Autoregressive process with random walk perturbation (with drift)

Question

Suppose we have an autoregressive process, $$y_t=\phi y_{t-1} +u_t$$ where $|\phi|<1$. If $u_t$ is an i.i.d random variable this process is stationarity. What if $$u_t=u_{t-1}+g+\epsilon_t$$ where $\epsilon_t$ is white noise and $g$ is a constant? I tried to use the lag operator to show that it's still stationary (due to the presence of $g$) , but I am having some issues. Is $y$ still stationary in that case? Thanks in advance.

Answer 1

No, it is not. As you recognize, $u_t$ is nonstationary, and adding $g$ creates an additional drift, and certainly does not restore stationarity. By recursive substitution, you can write (assuming the process starts at 0) $$ u_t=\sum_{s=1}^t\epsilon_s+g\cdot t, $$ and using this process to drive $y_t$ "translates" the nonstationarity to $y_t$.

Illustration:

n <- 100
g <- .2
eps <- rnorm(n)
u <- cumsum(eps) + g*(1:n)
y <- arima.sim(n=n, list(ar=0.5), innov = u)
plot(y)