ECON20003 · Quantitative Methods 2
Stationarity, Spurious Regression & the Dickey-Fuller Test
Stationarity, Spurious Regression & the Dickey-Fuller Test closes the subject by asking whether a time series is well-behaved enough to regress at all. A weakly (covariance) stationary series has a constant finite mean and autocovariances that depend only on the lag, not on time; white noise is the simplest case. A random walk has a unit root and is nonstationary, and regressing one nonstationary (I(1)) series on an unrelated one produces a spurious regression — a high R² and significant t with no real relationship. The (Augmented) Dickey-Fuller test checks for a unit root, and crucially uses special DF critical values, not the standard t-table.
What this chapter covers
- 01Weak (covariance) stationarity: constant mean, autocovariance depends on lag not time
- 02White noise: mean 0, constant variance, zero autocorrelation
- 03Nonstationarity: random walk yₜ = yₜ₋₁ + εₜ (unit root); random walk with drift
- 04Spurious regression: high R² and significant t between unrelated I(1) series
- 05(Augmented) Dickey-Fuller test: Δyₜ = α + γyₜ₋₁ + … + εₜ; H₀: γ = 0 (unit root)
- 06Use DF critical values, NOT standard t; reject H₀ ⇒ stationary
Augmented Dickey-Fuller unit-root test
- 2 marksState the hypotheses: H₀: γ = 0 (a unit root, so the series is nonstationary) versus H₁: γ < 0 (the series is stationary). This is a left-tailed test.
- 1 markUse the correct critical values: the ADF statistic does NOT follow the standard t-distribution, so compare against the Dickey-Fuller critical value −2.88, not a t-table value.
- 1 markDecision rule: reject H₀ only if the ADF statistic is MORE negative than −2.88.
- 1 markCompare: the ADF statistic −2.10 is greater than (not as negative as) −2.88, so do NOT reject H₀.
- 1 markConclude: the series has a unit root and is nonstationary (I(1), random-walk-like).
- 1 markImplication: regressing this series on another I(1) series risks a spurious regression (high R², significant t, no real relationship), so difference the series to I(0) before modelling.
Key terms
- Weak (covariance) stationarity
- A time series whose mean is constant and finite and whose autocovariances depend only on the lag between observations, not on the time point. Stationarity is the precondition for standard time-series inference.
- Random walk (unit root)
- A nonstationary process yₜ = yₜ₋₁ + εₜ where the current value equals the last plus a shock. Its variance grows over time; adding a constant gives a random walk with drift.
- Spurious regression
- Regressing one nonstationary (I(1)) series on an unrelated nonstationary series yields a high R² and significant t-statistics despite there being no real relationship — a trap differencing the data avoids.
- Dickey-Fuller test
- A unit-root test using Δyₜ = α + γyₜ₋₁ + … + εₜ with H₀: γ = 0 (unit root, nonstationary) versus H₁: γ < 0 (stationary). It must be compared against special Dickey-Fuller critical values, not the standard t-table.
Stationarity, Spurious Regression & the Dickey-Fuller Test FAQ
Why can't I use the normal t-table for the Dickey-Fuller test?
Under a unit root the usual t-distribution theory breaks down, so the ADF statistic follows a non-standard distribution. Dickey and Fuller tabulated special critical values (more negative than the ordinary t-values), and you must compare your statistic against those — using a t-table would wrongly reject the unit root too often.
What does failing to reject the Dickey-Fuller H₀ tell me?
It means the series has a unit root and is nonstationary (I(1)). You should not regress it directly on another nonstationary series — that risks a spurious regression — so difference the series to make it stationary (I(0)) before modelling, and correct any remaining autocorrelation in the standard errors.
Exam move
Lock in the two exam traps: the Dickey-Fuller test uses its own (more negative) critical values, and 'fail to reject' means nonstationary. Practise stating the ADF hypotheses (γ = 0 unit root vs γ < 0 stationary) and the spurious-regression warning so you can chain this question after a time-series diagnostic, exactly as the sample exam does.
