Question 2

Consider the following ARMA(2,1)-ARCH(2) model,rt=0+1rt1+2rt2+et+1et1,et=tzt,2t=+1e2t1+2e2t2.

(a) For given information available at time t, derive the 1-step, 2-step and 3-step ahead forecasts of variance of rt(show all necessary steps and conditions).

(b) If we estimate the model, we obtain0= 0.1,1= 0.3,2=0.15,1= 0.6,= 0.15,1= 0.3 and2= 0.18.Compute the var(rt=4|It=3),var(rt=5|It=3) and var(rt=6|It=3) based on the information provided in the Table 2.Table 2: Monthly returnstrtet11.512-1.30.832.2-Consider the following GARCH (2,1) model,rt=+et,et=tzt,2t=+1e2t1+2e2t2+12t1,(c) Derive the unconditional variance of rt,var(rt) (show all necessary steps and conditions).(d) For given information available at time t, derive the 1-step, 2-step and 3-step ahead forecasts of variance of rt(show all necessary steps and conditions).(e) If we estimate the model, we obtain= 0.85,= 0.28,1= 0.3,2= 0.2 and1= 0.65.

Compute the var(rt=4|It=3),var(rt=5|It=3) and var(rt=6|It=3) based on the information provided in the Table 3.(f) Comment on the estimated GARCH(2,1) model in (e), is there any issue with the estimated model? If yes, what is the issue?

Table 3: Monthly returns trtett11.5-1.52-1.3-1.832.2-2.5