Serving the Quantitative Finance Community

 
User avatar
GuitarTrader
Topic Author
Posts: 1
Joined: July 13th, 2011, 1:01 pm

about the unconditional variance of th EGARCH(1,1) model

March 4th, 2014, 2:14 am

I need an EGARH expert.how to calculate the unconditional variance of th EGARCH(1,1) model..I find a reference in a matlab website, but I think it's wrong. http://www.mathworks.cn/cn/help/econ/as ... ns.htmlany idea ?
Last edited by GuitarTrader on March 3rd, 2014, 11:00 pm, edited 1 time in total.
 
User avatar
Alan
Posts: 2958
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

about the unconditional variance of th EGARCH(1,1) model

March 4th, 2014, 7:19 pm

Not an EGARCH expert, but just thinking out loud.If [$]y_t = \log \sigma_t[$], the recursion (in non-standard notation) is equivalent to(*) [$] y_t = \omega + \alpha y_{t-1} + \beta \epsilon_t[$], where [$]\epsilon_t \sim N(0,1)[$].Take exp(iu ...) of both sides of (*) and take the time-0 expectation. Let [$]t \rightarrow \infty[$]. The result is(**) [$]\Phi(u) = e^{i \omega u - u^2 \beta^2/2} \Phi(u \alpha)[$], where [$]\Phi(u)[$] is the char. func. of the stationary density of [$]\log \sigma_t[$].If you can solve (**) for [$]\Phi(u)[$], then [$]\lim_{t \rightarrow \infty} E[\sigma_t^2] = \Phi(-2 i)[$],assuming that exists. =================================================================p.s. No guarantees, but if my algebra is right, carrying it all through, I get [$]\lim_{t \rightarrow \infty} E[\sigma_t^2] = \exp \left\{ \frac{2 \omega (1 + \alpha) + 2 \beta^2}{1-\alpha^2} \right\}[$], [$]\quad (|\alpha| < 1) [$] Definitely deserves a Monte Carlo check, which I leave to you.
Last edited by Alan on March 3rd, 2014, 11:00 pm, edited 1 time in total.
 
User avatar
Alan
Posts: 2958
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

about the unconditional variance of th EGARCH(1,1) model

March 5th, 2014, 4:08 pm

Update: I have spot checked my formula now, and it looks fine -- so it is very likely correct.