Εισαγωγή στην Οικονομετρία Σημειώσεις

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Σηεμιεώσεις στην Οικονομετρία

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, , ( 2007). 1.1

(economic models) . . : QX = f(PX, Yd) () (). ( ), ( , ). : (1.1) QX = 0 - 1X + 2d 1,2,3

. ( QX) . (econometric model): (1.2) QX = 0 - 1X + 2d + , (stochastic term) . (1.1) (deterministic) , , . , . (1.2) , . (0 - 1X + 2d) (Q) . , () .

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1.2

. . : (1.3) C = f(Y) = 0 + 1 (coefficients). (C) , () 0, 1

:Consumption

C = f(Y) = 0 + 1

C4 C3 C2 C1 C0 0 1 1 1

. , , C . Ci = 0 + 1i + : 4

1

2

3

4

Income

C = f(Y, ) Ci = 0 + 1i + :

3 2 1 0 1

3 2

4

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

Y1

Y2

Y3

Y4

Y

2

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1.3

. (time series data): . i i k . , 1990 2005:

i1: Xi2: Xi3: Xi4: Xi5: Xi6: . (cross-sectional data): . . (pooled data): . . (panel data): . . : (simultaneity bias). ii. , . iii. . i.

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1.4

(linear) (non-linear) : = 0 + 1 + = 0 + 1logX + Y = 011 + 22 + = 0 + 1 + 2 + log = 0 + 1logX + log ) = 0 + 1 + logY = 0 + 1logX + = 0 + 1 + = 0e1

1.5 x

, , (system equation). : Qd = 0 + 1 + 2 + (1.5) Qs = 0 + 1 + u (1.5) 0 = P = 0 + 1 + v (1.6) (1.6) : Q = 2 + 3 + z (1.7) 2 = , 3 = z = , 1= v = : 0 + 1 + 2 + = 0 + 1 + u P = +

(1.6) (1.7) (reduced-form system). 1.6

, (dynamic). . t m :

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Yt = + 0t + 1Xt-1 ++ mXt-m + t Yt = + t) (1.8)

(1.8) (distributed lag model). t-m . = 0 m). , , OLS. (multicollinearity), OLS . (autoregressive distributed lag model) AR : Yt = 0t + 1t-1 ++ mt-m + t (1.9)

(regression analysis). , . 2.1

i = 0 + 1i + i : . . . () .

. n , , 2

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. i (i) = 0 + 1i. . (1) = 0 + 1i i, (1),E(Y2),,E(Yn) (i) = 0 + 1i. i (4) (3) . i + (2) (1) (0,2 ), - (mean) . 0 ( ) 1 2 3 4 1 ( ) . , (0,1), , (i), , i = i E(Y/Xi). 2.2 E T (OLS)

(2.1) 0 = 0 1 = 1. . : Yi eii = 0 + 1Xi i=0, 1, 2, , n

i (). i i (residuals). (ei) (i).: ei = Yi - i ei = Yi - 0 + 1Xi

0 i

OLS 0 1 ei2 ): Yi - 0 + 1Xi i)2

min

ei2 ) = min

Yi - i)2 = min

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: 1=

=

0 = - 1

: . (unbiased) . ( 1)=1, ( 2) . (consistent) , 0 0 0 1 . . (efficient) (standard error) . . (ei) i . i=

(ei ), i . (), i . 2.3 .

i = 0 + 1Xi : . ( , : = = 0 + 1 . (i) ( i): i= i

. : I= i i

. i :

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iXi = 0

. ei i : 2.4 i I=0 E(i) = 0 (i = 1, 2, , n)

. i, :

. Yi, Xi (linear). . i, 2 : Var(i) = E(i2) = 2 () . i j Cov(i, j) = E(i, j) = 0 , :

. i i : Cov(Xi, i ) = E(Xi, i) = 0 . i E(Yi) = 0 + 1i 2 2.5

Gauss-Markov (Gauss-Markov Theorem) To Gauss-Markov , 0 , 1 0 1, . , 0 , 1, BLUE (Best Linear Unbiased Estimator). (Central Limit Theorem) yi 0, 1 . , yi , xi , . Yi - = yi i - = xi :

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. 1 : E( 1) = 1 . ( 1 : Var( 1) = =

) se2 = =

. se2 ( ei = Yi -

=

)

. n-2 . . (standard error) : . : ( se = =

=

.

:( 0) =

=

=

. 0 , 2.6

1 :

Cov( 0,

1) =

=

, i = 0 + 1Xi , : . 0 1 t-. tn-2 = ( n-2 ,

) . f-. . 2-. 2.7 (hypothesis testing)

: (false) (true).

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0 : (null hypothesis) . (calculated price) tn-2 tc (critical price) 0. 1 : 0 (alternative hypothesis). , 1- . 1- (confidence interval) . 1. 95% i 1.96 . 2. 95% i. 3. 100 ( 95% ) i 100 95 100 i. 4. = (level of significance) . 2.8

: I (type I error) : . II. (type II error) : . I. R2

2.9

O (coefficient of determination) R2 Yi Xi.

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Yi Yi - i ( ) i ( ). : Yi - = ( i - ) + (Yi - i) Yi , ei Xi. (TSS) (RSS) (RSS). TSS = = RSS + + ESS

R2 : R2 = =1-

2.10

(r) -1 +1. r=,

=

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t P 0,25 0,50 1,000 0,816 0,765 0,741 0,727 0,718 0,711 0,706 0,703 0,700 P 0,10 0,20 3,078 1,886 1,638 1,533 1,476 1,440 1,415 1,397 1,383 1,372 P 0,05 0,10 6,314 2,920 2,353 2,132 2,015 1,943 1,895 1,860 1,833 1,812 P 0,025 0,05 12,706 4,303 3,182 2,776 2,571 2,447 2,365 2,306 2,262 2,228 P 0,01 0,02 31,821 6,965 4,541 3,747 3,365 3,143 2,998 2,896 2,821 2,764 P 0,005 0,010 63,657 9,925 5,841 4,604 4,032 3,707 3,499 3,355 3,250 3,169 P 0,001 0,002 318,31 22,327 10,214 7,173 5,893 5,208 4,785 4,501 4,297 4,144

df 1 2 3 4 5 6 7 8 9 10

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11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 0,697 0,695 0,694 0,692 0,691 0,690 0,689 0,688 0,688 0,687 0,686 0,686 0,685 0,685 0,684 0,684 0,684 0,683 0,683 0,683 0,681 0,679 0,677 0,674 1,363 1,356 1,350 1,345 1,341 1,337 1,333 1,330 1,328 1,325 1,323 1,321 1,319 1,318 1,316 1,315 1,314 1,313 1,311 1,310 1,303 1,296 1,289 1,282 1,796 1,782 1,771 1,761 1,753 1,746 1,740 1,734 1,729 1,725 1,721 1,717 1,714 1,711 1,708 1,706 1,703 1,701 1,699 1,697 1,684 1,671 1,658 1,645 2,201 2,179 2,160 2,145 2,131 2,120 2,110 2,101 2,093 2,086 2,080 2,074 2,069 2,064 2,060 2,056 2,052 2,048 2,045 2,042 2,021 2,000 1,980 1,960 2,718 2,681 2,650 2,624 2,602 2,583 2,567 2,552 2,539 2,528 2,518 2,508 2,500 2,492 2,485 2,479 2,473 2,467 2,462 2,457 2,423 2,390 2,358 2,326 3,106 3,055 3,012 2,977 2,947 2,921 2,898 2,878 2,861 2,845 2,831 2,819 2,807 2,797 2,787 2,779 2,771 2,763 2,756 2,750 2,704 2,660 2,617 2,576 4,025 3,930 3,852 3,787 3,733 3,686 3,646 3,610 3,579 3,552 3,527 3,505 3,485 3,467 3,450 3,435 3,421 3,408 3,396 3,385 3,307 3,232 3,160 3,090

: . df (degrees of freedom). , df = 25 = 0,05 t0.os = 1,708. -, df = 15 = 0,025 t0025 2,131. Gujarati (2003).

(multiple regression models). = . i

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3.1

. : ( ) = 0 . , .

. (2) : Var( ) = ( 2) = 2 . : Cov( , ) = ( )=0 . : Cov( , ) = E( )=0 . : ~ ( , ) . . . : n>k 1. = (partial regression coefficients)

=

= : y= ,

x1 =

,

x2 =

,

=

,

=

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3.2 , , ,

, , : = = = = =

( ) ( ) [ ]

3.3

=

=

=

=

(coefficient of multiple determination), , , , . =

=

R2 = 1

=

0 1 . . (adjusted coefficient of multiple determination).

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3.4

. , R2 . = = 3.5

, (partial correlation coefficients). , .