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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¥¼TTYlánedayGñkniBn§ 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
edátWm¨gKNitviTüa 154 MATHEMATICS DEPARTMENT
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
edátWm¨gKNitviTüa 155 MATHEMATICS DEPARTMENT
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)
saklviTüal&yPUminÞPñMeBj ROYAL UNIVERSITY OF PHNOM PENH
edátWm¨gKNitviTüa 156 MATHEMATICS DEPARTMENT
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
Formatted
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Formatted: Bullets and Numbering
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