R R 16 ( 3 )

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1 (017 ) ( ) ( 3 ) ( 010 ) 1 (P3) 1 11 (P4) 1 1 (P4) 1 (P15) 1 (P16) (P0) 3 (P18) 3 4 (P3) α β 9 63 û R

2 R R 16 ( 3 )

3 (P1) 1 ( ) 1, (a) (b) (c) ( ) (d) ( ) ( ) 1 (P3) 11 (P4) (P4) 1 ( ) ( ) ( 009 GDP GNP ) 1 (P4) 13 (P7) (0 ): (P8) ( ) 11 0 (P11) (P9,10) 1 (P6 11 1, P9 14) ( ) (P10 16)

4 (P15) ( ) ( ) (P16) ( ) ( ) m 1 a 0 a 1 f 1 m a 1 a f m k a k 1 a k f k m 1 a 0 + a 1, m a 1 + a,, m k a k 1 + a k x 1 (f 1m 1 + f m + + f k m k ) 1 k f m ( ) x k f m f x 1 0 ( ) (P16) x 1, x,, x (P16) x 1 (x 1 + x + + x ) 1 13 (P7) x 1 ( ) x (P0) (s ) s 1 ((x 1 x) + (x x) + + (x x) ) 1 (x x) x 1 x

5 s s 1 x x s 1 1 (x x) (x xx + x ) 1 ( x x x + x ) 1 ( x x + x ) 1 ( x x ) 1 x x 13 (P7) s 1 ( (43 553) + (5 553) + 0 +(59 553) ) s 1 0 ( ) s (P17) s 1 k f (m x) x 1 s 1 k f m k f m x s 1 1 k f (m x) k f (m xm + x ) 1 ( k f m x k f m + x 1 ( k f m x + x ) 1 ( k f m x ) 1 k f m x k ) f s 1 ( ( ) + 3( ) 0 +8( ) + 5( ) +( ) ) 119 s 1 0 ( ) s (P18) 5 ( 1 ) 50 ( ) 75 ( 3 ) 3 1 3

6 ( ) 13 (P7) ( )/ (P7) (P3) 100 (x 1, y 1 ), (x, y ),, (x, y ) s xy s xy 1 ( (x 1 x)(y 1 y) + (x x)(y y) ) + + (x x)(y y) 1 1 (x x)(y y) x y xy s xy > 0 (x y ) s xy < 0 (x y ) s xy 0 (x y ) ( ( ) 1988) r r s xy s x s y s x 1 (x x), s y 1 (y y), s x, s y x y r > 0 (x y ) r < 0 (x y ) r 0 (x y ) r 1 r 1 t f(t) 1 (, (x x)t (y y)) t f(t) 0 t f(t) t 1 (x x) + t 1 (x x)(y y) + 1 (y y) s xt + s xy t + s y 0 D 4 s xy s xs y 0 s xy s xs y 1, 1 s xy s x s y 1, r 1 (x y ) r 1 (x y ) r 1, 1 x y (r 1 r 1 ) 3 () 4

7 (GNP ) 31 1 C f(y ) C Y a, b a, b 3 a > 0 1 > b > 0 1 Y C dc 1 dy 3 dc dy > 0 1 C a + by b dc dy 3 a (Y 0 ) 4 a > 0 b > 0 (1 > b) 1 (C ) (Y ) Q f(y, P 1, P ) Q Y P 1 P 1 Y Q, P 1 Q, P Q Q Y > 0, Q < 0, P (A) Q P > Q a + b 1 Y + b P 1 + b 3 P 000 C Q, Y, P 1, P a, b 1, b, b 3 ( ) 6 b 1 > 0, b < 0, b 3 > 0, a? t (Q, Y, P 1, P ) 500 8, e, 1,,, Y 1 (Y, C ) t, e, 1,,, 9 9 (B) Q a + b 1 Y + b P 1 P b 1 > 0, b < 0 5

8 10 (C) log(q) a + b 1 log(y ) + b log( P 1 P ) b 1 > 0, b < 0 11 (A), (B), (C) 4 A ( ) a11 a 1 A a 1 a a j A j a 1 ( ) ( ) a1 a a a a a 1 ( ) a ( a 1 a ) a a A k a 11 a 1k A a 1 a k a j A j (j ) a 1 ( ) a a 1 a a a a 1 k ( ) a ( a 1 a k ) a a A B k A B 1,,, j 1,, k a j b j a j, b j A, B j x 3, y ( ) ( ) x 3 ( x y ) ( 3 ) y A, B k a 11 a 1k b 11 b 1k A + B + a 1 a k b 1 b k a 11 + b 11 a 1k + b 1k a 1 + b 1 a k + b k A + B j a j + b j ( ) ( ) A B ( ) ( ) A + B ( ) ( ) A B (1 1 ) a 11 a 1k ca c A k c ca 11 ca 1k a 1 a k ca 1 ca k ( ) 1 A 3 4 c 5 ( ) ( ) ( ) ca

9 A, B k k a 11 a 1k b 11 b 1 AB a 1 a k b k1 b k k m1 a k 1mb m1 m1 a 1mb m k m1 a mb m1 k m1 a 1mb m AB AB j a 1 b 1j + a b j + + a k b kj k m1 a kb kj b 11 b 1 a 11 a 1k BA b k1 b k a 1 a k m1 b 1ma m1 m1 b 1ma mk m1 b kma m1 m1 b 1ma mk BA k k BA j b 1 a 1j + b a j + + b k a kj k m1 a kb kj AB BA ( ) 1 A 3 4 ( ) 5 6 B 7 8 ( ) ( ) AB ( ) ( ) ( ) ( ) BA ( ) ( ) AB BA c cab AcB (Ac)B A(cB) ABc c { x + y 3 4x + 5y 6 ( ) ( ) ( ) 1 x y 6 x + y + 3z 4 5x + 6y + 7z 8 9x + 10y + 11z x y z I I I I A x 1 () I A AI A I x x 1 0 a 11 a a 1 a a 11 a a 1 a 0 1 a 11 a 1 a 1 a 1 0 x x x 1 x 7

10 A A AB I BA I B A B B A 1 A A 1 A 1 A ( ) a b A c d A 1 1 ad bc ( ) d b c ( ) ( ) A 1 1 d b a b A ad bc c a c d ( ) 1 da bc db bd ad bc ca + ac bc + ad ( ) 1 0 I 0 1 ( ) ( ) a b AA 1 1 d b c d ad bc c a ( ) 1 ad bc ab + ba ad bc cd dc cb + da ( ) 1 0 I 0 1 () Ax b A 1 A 1 Ax A 1 b A 1 A I I x A 1 b I x x x A, b x A 1 b a A x b 1 { x + y 3 4x + 5y 6 ( ) ( ) ( ) 1 x y 6 x, y ( ) 1 ( ) ( ) ( ) 1 ( ) 1 1 x y ( ) ( ) ( ) 1 ( ) 1 0 x y ( ) ( ) 1 ( ) x 1 3 y ( ) ( ) ( ) ( ) x + y + 3z 4 5x + 6y + 7z 8 9x + 10y + 11z x y z 1 x, y, z x 1 3 y z

11 A k A j a j A (A t A) j a j a 11 a 1k A a 1 a k a 11 a 1 A a 1k a k A k (A ) A 5 1 ( ) ( ) t ( ) ( ) 6 61 x 1 x x x X X (X X) 0 x ( x 1 x x ) (X 1, Y 1 ), (X, Y ),, (X, Y ) X Y Y α + βx, X Y α, β α β {(X, Y ), 1,,, } α β ˆα ˆβ {(X, Y ), 1,,, } Y ˆα + ˆβX + û, Y ˆα + ˆβX û 3 (X X) X X 6 α β 4 (X X)(Y Y ) X)Y (Y Y )X X Y X Y 5 1 ( d ) b ad bc c a (X ( ) 1 a b c d α, β ˆα, ˆβ S(ˆα, ˆβ) S(ˆα, ˆβ) û (Y ˆα ˆβX ) m S(ˆα, ˆβ) ˆα, ˆβ 9

12 ˆα, ˆβ S(ˆα, ˆβ) ˆα 0, ˆα, ˆβ S(ˆα, ˆβ) ˆβ 0 ˆα, ˆβ (Y ˆα ˆβX ) 0, (1) X (Y ˆα ˆβX ) 0, () Y ˆα + ˆβ X (3) X Y ˆα X + ˆβ (3) 1 Y ˆα + ˆβ 1 X X (4) Y ˆα + ˆβX (5) X 1 X, Y 1 Y, X X (5) ˆα X Y (Y ˆβX)X + ˆβ ˆβ ˆβ X Y XY X X (X X)(Y Y ) (X X) ˆα (5) X S XY S X (6) ˆα Y ˆβX (7) S XY 1 (X X)(Y Y ) S X 1 (X X) ( Y ) ( X X ) ( ) Y X, ˆαˆβ ˆα ˆβ ( ) ( ˆαˆβ X X X 1 X ( X ) ( X X ˆβ X ) 1 ( Y ) X Y X ˆβ X Y ( X )( Y ) X ( X ) X Y X Y X X ˆα (X X)(y Y ) (X X) ) ( Y ) X Y ˆα ( X )( Y ) ( X )( X Y ) X ( X ) Y X X X Y X X Y ( X X ) X( X Y Y X) X X Y X Y Y X X X X Y ˆβX Ŷ ˆα + ˆβX, Ŷ X Y 10

13 Y α + βx α β ˆα ˆβ X Y ˆα ˆβ ˆβ X Y X Y X, ˆα Y ˆβX, X X Y X X Y X Y X X Y X Y X X Y X Y ˆβ ˆα , Y Y 4 { û 4 Ŷ 4 1: Y X Ŷ û X 4 Ŷ ˆα + ˆβX X X Y X X Y Ŷ X Y X X Y Ŷ X Y 3 6 Ŷ Y û Y Ŷ, û Y, Ŷ, û Ŷ, X, ˆα, ˆβ Y Ŷ + û ˆα + ˆβX + û, 1 α, β ˆα, ˆβ α, β Ŷ ˆα + ˆβX Ŷ X, Ŷ1, Ŷ,, Ŷ5 Y X Ŷ û 1 63 û û Y ˆα ˆβX (1) () û 0, X û 0, Ŷ ˆα + ˆβX Ŷ û 0, 11

14 Ŷ û (ˆα + ˆβX )û ˆα û + ˆβ X û 0 X Y Ŷ û X û Ŷ û X Y Ŷ û X û Ŷ û X Y (Ŷ Y ) (Y Y ) + û (Y Y ), 1 3 (Y Y ) Y (Ŷ Y ) Ŷ () û Ŷ () R R (Ŷ Y ) (Y Y ), (8) 64 R Y, Ŷ, û Y Ŷ + û, Y (Y Y ) (Ŷ Y ) + û, (Y Y ) ( ) ( Ŷ Y ) + û (Ŷ Y ) + (Ŷ Y )û + (Ŷ Y ) + û Ŷû Y û 0 (Y Y ) (Ŷ Y ) + û û R Y Ŷ X R 1 û (Y Y ), (9) R : R (8) R 0 (9) 1 R 1 R 0 R 1, R 1 û 0 (X, Y ) R 0 Y X ˆβ 0 1

15 R 0 R 1 R 1 R 0 09 R 65 R : R R (Ŷ Y ) (Ŷ Y )(Y Y û ) (Ŷ Y )(Y Y ) (Ŷ Y )(Y Y ), (Ŷ Y )û Ŷ Y û R R (Ŷ Y ) (Y Y ) ( ) (Ŷ Y ) ( )( (Y ) Y ) (Ŷ Y ) ( ) (Ŷ Y )(Y Y ) (Y, Y ) (Ŷ Y ) R Y Ŷ (Ŷ Y ) Ŷ ˆα + ˆβX Y ˆα + ˆβX (Ŷ Y ) ˆβ ˆβ (X X) (X X)(Y Y ), R (Ŷ Y ) (Y Y ) ˆβ (X X) (Y Y ) ( (X X)(Y Y ) (Y Y ) (X X) S XY SX, S Y X Y R 1 ) û Y Y û Y Y X Y Ŷ û û Y X Y Ŷ û û Y X Y 3 6 Y 6 û 43 R 1 65 Y X Y (a) (d) 13

16 5 4 3 Y (a) 1 Ŷ X R X : Y (b) Ŷ X R X 66 ˆα ˆβ ˆβ X Y X Y X X ˆα Y ˆβX X Y X X Y 5 4 Y (c) 3 1 Ŷ X R X Y (d) 1 R X R 1 û Y Y û Y ˆα ˆβX û Y Y 7 (a) (b) (c) (d) X Y X Y X Y X Y (a) (b) ˆα 0 ˆβ 1 (a) 075 (b) 093 (a) (b) X X 4 (b) (a) Y (b) (a) (c) 1 (d) X Y (d) X Y k Y β 1 X 1 + β X + + β k X k X j j β 1, β,, β k X 1 1 β 1 (Y, X 1, X,, X k ), 1,,, β 1, β,, β k β 1, β,, β k ˆβ 1, ˆβ,, ˆβ k {(X, Y ), 1,,, } Y ˆβ 1 X 1 + ˆβ X + + ˆβ k X k + û Ŷ + û, Y Ŷ ˆβ 1 X 1 + ˆβ X + + ˆβ k X k û S( ˆβ 1, ˆβ,, ˆβ k ) S( ˆβ 1, ˆβ,, ˆβ k ) (Y ˆβ 1 X 1 ˆβ X ˆβ k X k ) u m ˆβ 1, ˆβ,, ˆβ k S( ˆβ 1, ˆβ,, ˆβ k ) 14

17 ˆβ 1, ˆβ,, ˆβ k S( ˆβ 1, ˆβ,, ˆβ k ) ˆβ 1 0 S( ˆβ 1, ˆβ,, ˆβ k ) ˆβ 0 S( ˆβ 1, ˆβ,, ˆβ k ) ˆβ k 0 ˆβ 1, ˆβ,, ˆβ k ˆβ 1, ˆβ,, ˆβ k (Y ˆβ 1 X 1 ˆβ X ˆβ k X k )X 1 0, (Y ˆβ 1 X 1 ˆβ X ˆβ k X k )X 0, (Y ˆβ 1 X 1 ˆβ X ˆβ k X k )X k 0, X jx l X jy X j X l Xj Y j 1, l 1, 71 k Y β 1 X 1 + β X + u, β 1, β m β 1, β (Y β 1 X 1 β X ) 1,,, ˆβ 1 ˆβ ( ) ( ˆβ1 X ) 1 ( ) 1 X1X X1Y ˆβ X1X X XY X 1 Y ˆβ 1 X Y ˆβ 1 X k Y ˆβ 1 X 1 + ˆβ X 1 X + ˆβ X 1 X k + ˆβ X 1 X + + ˆβ k X + + ˆβ k X X k + + ˆβ k X1Y X 1 X1X XY X1X X Xk Y X1X k XX k X k X1X k XX k X 1 X k X X k Xk ˆβ 1 ˆβ ˆβ k 1 ( X1 )( X ) ( X 1X ) ( X X 1X X 1X X 1 ) ( ) X1Y XY ( X )( X 1Y ) ( X 1X )( X Y ) ( X 1 )( X ) ( X 1X ) ( X 1X )( X 1Y ) + ( X 1)( X Y ) ( X 1 )( X ) ( X 1X ) Y α 1 X + v X 1 α X + w α 1 α X Y X X ˆα 1 X, ˆα 1 X ˆβ 1, ˆβ,, ˆβ k ˆβ 1 X 1 X1X ˆβ X1X X ˆβ k X1X k XX k X k X1X k XX k 1 X1Y XY Xk Y ˆα 1 ˆα ˆv ŵ ˆv Y ˆα 1 X, ŵ X 1 ˆα X 15

18 ˆv ŵ Y X 1 X ˆv γŵ + ɛ γ ˆγ ˆβ 1 ŵˆv (X1 ˆα ˆγ X )(Y ˆα 1X ) ŵ (X1 ˆα X ) X1Y ˆα 1 X1X ˆα XY + ˆα 1 ˆα X X 1 ˆα X1X + ˆα X X1Y ( X Y )( X 1X ) X X 1 ( X 1X ) X ( X )( X 1Y ) ( X 1X )( X Y ) ( X 1 )( X ) ( X 1X ) ˆβ 1, Y X X 1 X β 1 Y β 1 X 1 + β X + + β k X k j β j Y X 1,, X j 1,, X j+1,,, X k ( X j ) X j X 1,, X j 1,, X j+1,,, X k ( X j ) 7 R R R R (Ŷ Y ) (Y Y ) 1 û (Y Y ) Ŷ ˆβ 1 X 1 + ˆβ X + + ˆβ k X k Y Ŷ + û R û R R R 1 û /( k) (Y Y ) /( 1), û /( k) u σ (Y Y ) /( 1) Y R R R 1 (1 R ) 1 k, 1 R 1 R 1 k 1, R R (k 1 ) R X Y Ŷ û û Y X Y Ŷ û û Y X Y 3 6 Y 6 û 43 û Y 43 R 1 Y Y R R û 1 /( k) ( Y Y )/( 1) 43/(5 ) 1 9/(5 1)

19 û ˆβ 1, ˆβ,, ˆβ k k k (degree of freedom) X 1 Y X, X 3,, X k β β 3 β k 0 û Y ˆβ 1 û ˆβ 1 ˆβ 1 Y 1 R R u () 17

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