s = 1.15 (s = 1.07), R = 0.786, R = 0.679, DW =.03 5 Y = 0.3 (0.095) (.708) X, R = 0.786, R = 0.679, s = 1.07, DW =.03, t û Y = 0.3 (3.163) + 0

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7 DW 7.1 DW u 1, u,, u (DW ) u u 1 = u 1, u,, u + + + - - - - + + - - - + + u 1, u,, u + - + - + - + - + u 1, u,, u u 1, u,, u u +1 = u 1, u,, u Y = α + βx + u, u = ρu 1 + ɛ, H 0 : ρ = 0, H 1 : ρ 0 ɛ 1, ɛ,, ɛ = DW = (û û 1 ) û DW = DW = (û û 1 ) û = = û = ûû 1 + = û 1 û = û (û 1 + û ) = ûû 1 û û (1 ρ), = ρ, = ûû 1 = û 1 ρ û û 1 u = ρu 1 + ɛ u, u 1 û, û 1 ρ ρ 1. DW ( ρ = 0 DW ). DW 3. DW k 1 1 k DW Y X X Y X Ŷ û 1 6 10 60 100 6.8 0.8 9 1 108 144 8.1 0.9 3 10 14 140 196 9.4 0.6 4 10 16 160 56 10.7 0.7 Y X XY X Ŷ û 35 5 468 696 35 0 Y X 8.75 13 DW = = (û û 1 ) û = ( 0.8 0.9) + (0.9 0.6) + (0.6 ( 0.7)) ( 0.8) + 0.9 + 0.6 + ( 0.7) = 4.67.30 =.03 û 1 + û û 0, Y = α + βx + u, = ûû 1 û = = ûû 1 = û 1 + û α = 0.3, β = 0.65, s α = 10.0005 = 3.163, = α β 0.0575 = 0.40, = 0.095, =.708, s β s α s β 3

s = 1.15 (s = 1.07), R = 0.786, R = 0.679, DW =.03 5 Y = 0.3 (0.095) + 0.65 (.708) X, R = 0.786, R = 0.679, s = 1.07, DW =.03, t û Y = 0.3 (3.163) + 0.65 (0.40) X, R = 0.786, R = 0.679, s = 1.07, DW =.03, s = 1.15 = 1.07 7. Y = α + βx + u, = 1,,, E(u ) = 0 V(u ) = E(u ) = σ j Cov(u, u j ) = E(u u j ) = σ j û 4 β = ω Y = β + ω u X X ω = j (X j X) E( β) E( β) = E(β + ω u ) = β + ω E(u ) = β u 1, u,, u β 33

1: 5 % (1) k = 1 A B C D E 0 dl dl du du 4 du 4 du 4 dl 4 dl 4 15 0 1.08 1.08 1.36 1.36.64.64.9.9 4 0 0 1.0 1.0 1.41 1.41.59.59.80.80 4 5 0 1.9 1.9 1.45 1.45.55.55.71.71 4 30 0 1.35 1.35 1.49 1.49.51.51.65.65 4 () k = A B C D E 0 dl dl du du 4 du 4 du 4 dl 4 dl 4 15 0 0.95 0.95 1.54 1.54.46.46 3.05 3.05 4 0 0 1.10 1.10 1.54 1.54.46.46.90.90 4 5 0 1.1 1.1 1.55 1.55.45.45.79.79 4 30 0 1.8 1.8 1.57 1.57.43.43.7.7 4 (3) k = 3 A B C D E 0 dl dl du du 4 du 4 du 4 dl 4 dl 4 15 0 0.8 0.8 1.75 1.75.5.5.5 3.18 4 0 0 1.00 1.00 1.68 1.68.3.3.3 3.00 4 5 0 1.1 1.1 1.66 1.66.34.34.34.88 4 30 0 1.1 1.1 1.65 1.65.35.35.35.79 4 (4) k = 4 A B C D E 0 dl dl du du 4 du 4 du 4 dl 4 dl 4 15 0 0.69 0.69 1.97 1.97.03.03 3.31 3.31 4 0 0 0.90 0.90 1.83 1.83.17.17 3.10 3.10 4 5 0 1.04 1.04 1.77 1.77.3.3.96.96 4 30 0 1.14 1.14 1.74 1.74.6.6.86.86 4 (5) k = 5 A B C D E 0 dl dl du du 4 du 4 du 4 dl 4 dl 4 15 0 0.56 0.56.1.1 3.44 3.44 4 0 0 0.79 0.79 1.99 1.99.01.01 3.1 3.1 4 5 0 0.95 0.95 1.89 1.89.11.11 3.05 3.05 4 30 0 1.07 1.07 1.83 1.83.17.17.93.93 4 A: B: C: D: E: : 5 % k = 1 k = k = 3 k = 4 k = 5 k = 6 k = 7 k = 8 k = 9 k = 10 k = 11 k = 1 k = 13 dl du dl du dl du dl du dl du dl du dl du dl du dl du dl du dl du dl du dl du 6 0.610 1.400 7 0.700 1.356 0.467 1.896 8 0.763 1.33 0.559 1.777 0.367.87 9 0.84 1.30 0.69 1.699 0.455.18 0.96.588 10 0.879 1.30 0.697 1.641 0.55.016 0.376.414 0.43.8 11 0.97 1.34 0.758 1.604 0.595 1.98 0.444.83 0.315.645 0.03 3.004 1 0.971 1.331 0.81 1.579 0.658 1.864 0.51.177 0.380.506 0.68.83 0.171 3.149 13 1.010 1.340 0.861 1.56 0.715 1.816 0.574.094 0.444.390 0.38.69 0.30.985 0.147 3.66 14 1.045 1.350 0.905 1.551 0.767 1.779 0.63.030 0.505.96 0.389.57 0.86.848 0.00 3.111 0.17 3.360 15 1.077 1.361 0.946 1.543 0.814 1.750 0.685 1.977 0.56.0 0.447.471 0.343.77 0.51.979 0.175 3.16 0.111 3.438 16 1.106 1.371 0.98 1.539 0.857 1.78 0.734 1.935 0.615.157 0.50.388 0.398.64 0.304.860 0. 3.090 0.155 3.304 0.098 3.503 17 1.133 1.381 1.015 1.536 0.897 1.710 0.779 1.900 0.664.104 0.554.318 0.451.537 0.356.757 0.7.975 0.198 3.184 0.138 3.378 0.087 3.557 18 1.158 1.391 1.046 1.535 0.933 1.696 0.80 1.87 0.710.060 0.603.57 0.50.461 0.407.668 0.31.873 0.44 3.073 0.177 3.65 0.13 3.441 0.078 3.603 19 1.180 1.401 1.074 1.536 0.967 1.685 0.859 1.848 0.75.03 0.649.06 0.549.396 0.456.589 0.369.783 0.90.974 0.0 3.159 0.160 3.335 0.111 3.496 0 1.01 1.411 1.100 1.537 0.998 1.676 0.894 1.88 0.79 1.991 0.691.16 0.595.339 0.50.51 0.416.704 0.336.885 0.63 3.063 0.00 3.34 0.145 3.395 1 1.1 1.40 1.15 1.538 1.06 1.669 0.97 1.81 0.89 1.964 0.731.14 0.637.90 0.546.461 0.461.633 0.380.806 0.307.976 0.40 3.141 0.18 3.300 1.39 1.49 1.147 1.541 1.053 1.664 0.958 1.797 0.863 1.940 0.769.090 0.677.46 0.588.407 0.504.571 0.44.735 0.349.897 0.81 3.057 0.0 3.11 3 1.57 1.437 1.168 1.543 1.078 1.660 0.986 1.785 0.895 1.90 0.804.061 0.715.08 0.68.360 0.545.514 0.465.670 0.391.86 0.3.979 0.59 3.19 4 1.73 1.446 1.188 1.546 1.101 1.656 1.013 1.775 0.95 1.90 0.837.035 0.750.174 0.666.318 0.584.464 0.506.613 0.431.761 0.36.908 0.97 3.053 5 1.88 1.454 1.06 1.550 1.13 1.654 1.038 1.767 0.953 1.886 0.868.013 0.784.144 0.70.80 0.61.419 0.544.560 0.470.70 0.400.844 0.335.983 6 1.30 1.461 1.4 1.553 1.143 1.65 1.06 1.759 0.979 1.873 0.897 1.99 0.816.117 0.735.46 0.657.379 0.581.513 0.508.649 0.438.784 0.373.919 7 1.316 1.469 1.40 1.556 1.16 1.651 1.084 1.753 1.004 1.861 0.95 1.974 0.845.093 0.767.16 0.691.34 0.616.470 0.544.600 0.475.730 0.409.860 8 1.38 1.476 1.55 1.560 1.181 1.650 1.104 1.747 1.08 1.850 0.951 1.959 0.874.071 0.798.188 0.73.309 0.649.431 0.578.555 0.510.680 0.445.805 9 1.341 1.483 1.70 1.563 1.198 1.650 1.14 1.743 1.050 1.841 0.975 1.944 0.900.05 0.86.164 0.753.78 0.681.396 0.61.515 0.544.634 0.479.754 30 1.35 1.489 1.84 1.567 1.14 1.650 1.143 1.739 1.071 1.833 0.998 1.931 0.96.034 0.854.141 0.78.51 0.71.363 0.643.477 0.577.59 0.513.708 31 1.363 1.496 1.97 1.570 1.9 1.650 1.160 1.735 1.090 1.85 1.00 1.90 0.950.018 0.879.10 0.810.6 0.741.333 0.674.443 0.608.553 0.545.665 3 1.373 1.50 1.309 1.574 1.44 1.650 1.177 1.73 1.109 1.819 1.041 1.909 0.97.004 0.904.10 0.836.03 0.769.306 0.703.411 0.638.518 0.576.65 33 1.383 1.508 1.31 1.577 1.58 1.651 1.193 1.730 1.17 1.813 1.061 1.900 0.994 1.991 0.97.085 0.861.181 0.796.81 0.731.38 0.667.484 0.606.588 34 1.393 1.514 1.333 1.580 1.71 1.65 1.08 1.78 1.144 1.808 1.079 1.891 1.015 1.978 0.950.069 0.885.16 0.81.57 0.758.355 0.695.454 0.634.553 35 1.40 1.519 1.343 1.584 1.83 1.653 1. 1.76 1.160 1.803 1.097 1.884 1.034 1.967 0.971.054 0.908.144 0.845.36 0.783.330 0.7.45 0.66.51 36 1.411 1.55 1.354 1.587 1.95 1.654 1.36 1.74 1.175 1.799 1.114 1.876 1.053 1.957 0.991.041 0.930.17 0.868.16 0.808.306 0.748.398 0.689.49 37 1.419 1.530 1.364 1.590 1.307 1.655 1.49 1.73 1.190 1.795 1.131 1.870 1.071 1.948 1.011.09 0.951.11 0.891.197 0.831.85 0.77.374 0.714.464 38 1.47 1.535 1.373 1.594 1.318 1.656 1.61 1.7 1.04 1.79 1.146 1.864 1.088 1.939 1.09.017 0.970.098 0.91.180 0.854.65 0.796.351 0.739.438 39 1.435 1.540 1.38 1.597 1.38 1.658 1.73 1.7 1.18 1.789 1.161 1.859 1.104 1.93 1.047.007 0.990.085 0.93.164 0.875.46 0.819.39 0.763.413 40 1.44 1.544 1.391 1.600 1.338 1.659 1.85 1.71 1.30 1.786 1.175 1.854 1.10 1.94 1.064 1.997 1.008.07 0.95.150 0.896.8 0.840.309 0.785.391 45 1.475 1.566 1.430 1.615 1.383 1.666 1.336 1.70 1.87 1.776 1.38 1.835 1.189 1.895 1.139 1.958 1.089.0 1.038.088 0.988.156 0.938.5 0.887.96 50 1.503 1.585 1.46 1.68 1.41 1.674 1.378 1.71 1.335 1.771 1.91 1.8 1.46 1.875 1.01 1.930 1.156 1.986 1.110.044 1.064.103 1.019.163 0.973.5 55 1.58 1.601 1.490 1.641 1.45 1.681 1.414 1.74 1.374 1.768 1.334 1.814 1.94 1.861 1.53 1.909 1.1 1.959 1.170.010 1.19.06 1.087.116 1.045.170 60 1.549 1.616 1.514 1.65 1.480 1.689 1.444 1.77 1.408 1.767 1.37 1.808 1.335 1.850 1.98 1.894 1.60 1.939 1. 1.984 1.184.031 1.145.079 1.106.17 65 1.567 1.69 1.536 1.66 1.503 1.696 1.471 1.731 1.438 1.767 1.404 1.805 1.370 1.843 1.336 1.88 1.301 1.93 1.66 1.964 1.31.006 1.195.049 1.160.093 70 1.583 1.641 1.554 1.67 1.55 1.703 1.494 1.735 1.464 1.768 1.433 1.80 1.401 1.838 1.369 1.874 1.337 1.910 1.305 1.948 1.7 1.987 1.39.06 1.06.066 75 1.598 1.65 1.571 1.680 1.543 1.709 1.515 1.739 1.487 1.770 1.458 1.801 1.48 1.834 1.399 1.867 1.369 1.901 1.339 1.935 1.308 1.970 1.77.006 1.47.043 80 1.611 1.66 1.586 1.688 1.560 1.715 1.534 1.743 1.507 1.77 1.480 1.801 1.453 1.831 1.45 1.861 1.397 1.893 1.369 1.95 1.340 1.957 1.31 1.990 1.83.04 85 1.63 1.671 1.600 1.696 1.575 1.71 1.550 1.747 1.55 1.774 1.500 1.801 1.474 1.89 1.448 1.857 1.4 1.886 1.396 1.916 1.369 1.946 1.34 1.977 1.315.008 90 1.635 1.679 1.61 1.703 1.589 1.76 1.566 1.751 1.54 1.776 1.518 1.801 1.494 1.87 1.469 1.854 1.445 1.881 1.40 1.909 1.395 1.937 1.369 1.966 1.344 1.995 95 1.645 1.687 1.63 1.709 1.60 1.73 1.579 1.755 1.557 1.778 1.535 1.80 1.51 1.87 1.489 1.85 1.465 1.877 1.44 1.903 1.418 1.930 1.394 1.956 1.370 1.984 100 1.654 1.694 1.634 1.715 1.613 1.736 1.59 1.758 1.571 1.780 1.550 1.803 1.58 1.86 1.506 1.850 1.484 1.874 1.46 1.898 1.439 1.93 1.416 1.948 1.393 1.974 150 1.70 1.747 1.706 1.760 1.693 1.774 1.679 1.788 1.665 1.80 1.651 1.817 1.637 1.83 1.6 1.846 1.608 1.86 1.593 1.877 1.579 1.89 1.564 1.908 1.549 1.94 00 1.758 1.779 1.748 1.789 1.738 1.799 1.78 1.809 1.718 1.80 1.707 1.831 1.697 1.841 1.686 1.85 1.675 1.863 1.665 1.874 1.654 1.885 1.643 1.897 1.63 1.908 k http://www.staford.edu/ clt/bech/dwcrt.htm 34

V( β) V( β) = V(β + ω u ) = V( ω u ) Y = α + βx + ɛ, = E(( ω u ) ) = E(( ω u )( ω j u j )) j = E( ω ω j u u j ) j = ω ω j E(u u j ) j = ω E(u ) + ω ω j E(u u j ) j = σ σ ω + ω j σ j ω ω j j j u 1, u,, u β s s j ω ω j j j s, s j σ, σ j s ω 7.3 Y = α + βx + u, u = ρu 1 + ɛ, ω + ɛ 1, ɛ,, ɛ u (Y ρy 1 ) = α(1 ρ) + β(x ρx 1 ) + ɛ, Y = (Y ρy 1 ), X = (X ρx 1 ) ɛ 1, ɛ,, ɛ α = α(1 ρ) Y = β 1 X 1 + β X + + β k X k + u, u = ρu 1 + ɛ, ɛ 1, ɛ,, ɛ u (Y ρy 1 ) = β 1 (X 1 ρx 1, 1 ) + β (X 1 ρx, 1 ) + Y = (Y ρy 1 ), X 1 = (X 1 ρx 1, 1 ), X = (X ρx, 1 ),, X k = (X k ρx k, 1 ) Y + β k (X 1 ρx k, 1 ) + ɛ, = β 1 X 1 + β X + + β k X k + ɛ ɛ 1, ɛ,, ɛ ρ : DW DW (1 ρ) DW ρ ρ Y = (Y ρy 1 ), X 1 = (X 1 ρx 1, 1 ), X = (X ρx, 1 ),, X k = (X k ρx k, 1 ) Y = β 1 X 1 + β X + + β k X k + ɛ, 35

8 () 8.1 () Y = α + βx + u X Y u ( ) u 1, u,, u σ (z ) u σ z Y = α + βx + u Y = α 1 + β X + u z z z z = α 1 z + β X z + u u σ () ( ) ( ) E(u u 1 ) = E = E(u ) = 0 z u E(u ) = 0 ( ) ( ) V(u u 1 ) = V = V(u ) = σ z z z u V(u ) = σ z Y 1,, X z z z û = γz + ɛ γ γ ( t ) z u σ X X Y = α 1 + β + u X X X β () 8. Y = α + βx + u, E(u ) = 0 = 1,,, V(u ) = E(u ) = σ j Cov(u, u j ) = E(u u j ) = 0 β = ω Y = β + ω u X X ω = j (X j X) E( β) E( β) = E(β + = β + ω u ) ω E(u ) = β u 1, u,, u β V( β) V( β) = V(β + = E(( ω u ) = V( ω u ) ) ω u ) = E(( ω u )( ω j u j )) j = E( ω ω j u u j ) j = ω ω j E(u u j ) j = ω E(u ) + ω ω j E(u u j ) j j = α 1 X + β + u = σ ω σ ω 36

u 1, u,, u β s ω s σ s ω 9 Y = αw + βx + u W, X Y u W = 1 α W X W X α, β W X 1 ( ) W = γx Y = αw + βx + u = (αγ + β)x + u αγ + β α, β Y = αw + βx + u αγ + β α, β α, β u = (Y αw βx ) α, β α, β u α = (Y αw βx )W = 0 Y W α W β X W = 0 Y X α W X β X = 0 ( ) ( Y W W ) ( ) = X W Y X W X X α β α, β ( ) ( W ) 1 ( ) = X W Y W α β W X X Y X ( ) = α β 1 ( W )( X ) ( W X ) ( X X W W X W ) ( ) Y W Y X (W = γx ) ( W )( X ) ( W X ) = 0 ( ) ( ) V( α) Cov( α, β) V = α β Cov( α, β) V( β) ( W ) 1 = σ X W W X X σ = W )( X ) ( W X ) ( X ) X W W X W W = 1 W = 0X = 1 X = 0 W X r (W w)(x X) r = (W W ) (X X) u β = (Y αw βx )X = 0 W X = W X 37

r V( α)v( β) V( α) = V( β) = σ X ( W )( X ) ( W X ) σ = (1 r ) W σ W ( W )( X ) ( W X ) σ = (1 r ) X r 1 1 (r 1 )V( α)v( β) = = W X Y β + β 3 = 1 H 0 : β + β 3 = 1 H 1 : β + β 3 1 0 0 + 1 Y = α + βx + γd + δd X + u, D = { 0, = 1,,, 0 1, = 0 + 1, 0 +,, 0 + 1 H 0 : γ = δ = 0 H 1 : γ 0δ 0 1.. (R R ) t 3. () 4. 10 F F 10.1 1 Q K L log(q ) = β 1 + β log(k ) + β 3 log(l ) + u, 3 Y = α + βx + γz + u, X Z Y H 0 : β = γ = 0 H 1 : β 0γ 0 10. U χ ()V χ (m)u V F = U/ F (, m) V/m 38

10.3 Y = β 1 X 1 + β X + + β k X k + u, β 1, β,, β k G û ũ H 0 : β k G+1 = = β k = 0 H 1 : H 0 Y = β 1 X 1 + β X + + β k G X k G, + β k G+1 X k G+1, + + β k X k + u, û () Y = β 1 X 1 + β X + + β k G X k G, + u, ũ () ũ 1. H 0 û χ (G) ( ) û. χ ( k) () σ ũ 3. û () σ σ 4. ( ũ û )/G û /( k) () û σ 1 F (G, k), log(q ) = β 1 + β log(k ) + β 3 log(l ) + u, β + β 3 = 1 log( Q L ) = β 1 + β log( K L ) + u, Y = α + βx + γd + δd X + u, γ = δ = 0 Y = α + βx + u, 3 Y = α + βx + γz + u, β = γ = 0 Y = α + u, 11 11.1 1. (Y ) (C). C = α + βy () 3. α β 4. α () β 1 5. α, β 6. C Y () 7. 10 1. 1970 1996 ( ) 39

1: () () 1970 1996 3: 1970 37784.1 45913. 35. 1971 4571.6 51944.3 37.5 197 4914.1 6045.4 39.7 1973 59366.1 7494.8 44.1 1974 7178.1 93833. 53.3 1975 83591.1 10871.8 59.4 1976 94443.7 13540.9 65. 1977 105397.8 135318.4 70.1 1978 115960.3 14744. 73.5 1979 17600.9 157071.1 76.0 1980 138585.0 169931.5 81.6 1981 147103.4 181349. 85.4 198 157994.0 190611.5 87.7 1983 166631.6 199587.8 89.5 1984 175383.4 09451.9 91.8 1985 185335.1 0655.6 93.9 1986 193069.6 9938.8 94.8 1987 007.8 3594.0 95.3 1988 1939.9 47159.7 95.8 1989 71. 63940.5 97.7 1990 43035.7 80133.0 100.0 1991 55531.8 9751.9 10.5 199 65701.6 30956.6 104.5 1993 7075.3 31701.6 105.9 1994 79538.7 35655.7 106.7 1995 8345.4 331967.5 106. 1996 91374.8 34303.0 106.0 C 300 00 100 0 100 00 300 Y. 3. () 4. = ( 1 100 = /100 ) 5. 6. C Y 7. 1990 100 1990 8. 1990 1990 40

C = α + βy + u, u N(0, σ ), u C = 316.7 (3844.54) +.93354 (.016333) Y, R =.99406, R =.9910, s = 4557.04, DW =.89838, 1. α 316.7 β 0.93354. α, β α, β (a) 7 7 = 5 (b) α = 0.05 t α/ (5) =.060α = 0.01 t α/ (5) =.787 (c) 0.05 H 0 : α = 0H 1 : α 0 316.7 3844.54 = 6.039 < t 0.05(5) =.060, 0.05 H 0 : β = 0H 1 : β 0.93354.016333 = 57.16 > t 0.05(5) =.060, (d) α β (e) β > 0 β < 1 H 0 : β = 1H 1 : β 1 () (f) (g) = 3. s = 4557.04 u σ () 4. R = 0.9910 1 5. DW = 7, k = 5% dl = 1.3, du = 1.47 5% (a) DW < 1.3 (b) 1.3 DW < 1.47 (c) 1.47 DW <.53 (d).53 DW <.68 (e).68 DW DW = 0.89838 = = t(5) () C = α + βy + u, u = ρu 1 + ɛ, ɛ ρ ρ DW ρ = 1 DW = 1.898375 =.855081 C = C ρc 1, Y = Y ρy 1, 41

C = α + βy + ɛ, ɛ N(0, σ ɛ ), α = α(1 ρ) C = 3679.78 (183.63) +.930350 (.05401) Y, R =.988, R =.919180, s = 3.08, DW = 1.35684, 1. 6 6 = 4. α α = α 1 ρ 3679.78 = 539.0 1.855081 β.930350 3. α α α α 0.05 3679.78 183.63 = 1.685 > t 0.05(4) =.064, α 4. α 5. 6. β 0.05.930350.05401 = 17.5 > t 0.05(4) =.064, 7. s = 3.08 ɛ σ ɛ () 8. R 0.9910.919180 = R = (4557.04 3.08) 9. DW 1.35684 1.3 DW < 1.47 = = Stata http://www.eco.osaka-u.ac.jp/~tazak/class/011/ecoome/cos.csv ----------------------------------------------------------. tsset year tme varable: year, 1970 to 1996 delta: 1 ut. ge ryd=yd/(pcos/100). ge rcos=cos/(pcos/100). reg rcos ryd Source SS df MS Number of obs = 7 -------------+------------------------------ F( 1, 5) = 367.05 Model 6.7845e+10 1 6.7845e+10 Prob > F = 0.0000 Resdual 51916448 5 0766569.9 R-squared = 0.994 -------------+------------------------------ Adj R-squared = 0.991 Total 6.8365e+10 6.694e+09 Root MSE = 4557 rcos Coef. Std. Err. t P> t [95% Cof. Iterval] ryd.93354.016336 57.16 0.000.8999045.9671799 _cos -316.75 3844.539-6.04 0.000-31134.7-1598.77 Durb-Watso d-statstc(, 7) =.898375. ge rho=1-0.5*.898375. ge drcos=rcos-rho*l.rcos (1 mssg value geerated). ge dryd=ryd-rho*l.ryd (1 mssg value geerated). reg drcos dryd Source SS df MS Number of obs = 6 -------------+------------------------------ F( 1, 4) = 61.9 Model 1.3558e+09 1 1.3558e+09 Prob > F = 0.0000 Resdual 14530197 4 5188758.1 R-squared = 0.9159 -------------+------------------------------ Adj R-squared = 0.914 Total 1.4803e+09 5 591957 Root MSE = 77.9 drcos Coef. Std. Err. t P> t [95% Cof. Iterval] 4

dryd.9315055.05766 16.16 0.000.815708 1.05044 _cos -3731.757 353.61-1.59 0.16-8588.649 115.134 DRCONS = RCONS ρ RCONS 1 DRY = RY ρ RY 1 ( ρ = 1.5DW = 1.5.898375) 11. 1kg 1kg 1g 100g 1g 100g () Q 1 P 1 Q P Q 3 P 3 E 198 164. 435.98 4161 54.04 36854 45.3 303804 0.811 1983 160.14 448.0 41745 55.88 3649 47.97 311447 0.85 1984 158.06 461.69 40890 57.6 36500 49.1 319589 0.844 1985 154.51 477.41 39545 59.4 36099 50.0 377373 0.861 1986 150.96 48.80 3953 60.86 35859 50.74 3316493 0.867 1987 14.60 48.67 38710 61.53 34576 51.83 337136 0.868 1988 13.04 478.40 3918 61.75 33971 5.65 3493468 0.874 1989 18.40 486.37 3997 63.99 33603 54.71 35905 0.893 1990 15.78 497.33 39157 66.71 3890 57.14 3734084 0.91 1991 13.8 499.36 39659 69.57 3615 61.44 395358 0.951 199 10.58 516.05 39697 70.75 33401 61.06 4003931 0.967 1993 11.93 536.85 4009 70.51 35085 59.80 40955 0.979 1994 107.99 587.50 40458 71.08 35760 58.37 4006086 0.986 1995 106.4 496.64 38766 71.97 35096 56.77 3948741 0.985 1996 104.91 476.6 38436 7.74 34804 55.90 3946187 0.986 1997 10.81 460.70 38333 74.39 35061 56.77 3999759 1.004 1998 103.53 439.4 3887 74.10 34956 55.98 393835 1.010 1999 101.99 47.60 3946 73.00 34963 55.37 3876091 1.007 000 100.40 406.8 38480 71.47 347 53.83 3805600 1.000 001 97.83 394.67 37554 70.1 34753 5.5 370498 0.993 00 95.15 391.8 4377 61.34 36493 50.11 3673550 0.984 003 94.83 398.37 45876 60.1 3730 48.93 3631473 0.981 004 89.0 46.1 46653 59.9 37957 47.7 3650436 0.981 ( 16 )() ( 16 )() 43

Q 1 = 397.0 (8.89) + 0.184 (0.9) 0.00018 (9.35) P + 5.614 (5.57) E + 0.194 (4.39) P 3 P 1 s = 6.86, R = 0.93, R = 0.917, DW = 1.53 Q = 61946.7 (13.9) 710.4 (11.4) + 0.00893 (4.6) P 15.5 (1.51) E + 4.014 (0.91) P 3 P 1 s = 685.1, R = 0.931, R = 0.915, DW = 1.334 Q 3 = 5664.5 (16.9) 48.8 (1.04) 0.00098 (0.67) P 403.6 (5.3) E + 17.9 (5.40) P 3 P 1 s = 515.7, R = 0.885, R = 0.860, DW = 1.044 log Q 1 = 70.9 (10.4) 0.018 (0.06) 5.35 (10.4) log P +.68 (6.46) log E + 0.705 (4.53) log P 3 log P 1 s = 0.048, R = 0.949, R = 0.937, DW = 1.713 log Q = 4.35 (1.88) 1.135 (10.9) + 0.753 (4.33) log P 0.184 (1.31) log E + 0.063 (1.1) log P 3 log P 1 s = 0.016, R = 0.931, R = 0.916, DW = 1.317 log Q 3 = 14.5 (6.87) 0.05 (0.7) 0.188 (1.19) log P 0.68 (5.30) log E + 0.66 (5.51) log P 3 log P 1 s = 0.015, R = 0.883, R = 0.856, DW = 1.067 () log Q 1 = 70.9 (10.9) +.67 (7.69) 5.36 (11.) log P 3 log E + 0.709 (5.3) log P 1 s = 0.047, R = 0.949, R = 0.941, DW = 1.74 log Q 3 = 14.7 (7.4) 0.700 (6.49) 0.00 (1.35) log P 3 log E + 0.71 (6.45) log P 1 s = 0.014, R = 0.88, R = 0.863, DW = 1.1 Stata http://www.eco.osaka-u.ac.jp/~tazak/class/011/ecoome/demad.csv ----------------------------------------------------------. tsset year tme varable: year, 198 to 004 delta: 1 ut. ge re=e/p. ge rp1=p1/p. ge rp=p/p. ge rp3=p3/p. reg q1 re rp1 rp rp3 Source SS df MS Number of obs = 3 -------------+------------------------------ F( 4, 18) = 6.3 Model 11605.1437 4 901.859 Prob > F = 0.0000 Resdual 837.9394 18 46.551941 R-squared = 0.937 -------------+------------------------------ Adj R-squared = 0.9177 Total 1443.067 565.593953 Root MSE = 6.89 q1 Coef. Std. Err. t P> t [95% Cof. Iterval] re -.0001813.000019-9.43 0.000 -.00017 -.0001409 rp1.193549.043943 4.40 0.000.10104.858457 rp.185145.61349 0.9 0.77-1.1891 1.4879 rp3 5.63917 1.003036 5.61 0.000 3.516618 7.73117 _cos 397.3071 44.31083 8.97 0.000 304.135 490.4007 Durb-Watso d-statstc( 5, 3) = 1.538178. reg q re rp1 rp rp3 Source SS df MS Number of obs = 3 -------------+------------------------------ F( 4, 18) = 60.6 Model 113479303 4 836985.8 Prob > F = 0.0000 Resdual 844108.48 18 468006.07 R-squared = 0.9309 -------------+------------------------------ Adj R-squared = 0.9155 Total 1190341 5541064.17 Root MSE = 684.11 q Coef. Std. Err. t P> t [95% Cof. Iterval] re.008994.001988 4.63 0.000.004877.019817 rp1 3.975941 4.40605 0.90 0.379-5.8083 13.371 rp -710.4687 6.30106-11.40 0.000-841.3584-579.5791 rp3-15.137 100.5718-1.51 0.148-363.417 59.16979 _cos 61930.6 444.933 13.94 0.000 5596.36 7164.87 Durb-Watso d-statstc( 5, 3) = 1.340373. reg q3 re rp1 rp rp3 Source SS df MS Number of obs = 3 -------------+------------------------------ F( 4, 18) = 34.8 Model 37006848.3 4 95171.07 Prob > F = 0.0000 Resdual 478689.6 18 65704.979 R-squared = 0.8856 -------------+------------------------------ Adj R-squared = 0.8601 Total 41789537.9 189954.45 Root MSE = 515.47 q3 Coef. Std. Err. t P> t [95% Cof. Iterval] 44

re -.00097.0014534-0.67 0.513 -.004034.000834 rp1 17.8958 3.31989 5.39 0.000 10.91775 4.86741 rp -48.87487 46.9486-1.04 0.31-147.498 49.74843 rp3-403.4164 75.7795-5.3 0.000-56.67-44.101 _cos 56606.16 3347.679 16.91 0.000 4957.95 63639.37 Durb-Watso d-statstc( 5, 3) = 1.04784. ge lq1=log(q1). ge lq=log(q). ge lq3=log(q3). ge lre=log(re). ge lrp1=log(rp1). ge lrp=log(rp). ge lrp3=log(rp3). reg lq1 lre lrp1 lrp lrp3 Source SS df MS Number of obs = 3 -------------+------------------------------ F( 4, 18) = 84.4 Model.76657485 4.191643713 Prob > F = 0.0000 Resdual.04086074 18.0070041 R-squared = 0.9494 -------------+------------------------------ Adj R-squared = 0.9381 Total.80743559.036701618 Root MSE =.04764 lq1 Coef. Std. Err. t P> t [95% Cof. Iterval] lre -5.364744.475081-11.9 0.000-6.359103-4.370386 lrp1.7074307.1347484 5.5 0.000.45399.989463 lrp3.6747.3456 7.75 0.000 1.95169 3.39691 _cos 70.98983 6.47565 10.96 0.000 57.43613 84.5435 Durb-Watso d-statstc( 4, 3) = 1.73314. reg lq3 lre lrp1 lrp3 Source SS df MS Number of obs = 3 -------------+------------------------------ F( 3, 19) = 47.40 Model.0983539 3.009945131 Prob > F = 0.0000 Resdual.00398684 19.00009833 R-squared = 0.881 -------------+------------------------------ Adj R-squared = 0.8635 Total.033816.001537373 Root MSE =.01449 lq3 Coef. Std. Err. t P> t [95% Cof. Iterval] lre -.1989407.148381-1.34 0.196 -.509508.111667 lrp1.709758.040859 6.44 0.000.18889.359065 lrp3 -.6995357.1078355-6.49 0.000 -.9538 -.4738335 _cos 14.63644.0536 7.4 0.000 10.403 18.86965 Durb-Watso d-statstc( 4, 3) = 1.17541 lq1 Coef. Std. Err. t P> t [95% Cof. Iterval] lre -5.35563.509073-10.5 0.000-6.4548-4.85818 lrp1.7030891.1547336 4.54 0.000.3780058 1.0817 lrp -.01919.3056177-0.06 0.951 -.661689.6889 lrp3.687511.416367 6.51 0.000 1.80594 3.55449 _cos 70.9068 6.78413 10.45 0.000 56.65335 85.1591 Durb-Watso d-statstc( 5, 3) = 1.70916. reg lq lre lrp1 lrp lrp3 11.3 Source SS df MS Number of obs = 3 -------------+------------------------------ F( 4, 18) = 61.41 Model.06470875 4.01618019 Prob > F = 0.0000 Resdual.00474568 18.00063476 R-squared = 0.9317 -------------+------------------------------ Adj R-squared = 0.9166 Total.06946344.00315749 Root MSE =.0163 lq Coef. Std. Err. t P> t [95% Cof. Iterval] lre.75305.1734795 4.34 0.000.388558 1.11749 lrp1.0633603.057155 1.0 0.45 -.0473909.1741114 lrp -1.135159.1041195-10.90 0.000-1.353906 -.9164119 lrp3 -.1838809.1405793-1.31 0.07 -.47971.1114654 _cos 4.34689.31155 1.88 0.076 -.5094776 9.0055 Durb-Watso d-statstc( 5, 3) = 1.33931. reg lq3 lre lrp1 lrp lrp3 Source SS df MS Number of obs = 3 -------------+------------------------------ F( 4, 18) = 33.83 Model.0985108 4.0074680 Prob > F = 0.0000 Resdual.003971008 18.000061 R-squared = 0.886 -------------+------------------------------ Adj R-squared = 0.8565 Total.033816.001537373 Root MSE =.01485 lq3 Coef. Std. Err. t P> t [95% Cof. Iterval] lre -.1868154.15874-1.18 0.55 -.503.146689 lrp1.65044.04837 5.50 0.000.1638618.366547 lrp -.055097.095743-0.7 0.79 -.56736.174654 lrp3 -.681934.186368-5.30 0.000 -.9519 -.4116784 _cos 14.5537.114908 6.87 0.000 10.081 18.96863 Durb-Watso d-statstc( 5, 3) = 1.071649. reg lq1 lre lrp1 lrp3 Source SS df MS Number of obs = 3 -------------+------------------------------ F( 3, 19) = 118.79 Model.76656590 3.5551967 Prob > F = 0.0000 Resdual.040869691 19.00151036 R-squared = 0.9494 -------------+------------------------------ Adj R-squared = 0.9414 Total.80743559.036701618 Root MSE =.04638 45

_cos -1883.071 730.508 -.58 0.010-3315.696-450.4449 Durb-Watso d-statstc( 3, 08) =.01141. reg kabu exrate r l.kabu Source SS df MS Number of obs = 07 -------------+------------------------------ F( 3, 03) =. Model.4997e+10 3 8.333e+09 Prob > F = 0.0000 Resdual 939878.6 03 46119.05 R-squared = 0.9963 -------------+------------------------------ Adj R-squared = 0.9963 Total.5090e+10 06 1384078. Root MSE = 14.75 kabu Coef. Std. Err. t P> t [95% Cof. Iterval] exrate.1643596.595945 0.31 0.756 -.874481 1.0967 r 36.51834 18.651 1.96 0.050 -.060989 73.09767 kabu L1..9945113.00064 481.84 0.000.9904636.998559 _cos -.35914 68.05383-0.03 0.97-135.8 131.1038 Kabu = 1883.1 (.58) + 46.3 (8.7) ExRate + 680.6 (51.7) s = 309.9, R = 0.570, R = 0.570, DW = 0.01 Kabu =.36 (0.03) 0.995 (48) Kabu 1 + 0.164 (0.31) ExRate + 36.5 (1.96) R R + s = 14.8, R = 0.996, R = 0.996, DW =.100 Kabu = 8.11 (0.119) 0.091 (0.173) ExRate 0.86 (0.070) s = 15.1, R = 0.000015, R = 0.00097, DW =.106 Stata http://www.eco.osaka-u.ac.jp/~tazak/class/011/ecoome/kke.csv ----------------------------------------------------------. ge tme=_. tsset tme tme varable: tme, 1 to 08 delta: 1 ut. reg kabu exrate r Source SS df MS Number of obs = 08 -------------+------------------------------ F(, 05) = 1341.74 Model 1.4318e+10 7.1590e+09 Prob > F = 0.0000 Resdual 1.0805e+10 05 5335630.99 R-squared = 0.5699 -------------+------------------------------ Adj R-squared = 0.5695 Total.513e+10 07 1394036.8 Root MSE = 309.9 kabu Coef. Std. Err. t P> t [95% Cof. Iterval] exrate 46.305 5.601817 8.7 0.000 35.3193 57.9115 r 680.61 131.6017 51.69 0.000 6544.51 7060.699 R Durb-Watso d-statstc( 4, 07) =.10048. reg dkabu exrate r f t(,08) Source SS df MS Number of obs = 07 -------------+------------------------------ F(, 04) = 0.0 Model 1443.3541 71.617703 Prob > F = 0.9845 Resdual 936497.9 04 4657.3754 R-squared = 0.0000 -------------+------------------------------ Adj R-squared = -0.0010 Total 9366371.1 06 461.44 Root MSE = 15.08 dkabu Coef. Std. Err. t P> t [95% Cof. Iterval] exrate -.09060.51644-0.17 0.86-1.113579.933746 r -.864553 1.7849-0.07 0.944-4.945 3.1734 _cos 8.109931 68.0417 0.1 0.905-15.391 141.549 Durb-Watso d-statstc( 3, 07) =.106308 1 1.1 () x 1, x,, x X 1, X,, X E(X ) = µv(x ) = σ (µ, σ ) x 1, x,, x µ m µ (x µ) µ µ µ = 1 x µ x 46

d (x µ) = 0 dµ µ µ x X µ = 1 X µ X ( µ ) (Y α βx ) m α,β (Y α βx ) = 0 α (Y α βx ) = 0 β α, β 1. X 1, X,, X f(x) f(x; θ) θ θ = (µ, σ ) X 1, X,, X f(x 1, x,, x ; θ) f(x ; θ) x 1, x,, x f(x ; θ) θ l(θ) = f(x ; θ) l(θ) max θ l(θ) θ θ = θ(x 1, x,, x ) x 1, x,, x X 1, X,, X θ = θ(x 1, X,, X ) max θ max θ l(θ) log l(θ) θ log l(θ) θ N(θ, σ θ) σθ 1 = [( d log E f(x ; θ) ) ] dθ 1 = [ d E log f(x ; θ) ] dθ θ (k 1) θ N(θ, Σ θ ) ( [( log f(x ; θ) )( log f(x ; θ) ) ]) 1 Σ θ = E θ θ ( [ log f(x ; θ) = E θ θ ]) 1 47

1 N(µ, σ ) x 1, x,, x (1) σ µ () σ µ σ N(µ, σ ) f(x; µ, σ 1 ( ) = exp 1 (x µ)) πσ σ X 1, X,, X f(x 1, x,, x ; µ, σ ) = f(x ; µ, σ ) 1 ( exp 1 πσ σ (x µ) ) = (πσ ) exp ( 1 σ (x µ) ) (1) σ l(µ) l(µ) = (πσ ) exp ( 1 σ (x µ) ) l(µ) µ log l(µ) µ log l(µ) = log(πσ ) 1 σ d log l(µ) dµ = 1 σ (x µ) = 0 (x µ) µ µ µ µ µ = 1 x x x 1, x,, x X 1, X,, X µ µ = 1 X X µ log f(x ; µ) = 1 log(πσ ) 1 σ (X µ) d log f(x ; µ) = 1 dµ σ (X µ) ( d log f(x ; µ) ) 1 = dµ σ 4 (X µ) [( d log f(x ; µ) ) ] E = 1 dµ σ 4 E[(X µ) ] = 1 σ µ N(µ, σ µ) σ µ = 1 [( d log E f(x ; µ) ) ] = σ dµ µ N(µ, σ µ) () σ µ σ ( l(µ, σ ) = (πσ ) exp 1 σ (x µ) ) log l(µ, σ ) = log(π) log σ 1 σ (x µ) µ σ log l(µ, σ ) µ = 1 σ (x µ) = 0 log l(µ, σ ) σ = 1 σ + 1 σ 4 (x µ) = 0 48

µ, σ µ, σ µ = 1 σ = 1 x x (x µ) 1 (x x) x 1, x,, x X 1, X,, X µ, σ µ = 1 σ = 1 X X (X µ) 1 (X X) σ σ σ S = 1 (X X) 1 θ = (µ, σ ) θ N(θ, Σ θ ) ( [ log f(x ; θ) Σ θ = E θ θ ]) 1 log f(x ; θ) θ θ log f(x ; θ) log f(x ; θ) µ = µ σ log f(x ; θ) log f(x ; θ) σ µ (σ ) ( 1 = σ 1 σ 4 (X ) µ) 1 σ 4 (X 1 µ) σ 4 1 σ 6 (X µ) [ log f(x ; θ) ] E θ θ ( 1 = σ 1 σ 4 E(X ) µ) 1 σ 4 E(X 1 µ) σ 4 1 σ 6 E[(X µ) ] ( 1 ) = σ 0 0 1 σ 4 ( [ log f(x ; θ) ]) 1 Σ θ = E θ θ σ 0 = σ 4 0 µσ µ = (1/) X σ = (1/) (X X) ( ) ( ( ) σ ) µ µ 0 σ N σ, σ 4 0 log f(x ; θ) = 1 log(π) 1 log(σ ) 1 σ (X µ) log f(x ; θ) θ = = log f(x ; θ) µ log f(x ; θ) σ ( 1 σ (X µ) 1 σ + 1 σ 4 (X µ) ) X 1, X,, X p X f(x; p) = p x (1 p) 1 x x = 0, 1 l(p) = f(x ; p) = p x (1 p) 1 x 49

log l(p) = log f(x ; p) = log(p) x + log(1 p) (1 x ) = log(p) x + log(1 p)( x ) log l(p) p d log l(p) = 1 x 1 dp p 1 p ( x ) = 0 p p p p = 1 x x X p p p = 1 X p log f(x ; p) = X log(p) + (1 X ) log(1 p) d log f(x ; p) dp [( d log f(x ; p) ) ] E dp E[(X p) ] = = X p 1 X 1 p = X p p(1 p) = = E[(X p) ] p (1 p) 1 (x p) f(x ; p) x =0 1 (x p) p x (1 p) 1 x x =0 = p (1 p) + (1 p) p = p(1 p) 1 [( d log E f(x ; p) ) ] = dp p N ( p, p(1 p) ) p(1 p) 3 X 1, X,, X λ X f(x; λ) = λx e λ x! l(λ) = f(x ; λ) = log l(λ) = x = 0, 1,, λ x e λ log f(x ; λ) = log(λ) x! x λ log l(λ) p d log l(λ) dλ = 1 λ x = 0 log(x!) λ λ λ λ = 1 x x X λ λ λ = 1 X λ log f(x ; λ) = X log(λ) λ log(x!) d log f(x ; λ) dλ d log f(x ; λ) dλ = X λ 1 = X λ ( d log f(x ; λ) ) E dλ = E(X ) λ 50

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