Multivariate GARCH Models. Eduardo Rossi University of Pavia

Transcription

1 Multivariate GARCH Models Eduardo Rossi University of Pavia

2 Multivariate GARCH The extension from a univariate GARCH model to an N-variate model requires allowing the conditional variance-covariance matrix of the N-dimensional zero mean random variables ε t depend on the elements of the information set. Let {z t } be a sequence of (N 1) i.i.d. random vector with the following characteristics: E [z t ] = 0 E [z t z t] = I N z t G(0,I N ) with G continuous density function. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 2

3 Multivariate GARCH Let {ǫ t } be a sequence of (N 1) random vectors generated as: where ǫ t = H 1/2 t z t E t 1 (ǫ t ) = 0 E t 1 (ǫ t ǫ t) = H t where H t is a matrix (N N) positive definite and measurable with respect to the information set Φ t 1, that is the σ-field generated by the past observations: {ǫ t 1, ǫ t 2,...}. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 3

4 Multivariate GARCH The parametrization of H t as a multivariate GARCH, which means as a function of the information set Φ t 1, allows each element of H t to depend on q lagged of the squares and cross-products of ǫ t, as well as p lagged values of the elements of H t. So the elements of the covariance matrix follow a vector of ARMA process in squares and cross-products of the disturbances. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 4

5 Multivariate GARCH-Vech representation Let vech denote the vector-half operator, which stacks the lower triangular elements of an N N matrix as an [N (N + 1)/2] 1 vector. Since the conditional covariance matrix H t is symmetric, vech(h t ) contains all the unique elements in H t. Following Bollerslev et al., a natural multivariate extension of the univariate GARCH(p,q) model is q vech(h t ) = W + A i vech ( ǫ t i ǫ ) p t i + B jvech(h t j ) i=1 j=1 = W + A (L) vech(ǫ t ǫ t) + B (L) vech(h t ) (1) Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 5

6 ¾ ¾ ¾ ¾ Multivariate GARCH-Vech representation W : [N (N + 1) /2] 1 A i, B j : [(N (N + 1) /2) (N (N + 1) /2)] N N(N + 1) 2 N = 3, Vec-GARCH(1,1): h 11,t w 1 a 11 a 12 a 13 h 21,t = w 2 + ¾ a 21 a 22 a 23 h 2,t w 3 a 31 a 32 a 33 ǫ 2 1,t 1 b 11 b 12 b 13 h 11,t 1 ǫ 1,t 1 ǫ 2,t 1 + ¾ b 21 b 22 b 23 h 21,t 1 ǫ 2 2,t 1 b 31 b 32 b 33 h 2,t 1 h ij,t = w i + N j=1 a ijǫ i,t 1 ǫ j,t 1 + N j=1 b ijh ij,t 1 Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 6

7 Multivariate GARCH-Vech representation This general formulation is termed vec [ representation by Engle and Kroner. The number of parameters is 1 + (p + q) [N (N + 1) /2] 2]. Even for low dimensions of N and small values of p and q the number of parameters is very large; for N = 5 and p = q = 1 the unrestricted version of (1) contains 465 parameters. For any parametrization to be sensible, we require that H t be positive definite for all values of ε t in the sample space in the vech representation this restriction can be difficult to check, let alone impose during estimation. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 7

8 Multivariate GARCH-Diagonal vech model A natural restriction is the diagonal representation, in which each element of the covariance matrix depends only on past values of itself and past values of ε jt ε kt. In the diagonal model the A i and B j matrices are all taken to be diagonal. For N = 2 and p = q = 1, the diagonal model is written as: h 11,t w 1 a ε 2 1,t 1 h 21,t = w a 22 0 ε 1,t 1 ε 2,t 1 h 22,t + w a 33 b b 22 0 h 11,t 1 h 21,t 1 ε 2 2,t b 33 h 22,t 1 h ij,t = w i + a iiǫ i,t 1 ǫ j,t 1 + b iih ij,t 1 Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 8

9 Multivariate GARCH-Diagonal vech model Thus the (i, j) th element in H t depends on the corresponding (i, j)th element in ε t 1 ε t 1 and H t 1. This restriction reduces the number of parameters to [N (N + 1) /2](1 + p + q). This model does not allow for causality in variance, co-persistence in variance and asymmetries. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 9

10 Multivariate GARCH-BEKK representation Engle and Kroner (1995) propose a parametrization that imposes positive definiteness restrictions. Consider the following model H t = CC + K q A ik ǫ t i ǫ t ia ik + k=1i=1 K p B ik H t i B ik (2) k=1i=1 where C, A ik and B ik are (N N). The intercept matrix is decomposed into CC, where C is a lower triangular matrix. Without any further assumption CC is positive semidefinite. This representation is general, it includes all positive definite diagonal representations and nearly all positive definite vech representations. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 10

11 Multivariate GARCH-BEKK representation For exposition simplicity we will assume that K = 1: H t = CC + q A i ǫ t i ǫ t ia i + i=1 p B i H t i B i (3) i=1 Consider the simple GARCH(1,1) model: H t = CC + A 1 ǫ t 1 ǫ t 1A 1 + B 1 H t 1 B 1 (4) Proposition 1. (Engle and Kroner) Suppose that the diagonal elements in C are restricted to be positive and that a 11 and b 11 are also restricted to be positive. Then if K = 1 there exists no other C, A 1, B 1 in the model (4) that will give an equivalent representation. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 11

12 Multivariate GARCH-BEKK representation The purpose of the restrictions is to eliminate all other observationally equivalent structures. For example, as relates to the term A 1 ǫ t 1 ǫ t 1A 1 the only other observationally equivalent structure is obtained by replacing A 1 by A 1. The restriction that a 11 (b 11 ) be positive could be replaced with the condition that a ij (b ij ) be positive for a given i and j, as this condition is also sufficient to eliminate A 1 from the set of admissible structures. In the bivariate case the BEKK becomes H t = CC + + a 11 a 12 a 21 a 22 b 11 b 12 b 21 b 22 h 11t 1 ε2 1t 1 h 21t 1 ε 2t 1 ε 1t 1 h 12t 1 h 22t 1 ε 1t 1 ε 2t 1 ε 2 2t 1 b 11 b 12 b 21 b 22 a 11 a 12 a 21 a 22 Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 12

13 Multivariate GARCH-BEKK representation Sufficient condition for positive definiteness of H t in BEKK-GARCH(p,q) model. Proposition 2. (Engle and Kroner) (Sufficient condition) If H 0, H 1,...,H p+1 are all positive definite, then the BEKK parametrization (with K = 1) yields a positive definite H t for all possible values of ε t if C is a full rank matrix or if any B i i = 1,...,p is a full rank matrix. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 13

14 Multivariate GARCH-BEKK representation For simplicity consider the GARCH(1,1) model. The BEKK parameterization is H t = CC + +A 1 ǫ t 1 ǫ t 1A 1 + B 1 H t 1 B 1 The proof proceeds by induction. First H t is p.d. for t = 1: The term A 1 ǫ 0 ǫ 0A 1 is positive semidefinite because ǫ 0 ǫ 0 is positive semidefinite. Also if the null spaces of the matrices of C and B 1 intersect only at the origin, that is at least one of two is full rank then CC + B 1 H 0 B 1 is positive definite. This is true if C or B 1 has full rank. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 14

15 Multivariate GARCH-BEKK representation To show that the null space condition is sufficient CC + B 1 H 0 B 1 is p.d. if and only if x (CC + B 1 H 0 B 1) x > 0 x 0 or (C x) (C x) + ( ) ( ) H 1/2 0 B 1x H 1/2 0 B 1x > 0 x 0 (5) where H 0 = H 1/2 0 H 1/2 0 and H 1/2 0 is full rank. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 15

16 Multivariate GARCH-BEKK representation Defining N (P) to be the null space of the matrix P, (5) is true if and only if ( ) N (C ) N H 1/2 0 B 1 =. ( ) N H 1/2 0 B 1 = N (B 1) because H 1/2 0 is full rank. This implies that CC + B 1 H ( 0 B 1 is positive ) definite if and only if N (C ) N H 1/2 0 B 1 =. Now suppose that H t is positive definite for t = τ. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 16

17 Multivariate GARCH-BEKK representation Then, H τ+1 = CC + A 1 ǫ τ ǫ τa 1 + B 1 H τ B 1 is positive definite if and only if, given that A 1 ǫ τ ǫ τa 1 is positive semidefinite, the null space condition holds, because H τ is positive definite by the induction assumption. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 17

18 Multivariate GARCH-BEKK representation We now examine the relationship between the BEKK and vech parametrizations. The mathematical relationship between the parameters of the two models can be found simply vectorizing the equation (3): vec(h t ) = vec(cc )+ q vec(a i ǫ t i ǫ t ia i)+ i=1 p vec(b i H t i B i) i=1 where vec () is an operator such that given a matrix A (n n), vec(a) is a ( n 2 1 ) vector. The vec () satisfies vec (ABC) = (C A)vec (B) Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 18

19 BEKK and vec representations For A (n n) symmetric, then vech(a) contains precisely the n (n + 1)/2 distinct elements of A and the elements of vec (A) are those of vech(a) with some repetitions. Hence there exists a unique n 2 n (n + 1)/2 which transforms, for symmetric A, vech(a) into vec (A). This matrix is called the duplication matrix and is denoted D n : vec (A) = D n vech(a) where D n is the duplication matrix. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 19

20 BEKK and vec representations Then q vec(h t ) = vec(cc ) + (A i A i ) vec(ε t i ǫ t i) i=1 p + (B i B i )vec(h t i ) i=1 q D N vech(h t ) = D N vech(cc ) + (A i A i )D N vech(ǫ t i ǫ t i) i=1 p + (B i B i )D N vech(h t i ) i=1 Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 20

21 BEKK and vec representations If D N is a full column rank matrix we can define the generalized inverse of D N as: D + N = (D ND N ) 1 D N that is a (N (N + 1) /2) ( N 2) matrix, where D + N D N = I N Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 21

22 BEKK and vec representations This implies that premultiplying by D + N ( q ) vech(h t ) = vech(cc ) + D + N (A i A i ) D N vech(ǫ t i ǫ t i) +D + N i=1 ( p ) (B i B i ) D N vech(h t i ) i=1 The vech model implied by any given BEKK model is unique, while the converse is not true. The transformation from a vech model to a BEKK model (when it exists) is not unique, because for a given A 1 the choice of A 1 is not unique. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 22

23 BEKK and vec representations This can be seen recognizing that (A i A i ) = ( A i A i ) so while A i = D+ N (A i A i )D N is unique, the choice of A i is not unique. It can also be shown that all positive definite diagonal vech models can be written in the BEKK framework. Given A i diagonal matrix, then D + N (A i A i )D N is also diagonal, with diagonal elements given by a ii a jj (1 j i N) (See Magnus). Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 23

24 Multivariate GARCH-Covariance Stationarity Given the vech model vech(h t ) = W + A (L) vech(ε t ε t) + B (L) vech(h t ) the necessary and sufficient condition for covariance stationary of {ε t } is that all the eigenvalues of A (1) + B (1) are less than one in modulus. But defining ( q ) A (1) B (1) = D + N = D + N i=1 (A i A i ) ( q ) (B i B i ) i=1 D N D N Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 24

25 Multivariate GARCH-Covariance Stationarity This implies also that in the BEKK model, {ǫ t } is covariance stationary if and only if all the eigenvalues of ( q ) ( p ) (A i A i ) D N + D + N (B i B i ) D N D + N i=1 i=1 are less than one in modulus. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 25

26 Multivariate GARCH-Covariance Stationarity Let λ 1,...,λ N be the eigenvalues of A i, the eigenvalues of ( q ) (A i A i ) D + N i=1 D N are λ i λ j (1 j i N) (Magnus). Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 26

27 Multivariate GARCH-Covariance Stationarity Given that ǫ t ǫ t = H t + V t with E (V t ) = 0. vech(ǫ t ǫ t) = vech(h t ) + vech(v t ) with E (vech(v t )) = vech(e (V t )) = 0. For a GARCH(1,1), the unconditional covariance matrix, when it exists, is given by vech(ǫ t ǫ t) = W + A 1vech (( ǫ t 1 ǫ )) t 1 [ ( ) +B 1 vech ǫt 1 ǫ t 1 vech(vt 1 ) ] + vech(v t ) Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 27

28 Multivariate GARCH-Covariance Stationarity E (ǫ t ǫ t) = Σ vech(e (ǫ t ǫ t)) = W + (A 1 + B 1) [ vech ( E ( ǫ t 1 ǫ ))] t 1 vech(σ) = [I N A 1 B 1] 1 W and in the BEKK model vec (ǫ t ǫ t) = vec (CC ) + (A 1 A 1 ) vec ( ǫ t 1 ǫ ) t 1 + (B 1 B 1 ) [ vec ( ǫ t 1 ǫ t 1) vec (Vt 1 ) ] + vec (V t ) Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 28

29 Multivariate GARCH-Covariance Stationarity E [vec (ǫ t ǫ t)] = vec (CC )+[(A 1 A 1 ) + (B 1 B 1 )]E [ vec ( )] ǫ t 1 ǫ t 1 vec (Σ) = [I N 2 (A 1 A 1 ) (B 1 B 1 )] 1 vec (CC ) or in vech representation as D N vech(e (ǫ t ǫ t)) = D N vech(cc ) + (A 1 A 1 )D N vech ( E ( ǫ t 1 ǫ )) t 1 + (B 1 B 1 )D N vech ( E ( ǫ t 1 ǫ )) t 1 vech(σ) = [ I N D + N (A 1 A 1 )D N D + N (B 1 B 1 )D N ] 1 vech(cc ) N = N (N + 1) /2. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 29

30 Multivariate GARCH-Covariance Stationarity The diagonal vech model is stationary if and only if the sum a ii + b ii < 1 for all i. In the diagonal BEKK model the covariance stationary condition is that a 2 ii + b2 ii < 1. Only in the case of diagonal models the stationarity properties are determined solely by the diagonal elements of the A i and B i matrices. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 30

31 Multivariate GARCH-Covariance Stationarity For a GARCH(p,q) model vech(e (ǫ t ǫ t)) = vech(σ) = [I N A (1) B (1)] 1 W Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 31

32 Factor-GARCH The Factor GARCH model, introduced by Engle et al. (1990), can be thought of as an alternative simple parametrization of the BEKK model. Suppose that the (N 1) y t has a factor structure with K factors given by the K 1 vector f t and a time invariant factor loadings given by the N K matrix B: y t = Bf t + ǫ t (6) Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 32

33 Factor-GARCH Assume that the idiosyncratic shocks ǫ t have conditional covariance matrix Ψ which is constant in time and positive semidefinite, and that the common factors are characterized by E t 1 (f t ) = 0 E t 1 (f t f t) = Λ t Λ t = diag (λ 1,...,λ K ) and positive definite. The conditioning set is {y t 1,f t 1,...,y 1,f 1 }. Also suppose that E (f t ǫ t) = 0. The conditional covariance matrix of y t equals E t 1 (y t y t) = H t = Ψ + BΛ t B = Ψ + K β k β kλ kt k=1 where β k denotes the kth column in B. Thus, there are K statistics which determine the full covariance matrix. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 33

34 Factor-GARCH Forecasts of the variances and covariances or of any portfolio of assets, will be based only on the forecasts of these K statistics. There exists factor-representing portfolios with portfolio weights that are orthogonal to all but one set of factor loadings: r kt = φ ky t φ kβ j = 1 k = j 0 otherwise the vector of factor-representing portfolios is r t = Φ y t where the columns of matrix Φ are the φ k vectors. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 34

35 Factor-GARCH The conditional variance of r kt is given by V ar t 1 (r kt ) = φ ke t 1 (y t y t) φ k = φ kh t φ k = φ k (Ψ + BΛ t B )φ k = ψ k + λ kt where ψ k = φ k Ψφ k. The portfolio has the exact time variation as the factors, which is why they are called factor-representing portfolios. In order to estimate this model, the dependence of the λ kt s upon the past information set must also be parameterized: θ kt φ kh t φ k = V ar t 1 (r kt ) = ψ k + λ kt So we get that K β k β kθ kt = k=1 K β k β kψ k + k=1 K β k β kλ kt k=1 Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 35

36 Factor-GARCH K β k β kλ kt = k=1 K β k β kθ kt k=1 K β k β kψ k k=1 H t = Ψ + where Ψ = = Ψ + K β k β kλ kt = Ψ + k=1 ( Ψ K K β k β kθ kt k=1 k=1 β k β k ψ k ). K β k β kθ kt k=1 K β k β kψ k k=1 Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 36

37 Factor-GARCH The simplest assumption is that there is a set of factor-representing portfolios with univariate GARCH(1,1) representations. The conditional variance θ kt follows a GARCH(1,1) process θ kt = ω k + α k (φ kǫ t 1 ) 2 + γ k E t 2 ( r 2 kt 1 ) θ kt = ω k + α k φ k θ kt = ω k + α k φ k θ kt = ω k + α k φ k ( ǫt 1 ǫ ) t 1 φk + γ k E t 2 [(φ ky t ) (φ ky t )] ( ) ǫt 1 ǫ t 1 φk + γ k [φ ke t 2 (y t y t) φ k ] ( ) ǫt 1 ǫ t 1 φk + γ k [φ kh t 1 φ k ] Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 37

38 Factor-GARCH The conditional variance-covariance matrix of y t can be written as H t = Ψ + = Ψ + ( = Ψ + K β k β kθ kt k=1 K β k β k { [ ( ωk + α k φ k ǫt 1 ǫ ) ] t 1 φk + γk [φ kh t 1 φ k ] } k=1 ) K β k β kω k + k=1 k=1 K { [ ( ) ] β k β k αk φ k ǫt 1 ǫ t 1 φk + γk [φ kh t 1 φ k ] } H t = Γ + K β k β kθ kt k=1 Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 38

39 Factor-GARCH where Γ = Ψ + K β k β k ω k, therefore H t = Γ + k=1 K [ α k βk φ ( k ǫt 1 ǫ ) t 1 φk β k] K + γ k [β k φ kh t 1 φ k β k] k=1 so that the factor GARCH model is a special case of the BEKK parametrization. Estimation of the factor GARCH model is carried out by maximum likelihood estimation. It is often convenient to assume that the factor-representing portfolios are known a priori. k=1 Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 39

40 Multivariate GARCH - The Constant Correlations Model The time-varying conditional covariances are parameterized to be proportional to the product of the corresponding conditional standard deviations. The model assumptions are: E t 1 [ǫ t ] = 0 E t 1 [ǫ t ǫ t] = H t {H t } ii = σit 2 {H t } ij = σ ijt = ρ ij σ it σ jt i j D t = diag { } σ1t, 2...,σNt 2 Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 40

41 Multivariate GARCH - The Constant Correlations Model Let Γ t denote the matrix of constant correlations {Γ t } ij = {H t } ij [{H t } ii {H t } jj ] 1/2 i, j = 1,...,N the model assumes Γ t = Γ H t = D 1/2 t ΓD 1/2 t H t = ¾ σ 1t σ Nt 1 ρ ρ 1N ρ ρ N 1N ρ N1... ρ NN 1 1 ¾ σ 1t σ Nt. ¾ Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 41

42 Multivariate GARCH - The Constant Correlations Model When N = 2 H t = σ 1t 0 0 σ 2t ρ 12 ρ 21 1 σ 1t 0 0 σ 2t = σ2 1t ρ 12 σ 1t σ 2t ρ 12 σ 1t σ 2t σ 2 2t. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 42

43 Multivariate GARCH - The Constant Correlations Model The sequence of conditional covariance matrices {H t } is guaranteed to be positive definite a.s. for all t, If the conditional variances along the diagonal in the D t matrices are all positive, and the conditional correlation matrix Γ is positive definite Furthermore the inverse of H t is given by H 1 t = D 1/2 t Γ 1 D 1/2 t. When calculating the log-likelihood function only one matrix inversion is required for each evaluation. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 43

44 Asymmetric Multivariate GARCH-in-mean model A general multivariate model can be written as: y t = µ + Π (L)y t 1 + Ψx t 1 + Λvech(H t ) + ǫ t (7) y t : (N 1) Π (L) = Π 1 + Π 2 L + + Π k L k 1 Ψ : (N k) Λ : (N N (N + 1) /2) Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 44

45 Asymmetric Multivariate GARCH-in-mean model x t 1 contains predetermined variables. ǫ t is the vector of innovation with respect to the information set formed exclusively of past realizations of y t. H t = E t 1 (ǫ t ǫ t) H t = CC + q A i (ǫ t i + γ)(ǫ t i + γ) A i + i=1 p B j H t j B j (8) j=1 Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 45

46 Asymmetric Multivariate GARCH-in-mean model We can consider a multivariate generalization of the size effect and sign effect: H t = CC +A 1 ǫ t 1 ǫ t 1A 1+B 1 H t 1 B 1+Dv t 1 v t 1D +G 1ǫ t 1 ǫ t 1G where v t = z t E z t, with z it = ε it / h ii,t and I (ε 1t 1 < 0) g G = I (ε Nt 1 < 0) g NN Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 46

47 Asymmetric Multivariate GARCH-in-mean model When N = 2 v t 1 v t 1 = ¾ ε 1t 1 / h 11,t 1 E ε Ô 2t 1 / h 22,t 1 E Ô ε 1t 1 / h 11,t 1 ε Ô 2t 1 / h 22,t 1 Ô ε 1t 1 / Ô h 11,t 1 E ε 1t 1 / Ô h 11,t 1 ¾ ε 2t 1 / Ô h 22,t 1 E ε 2t 1 / Ô h 22,t 1 = ¾ ( z 1t E z 1t ) 2 ( z 1t E z 1t ) ( z 2t E z 2t ) ( z 2t E z 2t ) ( z 1t E z 1t ) ( z 2t E z 2t ) 2 Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 47

48 Asymmetric Multivariate GARCH-in-mean model G 1ǫ t 1 ǫ t 1G = = g 2 11ε 2 1t 1 g 11g 22ε 1t 1 ε 2t 1 I (ε 1t 1 < 0) g11ε 2 2 1t 1 δ 12 g 11 g 22 ε 1t 1 ε 2t 1 g 11g 22ε 1t 1 ε 2t 1 g 2 22ε 2 2t 1 δ 12 g 11 g 22 ε 1t 1 ε 2t 1 I (ε 2t 1 < 0)g 2 22ε 2 2t 1 δ 12 = I (ε 1t 1 < 0)I (ε 2t 1 < 0) Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 48

49 Estimation procedure Given the model (7)-(8), the log-likelihood funcion for {ε T,...,ε 1 } obtained under the assumption of conditional multivariate normality is: log L T (ε T,...,ε 1 ; θ) = 1 2 [ TN log (2π) + T t=1 The assumption of conditional normality can be quite resctrictive. ( log Ht + ǫ th 1 ) ] t ǫ t The symmetry imposed under normality is difficult to justify, and the tails of even conditional distributions often seem fatter than that of normal distribution. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 49

50 Estimation procedure Let {(y t, x t ) : t = 1, 2,...} be a sequence of observables random vectors with y t (N 1) and x t (L 1). The vector y t contains the endogenous variables and x t contains contemporaneous exogenous variables. w t = (x t, y t 1, x t 1,...,y 1, x 1 ). The conditional mean and variance functions are jointly parametrized by a finite dimensional vector θ: {µ t (w t, θ), θ Θ} {H t (w t, θ), θ Θ} where Θ R P and µ t and H t are known functions of w t and θ. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 50

51 Estimation procedure The validity of most of the inference procedures is proven under the null hypothesis that the first two conditional moments are correctly specified, for some θ 0 Θ, E (y t w t ) = µ t (w t, θ 0 ) (9) V ar (y t w t ) = H t (w t, θ 0 ) t = 1, 2,... (10) The procedure most often used to estimate θ 0 is the maximization of a likelihood function that is constructed under the assumption that y t w t N (µ t,h t ). The approach taken here is the same, but the subsequent analysis does not assume that y t has a conditional normal distribution. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 51

52 Estimation procedure For observation t the quasi-conditional log-likelihood is l t (θ; y t, w t ) = N 2 ln(2π) 1 2 ln H t (w t, θ) 1 2 (y t µ t (w t, θ)) H 1 t (w t, θ) (y t µ t (w t, θ)) Letting ǫ t (y t, w t, θ 0 ) y t µ t (w t, θ) : (N 1) denote the residual function l t (θ) = N 2 log (2π) 1 2 log H t (θ) 1 2 ǫ t (θ)h 1 t (θ) ǫ t (θ) (11) log L T (θ) = T l t (θ) t=1 Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 52

53 Estimation procedure If µ t (w t, θ) and H t (w t, θ) are differentiable on Θ for all relevant w t, and if H t (w t, θ) is nonsingular with probability one for all θ Θ, then the differentiation of (11) yields the (1 P) score function s t (θ): s t (θ) = θ l t (θ) θ µ t (θ) H 1 t (θ)ǫ t (θ) θh t (θ) [ H 1 t (θ) H 1 t (θ) ] vec [ ǫ t (θ) ǫ t (θ) H t (θ) ] where θ µ t (θ) : (N P) θ H t (θ) : ( N 2 P ) Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 53

54 Estimation procedure If the first conditional two moments are correctly specified, the true error vector is defined as ǫ 0 t ǫ t (θ 0 ) = y t µ t (w t, θ 0 ) and E ( ǫ 0 t w t ) = 0, E ( ǫ 0 tǫ 0 t w t ) = Ht (w t, θ 0 ) It follows that under correct specification of the first two conditional moments of y t given w t : E [s t (θ 0 ) w t ] = 0 The score evaluated at the true parameter is a vector of martingale difference with respect to the σ fields {σ (y t, w t ) : t = 1, 2,...}. This result can be used to establish weak consistency of the quasi-maximum likelihood estimator (QMLE). Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 54

55 Estimation procedure For robust inference we also need an expression for the hessian h t (θ) of l t (θ). Define the positive semidefinite matrix a t (θ 0 ) = E [ θ s t (θ 0 ) w t ] = E [ h t (θ 0 ) w t ] : (P P) a t (θ 0 ) = θ µ t (θ 0 ) H 1 t (θ 0 ) θ µ t (θ 0 ) θh t (θ) [ H 1 t (θ) H 1 t (θ) ] θ H t (θ) When the normality assumption holds the matrix a t (θ 0 ) is the conditional information matrix. However, if y t does not have a conditional normal distribution then V ar [s t (θ 0 ) w t ] a t (θ 0 ) and the information matrix equality is violated. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 55

56 Estimation procedure The QMLE has the following properties: [ A 0 1 T B 0 TA 0 1 ] 1/2 ) d T T ( θt θ 0 N (0,IP ) where and A 0 T 1 T B 0 T V ar in addition T E [h t (θ 0 )] = 1 T t=1 [ ] T 1/2 S T (θ 0 ) T E [a t (θ 0 )] t=1 = 1 T T E [ s t (θ 0 ) s t (θ 0 ) ] t=1 Â T A 0 T B T B 0 T p 0 p 0 Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 56

57 Estimation procedure The matrix  1 B T T  1 T is a consistent estimator od the robust asymptotic covariance matrix of ) T ( θt θ 0. In practice, θ T N ( θ,â 1 B ) T T  1 T /T Under normality, the variance estimator can be replaced by  1 T /T (Hessian form) or /T (outer product of the gradient form). B 1 T Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 57

58 Wald Test The null hypothesis is H 0 : r (θ 0 ) = 0 where r : Θ R Q is continuously differentiable on int (Θ) and Q < P. Let R (θ) = θ r (θ) : (Q P) be the gradient of r on int (Θ). If θ 0 int (Θ) and rank (R (θ 0 )) = Q then the Wald statistic ξ W = Tr ) [ ( θt R ( θt )Â 1 B ) ] 1 ) T T Â 1 T ( θt R d H0 r ( θt χ 2 Q. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 58

59 Orthogonal-GARCH model The orthogonal models are particular factor models. They are based on the assumption that the observed data can be obtained by a linear transformation of a set of uncorrelated components. The components are the principal components of the data, or a subset of them. The diagonal matrix V contains the empirical variances of y t : V = diag{s 2 1,...,s 2 N} the standardized returns are u t = V 1/2 y t E[u t ] = 0 E[u t u t] = R Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 59

60 Orthogonal-GARCH model The sample correlation matrix can be decomposed as: R = PΛP P is the orthonormal eigenvectors matrix, Λ is the diagonal matrix of the eigenvalues: Λ = diag{λ 1,...,λ N } ranked in descending order. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 60

61 Orthogonal-GARCH model P satisfies: P = P 1 P P = I N PP = I N R = PΛ 1/2 Λ 1/2 P = LL L = PΛ 1/2 Principal components f t = L 1 u t E[f t f t] = L 1 E[u t u t]l 1 = L 1 RL 1 Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 61

62 Orthogonal-GARCH model E[f t f t] = L 1 PΛ 1/2 Λ 1/2 P L 1 = L 1 LL L 1 = I N Assuming E t 1 [f t f t] = Q t Q t is a diagonal matrix. q ii,t GARCH(p,q) E t 1 [u t u t] = E t 1 [Lf t f tl ] = LQ t L E t 1 [y t y t] = E t 1 [V 1/2 u t u tv 1/2 ] = V 1/2 LQ t L V 1/2 Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 62

63 Orthogonal-GARCH model We can work with a reduced number m < N of principal components (eigenvalues), those which explain most of the variation in the data. L 1 is replaced by a matrix (m N): L 1 m = Λ 1/2 P m P m is a matrix (N m) containing the m eigenvalues of P corresponding to the largest eigenvalues. V 1/2 y t = u t u t = L m f t E t 1 [f t ] = 0 E t 1 [f t f t] = Q t = diag(σ 2 f 1,t,...,σ 2 f m,t) Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 63

64 Generalized Orthogonal-GARCH model Assume that y t Φ t 1 N(0,V t ) The observed economic process y t is governed by a linear combination of independent economic components {f t } y t = Zf t f t uncorrelated components, Z 0. The unobserved components are normalized such that: E[f t f t] = I N V = E[y t y t] = ZZ Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 64

65 Generalized Orthogonal-GARCH model V t = E t 1 [y t y t] = ZE t 1 [f t f t]z = ZH t Z H t = diag{h 1,t,...,h N,t } h i,t = (1 α 1 β i ) + α i y 2 i,t i + β i h i,t 1 i = 1,...,N V = E[V t ] = ZHZ H diagonal. The diagonal decomposition of the unconditional covariance matrix: V = PΛP P: orthogonal matrix, O-GARCH estimator for Z, is only guaranteed to coincide with Z, when the diagonal elements of H are all distinct. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 65

66 Generalized Orthogonal-GARCH model Suppose that H = I V = E[V t ] = ZIZ = ZZ The matrix Z is no longer identified by the eigenvector matrix of V as for every orthogonal matrix Q we have (ZQ)(ZQ) = I The matrix Z is well identified when conditional information is taken into account. Based on singular value decomposition: PΛ 1/2 U 0 = Z Let the estimator of U 0 be U. We restrict U = 1. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 66

67 Generalized Orthogonal-GARCH model The matrices P and Λ have N(N 1) 2 and N parameters, so we have N 2 parameters for the invertible matrix Z. The matrices P and Λ will be estimated by means of unconditional information, as they will extracted from the sample covariance matrix V ( N(N+1) 2 parameters). Conditional information is required to estimate U 0 ( N(N 1) 2 free parameters). The orthogonal U 0 is parameterized by means of rotation matrices. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 67

68 Generalized Orthogonal-GARCH model Rotation matrix: G rs, (N N), r, s N, r < s N: G rs = {g ij } g rr = g ss = cos(θ) g ii = 1 i = 1,...,N i r, s g sr = sin(θ) g rs = sin(θ) and all other elements are zero. N = 3: cos(θ) sin(θ) 0 G 12 = sin(θ) cos(θ) Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 68

69 Generalized Orthogonal-GARCH model Every N-dimensional orthogonal matrix U with det(u) = 1 can be represented as a product of N(N 1) 2 rotation matrices: U = i<j G ij (θ ij ) π θ ij π G ij (θ ij ) performs a rotation in the plane spanned by the i-th and the j-th vectors of the canonical basis of R N over an angle δ ij. N = 3: U = G 12 G 13 G 23 Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 69

70 Generalized Orthogonal-GARCH model Time-varying conditional correlations: R t = D 1 t V t D 1 t D t = (V t I m ) 1/2 N = 2 Z = 1 0 cosθ sinθ θ measures the extent to which the uncorrelated components are mapped in the same direction. For θ = 0 the mapping in not invertible, yielding perfect correlation between the observed variables, whereas θ = π 2, Z = I so that observed variables are completely uncorrelated. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 70

71 Generalized Orthogonal-GARCH model ρ t = Cov t 1 (y 1t, y 2t ) V art 1 (y 1t ) V ar t 1 (y 2t ) In the GO-GARCH model, the conditional correlations matrix is: R t = D 1 t V t D 1 t = D 1 t = ZH t Z D 1 t h 1/2 1t 0 0 h 1/2 1 cosθ 0 sin θ 2t 1 0 cosθ sinθ h 1/2 1t 0 0 h 1/2 2t h 1t h 12t h 12t h 2t Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 71

72 Generalized Orthogonal-GARCH model ρ t = = = h 1t cos(θ) h1t h 1t cos 2 (θ) + h 2t sin 2 (θ) cos (θ) cos 2 (θ) + h 1t h 2t sin 2 (θ) zt tan 2 (θ) where z t = h 1t h 2t. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 72

73 Generalized Orthogonal-GARCH model Suppose GO-GARCH(1,1), parameters to be estimated by means of the conditional information are (α, β, θ ) α = (α 1,...,α N ) β = (β 1,...,β N ) θ = (θ 1,...,θ m ) m = The log-likelihood can be expressed: l t = 1 2 = 1 2 = 1 2 N(N 1) 2 ( N log(2π) + log Vt + x tv 1 t x t ) ( N log(2π) + log Zθ H t Z θ + x t(z θ H t Z θ) 1 x t ) ( N log(2π) + log Zθ Z θ + log H t + x t(z θ H t Z θ) 1 x t ) log L T = t l t Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 73

74 Generalized Orthogonal-GARCH model Two-step estimation 1. Z θ = PΛ 1/2 U 0 where P and Λ from the sample covariance matrix: V = P Λ P Z θ = P Λ 1/2 U 2. Estimate (α, β, θ ) by maximizing the log L T. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 74

75 The Dynamic Conditional Correlation GARCH Model The conditional correlation between two random variables, X t and Y t is defined as: ρ Y X,t = E t 1 (X t µ x,t )(Y t µ y,t ) Et 1 (X t µ x,t ) 2 E t 1 (Y t µ y,t ) 2 Assets returns conditional distribution: r t I t 1 N(0,H t ) H t = D t R t D t. where the standardized returns are: ǫ t = D 1 t r t E t 1 [ǫ t ] = 0 E t 1 [ǫ t ǫ t] = D 1 t H t D 1 t = R t = {ρ ij,t } Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 75

76 The Dynamic Conditional Correlation GARCH Model The conditional correlation estimator is ρ ij,t = q ij,t qii,t q jj,t. A possible specification of the conditional correlation matrix is based on the Exponential Smoothing Estimator: ˆρ ij,t = t 1 s=1 λs ǫ i,t s ǫ j,t s ( t 1 s=1 λs ǫ 2 i,t s )( t 1 s=1 λs ǫ 2 j,t s ) = [R t ] ij, q ij,t = (1 λ)(ǫ i,t 1 ǫ j,t 1 ) + λ(q ij,t 1 ) (12) where q ijt = E t 1 [ǫ it ǫ jt ]. The process of q ij,t is I(1), so nonstationary. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 76

77 The Dynamic Conditional Correlation GARCH Model Since the conditional correlation between ǫ it and ǫ jt is equal to Corr t 1 [r it, r jt ], and E t 1 [ǫ t ǫ t] = R t, we can use the conditional variance of ǫ t to describe the conditional correlation of r t. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 77

78 The Dynamic Conditional Correlation GARCH Model Engle (2003) extends the Bollerslev s Constant Conditional Correlation Model, Dynamic Conditional Correlation Model. Where q ij,t, the conditional covariances, are modelled with a GARCH(1,1) model q ij,t = ρ ij + α(ǫ i,t 1 ǫ j,t 1 ρ ij ) + β(q ij,t 1 ρ ij ) (13) q ij,t = ( ) 1 α β ρ 1 β ij + α β s ǫ i,t (s+1) ǫ j,t (s+1). (14) s=0 The unconditional expectation for the cross-products is ( ) 1 α β E(q ij,t ) = ρ ij + α 1 β 1 β E(ǫ i,t 1ǫ j,t 1 ) = 1 α β 1 α β ρ ij = ρ ij The unconditional variances of ǫ i,t is ρ ii = 1. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 78

79 The Dynamic Conditional Correlation GARCH Model ρ ij represents the unconditional correlation between ǫ it and ǫ jt ; equivalently it can be interpreted as the unconditional expected value of q ij,t. ρ ij = E(ǫ i,t ǫ j,t ) E(ǫ 2 i,t )E(ǫ2 j,t ) = E(ǫ i,t ǫ j,t ) with E[ǫ 2 it] = 1 Moreover ρ ii = 1 E(ǫ i,t ǫ j,t ) = E [ E t 1 [ǫ i,t ǫ j,t ] ] = E[q ij,t ]. ρ ii = E[q iit ] = E [ E t 1 [ǫ 2 i,t] ] = E[1] = 1 The model is mean-reverting if α + β < 1, while it coincides with the integrated exponential smoothing when α + β = 1. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 79

80 The Dynamic Conditional Correlation GARCH Model The conditional covariance matrix is definite positive, Q t, as long as it is a weighted average of definite matrices and semidefinite matrices. In matrix from: Q t = Q(1 α β) + α(ǫ t 1 ǫ t 1) + β(q t 1 ) (15) where Q is the unconditional covariance matrix of ǫ t. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 80

81 The Dynamic Conditional Correlation GARCH Model Clearly more complex positive definite multivariate GARCH models could be used for the correlation parameterization as long as the unconditional moments are set to the sample correlation matrix. For example, the MARCH family of Ding and Engle (2001) can be expressed in first order form as: Q t = Q (ιι A B) + A ǫ t 1 ǫ t 1 + B Q t 1 (16) where denotes the Hadamard product ({A B} ij = a ij b ij ). Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 81

82 The Dynamic Conditional Correlation GARCH Model The DCC model specification: D 2 t = diag{ω i } + diag{κ i } r t 1 r t 1 + diag{λ i } D 2 t 1 Q t = S (ιι A B) + A ǫ t 1 ǫ t 1 + B Q t 1 R t = diag{q t } 1 Q t diag{q t } 1. The assumption of normality gives rise to a likelihood function. Without this assumption, the estimator will still have the QML interpretation. The second equation simply expresses the assumption that each of the assets follows a univariate GARCH process. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 82

83 The Dynamic Conditional Correlation GARCH Model A real square matrix A, is positive definite if and only if B = A 1 AA 1 is positive definite, with A = diag{a}. In order to ensure that H t is positive definite we must have that D 1 t H t D 1 is positive definite. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 83

84 The Dynamic Conditional Correlation GARCH Model If the following restrictions on the univariate GARCH parameters are satisfied for all series i [1,...,N] : 1. ω i > 0 2. κ i and λ i such that D 2 ii,t > 0 with probability 1 3. D 2 ii,0 > 0 4. The roots of 1 κ i Z λ i Z are outside the unit circle. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 84

85 The Dynamic Conditional Correlation GARCH Model and the parameters in the DCC (with A = αi and B = βi) satisfy: 1. α 0 2. β 0 3. α + β 1 4. The minimum eigenvalue of Q 0 > δ > 0 (where Q 0 must be positive definite) then H t is positive definite t T. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 85

86 The Dynamic Conditional Correlation GARCH Model The log-likelihood function can be written as: log L T = 1 2 = 1 2 = 1 2 T (N log(2π) + log H t + r th 1 t r t ) t=1 T t=1 (N log(2π) + log D t R t D t + r td 1 t R 1 t D 1 t r t ) T (N log(2π) + 2 log D t + log R t + ǫ tr 1 t ǫ t ) t=1 Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 86

87 The Dynamic Conditional Correlation GARCH Model Adding and subtracting r td 1 t T log L T = 1 2 t=1 D 1 t r t = ǫ tǫ t (N log(2π) + 2 log D t + r td 1 t ǫ tǫ t + log R t + ǫ tr 1 t ǫ t ) = 1 T (N log(2π) + log D t 2 + r 2 td 2 t r t ) 1 2 t=1 T t=1 (ǫ tr 1 t ǫ t ǫ tǫ t + log R t ) D 1 t r t Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 87

88 The Dynamic Conditional Correlation GARCH Model Volatility component: L V (θ) log L V,T (θ) = 1 2 Correlation component: T t=1 (N log(2π) + log D t 2 + r td 2 t r t ) L C (θ, φ) log L C,T (θ, φ) = 1 2 T t=1 (ǫ tr 1 t ǫ t ǫ tǫ t + log R t ) θ denotes the parameters in D t and φ the parameters in R t. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 88

89 The Dynamic Conditional Correlation GARCH Model L(θ, φ) = L V (θ) + L C (θ, φ) ( ) L V (θ) = 1 T N log(2π) + log(h i,t ) + r2 i,t. 2 h i,t t=1 i=1 The likelihood is apparently the sum of individual GARCH likelihoods, which will be jointly maximized by separately maximizing each term. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 89

90 The Dynamic Conditional Correlation GARCH Model Two-step procedure: ˆθ = arg max{l V (θ)} max φ {L C(ˆθ, φ)}. Under regularity conditions, consistency of the first step will ensure consistency of the second step. The maximum of the second step will a function of the first step parameter estimates. If the first step is consistent then the second step will be too as long as the function is continuous in a neighborhood of the true parameters. Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 90

91 The Dynamic Conditional Correlation GARCH Model Two step GMM problem (Newey and McFadden, 1994). Consider the moment condition corresponding to the first step θ L V (θ) = 0 The moment corresponding to the second step is φ L(θ, φ) = 0 Under regularity conditions the parameter estimates will be consistent, and asymptotically normal, with asymptotic covariance matrix V (φ) = [ ] 1 [ ( E( φφ L C ) E { φ L C E( φθ L C )[E( θθ L V ] 1 θ L V } { φ L C E( φθ L C )[E( θθ L V ] 1 θ L V } )][ ] 1 E( φφ L C ) Eduardo Rossi - Econometria dei mercati finanziari (avanzato) 91