A Appendix: Proofs The proofs of Theore 1-3 are along the lines of Wied and Galeano (2013) Proof of Theore 1 Let D[d 1, d 2 ] be the space of càdlàg functions on the interval [d 1, d 2 ] equipped with the supreu nor Denote the tie invariant vector of variances by σ 2 = ( σ 2 1,, σ2 p) and define {P (d), d [c, B]} by P (d) = ˆD 1 2 [ d] [ c] ([ ] ) ˆσ 2 [ d] [ c] σ2 = ˆD [ d] [ c] 2 1 [ ] [ d] ˆσ 2 1 [ c] σ2 1 [ ] [ d] ˆσ 2 p [ c] σ2 p To obtain convergence on D[c, B], we use the invariance principle of Theore 2918 in Davidson (1994) and argue analogously to Lea A1 in Wied et al (2012b) For fixed c 0 and assuing, {P (d), d [c, B]} converges in distribution to {W p (d) W p (c), d [c, B]} with W p ( ) being a p diensional Brownian Motion S (b) := ˆD 1 2 [ b]+2 ([ ] ) +[ b]+2 ˆσ 2 σ 2 +1 ([ ˆD ] 1 2 ˆσ 2 σ2) 1 d W p (b + 1) W p (1), for b [0, B] W p (1) Consequently, V [ b]+2 = ˆD 1 2 [ b] + 2 ([ ] ) ˆσ 2 +[ b]+2 σ 2 ˆD 2 1 +1 [ b] + 2 ([ ˆσ 2 ] 1 σ2) converges to the process {W p (b+1) (b+1)w p (1), b [0, B]} Applying the continuous apping theore and calculating the covariance structure of the liit process proves the result Proof of Theore 2 The proof uses the sae arguents as the one of Theore 1 and is ainly based on the fact 1
that for fixed c 0, and the process {P (d), d [c, B]} converges in distribution to W p (d) W p (c) + H d c d c g 1(z)dz, d [c, B] g p(z)dz on D[c, B] The constant H is, up to a constant, the liit of ˆD under the null hypothesis, see the proof of Theore 2 in Wied et al (2012a) This result is a generalization of arguents used in Theore 2 in Wied et al (2012a), executed along the lines of to the proof of Theore 1 Proof of Theore 3 Assue wlog g 1 ( ) Mh( ) Then, the detector converges in the following way: sup b [0,B] V 2 b +2 M b+1 h(u)du 1 G d 1 (b) b+1 sup g b [0,B] + D 1 1 2 (u)du 2 (1) G p (b) b+1 g p (u)du 1 2 Denote D 1 2 ( di j ) and define constants c 1(b) b+1 i, j=1,,p 1 i = 2,, p Thus, (1) has asyptotically the sae distribution as h(u)du and c i (b) b+1 g 1 i (u)du, sup b [0,B] p G j(b) + Md j1 c 1 (b) + j=1 2 p d ji c i (b) (2) i=2 Since D p is positive definite there exists j {1,, p} with d j1 0 Assuing M, we have G j (b) + Md j1 c 1 (b) p Thus, Jensen s inequality iplies that for all b (0, B] the square root of the su in (2) tends to The fact that the ter degenerates for b = 0 does not affect this result, since this event occurs with zero probability This iplies that (2) will exceed every quantile of the asyptotic null distribution for M 2
Proof of Lea 1 Hafner (2003) provides conditions to establish the existence of the atrix of fourth oents and cross oents of a ultivariate GARCH(1, 1) odel in vech representation: vech (H t ) = C 0 + A 1 vech ( X t 1 X t 1) + B1 vech (H t 1 ), (3) where C 0 is a d diensional paraeter vector and A 1 and B 1 are paraeter atrices of diension d d The closely related vec representation is given as vec (H t ) = C 0 + A 1 vec ( X t 1 X ) t 1 + B 1 vec (H t 1) (4) and contains several redundant equations that lead to inflated paraeter atrices A 1 and B 1 of diension p 2 p 2 and a paraeter vector C 0 of diension p2 Following Engle and Kroner (1995), the odel in (7) in the ain paper can be given in vec representation by choosing C 0 = (1 α β) [ H 1 2 H 1 2 ] vec ( Ip ), A 1 = αi p 2 and B 1 = βi p2 (5) in (4) Thus, (7) can be given in vech representation by transforing it first to its vec and then to its vech representation Substituting (5) in (4) and ultiplying D p and L p gives C 0 = (1 α β) L p H 1 2 H 1 2 vec ( Ip ), A1 = αi d and B 1 = βi d (6) in odel (3) Note that A 1 A 1 = α 2 I d 2, B 1 B 1 = β 2 I d 2 and A 1 B 1 = B 1 A 1 = αβi d 2 Using (6) and G p fro Hafner (2003) to construct Z allows to check the existence of Γ according to Hafner (2003) 3
B Appendix: Further Siulations: Monitoring Tie Series of IID Rando Vectors To begin with, we investigate the size of the proposed procedure under the null hypothesis of no structural break First, we siulate tie series that capture neither serial nor cross-sectional dependence to gain reference values to which the perforance in ore coplex scenarios can be copared As siplest possible case, realizations of processes of iid rando vectors are siulated These siulation results can work as a benchark for ore coplex siulation scenarios The rando vectors under consideration are iid ultivariate noral and ultivariate t distributed with ν = 8 degrees of freedo As covariance atrix the atrices 1 01 011 012 015 02 025 03 035 04 1 03 04 05 06 03 1 03 04 05 01 1 01 035 1 07 Σ 2 = 07 1, Σ 5 = 04 03 1 03 04 and Σ 10 = 05 04 03 1 03 035 01 1 01 06 05 04 03 1 04 035 03 025 02 015 012 011 01 1 are used as well as identity atrices I p of corresponding diension p To enable a reasonable coparison of the results, the covariance atrices in the case of the t distribution are standardized by ultiplying ν 2 The results are given in Tables 4 and 5 in Appendix B and illustrated ν in Figure 1 For the sake of clarity and since the results differ only slightly for the different values of the tuning paraeter γ and the different types of covariance atrix, the figure only shows the epirical sizes for γ = 0 and an identity covariance atrix The fact that there is hardly a difference in the epirical sizes depending on whether the covariance atrix is diagonal or not was expected as our procedure is only based on estiates of the ain diagonal eleents of the covariance atrix and not of the reaining entries In general, the epirical size increases with the diension In order to deterine the source of this, we use the actual atrix D p that can be easily calculated for an identity covariance atrix I p and its standardized analogue ν 2I ν p, respectively The atrix is given as D p = 2I p for noral distributed rando vectors and as D p = 2(ν 1) I ν 4 p for t distributed ones The results are also given in Tables 4 and 5 in Appendix B and illustrated in Figure 1 They state that the ain fraction of 4
Noral distribution Standardized t distribution Epirical size 000 005 010 015 020 025 For both graphics: iid: D known iid: D estiated scalar BEKK Epirical size 000 005 010 015 020 025 Figure 1: Size coparison: iid rando vectors with atrix D p known and estiated, respectively, and scalar BEKK tie series the increased size is caused by an insufficient estiation of the atrix D p Furtherore, heavy tails in the distribution of the rando vectors entail an additional size increase Unfortunately, the estiation of the atrix D p could not be iproved by using an alternative bandwidth or estiation procedure The epirical size is distinctly larger in the case of the t distributed rando vectors and decreases with growing length of the historical period This convergence to the theoretical size goes back to the fact that all of the asyptotic stateents are established for While for the different values of B - indicating different lengths of the onitoring period - no tendency in the epirical sizes can be recognized, larger values of γ result in a slight increase of the sizes This is a plausible result as larger values of this paraeter tend to sensitize the procedure for changes that are expected early in the onitoring period at the expense of increased probabilities of false alars In the following, the power of the onitoring procedure is investigated considering two different types of scenario In both cases the covariance atrix in the pre-break period equates the atrix Σ p whose diagonal eleents are affected by a structural break later in the series In the first setting, the variances of all coponents increase fro 1 to 13 In the second one, the variance of only one of the coponents jups to 15 In both scenarios the power to detect 5
an early and a later occurring change are copared Since the length of the onitoring period depends on the paraeters and B, we assue that, independent of the length of the tie series, the change happens at the sae fraction of the onitoring period indicated by λ (0, 1) We choose λ {005, 05} to ark changepoints located at the beginning (k = 005B) or in the iddle (k = 05B) of the onitoring period The results for the first scenario are given in Tables 6 and 7 in Appendix B and illustrated in Figure 2, while those for the second scenario are presented in Tables 8 and 9 and illustrated in Figure 3 They state that the power increases considerably with growing length of the historical and the onitoring period If all of the variances are affected by a change, the power increases with growing diension of the rando vectors If only one of the variances experiences a change, the frequency of detecting the change decreases for growing diension p since the portion of variance coponents that are not struck by the change increases Early changes can be detected reliably in both scenarios However, the power gets quite low if the changepoint is located in the advanced series, especially for t distributed rando vectors as the rejection fractions in Tables 7 and 9 state The direct coparison of the two scenarios shows that a ajor change in just one of the variances can be detected ore frequently than a inor change that affects all of the variances only when Early change, noral distribution Late change, noral distribution Late change, standardized t distribution Power Power 05 06 07 08 09 10 05 06 07 08 09 10 Early change, standardized t distribution Power 00 02 04 06 08 10 Power 00 02 04 06 08 10 For all graphics: B=05 B=1 B=2 sudden change contin change Figure 2: Power: iid rando vectors when all of the variances are affected by a change 6
the diension is rather sall and the change occurs not too late in the onitoring period In all of the settings the procedure perfors worse in the case of t distributed rando vectors, but the differences to the noral distribution results are declining with Also, in ost cases the power is lower for the higher value of γ While for a later change this is a plausible result, it contradicts the expectation that early changes can be detected ore frequently using a higher value of γ An explanation for this result is that in both cases the values of the detector are copared to the values of the scaled threshold function that has a higher slope in the case of the larger tuning paraeter Since both functions intersect the down scaling of the differences by ultiplying k can cause an earlier crossing of the threshold function for γ = 0 than for γ = 025 Overall, changes that occur right after the beginning of the onitoring period can be detected uch ore frequently than those located in the advanced onitoring period no atter how the tuning paraeter was chosen Now, the results can be copared to scenarios of continuously appearing changes, ie, a slow linear increase of the affected variances that starts at λ 1 = 005 and 05, respectively, and is copleted at λ 2 = 03 and 075, respectively The results are also illustrated in Figures 2 and 3 and presented in Tables 6-9 in Appendix B as values in parentheses The ipact of variations Early change, noral distribution Late change, noral distribution Late change, standardized t distribution Power Power 05 06 07 08 09 10 05 06 07 08 09 10 Early change, standardized t distribution Power 00 02 04 06 08 10 Power 00 02 04 06 08 10 For all graphics: B=05 B=1 B=2 sudden change contin change Figure 3: Power: iid rando vectors when only one of the variances is affected by a change 7
in the paraeters reains the sae as in the situation of a sudden variance change However, the power is considerably lower in the case of a slow increase Since the power siulations for sudden changes suggest that later changes can be detected less frequently it is clear that changes that are copleted later in the onitoring period are ore difficult to be detected Although in our siulations the detectability of changes that start in the advanced onitoring period is kind of low especially for short historical periods, the power increases quickly with growing length of the historical period 8
C Appendix: Tables D is known Σ = Ip Σ = Hp p = 2 γ B = = = 05 00476 00431 00451 00493 00518 00535 00 00502 00515 0 1 00432 00465 00496 00485 00501 00 00487 00488 00474 2 00459 00435 00507 00507 00520 00536 00527 00533 00493 05 00522 00489 00434 00550 00480 00518 00499 00496 00499 025 1 00487 00473 00498 00470 00474 00457 00473 00485 00487 2 00516 00462 00478 00528 00506 00539 00529 00522 00517 05 00659 00558 00574 00740 00641 00607 01041 00726 00652 0 1 00608 00605 00528 00802 00659 00588 01080 00781 00650 2 00689 00605 00598 00852 00657 00573 01016 00826 00628 05 00639 00613 00545 00839 00677 00613 01034 00752 00633 025 1 00674 00623 00573 00879 00668 00584 01144 00884 00755 2 00702 00564 00539 00820 00747 00605 01132 00813 00666 05 00636 00588 00569 00788 00652 00607 01005 00741 00645 0 1 00632 00611 00580 00845 00669 00607 01083 00814 00696 2 00662 00588 00589 00844 00735 00644 01024 00787 00642 05 00620 00672 00576 00844 00732 00646 01134 00818 00680 025 1 00646 00570 00518 00900 00800 00652 01128 00850 00760 2 00696 00640 00508 00926 00664 00688 01190 00946 00690 Table 4: Size when onitoring a sequence of realizations of iid N ( 0, Σ p ) distributed rando vectors D is known Σ = Ip Σ = Hp p = 2 γ B = = = 05 00492 00509 00520 00713 00614 00610 00777 00719 00695 0 1 00520 00512 00523 00633 00634 00641 00767 00696 00690 2 00533 00526 00555 00634 00631 00639 00746 00771 00724 05 00631 00590 00582 00831 00754 00657 01073 00911 00907 025 1 00636 00601 00586 00810 00762 00706 00945 00912 00813 2 00625 00621 00528 00845 00749 00706 01033 00914 00872 05 00914 00796 00711 01356 01066 00830 01943 01430 01053 0 1 00942 00821 00664 01405 01045 00832 02037 01386 01069 2 01027 00800 00701 01475 01085 00836 02013 01384 00961 05 01095 00906 00775 01632 01239 00993 02329 01635 01169 025 1 01199 00906 00815 01713 01279 01020 02487 01810 01324 2 01079 00967 00738 01741 01259 00961 02490 01670 01210 05 00966 00818 00717 01382 01067 00900 02008 01382 01006 0 1 01003 00792 00686 01413 01101 00848 02107 01459 01057 2 00978 00791 00745 01413 01112 00860 01962 01405 01013 05 01140 00918 00836 01622 01212 00924 02472 01706 01162 025 1 01138 00878 00712 01850 01316 01026 02522 01662 01224 2 01174 00918 00718 01796 01334 01036 02702 01832 01214 Table 5: Size when onitoring a sequence of realizations of iid t ν ( 0, ν 2 ν Σ p) distributed rando vectors with ν=8 degrees of freedo 9
p = 2 k = 005 k = 05 γ B = = 05 07775 (06665) 09597 (08941) 09995 (09938) 03398(02276) 05127 (03389) 07670 (05243) 0 1 08923 (07960) 09939 (09742) 10000 (09998) 04174 (02798) 06592 (04258) 09101 (06819) 2 09549 (08972) 09992 (09929) 10000 (10000) 05121 (03336) 07742 (05253) 09703 (08163) 05 07664 (06482) 09552 (08818) 09995 (09929) 03137 (02073) 04846 (03123) 07483 (04987) 025 1 08876 (07848) 09917 (09644) 10000 (09997) 03930 (02599) 06095 (03750) 08885 (06337) 2 09432 (08689) 09987 (09901) 10000 (10000) 04585 (02890) 07364 (04771) 09604 (07841) 05 09568 (08929) 09987 (09911) 10000 (10000) 05393 (03564) 07563 (05098) 09642 (07866) 0 1 09926 (09671) 10000 (09997) 10000 (10000) 06677 (04431) 08977 (06687) 09967 (09264) 2 09991 (09934) 10000 (10000) 10000 (10000) 07540 (05245) 09590 (07791) 09994(09782) 05 09553 (08852) 09987 (09900) 1000 (09999) 05155 (03344) 07431 (04934) 09546 (07501) 025 1 09919 (09630) 10000 (09997) 10000 (10000) 06409 (04137) 08752 (06294) 09951 (09044) 2 09990 (09918) 10000 (10000) 10000 (10000) 07286 (04893) 09508 (07509) 09994 (09725) 05 09998 (09977) 10000 (10000) 10000 (10000) 08464 (06361) 09821 (08518) 10000 (09870) 0 1 10000 (10000) 10000 (10000) 10000 (10000) 09491 (07749) 09989 (09529) 10000 (09994) 2 10000 (10000) 10000 (10000) 10000 (10000) 09820 (08546) 10000 (09885) 10000 (10000) 05 09997 (09969) 10000 (10000) 10000 (10000) 08270 (06084) 09736 (08174) 1000 (09840) 025 1 10000 (10000) 10000 (10000) 10000 (10000) 09349 (07433) 09986 (09434) 10000 (09994) 2 10000 (10000) 10000 (10000) 10000 (10000) 09774 (08499) 09999 (09868) 10000 (10000) Table 6: Power when onitoring a sequence of realizations of iid N ( 0, Σ p ) distributed rando vectors when all of the variances are affected by a change p = 2 k = 005 k = 05 γ B = = 05 06063 (05079) 07991 (07059) 09636 (09083) 02868 (02222) 03797 (02635) 05689 (03786) 0 1 07377 (06359) 09112 (08363) 09921 (09746) 03566 (02595) 04842 (03329) 07015 (04742) 2 08173 (07291) 09568 (09032) 09984 (09933) 04045 (02834) 05 (03715) 08021 (05726) 05 06154 (05066) 07883 (06833) 09547 (08912) 02821 (02216) 03584 (02454) 05247 (03401) 025 1 07339 (06217) 08980 (08100) 09902 (09670) 03479 (02564) 04371 (02967) 06646 (04337) 2 08026 (07045) 09513 (08896) 09980 (09916) 03775 (02677) 05176 (03450) 07730 (05302) 05 07924 (07084) 09349 (08636) 09971 (09836) 04229 (03303) 05270 (03801) 07286 (05106) 0 1 08999 (08297) 09860 (09547) 09996 (09981) 05232 (03931) 06607 (04719) 08745 (06661) 2 09515 (08975) 09965 (09827) 10000 (09996) 05944 (04377) 07575 (05446) 09353 (07727) 05 08056 (07119) 09358 (08603) 09962 (09803) 04245 (03355) 05147 (03700) 07008 (04799) 025 1 08995 (08250) 09841 (09503) 09996 (09977) 05094 (03898) 06412 (04536) 08501 (06338) 2 09493 (08881) 09963 (09807) 10000 (09995) 05757 (04201) 07412 (05294) 09227 (07483) 05 09326 (08790) 09889 (09687) 09999 (09990) 06205 (04834) 07074 (05246) 08861 (06824) 0 1 09763 (09506) 09989 (09939) 10000 (1000) 07016 (05559) 08513 (06592) 09639 (08228) 2 09929(09806) 09999 (09990) 10000 (10000) 07831 (06117) 09144 (07545) 09899 (09103) 05 09355 (08776) 09879 (09651) 09999 (09988) 06117 (04798) 06832 (04983) 08769 (06645) 025 1 09781 (09510) 09990 (09919) 10000 (10000) 06956 (05564) 08293 (06292) 09581 (08033) 2 09951 (09771) 09999 (09986) 10000 (10000) 07683 (06135) 08965 (07175) 09907 (08983) Table 7: Power when onitoring a sequence of realizations of iid t ν ( 0, ν 2 ν Σ p) distributed rando vectors with ν=8 degrees of freedo when all of the variances are affected by a change 10
p = 2 k = 005 k = 05 γ B = = 05 09695 (09298) 09991 (09975) 10000 (10000) 06006 (04040) 08514 (06364) 09853 (08923) 0 1 09972 (09843) 10000 (10000) 10000 (10000) 07483 (05150) 09573 (07892) 09995 (09784) 2 09997 (09982) 10000 (10000) 10000 (10000) 08618 (06383) 09894 (08994) 10000 (09970) 05 09656 (09196) 09989 (09967) 10000 (10000) 05672 (03673) 08248 (05966) 09825 (08792) 025 1 09967 (09809) 10000 (10000) 10000 (10000) 07167 (04821) 09520 (07731) 09994 (09712) 2 09997 (09968) 10000 (10000) 10000 (10000) 08280 (05847) 09861 (08746) 10000 (09953) 05 09310 (08578) 09972 (09877) 10000 (09996) 04833 (03255) 07406 (04878) 09518 (07774) 0 1 09883 (09588) 10000 (09997) 10000 (10000) 06319 (04153) 08966 (06611) 09950 (09283) 2 09995 (09908) 10000 (10000) 10000 (10000) 07430 (07) 09621 (07900) 09995 (09792) 05 09244 (08443) 09968 (09853) 10000 (09996) 04566 (02945) 07144 (04595) 09444 (07568) 025 1 09874 (09533) 10000(09996) 10000 (10000) 06135 (03952) 08707 (06152) 09936 (09140) 2 09988 (09890) 10000 (10000) 10000 (10000) 07191 (04722) 09530 (07582) 09994 (09720) 05 08507 (07481) 09878 (09524) 10000 (09996) 03891 (02557) 05842 (03674) 08753 (06334) 0 1 09600 (08996) 09993 (09941) 10000 (10000) 05139 (03417) 07631 (05013) 09753 (08131) 2 09896 (09530) 10000 (09996) 10000 (10000) 06127 (04093) 08885 (06395) 09963 (09210) 05 08541 (07444) 09879 (09504) 10000 (09996) 03814 (02543) 05727 (03550) 08574 (05955) 025 1 09575 (08892) 09993 (09934) 10000 (10000) 04898 (03244) 07453 (04769) 09702 (07898) 2 09886 (09477) 10000 (09995) 10000 (10000) 05843 (03821) 08730 (06110) 09950 (09082) Table 8: Power when onitoring a sequence of realizations of iid N ( 0, Σ p ) distributed rando vectors when only one of the variances is affected by a change p = 2 k = 005 k = 05 γ B = = 05 08771 (07992) 09861 (09606) 09994 (09989) 04695 (03303) 06917 (04828) 09132 (07289) 0 1 09597 (09155) 09976 (09921) 09998 (09997) 05990 (04133) 08311 (06079) 09802 (08709) 2 09873 (09641) 09998 (09976) 09999 (09999) 07014 (04957) 09167 (07339) 09965 (09512) 05 08717 (07878) 09836 (09544) 09993 (09986) 04522 (03206) 06599 (04538) 08969 (06931) 025 1 09564 (09064) 09972 (09904) 09998 (09997) 05784 (03974) 08124 (05769) 09731 (08427) 2 09848 (09551) 09995 (09972) 09999 (09998) 06687 (04624) 09015 (07018) 09945 (09395) 05 08261 (07345) 09710 (09238) 09995 (09968) 04155 (03161) 05889 (03971) 08428 (06091) 0 1 09327 (08623) 09955 (09846) 09999 (09997) 05220 (03706) 07521 (05402) 09508 (07864) 2 09748 (09352) 09993 (09958) 09999 (09998) 06300 (04340) 08600 (06618) 09851 (08855) 05 08295 (07370) 09686 (09165) 09996 (09961) 04205 (03272) 05665 (03805) 08277 (05871) 025 1 09307 (08552) 09947 (09801) 09999 (09997) 05102 (03687) 07196 (05018) 09423 (07646) 2 09728 (09267) 09993 (09950) 09999 (09998) 06135 (04242) 08380 (06273) 09832 (08746) 05 07508 (06610) 09238 (08528) 09961 (09843) 03973 (03184) 04922 (03394) 07157 (04896) 0 1 0875 (08071) 09815 (09522) 09998 (09989) 04815 (03773) 06395 (04323) 08718 (06423) 2 09469 (08809) 09958 (09849) 10000 (09994) 05613 (04238) 07520 (05393) 09498 (07700) 05 07605 (06705) 09207 (08456) 09954 (09806) 04150 (03410) 04838 (03398) 06851 (04607) 025 1 08764 (08014) 09819 (09512) 09998 (09983) 04830 (03883) 06348 (04341) 08552 (06176) 2 09462 (08787) 09950 (09822) 10000 (09993) 05660 (04325) 07358 (05250) 09416 (07451) Table 9: Power when onitoring a sequence of realizations of iid t ν ( 0, ν 2 ν Σ p) distributed rando vectors with ν=8 degrees of freedo when only one of the variances is affected by a change 11
ultivariate procedure univariate procedures p = 2 γ B = = = 0 025 0 025 05 00733 00694 00612 00984 00778 00662 01325 01031 00834 1 00788 00706 00602 01056 00789 00739 01408 01051 00876 2 00762 00716 00692 01058 00861 00723 01326 00973 00785 05 00751 00704 00578 01103 00822 00734 01362 00987 00798 1 00879 00693 00666 01147 00888 00787 01540 01128 00925 2 00836 00694 00596 01149 00810 00731 01477 01113 00908 05 00645 00596 00524 00912 00789 00683 01031 00812 00682 1 00705 00634 00607 00905 00803 00694 01101 00897 00733 2 00653 00602 00504 00961 00759 00738 00953 00732 00639 05 00706 00624 00555 01065 00884 00757 01226 00884 00699 1 00753 00701 00567 01001 00767 00687 01306 00992 00852 2 00844 00743 00644 01048 00860 00689 01428 01068 00870 Table 10: Size when onitoring scalar BEKK tie series with paraeters α = 003, β = 045 and N ( 0, Σ p ) distributed innovations ultivariate procedure univariate procedures p = 2 γ B = = = 0 025 0 025 05 01091 00877 00798 01533 01207 00987 02430 01674 01288 1 01045 00871 00754 01703 01294 00978 02492 01816 01319 2 01133 00938 00747 01684 01323 01001 02434 01663 01194 05 01278 01005 00831 02012 01462 01173 02771 01974 01417 1 01336 01082 00894 02101 01567 01091 03112 02143 01568 2 01361 01018 00862 02017 01540 01160 03068 02131 01525 05 00910 00713 00663 01443 01163 00924 01825 01369 01060 1 01004 00812 00700 01448 01144 00896 02025 01468 01158 2 00988 00853 00661 01508 01217 00895 01941 01369 01079 05 01069 00919 00785 01761 01475 01107 02785 02102 01589 1 01199 01022 00825 01769 01392 01046 02782 02147 01570 2 01237 01015 00842 01932 01481 01157 02714 01964 01417 Table 11: Size when onitoring scalar BEKK tie series with paraeters α = 003, β = 045 and t ν ( 0, ν 2 ν Σ p) distributed innovations with ν=8 degrees of freedo p = 2 k = 005 k = 05 γ B = = 05 07516 (06546) 09384 (08732) 09982 (09900) 03388 (02451) 04866 (03405) 07274 (05131) 0 1 08748 (07820) 09908 (09599) 10000 (09990) 04066 (02876) 06120 (04159) 08716 (06522) 2 09390 (08707) 09978 (09879) 10000 (10000) 04944 (03413) 07308 (05197) 09436 (07794) 05 07 (06408) 09312 (08602) 09972 (09869) 03172 (02340) 04578 (03120) 06946 (04760) 025 1 08708 (07608) 09888 (09517) 10000 (09988) 03828 (02639) 05762 (03816) 08522 (06150) 2 09274 (08518) 09978 (09847) 10000 (10000) 04586 (03109) 06936 (04796) 09322 (07428) 05 09416 (08757) 09974 (09891) 10000 (09999) 05108 (03782) 07194 (05140) 09412 (07645) 0 1 09902 (09630) 10000 (09992) 10000 (10000) 06532 (04592) 08756 (06433) 09904 (09010) 2 09978 (09878) 10000 (10000) 10000 (10000) 07626 (05333) 09464 (07625) 09990 (09622) 05 09402 (08654) 09968 (09869) 10000 (09999) 04838 (03544) 06826 (04795) 09248 (07298) 025 1 09900 (09586) 10000 (09991) 10000 (10000) 06250 (04360) 08610 (06146) 09884 (08840) 2 09970 (09852) 10000 (10000) 10000 (10000) 07284 (08) 09314 (07305) 09986 (09536) 05 09996 (09964) 10000 (10000) 10000 (10000) 08292 (06333) 09688 (08231) 09998 (09795) 0 1 09998 (09999) 10000 (10000) 10000 (10000) 09344 (07639) 09968 (09412) 10000 (09990) 2 10000 (10000) 10000 (10000) 10000 (10000) 09786 (08492) 09996 (09802) 10000 (10000) 05 09996 (09957) 10000 (10000) 10000 (10000) 08106 (06086) 09624 (08032) 09998 (09743) 025 1 09998 (09999) 10000 (10000) 10000 (10000) 09234 (07415) 09962 (09288) 10000 (09983) 2 10000 (10000) 10000 (10000) 10000 (10000) 09734 (08321) 09996 (09765) 10000 (10000) Table 12: Power when onitoring scalar BEKK tie series and all of the variances increase (N ( ) 0, Σ p distributed innovations) 12
p = 2 k = 005 k = 05 γ B = = 05 05999 (05036) 07869 (06778) 09509 (08787) 03110 (02225) 03778 (02525) 05388 (03506) 0 1 07255 (06175) 08954 (08044) 09885 (09580) 03634 (02602) 04692 (03140) 06691 (04451) 2 08115 (07052) 09474 (08835) 09973 (09881) 04022 (02922) 05604 (03749) 07846 (05585) 05 06115 (05168) 07820 (06812) 09462 (08763) 03078 (02418) 03609 (02598) 05063 (03347) 025 1 07241 (06213) 08873 (07956) 09858 (09558) 03517 (02743) 04434 (03116) 06342 (04272) 2 08035 (06948) 09412 (08755) 09966 (09847) 03815 (02835) 05256 (03575) 07514 (05266) 05 07884 (07018) 09277 (08550) 09924 (09762) 04446 (03476) 05299 (03916) 07123 (05033) 0 1 08906 (08171) 09796 (09466) 09998 (09971) 05303 (04188) 06547 (04707) 08440 (06412) 2 09399 (08823) 09946 (09774) 10000 (09997) 05997 (04534) 07450 (05464) 09270 (07491) 05 07994 (07103) 09285 (08517) 09915 (09716) 04467 (03645) 05117 (03797) 06918 (04745) 025 1 08917 (08188) 09777 (09348) 09995 (09953) 05206 (04077) 06295 (04593) 08207 (06193) 2 09402 (08838) 09942 (09751) 09999 (09992) 05874 (04555) 07219 (05378) 09138 (07240) 05 09279 (08772) 09856 (09609) 09995 (09988) 06379 (05284) 07077 (05673) 08697 (06859) 0 1 09784 (09403) 09979 (09928) 10000 (09998) 07284 (05768) 08357 (06494) 09590 (08109) 2 09902 (09798) 09999 (09979) 10000 (10000) 07953 (06480) 09071 (07476) 09877 (09011) 05 09328 (08755) 09852 (09549) 09996 (09977) 06318 (05190) 06924 (05357) 08549 (06618) 025 1 09780 (09463) 09977 (09932) 10000 (10000) 07121 (05770) 08171 (06474) 09485 (07978) 2 09903 (09730) 09999 (09981) 10000 (10000) 07855 (06399) 08948 (07114) 09852 (08835) Table ( 13: Power when onitoring scalar BEKK tie series and all of the variances increase (t ν 0, ν 2 ν Σ p) distributed innovations with ν=8 degrees of freedo) p = 2 k = 005 k = 05 γ B = = 05 09582 (09063) 09986 (09950) 10000 (10000) 05642 (03937) 08276 (06026) 09776 (08623) 0 1 09938 (09794) 10000 (09995) 10000 (10000) 07120 (05065) 09364 (07684) 09990 (09646) 2 09992 (09954) 10000 (10000) 10000 (10000) 08358 (06141) 09808 (08814) 10000 (09938) 05 09552 (08979) 09986 (09926) 10000 (10000) 05410 (03747) 08000 (05614) 09728 (08442) 025 1 09918 (09721) 10000 (09996) 10000 (10000) 06862 (04712) 09248 (07320) 09990 (09583) 2 09988 (09948) 10000 (10000) 10000 (10000) 08102 (05855) 09752 (08482) 10000 (09906) 05 09120 (08341) 09954 (09784) 10000 (10000) 04640 (03264) 06984 (04779) 09378 (07516) 0 1 09804 (09454) 09996 (09985) 10000 (10000) 06216 (04172) 08746 (06438) 09914 (09058) 2 09962 (09862) 10000 (10000) 10000 (10000) 07292 (05058) 09454 (07637) 09990 (09696) 05 09028 (08256) 09950 (09760) 10000 (09999) 04368 (03021) 06646 (04463) 09226 (07299) 025 1 09782 (09364) 09996 (09985) 10000 (10000) 05964 (03930) 08588 (06079) 09894 (08873) 2 09960 (09811) 10000 (10000) 10000 (10000) 07004 (04869) 09330 (07326) 09986 (09614) 05 08280 (07390) 09804 (09396) 10000 (09985) 03942 (02859) 05632 (03695) 08462 (06011) 0 1 09480 (08844) 09990 (09919) 10000 (10000) 05062 (03666) 07442 (05067) 09612 (07968) 2 09878 (09499) 10000 (09995) 10000 (10000) 06328 (04210) 08622 (06228) 09936 (08966) 05 08266 (07231) 09772 (09302) 09998 (09980) 03790 (02779) 05384 (03554) 08232 (05619) 025 1 09464 (08729) 09988 (09882) 10000 (10000) 04914 (03485) 07208 (04888) 09528 (07668) 2 09860 (09483) 10000 (09982) 10000 (10000) 06102 (03947) 08396 (05893) 09902 (08758) Table 14: Power when onitoring scalar BEKK tie series and just one of the variances increases (N ( 0, Σ p ) distributed innovations) 13
p = 2 k = 005 k = 05 γ B = = 05 08604 (07823) 09778 (09444) 09998 (09982) 04666 (03340) 06556 (04609) 08922 (07014) 0 1 09487 (09004) 09965 (09895) 09999 (09997) 05838 (04263) 08107 (06036) 09716 (08481) 2 09842 (09540) 09988 (09972) 10000 (10000) 06880 (04919) 08969 (07163) 09913 (09296) 05 08574 (07725) 09756 (09365) 09997 (09977) 04532 (03186) 06304 (04463) 08750 (06724) 025 1 09458 (08842) 09960 (09870) 09999 (09996) 05645 (04001) 07889 (05738) 09643 (08206) 2 09818 (09450) 09988 (09970) 10000 (09998) 06656 (04827) 08810 (06907) 09892 (09206) 05 08043 (07142) 09612 (09085) 09989 (09929) 04229 (03175) 05673 (03925) 08130 (05881) 0 1 09135 (08439) 09933 (09759) 09996 (09997) 05342 (03904) 07229 (05086) 09314 (07587) 2 09673 (09233) 09989 (09946) 10000 (10000) 06248 (04487) 08317 (06239) 09774 (08684) 05 08046 (07168) 09599 (08983) 09988 (09940) 04274 (03304) 05554 (03863) 07947 (05621) 025 1 09109 (08507) 09921 (09773) 09996 (09994) 05316 (04003) 07030 (05138) 09199 (07378) 2 09656 (09207) 09986 (09925) 10000 (09999) 06135 (04460) 08176 (05954) 09744 (08456) 05 07439 (06705) 09084 (08354) 09925 (09750) 04308 (03587) 04926 (03673) 07025 (04872) 0 1 08722 (07969) 09784 (09419) 09996 (09984) 05129 (03983) 06390 (04642) 08607 (06375) 2 09304 (08736) 09932 (09747) 10000 (09993) 05776 (04425) 07384 (05159) 09330 (07415) 05 07505 (06736) 09033 (08235) 09915 (09700) 04482 (03761) 04862 (03648) 06792 (04611) 025 1 08728 (07939) 09749 (09367) 09996 (09960) 05220 (04322) 06207 (04526) 08381 (06152) 2 09326 (08691) 09924 (09741) 10000 (09993) 05859 (04655) 07270 (05185) 09238 (07241) Table 15: Power when onitoring scalar BEKK tie series and just one of the variances increases (t ν ( 0, ν 2 ν Σ p) distributed innovations with ν=8 degrees of freedo) p = 2 noral distribution t distribution k = 005 k = 05 k = 005 k = 05 γ B = = = = 05 07570 09416 09987 03344 04931 07526 05714 07736 09479 02615 03506 05175 0 1 08772 09886 10000 04108 06381 08932 07161 08947 09892 03477 04684 06811 2 09477 09978 10000 05164 07538 09583 08052 09508 09960 04051 05672 07915 05 07495 09351 09984 03119 04591 07204 05846 07773 09474 02691 03455 05025 025 1 08716 09870 10000 03797 05985 08686 07056 08849 09875 03338 04399 06480 2 09404 09972 10000 04822 07214 09491 07843 09414 09952 03752 05244 07544 05 09400 09966 10000 05042 07155 09250 07913 09293 09957 04323 05455 07252 0 1 09850 09997 10000 06220 08525 09851 08905 09839 09992 05141 06556 08657 2 09962 10000 10000 06988 09099 09960 09375 09932 10000 05767 07372 09157 05 09385 09962 10000 04826 06860 09078 08023 09288 09952 04331 05283 07040 025 1 09834 09995 10000 05850 08201 09799 08859 09820 09989 04984 06240 08391 2 09955 10000 10000 06542 08836 09942 09334 09925 09999 05514 07052 08952 05 09953 10000 10000 07139 08957 09934 09227 09872 09999 06011 07131 08720 0 1 09996 10000 10000 07678 09476 09990 09599 09977 10000 06367 07810 09346 2 10000 10000 10000 08812 09903 09999 09895 09999 10000 07353 08737 09812 05 09947 10000 10000 06850 08697 09896 09223 09868 09996 05964 06914 08481 025 1 09993 10000 10000 07346 09293 09988 09620 09973 10000 06287 07620 09199 2 10000 10000 10000 08425 09802 09999 09888 09997 10000 07054 08446 09732 Table 16: Power when using univariate onitoring procedures: scalar BEKK tie series and all of the variances increase 14
p = 2 noral distribution t distribution k = 005 k = 05 k = 005 k = 05 γ B = = = = 05 09095 09958 10000 04507 07100 09453 07367 09279 09962 03457 05112 07653 0 1 09785 10000 10000 06033 08653 09945 08710 09813 09996 04473 06616 08949 2 09957 10000 10000 07160 09497 09999 09285 09932 09995 05292 07599 09532 05 08975 09944 10000 04309 06846 09381 07297 09196 09947 03366 04876 07357 025 1 09750 09999 10000 05588 08394 09902 08592 09770 09994 04299 06328 08790 2 09955 10000 10000 06964 09356 09993 09224 09919 09995 05083 07353 09455 05 08684 09908 10000 03904 06333 09047 06736 08966 09909 03282 04417 06878 0 1 09695 09999 10000 05594 08340 09863 08340 09715 09978 04315 06155 08659 2 09910 10000 10000 06351 09132 09980 08934 09879 09987 04846 06865 09221 05 08554 09867 10000 03653 05873 08846 06773 08923 09898 03387 04318 06681 025 1 09585 09994 10000 05095 07948 09796 08219 09664 09973 04203 05829 08399 2 09907 09999 10000 06262 09021 09980 08898 09872 09986 04856 06712 09126 05 08388 09879 10000 03676 05919 08820 06612 08711 09903 03401 04302 06469 0 1 09599 09993 10000 05167 07915 09787 08167 09611 09984 04340 05860 08226 2 09854 10000 10000 05763 08748 09953 08709 09800 09994 04640 06514 08976 05 08393 09857 10000 03716 05752 08677 06824 08723 09901 03845 04463 06354 025 1 09577 09989 10000 05168 07788 09756 08277 09602 09983 04735 05954 08143 2 09838 10000 10000 05572 08563 09942 08696 09783 09993 04795 06418 08846 Table 17: Power when using univariate onitoring procedures: scalar BEKK tie series and only one of the variances increases p = 2 noral distribution t distribution k = 005 k = 05 k = 005 k = 05 γ B = = = = 05 07807 09559 09990 03618 05348 07816 06241 08071 09614 03136 03901 05716 0 1 08966 09933 10000 04497 06823 09162 07451 09112 09931 03738 05050 07194 2 09556 09994 10000 05335 07911 09715 08385 09619 09980 04442 06063 08187 05 07766 09521 09989 03408 05036 07546 06279 08001 09563 03127 03699 05420 025 1 08895 09910 10000 04205 06440 08970 07410 09043 09911 03607 04759 06876 2 09498 09988 10000 04972 07612 09651 08313 09570 09975 04228 05738 07890 05 09589 09994 10000 05760 07803 09645 08156 09416 09962 04771 05696 07556 0 1 09914 09999 10000 07058 09143 09969 09115 09866 09997 05624 06920 08862 2 09989 10000 10000 08065 09653 09997 09570 09969 10000 06372 07955 09505 05 09568 09994 10000 05514 07541 09564 08212 09397 09957 04742 05493 07296 025 1 09913 09999 10000 06816 08998 09964 09144 09861 09997 05556 06740 08684 2 09986 10000 10000 07865 09581 09997 09568 09968 10000 06228 07747 09403 05 09998 10000 10000 08720 09845 10000 09453 09906 09997 06657 07525 09024 0 1 10000 10000 10000 09604 09994 10000 09858 09998 10000 07544 08717 09721 2 10000 10000 10000 09888 10000 10000 09955 09998 10000 08266 09287 09936 05 09997 10000 10000 08578 09814 10000 09487 09902 09997 06690 07395 08920 025 1 10000 10000 10000 09535 09991 10000 09860 09998 10000 07472 08574 09665 2 10000 10000 10000 09854 09999 10000 09958 09998 10000 08191 09190 09920 Table 18: Power when the variance of the innovations increases 15
References Davidson, J (1994): Stochastic Liit Theory, Oxford University Press Engle, R and K Kroner (1995): Multivariate Siultaneaous Generalized Arch, Econoetric Theory, 11, 122 150 Hafner, C (2003): Fourth Moent Structure of Multivariate GARCH Models, Journal of Financial Econoetrics, 1(1), 26 54 Wied, D, M Arnold, N Bissantz, and D Ziggel (2012a): A New Fluctuation Test for Constant Variances with Applications to Finance, Metrika, 75(8), 1111 1127 Wied, D and P Galeano (2013): Monitoring Correlation Change in a Sequence of Rando Variables, Journal of Statistical Planning and Inference, 143(1), 186 196 Wied, D, W Kräer, and H Dehling (2012b): Testing for a Change in Correlation at an Unknown Point in Tie Using an Extended Functional Delta Method, Econoetric Theory, 68(3), 570 589 16