Upload
dokhuong
View
249
Download
0
Embed Size (px)
Citation preview
PowerPoint® Slides
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
Chapter Analisis Regresi Linier Berganda : Persoalan Estimasi dan Pengujian Hipotesis
Penerapan pada Kasus 2
3
2
32 .3123 .122
i
iiii
y
yxbyxbR
9387,0
889,260.1
0028,20533,203.1
Chapter Analisis Regresi Linier Berganda : Persoalan Estimasi dan Pengujian Hipotesis
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
Chapter Analisis Regresi Linier Berganda : Persoalan Estimasi dan Pengujian Hipotesis
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
Chapter Analisis Regresi Linier Berganda : Persoalan Estimasi dan Pengujian Hipotesis
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
Chapter Analisis Regresi Linier Berganda : Persoalan Estimasi dan Pengujian Hipotesis
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
Chapter Analisis Regresi Linier Berganda : Persoalan Estimasi dan Pengujian Hipotesis
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
Chapter Analisis Regresi Linier Berganda : Persoalan Estimasi dan Pengujian Hipotesis
QUIZ
1
2
3
4
23
Chapter Analisis Regresi Linier Berganda : Persoalan Estimasi dan Pengujian Hipotesis
TERIMA KASIH
24