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Exercise 14.1 a, First we check the stationary of variables by unit root test Augmented Dickey-Fuller Unit Root Test on R_STOCK3 Null Hypothesis: RLSTOCK3 has a unit root Exogenous: Constant Lag Length: 1 (Automatic - based on AIC, maxlag=27) t-Statistic Prob.* Augmented Dickey-Fuller test statistic -36.08954 0.0000 Test critical values: 1% level -3.432663 -2.862447 5% level -2.567298 10% level *MacKinnon (1996) one-sided p-values Augmented Dickey-Fuller Test Equation Dependent Variable: D(R_STOCK3) Method: Least Squares Date: 09/18/23 Time: 21:07 Sample: 1/01/1990 12/31/1999 Included observations: 2610 Variable Coefficient Std Error t-Statistic Prob R_STOCK3(-1) -0.928560 0.025729 -36.08954 0.0000 D(R_STOCK3(-1)) 0.070623 0.019535 3.615189 0.0003 0.000524 0.000358 1.463741 0.1434 Cc R-squared 0.436515 Mean dependent var -4.73E-06 Adjusted R-squared 0.436083 S.D dependent var 0.024328 S.E of regression 0.018269 Akaike info criterion -5.166069 Sum squared resid 0.870107 Schwarz criterion -5.159325 Log likelihood 6744.720 Hannan-Quinn criter -5.163626 F-statistic 1009.783 Durbin-Watson stat 2.001773 Prob(F-statistic) 0.000000 Augmented Dickey-Fuller Unit Root Test on R_STOCK2 Null Hypothesis: R_LSTOCK2 has a unit root Exogenous: Constant Lag Length: 18 (Automatic - based on AIC, maxlag=27) t-Statistic Prob.” Augmented Dickey-Fuller test statistic -14.42998 0.0000 -3.432663 Test critical values: 1% level -2.862447 -2.567298 5% level 10% level *MacKinnon (1996) one-sided p-values Augmented Dickey-Fuller Test Equation Dependent Variable: D(R_STOCK2) Method: Least Squares Date: 09/18/23 Time: 21:07 Sample: 1/01/1990 12/31/1999 Included observations: 2610 Variable Coefficient Std Error t-Statistic Prob R_STOCK2(-1) -1.414732 0.098041 -14.42998 0.0000 D(R_STOCK2(-1)) 0.421673 0.094579 4458432 0.0000 D(R_STOCK2(-2)) 0388858 0.091168 4.265289 0.0000 D(R_STOCK2(-3)) 0354007 0.087637 44039460 0.0001 D(R_STOCK2(-4)) 0333779 0084272 3960715 0.0001 D(R_STOCK2(-5)) 0353001 0.080733 44372432 0.0000 D(R_STOCK2(-6)) 0354893 0.077472 44580919 0.0000 D(R_STOCK2(-7)) 0300115 0.073905 4.060807 0.0001 D(R_STOCK2(-8)) 0262105 0.070237 3731740 0.0002 Augmented Dickey-Fuller Unit Root Test on R_STOCK1 Null Hypothesis: R_STOCK1 has a unit root Exogenous: Constant on AIC, maxlag=27) Lag Length: 2 (Automatic - based t-Statistic Prob.* Augmented Dickey-Fuller test statistic -31.52324 0.0000 -3.432663 Test critical values: 1% level -2.862447 -2.567298 5% level 10% level *MacKinnon (1996) one-sided p-values Augmented Dickey-Fuller Test Equation Dependent Variable: D(R_STOCK1) Method: Least Squares Date: 09/18/23 Time: 21:08 Sample: 1/01/1990 12/31/1999 Included observations: 2610 Variable Coefficient Std Error t-Statistic Prob R_STOCK1(-1) -1.042949 0.033085 -31.52324 0.0000 D(R_STOCK1(-1)) 0.088438 0.027006 3.274785 0.0011 D(R_STOCK1(-2)) 0.066132 0.019577 3.378070 0.0007 4.14E-05 0.000297 0.139533 0.8890 Cc R-squared 0.479684 Mean dependent var -4.65E-06 Adjusted R-squared 0.479085 S.D dependent var 0.021008 S.E of regression 0.015162 Akaike info criterion -5.538508 Sum squared resid 0.599089 Schwarz criterion -5.529516 Log likelihood 7231.753 Hannan-Quinn criter -5.535250 F-statistic 800.8308 Durbin-Watson stat 2.003442 Prob(F-statistic) 0.000000 Augmented Dickey-Fuller Unit Root Test on R_FTSE Null Hypothesis: R_LFTSE has a unit root Exogenous: Constant Lag Length: 7 (Automatic - based on AIC, maxlag=27) t-Statistic Prob.* Augmented Dickey-Fuller test statistic -19.40071 0.0000 Test critical values: 1% level -3.432663 5% level -2.862447 10% level -2.567298 *MacKinnon (1996) one-sided p-values Augmented Dickey-Fuller Test Equation Dependent Variable: D(R_FTSE) Method: Least Squares Date: 09/18/23 Time: 21:08 Sample: 1/01/1990 12/31/1999 Included observations: 2610 Variable Coefficient Std.Error t-Statistic Prob R_FTSE(-1) -1.089811 0.056174 -19.40071 0.0000 D(R_FTSE(-1)) 0.160701 0.051654 3.111103 0.0019 D(R_FTSE(-2)) 0.119276 0.047337 2.519714 0.0118 D(R_FTSE(-3)) 0.084025 0.042922 1.957589 0.0504 D(R_FTSE(-4)) 0.093398 0.038246 2.442047 0.0147 D(R_FTSE(-5)) 0.067515 0.032814 2.057507 0.0397 D(R_FTSE(-6)) 0.025725 0.026757 0.961458 0.3364 D(R_FTSE(-7)) -0.037158 0.019587 -1.897076 0.0579 0.000426 0.000184 2.315359 0.0207 Cc R-squared 0.470579 Mean dependent var -6.70E-06 Adjusted R-squared 0.468950 S.D dependent var 0.012811 S.E of regression 0.009336 Akaike info criterion -6.506417 Sum squared resid 0.226710 Schwarz criterion -6.486186 Log likelihood 8499.874 Hannan-Quinn criter -6.499088 F-statistic 288.9890 Durbin-Watson stat 1.999821 mes -^^^^~^ As we can see, all 4 variables are stationary Then we estimate an AR(1) up to AR(15): [view| Proc] Object[PriNnamte| |Ereeze ÏÏ£simate |Forecast|Stats| Resids |222201Vi)ew | Pr| Oobjcect] Print| Na|mFreeeze || Estimate Forecast] Stats | Res| ids200¡ Dependent Variable: R_FTSE ˆ Dependent Variable: R_STOCK1 product of gradients Method: ARMA Maximum Likellhood (BFGS) Method: ARMA Maximum Likelihood (BFGS) Date: 09/18/23 | Time: 21:10 Date: 09/18/23 Time: 21:11 reload thoeneone 261 m Sample: 1/01/1990 12/31/1999 Failure to improve objective (non-zero gradients) after 0 iterations Included observations: 2610 Coefficient covariance computed using outer product of gradients Convergence achieved after 74 iterations Coefficient covariance computed using outer Variable Coefficient Std Error t-Statistic Prob Variable Coefficient Std.Error t-Statstic Prob ° Q000392 0.000186 2402685 00356 AR(1) 0.071225 0.016328 + 4.362240 0.0000 AR(2) -0042617 0.015716 -2711643 0.0067 Cc 3.85E-05 0.000280 0.137745 0.8905 -0.034858 0015844 -2/200122 0.0279 AR(3) 0.011813 0.016681 0.708153 0.4789 AR(1) 0.043786 0.015629 2.801581 0.0051 0.024398 = 0.016355 -1.491748 0.1359 AR(4) -0.039511 0.016886 -2.339893 0.0194 AR(2) -0.022737 0.016396 -1.386766 0.1656 7.060914 0016927 -3.598675 0.0003 AR(5) 00 000338313377 00.001167472882 20 316827360658 00 80511723 AR(3) -0.064752 0.014778 -4.381548 0.0000 0030328 0016063 1.888068 0.0591 AR(6) 0029102 0.015988 1.820166 0.0688 AR(4) -0026803 0016694 -1605486 0.1085 AR(7) -0.003232 0.016486 -0.196041 0.8446 AR(5) 0005892 0.016081 0.366400 0.7141 0.033960 0.016363 2.075374 0.0381 AARR(G9)) 0006689 0018038 0.370637 0.7108 AR(6(6) ¬0003079 00: 17248 7-0178526 0.ở 8583 -0.006784 0.016508 -0.410946 0.6811 AR(10) 8.66E-05 1.84E-06 47.15898 0.0000 AR(7) -0.006745 0.017449 -0.386573 0.6991 AR(11) - _ ` - AR(8) -0.005816 0.017867 -0.325500 0.7448 0.019158 Meandependent var 0.000391 AR(12) 0.013106 S.D dependent var 0.009398 AR(9) -0.015367 0.018584 -0.826913 0.4084 0.009336 Akaike info criterion 6.503292 AR(13) 0.226020 Schwarz criterion -6.465077 AR(10) 0.001133 0.017264 0.065632 0.9477 AR(14) AR(11) 0012012 0.017830 04673690 0.5006 AR(15) AR(12) 4 0.000535 0017488 7 -0.030605 0.9756 SIGMASQ AR(13) 0.016900 0.019189 0.880685 0.3786 R-squared AR(14) ~0.004491 0.017551 -0.255883 0.7981 Adjusted R-squared AR(15) 0.004741 0.015423 -0.307358 0.7586 S.E of regression SIGMASQ 0.000229 3.31E-06 69.14144 0.0000 Sum squared resid Log likelihood 8503.796 Hannan-Quinn criter -6.489449 a 3.165465 _ Durbin-Watson stat 2.002491 0.008680 Mean dependent var 3.98E-05 F-statistic 0.000021 Resquared 0.002563 S.D dependent var 0.015208 Prob(F-statistic) Adjusted R-squared 0.015188 Akaike info criterion -8.530115 S.E of regression 0.598144 Schwarz criterion -5.491900 Inverted AR Roots 69+45i — 69-.45i 48 Sum squared resid 7233.800 Hannan-Quinn criter -8.516272 .37+.64i _ 10-.79i 10+.79i Log likelihood 1.419040 Durbin-Watson stat 2.000101 -24+.72i -35 -.58-.56Ï F-statistic -7§-16i -.75+.16i 0.122937 Prob(F-statistic) View| Proc| Object]] Print| Na| mFreeeze jl Estimate |Forecast| stats | Resids SG View| Proc] Object] Print| Name] Freezfej Estimate| Forecast] Stats Resids JINN Dependent Variable: R_STOCK2 product of gradients Dependent Variable: R_STOCK3 Method: ARMA Maximum Likelihood (BFGS) Method: ARMA Maximum Likelihood (BFGS) Date: 09/18/23 Time: 21:11 Date: 09/18/23 Time: 21:12 Sample: 1/01/1990 12/31/1999 Sample: 1/01/1990 12/31/1999 Included observations: 2610 Included observations: 2610 Convergence achieved after 22 Iterations Convergence achieved after 11 iterations Coefficient covariance computed using outer Coefficient covariance computed using outer product of gradients Variable Coefficient Std Error t-Statistic Prob Variable Coefficient Std Error t-Statistic Prob c 0.000142 0.000265 0.535982 0.5920 c 0.000563 0.000365 1.544148 0.1227 AR(1) 0.008709 0.012208 0.713339 0.4757 AR(1) 0.140578 0.014207 9.894839 0.0000 AR(2) -0026409 0.015453 -1.708997 0.0876 AR(2) 0.067243 0.015060 -4.464972 0.0000 AR(3) 0.033693 0.015741 -2.140454 0.0324 AR(3) -0.016377 0.015892 -1.030496 0.3029 AR(4) -0.016439 0.015758 -1.043240 0.2969 AR(4) 0.008132 0016245 0.500565 0.6167 AR(5) 0.021065 0.015996 1.316930 0.1880 AR(S) -0.035340 0.016104 -2.194532 0.0283 AR(6) 0.001996 0017903 0.111506 0.9112 AR(6) -0.015469 0.016089 -0.961511 0.3364 AR(7) -0.051193 0.018226 -2.808746 0.0050 AR(7) 70.023646 0.015568 -1.518846 0.1289 AR(8) -0035697 0017572 -2031484 - 0.0423 AR(8) 0.007297 0.015915 0.6466 AR(9) -0032041 0.018552 -1.727084 0.0843 AR(9) 0.015612 0.016254 0.458526 0.3369 AR(10) 0016793 0.018214 0.921950 0.3566 AARRi(10) 0017603 0.015011 0.960502 0.2410 AR(1(111) -0019200 0016387 -1.171668 0.2414 AR(1(121)) 00 003043605546 00 001155852176 1.172702 00 08248369 AR(12) 0.022337 0.016605 -1.345177 0.1787 AR(13) 0.021867 0.017479 -02 119896597310 0.2174 AR(13) 0.012741 0.018048 0.705926 0.4803 AR(14) -0.012107 0.015718 + 1.233880 0.4412 AR(14) 0.054170 0.016946 -3.196546 0.0014 AR(15) "0.015060 0016491 -0.770285 03612 AR(15) 0.000899 0.017368 0.051775 0.9587 SIGMASQ 0000332 575E.06 -0913204 00000 SIGMASQ 0.000258 3.72E-06 + 69.26567 0.0000 5764115 _ R-squared _ _ R-squared ` 0.000565 pgiusted R-squared 0.027757 Adjusted R-squared 0.011610 Mean dependent var 0.000148 Mean dependent var 0.018471 S.E of regression SE ofregression 0.021758 0.005511 S.D dependent var 0.016160 sum squared resid S.D dependent var -5.160693 Sum squared resid 0.016115 Akaike info criterion -5.411582 0.018269 -5.122478 Log likelihood 0.865438 Akaike info criterion Log likelihood 0.673398 Schwarzcrierion ~5.373367 Schwarz criterion -5.146849 F-statistic F-statistic 6751.704 7079.114 _ Hannan-Quinn criter -5.397738 _ Prob(F-statistic) Hannan-Quinn criter 2.000370 Prob(F-statistic) 1.903618 Durbin-Watson stat 1.999793 4.626763 Inverted AR Roots Inverted AR Roots 0.000000 Durbin-Watson stat 0.016213 .62+.49i 03-77i .74+.17i .74-.17i 83+.46i _ 63-46i 79+.22i 79-22i 62-49i -65+.B3i 40+.64i ‘40-641 .10-82i — 10+.82i 33-.75i 334.751 03+77i -29+76Ì -/29-761 -56-49i - -86+.49i -33-75i -.33+75i -60+23i -.80-23i -73 02 -80-17i -.80+.17i ~65-.53i