y_t = \left\{

\begin{array}{l l}

a + by_{t-1} + e_t & \text{if } t \geq 1 \\

a + e_t & \text{if t = 0}\\

\end{array} \right.

\\ . \\

y_t = a + by_{t-1} + e_t \\

y_t = a + e_t + b(a + by_{t-2} + e_{t-1})

y_t = a(1+b) + e_t + be_{t-1} + b^2y_{t-2} )

\[ \text{ ... for any k less than t} \]

\displaystyle

y_t = a\sum_{n=0}^k b^n + \sum_{n=0}^k b^n e_{t-n} + b^{k+1} y_{t-k-1} \\

\\

\[ \text {let k=t-1} \]

\\

\displaystyle

y_t = a\sum_{n=0}^{t-1} b^n + \sum_{n=0}^{t-1} b^n e_{t-n} + b^{t} y_{0} \\

\displaystyle

y_t = a\sum_{n=0}^{t-1} b^n + \sum_{n=0}^{t-1} b^n e_{t-n} + b^{t}(a + e_0) \\

\[ \text{incorporate last terms into the existing sums} \]

\displaystyle

y_t = a\sum_{n=0}^{t} b^n + \sum_{n=0}^{t} b^n e_{t-n} \\

\[ \text{ first sum is a geometirc progression} \]

\displaystyle

y_t = a \frac{1-b^{t+1}}{1-b} + \sum_{n=0}^{t} b^n e_{t-n} \\

\[ \text{ take expectation \]

\displaystyle

E(y_t) = a \frac{1-b^{t+1}}{1-b} + 0 \\

\[ \text{ which varies with $t$, so the series is not stationary unless a=0} \]