Intro VEC and BEKK Example Factor Models Cond Var and Cor Application Ref 4. MGARCH

Size: px

Start display at page:

Download "Intro VEC and BEKK Example Factor Models Cond Var and Cor Application Ref 4. MGARCH"

Edith Harvey
5 years ago
Views:

1 ntro VEC and BEKK Example Factor Models Cond Var and Cor Application Ref 4. MGARCH JEM 140: Quantitative Multivariate Finance ES, Charles University, Prague Summer 2018 JEM 140 () 4. MGARCH Summer / 26

2 ntro VEC and BEKK Example Factor Models Cond Var and Cor Application Ref Motivation Multivatiate volatility models add cross-covariances of asset price movements in a portfolio to univariate modeling Covariances are important for potfolio analysis Asset allocation and risk management Asset pricing Volatility transition and spill-over e ects Drawback: number of parameters increases rapidly with portfolio size We need to trade o exibility and accuracy vs parsimony and implementation feasibility JEM 140 (ES) 4. MGARCH Summer / 26

3 ntro VEC and BEKK Example Factor Models Cond Var and Cor Application Ref Notation Framework Consider an (N 1) stochastic vector process fr t g of asset returns, with E [r t ] = 0 Denote by F t 1 the information set at t 1 Let where: r t = H 1/2 t η t jf t 1 H t = [h ij ] is the (N N) covariance matrix of r t η t is an (N 1) iid vector error process s.t. E η t η 0 t = JEM 140 (ES) 4. MGARCH Summer / 26

4 ntro VEC and BEKK Example Factor Models Cond Var and Cor Application Ref Model Overview 1. The covariance matrix H t is modeled directly: VEC and BEKK type models 2. Factor models 3. Model conditional variances and correlations: CCC type models JEM 140 (ES) 4. MGARCH Summer / 26

5 ntro VEC and BEKK Example Factor Models Cond Var and Cor Application Ref Models of Conditional Covariance Matrix The VEC-GARCH model (Bollerslev et al, 1998): vech(h t ) = c + q j=1 A j vech(r t j r 0 t j ) + p j=1 B j vech(h t j ) Straightforward generalization of the univariate GARCH model Dimensionality: c is N(N + 1)/2, A j and B j are N(N + 1)/2 N(N + 1)/2 Every conditional covariance is a function of all lagged: variances covariances squared returns cross-products of returns JEM 140 (ES) 4. MGARCH Summer / 26

6 ntro VEC and BEKK Example Factor Models Cond Var and Cor Application Ref VEC-GARCH Likelihood function: L(θ) = T `t(θ) = c 1 T t=1 2 ln jh t j t=1 T 1 2 rth 0 t 1 r t (1) t=1 Assuming η t is a Gaussian process Need to evaluate H 1 t for each t JEM 140 (ES) 4. MGARCH Summer / 26

7 ntro VEC and BEKK Example Factor Models Cond Var and Cor Application Ref VEC-GARCH Advantages Very exible Potentially very accurate Disadvantages Curse of dimensionality (e.g. when p = 1, q = 1, N = 10, the model has 3080 parameters) Computationally very demanding Potentially unstable numerical inversion of H t in likelihood evaluation Di cult to ensure H t stays positive de nite JEM 140 (ES) 4. MGARCH Summer / 26

8 ntro VEC and BEKK Example Factor Models Cond Var and Cor Application Ref Diagonal VEC Simpli ed version: A j and B j are diagonal Only (p + q + 1)N(N + 1)/2 parameters (e.g. when p = q = 1, N = 10, the model has 55 parameters) Disadvantage: Very restrictive model No interaction allowed between conditional variances and covariances JEM 140 (ES) 4. MGARCH Summer / 26

9 ntro VEC and BEKK Example Factor Models Cond Var and Cor Application Ref BEKK BEKK (Engle and Kroner, 1995): H t = CC 0 + q K j=1 k=1 A 0 kj r t jr 0 t ja kj + p K j=1 k=1 B 0 kj H t jb kj where: C, A, B are (N N) parameter matrices C is lower-triangular The constant term CC 0 ensures positive de niteness of H t JEM 140 (ES) 4. MGARCH Summer / 26

10 ntro VEC and BEKK Example Factor Models Cond Var and Cor Application Ref BEKK Advantages Still quite exible Potentially accurate Positive de niteness of H t automatically satis ed Disadvantages N(N + 1)/2 + (p + q)k (N N) parameters (e.g. when p = q = K = 1, N = 10, the model has 255 parameters) Fewer parameters than VEC but still many overall Matrix inversion of H t required at each t for likelihood evaluation JEM 140 (ES) 4. MGARCH Summer / 26

11 ntro VEC and BEKK Example Factor Models Cond Var and Cor Application Ref Diagonal and Scalar BEKK Diagonal BEKK: A, B are diagonal matrices Advantage: parsimonious (N(N + 1)/2 + (p + q)kn parameters) Disadvantage: restrictive - no covariance dynamics Scalar BEKK: A = a, B = b, where a, b 2 R Advantage: very parsimonious (N(N + 1)/2 + (p + q)k parameters) Disadvantage: very restrictive - no variance or covariance dynamics Both are special cases of VEC and BEKK JEM 140 (ES) 4. MGARCH Summer / 26

12 ntro VEC and BEKK Example Factor Models Cond Var and Cor Application Ref Example Consider the monthly simple returns, including dividends, of two major drug companies, P zer and Merck, from January 1965 to December 2003 for 468 observations. JEM 140 (ES) 4. MGARCH Summer / 26

13 ntro VEC and BEKK Example Factor Models Cond Var and Cor Application Ref Estimated Volatilities with BEKK(1,1) JEM 140 (ES) 4. MGARCH Summer / 26

14 ntro VEC and BEKK Example Factor Models Cond Var and Cor Application Ref Factor GARCH Assumption: returns are generated by a (relatively small) number of "factors" with a GARCH-type structure Advantage: dimensionality reduction when K dim(factors) < dim(r t ) = N Engle (1990): H t = Ω + K k=1 w k w 0 k f k,t k = 1,..., K, where: Ω is an (N N) PSD matrix w k are (N 1) linearly independent vectors of factor weights f k,t are factors with the GARCH structure with ω k, α k, β k 2 R f k,t = ω k + α k (γ 0 k r t 1) 2 + β k f k,t 1 JEM 140 (ES) 4. MGARCH Summer / 26

15 ntro VEC and BEKK Example Factor Models Cond Var and Cor Application Ref Factor GARCH Engle (1990) application Factor 1: value-weighted stock index returns Factor 2: average T-bill returns Disadvantage: possible correlation among factors leads to loss of e ciency Follow-up papers: uncorrelated factor structure r t = W t z t where: z t is an unobservable vector of uncorrelated factors estimated from data W t is a linear transformation matrix JEM 140 (ES) 4. MGARCH Summer / 26

16 ntro VEC and BEKK Example Factor Models Cond Var and Cor Application Ref GO-GARCH Generalized Orthogonal (GO-)GARCH (van der Weide, 2002): H z t = ( A B) + A (z t 1 z 0 t 1) + BH z t 1 where: E [z t zt 0 ] = (normalization) A, B are (N N) diagonal parameter matrices denotes Hadamard (i.e. elementwise) product Then, H t = WH z t W 0 N = w (k) w(k) 0 hz k,t k=1 where: w (k ) are the columns of the matrix W hk,t z are the diagonal elements of the matrix Hz t Disadvantage: in the GO-GARCH K dim(factors) = dim(r t ) = N JEM 140 (ES) 4. MGARCH Summer / 26

17 ntro VEC and BEKK Example Factor Models Cond Var and Cor Application Ref GOF-GARCH Generalized Orthogonal Factor (GOF-) GARCH (Lanne and Saikkonen, 2007): W = CV obtained via the polar decomposition, where: C is SPD (N N) matrix V is an orthogonal (N N) matrix Since E [r t r 0 t] = WW 0 = CC 0 the matrix C is estimated using the spectral decomposition C = UΛ 1/2 U 0 where the columns of U are the eigenvectors of E [r t r 0 t] and the diagonal matrix Λ contains its eigenvalues. Combines advantages of both factor models (reduced number of factors) and orthogonal models (ease of estimation) JEM 140 (ES) 4. MGARCH Summer / 26

18 ntro VEC and BEKK Example Factor Models Cond Var and Cor Application Ref CCC Based on the decomposition of H t into conditional standard deviations and correlations Constant Conditional Correlation (CCC-) GARCH (Bollerslev, 1990): where: H t = D t PD t (2) D t = diag(h1t 1/2,..., h 1/2 Nt ) P = [ρ ij ], PD with ρ ii = 1 for i = 1,..., N (CCC matrix) For the o -diagonal elements of H t this implies [H t ] ij = h 1/2 it h 1/2 jt ρ ij, i 6= j JEM 140 (ES) 4. MGARCH Summer / 26

19 ntro VEC and BEKK Example Factor Models Cond Var and Cor Application Ref CCC Each r t follows a univariate GARCH model, yielding h t = ω + q j=1 A j r (2) p t j + B j h t j=1 j where: h t is an (N 1) vector of conditional variances ω is an (N 1) vector of parameters A, B are diagonal (N N) matrices r (2) t = r t r t JEM 140 (ES) 4. MGARCH Summer / 26

20 ntro VEC and BEKK Example Factor Models Cond Var and Cor Application Ref ECCC Extended CCC (ECCC-) GARCH (Jeantheanu, 1998) A, B are full (N N) matrices (no longer diagonal) Thus, h it = ω i + a 11 r 2 1,t a 1N r 2 N,t 1 + b 11 h 1,t b 1N h N,t 1 Advantages: Past squared returns and variances of all series enter individual variance equations Richer autocorrelation structure than CCC JEM 140 (ES) 4. MGARCH Summer / 26

21 ntro VEC and BEKK Example Factor Models Cond Var and Cor Application Ref CCC Estimation The CCC structure greatly simpli es estimation Due to the decomposition (2) the likelihood (1) has the following simple structure: T `t(θ) = c t=1 1 2 T t=1 N i=1 ln jh it j T 1 2 ln jpj t=1 T 1 2 rtd 0 t 1 P 1 Dt 1 t=1 Advantages: P has to be inverted only once per maximization iteration Dt 1 = diag(1/h1t 1/2,..., 1/h 1/2 Nt ) Easy numerical estimation Disadvantages: The assumption of constant conditional correlations is too restrictive in many applications JEM 140 (ES) 4. MGARCH Summer / 26 r t

22 ntro VEC and BEKK Example Factor Models Cond Var and Cor Application Ref DCC Dynamic Conditional Correlation (DCC) GARCH (Engle, 2002): Q t = (1 a b)s + aε t 1 εt bq t 1 P t = ( Q t ) 1/2 Q t ( Q t ) 1/2 (rescaling) where: a, b 2 R, s.t. a + b < 1 S is the unconditional correlation matrix of the standardized errors ε t Advantage: Dynamics in correlations relative to CCC Disadvantage: All correlations are restricted to have the same dynamic structure JEM 140 (ES) 4. MGARCH Summer / 26

23 ntro VEC and BEKK Example Factor Models Cond Var and Cor Application Ref Application Model comparison of several (bivariate) MGARCH models diag BEKK, GOF, DCC Data: daily returns of S&P 500 index futures and 10-year bond futures, January August 2003 No theoretical consensus on how stocks and long-term bonds are related The long-run correlation is state-dependent and varies with other macro-economic variables JEM 140 (ES) 4. MGARCH Summer / 26

24 ntro VEC and BEKK Example Factor Models Cond Var and Cor Application Ref Estimated Conditional Correlations JEM 140 (ES) 4. MGARCH Summer / 26

25 ntro VEC and BEKK Example Factor Models Cond Var and Cor Application Ref Estimates Sample Correlations of the Estimated Conditional Correlations JEM 140 (ES) 4. MGARCH Summer / 26

26 ntro VEC and BEKK Example Factor Models Cond Var and Cor Application Ref Comparison Assess model t by likelihood value at the parameter estimates Models that are relatively easy to estimate t the data less well than the other models Models with more complicated structure attain higher likelihood values JEM 140 (ES) 4. MGARCH Summer / 26

27 ntro VEC and BEKK Example Factor Models Cond Var and Cor Application Ref Reference Reference: Silvennoinen, A. and Teräsvirta, T. "Multivariate GARCH Models", in Hanbook of Financial Time Series, Andersen et al (Eds.), Springer, JEM 140 (ES) 4. MGARCH Summer / 26

Financial Times Series. Lecture 12

Financial Times Series. Lecture 12 Financial Times Series Lecture 12 Multivariate Volatility Models Here our aim is to generalize the previously presented univariate volatility models to their multivariate counterparts We assume that returns