I have a question.
We have the following model:
$y_t=\delta+\phi y_{t-1}+\epsilon_t$
$\epsilon^2_t=\varpi+\alpha \epsilon^2_{t-1}+v_t$
with $|\phi|<1, \varpi>0, \alpha \geq0$
and that:
E($\epsilon_t$|$I_{t-1}$)=E($v_t$|$I_{t-1}$)=0. (1.1)
I have to show that:
E($\epsilon_t$)=E($\epsilon_t$$y_{t-i})$=E($\epsilon_t$$y^2_{t-i}$)=0 (1.2)
E($v_t$)=E($v_t$$y_{t-i}$)=E($v_t$$y^2_{t-i}$)=0 (1.3)
for all i=1,2,... given (1.2)
Furthermore, I have need to find how this result can be used to derive a generalized method of moments, GMM, estimator for $\theta$. I have to write the population moment conditions using instruments $z_t=(1,y_{t-1},y^2_{t-1})'$ as well as the corresponding sample moment conditions. How many moment conditions, R, and number of parameters, K, would there be?
My thoughts so far:
Since condition (1.1) is equal to zero, I thought that one might be able to do this:
$E(\epsilon_t)=E(\epsilon_t y_{t-i})=E(\epsilon_t y^2_{t-i})=0$ <=> $E(\epsilon_t)=E(\epsilon_t)*E(y_{t-i})=E(\epsilon_t)*E(y^2_{t-i})= 0*E(y^2_{t-i})=0$
Regarding the GMM part of the question, I have no idea how to proceed.
Can anyone help me with this, please?