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 $$\mathrm{E}[\mathbf{x}_{t+1} \mathbf{x}_{t+1}^{T}] = ?$$

  • 1
    $\begingroup$ As stated here by Bauwens et al. on p.11: "The unconditional variances are easily obtained, as in the univariate case, but the unconditional covariances are difficult to calculate because of the nonlinearity in $H_{t+1}$." $\endgroup$ – skoestlmeier Sep 27 '18 at 15:55

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