Technical Appendix for "Robust Non-Parametric Tests of Extremal Dependence" Jonathan B. Hill¤ Dept. of Economics University of North Carolina - Chapel Hill June 4, 2008 In the following appendix omitted proofs of Lemmas A.5-A.7 are given, and Lemma A.8 proves population NED implies Extremal-NED (Section A.1). Tail array properties are detailed (Section A.2) and omitted tables and all …gures are also presented in the end. Recall 1) with probability one, and take values on [0 denotes either or and ¡ ( ) = ( ) as ! 1, slowly varying
and
( ) ! 1 The tail arrays used throughout are := (ln )+ ¡
(A.1) (A.2)
¡1 and ( := ) ¡ ( )
A.1 SUPPORTING LEMMAS A.5-A.7 We present an expanded version of Lemma A.5. The proof of Lemma A.3 uses claim ( ) which is proved by the following claims ( )-( ). De…ne ) = ( ( ) ¡ ( ), 2 R+
LEMMA A.5 Let Assumption A hold. p p f () g and f g are 4 -NED on fzg with con stants ½ Z ¾ ¡14 ¡1 ( ) ( ) = (1 ) ¤ Dept . of Eco no mics, Univers ity of N orth Car olina -Chapel H ill, w ww.unc.edu/»jbhill,
[email protected].
1
¡12 and common coe¢cients = (( )14¡1 ). p p For any ¸ 1, f ) ) g and f ¡( ( ¡ g are 2-NED on fzg with constants ½ ¾ Z 1 ( ) ( ) ¡12 ¡1 ) ) = (1 ) ( ( 0
¡12 and common coe¢cients = (( )12¡1 ). P p For any 2 R, 0 = 1, f () ) g and ¡ ( =1 P p f =1 g with constants ¡ g are 2 -NED on fz ½ n o n o Z 1 n o ¾ () ( ) ( ) ( ) ( ) ( ) = max ( ) max ( ) 1· ·
=
0
¡12 ¡1 (1 )
1· ·
¡12 and common coe¢cients () = (( )12¡1 ) P p . Under the conditions of ( ), f () () zg ¡ =1 P p and f =1 zg form 2-mixingale sequences with ¡ ¤ g, where sup f ¤ g = ( ¡12). size 1 2 and constants f
LEMMA A.6 Under Assumptions A, B and D, j~ ) ¡ )j = ( ( (1) for each = 1 P ^ LEMMA A.7 Under Assumptions A, B and D 1 ) 2) ( 1 ) =1 ((¡ = (1) for all = 1 . LEMMA A.8 Consider a univariate stochastic process that takes vales on [0 1), and has tail (A.1), with threshold sequence f g as la (A.2). Let f g be -NED on fz g with constants f g, sup ¸1 1, and coe¢cients fg of size 0. Then for some displacement sequence f g, ! 1, f g is -E-NED for any ¸ 2 with constants ( ) Lebesgue integrable on R +, and coe¢cients of size £ minf 1g (4 ). Proof of Lemma A.5. See Lemma A.1 of Hill (2005). p Lemma 5.3 of Hill (2007) implies f ) () g is 2 -E-NED ¡( on fzg with constants () ¡12 ) = maxf (1 ) ( )1) ¡( 2)g = ( (
and coe¢cients ¡12 = maxf g= (( )12¡1£ )
2
An argument identical to Lemma A.1 of Hill (2005) therefore implies p () f g with constants s01( ) and ¡ g is 2-E-NED on fz ( coe¢cients . From Theorem 17.8 of Davidson (1994) and ( ) it follows that P P p p () ) and are 2 -NED ¡ ( ¡ =1 =1 P p on fzg. In particular, ( ) := () () satis…es ¡ =1 ° h i° X ° ( ( ) + ° ) ¡ ( )jz · () £ ° ° ¡ =1 2 n o ¡ ¢ ( ) · max ( ) £ £ 1· · ()
()
= ) £ (
() 2 ) ¡12 where ) = (¡1 ( )1) and ( = (( )12¡1 ) under ( Claim ( ). An argument identical to Lemma A.1 of Hill (2005) applies to P p . ¡ =1
The …nal claim follows from Claim ( ) and arguments in Davidson (1994: P p p. 264-265). Consider ( ) := () ) , the proof ¡ ( =1 P p for =1 being similar. ¡ If the base f then Davidson (1994: g is strong mixing with coe¢cients eq. (17.18)) ´ © ª ³ 12¡1 () k[ )jz¡ ]k2 · max k ( )k () £ + ( The Minkowski and the Cauchy-Schwartz inequalities, 0 = 1, and jj )jj ( = (( )1) by Lemma C.1, below, imply for any ¸ 2 ³ ´ 2 2 k ( )k· £ ¡1 k k2k k2= ¡1 ( )1
() 2 ) ¡12 12¡1 Since ) = (¡1 ( )1), ( = (( )12¡1 ), and ( ¡(12¡1)£(¡2)
= (( )12¡1£
¡12
)= (( )12¡1£
) it follows
k[ )jz¡ ( ]k2
³ ´ 1 12¡1 () 2 · ¡1 ( ) + ³ ´ 1 2 ¡12 ¡12 = ¡1 ( ) (( )12¡1 )+ (( )12¡1 ) ³ ´ ¡12 12¡1 ¡12 = ¡12 ( )1¡12 (( )12¡1 ) + (( ) ) ¡12 = ¡12 £ ( )
A similar argument applies to the remaining mixingale inequality jj ( ) ¡12 ¡1 2 ¡ [ )jz+ ]jj2 · £ ( ), and in the uniform mixing case, ( cf. Davidson (1994: eqs. (17.19)-(17.20)). 3
Proof of Lemma A.6. Write ) := ( ¡ . We will prove ¯ à !2¯¯ ¯ X ¯ ¯ 1 ¯~ ) ¡ p ) ¯¯ ! 0 ( ¯ ( =1 ¯ ¯
by verifying Assumptions 1-3 of de Jong and Davidson’s (2000) [JD] Theorem 2.1 hold. JD’s Assumption 1 holds by the statement of the lemma. Moreover, p zg forms an 2-mixingale array with size 1 2 and constants () ¡12 = ( ) by Lemma A.5. Thus JD’s Assumption 2 is satis…ed. Finally, 2 JD’s Assumption 3 is satis…ed by max = (1) given = (). 1· · Proof of Lemma A.7. Write ^ ^ ¡ ) ) and ¢ := (( := ¡ and (A.3) ^ ( 1 2) ^ ^ ^ ^¡ = ¡1 ¡2 ¡1 ¡2 ^ ^ ^ = ¡1 £ ¢ £ ¢ ¡2 £ ¢ ^ ^ ^ + ¢ ¡1 £ £ ¢ ¡2 £ ¢ ^ ^ ^ + ¢ ¡1 £ ¢ £ ¡2 £ ¢ ^ ^ ^ + ¢ ¡1 £ ¢ £ ¢ ¡2 £ ^ ^ + £ £ ¢ £ ¢ ¡1 ¡2
+ ¢¢ ¢ ^ ^ ^ ^ + ¢ ¡1 £ ¢ £ ¢ ¡2 £ ¢ We will prove ¯ ¯ 1 X ¯ ¯ = (1) ^ ¢ ¯ ¯ ¡1 ¡2 =1
Nearly identical arguments su¢ce to show (1). p all remaining terms in (A.3) are ¡1 Exploiting ^ ¡1 = + (1 ), cf. Theorem, 5 of Hill (2005), As sumption B (Hsing 1991: p. 1554) ¡ ¢ ^ ¡ = ln ( +1) + ¡ (ln )+ ¡ ¡1 ¢ ¡ ( ) ¡ ( )(ln )+ ¡ ¡1 ¢ ¡ ( ) ^ ) ¡ ¡1 ( ¡ ¢ p = ln ( ) ( +1) + ¡ (ln )+ + 4
By cases it is straightforward to show ¯¡ ¯ ¯ ¯ ¢ ¯ ¯ ¯ ¯ ¯ ln (+1) + ¡ (ln )+ ¯ · ln (+1) Note
³ ´ 1 X ^ ¡1 ¡2 ¡ =1
=
1 X ¡1 ¡2 =1 ³¡ ´ ¢ £ ln )+ (+1) + ¡ (ln
+
=
1 X (( )12) £ ¡12 ¡1 ¡2 £ =1
1 X ¡1 ¡2 =1 ³¡ ´ ¢ £ ln ¡ (ln ) + + (1) (+1) +
The last line follows from from applications of the Minkowski’sPthe CauchySchwartz inequalities, and Lemma C.1, below: exploiting 1 j =1 j 1 2 = ( ) under Assumption D, ° ° ° 1 X ° ° ° ° ¡1 ¡2 ° ° =1 ° 1
1 X 4 · j j k k2 k k2 =1 µ ³ ´2 ¶ ³ ´ = 12( )12 ( )14 = 12
and ( 12 ( )12 ¡12 ) = (( )12) = (1).
5
Therefore ° ° ° 1 X ³¡ ´° ¢ ° ° ° ln ¡1 ¡2 (+1) + ¡ (ln )+ ° ° ° =1
1
° ° 1 X ° (+1) ° · j j k ¡1 ¡2 k2 ln 2 =1
1 1 X j (( )12 ) j k ¡1 ¡2 k2 £ 12 =1 °p ° ° · ° ln (+1) 2 1 1 X £ 12 j k4 j £ k k8 £ k k8 k =1
+
+ (( )12) 1 1 X = (1) £ 12 j j + (( )12) = (1) =1
p Similarly, because ln (+1) = (1 ) by Lemma 4 of Hill (2005), ¡1
X
=1
³ ´³ ´ ^ ^ ¡1 ¡ ¡1 ¡
³ ´³ ´ ^ ^ £ ¡2 ¡ ¡2 ¡
¯ ¯2 ¯ ¯2 ¡1 X ¯ ¯ln ¯ · ¯ln (+1) j j (+1) =1
1 XX = (1 ) £ j j =1 =1
= (12 ) £ = (1)
1 1 XX £ j j 12 =1 =1
6
¹ Proo of Lemma A.8. Write ) := ( ( ). For some 0 to be + ¹ ¹ chosen below and any ¸ 2, since j () ¡ [ )jz ]j · 1 ( ¡
¯ h i¯ + ¯ ¹() ¡ ¹ ¯¯ ()jz ¯ ¡ ³ h i´2 + ¹ ¹ · )¡ )jz ( ( ¡ ³ ³ h i ´´2 ³¯ h i¯ ´ + + ¯ ¹ ¹ · )¡ )jz ¯¯ ¯· ( ( ¡ jz ¡ ¡ ³ ³ h i ´´2 ³¯ h i¯ ´ + + ¯ ¹ ¹ + )¡ ()jz ¯¯ ¯ ( ¡ jz ¡ ¡ ³¯ h i¯ ´ + ¯ · [( )] + ¯¯ jz ¯ ¡ + ¡ ¡ ³¯¯ h i¯¯ ´ £¹ ¤ + ¹( = ( )¡ ) + ¯ ¯ ¡ + ¡ jz ¡ ³¯ h i¯ ´ £ ¤ + ¯ ¹( ¹( · )¡ ) + ¯¯ ¯ ¡ + ¡ jz ¡ £ ¤ ¹( ¹( · )¡ ) + ¡ +
The second inequality is due to the conditional expectations minimizing the mean-squared-error, and a trivial identity. The third follows from basic logic, and a trivial inequality involving the indicator function. The fourth is Markov’s inequality, and the …fth follows from the -NED property. 12 De…ne ¹ := sup ¸1 2 [0 1). Put = , use (A.1) and the mean-value-theorem to deduce ³ ³ ´ ´ 12 ¹( ¹ § ) = 1 § ³ ´¡ 12 ¡ ¡ ¡ 12 ¡ = 1 § = £ (1) Therefore for any
³ h i´2 ¹ ¹()jz+ )¡ ( ¡ h ³ ³ ´ ´ ³ ³ ´ ´i 1 2 12 ¡ ¡ 2 ¹ ¹ · ¡ + 1¡ 1 + h ³ ³ ´ ´ ³ ³ ´ ´i 12 12 ¡ ¡ 2 ¹ ¹ = ¡ + 1 ¡ 1 + ³ ´ 12 ¡ ¡ 2 · £ (1) + 1g2 ¡ ¡ minf ·
hence as ! 1 ° h i° ° + ° ¹ )¡ )jz ° ¹ ° ( ( ¡
· =
³ ´ nf 1g(2) ¡ ¡ £ mi () £ 7
where for arbitrary ¸
¡ () = ( )1 ( "µ #) ¶ n o 1 ¡ ¡mi nf1g(4) 1g(2) 1¡1 minf 1g(4) minf = (( ) ) £
Clearly sup 2N ) = (( )1) if ) ( is uniformly bounded, and ( is Lebesgue integrable on R +. The size is minf 1g (4 ) su¢ciently if ( ) minf 1g4
A.2
= (1)
TAIL ARRAY PROPERTIES
LEMMA B.1 Under (A.1)-(A.2), jj jj= (( )1) and jj jj= 1 (( ) ) for any ¸ 1. Proof. Use Minkowski’s and Liaponov’s inequalities and arguments in Hsing (1991: p. 1548), cf. (A.1)-(A.2), to deduce for all ¸ 1 k k · »
¡ ¢ 1 2 (ln )+ ¡ ¢ 1 2 ( ) £ ! £ ¡1 = (( )1)
Similarly, the construction of implies for any ¸ 1
1 k k · ( + ( ) )
» ( )1+ ( ) = (( )1)
8
Appendix 1: Omitted Tables and Figures
(1)
h 1 2 3 4
TABLE 1 - EVAR (n = 500) Two-Tailed med{ ^ § } (weak)
i. 9 ^§ %rej KS -.006 §.06 .47 .01 .022 -.002§.07 .52 .01 .000 .007 §.07 .63 .02 .000 .002 §.07 .67 .03 .000
ii. ! § %rej ^ .285§.22 .01 .94 .281§.21 .04 .67 .240§.20 .06 .55 .193§.19 .07 .31
(strong)
iii. ! § ^ .336 § 20 .00 .300§.19 .02 .235§.17 .03 .178§.15 .05
%rej .98 .83 .76 .65
TABLE 1 - SAV (n = 500) Two-Tailed med{ ^ § }
(h)
(1)
h 1 2 3 4
h 1 2 3 4 h
i. 9 ^§ %rej .002§.07 .46 .01 .002§.07 .59 .01 -.004 §.06 .66 .02 -.006 §.06 .73 .02
( 1)
KS .008 .000 .000 .000
ii. ! ^ § %rej .126 §.10 .04 .51 .102 §.09 .08 .33 .083 §.08 .11 .26 .065 §.08 .18 .17
TABLE 1 - E-VAR (n = 1000) Two-Tailed med{^ § }
i. 9 ^§ rej % KS -.005§.056 .424 .06 .223 .001 §.053 .543 .04 .152 .000 §.049 .631 .05 .119 -.002 §.052 .708 .03 .071 Two-Tailed
(weak)
ii. ! ^§ rej % .303§.166 .003 .97 .289§.163 .024 .88 .234§.152 .038 .67 .183§.140 .057 .51 med{^ § } (weak)
(1)
i. 9 1 ^§ rej % -.021 §.039 .406 .01 -.002 §.039 .539 .01 .000 §.039 .640 .02 .001 §.039 .691 .02
iii. ^§ .382§.182 .327§.167 .248§.148 .187§.132
(strong)
!
.001 .008 .010 .022 (strong)
rej % 1.0 .97 .96 .89
ii. ! iii. ! KS ^§ rej % ^§ rej % 1 .094 .191§.055 .001 1.0 .270§.058 .000 1.0 2 .023 .176§.053 .000 1.0 .213§.055 .000 1.0 3 .006 .128§.051 .000 .96 .149§.052 .000 1.0 4 .000 .089§.050 .001 .94 .105§.050 .000 1.0 Note s: a. Me dian 95% con…dence bands over 2 . b. Median Wald statistic p -value (under the null ( ) ) 20 ()). c. Rejection frequenc y at the 5%-level. d. The Kolmogorov-Smirnov p-value for a test W(h) » 2(h).
9
TABLE 1 - SV (n = 1000) Two-Tailed med{ ^ § } (1)
h 1 2 3 4
i. 9 ^§ %re j .005 §.051 ..445 .000 .008 §.053 .552 .000 -.008§.043 .564 .000 .001 §.046 .617 .000 Two-Tailed
h 1 2 3 4
i. 9 1 ^§ %re j -.002§.04 .556 .000 -.000§.04 .628 .000 -.001§.04 .713 .000 .003 §.04 .756 .000
(1)
()
ii. ! KS § ^ .046 .123 §.078 .016 .012 .102 §.072 .029 .007 .083 §.069 .053 .000 .054 §.057 .092 med{^ § }
%rej .975 .754 .612 .322
()
KS .025 .011 .004 .000
ii. ! § ^ .119§.06 .002 .105§.06 .002 .082§.05 .003 .064§.05 .005
%rej .978 .978 .921 .843
Table 2: Bivariate Extremal Spillover Two-Tailed med{^ § } h 1 2 3 4 h 1 2 3 4 h 1 2 3 4
Yen ! BP
.001§.036 -.001 §.039 -.006 §.040 -.007 §.038
.771 .669 .805 .866
.011§.036 -.019 §.037 .009§.039 .001§.038
.678 .716 .853 .910
.007§.040 -.004 §.037 .016§.040 .013§.037
.76 .84 .91 .90
BP ! Yen
NAS ! Yen
Yen ! NAS
Yen ! SP
Yen ! LSE
.009 §.041 -.002§.039 .010 §.040 .008 §.040
.68 .84 .88 .85
.012 §.038 .010 §.038 -.000 §.039 .016 §.042
.69 .83 .89 .91
.021§.040 .013§.037 - .008 §.037 - .008 §.039
.524 .737 .740 .806
-.010§.039 -.005§.040 -.007§.036 -.008§§.038
.719 .784 .895 .947
-.016 §.042 -.005 §.041 -.024 §.036 -.004 §.038
.577 .783 .801 .902
.004§.039 - .005 §.038 - .008 §.040 - .022 §.036
.783 .855 .909 .889
-.023§.039 -.010§.038 -.007§.040 -.015§.039
.366 .449 .633 .720
.140 §.044 .058 §.043 .057 §.042 .084 §.042
.00 .00 .01 .01
.117 §.05 .052 §.04 .070 §.04 .081 §.04
.004 .013 .018 .015
BP! NAS
NAS ! BP
BP ! SP
NAS ! SP
Notes: a. Wald statistic W(h) p-value .
10
BP ! LSE
NAS ! LSE
Yen ! NIK
.043§.036 .006§.037 .023§.041 -.002 §.039
.229 .459 .525 .651
-.009 §.037 -.003 §.041 .008§.040 -.014 §.041
.674 .832 .899 .907
.074 §.04 .001 §.04 .012 §.04 .040 §.04
.022 .058 .123 .177
BP ! NIK
NAS ! NIK
Table 2 - Continued Two-Tailed med{ ^ § } h 1 2 3 4 h 1 2 3 4 h 1 2 3 4
SP! Yen -.003 §.039 .000§.040 .001§.039 .015§.039
.76 .89 .92 .94
LSE ! Yen .000§.037 .838 .022§.038 .584 .028§.038 .567 .008§.039 .727 NIK! Yen -.012 §.039 .661 .007§.038 .745 -.007 §.041 .821 .002§.037 .896
SP ! BP
-.029 §.041 -.010 §.040 -.010 §.036 -.008 §.038
.271 .423 .509 .605
LSE ! BP -.021 §.035 .414 .019§.037 .527 .005§.038 .666 -.016 §.038 .679 NIK ! BP -.012 §.042 .660 -.004 §.041 .843 .005§.039 .906 -.023 §.041 .817
11
SP ! NAS
.049§.044 .059§.042 .078§.041 .066§.041
.15 .14 .09 .12
LSE ! NAS .087§.042 .022 .055§.039 .052 .075§.040 .043 .069§.041 .057 NIK ! NAS .013§.042 .679 .028§.043 .557 .016§.039 .635 .040§.043 .654
SP ! LSE
.137 §.047 .101 §.049 .093 §.051 .126 §.053
.001 .003 .004 .003
LSE ! SP .096 §.044 .011 .069 §.044 .032 .108 §.043 .014 .123 §.044 .011 NIK ! SP .012 §.040 .597 .014 §.039 .681 -.004§.038 .816 .034 §.043 .750
SP ! NIK
.076 §.038 -.004§.036 .001 §.036 .040 §.038
.024 .069 .144 .189
LSE ! NIK .052 §.041 .154 .027 §.039 .283 .041 §.039 .267 .048 §.042 .309 NIK ! LSE .019 §.041 .713 .015 §.040 .757 .026 §.041 .737 .020 §.039 .737
Figure 1: Rolling Fractile Two-Tailed ^() : n = 500, h = 1...4 E-VAR: No S pil love r Two-Tai le d Tai l De pe n de nce C oe f. r(h) an d Robu s t C on fide n ce B an ds med r(1) = -.014 ± .050
E-VAR: S tron g S pi ll ove r Two-Tai le d Tail De pe n de n ce C oe f. r(h) an d Robu s t C on fi de n ce B ands med r(1) = .273 ± .073
0 .10
0 .50
0 .0 8
0 .4 0
0 .0 6
0 .3 0
0 .0 4
0 .2 0
0 .0 2
0 .10
0 .0 0
0 .0 0
-0 .0 2
-0 .10
-0 .0 4
-0 .2 0
-0 .0 6
-0 .3 0
-0 .0 8
-0 .4 0
-0 .10
-0 .50
1
21
41
61
81
10 1 12 1
14 1 16 1
18 1 2 0 1
2 21 24 1
26 1 2 81 3 01
32 1 3 41
3 61 38 1
m
25
45
65
85
10 5
12 5 14 5 16 5
18 5 2 0 5 2 2 5 2 4 5 2 6 5 2 8 5
30 5 3 25
3 45 3 65
38 5
-k(1)
m r(1)
k(1)
-k(2 )
r(2 )
k(2 )
-k(1)
m r(1)
k(1)
-k(2 )
r(2 )
k(2 )
-k(3 )
r(3 )
k(3 )
-k(4 )
r(4 )
k(4 )
-k(3 )
r(3 )
k(3 )
-k(4 )
r(4 )
k(4 )
S AV: S pi ll ove r
S AV: No S pill ove r Two-Tail e d Tail De pe n de n ce C oe f. r(h ) an d Robu st C on fi de n ce B an ds med r(1) = -.001 ± .063
Two-Tai le d Tai l De pe n de nce C oe f. r(h ) an d Robus t C on fide n ce B ands med r(1) = .115 ± .071
0 .10
0 .2 0
0 .0 8
0 .16
0 .0 6
0 .12
0 .0 4
0 .0 8
0 .0 2
0 .0 4
0 .0 0
0 .0 0
-0 .0 2
-0 .0 4
-0 .0 4
-0 .0 8
-0 .0 6
-0 .12
-0 .0 8
-0 .16
-0 .10
-0 .2 0
6
21
36
51
66
81
96
111
12 6
14 1
156
171
18 6 2 0 1
2 16 2 3 1 2 4 6
2 6 1 2 76
6
21
36
51
66
81
96
111
12 6
14 1 156
171
18 6 2 0 1
2 16
23 1 24 6
2 6 1 2 76
-k(1)
m r(1)
k(1)
-k(2 )
r(2 )
k(2 )
-k(1)
m r(1)
k(1)
-k(2 )
r(2 )
k(2 )
-k(3 )
r(3 )
k(3 )
-k(4 )
r(4 )
k(4 )
-k(3 )
r(3 )
k(3 )
-k(4 )
r(4 )
k(4 )
Figure 2: Rolling Fractile Two-Tailed ^() : n = 500, h = 1...4 E-VAR: No S pi ll ove r Two-Tai l e d Exce e de n ce C orre l ation q(h ) an d Robu st C on fide n ce B an ds med q(1) = -.006 ± .064
E-VAR: S tron g S pi ll ove r Two-Tai le d Exce e de n ce C orre lati on q(h ) an d Robu s t C on fide n ce B an ds med q(1) = .336 ± .199 1.0 0
0 .10 0 .0 8
0 .75
0 .0 6 0 .50 0 .0 4 0 .2 5
0 .0 2
0 .0 0
0 .0 0 -0 .0 2
-0 .2 5
-0 .0 4 -0 .50 -0 .0 6 -0 .75
-0 .0 8 -0 .10
-1.0 0 6
26
46
66
86
10 6 12 6 14 6
16 6 18 6 2 0 6 2 2 6 2 4 6 2 6 6 2 8 6 3 0 6 3 2 6
34 6 3 6 6 3 86
6
26
46
66
86
10 6 12 6 14 6
16 6 18 6
m
2 06 22 6
2 46 2 6 6 28 6
3 0 6 32 6
3 4 6 36 6
386
-k(1)
q (1)
k(1)
-k(2 )
q (2 )
k(2 )
-k(1)
m q (1)
k(1)
-k(2 )
q (2 )
k(2 )
-k(3 )
q (3 )
k(3 )
-k(4 )
q (4 )
k(4 )
-k(3 )
q (3 )
k(3 )
-k(4 )
q (4 )
k(4 )
S AV: No Spi ll ove r
S AV: Spil love r Two-Tail e d Exce e de n ce C orre l ation q(h ) an d Robu s t C on fide n ce B an ds med q(1) = .126 ± .101
Two-Taile d Exce e de n ce C orre l ati on q(h ) an d Robu s t C on fide n ce B an ds med q(1) = -.002 ± .071 0 .10
0 .2 0
0 .0 5
0 .10
0 .0 0
0 .0 0
0 .15
0 .0 5
-0 .0 5
-0 .0 5
-0 .10 -0 .15
-0 .10
-0 .2 0
6
21
36
51
66
81
96
111
12 6
14 1 156
-k(1)
q (1)
-k(3 )
q (3 )
m
171
18 6
2 0 1 2 16
23 1 2 46
2 6 1 2 76
6
21
36
51
66
81
96
111
12 6
14 1 156
k(1)
-k(2 )
q (2 )
k(2 )
-k(1)
q (1)
k(3 )
-k(4 )
q (4 )
k(4 )
-k(3 )
q (3 )
12
m
171
18 6
2 0 1 2 16 2 3 1
246
2 6 1 2 76
k(1)
-k(2 )
q (2 )
k(2 )
k(3 )
-k(4 )
q (4 )
k(4 )
Figure 3: Two-Tailed med{ ^ § } : n = 500, h = 1...20 E -VA R: N o Sp illo v er M edian T wo T ailed E x ceedan ce Co rrelat io n
E -VA R: St ro n g Sp illo v er M edian T wo T ailed E x ceedan ce Co rrelat io n
0 .10
0 .4 0
0 .0 8 0 .0 6 0 .0 4 0 .0 2
0 .3 0 0 .2 0 0 .10
0 .0 0 - 0 .0 2 - 0 .0 4 - 0 .0 6 - 0 .0 8 - 0 .10
0 .0 0 -0 .10 -0 .2 0 -0 .3 0
1
2
3
4
5
6
7
8
9 10 11 12 13 14
15 16 17 18 19 2 0
h
1 2
3
4
5
6
7
8
9 10
11 12 13 14
15 16
17 18
h
-k q ( h) k
SA V: N o Sp illo v er
19 2 0
-k q ( h) k
SA V: Sp illo v er M edian T wo T ailed E x ceedan ce Co rrelat io n
M edian T wo T ailed E x ceedan ce Co rrelat io n 0 .2 0
0 .10 0 .0 8 0 .0 6 0 .0 4 0 .0 2 0 .0 0 -0 .0 2 -0 .0 4
0 .15 0 .10 0 .0 5 0 .0 0 -0 .0 5
-0 .0 6 -0 .0 8 - 0 .10
- 0 .10 - 0 .15
h
1 2
3
4
5
6
7
8
9 10
11 12
13 14 15 16
h
1
17 18 19
2
3
4
5
6
7
8
9 10 11 12 13 14
15 16 17 18 19
h
-k
2
-k q ( h)
q ( h) k
k
Figure 4: Rolling Two-Tailed med{^ (1) § } : n = 500 E - V A R , n o s p i ll o v e r , t w o - t a il e d r ( 1 )
E - V A R , st r o n g s p ill o v e r , t w o - t a i le d r ( 1 )
m = { 5 ,… ,2 5 } … { 5 ,… ,4 5 0 }
SA V , s p ill o v e r , t w o - t a i le d r ( 1 ) m = { 5 ,… ,2 5 } … { 5 ,… ,4 5 0 }
m = {5 ,… ,2 5 } … {5 ,… ,4 5 0 }
0 . 10
0 .8 0
0 .2 5 0 .2 0
0 .6 0
0 .0 5
0 .15
0 .4 0
0 .0 0
0 .10
0 .2 0
0 .0 5 0 .0 0
0 .0 0
- 0 .0 5
- 0 .0 5
- 0 .2 0
- 0 .10
- 0 . 10 - 0 .4 0
- 0 .15
- 0 .15
- 0 .6 0
25
65
10 5
14 5
18 5
225
265
305
m ax {m }
345
-k
385
425
r ( 1)
- 0 .2 0
25
65
10 5
14 5
18 5
225
265
305
m ax {m }
k
345
-k
385
425
r ( 1)
25
65
E - V A R , n o s p il lo v e r , t w o - t a ile d r ( 1 )
18 5
225
265
305
345
-k
385
425
r ( 1)
k
SA V , s p ill o v e r , t w o - t a il e d r ( 1 ) m = {3 2 5 ,… ,3 5 0 } … { 5 ,… ,3 5 0 }
m = {3 2 5 ,… ,3 5 0 } … { 5 ,… ,3 5 0 }
m = {3 2 5 ,… ,3 5 0 } … { 5 ,… ,3 5 0 }
14 5
m ax {m }
E - V A R , st r o n g s p il lo v e r , t w o - t a ile d r ( 1 ) 0 .3 0
0 .10
10 5
k
0 .2 0
0 .2 5 0 .15
0 .2 0
0 .0 5
0 .10
0 . 15 0 .10
0 .0 0
0 .0 5
0 .0 5 0 .0 0
0 .0 0
- 0 .0 5
- 0 .0 5
- 0 .0 5 - 0 .10
- 0 .10 325
295
265
235
205
17 5
m in { m }
14 5
115
85 -k
55 r ( 1)
- 0 .10
325
25
295
265
235
205
17 5
m in { m }
k
14 5
115
85 -k
55 r ( 1)
25
325
295
265
k
The median medf (1)g is computed over rolling fractile sets from low upper bound = f5 25g to high upper bound = f5 450g, and high lower bound = f345 350g to low lower bound = f5 350g. Clearly the median is robust to the choice of window for E-VAR processes with or without spillover, and more sharp for SAV processes when the window contains only intermediate to large tail observations (small to intermediate ). 13
235
205
17 5
m in { m }
14 5
115
85 -k
55 r ( 1)
25 k
Figure 5: Equity and FX Returns and Two-Tailed Tail Indices NAS DAQ Daily Returns
NAS DAQ B oots trapped Hill-Es timator and KS -Dis tance a(m* ) = 1 .9 2 ± .27 3 , m* =arg min{D(m)}=33 1
0 .12 0 .10 5
0 .3 5
0 .0 8 0 .3 0
0 .0 6
4
0 .0 4
0 .2 5
0 .0 2
3
0 .2 0
0 .0 0
0 .15
2
-0 .0 2 0 .10
-0 .0 4
1 0 .0 5
-0 .0 6 0
-0 .0 8
0 .0 0 15
65
115
16 5
2 15
265
3 15
365
-0 .10 J a n-01
Ma y- 0 1
Oc t - 01
Ma r - 0 2
Aug - 0 2
J a n -0 3
4 15
465
SP500 Daily Returns
56 5
a(m ) a(m )-k D(m )
m
Ma y - 03
515
6 15
665
m ean [a(j,m )] a(m )+k
SP500 B oots tr appe d Hill-Es timator an d KS -Dis tanc e a(m* ) = 1 .9 6 ± .2 9 7 , m* =ar g min{D(m)}= 3 3 6
0 .0 5 0 .0 4 5
0 .3 5
0 .0 3
0 .3 0 4
0 .0 2
0 .2 5
0 .0 1
3
0 .2 0
0 .0 0
0 .15
2
-0 .0 1
0 .10 1
-0 .0 2
0 .0 5
-0 .0 3
0
0 .0 0 15
65
115
16 5
2 15
265
3 15
-0 .0 4 J a n- 01
Ma y -01
Oc t - 01
Ma r-0 2
Aug -02
J a n- 03
365
4 15
m
Ma y- 03
465
5 15
56 5
a (m ) a (m )- k D(m )
6 15
665
m e a n [a ( j,m ) ] a( m )+k
LS E B oots tr appe d Hill-Es ti mator and KS -Dis tanc e a(m* ) = 1 .9 6 ± .3 3 8 , m* =ar g min{D (m)}=2 9 6
LSE Daily Returns 0 .0 6
5
0 .0 4
1.2 0 1.0 0
4 0 .0 2
0 .8 0 0 .0 0
0 .6 0
E DD
alpha
3
2 0 .4 0
-0 .0 2
1
-0 .0 4
0 .0 0 0 .0 1
0 .0 6
0 .11
0 .16
0 .2 1
0 .2 6
0 .3 1
0 .3 6
0 .4 1
m/ nth quantile
-0 .0 6 J a n- 01
0 .2 0
0
J u n- 01
Oc t - 01
Ma r- 02
Aug- 0 2
J a n- 0 3
J un -0 3
Oc t -0 3
Ma r -0 4
Aug -0 4
De c -0 4
0 .4 6
0 .51
0 .56
0 .6 1
0 .6 6
a(m)
m e a n[ a ( j,m ) ]
a(m)-k
a ( m ) +k
E DD (m)
The plots on the right are Hill-estimates ^ based on a Minimum Empirical Distribution Distance bootstrap method. The bootstrap average is ¹ P ¡1 ¡1 = (¹¡1 ) , where ¹ = 1 ^ and ^ is computed from a =1 bootstrapped series f g with the same tail index as (Hill, J.B. 2007, =1 Empirical Distribution Distance Tail Index Inference for Dependent, Heterogeneous Data, working paper, University of North Carolina - Chapel Hill). The Kolmogorov-Smirnov distance () between the empirical distribution of ¹ ¡1 and the asymptotic normal distribution is minimized over a window of fractiles .
14
Figure 6: Equity and FX Returns and Two-Tailed Tail Indices Continued NIKKEI Daily Returns
N IKKEI B oots tr appe d Hill-Es tim ator and KS -D is tanc e a(m * ) = 2 .2 3 ± .2 7 0 , m * =ar g m in {D (m )}= 3 9 2
0 .0 5 0 .0 4
6
1.0 0
0 .0 3
5
0 .0 2
0 .8 0
4
0 .0 1
0 .6 0 0 .0 0
3 0 .4 0
-0 .0 1
2 -0 .0 2
0 .2 0
1 -0 .0 3
0
-0 .0 4
0 .0 0 1
51
10 1
15 1
20
2 51
30
3 51
-0 .0 5 J a n-01
J u n- 0 1
Oc t - 0 1
Ma r - 0 2
Au g- 0 2
J a n-03
J un-03
No v- 0 3
Ap r - 0 4
40
4 51
YEN/USD Daily Returns
50 1
5 51
60
a(m )-k m e a n [ a ( j,m ) ] E D D (m )
m
S e p-04
6 51
a(m ) a(m )+k
YEN/US D Boots trapped Hill-Es timator and KS -Distance a(m*) = 2.23 ± .260, m*=argmin{D(m)}=348
0 .0 3
0 .0 2
5
0 .0 1
4
0 .0 0
3
0 .3 5 0 .3 0
0 .2 5 0 .2 0
-0 .0 1
0 .15
2
0 .10
-0 .0 2 1
0 .0 5
-0 .0 3 0
0 .0 0 15
65
115
16 5
2 15
265
3 15
-0 .0 4 J a n-01
Ma y - 0 1
Oc t - 0 1
Ma r - 0 2
Au g - 0 2
J a n- 03
Ma y - 0 3
Oc t - 0 3
Ma r - 0 4
Au g - 0 4
365
4 15
B P/US D Daily Returns
515
56 5
a(m) a(m)-k D(m )
m
De c - 0 4
465
6 15
665
mean[a(j,m)] a(m )+k
YEN/US D Bootstrapped Hill-Estimator and KS -Dis tance a(m*) = 2.3 1 ± .296 , m* =arg min{D(m)}=3 29
0 .0 3 0 .0 2 7
1.0 0
0 .0 2
0 .9 0 6 0 .8 0
0 .0 1
5
0 .0 1
0 .70 0 .6 0
4
0 .50
0 .0 0
3
-0 .0 1
2
-0 .0 1
1
-0 .0 2
0
0 .4 0 0 .3 0 0 .2 0 0 .10 0 .0 0 15
-0 .0 2 J a n-00
Ap r - 0 0
J u l- 0 0
Oc t - 0 0
Fe b - 0 1
Ma y - 0 1
Aug-01
De c - 0 1
Ma r - 0 2
J un -02
15
65
115
16 5
2 15
2 65
3 15
3 65
m
4 15
46 5
a(m ) a(m )-k D(m )
515
56 5
6 15
66 5
m ean [a(j,m )] a(m )+k
Figure 7.1: Rolling Fractile First-Order Two-Tailed Serial ^ SP500 Two-Tailed First-Order Tail Dependence Coefficient Aver = .13 ± .09
NASDAQ Two-Tailed First-Order Tail Dependence Coefficient Median = .10 ± .08
0 .3 0
0 .3 0
0 .25
0 .2 5
0 .2 0
0 .2 0 0.15
0 .15 0.10
0 .10
0 .05
0 .0 5
0 .0 0
0 .0 0
-0 .05
-0 .0 5
-0.10
-0 .10
-0 .15
-0.15
-0 .2 0
-0 .2 0
-0 .25
-0 .2 5 -0 .3 0 0.01
-0 .3 0 0 .0 1
0 .06
0.11
0 .16
0 .2 1
0.2 6
0.31
0 .3 6
0 .4 1
0.4 6
0 .51
0 .56
-k
0 .6 1
0 .6 6
r(1,m)
0 .0 6
0 .11
0 .16
0.21
0 .2 6
k
LSE Two-Tailed First-Order Tail Dependence Coefficient Median = .18 ± .05
0 .4 0
0.3 1
0 .3 6
0.4 1
0 .4 6
0 .51
0 .56
0 .6 1
-k
mnth quantile
m/nth quantile
0 .6 6
r(1,m)
k
NIKKEI Two-Tailed First-Order Tail Dependence Coefficient Median = .02 ± .04 0 .15
0 .3 5 0 .3 0
0 .10
0 .2 5 0 .2 0 0 .0 5
0 .15 0 .10 0 .0 5
0 .0 0
0 .0 0 -0 .0 5 -0 .10
-0 .0 5
-0 .15 -0 .2 0
-0 .10
-0 .2 5 -0 .3 0 0 .0 1
0 .0 5
0 .0 9
0.13
0 .17
0 .2 1
0 .2 5
0 .2 9
0.33
0 .3 7
0 .4 1
0 .4 5
-k
0 .4 9
-0 .15 0 .0 1
0.53
r(1,m)
0 .0 5
0 .0 8
0 .12
0 .16
0 .19
0 .2 6
0 .3 0
0 .3 3
0 .3 7
0 .4 1
-k
0 .4 4
0 .4 8
r(1,m)
k
m/nth quantile
EURO/US D Two-Tailed Firs t-Order Tail Dependence Coefficient Median = -.020 ± .04
0 .10
0 .2 3
k
m/nth quantile
Pound Two-Tailed First-Order Tail Dependence Coefficient Median = .01 ± .04
0 .10
0 .0 8
0 .0 8
0 .0 6
0 .0 5 0 .0 4
0 .0 3
0 .0 2 0 .0 0
0 .0 0
-0 .0 2
-0 .0 3
-0 .0 4
-0 .0 5 -0 .0 6
-0 .0 8
-0 .0 8
-0 .10
-0 .10 0 .0 1
0 .0 4
0 .0 7
0 .10
0 .14
0 .17
0 .2 0
0 .2 3
0 .2 6
m/nth quantile
0 .3 0
0 .3 3
0 .3 6
0 .3 9
0 .0 1
0 .4 2
0 .0 5
0 .0 8
0 .12
0 .15
0 .19
0 .2 2
0 .2 6
0 .2 9
0 .3 3
0 .3 6
-k -k
r(1,m)
k
16
m/nth quantile
0 .4 0
0 .4 3
r(1,m)
0 .4 7
k
1 Figure 7.2: Two-Tailed Serial med{^ § } SP 5 0 0 M e d i a n T w o T a ile d Se r ia l
N A SD A Q M e d i a n T w o T a ile d Se r i a l
T a il D e p e n d e n c e C o e f f ic ie n t a n d 9 5 % B a n d
T a il D e p e n d e n c e C o e f f ic i e n t a n d 9 5 % B a n d
0 .1 5
0 .2 0 0 .1 5
0 .1 0
0 .1 0
0 .0 5
0 .0 5 0 .0 0
0 .0 0
- 0 .0 5
- 0 .0 5
- 0 .1 0 1
11
21
31
41
51
61
71
81
1
91
11
21
31
41
-k r ( h ,0 ,0 ) k
h
51
61
71
81
91
-k r( h ,0 ,0 ) k
h
N I K K I E M e d i a n T w o T a i le d Se r i a l
L SE M e d ia n T w o T a i le d Se r i a l
T a il D e p e n d e n c e C o e f f ic ie n t a n d 9 5 % B a n d
T a il D e p e n d e n c e C o e f f ic ie n t a n d 9 5 % B a n d 0 .2 0
0 .0 8
0 .1 5
0 .0 6 0 .0 4 0 .0 2
0 .1 0 0 .0 5
0 .0 0
0 .0 0
- 0 .0 2 - 0 .0 4 - 0 .0 6
- 0 .0 5 - 0 .1 0 1
11
21
31
41
51
61
71
81
1
91
11
21
31
41
-k r ( h ,0 ,0 ) k
h
51
61
71
81
91
-k r( h ,0 ,0 ) k
h
E U R O M e dia n T wo T a ile d Se r ia l
Y E N M e d i a n T w o T a il e d Se r i a l
T a il D e p e n de n c e C o e f f ic ie n t a n d 9 5 % B a n d
T a il D e p e n d e n c e C o e f f i c ie n t a n d 9 5 % B a n d 0 .0 6
0 .0 6
0 .0 4
0 .0 4
0 .0 2
0 .0 2
0 .0 0
0 .0 0
- 0 .0 2
- 0 .0 2 - 0 .0 4
- 0 .0 4
- 0 .0 6
- 0 .0 6 1
11
21
31
41
51
61
71
81
1
91
11
21
31
h
41
51 h
-k r(h ,0 ,0 ) k
61
71
81
91
-k r ( h ,0 , 0 ) k
B P M e d i a n T w o T a il e d Se r ia l T a il D e p e n d e n c e C o e f f ic ie n t a n d 9 5 % B a n d 0 .0 6 0 .0 4 0 .0 2 0 .0 0 - 0 .0 2 - 0 .0 4 - 0 .0 6 1
11
21
31
41
51 h
61
71
81
91
-k r ( h ,0 ,0 ) k
1 Note: ( ) denotes t he tail event bas ed coe¢cient fo r tail-o f ¡ and tail-o f , 2 f0 1 2g, where 0 denot es two -t ails, 1 deno tes left tail, and 2 denotes r ight tail.
17
Figure 8: Equity-to-Equity Rolling Fractile Plots Two-Tailed ^(1) § NASDAQ(t-1) --> SP500(t) Firs t Order Two-Tailed Extremal r (1,m) Spillover Median = .17 ± .08
0 .3 0
NASDAQ(t-1) --> LSE(t) First Order Two-Tailed Extremal r (1,m) Spillover Median = .12 ± .05
0 .3 0
0 .2 5
0 .2 5
0 .2 0
0 .2 0
0 .15
0 .15
0 .10
0 .10
0 .0 5
0 .0 5
0 .2 5 0 .2 0 0 .15 0 .10
0 .0 0
0 .0 0
-0 .0 5
-0 .0 5
-0 .10
-0 .10
0 .0 5 0 .0 0 -0 .0 5 -0 .10
-0 .15
-0 .15
-0 .15
-0 .2 0
-0 .2 0
-0 .2 0
-0 .2 5
-0 .2 5
-0 .3 0
-0 .2 5
-0 .3 0
0 .0 1 0 .0 6
0 .11
0 .16
0 .2 1 0 .2 6
0 .3 1
0 .3 6
0 .4 1
0 .4 6
m/nth quantile
0 .51
0 .56
-k
0 .6 1 0 .6 6
r(1,m)
NASDAQ(t-1) --> Nikkei(t) First Order Two-Tailed Extremal r(1,m) Spillover Median = .05± .04
0 .3 0
-0 .3 0
0 .0 1
0 .0 6
0 .11
0 .16
0 .2 2
0 .2 7
0 .3 2
0 .3 7
0 .4 2
0 .4 7
0 .52
m/nth quantile
k
SP500(t-1) --> NASDAQ(t) First Order Two-Tailed Extremal Volatility r(1,m) Spillover Median = .093 ± .08
0 .57
-k
0 .6 2
0 .6 7
0 .0 1
r(1,m)
0.30
0 .25
0.12
0.25
0.2 0
0.20
0.05
0 .05
0.00
0.0 0
-0.05
-0 .05
-0.10
-0.08
0 .2 8
0 .3 3
0 .3 8
0 .4 4
0 .4 9
0 .54
0 .6 0
-k
0 .6 5
0 .70
r(1 ,m)
k
0.10
0.10
-0.04
0 .2 2
0.15
0.15
0.00
0 .17
SP500(t-1) --> Nikkei(t) First Order Two-Tailed Extremal Volatility r(1,m) Spillover Median = .08± .054
0.16
0.04
0 .12
m/nth quantile
SP500(t-1) --> LSE(t) First Order Two-Tailed Extremal Volatility r(1,m) Spillover Median = .14± .05
0.08
0 .0 7
k
-0.10
-0.15 -0.15
-0.20
-0.12 -0.16 0.01
0.06
0.11
0.16
0 .21
0 .26
0.31
0 .36
0.4 1
0.46
m/nth quantile
0 .2 5
0.51
0.56
-k
0.61
0.66
r(1,m)
-0.25
-0.2 0
-0.30
-0 .25 0.01
0 .06
0.11
0.16
0 .22
0.27
0.32
0 .37
0 .42
0.47
m/nth quantile
k
LS E(t-1) --> NASDAQ(t) First Order Two-Tailed Extremal r(1,m) Spillover Median = .09 ± .04
0.52
0.57
-k
0.62
0.6 7
r(1,m)
0 .01
0.12
0.17
0 .22
0.2 8
0.33
0.3 8
0.44
0 .49
0 .54
m/nth quantile
LSE(t-1) --> SP500(t) First Order Two-Tailed Extremal r(1,m) Spillover Median = .10 ± .04
0 .3 5
0.07
k
-k
0.6 0
0.6 5
r(1,m)
0.70
k
LS E(t-1) --> Nik k ei(t) First Order Two-Tailed Extremal r (1,m) Spillover Median = .10 ± .04
0 .15
0 .3 0
0 .2 0
0 .2 5
0 .10
0 .2 0
0 .15
0 .15
0 .10
0 .0 5
0 .10 0 .0 5
0 .0 5 0 .0 0
0 .0 0
0 .0 0
-0 .0 5 -0 .0 5
-0 .10
-0 .10
-0 .15
-0 .15
-0 .2 0
-0 .0 5
-0 .10
-0 .2 5 -0 .2 0 -0 .2 5 0 .0 1
-0 .3 0 0 .0 6
0 .11
0 .16
0 .2 2
0 .2 7
0 .3 2
0 .3 7
0 .4 2
m/nth quantile
0 .15
0 .4 7
0 .52
-k
0 .57
0 .6 2
r(1,m)
0 .6 7
-0 .3 5 0 .0 1
-0 .15
0 .0 6
0 .11
0 .16
0 .2 2
Nik kei(t-1) --> NAS DAQ(t) Firs t Order Two-Tailed Extremal r(1,m) S pillover Median = .01 ± .04
0 .2 7
0 .3 2
0 .3 7
0 .4 2
m/nth quantile
k
0 .4 7
0 .52
-k
0 .57
0 .6 2
0 .6 7
r(1,m)
0 .0 1
Nik kei(t-1) --> S P500(t) First Order Two-Tailed Extremal r (1,m) S pillover Median = .01 ± .04
0 .10
0 .0 6
0 .12
0 .17
0 .2 2
0 .2 7
0 .3 3
0 .3 8
0 .4 3
m/nth quantile
k
0 .2 0
0 .4 8
0 .54
-k
0 .59
0 .6 4
r(1 ,m )
0 .6 9
k
Nik k ei(t-1) --> NLS E(t) Firs t Order Two-Tailed Extremal r (1 ,m) S pillover Median = .01 ± .0 4
0 .13 0 .0 8
0 .10 0 .0 8
0 .15
0 .0 5
0 .0 5
0 .10
0 .0 3
0 .0 5
0 .0 3 0 .0 0 -0 .0 3
0 .0 0
0 .0 0
-0 .0 3
-0 .0 5
-0 .0 5
-0 .10
-0 .0 5 -0 .0 8 -0 .10 -0 .0 8
-0 .13 -0 .15 0 .0 1
-0 .15
-0 .10
0 .0 7
0 .12
0 .17
0 .2 2
0 .2 8
0 .3 3
0 .3 8
0 .4 4
m/nth quantile
0 .4 9
-k
0 .54
0 .6 0
0 .6 5
r(1,m)
0 .70
-0 .2 0
0 .0 1
0 .0 7
0 .12
0 .17
0 .2 2
0 .2 8
0 .3 3
0 .3 8
0 .4 4
m/nth quantile
k
18
0 .4 9
-k
0 .54
0 .6 0
0 .6 5
r(1,m)
0 .70
k
0 .0 1
0 .0 7
0 .12
0 .17
0 .2 2
0 .2 8
0 .3 3
0 .3 8
0 .4 4
m/nth quantile
0 .4 9
-k
0 .54
0 .6 0
0 .6 5
r(1 ,m )
0 .70
k
Figure 9: Equity-to-Equity Two-Tailed med{^ § } NASDAQ --> SP500 Median Two Tailed Extremal Volatility r(h,0,0) Spillover
NASDAQ --> NIKKEI
NASDAQ --> LSE Median T wo T ailed Extremal
Median Two Tailed Extremal Volatility r(h,0,0) Spillover
Volatility r(h,0,0) Spillover
0 .15
0 .0 8
0 .14 0 .12
0.10
0 .0 6
0 .10 0 .0 4
0 .0 8
0 .05
0 .0 6
0 .0 2
0 .0 4
0.0 0
0 .0 0
0 .0 2 0 .0 0
-0 .05
-0 .0 2
-0 .0 2
-0 .0 4
-0 .0 4
-0.10 1
6
11
16 2 1 26 31 3 6 41 4 6
51 56 6 1 66 71 76 h
81 8 6 91 9 6
-0 .0 6
-0 .0 6 1
6
11
16 2 1 2 6
31 3 6 4 1 4 6
51 56
6 1 6 6 71 76
h
-k r(h,0,0 ) k
81 86 91 96
1
6
11
81 86 91 96 -k r(h,0 ,0 ) k
SP500 --> NIKKEI Median Two Tailed Extremal Volatility r(h,0,0) Spillover
0.15
0 .10
51 56 6 1 6 6 71 76 h
SP500 --> LSE Median Two Tailed Extremal Volatility r(h,0,0) Spillover
SP500 --> NASDAQ Median Two T ailed Extremal Volatility r(h,0,0) Spillover
16 2 1 2 6 3 1 3 6 4 1 4 6
-k r(h,0 ,0 ) k
0 .10 0 .08
0 .0 8
0.10 0 .06
0 .0 6 0 .0 4
0 .04
0.05
0 .02
0 .0 2 0.00
0 .0 0 -0 .0 2
0 .00 -0 .02
-0.05
-0 .0 4
-0 .04
-0 .0 6
-0.10
1
6
11 16
21 26 31 36 41 46
51 56
-0 .06
61 66 71 76 8 1 8 6 9 1 9 6
h
1
6
11
16 21 26 31 36 41 46
51 56
6 1 66
71 76 81 86 91 9 6
h
-k r(h,0 ,0) k
LSE --> NASDAQ Median T wo T ailed Ext remal Volatility r(h,0,0) Spillover
6
11
0 .10
0 .14 0 .12
0 .0 6
0 .0 2
0 .0 4
0 .0 0
0 .0 2
-0 .0 4
-0 .0 6
-0 .0 6
1 6 11 16 2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6 71 76 8 1 8 6 9 1 9 6 -k h r(h,0 ,0 ) k
0 .0 0
c
-0 .0 2 -0 .0 4 -0 .0 6
1
6
11 16 2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6 71 76 8 1 8 6 9 1 9 6 -k h r(h,0 ,0 )
1
6
k
NIKKEI --> LSE Median Two Tailed Extremal Volatility r(h,0,0) Spillover
NIKKEI --> SP500 Median Two Tailed Extremal Volatility r(h,0,0) Spillover 0.05 0.04 0.03 0.02 0.01 0.00 -0.01 -0.02 -0.03 -0.04 -0.05
0.04 0.02 0.00 -0.02 -0.04 -0.06 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96
11 16 2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6 71 76 8 1 8 6 9 1 9 6 -k h r(h,0 ,0 )
k
NIKKEI --> NASDAQ Median Two Tailed Extremal Volatility r(h,0,0) Spillover 0.06
91 96
-k r(h,0 ,0 ) k
0 .0 2
-0 .0 2
-0 .0 4
76 8 1 8 6
0 .0 4
0 .0 0
-0 .0 2
6 1 6 6 71
0 .0 6
0 .0 8
0 .0 4
51 56
LSE --> NIKKEI Median T wo T ailed Ext remal Volat ilit y r(h,0,0) Spillover
0 .10
0 .0 6
16 2 1 2 6 3 1 3 6 4 1 4 6
h
LSE --> SP 500 Median T wo T ailed Ext remal Volatility r(h,0,0) Spillover
0 .0 8
h
1
-k r(h,0,0) k
0.06 0.04 0.02 0.00 -0.02 -0.04 -0.06
1
6
11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 h
-k r(h,0,0) k
19
-k r(h,0,0) k
1
6
11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 h
-k r(h,0,0) k
Figure 10: Equity-to-Equity Cross-Tailed med{^ § } NASDAQ
NASDAQ --> SP 500 Median Cross-T ailed Ext remal Volat ility r(h,1,2) Spillover
NASDAQ --> SP 500 Median Cross-T ailed Ext remal Volat ilit y r(h,2,1) Spillover
0 .15
0 .0 8 0 .0 6
0 .10
0 .0 4 0 .0 2
0 .0 5
0 .0 0 -0 .0 2
0 .0 0
-0 .0 4 -0 .0 6
-0 .0 5
-0 .0 8
-0 .10
-0 .10 -0 .12
-0 .15
-0 .14
1
6
11
16
21 26 31 36 41 46
51 56
61 66
71 76
81 86 91 96
h
1
6
11 16
2 1 2 6 3 1 3 6 4 1 4 6 51
56
61 66
71
76
81 86
h
-k
91 96
-k
r(h,1,2 )
r(h,2 ,1)
k
k
NASDAQ --> LSE Median Cross-T ailed Ext remal Volatilit y r(h,2,1) Spillover
NASDAQ --> LSE Median Cross-T ailed Extrem al Volatilit y r(h,1,2) Spillover 0 .10
0 .10
0 .0 5
0 .0 5
0 .0 0 0 .0 0 -0 .0 5 -0 .0 5 -0 .10 -0 .10
-0 .15 -0 .2 0
-0 .15 1
6
11 16
21 26 31 36 41 46
51 56
61 66
71
76
81 86
91 96
1
6
11 16
21
26
31 36
41
46
h
51 56
61 66
71
76
r(h,2 ,1) k
NASDAQ --> NIKKEI Median Cross-T ailed Ext remal Volat ilit y r(h,2,1) Spillover
0 .10
0 .10
0 .0 5
0 .0 5
0 .0 0
0 .0 0
-0 .0 5
-0 .0 5
-0 .10
-0 .10
-0 .15
-0 .15
-0 .2 0
-0 .2 0
11
16
21 26
31 36
41
46
51 h
91 96
r(h,1,2 )
NASDAQ --> NIKKEI Median Cross-T ailed Extremal Volatilit y r(h,1,2) Spillover
6
86 -k
k
1
81
h -k
56
61
66
71 76
81
86
91
96
1
6
11
16
21 26
31 36
41 46
51 h
-k
56 6 1
66
71
76
81 86 -k
r(h,1,2 )
r(h,2 ,1)
k
k
20
91 96
SP500 SP 500 --> NASDAQ Median Cross-T ailed Ext remal Volat ilit y r(h,1,2) Spillov er
SP 500 --> NASDAQ Median Cross-T ailed Ext remal Volatility r(h,1 ,2) Spillover
0 .12
0 .12
0 .10
0 .10
0 .0 8
0 .0 8
0 .0 6
0 .0 6
0 .0 4
0 .0 4
0 .0 2
0 .0 2
0 .0 0
0 .0 0
-0 .0 2
-0 .0 2
-0 .0 4
-0 .0 4 -0 .0 6
-0 .0 6 -0 .0 8
-0 .0 8 1
6
11 16
21 26
31 36 41 46
51
56
61 66
71
76
h
81 86
91 96
1
6
11
16
21 26 31 36 41 46
51
56
6 1 6 6 71
76
h
-k
81 86 91 96 -k
r(h,1,2 )
r(h,1,2 )
k
k
SP 500 --> LSE Median Cross-T ailed Ext rem al Volatility r(h,1,2) Spillover
SP 500 --> LSE Median Cross-T ailed Extremal Volatilit y r(h,2,1) Spillover
0 .15
0 .10
0 .10
0 .0 5
0 .0 5
0 .0 0
0 .0 0
-0 .0 5
-0 .0 5
-0 .10
-0 .10
-0 .15
-0 .15
-0 .2 0
1
6
11
16
21 26
31 36
41 46
51 56
61 66
71
76
81 86
h
91 96
1
6
11
16
21 26
31 36
41
46
51
56
61
66
71 76
h
-k
81
86
91
96
-k
r(h,1,2 )
r(h,2 ,1)
k
k
SP 500 --> NIKKEI Median Cross-T ailed Extremal Volatility r(h,2,1) Spillover
SP 500 --> NIKKEI Median Cross-T ailed Ext remal Volat ilit y r(h,1,2) Spillover 0 .10
0 .10
0 .0 5
0 .0 5 0 .0 0
0 .0 0
-0 .0 5
-0 .0 5
-0 .10
-0 .10 -0 .15
-0 .15
-0 .2 0
-0 .2 0 1
6
11 16
21
26
31
36
41
46
51
56
61 66
71 76
81 86
1
91 96
6
11
16
21 26 31 36
41 46
51
56
61 66
71 76
81 86
h
h -k
-k
r(h,1,2 )
r(h,2 ,1) k
k
21
91 96
LSE L SE - -> N A SD A Q
L SE -- > N A SD A Q
M edian Cr o ss-T ailed E x t r em a l Vo lat ilit y r( h ,1 ,2 ) Sp illo v er
M edian Cro ss- T a iled E x t rem al Vo lat ilit y r (h ,2 ,1 ) Sp illo v er
0 .12
0 .0 8
0 .10
0 .0 6
0 .0 8 0 .0 4
0 .0 6
0 .0 2
0 .0 4 0 .0 2
0 .0 0
0 .0 0
- 0 .0 2
- 0 .0 2
- 0 .0 4
- 0 .0 4
- 0 .0 6
- 0 .0 6 - 0 .0 8
- 0 .0 8
1
6
11 16 2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6 71 76 8 1 8 6 9 1 9 6 -k h r ( h,1,2 )
1
6
11 16 2 1 2 6 3 1 3 6 4 1 4 6
51 56 6 1 6 6 71 76 8 1 8 6 9 1 9 6 -k
h
r ( h,2 ,1)
k
k
L SE --> SP 5 0 0 M edian Cr o ss-T ailed E x t rem al Vo lat ilit y r (h ,1 ,2 ) Sp illo v er
L SE -- > SP 5 0 0 M edian Cr o ss-T ailed E x t rem al Vo lat ilit y r (h ,2 ,1 ) Sp illo v er
0 .12
0 .0 8
0 .10
0 .0 6
0 .0 8
0 .0 4
0 .0 6
0 .0 2
0 .0 4
0 .0 0
0 .0 2
- 0 .0 2
0 .0 0 - 0 .0 4
- 0 .0 2
- 0 .0 6
- 0 .0 4 - 0 .0 6
- 0 .0 8
- 0 .0 8
- 0 .10
1
6
11 16 2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6 71 76 8 1 8 6 9 1 9 6 -k h r( h ,1,2 ) k
1
6
11 16 2 1 2 6 3 1 3 6 4 1 4 6 51 5 6 6 1 6 6 h
L SE --> N I K K E I
71 76 8 1 8 6 9 1 9 6 -k r ( h,2 ,1) k
L SE --> N IK K E I M e dia n Cro ss-T a ile d E x t re m al Vo la t ilit y r(h ,2 ,1 ) Sp illo v e r
M edian Cro ss-T ailed E x t r em al Vo lat ilit y r( h ,1 ,2 ) Sp illo v er 0 .0 8
0 .10
0 .0 6 0 .0 4 0 .0 2
0 .0 5
0 .0 0
0 .0 0
- 0 .0 2
c
- 0 .0 4
c -0 .0 5
- 0 .0 6 - 0 .0 8 - 0 .10
- 0 .10
- 0 .12 - 0 .14
- 0 .15
1
6
11 16 2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6 71 76 8 1 8 6 9 1 9 6 -k h r (h ,1,2 ) k
1
22
6
11 16 2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6 71 76 8 1 8 6 9 1 9 6 -k h r( h,2 ,1) k
Nikkei NIKKEI --> NASDAQ
NIKKEI --> NASDAQ Median Cross-T ailed Extremal
Median Cross-T ailed Extremal Volatility r(h,1,2) Spillover
Volatility r(h,1,2) Spillover 0 .0 8
0 .08
0 .0 6
0 .06
0 .0 4
0 .04
0 .0 2
0 .02
0 .0 0
0 .00
-0 .0 2
-0 .02
-0 .0 4
-0 .04 -0 .06
-0 .0 6
-0 .08
-0 .0 8 1
6
1
11 16 2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6 71 76 8 1 8 6 9 1 9 6 h
6
11 16 2 1 2 6 3 1 3 6 41 4 6 51 56 6 1 6 6 71 76 8 1 86 9 1 9 6 h
-k
-k
r(h,1,2 )
r(h,2 ,1)
k
k
NIKKEI --> SP500
NIKKEI --> SP500
Median Cross-T ailed Extremal Volatility r(h,1,2) Spillover
Median Cross-Tailed Extremal Volatility r(h,2,1) Spillover
0 .0 8
0 .0 8
0 .0 6
0 .0 6
0 .0 4
0 .0 4
0 .0 2
0 .0 2
0 .0 0
0 .0 0
-0 .0 2
-0 .0 2
-0 .0 4
-0 .0 4 -0 .0 6
-0 .0 6
-0 .0 8
-0 .0 8 1
6
1
11 16 2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6 71 76 8 1 8 6 9 1 9 6 h
6
11 16 2 1 2 6 3 1 3 6 41 4 6 51 56 6 1 6 6 71 76 h
-k
81 86 91 96 -k
r(h,1,2 )
r(h,2 ,1)
k
k
NIKKEI --> LSE
NIKKEI --> LSE
Median Cross-T ailed Extremal Volatility r(h,1,2) Spillover
Median Cross-T ailed Ext remal Volat ilit y r(h,2,1) Spillover
0 .0 8
0 .0 8
0 .0 6
0 .0 6
0 .0 4
0 .0 4 0 .0 2
0 .0 2
0 .0 0
0 .0 0
-0 .0 2
-0 .0 2
-0 .0 4
-0 .0 4
-0 .0 6
-0 .0 6
-0 .0 8
-0 .0 8
-0 .10
1
6
11 16 2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6 71 76 8 1 8 6 9 1 9 6 h
1
6
11
16
21 26
31 36 41 46
51 56 6 1 6 6 h
-k r(h,1,2 ) k
71 76 8 1 8 6 9 1 9 6 -k r(h,2 ,1) k
23
Figure 11: FX-to-FX BP --> YEN Median T wo T ailed Extremal Volatility r(h,0,0) Spillover
YEN --> BP Median Two Tailed Extremal Volatility r(h,0,0) Spillover
0 .0 5
0.05 0.04 0.03 0.02 0.01 0.00 -0.01 -0.02 -0.03 -0.04 -0.05
0 .0 4 0 .0 3 0 .0 2 0 .0 1 0 .0 0 -0 .0 1 -0 .0 2 -0 .0 3 -0 .0 4
1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 h
-0 .0 5 1
-k r(h,0,0) k
6
11 16 2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6 71 76 8 1 8 6 9 1 9 6 -k h r(h,0 ,0 ) k
BP --> YEN Median Cross-T ailed Extremal Volatility r(h,1,2) Spillover
YEN --> BP Median Cross-Tailed Extremal Volatility r(h,1,2) Spillover
0 .0 8
0 .08 0 .06
0 .0 6
0 .04
0 .0 4
0 .02
0 .0 2
0 .00
0 .0 0
-0 .02 -0 .0 2
-0 .04
-0 .0 4
-0 .06
-0 .0 6
-0 .08 1
6 11 16 21 26 31 36 41 46 51 56 6 1 66 71 76 81 86 91 9 6 h
-0 .0 8 1
-k
6
r(h,1,2) k
11 16 2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6 71 76 8 1 8 6 9 1 9 6 -k h r(h,1,2 ) k
YEN --> BP
BP --> YEN Median Cross-T ailed Ext remal Volat ilit y r(h,2,1) Spillover
Median Cross-Tailed Extremal Volatility r(h,2,1) Spillover 0.06
0 .0 6
0.04
0 .0 4
0.02
0 .0 2
0.00
0 .0 0
-0.02 -0 .0 2
-0.04 -0 .0 4
-0.06 1
6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 h
-0 .0 6 1
-k r(h,2,1) k
6
11 16 2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6 71 76 8 1 8 6 9 1 9 6 -k h r(h,2 ,1) k
24
Figure 12: FX-to-Equity BP --> NA SDA Q M edian Cro ss-T ailed Ex t rem al
BP --> N A SD A Q M edian T wo T ailed E x t rem al Vo lat ilit y r(h ,0 ,0 ) Sp illo v er
BP --> NASDAQ M edian Cro ss-T ailed E x t rem al Vo lat ilit y r(h ,2 ,1 ) Sp illo v er
Vo lat ilit y r(h ,1 ,2 ) Sp illo v er
0 .0 6
0 .0 6
0 .0 6
0 .0 4
0 .0 4
0 .0 4
0 .0 2
0 .0 2
0 .0 2 0 .0 0
0 .0 0
0 .0 0
-0 .0 2
- 0 .0 2
- 0 .0 2 - 0 .0 4
-0 .0 4
- 0 .0 6
- 0 .0 4
-0 .0 6
- 0 .0 8 1
6
- 0 .0 6
1
11 16 2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6 71 76 8 1 8 6 9 1 9 6 -k h r ( h,0 ,0 )
6
11 16
2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6 71 76 8 1 8 6 9 1 9 6
1
6
-k
h
r(h,1,2 )
k
11 16 2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6 71 76 8 1 8 6 9 1 9 6 -k h r ( h,2 ,1) k
k
BP --> SP 5 0 0
BP --> SP 5 0 0 M edian Cro ss-T ailed E x t rem al Vo lat ilit y r(h ,1 ,2 ) Sp illo v er
M edian T wo T ailed E x t rem al Vo lat ilit y r(h ,0 ,0 ) Sp illo v er 0 .0 6 0 .0 4
BP --> SP 5 0 0 M edian Cro ss-T ailed E x t rem al Vo lat ilit y r(h,2 ,1 ) Sp illo v er
0 .0 6
0 .0 6
0 .0 4
0 .0 4
0 .0 2
0 .0 2
0 .0 2 0 .0 0 0 .0 0
0 .0 0
- 0 .0 2
-0 .0 2
-0 .0 6
- 0 .0 4
-0 .0 4
-0 .0 8
- 0 .0 6
-0 .0 2 -0 .0 4
1
6
11 16 2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6 71 76 h
8186 9196 -k
-0 .0 6
1
6
r(h,0 ,0 )
11 16 2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6 71 76 8 1 8 6 9 1 9 6 -k h r (h,1,2 )
k
0 .0 6 0 .0 4 0 .0 2
BP --> L SE M edian Cro ss-T ailed E x t rem al Volat ilit y r(h ,2 ,1 ) Sp illo v er
M edian Cr o ss-T ailed E x t rem al Vo lat ilit y r (h ,1 ,2 ) Sp illo v er 0 .0 8
0 .0 8
0 .0 6
0 .0 6
0 .0 4
0 .0 4
0 .0 0
0 .0 0
-0 .0 2
-0 .0 2
0 .0 2 0 .0 0 -0 .0 2
-0 .0 4
-0 .0 4
-0 .0 6
-0 .0 6
-0 .0 6
-0 .0 8
-0 .0 8
-0 .10
-0 .0 8 1
6
11 16
21 26 31 36 41 46
11 16 2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6 71 76 8 1 8 6 9 1 9 6 -k h r (h,2 ,1) k
0 .0 2
-0 .0 4
6
BP --> L SE
BP - -> L SE M edian T wo T ailed E x t r em al Vo lat ilit y r( h ,0 ,0 ) Sp illo v er
1
k
1
51 5 6 6 1 6 6 71 7 6 8 1 8 6 9 1 9 6 -k h r (h ,0 ,0 ) k
6 11 16 2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6 h
1
71 76 8 1 8 6 9 1 9 6 -k r( h,1,2 ) k
BP - -> N IK K E I
6 11 16 2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6 71 76 8 1 8 6 9 1 9 6 -k h r(h,2 ,1) k
BP --> N IK K E I M edian Cro ss-T ailed E x t re m al
M e dia n T wo T ailed E x t rem a l Vo la t ilit y r(h ,0 ,0 ) Sp illo v er
BP --> N IK K E I M edian Cro ss-T ailed E x t rem al Vo lat ilit y r(h ,2 ,1 ) Sp illo v er
Vo lat ilit y r( h ,1 ,2 ) Sp illo v er
0 .0 5
0 .0 8
0 .0 6
0 .0 4
0 .0 6
0 .0 4
0 .0 3 0 .0 2
0 .0 4
0 .0 2
0 .0 2
0 .0 1
0 .0 0
0 .0 0
0 .0 0 -0 .0 1
-0 .0 2
- 0 .0 2
-0 .0 2
-0 .0 4
- 0 .0 4
-0 .0 3 -0 .0 4
-0 .0 6
- 0 .0 6 - 0 .0 8
-0 .0 5 1
6
11 16 2 1 2 6 3 1 3 6 4 1 4 6
51 56 6 1 6 6 7 1 7 6 8 1 8 6 9 1 9 6 -k h r(h ,0 ,0 ) k
-0 .0 8
1
6 11 16
25
2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6 71 76 8 1 8 6 9 1 9 6 -k h r(h,1,2 ) k
1
6
11 16 2 1 2 6 3 1 3 6 4 1 4 6
51 56 h
6 1 6 6 71 76 8 1 8 6 9 1 9 6 -k r(h,2 ,1) k
Figure 12: FX-to-Equity - Continued YEN --> NASDAQ Median T wo T ailed Extremal Volatilit y r(h,0,0) Spillover
YEN --> NASDAQ Median Cross-T ailed Extremal Volatility r(h,1,2) Spillover
0 .0 6 0 .0 5 0 .0 4 0 .0 3 0 .0 2 0 .0 1 0 .0 0 -0 .0 1 -0 .0 2 -0 .0 3 -0 .0 4 -0 .0 5
YEN --> NASDAQ Median Cross-T ailed Extremal Volat ility r(h,2,1) Spillover
0 .0 8
0 .0 8
0 .0 6
0 .0 6
0 .0 4
0 .0 4
0 .0 2
0 .0 2
0 .0 0
0 .0 0
-0 .0 2
-0 .0 2
-0 .0 4
-0 .0 4
-0 .0 6
-0 .0 6
-0 .0 8
1
6
11 16 2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6
-0 .0 8
71 76 8 1 8 6 9 1 9 6
h
1
6
11 16 2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6 71 76 8 1 8 6 9 1 9 6 h
-k r(h,0 ,0 ) k
1
6 11 16
51 56 6 1 6 6
71 76 8 1 8 6 9 1 9 6
h
-k r(h,2 ,1)
k
YEN --> SP 500 Median T wo T ailed Ext remal Volat ilit y r(h,0,0) Spillover
k
YEN --> SP 500 Median Cross-T ailed Ext remal Volat ilit y r(h,1,2) Spillover
0 .0 5 0 .0 4 0 .0 3 0 .0 2 0 .0 1 0 .0 0 -0 .0 1 -0 .0 2 -0 .0 3 -0 .0 4 -0 .0 5
21 26 31 36 41 46
-k r(h,1,2 )
YEN --> SP 500 Median Cross-T ailed Ext remal Volat ilit y r(h,2,1) Spillover
0 .0 6
0 .0 8 0 .0 6
0 .0 4
0 .0 4
0 .0 2
0 .0 2
0 .0 0
0 .0 0 -0 .0 2
-0 .0 2
-0 .0 4
-0 .0 4
-0 .0 6
-0 .0 6
1
6
11 16
2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6
71 76 8 1 8 6 9 1 9 6
h
-0 .0 8
1
6
11 16 2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6 71 76 h
-k r(h,0 ,0 ) k
YEN --> LSE Median T wo T ailed Ext remal Volat ilit y r(h,0,0) Spillover
81 86 91 96
1
6
h
YEN --> LSE Median Cross-T ailed Ext remal Volat ilit y r(h,1,2) Spillover
0 .0 5 0 .0 4 0 .0 3 0 .0 2
11 16 2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6 71 76
-k r(h,1,2 ) k
81 86 91 96 -k r(h,2 ,1) k
YEN --> LSE Median Cross-T ailed Ext remal Volat ilit y r(h,2,1) Spillover
0 .0 8
0 .0 6
0 .0 6
0 .0 4
0 .0 4 0 .0 2
0 .0 2
0 .0 1 0 .0 0 -0 .0 1 -0 .0 2 -0 .0 3 -0 .0 4 -0 .0 5
0 .0 0
0 .0 0 -0 .0 2
-0 .0 2
-0 .0 4 -0 .0 4
-0 .0 6 -0 .0 8
1
6
11 16
2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6 71 76 8 1 8 6 9 1 9 6 h
-0 .0 6 1
6 11 16 2 1 2 6 3 1 3 6 4 1 4 6 51 56
61 66
71 76 8 1 8 6 9 1 9 6
h
-k r(h,0 ,0 ) k
YEN --> NIKKEI Median T wo Tailed Extremal Volatility r(h,0,0) Spillover
1
YEN --> NIKKEI
YEN --> NIKKEI Median Cross-T ailed Extremal Volatility r(h,2,1) Spillover
0 .0 8
0 .0 8
0 .0 6
0 .0 6
0 .0 4
0 .0 4
0 .0 0 -0 .0 1 -0 .0 2 -0 .0 3
0 .0 0
0 .0 0
-0 .0 2
-0 .0 2
-0 .0 4
-0 .0 4
0 .0 2
0 .0 2
-0 .0 6
-0 .0 6
-0 .0 8
6
11 16
21 26 3 1 36
41 46
51 56 6 1 6 6 71 h
76 8 1 8 6 9 1 9 6
76 8 1 8 6 9 1 9 6 -k r(h,2 ,1) k
Median Cross-T ailed Extremal Volatility r(h,1,2) Spillover
0 .0 4 0 .0 3 0 .0 2 0 .0 1
1
11 16 2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6 71 h
0 .0 5
-0 .0 4 -0 .0 5
6
-k r(h,1,2 ) k
-0 .0 8
1
6
11 16 21 2 6 31 3 6 41 4 6 51 56 6 1 6 6 71 76 8 1 8 6 9 1 9 6 h
-k r(h,0 ,0 ) k
26
-k
1
6
11 16 2 1 2 6 3 1 3 6 4 1 4 6
51 56 h
61 66
71 76
81 86 -k
r(h,1,2 )
r(h,2 ,1)
k
k
91 96
Figure 13: Equity-to-FX NASDAQ --> YEN Median T wo T ailed Extremal Volatility r(h,0,0) Spillover
NASDAQ --> YEN Median Cross-T ailed Extremal Volatility r(h,1,2) Spillover
0 .0 5
NASDAQ --> YEN Median Cross-T ailed Extremal Volatility r(h,2,1) Spillover
0 .0 8
0 .0 4
0 .0 6
0 .0 6 0 .0 4
0 .0 3 0 .0 4
0 .0 2
0 .0 2
0 .0 2
0 .0 1 0 .0 0
0 .0 0
-0 .0 1
0 .0 0
-0 .0 2
-0 .0 2
-0 .0 2 -0 .0 4
-0 .0 3
-0 .0 4
-0 .0 6
-0 .0 4 -0 .0 5
-0 .0 8
1
6
11
16
21 26 31 36 41 46
51 56 6 1 6 6 71 76 8 1 8 6 h
91 9 6
-0 .0 6
1
6
11 16
21 26 31 36
41 46
51 56 6 1 6 6
71 76 8 1 8 6
h
-k
91 96
r(h,0 ,0 )
-k r(h,1,2 )
k
k
1
6
11
16
21 2 6
41 4 6
51 56 6 1 6 6
71 76 8 1 8 6
h
91 96
-k r(h,2 ,1) k
NASDAQ --> BP Median Cross-T ailed Ext remal Volat ilit y r(h,1,2) Spillover
NASDAQ --> BP Median T wo T ailed Ext remal Volat ilit y r(h,0,0) Spillover
31 3 6
NASDAQ --> BP Median Cross-T ailed Ext rem al Volat ilit y r(h,2,1) Spillover
0 .0 6
0 .0 6
0 .0 6
0 .0 4
0 .0 4
0 .0 4
0 .0 2
0 .0 2
0 .0 0
0 .0 0
-0 .0 2
-0 .0 2
-0 .0 4
-0 .0 4
0 .0 2 0 .0 0 -0 .0 2 -0 .0 4
-0 .0 6
-0 .0 6
-0 .0 6
1
6
11
16
21 26
31 36
41 46
51 56
61 66
71 76
81 86
91 96
-0 .0 8
1
6
11
16
21 26
31 36
41 46
h
51 56 6 1 6 6
71 76 8 1 8 6 9 1 9 6
1
6
11
16
21 26 31 36
41 46
51 56
h
-k r(h,0 ,0 )
-k r(h,1,2 )
k
k
0 .0 4 0 .0 3
76
81 86 91 96 -k r(h,2 ,1)
SP 500 --> YEN Median Cross-T ailed Extremal Volatility r(h,2,1) Spillover
Median Cross-T ailed Extremal Volatility r(h,1,2) Spillover
0 .0 5
71
k
SP 500 --> YEN
SP 500 --> YEN Median T wo T ailed Extremal Volatility r(h,0,0) Spillover
61 66
h
0 .0 8
0 .0 8
0 .0 6
0 .0 6
0 .0 4
0 .0 4
0 .0 2 0 .0 2
0 .0 1
0 .0 2
0 .0 0
0 .0 0
0 .0 0
-0 .0 1
-0 .0 2
-0 .0 2
-0 .0 4
-0 .0 4
-0 .0 2 -0 .0 3
-0 .0 6
-0 .0 4 -0 .0 5
-0 .0 6
-0 .0 8
1
6
11
16
21 26
31 36 4 1 46
51 56
61 66
71 76
h
-0 .0 8
1
81 86 9 1 9 6
6
11
16
2 1 26
31 36 41 46
51 56 6 1 6 6
71 76 8 1 8 6
1
6
11
16
51 56 6 1 6 6
71 76
81 86
91 96
-k r(h,2 ,1)
k
SP 500 --> BP Median T wo T ailed Ext remal Volat ilit y r(h,0,0) Spillover
k
SP500 --> BP Median Cross-T ailed Ext remal Volatilit y r(h,2,1) Spillover
SP 500 --> BP Median Cross-T ailed Ext remal Volat ilit y r(h,1,2) Spillover
0 .0 4
41 46
-k r(h,1,2 )
r(h,0 ,0 ) k
0 .0 6
21 26 31 36
h
h
-k
9 1 96
0 .0 8
0 .0 6
0 .0 6
0 .0 4
0 .0 4
0 .0 2
0 .0 2 0 .0 2
0 .0 0
0 .0 0
0 .0 0 -0 .0 2
-0 .0 2
-0 .0 2
-0 .0 4
-0 .0 4
-0 .0 4 -0 .0 6
-0 .0 6
-0 .0 6
-0 .0 8
1
6
11
16
21 2 6 3 1 3 6
41 46
51 56 h
6 1 66
71 76
81 86 91 96
1
6
11
16
2 1 26 3 1 3 6
4 1 46
51 56 h
-k r(h,0 ,0 ) k
27
6 1 66
71
76
81 86
91 96
1
6
11
16
21 26
31 36
41 46
51 56
61 66
71 76 8 1 8 6
h -k r(h,1,2 )
-k r(h,2 ,1)
k
k
91 96
Figure 13: Equity-to-FX - Continued LSE --> YEN M edian T wo T ailed E xt rem al Volat ilit y r(h,0,0 ) Sp illo ver
L SE --> YEN M edian Cross-T ailed Ex t rem al Vo lat ilit y r(h,1,2 ) Spillov er
0 .0 8
0 .0 5
0 .0 6
0 .0 6
0 .0 4 0 .0 3
0 .0 4
0 .0 4
0 .0 2
L SE --> YE N M edian Cro ss-T ailed Ex t rem al Vo lat ilit y r(h ,2 ,1 ) Sp illo v er
0 .0 2
0 .0 2
0 .0 1 0 .0 0
0 .0 0
0 .0 0
-0 .0 1
-0 .0 2
-0 .0 2 -0 .0 3
-0 .0 4
-0 .0 2 -0 .0 4
-0 .0 6
-0 .0 4
-0 .0 6
-0 .0 8
-0 .0 5 1
-0 .0 8
1
6 11 16 2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6 71 76 8 1 8 6 9 1 9 6 -k h r(h,0 ,0 ) k
6 11 16 2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6 71 76 8 1 8 6 9 1 9 6 -k h r( h,1,2 )
1
LSE --> BP M edian T wo T ailed E x t rem al Vo lat ilit y r(h ,0 ,0 ) Sp illo v er 0 .0 6
0 .0 8
0 .0 6
0 .0 4
0 .0 6
0 .0 4
0 .0 4
0 .0 2
0 .0 2
0 .0 0
51 56 6 1 6 6 71 76 8 1 8 6 9 1 9 6 -k r(h,2 ,1) k
h
L SE --> BP M edian Cro ss-T ailed E x t rem al Vo lat ilit y r(h ,2 ,1 ) Sp illo v er
L SE - -> BP M edian Cro ss-T ailed E x t rem al Vo lat ilit y r(h ,1 ,2 ) Sp illo v er
0 .0 2
6 11 16 2 1 2 6 3 1 3 6 4 1 4 6
k
0 .0 0
0 .0 0
-0 .0 2
-0 .0 2
- 0 .0 2
-0 .0 4
-0 .0 4
- 0 .0 4
-0 .0 6
-0 .0 6
- 0 .0 6
-0 .0 8
-0 .0 8
- 0 .0 8
1
6 11 16 2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6 71 76 8 1 8 6 9 1 9 6 -k h r(h,0 ,0 ) k
1
6
11 16 2 1 2 6 3 1 3 6 4 1 4 6
NIKKEI --> YEN Median T wo Tailed Extremal Volatility r(h,0,0) Spillover
-0.02 -0.03 -0.04 -0.05
Volatility r(h,2,1) Spillover
0 .0 8
0 .0 8
0 .0 6
0 .0 6
0 .0 4
0 .0 4
0 .0 2
0 .0 2
0 .0 0
0 .0 0
-0 .0 2
-0 .0 2
-0 .0 4
-0 .0 4 -0 .0 6
-0 .0 8
21 26 31 36 41 46 51 56
61 66 71 76
h
81 86 91 96
71 76 8 1 8 6 9 1 9 6 -k r( h,2 ,1) k
NIKKEI --> YEN Median Cross-T ailed Extremal
-0 .0 6
11 16
h
NIKKEI --> YEN
0.03 0.02 0.01 0.00 -0.01
6
11 16 2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6
Median T wo Tailed Extremal Volatility r(h,1,2) Spillover
0.05 0.04
1
1 6
51 56 6 1 6 6 71 76 8 1 8 6 9 1 9 6 -k h r(h,1,2 ) k
-0 .0 8
1
6
11 16 2 1 2 6 3 1 3 6 4 1 4 6 51 56 6 1 6 6
71 76
h
-k r(h,0,0) k
NASDAQ --> BP Median T wo T ailed Extremal Volatility r(h,0,0) Spillover
81 86 9 1 96
1
6
11 16
21 2 6
3 1 3 6 4 1 4 6 51 56
6 1 6 6 71 76 8 1 8 6 9 1 9 6
h
-k r(h,1,2 ) k
-k r(h,2 ,1) k
NASDAQ --> BP Median Cross-T ailed Extremal Volatility r(h,2,1) Spillover
NASDAQ --> BP Median Cross-T ailed Extremal Volatilit y r(h,1,2) Spillover
0 .0 6
0 .0 6
0 .0 6
0 .0 4
0 .0 4
0 .0 4
0 .0 2
0 .0 2
0 .0 2 0 .0 0
0 .0 0
0 .0 0 -0 .0 2
-0 .0 2
-0 .0 2 -0 .0 4
-0 .0 4
-0 .0 6
-0 .0 6
-0 .0 8
1
6
11
16
21 26
31 36
41 46
51 56 h
61 66
71 76 8 1 8 6 9 1 9 6
-0 .0 4 -0 .0 6 1
6
11
16
21 26
31 36
41 46
51 56 h
-k r(h,0 ,0 ) k
61 66
71 76
81 86 -k r(h,2 ,1) k
28
91 96
1
6
11
16
21
26 31 36
41 46
51 56 6 1 6 6 h
71
76 8 1 8 6 -k r(h,1,2 ) k
91 96
Figure 14: Rolling Windows : NASDAQ and LSE NASDAQ(t-1) --> LSE(t) Two Tailed Tail Dep endence r(1) In creas ing Rolling W in dows
LSE(t-1) --> NA SDA Q(t) Two Tailed Tail Dependence r(1) Increas ing Rolling W indows .06
.3 0
.04
.2 5
.0 5
.2 0 .0 0
.1 5
-.0 5 p-value
0 .3 5
0.45
0.55
0 .6 5
0.75
0.85
.60
.02
.40
-.04
.0 5
-.06
.0 0
-.08 0.25
.20 .10 end: Oct . 19, 04 0.35
0.45
0.65
0.75
0.85
.00 0.95
-k k
Sam ple Fract ion
r(1) p-value
LSE(t-1) --> NA SDA Q(t) Two T ailed T ail Dep en d en ce r(1) Fixed Ro lling W in d ows : .50n
.20
.25
.1 0
.8 0
.0 8
r(1) .20
r(1 )
.7 0
.0 6
.10
.6 0
.15 .05
st art : Oct . 12, 01 .10
end: Oct . 10, 03
r(1 ,0 ,0)
.0 4 p -va lu e
r(1 ,0 ,0 )
0.55
r(1) p-v alue
NASDA Q(t-1) --> LSE(t)
.00
st art : Nov. 26 , 01 end: No v. 21, 03
1
6
11
16
21
26
p - value
.1 0
-.0 8
.00 31
36
41 -k k
.0 0 1
46
6
11
21
26
31
W in do w
r(1 ) p- value
0 .9 0
r(1)
0 .50 0 .0 0
0 .4 0
-0 .0 2 0 .3 0
-0 .0 4 -0 .0 6
0 .2 0
-0 .0 8
0 .10
0 .70 0 .6 0
p-valu e
0 .6 0
r (1, 0,0)
0 .0 4
0 .8 0
r(1) 0 .10
0 .70
0 .0 5
0 .50 0 .4 0
0 .0 0
0 .3 0
s ta rt: F e b. 6, 02
p-value
-0 .0 5
0 .2 0
e nd: F e b. 20, 04
0 .10 -0 .10
0 .0 0
0 .0 0 1
0 .55 0 .6 0 0 .6 5 0 .70 0 .75 0 .8 0 0 .8 5 0 .9 0 0 .9 5
Sample Fract ion
46
-k k
0 .8 0
p-value
0 .4 5 0 .50
41
0 .15
0 .9 0
0 .0 8
0 .3 0 0 .3 5 0 .4 0
36
BP(t-1) --> YEN(t) Cross Tailed (negative to positive) Fixed Rolling W indows: .50n
0 .10
-0 .10 0 .2 5
16
r(1 ) p-value
BP(t-1) --> YEN(t) Cross Tailed (negative to positive) Increasing Rolling W indows
0 .0 2
.3 0 .2 0
-.0 6
W indow
0 .0 6
st art :Sep . 2 0 , 0 2 en d:Sep. 1 4 , 0 4
-.0 4
.05
end: Oct . 12, 04 start: June 8 , 01
.4 0 .0 0
st art : Oct. 19, 02
-.05 -.10 end: June 17 , 03
.5 0
.0 2
-.0 2
p-value
r (1,0 ,0)
.30
p-value
Two Tailed Tail Dependence r(1) Fixed Rolling W indows : .50n
.15
.50
.00 -.02
.1 0
0 .9 5
-k k
Sam ple Fraction
.70
r(1)
p- va lue
end: Nov. 19, 0 2
-.1 0 0.25
.80 end: Aug. 22, 03
p-va lue
.08
.3 5
p -value
r (1,0,0)
.4 0
p-value
r(1)
.1 0
r(1,0,0)
.1 5
6
11
16
21
26
Window
-k
r(1)
k
p -value
31
36
41
46 -k
r(1)
k
p -value
Notes: Fixed window "start" and "end" dates denote the window span. In creasing windows all start at Jan. 2, 2001.
A rolling window analysis provides strikingly di¤erent results given Sept. 11, 2001 and the U.S. economic slowdown over 2000-2002. We comment here only on NASDAQ and LSE dependence. Two-tailed tail dependence tests were performed over …xed windows of roughly 500 days ( 5 £ ), and increasing windows beginning with (roughly) 250 days. Once the increasing sample contains late 2002 (i.e. sample size 500, sample fraction .50) two-tailed extremal dependence NASDAQ ! LSE is signi…cant at the 5% level. Conversely, spillover is highly dependent during …xed windows containing Sept. 11, 2001 with a sharp decline as the window passes by the date. Signi…cance increases again over windows containing 2002. Finally, tail dependence from lagged LSE to contemporary NASDAQ becomes precipitously more signi…cant in increasing windows starting in the third quarter of 2003
29
