nig EdlTTYlán nwgCasnÞsßn_á¨n´

RbmaNlIenEG‘rminlMeGoglðbMputèn nig 2 erogKña .

lkçN£KYrcabGarmµN_énKMrUelakarIt-elakarIt EdleFVIeGaymankareRbIRásCaTUeTAkñúgkargarGnuvtþ KWfa

emKuNRábTis 2 vasbMlas´bþÚrén Y eFobnwg X (bMErbMrYlCaPaKryén Y sMrab´bMErbMrYltMéltUcén X) . dUecñ¼

RbsinebI Y tageGaybrimaNtMrUvkarTMnij nig X tageGaytMélmYyÉktarbs´va ena¼ 2 vas´bMlas´bþÚrtMrUvkaréfø

(Price Elasticity of Demand) EdlCatMéláraEmtmYymankarcab´GarmµN_y¨agxøaMgkñúgvis&yesdækic©. Rbsin

ebITMnak´TMngrvagbrimaNtMrUvkar nigtMélrbs´vadUcánbgHajkñúgrUb 6.3a ena¼bMElgelakarItDubdUcbgHajkñúgrUb

6.3b nwgpþl´eGaynUvtMélánRbmaNénbMlas´bþÚréfø (-2) .

lkçN£BiessBIrènKMrUlIenEG‘relakarIt GacRtUvktsMKal´ £ KMrUsnµtfa emKuNbMlas´bþÚrrvag Y nig X

(2) rkßatMélefr (ehtuGVI?) ehtudUecñ¼eKGacehAvaánmü¨ageTotfa KMrUbMlas´bþÚrefr (Constant Elasticity

Model) . mü¨ageTot dUckñúgrUb 6.3b bgHaj bMErbMrYlelI lnY kñúgbMErbMrYlmYyÉktaén lnX (bMlas´bþÚr 2)

saklviTüal&yPUminÞPñMeBj ROYAL UNIVERSITY OF PHNOM PENH

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rkßatMélefr eTa¼CaRtg´tMél lnX EdleyIgvas´bMlas´bþÚrNak¾eday . lkçN£mü¨ageToténKMrUKWfa eTa¼Ca

nig 2 CasnÞsßn_ánRbmaNminlMeGogén nig 2 k¾eday 1 (tMéláraEmtEdlbBa©ÚlkñúgKMrUedIm) enA

eBlRtUvánRbmaNeday 1 =antilog( ) CasnÞsßn_ánRbmaNlMeGogedayxøÜnva . eTa¼Cay¨agNak¾eday

kñúgcMeNaTGnuvtþPaKeRcIn tYcMnuckatG&kßmansar£sMxan´bnÞab´bnßM nigeKmincaMác´ármÖGMBIkarrktMélánRbmaN

minlMeGogrbseT .

kñúgKMrUBIrGefr viFIgaybMputedIm,InwgsMerccitþfa etIKMrUlIenEG‘relakarItNaRtUvtMrUvTinñn&y eKRtUvedAkñúg

düaRkamBRgayén lnYi Tlnwg lnXi nigemIlfaetIcMnucBRgayTMngCartenACitbnÞat´ dUckñúgrUb 6.3b .

rUb 6.3 : KMrUbMErbMrYlefr

«TahrN_cgðúlbgHaj £ GnuKmn_tMrUvkarkaehV (An Illustrative Example: The Coffee

Demand Function Revisited)

eyageTAGnuKmn_tMrUvkaehVEpñk 3.7 GñkCMnYykarRsavRCavànpþl´Bt’manfa enAeBlTinñn&yRtUvànedA

edayeRbI kMrit lnY nig lnX düaRkamBRgayTMngCacgðúlbgHajfa KMrUelakarIt-elakarItGacpþl´eGaykartMrUv

Tinñn&yànlðdUcCaKMrUrWERh:ssüúglIenEG‘r (3.7.1) Edr . edayeRbIkarKNna GñkCMnYykarRsavRCavTTYlán

lT§plxageRkam £

lnYt = 0,7774 – 0,2530lnXt r2 = 0,7448

(0,0152) (0,04494) F1,9 = 26,27 (6.4.5)

t = (51,1447) (-5,1214)

tMél p =( 0,000) (0,0003)

Edl Yt = kareRbIRás´kaehV (cMnYnEBgkññúgmnusßmñakkñúgmYyéf¶) nig Xt = tMélBiténkaehV (duløakñúgmYyepan) .

BIlT§plTaMgen¼ eyIgeXIjfa emKuNbMlas´bþÚréfø KW –0.25 Edlmann&yfa sMrab´kMenIn 1 % elI

tMélBiténkaehVkñúg 1epan/ tMrUvkaehV (vasCacMnYnEBgeRbIRás´kñúgmYyéf¶) CamFümfycu¼RbEhl 0,25% .

edayehtufabMlas´bþÚréfø 0,25 tUcCag 1 KitCatMéldacxat eyIgGacniyayfa tMrUvkarkaehVKµanbMlas´bþÚr-éfø

( Price-Inelastisity) .

brim

aNtMrUvkar

ekak

arItén

brim

aNtMrUvkar

éfø elakarIténéfø X lnX

Y = 12

iX lnY = ln1 - 2lnXi

Y lnY

(a) (b)

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sMnYrKYrcabGarmµN_£ edayeRbobeFoblT§plènGnuKmn_tMrUvkarlIenEG‘relakarIt nwgGnuKmn_tMrUvkarlIen-

EG‘r (3.7.1) etIeyIgsMerccitþfa mYyNaCaKMrUlðCageK ? etIeyIgGacniyayfa (6.4.5) RbesIrCag (3.7.1)

BIeRBa¼ tMél r2 rbs´vax<s´Cag (0,7448 Tlnwg0,6628)? eyIgminGacniyayfa dUcnwgbgHajkñúg CMBUk 7 enA

eBlGefrTakTgénKMrUTaMgBIr minesµIKña (lnY nig Y) , tMél r2 TaMgBIrminGaceRbob eFobKñaánedaypÞal´ . eyIg

k¾minGaceRbobemKuNTaMgBIredaypÞalánEdr BIeRBa¼kñúg (3.7.1) emKuNRábTis pþl´eGay}T§iBlénbMErbMrYl 1

ÉktaelItMélkaehV «¿ $ 1kñúgmYyepan KitCabrimaNfycu¼CatMéldacxat (min EmneFob) efrénkareRbIRás´

kaehVmantMélesµInwg0,4795 EBgkñúgmYyéf¶ . mü¨ageTot emKuN –0,2530 EdlTTYlánBI (6.4.5) pþl´eGay

karfycu¼CaPaKryefrelIkareRbIRás´kaehVCalT§plénkMenIn 1% éntMélkaehVkñúgmYyepan (KWvaeFVIeGayman

bMlas´bþÚréfø) .

etIeyIgGaceRbobeFoblT§plénKMrUTaMgBIry¨agdUcemþc ? sMnYren¼CaEpñkd¾tUcmYyénviPaKbBa¢ak´

(Specification Analysis) EdlCaRbFanbTnwgRtUvBiPakßakñúgCMBUk 13 . sMrabeBl}LÚven¼ viFImYyEdleyIg

GaceRbobKMrUTaMgBIr KWRtUvKNnargVas´ánRbmaNénbMlas´bþÚréføsMrab´KMrU (3.7.1) . eKGaceFVItamviFIxageRkam £

bMlas´bþÚr EénGefr Y («¿brimaNtMrUvkar) eFobnwgGefrmYyeTot X («¿ éfø) RtUvkMnteday £

E =

= 100)./(

100)./(

XX

YY

(6.4.6)

= Y

X

X

Y

= emKuNRábTis . (X/Y)

Edl kMnteGaybMErbMrYltMéltUc . RbsinebI tUclµm eyIgGacCMnYs Y/X edaykarKNnaedrIev

(dY/dX) .

}LÚven¼sMrab´KMrUlIenEG‘r (3.7.1) tMélánRbmaNénemKuNRábTisRtUvpþl´eGayedayemKuNán´

RbmaN 2 EdlsMrab´GnuKmn_tMrUvkarkaehV KW –0,4795 . dUc (6.4.6) bgHaj edIm,IKNnabMlas´bþÚr eyIgRtUv

EtKuNemKuNRábTisnwgpleFob X/Y (éfø /brimaN) . buEnþ eyIgerIsyktMél X nig Y Naxø¼ ? dUckñúg

tarag 3.4 bgHaj manKUtMél (X,Y) 11KU . RbsinebIeyIgeRbItMélTaMgGsen¼ eyIgnwgman 11tMélánRbmaN

énbMlas´bþÚr .

eTa¼Cay¨agNak¾eday kñúgkarGnuvtþ bMlas´bþÚrRtUvKNnaRtg´cMnuctMélmFümén Y nig X . eyIgKNna

tMélánRbmaNénbMlas´bþÚrmFüm (Mean Elasticity) . sMrab´«TahrN_rbseyIg Y =2,43 EBg nig

X =$1,11 . edayeRbItMélTaMgen¼ nigtMélánRbmaNemKuNRáb´Tis –0,4795 eyIgTTYlánBI (6.4.6)

nUvemKuNbMlas´bþÚrtMélmFüm –0,4795.1,1/2,43 = -0,219 -0,22 . lT§plen¼xusBIemKuNbMlas´bþÚr

bMErbMrYlelI Y (%)

bMErbMrYlelI X(%)

saklviTüal&yPUminÞPñMeBj ROYAL UNIVERSITY OF PHNOM PENH

edátWm¨gKNitviTüa 157 MATHEMATICS DEPARTMENT

–0,25 EdlTTYlànBIKMrUlIenEG‘relakarIt . kMntsMKal´fa bMlas´bþÚrTI2 rkßatMèldEdledayminKitBItMèl

EdleKvas´ (ehtuGVI?) rIÉbMlas´bþÚrTI1GaRs&yelItMélmFümCak´lak .

6¿5 KMrUBak´kNþalelakarIt£ KMrUlIenEG‘relakarIt nigKMrUelakarItlIenEG‘r (Semilog

Models: Log-Lin and Lin-Log Models)

viFIvas´GRtakMenIn £ KMrUlIenEG‘relakarIt (How to Measure the Growth Rate: The Log-Lin Model)

Gñkesdækic©/ GñkCMnYj nigrdæaPiálmankarcab´GarmµN_nwgkarvas´GRtakMenInénGefresdækic©Cak´lak´ dUc

Ca cMnYnRbCaCn/ GNP, karp:tp:g´Rák/ kargar/ RbsiT§iPaBBlkmµ/ {nPaBpvikaBaNiC¢kmµ . l .

kñúglMhat´ 3.22 eyIgánbgHajTinñn&yelI GDP Bitrbs´ US kñúgkMlugeBl 1972-1991 . «bma

fa eyIgcg´rkGRtakMenInén GDP Bit kñúgkMlugeBlen¼ . tag Yt = GDP Bit (RGDP) Rtg´xN£eBl t nig

Y0 = tMél GDP xN£edIm («¿ 1972) . }LÚven¼KYrrlwkfa rUbmnþkarRáksmasBIkarsikßa rUbiyvtSú FnaKar

nighirBaØvtSú KW £

Yt =Y0 (1+r)t (6.5.1)

Edl r CaGRtakarRáksmasénkMenInén Y . edaydak´elakarItelI (6.5.1) eyIgGacsresr £

lnYt = lnY0 + tln (1+r) (6.5.2)

}LÚven¼ tag £ 1 =ln Y0 (6.5.3)

2 =ln (1+r) (6.5.4)

eyIgGacsresr (6.5.2) Ca £

lnYt = 1 + 2t (6.5.5)

edaybUktYclkreTAkñúg (6.5.5) eyIgán £

lnYt = 1 + 2t + ut (6.5.6)

KMrUren¼dUcCaKMrUrWERh:ssüúglIenEG‘repßgeTotEdr ena¼KWtMèlàraEmt 1 nig 2 lIenEG‘r . PaBxusKñaEtmYy

yKtKW £ snÞsßn_rWERh:ssüúgKWCaelakarItén Y nigtMélrWERh:ssüúgKWCa ²eBl³ EdlyknwgtMél 1, 2, 3....

KMrUdUcCa (6.5.6) ehAfa KMrUBak´kNþalelakarIt BIeRBa¼manEtGefrEtmYyKt (snÞsßn_rWERh:ssüúg)

CatYènelakarIt . sMrab´eKalbMNgeRbIRàs KMrUEdlsnÞsßn_rWERh:ssüúgCatMrg´elakarItehAfa KMrUlIenEG‘r

elakarIt (log-lin model) . enAeBleRkay eyIgnwgseg;temIlKMrUEdlsnÞsßn_rWERh:ssüúglIenEG‘r buEnþ

tMélrWERh:ssüúgCatYénelakarIt nigeKehAKMrUen¼fa KMrUelakarItlIenEG‘r (lin-log model) .

munnwgeyIgtaglT§plrWERh:ssüúg eyIgBinitüemIllkçN£énKMrU (6.5.5) . kñúgKMrUen¼ emKuNRábTis

vas´bMErbMrYlsmamaRt rW eFobefr (Constant Proportional or Relative Change) elI Y sMrab´bMErbMrYlCa

tMéldac´xatelItMélrwERh:ssüúg (Gefr t) £

2 = (6.5.8)

RbsinebIeyIgKuNbMErbMrYleFobelI Y nwg 100, ena¼ (6.5.7) nwgpþl´eGaybMErbMrYlCaPaKry rWGaRtakMenInelIY

sMrabbMErbMrYlCatMéldacxatelI X (tMélrWERh:ssüúg) .

bMErbMrYleFobelIsnÞsßn_rWERh:ssüúg

bMEbrMrYlCatMéldac´xatelItMélrWERh:ssüúg

saklviTüal&yPUminÞPñMeBj ROYAL UNIVERSITY OF PHNOM PENH

edátWm¨gKNitviTüa 158 MATHEMATICS DEPARTMENT

KMrUlIenEG‘relakarItdUcCa (6.5.5) manRbeyaCn_CaBiesskñúgsSanPaBEdl Gefr X CaeBl dUckñúg

«TahrN_ GNP edayehtufa kñúgkrNIena¼ KMrUbgHajBIGRtakMenInefrCaPaKryefr (1001) rW GRtakMenIneFob

efr (=2) (RbsinebI 2 >0 ) rWGaRtafycu¼ (Rate of Decay) (2 <0) elIGefr Y . ena¼CamUlehtu Edl

KMrU (6.5.5) ehAfa KMrUkMenIn(efr) [(Constant) Growth Model] .

RtlbeTA «ThrN_GDPBit eyIgGacsresrlT§plrWERh:ssüúg edayEpðkelI (6.5.6)dUcxageRkam £

ln RGDPt = 8,0139 + 0,02469t

se = (0,0114) (0,00956) r2 = 0,9738

t = (700,54) (25,8643)

tMél p = (0,0000)* (0,0000)*

* : kMnteGaytMéltUc

bMNkRsayénrWERh:ssüúgen¼mandUcxageRkam £ kñúgkMlugeBl 1972-1991, GDP Bitkñúg US ekIneLIgeday

GRta 2,469% kñúgmYyqñaM . edayehtufa 8,0139 = lnY0 (ehtuGVI?) . RbsinebIeyIgKNna Gg´TIelakarItén

8,0139 eyIgeXIjfa 0Y 3022,7 KWfaenAedImqñaM 1972 GDP BitánRbmaNesµInwg 3023 Ban´landuløa .

bnÞat´rWERh:ssüúg EdlTTYlánBI (6.5.8) RtUvbgHajkñúgrUb 6.4 .

rUb 6.4: kMenInén GDP Bit (US)qñaM 1972-1991 (KMrUBak´kNþalelakarIt)

GRtakMenInxN£ nigGRtakMenInsmas (Instantaneous versus Compound Rate of Growth)

emKuNRábTis 0,02469 EdlTTYlánBI (6.5.8) rWCaTUeTACagen¼ehAfa emKuN 2 énKMrUkMenIn

(6.5.5) pþl´eGayGaRtakMenInxN£ (KWfaenAxN£eBlNamYy) nigminEmnCaGRtakMenInsmas (elIkMlug

eBl) . buEnþGaRtaTI2en¼ GacRtUvKNnaedayRsYlBI (6.5.4) : RKan´EtKNnaGg´TIelakarItén 0,02469

dknwg 1 nigKuNpldknwg 100 . dUecñ¼ kñúgkrNIen¼ anitlog (0,02469) –1 = 0,024997 2,499 % .

KWfa kñúgkMlugeBlsikßa GRtakMenInsmasén GDPBit mantMélRbEhl 2,499 % kñúgmYyqñaM . GRtakMenInen¼

x<s´bnþicCagGRtakMenInxN£EdlmantMél 2,469 % .

0 5 10 15 20 25

8,5

8,4

8,3

8,2

8,1

8,0

ln R

GD

P

eBl ( qñaM)

+

+ + +

+

+

+ +

+ +

+ +

+ + + +

saklviTüal&yPUminÞPñMeBj ROYAL UNIVERSITY OF PHNOM PENH

edátWm¨gKNitviTüa 159 MATHEMATICS DEPARTMENT

KMrUninñakarlIenEG‘r (Linear Trend Model)

CMnYseGayKMrUánRbmaN (6.5.6) GñkRsavRCavCYnkalánRbmaNKMrUxageRkam £

Yt = 1 + 2t + ut (6.5.9)

KWfa CMnYseGaykarrkrWERh:ssüúgén log Y elIeBl eyIgeFVIrWERh:ssüúg Y elI eBl . KMrUEbben¼ ehAfa

KMrUninñakarlIenEG‘r nigGefreBl t ehAfaGefrninñakar (Trend Variable). tamBakü²ninñakar³ eyIgcg´mann&y

lkçN£ERbRbYlénGefrRtUvrkßaán . RbsinebI emKuNRábTiskñúg (6.5.9) viC¢man eyIgminmanninñakarpt

(Upward Trend) elIGefr Y buEnþRbsinebIGviC¢man eyIgmanninñakareág (Downward Trend) elI Y .

sMrabTinñn&y GDP Bit lT§pl EdlEpðkelI (6.5.9) mandUcxageRkam £

RGDP 1 =2933,0538 + 97,6806t

se = (50,5913) (4,2233) r2 = 0,9674 (6.5.10)

t = (57,9754) (23,1291)

tMél p = (0,0000)* (0,0000)*

* £kMnttMéltUc

pÞúyBI (6.5.8) bMNkRsayénrWERh:ssüúgmandUcteTA . elIry£eBl 1972-1991 CamFüm , GDP Bit ekIn

eLIgedayGRtadacxat (ktsMKal´ £minEmneFob) RbEhlCa 97,68 landuløa . dUecñ¼ elIkMlugeBlena¼

eyIgmanninñakarptelI GDP Bit .

CMerIsrvagKMrUkMenIn (6.5.8) nigKMrUninñakarlIenEG‘r (6.5.10) nwgGaRs&yelIfa etIeKcab´GarmµN_ nwgbMEr

bMrYl eFob rWdac´xatén GDP Bit eTa¼CasMrab´eKalbMNgCaeRcIn vaCabMErbMrYleFobEdlsMxan´Cagk¾eday .

ktsMKal´fa eyIgminGaceRbobeFob r2énKMrU (6.5.8) nig (6.5.10) BIeRBa¼snÞsßn_rWERh:ssüúgelIKMrUTaMgBIr

xusKña .

ktcMNaMelIKMrUlIenEG‘relakarIt nigKMrUTinñakarlIenEG‘r (A Caution on log-lin and Linear Trend Models)

eTa¼CaKMrUTaMgen¼RtUveRbICajwkjab´edIm,IánRbmaNbMErbMrYleFob rWdac´xatelIGefrTak´TgelIGefreBl

k¾eday kareRbIRàsCaFmµtasMrabeKalbMNgen¼RtUvànecaTCasMnYredayGñkviPaK es‘rIeBl (Time Series

Analysist) . GMn¼GMnagrbs´eKKWfa KMrUEbben¼Gacsmrmü RbsinebIes‘rIeBlERbRbYlkñúgn&y EdlkMntkñúgEpñk

1.7 . sMrab´ GñkGankMritx<s´RbFanbTen¼RtUvBiPakßalMGitkñúgCMBUk 21 elIesdæmaRtviTüaes‘rIeBl (ktsMKal´ £

CMBUken¼CaCMBUkCMerIseRsccitþ ) .

KMrUelakarItlIenEG‘r (The Lin-Log Model)

«bmafa eyIgmanTinñn&ykñúgtarag 6.3 Edl Y CaGNP nig X Cakarp:tp:g´Rák´ (niymn&y M2) .

bnÞab´mk«bmafa eyIgcab´GarmµN_nwgrkkMenIn (tMéldac´xat) GNP RbsinebIkarp:tp:g´RákekIn 1 % .

mindUcCaKMrUkMenIn EdleTIbEtánBiPakßa EdleyIgcgrkkMenInCaPaKryelI Y sMrab´bMErbMrYl 1 Ékta

CatMéldacxatelI X eyIg}LÚven¼cab´GarmµN_nwgrkbMErbMrYldac´xatelI Y sMrabbMErbMrYl 1% elI X . KMrUEdl

GacsMerceKalbMNgen¼GacsresrCa £

Yi = 1 + 2ln Xi + ui (6.5.11)

saklviTüal&yPUminÞPñMeBj ROYAL UNIVERSITY OF PHNOM PENH

edátWm¨gKNitviTüa 160 MATHEMATICS DEPARTMENT

sMrabeKalbMNgeRbIRàs eyIgehAKMrUEbben¼fa KMrUelakarItlIenEG‘r (lin-log Model) .

bMNkRsayemKuNRábTis 2 £

2 =

=

CMhanTI 2 eKarBtamkrNIEdlbMErbMrYlelIelakarIténcMnYnmYyCabMErbMrYleFob .

CanimitþsBaØa eyIgman £

2 = XX

Y

/

(6.5.12)

Edl CaFmµta kMnteGaybMErbMrYltUc . smIkar (6.5.12) GacsresrCa £

Y = 2(X/X) (6.5.13)

smIkaren¼bgHajfa bMErbMrYldac´xatelI Y (=Y) esµInwg 2 KuNnwgbMErbMrYleFobelI X . RbsinebIbMErbMrYl

eRkayen¼ KuNnwg 100 ena¼ (6.5.13) pþl´eGaybMErbMrYldac´xatelI Y sMrab´bMErbMrYlCaPaKryelI X . dUecñ¼

RbsinebI X/X ERbRbYl 0,01 Ékta ( rW 1%) ena¼bMErbMrYldac´xatelI Y esµInwg 0,01 (2) . dUecñ¼

RbsinebIkñúgkarGnuvtþmYy eKeXIjfa 2 = 500 bMErbMrYldacxatelI Y esµInwg 0,01.500 = 5,0 . dUecñ¼ enA

eBlrWERh:ssüúgdUcCa (6.5.11) RtUvánRbmaNeday OLS/ KuNtMélénemKuNRábTisánRbmaN 2 nwg

0,01 rWEcknwg 100 .

RtlbeTATinñn&ypþl´eGaykñúgtarag 6.3 eyIgGacsresrlT§plKMrUrWERh:ssüúgdUcxageRkam £

tY = -16329,0 + 2584,8 Xt

t = (-23,494) (27,549) r2 =0,9832 (6.5.14)

tMél p = (0,0000) * (0,0000)* (* £ cgðúlbgHajfa tMéltUc )

tarag 6.3: GNP nigkarp:t´p:g´RákenA US (1973-1987)

qñaM GNP duløa (Banlan) M2

1973 1359,3 861,0

1974 1472,8 908,5

1975 1598,4 1023,2

1976 1782,8 1163,7

1977 1990,5 1286,7

1978 2249,7 1389,0

1979 1508,2 1500,2

1980 2723,0 1633,1

1981 3052,6 1795,5

1982 3166,0 1954,0

1983 3405,7 2185,2

1984 3772,2 2363,6

1985 4014,9 2562,6

1986 4240,3 2807,7

1987 4526,7 2901,0

ktsMKal´ £ tYelx GNP RtUvEktMrUvtamRtImas CaGRtaRbcaMqñaM

bMErbMrYlelI Y

bMErbMrYlelI lnX

bMErbMrYlelI Y

bMErbMrYleFobelI X

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saklviTüal&yPUminÞPñMeBj ROYAL UNIVERSITY OF PHNOM PENH

edátWm¨gKNitviTüa 161 MATHEMATICS DEPARTMENT

M2 : rUbIyvtSú + Rák´beBaØItMrUvkar + mUlb,Tanb&RteTscr

+ RákbeBaØICamUlb,Tanb&RtepßgeTot + RP ry£eBlmYyyb´ nigduløaGWrU

+ tulüPaB MMMF ( sg:hFnTIpßarRák´)

+ MMDA (KNnIbeBaØITIpßarRák´) + Rák´snßM nigRákbeBaØIepßgeTot

Tinñn&yTaMgen¼CatYelxRbcaMéf¶CamFüm (EktMrUvtamqñaM)

RbPB £ Economic Report of the President, 1989, GNP data from TableB-1,p.308, and M2 data

form Table B-67, p.385.

ktsMKal´fa eyIgminmanlMeGogKMrU (etIeyIgGacrktMélen¼eT ? ) .

bMnkRsaytamlkçN£dUcbgHaj emKuNRábTis 2585 mann&yfa kñúgkMlugeBlKMrUtagkMenInelIkar

p:tp:g´Rák´ 1% CamFümeFVIeGaymankMenInelI GNP 25,85 Banlanduløa (ktsMKal´ £ EckemKuNRábTis

ánRbmaNnwg 100 ) .

munnwgBnül´bnþ ktsMKal´fa RbsinebIeKcg´KNnaemKuNbMlas´bþÚrsMrab´KMrUlIenEG‘relakarIt rWKMrU

elakarItlIenEG‘r eKGaceFVIàntamniymn&yénemKuNbMlas´bþÚr pþl´eGaymun (dY/dX).(X/Y) . CakarBit enA

eBlTMrg´GnuKmn_énKMrUs:al´ eKGacKNnabMlas´bþÚredayeRbIniymn&ymun . tarag 6.5 Edlnwgpþl´eGayeBl

eRkay segçbemKuNbMlas´bþÚrsMrabKMrUepßg@ EdleyIgánseg;temIlkñúgCMBUken¼ .

6¿6 KMrURcas (Reciprocal Models)

KMrUénRbePTxageRkam ehAfaKMrURcas £

Yi = 1 + 2

iX

1+ ui (6.6.1)

eTa¼bICaKMrUen¼minEmnlIenEG ‘relIGefr X k¾eday (BIeRBa¼vaCacMras) KMrUen¼lIenEG‘relI 1nig 2 dUecñ¼ vaCa

KMrUrWERh:ssüúglIenEG‘r .

KMrUen¼manlkçN£TaMgen¼ £ enAeBl X ekIndl´Gnnþ tY 2

iX

1xitCitsUnü (ktsMKal´ £ 2 CacMnYn

efr) nig Y mantMéllImIt 1 . dUecñ¼ KMrUdUcCa (6.6.1) ánbeg;IteGaymantMéllImIt EdlGefrTak´Tgnwgyk

tMél enAeBl X ekIndl´Gnnþ .

rUbénExßekagRtUvKñanwg (6.6.1) RtUvbgHajkñúgrUb 6.5 . «TahrN_énrUb 6.5a RtUvpþl´eGaykñúgrUb 6.6

EdlTak´TgnwgcMNayefrmFüm (Average Fixed Cost (AFC)) énplitkmµelIkMritplitpl . dUcrUbán

bgHaj AFC cu¼Canic© enAeBl plitplekIn (BIeRBa¼cMNayefr RtUvrayelIcMnYnÉktaplitply¨ageRcIn)

nigCacugeRkayxiteTArkGasIumtUt Y =1 .

karGnuvtþd¾sMxan´énrUb 6.5b CaExßekag Phillip énmaRkUesdækic© . edayEpðkelITinñn&yelIGRtaPaKry

énbMErbMrYlénRák´Ex Y nig kMritKµankargareFVICaPaKry X rbs´RbeTsGg´eKøssMrab´kMlug eBl 1861-1957

elak Phillip ánTTYlExßekag EdlragTUeTArbsva dUcKñanwg rUb 6.5b nigRtUvbgHajkñúg rUb 6.7 .

saklviTüal&yPUminÞPñMeBj ROYAL UNIVERSITY OF PHNOM PENH

edátWm¨gKNitviTüa 162 MATHEMATICS DEPARTMENT

dUckñúgrUb 6.7 bgHaj eyIgminmanPaBqøú¼énbMErbMrYlRákebovtßeFobnwgGRtaKµankargareFVVI £ Rák´ebovtß

ekInelOnsMrabbMErbMrYlmYyÉktaelIPaBminmankargar RbsinebIGRtaKµankargareRkam UN EdlehAfa GRtaKµan

kargarFmµCati (Natural Rate of Unemployment)tamGñkesdækic© CagvaFøakcu¼sMrab´bMErbMrYlsmmUlenA

eBlGRtaKµankargarenAelIkMritFmµCati 1 EdlCabnÞat´GasIumtUtsMrab´bMErbMrYlRákebovtß . lkçN£Biess

énExßekag Phillip GacGaRs&ynwgktþasSab&n dUcCaGMNactv¨arbsshCIB/ RákebovtßGb,brma/ karTUTat´

PaBminmankargar . l .

rUb 6.5 : KMrURcas £ Y = 1 + 2 X

1

rUb 6.6: KMrURcas

rUb 6.7: Exßekag Phillip

plitpl

cMN

ayef

rCam

Füm

énpl

itkmµ

Y

X 1

U

N

-1

GRt

abMErbMrYl

énRá

k´eb

ovtß

(%)

GaRtaKµankargar (%)

GaRtaKµankargarFmµCati

0

Y

2 > 0

1 >0 2 > 0

1 <0

Y

1

0

-1

0

2 < 0

Y

1

1

2

(a) (b) (c)

X X X

Formatted

Formatted

Formatted

Formatted

Formatted

Formatted

Formatted

Formatted

Formatted

Formatted

Formatted

Formatted

Formatted

Formatted

Formatted

Formatted

Formatted

saklviTüal&yPUminÞPñMeBj ROYAL UNIVERSITY OF PHNOM PENH

edátWm¨gKNitviTüa 163 MATHEMATICS DEPARTMENT

karGnuvtþd¾sMxan´énrUb 6.5c KW ExßekagcMNay Engel (ehAtameQµa¼ GñksSitiGalWmg´ Ernst Engel

(1821-1896)) EdlP¢ab´TMnak´TMngrvagcMNayrbs´GtifiCnelITMnij nigcMNay rWcMnUlsrubrbs´Kat´ . Rbsin

ebIeyIgtag Y eGaycMNayelITMnij nig X CacMnUl ena¼TMnijCaklak´manlkçN£TaMgen¼ £ (a) mankMritcab´

epþImcMnUl (Threshold level of Income) xageRkam tMélEdlTMnijminRtUvánTij . kñúgrUb 6.5c cMnuccab

epþImcMnUlen¼ sSitenARtg´kMrit - 1

2

. (b) : mankMritkMBUlénkareRbIRás (Satiety Level of Assumption)

elIBItMélEdlGñkeRbIRásnwgmincMNayeTa¼CaKatmancMnUlx<s´y¨agNak¾eday . kMriten¼KµanGVIeRkABIGasIumtUt

1 EdlRtUvbgHajkñúgrUben¼ . sMrab´TMnijEbben¼ KMrURcas EdlbgHajkñúgrUb 6.5c KWCaKMrUsmrmübMput .

«TahrN_cgðúlbgHaj £ Exßekag Phillip sMrab´Gg´eKøs (1950-1966)

tarag 6.4 pþl´eGayTinñn&yelIbMErbMrYlCaPaKryRbcaMqñaMelIGRtaRákebovtß Y nigGaRtaKµankargar X

sMrabGg´eKøskñúgGMlugqñaM (1950-1966) .

eKalbMNgtMrUvKMrURcas (6.6.1) pþl´eGaylT§plxageRkam (emIllT§plkMuBüÚT&r SAS kñúgesckþIbEnSm

6A Epñk 6A.3):

tY = -1,4282 + 8,2743 tX

1 r

2 = 0,3848 (6.6.2)

(2,0675) (2,8478) F1, 15 =9,39

EdltYelxkñúgrgVg´RkckCalMeGogKMrUánRbmaN .

bnÞat´rWERh:ssüúgánRbmaNRtUvbgHajkñúgrUb 6.8 . BIrUben¼ vaCakarc,asfa GasIumtUtRákebovtß

esµInwg -1,43 KWfa enAeBl X ekIndl´Gnnþ Rákebovtßnwgfycu¼mineRcInCag 1,43 % .

tarag 6.4: kMenIntamqñaMelIGRtaRák´ebovtß nigGRtaKµankargarenAGg´eKøs (1950-1966)

qñaM kMenIntamqñaMelIGRtaRák´ebovtß (%)

Y

Kµankargar (%)

X 1950 1,8 1,4

1951 8,5 1,1

1952 8,4 1,5

1953 4,5 1,5

1954 4,3 1,2

1955 6,9 1,0

1956 8,0 1,1

1957 5,0 1,3

1958 3,6 1,8

1959 2,6 1,9

1960 2,6 1,5

1961 4,2 1,4

1962 3,6 1,8

1963 3,7 2,1

1964 4,8 1,5

1965 4,3 1,3

1966 4,6 1,4

saklviTüal&yPUminÞPñMeBj ROYAL UNIVERSITY OF PHNOM PENH

edátWm¨gKNitviTüa 164 MATHEMATICS DEPARTMENT

RbPB £ Cliff Pratten, Applied Macroeconomics. Oxford University Press, Oxford, 1985, p. 85

ktsMKal´fatMélánRbmaNr2 Tabbnþic eTa¼CaemKuNRáb´TisxusBIsUnüCasar£sMxan´sSitik¾eday nig

vamansBaØaRtwmRtUv . karBinitüseg;ten¼ (nwgRtUvBiPakßakñúgCMBUk 7) KWCaehtupl EdleKminKYrbBa¢ak´eGayRCul

BItMél r2 .

rUb 6.8: Exßekag Phillip sMrab´Gg´eKøs (1950-1966)

6¿7 segçbTMrg´GnuKmn_ (Summary of Functional Forms)

kñúgtarag 6.5 eyIgsegçblkçN£sMxan´@énTMrg´GnuKmn_epßg@ Edlánsikßamkdl´eBlen¼ .

tarag 6.5

KMrU smIkar emKuNRáb´Tis(=

dX

dY) bMlas´bþÚr (=

Y

X

dX

dY. )

lIenEG‘r Y = 1 + 2X 2 2 ( Y

X)*

lIenEG‘relakarIt rW

elakarIt-elakarIt

lnY = 1 + 2lnY 2 (

Y

X)

2

lIenEG‘relakarIt lnY = 1 + 2X 2(Y) 2 (X)*

elakarItlIenEG‘r Y = 1 +2lnX 2 (

X

1) 2 (

Y

1)*

Rcas Y = 1 + 2(

X

1) -2 (

2

1

X) -2 (

XY

1)*

kMnt´sMKal´£ * cgðúlbgHajfa emKuNbMlas´bþÚr CaGefr EdlGaRs&yelItMélKitelI X rW Y rWTaMgBIr . enAeBl

KµantMél X nig Y RtUvkMnt´ kñúgkarGnuvtþ bMlas´bþÚrTaMgen¼RtUvvas´Rtg´tMélmFüm X nig Y .

6¿8 ktsMKal´elIlkçN£éntYlMeGog{kas £ tYlMeGog{kasRbmaNviFIbUk nigRbmaNviFIKuN

(A Note on The Nature of the Stochastic Error Term: Additive versus Multiplicative Stochastic Error Term)

-1,43

bMErbMrYl

elIGRt

aRá

k´eb

ovtß

(%)

GaRtaKµankargar (%)

tY = -1,4282 + 8,7243(1/Xt)

0

saklviTüal&yPUminÞPñMeBj ROYAL UNIVERSITY OF PHNOM PENH

edátWm¨gKNitviTüa 165 MATHEMATICS DEPARTMENT

seg;temIlKMrUrWERh:ssüúgxageRkam EdldUcKñanwg (6.4.1) buEnþminmantYlMeGog £

Yi = 1 2X (6.8.1)

sMrabeKalbMNgánRbmaN eyIgGacsresrKMrUen¼CaTMrg´bIxusKña £

Yi = 12

iX ui (6.8.2)

Yi = 12

iX iue (6.8.3)

Yi = 12

iX + ui (6.8.4)

edayKNnaelarItelIGg:TaMgBIrénsmIkarTMagen¼ eyIgTTYlán £

ln Yi = + 2lnXi + lnui (6.8.2a)

lnYi = + 2lnXi + ui (6.8.3a)

lnYi = ln(12

iX + ui ) (6.8.4a)

Edl = ln1 .

KMrUdUcCa (6.8.2) CaKMrUrWERh:ssüúglIenEG‘rGtSiPaB (Intrinsically Linear Regression) (elI

tMéláraEmt) kñúgn&yEdlbMElg (elakarIt)smrmü KMrUGacRtUvbeg;ItCalIenEG‘relItMèlàraEmt nig 2 (kt

sMKal´£ KMrUTaMgen¼minlIenEG‘relI 1 ) . buEnþ KMrU (6.8.4)minlIenEG‘rGtSiPaB . KµanviFIKNnaelakarItèn

(6.8.2) BIeRBa¼ ln (A+B) lnA + lnB .

eTa¼Ca (6.8.2) nig (6.8.3) CaKMrUrWERh:ssüúglIenEG‘r nigGacRtUvánRbmaNeday OLS rW ML

k¾eday / eyIgRtUvEtykcitþTukdak´elIlkçN£éntYlMeGog{kas EdlbBa©ÚlkñúgKMrUTaMgen¼ . caMfa lkçN£

BLUE én OLS tMrUveGay ui mantMélmFümsUnü/ v¨arü¨gefr nigGUtUkUrWLasüúgsUnü . sMrab´karBinitüemIl

smµtikmµ eyIgsnµteTotfa ui eKarBtamráyn&rmal´edaymantMélmFüm nigv¨arü¨g´eTIbánBiPakßa . Casegçb

eyIgánsnµtfa ui N(0,2) .

}LÚven¼Binitüseg;tKMrU (6.8.2) . KMrUsSitiRtUvKñarbs´vapþl´eGaykñúg (6.8.2a) . edIm,IeRbIKMrUrWERh:ssüúg

lIenEG‘rn&rmal´køasik (CNLRM) eyIgRtUvsnµtfa £

lnui N(0,2) (6.8.5)

dUecñ¼ enAeBleyIgeFVIrWERh:ssüúg (6.8.2a) eyIgnwgRtUvEteRbIkarBinitüemIllkçN£n&rmal´ EdlánBiPakßakñúg

CMBUk 5 elIsMnl´ EdlKNnaBIrWERh:ssüúg . ktsMKal´fa RbsinebI lnui eKarBtamráyn&rmal´edaymFüm 0

nig v¨arü¨g´efr ena¼RTwsþIsSitibgHajfa ui kñúg (6.8.2) RtUvEteKarBtamráyn&rmal´elakarIt (log-normal

distribution) edaymFüm 2/2e nigv¨arü¨g´ 1e (

1e - 1 ) .

dUcviPaKrWERh:ssüúgmunbgHaj eKRtUvEtykcitþTukdak´nwgtYKMrUlMeGogkñúgkarbMElgKMrUsMrabviPaKrWERh:s-

süúg . GaRs&yeday (6.8.4) / KMrUen¼CaKMrUrWERh:ssüúgminlIenEG‘relItMèlàraEmt nigRtUveKeda¼Rsaytam

kmµviFIkMuBüÚT&r . eKminKYrykKMrU (6.8.3) eTAeFVIkaránRbmaNtMél .

Casegçb eKKYrykcitþTukdak´nwgtYclkr enAeBlGñkbMElgKMrUsMrabviPaKrWERh:ssüúg . eRkABIen¼ kar

Gnuvtþminc,aslas´én OLS elIKMrUbMElg nwgminbeg;ItKMrUEdlmanlkçN£sSiticg´án .

saklviTüal&yPUminÞPñMeBj ROYAL UNIVERSITY OF PHNOM PENH

edátWm¨gKNitviTüa 166 MATHEMATICS DEPARTMENT

6¿9 esckþIsegçb nigkarsnñidæan (Summary and Conclusions)

CMBUken¼ànbgHajeGays:alnUvcMnuclMGitènKMrUrWERh:ssüúglIenEG‘rkøasik (CLRM) CaeRcIn .

1. CYnkalKMrUrWERh:ssüúgGacminmantYcMnuckatG&kßc,as´las . KMrUEbben¼ehAfa rWERh:ssüúgkat´tam

G&kß . eTa¼CaBiCKNiténkaránRbmaNKMrUEbben¼ gayRsYlk¾eday eKKYreRbIKMrUEbben¼edayRbug

Rby&tñ . kñúgKMrUEbben¼ plbUksMnl´ iu minesµIsUnü . elIsBIen¼ r2 EdlKNnatamTMlab´ Gac

minmann&y . RbsinebIminmanehtuplRTwsþIKYrsmrmüeT eKKYrbBa¢ak´cMnuckat´G&kßeGayánc,as .

2. Ékta rWkMrit EdlsnÞsßn_rWERh:ssüúg rWtMélrWERh:ssüúgRtUvKit mansar£sMxanNas´ BIeRBa¼bMNk

RsayénemKuNrWERh:ssüúgGaRs&yy¨agxøaMgelIva . kñúgkarRsavRCavBiesaFn_ GñkRsavRCav minKYrRKan´Et

Rsg´RbPBénTinñn&ybueNÑa¼eT EtRtUvbBa¢ak´eGayc,asBIviFIEdlGefrRtUveKvas´ .

3. cMnucsMxanKWTMrg´GnuKmn_énTMnak´TMngrvagsnÞsßn_rWERh:ssüúg nigtMélrWERh:ssüúg . TMrg´GnuKmn_

sMxan´xø¼@EdlBiPakßakñúgCMBUken¼KW £ (a). KMrUbMlas´bþÚrefr rWKMrUlIenEG‘relakarIt (b). KMrUrWERh:ssüúg

BakkNþalelakarIt nig (c). KMrURcas .

4. kñúgKMrUlIenEG‘relakarIt snÞsßn_rWERh:ssüúg nigtMèlrWERh:ssüúg RtUvsresrCaTMrg ´elakarIt . emKuN

rWERh:ssüúgEdlP¢ab´eTAelakarIténtMélrWERh:ssüúg RtUvbkRsayCabMlas´bþÚrénsnÞsßn_rWERh:ssüúg

eFobnwgtMélrWERh:ssüúg .

5. kñúgKMrUBak´kNþalelakarIt snÞsßn_rWERh:ssüúg rWtMélrWERh:ssüúgCaTMrg´elakarIt . kñúgTMrg´Bak´

kNþalelakarIt EdlsnÞsßn_rWERh:ssüúgCaTMrg´elakarIt nig tMélrWERh:ssüúg X CaeBl emKuN

Ráb´TisánRbmaN (KuNnwg 100) vas´GRtakMenInénsnÞsßn_rWERh:ssüúg . KMrUEbben¼ RtUveKeRbICa

jwkjab´edIm,IvasGRtakMenInénátuPUtesdækic©CaeRcIn . kñúgKMrUBak´kNþalelakarIt RbsinebItMél

rWERh:ssüúgCaTMrg´elakarIt emKuNrbs´vavas´kMritbMErbMrYldac´xat elIsnÞsßn_rWERh:ssüúgsMrab

bMErbMrYlCaPaKryNamYy elItMélrWERh:ssüúg .

3.6.kñúgKMrURcas snÞsßn_rWERh:ssüúg rWtMélrWERh:ssüúgRtUvsresrCaTMrg´cMras´edIm,ITTYlánTMnak´TMng

minEmnlIenEG‘rrvagGefresdækic©dUckñúgkrNIExßekag Phillips .

4.7.kñúgkareRCIserIsTMrg´GnuKmn_epßg@ eKKYrykcitþTukdak´nwgtYclkr{kas ui . dUcánktsMKal´kñúgCMBUk

5 / CLRM snµtfa tYclkrmantMélmFümsUnü nigv¨arü¨gefr (GUm¨Us:IdasÞIsIuFI) nigsnµtfa vaminman

kUrWLasüúg nwgtMélrWERh:ssüúg . eRkamkarsnµtTaMgen¼ snÞsßn_ánRbmaN OLS KW BLUE. elIs

BIen¼ eRkam CNLRM snÞsßn_ánRbmaN OLS k¾CaGefrráyn&rm¨al´pgEdr . dUecñ¼eKKYrseg;t

emIlfa karsnµtTaMgen¼GaceRbIánkñúgTMrg´GnuKmn_EdláneRCIserIssMrab´viPaKBiesaFn_rWeT . bnÞab´BI

áneFVIrWERh:ssüúg GñkRsavRCavKYrEteRbIkarBinitüemIlsakl,g (Diagnostic Test) dUcCakarBinitü

emIllkçN£n&rmal´EdlánBiPakßakñúgCMBUk 5 . eKminGacbBa¢ak´eGayc,ashYsehtuena¼eT BIeRBa¼

Formatted: Bullets and Numbering

saklviTüal&yPUminÞPñMeBj ROYAL UNIVERSITY OF PHNOM PENH

edátWm¨gKNitviTüa 167 MATHEMATICS DEPARTMENT

karBinitüemIlsmµtikmµkøasik dUcCa t, F, nig 2 EpðkelIkarsnµtEdlfa tYclkrCaGefrn&rmal´ .

krNIen¼caMác´NasRbsinebITMhMKMrUtagtUc .

5.8.eTa¼CakarBiPakßamkdl´eBlen¼RtUvkMntRtwmKMrUrWERh:ssüúgBIGefrk¾eday CMBUkbnÞab´nwgbgHajfa kñúg

krNICaeRcIn karBRgIkeTAelIBhurWERh:ssüúgRKan´EtCab´Tak´TgnwgBiCKNitbEnSmedayminánbgHaj

bEnSmelIbBaØtiRKw¼ . ena¼CamUlehtusMxan´Nas´EdlGñkRtUvmankaryl´c,aselIKMrUrWERh:ssüúgBIr

Gefr . rts

Formatted