Volatility in Several Series The various xARCH models provide many - PowerPoint PPT Presentation

ST 810-006 Statistics and Financial Risk Volatility in Several Series The various xARCH models provide many ways to model volatility dynamics in a single series. Many areas of financial risk involve the joint behavior of several variables. Graphs show that volatility typically varies simultaneously across series. Individual xARCH models could be fitted to each variable, but would not be linked. 1 / 17

ST 810-006 Statistics and Financial Risk GARCH Recall the GARCH model: y t = financial variable, such as log-return on some asset or perhaps the residual of some variable from an ARIMA model for the conditional mean structure. Assume y t = σ t ǫ t , where { ǫ t } are independent and follow some fixed distribution (standard normal, standardized t , . . . ). GARCH(1 , 1): σ 2 t = ω + α y 2 t − 1 + βσ 2 t − 1 . σ − 1 The “standardized residuals” are z t = ˆ t y t . 2 / 17

ST 810-006 Statistics and Financial Risk Multivariate GARCH With several series, y t = vector of J financial variables . The analog of σ 2 t = var( y t | y t − 1 , y t − 2 , . . . ) is Σ t = var( y t | y t − 1 , y t − 2 , . . . ) . The purpose of a multivariate xARCH model is to provide a recursive expression for Σ t . 3 / 17

ST 810-006 Statistics and Financial Risk PC-GARCH : use PCA. First stage: fit univariate GARCH models to the individual series. Put standardized residuals for time t in a vector z t = S − 1 1 , t y t , where S 1 , t is the diagonal matrix of first stage conditional standard deviations. Set up a data matrix and use the SVD: Z = ( T × J ) data matrix with rows z ′ t , t = 1 , 2 , . . . , T � � � � � � V ′ = U D J × J T × J J × J , diagonal 4 / 17

ST 810-006 Statistics and Financial Risk Second stage: fit univariate GARCH models to the PC scores (here columns of U ; more conventionally, of UD ). Write ǫ t = S − 1 2 , t u t where S 2 , t is the diagonal matrix of second stage conditional standard deviations. Model ǫ t as: N J ( 0 , I ); t J ,ν ( 0 , I ); a meta t -distribution, combining marginal t -distributions and a t -copula, all potentially with different degrees of freedom; some other non-Gaussian distribution with chosen tail length, tail dependence, and shape. 5 / 17

ST 810-006 Statistics and Financial Risk The conditional distribution of y T +1 is represented as y T +1 = S 1 , T +1 z T +1 = S 1 , T +1 VDu T +1 = S 1 , T +1 VDS 2 , T +1 ǫ T +1 where: V and D come from the SVD; S 1 , T +1 and S 2 , T +1 come from the first and second stage GARCH recursions, respectively; ǫ T +1 follows the chosen model. Note that the distribution of ǫ T +1 does not depend on the past, so it is independent of the past. 6 / 17

ST 810-006 Statistics and Financial Risk If the chosen distribution of ǫ T +1 is Gaussian or t , the conditional distribution of y T +1 is in the same family. Otherwise, the conditional distribution of y T +1 is likely to be intractable except for simulation. Either way, use it for instance to compute VaR or ES of a portfolio of the assets on which these are the returns. 7 / 17

ST 810-006 Statistics and Financial Risk Reduced Rank PC-GARCH: recall that if d 2 1 + d 2 2 + · · · + d 2 k ≫ d 2 k +1 + · · · + d 2 J then � � � � � � V ( k ) ′ U ( k ) D ( k ) Z ≈ . k × k , diagonal T × k k × J So only k principal component score series need to be modeled, and ǫ t consists of only k variables whose distributions need to be modeled. If k ≪ J , modeling is much simplified. 8 / 17

ST 810-006 Statistics and Financial Risk PC-GARCH and Reduced Rank PC-GARCH are related to Orthogonal GARCH (OGARCH) and Generalized Orthogonal GARCH (GO-GARCH). These model the conditional covariance matrix of y t directly: Σ t = XD t X ′ where D t is a diagonal matrix of univariate GARCH conditional variances, and: in OGARCH, X is ( J × k ) with orthogonal columns, like V ( k ) ; in GO-GARCH, X is ( J × J ) with no orthogonality constraints. Then y t is modeled as Gaussian or t with covariance matrix Σ t . Note: PC-GARCH makes Σ t more complicated: Σ t = S 1 , t VDS 2 2 , t DV ′ S 1 , t . 9 / 17

ST 810-006 Statistics and Financial Risk Non-PCA Approaches MGARCH is a very general extension of univariate GARCH. MGARCH(1 , 1): vech( Σ t ) = A vech( y t − 1 y ′ t − 1 ) + B vech( Σ t − 1 ) + c where vech( · ) vectorizes the lower triangle of a symmetric matrix:   s 1 , 1 s 2 , 1   .   . .     s J , 1   vech( S ) =   s 2 , 2     s 3 , 2     . .   .   s J , J 10 / 17

ST 810-006 Statistics and Financial Risk MGARCH is over-parametrized: vech( S ) is ( J ( J + 1) / 2 × 1). So A and B are (( J ( J + 1) / 2) × ( J ( J + 1) / 2)), with ∼ J 4 / 4 entries each, and c is (( J ( J + 1) / 2) × 1). Constraining Σ t to be non-negative definite is a challenge. 11 / 17

ST 810-006 Statistics and Financial Risk In diag-MGARCH , A and B are constrained to be diagonal–some improvement. Note: diagonal multiplication of vech( · ) is equivalent to entrywise multiplication: Σ t = A ◦ ( y t − 1 y ′ t − 1 ) + B ◦ Σ t − 1 + C where A , B , and C are now ( J × J ), and “ ◦ ” denotes entrywise (Hadamard, or Schur) product. Constraining Σ t to be non-negative definite is still a challenge: requiring A , B , and C to be non-negative definite is sufficient, but not necessary. No “cross-talk” in diag-MGARCH. 12 / 17

ST 810-006 Statistics and Financial Risk BEKK is a different simplified form of MGARCH: Σ t = A ′ ( y t − 1 y ′ t − 1 ) A + B ′ Σ t − 1 B + C Here A and B are unrestricted, and C is positive definite symmetric. Off-diagonal entries in A and B introduce cross-talk: volatility in one variable can flow into another from either the y t − 1 y ′ t − 1 term or the Σ t − 1 term. 13 / 17

ST 810-006 Statistics and Financial Risk Constant Conditional Correlation (CCC) We can always decompose Σ t : Σ t = D t R t D t , where D t is the diagonal matrix of conditional standard deviations, and R t is the conditional correlation matrix. In CCC, assume that R t = R , constant. Use separate GARCH models to build D t , and estimate R from the standardized residuals z t . 14 / 17

ST 810-006 Statistics and Financial Risk Dynamic Conditional Correlation (DCC) As in CCC, use separate GARCH models to build D t . Then R t = (diag( Q t )) − 1 / 2 Q t (diag( Q t )) − 1 / 2 where Q t satisfies the recursion t − 1 + θ 2 Q t − 1 + (1 − θ 1 − θ 2 ) ¯ Q t = θ 1 z t − 1 z ′ Q . That is, Q t follows an even simpler MGARCH model with scalar multipliers, but driven by the standardized residuals z t instead of the original returns y t . R t simply extracts the correlation structure from Q t . 15 / 17

ST 810-006 Statistics and Financial Risk Role of the Copula In each case, we construct Σ t = var( y t | y t − 1 , y t − 2 , . . . ) . Write ǫ t = Σ − 1 / 2 y t t so that var( ǫ t ) = I J . Here Σ − 1 / 2 could be any inverse square root of Σ t ; e.g. t triangular (Choleski), symmetric positive definite (spectral decomposition). In PC-GARCH, a specific version was implied. 16 / 17

ST 810-006 Statistics and Financial Risk The distribution of ǫ t might be assumed to be independent normal (with no tail dependence), or multivariate t (with positive tail dependence). It could alternatively be constructed to have appropriate tail lengths and appropriate tail dependences by separately estimating: the marginal distribution of each component of ǫ t ; a copula to introduce nonlinear dependence. Note: if Σ − 1 / 2 is non-sparse, the tail properties of components t of ǫ t affect all the components of y t . The symmetric positive definite version relates each component of y t most closely to the corresponding component of ǫ t . 17 / 17