There appears to be a formatting issue. Please ignore the jumbled first paragraph and refer straight to the second paragraph.

Suppose that we have an ARCH(2) model for financial returns

ut = _tt

²_t= ₀+₁u²_t1+₂u²_t2tiidN(0,1)

where _tis independent of _t, ₁ 0,₂ 0 and ₀> 0. We assume that {u_t} is weakly

stationary.

a) Derive the unconditional mean and variance of {u_t}.

b) Derive a condition on the parameters which ensures weak stationarity of the {u_t} pro-

cess.

PLEASE NOTE: When ^ is written, it refers to a hat on the previous value (for example a^ would be ) , not as a "power"sign. I was unable to get the ^ to sit on top on the symbols.

c) Suppose that we have estimated the ARCH(2) model and obtained the parameter

estimates: ₀= 1, ₁= 0.5, and ₂= 0.1. Assume that u²_T= 1, and u²_T1= 0

are observed. Denote by ²_T-hITthe h-step-ahead forecast of the conditional variance.

Compute the one- and two-step-ahead forecasts of the conditional variance ²_t.