1

D. Ciołek

EKONOMETRIA – wykład 3

EKONOMETRIA

Wykład 3: Metoda Najmniejszych Kwadratów

dr Dorota Ciołek

Katedra Ekonometrii

Wydział Zarządzania UG

Konsultacje:

budynek Wydziału Zarządzania p. 115

wtorek 13:00-14:30

http://wzr.pl/~dciolek

[e-mail]

2

D. Ciołek

EKONOMETRIA – wykład 3

Liniowy model ekonometryczny:

y

–

x

–

-
-

dzy nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn

nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn
nnnnnnzmiennymi

x

i

y.

-

-

t

tk

k

t

t

t

x

x

x

y























...

2

2

1

1

0









ˆ

3

D. Ciołek

EKONOMETRIA – wykład 3

Oszacować

(estymować) model oznacza

kkkkkkkkkkkk

kkkkkkkkkkkkkkkkk

Metody szacowania parametrów strukturalnych:

-

-

-

-

i wiele innych…

Twierdzenie Gaussa-Markowa:

W klasycznym modelu regresji liniowej najlepszym

nieobciążonym estymatorem linowym parametrów jest
estymator uzyskany

4

D. Ciołek

EKONOMETRIA – wykład 3

Własności estymatorów

Nieobciążoność –

g

jest nieobciążonym estymatorem ,

jeżeli

E(g)=

, co znaczy, gdy wartość oczekiwana w

rozkładzie z próby

g

jest równa .

lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll

lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
llllllllllllllllllllllllllll

Efektywność –

llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll

llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll

Estymator z najmniejszą wariancją –







5

D. Ciołek

EKONOMETRIA – wykład 3

Własności estymatorów

Zgodność –

lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll

llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
lllllllllllllllllllllllllllllllllllllllllllllllllllllll

Można wykazać, że:

Metoda Najmniejszych Kwadratów jest estymatorem:

-
-
-

BLUE –Best Linear Unbiased Estimator





1

lim















g

P

n

6

D. Ciołek

EKONOMETRIA – wykład 3

Założenia MNK

Założenia numeryczne – warunki stosowalności:

1) T > (k+1),

llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll

llllllllllllllllllllllllllllllllllllllll

2) r(X)=(k+1),

llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll

lllllllllllllllllllllllllllllllll

7

D. Ciołek

EKONOMETRIA – wykład 3

Przykład współlinowości zmiennych:

X1-

X2-

X3-

X1=X2+X3, czyli X1-X2-X3=0

Rząd macierzy X=3 < k+1=4



































50

10

60

1

17

3

20

1

41

6

47

1

48

8

56

1

26

4

30

1

X

8

D. Ciołek

EKONOMETRIA – wykład 3

Założenia MNK

Założenia stochastyczne (dotyczą składnika losowego):

1)

lllllllllllllllllllllllllllllllllllllllllllllllllll

lllllllllllllllllllllllllllllllllllllllllllllllllll

2)

llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll

llllllllllllllllllllllllllllllllll

3) i są niezależne dla -

llllllllllllllllllllllllllllllllllllll lllllllllllllllllll

llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
lllllllllllllllllllllllllllllllllll

4) i są niezależne dla wszystkich

t

–

lllllllllllllllllllllllllllllllllll

llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
lllllllllllllllllllllllllllllll

5) -

lllllllllllllllllllllllllllllllllllllllllllll

 

0



t

E



 

2

2









t

i



j



t

x

t



j

i 





2

,

0

~





N

t

9

D. Ciołek

EKONOMETRIA – wykład 3

Założenia MNK

Jeżeli nie są spełnione

założenia numeryczne

–

lllllllllllllllllllllllllllllllllllllllll

Jeżeli nie są spełnione

stochastyczne założenia 1), 2), 3), 4)

llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
lllllllllllllllllllllllllllllllllllllll

Założenie 5)

10

D. Ciołek

EKONOMETRIA – wykład 3

Rzeczywisty rozkład punktów

x

y

y

t

11

D. Ciołek

EKONOMETRIA – wykład 3

Model z jedną zmienną objaśniającą:

lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll

lllllllllllllllllllllllllllllllllllllllllllllllllllll

MNK to metoda,

t

t

t

x

y













1

0

t

t

x

y

1

0

ˆ

ˆ

ˆ









12

D. Ciołek

EKONOMETRIA – wykład 3

Odległość rzeczywistego punktu od prostej nazywana jest

odchyleniem, albo :

t

t

t

y

y

ˆ

ˆ







13

D. Ciołek

EKONOMETRIA – wykład 3

Idea MNK

 

min

ˆ

1

2







T

t

t











min

ˆ

ˆ

ˆ

1

2

1

0

1

2



















T

t

t

t

T

t

t

t

x

y

y

y





14

D. Ciołek

EKONOMETRIA – wykład 3

Estymator MNK





y

X

X

X

T

T

1

ˆ







-

y

–

X

–



ˆ

15

D. Ciołek

EKONOMETRIA – wykład 3

Własności numeryczne oszacowania MNK











T

t

t

T

t

t

y

y

1

1

ˆ

0

ˆ

1







T

t

t



0

ˆ

' 



X

0

ˆ

'

ˆ 



y