Constant Conditional Correlation MGARCH on a panel of return series. The conditional covariance factorises as \(H_t = D_t R D_t\), with \(D_t = diag(\sqrt{h_{1t}}, \ldots, \sqrt{h_{kt}})\) carrying the time variation and \(R\) held constant.
Value
Named list with R (constant correlation matrix),
sigmas (T x k conditional standard deviations),
conditional_variance (T x k), loglik, n,
k, method.
Details
Because \(R\) does not move, the Gaussian likelihood separates: the
marginals are fitted series by series with
morie_garch_fit, and the maximum-likelihood \(R\) is
then simply the sample correlation of the standardised residuals. No
numerical search is needed for the second step. That closed form is
the point of the constant-correlation restriction, and it is what
morie_dcc_multivariate_garch relaxes.
Native throughout; no multivariate GARCH package is used.
References
Bollerslev T (1990). Modelling the coherence in short-run nominal exchange rates: a multivariate generalized ARCH model. Review of Economics and Statistics, 72(3), 498-505.
Examples
set.seed(1)
morie_ccc_multivariate_garch(x = matrix(rnorm(150), 50, 3))
#> $R
#> [,1] [,2] [,3]
#> [1,] 1.00000000 -0.04138272 0.03184470
#> [2,] -0.04138272 1.00000000 0.01908927
#> [3,] 0.03184470 0.01908927 1.00000000
#>
#> $sigmas
#> [,1] [,2] [,3]
#> [1,] 0.8313939 0.9688279 0.8997394
#> [2,] 0.8289829 0.9672447 0.8822589
#> [3,] 0.8265788 0.9656640 0.8613568
#> [4,] 0.8241818 0.9640860 0.8638983
#> [5,] 0.8217916 0.9625105 0.8484082
#> [6,] 0.8194085 0.9609376 0.8415537
#> [7,] 0.8170322 0.9593673 0.9562419
#> [8,] 0.8146628 0.9577995 0.9468356
#> [9,] 0.8123003 0.9562343 0.9516009
#> [10,] 0.8099446 0.9546717 0.9273754
#> [11,] 0.8075957 0.9531116 1.0082538
#> [12,] 0.8052537 0.9515541 0.9726275
#> [13,] 0.8029185 0.9499991 0.9383620
#> [14,] 0.8005900 0.9484466 0.9896688
#> [15,] 0.7982684 0.9468967 0.9576162
#> [16,] 0.7959534 0.9453493 0.9227041
#> [17,] 0.7936451 0.9438045 0.8955237
#> [18,] 0.7913435 0.9422621 0.8719559
#> [19,] 0.7890487 0.9407223 0.8520637
#> [20,] 0.7867604 0.9391850 0.8506718
#> [21,] 0.7844788 0.9376503 0.8339806
#> [22,] 0.7822038 0.9361180 0.8250376
#> [23,] 0.7799354 0.9345882 0.8951466
#> [24,] 0.7776736 0.9330610 0.8707722
#> [25,] 0.7754184 0.9315362 0.8505241
#> [26,] 0.7731697 0.9300139 0.8339390
#> [27,] 0.7709275 0.9284941 0.8483929
#> [28,] 0.7686918 0.9269768 0.8323210
#> [29,] 0.7664626 0.9254620 0.8194180
#> [30,] 0.7642399 0.9239496 0.8191672
#> [31,] 0.7620236 0.9224397 0.8093078
#> [32,] 0.7598137 0.9209323 0.8018963
#> [33,] 0.7576103 0.9194274 0.8015853
#> [34,] 0.7554132 0.9179249 0.8121257
#> [35,] 0.7532225 0.9164248 0.8723457
#> [36,] 0.7510382 0.9149272 0.8595233
#> [37,] 0.7488602 0.9134321 0.9098894
#> [38,] 0.7466885 0.9119394 0.8836361
#> [39,] 0.7445231 0.9104491 0.8662454
#> [40,] 0.7423640 0.9089613 0.8559683
#> [41,] 0.7402111 0.9074759 0.8386516
#> [42,] 0.7380645 0.9059930 0.9348707
#> [43,] 0.7359241 0.9045124 0.9630220
#> [44,] 0.7337899 0.9030343 1.0015713
#> [45,] 0.7316619 0.9015586 0.9626030
#> [46,] 0.7295401 0.9000853 0.9576881
#> [47,] 0.7274245 0.8986144 0.9347598
#> [48,] 0.7253149 0.8971460 1.0635884
#> [49,] 0.7232115 0.8956799 1.0126527
#> [50,] 0.7211142 0.8942162 1.0095521
#>
#> $conditional_variance
#> [,1] [,2] [,3]
#> [1,] 0.6912159 0.9386275 0.8095311
#> [2,] 0.6872126 0.9355623 0.7783807
#> [3,] 0.6832326 0.9325070 0.7419355
#> [4,] 0.6792756 0.9294618 0.7463203
#> [5,] 0.6753415 0.9264265 0.7197965
#> [6,] 0.6714302 0.9234011 0.7082126
#> [7,] 0.6675416 0.9203856 0.9143986
#> [8,] 0.6636754 0.9173799 0.8964976
#> [9,] 0.6598317 0.9143841 0.9055443
#> [10,] 0.6560102 0.9113980 0.8600251
#> [11,] 0.6522109 0.9084217 1.0165758
#> [12,] 0.6484336 0.9054552 0.9460043
#> [13,] 0.6446781 0.9024983 0.8805232
#> [14,] 0.6409444 0.8995510 0.9794444
#> [15,] 0.6372324 0.8966134 0.9170289
#> [16,] 0.6335418 0.8936853 0.8513828
#> [17,] 0.6298726 0.8907669 0.8019627
#> [18,] 0.6262246 0.8878580 0.7603071
#> [19,] 0.6225978 0.8849585 0.7260126
#> [20,] 0.6189920 0.8820686 0.7236425
#> [21,] 0.6154070 0.8791881 0.6955237
#> [22,] 0.6118428 0.8763169 0.6806870
#> [23,] 0.6082993 0.8734552 0.8012875
#> [24,] 0.6047763 0.8706028 0.7582443
#> [25,] 0.6012737 0.8677597 0.7233912
#> [26,] 0.5977914 0.8649259 0.6954543
#> [27,] 0.5943292 0.8621013 0.7197705
#> [28,] 0.5908871 0.8592860 0.6927583
#> [29,] 0.5874649 0.8564799 0.6714459
#> [30,] 0.5840626 0.8536829 0.6710348
#> [31,] 0.5806799 0.8508951 0.6549792
#> [32,] 0.5773169 0.8481163 0.6430377
#> [33,] 0.5739733 0.8453467 0.6425390
#> [34,] 0.5706491 0.8425861 0.6595482
#> [35,] 0.5673441 0.8398345 0.7609870
#> [36,] 0.5640583 0.8370919 0.7387803
#> [37,] 0.5607915 0.8343582 0.8278987
#> [38,] 0.5575437 0.8316335 0.7808128
#> [39,] 0.5543146 0.8289176 0.7503811
#> [40,] 0.5511042 0.8262107 0.7326817
#> [41,] 0.5479125 0.8235126 0.7033366
#> [42,] 0.5447392 0.8208232 0.8739832
#> [43,] 0.5415843 0.8181427 0.9274115
#> [44,] 0.5384477 0.8154710 1.0031451
#> [45,] 0.5353292 0.8128079 0.9266046
#> [46,] 0.5322288 0.8101536 0.9171664
#> [47,] 0.5291464 0.8075079 0.8737759
#> [48,] 0.5260818 0.8048709 1.1312203
#> [49,] 0.5230349 0.8022424 1.0254654
#> [50,] 0.5200057 0.7996226 1.0191954
#>
#> $loglik
#> [1] -194.5601
#>
#> $n
#> [1] 50
#>
#> $k
#> [1] 3
#>
#> $method
#> [1] "CCC(1,1) two-step Gaussian QMLE"
#>