Kuliah 3 ekonometrika S1 2015

(1)

Materi 2: Program S-1 Statistika, ITS, Surabaya

P ERTEMUAN 3:

E KONOMETRIKA

P ROGRAM S TUDI S -1 S TATISTIKA ITS S URABAYA

A NALISIS

“ R EGRESI”

ITS, Surabaya, 15 September 2015

(2)

B AHASAN

2

1. K ORELASI

2. A NALISIS R EGRESI

3. E STIMASI R EGRESI (OLS, ML, MM)

4. S TANDAR E RROR OLS

5. A SUMSI DALAM R EGRESI

6. K OEFISIEN D ETERMINASI

(3)

K ORELASI

3

1. H UBUN GAN 1 A RAH (X  Y), Y  X

BBM naik  Inflasi naik;

Anggaran kes. N aik  orang sakit naik 2. H UBUN GAN DUA ARAH (X  Y), Y  X

Jumlah barang naik  harga turun H arga naik  jumlah barang turun 3. H UB. T AK LAN GSUN G (X  Y)

Z

harga BBM  Investasi (ada perantara biaya produksi)

4. T IM E S ERIES ( Y T- Y T-1)

(4)

K ORELASI

4

1 r  0

r  r  0.1

0.5 r 

0.3 r 

0.85 r 

13 23

12 13 23

12.3 2 2

.

1 1

r r r

 

 

(5)

K ORELASI

Department of Statistics, ITS Surabaya Slide-6

Types of Relationships

Y

X Y

X

Y Y

X

X Linear relationships Curvilinear relationships

Department of Statistics, ITS Surabaya Slide-7

Types of Relationships

Y

X Y

X

Y Y

X

X Strong relationships Weak relationships

(continued)

   















n

i i n

i i

n

i i i xy

n xy

i i n

i

i i

xy

y n y x

n x

y x n y x r

r y

y x

x

y y x x r

1

2 2 1

2 2

1

2 1

2

1 ; -1 1

(6)

R EGRESI P OPULASI DAN S AMPEL

0 1

i i i

Y     X  

Model Jumlah Permintaan dan Harga Populasi

- Y

- 0 X

( ) ? i

E Y  Y i  E Y ( ) i   i

0 1

( )

_i _i

E Y     X



0



1

(7)

R EGRESI P OPULASI DAN S AMPEL

0 1

ˆ ˆ

ˆ i i

Y     X

Model Jumlah Permintaan dan Harga

-

Sampel

-

Y

-

0 Xi

X

( ) ? i

E Y  Y i   Y ˆ i  ˆ i

0 1

( )

_i _i

E Y     X A

( )

_i

E Y

⁰ ¹

ˆ ˆ

ˆ

i i

Y     X

ˆ

i

Y ˆ

_i

e e

i

Y i

Sampel populasi

Hasil estimasi Sampel sebelah kiri A

overestimate,

Sebelah kanan A under

estimate

(8)

E STIMASI R EGRESI OLS

Y

1

0 1

ˆ ˆ

ˆ

i i

Y     X

ˆ

1

e

Y

0 X1 X2 X3 X4 X5

Y

2 ³

Y

5

Y

4

ˆ

2

e

ˆ

3

e

ˆ

4

e

ˆ

5

e

(9)

E STIMASI R EGRESI OLS

0 1

ˆ ˆ

ˆ

i i

Y     X

Y

0 Xi

var iasi Total=( Y Y

_i

 )

Y

i

ˆ

_i

var residual e  iasi

var iasi regresi=( Y Y ˆ

_i

 )

Y

(10)

E STIMASI OLS

Bentuk matriks

2 1

ˆ ?

n i i

e



 

ˆ 0 ??

   ˆ 1  ??

2 1

1

ˆ

0

2 2 0

( )

n i i

e





  



    



  





^T ^T

T T T T T T

T T

S ε ε (Y - Xβ) (Y - Xβ) S = 0

β

Y Y Y Xβ β X Y β X Xβ β

X Y X Xβ

X Xβ X Y

β X X X Y

(11)

E STIMATOR OLS BLUE

Catatan :

2 2 2 2

2 2 2 2 2

1. 1 ;

2. ( ) 0

3. ( ) 2

2

4. ( )( )

5. ( )

6. ( )

i i

i i i

i i i i

i i

i i i i i i

i i i i i i i i i

i i i i i i i i i

X X X nX

n

x X X X nX nX nX

x X X X X X nX

X nX nX X nX

x y X X Y Y X Y nXY

x y x Y Y x Y Y x x Y

x y y X X y X X y y X

 

      

    

    

    

    

 

  

   

 

  

    

(12)

E STIMATOR OLS BLUE

BLUE= Best Linear Unbiased Estimator

1. Linear , b1 linear terhadap var Y

1 2 2

2

ˆ

; ( )

i i i i

i i

i

i i

i

x y x Y

x x

w Y w x

x

  

 

 

0 1

ˆ ˆ (1/ )

[(1/ ) ]

i i i

i i i i

Y X n Y X wY n Xw Y k Y

     

  

 

(13)

E STIMATOR OLS BLUE

2. Unbiased, nilai harapan sama dengan nilai sebenarnya

1 2 0 1

0 1

1

ˆ ( )

? ( ˆ ) ?

i i

i i i i i

i

i i i i i

x Y w Y w X

x

w w X w

E

   

  



    

  



   

  

0 0 1

0 1

0 0

ˆ [(1/ ) ] [(1/ ) ]( )

[(1/ ) ] [(1/ ) ] [(1/ ) ]

[(1/ ) ] ( ) ˆ

i i i i i

i i

n Xw Y n Xw X

n Xw n Xw X n Xw

n Xw E

   

  

 



     

     

  



 

  



  ˆ

¹

E   

=

= 1

0

(14)

E STIMATOR OLS BLUE

3. Best (Var Minimum)

2 2

1 1 1 1 1

1 1 1 1

2 1

2 2 2 2 2 2

1 1 2 2 1 2 1 2 1 1

2 2

1 1 1

2 2

1

ˆ ˆ ˆ ˆ

var( ) [ ( )] [ ]

ˆ ˆ

var( ˆ ) [ ]

[ ... 2 ... 2 ]

[ 2 ],

[ ] 2

i i i i

i i

n n n n n n

n n n

i i i j i j

i i j

n

i i i j

i

E E E

ingat w e w e

E w e

E w e w e w e w w e e w w e e

E w e w w e e i j

w E e w w

    

   



 

  



   

    



      

  

 

 



  



1 1

2

2 2 2 2 1 2

2 2 2

1 1 1 1

1 1

2 2

0 2

[ ]

2 (0)

( )

var( ˆ )

n n

i j

n

n n n n i

i

i i j i n n

i i j i

i i

E e e

x

w w w w

x x

X

n x

   

 

 



   

 

    



 

    

 

(15)

E STIMATOR OLS BLUE

Materi 2: Program S-1 Statistika, ITS, Surabaya 1

1

* 1

*

0 1

*

1

ˆ vs ˆ ; =

ˆ = ( )

( )( )

( ) ( )

( )

?, 0, 1

( ˆ )

i i i i i i i

i i i i i

i i i i

i i i i i

i i i

w Y c Y c w k

c Y w k Y

w k X e

w k w k X

w k e

ingat w w X

E

 



 

  

 

   

 

  



 

  

(16)

E STIMATOR OLS BLUE

1 1 1 1

1

* * * 2 * 2

1

*

1

* 2

2 2

2 2 2 2 2

2 2

1

ˆ ˆ ˆ ˆ

var( ) [ ( )] [ ] ,

ˆ ( )

var( ˆ ) [ ( ) ]

( ) [ ]

( )

2 var( ˆ )

i i

i

i i i

i i

E E E

ingat w k e

E w k e

w k E e w k

w w k k

k

    

 





  

 

   

  

 

  

 



   



(17)

E STIMATOR OLS BLUE

2 2

1 1 1 1 1

1 1 1 1

2 1

2 2 2 2 2 2

1 1 2 2 1 2 1 2 1 1

2 2

1 1 1

2 2

1

ˆ ˆ ˆ ˆ

var( ) [ ( )] [ ]

ˆ ˆ

var( ˆ ) [ ]

[ ... 2 ... 2 ]

[ 2 ],

[ ] 2

i i i i

i i

n n n n n n

n n n

i i i j i j

i i j

n

i i i j

i

E E E

ingat w e w e

E w e

E w e w e w e w w e e w w e e

E w e w w e e i j

w E e w w

    

   



 

  



   

    



      

  

 

 



  



1 1

2 2

1 1 1

2

2 2 1 2

2 2 2

1

1 1

[ ]

2 (0)

( )

n n

i j

n n n

i i j

n n i

i

i n n

i

i i

E e e

w w w

x w

x x



  

 

  



 

 

  

 

  

 

(18)

E STIMATOR OLS BLUE

2 2

0 0 0 0 0

2

2 2

2 2 2

2 2

2 2 2 2

2 2

2

2 2

ˆ ˆ ˆ ˆ

var( ) [ ( )] [ ]

[ 1 ) ]

( 1 ) ( )

( )

(2 )

( )

i i

i

i i

i

i i

E E E

E Xw e

n

x nX nX

n x n x

X X nX

n x

X nX nX nX

n x X

n x

    

 



   

 

   

 



  





  

 

(19)

E STIMATOR OLS BLUE

0 1 0 0 1 1

0 0 1 1

1 1 1 1

2

1 1 1

2 2

ˆ ˆ ˆ ˆ ˆ ˆ

cov( , ) [( ( ))( ( ))]

ˆ ˆ

[( )( )]

ˆ ˆ

[( ( )( )]

ˆ ˆ

[( )( )]

ˆ ˆ

[( )( )]

ˆ ˆ

[ ( )( )]

ˆ ˆ

( ) var( )

( i )

E E E

E

E Y X Y X

E X X

E X

XE X

X X X

     

   

  



  

    

    

   

   

  

 

 

(20)

E STIMATOR OLS BLUE

1

* 1

*

0 1

0 0 1 1

*

1

ˆ vs ˆ ; =

ˆ = ( )

( )( )

( ) ( ) ( )

( )

tak bias maka 0

( ˆ ) [ ( )

i i i i i i i

i i i i i

i i i i

i i i i i i i i

i i i i i i i i i

i i i

i i

wY c Y c w k

c Y w k Y

w k X e

w k w k X w k e

w k w X k X w k e

agar

k k X

E E w k

 



 

   

 

  

 

   

     

 

  

 

   

    

 

1 1

]

( ) ( )

i

i i i

e w k E e

 

   

 

(21)

E STIMATOR OLS BLUE

1 1 1 1

1

* * * 2 * 2

1

*

1

* 2

2 2

2 2 2 2 2 2 2

2 2 2 2

2 2

1

ˆ ˆ ˆ ˆ

var( ) [ ( )] [ ] ,

ˆ ( )

var( ˆ ) [ ( ) ]

( ) [ ]

( ) 2

( )

2 ( )

var( ) ˆ

i i

i

i i i

i i i i

i i

i

E E E

ingat

w k e E w k e

w k E e

w k w w k k

X X

k k

x X X

k

    

 



   

  

 

   

  

 

    

   



 



   

 



(22)

E STIMATOR OLS BLUE

0 1

0 1 0 1

1

1 1

2 2

1 1

2 2

1 1 1 1 1

var( )

( )

( ) ( )

ˆ ˆ

( );

ˆ ( ) ˆ

ˆ ( ( ˆ ) ( ))

ˆ ˆ

( ) 2( ) (

i

i i i

i i i i i

i i i

i i i i i i

i i i i

i i

i

Y X

w w X w

Y Y X X

y x y x

x x

x

x x



  

     

  

    

    

    

    

  

      

    

    

   

   

   

  

 

   )  (  i   )

²

 

(23)

E STIMATOR OLS BLUE

2 2 2

1 1 1 1 1

2

1 1

2

2 2

1 1

2 2

1 1

2 2

1 1

ˆ ˆ

( ˆ ) ( ) 2 ( ) ( )

( )

var( ˆ ) 2 ( ) ( )

( 1)

2 ( ) ( 1)

i i

i

i i i

i i i i i

E x E E x

E

x E w x

n

E w x w x n

      

 

   



     

    

 

  

 

    

  



 

  

 

(24)

E STIMATOR OLS BLUE

2 2

2

2 2

2 2 2

( ) ( )

( )

1 )

1 ( 1)

i

i i

E E

E n

n

E n

n n

   

 

  

  

 

   

 



 

(25)

E STIMATOR OLS BLUE

1

2

1 1

1 2 2 1

2 2

2 2 2 2

2 2 2

( )

( ˆ ) 2 ( 1)

2 ( 2)

i

i i i i

E w x E w x

E x x

x E

E n

n n

  



 

   

  



 

   

   

 



 

(26)

E STIMATOR OLS BLUE

2 2

( ˆ ) ( 2)

ˆ ˆ

( 2)

( ˆ ) ( 2)

( ˆ )

( 2) ( 2)

i

E n

n

E n

E n n

 

 

 

 

 

   

 

 



(27)

E STIMASI M AXIMUM L IKELIHOOD

0 1

2

0 1

2 2

2

1 2 0 1 1 2

2

0 1

2 2 1

2

0 1

2 /2 2

2

,

1 1

( ) exp[ ( ( ) ]

2 2

( , ,..., , , , ) ( ) ( )... ( )

1 1

exp[ ( ( )) ]

2 2

1 1

exp[ ( ( )) ]

(2 ) 2

ln ln ln(2 ) 1 (

2 2 2

i i i i

i i i

n n

n

i i

i

i i

n

Y X Y IIDN

p Y Y X

LF Y Y Y p Y p Y p Y

Y X

n n

LF

  

 

 

  

 

 

 

 

 





   

   



   

  





2

0 1

2

0 1

) pertama terhadap , dan

i i

Y X

diturunkan

 

  

 



ˆ

0

??

  ˆ

1

??

 

ˆ

2

??

 

(28)

E STIMASI M ETHOD OF M OMENT ( MM )

0 1

[ ] 0,

[ ( )] 0

[ ] 0

[ ( ( )] 0,

1 ( ˆ ˆ ) 0

ˆ ??; ˆ ??

i i i

i

i i

i i i

i i

i i i

Y X

E moment

E Y X

E X

E X Y X population moment sample moment

Y X

n

X Y X

n

  



 



 

  



  



  

 



(29)

S TANDAR E RROR DAN A SUMSI OLS

Asusmsi

1.

Hub Y dan X linear

2.

X var fix, bukan var random, antar var X independen

3.

Nilai rata-rata ekpekstasi ei=0

4.

Varians error sama (homoskedasitas)

5.

Antar error pengamatan independen

6.

Variabel error berdistribusi normal

2 2

0 2 2 0

2 2

1 2 2 1

2

2 1

ˆ ˆ

var( ) , ( ) ?

( )

1 1

ˆ ˆ

var( ) , ( ) ?

( )

ˆ

i i

i

i i

n

i

X X

n x n X X se

x X X se

n k

   





^

  



  



 

 



KOEFISIEN DETERMINASI

2 2 2

2 2

2

2 2

ˆ ˆ; ˆ ˆ

ˆ ˆ

( ) ( )

ˆ ˆ

( ) ( ) ( )

ˆ ˆ

( ) ( ) ( )

ˆ ˆ

( ) ( )

1 1

( ) ( )

i i i i i i

i i i

i i i i

i i i

i i

Y Y Y Y Y Y

Y Y Y Y

Y Y Y Y Y Y

SST SSR SSE

Y Y SSR Y Y SSE

R Y Y SST Y Y SST

 



     

   

    

      

 

   

     

   

(30)

P ERTEMUAN 3:

E KONOMETRIKA

P ROGRAM S TUDI S -1 S TATISTIKA ITS S URABAYA

A NALISIS

“ R EGRESI”

ITS, Surabaya, 15 September 2015