Consider the time series:
$X(t) = 2 + 3t + Z(t) $
where $Z(t)$ are Gaussian white noises from $\mathcal{N}(0,1)$.
- Is $X(t)$ stationary - why or why not?
- Is $Y(t) = X(t) - X(t-1)$ stationary, why or why not?
- Let $V(t)= \frac{1}{2q+1}\sum_{j=-q}^q X(t-j)$.
What is the mean and auto-covariance function of $V(t)$.
My approach is that: I know a stationary process is one in which the statistical properties of a given series are constant, such as constant mean, auto-covariance etc. I know that the expected value or mean of the noise component is zero. How do I compute the expectation of $X(t)$? How do I show the statistical properties are constant or not constant?
The series is not stationary because there is a linear trend which means that the process average is changing with time.
Y(t) is just Z(t)-Z(t-1) . So the linear trend is remove. It now has mean 0 because Z has mean 0. Z is stationary because it is Gaussian white noise.
Expressed in terms of past Ys Y(t)= Y(t-1)-Y(t-2)+Y(t-3)-...+Z(t). It has a constant mean but to be statioanry it must have a constant variance too. Does it?
V(t) = ∑$_j$ Z(t-j)/(2q-1) +(2q+1)(2+3t)/(2q-1) - ∑$_j$ j/(2q-1) But the sum of j from -q to q is 0 so we get
V(t) = ∑$_j$ Z(t-j)/(2q-1) + (2q+1)(2+3t)/2q-1). From this you should be able to compute the mean function and autocovariance function+