Isogai, T., Building a dynamic correlation network for fat-tailed financial asset returns, Applied Network Science (7):-24, 206,

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2 3 4 5 6 Isogai, T., Building a dynamic correlation network for fat-tailed financial asset returns, Applied Network Science (7):-24, 206, http://link.springer.com/article/0.007/s409-06-0008-x (TMU) 206 8 29 2 / 34

:? : : spurious correlation (TMU) 206 8 29 3 / 34

Static : Correlation matrix Adjacency matrix 0 Correlation network 0.2 0. 0 0.2 0. Data Preprocess 0.2 0.2 0 = 0. 0. 0 Asset price e.g. log returns 0 Data filtering for network building Network analysis Dynamic : Conditional (dynamic) correlation Data Correlation matrix Correlation matrix 2 Correlation matrix 3 Correlation matrix 4 Time series Unconditional (static) correlation Correlation matrix (TMU) 206 8 29 4 / 34

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: ρ X,Y = Cov (X, Y ) Var (X) Var (Y ) ρ X,Y?. White noise x -20-5 -0-5 0 x2-25 -20-5 -0-5 0 2. Large shock 0 50 00 50 200 250 300 0 50 00 50 200 250 300 Index N (0, ) cor(x, x2) = 0 N (0, 0) cor(x, x2) = 0.8 Index (TMU) 206 8 29 6 / 34

GARCH : i.i.d.; variance= Fat-tailed return = mean + Volatility Standardized residual Data Location Scaling factor (Deterministic) Probabilistic variable Distorted correlation Reliable correlation (TMU) 206 8 29 7 / 34

ARMA GARCH (M)GARCH: r t = µ t + ε t = µ t + H /2 t z t () µ t = E (r t F t ), E (z t ) = 0, Var (z t ) = I n where H t is a conditional variance covariance matrix (volatility), I n is an identity matrix of order n, and F t is the information set at time t. Mean model: conditional means modeled by ARMA(P, Q), separately P Q µ t = µ + a i r t i + b j ε t j (2) i= j= Volatility model: conditional volatilities as a vector form of GARCH(p,q) q p h t = ω + A i ε t i ε t i + B i h t i (3) i= j= where denotes the Hadamard operator (the entrywise product). (TMU) 206 8 29 8 / 34

: Data filtering 0.2 0. 0 0 0.2 0. stock GARCH filtering 0.2 0. 0.2 0 0. 0 0 Return data stock2 stock3 stock4 GARCH filtering GARCH filtering.. Correlation matrix 0.2 0. 0.2 0. 0.2 0.2 0. 0.2 0. 0.2 0. 0. Adjacency matrix Static network 0 0 0 0 0.2 0. 0 0.2 0. 0.2 0 0 0.2 0. 0.2 0 0. 0.2 0 0 0. 0 0. 0 0 0 0 Correlation matrices Adjacency matrices Dynamic network Moving window (- - ->) ( >) (TMU) 206 8 29 9 / 34

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() Moving window ( ) Unfiltered log-returns R t : R t t window (e.g., 30 ). Time Non-overlapping window Time Overlapping window Type sample Type2 sample 0.2 0. 0.2 0. 0.2 0.2 0. 0.2 0. 0.2 0. 0. Correlation matrices Spurious correlation problem (lagged effect) Window. Moving window (TMU) 206 8 29 / 34

(2) DCC GARCH: GARCH stock stock2 GARCH filtering volatility volatility Dynamic Conditional Correlation residual residual Joint distribution modeling Correlation dynamics modeling Multiple correlation matrices t-copula f ( ) r t µ t, h t, R t, η = c S t (u t,..., u N t R t, η) N i= hi t f i t (z i t θ i ) (4) where u i t = F i (r i t µ i t, h i t, θ i ), θ i is a parameter set including the ARMA GARCH parameters and distributional parameters of i.i.d. residual z i, c S t ( ) is the Student t-copula density function, and η is the shape parameter of the Student t-copula. (TMU) 206 8 29 2 / 34

(2) DCC GARCH: R t : Q R t = diag (Q t ) 2 Q t diag (Q t ) 2 (5) m ) n ( Q t = Q + a i (z t i z t i Q + b j Qt i Q ) (6) i= j= Q t R t proxy z t standardized residuals (shocks). (TMU) 206 8 29 3 / 34

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2 : 50 2 : ( ) ) : 2008 205 6 ; (2008 ) (20 ). : (DCC) moving window ( 200 ) Comparative analysis Stock returns. Transportation 50 2. Banks 50 Correlation. Dynamic 2. Moving window Observation. Intensity 2. Structure (TMU) 206 8 29 5 / 34

DCC-GARCH : DCC GARCH R t DCC Estimation Result Sector m, n a b b2 b+b2 η Transportation, 2 0.006 0.46 0.4493 0.8654 30.460 equipment (0.0007) (0.0808) (0.0805) (.2898) Banks, 2 0.0094 0.3940 0.446 0.8402 2.5952 (0.0009) (0.0686) (0.0695) (.0033) Note: DCC order (m, n) and parameters a, b, and b 2 are defined in (6). η is the shape parameter of the Student t-copula in (4). (TMU) 206 8 29 6 / 34

( ): Correlation matrix Eigenvalue decomposition Largest eigenvalue (λmax scalar) Intensity Largest eigenvector (qmax vector) Direction observe systematic changes in correlation observe divergence from usual relationship q max t q max (benchmark). normalized cosine distance) cos(θ) = x y x y = xi y i, γ(x, y) = cos(θ) (7) x 2 i y 2 i ν max t = γ(q max t, q max ) std(γ max ) (TMU) 206 8 29 7 / 34 (8)

: Eigenvalues of Dynamic Correlation Matrix Eigenvalue 99 percentile Largest 2nd largest Tracy-Widom (min - max) (min - max) Transportation equipment Dynamic 24.28-27.7 0.97 -.40 Moving Average 9.97-3.39.32-2.67 Banks Dynamic 3.56-35.20.25 -.73 Moving Average 25.89-39.60 0.84-2.70.38.38 Note: Eigenvaues of R t are calculated on every trading day during the observation period. The min and max represent the minimum and maximum of the vector of corresponding eigenvalues, respectively. (TMU) 206 8 29 8 / 34

: 28 27 Moving window (right) 3 29 27 26 25 25 23 Dynamic correlation (left) 24 2008/0 2009/0 200/0 20/0 202/0 203/0 204/0 205/0 2 9 DCC : (2008, 20). DCC moving window. Moving window lagged effects.. (TMU) 206 8 29 9 / 34

: 36 35 Moving window (right) 39 37 34 35 33 33 3 32 Dynamic correlation (left) 29 27 3 2008/0 2009/0 200/0 20/0 202/0 203/0 204/0 205/0 25 DCC :. Moving window lagged effect.. (TMU) 206 8 29 20 / 34

: 4 2 0 8 6 Dynamic correlation (left) Moving window (right) 6 5 4 3 4 2 2 0 2008/0 2009/0 200/0 20/0 202/0 203/0 204/0 205/0 0 DCC :. DCC. Moving window lagged effects. (TMU) 206 8 29 2 / 34

: 22 20 8 6 Dynamic correlation (left) 5 4 4 2 0 8 6 4 2 Moving window (right) 3 2 0 2008/0 2009/0 200/0 20/0 202/0 203/0 204/0 205/0 0 DCC : 2008. Moving window lagged effects (TMU) 206 8 29 22 / 34

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: D(A).. i j>i D(A) = A ij mean (k) mean (k) = (9) n (n ) /2 n n k i = j i A ij (0) k i (TMU) 206 8 29 24 / 34

: C(A). 0. C(A) = n ( max (k) n 2 n = n n 2 ( max (k) n mean (k) n ) D(A) ) max (k) n D(A) () H(A).. H(A) = var (k) mean (k) = n i k2 i ( i k i) 2 (2) (TMU) 206 8 29 25 / 34

: 0.55 0.53 B A 0.5 0.49 0.47 0.45 2008/ 2009/ 200/ 20/ 202/ 203/ 204/ 205/ A B. (TMU) 206 8 29 26 / 34

: 0.3 A 0.2 0. B 0.0 2008/ 2009/ 200/ 20/ 202/ 203/ 204/ 205/. (TMU) 206 8 29 27 / 34

: 0.4 0.3 0.2 0. A B 0.0 2008/ 2009/ 200/ 20/ 202/ 203/ 204/ 205/ A B. (TMU) 206 8 29 28 / 34

: 0.70 B 0.68 A 0.66 0.64 0.62 C 0.60 2008/ 2009/ 200/ 20/ 202/ 203/ 204/ 205/ A B.. (TMU) 206 8 29 29 / 34

: 0.08 C A 0.07 0.06 B 0.05 2008/ 2009/ 200/ 20/ 202/ 203/ 204/ 205/. (TMU) 206 8 29 30 / 34

: 0. C 0.0 0.09 0.08 A B 0.07 2008/ 2009/ 200/ 20/ 202/ 203/ 204/ 205/ A B. (TMU) 206 8 29 3 / 34

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: DCC GARCH.. moving window. 2.. :.. (TMU) 206 8 29 33 / 34

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