The proofs of Theorem 1-3 are along the lines of Wied and Galeano (2013).

Transcription

1 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 [ 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

2 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 (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

3 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

4 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 Σ 2 = 07 1, Σ 5 = and Σ 10 = 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

5 Noral distribution Standardized t distribution Epirical size For both graphics: iid: D known iid: D estiated scalar BEKK Epirical size 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

6 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 Early change, standardized t distribution Power Power 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

7 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 Early change, standardized t distribution Power Power 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

8 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

9 C Appendix: Tables D is known Σ = Ip Σ = Hp p = 2 γ B = = = 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 = = = 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

10 p = 2 k = 005 k = 05 γ B = = (06665) (08941) (09938) 03398(02276) (03389) (05243) (07960) (09742) (09998) (02798) (04258) (06819) (08972) (09929) (10000) (03336) (05253) (08163) (06482) (08818) (09929) (02073) (03123) (04987) (07848) (09644) (09997) (02599) (03750) (06337) (08689) (09901) (10000) (02890) (04771) (07841) (08929) (09911) (10000) (03564) (05098) (07866) (09671) (09997) (10000) (04431) (06687) (09264) (09934) (10000) (10000) (05245) (07791) 09994(09782) (08852) (09900) 1000 (09999) (03344) (04934) (07501) (09630) (09997) (10000) (04137) (06294) (09044) (09918) (10000) (10000) (04893) (07509) (09725) (09977) (10000) (10000) (06361) (08518) (09870) (10000) (10000) (10000) (07749) (09529) (09994) (10000) (10000) (10000) (08546) (09885) (10000) (09969) (10000) (10000) (06084) (08174) 1000 (09840) (10000) (10000) (10000) (07433) (09434) (09994) (10000) (10000) (10000) (08499) (09868) (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 = = (05079) (07059) (09083) (02222) (02635) (03786) (06359) (08363) (09746) (02595) (03329) (04742) (07291) (09032) (09933) (02834) 05 (03715) (05726) (05066) (06833) (08912) (02216) (02454) (03401) (06217) (08100) (09670) (02564) (02967) (04337) (07045) (08896) (09916) (02677) (03450) (05302) (07084) (08636) (09836) (03303) (03801) (05106) (08297) (09547) (09981) (03931) (04719) (06661) (08975) (09827) (09996) (04377) (05446) (07727) (07119) (08603) (09803) (03355) (03700) (04799) (08250) (09503) (09977) (03898) (04536) (06338) (08881) (09807) (09995) (04201) (05294) (07483) (08790) (09687) (09990) (04834) (05246) (06824) (09506) (09939) (1000) (05559) (06592) (08228) (09806) (09990) (10000) (06117) (07545) (09103) (08776) (09651) (09988) (04798) (04983) (06645) (09510) (09919) (10000) (05564) (06292) (08033) (09771) (09986) (10000) (06135) (07175) (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

11 p = 2 k = 005 k = 05 γ B = = (09298) (09975) (10000) (04040) (06364) (08923) (09843) (10000) (10000) (05150) (07892) (09784) (09982) (10000) (10000) (06383) (08994) (09970) (09196) (09967) (10000) (03673) (05966) (08792) (09809) (10000) (10000) (04821) (07731) (09712) (09968) (10000) (10000) (05847) (08746) (09953) (08578) (09877) (09996) (03255) (04878) (07774) (09588) (09997) (10000) (04153) (06611) (09283) (09908) (10000) (10000) (07) (07900) (09792) (08443) (09853) (09996) (02945) (04595) (07568) (09533) 10000(09996) (10000) (03952) (06152) (09140) (09890) (10000) (10000) (04722) (07582) (09720) (07481) (09524) (09996) (02557) (03674) (06334) (08996) (09941) (10000) (03417) (05013) (08131) (09530) (09996) (10000) (04093) (06395) (09210) (07444) (09504) (09996) (02543) (03550) (05955) (08892) (09934) (10000) (03244) (04769) (07898) (09477) (09995) (10000) (03821) (06110) (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 = = (07992) (09606) (09989) (03303) (04828) (07289) (09155) (09921) (09997) (04133) (06079) (08709) (09641) (09976) (09999) (04957) (07339) (09512) (07878) (09544) (09986) (03206) (04538) (06931) (09064) (09904) (09997) (03974) (05769) (08427) (09551) (09972) (09998) (04624) (07018) (09395) (07345) (09238) (09968) (03161) (03971) (06091) (08623) (09846) (09997) (03706) (05402) (07864) (09352) (09958) (09998) (04340) (06618) (08855) (07370) (09165) (09961) (03272) (03805) (05871) (08552) (09801) (09997) (03687) (05018) (07646) (09267) (09950) (09998) (04242) (06273) (08746) (06610) (08528) (09843) (03184) (03394) (04896) (08071) (09522) (09989) (03773) (04323) (06423) (08809) (09849) (09994) (04238) (05393) (07700) (06705) (08456) (09806) (03410) (03398) (04607) (08014) (09512) (09983) (03883) (04341) (06176) (08787) (09822) (09993) (04325) (05250) (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

12 ultivariate procedure univariate procedures p = 2 γ B = = = 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 = = = 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 = = (06546) (08732) (09900) (02451) (03405) (05131) (07820) (09599) (09990) (02876) (04159) (06522) (08707) (09879) (10000) (03413) (05197) (07794) (06408) (08602) (09869) (02340) (03120) (04760) (07608) (09517) (09988) (02639) (03816) (06150) (08518) (09847) (10000) (03109) (04796) (07428) (08757) (09891) (09999) (03782) (05140) (07645) (09630) (09992) (10000) (04592) (06433) (09010) (09878) (10000) (10000) (05333) (07625) (09622) (08654) (09869) (09999) (03544) (04795) (07298) (09586) (09991) (10000) (04360) (06146) (08840) (09852) (10000) (10000) (08) (07305) (09536) (09964) (10000) (10000) (06333) (08231) (09795) (09999) (10000) (10000) (07639) (09412) (09990) (10000) (10000) (10000) (08492) (09802) (10000) (09957) (10000) (10000) (06086) (08032) (09743) (09999) (10000) (10000) (07415) (09288) (09983) (10000) (10000) (10000) (08321) (09765) (10000) Table 12: Power when onitoring scalar BEKK tie series and all of the variances increase (N ( ) 0, Σ p distributed innovations) 12

13 p = 2 k = 005 k = 05 γ B = = (05036) (06778) (08787) (02225) (02525) (03506) (06175) (08044) (09580) (02602) (03140) (04451) (07052) (08835) (09881) (02922) (03749) (05585) (05168) (06812) (08763) (02418) (02598) (03347) (06213) (07956) (09558) (02743) (03116) (04272) (06948) (08755) (09847) (02835) (03575) (05266) (07018) (08550) (09762) (03476) (03916) (05033) (08171) (09466) (09971) (04188) (04707) (06412) (08823) (09774) (09997) (04534) (05464) (07491) (07103) (08517) (09716) (03645) (03797) (04745) (08188) (09348) (09953) (04077) (04593) (06193) (08838) (09751) (09992) (04555) (05378) (07240) (08772) (09609) (09988) (05284) (05673) (06859) (09403) (09928) (09998) (05768) (06494) (08109) (09798) (09979) (10000) (06480) (07476) (09011) (08755) (09549) (09977) (05190) (05357) (06618) (09463) (09932) (10000) (05770) (06474) (07978) (09730) (09981) (10000) (06399) (07114) (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 = = (09063) (09950) (10000) (03937) (06026) (08623) (09794) (09995) (10000) (05065) (07684) (09646) (09954) (10000) (10000) (06141) (08814) (09938) (08979) (09926) (10000) (03747) (05614) (08442) (09721) (09996) (10000) (04712) (07320) (09583) (09948) (10000) (10000) (05855) (08482) (09906) (08341) (09784) (10000) (03264) (04779) (07516) (09454) (09985) (10000) (04172) (06438) (09058) (09862) (10000) (10000) (05058) (07637) (09696) (08256) (09760) (09999) (03021) (04463) (07299) (09364) (09985) (10000) (03930) (06079) (08873) (09811) (10000) (10000) (04869) (07326) (09614) (07390) (09396) (09985) (02859) (03695) (06011) (08844) (09919) (10000) (03666) (05067) (07968) (09499) (09995) (10000) (04210) (06228) (08966) (07231) (09302) (09980) (02779) (03554) (05619) (08729) (09882) (10000) (03485) (04888) (07668) (09483) (09982) (10000) (03947) (05893) (08758) Table 14: Power when onitoring scalar BEKK tie series and just one of the variances increases (N ( 0, Σ p ) distributed innovations) 13

14 p = 2 k = 005 k = 05 γ B = = (07823) (09444) (09982) (03340) (04609) (07014) (09004) (09895) (09997) (04263) (06036) (08481) (09540) (09972) (10000) (04919) (07163) (09296) (07725) (09365) (09977) (03186) (04463) (06724) (08842) (09870) (09996) (04001) (05738) (08206) (09450) (09970) (09998) (04827) (06907) (09206) (07142) (09085) (09929) (03175) (03925) (05881) (08439) (09759) (09997) (03904) (05086) (07587) (09233) (09946) (10000) (04487) (06239) (08684) (07168) (08983) (09940) (03304) (03863) (05621) (08507) (09773) (09994) (04003) (05138) (07378) (09207) (09925) (09999) (04460) (05954) (08456) (06705) (08354) (09750) (03587) (03673) (04872) (07969) (09419) (09984) (03983) (04642) (06375) (08736) (09747) (09993) (04425) (05159) (07415) (06736) (08235) (09700) (03761) (03648) (04611) (07939) (09367) (09960) (04322) (04526) (06152) (08691) (09741) (09993) (04655) (05185) (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 = = = = Table 16: Power when using univariate onitoring procedures: scalar BEKK tie series and all of the variances increase 14

15 p = 2 noral distribution t distribution k = 005 k = 05 k = 005 k = 05 γ B = = = = 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 = = = = Table 18: Power when the variance of the innovations increases 15

16 References Davidson, J (1994): Stochastic Liit Theory, Oxford University Press Engle, R and K Kroner (1995): Multivariate Siultaneaous Generalized Arch, Econoetric Theory, 11, Hafner, C (2003): Fourth Moent Structure of Multivariate GARCH Models, Journal of Financial Econoetrics, 1(1), Wied, D, M Arnold, N Bissantz, and D Ziggel (2012a): A New Fluctuation Test for Constant Variances with Applications to Finance, Metrika, 75(8), Wied, D and P Galeano (2013): Monitoring Correlation Change in a Sequence of Rando Variables, Journal of Statistical Planning and Inference, 143(1), 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),