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Statistical issues with financial market data A: Cross-Section Data: - deviations from multivariate normality - tail dependence - copulas - default predictions.

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Präsentation zum Thema: "Statistical issues with financial market data A: Cross-Section Data: - deviations from multivariate normality - tail dependence - copulas - default predictions."—  Präsentation transkript:

1 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



4 Selected bond rating agencies Namein business since Moodys investors service (Moodys) 1900 Fitch investor service (Fitch) 1922 Standard and Poors Corporation (S&P) 1923 Thomson Bank Watch1974 Dominion bond rating service (DBRS) 1976 Japanese bond rating institute 1977 Duff and Phelps Credit Rating 1986.


6 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 ) Graderel. frequencies of default (%) AAA - AA0,01 A0,04 BBB0,21 BB+0,75 BB-1,14 B5,16 CCC10,00 CC20,00 D100,00

7 Rated A by S+P, Sept. 9, 2008 Default, Sep. 15, 2008

8 Filed for bancruptcy Dec Was rated investment status by both Moodys and S+P 1 month before Formerly Worldcom; defaults on credit payments in July 2002, rated A by S+P In April 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

9 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

10 Credit ratings Rated the same by Moody and S&P SovereignCorporate AA/Aa or above67%53% Other investment grade 56%36% Below investment grade 29%41% Sources: Moodys; Federal Reserve Bank of New York

11 Example: Assigning default probabilities to 800 borrowers Predicted Default probability Distribution of borrowers across default probabilities according to different probability forecasters ABCD 0,5%00200 (1)160 1%400 (4) ,5%00400 (6)0 2%0800(16)00 3%400(12) ,5%00200 (9)0

12 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 Bs. Likewise for non-defaults

13 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.

14 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.

15 overallDefaults bad10%50% medium70%45% good20%5% Lorenz-curve, power curve, cumulative accuracy profile

16 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

17 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) => As ROC and power curves are nowhere below those of B The converse does not hold


19 California Edison: rated A+ in 1999, default 2001 (has recovered in the meantime)


21 W. Krämer: Strukturbruchtests bei Renditekorrelationen Gemeinsame Arbeiten mit Jonas KaiserDominik Wied Maarten van Kampen

22 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 %

23 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 %

24 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, Markov-Switching: D. Pelletier: Regime switching for dynamic correlations, Journal of Econometrics, 2006, M. Haas: Covariance forecasts and long-run correlations in a Markov-switching model for dynamic correlations, Finance Research Letters, 7, 2010, Dynamische Copulas: A. Patton: Modelling asymmetric exchange rate dependence, International Economic Review 47, 2006, D. Totouom: Dynamic Copulas: Applications to finance and economics, Paris E. Giacomini, W. Härdle, und V. Spokoiny: Inhomogeneous dependency modelling with time varying copulae, Journal of Business and Economic Statistics 27, 2009, D. Guegan und J. Zhang: Change analysis of a dynamic copula for measuring dependende in multivariate data, Quantitative Finance 10, 2010,

25 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,

26 Randabhängigkeit (tail dependence): Links: 3000 tägliche BMW- und VW-Renditen, Rechts: 3000 bivariat normalverteilte Zufallsvektoren = ?

27 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 Bücher/Dette/Volgushev: A new estimator of the Pickands dependence function and a test for extrem-value correlation, Dortmund 2010 (SFB 823 Diskussionspapier).

28 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)


30 bedingte Korrelation von X und Y, gegeben XA Bei bivariater Normalverteilung gilt (Boyer et al. (1999) Pitfalls in tests for changes in correlations, International Finance Discussion Papers Number 597):

31 Theorem: E(XZ|XA) = 0

32 Gemeinsame Verteilung konstant? Copula konstant? Dias/Embrechts 2004 R lard/Scaillet 2009 Zweite Momente konstant ? Bartlett 1949 Aue et al Copula constant in einem Punkt? Harvey/Busetti 2009 Krämer/v.Kampen 2010 Spearman ρ, Kendall τ konstant? Dobric/Frahm Schmid 2007 Schmid/Gaisser 2010 Varianzen konstant? Riesige Literatur Korrelationen konstant? Kullback 1967 Jennrich 1970 Fischer 2007 Wied 2009 Wied/Krämer/ Dehling 2010

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

34 Grundidee (Busetti & Harvey 2009): Betrachte empirische Copula C*(τ, τ) und 1 (sowohl X t wie Y t links vom I T,t (τ, τ) := empirischen τ-Quantil 0 sonst Unter H 0 :

35 Typische Zeitverläufe der Teststatistik

36 Grundidee: Lehne H o ab bei extremer Fluktuation von empirische Korrelation der Datenpaare 2, …, t Emp. Korrelation unter Nutzung aller Datenpunke = Approximation für wahres ρ

37 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).

38 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, Nonparametric inference on multivariate versions of Blomqvist's beta and related measures of tail-dependence (with R. Schmidt), Metrika, Vol. 66, , 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, , Multivariate Extensions of Spearman's Rho and Related Statistics (with R. Schmidt), Statistics and Probability Letters, Vol. 77, No. 4, 2007.

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