Q8-1 テキスト P131 Engle-Granger 検定 Dependent Variable: RM2 Date: 11/04/05 Time: 15:15 Sample: 1967Q1 1999Q1 Included observations: 129 RGDP 0.012792 0.000194 65.92203 0.0000 R -95.45715 11.33648-8.420349 0.0000 C -443.4312 124.8461-3.551821 0.0005 R-squared 0.989294 Mean dependent var 3124.543 Adjusted R-squared 0.989124 S.D. dependent var 1529.608 S.E. of regression 159.5169 Akaike info criterion 13.00516 Sum squared resid 3206149. Schwarz criterion 13.07166 Log likelihood -835.8327 F-statistic 5821.739 Durbin-Watson stat 0.677246 Prob(F-statistic) 0.000000 共和分なしが棄却 上記パラメーターの係数推定値は 超一致性 という性質を有する ( つまり共和分あり ) ( 上記パラメーターの計数推定値を用いたときの残差系列が定常である (= 共和分あり ) かどうか以下にて確認を行う ) 283
Null Hypothesis: RESID01 has a unit root Exogenous: None Lag Length: 4 (Automatic based on SIC, MAXLAG=12) t-statistic Prob.* Augmented Dickey-Fuller test statistic -3.861180 0.0002 Test critical values: 1% level -2.583744 5% level -1.943427 10% level -1.615011 *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(RESID01) Date: 11/04/05 Time: 15:19 Sample (adjusted): 1968Q2 1999Q1 Included observations: 124 after adjustments この臨界値は利用不可 Mackinnon 表 (p6) を用いて正しい臨界値を算出 RESID01(-1) -0.189323 0.049032-3.861180 0.0002 D(RESID01(-1)) -0.096813 0.083308-1.162111 0.2475 D(RESID01(-2)) 0.224583 0.081044 2.771140 0.0065 D(RESID01(-3)) -0.038088 0.083831-0.454342 0.6504 D(RESID01(-4)) 0.524326 0.080090 6.546694 0.0000 R-squared 0.740254 Mean dependent var 1.251927 Adjusted R-squared 0.731523 S.D. dependent var 132.5547 S.E. of regression 68.68298 Akaike info criterion 11.33637 Sum squared resid 561364.9 Schwarz criterion 11.45009 Log likelihood -697.8548 Durbin-Watson stat 1.857663 τ= -3.7429 +(1/124) (-8.352) +(1/(124 124)) (-13.41)= -3.81-3.86<τ=-3.81 であり 有意水準 5 パーセントで H 0 : 共和分なしを棄却 284
Q8-2 テキストP139 Johansenタイプの共和分検定 Date: 11/04/05 Time: 17:45 Sample (adjusted): 1967Q3 1999Q1 Included observations: 127 after adjustments Trend assumption: Linear deterministic trend Series: RM2 RGDP R Lags interval (in first differences): 1 to 1 トレース検定 Unrestricted Cointegration Rank Test (Trace) Hypothesized Trace 0.05 No. of CE(s) Eigenvalue Statistic Critical Value Prob.** None * 0.179662 31.90042 29.79707 0.0282 At most 1 0.047958 6.749453 15.49471 0.6069 At most 2 0.003992 0.507944 3.841466 0.4760 Trace test indicates 1 cointegrating eqn(s) at the 0.05 level * denotes rejection of the hypothesis at the 0.05 level 共和分ベクトルは1 本 **MacKinnon-Haug-Michelis (1999) p-values 最大固有値検定 Unrestricted Cointegration Rank Test (Maximum Eigenvalue) Hypothesized Max-Eigen 0.05 No. of CE(s) Eigenvalue Statistic Critical Value Prob.** None * 0.179662 25.15096 21.13162 0.0128 At most 1 0.047958 6.241509 14.26460 0.5824 At most 2 0.003992 0.507944 3.841466 0.4760 Max-eigenvalue test indicates 1 cointegrating eqn(s) at the 0.05 level * denotes rejection of the hypothesis at the 0.05 level **MacKinnon-Haug-Michelis (1999) p-values Unrestricted Cointegrating Coefficients (normalized by b'*s11*b=i): RM2 RGDP R 0.004483-5.93E-05 0.788142 0.001070-2.72E-05-0.633378 0.005624-6.86E-05 0.344075 Unrestricted Adjustment Coefficients (alpha): D(RM2) -23.48677 6.756824 2.457682 D(RGDP) 581.0812 385.6731 146.5167 D(R) -0.008676 0.061241-0.011492 285
1 Cointegrating Equation(s): Log likelihood -1955.714 共和分ベクトル Normalized cointegrating coefficients (standard error in parentheses) 符号は正負反対 RM2 RGDP R 1.000000-0.013220 175.8026 (0.00061) (36.2769) Adjustment coefficients (standard error in parentheses) D(RM2) -0.105294 (0.02865) D(RGDP) 2.605050 (1.28488) D(R) -3.89E-05 (0.00014) 2 Cointegrating Equation(s): Log likelihood -1952.593 Normalized cointegrating coefficients (standard error in parentheses) RM2 RGDP R 1.000000 0.000000 1007.103 (241.523) 0.000000 1.000000 62883.50 (17959.0) Adjustment coefficients (standard error in parentheses) D(RM2) -0.098061 0.001208 (0.02932) (0.00041) D(RGDP) 3.017890-0.044934 (1.31116) (0.01855) D(R) 2.67E-05-1.15E-06 (0.00014) (1.9E-06) 286
Q8-3 テキスト P142 誤差修正モデルの集計 Model A Dependent Variable: D(RM2) Date: 11/11/05 Time: 12:13 Sample (adjusted): 1967Q3 1999Q1 Included observations: 127 after adjustments D(RM2(-1)) -0.766421 0.059619-12.85523 0.0000 D(RGDP(-1)) 0.003851 0.002078 1.853156 0.0663 D(R(-1)) 10.58822 20.41298 0.518700 0.6049 RESID01(-1) -0.087977 0.048281-1.822198 0.0709 C 58.21688 8.907505 6.535711 0.0000 R-squared 0.639576 Mean dependent var 39.60394 Adjusted R-squared 0.627759 S.D. dependent var 122.7610 S.E. of regression 74.89842 Akaike info criterion 11.50872 Sum squared resid 684392.3 Schwarz criterion 11.62069 Log likelihood -725.8035 F-statistic 54.12250 Durbin-Watson stat 1.210813 Prob(F-statistic) 0.000000 Model B Dependent Variable: D(RM2) Date: 11/17/05 Time: 18:46 Sample (adjusted): 1967Q3 1999Q1 Included observations: 127 after adjustments D(RGDP) 0.007428 0.003139 2.366266 0.0196 D(R) 30.24476 29.81975 1.014253 0.3125 D(RGDP(-1)) -0.000973 0.003150-0.309038 0.7578 D(R(-1)) 11.86488 31.01801 0.382516 0.7027 RESID01(-1) -0.342852 0.068291-5.020489 0.0000 C 21.67259 15.36925 1.410127 0.1611 R-squared 0.196848 Mean dependent var 39.60394 Adjusted R-squared 0.163659 S.D. dependent var 122.7610 S.E. of regression 112.2669 Akaike info criterion 12.32573 Sum squared resid 1525068. Schwarz criterion 12.46010 Log likelihood -776.6837 F-statistic 5.931264 Durbin-Watson stat 3.023075 Prob(F-statistic) 0.000061 287
Model C Dependent Variable: D(RM2) Date: 11/17/05 Time: 18:47 Sample (adjusted): 1967Q3 1999Q1 Included observations: 127 after adjustments D(RM2(-1)) -0.762839 0.056712-13.45114 0.0000 D(RGDP) 0.007324 0.001990 3.679793 0.0004 D(R) 20.14072 18.92361 1.064317 0.2893 D(RGDP(-1)) 0.002569 0.002015 1.275243 0.2047 D(R(-1)) 8.047629 19.67056 0.409120 0.6832 RESID01(-1) -0.114351 0.046516-2.458315 0.0154 C 41.97706 9.861847 4.256511 0.0000 R-squared 0.679735 Mean dependent var 39.60394 Adjusted R-squared 0.663722 S.D. dependent var 122.7610 S.E. of regression 71.18844 Akaike info criterion 11.42208 Sum squared resid 608135.4 Schwarz criterion 11.57885 Log likelihood -718.3020 F-statistic 42.44834 Durbin-Watson stat 1.243786 Prob(F-statistic) 0.000000 288
演習 テキスト P143 問 2) 各変数の単位根検定 WDOTについてのADF 検定 ~ 和分次数は1 注 ) レベル系列では単位根ありの帰無仮説を棄却できず Null Hypothesis: D(WDOT) has a unit root Exogenous: None Lag Length: 0 (Automatic based on SIC, MAXLAG=9) t-statistic Prob.* Augmented Dickey-Fuller test statistic -6.868380 0.0000 Test critical values: 1% level -2.625606 5% level -1.949609 10% level -1.611593 *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(WDOT,2) Date: 11/17/05 Time: 14:13 Sample (adjusted): 1962 2000 Included observations: 39 after adjustments D(WDOT(-1)) -1.085957 0.158110-6.868380 0.0000 R-squared 0.553713 Mean dependent var -0.086667 Adjusted R-squared 0.553713 S.D. dependent var 4.868098 S.E. of regression 3.252120 Akaike info criterion 5.221798 Sum squared resid 401.8989 Schwarz criterion 5.264453 Log likelihood -100.8251 Durbin-Watson stat 2.034670 289
CPIDOTについてのADF 検定 ~ 和分次数は1 注 ) レベル系列では単位根ありの帰無仮説を棄却できず Null Hypothesis: D(CPIDOT) has a unit root Exogenous: None Lag Length: 0 (Automatic based on SIC, MAXLAG=9) t-statistic Prob.* Augmented Dickey-Fuller test statistic -6.238175 0.0000 Test critical values: 1% level -2.625606 5% level -1.949609 10% level -1.611593 *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(CPIDOT,2) Date: 11/17/05 Time: 14:16 Sample (adjusted): 1962 2000 Included observations: 39 after adjustments D(CPIDOT(-1)) -1.005614 0.161203-6.238175 0.0000 R-squared 0.505860 Mean dependent var -0.053590 Adjusted R-squared 0.505860 S.D. dependent var 4.079513 S.E. of regression 2.867699 Akaike info criterion 4.970203 Sum squared resid 312.5004 Schwarz criterion 5.012859 Log likelihood -95.91896 Durbin-Watson stat 2.010199 290
INVRUについてのADF 検定 ~ 和分次数は1 注 ) レベル系列では単位根ありの帰無仮説を棄却できず Null Hypothesis: D(INVRU) has a unit root Exogenous: None Lag Length: 0 (Automatic based on SIC, MAXLAG=9) t-statistic Prob.* Augmented Dickey-Fuller test statistic -4.442401 0.0000 Test critical values: 1% level -2.625606 5% level -1.949609 10% level -1.611593 *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(INVRU,2) Date: 11/17/05 Time: 14:17 Sample (adjusted): 1962 2000 Included observations: 39 after adjustments D(INVRU(-1)) -0.663213 0.149292-4.442401 0.0001 R-squared 0.341306 Mean dependent var -0.001700 Adjusted R-squared 0.341306 S.D. dependent var 0.061669 S.E. of regression 0.050050 Akaike info criterion -3.126265 Sum squared resid 0.095192 Schwarz criterion -3.083610 Log likelihood 61.96217 Durbin-Watson stat 1.869526 291
問 3)Engle-Granger 検定 Engle-Granger 検定 : 帰無仮説 ( 共和分なし ) が棄却 共和分あり Null Hypothesis: RESID01 has a unit root Exogenous: None Lag Length: 0 (Automatic based on SIC, MAXLAG=9) t-statistic Prob.* Augmented Dickey-Fuller test statistic -4.797174 0.0000 Test critical values: 1% level -2.624057 5% level -1.949319 10% level -1.611711 *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(RESID01) Date: 11/17/05 Time: 14:21 Sample (adjusted): 1961 2000 Included observations: 40 after adjustments この臨界値は利用不可 Mackinnon 表 (p132) を用いて正しい臨界値を算出 RESID01(-1) -0.762076 0.158859-4.797174 0.0000 R-squared 0.370880 Mean dependent var 0.036024 Adjusted R-squared 0.370880 S.D. dependent var 1.960219 S.E. of regression 1.554788 Akaike info criterion 3.745238 Sum squared resid 94.27728 Schwarz criterion 3.787460 Log likelihood -73.90476 Durbin-Watson stat 1.987382 Mackinnon 表 (p132) での有意水準 5% における臨界値を算出 τ= -3.7429 +(1/40) (-8.352) +(1/(40 40)) (-13.41)= -3.96-4.79<τ=-3.96 であり帰無仮説は有意水準 5% にて棄却される 292
問 4)Error Correction Model の推定 Model A(P142) に相当 ~RESID01 が有意でない Dependent Variable: D(WDOT) Date: 11/11/05 Time: 16:42 Sample (adjusted): 1962 2000 Included observations: 39 after adjustments D(WDOT(-1)) -0.796663 0.288256-2.763738 0.0092 D(INVRU(-1)) 36.01574 9.973617 3.611101 0.0010 D(CPIDOT(-1)) 0.646496 0.291678 2.216471 0.0335 RESID01(-1) 0.054406 0.358856 0.151611 0.8804 C -0.100094 0.446824-0.224012 0.8241 R-squared 0.372055 Mean dependent var -0.365641 Adjusted R-squared 0.298179 S.D. dependent var 3.243661 S.E. of regression 2.717369 Akaike info criterion 4.956414 Sum squared resid 251.0591 Schwarz criterion 5.169691 Log likelihood -91.65008 F-statistic 5.036220 Durbin-Watson stat 1.533185 Prob(F-statistic) 0.002688 Model B(P143) に相当 ~RESID01 が有意 Dependent Variable: D(WDOT) Date: 11/11/05 Time: 16:47 Sample (adjusted): 1962 2000 Included observations: 39 after adjustments D(INVRU) 17.08327 5.490076 3.111663 0.0038 D(CPIDOT) 0.876423 0.097327 9.004890 0.0000 D(INVRU(-1)) -6.327187 5.780843-1.094509 0.2817 D(CPIDOT(-1)) 0.211404 0.097709 2.163624 0.0378 RESID01(-1) -0.758395 0.158484-4.785316 0.0000 C -0.098060 0.255199-0.384251 0.7033 R-squared 0.809647 Mean dependent var -0.365641 Adjusted R-squared 0.780805 S.D. dependent var 3.243661 S.E. of regression 1.518625 Akaike info criterion 3.814127 Sum squared resid 76.10537 Schwarz criterion 4.070059 Log likelihood -68.37547 F-statistic 28.07234 Durbin-Watson stat 1.887822 Prob(F-statistic) 0.000000 293
Model C(P143) に相当 ~RESID01 が有意 Dependent Variable: D(WDOT) Date: 11/17/05 Time: 14:30 Sample (adjusted): 1962 2000 Included observations: 39 after adjustments D(WDOT(-1)) -0.075629 0.184465-0.409990 0.6845 D(INVRU) 16.48883 5.746530 2.869354 0.0072 D(CPIDOT) 0.859512 0.106860 8.043359 0.0000 D(INVRU(-1)) -4.396006 7.514612-0.584994 0.5627 D(CPIDOT(-1)) 0.268849 0.171537 1.567293 0.1269 RESID01(-1) -0.696040 0.221128-3.147675 0.0035 C -0.096928 0.258493-0.374973 0.7102 R-squared 0.810641 Mean dependent var -0.365641 Adjusted R-squared 0.775136 S.D. dependent var 3.243661 S.E. of regression 1.538137 Akaike info criterion 3.860169 Sum squared resid 75.70768 Schwarz criterion 4.158757 Log likelihood -68.27330 F-statistic 22.83189 Durbin-Watson stat 1.842557 Prob(F-statistic) 0.000000 294
問 5)Johansen タイプの共和分検定 Date: 11/17/05 Time: 14:35 Sample (adjusted): 1962 2000 Included observations: 39 after adjustments Trend assumption: No deterministic trend Series: WDOT INVRU CPIDOT Lags interval (in first differences): 1 to 1 トレース検定 Unrestricted Cointegration Rank Test (Trace) Hypothesized Trace 0.05 No. of CE(s) Eigenvalue Statistic Critical Value Prob.** None 0.317440 22.90871 24.27596 0.0736 At most 1 0.151451 8.014409 12.32090 0.2355 At most 2 0.040430 1.609526 4.129906 0.2401 Trace test indicates no cointegration at the 0.05 level * denotes rejection of the hypothesis at the 0.05 level **MacKinnon-Haug-Michelis (1999) p-values 最大固有値検定 Unrestricted Cointegration Rank Test (Maximum Eigenvalue) Hypothesized Max-Eigen 0.05 No. of CE(s) Eigenvalue Statistic Critical Value Prob.** None 0.317440 14.89430 17.79730 0.1297 At most 1 0.151451 6.404883 11.22480 0.3059 At most 2 0.040430 1.609526 4.129906 0.2401 Max-eigenvalue test indicates no cointegration at the 0.05 level * denotes rejection of the hypothesis at the 0.05 level **MacKinnon-Haug-Michelis (1999) p-values Unrestricted Cointegrating Coefficients (normalized by b'*s11*b=i): WDOT INVRU CPIDOT -0.259022-0.181387 0.561796 0.394924-3.722120-0.259232-0.065622 2.877015-0.021370 Unrestricted Adjustment Coefficients (alpha): D(WDOT) -1.229669-0.354196 0.186531 D(INVRU) -0.015846-0.001795-0.006875 D(CPIDOT) -1.106914 0.334827 0.250378 295 次頁に続く
1 Cointegrating Equation(s): Log likelihood -90.71356 Normalized cointegrating coefficients (standard error in parentheses) WDOT INVRU CPIDOT 1.000000 0.700275-2.168908 (3.13575) (0.32597) Adjustment coefficients (standard error in parentheses) D(WDOT) 0.318512 (0.09731) D(INVRU) 0.004105 (0.00182) D(CPIDOT) 0.286715 (0.09717) 2 Cointegrating Equation(s): Log likelihood -87.51112 Normalized cointegrating coefficients (standard error in parentheses) WDOT INVRU CPIDOT 1.000000 0.000000-2.064301 (0.16164) 0.000000 1.000000-0.149380 (0.02490) Adjustment coefficients (standard error in parentheses) D(WDOT) 0.178631 1.541406 (0.17517) (1.38216) D(INVRU) 0.003396 0.009556 (0.00332) (0.02618) D(CPIDOT) 0.418946-1.045486 (0.17514) (1.38194) Johansen タイプの検定では 共和分ベクトルなし の結果 296