Unconditional Covariance in a CCC GARCH Model

Summary

The document asks how to derive the unconditional covariance matrix for a constant conditional correlation (CCC) GARCH model. It specifies a multivariate process with a fixed correlation matrix and time-varying component variances, whose dynamics depend on a constant term and lagged variance and squared returns.

It gives the unconditional variance expression for each component, obtained from the model’s variance recursion, then asks how to extend that result to cross-covariances. The text does not supply a derivation, answer, empirical evidence, or conditions for existence and stationarity. Readers should treat it as a focused theoretical question rather than a complete explanation; the covariance result would require additional assumptions or derivation beyond what is presented.

Key ideas

• The model uses a fixed conditional correlation matrix while individual conditional variances change over time.
• The variance recursion combines a constant term with lagged variances and squared returns.
• The document states an expression for unconditional component variances.
• It leaves the unconditional cross-covariance matrix unresolved.

Tags

# Unconditional correlation in CCC GARCH


# Unconditional correlation in CCC GARCH












What is the unconditional correlation (covariance) in CCC GARCH model

$$\mathbf{x}_{t+1} = \mathbf{H}_{t+1}^{1/2} \mathbf{z}_{t+1}$$ $$\mathbf{H}_{t+1} = \mathbf{D}_{t+1}^{1/2} \mathbf{R} \mathbf{D}_{t+1}^{1/2}$$ $$\mathbf{D}_{t+1} = \mathrm{diag}(\mathbf{h}_{t+1})$$ $$\mathbf{h}_{t+1} = \mathbf{w} + \mathbf{A} \mathbf{h}_{t} + \mathbf{B} \left(\mathbf{x}_{t} \odot \mathbf{x}_{t} \right)$$ $$\mathrm{E}[\mathbf{z}_{t+1}\mathbf{z}_{t+1}^{T}] = \mathbf{I}$$

where ($\mathbf{a} \odot \mathbf{b})_{i} = a_ib_i$ is the Hadamard element-wise product.

Since $\mathrm{E}[\mathbf{x}_{t+1}] = \mathbf{0}$, the unconditional variance can be found to be given by

$$\mathrm{E}[\mathbf{x}_{t+1} \odot \mathbf{x}_{t+1}] = \left( \mathbf{I - A -B}\right)^{-1} \mathbf{w}$$

However, I can't work out (or find anywhere in the literature) what the expression is for the unconditional covariance matrix $$\mathrm{E}[\mathbf{x}_{t+1} \mathbf{x}_{t+1}^{T}] = ?$$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.