Q9-1【テキスト P166】
２）VARの推定
注）各変数についてADF検定を行った結果、和文の次数はすべて１である。
作業手順④ 情報量基準（ＡＩＣ）によるラグ次数の選択
VAR Lag Order Selection Criteria
Endogenous variables: D(IG90S) D(IP90S) D(CP90S) Exogenous variables: C
Date: 11/18/05 Time: 13:53 Sample: 1955Q1 1985Q4 Included observations: 114
Lag LogL LR FPE AIC SC HQ
0 -2814.521 NA 5.89e+17 49.43020 49.50220 49.45942 1 -2786.326 54.41230 4.20e+17 49.09344 49.38146* 49.21033* 2 -2776.753 17.97021 4.16e+17 49.08339 49.58742 49.28795 3 -2761.367 28.07214 3.73e+17* 48.97136* 49.69141 49.26358 4 -2753.898 13.23478 3.83e+17 48.99821 49.93428 49.37811 5 -2744.895 15.47873 3.84e+17 48.99816 50.15024 49.46573 6 -2742.797 3.497094 4.35e+17 49.11924 50.48734 49.67448 7 -2733.586 14.86749 4.36e+17 49.11554 50.69965 49.75844 8 -2721.523 18.83405* 4.16e+17 49.06181 50.86194 49.79238
* indicates lag order selected by the criterion
LR: sequential modified LR test statistic (each test at 5% level) FPE: Final prediction error
AIC: Akaike information criterion SC: Schwarz information criterion HQ: Hannan-Quinn information criterion
作業手順⑤ ＶＡＲの推定
Vector Autoregression Estimates Date: 11/15/05 Time: 14:39 Sample (adjusted): 1956Q2 1985Q4
Included observations: 119 after adjustments Standard errors in ( ) & t-statistics in [ ]
D(IG90S) D(IP90S) D(CP90S) D(IG90S(-1)) -0.015265 0.045663 0.149928 (0.09991) (0.07607) (0.16963) [-0.15279] [ 0.60029] [ 0.88387] D(IG90S(-2)) -0.018514 0.165416 0.246438 (0.09970) (0.07591) (0.16927) [-0.18570] [ 2.17914] [ 1.45588] D(IG90S(-3)) -0.030582 0.093583 0.420824 (0.10168) (0.07742) (0.17263) [-0.30077] [ 1.20882] [ 2.43768] D(IP90S(-1)) -0.157773 0.399435 0.241013 (0.13609) (0.10361) (0.23105) [-1.15935] [ 3.85505] [ 1.04312] D(IP90S(-2)) -0.291755 0.237038 -0.073353 (0.14210) (0.10819) (0.24125) [-2.05321] [ 2.19097] [-0.30405] D(IP90S(-3)) 0.090387 0.039588 -0.216289 (0.13804) (0.10510) (0.23436) [ 0.65479] [ 0.37667] [-0.92288]
次頁に続く
被説明変数
説明変数
過去の公共投資（↑）により 現在の民間消費（↑） ～乗数効果？ 過去の民間投資（↑）により 現在の公共投資（↓）D(CP90S(-1)) -0.051571 0.018786 -0.227972 (0.05769) (0.04393) (0.09795) [-0.89388] [ 0.42767] [-2.32740] D(CP90S(-2)) 0.098860 0.025561 0.018141 (0.05887) (0.04482) (0.09994) [ 1.67943] [ 0.57032] [ 0.18152] D(CP90S(-3)) 0.056896 -0.003718 0.339380 (0.05843) (0.04448) (0.09919) [ 0.97382] [-0.08358] [ 3.42136] C 189.7315 38.35463 1125.562 (152.438) (116.062) (258.809) [ 1.24464] [ 0.33047] [ 4.34901] R-squared 0.113416 0.396779 0.252110 Adj. R-squared 0.040212 0.346972 0.190358 Sum sq. resids 64169188 37198005 1.85E+08 S.E. equation 767.2732 584.1798 1302.672 F-statistic 1.549312 7.966285 4.082601 Log likelihood -954.1293 -921.6859 -1017.120 Akaike AIC 16.20385 15.65859 17.26251 Schwarz SC 16.43739 15.89213 17.49605 Mean dependent 175.6050 430.8067 1439.235 S.D. dependent 783.1814 722.9038 1447.733
Determinant resid covariance (dof adj.) 2.62E+17 Determinant resid covariance 2.01E+17
Log likelihood -2877.196
Akaike information criterion 48.86044
Q9-2【テキスト P168】
◆ Granger因果性の検定
VAR Granger Causality/Block Exogeneity Wald Tests Date: 11/17/05 Time: 17:25
Sample: 1955Q1 1985Q4 Included observations: 119
Dependent variable: D(IG90S)
Excluded Chi-sq df Prob.
D(IP90S) 8.951706 3 0.0299
D(CP90S) 5.004284 3 0.1715
All 13.71310 6 0.0330
Dependent variable: D(IP90S)
Excluded Chi-sq df Prob.
D(IG90S) 6.195911 3 0.1025
D(CP90S) 0.466446 3 0.9262
All 8.793599 6 0.1855
Dependent variable: D(CP90S)
Excluded Chi-sq df Prob.
D(IG90S) 8.276349 3 0.0406 D(IP90S) 1.638052 3 0.6508 All 14.21602 6 0.0273
民間消費（ＣＰ）
⇒公共投資（ＩＧ）
公共投資（ＩＧ）
⇒民間消費（ＣＰ）
Q9-3【テキスト P170】
◆ Toda and Yamamoto(1995)による Lag Augmented VAR
①レベル項での最適ラグ次数は『４』（ＡＩＣ基準）
VAR Lag Order Selection Criteria
Endogenous variables: IG90S IP90S CP90S Exogenous variables: C
Date: 11/28/05 Time: 17:18 Sample: 1955Q1 1985Q4 Included observations: 115
Lag LogL LR FPE AIC SC HQ
0 -3523.909 NA 8.73e+22 61.33754 61.40915 61.36661 1 -2830.772 1338.055 5.94e+17 49.43952 49.72594 49.55578 2 -2802.561 52.98754 4.26e+17 49.10541 49.60666* 49.30887 3 -2787.773 27.00426 3.85e+17 49.00475 49.72082 49.29540 4 -2772.222 27.58693 3.44e+17* 48.89081* 49.82170 49.26866* 5 -2766.912 9.142496 3.68e+17 48.95499 50.10070 49.42002 6 -2758.593 13.88869 3.74e+17 48.96683 50.32737 49.51907 7 -2754.352 6.858779 4.08e+17 49.04961 50.62496 49.68903 8 -2741.999 19.33511* 3.88e+17 48.99129 50.78147 49.71791
* indicates lag order selected by the criterion
LR: sequential modified LR test statistic (each test at 5% level) FPE: Final prediction error
AIC: Akaike information criterion SC: Schwarz information criterion HQ: Hannan-Quinn information criterion
②Q8-1より利用する全ての変数の和分次数は『１』である
⇒Lag Augmented VAR のラグ次数は ４＋１＝５
③ラグ次数を５としたレベル項でのLag Augmented VARのもとでのGranger因果性検定結果
VAR Granger Causality/Block Exogeneity Wald TestsDate: 02/05/06 Time: 14:15 Sample: 1955Q1 1985Q4 Included observations: 118
Dependent variable: IG90S
Excluded Chi-sq df Prob.
IP90S 14.97560 5 0.0105
CP90S 8.493853 5 0.1310
All 20.07303 10 0.0286
Dependent variable: IP90S
Excluded Chi-sq df Prob.
IG90S 17.69852 5 0.0033
CP90S 11.78321 5 0.0379
All 23.76357 10 0.0083
Dependent variable: CP90S
Excluded Chi-sq df Prob.
IG90S 10.68694 5 0.0580
IP90S 2.221070 5 0.8178
All 16.61671 10 0.0833
【乗数効果】：公共投資（ＩＧ）⇒ 民間消費（ＣＰ）
Q9-4【テキスト P175】
◆ インパルス応答関数
-400 -200 0 200 400 600 800 1000 1 2 3 4 5 6 7 8 9 10Response of D(IG90S) to D(IG90S)
-400 -200 0 200 400 600 800 1000 1 2 3 4 5 6 7 8 9 10
Response of D(IG90S) to D(IP90S)
-400 -200 0 200 400 600 800 1000 1 2 3 4 5 6 7 8 9 10 Response of D(IG90S) to D(CP90S) -200 -100 0 100 200 300 400 500 600 700 1 2 3 4 5 6 7 8 9 10
Response of D(IP90S) to D(IG90S)
-200 -100 0 100 200 300 400 500 600 700 1 2 3 4 5 6 7 8 9 10
Response of D(IP90S) to D(IP90S)
-200 -100 0 100 200 300 400 500 600 700 1 2 3 4 5 6 7 8 9 10 Response of D(IP90S) to D(CP90S) -800 -400 0 400 800 1200 1600 1 2 3 4 5 6 7 8 9 10 Response of D(CP90S) to D(IG90S) -800 -400 0 400 800 1200 1600 1 2 3 4 5 6 7 8 9 10 Response of D(CP90S) to D(IP90S) -800 -400 0 400 800 1200 1600 1 2 3 4 5 6 7 8 9 10 Response of D(CP90S) to D(CP90S) Response to Cholesky One S.D. Innovations ± 2 S.E.
公共投資にショックが加わったと
きの民間消費の反応
◆ 累積インパルス応答関数
-1000 -500 0 500 1000 1 2 3 4 5 6 7 8 9 10Accumulated Response of D(IG90S) to D(IG90S)
-1000 -500 0 500 1000 1 2 3 4 5 6 7 8 9 10
Accumulated Response of D(IG90S) to D(IP90S)
-1000 -500 0 500 1000 1 2 3 4 5 6 7 8 9 10
Accumulated Response of D(IG90S) to D(CP90S)
-500 0 500 1000 1500 2000 2500 1 2 3 4 5 6 7 8 9 10
Accumulated Response of D(IP90S) to D(IG90S)
-500 0 500 1000 1500 2000 2500 1 2 3 4 5 6 7 8 9 10
Accumulated Response of D(IP90S) to D(IP90S)
-500 0 500 1000 1500 2000 2500 1 2 3 4 5 6 7 8 9 10
Accumulated Response of D(IP90S) to D(CP90S)
-1000 -500 0 500 1000 1500 2000 2500 1 2 3 4 5 6 7 8 9 10
Accumulated Response of D(CP90S) to D(IG90S)
-1000 -500 0 500 1000 1500 2000 2500 1 2 3 4 5 6 7 8 9 10
Accumulated Response of D(CP90S) to D(IP90S)
-1000 -500 0 500 1000 1500 2000 2500 1 2 3 4 5 6 7 8 9 10 Accumulated Response of D(CP90S) to D(CP90S)
Q9-5【テキスト P176】
◆ インパルス関数
-500 0 500 1000 1500 1 2 3 4 5 6 7 8 9 10 Response of D(CP90S) to D(CP90S) -500 0 500 1000 1500 1 2 3 4 5 6 7 8 9 10 Response of D(CP90S) to D(IP90S) -500 0 500 1000 1500 1 2 3 4 5 6 7 8 9 10 Response of D(CP90S) to D(IG90S) -100 0 100 200 300 400 500 600 700 1 2 3 4 5 6 7 8 9 10 Response of D(IP90S) to D(CP90S) -100 0 100 200 300 400 500 600 700 1 2 3 4 5 6 7 8 9 10Response of D(IP90S) to D(IP90S)
-100 0 100 200 300 400 500 600 700 1 2 3 4 5 6 7 8 9 10
Response of D(IP90S) to D(IG90S)
-400 -200 0 200 400 600 800 1000 1 2 3 4 5 6 7 8 9 10 Response of D(IG90S) to D(CP90S) -400 -200 0 200 400 600 800 1000 1 2 3 4 5 6 7 8 9 10
Response of D(IG90S) to D(IP90S)
-400 -200 0 200 400 600 800 1000 1 2 3 4 5 6 7 8 9 10
Response of D(IG90S) to D(IG90S)
Response to Cholesky One S.D. Innovations ± 2 S.E.
公共投資にショックが生じたと
きの民間消費の反応
◆ 累積インパルス応答関数
-2000 -1000 0 1000 2000 3000 1 2 3 4 5 6 7 8 9 10 Accumulated Response of D(CP90S) to D(CP90S) -2000 -1000 0 1000 2000 3000 1 2 3 4 5 6 7 8 9 10Accumulated Response of D(CP90S) to D(IP90S)
-2000 -1000 0 1000 2000 3000 1 2 3 4 5 6 7 8 9 10
Accumulated Response of D(CP90S) to D(IG90S)
-400 0 400 800 1200 1600 2000 1 2 3 4 5 6 7 8 9 10
Accumulated Response of D(IP90S) to D(CP90S)
-400 0 400 800 1200 1600 2000 1 2 3 4 5 6 7 8 9 10
Accumulated Response of D(IP90S) to D(IP90S)
-400 0 400 800 1200 1600 2000 1 2 3 4 5 6 7 8 9 10
Accumulated Response of D(IP90S) to D(IG90S)
-1000 -500 0 500 1000 1 2 3 4 5 6 7 8 9 10
Accumulated Response of D(IG90S) to D(CP90S)
-1000 -500 0 500 1000 1 2 3 4 5 6 7 8 9 10
Accumulated Response of D(IG90S) to D(IP90S)
-1000 -500 0 500 1000 1 2 3 4 5 6 7 8 9 10
Accumulated Response of D(IG90S) to D(IG90S)
演習
◆ Johansen型の共和分検定
Date: 02/05/06 Time: 14:20 Sample (adjusted): 1989M04 2004M10Included observations: 187 after adjustments
Trend assumption: Linear deterministic trend (restricted) Series: LER LUSEXP LJPEXP
Lags interval (in first differences): 1 to 2
Unrestricted Cointegration Rank Test (Trace)
Hypothesized Trace 0.05
No. of CE(s) Eigenvalue Statistic
Critical
Value Prob.** None * 0.163443 56.60030 42.91525 0.0013 At most 1 0.088767 23.22809 25.87211 0.1030 At most 2 0.030774 5.845234 12.51798 0.4802 Trace 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 Cointegration Rank Test (Maximum Eigenvalue)
Hypothesized Max-Eigen 0.05
No. of CE(s) Eigenvalue Statistic
Critical
Value Prob.** None * 0.163443 33.37221 25.82321 0.0042 At most 1 0.088767 17.38285 19.38704 0.0955 At most 2 0.030774 5.845234 12.51798 0.4802 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):
最大固有値検定
トレース検定
-29.72130 -53.50830 78.19023 0.127677 32.47143 77.89612 -48.65629 -0.083307 -11.74369 15.16690 21.27877 0.024376 Unrestricted Adjustment Coefficients (alpha):
D(LER) -0.009810 0.002047 0.003644 D(LUSEXP) -0.000163 -0.001016 2.27E-05 D(LJPEXP) -0.005205 0.000350 6.57E-05 1 Cointegrating Equation(s): Log likelihood 1824.311
Normalized cointegrating coefficients (standard error in parentheses)
LER LUSEXP LJPEXP @TREND(89M02)
1.000000 1.800335 -2.630781 -0.004296 (0.23293) (0.14230) (0.00026) Adjustment coefficients (standard error in parentheses)
D(LER) 0.291562 (0.06923) D(LUSEXP) 0.004847 (0.00763) D(LJPEXP) 0.154710 (0.02630)
2 Cointegrating Equation(s): Log likelihood 1833.003
Normalized cointegrating coefficients (standard error in parentheses)
LER LUSEXP LJPEXP @TREND(89M02)
1.000000 0.000000 -6.036534 -0.009500 (0.77392) (0.00147) 0.000000 1.000000 1.891733 0.002891 (0.37858) (0.00072) Adjustment coefficients (standard error in parentheses)
D(LER) 0.358021 0.684338 (0.10231) (0.21965) D(LUSEXP) -0.028156 -0.070445 (0.01079) (0.02317) D(LJPEXP) 0.166077 0.305799 (0.03894) (0.08360)
3）Error Correction VAR
Vector Error Correction Estimates Date: 02/05/06 Time: 14:23 Sample (adjusted): 1989M04 2004M10
Included observations: 187 after adjustments Standard errors in ( ) & t-statistics in [ ]
Cointegrating Eq: CointEq1
LER(-1) 1.000000 LUSEXP(-1) 1.800335 (0.23293) [ 7.72900] LJPEXP(-1) -2.630781 (0.14230) [-18.4880] @TREND(89M01) -0.004296 (0.00026) [-16.3838] C -0.225260
Error Correction: D(LER) D(LUSEXP) D(LJPEXP)
CointEq1 0.291562 0.004847 0.154710 (0.06923) (0.00763) (0.02630) [ 4.21152] [ 0.63540] [ 5.88158] D(LER(-1)) -0.281801 -0.030154 0.266593 (0.13165) (0.01451) (0.05002) [-2.14050] [-2.07879] [ 5.32954] D(LER(-2)) -0.121851 -0.021612 0.004925 (0.14446) (0.01592) (0.05489) [-0.84350] [-1.35782] [ 0.08973] D(LUSEXP(-1)) -1.529970 0.178081 -0.228801
[-2.19504] [ 2.31886] [-0.86395] D(LUSEXP(-2)) -0.769022 0.111364 -0.196671 (0.68633) (0.07562) (0.26078) [-1.12048] [ 1.47267] [-0.75418] D(LJPEXP(-1)) 0.561962 0.030088 -0.082251 (0.30327) (0.03341) (0.11523) [ 1.85300] [ 0.90045] [-0.71380] D(LJPEXP(-2)) 0.355460 0.030314 0.138639 (0.19690) (0.02169) (0.07481) [ 1.80531] [ 1.39732] [ 1.85315] C 0.000903 0.000404 -0.001004 (0.00243) (0.00027) (0.00092) [ 0.37175] [ 1.50970] [-1.08749] R-squared 0.105521 0.120659 0.504210 Adj. R-squared 0.070542 0.086271 0.484821 Sum sq. resids 0.181611 0.002205 0.026218 S.E. equation 0.031853 0.003510 0.012103 F-statistic 3.016653 3.508792 26.00570 Log likelihood 383.2677 795.7219 564.2280 Akaike AIC -4.013558 -8.424834 -5.948962 Schwarz SC -3.875329 -8.286604 -5.810733 Mean dependent -0.001202 0.000498 -0.001485 S.D. dependent 0.033039 0.003671 0.016862
Determinant resid covariance (dof
adj.) 7.69E-13
Determinant resid covariance 6.74E-13
Log likelihood 1824.311
Akaike information criterion -19.21189 Schwarz criterion -18.72808
【参考】Engle-Granger 検定で得られた残差（誤差修正項）に立脚したECMの推定
※ 定式化は Granger の表現定理に準拠
Estimation Equation:
=====================
D(LER) = C(1)*D(LER(-1)) + C(2)*D(LJPEXP(-1)) + C(3)*D(LUSEXP(-1)) + C(4)*RESID01(-1) + C(5)
Dependent Variable: D(LER) Method: Least Squares Date: 11/17/05 Time: 16:53 Sample (adjusted): 1989M03 2004M10
Included observations: 188 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
D(LER(-1)) -0.183124 0.116728 -1.568804 0.1184 D(LJPEXP(-1)) 0.317266 0.193871 1.636479 0.1035 D(LUSEXP(-1)) -0.938010 0.654983 -1.432115 0.1538 RESID01(-1) 0.244148 0.102017 2.393210 0.0177 C -0.000223 0.002413 -0.092412 0.9265
R-squared 0.045831 Mean dependent var -0.000974 Adjusted R-squared 0.024974 S.D. dependent var 0.033098 S.E. of regression 0.032682 Akaike info criterion -3.977729 Sum squared resid 0.195469 Schwarz criterion -3.891654 Log likelihood 378.9066 F-statistic 2.197461 Durbin-Watson stat 2.004890 Prob(F-statistic) 0.070989
