y + gamma t

Problem

Suppose that \(y_1, \ldots ,y_T\) is a stationary process. Define a new process \(x_1, \ldots ,x_T\) from

\[x_t=y_t+γt\]

where \(γ>0\) . Is \(x_1, \ldots ,x_T\) a stationary process?

Solution

No.

\[E\left( x_t \right)=E\left( y_t+γt \right)=E\left( y_t \right)+E\left( γt \right)=E\left( y_t \right)+γt\]

\(E\left( y_t \right)\) does not depend on \(t\) but the second term, \(γt\) , increases with \(t\) . Thus, \(E\left( x_t \right)\) increases with \(t\) and it is not stationary.