Hey I need some help with this exercise:
Assume $z_{ t }$ be a sequence of independent normal random variables, each
with mean 0 and time independent variance $\sigma^{ 2 }$, and let c be a
constant. Is the following process for a time series stationary? If it is
stationary specify the mean and the autocovariance function:
$$
x_{ t }=z_{ 1 }*cos\left( ct \right)+z_{ 2 }*sin\left( ct \right)
$$
So for a weak stationarity time series the mean and covariance function should be independent of time. I'm not sure how to calculate them by hand without R.
Thanks for help.
Best Answer
The model is $$x_t = z_1 \cdot \cos(ct) + z_2 \cdot \sin(ct).$$ Thus the mean function of the series will be $$m(t) = E(z_1)\cdot \cos(ct) + E(z_2) \cdot \sin(ct) = 0$$ and the covariance function,
\begin{align} cov(x_j, x_t) &= cov\left(z_1 \cdot \cos(cj) + z_2 \cdot \sin(cj), ~z_1 \cdot \cos(ct) + z_2 \cdot \sin(ct)\right)\\ &= \cos(cj)\cdot \cos(ct)\cdot var(z_1) + \sin(cj)\cdot\sin(ct) \cdot var(z_2)\\ &=\sigma^2 \cdot cos(c(j-t)). \end{align}
We can see the mean function is independent of time and the covariance function depends on the time difference implying the series weak stationary.