- #1
Usagi
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Question: Assume a bivariate GARCH process as follows:
\begin{align} r_{mt} &= \sigma_{mt}\epsilon_{mt} \ \ \ \cdots \ \ \ \text{(1)} \\ r_{it}&=\sigma_{it}\rho_{it}\epsilon_{mt}+\sigma_{it}\sqrt{1-\rho_{it}^2}\xi_{it} \ \ \ \cdots \ \ \ \text{(2)} \\ (\epsilon_{mt}, \xi_{it}) & \sim S \end{align}
where:
$\sigma_{mt}$ is the conditional standard deviation of $r_{mt}$
$\sigma_{it}$ is the conditional standard deviation of $r_{it}$
$\rho_{it}$ is the conditional correlation between $r_{it}$ and $r_{mt}$
$(\epsilon_{mt}, \xi_{it})$ are the shocks that drive the system. The shocks $(\epsilon_{mt}, \xi_{it})$ are independent and identically distributed and have zero mean, unit variance and zero covariance. However they are not necessarily independent of each other. The distribution $S$ is unspecified.
From Eqns. $(1)$ and $(2)$ we have:
$$r_{mt} = \frac{\sigma_{mt}}{\sigma_{it}\rho_{it}}r_{it} - \frac{\sigma_{mt}\sqrt{1-\rho_{it}^2}}{\rho_{it}}\xi_{it}$$
Define $CoVaR$ as:
$$Pr\left(r_{mt} \le CoVaR \mid r_{it} = C \right)=\alpha$$
or equivalently,
$$1-Pr\left(r_{mt} \le CoVaR \mid r_{it} = C \right)=1-\alpha \\ Pr\left(\xi_{it} \le \frac{\rho_{it}}{\sigma_{mt}\sqrt{1-\rho_{it}^2}} \left(\frac{\sigma_{mt}}{\sigma_{it}\rho_{it}}C - CoVaR \right)\mid r_{it} = C \right)=1-\alpha \ \ \ \cdots \ \ \ (3)$$----------
Need clarification #1: The question then goes on to state: In the special case where the conditional mean function of $\xi_{it}$ is linear in $r_{it}$, the first two conditional moments of $\xi_{it}$, given $r_{it} = C$, can be expressed as:
$$\displaystyle \mathbb{E}\left(\xi_{it} \mid r_{it} = C\right) = \frac{cov(\xi_{it}, r_{it})}{\sigma_{it}^2} \times C \\ \mathbb{V}(\xi_{it} \mid r_{it} ) = \mathbb{V}(\xi_{it}) - \mathbb{V}_{r_{it}}\left[\mathbb{E}\left(\xi_{it} \mid r_{it} \right) \right] = \mathbb{V}(\xi_{it})\times\left[1-\left( \frac{cov(\xi_{it}, r_{it})}{\sigma_{it}^2}\right)^2 \sigma_{it}^2 \right]$$
Where does the expression $\displaystyle \mathbb{V}(\xi_{it})\times\left[1-\left( \frac{cov(\xi_{it}, r_{it})}{\sigma_{it}^2}\right)^2 \sigma_{it}^2 \right]$ come from? How do you derive this expression?
Need clarification #2: The question continues to state: Consider $G(\cdot)$, the conditional (location-scale) demeaned and standardized cdf of $\xi_{it}$ such that:
$$ \mathbb{E}\left[\frac{1}{\rho_{it}}\left(\xi_{it} - \frac{\sqrt{1-\rho_{it}^2}}{\sigma_{it}} \times C \right) \mid r_{it} = C\right] =0 \\ \mathbb{V}\left[\frac{1}{\rho_{it}}\left(\xi_{it} - \frac{\sqrt{1-\rho_{it}^2}}{\sigma_{it}} \times C \right) \mid r_{it} = C \right] =1$$
Then Eqn. $(3)$ can be expressed as:
$$\frac{1}{\rho_{it}} \left[ \frac{\rho_{it}}{\sigma_{mt}\sqrt{1-\rho_{it}^2}} \left(\frac{\sigma_{mt}}{\sigma_{it}\rho_{it}}C - CoVaR \right) - \frac{\sqrt{1-\rho_{it}^2}}{\sigma_{it}} \times C \right] = G^{-1} \left(1-\alpha\right) \ \ \ \cdots (4)$$
Where does Eqn. $(4)$ come from? How do you derive it?
Thanks in advance!
\begin{align} r_{mt} &= \sigma_{mt}\epsilon_{mt} \ \ \ \cdots \ \ \ \text{(1)} \\ r_{it}&=\sigma_{it}\rho_{it}\epsilon_{mt}+\sigma_{it}\sqrt{1-\rho_{it}^2}\xi_{it} \ \ \ \cdots \ \ \ \text{(2)} \\ (\epsilon_{mt}, \xi_{it}) & \sim S \end{align}
where:
$\sigma_{mt}$ is the conditional standard deviation of $r_{mt}$
$\sigma_{it}$ is the conditional standard deviation of $r_{it}$
$\rho_{it}$ is the conditional correlation between $r_{it}$ and $r_{mt}$
$(\epsilon_{mt}, \xi_{it})$ are the shocks that drive the system. The shocks $(\epsilon_{mt}, \xi_{it})$ are independent and identically distributed and have zero mean, unit variance and zero covariance. However they are not necessarily independent of each other. The distribution $S$ is unspecified.
From Eqns. $(1)$ and $(2)$ we have:
$$r_{mt} = \frac{\sigma_{mt}}{\sigma_{it}\rho_{it}}r_{it} - \frac{\sigma_{mt}\sqrt{1-\rho_{it}^2}}{\rho_{it}}\xi_{it}$$
Define $CoVaR$ as:
$$Pr\left(r_{mt} \le CoVaR \mid r_{it} = C \right)=\alpha$$
or equivalently,
$$1-Pr\left(r_{mt} \le CoVaR \mid r_{it} = C \right)=1-\alpha \\ Pr\left(\xi_{it} \le \frac{\rho_{it}}{\sigma_{mt}\sqrt{1-\rho_{it}^2}} \left(\frac{\sigma_{mt}}{\sigma_{it}\rho_{it}}C - CoVaR \right)\mid r_{it} = C \right)=1-\alpha \ \ \ \cdots \ \ \ (3)$$----------
Need clarification #1: The question then goes on to state: In the special case where the conditional mean function of $\xi_{it}$ is linear in $r_{it}$, the first two conditional moments of $\xi_{it}$, given $r_{it} = C$, can be expressed as:
$$\displaystyle \mathbb{E}\left(\xi_{it} \mid r_{it} = C\right) = \frac{cov(\xi_{it}, r_{it})}{\sigma_{it}^2} \times C \\ \mathbb{V}(\xi_{it} \mid r_{it} ) = \mathbb{V}(\xi_{it}) - \mathbb{V}_{r_{it}}\left[\mathbb{E}\left(\xi_{it} \mid r_{it} \right) \right] = \mathbb{V}(\xi_{it})\times\left[1-\left( \frac{cov(\xi_{it}, r_{it})}{\sigma_{it}^2}\right)^2 \sigma_{it}^2 \right]$$
Where does the expression $\displaystyle \mathbb{V}(\xi_{it})\times\left[1-\left( \frac{cov(\xi_{it}, r_{it})}{\sigma_{it}^2}\right)^2 \sigma_{it}^2 \right]$ come from? How do you derive this expression?
Need clarification #2: The question continues to state: Consider $G(\cdot)$, the conditional (location-scale) demeaned and standardized cdf of $\xi_{it}$ such that:
$$ \mathbb{E}\left[\frac{1}{\rho_{it}}\left(\xi_{it} - \frac{\sqrt{1-\rho_{it}^2}}{\sigma_{it}} \times C \right) \mid r_{it} = C\right] =0 \\ \mathbb{V}\left[\frac{1}{\rho_{it}}\left(\xi_{it} - \frac{\sqrt{1-\rho_{it}^2}}{\sigma_{it}} \times C \right) \mid r_{it} = C \right] =1$$
Then Eqn. $(3)$ can be expressed as:
$$\frac{1}{\rho_{it}} \left[ \frac{\rho_{it}}{\sigma_{mt}\sqrt{1-\rho_{it}^2}} \left(\frac{\sigma_{mt}}{\sigma_{it}\rho_{it}}C - CoVaR \right) - \frac{\sqrt{1-\rho_{it}^2}}{\sigma_{it}} \times C \right] = G^{-1} \left(1-\alpha\right) \ \ \ \cdots (4)$$
Where does Eqn. $(4)$ come from? How do you derive it?
Thanks in advance!