Let \(x_{t}\) be a time series with linear and quadratic trend.

\[ x_{t}=\beta_{0}+\beta_{1} t+\beta_{2} t^{2}+w_{t}, \quad \text { for } t=1,2, \ldots \]

where \(\beta_{0}, \beta_{1}\), and \(\beta_{2}\) are constants, and \(w_{t}\) is a white noise with variance \(\sigma^{2}\).

Let \(\left\{y_{t}\right\}\) be a first-order difference process on \(\left\{x_{t}\right\}\) :

\[ y_{t}=\triangle\left(x_{t}\right)=x_{t}-x_{t-1}, \quad \text { for } t=2,3, \ldots \]

and let \(\left\{z_{t}\right\}\) be a second-order difference process on \(\left\{x_{t}\right\}\) :

\[ z_{t}=\triangle^{2}\left(x_{t}\right)=y_{t}-y_{t-1}, \quad \text { for } t=3,4, \ldots \]

In answering the following questions, be sure you note which series the question concerns. " \(x_{t}\) ", " \(y_{t}\) ", and " \(z_{t}\) " can look very much alike!

(a) Is \(\left\{y_{t}\right\}\) (weakly) stationary? Why or why not?

(b) Is \(\left\{z_{t}\right\}\) stationary? Why or why not?