Dimension Reduction with Heavy Tails Gabriel Kuhn Munich University - PowerPoint PPT Presentation

Dimension Reduction with Heavy Tails Gabriel Kuhn Munich University of Technology http/www.ma.tum.de/stat 4th Conference on Extreme Value Analysis Gothenburg, August 15 . – 19 . , 2005 Kl¨ uppelberg, C. and Kuhn, G. (2005) Dimension Reduction with Reference: Heavy Tails. In preparation . � (Gabriel Kuhn, TU Munich) c EVA2005, Gothenburg – 1 –

Factor Model • Observable d -dimensional random vector X X X • Model: X X X = µ µ + L µ L f L f + V f V V e e e L f f L – L L f f : k -dimensional non-observable common factors f f , loading matrix L L V e e V – V V e e : specific factors e e , diagonal matrix V V – ( f f f,e e e ) are uncorrelated (independent) X Distribution of X X described by linear combination of k factors with Idea: componentwise extra source of randomness. L T + V V 2 • Classical model: ( f f e e ) ∼ N k + d (0 0 I X Σ L L V f,e 0 , I I) and Cov( X X ) =: Σ Σ = L LL Disadvantages: – Data may not be normal – No heavy-tailed model – Dependence in extremes cannot be modeled – Margins of the same type • Task: Overcome disadvantages above � (Gabriel Kuhn, TU Munich) c EVA2005, Gothenburg – 2 –

Distribution Models d U ( k ) • Elliptical Distribution: X µ A U X X = µ µ + G A A U U ( k ) ∼ unif { s A T A ∈ R d × k , Σ U s s : � s s s � = 1 } , A A Σ A A G > 0 independent of U Σ := A AA Σ) − 1 / 2 =: R X ) = EG 2 Σ Σ) − 1 / 2 Σ E ( X X X ) = µ µ µ , Cov( X X Σ Σ /k , Corr( X X X ) = diag(Σ Σ Σ Σdiag(Σ Σ R R U ( k ) ∼ N d ( µ d � χ 2 • Example: – normal: X X µ A U µ Σ X = µ µ + k A A U µ, Σ Σ) d U ( k ) d � � νχ 2 X µ k /χ 2 A U µ ν/χ 2 0 Σ – multivariate t ν : X X = µ µ + ν A A U = µ µ + ν N d (0 0 , Σ Σ) � (Gabriel Kuhn, TU Munich) c EVA2005, Gothenburg – 3 –

Extended Factor Model L T + V V 2 = Σ L L V Σ Drop normal assumption and consider (with L LL Σ ): � � f f f d d U ( k + d ) , X X X = µ µ µ + L L L f f f + V V V e e e is elliptical: L L L f f f + V V V e e = (L e L L , V V V) = G (L L , V L V) U V U e e e e.g. choose multivariate t ν -distribution f e Remark: – f f and e e are uncorrelated but not independent – alternatively: same dependence structure, but arbitrary margins (copula approach) – P ( G > x ) ∼ Cx − ν needed for modelling tail dependence � (Gabriel Kuhn, TU Munich) c EVA2005, Gothenburg – 4 –

Factorization L T +V V 2 • Standard approach uses (normal) ml algorithm for decomposition Σ Σ L L V Σ = L LL L T + V A)) 2 := (L V) T = L f ( k ) ( f ( k ) V 2 Definition: ml (A A A) = (L L L , V V V) , ml (A A L L , V V V)(L L L , V V L LL L V L T +V V 2 and neighborhood of Σ P Σ decomposable by f ( k ) • Lemma: Σ Σ Σ n − → Σ Σ Σ = L LL L L V Σ ml � 2 � P f ( k ) ⇒ ml (Σ Σ Σ n ) − → Σ Σ Σ • Interpretation: Given some consistent and composable covariance (correlation) estimator, the algorithm computes a consistent decomposition (independent of distribution model) • Remark: In application algorithm almost always produces a decomposition � (Gabriel Kuhn, TU Munich) c EVA2005, Gothenburg – 5 –

Dependence Concepts iid ∼ ( ˜ X, ˜ • Kendall’s τ : Let ( X, Y ) Y ) � � � � ( X − ˜ X )( Y − ˜ ( X − ˜ X )( Y − ˜ − P τ := P Y ) > 0 Y ) < 0 Elliptical distribution ⇒ R R T T R = sin( π T T / 2) , T T = ( τ ij ) 1 ≤ i,j ≤ d • Tail Dependence: ( X, Y ) with margins F X , F Y λ := lim u ց 0 P ( Y < F ← Y ( u ) | X < F ← X ( u )) Elliptical distribution and P ( G > x ) ∼ Cx − ν , ν ∈ (0 , ∞ ) � 2 � F ← t,ν +1 (1 − Λ / 2) ⇒ R R R = 1 − 2 � 2 � F ← t,ν +1 (1 − Λ / 2) ν + 1 + Λ Λ = ( λ ij ) 1 ≤ i,j ≤ d and F t,ν : df of 1 -dim t ν Λ • Remark: Both dependence concepts independent of margins � (Gabriel Kuhn, TU Munich) c EVA2005, Gothenburg – 6 –

Example • Choose d = 10 , k = 2 factors with loadings component 1 2 3 4 5 6 7 8 9 10 L L L · , 1 . 85 . 76 . 67 . 58 . 49 . 41 . 32 . 23 . 14 . 05 L L L · , 2 . 17 . 41 . 55 . 64 . 71 . 77 . 81 . 84 . 85 . 86 V 2 ) diag(V V . 25 . 25 . 25 . 25 . 25 . 25 . 25 . 25 . 25 . 25 L T + V V 2 = R ( ⇒ L L L V R LL R is a correlation matrix) • consider factor model(s) (given before) d 1. X X X =L L L f f f + V V Ve e e ∼ t d (0 0 0 , R R R , ν ) with ν = 6 2. X X has same t (0 X 0 , R 0 R R , ν ) dependence structure, ν but different margins F i = t ν i , ν ν = (3 , . . . , 10) • Simulation length n = 2000 , repeat 500 times Plots of different estimation methods and 95% -CI’s of loadings � (Gabriel Kuhn, TU Munich) c EVA2005, Gothenburg – 7 –

factor 1 factor 2 specific factor R R R n R R R n R R R n 1 1 1 loadings loadings loadings . 6 . 6 . 6 . 2 . 2 . 2 0 0 0 2 4 6 8 10 2 4 6 8 10 2 4 6 8 10 components components components R mle R mle R mle R R R R R R n n n 1 1 1 loadings loadings loadings . 6 . 6 . 6 R , ν ) . 2 . 2 . 2 0 0 0 R 0 , R 2 4 6 8 10 2 4 6 8 10 2 4 6 8 10 components components components 0 X ∼ t d (0 T T T R T T R T T R T T R R R R R R n n n 1 1 1 loadings loadings loadings . 6 . 6 . 6 X X . 2 . 2 . 2 0 0 0 2 4 6 8 10 2 4 6 8 10 2 4 6 8 10 components components components R Λ Λ Λ R Λ Λ Λ Λ R Λ Λ R R R R R R n n n 1 1 1 loadings loadings loadings . 6 . 6 . 6 . 2 . 2 . 2 0 0 0 2 4 6 8 10 2 4 6 8 10 2 4 6 8 10 components components components � (Gabriel Kuhn, TU Munich) c EVA2005, Gothenburg – 8 –

factor 1 factor 2 specific factor R R R n R R R n R R R n 1 1 1 X has t -dependence, different margins loadings loadings loadings . 6 . 6 . 6 . 2 . 2 . 2 0 0 0 2 4 6 8 10 2 4 6 8 10 2 4 6 8 10 components components components R mle R mle R mle R R R R R R n n n 1 1 1 loadings loadings loadings . 6 . 6 . 6 . 2 . 2 . 2 0 0 0 2 4 6 8 10 2 4 6 8 10 2 4 6 8 10 components components components R T T T R T T T R T T T R R R R R R n n n 1 1 1 loadings loadings loadings . 6 . 6 . 6 . 2 . 2 . 2 0 0 0 2 4 6 8 10 2 4 6 8 10 2 4 6 8 10 X X components components components R Λ Λ Λ R Λ Λ Λ R Λ Λ Λ R R R R R R n n n 1 1 1 loadings loadings loadings . 6 . 6 . 6 . 2 . 2 . 2 0 0 0 2 4 6 8 10 2 4 6 8 10 2 4 6 8 10 components components components � (Gabriel Kuhn, TU Munich) c EVA2005, Gothenburg – 9 –

Example • Consider 8-dimensional set of data: oil, s&p500, gbp, usd, chf, jpy, dkk and sek (exchange rates w.r.t. euro) • Daily log-returns between May, 1985 to June, 2004 (n=4904) • Apply factor analysis with different estimators as before � (Gabriel Kuhn, TU Munich) c EVA2005, Gothenburg – 10 –

factor 1 factor 2 1 . 0 1 . 0 4 2 3 1 3 1 2 4 4 4 4 1 0 . 5 0 . 5 4 1 3 1 2 3 2 4 loadings 2 2 1 1 2 4 3 1 4 4 1 3 4 2 2 1 3 4 1 3 3 3 2 2 1 0 . 0 0 . 0 3 3 2 2 2 1 1 1 3 2 3 3 1 4 2 3 emp emp 4 4 1 4 1 mle mle 2 2 − 0 . 5 − 0 . 5 tau tau 3 3 taild taild 4 4 p p 0 d y k k 0 d y k k f f l l h h i 0 b p k e i 0 b p k e s s o o u c u c 5 g j d s 5 g j d s p p & & s s factor 3 specific factors 1 . 0 1 . 0 2 1 3 4 1 2 3 3 emp 1 4 2 1 2 mle 3 2 2 tau 3 3 1 2 3 1 1 taild 0 . 5 4 0 . 5 4 4 loadings 4 4 3 4 3 2 3 3 3 1 1 2 1 3 3 2 4 2 4 0 . 0 0 . 0 1 1 4 4 4 4 2 1 3 3 1 2 2 3 1 4 1 2 2 emp 4 1 2 1 4 2 mle − 0 . 5 − 0 . 5 tau 3 4 taild 0 p k k 0 p k k d f y d f y l l i h i h 0 b s p k e 0 b s p k e o o 5 u c s 5 u c s g j d g j d p p & & s s � (Gabriel Kuhn, TU Munich) c EVA2005, Gothenburg – 11 –