) 041 (CIP ),2003.9 ISBN 5037 3561 9 /,.:...-.F10 CIP (2003) 032658 / / 75 / 100826 / 6 / ( 010) 63459084 63266600-22500( ) / / / 787960mm 1/ 16 / 301

) 041 (CIP ),2003.9 ISBN 5037 3561 9 /,.:...-.F10 CIP (2003) 032658 / / 75 / 100826 / 6 / ( 010) 63459084 63266600-22500( ) / / / 787960mm 1/ 16 / 301 / 16.75 / 1-5000 / 2003 9 1 / 2003 9 1 / ISBN 7-5037-3561-9/ F1261 / 26.80,,,,

,,,,,,,,, 20,,,, :,,,,, ;,,,, SAS SPSS T SP, Excel ;,,,,, : ( ), (), ( ), ( ), ( ),,,,,!,,,,! 2003 9

( 1) ( 1) ( 4) (10) (13) (Delphi poll) (13) (17) (20) (24) (25) (40) (53) (74) (83) (83) (87) (92) ( 111) ( 115) ( 115) ( 116) ( 126) ( 128) ( 128) ( 133) ( 136)

2 ( 144) ( 147) ( 150) ( 152) ( 153) ( 155) ( 159) ( 163) X - 11 ( 166) ( 170) ( 170) ( 175) ( 182) ( 188) ( 188) ( 193) ( 197) AR MA ( 202) ( 210) ( 219) ( 223) ( 232) ( 232) ( 237) 1 ( 241) 2 X 2 ( 243) 3 X 2 () ( 245) 4 F ( 247) 5 D.W ( 255) 6 ( r z ) ( 259) ( 260)

( forecasting ),,,,,,,,,,,,,,, 20 30 40,,,,,,,,,,,,,

2 ), (economic forecasting),, 4,,,,,,,,, (),,,,,,,,,,,,,, :,,

3 ),,,,,, :,,,,,,, ( ),,,,,,,, ( ),,,,,,,,,,,,, (),, 5 ; 1 5, 1,

4,,,,,,,,,,,, :, 1.1 Box - Jenkins 1.1, : () :

5 ;,,, ( ) :, ;, (),,,,,, (),, ;, (),,,, ;,,, ( ), : = + = -

6 ( ) : 1.,, 2.,, (),,,,,,,,,,,,, ;,, (),,,,

7 ),,,,,,, TSP, E-views, SPSS, SAS, Excel, () 1.,, :,,,,,,,,,,,,,,,,,,,,,,, 2.,,

8,,,,,,,,,,, 3.,,, 4.,,,,,,, 5.,,,,,, ( ) 1., 2. 3. () (),,,,,,

9,, ( ),, ;,,,, 1.2 : 1.2 (),, :,,,,,,,,,,,,,,,,,,,,

10,,,,, ;, : 1.,,, ( + ),,, 2., 3. : ;, ;,,,,,,, : (1 ) : (2 ) :,,,,, :

11,,,,,,,,,,,,,,,,,,,,,,,,,, ( ),,,,,,,,,,,,,,,,,

12 ),,,,,,, (),,,,,

,,,, ;,, :,,,,,, :,,, ( ), :, ; ; :,,,,,,,,,,, (Delphi Poll),,

14,,,,,,,,, 20 60 O. N., T.J., 1964,,,, :,,, 50,,,,,,,, 1964, T.J.O. (1015 ), : ; ; ; ; ;, 50%,,,,,,,, () ( ),,, : ( ),,

( Delphi Poll) 15 20,, ( ),,,,,,,,, (),, (),,,,,,,, (),,,,,,, (),,,,,, 2.1,, B,

16,, ( ),, 26, 60,46 2.1 A 25 60 85 25 70 80 25 75 80 B 35 50 75 35 50 75 35 50 75 C 50 60 70 40 50 60 50 70 75 A 5 15 37 9 22 47 9 24 47 B 30 55 85 35 50 70 25 68 75 A 40 55 80 35 45 70 25 35 60 B 10 25 55 22 35 60 20 35 60 A 19 22 31 22 28 34 22 28 34 B 20 30 45 22 34 44 22 34 44 A 16 22 31 12 25 31 28 37 62 B 20 35 50 20 35 50 25 45 50 20 35 55 20 35 50 25 45 50 23 39 58 24 40 57 26 46 60 37 57 77 33 57 72 37 65 77 18 34 61 17 36 59 17 46 60 22 42 65 25 41 63 22 38 60 20 26 38 22 31 39 22 31 39 18 29 41 16 30 41 27 41 56 20 35 55 20 35 50 25 45 50,,,,,,,,, ;

17 ;,,,,, : ; ;, : (1 ),, ; (2 ),,, : (1 ) ; (2 ), ; (3 ), (Subjective Probability),,,, : P( Ai ) Ai (1 )0P( Ai )1 (2 ) P(= 1), : (3 )Ai Aj, ij, j = 1, 2, Ai, Aj, : P i = 1 Ai = P ( i = 1 Ai ),,, 1/ 2,

18,,,,,, :,, : = :, 2.2 ( ) * 1000 0.3 30 800 0.5 400 600 0.2 300 820 1200 0.2 240 1000 0.6 600 800 0.2 160 1000 900 0.2 180 700 0.5 350 500 0.3 150 680,, 1/ 3, (820 + 1000 + 680)/ 3 = 833.4( ) 950, 750, 50%,

19 950 + 750)/ 2 = 850( ) 60%, 40%, : 833.460 + 850 + 40 100 = 840.04,,, 8 : 2.3 1 2 3 4 5 6 7 8 0.98 1.03 1.02 0.86 0.97 1.01 0.93 1.04 0.98 98 % - 1 = - 2 % 840.04 98 % = 823.6( ),,, :, 205, 6 2.4 ( % ) 1.0 12.5 25 37.5 50 62.5 75 87.5 99 ( ) 1%,, 99%,,

20.5 1 12.5 25 37.5 50 62.5 75 87.5 99 1 2 3 4 5 6 7 8 9 10 190 178 184 194 198 168 194 180 188 200 192 190 190 195 199 179 198 185 189 201 194 192 192 196 200 180 200 186 190 202 198 194 193 196 202 184 206 189 191 205 200 198 202 193 205 190 208 192 192 207 202 200 204 199 208 192 212 195 193 209 204 204 206 200 210 194 216 198 194 212 205 205 208 201 212 196 219 200 195 213 208 225 220 202 216 198 224 205 196 220 187.4 191.8 193.2 195.9 199.2 201.4 203.8 205.4 211.4,,, 199.2, 50%,, (199.2-6, 199.2 + 6 )( 193.2, 205.2),, 62.5% (87.5% 25% ),,,,,,,,,,,, ( Leading Indicators ),,,,

21 10 M1 18 ( Coincident Indicators ),,,, : M2 10 ( Lagging Indica tors),,,, : 6,,,,, : :, yt xt tth y y th yt = f ( xt - ), yth = f ( xt + h - ), = g(x th - ),,, 2.1

22,, t, t + 2.1 2.6 ( ) Money supply H ousing Permits Stock market Mortgage debt Residential Investmen t Busines s Loans Capital goods orders Inven tories Busines s invent ory Cr edit outstanding Industr ial construction Unemployed Rate Automobile sales Free r eserves 28.9 14.8 14.5 13.7 11.9 11.5 10.4 9.3 8.5 8.5 6.6 4.2 3.9 2.9

23,,,,,,?, (Diffusion Index ), 1, 0.5, 0, : ( DI t ) = + 0.5 100 DIt t, 1 6 9,,,, :, 0 < DIt < 50 %, 50%,,,,, 50% < DIt < 100 %,,,, DIt 100 %,, 100% > DIt > 50 %,,,,,,, 50% > DIt > 0,,,,,,, ;,

,,,, ( ), :,,, : 3.1 : (1 ); (2), ; (3 ),, ; (4 )

25 ) Y, X, 1. ( ) Y = +X +, Y X ;,,, X,, : 1.Y1 =+X1 + 1 2.Y2 =+X2 + 2 3.Y3 =+ X3 + 3, : N.Y N =+ X N + N Y i = +Xi + i, E( i ) = 0 Cov( i, j ) = 2 i = j 0 ij, 50,, : Y = + X +,,, Y =+ X,

26.,, n,, : y x 1 y1 x1 : 2 y2 x2 n yn xn a,, b, Y X : y = a + bx + e(, ), a b,, y = a + bx e,, y^ = a + bx y ei = yi - y^i ( i = 1, 2n) : y^1 = a + bx e1 = y1 - y^1 y^2 = a + bx e2 = y2 - y^2 y^3 = a + bx e3 = y3 - y^3 y^ n = a + bx en = yn - y^ n,,, (),, :, a b; a b

27,,, y x 1 y1 x1 2 y2 x2 n y n x n y^ = a + bx y, x, a b, yi y^i, n n ( y i = 1 i - y^i ) 2 = i = 1 e2 i n = ( i = 1 yi - a - bx) 2, a b : : b= n n i = 1 xi y i n n i = 1 x2 i a= y - b x b = n i = 1 a = y - b x n n - x i = 1 i i = 1 yi n - ( i = 1 xi ) 2 ( xi - x) ( yi - y) n ( x i = 1 i - x) 2,, a b, : 1.,,, b,, b, 2., : (1 ); (2 ) ; (3 )

28 a b, a b,, a b,, a b, a b y, y, a b,, : 2 bn [, n ] ( x i = 1 i - x) 2 an [, ( 1 n + x 2 n ) 2 ] ( x i = 1 i - x) 2 cov( a, b) = - n ( i = 1 xi x 2 - x) 2 : a, b, 2, S y 2 = n i = 1 e2 i / ( n - 2 ), 2, yi y^i, ( ) n b - 2 S y 2 ( xi - x) i = 1 a - ( 1 n + x 2 2 n ) Sy - x) 2 i = 1 ( xi t( n - 2 ) t ( n - 2), a b,: 1. ; 2., E( ei, ej ) = 0 ( ij) ; 3., i N( 0, 2 )

29,, ( ) 1., X Y 0, X Y b - ( Sb b ) t, t Sb, H0 : = 0, H1 0 tb = b Sb = b ( x - x) 2 Sy S y 2 = ( y - y^ ) 2 / ( n - 2 ) t tc ( n - 2) t > tc ( n - 2), H0, H1 : 0, 2., y^i = a + bxi y i = a + bx i + ei,,,,, y x,, b 0,,,,,, : H0 : b= 0, F,,, = +, : n n n ( i = 1 yi - y) 2 = ( i = 1 y^i - y) 2 + ( i = 1 yi - y^i ) 2 : n - 1 1 n - 2 : ST = SR + SE,

30 ST 2 2 ( n - 1) SR/ 2 2 ( 1) SE/ 2 2 ( n - 2) SR S E, b= 0, F = SR S E/ ( n - 2) = ( y^ - y) 2 / 1 F( 1, n - 2 ) ( y - y^)/ ( n - 2 ),, F F(1, n - 2 ), F > F( 1, n - 2 ),,,, F, 3.1 F SR SE 1 n - 2 S T n - 1 SR SE/ ( n - 2) S R SE/ ( n - 2 ) F : (1 ), ; (2 ) Y X ; (3 ) Y X 3. D.W, : Cov( i, j ) = 0,,,,, t F, D. W, ij : H0 := 0,,

31 d n i = 2 ( ei - ei - 1 ) 2 n i = 1 ei 2, n p, Durbin - Wa tson D.W, dl du, d d L du, 3.2 : 3.2 D.W du < d < 4 - du,,, d 2,, 0 < d < dl,, 4 - dl < d < 4,, dl ddu 4 - du d4 - dl, : (1 ); (2 ),, ; (3 ) ; (4 ) ; (5 ) 4. ( S.E. of regression ) S y

32 Sy n i = 1 ( yi - y^) 2 n - 2 Sy,, ; Sy 0,,,, ;, Sy, Sy/ y S y/ y 15 %, 5.,, = +,,, R 2, : R 2 =,0R 2 1R 2 n i = 1 ( y^i - y) 2 n = 1 - ( i = 1 yi - y) 2 R 2,, R 2, R 2 n ( y i = 1 i - y^i ) 2 n ( y i = 1 i - y) 2 = 1,, = 0, 1, 0.8,,,,, ( ) 1. ( Mean Error) n ME = ei/ n i = 1 2. ( Mean Absolute Error) n MAE = ei / n i = 1

33. ( Sum of Squared Error ) SSE = 4. ( Mean Squared Error) n 2 i = 1 ei 2 MSE = / n n i = 1 ei 5. ( Standard devia tion of Error) SDE = n i = 1 ei 2 / ( n - 1 ),, ;,, ;,,,,,,,,,, ( ), : 1. ( Percentage Error) P Ei = ei/ y100 % 2. ( Mean Percentage Error ) n MP E = P Ei/ n i = 1 3. ( Mean Absolute Percentage Error ) n MA PE = PEi / n i = 1,,, MAP E < 10, x, y (), x,

34 y a N (, 1 n + x2 Lx x b N, 2 cov( a, b) = -, Lx x Lx x x L x x 2 ) 2 n 2 = ( xi - x) i = 1, x0,, y^0 = a + bx 0 y^0 N + x0, 1 n + ( x0 - x) 2 Lx x 2 y^0 E( y0 ) = + x0, y^0 y^0,, y0 - y^0 ( ) x0 y^0, 1 - ( ), x0, y0, P( y0 - y^0 ) = 1 - P( y^0 - y0 y^0 + ) = 1 - y^0 -, y^0 + y0 1 -?, x = x0, y0 y^0, y0, y^0 x0, y0 y^0, N (0, 2 ), y0 - y^0, E( y0 - y^0 ) = 0 var ( y0 - y^0 ) = var ( y0 ) + var ( y^0 ) = 1 + 1 n + ( x0 - x) 2 Lx x 2

35 y - y^0 N 0, 1 + 1 2 ( x0 - x) + n Lx x 2 Sy/ 2 2 ( n - 2 ), S y y0 - y^0, t y0 - y^0 1 + 1 n + ( x0 - x) 2 Lx x S y ( n - 2) 2 y0 - y^0 = ^1 + 1 n + ( x0 - x) 2 Lx x t( n - 2 ) ^2 = S y ( n - 2 ) ^ = ^ 2, P y0 - y^0 = P y0 ^1 + 1 n - y^0 t + ( x0 - x) 2 Lx x 1 + 1 n ^1 + 1 2 ( x0 - x) + n Lx x ^ + ( x0 - x) 2 Lx x = t1 - a 2 ( n - 2) = 1 - : = t1 - a ( n - 2) 2 ^ 1 + 1 n + ( x0 - x) 2 Lx x y^0 - t1 - a ( n - 2) 2 ^ 1 + 1 n + ( x 0 - x) 2 Lx x, y^0 - t1 - a ( n - 2) 2 ^ 1 + 1 n + ( x0 - x) 2 Lx x ^2 = Sy ( n - 2) = y^0 ts y 1 + 1 n + ( x0 - x) 2 ( x - x) 2

36 Lx x, x0 Lx x x0 x x, Lx x, x = x, x0, : x 3.3 n, 1 + 1 n + ( X0 - x) 2 Lx x 1, :, Sy = y^tsy,, Excel, : :, 3.2 :,, 3.4, :,,, 3.5

37.2 3.4 3.5

38, y x,, 3.6, 3.6 : 3.7

39 : (1 ) y^ = 21. 22187 + 0. 086229 (2 ) R 2 = 0. 822409, (3 ) t ta = 4.058403, tb = 9.38015 5 %, 21-2 = 19, t, t0. 02 5 = 2. 093,,, 10.27719 < a < 32.16654 0.066989 < b< 0.10547 (4 ) F ( ) 3.3 df SS MS F SignificanceF 1 20792.16 20792.16 87.98721 1.46E - 08 19 4489.87 236.3089 20 25282.03 = 5 %, ( 1, 19 ), F, F0. 0 5 (1. 19) = 4. 38, F, (5 ) S y = 15.37234, S y y = 59.49% > 15%, (6 )D.W d = 0.316878, = 1%, n = 21 p = 1, D.W, dl = 0.97 du = 1.16,0 < d < dl,,,,, 19862000,, 3.4:

40 3.4 Dependent Variable : Y Met hod: Least Squares Sample : 1986 2000 Included obse rvations : 15 Variable Coefficient Std. Error t - Statistic P rob. C X 42.89438 0.059689 3.508056 0.005232 12.22739 11.40890 0.0000 0.0000 R - squared Adjusted R - squared S.E. of regression Sum squared resid Log likelihood Durbin - Watson stat 0.909194 0.902209 6.996619 636.3848-49.39223 1.179151 Mean dependen t var S.D. dependent var Akaike info c riterion Sch warz criterion F - statistic Prob( F - statistic) 77.20264 22.37379 6.852297 6.946703 130.1630 0.000000, F d = 1.179, = 1%, n = 15 p = 1, D.W, d1 = 0.81 du = 1.07, du < d < 4 - du, : 2001 x0 = 1100, y^ 0 = 108.55228 95 % y^0 tsy 1 + 1 n 2 ( x0 - x) + ( x - x) 2 = 108.551.77096.99661 + 1 15 = 108.5513.70886 + ( 1100-436.3606 ) 2 2796352 2001 1100, 95 %, 94.84114122.25886,

41,,,,,, ( ) Y, X1, X2, X p,, Y = 0 + 1 X1 + 2 X2 + + p X p + j, X Y ; 0, j ( j = 1, 2p),, X,, : 1. Y1 = 0 + 1 X11 + 2 X12 + + p X 1 p 2.Y2 = 0 + 1 X21 + 2 X22 + + p X 2 p N. Y N = 0 + 1 X N 1 + 2 X N2 + + i X N p, : Y i = 0 + 1 Xi1 + 2 Xi2 + + p X ip, : + i + 1 + 2 + N (1 )X1, X2, X p,, (2 ) Y, (3 )0, E( i ) = 0 Cov( i, j ) = 2 i = j 0 ij : E( Y ) = 0 + 1 X1 + 2 X2 + + p X p va r( Y ) = 2 I

42 YN 0 + 1 X1 + 2 X2 + + p X, 2 I) Y = 0 + 1 X1 + 2 X2 + + p X p ( ),, n,, : y x1 x2 xp 1 y1 x11 x12 x1 p : 2 y2 x21 x22 x2 p n y n x n1 xn2 xnp j, j, b0 : ) bj ( j = 1p), Y X y = b0 + b1 x1 + b2 x2 + + bp x p + e(,, b0 bj 0 j,, y^ = b0 + b1 x1 + b2 x2 + + bp x p e,, y^ = b0 + b1 x1 + b2 x2 + + bp x p y ei = y i - y^i ( i = 1, 2n) (),,, n i = 1 e2 1 = ( yi - b0 - b1 x1 - b2 x2 - bp x p ) 2, 0, : yi = nb0 + b1 xi1 + b2 x i2 + + bp xip x i1 y i = b0 xi1 + b1 x 2 i2 + b2 x i1 xi2 + + bp xi1 xi p x ip y i = b0 xip + b1 x ip x i1 + b2 xip x i2 + + bp x 2 ip

43 b0, b1 bp Y = Xb+ e y1 b1 e1 : Y = y2 b= b2 e = e2 yn bp en 1 x11 x1 2 x1 p X = 1 x21 x2 2 x2 p 1 xn1 xn2 xnp A = X X Ab = B, ( X X ) b= ( X Y ) b= A - 1 B = ( X X ) - 1 B = X Y X Y, b : (1 )b0, b1, bp y i ( ) yi, b0, b1, bp, bi, bj N [ j, 2 ( X X ) - 1 ] (2 ), b0, b1, bp 0, 1, p, 2 ( X X ) - 1 2, ^2, bj t, n - p - 1, : t = bj - j Sb, Sb bj (3 ) b 2 A - 1, Cov( bj, bk ) = 2 Cjk ( j, k = 1, 2p) C= A - 1, A, : 1.,,

44 bj xi, bj, 2., : (1 ) ; (2 ); (3 ) ( ),, bj, 0,,,,, bj 0,,,, t H0 : j = 0, H1 : j 0 bj, t = bj Sb bj, t Cj j = S ( n - p - 1) ( j = 1, 2p), Cj j C = A - 1 j S, t tc ( n - p - 1) t > tc ( n - p - 1 ), H0, H1 : 0,, t F, F, : H0 F = ( bj - j ) 2 Cjj ( S ( n - p - 1) ) F(1, n - p - 1 ), F = b 2 j Cjj S ( n - p - 1 )

45 : ; ;, ( ),,, : H0 b1 = b2 = b3 = = bp = 0 F F( p, n - p - 1 ) = ( y^i - y) 2 / p ( yi - y^i ) 2 / n - p - 1 F( p, n - p - 1 ) > F F( p, n - p - 1 ) F,,,, : (1 ); (2 ) (), : R 2, R 2 R 2 = ( y^i - y) 2 = 1 - ( yi - y^i ) 2 ( yi - y) 2 ( yi - y) 2 1,,, ( y^i - y) 2, ( y^i - y) 2,, R 2 R 2,, : R 2 = 1 - ( yi - y^i ) 2 / n - p - 1 ( yi - y) 2 / n - 1,,,,, R 2 R 2,, R 2 R 2

46 R 2 R = 1 - (1 - R 2 ) n - 1 n - p - 1,, F R 2 : F( p, n - p - 1) = R 2 n - p - 1 1 - R 2 p y F y x j ( j = 1, 2p), ( ) (D.W ),D.W, Sy = n ( y - y^) 2 i = 1 n - p - 1,,,,,,,,,, () 1.,,, y x 1 b1 ( 1 ), y x2 b2 ( 1 ), y x 1, x2 b1 ( 2 ), b2 ( 2 ), x1 x2,, b1 ( 1 ) b1 ( 2 ), b2 ( 1 ) b2 ( 2 ), x1 x2, x2 x1,, x1 x2, x1 x2, r12 0, b1 ry1 r1 2, x2 x1,, x1 x2,,,

47 2., R 2, R 2 y, ( yi - y) 2 y,, ( y^i - y) 2, ( yi - y^) 2, R 2, S( x1 ) y x1, S( x2 ) y x2, S( x1 x2 ) y x1, x2, S( x1 x2 ) x2, x1, S( x1 x2 ) = S( x2 ) - S( x1 ) x1 x2,, S( x1 x2 ) = S( x1 ) S( x1 ) y x1, x1 x1 x2, x1 x2, x1 S( x2 ), S( x1 x2 ) < S( x1 ),,, y ( ), (1 ) F, t ; (2 ), ; (3 ); (4 ),, r > 0.7, (),, ; ;,,,,

48,, : ( ),,,,, () AIC AIC ( An information criterion ), 1973 ( Akaike) ARMA, AR, MA AIC AIC = nlog ^2 + 2 p ^2 2, S p, n, p ^2 = 1 ^ ^ ( Y - X ) ( Y - X ) n ^ = ( X X ) - 1 X Y AIC,, AIC, AIC () Cp ( Mallows), Cp Cp Cp = S^2 + 2 p ^2 = S ( n - p - 1 ), p, p ( S p - 1SS ) Cp,, Cp, Cp,,,,

49,, : 3.5 :2000 Excel :, : 3.8

50 3.9, 3.6 3.6 M ultiple R R Square Adjusted R Square 0.983782 0.967828 0.957104 873.306 17 3.7 df SS MS F Significance F 4 2.75E + 08 68828824 90.24798 7.55E - 09 12 9151960 762663.3 16 2.84E + 08 3.8 Coefficients ts t a t P - v alu e Lo wer95 % Upper 95 % 95.0 % 95.0% In t erce pt - 47482.9 13258.96-3.58119 0.003774-76371.7-18594.1-76371.7-18594.1 X Variabl e1 0.71964 0.116885 6.156799 4.9 E - 05 0.464968 0.974311 0.464968 0.974311 X Variabl e2 4.558464 0.308989 14.75284 4.7 E - 09 3.885235 5.231693 3.885235 5.231693 X Variabl e3-0.08521 0.059508-1.43192 0.177695-0.21487 0.044446-0.21487 0.044446 X Variabl e4 0.072091 0.249412 0.289045 0.77748-0.47133 0.615514-0.47133 0.615514

51, (1 ) y^ = - 47482.9 + 0.71964 x1 + 4.558464 x2-0.08521 x3 + 0.07209 x4 (2 ) R 2 = 0.967828 R 2 = 0.957104, (3 ) t tb 0 = - 3.58119, tb 1 = 6.156799, tb 2 = 14.75284, tb 3 = - 1.43192, tb 4 = 0.289045 5 %, 17-4 - 1 = 12, t, t0.025 = 2.1788,, b3, b4 (4 ) F ( ), = 5%, ( 1, 15 ), F, F0.05 (1, 15) = 4.54, F, (5 ) S y = 873.306, Sy y = 1.953% < 15%, (6 )D.W d = 1.179151, = 1%, n = 17 p = 4, D.W, dl = 0.68 du = 1. 77, dl < d < du,,,,, F,,, 3.8 3.8 x 1 x1 x2 x3 x4 1 x 2-0.01885 1 x 3-0.28563 0.49574 1 x4 0.221964 0.487609 0.397576 1,,,,, X4,,, 3.9

52.9 Dependent Variable : Y Met hod: Least Squares Sample : 1985 2001 Included obse rvations : 17 Variable Coefficient Std. Error t - Statistic P rob. C - 46841.71 12602.95-3.716726 0.0026 X1 0.731662 0.105311 6.947652 0.0000 X2 4.588584 0.280450 16.36152 0.0000 X3-0.079758 0.054413-1.465795 0.1665 R - squared 0.967604 Mean dependent var 44716.60 Adjusted R - squared 0.960128 S.D. dependent var 4216.539 S.E. of regression 841.9587 Akaike info criter ion 16.51166 Sum squared resid 9215628. Schwarz criterion 16.70771 Log likelihood - 136.3491 F - statistic 129.4277 Durbin - Watson stat 1.162342 Prob( F - statistic) 0.000000 b3, D.W x3,, 3.10 3.10 Dependent Variable : Y Met hod: Least Squares Sample : 1985 2001 Included obse rvations : 17 Variable Coefficient Std. Error t - Statistic Prob. C - 54840.94 12706.85-4.315855 0.0008 X1 0.785732 0.111642 7.037864 0.0000 X2 4.403592 0.300906 14.63444 0.0000 X4-0.041124 0.245919-0.167226 0.8698 R - squared 0.962331 Mean dependent var 44716.60 Adjusted R - squared 0.953638 S.D. dependent var 4216.539 S.E. of regression 907.9002 Akaike info criterion 16.66247 Sum squared resid 10715676 Schwarz criterion 16.85852 Log likelihood - 137.6310 F - statistic 110.7029 Durbin - Watson stat 1.603753 Prob( F - statistic) 0.000000

53 D W, x4 x3, x4,, : 3.11 Dependent Variable : Y Met hod: Least Squares Sample : 1985 2001 Included obse rvations : 17 Variable Coefficient Std. Error t - Statistic Prob. C - 55546.06 1156.24-4.803678 0.0003 X1 0.780779 0.103852 7.518179 0.0000 X2 4.378207 0.250630 17.46878 0.0000 R - squared 0.962250 Mean dependent var 44716.60 Adjusted R - squared 0.956857 S.D. dependent var 4216.539 S.E. of regression 875.8150 Akaike info c riterion 16.54697 Sum squared resid 10738727 Schwarz criterion 16.69401 Log likelihood - 137.6493 F - statistic 178.4289 Durbin - Watson stat 1.612149 P rob( F - statistic) 0.000000,,,, y^ = - 55546.06 + 0.780779 x1 + 4.378207 x2, x1, x2,,,, ( ),,,, y, y

54 1. :, ( 2 n 2. - 1 ),,,,,,,,,,, ;, 3.,,,,,,,,,,,,,, ;, 4. (2 ) ( 3 ),,,, ( ),,,,,,,,,,,

55,,, :??,,,, F ;,,,,,,, : l, bj l S= bj L jy j = 1 = S- S, L jy l x j, : : V j : V j = b2 j C jj S= S- S = S- S = S- S Cj j ( A) Cij j : V ( l) j = ( l) b( j ) 2 C ( j j l) (1 ),, k, V ( l) k V ( l) l) k = min j V( j, F= ( n - l - 1) V k ( l) S ( l) FFa 2,, (2 ),,

56 ( l + 1 ) V k k, l + 1 ) = max j V( j ( l + 1 ) V k, F= ( n - ( l + 1 ) - 1 ) V k ( l + 1 ) S ( l + 1 ) = ( n - l - 2) V k ( l + 1 ) ( S ( l) - V k ( l + 1 ) ) FF 1,, ( ) y^ = b0 + b1 x1 + b2 x2 + + bp x p y^ - y = b1 ( x1 - x1 ) + b2 ( x2 - x2 ) + + bp ( x p - xp ) : x i = xi - xi S x i Sx i = n 1 n - 1 ( x i = 1 i - xi ) 2 y^ - y S y = b1 x1 - x1 S y + b2 x2 - x2 S y + + bp x p - xp S y = b1 x1 - x1 Sx 1 S x 1 Sy + b2 x2 - x2 S x 2 S x 2 Sy + + bp x p S x p - x p S x p S y b j = bj S x j S y = bj L j j L0 0, Lj j = ( x j - x j ) 2, L0 0 = ( yi - y) 2 : y^ - y S y x 1 = b1 - x1 Sx 1 x2 + b 2 - x2 Sx 2 + + b p x p - xp S x p y^ = b 1 x 1 + b 2 x 2 + + b p x p n ( x i1 i = 1 n n - x 1 ) 2 b 1 + + i1 i = 1 - x 1 ) ( x ip - x p ) b p = i1 i = 1 - x 1 ) ( y i - y) ( x ip - x p ) ( x i1 - x 1 ) b 1 + + ( x ip - x p ) 2 = ( x ip - x p ) ( y i - y)

57 L 1 L00 L 11 b 1 + L 12 L00 L00 b L 2 + + L 1 p b 22 L p = L 10 pp Lp1 L 00 b 1 + Lp2 L 11 L 00 b 2 + + Lpp L 22 L 00 b p = Lp0 L pp k ( k = 1,2, p) L00, Lj j ( j = 1,2p) L 11 L 11 b 1 + L 11 L 11 L 12 b 2 + + L 22 L 11 L 1 p b p = L pp L 10 L 00 L 11 Lpp Lp1 L11 b 1 + Lpp L p2 L22 b 2 + + L pp Lpp = r10 r11 b 1 + r1 2 b 2 + + Vip b p L pp b p = r21 b 1 + r2 2 b 2 + + r2 pb p = r2 0 rp1 b 1 + rp2 b 2 + + rpp b p = rp0 RX = r1 1 r1 2 r1 p r2 1 r2 2 r2 p rp1 rp1 rpp b= Rx b= Ry b 1 b 2 b p b= R - 1 x R y : RY = r10 r20 L00 L p0 rp0 Lpp b j = bj L jj L00 bj = b j Sij Cij : Cij = r ( - 1 ) i j : L ii L j j L00 L jj, S= L0 0, S= L0 0 S S ( ), V = L00 V, R = R, S = L00 S t, ti = t i, Ry x i = R y x i,,

58 ), r11 r12 r1 p r1 y r21 r22 r2 p r2 y R ( 0 ) =, rp1 rp2 rpp rpy ry1 ry2 ryp ry y,, R m = r ( m) i j m, m + 1 k, m + 1, : r ( m + 1 ) i j = ( m) rkj ( m) rkk i = k, jk rij ( m) - rik ( m) rkj ( m) rkk ( m) ik, jk 1 rkk ( m) i = k, j = k - rik ( m ) rkk ( m) ik, j = k m : ( m ) r11 ( m) r12 ( m) r1 p ( m) r1 y r21 ( m ) r22 ( m) r2 p ( m) r2 y ( m) R ( m) = ryi ( m) : rp1 ( m) rp2 ( m) rpp ( m ) rpy ( m) ry1 ( m) ry2 ( m ) ryp ( m) ry y ( m ) x1, x2 x1, ( l < p), : (1 ) ( m) ( m) (2 )xi, ryi = - riy, = riy ( m), ( m) (3 ) ry y (4) R ( m) k1, k2 k1 k1, k2 k1 ( ),R 0, Cij = r ( ij - 1) Sii Sjj (5 ) : m + 1, xk m) V k ( m + 1 ) = r( k y r ( kk m) 2

59 y( / ) : x1 3CaOAl2 O3 ( % ) ; x2 3CaO SiO2 ( % ) ; x3 4CaO Al2 O3 Fe2 O3 ( % ) ;x2 4CaOSiO2 ( % )3.12 3.12 1 2 3 4 5 6 7 8 9 10 11 12 13 x 1 x 2 x 3 x 4 y 7 26 6 60 78.5 1 29 15 52 74.3 11 56 8 20 104.3 1 31 8 47 87.6 7 52 6 33 95.9 11 55 9 22 109.2 3 71 17 6 102.7 1 31 22 44 72.5 2 54 18 22 93.1 21 47 4 26 115.9 1 40 23 34 83.8 11 66 9 12 113.3 10 68 8 12 109.4 y x1, x2, x3, x4 ( ) 1. 0 r11 ( 0 ) r12 ( 0 ) r1 p ( 0 ) r1 y ( 0 ) R ( 0 ) = ( 0 ) r21 ( 0 ) r22 ( 0 ) r2 p ( 0 ) r2 y rp1 ( 0 ) rp2 ( 0 ) rpp ( 0 ) rpy ( 0 ) ( 0 ) ( 0 ) ( 0 ) ( 0 ) ry1 ry2 ryp ryy 1 0.228579-0.824134-0.245446 0.73071 0.228579 1-0.139242-0.97295 0.816253 = - 0.824134-0.1392 1 0.029537-0.534671-0.245445-0.9729 0.029537 1-0.821305 0.73071 0.81625-0.53471-0.82105 1 2., F

60 F,, F ( ) ; F,,,,, n m, ( n - m - 1), = 0.10, 4 2-3, f1 f2 = 10, F F( 1, 10 ) = 3.28 ( ) : = 1 ( 1 ), (1 )( ), (, ) : V j ( 1 ) = ( rjy ( 0 ) ) 2 rjj ( 0 ) = U = blx y, F, ( 1 ) V1 = ( r1 y ) 2 r11 V2 ( 1 ) = ( r2 y ) 2 ( 1 ) V3 x4, : x4 ( 1 ) F4 r22 = 0.285873; V4 = ( 1 ) Fj = (0.730717) 2 = 0.533947 = (0816253 ) 2 = 0.666269 ( 1 ) = ( n - 2 ) V j 1 - V j 110.674542 1-0.674542 = 0.67542 * ( 1 ) ( 1 ) = 22.7986 > 3.28 (2 ), 1 R ( 1 ) R ( 1 ) = = ( rij ( 1 ) ) 0.9377-0.0102-0.8168 0.2454 0.5291-0.0102 0.0533-0.1105 0.9729 0.0171-0.8186-0.1105 0.9991-0.0295-0.5104-0.2454-0.9729 0.0295 1.0000-0.8213 0.5291 0.0171-0.5104 0.8213 0.3254

61 x : R ( 0 ) x4, x4-0.8213 0.3254 4. : 3.13 F 1 11 10.3254 = 0.6745 0.3254 12 1 0.6745/ 1 = 22.7986 0.3254/ 11 : (1 ), 1 R ( 1 ), V1 Fj ( 2 ) = = ( 2 ) ( 2 ) V j ( 2 ) V 1 ( 2 ) V2 = ( rjy = ( 1 ) ) 2 ( 1 ) rjj (0.5291) 2 0.9397 = 0.0055 V3, F, ( n - ( l + 1) - 1) Vj (13-2 - 1) V 1 S ( l + 1 ) ( 2 ) ( 2 ) = S- V1 ( l + 1 ) = 0.2979 * ( 2 ) (13-3 )0.2979 0.3254-0.2979 x1, x1 = 0.2607 = 108.22 > 3.28 (2 ) x1 2 : R ( 2) = (3 ) R ( 2 ) : 1.064105-0.010884-0.86925 0.261179 0.563052 0.010833 0.053248-0.119395 0.975626 0.022919 0.869250-0.119395 0.289051 0.183816-0.050463 0.261179-0.975626-0.183816 1.064105-0.683107-0.563052 0.022919-0.050463 0.683107 0.027528

62 x, x4 : b4 ( 2 ) ( 2 ) b1 = - 0.681307 = 0.563052 0.027528 : 3.14 F 2 0.972471 176.6265 10 0.027528 12 1 x1, x4 x1, x4 F, F,, V4 F4 ( 2 ) x4 : F (2 ) = ( r4 y ( 2 ) ) 2 ( 2 ) = ( - 0.683107) 2 r44 1.064105 = ( n - l - 1) V k ( l) ( l) = S = 159.29 > 3.28 = 0.4385 ( 13-2 - 1 )0.4385 0.027528 (1 ), 2 R ( 2 ), V2 ( 3 ) ( 3 ) V j V2 ( 3 ) = ( 3 ) V3, F, Fj ( 3 ) = = = ( rj y ( 2 ) ) 2 ( 2 ) rj j (0.0229119 ) 2 0.053248 = 0.008810 ( n - ( l - 1) - 1) Vj S ( l + 1 ) ( l + 1 ) (13-3 - 1)0.009865 0.027528-0.009865 = 0.009865 * = 5.026 > 3.28

x, x2 (2 ) x2 3 : R ( 3 ) = 1.06633 0.204391-0.8936 0.660589 0.567737 0.204391 18.780350-2.242271 18.3226 0.430415 0.893654 0.242271 0.021336 2.371435 3.000926 0.460589 18.322604-2.371435 18.940119-0.263182-0.567737-0.430415 0.000926 0.263182 0.017664 (3 ) R ( 3 ) : x1, x4, x2 : b ( 3 ) 4 = - 0.263182 ( 3 ) b1 = 0.567737 b2 ( 3 ) = 0.430415 0.017664 : 63 3.15 F 3 0.982336 166.8340 9 0.017664 12 1 x1, x4, x2 x2 F, F,,, x1, x4 ( 3 ) V4 ( 3 ) F4 ) 2 = ( ( 2 ) r4 y ( 2 ) = 0.003657 r4 4 ( 3 ) ( 3 ) = S = ( n - 3-1 ) V4 = 1.863 < 3.28 F ( 13-3 - 1 )0.003657 0.017664 x2, x4, x4 : (1 ) F x4 (2 )x4,,

64 1.05512-0.24118-0.835985 0.024318 0.574137-0.24118 1.055129 0.051847-0.967396 0.688507 R 4 ) = 0.855985-0.051857 0.318256-0.125207 0.033878 0.024318 0.967396-0.125207 0.052798-0.013896-0.574137-0.68501 0.033878-0.013896 0.021322 (3 )R ( 4 ) : x1, x4 : b1 ( 4 ) ( 4 ) b2 = 0.574137 = 0.685017 0.021322 : 3.16 F 2 0.978678 229.5211 10 0.021322 12 1 x1, x2 F, F1 ( 4 ) = 146.52 > 3.28 F2 ( 4 ) = 208.578 > 3.28 :, ( 5 ) V 3 ) 2 = ( ( 4 ) r3 y ( 4 ) = 0.00360628 r33 V 4 ( 5 ) = 0.00363731 x4, x4,, ( ) 1. 4 x3, x4 3 4 3 4, R ( 4 ) = 1.055129-0.241181 0.574137-0.241181 1.055129 0.685017-0.574137-0.685017 0.021322

65 x, x2 x1, x2, 1.055129-0.241181 C ( 2 ) = - 0.241181 1.055129 x1, x2 2. bj = b j L00 L jj L0 0 = 2715.7635, L11 = 415.2308, L22 = 2905.6923 b 1 = 0.574137 b 2 = 0.685017 b1 = 1.4683 b2 = 0.6623 b0 = y - b1 x1 - b2 x2 = 52.5742 : y = 52.5742 + 1.4683 x1 + 0.6623 x2 3.17 F 2 2657.8889 1328.9290 10 5.7906 5.7906 229.2511 12 2715.7635 s = R 2 = 0.9893 n ( y i = 1 i - y^i ) 2 13-2 - 1 = 2.4064 x1, x2, y x 3, x4 : r3 y 12 2 = ( r3 y ( 4 ) 2 r4 y 12 ) 2 r33 ( 4 ) ry y ( 4 ) = 0.1691 = ( r4 y ( 4 ) r44 ( 4 ) ry y ) 2 ( 4 ) = 0.1715 SPSS : : SPSS, :

66.10 : (1 ) Analyze Regression Linear (Dependent), ( Independent), 3.11 3.11

67 2 ) Statistics, 3.12 (3 ) Continue Plots, 3.13 (4 ) Continue Save,

68.14 (5 ) Continue Option s 3.15

69, 3.16 3.16 : 3.18 ( Descriptive Statistics) Mean Std. Deviation N Y 95.4231 15.0437 13 X1 7.4615 5.8824 13 X2 48.1538 15.5609 13 X3 11.7692 6.4051 13 X4 30.0000 16.7382 13 3.19 ( Correlations ) Y X1 X2 X3 X4 Pea rson Corr elation Y 1.000.731.816 -.535 -.821 X1.731 1.000.229 -.824 -.245 X2.816.229 1.000 -.139 -.973 X3 -.535 -.824 -.139 1.000.030 X4 -.821 -.245 -.973.030 1.000 Sig. ( 1 - tailed) Y..002.000.030.000 X1.002..226.000.209 X2.000.226..325.000 X3.030.000.325..462 X4.000.209.000.462.

70 3.20 ( Variables Entered/ Removed) Varia bles Va riables Model E ntered Removed M e thod 1 X4 Stepwise (Criteria: F-to-enter > = 3.840, F-to-remove < = 2.710). 2 X1 Stepwise (Criteria: F-to-enter > = 3.840, F-to-remove < = 2.710). 3 X2 Stepwise (Criteria: F-to-enter > = 3.840, F-to-remove < = 2.710). 4 X4 Stepwise (Criteria: F-to-enter > = 3.840, F-to-remove < = 2.710). a Dependen t Variable: Y 3.21 ( Model Summary) R Adju st ed Std. E rror of Mod el R R Square Squ are R Square the Es tima te Cha nge Ch a ng e St atistics F df1 df2 Ch a ng e Dur bin Sig. F -Wa tson Ch ang e 1.821.675.645 8.9639.675 22.799 1 11.001 2.986.972.967 2.7343.298 108.224 1 10.000 3.991.982.976 2.3087.010 5.026 1 9.052 4.989.979.974 2.4063 -.004 1.863 1 11.205 1.922 a b c d e Predictors : ( Constant), X4 Predictors : ( Constant), X4, X1 Predictors : ( Constant), X4, X1, X2 Predictors : ( Constant), X1, X2 Dependent Variable : Y 3.22 ( ANOVA ) Model Sum of Squares df Mean Squa re F Sig. 1 Regression 1831.896 1 1831.896 22.799.001 R esidual 883.867 11 80.352 To tal 2715.763 12 2 Regression 2641.001 2 1320.500 176.627.000 R esidual 74.762 10 7.476 To tal 2715.763 12 3 Regression 2667.790 3 889.263 166.832.000 R esidual 47.973 9 5.330 To tal 2715.763 12 4 Regression 2657.859 2 1328.929 229.504.000 R esidual 57.904 10 5.790 To tal 2715.763 12 a b c d e Predictors : ( Constant), X4 Predictors : ( Constant), X4, X1 Predictors : ( Constant), X4, X1, X2 Predictors : ( Constant), X1, X2 Dependent Variable : Y

71.23 (Coefficients ) Model Unstanda rdized Coefficie nts Std. B E rror Standardized Coefficie nts Be ta t Sig. 95 % Confide nce Int erval for B Lower U pper Bound Bound Correla tion s Zero- order Partial Part 1 ( C) 117.568 5.262 22.342.000 105.986 129.150 X4 -. 738.155 -. 821-4.775.001-1.078 -. 398 -. 821 -. 821-. 821 2 ( C) 103.097 2.124 48.540.000 98.365 107.830 X4 -. 614.049 -. 683-12.621.000 -. 722 -. 506 -. 821 -. 970-. 662 X1 1.440.138.563 10.403.000 1.132 1.748.731.957.546 3 ( C) 71.648 14.142 5.066.001 39.656 103.641 X4 -. 237.173 -. 263-1.365.205 -. 629.155 -. 821 -. 414-. 060 X1 1.452.117.568 12.410.000 1.187 1.717.731.972.550 X2.416.186.430 2.242.052 -. 004.836.816.599.099 4 ( C) 52.577 2.286 22.998.000 47.483 57.671 X1 1.468.121.574 12.105.000 1.198 1.739.731.968.559 X2.662.046.685 14.442.000.560.764.816.977.667 a Depe ndent Variable : Y 3.24 ( Ex clude d Varia bles ) Model Beta In t Sig. Partial Correlation Collinearity Statistics Tolera nce 1 X1.563 10.403.000.957.940 X2.322.415.687.130 5.336 E - 02 X3 -. 511-6.348.000 -. 895.999 2 X2.430 2.242.052.599 5.325 E - 02 X3 -. 175-2.058.070 -. 566.289 3 X3.043.135.896.048 2.134 E - 02 4 X3.106 1.354.209.411.318 X4 -. 263-1.365.205 -. 414 5.280E - 02 a b c d e P redictors in the Model: (Constant), X4 P redictors in the Model: (Constant), X4, X1 Predictors in the Model : (Constant), X4, X1, X2 Predictors in the Model : (Constant), X1, X2 Dependent Va ria ble : Y

72 3.25 Coefficient Correlations Model X4 X1 X2 a 1 Correlations X4 1.000 Covariances X4 2.390E - 02 2 Correlations X4 1.000.245 X1.245 1.000 Correlations X4 2.366E - 03 1.653E - 03 X1 1.653E - 03 1.916E - 02 X4 1.000.102.972 Correlations X1.102 1.000.046 3 X2.972.046 1.000 X4 3.003E - 02 2.078E - 03 3.125E - 02 Correlations X1 2.078E - 03 1.369E - 02 9.918E - 04 X2 3.125E - 02 9.918E - 04 3.445E - 02 4 Correlations X1 1.000 -.229 X2 -.229 1.000 Correlations X1 1.471E - 02-1.271E - 03 X2-1.271E - 03 2.103E - 03 Dependent Va riable : Y 3.26 ( Residuals Sta tistics ) Minim um Maximu m Mean Std. Deviation N P redicted Value 73.2509 114.5375 95.4231 14.8825 13 Std. P redicted Value -1.490 1.284.000 1.000 13 Standard Error of Predicted Value.6958 1.7846 1.1238.2819 13 Adjusted Predicted Value 72.8787 113.0821 95.3974 14.7715 13 Residual -2.8934 4.0475-1.2025E-14 2.1967 13 St d. Residual -1.202 1.682.000.913 13 Stud. R esidual -1.356 1.788.002 1.024 13 Deleted Residual -3.6821 4.5741 2.566 E - 02 2.7969 13 Stud. Deleted Residual -1.425 2.057.029 1.078 13 Mahal. Dista nce.080 5.677 1.846 1.447 13 Cook s Dista nce.011.290.093.081 13 Cen ter ed Lever age Value.007.473.154.121 13 a Dependent Va riable : Y

73.17 3.18 3.19

74.20 3.27,,, ( ),

75,,,,,, :, ( ), : 1. y = b0 + b1 x + b2 x 2 + + bk x k + e y = b0 + b1 x1 + b2 x2 2 + + bk x k k + e 3.21 3. 22 2. y = ab x e y = ae bx e 3.23

76. y = ax b e 3.24 4. y = a + b x + e 1 y = a + b x + e 3.25 5. y = a + bln x + e 3.26

77. y = a + bsin x + e ( ), : 1. y = b0 + b1 [ e x p( b2 x) ] + e y = L + ab x y = L + ae bx 3.27 y = L 3.27 2., ^ y = Le - ae - bx y = La bx 3.28,y = L/ e, x = lna/ b, 3.28

78., ^ y = L 1 + ae - bx, y = L/ 2, x = ln a/ b, 3.29 S, S, ( ),,, : y = b0 + b1 x + b2 x 2 + + bk x k + e X1 = x, X2 = x 2, Xk = x k y = b0 + b1 X1 + b2 X2 + + bk X k + e y = a+ b x + e X = 1 x, y = a + b X + e

79 y a + bln x + e X = ln x, y = a + b X + e y = a + bsin x + e X = sin x, y = a + b X + e,,,, ( ), y = ae bx e ln y = ln a + bx + lne Y = ln y, A = ln a, B = b, E = ln e Y = A + Bx + E,, y = ax b e ln y = ln a + bln x + ln e Y = ln y, A = ln a, B = b, X = ln x, E = lne Y = A + B X + E,, - ae - bx y = Le : y L = e- ae - bx ln y L = - aebx ln L y = aebx ln( ln L y ) = ln a - bx

80 L y L y > 1, ln L y > 0, Y = ln( ln L y ), A = ln a, Y = A - bx, L,,, y = L 1 + ae - bx : L y - 1 = ae- bx ln( L y - 1) = ln a - bx Y = ln( L y - 1), A = ln a, Y = A - bx, L,,,,,, y,,,,,,,,,,, ( x) ( y),

81.28 :,, y^ = a+ b x + e, X = 1 x y = a + b X + e 3.30 :, Excel,,, 3.31

82.31 a= 2.225407, b= 7.621271, y^ = 2.225407 + 7.621271 x 28.5, 29 30,, :2.49282 % 2.488209% 2.479449%

( ) Y t, {Y t, t = t0, t1 }Y t, t = 1, 2,, 20 60 70,,, Warren Persons (),,,,,,,,,,, ( T) ( S) ( C) ( I)

84. (T ),,,,,, 2. (S ), 5, 3. (C ),,,, 12 4 1 1, 4. (I ),,,,, ( ),, : 1.,,,,,,,,

85.,,,,,, 12 4 7, 3.,, 4., ( ),,, 5.,,,,,,,, t, t, t,, ( ),

86,,,, : Y t = f( T1, S1, C1, I1 ), : : Y t = T1 St Ct It : Y t = Tt + St + Ct + It Y t Y t = Tt St + Ct It,, ;, ;,,,, ( ),,,,,,, Box - Jenkins 4.1, Box - Jenkins 4.1

87,,,,, : n : y1, y2, y3 yn - 1, yn t, t + 1 : Ft + 1 = y = i = 1 t, y t, Ft + 1 t + 1 : t yi et + 1 = yt + 1 - Ft + 1 t + 2, : Ft + 2 t + 1 yi = y = i = 1 t + 1 et + 2 = yt + 2 - Ft + 2, : ;, Ft + 1 = y = i = 1 t t yi

88 t + 1 yi Ft 2 = y = i = 1 t + 1 = y1 + y2 + + yi + y i + 1 t + 1 = t Ft + 1 + yt + 1 t + 1 Ft + 3 = t + 1 t + 2 Ft + 2 + 1 t + 2 yt + 2 = t t + 1 Ft + 1 + 1 t + 1 yt + 1,,,,,,,,,, t,, : t + 1 t t + 2 Ft + 1 = Ft + 2 = y1 + y2 + + yt t y2 + y3 + + yt + 1 t, = 1 t t i = 1 yi = 1 t + 1 t i = 2 yi, Excel, : : : C5 = AVERAGE( B2: B4 ), : C5, C14,, 4.2

89 4.1 4.2,,,,,, :, t, 12, 4,

90 t + 1 Ft 2 = 1 t i = 2 yi = 1 t ( y1 + y2 + yt + yt + 1 - y1 ) = Ft + 1 + 1 t ( yt + 1 - y1 ),, ( ),,,,,,, : Ft + 1 =, i, wi Ft + 1 t i = 1 i y i t i i = 1 t = i = 1 wi y i t, w1 w2 wt, wi = 1 i = 1 Excel,, : :, 0 = 1.5, 1 = 1, 2 = 0.5 : D5 : = ( B41.5 + B31 + B20.5)/ 3, : D5, D14, : 4.2

91.3,,,,,,, : ( t ), ;, ;,,,,,, 4.3

92 4.3,,,,,,,,,, Ft + 1 = Ft + 1 n ( y t - yt - n ) n, t Ft y t - n, Ft + 1 = Ft + 1 ( yt - Ft ) n Ft + 1 = 1 n y t + 1-1 n Ft = 1, 0 < < 1, n Ft + 1 =yt + (1 - ) Ft, Ft t Ft + 1 t + 1, t + 1 : St + 1 =yt + (1 - ) St

93, : St = y t + ( 1 - a) St - 1, 0.2, 0.5, 0.7, 4.3 4.3 ( ) y t = 0.2 = 0.5 = 0.7 1 371.5 2 267.4 371.5 371.5 371.5 3 372.4 350.68 319.45 298.63 4 368.2 355.02 345.93 350.27 5 349.4 357.66 356.06 362.82 6 362.8 356.01 353.23 353.43 7 420.9 357.37 358.02 359.99 8 380.4 370.07 389.46 402.63 9 385.6 372.14 384.93 387.07 10 335.0 374.83 285.27 386.04 11 338.5 366.86 360.13 350.31 12 306.6 361.19 349.32 342.04 1 350.27 327.96 317.23 : = 0.2 S1 = 0.2371.5 + 0.8371.5 = 371.5 S2 = 0.2267.4 + 0.8371.5 = 350.68 : 4.4

94 0.2,,= 0.7,,, (1 - ) n S t - n + 1 St = yt - 1 St =yt + (1 - ) St - 1 =yt + ( 1 - ) St - 1 St - 1 =yt - 2 + (1 - ) St - 2 + (1 - ) yt - 1 + ( 1 - ) 2 yt - 2 + + ( 1 - ) n y t - n + 1 +,, t,, 4.4 = 0.3 = 0.2 = 0.1 1 0.012106 0.026844 0.038742 2 0.01729 0.033554 0.043047 3 0.02471 0.041943 0.047830 4 0.03529 0.052429 0.053144 5 0.05042 0.065536 0.059049 6 0.07203 0.08192 0.06561 7 0.1029 0.1024 0.0729 8 0.147 0.128 0.081 9 0.21 0.16 0.09 10 0.3 0.2 0.1,,,, ;,, : ( ),

95 0.100.30, 0.30,,,,,,,, SSE MSE MAE ( ), n, (1 - ) n,,, ;,, ;,,, :,,,, 8 3,,,,,, ( ),,, (), : St + 1 = St + ( yt - St ) St + 1 = St + ( et ) = +,,,, ( 4.5 ),

96, : S ( 1 ) t = Yt + (1 - ) S ( 1 ) t - 1 S ( 2 ) t = S ( 2 ) t + ( 1 - ) S ( 2 ) t - 1 4.5 1 2 1.00 0.500 2 4 2.50 0.375 3 6 4.25 2.313 4 8 6.13 4.219 5 10 8.06 6.141 6 12 10.03 8.086 7 14 12.02 10.053 8 16 14.01 12.031 9 18 16.00 14.016,,,,,,, : : at bt Ft + m = at + bt m t ; t ; m at bt, () ( Brown ) 1. (1 ), : S ( t 1 ) = Yt + (1 - ) S ( t 1 - ) 1 S ( t 2 ) = S ( t 1 ) + ( 1 - ) S ( t 2 - ) 1,,,, 1 - bt, bt t

97,, (2 ), : t bt Ft + m t t m,, 2. = t + bt m at = S ( 1 ) t + S ( 1 ) t - S ( 2 ) t = 2 S ( 1 ) t - S ( 2 ) t bt = 1 - ( S( t 1 ) - S ( 2 ) t ) 4.6 4.6 ( ) 1 1 143 2 2 152 3 3 161 4 4 139 5 5 137 6 6 174 7 7 142 8 8 141 9 9 162 10 10 180 11 11 164 12 12 171 1 13 206 2 14 193 3 15 207 4 16 218 5 17 229 6 18 225 7 19 204 8 20 227 9 21 223 10 22 242 11 23 239 12 24 266

98 Excel : : S ( 1 ) 0 = S ( 2 ) 0 = 143,= 0.2 0, D2, E2 143 D3 : = 0.2 * C3 + 0.8 * D2, D2 6, : E3 E26, : F3 F26, : G3 G2 6, b : = 0.2 * D3 + 0.8 * E2, : = 2 * D3 - E3, : = 0.2 * (D3 - E3 )/ 0.8, : 1 ( m = 1 ), H4 H27, 225 : = F3 + G3, 2, Ft + m = at + bt m m = 2, F26 = a24 + b24 2 = 252.246 + 5.5142 = 263.27 ( ) m = 3, F27 = a24 + b24 3 = 252.246 + 5.5143 = 268.79 ( ), : 4.7

99 4.5 3. ( ) ( H OLT ) 1., ( ) : St =yt + (1 - ) ( St - 1 + bt - 1 ) bt - 1 St - 1,, St, yt ( ) bt = ( St - St - 1 ) + ( 1 - ) bt - 1, St,,, : : St bt Ft + m = St + bt m t ; t ; m - St - 1,, 2., :

100 b0 b0 = (152-143) + (161-152) + (139-161) 3 S0 = 143, b0 = - 1.33, = - 1.33 = 0.2,= 0.3 D2 143, E2-1.33 D3 : = 0.2 * C3 + 0.8 * (D2 + E2), : E3 : = 0.3 * (D3 - D2) + 0.7 * E2, : D3, E3, E26, : 1 ( m = 1 ), F3 : = D2 + E2, H27, 225 2, Ft + m = at + bt m : m = 2, F26 = a24 + b24 2 = 254.6857 + 6.0035472 = 266.693( ) m = 3, F27 = a24 + b24 3 = 254.6857 + 6.0035473 = 272.696( ), 4.8 4.5: 4.8

101 3..6 ( H OLT ),,,,, bt 0,,,,, h, b h = 1 - (1 - b ) 2,= 1 - b,, 2(1 - b ), 2 - b, ( H OLT ) 4.64.7 4.7

102.8 : ( ) 1.,, S ( 1 ) t = Yt + (1 - ) S ( 1 ) t - 1 S ( t 2 ) = S ( t 1 ) S ( 3 ) t = S ( 2 ) t + ( 1 - ) S ( t 2 - ) 1 + ( 1 - ) S ( 3 ) t - 1,, Ft + m = at + bt m + 1 2 ct m2, at = 3 S ( t 1 ) - 3 S ( t 2 ) + S ( t 3 ) bt = ct = 2 (1 - ) S ( 1 ) t ) 2 [ ( 6-5 2 ( S ( 1 ) (1 - ) 2 t - 2 S ( 2 ) t + S ( 3 ) t ) - (10-8 ) S ( 2 ) t + (4-3 ) S ( 3 ) t ],,,

103. : 4.9 ( ) 1 1 3.4 2 2 3 3 3 3.4 4 4 3.7 5 5 3.3 6 6 4.6 7 7 3.2 8 8 3.8 9 9 4.2 10 10 4 11 11 6.1 12 12 7.3 1 13 5.7 2 14 4.5 3 15 6 4 16 7 5 17 7.6 6 18 9.3 7 19 11.8 8 20 19.9 9 21 15.5 10 22 20.1 11 23 16.1 12 24 18.4 Excel, : : S ( 1 ) 0 = S ( 2 ) 0 = S ( 3 ) 0 = 3.4,= 0.250, D2, E2, F2 3.4D3 : = 0.25 * C3 + 0.75 * D2, D26, : E3 E26, : F3 F26, : G3 G2 6, a : H3 : = 0.25 * D3 + 0.75 * E2, : = 0.25 * E3 + 0.75 * F2, : = 3 * D3-3 * E3 + F3, :

104 0.25 * ( ( 6-5 * 0.25 ) * D3-2 * (5-4 * 0.25 ) * E3 + ( 4-3 * 0.25) * F3 )/ (2 * 0.75 * 0.75 ), H26, b : I3 : = 0.25 * 0.25 * ( D3-2 * E3 + F3 )/ (2 * 0.75 * 0.75 ), I2 6, c : 1 ( m = 1 ), J4 : = G3 + H3 + I3, J27, 225 2, Ft + m = at + bt m + 1 2 ct m2, : m = 2, F2 6 = a2 4 + b2 4 2 + 1 2 c24 22 = 19.2130 + 1.3362 + 1 2 0.01858422 = 21.922168 ( ), : 4.10 :

105.9 3. ( ) ( ) 1., St = y t b1 It - L + ( 1 - ) ( St - 1 + bt - 1 ) = ( St - St - 1 ) + ( 1 - ) bt - 1 It = y t S t + ( 1 - ) It - L L, I,, Ft + m = ( St + bt m) It - L + m : ( St ), ( bt ) ( It ) 2. : 4.11 ( ) 1996 1997 1998 1999 2000 2001 2002 115 109.9 110.1 136.6 151.6 212.2 218 69.9 68.2 90.3 93 94.6 97.6 103 64.5 84.2 117.3 51.7 123.6 99.6 105.6 92.7 89.4 90.9 74 144.8 156.6 135.2

106 4.9,, :,,,,,,, : :, S0 = 77.254, b0 = 2.335, : I - 3 = 1.37, I - 2 = 0.81, I - 1 = 0.851, I0 = 0.964 = 0.2,= 0.2,= 0.3, : D6 : = 0.2 * ( C6/ F2) + 0.8 * ( D5 + E5), E6 : = 0.2 * (D6 - D5) + 0.8 * E5, F6 : = 0.3 * ( C6/ D6) + 0.7 * F2, D7 : = 0.2 * ( C7/ F3) + 0.8 * ( D6 + E6), E7 : = 0.2 * (D7 - D6) + 0.8 * E6, F7 : = 0.3 * ( C7/ D7) + 0.7 * F3, D8 : = 0.2 * ( C8/ F4) + 0.8 * ( D7 + E7), E8 : = 0.2 * (D8 - D7) + 0.8 * E7, F8 : = 0.3 * ( C8/ D8) + 0.7 * F4, D9 : = 0.2 * ( C9/ F5) + 0.8 * ( D8 + E8), E9 : = 0.2 * (D9 - D8) + 0.8 * E8, F9 : = 0.3 * ( C9/ D9) + 0.7 * F5, : D6, E6, F6, D9, E9, F9, F33, : 1 ( m = 1), G6 : = ( D5 + E5) * F2, G7 : = ( D6 + E6) * F3, G8 : = ( D7 + E7) * F4, G9 : = ( D8 + E8) * F5, G6G9 G34, 229 2, Ft + m = ( St + bt m) It - L + m : m = 2, F30 = ( S28 + 2b28 )I26 = (146.8142 + 21.8930 )0.78

107 117.4681 ( ) m = 3, F31 = ( S28 + 3b28 )I27 = (146.8142 + 31.8930 )0.80 = 121.9946 ( ), : 4.12 ( ) 4.10 3.

108 1. ) ( ), St = ( yt - It - L ) + ( 1 - ) ( St - 1 + bt - 1 ) bt = ( St - St - 1 ) + ( 1 - ) bt - 1 It = ( yt - St ) + (1 - ) It - L L, I,, 2. Ft + m = ( St + bt m ) + It - L + m, : :, S0 = 77.254, b0 = 2.335, : I - 3 = 39.76786, I - 2 = - 22.6321, I- 1 = - 18.3607, I0 = 1.225 = 0.2,= 0.2,= 0.3, : D6 : = 0.2 * ( C6 - F2 ) + 0.8 * (D5 + E5 ), E6 : = 0.2 * (D6 - D5) + 0.8 * E5, F6 : = 0.3 * ( C6 - D6 ) + 0.7 * F2, D7 : = 0.2 * ( C7 - F3 ) + 0.8 * (D6 + E6 ), E7 : = 0.2 * (D7 - D6) + 0.8 * E6, F7 : = 0.3 * ( C7 - D7 ) + 0.7 * F3, D8 : = 0.2 * ( C8 - F4 ) + 0.8 * (D7 + E7 ), E8 : = 0.2 * (D8 - D7) + 0.8 * E7, F8 : = 0.3 * ( C8 - D8 ) + 0.7 * F4, D9 : = 0.2 * ( C9 - F5 ) + 0.8 * (D8 + E8 ), E9 : = 0.2 * (D9 - D8) + 0.8 * E8, F9 : = 0.3 * ( C9 - D9 ) + 0.7 * F5, : D6, E6, F6, D9, E9, F9, F33, X : 1 ( m = 1),

109 G : = ( D5 + E5) + F2, G7 : = ( D6 + E6) + F3, G8 : = ( D7 + E7) + F4, G9 : = ( D8 + E8) + F5, G6G9 G34, 229 2, Ft + m = ( St + bt m) + It - L + m : m = 2, F30 = ( S28 + 2b28 + I2 6 = ( 146.8355 + 21.908335) + 53.08 = 203.7322 ( ) m = 3 F31 = ( S2 8 + 3 b28 ) + I27 = ( 146.8355 + 3 1.908335 ) - 29.32 = 123.2405 ( ), 4.13, 4.11 4.13 ( ) 4.12, 4.13

110.11 4.12 4.13,,

111.,,,,,,,,,, :,,,,,,,,, ( ),,,,,,,,,,,,,,,

112,,,,, (, ) ; : 4.14 1 2 3 A ( ) B ( ) C( ) A1 B1 C1 A2 B2 C2 A3 B3 C3 : 4.15 A - 1 A - 2 A - 3 B - 1 B - 2 B - 3 :, 1996 1

113 C 1 C - 2 C - 3 : 4.16 A1 A2 A3 B1 B1 S t + 1 = y t + ( 1 - ) S t F t + 1 = S t + 1 S t = ( y t - I t - L ) + ( I - ) S t - 1 F t + m = S t + I t - m + L I t = ( y t - S t ) + ( 1 - ) I t - L S t =( y t / I t - L ) + ( 1 - ) S t - 1 F t + m = S t I t - m+ L I t =( y t / S t ) + ( 1 - ) I t - L S (1 ) t =Y t + ( 1 - ) S ( t 1 - ) 1 S ( t 2 ) =S ( t 1 ) + ( 1 - ) S ( t 2 - ) 1 S t =y t + (1 - ) ( S t - 1 + b t - 1 ) a t = S (1) t + ( S ( t 1) - S (2) t ) = 2 S (1 t ) - S ( t 2) b t = 1 - ( S( t 1 ) - S ( t 2 ) F t + m = a t + b t m F t + m = S t + b t m b t = ( S t - S t- 1 ) + (1 - ) b t - 1 C1 S (1 ) t =Y t + ( 1 - ) S ( t 1 - ) 1 S ( t 2 ) =S ( t 1 ) S ( t 3 ) =S ( t 2 ) + ( 1 - ) S ( t 2 - ) 1 + ( 1 - ) S ( t 3 - ) 1 a t = 3 S ( t 1) - 3 S ( t 2 ) + S ( t 3) b t = 2( 1 - ) 2 [(6-5 ) S (1) t - (10-8 )S (2) t + (4-3 ) S (3) t ] c t = 2 (1 - ) 2 ( S( t 1 ) - 2 S ( t 2 ) + S ( t 3 ) ) F t + m = a t + b t m + 1 2 c t m 2

114 B B3 S t = (y t - I t- L ) + (1 - )(S t - 1 + b t- 1 ) b t = ( S t - S t- 1 ) + (1 - ) b t - 1 F t + m = ( S t + b t m ) + I t - L + m I t = ( y t - S t ) + ( 1 - ) I t - L S t = y t + (1 - ) ( S I t - 1 + b t - 1 ) t- L b t = ( S t - S t- 1 ) + (1 - ) b t - 1 F t + m = ( S t + b t m ) I t - L + m I t = y t S t + ( 1 - ) I t - L

( self - adaptive filtering) -,,,,,,, y1, y2 yt, y^t + 1 = w1 yt + w2 yt - 1 + + wn y t - n + 1 y^t + 1 n = i = 1 wi y t - i + 1 t + 1, W i, n yt - i + 1 t - i + 1,,,,, yt + 1 t + 1, yt + 1 = y^t + 1 + et + 1 = w1 yt + w2 yt - 1 + + wn y t - n + 1 + et + 1 et + 1 et + 1 = yt + 1 - y^ t + 1,,,,

116 et + 1 ( ),, : w i = wi - ke 2 t + 1, W i, W i, e 2 t + 1 e 2 t + 1 ; k,,,, B.Widrow k : k n 1 i = 1 y2 i n, n ( ) ma x wi et + 1 = y t + 1 - y^t + 1 = y t + 1 - w1 yt - w2 yt - 1 - - wn y t - n - 1 e 2 t + 1 = ( yt + 1 - w1 yt - w2 y t - 1 - - wn y t - n - 1 ) 2 : e 2 t + 1 = e2 t + 1 wi w i = wi et + 1 = 2 et + 1 wi = 2 et + 1 ( - yt - i + 1 ) = - 2 et + 1 yt - i + 1 w i = wi + 2 ket + 1 yt - i + 1 - ke 2 t + 1 ( i = 1, 2n), n yt - i + 1 t - i + 1 ( ),, : 5.1 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20

117 :, n 4 : k k n 1 i = 1 y2 i max = 1 14 2 + 16 2 + 18 2 + 20 2 = 0.00085 :, W i = 1 n = 1 4 = 0.25, n = 4, t t = 4 y^t + 1 = y^5 = w1 y4 + w2 y3 + w3 y2 + w4 y1 = 0.258 + 0.256 + 0.254 + 0.253 = 5 et + 1 = e5 = y5 - y^ 5 = 10-5 = 5 W i = wt, + 2 ket + 1 y t - i + 1 : w 1 = 0.25 + 20.0008558 = 0.314 w 2 = 0.25 + 20.0008556 = 0.296 w 3 = 0.25 + 20.0008554 = 0.282 w 4 = 0.25 + 20.0008552 = 0.266 :,,,, : t = 5, y^t + 1 = y^6 = w1 y5 + w2 y4 + w3 y3 + w4 y2 = 0.31410 + 0.2988 + 0.2826 + 0.2664 = 8.28 et + 1 = e6 = y6 - y^6 = 12-8.28 = 3.72 w i = wi + 2 ket + 1 yt - i + 1 : w 1 = 0.314 + 20.000853.7210 = 0.374 w 2 = 0.298 + 20.000853.728 = 0.349 w 3 = 0.282 + 20.000853.726 = 0.320 w 4 = 0.266 + 20.000853.724 = 0.291, t = 6, t = 7, t = 8,, t = 10, y1 1, e11 wi,,, t = 4, ( ),

118, : w1, w2, w3, w4 11, 12 : 11 y^1 1 = w1 y1 0 + w2 y9 + w3 y8 + w4 y7 12 y^1 2 = w1 y^1 1 + w2 y10 + w3 y9 + w4 y8, Excel, : :, 4,, 5.2 5.2 : K6 : = G6 * C6 + H6 * D6 + I6 * E6 + J6 * F6, : L6 : = B6 - K6, : G7 : = G6 + 2 * 0.0008 * L6 * B5, w1 H7 : = H6 + 2 * 0.0008 * L6 * B4, w2 I7 : = I6 + 2 * 0.0008 * L6 * B3, w3 J7 : = J6 + 2 * 0.0008 * L6 * B2, w4 K6, L6, L7,

119 5.3 : G7, H7, I7, J7, K7, L7, L11, : 5.4 0.384368 0.353706 0.323044 0.292383 word,, G6 J6 (, ),, 5.5,, :

120 5.6 5.7,,,,, : ( ) ( n),, 4 8, 12 n,,, n 2 6 n, n (),,, wi = 1 n r1, r2 Yule - Walker

121 n, w1 = ( ) k r1 ( 1 - r2 ) 1 - r 2 1, w2 = r2 - r2 1 1 - r 2 1, k, n,,,, k, k 1 n : 5.8 ( ) 1 1 3.4 2 2 3 3 3 3.4 4 4 3.7 5 5 3.3 6 6 4.6 7 7 3.2 8 8 3.8 9 9 4.2 10 10 4 11 11 6.1 12 12 7.3 1 13 5.7 2 14 4.5 3 15 6 4 16 7 5 17 7.6 6 18 9.3 7 19 11.8 8 20 19.9 9 21 15.5 10 22 20.1 11 23 16.1 12 24 18.4 Excel, : :, 1 3, wi = 1/ 3

122 k 1 15.5 2 + 20.1 2 + 16.1 2 = 0.0000095271 : J5 : = G5 * D5 + H5 * E5 + I5 * F5, : K5 : = C5 - J5, L5 : = K5 * K5, : G6 : = G5 + 2 * 0.0000095271 * K5 * C5, w1 H6 : = H5 + 2 * 0.0000095271 * K5 * C4, w2 I6 : = I5 + 2 * 0.0000095271 * K5 * C3, w3 J5, K5, L5, L6, : G6, H6, I6, J6, K6, L6, L25, L26 176.159/ 21 = 8.388526 : 5.9

123.3408 0.3382 0.33769 word,, G5 I5 (, ),, 5.10: 5.10 21, :

124.11 8.388526 8.040016 7.753617 7.520465 7.330499 7.175578 7.051652 6.952486 6.874098 6.813025 6.765576 6.729515 6.703358 6.68437 6.671673 6.664001 6.660141 6.659421 6.661035 6.664456 6.66923,,, 18, :

125 5.11 18 : : w1 = 0.4178, w2 = 0.3733, w3 = 0.37066 18 5.1 : y25 = 0.417818.4 + 0.373316.1 + 0.370620.1 = 20.33( ) y26 = 0.417820.33 + 0.373318.4 + 0.370616.1 = 21.33( )

126 (1 ),, (2 ),,,,,,,,,,,,,,,,, n wi = 1,, i = 1,,,,, 1,,, ( Weight ed Sum) ( ),,,,,,,,,

127, ;, 1, ( ) ARMA AR MA : y^t + 1 = 1 yt + 2 yt - 1 + + n y t - n + 1 n : y^t + 1 = w1 yt + w2 yt - 1 + + wn y t - n + 1,, ARMA,, 1973,,,,,,

,,,,, t, y, y^ = F( t), t, (trend projection ) t,,, : y^ = F( t), n n Q = ( y - y^) 2 i = 1 n 2 = [ y - F( t) ] i = 1 (),, : Q = 0 ( i = 1,2p) i p

129 :,,, t, y, : ( ) yt ^ = a+ bt 6.1 () yt ^ = b0 + b1 t + b2 t 2 6.2

130 ) ^ yt = at b 6.3 ( 1) ( b > 0 ) 6.4 (2 ) ( a < 0) () yt ^ = b0 + b1 t + b2 t 2 + b3 t 3 6.5 () yt ^ = a+ blnt

131.6 ( ) yt ^ = a+ b 1 t 1 ^ = a + b1 t yt 6.7 () yt ^ = ae bt 6.8

132 ) ^ yt = L + ae bt ( a< 0, b< 0 ) yt ^ = L + ab t ( a< 0, 0 < b < 1) 6.9 () yt ^ = La bt - ae - bt yt ^ = Le 6.10 () yt ^ = 1 L + ab t ( )

133 yt = L 1 + ae - bt 6.11, t,,, 1991 2002 : 6.1 19911992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 704 846 ( ) 1093 1444 1812 2299 2971 4123 5553 7621 9575 13131 t, :

134.12,,,,,,,,,,, : 6.2,, 6.2 ( t) yt = a + bt ( yt - yt - 1 ) 1 2 3 4 a + b a + 2 b a + 3 b a + 4 b - b b b t - 1 t a + ( t - 1) b a + tb b b 6.3,,

135 6.3 yt b0 b1 b2 ( yt - yt - 1 ) ( t) = + t + t 2 1 b 0 + b 1 + b 2-2 b 0 + 2 b 1 + 4b 2 b 1 + 3 b 2 3 b 0 + 3 b 1 + 9b 2 b 1 + 5 b 2 4 b0 + 4 b1 + 16b2 b1 + 7 b2 t - 1 b 0 + ( t - 1 ) b 1 + ( t - 1 ) 2 b 2 b 1 + ( 2t - 3) b 2 t b0 + b1 t + b2 t 2 b 1 + ( 2t - 1) b 2 [ ( y t - y t - 1 ) - ( yt - 1 - yt - 2 ) ] - - 2 b 2 2 b2 2 b 2 2 b2 6.4,, 6.4 ( t) y t = b 0 + b 1 t + b 2 t 2 + b 3 t 3 1 b 0 + b 1 + b 2 + b 3 - - 2 b 0 + 2 b 1 + 4 b 2 + 8b 3 - - 3 b 0 + 3 b 1 + 9 b 2 + 27b 3 2 b 2 + 12b 3-4 b 0 + 4b 1 + 16 b 2 + 64 b 3 2 b 2 + 18b 3 6 b3 t - 1 b0 + ( t - 1) b1 + ( t - 1) 2 b2 + ( t - 1) 3 b3 2 b 2 + 6( t - 1 ) b 3 6 b 3 t b0 + b1 t+ b2 t 2 + b3 t 3 2 b 2 + 6tb 3 6 b3 6.5,, 6.5 ( t) y t = ae bt ( y 1 / y t - 1 ) 1 2 3 4 t - 1 t ae b ae 2b ae 3b ae 4b ae ( t - 1) b ae tb - e b e b e b e b e b

136.6,, 6.6 ( t) yt = a + bc t ( y t - y t - 1 ) y t - y t - 1 1 2 3 4 t - 1 t y t = a + bc yt = a + bc 2 y t = a + bc 3 y t = a + bc 4 yt = a + bc ( t - 1) y t = a + bc t - bc( c - 1 ) bc 2 ( c - 1) bc 3 ( c - 1) bc t - 2 ( c - 1) bc t - 1 ( c - 1) - - c c c c yt - 1 - yt - 2 6.7,, 6.7 ( t) y t = La bt lg y t = lg L + b t lg a lg y t - lg y t - 1 lg yt - lg yt - 1 lg y t - 1 - lg y t - 2 1 2 3 4 t - 1 t La b La b2 La b3 La b4 La bt - 1 La bt lg L + blga lg L + b 2 lga lg L + b 3 lga lg L + b 4 lga lg L + b t - 1 lg a lg L + b t lg a - b( b - 1) lga b 2 ( b - 1) lga b 3 ( b - 1) lga b t - 1 ( b - 1)lg a b t ( b - 1) lga - - b b b b, : ;,

137,,,, t 6.1, y^t = ae bt ln y^ t = ln a + bt, ln y, t ln yt, ln a b Excel : : ln yt D2 : = LN ( C2 ), D13 6.8 : t ln y t, : 6.13

138 :.14 6.9 M ultiple R R Square Adjusted R Square 0.998627 0.997256 0.996981 0.053461 12 6.10 df SS MS F Significance F 1 10.38541 10.38541 3633.75 3.84E - 14 10 0.02858 0.002858 11 10.41399 6.11 Coefficie nts t Sta t P - value Lower 95 % Upper 95 % 95.0 % 95.0 % Int ercept 6.193556 0.0329031 88.2382 4.4 E - 19 6.120245 6.266868 6.120245 6.266868 XVariable1 0.269491 0.0044716 0.18059 3.84 E - 14 0.25953 0.279452 0.25953 0.279452 : ln a = 6.193556, b= 0.269491 :

139 y t = 489.5842 0.2 694 91t e 0.2 6949 1t y^ t = 489.58422.718282 :, ln yt E, F2 : = EXP ( E2 ),, F13, yt F14 : = 489.5842 * ( 2.718282 )^(0.269491 * 13 ), 2003 y^2 003 = 16267.75 : 6.12 6.15,,, :

140 6.13 y = a + bx y = a + b/ x y = L + ab x y = L + ax b y = 1/ (1/ L + ab x ) y = a + bx y = a + b( 1/ x) lg( y - L) = lg a + lgbx lg( y - L) = lg a + blg x lg (1/ y - 1/ L) = lg a + lgbx ( ),, (), () 1. y^t = L + ae bt ( a< 0, b< 0 ),,,, y0, y1, yn - 1,, y3 ( n - 1 ) 3 n,, yi, yn - 1 y3 n - 1 y0 = L + ae 0 y1 = L + ae b = L + ae ( n - 1 ) b = L + ae ( 3 n - 1 ) b, n,, n - 1 1 y = yi = nl + a( e0 + i = 0 e b + + e ( n - 1 ) b ) = nl + a enb - 1 e b - 1

2 n - 1 y = y i = nl + aenb i = n e nb - 1 ( e 0 + e b + e ( n - 1 ) b ) = nl + ae nb e b - 1 3 n - 1 3 y = y i = 2 n i = nl + nb ae2 ( e 0 + e b + e ( n - 1 ) b 2 nb enb - 1 ) = nl + ae e b - 1 nb ( e nb - 1) 2 ae 3 y - 2 y 2 y - 1 y = ( e b - 1 ) a ( ebn - 1) 2 ( e b - 1 ) 2 y - 1 y = a ( enb - 1 ) 2 ( e b - 1 ) a = (2 y - 1 y) ( eb - 1 ) ( e nb - 1 ) 2 b= ln n 3 y - 2 y 2 y - 1 y L = 1-1 (1 y - aenb n e b - 1 ) = 1 n [ 1 y3 y - (2 y) 2 1 y + 3 y - 22 y ], 3,,,,,, y^t = L + ab t ( a< 0, 0 < b < 1) 2. a = ( 2 y - 1 y) b - 1 ( b n - 1) 2 b= n 3 y - 2 y 2 y - 1 y L = 1 n [ 1 y - a( bn - 1 b - 1 ) ] y^t = La b t 141

142 ln yt = ln L + ( ln a) b t, 3. ln a = (2 ln y - 1 ln y) b - 1 ( b n - 1) 2 b= n 3 ln y - 2 ln y 2 ln y - 1 ln y ln L = 1 n [1 ln y - ln a( bn - 1 b - 1 ) ] ^ y t = 1 L + ab t 1 ^ = L + abt yt, a = (2 b= n 3 2 4. L ^ yt = 1 + ae - bt 1 ^ = 1 L + a L e - bt yt : 1 b c = e- ( 1 - e - nb ) 1 - e - b 1 y - 1 1 y ) b - 1 ( b n - 1 ) 2 1 y - 1 2 y 1 y - 1 1 y L = 1 n [1 1 y - a( bn - 1 b - 1 ) ] 1 y, 1 2 y, 1 3 y

143 1 y = 1 ( n + ac) ( t = 1, t = 0 ) L 2 3 1 2 1 y = 1 L ( n + ace- nb ), 1 y = 1 L ( n + ace- 2 nb ) 1 y - 2 1 y = ac L ( 1 - e- nb ) 1 y - 3 1 y = ac L e - nb (1 - e - nb ) 2 e - nb = 1 1 y - 1 3 y 1 y - 1 2 y b = 1 n [ln(1 1 y - 2 1 y ) - ln (2 1 y - 3 1 y ) ] 1 a = L (1 c y - 1 2 y ) 2 1 1 y + 1 3 y - 1 22 y L = n 1 1 1 y + 1 3 y - 1 22 y 1 1 y 3 y - 1 (2 y ) 2 :, 6.14 : : I10 = P OWE R ( ( E14 - E8 )/ ( E8 - E2), 1/ 6 ), b; I9 = ( E8 - E2 ) * ( ( I10-1)/ ( I10^6-1)^2 ), a; I9 = 1/ 6 * ( E2 - I9 * ( ( I10^6 ) - 1 )/ (I10-1) ), L : y^t = 1 0.0000937061 + 0.0001165570.965171532 t : F 2 : 1/ ( 0.0 0 0 0 9 3 7 0 6 1 + 0.0 0 0 1 1 6 5 5 7 * (0.965171532 )^ B2 ), F21, 2003 2004, 6.14

144 6.14,,, ;,,,,,, ( ) 1.,,, 6.15,,,

145 A B C yt = ae bt yt = b0 + b1 t + b2 t 2 y^ t = b0 + b1 t + b2 t 2 + b3 t 3, y^t 0.12 077 8t = 93.88849e y^t = 182.9483-25.1507 t + 3.556453 t 2 y^t = 160.016-15.1493 t + 2.57644 t 2 + 0.026134 t 3 6.15 6.15,, B C,

146.16 2.,, : 6.16 A B C 4.96 1.11318E - 13-6.15804E - 14 55.69197 59.611699 57.51389 7.780241 14.40331 12.7175 7208.743 5552.718 5507.231 86.73042 76.11925 75.80683, C,,, ( ),,, : ;, :

147,, ;,,,,, ;,,,,,,,,,,,,, 6.17, 6.17,,, A B C,, y^t 0.13 628 5t = 173416.6e yt = 327697.2-51132.6 t + 8505.003 t 2 y^t = 269234.1-25635.4 t + 6006.579 t 2 + 66.62464 t 3

148.17 ( ) ( y) ( ) ( y) 1978 207222 1990 961640 1979 226592 1991 1061667 1980 264779 1992 1282631 1981 291566 1993 1561087 1982 311925 1994 1876878 1983 344704 1995 2299106 1984 402855 1996 2595144 1985 518104 1997 2785205 1986 619019 1998 3037155 1987 721103 1999 3315515 1988 927708 2000 3627022 1989 1006533 2001 3954305 6.17

149,,,, 6.18,,,,,,, 6.18 6.18 A B C 7.0888 8.912377 6.685767 32784099801 9135759662 8840122771

150, 6.19: 6.19 ( ) ( ) 2002 25 4365009 4423470.975 2003 26 4747632 4834155.777 2004 27 5147264 5267247.18 2005 28 5563907 5723144.933,,,,,,,,, :

151 6.20 F t + m = a t + b t m ^t, t y = a + bt, t, m F t + m = a t + b t m + 1 2 c t m 2 ^t y = b0 + b1 t + b2 t 2, t, t, m,,,,,,,,

,,,,,,,, (),, 12, 4, ;, ;,,,, :, ( S ) ; (T ) ;,, (C ),, ( I ) : Y = TCI S, ; X - 11 X - 11

153 y^t = yf i y ; f i, 35 ( ) 1991 2001, 7.1 : (), : 7.1

154 ( ) :,, B13 : = AVERAGE( B2: B12),, M13 N2 : = A VE RAGE ( A2: M2 ),, N13, 112 (),, 12 1200%,,, 1200, : f i = ( ) ( ) B14 : = B13/ 92.40909 100,, M14 B15 : = B14 * 1200/ 1200.002,, M15 7.2

155.2 : 7.2 ( ) 2001 135.583, : y^i = ( 135.583) f i B16 : = 135.583 * B15/ 100,, M162002 7.3 26.143 33.479 86.165 138.851 285.4381 381.207 335.99 166.995 80.029 38.1473 26.676 27.8769 y^t = ( a + bt) f i : ( a + bt) ; f i,

156 997 1 2001 12 : 7.4 ( ) 7.3,,,, 7.3 : (), (1 ), Ft = a + bt,,, 12 ; EXCEL :

157.5 Coefficie nts t St a t P - value Lower 95 % U pper 95 % 95.0 % 95.0 % Intercept 190.1555361 5.223376 36.40472 1.2E -41 179.6998 200.6113 179.6998 200.6113 XVariable1 3.577468782 0.148926 24.02185 7.88 E-32 3.279362 3.875576 3.279362 3.875576 7.6 M ultiple R R Square Adjusted R Square 0.95324 0.90867 0.90709 19.9777 60 7.7 df SS MS F Significance F 1 230305.1 230305.1 577.05 7.87805E - 32 58 23148.27 399.108 59 253453.4 : Ft = 190.1555 + 3.5774 t, 7.8 7.8 (2 ), ; E2 : = C2/ D2,, E61

158 7.9 (3 ),, 1200 %,, f i = Si = yi F i, Si + Si + T + Si + 2 T + + Si + ( m - 1 ) T m m, T,,, : 7.10 () : y^t = ( 190.1555 + 3.5774 t)f i

159,, C2 : = 190.1555 + 3.5774 * B2,, C13,, E2 : = C2 * D2,, E137.11 : 7.11 yt + m = ( at + bt m)f t + m at, bt f t + m (1) y1 (2 ): yj b0 = yj - y1 T - 12 T (3 ) t = 0, a0 : (4 ) t a0 = y1-6.5 b0 at = a0 + b0 t

160 5 ) f t f t = yt (6 ) at ( t = 1, 2,, T) f = ^1 f = ^2 1 J ( f 1 + f 13 + + f T - 1 1 ) 1 J ( f 2 + f 14 + + f T - 1 0 ) f = ^12 1 J ( f12 + f24 + + f T ) (7 ),,, at = yt bt ^t f + (1 - ) ( at - 1 + bt - 1 ) = ( at - at - 1 ) + ( 1 - ) bt - 1 f t + 12 = yt at + (1 - ) ^f 1997 1 2001 12 7.12 1997 203.8 214.1 229.9 223.7 220.7 198.4 207.8 228.5 206.5 226.8 247.8 295.5 1998 240.3 222.8 243.1 222.2 220.6 218.7 234.5 248.6 261 275.3 269.4 291.2 1999 301.9 285.5 286.6 260.5 298.5 291.8 267.3 277.9 303.5 313.3 327.6 338.3 2000 340.37 318.51 336.85 326.64 342.9 337.53 320.9 332.17 344.01 355.79 350.67 367.37 2001 354.39 359.41 390.93 363.78 342.92 380.94 379.92 398.01 404.99 413.77 419.57 469.86 - (1 ) y1 - y1 - = 225.2917, yj (2 )b0 = = 389.8742, J = 5 - yj - y1 T - 12 - (3 ) t = 0, a0 - yj = 389.8742-225.2917 60-12 a0 = y1-6.5 b0 = 225.2917-6.53.428802 = 203.0045 = 3.428802 ;

161 4 ) t at = a0 + b0 t = 203.00 45 + 3.428 802 t, 7.13 7.1 4 D3 : = 203.0045 + 3.428802 * B3,, D62 (5 ) f t E3 : = C3/ D3,, E62 (6 ) F3 : = ( E3 + E15 + E27 + E39 + E51)/ 5,, F14 (7 ) = 0.1,= 0.2,= 0.3 at, bt a0, b0 ; G3 : = 0.1 * (C3/ F3) + 0.9 * ( G2 + H2),, H3 : = 0.2 * (G3 - G2) + 0.8 * H2, G14, H14 G15 : = 0.1 * ( C15/ F3) + 0.9 * (G14 + H14),, G26, H26 G27 : = 0.1 * ( C27/ F3 ) + 0.9 * ( G26 + H26 ),, G38, H38 G39 : = 0.1 * ( C39/ F3 ) + 0.9 * ( G38 + H38 ),, G50, H50 G51 : = 0.1 * ( C51/ F3 ) + 0.9 * ( G50 + H50 ),, G62, H62 f t + 12 I51 : = 0.3 * ( C51/ G51 ) + 0.7 * F3,, I6212, 7.13

162 7.14 t y a at bt f ( t + 12 ) f ( t + 12 ) 0 203.0045 203.0045 3.428802 1997.01 1 203.8 206.4333 0.987244 0.998099063 206.2088 3.383899 2 2 214.1 209.8621 1.020194 0.960103829 210.9331 3.6519797 3 3 229.9 213.2909 1.077871 1.007568598 215.9439 3.9237392 4 4 223.7 216.7197 1.032209 0.93588291 221.7834 4.3068996 5 5 220.7 220.1485 1.002505 0.944613414 226.8453 4.4579044 6 6 198.4 223.5773 0.887389 0.924321826 229.6373 4.1247167 7 7 207.8 227.0061 0.915394 0.908003123 233.2712 4.0265532 8 8 228.5 230.4349 0.991603 0.948374301 237.6618 4.0993707 9 9 206.5 233.8637 0.882993 0.954249937 239.2251 3.592153 10 10 226.8 237.2925 0.955782 0.988151063 241.4875 3.3261988 11 11 247.8 240.7213 1.029406 0.99950753 245.1245 3.3883668 12 12 295.5 244.1501 1.210321 1.083261236 250.9404 3.8738563 1998.01 13 240.3 247.5789 0.9706 253.4086 3.5927255 2 14 222.8 251.0077 0.887622 254.507 3.0938646 3 15 243.1 254.4365 0.955445 255.9681 2.7673258 4 16 222.2 257.8653 0.86169 256.6042 2.3410735 5 17 220.6 261.2941 0.844259 256.4042 1.8328617 6 18 218.7 264.7229 0.826147 256.074 1.4002379 7 19 234.5 268.1517 0.874505 257.5527 1.4159342 8 20 248.6 271.5805 0.915382 259.285 1.4792176 9 21 261 275.0093 0.949059 262.0392 1.7341972 10 22 275.3 278.4381 0.988729 265.2561 2.0307525 11 23 269.4 281.8669 0.95577 267.5115 2.0756694 12 24 291.2 285.2957 1.020695 269.5102 2.0602847 1999.01 25 301.9 288.7246 1.045633 274.6609 2.6783744 2 26 285.5 292.1534 0.977227 279.3418 3.0788614 3 27 286.6 295.5822 0.969612 282.6233 3.1193919 4 28 260.5 299.011 0.871206 285.0031 2.9714744 5 29 298.5 302.4398 0.986973 290.7773 3.5320292 6 30 291.8 305.8686 0.954005 296.4475 3.9596601 7 31 267.3 309.2974 0.864217 299.8047 3.8391615 8 32 277.9 312.7262 0.888637 302.5822 3.6268399 9 33 303.5 316.155 0.959972 307.3932 3.8636753 10 34 313.3 319.5838 0.980338 311.8369 3.9796724 11 35 327.6 323.0126 1.014202 317.0111 4.2185692 12 36 338.3 326.4414 1.036327 320.3364 4.0399306 2000.01 37 340.37 329.8702 1.03183 326.0406 4.3727682 2 38 318.51 333.299 0.955629 330.5465 4.3994093 3 39 336.85 336.7278 1.000363 334.8833 4.3868841 4 40 326.64 340.1566 0.960264 340.245 4.5818409 5 41 342.9 343.5854 0.998005 346.6447 4.9454172 6 42 337.53 347.0142 0.972669 352.9476 5.2169152 7 43 320.9 350.443 0.915698 357.6894 5.1218822 8 44 332.17 353.8718 0.938673 361.5553 4.8706977 9 45 344.01 357.3006 0.962803 365.8337 4.752238 10 46 355.79 360.7294 0.986307 369.533 4.5416447

163 t y a at bt f t + 12 ) f ( t + 12 ) 11 47 350.67 364.1582 0.962961 371.7515 4.0770079 12 48 367.37 367.587 0.99941 372.1589 3.3431054 2001.01 49 354.39 371.0158 0.955188 373.4583 2.93436380.9833521.008415 2 50 359.41 374.4446 0.959848 376.1879 2.89340830.9586930.983128 3 51 390.93 377.8734 1.034553 379.9725 3.07165 1.0139491.039792 4 52 363.78 381.3022 0.954064 383.61 3.1848155 0.93961 0.963559 5 53 342.92 384.731 0.891324 384.418 2.70945530.9288440.952518 6 54 380.94 388.1598 0.9814 389.6277 3.20948920.9403360.964303 7 55 379.92 391.5886 0.970202 395.3947 3.72099950.9238610.947408 8 56 398.01 395.0174 1.007576 401.1717 4.13220710.9614980.986004 9 57 404.99 398.4462 1.016423 407.2142 4.51426020.9663360.990966 10 58 413.77 401.875 1.029599 412.4288 4.65432090.9926811.017982 11 59 419.57 405.3038 1.035199 417.3525 4.70819361.0012491.026769 12 60 469.86 408.7326 1.149553 423.2292 4.941896 1.0913361.119152 11.70175 : y6 0 = ( 423.2292 + 4.941896 m) f6 0 + m m = 1,212, 2002 7.15 7.15 y^t = a + bt + di : ( a + bt) ; di (),

164 ( ), : y1 yj ; - y1 : b= yj t - 12 : t : a = y1-6.5b : Ft = a + bt Ft () di dt dt = yt - Ft () di di = : i = 1, 2, 12 1, 2,, 4; T ; di + di + T + + di + ( m - 1 ) T m m di,, 1997 1 2001 12 : 7.16 1997 1 2001 12 1997 203.8 214.1 229.9 223.7 220.7 198.4 207.8 228.5 206.5 226.8 247.8 295.5 1998 240.3 222.8 243.1 222.2 220.6 218.7 234.5 248.6 261 275.3 269.4 291.2 1999 301.9 285.5 286.6 260.5 298.5 291.8 267.3 277.9 303.5 313.3 327.6 338.3 2000 340.37 318.51 336.85 326.64 342.93 37.53 320.9 332.17 344.01 355.79 350.67 367.37 2001 354.39 359.41 390.93 363.78 342.92 380.94 379.92 398.01 404.99 413.77 419.57 469.86

165 ),, : Ft = 190.1555 + 3.5774 t, 7.17 7.17 () di E2 : = C2 - D2,, E61 di F2 : = ( E2 + E14 + E26 + E38 + E50 )/ 5,, F137.18 7.18 : Ft = 190.1555 + 3.5774 t + di () 7.19,,, C2 : = 190.1555 + 3.5774 * B2,, C13, E2 : = C2 + D2,

166 E13 : 7.19 X - 11 X - 11 1954 ( Bureau of Census Department of Commerce) (NBE R ) ( The Ratio - Moving Average Method),,,, X 1960, X - 3, X - 3, 1961, X - 10 1965 10 X - 11,, X - 11,,,,,,,, X - 11,, X - 11, ( IM F ),

X - 11 167 X - 11, : ( TC) ; ( S ) ; ( I) ; (D),,, : Dp: ; Dr: ; P : ; E : ; I: : T = T CSID T = T CSID T = T CSID T = T CSI : D= Dp Dr, D= Dr, I= PEI, I= EI : T = T C+ S + I+ D T = T C+ S + I+ D T = T C+ S + I+ D T = T C+ S + I : I = P + E + I, I= E + I : Y = T CSI, ( I= EI) : Y = T C+ S + I, ( I= E + I)

168 X 11 ( ) X - 11 TC S, 1. : { xt, t = 1, 2, n}, y L - 1 2 = 1 L - 1 + t L xt + i t = 1, 2,, N - L + 1 i = 0 L, M AL, yt x t y L - 1 = M AL ( xt ) + t 2, x t, xt, xt + 1,, x t + L - 1 L, y L - 1 2 + t t, t + 1,, t + L - 1, 2.,, ( 12, 4 ), M A2 m y 2 m - 1 2 + t, y S - 1 1 + t 2 t = 1, 2,, N - S1 + 1 { xt }, t = 1, 2, n M As1 : Z S2-1 2 + S 1-1 + t 2 = M As 2 y S1-1 + t 2, = M As 2 ( M As 1 ( xt ) ) t = 1, 2,, N - S1 - S2 + 2 2, M As s 2 1 3 4, M A, M As 2 s 1, Z S2-1 2, S1, S2 + S1-1 2 + t M A M A2 m + 1 3. X - 11 M A M A212, TC, TCI

X - 11 169 M A 3, St, St 4.H enderson ( MA H ) TCI, TC H enderson 9,13, 23,, X - 11, TCI, MAH9, MA H13 MAH23, : (1 ) T C, MA H13 (2 ) I, TCI/ T C (3 ) I/ - TC -, I, - TC - I TC ( ),, MAH 7.20 MAH I/ T - C - MA H 0.00-0.99 1.00-3.49 4.00 MAH9 MA H13 MA H23 ( ), S S2, y s - 1 2 S 2 N - S 2 + t 1 + 1 N, ( ),,,

( ), Y ( t) t ti, Y ( ti ) Y ( t) t = ti,,, Y, Y () tk, t > tk tk, tk,,,,,,,, ( ),,, () t, () Y ( t) t = ti Y ( ti )

171 t ti,,, ;,,,, : ;,,,,,, ( ),,,,, () AB, P( B A) A, B A, P( B A) A, B A B, AB =, P( B A) A B () Pij n,, n ( ),,,, A B n,, n, Pi1, Pi2, Pii, Pin () n, nn, P1 1 P12 P1 n P = P2 1 P22 P2 n Pn1 Pn2 Pnn

172 () n Pij Pij = 1 j = 1 1, 2,, n n P, k P ( k ), P ( k) = P ( k - 1 ) P = PPP ( k ), 1000 (,,), 2002 1 500 400 100,2 500, 50,50 ; 400,20,80 ; 100,10,10, : 8.1, P11 = 400 500 = 0.8 ; P12 = 50 500 = 0.1,,, : 8.2, P ( 3 ) : = PPP

173 C 5: E17, C15 =, MMULT,, 8.1, : 8.2 Ctrl + Shift + Enter,, 8.3: 8.3

174 C 0: E22, C20 =, MMULT,, : 8.3 Ctrl + Shift + Enter,, 8.4: 8.4 8.5: 8.5

175, 55%, 21 %, 24 % ;,, 48%, 14%, 38% ;,, 58%, 21 %, 21% ( ) Y t t = k + 1 Y t t = k, t = k,,, Y t t = k + 1 Y t t = k t = k - 1, t = k - 1, (),,,, () 1.,,,,, 2., Ei ( i = 1, 2,, n),, Ei M i, Ei : Pi Fi = Mi N N = Mi n i = 1

176 :. Ei E j M ij, Pij = P( Ei Ej ) Fi j = Mij Mi : p11 p1 2 p1 n P = p21 p2 2 p2 n pn1 pn2 pnn,, 4.,,,, ;,, ( ) 20, 21 8.6 1.,,

177,,, 2. 8.4,,,,,,,,, 8.4 : 3.,,, < 60 ; 60 100 ; > 100 4.,, 8.7: 8.7 60 60 100 100 7 5 8 0.35 0.25 0.4

178 Mij., 6. 8.4, M11 = 3, M12 = 4, M13 = 0, M21 = 1, M22 = 1, M23 = 3, M31 = 2, M32 = 0, M33 = 5, P = 3 7 1 5 2 7 20, : 4 7 1 5 0 0 3 5 5 7 P31 = 2 7, P3 2 = 0, P33 = 5 7 P33 P3 1 P32, 21 100 21 ( ) : S ( k+ 1 ) = S k P k+ 1 P11 P12 P1 n P21 P22 P2 n = S ( 0 ) Pn1 Pn2 Pnn, S ( k) t = k; P ; S ( k+1 ) t = k + 1, S ( 0 ), (),,

179 ( ) (), 8 9, 1. : 8.8 8.9 9, B17 : = B14/ 1839,, E17 : 8.10 2. B21 : = B10/ 545,, E21; B22 : = B11/ 495,, E22; B23 : = B12/ 417,, E23; B24 : = B13/ 382,, E24 8.11:

180.11 3. : S ( k+ 1 ) = S ( k) P 10 B27: E27, B27 =, MMULT,, : 8.5 Ctrl + Shift + Enter,, 10 11 B28: E28, C28 =, MMULT,, :

181.6 Ctrl + Shift + Enter,, 11 12 B29: E29, C29 =, MMULT,, : 8.7 Ctrl + Shift + Enter,, 12 8.12:

182 8.12,,,,,,,,,,,,, ( ),,, n + 1, : S ( n+ 1 ) = S ( n),,,,,

183 ),,,, (),,,,,, (), S ( k+ 1 ) = S ( k), S ( k + 1 ) = S ( k) P = S ( k) S ( k) = ( x1, x2, xn ) x i : n i = 1 = 1, k, p11 p12 p1 n p = p21 p22 p2 n pn1 pn2 pnn S ( k+ 1 ) = S ( k) p = S ( k) : p11 x1 + p21 x2 + + pn1 xn = x1 p12 x1 + p22 x2 + + pn2 xn = x2 p1 n x1 + p2 n x2 + + pnn x n = xn x1 + x2 + + xn = 1 n, ( n + 1 ), n,, ( p11-1 ) x1 + p21 x2 + + pn1 xn = 0 p1 2 x1 + ( p22-1 ) x2 + + pn2 xn = 0 p1 ( n - 1 ) x1 + p2 ( n - 1 ) x2 + + ( pn( n - 1 ) - 1 ) xn = 0 x1 + x2 + + xn = 1

184 x1 x2 = xn p1 1-1 p21 pn1 p1 ( n - 1 ) p2 ( n - 1 ) pn( n - 1 ) - 1 1 1 1-1 0 0 1,, Excel : k S ( k) = ( x1, x2, xn ) (1 ) (2 ), 8.13 B9: E12, B9 =, T RANSPOSE, 8.8, 8.8, :

185.9 Ctrl + Shift + Enter,, (3 ), 1, 8.13 (4 ) 8.14 B21: E24, B21 =, MIN VE RSE8.10

186.10, : 8.11 Ctrl + Shift + Enter,, (5 ), 8.14:

187 8.14, 27.24%, 33.30%, 21.83%, 17.63% :,,,,,,,,

- -, B - J,, : ( Autoregressive), A R ; ( Moving Average), MA ; ( Autoregressive Moving Average), AR MA -, 20 30 (G.U.Yule) ( H.Wold) ( G.U.Yule) 1926 ( AR ), ( Walker) 1934, 1937 ( Slutzky ) ( MA ), 1938 ( H.Wold) ARMA A RMA,, ( ARMA), ARMA,,, ( G.E.P.Box) ( G.M.Jenkins ),1970,,,, B - J ( ) - -, ARMA

188 -. { x( t), t T},, T, t, x( t), t T,, t T = {- 2, - 1,0,1,2, }, x( t),, 2.,,, { xt } : Ft 1, t 2 t n ( x1, x2, xn ) = F t1 + r, t2 + r t n+ r ( x1, x2, xn ), n, r, t1, tn, { x1 }, ( xt 1, xt n ) ( xt xt 1+ r n+ r ), 3.,,,,,,,, ( x t 1, xt n ) ( xt xt 1+ r n+ r ),,,, m : n, r, t1, t2, tn, ( xt 1, x t n ) m, E[ x m t 1 1 xt m 2 2 x t m n n ] = E[ xt m 1 1 + r x m t 2 2 + r x t m n n+ r ], m1, mn, m1 + m2 + + mn m, { x1 } m { x1 }, : t,, Ex t = ; E[ x 2 t ] = 2, 2 t, 2 < + ; t, s, E[ x t, xs ] t - s :,, t, s

189 t, s,,,,,,,, ( ),, ( ) - - :,,,,,,,,, B - J (),,, : y = b0 + b1 x1 + b2 x2 + + bk x k + e, y, x1 xk, b1 bk, b0, e,,, yt yt- 1, yt- 2, yt- k,, yt = b0 + b1 y t- 1 + b2 yt - 2 + + bk y t- k + et,,,,,,,ar, yt = 1 yt- 1 + 2 yt- 2 + + p y t - p + et

190 - yt 1 yt- 1 + 2 yt- 2 + + p y t - p + et, et, et, (et et - 1, et- 2 ), et y k ( k < t) ( et yt - 1, yt - p ), yt - 1, y t- p, 1 yt- 1 + 2 yt - 2 + + p y t- p ; yt - 1, yt - p, et,, et, AR, et, et, et- 1, et- 2 et- q, - 1 et- 1-2 et - 2 - - qet- q,, et,, et e t,, : e t = et - 1 et- 1-2 et- 2 - - q et- q e t ( ), : yt = 1 yt- 1 + 2 yt- 2 + + p y t - p = 1 y t- 1 + 2 y t- 2 + + p y t- p yt = 1 yt- 1 + 2 yt- 2 + + p y t- p + e t - 1 et - 1-2 et - 2 - - q et - q - 1 et- 1-2 et- 2 - - q et- q, p q, ARMA( p, q) p, q 1, 2 q, 1, 2 p : et, q = 0,, AR ( p), AR MA( p, 0) ; AR ( p) y1 = 1 yt- 1 + 2 yt- 2 + + p y t - p p = 0,, MA( q), A RMA( 0, q) MA( q) + et y t = et - 1 et- 1-2 et- 2 - - q et- q ARMA( p, q) Y t AR ( p) MA( q) ( ) + et + et A RMA( p, q),, ARMA( p, q) B k k, By t = yt- 1

191 B y t = yt- 2 B k y t = yt- k B k e t = et- k B k C C( C ) ( B) = 1-1 B - 2 B 2 - - p B P ( B) = 1-1 B - 2 B 2 - - q B q AR M A ( p, q) B - J ( B) Y t = ( B) et 9.1 B - J

192 - -,,,, yt y t - 1, yt - 2,,, rk n t = k+ 1 = ( yt - y) ( yt - k - y) n ( yt - y) 2 t = 1 n, k, y,,,, - 11, - 1 rk 1, rk 1,, r1, r2, r3 9.1 t 1 2 3 4 5 6 7 8 9 10 yt 13 16 5 8 4 20 18 15 10 7 : :

193 9.2 ( t) y t y t- 1 y t- 2 y t- 3 1 13 2 16 13 3 5 16 13 4 8 5 16 13 5 4 8 5 16 6 20 4 8 5 7 18 20 4 8 8 15 18 20 4 9 10 15 18 20 10 7 10 15 18, y = 1 1 0 t = 1 10 t = 2 r1 r2 1 0 10 t = 1 y i = 11. 6 ( yt - y) 2 = 282.4 10 t = 3 10 t = 4 ( yt - y) ( yt- 1 - y) ( y t t = 1 - y) 2 10 = 0.148 ( yt - y) ( yt- 2 - y) ( y t t = 1 - y) 2 10 = - 0.181 ( yt - y) ( yt- 3 - y) r3 10 = - 0.658 ( y t - y) 2 t = 1, ; ( ),,, k,, k

194 -, 0, = 1 n ( ), F( t), ( - t, t ),, ( - t, + t) 95% ( - 1. 96, 1. 96 ), yt, yt- 1, yt- 2 yt- k+ 1, yt k yt- k yt- 1, yt- 2 yt- k+ 1, yt yt- k kk,, - 1 kk 1 : kk = r1 k = 1, r k kk k - 1 - j = 1 k - 1, j rk - j k - 1 1 - k - 1, j rj j = 1 k, j = k - 1, j k = 2, 3,, - kk k - 1, k - j k = 2, 3, j = 1, 2, k - 1, r1 = 0.148, r2 = - 0.181, r3 = - 0.658, : 11 = r1 = 0.418 22 = r2 - ( 11 r1 ) 1 - ( 1 1 r1 ) = - 0.208 21 = 11-2 2 11 = 0.178784 33 = r3 - ( 21 r2 ) - ( 2 2 r1 ) 1 - ( 1 1 r1 ) - ( 22 r2 ) 31 = 21-3 3 22 = 0.046704 32 = 22-3 3 21 = - 0.09447 = - 0.635,,,,

195,, E - Views : 1. ( 1952 2000 ) 9.3 2. View Correlogram, : 9.2 3.,,OK,, 9.3

196 -.3 AC, PAC,, -,,, ( ),,, (),

197 ;, 9.4, 9.4 ( ) B - J B - J ( ), (),,, (), k = 2(k = 3 ),, 0, ;, 9.5, k = 1,,, 0, 9.6,

198 -.5 9.6 ( ) B - J B - J, ARMA, B - J (),, ARMA, AR MA, B - J

199 ARMA,, yt yt = yt - yt- 1 ( t > 1 ) (y t ) = 2 yt = ( yt - yt- 1 ) = yt - yt- 1 2 yt = ( yt - yt- 1 ) - ( yt- 1 - yt- 2 ) = yt - 2 yt- 1 + yt- 2 ( t > 2), Y t, d, Zt, : zt = d y t ( t < d),, 9.3 ;, 9.6;, 9.5 k = 1,,,, ( ) ( ), (),, k = 12, 24, 36, 48 0,,, 0,,, 9.7

200 -.7, k = 12, 24 0, ( ),,, 12, yt 1 2 yt = y t - yt- 12 ( t > 12 ) 2 12 2 12 ( 1 2 y t ) = 12 y t = ( yt - yt- 12 ) = y t - yt- 12 yt = ( yt - yt- 12 ) - ( yt- 12 - yt- 24 ) = yt - 2 yt- 12 + yt- 24 ( t > 24), Y t, D, W t, : wt = s D y t ( t > Ds ), s,,

ARMA 201.8 9.8 ARMA,,, -,,, ARMA ARMA ( ) MA( q) MA( q) yt = et - 1 et - 1-2 et - 2 - - qe t- q Y t = ( B) et MA( q) rk = - k + 1 k+ 1 + + q - k q 1 + 2 1 + 2 2 + + 2 q 1 k q 0 k > q, MA( q) y t, yt y t- k, k > q

202 -, rq 0, -,, MA( q) : 9.9 MA(1) MA( q),, k,, 0 MA( q) 9.119.12 9.10 MA (2)

ARMA 203.11 MA(q) 9.12 MA( q) ( ) AR ( p) AR ( p) AR ( p) rk yt = 1 yt- 1 + 2 yt- 2 + + p y t - p ( B) yt = et : B ( rk ) = 0 k > 0, AR( p),, + et 9.139.14

204 -.13 AR ( p) 9.14 AR ( p) AR ( p) : kj = j 1 j p 0 p + 1 j k AR ( p) kk p, 0 k p k k = 0 k > p,, A R( p), kj j, AR ( p), kj k j, k k 9.15, 9.16 AR ( p)

ARMA 205.15 AR(1) 9.16 A R(2 ) A R( p),, -,, ( ) ARM A( p, q) ARMA( p, q), AR MA( p, q) AR MA, ARMA(1,1 )

206 -.17 A RMA( 1,1 ), ARMA ( p, q), ARMA( p, q), 9.4 ARMA ( p, q) AR( p) MA ( q) ARMA( p, q) ( B) y t = e t y t = ( B) e t ( B) y t = ( B) e t () ( p) ( q) ( ) ( ) ( ) ARMA( p, q),, AR MA( p, q),,, A RMA( p, q),,

ARMA 207 ) ARIMA( p, d, q) ARIMA( p, d, q),, d, ARMA( p, q), ARMA( p, q), ARIMA ( p, d, q) yt, d zt zt = d y t t > d zt ARMA( p, q), yt A RMA d, ARIMA( p, d, q),, d, p q z t B, yt = yt - yt- 1 = yt - By t = (1 - B) yt 2 yt = y t - 2 yt- 1 + yt- 2 = yt - 2 By t + B 2 yt = ( 1 - B) 2 yt ARMA( p, q) zt = d y t = ( 1 - B) d y t ( B) yt = ( B) et ARIMA ( p, d, q) ( B) (1 - B) d y t = ( B) et ( B) d y t = ( B) et, ARIMA( 0, 1, 0) (1 - B) yt = et ARIMA(0, d, 0) ( 1 - B) d y t = et ARIMA(1,1,1 ) ( B) (1 - B) yt = ( B) et (1 - B) ( 1-1 B) yt = ( 1-1 B) et AR( 1) MA ( 1) yt () ARIMA( p, d, q) ( P, D, Q) s = ( 1 + 1 ) y t- 1-1 y t- 2 + et - 1 et- 1, ARMA,,, ARMA, ARIMA( p, d, q) ( P, D, Q) s ( P, D, Q) s, s, P D Q (1,1,2) 12

208-2, 12 12, 12 ARIMA( p, d, q) ( P, D, Q) s,4,, Y t 4 y t = yt - yt - 4 = ( 1 - B 4 ) yt 12, 1 2 yt = yt - yt - 12 = ( 1 - B 12 ) yt s D W t, wt = s D y t w t = ( 1 - B s ) D y t t > Ds 4, ARIMA( p, d, q) ( P, D, Q) s ARIMA(1,1,1 ) ( 1, 1, 1) 4 (1-1 B) ( 1-1 B) 4 ( 1 - B) (1 - B 4 ) y t = (1-1 B) ( 1-1 B 4 ) et AR( 1) AR (1 ) MA (1 ) MA( 1) : yt = ( 1 + 1 ) yt - 1 + ( 1 + 1 ) y t - 4 - (1 + 1 + 1 + 1 + ( 1 + 1 1 ) yt - 6-1 y t - 8 + ( 1 + 1-1 1 yt- 10 + et - 1 et - 1-1 et- 4 + 1 1 ) yt - 9 1 et- 5 1 ) yt - s, 1, 1, 1, 1,, ARIMA( p, d, q) ( P, D, Q) s p ( B) p ( B s ) ( 1 - B) d ( 1 - B s ) D y t = q ( B) Q ( B s ) et p ( B) p ( B s ) d D s y t = q ( B) Q ( B s ) et p ( B s ) = 1-1 B s - 2 B 2 s - - p B p s, P Q ( B s ) = 1-1 B s - 2 B 2 s - - p B Qs, Q p ( B), q ( B), p ( B s ), Q ( B s ) ARMA, ARMA,,, ARMA,

209 ARIMA( p, d, q) ( P, D, Q) s,, -, AR, MA, ARMA, ARIMA : :,, d, D, p, q, P, Q : 1, 2, p 1, 2 q 1, 2, p 1, 2 Q : et ), ( ) (, k = 2(k = 3),, 0, ;,, k = 12, 24, 36,48 0,,, 0, (),,, d, D d, d D,, D, d, D 0, 1 2, d,, D () ( ) p, q, p,, p q

210 -,, q,,, 0, 0, 0,,,, ARMA,,,,, : p,, ;,, p ;, p q,, 0 ;, k = q0, q = q0 P, Q ( ), p, q, k = 12(4),24 (8),, P,, Q0,, P, Q,, P, Q 0 12,, () ( )

211.AR AR ( p) yt yt- k, k = 1,2,, p yt yt- k = 1 yt - 1 + 2 yt - 2 + + p y t- p + e1 = 1 y t- 1 yt- k + 2 yt- 2 yt - k + + p y t - p yt - k + e1 yt - k E( yt yt- k ) = 1 E( yt- 1 y t- k ) + 2 E( yt- 2 y t- k ) + + p E ( yt - py t- k ) + E( e1 yt - k ) et, yt- k, E( yt- ket ) 0, k yt, : k = 1 k - 1 + 2 k - 2 + + p k - p 0, rk = 1 rk - 1 + 2 rk - 2 + + p rk - p k = 1, 2,, p, r1 r2 rp = 1 + 2 r1 + + p rp - 1 = 1 r1 + 2 + + p rp - 2 = 1 rp - 1 + 2 rp - 2 + + p Yule - Walker, 1 2 = p, AR( 1) 1 = r1 AR (2 ), 1 r1 rp - 1 r1 1 rp - 2 rp - 1 rp - 2 1 1 2 1 r1 r1 1 1 = 2.MA( q) MA( q) - 1 r1 r2 r1 (1 - r2 ) 1 - r1 2 = r2 - r2 1 1 - r 2 1-1 r1 r2 rp

212 - y t et - 1 et- 1-2 et- 2 - - q et- q t - k, y t- k = et- k - 1 et- k - 1-2 et- k - 2 - - q et - k - q, yt yt- k = ( et - 1 et- 1-2 et- 2 - - q et- q ) ( et- k - 1 et - k - 1-2 et- k - 2 - - q et - k - q ) E( y t yt - k ) = E ( et - 1 et - 1-2 et - 2 - - q et - q ) ( et - k - 1 et- k - 1-2 et - k - 2 - - q et- k - q ) k = 0,,, k = k, rk 0 = 1 + 2 1 + 2 2 + + = - k E( et- k et - k ) + 1 k+ 1 E( et- k - 1 et- k - 1 ) + 2 k+ 2 E( et - k - 2 et- k - 2 ) + + q - k q E( et- q et - q ) = ( - k + 1 k + 1 + 2 k+ 2 + + q - k q ) 2 e 2 q 2 e, MA( 1), q = 1, rk - k + 1 k+ 1 + 2 k+ 2 + + q - k q 1 + 2 1 + 2 2 + + 2 q r1 = - 1 1 + 2 1 r1 2 1 + 1 + r1 = 0, 1 = - 1 1-4 r2 1 2 r1 1 < 1, 1, MA( 1) MA( 2), q = 2, MA( 3), q = 3, r1 = - 1 + 1 2 1 + 2 1 + 2 2 r2 = - 2 1 + 2 1 + 2 2

r = - 1 + 1 2 + 2 3 1 + 2 1 + 2 2 + 2 3 r2 = r3 = - 2 + 1 3 1 + 2 1 + 2 2 + 2 3-3 1 + 2 1 + 2 2 + 2 3 MA,,,, (50),,, 0 3.AR MA AR MR,, ARM R,,, ( ), ARMA, AR,,, MA ARMA,,,,,,,,,,,, :, () E - Views ARMA,,,,,,, d = 2 213

214-9.18,,, p = 1, q = 1, AR MA( 1, 2, 1) 9.18 (1-1 B) 2 y t = (1-1 ) Bet, Genr,, : dy1 = y - y( - 1), dy2 = dy1 - dy1( - 1) dy1 dy2 9.19 9.19 QuickEstimate Equation,

215.20 9.5: 9.5 ARMA Dependen t Variable :DY2 Method: Least Squares Sample(adjusted ) :1955 2000 Included obse rvations : 46 afte r adjusting endpoints Convergence achieved after 7 iterations Backcast : 1954 Va riab le Coefficient Std. Error t - Statistic Prob. C 1.096814 2.024689 0.541720 0.5908 AR (1) - 0.519881 0.361456-1.438300 0.1576 MA( 1) 0.172750 0.423367 0.408039 0.6853 R - squared 0.141456 Mean dependent var 1.063913 Adjusted R - squared 0.101524 S.D. dependent var 18.80812 S.E. of regression 17.82783 A kaike info criterion 8.662392 Sum squared resid 13666.76 Sch warz criterion 8.781651 Log likelihood - 196.2350 F - statistic 3.542407 Durbin - Watson stat 1.950018 P rob( F - statistic) 0.037661 Inverted AR Roots -.52 Inverted MA Roots -.17 : 1 = - 0.519881, 1 = 0.172750

216-1 + 0.519881 B) 2 y t = (1-0.172750) Be1, et (),,, ( ) et ( ),, (),, et,,, : 0,, ;, 0,, 9.21 ViewResidual testscorrelogram - Q - Statistics, 9.22,,,

217.22 ( ) 1970, m N (0, 1 ) m 2 n, n - k t - 1 rk ( e) = et et+ k e2 t nrk ( e) N (0,1 ) K = 1, 2, m, 2 2 m Q = nr k ( e) k = 1 m = nr 2 k ( e) k = 1 n ( ) ; m 1 -, 2 m 2 x 2 a ( m), Q 2 a ( m) Q 2 a ( m),,, ; Q > 2 a ( m),,, 2

218 - m Q = nr 2 k ( e) = 46 0.11342 = 5.21732 k = 1 = 0.05, 45 2 61.656, Q < 2 a ( m),,,,, AR Y t, AR, : y^t (1 ) = ^1 yt + ^2 yt - 1 + + ^p y t - p + 1 y^t (2 ) = ^1 y^t (1 ) + ^2 yt + + ^ p y t- p + 2 ^ p - 1 y^t ( p) = ^1 y^t ( p - 1) + ^2 y^t ( p - 2 ) + + yt ( 1) + p y t y^t ( L) = ^1 y^ t ( L - 1 ) + ^2 y^t ( L - 2 ) + + ^ p - 1 yt ( L - p + 1) + p y t ( L - p) ( L > p), L MA y, MA, : y^t + 1 ( 1) y^t+ 1 (2 ) y^t+ 1 ( q) = ^ 1 0 0 0 y^t (1 ) ^2 0 1 0 y^t (2 ) - ^q - 1 0 0 1 y^t ( q) ^q 0 0 0 ^1 ^2 y t+ 1 ^q,t y^t ( q) = y^t (1) y (2) ^t yt+ 1, t + 1 y ^ ( q) t+ 1 = ^y t+ 1 ( 1 ) ^y t+ MA( 1) y (q) ^t T ^ t+ T 1 (2 ) y 1 ( q)

219 ^t ^ ^1 yt yt+ 1 ( 1) = ^1 y (1 ) - ^ t+ MA( 2) t+ 1 y^ ( 1) = y^ t+ 1 ( 2) ^t ^y t ^1 yt + 1 y 1 ( 1) = (1 ) - ^1 1 ^2 0 y + 1 (1 ) = ^1 y^ t ( 1) + y ^1 yt + 1 y^ t ( 1) y^ t ( 2) - ^ t + 1 (2 ) = ^2 t y^ ( 1) - ARMA ^1 yt + 1 ^2 (2 ) - ^2 y t+ 1 ^1 yt+ 1 ARMA MA,,,, d, wt, wt d, d = 1, yt L, y^ k ( L) = yk + w^ k (1 ) + w^ k ( 2) + + w^ k ( L) U L = y^ t ( L) + T V( L ) Se LL = y^t ( L) - T V ( L) Se, U L, LL, T k, Se, Se 2 = 1 = 1-1 2 = 1 1 + 2-2 3 = 1 2 + 2 1-3 t = 1 L - 1 V( L ) = 1 + 2 2 j 4 = 1 3 + 2 2 + 3 1-4 j = 1 k - p - q V ( L) 1 e^2 t

220 - j 1 j - 1 + 2 j - 2 + + p j - p, - j,, 9.23,,, : 9.24

221,, 9.6, d = 2, y^ k ( L) = yk + w^ k (1 ) + w^ k ( 2) + + w^ k ( L), dy1 (2001) = dy1( 2000 ) + d y2^(2001) = 51. 38 + 3. 21283 dy1 (2002) = dy1 (2000) + dy2^( 2001 ) + dy2^(2002) = 51. 38 + 3. 21283-0. 00326, : 9.7 dy2^ dy1^ ^y 2000 51.38 983.36 2001 3.21283 54.59283 1037.953 2002-0.00326 54.58957 1092.542 2003 1.668724 56.25829 1148.801 2004 0.799489 57.05778 1205.858 2005 1.251388 58.30917 1264.168

222-2003 3, : 993 1 9.8 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 1 977.5 1192.2 1602.2 1909.1 2288.5 2549.5 2662.1 2962.9 3332.8 3596.1 3907.4 2 892.5 1162.7 1491.5 1911.2 2213.5 2306.4 2538.4 2805 3047.1 3324.4 3706.4 3 942.3 1167.5 1533.3 1860.1 2130.9 2279.7 2403.1 2627 2876.1 3114.8 3494.8 4 941.3 1170.4 1548.7 1854.8 2100.5 2252.7 2356.8 2572 2820.9 3052.2 5 962.2 1213.7 1585.4 1898.3 2108.2 2265.2 2364 2637 2929.6 3202.1 6 1005.7 1281.1 1639.7 1996 2164.7 2326 2428.8 2645 2908.7 3158.8 7 963.8 1251.5 1623.6 1888.7 2102.5 2286.1 2380.3 2597 2851.4 3096.6 8 959.8 1286 1637.1 1916.4 2104.4 2314.6 2410.9 2636 2889.4 3143.7 9 1023.3 1396.2 1756 2083.5 2239.6 2443.1 2604.3 2854 3136.9 3422.4 10 1051.1 1444.1 1818 2148.3 2348 2536 2743.9 3029 3347.3 3666.9 11 1102 1553.8 1935.2 2290.1 2454.9 2652.2 2859.1 3108 3421.7 3733.1 12 1415.5 1932.2 2389.5 2848.6 2881.7 3131.4 3383 3680 4033.3 4404.4 : :, : 9.25 9.25, y,,

223.26, 9.27,, r = 12, 24, 26 0,, 9.28 9.27

224 -.28, (12 ), (9.29) ( 9.30 ) : 9.29

225.30,, 0, k = 1,,,, d = 1, D = 1,,,, 0,,, p = 1, q = 1, k = 12,24, 36,,, ( 1), (12) (24), P = 1, Q = 1 ARIMA (1, 1, 1 ) ( 1, 1, 1) 12, (1-1 B) ( 1-1 B 12 ) 12 yt = (1-1 B) (1-1 B 12 ) et :

226 - Genr, : dy1 = y - y( - 1), dy2 = dy1 - dy1( - 12 )dy1, dy2 9.31: 9.31 QuickEstimate Equation, 9.32 9.9 ARMA Dep enden t Variable: DY2, ncl uded obse rvations: 97 af ter adj usting endpoints Sample(adjusted) : 1995 :03 2003 :03, Method: Least Squares Failure to improve SSR after 23 iterations, Backcast : 1994:02 1995:02 Variable Coefficient Std. Error t - Statistic P rob. C A R(1 ) SAR(12) MA( 1) SMA( 12 ) 1.273489-0.424551 0.728945-0.107668-0.885821 2.969703 0.129294 0.044957 0.140784 0.000266 0.428827-3.283613 16.21411-0.764773-3329.224 0.6691 0.0015 0.0000 0.4464 0.0000

227 Va riab le Coefficien t Std Error t - Statistic P rob. R - squared Adjusted R - squared S.E. of regression Sum squared resid Log likelihood Durbin - Watson stat 0.384165 0.357389 46.44617 198466.7-507.3848 2.140794 Mean dependent var S.D. dependent var Akaike info c riterion Sch warz criterion F - statistic Prob( F - statistic) 0.527835 57.93966 10.56464 10.69735 14.34765 0.000000 Inverted AR Roots.97.84 +.49i.84 -.49i.49 +.84i.49 -.84i -.00 -.97i -.00 +.97i -.42 -.49 -.84i -.49 +.84i -.84 +.49i -.84 -.49i Inverted MA Roots -.97.99.86 +.49i.86 -.49i.49 +.86i.49 -.86i -.49 -.86i -.99.11 -.49 +.86 -.00 -.99i -.86 +.49i -.00 +.99i -.86 -.49i = - 0.885821 : 1 = - 0.424551, 1 = 0.728945, 1 = - 0.107668, 1 (1 + 0.424551 B) (1-0.728945 B 12 ) 12 y t = (1 + 0.107668 B) ( 1 + 0.885821 B 12 ) et 1. : 9.33

228 - ViewResidual testscorrelogram Q - Statistics, 9.14,,, 9.34 2. m Q = nr 2 k ( e) = 110 0.509368 = 56.03048 k = 1 = 0.05, 45 2 61.656, 95, Q < ( 2 m),,, :, (9.35), 9.35

229 9.36 :,, 9.36,,, ( 9.10) 9.10, d = 1, D = 1, y^ k ( L) = yk + w^ k (1 ) + w^ k ( 2) + + w^ k ( L), dy1 (03.04 ) = dy1 (02.04 ) + d y2^(03.04 ) = - 55.9345

230 - dy (03.05 ) = dy1 (02.05 ) + d y2^(03.05 ) = 144.8889 y( 03.04) = y(03.03 ) + d y1^(03.04 ) = 3438.865 y( 03.05) = y(03.03 ) + d y1^(03.04 ) + dy1^ (03.05 ) = 3583.754, 9.11 d y2^ d y1^ y^ 2003.04 2003.05 2003.06 2003.07 2003.08 2003.09 2003.01 2003.11 2003.12 6.665497-5.01108 9.973344 4.482638 2.335643-2.80724-33.337 6.079988 0.485146-55.9345 144.8889-33.3267-57.7174 49.43564 275.8928 241.1663 72.27999 671.7851 3438.865 3583.754 3550.428 3492.71 3542.146 3818.039 4059.205 4131.485 4803.27 2000 9.37,, 9.37

, ;,,,,,,,, ( ),,,,,,,,, : (1 ) ; (2 ),,,, ; (3 ); (4 ), : (1 ),,, ;

232 2 ),,,,,, ; (3 ),,,,,,,,,, : (1 ),,, ; (2 ); (3 ), ;,,,,,,,,,, ( ),,,,,,,,,,,,,,, 73%,

233 57%,,,,,,,,,,,,,,,, ( ),,, : 1.,,,,,,,,,,,,,,, 2.,,,,,,,,

234.,,,, 4., :,,,,,,, 5.,,,, 6.,,,,,,, ( ) 1.,,,,,,,,,,,,,

235.,,,, 3.,,,,,,,,,,,, 4.,,, ;,,,,,,,,,,,,,, 5. : ;,,,,,,

236 ) :,, ( ), d1, ( ), d2 ; ( ), d2, ( ), s2, m2, r,, : = d1 + s1 = d2 + s2 = r + m + s2 : = d1 + s1 i + d2 j + s2 + ( r + m) : i ; j,,,,,,, ( AR ),,,,,,

237, AR,, B - J f ( t),,,, B - J,,, D.W, B - J,, B - J : y = b0 + b1 x1 + + bm x m + et e1 = 1 et - 1 + + p et - p - 1 ut - 1 - - q ut - q,,,,,,,,,,,, + ut,,,,,,,,,,,,,, :

238 k yi = j = 1 wj y^j i ^ wj = n 1 i = 1 e 2 j i k n 1 j = 1 i = 1 e 2 ji, k k, w j j, ^yj i j, e 2 ji j, wj, y^ ai y^bi y i, w y^ ai, ( 1 - w)y^ bi,,, y^ i = w y^ ai + ( 1 + w) y^ bi w, y^i y^ ai y^bi eai = y i - y^ ai, ebi = yi - y^ bi,, E( eai ) = 0, E( ebi ) = 0, ei = y i - y^i = yi - [ w y^ai + ( 1 - w) y^bi ] = w( yi - y^ ai ) + (1 - w) ( yi - y^ bi ) = weai + (1 - w) ebi, E( ei ) = 0, Var ( ei ) = E( e 2 i ) - [ E( ei ) ] 2 = E( e 2 i ) = MS E, MS E Var ( ei ), Var ( ei ) = Var [ weai + (1 - w) ebi ] = w 2 V ar( eai ) + (1 - w) 2 Var ( ebi ) + 2 w(1 - w) Cov( eai, ebi ) Cov( eai, ebi )eai ebi w, 0, dvar( ei ) d w w = = 2 wvar( eai ) - 2(1 - w) Var( ebi ) + 2(1-2 w) Cov( eai, ebi ) = 0 : V ar( ebi ) - Cov( eai, ebi ) Var ( eai ) + V ar( ebi ) - 2 Cov( eai, ebi ) eai ebi,

w Var ( ebi ) Var ( eai ) + V ar( ebi ), w Var ( ei ), Var ( ei ) Var ( eai )Va r( ebi ),, Var ( eai ) = E( e 2 ai ) - [ E( eai ) ] 2 = E( e 2 ai ) = Var ( ebi ) = E( e 2 bi ) - [ E( ebi ) ] 2 = E( e 2 bi ) = w w = n i = 1 e2 ai n i = 1 e2 ai n + i = 1 e2 bi n i = 1 e2 ai n n i = 1 e2 bi,,,, 239 n

240 1 t P{ t( n) > ta ( n) } = a n a = 0.25 0.1 0.05 0.025 0.01 0.005 1 1.0000 3.0777 6.3138 12.7062 31.8207 63.6574 2 0.8165 1.8856 2.9200 4.3027 6.9646 9.9248 3 0.7649 1.6377 2.3534 3.1824 4.5407 5.8409 4 0.7407 1.5332 2.1318 2.7764 3.7469 4.6041 5 0.7267 1.4759 2.0150 2.5706 3.3649 4.0322 6 0.7176 1.4398 1.9432 2.4469 3.1427 3.7074 7 0.7111 1.4149 1.8946 2.3646 2.9980 3.4995 8 0.7064 1.3968 1.8595 2.3060 2.8965 3.3554 9 0.7027 1.3830 1.8331 2.2622 2.8214 3.2498 10 0.6998 1.3722 1.8125 2.2281 2.7638 3.1693 11 0.6974 1.3634 1.7959 2.2010 2.7181 3.1058 12 0.6955 1.3562 1.7823 2.1788 2.6810 3.0545 13 0.6938 1.3502 1.7709 2.1604 2.6503 3.0123 14 0.6924 1.3450 1.7613 2.1448 2.6245 2.9768 15 0.6912 1.3406 1.7531 2.1315 2.6025 2.9467 16 0.6901 1.3368 1.7459 2.1199 2.5835 2.9208 17 0.6892 1.3334 1.7396 2.1098 2.5669 2.8982 18 0.6884 1.3304 1.7341 2.1009 2.5524 2.8784 19 0.6876 1.3277 1.7291 2.0930 2.5395 2.8609 20 0.6870 1.3253 1.7247 2.0860 2.5280 2.8453 21 0.6864 1.3232 1.7207 2.0796 2.5177 2.8314 22 0.6858 1.3212 1.7171 2.0739 2.5083 2.8188 23 0.6853 1.3195 1.7139 2.0687 2.4999 2.8073 24 0.6848 1.3178 1.7109 2.0639 2.4922 2.7969 25 0.6844 1.3163 1.7081 2.0595 2.4851 2.7874

1 241 26 0.6840 1.3150 1.7056 2.0555 2.4786 2.7787 27 0.6837 1.3137 1.7033 2.0518 2.4727 2.7707 28 0.6834 1.3125 1.7011 2.0484 2.4671 2.7633 29 0.6830 1.3114 1.6991 2.0452 2.4620 2.7564 30 0.6828 1.3104 1.6973 2.0423 2.4573 2.7500 31 0.6825 1.3095 1.6955 2.0395 2.4528 2.7440 32 0.6822 1.3086 1.6939 2.0369 2.4487 2.7385 33 0.6820 1.3077 1.6924 2.0345 2.4445 2.7333 34 0.6818 1.3070 1.6909 2.0322 2.4411 2.7284 35 0.6816 1.3062 1.6896 2.0301 2.4377 2.7238 36 0.6814 1.3055 1.6883 2.0281 2.4345 2.7195 37 0.6812 1.3049 1.6871 2.0262 2.4314 2.7154 38 0.6810 1.3042 1.6860 2.0244 2.4286 2.7116 39 0.6808 1.3036 1.6849 2.0227 2.4258 2.7079 40 0.6807 1.3031 1.6839 2.0211 2.4233 2.7045 41 0.6805 1.3025 1.6829 2.0195 2.4208 2.7012 42 0.6804 1.3020 1.6820 2.0181 2.4185 2.6981 43 0.6802 1.3016 1.6811 2.0167 2.4263 2.6951 44 0.6801 1.3011 1.6802 2.0154 2.4141 2.6923 45 0.6800 1.3006 1.6794 2.0141 2.4121 2.6806

242 2 2 P{ 2 ( n) > 2 ( n)} = a n a = 0.995 0.99 0.975 0.95 0.90 0.75 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0.010 0.072 0.207 0.412 0.676 0.989 1.344 1.735 2.156 2.603 3.074 3.565 4.075 4.601 5.142 5.697 6.265 6.844 7.434 8.034 8.643 9.260 9.886 10.520 0.020 0.115 0.297 0.554 0.872 1.239 1.646 2.088 2.558 3.053 3.571 4.107 4.660 5.229 5.812 6.408 7.015 7.633 8.260 8.897 9.542 10.196 10.856 11.524 0.001 0.051 0.216 0.484 0.831 1.237 1.690 2.180 2.700 3.247 3.816 4.404 5.009 5.629 6.262 6.908 7.564 8.231 8.907 9.591 10.283 10.982 11.689 12.401 13.120 0.004 0.103 0.352 0.711 1.145 1.635 2.167 2.733 3.325 3.940 4.575 5.226 5.892 6.571 7.261 7.962 8.672 9.390 10.117 10.851 11.591 12.338 13.091 13.848 14.611 0.016 0.211 0.584 1.064 1.610 2.204 2.833 3.490 4.168 4.865 5.578 6.304 7.042 7.790 8.547 9.312 10.085 10.865 11.651 12.443 13.240 14.042 14.848 15.659 16.473 0.102 0.575 1.213 1.923 2.675 3.455 4.255 5.071 5.899 6.737 7.584 8.438 9.299 10.165 11.037 11.912 12.792 13.675 14.562 15.452 16.344 17.240 18.137 19.037 19.939

2 243 26 11.160 12.198 13.844 15.379 17.292 20.843 27 11.808 12.879 14.573 16.151 18.114 21.749 28 12.461 13.565 15.308 16.928 18.939 22.657 29 13.121 14.257 16.047 17.708 19.768 23.567 30 13.787 14.954 16.791 18.493 20.599 24.478 31 14.458 15.655 17.539 19.281 21.434 25.390 32 15.134 16.362 18.291 20.072 22.271 26.304 33 15.815 17.074 19.047 20.807 23.110 27.219 34 16.501 17.789 19.806 21.664 23.952 28.136 35 17.192 18.509 20.569 22.465 24.797 29.054 36 17.887 19.233 21.336 23.269 25.613 29.973 37 18.586 19.960 22.106 24.075 26.492 30.893 38 19.289 20.691 22.878 24.884 27.343 31.815 39 19.996 21.426 23.654 25.695 28.196 32.737 40 20.707 22.164 24.433 26.509 29.051 33.660 41 21.421 22.906 25.215 27.326 29.907 34.585 42 22.138 23.650 25.999 28.144 30.765 35.510 43 22.859 24.398 26.785 28.965 31.652 36.430 44 23.584 25.143 27.575 29.787 32.487 37.363 45 24.311 25.901 28.366 30.612 33.350 38.291

244 3 2 () n a = 0.25 0.10 0.05 0.025 0.01 0.005 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 1.323 2.773 4.108 5.385 6.626 7.841 9.037 10.219 11.389 12.549 13.701 14.845 15.984 17.117 18.245 19.369 20.489 21.605 22.718 23.828 24.935 26.039 27.141 28.241 29.339 2.706 4.605 6.251 7.779 9.236 10.645 12.017 13.362 14.684 15.987 17.275 18.549 19.812 21.064 22.307 23.542 24.769 25.989 27.204 28.412 29.615 30.813 32.007 33.196 34.382 3.841 5.991 7.815 9.488 11.071 12.592 14.067 15.507 16.919 18.307 19.675 21.026 22.362 23.685 24.996 26.296 27.587 28.869 30.144 31.410 32.671 33.924 35.172 36.415 37.652 5.024 7.378 9.348 11.143 12.833 14.449 16.013 17.535 19.023 20.483 21.920 23.337 24.736 26.119 27.488 28.845 30.191 31.526 32.852 34.170 35.479 36.781 38.076 39.364 40.646 6.635 9.210 11.345 13.277 15.086 16.812 18.475 20.090 21.666 23.209 24.725 26.217 27.688 29.141 30.578 32.000 33.409 34.805 36.191 37.566 38.932 40.289 41.638 42.980 44.314 7.879 10.597 12.838 14.860 16.750 18.548 20.278 21.955 23.589 25.188 26.757 28.299 29.819 31.319 32.801 34.267 35.718 37.156 38.582 39.997 41.401 42.796 44.181 45.559 46.928

3 245 26 30.435 35.563 38.885 41.923 45.642 48.290 27 31.528 36.741 40.113 43.194 46.963 49.645 28 32.620 37.916 41.337 44.461 48.278 50.993 29 33.711 39.087 42.557 45.722 49.588 52.336 30 34.800 40.256 43.773 46.979 50.892 53.672 31 35.887 41.422 44.985 48.232 52.191 55.003 32 36.973 42.585 46.194 49.480 53.486 56.328 33 38.053 43.745 47.400 50.725 54.776 57.648 34 39.141 44.903 48.602 51.966 56.061 58.964 35 40.223 46.059 49.802 53.203 57.342 60.275 36 41.304 47.212 50.998 54.437 58.619 61.581 37 42.383 48.363 52.192 55.668 59.892 62.883 38 43.462 49.513 53.384 56.896 61.162 64.181 39 44.539 50.660 54.572 58.120 62.428 65.476 40 45.616 51.805 55.758 59.342 63.691 66.766 41 46.692 52.949 53.942 60.561 64.950 68.053 42 47.766 54.090 58.124 61.777 66.206 69.336 43 48.840 55.230 59.304 62.990 67.459 70.616 44 49.913 56.369 60.481 64.201 68.710 71.893 45 50.985 57.505 61.656 65.410 69.957 73.166

246 4 F n2 P{ F( n1, n2 ) > Fa ( n1, n2 )} = a 1 %, 5% 1 2 1 2 3 4 5 6 7 8 9 10 11 12 161 200 216 225 230 234 237 239 241 242 243 244 1 4052 4999 5403 5625 5764 5859 5928 5981 6022 6056 6082 6106 18.5119.0019.1619.25 19.3019.3319.3619.37 19.3819.3919.4019.41 2 98.4999.0099.1799.25 99.3099.3399.3499.36 99.3899.4099.4199.42 10.13 9.55 9.28 9.12 9.01 8.94 8.88 8.84 8.81 8.78 8.76 8.74 3 34.1230.8229.4628.71 28.2427.9127.6727.49 27.3427.2327.1327.05 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5.96 5.93 5.91 4 21.2018.0116.6915.98 15.5215.2114.9814.80 14.6614.5414.4514.37 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.78 4.74 4.70 4.68 5 16.2613.2712.0611.39 10.9710.6710.4510.27 20.1510.05 9.96 9.89 5.99 2.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 4.06 4.03 4.00 6 13.7410.92 9.78 9.15 8.75 8.47 8.26 8.10 7.98 7.87 7.79 7.72 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 3.63 3.60 3.57 7 12.25 9.55 8.45 7.85 7.46 7.19 7.00 6.84 6.71 6.62 6.54 6.47 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 3.34 3.31 3.28 8 11.26 8.65 7.59 7.01 6.63 6.37 6.19 6.03 5.91 5.82 5.74 3.67 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18 3.13 3.10 3.07 9 10.56 8.02 6.99 6.42 6.06 5.80 5.62 5.47 5.35 5.26 5.18 5.11 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02 2.97 2.94 2.91 10 10.04 7.56 6.55 5.99 5.64 5.39 5.21 5.06 4.95 4.85 4.78 4.71

4 247 1 1 2 3 4 5 6 7 8 9 10 11 12 2 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90 2.86 2.82 2.79 11 9.65 7.20 6.22 5.67 5.32 5.07 4.88 4.74 4.63 4.54 4.46 4.40 4.75 3.88 3.49 3.26 3.11 3.00 2.92 2.85 2.80 2.76 2.72 2.69 12 9.33 6.93 5.95 5.41 5.06 4.82 4.65 4.50 4.39 4.30 4.22 4.16 4.67 3.80 3.41 3.18 3.02 2.92 2.84 2.77 2.72 2.67 2.63 2.60 13 9.07 6.70 5.74 5.20 4.86 4.62 4.44 4.30 4.19 4.10 4.02 3.96 4.60 3.74 3.34 3.11 2.96 2.85 2.77 2.70 2.65 2.60 2.56 2.53 14 8.86 6.51 5.56 5.03 4.69 4.46 4.28 4.14 4.03 3.94 3.86 3.80 4.54 3.68 3.29 3.06 2.90 2.79 2.70 2.64 2.59 2.55 2.51 2.48 15 8.68 6.36 5.42 4.89 4.56 4.32 4.14 4.00 3.89 3.80 3.73 3.67 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.54 2.49 2.54 2.42 16 8.53 6.23 5.29 4.77 4.44 4.20 4.03 3.89 3.78 3.69 3.61 3.55 4.45 3.59 3.20 2.96 2.81 2.70 2.62 2.55 2.50 2.45 2.41 2.38 17 8.40 6.11 5.18 4.67 4.34 4.10 3.93 3.79 3.68 3.59 3.52 3.45 4.41 3.55 3.16 2.93 2.77 2.26 2.58 2.51 2.46 2.41 2.37 2.34 18 8.28 6.01 5.09 4.58 4.25 4.01 3.85 3.71 3.60 3.51 3.44 3.37 4.38 3.52 3.13 2.90 2.74 2.63 2.55 2.48 2.43 2.38 2.34 2.31 19 8.18 5.93 5.01 4.50 4.17 3.94 3.77 3.63 3.52 3.43 3.36 3.30 4.35 3.49 3.10 2.87 2.71 2.60 2.52 2.45 2.40 2.35 2.31 2.28 20 8.10 5.85 4.94 4.43 4.10 3.87 3.71 3.56 3.45 3.37 3.30 3.23 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37 2.32 2.23 2.25 21 8.02 5.78 4.87 4.37 4.04 3.81 3.65 3.51 3.40 3.31 3.24 3.17 4.30 3.44 3.05 2.82 2.66 2.55 2.47 2.40 2.35 2.30 2.26 2.23 22 7.94 5.72 4.82 4.31 3.99 3.76 3.59 3.45 3.35 3.26 3.18 3.12 4.28 3.42 3.03 2.80 2.64 2.53 2.45 2.38 2.32 2.28 2.24 2.20 23 7.88 5.66 4.76 4.26 3.94 3.71 3.54 3.41 3.30 3.21 3.14 3.07 4.26 3.40 3.01 2.78 2.62 2.51 2.43 2.36 2.30 2.26 2.22 2.18 24 7.82 5.61 4.72 4.22 3.90 3.67 3.50 3.36 3.25 3.17 3.09 3.03

248 4 1 1 2 3 4 5 6 7 8 9 10 11 12 2 4.24 3.38 2.99 2.76 2.60 2.49 2.41 2.32 2.28 2.24 2.20 2.16 25 7.77 5.57 4.68 4.18 3.86 3.63 3.46 3.34 3.21 3.13 3.05 2.99 4.22 3.37 2.98 2.74 2.59 2.47 4.39 2.32 2.27 2.22 2.18 2.15 26 7.72 5.53 4.64 4.14 3.82 3.59 3.42 3.29 3.17 3.09 3.02 2.96 4.21 3.35 2.96 2.73 2.57 2.46 2.37 2.30 2.25 2.20 2.16 2.13 27 7.68 5.49 4.60 4.11 3.79 3.56 3.39 3.26 3.14 3.06 2.98 2.93 4.20 3.34 2.95 2.71 2.56 2.44 2.36 2.29 2.24 2.19 2.15 2.12 28 7.64 5.45 4.57 4.07 3.76 3.53 3.36 3.28 3.11 3.03 2.95 2.90 4.18 3.33 2.93 2.70 2.54 2.43 2.35 2.28 2.22 2.18 2.14 2.10 29 7.60 5.42 4.54 4.04 3.73 3.50 3.33 3.20 3.08 3.00 2.92 2.87 4.17 3.32 2.92 2.69 2.53 2.42 2.34 2.27 2.21 2.16 2.12 2.09 30 7.56 5.39 4.51 4.02 3.70 3.47 3.30 3.17 3.06 2.98 2.90 2.84 4.15 3.30 2.90 2.67 2.51 2.40 2.32 2.25 2.19 2.14 2.10 2.07 32 7.50 5.34 4.46 3.97 3.66 3.42 3.25 3.12 3.01 2.94 2.86 2.80 4.13 3.28 2.88 2.65 2.49 2.38 3.30 2.23 2.17 2.12 2.08 2.50 34 7.44 5.29 4.42 3.93 6.61 3.38 3.21 3.08 2.97 2.89 2.82 2.76 4.11 3.26 2.86 2.63 2.48 2.36 2.28 2.21 2.15 2.10 2.06 2.03 36 7.39 5.25 4.38 3.80 3.58 3.35 3.18 3.04 2.94 2.86 2.78 2.72 4.10 3.25 2.85 2.62 2.46 2.35 2.26 2.19 2.14 2.09 2.05 2.02 38 7.35 5.21 4.34 3.86 3.54 3.32 3.15 3.02 2.91 2.82 2.75 2.69 4.08 3.23 2.84 2.61 2.45 3.34 2.25 2.18 2.12 2.07 2.04 2.00 40 7.31 5.18 4.31 3.83 3.51 3.29 3.12 2.99 2.88 2.80 2.73 2.66 4.07 3.22 2.83 2.59 2.44 2.32 2.24 2.17 2.11 2.06 2.02 2.99 42 7.27 5.15 4.29 3.80 3.49 3.26 3.10 2.96 2.86 2.77 2.70 2.64 4.06 3.21 2.82 2.58 2.43 2.31 2.23 2.16 2.10 2.05 2.01 2.98 44 7.24 5.12 4.26 3.78 3.46 3.24 3.07 2.94 2.84 2.75 2.68 2.62 4.05 3.20 2.81 2.57 2.42 2.30 2.22 2.14 2.09 2.04 2.00 2.97 46 7.21 5.10 4.24 3.76 3.44 3.22 3.05 2.92 2.82 2.73 2.66 2.60

4 249 1 2 1 2 3 4 5 6 7 8 9 10 11 12 48 50 55 60 65 4.04 4.03 4.02 4.00 3.99 3.19 3.18 3.17 3.15 3.14 2.80 2.79 2.78 2.76 2.75 2.56 2.56 2.54 2.52 2.51 2.41 2.40 2.38 2.37 2.36 2.30 2.29 2.27 2.25 2.24 2.21 2.20 2.18 2.17 2.15 2.14 2.08 2.03 2.99 2.96 2.13 2.07 2.02 2.98 2.95 2.11 2.05 2.00 2.97 2.93 2.10 2.04 1.99 1.95 1.92 2.08 2.02 1.98 1.94 1.90 7.19 7.17 7.12 7.08 7.04 5.08 5.06 5.01 4.98 4.95 4.22 4.20 4.16 4.13 4.10 3.74 3.72 3.68 3.65 3.62 3.42 3.41 3.37 3.34 3.31 3.20 3.18 3.15 3.12 3.09 3.04 3.02 2.98 2.95 2.93 2.90 2.80 2.71 2.64 2.58 2.88 2.78 2.70 2.62 2.56 2.85 2.75 2.66 2.59 2.53 2.82 2.72 2.63 2.56 2.50 2.79 2.70 2.61 2.54 2.47 3.98 3.13 2.72 2.50 2.35 2.23 2.14 2.07 2.01 1.97 1.93 1.89 70 7.01 4.92 4.08 3.60 4.29 3.07 2.91 2.77 2.67 2.59 2.51 2.45 3.96 3.11 2.72 2.48 2.33 2.21 2.12 2.05 1.99 1.95 1.91 1.88 80 6.96 4.88 4.04 3.56 3.25 3.04 2.87 2.74 2.64 2.55 2.48 2.41 3.94 3.09 2.70 2.46 2.30 2.19 2.10 2.03 1.97 1.92 1.88 1.85 100 6.90 4.82 3.98 3.51 3.20 2.99 2.82 2.69 2.59 2.51 2.43 2.36 3.92 3.07 2.68 2.44 2.29 2.17 2.08 2.01 1.95 1.90 1.86 2.83 125 6.84 4.78 3.94 3.47 3.17 2.95 2.79 2.65 2.56 2.47 2.40 2.33 150 200 400 3.91 3.89 3.86 3.06 3.04 3.02 2.67 2.65 2.62 2.43 2.41 2.39 2.27 2.26 2.23 2.16 2.14 2.12 2.07 2.05 2.03 2.00 1.94 1.89 1.85 2.82 1.98 1.92 1.87 1.83 1.80 1.96 1.90 1.85 1.84 1.78 6.81 6.76 6.70 4.75 4.71 4.66 3.91 3.88 3.83 3.44 3.41 3.36 3.14 3.11 3.06 2.92 2.90 2.85 2.76 2.73 2.69 2.62 2.53 2.44 2.37 2.30 2.60 2.50 2.41 2.34 2.28 2.55 2.46 2.37 2.29 2.33 3.85 3.00 1.61 2.38 2.22 2.10 2.02 1.95 1.89 1.84 1.80 2.76 1000 6.66 4.62 3.80 3.34 3.04 2.82 2.66 2.53 2.43 2.34 2.26 2.20 3.84 2.99 2.60 2.37 2.21 2.09 2.01 1.94 1.88 1.83 1.79 1.75 6.64 1.60 3.78 3.32 3.02 2.80 2.64 2.51 2.41 2.32 2.24 2.18

250 4 1 2 14 16 20 24 30 40 50 75 100 200 500 1 2 3 4 5 245 246 248 249 250 251 252 253 253 254 254 254 6145 6169 6208 6234 6258 6286 6302 6323 6334 6352 6361 6366 19.4219.4319.4419.4519.4619.4719.4719.48 19.4919.4919.5019.50 99.4399.4499.4599.4699.4799.4899.4899.49 99.4999.4999.5099.50 8.71 8.69 8.66 8.64 8.62 8.60 8.58 8.57 8.56 8.54 8.54 8.53 26.9226.8326.6926.6026.5026.4126.3526.27 26.2326.1826.1426.12 5.87 5.84 5.80 5.77 5.74 5.71 5.70 5.68 5.66 5.65 5.64 5.63 14.2414.1514.0213.9313.8313.7413.6913.61 13.5713.5213.4813.46 4.64 4.60 4.56 4.53 4.50 4.46 4.44 4.42 4.40 4.38 4.37 4.36 9.77 9.68 9.55 9.47 9.38 9.29 9.24 9.17 9.13 9.07 9.04 9.02 4.96 3.92 3.87 3.84 3.81 3.77 3.75 3.72 3.71 3.69 3.68 3.67 6 7.60 7.52 7.39 7.31 7.23 7.14 7.09 7.02 6.99 6.94 6.90 6.88 3.52 3.49 3.44 3.41 3.28 3.34 3.32 3.29 3.28 3.25 3.24 3.23 7 6.35 6.27 6.15 6.07 5.98 5.90 5.85 5.78 5.75 5.70 5.67 5.65 3.23 3.20 3.15 3.12 3.08 3.05 3.03 3.00 3.98 2.96 2.94 2.93 8 5.56 5.48 5.36 5.28 5.20 5.11 5.06 5.00 4.96 4.91 4.88 4.86 3.02 2.98 2.93 2.90 2.86 2.82 2.80 2.77 2.76 2.73 2.72 2.71 9 5.00 4.92 4.80 4.73 4.64 4.56 4.51 4.45 4.41 4.36 4.33 4.31 10 11 12 13 14 2.86 2.74 2.64 2.55 2.48 2.82 2.70 2.60 2.51 2.44 2.77 2.65 2.54 2.46 2.39 2.74 2.61 2.50 2.42 2.35 2.70 2.57 2.46 2.38 2.31 2.67 2.53 2.42 2.34 2.27 2.64 2.50 2.40 2.32 2.24 2.60 2.59 2.56 2.55 2.54 2.47 2.45 2.42 2.41 2.40 2.36 2.35 2.32 2.31 2.30 2.28 2.26 1.24 2.22 2.21 2.21 2.19 2.16 2.14 2.13 1.60 4.29 4.05 1.85 3.70 4.52 4.21 3.98 3.78 3.62 4.41 4.10 3.86 3.67 2.51 4.33 4.02 3.78 3.59 3.45 4.25 3.94 3.70 3.15 3.34 4.17 3.86 3.61 3.42 3.26 4.12 3.80 3.56 3.37 3.21 4.05 4.01 3.96 3.93 3.94 3.74 3.70 3.66 3.62 3.60 3.49 3.46 3.41 3.38 3.36 3.30 3.27 3.21 3.18 3.16 3.14 3.11 3.06 3.02 3.00

4 251 1 2 14 16 20 24 30 40 50 75 100 200 500 15 16 17 18 19 2.43 2.37 2.33 2.29 2.26 2.39 2.33 2.29 2.25 2.21 2.33 2.28 2.23 2.19 2.15 2.29 2.24 2.19 2.15 2.11 2.25 2.20 2.25 2.11 2.07 2.21 2.16 2.11 2.07 2.02 2.18 2.13 2.08 2.04 2.00 2.15 2.12 2.10 2.08 2.07 2.09 2.07 2.04 2.02 2.01 2.04 2.02 1.99 1.97 1.96 2.00 1.98 1.95 1.93 1.92 1.96 1.94 1.91 1.90 1.88 3.56 3.45 2.35 3.27 3.19 3.48 3.37 3.27 3.19 3.12 3.36 3.25 3.16 3.07 3.00 3.29 3.18 3.08 3.00 2.92 3.20 3.10 3.00 2.91 2.84 3.12 3.01 2.92 2.93 2.76 3.07 2.96 2.86 2.78 2.70 3.00 2.97 2.92 2.89 2.87 2.89 2.86 2.80 2.77 2.75 2.79 2.76 2.70 2.67 2.65 2.71 2.68 2.62 2.59 2.57 2.63 2.60 2.54 2.51 2.49 2.23 2.18 2.12 2.08 2.04 1.99 1.96 1.92 1.90 1.87 1.85 1.84 20 3.13 3.05 2.94 2.86 2.77 2.69 2.63 2.56 2.53 2.47 2.44 2.42 2.20 2.15 2.09 2.05 2.00 1.96 1.93 2.89 1.87 1.84 1.82 1.81 21 3.07 2.99 2.88 2.80 2.72 2.63 2.58 2.51 2.47 2.52 2.38 2.36 2.18 2.13 2.07 2.03 1.98 1.93 1.91 1.87 1.87 1.81 1.80 1.78 22 3.02 2.94 2.83 2.75 2.67 2.58 2.53 2.46 2.42 2.37 2.33 2.31 2.14 2.10 2.04 3.00 1.96 1.91 1.88 1.84 1.82 1.79 1.77 1.76 23 2.97 2.89 2.78 2.79 2.62 2.53 2.48 2.41 2.37 2.32 2.28 2.26 24 25 26 27 28 2.13 2.11 2.10 2.08 2.06 2.09 2.06 2.05 2.03 2.02 2.02 2.00 1.99 1.97 1.96 1.98 1.96 1.95 1.93 1.91 1.94 1.92 1.90 1.88 1.87 1.89 1.87 1.85 1.84 1.81 1.86 2.84 1.82 1.80 1.78 1.82 1.80 1.76 1.74 1.73 2.80 1.77 1.74 1.72 1.71 1.78 1.76 1.72 1.70 1.69 1.76 1.74 1.71 1.68 1.67 1.75 1.72 1.69 1.67 1.65 2.93 2.89 2.86 2.83 2.80 2.85 281 2.77 2.74 2.71 2.74 2.70 2.66 2.63 2.60 2.66 2.62 2.58 2.55 2.52 2.58 2.54 2.50 2.47 2.44 2.49 2.45 2.41 2.38 2.35 2.44 2.40 2.36 2.33 2.30 2.36 2.33 2.27 2.23 2.21 2.32 2.29 2.23 2.19 2.17 2.28 2.25 2.19 2.15 2.13 2.25 2.21 2.16 2.12 2.10 2.22 2.18 2.13 2.09 2.06

252 4 1 2 14 16 20 24 30 40 50 75 100 200 500 29 2.05 2.77 2.00 2.68 1.94 2.57 1.90 2.49 1.85 2.41 1.80 2.32 1.77 2.27 1.73 2.19 1.71 2.15 1.68 2.10 1.65 2.06 1.64 2.03 30 2.04 2.74 1.99 2.66 1.93 2.55 1.89 2.47 1.84 2.38 1.79 2.29 1.76 2.24 1.72 2.16 1.69 2.13 1.66 2.07 1.64 2.03 1.62 2.01 32 2.02 2.70 1.97 2.62 1.91 2.51 1.86 2.42 1.82 2.34 1.76 2.25 1.74 2.20 1.69 2.12 1.67 2.08 1.64 2.02 1.61 1.98 1.59 1.96 34 2.00 2.66 1.95 2.58 1.89 2.47 1.84 2.38 1.80 2.30 1.74 2.21 1.71 2.15 1.67 2.08 1.64 2.04 1.61 1.98 1.59 1.94 1.57 1.91 36 1.98 2.62 1.93 3.54 1.87 2.43 1.82 2.35 1.78 2.26 1.72 2.17 1.69 2.12 1.65 2.04 1.62 2.00 1.59 1.94 1.56 1.90 1.55 1.87 38 1.96 2.59 1.92 2.51 1.85 2.40 1.80 2.32 1.76 2.22 1.71 2.14 1.67 2.08 1.63 2.00 1.60 1.97 1.57 1.90 1.54 1.86 1.53 1.84 40 1.95 2.56 1.90 2.49 1.84 2.37 1.79 2.29 1.74 2.20 1.69 2.11 1.66 2.05 1.61 1.97 1.59 1.94 1.55 1.88 1.53 1.84 1.51 1.81 42 1.94 2.54 1.89 2.46 1.82 2.35 1.78 2.26 1.73 2.17 1.68 2.08 1.64 2.02 1.60 1.94 1.57 1.91 1.54 1.85 1.51 1.80 1.49 1.78 44 1.92 2.52 1.88 2.44 1.81 2.32 1.76 2.24 1.72 2.15 1.66 2.06 1.63 2.00 1.58 1.92 1.56 1.88 1.52 1.82 1.50 1.78 1.48 1.75 46 1.91 2.50 1.87 2.42 1.80 2.30 1.75 2.22 1.71 2.13 1.65 2.04 1.62 1.98 1.57 1.90 1.54 1.86 1.51 1.80 1.48 1.76 1.46 1.72 48 1.90 2.48 1.86 2.40 1.79 2.28 1.74 2.20 1.70 2.11 1.64 2.02 1.61 1.96 1.56 1.88 1.53 1.84 1.50 1.78 1.47 1.73 1.45 1.70 50 1.90 2.46 1.85 2.39 1.78 2.26 1.74 2.18 1.69 2.10 1.63 2.00 1.60 1.94 1.55 1.86 1.52 1.82 1.48 1.76 1.46 1.71 1.44 1.68 55 1.88 2.43 1.83 2.35 1.76 2.23 1.72 2.15 1.67 2.06 1.61 1.96 1.58 1.90 1.52 1.82 1.50 1.78 1.46 1.71 1.43 1.66 1.41 1.64

4 253 1 2 14 16 20 24 30 40 50 75 100 200 500 60 1.86 2.40 1.81 2.32 1.75 2.20 1.70 2.12 1.65 2.03 1.59 1.93 1.56 1.87 1.50 1.79 1.48 1.74 1.44 1.68 1.41 1.63 1.39 1.60 65 1.85 2.37 1.80 2.30 1.73 2.18 1.68 2.09 1.63 2.00 1.57 1.90 1.54 1.84 1.49 1.76 1.46 1.71 1.42 1.64 1.39 1.60 1.37 1.56 70 1.84 2.35 1.79 2.28 1.72 2.15 1.67 2.07 1.62 1.98 1.56 1.88 1.53 1.82 1.47 1.74 1.45 1.69 1.40 1.62 1.37 1.56 1.35 1.53 80 1.82 2.32 1.77 2.24 1.70 2.11 1.65 2.03 1.60 1.94 1.54 1.84 1.51 1.78 1.45 1.70 1.42 1.65 1.38 1.57 1.35 1.52 1.32 1.49 100 1.79 2.26 1.75 2.19 1.68 2.06 1.63 1.98 1.57 1.89 1.51 1.79 1.48 1.73 1.42 1.64 1.39 1.59 1.34 1.51 1.30 1.46 1.28 1.43 125 1.77 2.23 1.72 2.15 1.65 2.03 1.60 1.94 1.55 1.85 1.49 1.75 1.45 1.68 1.39 1.59 1.36 1.54 1.31 1.14 1.27 1.40 1.25 1.37 150 1.76 2.20 1.71 2.12 1.64 2.00 1.59 1.91 1.54 1.83 1.47 1.72 1.44 1.66 1.37 1.56 1.34 1.51 1.29 1.43 1.25 1.37 1.22 1.33 200 1.74 2.17 1.69 2.09 1.62 1.97 1.57 1.88 1.52 1.79 1.45 1.69 1.42 1.62 1.35 1.53 1.32 1.48 1.26 1.39 1.22 1.33 1.19 1.28 400 1.72 2.12 1.67 2.04 1.60 1.92 1.54 1.84 1.49 1.74 1.42 1.64 1.38 1.57 1.32 1.47 1.28 1.42 1.22 1.32 1.16 1.24 1.13 1.19 1.70 1000 2.09 1.65 2.04 1.58 1.89 1.53 1.81 1.47 1.71 1.41 1.61 1.36 1.54 1.30 1.44 1.26 1.38 1.19 1.28 1.13 1.19 1.08 1.11 1.67 2.07 1.64 1.99 1.57 1.87 1.52 1.79 1.46 1.69 1.40 1.59 1.35 1.52 1.28 1.41 1.24 1.36 1.17 1.25 1.11 1.15 1.00 1.00

254 5 D.W 5 % n 15 k = 2 k = 3 k = 4 k = 5 k = 6 d L d V d L d V d L d V d L d V d L d V 1.08 1.36 0.95 1.54 0.82 1.75 0.69 1.97 0.56 2.21 16 1.10 1.37 0.98 1.54 0.86 1.73 0.74 1.93 0.62 2.15 17 1.13 1.38 1.02 1.54 0.90 1.71 0.78 1.90 0.67 2.10 18 1.16 1.39 1.05 1.53 0.93 1.69 0.82 1.87 0.71 2.06 19 1.18 1.40 1.08 1.53 0.97 1.68 0.86 1.85 0.75 2.02 20 1.20 1.41 1.10 1.54 1.00 1.68 0.90 1.83 0.79 1.99 21 1.22 1.42 1.13 1.54 1.03 1.67 0.93 1.81 0.83 1.96 22 1.24 1.43 1.15 1.54 1.05 1.66 0.96 1.80 0.86 1.94 23 1.26 1.44 1.17 1.54 1.08 1.66 0.99 1.79 0.90 1.92 24 1.27 1.45 1.19 1.55 1.10 1.66 1.01 1.78 0.93 1.90 25 1.29 1.45 1.21 1.55 1.12 1.66 1.04 1.77 0.95 1.89 26 1.30 1.46 1.22 1.55 1.14 1.65 1.06 1.76 0.98 1.88 27 1.32 1.47 1.24 1.56 1.16 1.65 1.08 1.76 1.01 1.86 28 1.33 1.48 1.26 1.56 1.18 1.65 1.10 1.75 1.03 1.85 29 1.34 1.48 1.27 1.56 1.20 1.65 1.12 1.74 1.05 1.84 30 1.35 1.49 1.28 1.57 1.21 1.65 1.14 1.74 1.07 1.83 31 1.36 1.50 1.30 1.57 1.23 1.65 1.16 1.74 1.09 1.83 32 1.37 1.50 1.31 1.57 1.24 1.65 1.18 1.73 1.11 1.82 33 1.38 1.51 1.32 1.58 1.26 1.65 1.19 1.73 1.13 1.81 34 1.39 1.51 1.33 1.58 1.27 1.65 1.21 1.73 1.15 1.81 35 1.40 1.52 1.34 1.58 1.28 1.65 1.22 1.73 1.16 1.80 36 1.41 1.52 1.35 1.59 1.29 1.65 1.24 1.73 1.18 1.80 37 1.42 1.53 1.26 1.59 1.31 1.65 1.25 1.72 1.19 1.80 38 1.43 1.54 1.37 1.59 1.32 1.66 1.26 1.72 1.21 1.79 39 1.43 1.54 1.38 1.60 1.33 1.66 1.27 1.72 1.22 1.79 40 1.44 1.54 1.39 1.60 1.34 1.66 1.29 1.72 1.23 1.79 :n ; k,

5 255 n k = 2 k = 3 k = 4 k = 5 k = 6 d L d V d L d V d L d V d L d V d L d V 45 1.48 1.57 1.43 1.62 1.38 1.66 1.34 1.72 1.29 1.78 50 1.50 1.59 1.46 1.63 1.42 1.67 1.38 1.72 1.34 1.77 55 1.53 1.60 1.49 1.64 1.45 1.67 1.41 1.2 1.38 1.77 60 1.55 1.62 1.51 1.65 1.48 1.68 1.44 1.73 1.41 1.77 65 1.57 1.63 1.54 1.66 1.50 1.69 1.47 1.73 1.44 1.77 70 1.58 1.64 1.55 1.67 1.52 1.70 1.49 1.74 1.46 1.77 75 1.60 1.65 1.57 1.68 1.54 1.70 1.51 1.74 1.49 1.77 80 1.61 1.66 1.59 1.69 1.56 1.71 1.53 1.74 1.51 1.77 85 1.62 1.67 1.60 1.70 1.57 1.72 1.55 1.75 1.52 1.77 90 1.63 1.68 1.61 1.70 1.59 1.72 1.57 1.75 1.54 1.78 95 1.64 1.69 1.62 1.71 1.60 1.73 1.58 1.75 1.56 1.78 100 1.65 1.69 1.63 1.72 1.61 1.74 1.59 1.76 1.57 1.78 :n ; k,

256 5 1% n k = 2 dl dv k = 3 dl dv k = 4 dl dv k = 5 dl dv k = 6 dl dv 15 0.81 1.07 0.70 1.25 0.59 1.46 0.49 1.70 0.39 1.96 16 0.84 1.09 0.74 1.25 0.63 1.44 0.53 1.66 0.44 1.90 17 0.87 1.10 0.77 1.25 0.67 1.43 0.57 1.63 0.48 1.85 18 0.90 1.12 0.80 1.26 0.71 1.42 0.61 1.60 0.52 1.80 19 0.93 1.13 0.83 1.26 0.74 1.41 0.65 1.58 0.56 1.77 20 0.95 1.15 0.86 1.27 0.77 1.41 0.68 1.57 0.60 1.74 21 0.97 1.16 0.89 1.27 0.80 1.41 0.72 1.55 0.63 1.71 22 1.00 1.17 0.91 1.28 0.83 1.40 0.75 1.54 0.66 1.69 23 1.02 1.19 0.94 1.29 0.86 1.40 0.77 1.53 0.70 1.67 24 1.04 1.20 0.96 1.30 0.88 1.41 0.80 1.53 0.72 1.66 25 1.05 1.21 0.98 1.30 0.90 1.41 0.83 1.52 0.75 1.65 26 1.07 1.22 1.00 1.31 0.93 1.41 0.85 1.52 0.78 1.64 27 1.09 1.23 1.02 1.32 0.95 1.41 0.88 1.51 0.81 1.63 28 1.10 1.24 1.04 1.32 0.97 1.41 0.90 1.51 0.83 1.62 29 1.12 1.25 1.05 1.33 0.99 1.42 0.92 1.51 0.85 1.61 30 1.13 1.26 1.07 1.34 1.01 1.42 0.94 1.51 0.88 1.61 31 1.15 1.27 1.08 1.34 1.02 1.42 0.96 1.51 0.90 1.60 32 1.16 1.28 1.10 1.35 1.04 1.43 0.98 1.51 0.92 1.60 33 1.17 1.29 1.11 1.36 1.05 1.43 1.00 1.51 0.94 1.59 34 1.18 1.30 1.13 1.36 1.07 1.43 1.01 1.51 0.95 1.59 35 1.19 1.31 1.14 1.37 1.08 1.44 1.03 1.51 0.97 1.59 36 1.21 1.32 1.15 1.38 1.10 1.44 1.04 1.51 0.99 1.59 37 1.22 1.32 1.16 1.38 1.11 1.45 1.06 1.51 1.00 1.59 38 1.23 1.33 1.18 1.39 1.12 1.45 1.07 1.52 1.02 1.58 39 1.24 1.34 1.19 1.39 1.14 1.45 1.09 1.52 1.03 1.58 40 1.25 1.34 1.20 1.40 1.15 1.46 1.10 1.52 1.05 1.58

5 257 n k = 2 k = 3 k = 4 k = 5 k = 6 d L d V d L d V d L d V d L d V d L d V 45 1.29 1.38 1.24 1.42 1.20 1.48 1.16 1.53 1.11 1.58 50 1.32 1.40 1.28 1.45 1.24 1.49 1.20 1.54 1.16 1.59 55 1.36 1.43 1.32 1.47 1.28 1.51 1.25 1.55 1.21 1.59 60 1.38 1.45 1.35 1.48 1.32 1.52 1.28 1.56 1.25 1.60 65 1.41 1.47 1.38 1.50 1.35 1.53 1.31 1.57 1.28 1.61 70 1.43 1.49 1.40 1.52 1.37 1.55 1.34 1.58 1.31 1.61 75 1.45 1.50 1.42 1.53 1.39 1.56 1.37 1.59 1.34 1.62 80 1.47 1.52 1.44 1.54 1.42 1.57 1.39 1.60 1.36 1.62 85 1.48 1.53 1.46 1.55 1.43 1.58 1.41 1.60 1.39 1.63 90 1.50 1.54 1.47 1.56 1.45 1.59 1.43 1.61 1.41 1.64 95 1.51 1.55 1.49 1.57 1.47 1.60 1.45 1.62 1.42 1.64 100 1.52 1.56 1.50 1.58 1.48 1.60 1.46 1.63 1.44 1.65

258 6 ( r z ) r 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.00000 0.01000 0.02000 0.03001 0.04002 0.05004 0.06007 0.07012 0.08017 0.09024 0.1 0.10034 0.11045 0.12058 0.13074 1.14093 0.15114 0.16139 0.17167 0.18198 0.19234 0.2 0.20273 0.21317 0.22366 0.23419 0.24477 0.25561 0.26611 0.27686 0.28768 0.29857 0.3 0.30952 0.32005 0.33165 0.34283 0.35409 0.36544 0.37689 0.38842 0.40006 0.41180 0.4 0.47365 1.43561 0.44769 0.45990 0.47223 0.48470 0.49731 0.51007 0.52298 0.53606 0.5 0.54931 0.56273 0.57634 0.59014 0.60415 0.61838 0.63283 0.64752 0.66246 0.67767 0.6 0.69315 0.70892 0.72500 0.74142 0.75817 0.77530 0.79281 0.81074 0.82911 0.84795 0.7 0.86730 0.88718 0.90764 0.92783 0.95048 0.97285 0.99621 1.02033 1.04537 1.07143 0.8 1.09861 1.12703 1.15682 1.18813 1.22117 1.25615 1.29334 1.33308 1.37577 1.42192 0.9 1.47222 1.52752 1.58902 1.65839 1.73805 1.83178 1.94591 2.09229 2.29756 2.64695 : r,, z, r = 0.34 z, r = 0.3 r = 0.04 z 0.35409

1 ] 1 :,1990 [2 ] 1 :,1986 [3 ]1 :,1994 [4 ] 1 :, 1991 [5 ] 1 :, 1989 [6 ] 1 :, 2001 [7 ] Excel 1 :,2001 [8 ] 1 :, 1998 [9 ] George E.P.Box, Gwilym M. Jenkins Time Series Analysis: Forecasting and Control Rev. Ed.Holden - Day. ( T hird Edition ) [10] S.Makridakis, S.C.W heelwright, V.E.Mcqee Forecasting: Methods and Applications, John Wily and Sons, 1983