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

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1 H28. (TMU) / 34

2 Isogai, T., Building a dynamic correlation network for fat-tailed financial asset returns, Applied Network Science (7):-24, 206, (TMU) / 34

3 :? : : spurious correlation (TMU) / 34

4 Static : Correlation matrix Adjacency matrix 0 Correlation network Data Preprocess = 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) / 34

5 (TMU) / 34

6 : ρ X,Y = Cov (X, Y ) Var (X) Var (Y ) ρ X,Y?. White noise x x Large shock Index N (0, ) cor(x, x2) = 0 N (0, 0) cor(x, x2) = 0.8 Index (TMU) / 34

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

8 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) / 34

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

10 (TMU) / 34

11 () 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 Correlation matrices Spurious correlation problem (lagged effect) Window. Moving window (TMU) / 34

12 (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) / 34

13 (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) / 34

14 (TMU) / 34

15 2 : 50 2 : ( ) ) : ; (2008 ) (20 ). : (DCC) moving window ( 200 ) Comparative analysis Stock returns. Transportation Banks 50 Correlation. Dynamic 2. Moving window Observation. Intensity 2. Structure (TMU) / 34

16 DCC-GARCH : DCC GARCH R t DCC Estimation Result Sector m, n a b b2 b+b2 η Transportation, equipment (0.0007) (0.0808) (0.0805) (.2898) Banks, (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) / 34

17 ( ): 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) / 34 (8)

18 : Eigenvalues of Dynamic Correlation Matrix Eigenvalue 99 percentile Largest 2nd largest Tracy-Widom (min - max) (min - max) Transportation equipment Dynamic Moving Average Banks Dynamic Moving Average 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) / 34

19 : Moving window (right) Dynamic correlation (left) /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) / 34

20 : Moving window (right) Dynamic correlation (left) /0 2009/0 200/0 20/0 202/0 203/0 204/0 205/0 25 DCC :. Moving window lagged effect.. (TMU) / 34

21 : Dynamic correlation (left) Moving window (right) /0 2009/0 200/0 20/0 202/0 203/0 204/0 205/0 0 DCC :. DCC. Moving window lagged effects. (TMU) / 34

22 : Dynamic correlation (left) Moving window (right) /0 2009/0 200/0 20/0 202/0 203/0 204/0 205/0 0 DCC : Moving window lagged effects (TMU) / 34

23 (TMU) / 34

24 : 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) / 34

25 : 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) / 34

26 : B A / 2009/ 200/ 20/ 202/ 203/ 204/ 205/ A B. (TMU) / 34

27 : 0.3 A B / 2009/ 200/ 20/ 202/ 203/ 204/ 205/. (TMU) / 34

28 : A B / 2009/ 200/ 20/ 202/ 203/ 204/ 205/ A B. (TMU) / 34

29 : 0.70 B 0.68 A C / 2009/ 200/ 20/ 202/ 203/ 204/ 205/ A B.. (TMU) / 34

30 : 0.08 C A B / 2009/ 200/ 20/ 202/ 203/ 204/ 205/. (TMU) / 34

31 : 0. C A B / 2009/ 200/ 20/ 202/ 203/ 204/ 205/ A B. (TMU) / 34

32 (TMU) / 34

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

34 . (TMU) / 34

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