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1 黒住英司 [ 著 ] サピエンティア 計量経済学 訂正および練習問題解答 (206/2/2 版 ) 訂正 練習問題解答 3

2 .69, (X i X)U i i i (X i μ x )U i ( X μx ) U i. i E [ ] (X i μ x )U i i E[(X i μ x )]E[U i ]0. i V [ ] (X i μ x )U i i 2 i j E [(X i μ x )(X j μ x )U i U j ] 2 E [ (X i μ x ) 2] E [ Ui 2 ] i σ2 xσ 2 P ( ε) ε 2 V [ ( ) 2] ε 2 σ 2 xσ (i) ( X μx ) i U i (iii) i (X i X)U i i (X i X) 2 0 σ 2 x a 0.92, 4(d) ( ) ˆV i b 2 Ŵ i + Êi ( ) ˆV i b Ŵ i + Êi.0 (5-2) ( ) +0.X i ( ) +0.X i 4

3 .24 2 ( ) ( ) 0.25, 3 ( ) (6-2) 9.36, 6 ( )σi 2 X2 i ( )σ2 i cx i.4, 8. ( ) 4.3(iv) ( ) 4.3(v).49, ( ) 2 W i (8-0) Y i αw i + βx i + U i Ỹ ˆφ 2 Y, W ˆφ 2, Xi ˆφ 2 X, i 2,, Ỹ i Y i ˆφY i, Wi ˆφ, Xi X i ˆφX i i,, Ỹi W i X i ˆα ˆβ.66, (9 27) ( ) β 2i ( ) β 2.7, 2 ( ) ( ) 2

4 ( 3 4 ) (-3) (X i X) 2 i (X i X)(X i X) i X i (X i X) X i X i (X i X). i (X i X) i 3 (-4) (Y i Ȳ )2 i (Y i Ȳ )(Y i Ȳ ) i Y i (Y i Ȳ ) Ȳ i Y i (Y i Ȳ ). i (Y i Ȳ ) i 4 (-3) (-4) (X i X)(Y i Ȳ ) X i (Y i Ȳ ) X (Y i Ȳ ) i i i X i (Y i Ȳ ). i (X i X)(Y i Ȳ ) (X i X)Y i (X i X)Ȳ 3 i (X i X)Y i. i i i

5 (X i X)(Y i Ȳ ) X i Y i X Y i X i Ȳ + i i i i i X i Y i XȲ XȲ + XȲ i X i Y i XȲ. i XȲ 5 (-) (-7) 6(a) S(β) (Y i βx i ) 2 ds(β) 2 X i (Y i βx i )0. dβ β 2 i i b i X iy i i X2 i (b) (a) b Û i i (Y i bx i ) i Y i i i X iy i i X2 i 0 X i Û i X i (Y i bx i ) i i X i Y i i i X iy i i X2 i i X i Xi Ŷ i bx i X iy i X i i X2 i i 4 i i

6 Ȳ Ŷ i Û i b i X i Û i b X i X iy i i X2 i i X i Ȳ (c) 4 TSS ESS + RSS R 2 i 2 (a) (i) A {, 2, 3, 4}, A c {5, 6} P (A) 2 3, P(Ac ) 3 P (A)+P (A c ). B {4, 5, 6}, B c {, 2, 3} P (B) 2, P(Bc ) 2 P (B)+P (B c ). (ii) A B (iii) A B {, 2, 3, 4, 5, 6} P (A B) P (A)+P (B) P (A B) (b) A B {4} P (A B) /6 /2 3. (c) P (A B) /6 P (A B) /3 P (B)P (A B)

7 P (B A) /4 2 F (x) F (x) P (A)P (B A) x 0 : x a f(t)dt x a b a : a<x b : x>b a + b 2 (a b)2,. 2 3 X Y ( ) X Y f X (x) f Y (y) f X,Y (x, y) (i) E[a] (ii) E[aX + b] (iii) E[X + Y ] a af X (x)dx a (ax + b)f X (x)dx xf X (x)dx + b (x + y)f X,Y (x, y)dxdy ( ) x f X,Y (x, y)dy xf X (x)dx + E[X]+E[Y ]. (iv) E[cg(X)+dh(Y )] c c +d yf Y (y)dy. f(x)dx a. f X (x)dx ae[x]+b. dx + y ( ) f X,Y (x, y)dx dy (cg(x)+dh(y))f X,Y (x, y)dxdy ( ) g(x) f X,Y (x, y)dy dx ( ) h(y) f X,Y (x, y)dx dy g(x)f X (x)dx + d ce[g(x)] + de[h(y )]. 6 h(y)f Y (y)dy

8 (v) V [ax + b] {(ax + b) E[aX + b]} 2 f X (x)dx {a(x E[X])} 2 f X (x)dx a 2 (x E[X]) 2 f X (x)dx a 2 V [X]. (vi) V [X] (x E[X]) 2 f X (x)dx ( x 2 2xE[X]+(E[X]) 2) f X (x)dx E[X 2 ] 2E[X]E[X]+(E[X]) 2 E[X 2 ] (E[X]) ( a a 2 a 3 a 4 a 5 ) 5 E[ X] E[X] i μ μ. i V [ X] E [ ( X μ) 2] { } 2 E (X i μ) 2 i i j E[(X i μ)(x j μ)]. i j E[(X i μ)(x j μ)] σ 2 i j E[(X i μ)(x j μ)] 0 V [ X] 2 i σ 2 σ2. 6(a) 2.5 X 2 X +3 X 2 2μ +3μ 2. μ X 2 μ

9 (b) 2.6 ( σ X μ ) ( σ X 2 μ 2 ) 2 d σ N(0,σ 2 ) N(0, ), d σ 2 N(0,σ 2 2) N(0, ). X i X 2i ( ) 2 ( ) 2 ( σ X μ ) + ( σ X d 2 μ 2 ) χ (Excel ) (3-3) V [a] E[(a α) 2 ] X 2 V [b]+e[ū 2 ] 2 XE[(b β)ū]. E[Ū 2 ] σ2, [ E[(b β)ū] E i (X i X)U i i (X i X) 2 i (X i X)σ 2 i (X 0. i X) 2 V [a] σ 2 X2 i (X i X) 2 + σ2 ] U i i σ 2 i X2 i i (X i X) 2. Cov(a, b) E [{ (b β) X + Ū} (b β) ] XE[(b β) 2 ]+E[(b β)ū] σ 2 X i (X +0. i X) 2 8

10 3 (3-6) (3-7) (3-8) 4 a Ȳ b X Y i X i (X i X)Y i i (X i X) 2 i i c i Y i, c i X(Xi X) i (X i X) 2. 5 a i c i Y i a E[a ] α c i E[Y i ] i c i E[α + βx i + U i ] i c i + β i c i X i α a c i, c i X i 0 i i i a c i (α + βx i + U i )α + i V [a ]σ 2 i c 2 i. 2 a c i, c i X i 0 i i c i U i i V [a] σ 2 c 2 i i 9

11 (3-22) i c i(c i c i)0 i c 2 i {c i +(c i c i )} 2 i 6 X b V [a ] σ 2 i c 2 i σ 2 c 2 i + σ 2 i c 2 i + i (c i c i ) 2 i (c i c i ) 2 i σ 2 c 2 i V [a]. i μ x Ū β 56 (3-3) a α (b β) X + Ū α (β β)μ x +0α. 4 2 (4-6) (4-6) β,,β K b,,b K 2 K Ŷi b + b 2 X 2i + + bkx Ki X ki 2(a)R R (b) t 7 2,.25 0, , 0.22 (c) t (d) 0.5 ± [.284, 0.284] 3 [ 0.29, 0.097] [0.064, 0.42] [0.594, 0.960] [.47, 0.62] 0

12 4 (a) (4-6) b b 2 X i (Y i b X i b 2 X 2i ) 0 i X 2i (Y i b X i b 2 X 2i ) 0 i (b) b S 22S Y S 2 S 2Y S S 22 S 2 2, b 2 S S 2Y S 2 S Y S S 22 S 2 2 (c) b Y 2 S 2Y, b 2 S 2. S 22 S 22 (d) ˆV i Y i b Y 2 X 2i Ŵi X i b 2 X 2i i b Ŵi ˆV i i Ŵ i 2 (S 22S Y S 2 S 2Y )/S 22 (S S 22 S 2 2 )/S 22 S 22S Y S 2 S 2Y S S 22 S 2 2 b. 5 (4-2) X i X 2i S Y β S + β 2 S 2 + S U, S 2Y β S 2 + β 2 S 22 + S 2U b b β + S 22S U S 2 S 2U S S 22 S 2 2 X X 2 U E[b X,X 2 ]β + S 22E[S U ] S 2 E[S 2U ] S S 22 S 2 2 β. E[b ]E[E[b X,X 2 ]] β b β + S 22 S U S 2 S 2U S S 22 ( S 2) 2 β + lim S 22 0 lim S 2 0 lim S S 22 lim( S 2) β. 2

13 5 (5-8) , 0.95 (5-9) (5-20) i j D ji (j, 2, 3, 4) Y i α D i + α 2 D 2i + α 3 D 3i + α 4 D 4i + βx i + U i. 4 i D i Y i α + δ c D i + βx i + δ X D i X i + U i δ X 0 t 5 (5-2) (5-22) / ( s0 s 2 0.7s )/( s s )0.7/ b Ȳ b 2 X 2 (5-5) Ȳ β + β 2 X2 + β 3 X3 + Ū E[b ]β β 3 i (X 2i X 2 )(X 3i X 3 ) i (X 2i X 2 ) 2 X2 + β 3 X3, b β β 3 σ 23 σ 22 μ 2x + β 3 μ 3x. 7 (5-23) AIC 0.49 BIC 0.56 (5-24) AIC 0.2 BIC 0.23 (5-24) 8(a)g(Y i )logy i (b) g(y i ) Y i 2

14 6 (a)h 0 : β 2 β 3 0 H : β 2 β 3 0 (b) 2 (c) Y i β + β 4 X 4i + U i (d) F W (a)H 0 : β 2 β 3 β 4 0 H : β 2 β 3 β 4 0 (b) 3 (c) Y i β + U i (d) F W suw (a)e[a] E[Ȳ ] E[b] X (α + β X + E[Ū]) β X α. (b) 3 2, V [a] X 2 V [b]+e[ū 2 ] 2 XE[(b β)ū] X 2 i (X i X) 2 σi 2 { i (X i X) i 2 } 2 + σ2 i 2 2 X i (X i X)σ i 2 i (X i X) 2. 2(a) Y i,x i,x 2 i,x3 i Ûi 2 Û 2 i,x i,x 2 i X 3 i,x4 i,x5 i,x6 i R2 3 R 2 χ 2 6 (b) χ Y i /X i /Xi 8 (a)e[a] E[Ȳ ] E[b] X (α + β X + E[Ū]) β X α. 3

15 (b) 3 2, V [a] X 2 V [b]+e[ū 2 ] 2 XE[(b β)ū] X 2 i j (X i X)(X j X)σ ij { i (X i X) i j 2 } 2 + σ ij 2 2 X i j (X i X)σ ij i (X i X) (8-) 4 χ Y i (X i ) X 2i,X 3i Ûi 2 Ûi Ûi ˆφ 3 i Ỹ ˆφ 2 Y 9 X ˆφ 2 X 2 ˆφ 2 X 2 X 3 ˆφ 2 X 3 i 2,, Ỹ i Y i ˆφY i X i ˆφ X 2i X 2i ˆφX 2i X 3i X 3i ˆφX 3i i, 2,, Ỹi X i X 2i X 3i ˆβ ˆβ 2 ˆβ 3 a α + Ū (b β) X α γ xuμ x. σ 2 x 4

16 2 b 2 i (X 2i X 2 )(Y i Ȳ ) i (X 2i X 2 ) 2 i β 2 + β (X 2i X 2 )(X 3i X 3 ) 3 i (X 2i X + 2 ) 2 β 2 + β 3 σ 3x ρ 2,3 σ 2x. i (X 2i X 2 )(U i Ū) i (X 2i X 2 ) 2 3 b,iv β + Ū (b 2,IV β 2 ) X 2 (b K,IV β K ) X K β +0 (β 2 β 2 )μ 2x (β K β K )μ Kx β (8-3) i 2,, i.i.d.(0,σ 2 ε) i φ 2 φ 2 Y α φ 2 + β φ 2 X + φ 2 U V [ φ 2 U ]σε 2 φ 2 2 φ ˆφ 2 3 Cov(Y i,y i h ) σ 2 ( + θ 2 + θ2 2 ) : h 0 σ 2 (θ + θ θ 2 ) : h σ 2 θ 2 : h 2 0 : h 3 5

17 4 Y i Y + U U i E[Yi 2]Y 2 + σ2 (i ) [ ] 5 (0-9) E i2 Y 2 i ( )Y 2 + σ 2 ( 2)( ). 2 Y i φ Y i + + φ Y i + + φ Y i + φ Y i + + φ Y i + U i φ Y i + +(φ + φ )Y i + +( φ )ΔY i + + U i ( ) 2 TSS R 2 SSR/TSS( RSS ) 3 Y it X it U it t Ȳt i Y it X t i X it Ūt i U it Ȳ t α + β X t + γz t + Ūt (Y it Ȳt) β(x it X t )+(U it Ūt) 2 4(a) σu 2 (b) (c) 0 2 6

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