Financial Econometrics – Slides-14: Multivariate Volatility Models
Financial Econometrics
Slides-14: Multivariate Volatility Models
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School of Economics1
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Dr. School of Economics 1
Multivariate GARCH Models
• Multivariate GARCH models are used to estimate and to
forecast covariances and correlations.
• The basic formulation is similar to that of the GARCH model,
but where the covariances as well as the variances are
permitted to be time-varying.
• There are 3 main classes of multivariate GARCH formulation
that are widely used: VECH, diagonal VECH and BEKK.
VECH and Diagonal VECH
• e.g. suppose that there are two variables used in the model.
The conditional covariance matrix is denoted H t, and would
be 2× 2. Ht and VECH(Ht) are
, VEC (Ht) =
h11th22t
Dr. School of Economics 2
VECH and Diagonal VECH
• In the case of the VECH, the conditional variances and
covariances would each depend upon lagged values of all of
the variances and covariances and on lags of the squares of
both error terms and their cross products.
• In matrix form, it would be written
VECH(Ht) = C + AVECH(Ξt−1Ξ
t−1) + BVECH(Ht−1)
Ξt |ψt−1 ∼ N(0,Ht)
Dr. School of Economics 3
VECH and Diagonal VECH (Cont’d)
• Writing out all of the elements gives the 3 equations as
h11t = c11 + a11u
1t−1 + a12u
2t−1 + a13u1t−1u2t−1 + b11h11t−1
+ b12h22t−1 + b13h12t−1
h22t = c21 + a21u
1t−1 + a22u
2t−1 + a23u1t−1u2t−1 + b21h11t−1
+ b22h22t−1 + b23h12t−1
h12t = c31 + a31u
1t−1 + a32u
2t−1 + a33u1t−1u2t−1 + b31h11t−1
+ b32h22t−1 + b33h12t−1
Dr. School of Economics 4
VECH and Diagonal VECH (Cont’d)
• Such a model would be hard to estimate. The diagonal VECH
is much simpler and is specified, in the 2 variable case, as
h11t = α0 + α1u
1t−1 + α2h11t−1
h22t = β0 + β1u
2t−1 + β2h22t−1
h12t = γ0 + γ1u1t−1u2t−1 + γ2h12t−1
Dr. School of Economics 5
BEKK and Model Estimation for M-GARCH
• Neither the VECH nor the diagonal VECH ensure a positive
definite variance-covariance matrix.
• An alternative approach is the BEKK model (Engle & Kroner,
• The BEKK Model uses a Quadratic form for the parameter
matrices to ensure a positive definite variance / covariance
matrix H t.
• In matrix form, the BEKK model is
′W + A′Ht−1A + B
Dr. School of Economics 6
BEKK and Model Estimation for M-GARCH
• Model estimation for all classes of multivariate GARCH model
is again performed using maximum likelihood with the
following LLF:
log |Ht |+ Ξ′tH
where N is the number of variables in the system (assumed 2
above), θ is a vector containing all of the parameters, and T is
the number of obs.
Dr. School of Economics 7
Correlation Models and the CCC
• The correlations between a pair of series at each point in time
can be constructed by dividing the conditional covariances by
the product of the conditional standard deviations from a
VECH or BEKK model
• A subtly different approach would be to model the dynamics
for the correlations directly
• In the constant conditional correlation (CCC) model, the
correlations between the disturbances to be fixed through time
• Thus, although the conditional covariances are not fixed, they
are tied to the variances
• The conditional variances in the fixed correlation model are
identical to those of a set of univariate GARCH specifications
(although they are estimated jointly):
hii ,t = ci + ai�
i ,t−i + bihii ,t−1, i = 1, . . . ,N
Dr. School of Economics 8
More on the CCC
• The off-diagonal elements of Ht , hij ,t(i 6= j), are defined
indirectly via the correlations, denoted ρij :
hij ,t = ρijh
jj ,t , i , j = 1, . . . ,N, i < j • Is it empirically plausible to assume that the correlations are constant through time? • Several tests of this assumption have been developed, including a test based on the information matrix due and a • There is evidence against constant correlations, particularly in the context of stock returns. Dr. School of Economics 9 The Dynamic Conditional Correlation Model • Several different formulations of the dynamic conditional correlation (DCC) model are available, but a popular specification is due to Engle (2002) • The model is related to the CCC formulation but where the correlations are allowed to vary over time. • Define the variance-covariance matrix, Ht , as Ht = DtRtDt • Dt is a diagonal matrix containing the conditional standard deviations (i.e. the square roots of the conditional variances from univariate GARCH model estimations on each of the N individual series) on the leading diagonal • Rt is the conditional correlation matrix • Numerous parameterisations of Rt are possible, including an exponential smoothing approach Dr. School of Economics 10 The DCC Model – A Possible Specification • A possible specification is of the MGARCH form: Ht = S ◦ (ιι′ − A− B) + A ◦ ut−1u′t−1 + B ◦ Ht−1 • S is the unconditional correlation matrix of the vector of standardised residuals (from the first stage estimation), • ι is a vector of ones • Ht is an N × N symmetric positive definite variance-covariance matrix. • ◦ denotes the Hadamard or element-by-element matrix multiplication procedure. • This specification for the intercept term simplifies estimation and reduces the number of parameters. Dr. School of Economics 11 The DCC Model – A Possible Specification • Engle (2002) proposes a GARCH-esque formulation for dynamically modelling Ht with the conditional correlation matrix, Rt then constructed as Rt = diag{Q∗t } −1Htdiag{Q∗t } where diag(·) denotes a matrix comprising the main diagonal elements of (·) and Q∗ is a matrix that takes the square roots of each element in H. • This operation is effectively taking the covariances in Ht and dividing them by the product of the appropriate standard deviations in Q∗t to create a matrix of correlations. Dr. School of Economics 12 DCC Model Estimation • The model may be estimated in a single stage using ML although this will be difficult. So Engle advocates a two-stage procedure where each variable in the system is first modelled separately as a univariate GARCH • A joint log-likelihood function for this stage could be constructed, which would simply be the sum (over N) of all of the log-likelihoods for the individual GARCH models • In the second stage, the conditional likelihood is maximised with respect to any unknown parameters in the correlation Dr. School of Economics 13 DCC Model Estimation (Cont’d) • The log-likelihood function for the second stage estimation will be of the form `(θ2|θ1) = log |Rt |+ u′tR • where θ1 and θ2 denote the parameters to be estimated in the 1st and 2nd stages respectively. Dr. School of Economics 14 DCC Example Dr. School of Economics 15 Asymmetric Multivariate GARCH • Asymmetric models have become very popular in empirical applications, where the conditional variances and / or covariances are permitted to react differently to positive and negative innovations of the same magnitude • In the multivariate context, this is usually achieved in the Glosten et al. (1993) framework • Kroner and Ng (1998), for example, suggest the following extension to the BEKK formulation (with obvious related modifications for the VECH or diagonal VECH models) ′W + A′Ht−1A + B where zt−1 is an N-dimensional column vector with elements taking the value −�t−1 if the corresponding element of �t−1 is negative and zero otherwise. Dr. School of Economics 16 An Example: Estimating a Time-Varying Hedge Ratio for FTSE Stock Index Returns (Brooks, Henry and Persand, 2002). • Data comprises 3580 daily observations on the FTSE 100 stock index and stock index futures contract spanning the period 1 January 1985–9 April 1999. • Several competing models for determining the optimal hedge ratio (OHR) are constructed. Define the hedge ratio as β. – No hedge (β=0) – Näıve hedge (β=1) – Multivariate GARCH hedges: • Symmetric BEKK • Asymmetric BEKK In both cases, estimating the OHR involves forming a 1-step ahead forecast and computing Dr. School of Economics 17 OHR Results Symmetric Asymmetric Unhedged Naive hedge time-varying hedge time-varying hedge β = 0 β = −1 βt = (1) (2) (3) (4) (5) Return 0.0389 −0.0003 0.0061 0.0060 {2.3713} {−0.0351} {0.9562} {0.9580} Variance 0.8286 0.1718 0.1240 0.1211 Out-of-sample Symmetric Asymmetric Unhedged Naive hedge time-varying hedge time-varying hedge β = 0 β = −1 βt = Return 0.0819 −0.0004 0.0120 0.0140 {1.4958} {0.0216} {0.7761} {0.9083} Variance 1.4972 0.1696 0.1186 0.1188 Dr. School of Economics 18 Plot of the OHR from Multivariate GARCH – OHR is time-varying and less than 1 – M-GARCH OHR provides a better hedge, both in-sample and out-of-sample. – No role in calculating OHR for asymmetries Dr. School of Economics 19 程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com