unconditional expectation for GARCH model with normal inovations


New Member
Hi i have been asked to find unconditional expectations and then a recursive relation. I think I have started well but i am confused at how the recursive relation is made up.
this is then to be used with a GARCH model i am working on.

details of question:

d(t) = mu + sigma(t)Z(t)
(sigma(t))^2 = a + ( b(Z(t-1))^2 + c)(sigma(t-1))^2

Z(t) ~ N(0,1)

find unconditional moments for: E[d(t)] , Var[d(t)] , Kurtosis[d(t)]
and E[ (d(t)-mu)^2 * (d(t-1) - mu)^2 ]

for k = 2, 3, ..., derive a recursive formula for E[ (d(t)-mu)^2 * (d(t-k) - mu)^2 ]
in terms of E[ (d(t)-mu)^2 * (d(t-(k +1)) - mu)^2 ]

i found that
E[d(t)] = mu
Var[d(t)] = a/(1-b-c)
Kurtosis[d(t)] = 3(1-(b + c)^2) / (1 - (b + c)^2 - 2b^2)

its the last bit that eludes me.

Thank you for any help
Last edited:


New Member
No that's all. maybe that the Z's are iid.

like I said I can get started.

E[d(t)] = mu + E[Z(t)] * E[sigma(t)] = mu since Z~(0,1)

E[(sigma(t))^2] = E[a + ( b(Z(t-1))^2 + c)(sigma(t-1))^2] (calling the sigma value V for ease and using the fact its unconditional i can disregard the t constraint )
V = a + bV + cV

V = a/(1-b-c)

which if you look at it is the same as Var[d(t)]