ALRIGHT: Asymmetric LaRge-Scale (I)GARCH with Hetero-Tails Marc S. - PowerPoint PPT Presentation

ALRIGHT: Asymmetric LaRge-Scale (I)GARCH with Hetero-Tails Marc S. Paolella Swiss Banking Institute, University of Z¨ urich Marc S. Paolella ALRIGHT: Asymmetric LaRge-Scale (I)GARCH with Hetero-Tails

Do Asset Returns Have Different Tail Indices? Scatterplot of BoA and Wal−Mart 15 10 5 Wal−Mart 0 −5 −10 −15 −30 −20 −10 0 10 20 30 Bank of America Marc S. Paolella ALRIGHT: Asymmetric LaRge-Scale (I)GARCH with Hetero-Tails

Asset Returns Have Different Tail Indices Scatterplot of BoA and Wal−Mart Fitted Multivariate Student t 10 15 ˆ k = 2 . 014 10 5 5 Wal−Mart Wal−Mart 0 0 −5 −5 −10 −15 −10 −10 −5 0 5 10 −30 −20 −10 0 10 20 30 Bank of America Bank of America Marc S. Paolella ALRIGHT: Asymmetric LaRge-Scale (I)GARCH with Hetero-Tails

Asset Returns Have Different Tail Indices Estimated Parameter k and 95% Bootstrap C.I.s Estimated Parameter k and 95% Bootstrap C.I.s 6 12 5.5 11 5 10 Degrees of Freedom 4.5 Degrees of Freedom 9 4 8 3.5 7 3 6 2.5 5 2 4 1.5 1 3 0 5 10 15 20 25 30 0 5 10 15 20 25 30 The 30 individual stock return series The 30 individual stock return series Estimated Parameter θ and 95% Bootstrap C.I.s Estimated Parameter θ and 95% Bootstrap C.I.s 0.6 0.6 0.4 0.4 Noncentrality Parameter Noncentrality Parameter 0.2 0.2 0 0 −0.2 −0.2 −0.4 −0.4 −0.6 −0.6 0 5 10 15 20 25 30 0 5 10 15 20 25 30 The 30 individual stock return series The 30 individual stock return series Marc S. Paolella ALRIGHT: Asymmetric LaRge-Scale (I)GARCH with Hetero-Tails

Bank of America Percentage Returns Wal−Mart Percentage Returns 10 20 8 6 10 4 0 2 0 −10 −2 −4 −20 −6 −30 −8 2001 2002 2004 2005 2006 2008 2001 2002 2004 2005 2006 2008 Bank of America GARCH−Filtered Residuals Wal−Mart GARCH−Filtered Residuals 5 6 4 0 2 0 −5 −2 −4 −10 2001 2002 2004 2005 2006 2008 2001 2002 2004 2005 2006 2008 Marc S. Paolella ALRIGHT: Asymmetric LaRge-Scale (I)GARCH with Hetero-Tails

Meta-Elliptical t Distribution The pdf of the meta-elliptical t distribution is given by � d � � Φ − 1 k 0 (Φ k 1 ( x 1 )) , . . . , Φ − 1 f X ( x ; k , R ) = ψ k 0 (Φ k d ( x d )); R , k 0 φ k i ( x i ) , i =1 (1) where x = ( x 1 , . . . , x d ) ′ ∈ R d ; k = ( k 0 , k 1 , . . . , k d ) ′ ∈ R d +1 > 0 ; φ k ( x ) and Φ k ( x ) denote, respectively, the univariate Student’s t pdf and cumulative distribution function (cdf) with k degrees of freedom, evaluated at x ∈ R ; R is a d -dimensional correlation matrix, ... Marc S. Paolella ALRIGHT: Asymmetric LaRge-Scale (I)GARCH with Hetero-Tails

Meta-Elliptical t Distribution and, with z = ( z 1 , z 2 , . . . , z d ) ′ ∈ R d , the copula density function ψ ( · ; · ) = ψ ( z 1 , z 2 , . . . , z d ; R , k ) multiplicatively relating the joint distribution of X to their distribution under independence is given by � � − ( k + d ) / 2 Γ { ( k + d ) / 2 }{ Γ( k / 2) } d − 1 1 + z ′ R − 1 z ψ ( · ; · ) = � � d | R | 1 / 2 k Γ { ( k + 1) / 2 } � � ( k +1) / 2 d � 1 + z 2 i × . k i =1 Marc S. Paolella ALRIGHT: Asymmetric LaRge-Scale (I)GARCH with Hetero-Tails

FaK (Fang, Fang Kotz) We express a random variable T with location parameter µ = ( µ 1 , . . . , µ d ) ′ ∈ R d , scale terms σ = ( σ 1 , . . . , σ d ) ′ ∈ R d > 0 , and correlation matrix R , as T ∼ FaK ( k , µ , σ , R ), with FaK a reminder of the involved authors, and density � y 1 − µ 1 � f T ( y ; k , µ , σ , R ) = f X ( x ; k , R ) , . . . , y d − µ d , x = , σ 1 σ 2 · · · σ d σ 1 σ d (2) where f X ( x ; k , R ) is given in (1). From its construction as a copula, the marginal distribution of each ( T i − µ i ) /σ i is a standard Student’s t with k i degrees of freedom, irrespective of k 0 . If second moments exist for each T i , then the variance-covariance matrix of T is given by Σ = V ( T ) = MRM , where � M = diag ( σ ⊙ κ ), κ = ( κ 1 , . . . , κ d ) ′ , and κ i = k i / ( k i − 2), i = 1 , . . . , d . In particular, E [ T i ] = µ i and V ( T i ) = σ 2 i κ 2 i . Marc S. Paolella ALRIGHT: Asymmetric LaRge-Scale (I)GARCH with Hetero-Tails

FaK Parameter k 0 While the marginals are not influenced by k 0 , its value does alter the dependency structure of the distribution. Via comparison with scatterplots of actual financial returns data, one might speculate that only values of k 0 ≥ max i k i , i = 1 , . . . , d , are of interest, and one could entertain just setting k 0 = max i k i . In the empirical comparison, we indeed find that ˆ k 0 is very close to max(ˆ k 1 , ˆ k 2 ) when it is freely estimated jointly with all other model parameters; and its attained maximum log-likelihood is statistically indistinguishable from that of the model which imposes the restriction k 0 = max i k i . Marc S. Paolella ALRIGHT: Asymmetric LaRge-Scale (I)GARCH with Hetero-Tails

Effect of Parameter k 0 R = I , k 0 = 1 , k 1 = 2 , k 2 = 4 R = I , k 0 = 3 , k 1 = 2 , k 2 = 4 15 15 10 10 5 5 0 0 −5 −5 −10 −10 −15 −15 −80 −60 −40 −20 0 20 40 60 80 −80 −60 −40 −20 0 20 40 60 80 R = I , k 0 = 4 , k 1 = 2 , k 2 = 4 R = I , k 0 = 10 , k 1 = 2 , k 2 = 4 15 15 10 10 5 5 0 0 −5 −5 −10 −10 −15 −15 −80 −60 −40 −20 0 20 40 60 80 −80 −60 −40 −20 0 20 40 60 80 Marc S. Paolella ALRIGHT: Asymmetric LaRge-Scale (I)GARCH with Hetero-Tails

FaK with Asymmetric Marginals: AFaK Introduce noncentrality parameters θ i ∈ R , i = 1 , 2 , . . . , d , so that, with φ k ,θ ( x ) and Φ k ,θ ( x ) the pdf and cdf of the noncentral t distribution at x ∈ R , f X ( x ; k , R , θ ) is d � � � Φ − 1 k 0 ,θ 0 (Φ k 1 ,θ 1 ( x 1 )) , . . . , Φ − 1 ψ k 0 ,θ 0 (Φ k d ,θ d ( x d )); R , k 0 φ k i ,θ i ( x i ) , i =1 still in conjunction with (2), and with θ 0 = 0. The location-scale variant f T ( y ; k , µ , σ , R , θ ) is analogous to (2), and we write T ∼ AFaK ( k , µ , σ , R , θ ), for asymmetric FaK . We have V ( T ) = MRM , where M = diag ( σ ⊙ v 1 / 2 ), where v = ( V ( S 1 ) , . . . , V ( S d )) ′ , for S i = ( T i − µ i ) /σ i ∼ t ′ ( k i , θ i , 0 , 1), with the variance of S i computed from � k i � 1 / 2 Γ( k i / 2 − 1 / 2) � � E S i = θ i , k i > 1 , (3) 2 Γ( k i / 2) E [ S 2 i ] = [ k i / ( k i − 2)](1 + θ 2 i ) for k i > 2, V ( S ) = E [ S 2 ] − ( E [ S ]) 2 . Marc S. Paolella ALRIGHT: Asymmetric LaRge-Scale (I)GARCH with Hetero-Tails

Examples of Bivariate AFaK k 0 = k 1 = k 2 = 3, θ 0 = 0, θ 1 = − 0 . 7, θ 2 = − 0 . 7 k 0 = 4, k 1 = 1 . 5, k 2 = 3 . 5, θ 0 = 0, θ 1 = − 0 . 7, θ 2 = − 0 . 7 10 10 r = 0 r = 0 5 5 0 0 −5 −5 −10 −10 −10 −5 0 5 10 −10 −5 0 5 10 k 0 = k 1 = k 2 = 3, θ 0 = − 0 . 7, θ 1 = − 0 . 7, θ 2 = − 0 . 7 k 0 = 4, k 1 = 1 . 5, k 2 = 3 . 5, θ 0 = − 0 . 7, θ 1 = − 0 . 7, θ 2 = − 0 . 7 10 10 r = 0 r = 0 5 5 0 0 −5 −5 −10 −10 −10 −5 0 5 10 −10 −5 0 5 10 Marc S. Paolella ALRIGHT: Asymmetric LaRge-Scale (I)GARCH with Hetero-Tails

Bivariate Example: BoA and Wal-Mart loglik a FaK k 0 k 1 k 2 µ 1 µ 2 scale MLE − 7086 . 1 3.975 1.464 3.873 0.0331 0 . 0027 0.857 std err Hess (0.497) (0.067) (0.344) (0.026) (0.028) (0.028) std err NPB (0.562) (0.058) (0.376) (0.025) (0.031) (0.024) std err PB (0.526) (0.068) (0.349) (0.026) (0.028) (0.029) AFaK k 0 k 1 k 2 θ 1 θ 2 µ 1 µ 2 scale MLE − 7079 . 1 3.903 1.472 3.879 − 0 . 165 0.136 0.190 − 0 . 192 0.856 std err Hess (0.481) (0.068) (0.344) (0.055) (0.094) (0.057) (0.119) (0.028) std err NPB (0.551) (0.059) (0.374) (0.060) (0.094) (0.062) (0.115) (0.024) std err PB (0.486) (0.081) (0.330) (0.049) (0.096) (0.051) (0.122) (0.030) S-L 1 v 1 v 2 µ 1 µ 2 scale MLE − 7092 . 2 1.618 3.731 0.0275 − 0 . 0068 0.922 std err Hess (0.074) (0.306) (0.027) (0.028) (0.029) std err NPB (0.078) (0.317) (0.026) (0.029) (0.027) std err PB (0.082) (0.337) (0.035) (0.036) (0.033) S-L 2 v 1 v 2 µ 1 µ 2 scale MLE − 7142 . 7 1.601 4.813 0.0313 − 0 . 0057 0.926 std err Hess (0.077) (0.491) (0.027) (0.030) (0.030) std err NPB (0.072) (0.508) (0.027) (0.028) (0.027) std err PB (0.067) (0.498) (0.024) (0.024) (0.029) Marc S. Paolella ALRIGHT: Asymmetric LaRge-Scale (I)GARCH with Hetero-Tails

Data Scatterplot and the Fitted Densities Fitted Shaw−Lee Model #1 Fitted Shaw−Lee Model #2 10 10 5 5 Wal−Mart Wal−Mart 0 0 −5 −5 −10 −10 −10 −5 0 5 10 −10 −5 0 5 10 Bank of America Bank of America Fitted FaK Distribution Fitted AFaK Distribution 10 10 5 5 Wal−Mart Wal−Mart 0 0 −5 −5 −10 −10 −10 −5 0 5 10 −10 −5 0 5 10 Bank of America Bank of America Marc S. Paolella ALRIGHT: Asymmetric LaRge-Scale (I)GARCH with Hetero-Tails