Possible errata for formula II.4.28 (EGARCH() long-term-var)

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giancarp
Posts: 3
Joined: Wed Mar 12, 2014 6:05 pm

Possible errata for formula II.4.28 (EGARCH() long-term-var)

Postby giancarp » Wed Mar 12, 2014 6:33 pm

Dear Professor Alexander,

Let me start this email with my gratitude for writing the excellent series of books: “Market Risk Analysis”.
I have been studying the first two volumes and “Practical Financial Econometrics” is one of the most useful books on econometrics that I own.

I have been experimenting with the EGARCH(p,q) model and I wonder if there is a small errata in the formula for the long-term-variance at page 152.
Instead of computing the long-term-variance as you suggested by setting: sigma_t^2 = sigma_{t-1}^2, and use the fact that E(gz)) =0, I wrote the model as:

sigma_t^2 = exp(omega) * exp(g(z_{t-1}) * sigma_{t-1}^(2*beta)

If I now assume that sigma = sigma_t = sigma_{t-1} and take the expectation I find that:

Sigma^(2-2*beta) = exp(omega) * E[exp(g(z))]

or

log sigma^2 = ( omega + log(E[exp(g(z))]) ) / (1-beta)

If E[exp(g(z))] is zero then this matches formula (II.4.28), but this only happens if both gamma =1 and theta =0. In any other case the long term variance/volatility give in II.4.28 will give the wrong result.

Luckily you derive the value of E[exp(g(z))] just two pages later in equation (II.4.32), so I think the correct version is:

log sigma^2 = ( omega -gamma*sqrt(2/pi) + log(C ) ) / (1-beta)

Do you think the formula II.4.28 should be changed with the one above?
thanks
giancarlo pfeifer

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