Time Seriesadvanced

Stationarity and Unit Roots

A series is stationary when its distribution is stable over time and weakly dependent when correlations fade as observations move apart, the conditions that make OLS well behaved. A unit root breaks this: the random walk $y_t = y_{t-1} + \varepsilon_t$ is the leading example, where shocks never die out and the variance grows without bound. Such a series is integrated of order one, $I(1)$ , because its first difference $\Delta y_t = y_t - y_{t-1}$ is stationary, or $I(0)$ . The (Augmented) Dickey-Fuller test checks for a unit root, and because the statistic does not follow the usual $t$ distribution, it uses special Dickey-Fuller critical values.

Try it yourself

Random walk vs AR(1)

One AR(1) process, yₜ = φ·yₜ₋₁ + eₜ with y₀ = 0 and the same fixed shocks eₜ. Slide the persistence φ: for |φ| < 1 the path is mean-reverting to 0, at φ = 1 it is a random walk (unit root) whose shocks never fade.

φ = 0.50Stationary (mean-reverting)

Path ends at y₁₂₀ −0.41Var(yₜ) at φ = 1, t = 120 120 = t·σₑ²

Persistence φ0.50

With φ = 0.50 the process is Stationary (mean-reverting): shocks fade geometrically, so the path keeps returning to its mean of 0. The smaller φ is, the faster the pull back to 0. Raise φ toward 1 and the reversion gets weaker, so the path strays further before returning.

Why it matters

Think of a stationary series as a ball on a tether. Pull it away and it springs back to its mean. A unit-root series is a ball on ice. Give it a shove and it keeps that displacement forever, wandering off with no fixed level to return to. Differencing turns the wandering level into stationary changes, which is why $I(1)$ data are usually analyzed in first differences.

Formulas

Random walk (unit root)

y_t = y_{t-1} + \varepsilon_t

Shocks have permanent effects; the series is integrated of order one, I(1).

Dickey-Fuller regression

\Delta y_t = \alpha + \theta y_{t-1} + \varepsilon_t, \quad H_0:\ \theta = 0

A unit root corresponds to theta = 0; rejecting it favors stationarity. ADF adds lagged

\Delta y

terms.

Worked examples

Scenario

You need to know whether the log of an exchange-rate series has a unit root before modeling it.

Solution

Run an augmented Dickey-Fuller test allowing for autocorrelation: `dfuller lrate, lags(1)`. If the test statistic is not more negative than the Dickey-Fuller critical value, you fail to reject the unit-root null and treat the series as $I(1)$ . Then model it in first differences with `D.lrate`.

Note`dfuller` reports Dickey-Fuller critical values, not standard t critical values; `lags()` sets the augmentation.

Common mistakes

✗A random walk is stationary because it has no trend term. It is nonstationary: its variance grows over time and shocks have permanent effects.
✗You can read the Dickey-Fuller test against ordinary $t$ critical values. The statistic has a nonstandard distribution, so you must use the special Dickey-Fuller critical values.
✗Failing to reject the unit-root null proves there is a unit root. A non-rejection only means there is insufficient evidence against it; unit-root tests have limited power.
✗Differencing an $I(0)$ series is harmless. Over-differencing a stationary series introduces problems such as a non-invertible moving-average error.

Revision bullets

•Stationary + weakly dependent is what OLS needs
•A unit root (random walk) has permanent shocks and growing variance
•Random walk is $I(1)$ ; its first difference $\Delta y_t$ is $I(0)$
•Test with the (Augmented) Dickey-Fuller test
•ADF uses nonstandard Dickey-Fuller critical values

Quick check

The random walk $y_t = y_{t-1} + \varepsilon_t$ is best described as:

When conducting an Augmented Dickey-Fuller test, you must compare the statistic to:

Connected topics

TS Data Trends TS.1-TS.6 Serial Corr.Cointegration

Sources

Wooldridge (2019), §11.3 and §18.2
Wooldridge, Jeffrey M. Introductory Econometrics: A Modern Approach. 7th ed. Cengage, 2019.
Develops stationarity, weak dependence, highly persistent series, and the Dickey-Fuller unit-root test.
Dickey & Fuller (1979)
Dickey, D.A., and W.A. Fuller. Distribution of the Estimators for Autoregressive Time Series with a Unit Root. Journal of the American Statistical Association 74 (1979): 427-431.
Original derivation of the unit-root test and its nonstandard limiting distribution.

How to cite this page

Dr. Phil's Quant Lab. (2026). Stationarity and Unit Roots. Derivatives Atlas. https://phucnguyenvan.com/concept/efm-stationarity-unit-roots

← Back to the atlas See in the network →