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

Process Generator

AR(1) coefficient φ 1.00 (unit root)

Noise σ 0.60

Differencing order d

Live Diagnostics

— computing —

ADF stat (on d-diff series)

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Critical value (5%)

−2.89

Var(y) over window, raw series

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Rails shown

Raw series y (front rail)

d-times differenced (back rail)

How it works

Each tick generates one new sample of a first-order autoregressive process:

y[t] = φ · y[t-1] + ε[t],  ε[t] ~ N(0, σ²)

When |φ| < 1 the process is mean-reverting: shocks decay and the variance of y stays bounded — this is a stationary series. At exactly φ = 1 the process becomes a random walk: it has a unit root, shocks never decay, and its variance grows without bound over time.

Differencing (the "I" in ARIMA) removes a unit root: if y has one unit root, Δy = y[t] − y[t-1] is stationary. The d-selector applies differencing 0, 1 or 2 times before it is drawn on the back rail and before the test statistic below is computed.

The panel runs a simplified Augmented Dickey-Fuller regression on the currently displayed (d-differenced) series z:

Δz[t] = α + β·z[t-1] + ε[t]
ADF statistic = β̂ / SE(β̂)

If β is significantly negative (below the ≈ −2.89 critical value used here for illustration), the null hypothesis of a unit root is rejected — the series is treated as stationary. Otherwise the unit root cannot be rejected — the series is treated as non-stationary. This is exactly the diagnostic step every ARIMA / SARIMA / Box-Jenkins workflow runs before choosing the differencing order d and fitting the AR/MA terms.

φ slider — set the AR coefficient; push it to 1.00 to see a true unit-root random walk, or below to see mean reversion.
σ slider — controls the size of each random shock.
d buttons — choose how many times the back rail (and the ADF test) differences the raw series.
Reseed — restarts both series from a fresh random seed.

Raw y[t] d-differenced

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