I am self studying time series analysis with Shumway Stoffer text. I am stuck on Question 1.6.
Consider the time series
$x_t = \beta_1 + \beta_2 t + w_t$
where $\beta_1$ and $\beta_2$ are known constants and $w_t$ is a white noise process with variance $\sigma^2_w$.
Show that the mean of the moving average
$v_t=\frac{1}{(2q+1)} \sum_{j=-q}^{q} x_{t-j}$
is $\beta_1+\beta_2t$, and give a simplified expression for the autocovariance function.
What I have is
$\mu_{t-j} = E(x_{t-j}) = \beta_1 + \beta_2 t – \beta_2 j$
$E[v_t] =\frac{1}{(2q+1)} \sum_{j=-q}^{q} E(x_{t-j})=\frac{1}{(2q+1)} [2q (\beta_1 + \beta_2 t) + \sum_{j=-q}^{q} \beta_2 j]$
seems like it should be easy from here, but I just can't see it.
Best Answer
$\mu_{t-j} = E(x_{t-j}) = \beta_1 + \beta_2 t - \beta_2 j$
$\sum_{j=-q}^{q} \beta_2 j = 0$ because of symmetry
And note there are $2q + 1$ terms in series $j = -q,...,q$
So we have
$E[v_t] =\frac{1}{(2q+1)} \sum_{j=-q}^{q} E(x_{t-j})$
$=\frac{1}{(2q+1)} [(2q +1) (\beta_1 + \beta_2 t) + \sum_{j=-q}^{q} \beta_2 j]$
$= \frac{2q + 1}{2q+1} \beta_1+\beta_2 t$
$= \beta_1+\beta_2 t$