CHAPTER 5.b Koefisien Determinasi Dan Korelasi Berganda

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byYana RohmanaEducation University of Indonesian

© 2007 Laboratorium Ekonomi & Koperasi Publishing Jl. Dr. Setiabudi 229 Bandung, Telp. 022 2013163 - 2523

Koefisien Determinasi dan

Korelasi Berganda

Y = 1 + 2 X2 + 3 X3 +…+ k Xk + u

Chapter Analisis Regresi Linier Berganda : Persoalan Estimasi dan Pengujian Hipotesis

Koefisien Determiniasi dan Korelasi Berganda

Ingin diketahui berapa proporsi (presentase) sumbangan

X2 dan X3 terhadap variasi (naik turunnya) Y secara

bersama-sama.

Besarnya proporsi/persentase sumbangan ini disebut

koefisien determinasi berganda,dengan symbol R2.

Rumus R2 diperoleh dengan menggunakan definisi :

2

2

32 .3123 .122

2

2

2ˆ

i

iiii

i

i

y

yxbyxbR

y

y

TSS

ESSR


Penerapan pada Kasus 2

3

2

32 .3123 .122

i

iiii

y

yxbyxbR

9387,0

889,260.1

0028,20533,203.1


Persamaan garis regresi linier berganda (kasus 2)

Ŷ = b1.23 + b12.3 X2 + b13.2 X3

Ŷ = -17,8685 + 0,9277 X2 + 0,2532 X3

Standar error: (0,0972) (0,1464)

R2 = 0,9387

Se = 3,5907

4

The adjusted R2 (R2) as one of indicators of the overall fitness

R2 =ESS

TSS= 1 -

RSS

TSS= 1 -

ei2

yi2

^

R2 = 1 -

_ Se2

Sy2

R2 = 1 -

_ e2

y2

(n-1)

(n-k)

ei2 / (n-k)

yi2 / (n-1)

R2 = 1 -

_

k : # of independent

variables plus the

constant term.

n : # of obs.

n - 1R2 = 1 - (1 - R2)

_

n - k

R2 R2

_

Adjusted R2 can be negative: R2 0

0 < R2 < 1

5

Y

Yu

^

Y = 1 + 2 X2 + 3 X3 + u

TSS

n-1

6

Suppose X4 is not an explanatory

Variable but is included in regression

C

X2

X3

X4

7


Koefisien Korelasi Parsial Dan Hubungan Berbagai

Koefisien Korelasi dan Regresi

Y = b1.23 + b12.3 X2 + b13.2 X3 + ei

r12 = koefisien korelasi antara Y dan X2 (antara X2 dan Y)

r13 = koefisien korelasi antara Y dan X3 (antara X3 dan Y)

r23 = koefisien korelasi antara X2 dan X3 (antara X3 dan X2)

Antara X dan Y :

Antara X2 dan Y :

8

22

ii

ii

yx

yxr

22

2

212

ii

ii

yx

yxr


Koefisien Korelasi Parsial Dan Hubungan Berbagai

Koefisien Korelasi dan Regresi

Antara X3 dan Y :

Antara X2 dan X3 :

9

22

3

313

ii

ii

yx

yxr

2

3

2

2

3223

ii

ii

xx

xxr


Partial Correlation Coefficient

r12.3 = koefisien korelasi antara Y dan X2, kalau X3 konstan

r13.3 = koefisien korelasi antara Y dan X3, kalau X2 konstan

r23.1 = koefisien korelasi antara X2 dan X3, kalau Y konstan

10

)1()1( 2

23

2

13

2313123.12

rr

rrrr

)1()1( 2

23

2

12

231213

2.13

rr

rrrr

)1()1( 2

13

2

12

131223

1.23

rr

rrrr

1. Individual partial coefficient test

t =2 - 0^

Se (2)̂

=0.9277

0.0972

= 9.544

Compare with the critical value tc0.025, 6 = 2.447

Since t > tc ==> reject Ho

Answer : Yes, 2 is statistically significant and is

significantly different from zero.

^

H0 : 2 = 0

H1 : 2 0

holding X3 constant: Whether X2 has the effect on Y ?1

Y

X2

= 2 = 0?

11

1. Individual partial coefficient test (cont.)

holding X2 constant: Whether X3 has the effect on Y?2

H0 : 3 = 0

H1 : 3 0

Y

X3

= 3 = 0?

t =3 - 0^

Se (3)^

=0.2532 - 0

0.1464

= 1.730

Critical value: tc0.025, 6 = 2.447

Since | t | < | tc | ==> not reject Ho

Answer: Yes, 3 is statistically not significant and is

not significantly different from zero.

^

12

2. Testing overall significance of the multiple regression

3. Compare F and Fc , and

if F > Fc ==> reject H0

1. Compute and obtain F-statistics

2. Check for the critical Fc value (F c , k-1, n-k)

Y = 1 + 2X2 + 3X3 + u

H0 : 2 = 0, 3 = 0, (all variable are zero effect)

H1 : 2 0 or 3 0 (At least one variable is not zero)

3-variable case:

13

F =MSS of ESS

MSS of RSS

=ESS / k-1

RSS / n-k

= y2/(k-1)

u 2 /(n-k)

^

^

if F > Fck-1,n-k ==> reject HoH1 : 2 … k 0

H0 : 2 = … = k = 0

Analysis of Variance: Since y = y + u^

==> y2 = y2 + u2^ ^

TSS = ESS + RSS

Source of variation Sum of Square df Mean sum of Sq.

Due to regression(ESS) y 2 k-1

Due to residuals(RSS) u2 n-k

Total variation(TSS) y2 n-1

ANOVA TABLE

y2

k-1^

^

u2

(SS)

n-k= u

2^

^

^

(MSS)

Note: k is the total number of parameters including the intercept term.

14


Tabel Anavar, untuk Regresi Tiga Variabel

15

Sumber Variasi Jumlah Kuadrat (SS)

Derajat

Kebebasan

(df)

Rata-Rata

Jumlah Kuadrat

(MSS)*

Dari regresi

(ESS)

b12.3 Σ x2iyi + b13.2 Σ x3iyi 2

(k-1)

b12.3 Σ x2iyi + b13.2 Σ x3iyi

2

Kesalahan

pengganggu

(RSS)

Σ ei2 n-3

(n-k)

Σ ei2 / n - 3 = Se

2

TSSΣ yi

2 n-1

*Mean Sum of Squares.

Three-

variable

case

y = 2x2 + 3x3 + u^ ^ ^

y2 = 2 x2 y + 3 x3 y + u2^ ^ ^

TSS = ESS + RSS

F-Statistic =ESS / k-1

RSS / n-k=

(2 x2y + 3 x3y) / 3-1

u2 / n-3^

^ ^

ANOVA TABLE

Source of variation SS df(k=3) MSS

ESS 2 x2 y + 3 x3 y 3-1 ESS/3-1

RSS u2 n-3 RSS/n-3

TSS y2 n-1

^ ^

^

(n-k)

16

An important relationship between R2 and F

F =ESS / k-1

RSS / n-k

=ESS (n-k)

RSS (k-1)

=

TSS-ESS

ESS n-k

k-1

TSS

=ESS/TSS

ESS1 -

n-k

k-1

=R2

1 - R2

n-k

k-1

=R2 / (k-1)

(1-R2) / n-kF R2 =

(k-1)F + (n-k)

(k-1) F

Reverse :

For the three-variables case :

F =R2 / 2

(1-R2) / n-3

17

5.18

Overall significance test:

H1 : at least one coefficient

is not zero.

H0 : 2 = 3 = 4 = 0

2 0 , or 3 0 , or 4 0

Fc(0.05, 4-1, 20-4) = 3.24

k-1 n-k

Since F* > Fc ==> reject H0.

F* = =R2 / k-1

(1-R2) / n- k

= 179.13

0.9710 / 3

(1-0.9710) /16=

18

Construct the ANOVA Table (8.4) .(Information from EViews)

F* =MSS of regression

MSS of residual=

5164.3903

28.8288= 179.1339

Source ofvariation

SS Df MSS

Due toregression

(SSE)

R2(y

2)

=(0.971088)(28.97771)2x1

9 =15493.171

k-1

=3

R2(y

2)/(k-1)

=5164.3903

Due toResiduals

(RSS)

(1- R2)(y

2) or (

2)

=(0.0289112)(28.97771) )2x19=461.2621

n-k

=16

(1- R2)(y

2)/(n-k)

=28.8288

Total

(TSS)(y

2)

=(28.97771) 2x19=15954.446

n-1

=19

Since (y)2 = Var(Y) = y2/(n-1) => (n-1)(y)

2 = y2

19

Example:Gujarati(2003)-Table6.4, pp.185)

Fc(0.05, 3-1, 64-3) = 3.15

k-1 n-k

Since F* > Fc

==> reject H0.

F* =0.707665 / 2

(1-0.707665)/ 61=

R2 / k-1

(1-R2) / n- k

F* = 73.832

=ESS / k-

1RSS/(n- k)

H0 : 1 = 2 = 3 = 0

20

Construct the ANOVA Table (8.4) .(Information from EVIEWS)

F* =MSS of regression

MSS of residual=

130723.67

1770.547= 73.832

Source ofvariation

SS Df MSS

Due toregression

(SSE)

R2(y

2)

=(0.707665)(75.97807)2x6

4 =261447.33

k-1

=2

R2(y

2)/(k-1)

=130723.67

Due toResiduals

(RSS)

(1- R2)(y

2) or (

2)

=(0.292335)(75397807)2x64=108003.37

n-k

=61

(1- R2)(y

2)/(n-k)

=1770.547

Total

(TSS)(y

2)

=(75.97807)2x64=369450.7

n-1

=63

Since (y)2 = Var(Y) = y2/(n-1) => (n-1)(y)

2 = y2

21

Decision Rule:

Since F*= .73.832 > Fc = 4.98 (3.15) ==> reject Ho

Answer : The overall estimators are statistically significant

different from zero.

Fc0.01, 2, 61 = 4.98

Fc0.05, 2, 61 = 3.15Compare F* and Fc, checks the F-table:

H0 : 2 = 0, 3= 0,

H1 : 2 0 ; 3 0

Y = 1 + 2 X2 + 3 X3 + u

22


QUIZ

1

2

3

4

23


TERIMA KASIH

24

CHAPTER 5.b Koefisien Determinasi Dan Korelasi Berganda

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