Stochastic difference equation

Question

I am a newbie in studying time series. Could anyone help solve the following problem:

1. Consider the second-order stochastic difference equation: $y_t=1.5y_{t-1}-0.5y_{t-2}+\varepsilon_t$. Given initial conditions for $y_0$ and $y_1$, find $y_t$ in terms of $\{\varepsilon_t\}$ sequence.

2. Find the forecast function for $y_{T+s}$ given the value of $y_T$ and $y_{T-1}$.

Answer 1

\begin{matrix} y_t-y_{t-1}&=&0.5(y_{t-1}-y_{t-2})+\varepsilon_t\\ 0.5(y_{t-1}-y_{t-2})&=&0.5^2(y_{t-2}-y_{t-3})+0.5\varepsilon_{t-1}\\ &\vdots&\\ 0.5^{t-2}(y_2-y_1)&=&0.5^{t-1}(y_1-y_0)+0.5^{t-2}\varepsilon_2 \end{matrix} Summing up the LHS and RHS gives $$y_t-y_{t-1}+\sum_{i=1}^{t-2}0.5^i(y_{t-i}-y_{t-i-1})=\sum_{i=1}^{t-1}0.5^i(y_{t-i}-y_{t-i-1})+\sum_{i=0}^{t-2}0.5^i\varepsilon_{t-i}$$ so $$y_t-y_{t-1}=0.5^{t-1}(y_1-y_0)+\sum_{i=0}^{t-2}0.5^i\varepsilon_{t-i}$$ Then \begin{align*} y_t=\sum_{i=1}^t(y_{t-i+1}-y_{t-i})+y_0=&\sum_{i=1}^t\left(0.5^{t-i}(y_1-y_0)+\sum_{j=0}^{t-2}0.5^j\varepsilon_{t-j}\right)+y_0\\ =&(y_1-y_0)\sum_{i=1}^t0.5^{t-i}+t\sum_{i=0}^{t-2}0.5^i\varepsilon_{t-i}+y_0\\ =&\frac{1-0.5^t}{0.5}(y_1-y_0)+t\sum_{i=0}^{t-2}0.5^i\varepsilon_{t-i}+y_0\\ =&(2^{1-t}-1)y_0+(2-2^{1-t})y_1+t\sum_{i=0}^{t-2}0.5^i\varepsilon_{t-i} \end{align*}

Answer 2

I don't have time to give a full answer, but for part (1), it's just an AR(2) process, see e.g. Autoregressive model on wikipedia. So you can solve by writing in terms of the lag operator, etc.