Koefisien Determinasi dan Kor orel elas asii Be Berg rgan anda da Y =
1
+
2
X2 +
3
X3 +…+
PowerPoint® Slides
byYana by Yana Rohmana Education University of Indonesian
k
Xk + u
Koefisien Determiniasi dan Korelasi Berganda
Ingi Ingin n dik diketahu etahuii bera berapa pa prop propor orsi si (pr (presen esentas tase) e) sumb sumban anga gan n X2 dan dan X3 terh terhad adaap vari varias asii (nai (naik k turu turunn nnya ya)) Y secar secaraa bersama-sama. Besa Besarn rnya ya prop propor orsi si/p /per erse sent ntas ase e sumb sumban anga gan n ini ini dise disebu butt koefi oefisi sien en deter determi mina nasi si berga bergand nda, a,de deng ngan an symb symbol ol R2. Rumus R2 dipe diperroleh oleh deng dengaan meng menggu guna naka kan n def definis inisii :
R
2
R
2
yi TSS yi b x i yi b yi 2
ESS E SS
ˆ
2
1 2 .3
2
1 3. 2 2
x i yi 3
Penerapan pada Kasus 2 2
R
b1 2.3 x2i yi b1 3. 2 x3i yi
yi
2
1.203,533 20,0028 1.260,889
0,9387
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) R2 = 0,9387 Se = 3,5907
(0,1464)
The adjusted R 2 (R2) as one of indicators of the overall fitness R2 =
ESS
RSS =1-
=1-
TSS _ R2 = 1 -
TSS
_ R2 = 1 -
y i 2
ei2 / (n-k) y i 2 /
_ R2 = 1 -
e^i2
k: (n-1)
Se2
# of independent variables plus the constant term.
Sy2
e2
(n-1)
y 2
(n-k)
_ R2 = 1 - (1 - R2)
n : # of obs.
n-1 n-k
_
R2 R2 0 < R2 < 1 Adjusted R2 can be negative: R2 0
Y = 1 + 2 X2 + 3 X3 + u
Y ^
Y u
TSS n-1
Suppose X4 is not an explanatory Variable but is included in regression
C X2 X3 X4
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 :
r
r 1 2
xi yi 2
xi
2
yi
x2 i yi 2 x 2i
2 y i
Koefisien Korelasi Parsial Dan Hubungan Berbagai Koefisien Korelasi dan Regresi
Antara X3 dan Y :
r 13
Antara X2 dan X3 : r 23
x3i yi 2 3i
2 i
x
y
x2i x3i 2
2
x2i x3i
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 r r
12.3
12
(1
2 13
r
r r
13.2
13
(1
2 12
r r
23.1
23
(1
2 23
r
)
r r
12 23
) (1
2 12
r
13 23
) (1
r
r r
2 23
r
)
r r
12 13
) (1
2 13
r
)
1. Individual partial coefficient test 1
holding X3 constant: Whether X2 has the effect on Y ?
Y
H0 : 2 = 0
X2
H1 : 2 0
t=
^ 2 - 0
0.9277
=
Se (^ 2)
= 9.544 0.0972
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.
= 2 = 0?
1. Individual partial coefficient test (cont.) 2
holding X2 constant: Whether X3 has the effect on Y?
Y
H0 : 3 = 0
X3
H1 : 3 0
t=
^ 3 - 0
0.2532 - 0 =
Se (^3) Critical value:
= 1.730 0.1464
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.
= 3 = 0?
2. Testing overall significance of the multiple regression
3-variable case:
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)
1. Compute and obtain F-statistics 2. Check for the critical F c value (F 3. Compare F and Fc , and if F > Fc ==> reject H0
c
, k-1, n-k)
Analysis of Variance:
y = ^ y + u
Since
^ ==> y 2 = ^ y 2 + u2 TSS = ESS + RSS
ANOVA TABLE Source of variation
(SS) Sum of Square
Due to regression(ESS)
^2 y
Due to residuals(RSS)
^ u2
df
(MSS) Mean sum of Sq. ^ y 2
k-1
k-1 ^2 u
n-k
n-k
y 2
Total variation(TSS)
= ^u2
n-1
Note: k is the total number of parameters including the intercept term. MSS of ESS F=
= MSS of RSS
H0 : 2 = … = k = 0 H1 : 2
…
k 0
^ y 2/(k-1)
ESS / k-1 = RSS / n-k if F > Fck-1,n-k
^ u 2 /(n-k)
==> reject Ho
Tabel Anavar, untuk Regresi Tiga Variabel
Sumber Variasi
Dari regresi
Jumlah Kuadrat (SS)
b12.3 Σ x2iyi + b13.2 Σ x3iyi
(ESS) Kesalahan pengganggu
Derajat
Rata-Rata
Kebebasan
Jumlah Kuadrat
(df)
(MSS)*
2
b12.3 Σ x2iyi + b13.2 Σ x3iyi
(k-1)
Σ ei2
n-3 (n-k)
(RSS) TSS *Mean
Σ yi2 Sum of Squares.
n-1
2
Σ ei2 / n - 3 = Se2
^ ^ y = 2 x 2 + 3 x 3 + u^
Threevariable case
y 2
^ ^ ^ = 2 x 2 y + 3 x 3 y + u2
TSS =
ESS
+ RSS
ANOVA TABLE Source of variation ESS
SS ^ 2 x 2 y + ^ 3 x 3 y
df(k=3)
MSS
3-1
ESS/3-1 RSS/n-3
RSS
^ u2
n-3 (n-k)
TSS
y 2
n-1
ESS / k-1 F-Statistic = RSS / n-k
=
3 x 3y ) / 3-1 (^2 x 2y + ^ ^2 / n-3 u
An important relationship between R 2 and F ESS / k-1
ESS (n-k)
F=
= RSS / n-k
RSS (k-1)
ESS
n-k
TSS-ESS
k-1
ESS/TSS
n-k
=
=
ESS TSS
1-
For the three-variables case : R2 / 2 F=
k-1
R2
n-k
1 - R2
k-1
(1-R2) / n-3
=
F
R2 / (k-1)
=
Reverse : (1-R2) / n-k
R2 =
(k-1) F (k-1)F + (n-k)
Overall significance test: H 0 : 2 = 3 = 4 = 0 H1 : at least one coefficient is not zero. 0 , or
2
3
0 , or
4
0
R2 / k-1
F* =
(1-R2)
=
/ n- k
0.9710 / 3
=
(1-0.9710) /16
= 179.13 Fc(0.05, 4-1, 20-4) = 3.24
k-1
n-k Since F* > Fc ==> reject H0.
Construct the ANOVA Table (8.4) .(Information from EViews)
Source of SS variation 2 2 Due to R ( y ) regression =(0.971088)(28.97771)2x1 (SSE) =15493.171 9 Due to Residuals (RSS) Total (TSS)
2
2
MSS
k-1
R ( y )/(k-1)
=3
=5164.3903
2
n-k (1- R )( y ) or ( ) =(0.0289112)(28.97771) ) 2x19 =16 =461.2621 2
MSS of regression
2
2
=28.8288
=> (n-1)(y)2 = y 2
5164.3903 = 179.1339 28.8288
2
(1- R )( y )/(n-k)
=19
= MSS of residual
2
n-1
( y ) =(28.97771) 2x19 =15954.446 Since (y)2 = Var(Y) = y 2/(n-1)
F* =
Df
Example:Gujarati(2003)-Table6.4, pp.185)
H0 :
1
=
2=
3
=0 2
R / k-1 ESS / k* F = = 1 (1-R2) / n- k RSS/(nk)
0.707665 / 2
=
(1-0.707665)/ 61 F* = 73.832
Fc(0.05, 3-1, 64-3) = 3.15
k-1
n-k
Since F* > Fc ==> reject H0.
Construct the ANOVA Table (8.4) .(Information from EVIEWS)
Source of SS variation 2 2 Due to R ( y ) regression =(0.707665)(75.97807)2x6 (SSE) =261447.33 4 2
2
2
Due to Residuals (RSS)
(1- R )( y ) or ( ) =(0.292335)(75397807)2x64 =108003.37
Total (TSS)
( y ) =(75.97807)2x64 =369450.7
2
F* =
MSS
k-1
R ( y )/(k-1)
=2
=130723.67
n-k
(1- R )( y )/(n-k)
=61
2
2
=> (n-1)(y)2 = y 2
130723.67 = 73.832 1770.547
2
=1770.547
=63
= MSS of residual
2
n-1
Since (y)2 = Var(Y) = y 2/(n-1) MSS of regression
Df
Y = 1 + 2 X2 + 3 X3 + u H0 : 2 = 0, 3= 0, H1 : 2 0 ; 3 0 Compare F* and F c, checks the F-table:
Decision Rule: Since F*= .73.832 > Fc = 4.98 (3.15)
Fc0.01, 2, 61 = 4.98 Fc0.05, 2, 61 = 3.15
==> reject Ho
Answer : The overall estimators are statistically significant different from zero.
QUIZ
1
2
3
4
TERIMA KASIH
