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

▶
dimension reduction with heavy tails
SMART_READER_LITE
LIVE PREVIEW

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

Oct 30, 2022 345 likes •475 views

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

slide-1
SLIDE 1

c (Gabriel Kuhn, TU Munich) EVA2005, Gothenburg – 1 –

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 Reference: Kl¨ uppelberg, C. and Kuhn, G. (2005) Dimension Reduction with Heavy Tails. In preparation.

slide-2
SLIDE 2

c (Gabriel Kuhn, TU Munich) EVA2005, Gothenburg – 2 –

Factor Model

  • Observable d-dimensional random vector X

X X

  • Model: X

X X = µ µ µ + L L Lf f f + V V Ve e e – L L Lf f f: k-dimensional non-observable common factors f f f, loading matrix L L L – V V Ve e e: specific factors e e e, diagonal matrix V V V – (f f f,e e e) are uncorrelated (independent) Idea: Distribution of X X X described by linear combination of k factors with componentwise extra source of randomness.

  • Classical model: (f

f f,e e e) ∼ Nk+d(0 0,I I I) and Cov(X X X) =: Σ Σ Σ = L L LL L LT + V V V2 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
slide-3
SLIDE 3

c (Gabriel Kuhn, TU Munich) EVA2005, Gothenburg – 3 –

Distribution Models

  • Elliptical Distribution:

X X X

d

=µ µ µ + GA A AU U U (k) G > 0 independent of U U U (k) ∼ unif{s s s : s s s = 1}, A A A ∈ Rd×k, Σ Σ Σ := A A AA A AT E(X X X) = µ µ µ, Cov(X X X) = EG2Σ Σ Σ/k, Corr(X X X) = diag(Σ Σ Σ)−1/2Σ Σ Σdiag(Σ Σ Σ)−1/2 =: R R R

  • Example: – normal: X

X X

d

=µ µ µ +

  • χ2

kA

A AU U U (k) ∼ Nd(µ µ µ,Σ Σ Σ) – multivariate tν: X X X

d

=µ µ µ +

  • νχ2

k/χ2 νA

A AU U U (k) d =µ µ µ +

  • ν/χ2

νNd(0

0,Σ Σ Σ)

slide-4
SLIDE 4

c (Gabriel Kuhn, TU Munich) EVA2005, Gothenburg – 4 –

Extended Factor Model

Drop normal assumption and consider (with L L LL L LT + V V V2 = Σ Σ Σ): X X X

d

=µ µ µ + L L Lf f f + V V Ve e e is elliptical: L L Lf f f + V V Ve e e = (L L L,V V V)

  • f

f f e e e

  • d

=G(L L L,V V V)U U U (k+d), e.g. choose multivariate tν-distribution Remark: – f f f and e 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

slide-5
SLIDE 5

c (Gabriel Kuhn, TU Munich) EVA2005, Gothenburg – 5 –

Factorization

  • Standard approach uses (normal) ml algorithm for decomposition Σ

Σ Σ = L L LL L LT +V V V2 Definition: f (k)

ml (A

A A) = (L L L,V V V), (f (k)

ml (A

A A))2 := (L L L,V V V)(L L L,V V V)T = L L LL L LT + V V V2

  • Lemma:

Σ Σ Σn

P

− → Σ Σ Σ = L L LL L LT +V V V2 and neighborhood of Σ Σ Σ decomposable by f (k)

ml

⇒

  • f (k)

ml (Σ

Σ Σn) 2

P

− → 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

slide-6
SLIDE 6

c (Gabriel Kuhn, TU Munich) EVA2005, Gothenburg – 6 –

Dependence Concepts

  • Kendall’s τ: Let (X, Y )

iid

∼ ( ˜ X, ˜ Y ) τ := P

  • (X − ˜

X)(Y − ˜ Y ) > 0

  • − P
  • (X − ˜

X)(Y − ˜ Y ) < 0

  • Elliptical distribution ⇒ R

R R = sin(πT T T/2), T T T = (τij)1≤i,j≤d

  • Tail Dependence: (X, Y ) with margins FX, FY

λ := limuց0 P (Y < F ←

Y (u) |X < F ← X (u))

Elliptical distribution and P(G > x) ∼ Cx−ν, ν ∈ (0, ∞) ⇒ R R R = 1 − 2

  • F ←

t,ν+1(1 − Λ/2)

2 ν + 1 +

  • F ←

t,ν+1(1 − Λ/2)

2 Λ Λ Λ = (λij)1≤i,j≤d and Ft,ν : df of 1-dim tν

  • Remark:

Both dependence concepts independent of margins

slide-7
SLIDE 7

c (Gabriel Kuhn, TU Munich) EVA2005, Gothenburg – 7 –

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 diag(V V V2) .25 .25 .25 .25 .25 .25 .25 .25 .25 .25 (⇒ L L LL L LT + V V V2 = R R R is a correlation matrix)

  • consider factor model(s) (given before)
  • 1. X

X X

d

=L L Lf f f + V V Ve e e ∼ td(0 0,R R R, ν) with ν = 6

  • 2. X

X X has same t(0 0,R R R, ν) dependence structure, but different margins Fi = tνi, ν ν ν = (3, . . . , 10)

  • Simulation length n = 2000, repeat 500 times

Plots of different estimation methods and 95%-CI’s of loadings

slide-8
SLIDE 8

c (Gabriel Kuhn, TU Munich) EVA2005, Gothenburg – 8 –

X X X ∼ td(0 0,R R R, ν)

2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 6 6 6 6 6 8 8 8 8 8 8 8 8 8 8 8 8 10 10 10 10 10 10 10 10 10 10 10 10 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .6 .6 .6 .6 .6 .6 .6 .6 .6 .6 .6 .6 1 1 1 1 1 1 1 1 1 1 1 1 loadings loadings loadings loadings loadings loadings loadings loadings loadings loadings loadings loadings components components components components components components components components components components components components R R Rn R R RT

T T n

R R RΛ

Λ Λ n

R R Rmle

n

R R Rn R R RT

T T n

R R RΛ

Λ Λ n

R R Rmle

n

R R Rn R R RT

T T n

R R RΛ

Λ Λ n

R R Rmle

n

factor 1 factor 2 specific factor

slide-9
SLIDE 9

c (Gabriel Kuhn, TU Munich) EVA2005, Gothenburg – 9 –

X X X has t-dependence, different margins

2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 6 6 6 6 6 8 8 8 8 8 8 8 8 8 8 8 8 10 10 10 10 10 10 10 10 10 10 10 10 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .6 .6 .6 .6 .6 .6 .6 .6 .6 .6 .6 .6 1 1 1 1 1 1 1 1 1 1 1 1 loadings loadings loadings loadings loadings loadings loadings loadings loadings loadings loadings loadings components components components components components components components components components components components components R R Rn R R RT

T T n

R R RΛ

Λ Λ n

R R Rmle

n

R R Rn R R RT

T T n

R R RΛ

Λ Λ n

R R Rmle

n

R R Rn R R RT

T T n

R R RΛ

Λ Λ n

R R Rmle

n

factor 1 factor 2 specific factor

slide-10
SLIDE 10

c (Gabriel Kuhn, TU Munich) EVA2005, Gothenburg – 10 –

Example

  • Consider 8-dimensional set of data:
  • il, 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
slide-11
SLIDE 11

c (Gabriel Kuhn, TU Munich) EVA2005, Gothenburg – 11 –

−0.5 −0.5 −0.5 −0.5 0.0 0.0 0.0 0.0 0.5 0.5 0.5 0.5 1.0 1.0 1.0 1.0

factor 1 factor 2 factor 3 specific factors loadings loadings

emp emp emp emp mle mle mle mle tau tau tau tau taild taild taild taild

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4

  • i

l

  • i

l

  • i

l

  • i

l s & p 5 s & p 5 s & p 5 s & p 5 g b p g b p g b p g b p u s d u s d u s d u s d c h f c h f c h f c h f j p y j p y j p y j p y d k k d k k d k k d k k s e k s e k s e k s e k