AR(1) — Random Walk ($|\phi| = 1$)
Model ($\phi = 1$ case)
$$X_t = X_{t-1} + Z_t$$Expansion
$$X_t = X_0 + \sum_{j=1}^t Z_j$$Why Not Stationary
$$\text{Var}(X_t) = \text{Var}(X_0) + t\sigma^2$$Variance grows linearly with $t$ → not finite → cannot be (weakly) stationary.
Random Walk with Drift
$$X_t = \mu + X_{t-1} + Z_t \implies X_t = t\mu + X_0 + \sum_{j=1}^t Z_j$$Mean $E(X_t) = t\mu$ (depends on $t$) and $\text{Var}(X_t) = t\sigma^2$ (depends on $t$). Not stationary.
First difference: $\nabla X_t = X_t - X_{t-1} = \mu + Z_t$ — constant mean $\mu$, variance $\sigma^2$, uncorrelated → stationary (in fact, iid shifted by $\mu$).
ACF Behavior
Sample ACF of a random walk shows very slow linear decay — a visual indicator of non-stationarity and the need for differencing.