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(IG9S) D(IP9S) D(CP9S) Exogenous variables: C Date: 11/18/5 Time: 13:53 Sample: 1955Q1 1985Q4 Included observations: 114 Lag LogL LR FPE AIC SC HQ -2814.521 NA 5.89e+17 49.432 49.522 49.45942 1-2786.326 54.4123 4.2e+17 49.9344 49.38146* 49.2133* 2-2776.753 17.9721 4.16e+17 49.8339 49.58742 49.28795 3-2761.367 28.7214 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 5.1524 49.46573 6-2742.797 3.49794 4.35e+17 49.11924 5.48734 49.67448 7-2733.586 14.86749 4.36e+17 49.11554 5.69965 49.75844 8-2721.523 18.8345* 4.16e+17 49.6181 5.86194 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 情報量基準が最小になるものが最適ラグ次数 ゆえに最適ラグ次数は 3 297
作業手順 5 VAR の推定 Vector Autoregression Estimates Date: 11/15/5 Time: 14:39 Sample (adjusted): 1956Q2 1985Q4 Included observations: 119 after adjustments Standard errors in ( ) & t-statistics in [ ] D(IG9S) D(IP9S) D(CP9S) 被説明変数 D(IG9S(-1)) -.15265.45663.149928 (.9991) (.767) (.16963) [-.15279] [.629] [.88387] D(IG9S(-2)) -.18514.165416.246438 (.997) (.7591) (.16927) [-.1857] [ 2.17914] [ 1.45588] D(IG9S(-3)) -.3582.93583.42824 (.1168) (.7742) (.17263) [-.377] [ 1.2882] [ 2.43768] 過去の公共投資 ( ) により 現在の民間消費 ( ) ~ 乗数効果? 説明変数 D(IP9S(-1)) -.157773.399435.24113 (.1369) (.1361) (.2315) [-1.15935] [ 3.8555] [ 1.4312] D(IP9S(-2)) -.291755.23738 -.73353 (.1421) (.1819) (.24125) [-2.5321] [ 2.1997] [-.345] 過去の民間投資 ( ) により 現在の公共投資 ( ) D(IP9S(-3)).9387.39588 -.216289 (.1384) (.151) (.23436) [.65479] [.37667] [-.92288] 次頁に続く 298
D(CP9S(-1)) -.51571.18786 -.227972 (.5769) (.4393) (.9795) [-.89388] [.42767] [-2.3274] D(CP9S(-2)).9886.25561.18141 (.5887) (.4482) (.9994) [ 1.67943] [.5732] [.18152] D(CP9S(-3)).56896 -.3718.33938 (.5843) (.4448) (.9919) [.97382] [-.8358] [ 3.42136] C 189.7315 38.35463 1125.562 (152.438) (116.62) (258.89) [ 1.24464] [.3347] [ 4.3491] R-squared.113416.396779.25211 Adj. R-squared.4212.346972.19358 Sum sq. resids 64169188 371985 1.85E+8 S.E. equation 767.2732 584.1798 132.672 F-statistic 1.549312 7.966285 4.8261 Log likelihood -954.1293-921.6859-117.12 Akaike AIC 16.2385 15.65859 17.26251 Schwarz SC 16.43739 15.89213 17.4965 Mean dependent 175.65 43.867 1439.235 S.D. dependent 783.1814 722.938 1447.733 Determinant resid covariance (dof adj.) Determinant resid covariance 2.62E+17 2.1E+17 Log likelihood -2877.196 Akaike information criterion 48.8644 Schwarz criterion 49.5616 299
Q9-2 テキスト P168 Granger 因果性の検定 VAR Granger Causality/Block Exogeneity Wald Tests Date: 11/17/5 Time: 17:25 Sample: 1955Q1 1985Q4 Included observations: 119 Dependent variable: D(IG9S) Excluded Chi-sq df Prob. 民間消費 (CP) 公共投資 (IG) D(IP9S) 8.95176 3.299 D(CP9S) 5.4284 3.1715 All 13.7131 6.33 Dependent variable: D(IP9S) Excluded Chi-sq df Prob. D(IG9S) 6.195911 3.125 D(CP9S).466446 3.9262 All 8.793599 6.1855 Dependent variable: D(CP9S) 公共投資 (IG) 民間消費 (CP) Excluded Chi-sq df Prob. D(IG9S) 8.276349 3.46 D(IP9S) 1.63852 3.658 All 14.2162 6.273 3
Q9-3 テキスト P17 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: IG9S IP9S CP9S Exogenous variables: C Date: 11/28/5 Time: 17:18 Sample: 1955Q1 1985Q4 Included observations: 115 Lag LogL LR FPE AIC SC HQ -3523.99 NA 8.73e+22 61.33754 61.4915 61.36661 1-283.772 1338.55 5.94e+17 49.43952 49.72594 49.55578 2-282.561 52.98754 4.26e+17 49.1541 49.6666* 49.3887 3-2787.773 27.426 3.85e+17 49.475 49.7282 49.2954 4-2772.222 27.58693 3.44e+17* 48.8981* 49.8217 49.26866* 5-2766.912 9.142496 3.68e+17 48.95499 5.17 49.422 6-2758.593 13.88869 3.74e+17 48.96683 5.32737 49.5197 7-2754.352 6.858779 4.8e+17 49.4961 5.62496 49.6893 8-2741.999 19.33511* 3.88e+17 48.99129 5.78147 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 31
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 Tests Date: 2/5/6 Time: 14:15 Sample: 1955Q1 1985Q4 Included observations: 118 Dependent variable: IG9S Excluded Chi-sq df Prob. IP9S 14.9756 5.15 CP9S 8.493853 5.131 All 2.733 1.286 Dependent variable: IP9S Excluded Chi-sq df Prob. IG9S 17.69852 5.33 CP9S 11.78321 5.379 All 23.76357 1.83 Dependent variable: CP9S Excluded Chi-sq df Prob. IG9S 1.68694 5.58 IP9S 2.2217 5.8178 All 16.61671 1.833 乗数効果 : 公共投資 (IG) 民間消費 (CP) 帰無仮説 (Granger の意味での因果性のないこと ) が棄却される 因果性あり 32
Q9-4 テキスト P175 インパルス応答関数 Response to Cholesky One S.D. Innovations ± 2 S.E. 1 8 6 4 2-2 Response of D(IG9S) to D(IG9S) 1 8 6 4 2-2 Response of D(IG9S) to D(IP9S) 1 8 6 4 2-2 Response of D(IG9S) to D(CP9S) -4-4 -4 Response of D(IP9S) to D(IG9S) 7 6 5 4 3 2 1-1 -2 Response of D(IP9S) to D(IP9S) 7 6 5 4 3 2 1-1 -2 Response of D(IP9S) to D(CP9S) 7 6 5 4 3 2 1-1 -2 16 Response of D(CP9S) to D(IG9S) 16 Response of D(CP9S) to D(IP9S) 16 Response of D(CP9S) to D(CP9S) 12 12 12 8 8 8 4 4 4-4 -4-4 -8-8 -8 公共投資にショックが加わったと きの民間消費の反応 33
累積インパルス応答関数 Accumulated Response to Cholesky One S.D. Innovations ± 2 S.E. 1 Accumulated Response of D(IG9S) to D(IG9S) 1 Accumulated Response of D(IG9S) to D(IP9S) 1 Accumulated Response of D(IG9S) to D(CP9S) 5 5 5-5 -5-5 -1-1 -1 25 Accumulated Response of D(IP9S) to D(IG9S) 25 Accumulated Response of D(IP9S) to D(IP9S) 25 Accumulated Response of D(IP9S) to D(CP9S) 2 2 2 15 15 15 1 1 1 5 5 5-5 -5-5 Accumulated Response of D(CP9S) to D(IG9S) 25 2 15 1 5-5 -1 Accumulated Response of D(CP9S) to D(IP9S) 25 2 15 1 5-5 -1 Accumulated Response of D(CP9S) to D(CP9S) 25 2 15 1 5-5 -1 34
Q9-5 テキスト P176 インパルス関数 Response to Cholesky One S.D. Innovations ± 2 S.E. 15 Response of D(CP9S) to D(CP9S) 15 Response of D(CP9S) to D(IP9S) 15 Response of D(CP9S) to D(IG9S) 1 1 1 5 5 5-5 -5-5 7 6 5 4 3 2 1 Response of D(IP9S) to D(CP9S) 7 6 5 4 3 2 1 Response of D(IP9S) to D(IP9S) 7 6 5 4 3 2 1 Response of D(IP9S) to D(IG9S) -1-1 -1 1 8 6 4 2-2 Response of D(IG9S) to D(CP9S) 1 8 6 4 2-2 Response of D(IG9S) to D(IP9S) 1 8 6 4 2-2 Response of D(IG9S) to D(IG9S) -4-4 -4 公共投資にショックが生じたと きの民間消費の反応 35
累積インパルス応答関数 Accumulated Response to Cholesky One S.D. Innovations ± 2 S.E. 3 Accumulated Response of D(CP9S) to D(CP9S) 3 Accumulated Response of D(CP9S) to D(IP9S) 3 Accumulated Response of D(CP9S) to D(IG9S) 2 2 2 1 1 1-1 -1-1 -2-2 -2 2 Accumulated Response of D(IP9S) to D(CP9S) 2 Accumulated Response of D(IP9S) to D(IP9S) 2 Accumulated Response of D(IP9S) to D(IG9S) 16 16 16 12 12 12 8 8 8 4 4 4-4 -4-4 1 Accumulated Response of D(IG9S) to D(CP9S) 1 Accumulated Response of D(IG9S) to D(IP9S) 1 Accumulated Response of D(IG9S) to D(IG9S) 5 5 5-5 -5-5 -1-1 -1 36
演習 Johansen 型の共和分検定 Date: 2/5/6 Time: 14:2 Sample (adjusted): 1989M4 24M1 Included 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) トレース検定 Hypothesized Trace.5 Critical No. of CE(s) Eigenvalue Statistic Value Prob.** None *.163443 56.63 42.91525.13 At most 1.88767 23.2289 25.87211.13 At most 2.3774 5.845234 12.51798.482 Trace test indicates 1 cointegrating eqn(s) at the.5 level * denotes rejection of the hypothesis at the.5 level **MacKinnon-Haug-Michelis (1999) p-values 最大固有値検定 Unrestricted Cointegration Rank Test (Maximum Eigenvalue) Hypothesized Max-Eigen.5 Critical No. of CE(s) Eigenvalue Statistic Value Prob.** None *.163443 33.37221 25.82321.42 At most 1.88767 17.38285 19.3874.955 At most 2.3774 5.845234 12.51798.482 Max-eigenvalue test indicates 1 cointegrating eqn(s) at the.5 level * denotes rejection of the hypothesis at the.5 level **MacKinnon-Haug-Michelis (1999) p-values Unrestricted Cointegrating Coefficients (normalized by b'*s11*b=i): LER LUSEXP LJPEXP @TREND(89M2) 37
-29.7213-53.583 78.1923.127677 32.47143 77.89612-48.65629 -.8337-11.74369 15.1669 21.27877.24376 Unrestricted Adjustment Coefficients (alpha): D(LER) -.981.247.3644 D(LUSEXP) -.163 -.116 2.27E-5 D(LJPEXP) -.525.35 6.57E-5 1 Cointegrating Equation(s): Log likelihood 1824.311 Normalized cointegrating coefficients (standard error in parentheses) LER LUSEXP LJPEXP @TREND(89M2) 1. 1.8335-2.63781 -.4296 (.23293) (.1423) (.26) Adjustment coefficients (standard error in parentheses) D(LER).291562 (.6923) D(LUSEXP).4847 (.763) D(LJPEXP).15471 (.263) 2 Cointegrating Equation(s): Log likelihood 1833.3 Normalized cointegrating coefficients (standard error in parentheses) LER LUSEXP LJPEXP @TREND(89M2) 1.. -6.36534 -.95 (.77392) (.147). 1. 1.891733.2891 (.37858) (.72) Adjustment coefficients (standard error in parentheses) D(LER).35821.684338 (.1231) (.21965) D(LUSEXP) -.28156 -.7445 (.179) (.2317) D(LJPEXP).16677.35799 (.3894) (.836) 38
3)Error Correction VAR Vector Error Correction Estimates Date: 2/5/6 Time: 14:23 Sample (adjusted): 1989M4 24M1 Included observations: 187 after adjustments Standard errors in ( ) & t-statistics in [ ] Cointegrating Eq: CointEq1 LER(-1) 1. LUSEXP(-1) 1.8335 (.23293) [ 7.729] LJPEXP(-1) -2.63781 (.1423) [-18.488] @TREND(89M1) -.4296 (.26) [-16.3838] C -.22526 Error Correction: D(LER) D(LUSEXP) D(LJPEXP) CointEq1.291562.4847.15471 (.6923) (.763) (.263) [ 4.21152] [.6354] [ 5.88158] D(LER(-1)) -.28181 -.3154.266593 (.13165) (.1451) (.52) [-2.145] [-2.7879] [ 5.32954] D(LER(-2)) -.121851 -.21612.4925 (.14446) (.1592) (.5489) [-.8435] [-1.35782] [.8973] D(LUSEXP(-1)) -1.52997.17881 -.22881 (.6971) (.768) (.26483) 39
[-2.1954] [ 2.31886] [-.86395] D(LUSEXP(-2)) -.76922.111364 -.196671 (.68633) (.7562) (.2678) [-1.1248] [ 1.47267] [-.75418] D(LJPEXP(-1)).561962.388 -.82251 (.3327) (.3341) (.11523) [ 1.853] [.945] [-.7138] D(LJPEXP(-2)).35546.3314.138639 (.1969) (.2169) (.7481) [ 1.8531] [ 1.39732] [ 1.85315] C.93.44 -.14 (.243) (.27) (.92) [.37175] [ 1.597] [-1.8749] R-squared.15521.12659.5421 Adj. R-squared.7542.86271.484821 Sum sq. resids.181611.225.26218 S.E. equation.31853.351.1213 F-statistic 3.16653 3.58792 26.57 Log likelihood 383.2677 795.7219 564.228 Akaike AIC -4.13558-8.424834-5.948962 Schwarz SC -3.875329-8.28664-5.81733 Mean dependent -.122.498 -.1485 S.D. dependent.3339.3671.16862 Determinant resid covariance (dof adj.) Determinant resid covariance 7.69E-13 6.74E-13 Log likelihood 1824.311 Akaike information criterion -19.21189 Schwarz criterion -18.7288 31
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)*RESID1(-1) + C(5) Dependent Variable: D(LER) Method: Least Squares Date: 11/17/5 Time: 16:53 Sample (adjusted): 1989M3 24M1 Included observations: 188 after adjustments Variable Coefficient Std. Error t-statistic Prob. D(LER(-1)) -.183124.116728-1.56884.1184 D(LJPEXP(-1)).317266.193871 1.636479.135 D(LUSEXP(-1)) -.9381.654983-1.432115.1538 RESID1(-1).244148.1217 2.39321.177 C -.223.2413 -.92412.9265 R-squared.45831 Mean dependent var -.974 Adjusted R-squared.24974 S.D. dependent var.3398 S.E. of regression.32682 Akaike info criterion -3.977729 Sum squared resid.195469 Schwarz criterion -3.891654 Log likelihood 378.966 F-statistic 2.197461 Durbin-Watson stat 2.489 Prob(F-statistic).7989 やはり誤差修正項が望ましい符号条件を満たしていない 311