EGARCH模型无条件方差是什么
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EGARCH(1,1)模型的无条件方差是什么？

Chemist_MZ

[LaTex]ln(\sigma_t^2)=\omega+\beta ln(\sigma_{t-1}^2)+\alpha \left| \frac{u_{t-1}}{\sigma_{t-1}}\right|+Y(\frac{u_{t-1}}{\sigma_{t-1}})[/LaTex]

take expectation on both sides:

[LaTex]E(ln(\sigma_t^2))=\omega+\beta E(ln(\sigma_{t-1}^2))+\alpha \sqrt{\frac{2}{\pi}}[/LaTex] (note: [LaTex]z=\frac{u_{t-1}}{\sigma_{t-1}} is N(0,1), E(|z|)=\sqrt{\frac{2}{\pi}}[/LaTex])

let long term log variance [LaTex]E(ln(\sigma^2))=x[/LaTex]
[LaTex]x=\omega+\beta x+\alpha \sqrt{\frac{2}{\pi}}[/LaTex]

[LaTex]x=\frac{\omega+\alpha \sqrt{\frac{2}{\pi}}}{1-\beta}[/LaTex]

because E(f(y)) <>f(E(y)) (if f is not a linear function. in this case f=ln()), therefore we can use an approximation. If you really want a variance not log long term variance you can use approximation, given y is not too large. Usually variance is a small number.

[LaTex]ln(y) \approx y-1, E(ln(y))\approx E(y)-1, E(y)\approx E(ln(y))+1[/LaTex]

[LaTex]E(\sigma^2) \approx \frac{\omega+\alpha \sqrt{\frac{2}{\pi}}}{1-\beta}+1[/LaTex]
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