Transcript Folien vom 07.01.2011

Statistical issues with financial market data
A: Cross-Section Data:
- deviations from multivariate normality
- “tail dependence”
- copulas
- default predictions
B: Time series data
- heavy tails
- chaos?
- structural change in patterns of dependence
- Integration and Cointegration
- ARCH- and GARCH-effects
- Long memory and structural change
Selected bond rating agencies
Name
in business since
Moody’s investors
service (“Moody’s”)
1900
Fitch investor service
(“Fitch”)
1922
Standard and Poors’
Corporation (“S&P”)
1923
Thomson Bank Watch
1974
Dominion bond rating
service (“DBRS”)
1976
Japanese bond rating
institute
1977
Duff and Phelps
Credit Rating
1986
.
.
.
Correspondence between selected
S+P-grades and default probabilities
Grade
•
AAA - AA
rel. frequencies of default
(%)
0,01
A
0,04
BBB
0,21
BB+
0,75
BB-
1,14
B
5,16
CCC
10,00
CC
20,00
D
100,00
(from M. Carey: “Some evidence on the consistency of banks’ internal credit ratings,”
Federal Reserve Board 2001, Table 1, page 7 )
Rated A by S+P, Sept. 9, 2008
Default, Sep. 15, 2008
Filed for bancruptcy Dec. 1. 2001
Was rated investment status by
both Moody‘s and S+P 1 month before
Formerly Worldcom; defaults on credit
payments in July 2002, rated A by S+P
In April 2002.
Formerly the world's biggest dairy product
producer, had its credit rating cut to junk
after missing a payment in Dec. 2003;
rated A a couple of months before
Evaluating and comparing probability
forecasters (=rating agencies)
Case 1: Raters A and B rate different
obligors at different points in time
(“skill scores”)
Case 2: Raters A and B rate identical
obligors at identical points in time
Credit ratings
Rated the same by Moody’ and S&P
Sovereign
Corporate
AA/Aa or above
67%
53%
Other investment
grade
Below investment
grade
56%
36%
29%
41%
Sources: Moody’s; Federal Reserve Bank of New
York
Example: Assigning default probabilities to 800 borrowers
Predicted
Default
probability
Distribution of borrowers across default
probabilities according to different
probability forecasters
A
B
C
D
0,5%
0
0
200 (1)
160
1%
400 (4)
0
0
200
1,5%
0
0
400 (6)
0
2%
0
800(16)
0
0
3%
400(12)
0
0
440
4,5%
0
0
200 (9)
0
Default ordering
(S. Vardemann and G. Meeden, Journal of the American
Statistical Society 1983):
A is better than B if its cumulated percentage
of defaults (with cumulation starting in the
good grades) is nowhere above that of B‘s.
Likewise for non-defaults
Theorem 1 (Vardeman/Meeden 1983):
If A and B are both well calibrated, and
A dominates B in the default ordering,
then A is more refined than B.
Theorem 2 (Krämer 2005): Let
A and B be both well
calibrated. Then A and B
cannot be ordered according
to the Vardeman/
Meeden default ordering.
Lorenz-curve, power curve, cumulative accuracy profile
overall
Defaults
bad
10%
50%
medium
70%
45%
good
20%
5%
Theorem (independently by various
authors):
Consider all possible pairs of defaults and
non-defaults. The accuracy ratio (=area
underneath the ROC-curve) is then equal
to the probability that in one such
randomly chosen pair, the non-default is
ranked higher than the default
Theorem: Let A and B be (semi-)calibrated
probability forecasters. Then we have:
A dominates B in the Vardeman/ Meeden default
ordering sense
=>
A is more refined than B
(sufficient for)
=>
A’s ROC and power curves are nowhere below
those of B
The converse does not hold
California Edison: rated A+ in 1999,
default 2001
(has recovered in the meantime)
W. Krämer:
Strukturbruchtests bei Renditekorrelationen
Gemeinsame Arbeiten mit
Jonas Kaiser
Dominik Wied
Maarten van Kampen
Wertentwicklung globaler Aktienmärkte im Jahr 2007
USA
Japan
+ 6,4 %
Deutschland
+ 22,3
%
+ 3,8 %
GB
Frankreich
Spanien
Italien
China
Indien
- 11,1
%
+ 1,3 %
+ 7,3 %
- 7,0 %
+ 96,7
%
+ 47,1
Wertentwicklung der gleichen
Aktienmärkte 2008
USA (DJIA)
- 32,7 %
Japan (Nikkei 225)
- 29,7 %
Deutschland (DAX)
- 39,5 %
GB (FTSE 100)
- 30,9 %
Frankreich (CAC40)
- 42,0 %
Spanien (IBEX 35)
- 38,7 %
Italien (S+P Mib)
- 48,8 %
China (Shanghai Comp.)
- 65,4 %
Indien (Sensex 30)
- 52,9 %
Modellierung zeitvariabler Abhängigkeiten
1. Dynamische bedingte Korrelationen:
R. Engle: „Dynamic conditional correlation: A simple class of multivariate generalized
autoregressive conditional heteroskedasticity models,“ Journal of Business and
Economic Statistics 20, 2002, 339-350.
2. „Markov-Switching“:
D. Pelletier: „Regime switching for dynamic correlations,“ Journal of Econometrics,
2006, 445-473
M. Haas: „Covariance forecasts and long-run correlations in a Markov-switching model
for dynamic correlations“, Finance Research Letters, 7, 2010, 86-97
3. Dynamische Copulas:
A. Patton: „Modelling asymmetric exchange rate dependence,“ International Economic
Review 47, 2006, 527-556.
D. Totouom: Dynamic Copulas: Applications to finance and economics, Paris 2007.
E. Giacomini, W. Härdle, und V. Spokoiny: „Inhomogeneous dependency modelling
with time varying copulae,“ Journal of Business and Economic Statistics 27, 2009, 224234.
D. Guegan und J. Zhang: Change analysis of a dynamic copula for measuring
dependende in multivariate data,“ Quantitative Finance 10, 2010, 421-430.
Was ist eine Copula?
Ausgangspunkt ist folgendes ebenso zentrale wie elementare
Resultat der W-Theorie:
Sei X stetige Zve mit Verteilungsfunktion F. Dann ist die neue
Zve U:= F(X) auf [0,1] gleichverteilt
Def: die gemeinsame Verteilung von U=F(X) und V=G(Y) heißt
Copula von von X und Y
Satz: Die gemeinsame Verteilung von X und Y ist durch die
Copula und die beiden Randverteilungen eindeutig festgelegt
,
-6
-6
-4
-4
-2
-2
0
0
2
2
4
4
6
6
Randabhängigkeit („tail dependence“):
Links: 3000 tägliche BMW- und VW-Renditen,
Rechts: 3000 bivariat normalverteilte Zufallsvektoren
-6
-4
-2
0
2

4
limc0 P X  F
6
1
-6
-4
-2
0
2
c | Y  G c
1
4
6
=?
Ausgewählte Literatur zu Randabhängigkeiten
Longin/Solnik: „Extreme Correlation of international equity
markets,“ Journal of Finance 2001
R. Schmidt: „Tail dependence for elliptically contoured
distributions,“ Math. Meth. Oper. Research 2002
Falk/Michel: „Testing for tail dependence in extrem value models,“
AISM 2006
Hüsler/Li: „Testing asymptotic independence in bivariate
extremes,“ Journal of Statistical Planning and Inference 2009
F.Schmid/R.Schmidt/ J.Penzer: „Measuring Large Comovements in
Financial Markets“, erscheint in Quantitative Finance 2010.
Bücher/Dette/Volgushev: „A new estimator of the Pickands
dependence function and a test for extrem-value correlation,“
Dortmund 2010 (SFB 823 Diskussionspapier).
Signifikanztests auf konstante
Abhängigkeitsstruktur
A): endogene Brüche: mögliche Muster unter der
Alternative sind dateninduziert („truncated correlations“,
„excess correlations“)
B): exogene Brüche: Aufspaltung der Stichprobe nach
potentiell unterschiedlichen Abhängigkeitsmustern ohne
Ansicht der realisierten Werte von (X, Y)
 A :
bedingte Korrelation von X und Y, gegeben X
A
Bei bivariater Normalverteilung gilt (Boyer et al. (1999) „Pitfalls in
tests for changes in correlations“, International Finance
Discussion Papers Number 597):

A 
  (1   )
2
2

2
x
var(X | X  A)
Z :

(1  
2
2
) x
X
Theorem: E(XZ|X A) = 0
A 

1
1 
2
Y

2
2
 X Z : A
2
2
  (1   )
2
 X:A
Gemeinsame Verteilung
konstant?
Copula konstant?
Dias/Embrechts 2004
Remaillard/Scaillet 2009
Copula constant
in einem Punkt?
Harvey/Busetti 2009
Krämer/v.Kampen 2010
Spearman ρ,
Kendall τ
konstant?
Dobric/Frahm
Schmid 2007
Schmid/Gaisser 2010
Zweite Momente konstant?
Bartlett 1949
Aue et al. 2009
Korrelationen
konstant?
Varianzen
konstant?
Riesige Literatur
Kullback 1967
Jennrich 1970
Fischer 2007
Wied 2009
Wied/Krämer/
Dehling 2010

H0 : P X 
1
FX   und
Y

1
FY  
 konstant
Copula von X und Y an der
Stelle (τ, τ) =: C(τ, τ)
Grundidee (Busetti & Harvey 2009): Betrachte empirische
Copula C*(τ, τ) und
IT,t(τ, τ) :=
1 (sowohl Xt wie Yt links vom
empirischen τ-Quantil
0 sonst
Unter H0:
zT
1
C * τ, τ   IT, t τ τ 

TC * ττ 1  C * (τ τ   t 1

 Brownsche Brücke
Typische Zeitverläufe der Teststatistik
constant
1 break
1
2 breaks
0.5
3
0
0.0
1
-1
-0.5
0
1000
2000
t
0
1000
2000
t
0
1000
2000
t
H0 :  t : KorrXt , Yt   konstant,t  1, ..., T
Grundidee: Lehne Ho ab bei extremer Fluktuation von
S T t    rt  rT 
t
2
Emp. Korrelation unter Nutzung
aller Datenpunke =
Approximation für wahres ρ
empirische Korrelation der Datenpaare 2, …, t
Für Details siehe:
J. Kaiser, W. Krämer (2010): “A cautionary note on computing
conditional from unconditional correlations”. Erscheint in Economics
Letters.
W. Krämer, M. van Kampen (2010): „A simple nonparametric test for
structural change in joint tail probabilities“, Erscheint in Economics
Letters).
Dominik Wied: „A generalized functional delta method,“ Dortmund
2010 (SFB 823 Diskussionspapier).
D. Wied, W. Krämer, H. Dehling. "Testing for a change in correlation
at an unknown point in time", 2010, zur Veröffentlichung eingereicht.
M. van Kampen, D. Wied. "A nonparametric constancy test for
copulas under mixing conditions", Dortmund 2010 (SFB 823
Diskussionspapier).
M. Arnold, N. Bissantz, D. Wied, D. Ziggel. "A new online-test for
changes in correlations between assets", Dortmund 2010 (SFB 823
Diskussionspapier).
Verallgemeinerungen auf höhere Dimensionen
Copula-Based Measures of Multivariate Association (with T.
Blumentritt, S. Gaißer, M. Ruppert, R. Schmidt),
In: F. Durante, W. Härdle, P. Jaworski, T. Rychlik (eds.) Workshop on
Copula Theory and its Applications. Springer, 2010.
Nonparametric inference on multivariate versions of Blomqvist's
beta and related measures of tail-dependence (with R. Schmidt),
Metrika, Vol. 66, 323-354, 2007
Multivariate conditional versions of Spearman's rho and related
measures of tail dependence (with R. Schmidt), Journal of
Multivariate Analysis, Vol. 98, No. 6, 1123-1140, 2007.
Multivariate Extensions of Spearman's Rho and Related Statistics
(with R. Schmidt), Statistics and Probability Letters, Vol. 77, No. 4,
2007.