Microsoft Word - 計量研修テキスト_第5版）.doc

Q3-1-1 テキスト P59 10.8.3.2.1.0 -.1 -.2 10.4 10.0 9.6 9.2 8.8 -.3 76 78 80 82 84 86 88 90 92 94 96 98 R e s i d u al A c tual Fi tte d Dependent Variable: LOG(TAXH) Date: 10/26/05 Time: 15:42 Sample: 1975 1998 Included observations: 24 LOG(YNH) 1.264042 0.079310 15.93808 0.0000 C -5.594743 0.977449-5.723819 0.0000 R-squared 0.920296 Mean dependent var 9.977606 Adjusted R-squared 0.916673 S.D. dependent var 0.472175 S.E. of regression 0.136300 Akaike info criterion -1.068265 Sum squared resid 0.408708 Schwarz criterion -0.970094 Log likelihood 14.81918 F-statistic 254.0223 Durbin-Watson stat 0.203893 Prob(F-statistic) 0.000000 誤差項に 1 階の自己相関が発生している ( 視覚的には上記図参照 ) 224

Q3-1-2 テキスト P59 Omitted Variable の追加 : 新たに財産所得 [LOG(YAH)] を説明変数として加える 10.8 10.4 10.0.10.05 9.6 9.2 8.8.00 -.05 -.10 76 78 80 82 84 86 88 90 92 94 96 98 R e s i d ual A c tu a l Fi tted Dependent Variable: LOG(TAXH) Date: 10/26/05 Time: 15:47 Sample: 1975 1998 Included observations: 24 LOG(YNH) 0.916294 0.035528 25.79047 0.0000 LOG(YAH) 0.616127 0.044149 13.95566 0.0000 C -7.299619 0.335175-21.77856 0.0000 R-squared 0.992242 Mean dependent var 9.977606 Adjusted R-squared 0.991504 S.D. dependent var 0.472175 S.E. of regression 0.043523 Akaike info criterion -3.314578 Sum squared resid 0.039780 Schwarz criterion -3.167321 Log likelihood 42.77494 F-statistic 1343.016 Durbin-Watson stat 1.341033 Prob(F-statistic) 0.000000 DW 値が改善 Omitted Variable(=YAH) 225

Q3-1-3 テキスト P61 誤差項に 1 階の自己相関があることを考慮した最尤法による推定 10.8 10.4 10.0 9.6.08.04 9.2 8.8.00 -.04 -.08 -.12 76 78 80 82 84 86 88 90 92 94 96 98 R e s i d ual A c tu a l Fi tted Dependent Variable: LOG(TAXH) Date: 10/26/05 Time: 15:50 Sample (adjusted): 1976 1998 Included observations: 23 after adjustments Convergence achieved after 13 iterations LOG(YNH) 0.925516 0.058818 15.73535 0.0000 LOG(YAH) 0.565030 0.078030 7.241160 0.0000 C -6.917079 0.839160-8.242857 0.0000 AR(1) 0.423531 0.287921 1.470996 0.1577 R-squared 0.990946 Mean dependent var 10.02293 Adjusted R-squared 0.989517 S.D. dependent var 0.426075 S.E. of regression 0.043625 Akaike info criterion -3.269604 Sum squared resid 0.036160 Schwarz criterion -3.072126 Log likelihood 41.60044 F-statistic 693.1926 Durbin-Watson stat 1.572824 Prob(F-statistic) 0.000000 誤差項の1 階自己相関係数 Inverted AR Roots.42 BM 最尤法による推定 226

Q3-2-1 テキスト P61 データの季節性を考慮しない推定 76000 72000 6000 4000 2000 0 68000 64000 60000 56000-2000 -4000 90 91 92 93 94 95 96 97 98 99 00 01 02 R es i dua l A c tu a l Fi tte d Dependent Variable: C95 Date: 10/26/05 Time: 15:57 Sample: 1990Q1 2002Q4 Included observations: 52 RYLE 0.225107 0.035320 6.373288 0.0000 RDEPO 0.022780 0.002511 9.070765 0.0000 C 37201.96 2197.414 16.92988 0.0000 R-squared 0.825474 Mean dependent var 67935.34 Adjusted R-squared 0.818350 S.D. dependent var 4380.313 S.E. of regression 1866.907 Akaike info criterion 17.95791 Sum squared resid 1.71E+08 Schwarz criterion 18.07049 Log likelihood -463.9058 F-statistic 115.8799 Durbin-Watson stat 1.999396 Prob(F-statistic) 0.000000 一見誤差項の自己相関の問題はないように思えるが 227

Q3-2-2 テキスト P63 Breusch-Godfrey 検定 Breusch-Godfrey Serial Correlation LM Test: F-statistic 9.865135 Prob. F(8,41) 0.000000 Obs*R-squared 34.22164 Prob. Chi-Square(8) 0.000037 Test Equation: Dependent Variable: RESID Date: 11/11/05 Time: 14:50 Presample missing value lagged residuals set to zero. RYLE 0.006128 0.047806 0.128181 0.8986 RDEPO -0.000306 0.002158-0.141920 0.8878 C -207.4857 2373.977-0.087400 0.9308 RESID(-1) 0.087539 0.156865 0.558052 0.5798 RESID(-2) 0.029607 0.163283 0.181321 0.8570 RESID(-3) 0.146358 0.185603 0.788553 0.4349 RESID(-4) 0.747395 0.155621 4.802669 0.0000 RESID(-5) -0.154662 0.159225-0.971342 0.3371 RESID(-6) -0.050635 0.169050-0.299525 0.7661 RESID(-7) -0.254810 0.164733-1.546801 0.1296 RESID(-8) 0.086977 0.163270 0.532717 0.5971 4 期ラグとの相関関係強い R-squared 0.658108 Mean dependent var -4.31E-12 Adjusted R-squared 0.574720 S.D. dependent var 1829.935 S.E. of regression 1193.364 Akaike info criterion 17.19235 Sum squared resid 58388834 Schwarz criterion 17.60511 Log likelihood -436.0010 F-statistic 7.892108 Durbin-Watson stat 1.961748 Prob(F-statistic) 0.000001 228

Q3-2-3 テキスト P64 データの季節性を考慮した推定 1: 季節ダミー変数 (Q1~Q3) 導入 76000 72000 68000 3000 2000 1000 0 64000 60000 56000-1000 -2000-3000 90 91 92 93 94 95 96 97 98 99 00 01 02 R es i dua l A c tu a l Fi tte d Dependent Variable: C95 Date: 10/26/05 Time: 16:01 Sample: 1990Q1 2002Q4 Included observations: 52 RYLE 0.657721 0.102308 6.428806 0.0000 RDEPO 0.008269 0.003482 2.374602 0.0218 Q1 8719.269 2108.725 4.134853 0.0001 Q2 430.3446 1000.776 0.430011 0.6692 Q3 4517.505 1224.668 3.688760 0.0006 C 14050.98 5868.881 2.394149 0.0208 R-squared 0.954293 Mean dependent var 67935.34 Adjusted R-squared 0.949325 S.D. dependent var 4380.313 S.E. of regression 986.0591 Akaike info criterion 16.73348 Sum squared resid 44726378 Schwarz criterion 16.95862 Log likelihood -429.0704 F-statistic 192.0818 Durbin-Watson stat 1.789597 Prob(F-statistic) 0.000000 自由度修正済み決定係数の向上 229

Q3-2-3 続き 高階自己相関の検定 Breusch-Godfrey Serial Correlation LM Test: F-statistic 0.989192 Prob. F(8,38) 0.459711 Obs*R-squared 8.962582 Prob. Chi-Square(8) 0.345462 帰無仮説 : 高階の自己相関なし が棄却されない Test Equation: 高階自己相関なし Dependent Variable: RESID Date: 11/11/05 Time: 14:59 Presample missing value lagged residuals set to zero. RYLE -0.002311 0.131009-0.017644 0.9860 RDEPO 0.000190 0.004450 0.042659 0.9662 Q1-25.90941 2683.408-0.009655 0.9923 Q2 17.74427 1245.205 0.014250 0.9887 Q3-36.32161 1536.495-0.023639 0.9813 C 48.82814 7432.082 0.006570 0.9948 RESID(-1) 0.079197 0.162745 0.486633 0.6293 RESID(-2) 0.066109 0.167125 0.395565 0.6946 RESID(-3) 0.238338 0.167262 1.424935 0.1623 RESID(-4) 0.150854 0.175575 0.859199 0.3956 RESID(-5) -0.162026 0.184445-0.878451 0.3852 RESID(-6) -0.084119 0.178913-0.470166 0.6409 RESID(-7) 0.135445 0.172621 0.784640 0.4375 RESID(-8) -0.140330 0.174836-0.802634 0.4272 R-squared 0.172357 Mean dependent var 9.79E-13 Adjusted R-squared -0.110784 S.D. dependent var 936.4763 S.E. of regression 986.9871 Akaike info criterion 16.85200 Sum squared resid 37017458 Schwarz criterion 17.37733 Log likelihood -424.1519 F-statistic 0.608733 Durbin-Watson stat 1.921601 Prob(F-statistic) 0.831502 230

Q3-2-4 テキスト P65 高階自己相関を考慮した BM 最尤法 76000 72000 4000 2000 0 68000 64000 60000 56000-2000 -4000 91 92 93 94 95 96 97 98 99 00 01 02 R es i dua l A c tu a l Fi tte d Dependent Variable: C95 Date: 10/26/05 Time: 16:05 Sample (adjusted): 1991Q1 2002Q4 Included observations: 48 after adjustments Convergence achieved after 9 iterations RYLE 0.121915 0.103215 1.181180 0.2439 RDEPO 0.012172 0.009471 1.285122 0.2055 C 54243.32 12646.04 4.289353 0.0001 AR(4) 0.829315 0.081224 10.21025 0.0000 R-squared 0.935775 Mean dependent var 68521.24 Adjusted R-squared 0.931396 S.D. dependent var 3956.323 S.E. of regression 1036.252 Akaike info criterion 16.80426 Sum squared resid 47248042 Schwarz criterion 16.96020 Log likelihood -399.3023 F-statistic 213.6980 Durbin-Watson stat 2.024239 Prob(F-statistic) 0.000000 Inverted AR Roots.95.00-.95 i -.00+.95i -.95 231

Q3-2-5 テキスト P65 季節調整をしたデータによる推定 76000 72000 68000 64000 60000 56000 90 91 92 93 94 95 96 97 98 99 00 01 02 C 9 5 C 9 5_ SA 84000 80000 76000 72000 68000 64000 60000 56000 52000 48000 90 91 92 93 94 95 96 97 98 99 00 01 02 RY L E RY L E _ S A 900000 800000 700000 600000 500000 400000 90 91 92 93 94 95 96 97 98 99 00 01 02 RDEPO RDEPO_SA 232

76000 72000 3000 2000 1000 0 68000 64000 60000 56000-1000 -2000 90 91 92 93 94 95 96 97 98 99 00 01 02 R es i dua l A c tu a l Fi tte d Dependent Variable: C95_SA Date: 10/26/05 Time: 16:08 Sample: 1990Q1 2002Q4 Included observations: 52 RYLE_SA 0.717351 0.095644 7.500220 0.0000 RDEPO_SA 0.006412 0.003253 1.971218 0.0544 C 14652.88 4502.330 3.254510 0.0021 R-squared 0.947083 Mean dependent var 67939.80 Adjusted R-squared 0.944924 S.D. dependent var 3722.866 S.E. of regression 873.6954 Akaike info criterion 16.43930 Sum squared resid 37403835 Schwarz criterion 16.55187 Log likelihood -424.4218 F-statistic 438.4936 Durbin-Watson stat 1.655109 Prob(F-statistic) 0.000000 233

Q3-3-1 テキスト P67 誤差項の不均一分散 4.0E+07 3.0E+07 2.0E+07 8000000 4000000 1.0E+07 0.0E+00 0-4000000 -8000000 5 10 15 20 25 30 35 40 45 R e s i d u a l A c tu al Fi t t e d Dependent Variable: C95 Date: 10/26/05 Time: 16:11 Sample: 1 47 Included observations: 47 Y95 0.402304 0.016823 23.91344 0.0000 C 895493.3 302864.5 2.956746 0.0049 R-squared 0.927049 Mean dependent var 5354219. Adjusted R-squared 0.925428 S.D. dependent var 5991758. S.E. of regression 1636223. Akaike info criterion 31.49530 Sum squared resid 1.20E+14 Schwarz criterion 31.57403 Log likelihood -738.1396 F-statistic 571.8524 Durbin-Watson stat 2.456389 Prob(F-statistic) 0.000000 234

Q3-3-2 テキスト P68 不均一分散の有無を確認する検定 White Heteroskedasticity Test: F-statistic 28.26965 Prob. F(2,44) 0.000000 Obs*R-squared 26.43093 Prob. Chi-Square(2) 0.000002 Test Equation: Dependent Variable: RESID^2 Date: 10/26/05 Time: 16:13 Sample: 1 47 Included observations: 47 C -1.77E+12 1.21E+12-1.464800 0.1501 Y95 405573.2 132664.8 3.057128 0.0038 Y95^2-0.000501 0.001674-0.299661 0.7658 R-squared 0.562360 Mean dependent var 2.56E+12 Adjusted R-squared 0.542467 S.D. dependent var 7.05E+12 S.E. of regression 4.77E+12 Akaike info criterion 61.28715 Sum squared resid 1.00E+27 Schwarz criterion 61.40524 Log likelihood -1437.248 F-statistic 28.26965 Durbin-Watson stat 1.657246 Prob(F-statistic) 0.000000 H 0 : 不均一分散なし H 1 : 不均一分散 帰無仮説 : H 0 が棄却される 誤差項の不均一分散がある 235

Q3-3-3 テキスト P70 残差の 2 乗系列と POP の 2 乗系列の相関 Dependent Variable: RES^2 Date: 10/26/05 Time: 16:17 Sample: 1 47 Included observations: 47 POP^2 0.214883 0.018736 11.46915 0.0000 R-squared 0.705958 Mean dependent var 2.56E+12 Adjusted R-squared 0.705958 S.D. dependent var 7.05E+12 S.E. of regression 3.83E+12 Akaike info criterion 60.80437 Sum squared resid 6.73E+26 Schwarz criterion 60.84373 Log likelihood -1427.903 Durbin-Watson stat 1.672270 加重最小 2 乗法による推定 Dependent Variable: C95 Date: 10/26/05 Time: 16:20 Sample: 1 47 Included observations: 47 Weighting series: POP Y95 0.306956 0.013314 23.05541 0.0000 C 5017366. 637411.4 7.871472 0.0000 Weighted Statistics R-squared 0.978550 Mean dependent var 10778244 Adjusted R-squared 0.978073 S.D. dependent var 23941334 S.E. of regression 3545171. Akaike info criterion 33.04169 Sum squared resid 5.66E+14 Schwarz criterion 33.12042 Log likelihood -774.4798 F-statistic 531.5521 Durbin-Watson stat 1.997821 Prob(F-statistic) 0.000000 Unweighted Statistics R-squared 0.607595 Mean dependent var 5354219. Adjusted R-squared 0.598875 S.D. dependent var 5991758. S.E. of regression 3794846. Sum squared resid 6.48E+14 Durbin-Watson stat 0.417828 236

Q3-3-4 テキスト P71 Dependent Variable: C95 Date: 10/26/05 Time: 16:21 Sample: 1 47 Included observations: 47 White Heteroskedasticity-Consistent Standard Errors & Covariance Y95 0.402304 0.048281 8.332499 0.0000 C 895493.3 402135.4 2.226845 0.0310 R-squared 0.927049 Mean dependent var 5354219. Adjusted R-squared 0.925428 S.D. dependent var 5991758. S.E. of regression 1636223. Akaike info criterion 31.49530 Sum squared resid 1.20E+14 Schwarz criterion 31.57403 Log likelihood -738.1396 F-statistic 571.8524 Durbin-Watson stat 2.456389 Prob(F-statistic) 0.000000 係数推定値は Q3-3-1 と不変 t値 = bˆ が Q3-3-1 より低く算出されている s b 237

演習 3 テキスト P72 Dependent Variable: I90 Date: 10/26/05 Time: 16:43 Sample: 1 46 Included observations: 46 YF90 0.189574 0.052880 3.585007 0.0009 KP90 0.075103 0.003093 24.27810 0.0000 KG90 0.064635 0.005678 11.38287 0.0000 C 31.66028 42.89274 0.738127 0.4645 R-squared 0.997026 Mean dependent var 2704.154 Adjusted R-squared 0.996814 S.D. dependent var 2854.576 S.E. of regression 161.1305 Akaike info criterion 13.08525 Sum squared resid 1090448. Schwarz criterion 13.24426 Log likelihood -296.9607 F-statistic 4693.810 Durbin-Watson stat 2.292439 Prob(F-statistic) 0.000000 238

不均一分散検定 ( クロス項を考慮した場合 ) White Heteroskedasticity Test: F-statistic 6.478611 Prob. F(9,36) 0.000020 Obs*R-squared 28.44042 Prob. Chi-Square(9) 0.000805 Test Equation: Dependent Variable: RESID^2 Date: 10/26/05 Time: 16:43 Sample: 1 46 Included observations: 46 C 16992.09 18222.33 0.932487 0.3573 YF90 31.46843 36.63880 0.858883 0.3961 YF90^2 0.052062 0.018870 2.758903 0.0091 YF90*KP90-0.000374 0.001887-0.198314 0.8439 YF90*KG90-0.018688 0.007469-2.502186 0.0170 KP90-3.472903 3.345424-1.038106 0.3061 KP90^2-0.000230 0.000105-2.183258 0.0356 KP90*KG90 0.001338 0.000439 3.044910 0.0043 KG90-0.434575 4.673830-0.092981 0.9264 KG90^2 0.000191 0.000161 1.186791 0.2431 R-squared 0.618270 Mean dependent var 23705.39 Adjusted R-squared 0.522838 S.D. dependent var 40041.43 S.E. of regression 27659.40 Akaike info criterion 23.48298 Sum squared resid 2.75E+10 Schwarz criterion 23.88051 Log likelihood -530.1085 F-statistic 6.478611 Durbin-Watson stat 2.418430 Prob(F-statistic) 0.000020 239

不均一分散の問題を解決する方法の一例 対数変換 Dependent Variable: LOG(I90) Date: 10/26/05 Time: 16:44 Sample: 1 46 Included observations: 46 LOG(YF90) 0.129030 0.051881 2.487053 0.0169 LOG(KP90) 0.471989 0.052387 9.009568 0.0000 LOG(KG90) 0.415954 0.048082 8.650974 0.0000 C -1.640984 0.196237-8.362252 0.0000 R-squared 0.990954 Mean dependent var 7.589861 Adjusted R-squared 0.990308 S.D. dependent var 0.727104 S.E. of regression 0.071581 Akaike info criterion -2.353023 Sum squared resid 0.215203 Schwarz criterion -2.194011 Log likelihood 58.11954 F-statistic 1533.693 Durbin-Watson stat 2.418097 Prob(F-statistic) 0.000000 240

White Heteroskedasticity Test:( 対数変換したもの ) F-statistic 0.536864 Prob. F(9,36) 0.837797 Obs*R-squared 5.443347 Prob. Chi-Square(9) 0.794076 Test Equation: Dependent Variable: RESID^2 Date: 10/26/05 Time: 16:44 Sample: 1 46 Included observations: 46 帰無仮説 : 均一分散 は棄却されない C 0.099511 0.237257 0.419425 0.6774 LOG(YF90) 0.065050 0.111033 0.585862 0.5616 (LOG(YF90))^2 0.016748 0.015752 1.063247 0.2948 (LOG(YF90))*(LOG(KP90)) 0.003019 0.026549 0.113696 0.9101 (LOG(YF90))*(LOG(KG90)) -0.037112 0.026797-1.384943 0.1746 LOG(KP90) -0.020220 0.086660-0.233331 0.8168 (LOG(KP90))^2-0.007940 0.017536-0.452806 0.6534 (LOG(KP90))*(LOG(KG90)) 0.016769 0.025401 0.660176 0.5133 LOG(KG90) -0.050493 0.084294-0.599012 0.5529 (LOG(KG90))^2 0.008580 0.010766 0.796970 0.4307 R-squared 0.118334 Mean dependent var 0.004678 Adjusted R-squared -0.102083 S.D. dependent var 0.005723 S.E. of regression 0.006008 Akaike info criterion -7.201660 Sum squared resid 0.001300 Schwarz criterion -6.804129 Log likelihood 175.6382 F-statistic 0.536864 Durbin-Watson stat 2.014904 Prob(F-statistic) 0.837797 241