### Selected R comamnds and output of Chapter 4 > require(MTS) > kdx=c(2,1,1) ### A hypothetical set of Kronecker indices > Kronspec(kdx) Kronecker indices: 2 1 1 Dimension: 3 Notation: 0: fixed to 0 1: fixed to 1 2: estimation AR coefficient matrices: [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [1,] 1 0 0 2 0 0 2 2 2 [2,] 2 1 0 2 2 2 0 0 0 [3,] 2 0 1 2 2 2 0 0 0 MA coefficient matrices: [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [1,] 1 0 0 2 2 2 2 2 2 [2,] 2 1 0 2 2 2 0 0 0 [3,] 2 0 1 2 2 2 0 0 0 ### Flour index data set > da=read.table("flourc.txt") > head(da) V1 V2 V3 1 4.6738 4.6681 4.7086 2 4.7318 4.7221 4.7414 3 4.7247 4.7167 4.7493 4 4.7423 4.7300 4.7622 5 4.8154 4.8243 4.9053 6 4.8171 4.8227 4.8888 > zt=da > Kronid(zt,plag=3) h = 0 Component = 1 square of the smallest can. corr. = 0.9403009 test, df, & p-value: [1] 266.342 9.000 0.000 Component = 2 square of the smallest can. corr. = 0.8104609 test, df, & p-value: [1] 156.337 8.000 0.000 Component = 3 square of the smallest can. corr. = 0.761341 test, df, & p-value: [1] 133.959 7.000 0.000 ============= h = 1 Component = 1 Square of the smallest can. corr. = 0.04531181 test, df, p-value & d-hat: [1] 4.281 6.000 0.639 1.007 A Kronecker found Component = 2 Square of the smallest can. corr. = 0.03435539 test, df, p-value & d-hat: [1] 3.045 6.000 0.803 1.067 A Kronecker found Component = 3 Square of the smallest can. corr. = 0.02732133 test, df, p-value & d-hat: [1] 2.435 6.000 0.876 1.057 A Kronecker found ============ Kronecker indexes identified: [1] 1 1 1 > ## Estimation via Kronecker index approach > kdx=c(1,1,1) > m2=Kronfit(da,kdx) [,1] [,2] [,3] [,4] [,5] [,6] [1,] 1 0 0 2 2 2 [2,] 0 1 0 2 2 2 [3,] 0 0 1 2 2 2 [,1] [,2] [,3] [,4] [,5] [,6] [1,] 1 0 0 2 2 2 [2,] 0 1 0 2 2 2 [3,] 0 0 1 2 2 2 Number of parameters: 21 initial estimates: 0.2653 1.154 -0.5899 0.381 -1.1675 1.6282 -0.3749 0.2782 0.1966 0.3246 0.4207 -0.7387 1.3009 -0.438 0.3898 0.1355 -0.557 1.342 -0.3059 1.0568 -0.5455 Upper-bound: 0.5587 1.5616 -0.1305 0.5674 -0.2548 2.561 0.0583 0.5921 0.6324 0.8158 0.6201 0.2372 2.2983 0.0253 0.7244 0.6001 -0.0334 1.5545 0.7345 2.1202 -0.0516 Lower-bound: -0.0282 0.7464 -1.0492 0.1946 -2.0801 0.6955 -0.8082 -0.0356 -0.2393 -0.1665 0.2214 -1.7146 0.3035 -0.9013 0.0552 -0.3292 -1.0806 1.1295 -1.3463 -0.0066 -1.0394 Coefficient(s): Estimate Std. Error t value Pr(>|t|) [1,] 0.22161 0.13083 1.694 0.090299 . [2,] 1.20610 0.03675 32.821 < 2e-16 *** [3,] -0.64398 0.13673 -4.710 2.48e-06 *** [4,] 0.39135 0.09638 4.061 4.89e-05 *** [5,] -1.37906 0.38256 -3.605 0.000312 *** [6,] 1.80153 0.42583 4.231 2.33e-05 *** [7,] -0.41491 0.21580 -1.923 0.054519 . [8,] 0.22425 0.14376 1.560 0.118789 [9,] 0.26167 0.05258 4.976 6.48e-07 *** [10,] 0.25603 0.15168 1.688 0.091408 . [11,] 0.43460 0.10478 4.148 3.36e-05 *** [12,] -1.07730 0.41777 -2.579 0.009918 ** [13,] 1.56238 0.45348 3.445 0.000570 *** [14,] -0.47834 0.22940 -2.085 0.037050 * [15,] 0.31123 0.13413 2.320 0.020320 * [16,] 0.20986 0.09051 2.319 0.020418 * [17,] -0.63259 0.15831 -3.996 6.44e-05 *** [18,] 1.35840 0.09691 14.016 < 2e-16 *** [19,] -0.71976 0.46378 -1.552 0.120675 [20,] 1.34289 0.49117 2.734 0.006256 ** [21,] -0.62389 0.26935 -2.316 0.020541 * --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 --- Estimates in matrix form: Constant term: Estimates: 0.222 0.224 0.311 AR and MA lag-0 coefficient matrix [,1] [,2] [,3] [1,] 1 0 0 [2,] 0 1 0 [3,] 0 0 1 AR coefficient matrix AR( 1 )-matrix [,1] [,2] [,3] [1,] 1.206 -0.644 0.391 [2,] 0.262 0.256 0.435 [3,] 0.210 -0.633 1.358 MA coefficient matrix MA( 1 )-matrix [,1] [,2] [,3] [1,] 1.379 -1.802 0.415 [2,] 1.077 -1.562 0.478 [3,] 0.720 -1.343 0.624 Residuals cov-matrix: [,1] [,2] [,3] [1,] 0.001782420 0.001850524 0.001782202 [2,] 0.001850524 0.002053902 0.001960192 [3,] 0.001782202 0.001960192 0.002374990 ---- aic= -22.43436 bic= -21.88728 > MTSdiag(m2) [1] "Covariance matrix:" V1 V2 V3 V1 0.00180 0.00187 0.00180 V2 0.00187 0.00207 0.00198 V3 0.00180 0.00198 0.00240 CCM at lag: 0 [,1] [,2] [,3] [1,] 1.000 0.967 0.866 [2,] 0.967 1.000 0.888 [3,] 0.866 0.888 1.000 Simplified matrix: CCM at lag: 1 . . . . . . . . . CCM at lag: 2 . . . . . . . . . CCM at lag: 3 . . . . . . . . . CCM at lag: 4 . . . . . . . . . CCM at lag: 5 . . . . . . . . . CCM at lag: 6 . . . . . . . . . CCM at lag: 7 . . . . . . . . . CCM at lag: 8 . . . . . . . . . CCM at lag: 9 . . . . . . . . . CCM at lag: 10 . . . . . . . . . CCM at lag: 11 . . . . . . . . . CCM at lag: 12 . . . . . . . . . CCM at lag: 13 . . . . . . . . . CCM at lag: 14 . . . . . . . . . CCM at lag: 15 . . . . . . . . . CCM at lag: 16 . . . . . . . . . CCM at lag: 17 . . . . . . . . . CCM at lag: 18 . . . . . . . . . CCM at lag: 19 . . . . . . . . . CCM at lag: 20 . . . . . . . . . CCM at lag: 21 . . . . . . . . . CCM at lag: 22 . . . . . . . . . CCM at lag: 23 . . . . . . . . . CCM at lag: 24 . . . . . . . . . Hit Enter for p-value plot of individual ccm: ## Plot not shown Hit Enter to compute MQ-statistics: ## See Figure 4.3 of the text Ljung-Box Statistics: m Q(m) df p-value [1,] 1.00 6.04 9.00 0.74 [2,] 2.00 12.37 18.00 0.83 [3,] 3.00 19.75 27.00 0.84 [4,] 4.00 25.00 36.00 0.92 [5,] 5.00 31.92 45.00 0.93 [6,] 6.00 38.82 54.00 0.94 [7,] 7.00 52.00 63.00 0.84 [8,] 8.00 68.85 72.00 0.58 [9,] 9.00 72.87 81.00 0.73 [10,] 10.00 81.21 90.00 0.74 [11,] 11.00 86.22 99.00 0.82 [12,] 12.00 91.01 108.00 0.88 [13,] 13.00 96.36 117.00 0.92 [14,] 14.00 105.33 126.00 0.91 [15,] 15.00 112.02 135.00 0.93 [16,] 16.00 116.97 144.00 0.95 [17,] 17.00 124.22 153.00 0.96 [18,] 18.00 134.50 162.00 0.94 [19,] 19.00 146.12 171.00 0.92 [20,] 20.00 152.47 180.00 0.93 [21,] 21.00 163.40 189.00 0.91 [22,] 22.00 168.28 198.00 0.94 [23,] 23.00 179.54 207.00 0.92 [24,] 24.00 192.03 216.00 0.88 Hit Enter to obtain residual plots: ### See Figure 4.2 of the text > #### SCM approach > m2=SCMid2(da,maxp=1,maxq=1,crit=0.01) For (pi,qi) = ( 0 , 0 ) Tests: Eigvalue St.dev Test deg p-value [1,] 0.747 1 137.548 1 0 [2,] 0.815 1 306.056 4 0 [3,] 0.938 1 584.245 9 0 Summary: Number of SCMs detected: 0 For (pi,qi) = ( 1 , 0 ) Tests: Eigvalue St.dev Test deg p-value [1,] 0.003 1 0.294 1 0.588 [2,] 0.081 1 8.672 4 0.070 [3,] 0.271 1 39.922 9 0.000 Summary: Number of SCMs detected: 2 For (pi,qi) = ( 0 , 1 ) Tests: Eigvalue St.dev Test deg p-value [1,] 0.598 1 90.222 1 0 [2,] 0.799 1 249.032 4 0 [3,] 0.866 1 447.866 9 0 Summary: Number of SCMs detected: 0 For (pi,qi) = ( 1 , 1 ) Tests: Eigvalue St.dev Test deg p-value [1,] 0.003 0.998 0.343 1 0.558 [2,] 0.033 1.022 3.589 4 0.464 [3,] 0.057 1.448 7.491 9 0.586 [4,] 0.787 1.000 158.919 16 0.000 [5,] 0.874 1.000 362.131 25 0.000 [6,] 0.944 1.000 644.965 36 0.000 Summary: Number of SCMs detected: 3 wvector [,1] [,2] [1,] 0.332 -0.3376 [2,] -0.628 0.1248 [3,] 0.260 0.5881 [4,] -0.294 0.2914 [5,] 0.545 0.0233 [6,] -0.212 -0.6627 The number of newly found SCMs: 1 Vectors: [,1] V1 0.07998 V2 0.12880 V3 -0.00707 V1 -0.12213 V2 0.00758 V3 -0.07628 SUMMARY: Overall model: 1 1 Orders of SCM: [,1] [,2] [1,] 1 0 [2,] 1 0 [3,] 1 1 Transformation Matrix: [,1] [,2] [,3] V1 0.332 -0.338 0.07998 V2 -0.628 0.125 0.12880 V3 0.260 0.588 -0.00707 > > names(m2) [1] "Tmatrix" "SCMorder" > tmx=m2\$Tmatrix > print(tmx,digits=3) V1 V2 V3 [1,] 0.332 -0.628 0.25956 [2,] -0.338 0.125 0.58808 [3,] 0.080 0.129 -0.00707 ### Try VARMA(2,0) model > SCMid2(da,maxp=2,maxq=0,crit=0.01) For (pi,qi) = ( 0 , 0 ) Tests: Eigvalue St.dev Test deg p-value [1,] 0.747 1 137.548 1 0 [2,] 0.815 1 306.056 4 0 [3,] 0.938 1 584.245 9 0 Summary: Number of SCMs detected: 0 For (pi,qi) = ( 1 , 0 ) Tests: Eigvalue St.dev Test deg p-value [1,] 0.003 1 0.294 1 0.588 [2,] 0.081 1 8.672 4 0.070 [3,] 0.271 1 39.922 9 0.000 Summary: Number of SCMs detected: 2 For (pi,qi) = ( 2 , 0 ) Tests: Eigvalue St.dev Test deg p-value [1,] 0.001 1 0.070 1 0.791 [2,] 0.007 1 0.753 4 0.945 [3,] 0.034 1 4.180 9 0.899 Summary: Number of SCMs detected: 3 wvector [,1] [,2] [1,] 0.332 -0.3376 [2,] -0.628 0.1248 [3,] 0.260 0.5881 [4,] -0.294 0.2914 [5,] 0.545 0.0233 [6,] -0.212 -0.6627 The number of newly found SCMs: 1 Vectors: [,1] V1 -0.2076 V2 0.0385 V3 -0.0602 V1 -0.0265 V2 0.2068 V3 0.0553 V1 0.2485 V2 -0.3309 V3 0.0628 SUMMARY: Overall model: 2 0 Orders of SCM: [,1] [,2] [1,] 1 0 [2,] 1 0 [3,] 2 0 Transformation Matrix: [,1] [,2] [,3] V1 0.332 -0.338 -0.2076 V2 -0.628 0.125 0.0385 V3 0.260 0.588 -0.0602 > ### Estimation via the SCM approach > scms=matrix(c(1,1,1,0,0,1),3,2) > Tdx=c(2,3,1) > m1=SCMfit(da,scms,Tdx) Maximum VARMA order: ( 1 , 1 ) Locations of estimable parameters: Transformation Matrix [,1] [,2] [,3] [1,] 2 1 2 [2,] 2 0 1 [3,] 1 0 0 AR parameters [,1] [,2] [,3] [1,] 2 2 2 [2,] 2 2 2 [3,] 2 2 2 MA parameters [,1] [,2] [,3] [1,] 0 0 0 [2,] 0 0 0 [3,] 2 2 2 iniSCM p and q: 1 1 Number of parameters: 18 initial estimates: -0.0367 -0.7958 -0.2234 -0.7004 0.8665 -0.1789 0.0589 -0.966 -0.8392 -0.0937 0.9548 0.2642 1.1551 -0.5908 0.381 -1.1684 1.6292 -0.3749 Upper-bound: 0.0395 -0.6981 -0.136 -0.5726 0.9723 -0.0839 0.2355 -0.8514 -0.5961 0.1513 1.0602 0.5653 1.5691 -0.1276 0.5674 -0.2538 2.5638 0.0583 Lower-bound: -0.1129 -0.8936 -0.3108 -0.8281 0.7608 -0.2739 -0.1178 -1.0806 -1.0823 -0.3387 0.8493 -0.037 0.7411 -1.0541 0.1946 -2.0831 0.6946 -0.8081 Coefficient(s): Estimate Std. Error t value Pr(>|t|) [1,] 0.002667 1.000000 0.003 0.9979 [2,] -0.698080 1.000000 -0.698 0.4851 [3,] -0.222704 1.000000 -0.223 0.8238 [4,] -0.630929 1.000000 -0.631 0.5281 [5,] 0.861560 1.000000 0.862 0.3889 [6,] -0.152670 1.000000 -0.153 0.8787 [7,] 0.086371 1.000000 0.086 0.9312 [8,] -0.851433 1.000000 -0.851 0.3945 [9,] -0.732612 1.000000 -0.733 0.4638 [10,] -0.121506 1.000000 -0.122 0.9033 [11,] 0.985040 1.000000 0.985 0.3246 [12,] 0.204390 1.000000 0.204 0.8380 [13,] 1.249603 1.000000 1.250 0.2114 [14,] -0.631289 1.000000 -0.631 0.5279 [15,] 0.338744 1.000000 0.339 0.7348 [16,] -1.699326 1.000000 -1.699 0.0893 . [17,] 1.856212 1.000000 1.856 0.0634 . [18,] -0.148200 1.000000 -0.148 0.8822 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 --- Estimates in matrix form: Constant term: Estimates: 0.003 0.086 0.204 AR and MA lag-0 coefficient matrix [,1] [,2] [,3] [1,] -0.698 1 -0.223 [2,] -0.851 0 1.000 [3,] 1.000 0 0.000 AR coefficient matrix AR( 1 )-matrix [,1] [,2] [,3] [1,] -0.631 0.862 -0.153 [2,] -0.733 -0.122 0.985 [3,] 1.250 -0.631 0.339 MA coefficient matrix MA( 1 )-matrix [,1] [,2] [,3] [1,] 0.000 0.000 0.000 [2,] 0.000 0.000 0.000 [3,] 1.699 -1.856 0.148 Residuals cov-matrix: [,1] [,2] [,3] [1,] 0.001795279 0.001883551 0.001834701 [2,] 0.001883551 0.002114499 0.002050278 [3,] 0.001834701 0.002050278 0.002508061 ---- aic= -22.47114 bic= -22.08037 ### Multiple the data by 100 for numrical stability > zt=da*100 > m2=SCMfit(zt,scms,Tdx) Estimates in matrix form: Constant term: Estimates: 0.019 8.183 18.832 AR and MA lag-0 coefficient matrix [,1] [,2] [,3] [1,] -0.698 1 -0.222 [2,] -0.851 0 1.000 [3,] 1.000 0 0.000 AR coefficient matrix AR( 1 )-matrix [,1] [,2] [,3] [1,] -0.624 0.853 -0.150 [2,] -0.731 -0.122 0.985 [3,] 1.278 -0.662 0.345 MA coefficient matrix MA( 1 )-matrix [,1] [,2] [,3] [1,] 0.000 0.000 0.000 [2,] 0.000 0.000 0.000 [3,] 1.722 -1.886 0.156 Residuals cov-matrix: [,1] [,2] [,3] [1,] 17.95800 18.83915 18.34978 [2,] 18.83915 21.14626 20.50363 [3,] 18.34978 20.50363 25.08113 ---- aic= 5.15973 bic= 5.550505 > MTSdiag(m2) ### Model checking [1] "Covariance matrix:" V1 V2 V3 V1 18.1 19.0 18.5 V2 19.0 21.4 20.7 V3 18.5 20.7 25.3 CCM at lag: 0 [,1] [,2] [,3] [1,] 1.000 0.967 0.865 [2,] 0.967 1.000 0.890 [3,] 0.865 0.890 1.000 Simplified matrix: CCM at lag: 1 . . . . . . . . . CCM at lag: 2 . . . . . . . . . CCM at lag: 3 . . . . . . . . . CCM at lag: 4 . . . . . . . . . CCM at lag: 5 . . . . . . . . . CCM at lag: 6 . . . . . . . . . CCM at lag: 7 . . . . . . . . . CCM at lag: 8 . . . . . . . . . CCM at lag: 9 . . . . . . . . . CCM at lag: 10 . . . . . . . . . CCM at lag: 11 . . . . . . . . . CCM at lag: 12 . . . . . . . . . CCM at lag: 13 . . . . . . . . . CCM at lag: 14 . . . . . . . . . CCM at lag: 15 . . . . . . . . . CCM at lag: 16 . . . . . . . . . CCM at lag: 17 - . . . . . . . . CCM at lag: 18 . . . . . . . . . CCM at lag: 19 . . . . . + . . . CCM at lag: 20 . . . . . . . . . CCM at lag: 21 . . . . . . . . . CCM at lag: 22 . . . . . . . . . CCM at lag: 23 . . . . . . . . . CCM at lag: 24 . . . . . . . . . Hit Enter for p-value plot of individual ccm: ## Plot not shown Hit Enter to compute MQ-statistics: ## See Figure 4.5 of the text Ljung-Box Statistics: m Q(m) df p-value [1,] 1.0 11.1 9.0 0.27 [2,] 2.0 16.6 18.0 0.55 [3,] 3.0 22.9 27.0 0.69 [4,] 4.0 26.1 36.0 0.89 [5,] 5.0 31.9 45.0 0.93 [6,] 6.0 41.0 54.0 0.90 [7,] 7.0 54.4 63.0 0.77 [8,] 8.0 72.3 72.0 0.47 [9,] 9.0 75.2 81.0 0.66 [10,] 10.0 83.1 90.0 0.68 [11,] 11.0 88.0 99.0 0.78 [12,] 12.0 94.1 108.0 0.83 [13,] 13.0 98.2 117.0 0.90 [14,] 14.0 107.4 126.0 0.88 [15,] 15.0 113.9 135.0 0.91 [16,] 16.0 118.5 144.0 0.94 [17,] 17.0 126.0 153.0 0.95 [18,] 18.0 135.1 162.0 0.94 [19,] 19.0 151.2 171.0 0.86 [20,] 20.0 157.5 180.0 0.89 [21,] 21.0 168.9 189.0 0.85 [22,] 22.0 176.6 198.0 0.86 [23,] 23.0 189.8 207.0 0.80 [24,] 24.0 199.7 216.0 0.78 Hit Enter to obtain residual plots: ### See Figure 4.4 of the text > > m3=refSCMfit(m2,thres=0.3) ### refinement Maximum VARMA order: ( 1 , 1 ) Number of parameters: 13 initial estimates: -0.6982 -0.624 0.8534 8.1828 -0.8514 -0.7306 0.9847 18.8325 1.2779 -0.6624 0.3446 -1.7224 1.886 Upper-bound: 1.3018 1.376 2.8534 10.1828 1.1486 1.2694 2.9847 20.8325 3.2779 1.3376 2.3446 0.2776 3.886 Lower-bound: -2.6982 -2.624 -1.1466 6.1828 -2.8514 -2.7306 -1.0153 16.8325 -0.7221 -2.6624 -1.6554 -3.7224 -0.114 Coefficient(s): Estimate Std. Error t value Pr(>|t|) [1,] -0.92122 0.05056 -18.221 < 2e-16 *** [2,] -0.84029 0.06150 -13.662 < 2e-16 *** [3,] 0.91867 0.04202 21.863 < 2e-16 *** [4,] 10.18281 8.00245 1.272 0.2032 [5,] -0.75176 0.13215 -5.689 1.28e-08 *** [6,] -0.70629 0.11496 -6.144 8.05e-10 *** [7,] 0.93452 0.04639 20.145 < 2e-16 *** [8,] 16.83246 11.26423 1.494 0.1351 [9,] 1.20378 0.19163 6.282 3.35e-10 *** [10,] -0.38338 0.21094 -1.817 0.0691 . [11,] 0.14519 0.08217 1.767 0.0772 . [12,] -1.74540 0.37000 -4.717 2.39e-06 *** [13,] 1.75233 0.37067 4.727 2.27e-06 *** --- Estimates in matrix form: Constant term: Estimates: 0 10.183 16.832 AR and MA lag-0 coefficient matrix [,1] [,2] [,3] [1,] -0.921 1 0 [2,] -0.752 0 1 [3,] 1.000 0 0 AR coefficient matrix AR( 1 )-matrix [,1] [,2] [,3] [1,] -0.840 0.919 0.000 [2,] -0.706 0.000 0.935 [3,] 1.204 -0.383 0.145 MA coefficient matrix MA( 1 )-matrix [,1] [,2] [,3] [1,] 0.000 0.000 0 [2,] 0.000 0.000 0 [3,] 1.745 -1.752 0 Residuals cov-matrix: [,1] [,2] [,3] [1,] 18.22302 19.27914 18.80634 [2,] 19.27914 21.85261 21.17994 [3,] 18.80634 21.17994 25.66284 ---- aic= 5.11219 bic= 5.372707 > m4=refSCMfit(m3,thres=1.645) Maximum VARMA order: ( 1 , 1 ) Number of parameters: 11 initial estimates: -0.9212 -0.8403 0.9187 -0.7518 -0.7063 0.9345 1.2038 -0.3834 0.1452 -1.7454 1.7523 Upper-bound: -0.8201 -0.7173 1.0027 -0.4875 -0.4764 1.0273 1.587 0.0385 0.3095 -1.0054 2.4937 Lower-bound: -1.0223 -0.9633 0.8346 -1.0161 -0.9362 0.8417 0.8205 -0.8053 -0.0191 -2.4854 1.011 Coefficient(s): Estimate Std. Error t value Pr(>|t|) [1,] -0.91200 0.05417 -16.836 < 2e-16 *** [2,] -0.83869 0.06308 -13.295 < 2e-16 *** [3,] 0.92634 0.04203 22.041 < 2e-16 *** [4,] -0.77420 0.12756 -6.069 1.29e-09 *** [5,] -0.69176 0.11949 -5.789 7.07e-09 *** [6,] 0.91783 0.04265 21.520 < 2e-16 *** [7,] 1.29601 0.18541 6.990 2.75e-12 *** [8,] -0.42767 0.22005 -1.943 0.052 . [9,] 0.13054 0.08273 1.578 0.115 [10,] -1.78220 0.37326 -4.775 1.80e-06 *** [11,] 1.79590 0.37426 4.798 1.60e-06 *** --- Estimates in matrix form: Constant term: Estimates: 0 0 0 AR and MA lag-0 coefficient matrix [,1] [,2] [,3] [1,] -0.912 1 0 [2,] -0.774 0 1 [3,] 1.000 0 0 AR coefficient matrix AR( 1 )-matrix [,1] [,2] [,3] [1,] -0.839 0.926 0.000 [2,] -0.692 0.000 0.918 [3,] 1.296 -0.428 0.131 MA coefficient matrix MA( 1 )-matrix [,1] [,2] [,3] [1,] 0.000 0.000 0 [2,] 0.000 0.000 0 [3,] 1.782 -1.796 0 Residuals cov-matrix: [,1] [,2] [,3] [1,] 18.73704 19.80196 19.52138 [2,] 19.80196 22.37754 21.91817 [3,] 19.52138 21.91817 26.67848 ---- aic= 5.110271 bic= 5.318684 > ##### GDP example > da=read.table("q-gdp-ukcaus.txt",header=T) > gdp=log(da[,3:5]) > zt=diffM(gdp) > colnames(zt) <- c("uk","ca","us") > SCMid(zt) Column: MA order Row : AR order Number of zero canonical correlations 0 1 2 3 4 5 0 1 2 2 2 3 3 1 2 3 4 4 6 6 2 2 5 6 7 8 8 3 2 5 8 9 10 11 4 3 6 8 11 12 13 5 3 6 9 11 13 15 Diagonal Differences: 0 1 2 3 4 5 0 1 2 2 2 3 3 1 2 2 2 2 3 3 2 2 3 3 3 3 2 3 2 3 3 3 3 3 4 3 3 3 3 3 3 5 3 3 3 3 2 3 > SCMid2(zt,maxp=2,maxq=1) For (pi,qi) = ( 0 , 0 ) Tests: Eigvalue St.dev Test deg p-value [1,] 0.000 1 0.025 1 0.874 [2,] 0.098 1 12.863 4 0.012 [3,] 0.568 1 117.706 9 0.000 Summary: Number of SCMs detected: 1 For (pi,qi) = ( 1 , 0 ) Tests: Eigvalue St.dev Test deg p-value [1,] 0.000 1 0.015 1 0.903 [2,] 0.021 1 2.701 4 0.609 [3,] 0.138 1 21.049 9 0.012 Summary: Number of SCMs detected: 2 wvector [,1] [1,] 0.484 [2,] 0.287 [3,] -0.827 The number of newly found SCMs: 1 Vectors: [,1] uk 0.6690 ca -0.0942 us 0.3621 uk -0.4034 ca -0.1568 us -0.0690 For (pi,qi) = ( 0 , 1 ) Tests: Eigvalue St.dev Test deg p-value [1,] 0.006 1.008 0.758 1 0.384 [2,] 0.029 1.016 4.373 4 0.358 [3,] 0.293 1.000 47.283 9 0.000 Summary: Number of SCMs detected: 2 wvector [,1] [1,] 0.484 [2,] 0.287 [3,] -0.827 The number of newly found SCMs: 1 Vectors: [,1] uk -0.5305 ca 0.7068 us -0.0625 Exchangeable SCM found with order: 1 0 SUMMARY: Overall model: 1 1 Orders of SCM: [,1] [,2] [1,] 0 0 [2,] 1 0 [3,] 0 1 Transformation Matrix: [,1] [,2] [,3] uk 0.484 0.6690 -0.5305 ca 0.287 -0.0942 0.7068 <=== The maximum element of each column us -0.827 0.3621 -0.0625 is lcoated at 3, 1, 2. ### > scms=matrix(c(0,1,0,0,0,1),3,2) > Tdx=c(3,1,2) > m1=SCMfit(zt,scms,Tdx) Maximum VARMA order: ( 1 , 1 ) Locations of estimable parameters: Transformation Matrix [,1] [,2] [,3] [1,] 2 2 1 [2,] 1 2 0 [3,] 2 1 0 AR parameters [,1] [,2] [,3] [1,] 0 0 0 [2,] 2 2 2 [3,] 0 0 0 MA parameters [,1] [,2] [,3] [1,] 0 0 0 [2,] 0 0 0 [3,] 2 2 2 iniSCM p and q: 1 1 Number of parameters: 13 initial estimates: 0.0017 -0.3528 -0.4641 0.0018 -0.1253 0.3663 0.1463 0.0307 0.0039 -0.3808 0.2324 0.2629 0.4066 Upper-bound: 0.0031 -0.1865 -0.3224 0.0031 0.0518 0.5418 0.3109 0.22 0.0053 -0.209 0.4486 0.4777 0.6082 Lower-bound: 4e-04 -0.5191 -0.6058 4e-04 -0.3024 0.1908 -0.0182 -0.1586 0.0025 -0.5526 0.0162 0.048 0.2051 Coefficient(s): Estimate Std. Error t value Pr(>|t|) [1,] 0.0016050 0.0008439 1.902 0.05718 . [2,] -0.4298640 0.2137219 -2.011 0.04429 * [3,] -0.4421343 0.1946326 -2.272 0.02311 * [4,] 0.0017418 0.0009975 1.746 0.08078 . [5,] 0.0517923 0.3147652 0.165 0.86930 [6,] 0.4532115 0.0917254 4.941 7.77e-07 *** [7,] 0.2427513 0.1118700 2.170 0.03001 * [8,] 0.0147480 0.1151252 0.128 0.89807 [9,] 0.0032545 0.0016816 1.935 0.05295 . [10,] -0.5525652 0.2759256 -2.003 0.04522 * [11,] 0.0162190 0.1563863 0.104 0.91740 [12,] 0.3128596 0.1848953 1.692 0.09063 . [13,] 0.3311378 0.1047784 3.160 0.00158 ** --- Estimates in matrix form: Constant term: Estimates: 0.002 0.002 0.003 AR and MA lag-0 coefficient matrix [,1] [,2] [,3] [1,] -0.430 -0.442 1 [2,] 1.000 0.052 0 [3,] -0.553 1.000 0 AR coefficient matrix AR( 1 )-matrix [,1] [,2] [,3] [1,] 0.000 0.000 0.000 [2,] 0.453 0.243 0.015 [3,] 0.000 0.000 0.000 MA coefficient matrix MA( 1 )-matrix [,1] [,2] [,3] [1,] 0.000 0.000 0.000 [2,] 0.000 0.000 0.000 [3,] -0.016 -0.313 -0.331 Residuals cov-matrix: [,1] [,2] [,3] [1,] 2.884249e-05 3.603214e-06 7.358836e-06 [2,] 3.603214e-06 3.166129e-05 1.580864e-05 [3,] 7.358836e-06 1.580864e-05 3.829786e-05 ---- aic= -31.12173 bic= -30.91809 > > m2=refSCMfit(m1,thres=0.8) ## Refinement Maximum VARMA order: ( 1 , 1 ) Number of parameters: 10 initial estimates: 0.0016 -0.4299 -0.4421 0.0017 0.4532 0.2428 0.0033 -0.5526 0.3129 0.3311 Upper-bound: 0.0033 -0.0024 -0.0529 0.0037 0.6367 0.4665 0.0066 -7e-04 0.6827 0.5407 Lower-bound: -1e-04 -0.8573 -0.8314 -3e-04 0.2698 0.019 -1e-04 -1.1044 -0.0569 0.1216 Coefficient(s): Estimate Std. Error t value Pr(>|t|) [1,] 0.0014012 0.0008252 1.698 0.089488 . [2,] -0.4516302 0.2186715 -2.065 0.038891 * [3,] -0.4560473 0.1697517 -2.687 0.007219 ** [4,] 0.0017255 0.0006540 2.638 0.008329 ** [5,] 0.4120645 0.0710785 5.797 6.74e-09 *** [6,] 0.2440985 0.0610433 3.999 6.37e-05 *** [7,] 0.0017797 0.0013049 1.364 0.172618 [8,] -0.8265205 0.1968681 -4.198 2.69e-05 *** [9,] 0.2143750 0.1279605 1.675 0.093871 . [10,] 0.3405352 0.0886100 3.843 0.000121 *** --- Estimates in matrix form: Constant term: Estimates: 0.001 0.002 0.002 AR and MA lag-0 coefficient matrix [,1] [,2] [,3] [1,] -0.452 -0.456 1 [2,] 1.000 0.000 0 [3,] -0.827 1.000 0 AR coefficient matrix AR( 1 )-matrix [,1] [,2] [,3] [1,] 0.000 0.000 0 [2,] 0.412 0.244 0 [3,] 0.000 0.000 0 MA coefficient matrix MA( 1 )-matrix [,1] [,2] [,3] [1,] 0 0.000 0.000 [2,] 0 0.000 0.000 [3,] 0 -0.214 -0.341 Residuals cov-matrix: [,1] [,2] [,3] [1,] 2.905895e-05 3.423511e-06 7.379354e-06 [2,] 3.423511e-06 3.110141e-05 1.528242e-05 [3,] 7.379354e-06 1.528242e-05 3.792915e-05 ---- aic= -31.17982 bic= -31.04406 > m3=refSCMfit(m2,thres=1.5) Maximum VARMA order: ( 1 , 1 ) Number of parameters: 9 initial estimates: 0.0014 -0.4516 -0.456 0.0017 0.4121 0.2441 -0.8265 0.2144 0.3405 Upper-bound: 0.0031 -0.0143 -0.1165 0.003 0.5542 0.3662 -0.4328 0.4703 0.5178 Lower-bound: -2e-04 -0.889 -0.7956 4e-04 0.2699 0.122 -1.2203 -0.0415 0.1633 Coefficient(s): Estimate Std. Error t value Pr(>|t|) [1,] 0.0010864 0.0008854 1.227 0.219802 [2,] -0.4604395 0.2627118 -1.753 0.079664 . [3,] -0.4807831 0.1767058 -2.721 0.006512 ** [4,] 0.0021292 0.0005675 3.752 0.000175 *** [5,] 0.3703639 0.0627704 5.900 3.63e-09 *** [6,] 0.2429388 0.0570271 4.260 2.04e-05 *** [7,] -1.0519755 0.1219635 -8.625 < 2e-16 *** [8,] 0.1662339 0.1241518 1.339 0.180585 [9,] 0.3405549 0.0891370 3.821 0.000133 *** --- Estimates in matrix form: Constant term: Estimates: 0.001 0.002 0 AR and MA lag-0 coefficient matrix [,1] [,2] [,3] [1,] -0.460 -0.481 1 [2,] 1.000 0.000 0 [3,] -1.052 1.000 0 AR coefficient matrix AR( 1 )-matrix [,1] [,2] [,3] [1,] 0.00 0.000 0 [2,] 0.37 0.243 0 [3,] 0.00 0.000 0 MA coefficient matrix MA( 1 )-matrix [,1] [,2] [,3] [1,] 0 0.000 0.000 [2,] 0 0.000 0.000 [3,] 0 -0.166 -0.341 Residuals cov-matrix: [,1] [,2] [,3] [1,] 2.925214e-05 3.124567e-06 7.335133e-06 [2,] 3.124567e-06 3.124085e-05 1.525267e-05 [3,] 7.335133e-06 1.525267e-05 3.790759e-05 ---- aic= -31.18219 bic= -31.06906 > m4=refSCMfit(m3,thres=1.5) Maximum VARMA order: ( 1 , 1 ) Number of parameters: 7 initial estimates: -0.4604 -0.4808 0.0021 0.3704 0.2429 -1.052 0.3406 Upper-bound: 0.065 -0.1274 0.0033 0.4959 0.357 -0.808 0.5188 Lower-bound: -0.9859 -0.8342 0.001 0.2448 0.1289 -1.2959 0.1623 Coefficient(s): Estimate Std. Error t value Pr(>|t|) [1,] -0.6574063 0.2071234 -3.174 0.0015 ** [2,] -0.4306281 0.1675214 -2.571 0.0102 * [3,] 0.0021545 0.0005339 4.035 5.45e-05 *** [4,] 0.3434554 0.0585798 5.863 4.54e-09 *** [5,] 0.2645765 0.0507829 5.210 1.89e-07 *** [6,] -1.0863097 0.1141818 -9.514 < 2e-16 *** [7,] 0.4130114 0.0755228 5.469 4.53e-08 *** --- Estimates in matrix form: Constant term: Estimates: 0 0.002 0 AR and MA lag-0 coefficient matrix [,1] [,2] [,3] [1,] -0.657 -0.431 1 [2,] 1.000 0.000 0 [3,] -1.086 1.000 0 AR coefficient matrix AR( 1 )-matrix [,1] [,2] [,3] [1,] 0.000 0.000 0 [2,] 0.343 0.265 0 [3,] 0.000 0.000 0 MA coefficient matrix MA( 1 )-matrix [,1] [,2] [,3] [1,] 0 0 0.000 [2,] 0 0 0.000 [3,] 0 0 -0.413 Residuals cov-matrix: [,1] [,2] [,3] [1,] 2.947774e-05 2.557159e-06 6.978774e-06 [2,] 2.557159e-06 3.109328e-05 1.504319e-05 [3,] 6.978774e-06 1.504319e-05 3.826788e-05 ---- aic= -31.18843 bic= -31.12055 ### Model checking > MTSdiag(m4) [1] "Covariance matrix:" uk ca us uk 2.97e-05 2.59e-06 7.10e-06 ca 2.59e-06 3.13e-05 1.51e-05 us 7.10e-06 1.51e-05 3.85e-05 CCM at lag: 0 [,1] [,2] [,3] [1,] 1.0000 0.0849 0.210 [2,] 0.0849 1.0000 0.436 [3,] 0.2103 0.4363 1.000 Simplified matrix: CCM at lag: 1 . . . . . . . . . CCM at lag: 2 . . . . . . . . . CCM at lag: 3 . . . . . . . . . CCM at lag: 4 . . - . . . . . . CCM at lag: 5 . . . . . . . . . CCM at lag: 6 . . . . . . . - . CCM at lag: 7 . . . . . . . . . CCM at lag: 8 . . . . . . . . . CCM at lag: 9 . . + . . . . . . CCM at lag: 10 . . . . . . . . . CCM at lag: 11 . . . . . . . . . CCM at lag: 12 . . . . . . . . . CCM at lag: 13 . - . . . . . . . CCM at lag: 14 . - . . . . . . . CCM at lag: 15 . . . . . . . . . CCM at lag: 16 . . . . . . . . . CCM at lag: 17 . . . . + . . . . CCM at lag: 18 . . . . . . . . . CCM at lag: 19 . . . . . + . . . CCM at lag: 20 . . . . . . . . . CCM at lag: 21 . . . . . . . . . CCM at lag: 22 . . . . . . . . . CCM at lag: 23 . . . . . . . . . CCM at lag: 24 . . . . . . . . . Hit Enter for p-value plot of individual ccm: Hit Enter to compute MQ-statistics: ### See Figure 4.7 of the text Ljung-Box Statistics: m Q(m) df p-value [1,] 1.00 6.04 9.00 0.74 [2,] 2.00 13.47 18.00 0.76 [3,] 3.00 22.98 27.00 0.69 [4,] 4.00 39.58 36.00 0.31 [5,] 5.00 45.03 45.00 0.47 [6,] 6.00 51.69 54.00 0.56 [7,] 7.00 60.60 63.00 0.56 [8,] 8.00 75.89 72.00 0.35 [9,] 9.00 84.29 81.00 0.38 [10,] 10.00 93.99 90.00 0.37 [11,] 11.00 96.75 99.00 0.55 [12,] 12.00 107.30 108.00 0.50 [13,] 13.00 118.19 117.00 0.45 [14,] 14.00 125.84 126.00 0.49 [15,] 15.00 135.94 135.00 0.46 [16,] 16.00 139.97 144.00 0.58 [17,] 17.00 148.56 153.00 0.59 [18,] 18.00 154.28 162.00 0.65 [19,] 19.00 166.89 171.00 0.57 [20,] 20.00 176.78 180.00 0.55 [21,] 21.00 180.44 189.00 0.66 [22,] 22.00 188.51 198.00 0.67 [23,] 23.00 192.70 207.00 0.75 [24,] 24.00 206.38 216.00 0.67 Hit Enter to obtain residual plots: > ##### Kronecker index approach (GDP example) > Kronid(zt,plag=4) h = 0 Component = 1 square of the smallest can. corr. = 0.4780214 test, df, & p-value: [1] 76.715 12.000 0.000 Component = 2 square of the smallest can. corr. = 0.2498917 test, df, & p-value: [1] 33.786 11.000 0.000 Component = 3 square of the smallest can. corr. = 0.1130244 test, df, & p-value: [1] 14.033 10.000 0.172 A Kronecker index found ============= h = 1 Component = 1 Square of the smallest can. corr. = 0.07259501 test, df, p-value & d-hat: [1] 10.114 9.000 0.341 0.873 A Kronecker found Component = 2 Square of the smallest can. corr. = 0.08127047 test, df, p-value & d-hat: [1] 9.759 9.000 0.370 1.011 A Kronecker found ============ Kronecker indexes identified: [1] 1 1 0 > > kdx=c(1,1,0) ### Estimation > m1=Kronfit(zt,kdx) [,1] [,2] [,3] [,4] [,5] [,6] [1,] 1 0 0 2 2 0 [2,] 0 1 0 2 2 0 [3,] 2 2 1 0 0 0 [,1] [,2] [,3] [,4] [,5] [,6] [1,] 1 0 0 2 2 2 [2,] 0 1 0 2 2 2 [3,] 2 2 1 0 0 0 Number of parameters: 15 initial estimates: 0.0016 0.3827 0.2992 0.0467 -0.1849 0.0515 0.0023 0.1765 0.4634 0.1498 -0.1408 0.4883 0.0012 -0.3385 -0.5637 Upper-bound: 0.0032 0.7406 0.5637 0.449 0.1444 0.2321 0.0039 0.5442 0.7351 0.5631 0.1975 0.6738 0.0031 0.0753 -0.2608 Lower-bound: 0 0.0247 0.0346 -0.3557 -0.5143 -0.1292 7e-04 -0.1912 0.1917 -0.2635 -0.4791 0.3027 -6e-04 -0.7524 -0.8666 Coefficient(s): Estimate Std. Error t value Pr(>|t|) [1,] 0.0011779 0.0007821 1.506 0.132072 [2,] 0.4359168 0.2009422 2.169 0.030055 * [3,] 0.3115727 0.1457144 2.138 0.032497 * [4,] -0.0349292 0.2364575 -0.148 0.882565 [5,] -0.1574241 0.1607905 -0.979 0.327549 [6,] -0.0202607 0.0860754 -0.235 0.813911 [7,] 0.0031470 0.0010095 3.117 0.001825 ** [8,] 0.0550135 0.2587747 0.213 0.831645 [9,] 0.4642734 0.1883506 2.465 0.013703 * [10,] 0.2659062 0.2924782 0.909 0.363272 [11,] -0.0685226 0.2593701 -0.264 0.791634 [12,] 0.3804101 0.0947400 4.015 5.94e-05 *** [13,] 0.0014249 0.0008304 1.716 0.086173 . [14,] -0.2773537 0.2279209 -1.217 0.223648 [15,] -0.6050206 0.1708047 -3.542 0.000397 *** --- Estimates in matrix form: Constant term: Estimates: 0.001 0.003 0.001 AR and MA lag-0 coefficient matrix [,1] [,2] [,3] [1,] 1.000 0.000 0 [2,] 0.000 1.000 0 [3,] -0.277 -0.605 1 AR coefficient matrix AR( 1 )-matrix [,1] [,2] [,3] [1,] 0.436 0.312 0 [2,] 0.055 0.464 0 [3,] 0.000 0.000 0 MA coefficient matrix MA( 1 )-matrix [,1] [,2] [,3] [1,] 0.035 0.157 0.02 [2,] -0.266 0.069 -0.38 [3,] 0.000 0.000 0.00 Residuals cov-matrix: [,1] [,2] [,3] [1,] 2.831126e-05 3.164358e-06 7.422491e-06 [2,] 3.164358e-06 3.064330e-05 1.464591e-05 [3,] 7.422491e-06 1.464591e-05 3.747082e-05 ---- aic= -31.07752 bic= -30.73812 > m2=refKronfit(m1,thres=0.8) Number of parameters: 11 initial estimates: 0.0012 0.4359 0.3116 -0.1574 0.0031 0.4643 0.2659 0.3804 0.0014 -0.2774 -0.605 Upper-bound: 0.0027 0.8378 0.603 0.1642 0.0052 0.841 0.8509 0.5699 0.0031 0.1785 -0.2634 Lower-bound: -4e-04 0.034 0.0201 -0.479 0.0011 0.0876 -0.3191 0.1909 -2e-04 -0.7332 -0.9466 Coefficient(s): Estimate Std. Error t value Pr(>|t|) [1,] 0.0013289 0.0007140 1.861 0.062720 . [2,] 0.3953204 0.0730410 5.412 6.22e-08 *** [3,] 0.3226715 0.0963539 3.349 0.000812 *** [4,] -0.1747208 0.1184522 -1.475 0.140204 [5,] 0.0035365 0.0007979 4.432 9.33e-06 *** [6,] 0.4480783 0.0730643 6.133 8.64e-10 *** [7,] 0.3096088 0.0967649 3.200 0.001376 ** [8,] 0.3626396 0.0819954 4.423 9.75e-06 *** [9,] 0.0014695 0.0008233 1.785 0.074290 . [10,] -0.2832218 0.1938119 -1.461 0.143927 [11,] -0.5927226 0.1463937 -4.049 5.15e-05 *** --- Estimates in matrix form: Constant term: Estimates: 0.001 0.004 0.001 AR and MA lag-0 coefficient matrix [,1] [,2] [,3] [1,] 1.000 0.000 0 [2,] 0.000 1.000 0 [3,] -0.283 -0.593 1 AR coefficient matrix AR( 1 )-matrix [,1] [,2] [,3] [1,] 0.395 0.323 0 [2,] 0.000 0.448 0 [3,] 0.000 0.000 0 MA coefficient matrix MA( 1 )-matrix [,1] [,2] [,3] [1,] 0.00 0.175 0.000 [2,] -0.31 0.000 -0.363 [3,] 0.00 0.000 0.000 Residuals cov-matrix: [,1] [,2] [,3] [1,] 2.829157e-05 3.280528e-06 7.396183e-06 [2,] 3.280528e-06 3.087658e-05 1.474740e-05 [3,] 7.396183e-06 1.474740e-05 3.745909e-05 ---- aic= -31.13622 bic= -30.88733 > m3=refKronfit(m2,thres=1.5) Number of parameters: 9 initial estimates: 0.0013 0.3953 0.3227 0.0035 0.4481 0.3096 0.3626 0.0015 -0.5927 Upper-bound: 0.0028 0.5414 0.5154 0.0051 0.5942 0.5031 0.5266 0.0031 -0.2999 Lower-bound: -1e-04 0.2492 0.13 0.0019 0.3019 0.1161 0.1986 -2e-04 -0.8855 Coefficient(s): Estimate Std. Error t value Pr(>|t|) [1,] 0.0018025 0.0006393 2.819 0.004812 ** [2,] 0.4141640 0.0687453 6.025 1.70e-09 *** [3,] 0.2290879 0.0632500 3.622 0.000292 *** [4,] 0.0034814 0.0007941 4.384 1.17e-05 *** [5,] 0.4552175 0.0725277 6.276 3.46e-10 *** [6,] 0.3591799 0.0873271 4.113 3.90e-05 *** [7,] 0.3228186 0.0756491 4.267 1.98e-05 *** [8,] 0.0018911 0.0007950 2.379 0.017373 * [9,] -0.7707575 0.0967128 -7.970 1.55e-15 *** --- Estimates in matrix form: Constant term: Estimates: 0.002 0.003 0.002 AR and MA lag-0 coefficient matrix [,1] [,2] [,3] [1,] 1 0.000 0 [2,] 0 1.000 0 [3,] 0 -0.771 1 AR coefficient matrix AR( 1 )-matrix [,1] [,2] [,3] [1,] 0.414 0.229 0 [2,] 0.000 0.455 0 [3,] 0.000 0.000 0 MA coefficient matrix MA( 1 )-matrix [,1] [,2] [,3] [1,] 0.000 0 0.000 [2,] -0.359 0 -0.323 [3,] 0.000 0 0.000 Residuals cov-matrix: [,1] [,2] [,3] [1,] 2.902578e-05 3.650539e-06 7.956836e-06 [2,] 3.650539e-06 3.098021e-05 1.469478e-05 [3,] 7.956836e-06 1.469478e-05 3.786775e-05 ---- aic= -31.1301 bic= -30.92647 > MTSdiag(m3) [1] "Covariance matrix:" uk ca us uk 2.93e-05 3.68e-06 8.02e-06 ca 3.68e-06 3.12e-05 1.48e-05 us 8.02e-06 1.48e-05 3.82e-05 CCM at lag: 0 [,1] [,2] [,3] [1,] 1.000 0.122 0.240 [2,] 0.122 1.000 0.429 [3,] 0.240 0.429 1.000 Simplified matrix: CCM at lag: 1 . . . . . . . . . CCM at lag: 2 . . . . . . . . . CCM at lag: 3 . . . . . . . . . CCM at lag: 4 . . - . . . . . . CCM at lag: 5 . . . . . . . . . CCM at lag: 6 . . . . . . . . . CCM at lag: 7 . . . . . . . . . CCM at lag: 8 . . . . . . . . . CCM at lag: 9 . . . . . . . . . CCM at lag: 10 . . . . . . . . . CCM at lag: 11 . . . . . . . . . CCM at lag: 12 . . . . . . . . . CCM at lag: 13 . . . . . . . . . CCM at lag: 14 . - . . . . . . . CCM at lag: 15 . . . . . . . . . CCM at lag: 16 . . . . . . . . . CCM at lag: 17 . . . . . . . . . CCM at lag: 18 . . . . . . . . . CCM at lag: 19 . . . . . + . . . CCM at lag: 20 . . . . . . . . . CCM at lag: 21 . . . . . . . . . CCM at lag: 22 . . . . . . . . . CCM at lag: 23 . . . . . . . . . CCM at lag: 24 . . . . . . . . . Hit Enter for p-value plot of individual ccm: Hit Enter to compute MQ-statistics: ### See Figure 4.8 of the text Ljung-Box Statistics: m Q(m) df p-value [1,] 1.0 12.1 9.0 0.21 [2,] 2.0 19.3 18.0 0.38 [3,] 3.0 32.0 27.0 0.23 [4,] 4.0 49.6 36.0 0.07 [5,] 5.0 53.5 45.0 0.18 [6,] 6.0 58.5 54.0 0.31 [7,] 7.0 66.8 63.0 0.35 [8,] 8.0 82.8 72.0 0.18 [9,] 9.0 89.3 81.0 0.25 [10,] 10.0 99.1 90.0 0.24 [11,] 11.0 104.5 99.0 0.33 [12,] 12.0 116.6 108.0 0.27 [13,] 13.0 126.7 117.0 0.26 [14,] 14.0 134.0 126.0 0.30 [15,] 15.0 144.5 135.0 0.27 [16,] 16.0 148.9 144.0 0.37 [17,] 17.0 155.6 153.0 0.43 [18,] 18.0 161.7 162.0 0.49 [19,] 19.0 176.6 171.0 0.37 [20,] 20.0 188.1 180.0 0.32 [21,] 21.0 191.5 189.0 0.43 [22,] 22.0 201.1 198.0 0.43 [23,] 23.0 205.6 207.0 0.51 [24,] 24.0 218.3 216.0 0.44 Hit Enter to obtain residual plots: ### See Figure 4.9 of the text >