solutionJIS.dvi

May 0, 006 6 morimune@econ.kyoto-u.ac.jp /9/005 (7 0/5/006 1 1.1 (a) (b) (c) c + c + + c = nc (x 1 x)+(x x)+ +(x n x) =(x 1 + x + + x n ) nx = nx nx =0 c(x 1 x)+c(x x)+ + c(x n x) =c (x i x) =0 y i (x i x) = y i x i y i x = = = y i x i (y 1 x + y x + + y n x) y i x i x(y 1 + y + + y n ) y i x i x(ny) 1

x i (y i y) = y i x i x i y = y i x i y(nx) (x i x)(y i y) = y i x i nyx n 1 (x i x)(y i y) = 1 y i x i yx n n 1. (a) x = 1 3 (1 1+ +3 3+4 4 4.5.5) = 5 3, 5 x 3, y = 1 3 (1 4+ 3+3 +4 1 4.5.5) = 5 3, 5 3 /( 5 3 5 3 )= 1. (b) (a) = 1 3 (1 1+ +3 3+4 4 4.5.5) = 5 3, (c) x y = 1 3 (1 1+ +( 3) ( 3) + ( 4) ( 4) 4 ( 1) ( 1)) = 6 3, = 1 3 (1 ( 4) + ( 3) + 3 +4 1 4 ( 1).5) = 10 3 10 3 /( 6 3 5 3 )= 10 13 y =( 4, 3, 3, 4) -3.5 = 1 3 (1 ( 4) + ( 3) + 3 ( 3) + 4 ( 4) 4 ( 3.5).5) = 0

1.3 x : (1,),(,4),(-1,),(-,4) y : (1,),(,4),(1,-),(,-4) (1,),(,4),(-1,-),(-,-4), 1.4 1.5 t = 0 55.18 1685.36 = 5 110.36 113 5. t -.191 19 t 5% -1.73-1.65 t = 33.71 17 = 17 33.71 1849.79 135 17 1 4 (t 1.089) 16 t 1.75 1.65 1.5 t = 0.1 17 16 ( 0.1) 0.4 1 ( 0.1) 15 t t 0.389.1 y x x xy y 1 1 1 1 1 3+ 9 1=1.5 1 1.5 = 0.5 7 4 14 49 3+ 9 =6 7 6=1 10 3 9 30 100 3+ 9 3=10.5 10 10.5 = 0.5 18 6 14 45 150 18 0 3

b = 45 6 18/3 14 (6 6/3) = 9 a = (18/3) 9 (6/3) = 3. y x x xy y -5-1 1 5 5 1 0 0 0 1 4 1 1 4 16 0 0 9 4 b = 9 0 0 = 9, a =0 9 0=0..3 b = 00 980 (t t)y t (t t) 00 980 t t t = t 1979 t = t 1979 t t =(t 1979) (t 1979) = t t b = b 00 980 (t t )y t 00 980 (t t ) = 00 980 (t t)y t (t t) 00 980 4

.4 β = (x t x)(y t y) (x t x), γ = (y t y)(x t x) (y, t y) r = (y t y)(x t x) (y t y) n (x t x) r = β γ.5 r =0.567 1.31 = 0.698. 100 β = (x t x)(y t y) (x t x) α = y βx β = (x t x )(yt y n ) n = 100(x t x)10(y t y) (x t x ) n (100(x = 1 t x)) 10 β α = y β x =10y 1 10 β 100x =10 α ŷ i =1+x i 5

x 100 10ŷ i =10+0x i 10ŷ i =10+0 1 100 (100x i) =10+0. (100x i ).6 y i = µ + u i µ (y i µ) = = µ (y i µ) (y i µ) µ (y i µ) = =0 (y i µ) (y i µ) =0 ( ) µ y i n µ =0 µ = 1 n y i = y ( ) ŷ i = µ 6

ŷ i µ, RSS = = (y i µ) (y i y) TSS ESS = (ŷ i y) =0..7 y i = βx i + u i β (y i βx i ) = β (y i βx i ) = (y i βx i ) β (y i βx i ) = (y i βx i )x i ( ) = { =0 y i x i β x i x i } β = y ix i n. x i y i = βx i y i βx i, ( ) x i 0 7

.8 8.1 ln(439/479) = 0.087. 8. R ntt =0.0001(0.11) + 0.71699(0.6)R nikkei 0.00 0.% 8.3 0.005 0.087 0.005 = 0.09 (R ntt 0.005) = 0.0006(0.034) + 0.71699(0.6)(R nikkei 0.005). t P 0.97t P 0.047 3 3.1 x t =0, z t =0, x =0, z =0. z tx t =0, y t = n α y t x t = β x t y t z t = γ 3 z t 1 = 6 α =4 β 8

9=6 γ. y y c x z 7 1 1 1 1 1 + 1 ( 1) + 3 ( 1) = 3 1 1 4 7 4 1 1 1 + 1 (1) + 3 ( 1) = 5 1 1 4 7 3 9 1 0 1 + 1 (0) + 3 ( 1) = 4 1 1 7 4 16 1 1 1 + 1 ( 1) + 3 (1) = 9 1 1 4 7 5 5 1 1 1 + 1 (1) + 3 11 (1) = 1 1 4 7 6 36 1 0 1 + 1 (0) + 3 10 (1) = 1 1 1 91 0 0 1 0 3 TSS =91 1 6 1 1 = 35 R =1 35 3=9 35. y 7 1 x y 1 7 3 7 x y 1 7 3 7 (3.7) 0 y 3.5 3. x 1 z x,y 1 z y β = y t x t (x t ) θ = y t x t (y t ) β θ = ( y t x t ) (x t ) (y t ) 1.66 0.4 = 0.63 9

3.3 RES RES t =0 3.4 RES RESi = = RES t x t =0 RES t z t =0 RES i (y i α βx i γz i ) RES i y i α RES i β RES i x i γ RES i z i RES iy i RES i y i = 3.5 Φ (y i α βx i γz i )y i Φ α = y βx γz y i y = β(x i x)+ γ(z i z)+error (x i x) (z i z) (x i x) =δ(z i z)+error 10

δ {(x i x) δ(z i z)} ( ) δ x i =(x i x) δ(z i z) x i (x i ĉ ζz i ) ( ) ĉ ζ ζ ĉ ĉ = x ζz ( ) ( ) 4 15 (4.8) β K β K β K 4.1 x y y y x x xy 0 0-4 0 3 9-1 1-3 8 64 3 9 4 sum 11 73 0 14 1 y =3.6(5.9) + 1.5(5.)x, R =0.96,RSS =1.17,ESS =31.5 11

4. x z y,x,z y x z y 5/ 1/4 3/4 x 1/4 3/ 0 z 3/4 0 53/ y =0.4565(0.08446)x +0.0830(0.05564)z. t 5.41 0.51 b 0.967 0.9016 t 5% 1.71 4.3 K= K-1=5 DW=1.8 5% U=1.86 L=1.0 U L U F f = 6 3 0.88 0.660 =0.066 1 6 0.660 f = 6 3 0.955 0.94 =0.066 1 6 1 0.955 F 3 1 5% 3.34 3 16 5% 3.4 3 1 5% 3.0 3 4.4 118 log(l) b log(k) c b+c=1+d, b=1+d-c, log Y log L = a + d log L + c{log K log L} + e log H4+error log(y/l)=a + d log L + c log(k/l)+e log H4+error 1

log(y/l) log(k/l) log(y/l) =.01 + 0.045(4.73) log L + 0.31(7.74) log(k/l) + 0.16(6.1) log H4, R =0.9119,RSS=0.05380, t d d 4.5 y i = a + bx i + error x =(x 1 + x )/ (x 1 x) +(x x) = (x 1 x ) (x 1 x)y 1 +(x x)y = (x 1 x ) y 1 + (x x 1 ) y = (x 1 x ) (y 1 y ) b = (y 1 y ) (x 1 x ) y = c + (y 1 y ) (x 1 x ) x c y 1 = c + (y 1 y ) (x 1 x ) x 1 y = c + (y 1 y ) (x 1 x ) x ĉ = y (y 1 y ) (x 1 x ) x 13

4.6 (4.15) (4.5) m (4.5) x mi =1 {y i ( β 1 x 1i + + β K x Ki )} =0 (y i ŷ i )=0 y i = n 4.7 (4.5) ŷ i (y i ŷ i )x mi =0 4.8 ŷ i = β 1 x 1i + + β K x Ki (4.5) (y i ŷ i )ŷ i = β n 1 (y i ŷ i )x 1i + + β n K (y i ŷ i )x Ki =0 14

(y i ŷ i )ŷ i = y i ŷ i y i ŷ i = (ŷ i ) =0, (ŷ i ). n(y) y i = ŷi (y i y)(ŷ i y) = (ŷ i y). ± n (ŷ i y) 4.9 {y i ( β 1 x 1i + + β K x Ki )} β 1,, β K x 1i x i k l {y i ( β 1 x 1i + + β K x Ki )},i m,i l + {y m ( β 1 x 1m + β x m + + β K x Km )} + {y l ( β 1 x 1l + β x m + + β K x Kl )} x 1l =1,x m =0, x 1l =0,x m = 1, x 1l =0,x m =0,,i m,i l {y i ( β 3 x 3i + + β K x Ki )} + {y m ( β 1 + β 3 x 3m + + β K x Km )} + {y l ( β + β 3 x 3l + + β K x Kl )} 15

β 3 β K β 1 β β 1 = y m ( β 3 x 3m + + β K x Km ) β = y l ( β 3 x 3l + + β K x Kl ) 4.10 b=c d=1 b c = e d 1=f b = c + e d =1+f b d y t = a + ex t + c(x t + z t )+z t + fz t + u t (y t z t )=a + ex t + c(x t + z t )+fz t + u t e =0,f =0 F (y t z t )=a + c(x t + z t )+u t (y t z t )=a + c(x t + z t )+u t F 16

4.11 t V (b c) =V (b)+v (c) Cov(b, c) b c V (b)+v (c) Cov(b, c) t = 0.65 0.44 0.6 +0.0 ( 0.0075) =0.6 5% 5 5.1 1.1 Y 1 t 0.58 0.79 1 1. n=5 K=5 k=4 5% L=1.04,U=1.77 DW 1.7 L L 1.3 1.4 TSS=1350 1.5 r =1 dw =0.15. 0.8 =1 70 TSS f = 5 5 4 0.8 1 0.8 =0 4,0 F 1% 4.43 1.6 17

5..1 y x y = y x = x y x (x ) 16 1 (1/4) = 3 1 (1/4) 16 = 8 3 4 19 4 10 4 0.5 =3 10 0.5 16 = 6 4 6 4 6 0.5 4=4 4 0.5 10 = 1-4 1 7+ 3 1+8 3 6 197.1 (, 3), ( 1, 4) -1/3 (,3) 3=a + 1 3 a=11/3. b = 6 (7 + 3)(1 + 8 3)/3 197 (1 + 8 = 0.13969 3) /3 a = 7+ 3 3 +0.13969 1+8 3 3 5.3 3.1 y = 3 1 x, RSS =0 3 E(y t )=E(a + bx t + u t ) = a + bx t + E(u t ) = a + bx t =3.604 V (y t )=E{(a + bx t + u t ) E(a + bx t + u t )} = E{(a + bx t + u t ) (a + bx t )} = E(u t ) = V (u t ) t 1, t 3 3. 18

y x y/σ x/σ x*y* x* y* 14 1 7 7 49 1 4 10 5 10 5 4 6 4 6 4 4 16 36 sum 9 16 41 90 41 b = 41 9 16/3 90 16 16/3 = 1.5 a = a =3 ( 1.5) 16/3 =11. 3.3 y t = a + bx t + u t a = b y t = b(1 + x t )+u t b = (1 + x t )y t (1 + x t ) y t + x t yt = 3+ x t + x t a=0.8 3.4 = 9+41 3+ 16 + 90 =0.4 RSS a = (y t ) a y t b x t y t =41 (11) 9 ( 1.5) 41 = 3.5 RSS 0 = (y t ) b (1 + x t )y t =41 (0.4) (41 + 9) = 1 19

f = 3 1 1 3.5 =6 1,1 F 5% 1 t 3.8 14.44 t d=0 a = b + d y t = d + b(1 + x t )+u t d t 6 6.1 y t = α + β 0 x t + β 1 x t 1 + β x t + ε t β 0 = γ 0 β 1 = γ 0 + γ 1 β = γ 0 +γ 1 3β γ 3β β β β 0 β 1 γ 1 = β 1 β 0 γ 1 = β β 1 β β 1 = β 1 β 0 β =β 1 β 0 0

P=3 q= y t = α + β 0 x t + β 1 x t 1 + β x t + β 3 x t 3 + ε t β 0 = γ 0 β 1 = γ 0 + γ 1 + γ β = γ 0 +γ 1 +4γ β 3 = γ 0 +3γ 1 +9γ 1 4β 1 3β β 1 β 0 = γ 1 + γ β β 1 = γ 1 +3γ β 3 β = γ 1 +5γ (β β 1 ) (β 1 β 0 )=γ (β 3 β ) (β β 1 )=γ (β 3 β ) (β β 1 )=(β β 1 ) (β 1 β 0 ) 6. y t = α + β 0 x t + β 1 x t 1 + β x t + ε t, β 0 = γ 0 β 1 = γ 0 + γ 1 1

β = γ 0 +γ 1 β 1 = γ 0 γ 1 =0 γ 0 = γ 1 β 0 = γ 0 β 1 = γ 0 + γ 1 =β 0 β = γ 0 +γ 1 =3β 0 y t = α + β 0 (x t +x t 1 +3x t )+ε t, 6.3 6.4 6.5 6.6 6.7 6.8 SUR

6.9 6.10 6.11 1 1 W trend tax W g G X 1 C W p X P 0 0-1 0 0 0 0 0 1 W 0 0 0 1 0 0 0 1 0 C α 3 0 0 0 0 0-1 0 0 W p 0 γ 3 0 0 0 γ 0-1 γ 1 X 0 0 0 0 1 1 1 0-1 5 1 5 7 [ 1] (0,1) 1 P ( 1 3 < X< 3 ) 7.1 1/ 1/1 P ( 1 1 3 < X< 3 )=P{ 1 1 = P { 1 1 6 <Z<1 1 6 } = P { <Z<} 0.05 ( 1 3 1 1 ) < 1 1 (X 1 1 ) < ( 1 3 1 )} 1 X (a,b) 3

0.0454 1 [ ] 8 α=4 P (m <X<m) E(X) = = E(X )= m m = α α 1 m = m m = α α m x αm α x α 1 dx αm α x α dx x αm α x α 1 dx αm α x 1 α dx V (X) = α α α m ( α 1 m) α = (α )(α 1) m. P (m <X<m+ m ) = P (m α α 1 m<x α α 1 m<m + m α α 1 m) = P ( 1 α 1 m<x α α 1 m< α (α 1) m) = P ( n( 1 (α 1) (α ) m) <Z< n α = P ( n α 1 (α ) α m α <Z< n(α ) (α 1) (α ) α ) m = P ( <Z< 4 ) = P ( <Z<) (α 1) (α ) m m ) α 4

95 7.1 X,Y 7. X = 1 0 1 Y = 1 1 4 4 81 4 0 81 4 1 81 81 16 81 16 81 81 16 81 16 81 Z X = 1 0 1 Y = 1 1 0.5 0 0 0.5 0 0.5 1 0 0.5 1 P (Z = 1) = 1 81, P (Z = 0.5) = 8 81, P (Z =0)= 4 81, P (Z =0.5) = 3 81, P (Z =1)= 16 81 n=3 X 1 1 1 ( 1 3 )3, 1 3( 1 3 ) ( 3 ), 1 1 3( 1 3 )( 3 ), 1 0 ( 3 )3, E(Z) = ( 1 3 )3 +3( 1 3 ) ( 3 ) 3(1 3 )( 3 ) +( 3 )3 =( 1 3 )3 +3( 1 3 ) ( 3 )+3( 1 3 )( 3 ) +( 3 )3 =(( 1 3 )+( 3 ))3 =( 1 3 )3 5

E(Z )=( 1 3 )3 +3( 1 3 ) ( 3 )+3(1 3 )( 3 ) +( 3 )3 =(( 1 3 )+( 3 ))3 = (1) 3 =1 V (Z) =E(Z ) E(Z) =1 ( 1 3 )6 E(Z) =( 1 3 )n 7.3 V (Z) =1 ( 1 3 )n x 1 E(X) = 0 x xdx = 3 E(X )= x xdx =1 0 V (X) =1 ( 3 ) = 1 9 6

7.4 P(-1)=1/3,P(1)=/3. {X 1,X,X 3 } s = 1 3 1 {(X 1 X) +(X X) +(X 3 X) } = 1 {X 1 + X + X 3 3X } 1. -1 s =0, ( 1 3 )3.. 1-1 s = 4 3, 3( 1 3 ) ( 3 ). 3. 1 1-1 s = 4 3, 3( 3 ) ( 1 3 ). 4. 1 s =0, ( 3 )3. s 0 4 3 s =0, ( 1 3 )3 +( 3 )3 = 9 7 = 1 3. s = 4 3, 3( 3 ) ( 1 3 )+3(1 3 ) ( 3 )= 3. 4 3 3 = 8 9. 7.5 X s X s 1 (-1,-1) 3 1 3 = 1 9-1 0 (-1,1) or (1,-1) 9 + 9 = 4 9 0 (1,1) 3 3 = 4 9 1 0 X s X = 1 X =0 X =1 s s 1 4 5 =0 9 0 9 9 s 4 = 0 X 1 9 4 9 0 9 4 4 9 9 1 X s E(X) =( 1) 1 9 +0 4 9 + (1) 4 9 = 1 3 E(s )=0 5 9 + () 4 9 = 8 9 E(X s )=0. Cov(X,s )= 1 3 8 9 = 8 7. 7

7.6 A= A= B B= 0.5 0 0.5 B= 0 0.5 0.5 A 0.5 0.5 1 A= A= B B= 0 0.5 0.5 B= 0.5 0 0.5 A 0.5 0.5 1 A= A= B B= 0.4 0.1 0.5 B= 0.1 0.4 0.5 A 0.5 0.5 1 A= A= B B= 0.1 0.4 0.5 B= 0.4 0.1 0.5 A 0.5 0.5 1 A= A= B B= 0.5 0.5 0.5 B= 0.5 0.5 0.5 A 0.5 0.5 1 8

7.7 Z=XY Z Z 0 1 4 1 P 15 + 4 15 + 1 15 = 11 4 15 15 0+0 0 Z 0 1 P 11 15 4 15 7.8 3 1. V=0 ( 11 15 )3.. V=1 3( 11 15 ) ( 4 15 ). 3. V= 3( 11 15 )( 4 15 ). 4. V=3 ( 4 15 )3. X s X s 1 (-1,-1,-1) 7-1 0 (-1,-1,1) 3 7 = 6 1 4 7 3 3 (-1,1,1) 3 4 7 = 1 1 4 7 3 3 8 (1,1,1) 7 1 0 X s X = 1 X = 1 3 X = 1 3 X =1 s s 1 8 9 =0 7 0 0 7 7 s = 4 6 1 3 0 X 1 7 7 6 7 18 7 0 7 1 8 7 7 1 X s 9