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

• (from M. Carey: “Some evidence on the consistency of banks’ internal credit ratings,” Federal Reserve Board 2001, Table 1, page 7 )

Grade rel. frequencies of default (%)

AAA - AA 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

Rated A by S+P, Sept. 9, 2008

Default, Sep. 15, 2008

Filed for bancruptcy Dec. 1. 2001Was rated investment status byboth Moody‘s and S+P 1 month before

Formerly Worldcom; defaults on credit payments in July 2002, rated A by S+PIn April 2002.

Formerly the world's biggest dairy productproducer, had its credit rating cut to junkafter 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 ratingsRated the same by Moody’ and S&P

Sovereign Corporate

AA/Aa or above 67% 53%

Other investment grade

56% 36%

Below investment grade

29% 41%

Sources: Moody’s; Federal Reserve Bank of New York

Example: Assigning default probabilities to 800 borrowers

PredictedDefault

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.

overall Defaults

bad 10% 50%

medium 70% 45%

good 20% 5%

Lorenz-curve, power curve, cumulative accuracy profile

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 + 6,4 %

Japan - 11,1 %

Deutschland + 22,3 %

GB + 3,8 %

Frankreich + 1,3 %

Spanien + 7,3 %

Italien - 7,0 %

China + 96,7 %

Indien + 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, 445-473M. 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, 224-234.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

,

Randabhängigkeit („tail dependence“):Links: 3000 tägliche BMW- und VW-Renditen,

Rechts: 3000 bivariat normalverteilte Zufallsvektoren

-6 -4 -2 0 2 4 6

-6-4

-20

24

6

-6 -4 -2 0 2 4 6

-6-4

-20

24

6

cGYcFXPc11

0 |lim = ?

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):

)|var()1(

222

AXXx

A

YXZx

222 1

1

)1(:

Theorem: E(XZ|XA) = 0

AX

AZXA

:2

:22

)21(2

Gemeinsame Verteilung konstant?

Copula konstant?Dias/Embrechts 2004

Remaillard/Scaillet 2009

Zweite Momente konstant?Bartlett 1949

Aue et al. 2009

Copula constantin einem Punkt?Harvey/Busetti 2009

Krämer/v.Kampen 2010

Spearman ρ,Kendall τ

konstant?Dobric/FrahmSchmid 2007

Schmid/Gaisser 2010

Varianzenkonstant?

Riesige Literatur

Korrelationenkonstant?Kullback 1967Jennrich 1970Fischer 2007Wied 2009

Wied/Krämer/Dehling 2010

konstant

Y und :H0

11

YX FFXP

Copula von X und Y an der Stelle (τ, τ) =: C(τ, τ)

Grundidee (Busetti & Harvey 2009): Betrachte empirischeCopula C*(τ, τ) und

1 (sowohl Xt wie Yt links vom IT,t(τ, τ) := empirischen τ-Quantil

0 sonst

Unter H0:

Brücke Brownsche

I*C*C1*CT

1 zT

tT,

ττττ,τ(τττ

1t

Typische Zeitverläufe der Teststatistik

0 1000 2000

-0.5

0.0

0.5

t

constant

0 1000 2000

1

3

1 break

t0 1000 2000

-1

0

12 breaks

t

T ..., 1, t konstant, ,XKorr : :H t0 tt Y

Grundidee: Lehne Ho ab bei extremer Fluktuation von

t

2TtT rrtS

empirische Korrelation der Datenpaare 2, …, t

Emp. Korrelation unter Nutzung aller Datenpunke = Approximation für wahres ρ

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.