I have been given the following time series which has infinite history
$X_t = 0.4X_{t-1} + 0.2X_{t-2} + \epsilon_{t} + 0.025$
where $\epsilon_t$ is white noise distributed $N(0,\sigma^2)$
First I have been asked to find the expectation.
Second, explain how the infinite order moving average representation can be derived.
Given that it is stationary as I proved, would its expectation just be $\mu$, or $0.6X_{t-1} + 0.025$. If both of these are wrong, I would be very thankful if someone explained why.
For part 2, Its characteristic equation is $1-0.4z-.2z^2$, would I need to find an infinite expansion of the inverse of this?
Help greatly appreciated
Best Answer
If $E(X_t) = \mu_t$, then by linearity of expectation we have $\mu_t = 0.4 \mu_{t-1} + 0.2 \mu_{t-2} + 0.025$, which has general solution $\mu_t = 0.0625 + a r_1^t + b r_2^t$, where $a$ and $b$ are arbitrary constants and $r_1$ and $r_2$ are the roots of the polynomial $z^2 - 0.4 z - 0.2$: these happen to be approximately $.6898979486$ and $-.2898979486$. "Infinite history" doesn't really require stationarity, but if you don't want to allow $|E(X_t)|$ to go to $\infty$ as $t \to -\infty$ that makes $a=b=0$.
For the second, if we do assume stationarity and write $X_t = 0.0625 + \sum_{j=0}^\infty \beta_j \epsilon_{t-j}$, plug that in to the equation and you get $$ \eqalign{ \beta_0 &= 1\cr \beta_1 &= .4 \beta_0 = .4 \cr \beta_j &= .4 \beta_{j-1} + .2 \beta_{j-2} \ \text{otherwise}\cr}$$ which can be solved.