Koefisien Determinasi dan Korelasi Berganda

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1 Koefsen Detemnas dan Koelas Beganda Y = 1 + X + 3 X k X k + u PowePont Sldes byyana Rohmana Educaton Unvesty of Indonesan 007 Laboatoum Ekonom & Kopeas Publshng Jl. D. Setabud 9 Bandung, Telp

2 Koefsen Detemnas dan Koelas Beganda Ingn dketahu beapa popos (pesentase) sumbangan X dan X3 tehadap vaas (nak tuunnya) Y secaa besama-sama. Besanya popos/pesentase sumbangan n dsebut koefsen detemnas beganda,dengan symbol R. Rumus R dpeoleh dengan menggunakan defns : R ESS yˆ TSS y R b 1.3 y y b y Chapte Analss Reges Lne Beganda : Pesoalan Estmas dan Pengujan Hpotess

3 Peneapan pada Kasus R b y b y 3 y 1.03,533 0, ,889 0,9387 Chapte Analss Reges Lne Beganda : Pesoalan Estmas dan Pengujan Hpotess 3

4 Pesamaan gas eges lne beganda (kasus ) Ŷ = b b 1.3 X + b 13. X 3 Ŷ = -17, ,977 X + 0,53 X 3 Standa eo: (0,097) (0,1464) R = 0,9387 S e = 3,5907 Chapte Analss Reges Lne Beganda : Pesoalan Estmas dan Pengujan Hpotess 4

5 The adjusted R (R ) as one of ndcatos of the oveall ftness R = _ R = 1 - ESS TSS _ S e R = 1 - S y _ e R = 1 - y _ R = 1 - (1 - R ) _ = 1 - e / (n-k) y / (n-1) RSS TSS (n-1) (n-k) n - 1 n - k = 1 - k : n : e y # of ndependent vaables plus the constant tem. # of obs. R R 0 < R < 1 Adjusted R can be negatve: R 0 5

6 Y = 1 + X + 3 X 3 + u u Y Y TSS n-1 6

7 Suppose X4 s not an eplanatoy Vaable but s ncluded n egesson C X X3 X4 7

8 Koefsen Koelas Pasal Dan Hubungan Bebaga Koefsen Koelas dan Reges Y = b b 1.3 X + b 13. X 3 + e 1 = koefsen koelas antaa Y dan X (antaa X dan Y) 13 = koefsen koelas antaa Y dan X 3 (antaa X 3 dan Y) 3 = koefsen koelas antaa X dan X 3 (antaa X 3 dan X ) Antaa X dan Y : y y Antaa X dan Y : 1 y y Chapte Analss Reges Lne Beganda : Pesoalan Estmas dan Pengujan Hpotess 8

9 Chapte Analss Reges Lne Beganda : Pesoalan Estmas dan Pengujan Hpotess Koefsen Koelas Pasal Dan Hubungan Bebaga Koefsen Koelas dan Reges Antaa X 3 dan Y : Antaa X dan X 3 : y y 3 3 3

10 Patal Coelaton Coeffcent 1.3 = koefsen koelas antaa Y dan X, kalau X 3 konstan 13.3 = koefsen koelas antaa Y dan X 3, kalau X konstan 3.1 = koefsen koelas antaa X dan X 3, kalau Y konstan (1 13 ) 13 3 (1 3 ) (1 1 ) 1 3 (1 3 ) (1 1 ) 1 13 (1 13 ) Chapte Analss Reges Lne Beganda : Pesoalan Estmas dan Pengujan Hpotess 10

11 1. Indvdual patal coeffcent test 1 holdng X 3 constant: Whethe X has the effect on Y? H 0 : = 0 H 1 : 0 Y X = = 0? t = - 0 Se ( ) = = Compae wth the ctcal value t c 0.05, 6 =.447 Snce t > t c ==> eject H o Answe : Yes, s statstcally sgnfcant and s sgnfcantly dffeent fom zeo. 11

12 1. Indvdual patal coeffcent test (cont.) holdng X constant: Whethe X 3 has the effect on Y? H 0 : 3 = 0 H 1 : 3 0 Y X 3 = 3 = 0? t = 3-0 Se ( 3 ) = = Ctcal value: t c 0.05, 6 =.447 Snce t < t c ==> not eject H o Answe: Yes, 3 s statstcally not sgnfcant and s not sgnfcantly dffeent fom zeo. 1

13 . Testng oveall sgnfcance of the multple egesson 3-vaable case: Y = 1 + X + 3 X 3 + u H 0 : = 0, 3 = 0, (all vaable ae zeo effect) H 1 : 0 o 3 0 (At least one vaable s not zeo) 1. Compute and obtan F-statstcs. Check fo the ctcal F c value (F c, k-1, n-k ) 3. Compae F and F c, and f F > F c ==> eject H 0 13

14 Snce y = y + u Analyss of Vaance: ANOVA TABLE F = MSS of ESS MSS of RSS = ESS / k-1 RSS / n-k ==> y = y + u = TSS = ESS + RSS (SS) (MSS) Souce of vaaton Sum of Squae df Mean sum of Sq. Due to egesson(ess) y k-1 Due to esduals(rss) u n-k Total vaaton(tss) y n-1 y /(k-1) u /(n-k) y k-1 u n-k = u Note: k s the total numbe of paametes ncludng the ntecept tem. H 0 : = = k = 0 H 1 : k 0 f F > F c k-1,n-k ==> eject H o 14

15 Tabel Anava, untuk Reges Tga Vaabel Deajat Rata-Rata Sumbe Vaas Jumlah Kuadat (SS) Kebebasan Jumlah Kuadat (df) (MSS) * Da eges (ESS) Kesalahan pengganggu (RSS) TSS b 1.3 Σ y + b 13. Σ 3 y (k-1) Σ e n-3 (n-k) Σ y n-1 b 1.3 Σ y + b 13. Σ 3 y Σ e / n - 3 = S e * Mean Sum of Squaes. Chapte Analss Reges Lne Beganda : Pesoalan Estmas dan Pengujan Hpotess 15

16 Theevaable case y = u y = y y + u TSS = ESS + RSS ANOVA TABLE Souce of vaaton SS df(k=3) MSS ESS y y 3-1 ESS/3-1 RSS u n-3 (n-k) RSS/n-3 TSS y n-1 F-Statstc = ESS / k-1 RSS / n-k = ( y y) / 3-1 u / n-3 16

17 An mpotant elatonshp between R and F F = ESS / k-1 RSS / n-k = ESS (n-k) RSS (k-1) ESS = TSS-ESS ESS/TSS = ESS 1 - TSS = R 1 - R n-k k-1 n-k k-1 n-k k-1 Fo the thee-vaables case : R / F = (1-R ) / n-3 F = R / (k-1) Revese : R = (1-R ) / n-k (k-1) F (k-1)f + (n-k) 17

18 Oveall sgnfcance test: H 0 : = 3 = 4 = 0 H 1 : at least one coeffcent s not zeo. 0, o 3 0, o 4 0 R / k-1 F * = = (1-R ) / n- k = / 3 ( ) /16 = F c (0.05, 4-1, 0-4) = 3.4 k-1 n-k Snce F * > F c ==> eject H

19 Constuct the ANOVA Table (8.4).(Infomaton fom EVews) Souce of vaaton Due to egesson (SSE) SS Df MSS R (y ) =( )( ) 1 9 = k-1 =3 R (y )/(k-1) = Due to Resduals (RSS) (1- R )(y ) o ( ) =( )( ) ) 19 = n-k =16 (1- R )(y )/(n-k) =8.888 Total (TSS) (y ) =( ) 19 = n-1 =19 Snce ( y ) = Va(Y) = y /(n-1) => (n-1)( y ) = y F * = MSS of egesson MSS of esdual = =

20 Eample:Gujaat(003)-Table6.4, pp.185) H 0 : 1 = = 3 = 0 F * = ESS / k- 1 RSS/(n- k) = R / k-1 (1-R ) / n- k = / ( )/ 61 F * = F c (0.05, 3-1, 64-3) = 3.15 k-1 n-k Snce F * > F c ==> eject H 0. 0

21 Constuct the ANOVA Table (8.4).(Infomaton fom EVIEWS) Souce of vaaton Due to egesson (SSE) SS Df MSS R (y ) =( )( ) 6 4 = k-1 = R (y )/(k-1) = Due to Resduals (RSS) (1- R )(y ) o ( ) =(0.9335)( ) 64 = n-k =61 (1- R )(y )/(n-k) = Total (TSS) (y ) =( ) 64 = n-1 =63 Snce ( y ) = Va(Y) = y /(n-1) => (n-1)( y ) = y F * = MSS of egesson MSS of esdual = =

22 Y = 1 + X + 3 X 3 + u H 0 : = 0, 3 = 0, H 1 : 0 ; 3 0 Compae F* and F c, checks the F-table: F c 0.01,, 61 = 4.98 F c 0.05,, 61 = 3.15 Decson Rule: Snce F * = > F c = 4.98 (3.15) ==> eject H o Answe : The oveall estmatos ae statstcally sgnfcant dffeent fom zeo.

23 QUIZ Chapte Analss Reges Lne Beganda : Pesoalan Estmas dan Pengujan Hpotess 3

24 TERIMA KASIH Chapte Analss Reges Lne Beganda : Pesoalan Estmas dan Pengujan Hpotess 4