# Finding expected value and variance of time series

Consider a time series generated by the following model:

$$y_t = 0.8y_{t-1} + \epsilon_t$$

$$E(\epsilon_t) = 0$$ and $$Var(\epsilon_t) = 1$$. Note that for any s < t, $$y_s$$ is independent with $$\epsilon_t$$. The above time series is a strictly stationary sequence and each element is normally distributed. Hence, $$E(y_t)$$ and $$Var(y_t)$$ are both constants. In particular, $$E(y_t) = E(y_{t−1})$$ and $$Var(y_t) = Var(y_{t−1})$$. Denote $$E(y_t)$$ by µ and $$Var(y_t)$$ by $$σ^2$$. Find the values of µ and $$σ^2$$.

Here is what I have so far:

$$E(y_t) = 0.8E(y_{t-1}) + E(\epsilon_t) = 0.8y_{t-1}$$

$$Var(y_t) = 0.64Var(y_{t-1}) + Var(\epsilon_t) = 0.64σ^2_{y_{t-1}} + 1$$

I do not know how to find the specific values. I thought at first the question was asking for formula answers but the specific values are required for future questions. Also, how can $$E(y_t) = E(y_{t-1})$$ when $$E(y_t) = 0.8E(y_{t-1})$$?

\begin{align} E(y_t) &= E(y_{t-1}) = \mu\\ Var(y_t) &= Var(y_{t-1}) = \sigma^2 \end{align}
Can you make the substitutions and solve for $$\mu$$ and $$\sigma^2$$?
• Does $μ = 0$ and $σ^2 = 0.0278$? Oct 18, 2018 at 0:43
• @EthanSmith Your $\mu$ is correct. Did you misplace a decimal point? $\sigma^2 = 2.78$. Oct 18, 2018 at 2:22