Stationarity & the Unit Root: Random Walk vs AR(1) Differencing

Every ARIMA or SARIMA forecast starts with the same question: is this time series stationary? This simulator generates an AR(1) process live — y[t] = φ·y[t-1] + noise — and renders it in 3D as a scrolling rail of points next to its differenced counterpart. Push φ to exactly 1.00 and the series becomes a random walk with a unit root: its variance grows without bound and no amount of waiting makes it settle down. Pull φ below 1.00 and the process becomes mean-reverting and stationary. A live simplified Augmented Dickey-Fuller regression is computed every tick on the currently selected differencing order, so you can watch the test statistic cross the critical value in real time as you change φ, the noise level, or how many times the series has been differenced — exactly the diagnostic step every Box-Jenkins time-series workflow performs before fitting a model.