On a summation manipulation

34 Views Asked by At

I have $R_t= \frac{1}{h} \sum_{j=0}^{h-1} E_tr_{t+j} + \theta_t$

where $E_tr_{t+j} = E[r_{t+j}| I_t]$.

By subtracting $r_t$ from both sides and after some manipulations I should get: $$R_t - r_t= \frac{1}{h} \sum_{j=1}^{h-1} (1- \frac{j}{h}) E_t\Delta r_{t+j} + \theta_t$$

where $\Delta$ is the difference operator $\Delta r_t = (r_t - r_{t-1})$

Could someone show me the necessary operations? How does the j appear?