# Stationarity after differencing

I have the following two processes: \begin{align} x_t &= x_{t-1} + u_t \tag{1} \\ x_t &= {\beta}_0 + {\beta}_1t + u_t \tag{2} \end{align}

\begin{align} \Delta x_t &= u_t \tag{1} \\ \Delta x_t &= {\beta}_1 + u_t + u_{t-1} \tag{2} \end{align}

From my notes, I understand that process (1) is stationary as $u_t$ is a white noise process. However, my lecture notes also state that process (2) is not stationary.

Why is process (2) not stationary?

Assuming $E[u_t]=0$ and $\beta_1 \not = 0$ then in the second question $$E[x_t]-E[x_{t-1}] = E[\Delta x_t] = \beta_1 \not = 0$$ which imples the process is not stationary.
• If $\beta_1$ is a non-zero constant then $E[x_t]$ is increasing with $t$, so the original process is not constant in mean. – Henry May 3 '15 at 16:07