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Microeconomics · Macroeconomics · Econometrics & Finance

Most economic data drifts. GDP rises over time, stock prices wander, and exchange rates show no obvious tendency to return to a mean. Time series econometrics gives you the tools to handle that drift — and to avoid the regressions that look significant but are completely spurious. If you are studying this material at LSE, King’s or UCL, you already know it is the hardest part of the metrics course. A time series econometrics tutor in London works through it one test at a time.

1 · What stationarity means — and why it matters

A time series is stationary if its mean, variance and autocovariance do not depend on time. Shift the window forward ten periods and the distribution looks the same. That property is called covariance stationarity, and it is the foundation every standard estimator rests on.

Without stationarity, the law of large numbers does not apply. Sample means do not converge to population means. t-statistics do not follow the t-distribution. R² can be high even when the variables are completely unrelated — the classic spurious regression problem. Granger and Newbold showed this in 1974: regress one random walk on another and you get significant coefficients most of the time, despite zero true relationship.

Examiners test this distinction. They give you two trending series and ask whether a regression between them is meaningful. The answer turns on whether the trends are stochastic or deterministic — and whether the series share a common stochastic trend. That is the cointegration question, and Section 4 returns to it.

2 · The random walk and the unit root

The simplest non-stationary process is the random walk:

y_t = y_t−1 + ε_t, ε_t ~ IID(0, σ²)

Each period’s value is last period’s value plus a shock. The shock never fades. A one-unit impulse at time zero shifts the entire future path by one unit, permanently. The variance grows without bound: Var(y_t) = tσ². The series has infinite memory.

Write this as y_t = ρy_t−1 + ε_t and the condition for a unit root is ρ = 1. If |ρ| < 1, the process is stationary — shocks decay geometrically and the series reverts to its mean. If ρ = 1, it is a random walk. If ρ > 1, it is explosive, which is rare in economic data but possible in hyperinflation or bubbles.

The unit root is the knife-edge. Near it, standard asymptotics break down. The Dickey–Fuller test is designed precisely for this boundary.

3 · The Dickey–Fuller test

Start with the AR(1) model:

y_t = ρy_t−1 + ε_t

Subtract y_t−1 from both sides:

Δy_t = (ρ − 1)y_t−1 + ε_t = γy_t−1 + ε_t

The null hypothesis is γ = 0, which is equivalent to ρ = 1 — a unit root. The alternative is γ < 0, meaning ρ < 1 and the series is stationary.

The test statistic is the t-ratio on γ̂, but it does not follow the t-distribution. Under the null, the distribution is skewed left, with critical values around −2.86 at the 5% level for a sample of 100 — far more negative than the standard normal critical value of −1.65. Using the wrong distribution is the most common exam error. Marking schemes are strict about this.

Three variants exist. The pure random walk test has no deterministic terms. Adding an intercept allows for a non-zero mean under the alternative. Adding a linear trend allows for deterministic trend-stationarity. The choice matters: if the true process is trend-stationary but you omit the trend, the test loses power and you fail to reject the null too often.

4 · The augmented Dickey–Fuller test

The simple DF test assumes the errors are white noise. Real data rarely cooperate. Serial correlation in ε_t biases the test, so the augmented version adds lagged differences:

Δy_t = γy_t−1 + Σ_i=1^p δ_iΔy_t−i + ε_t

The extra terms soak up the serial correlation, leaving ε_t approximately white noise. The test on γ uses the same Dickey–Fuller critical values.

Choosing p is a practical problem. Too few lags and serial correlation remains, biasing the test. Too many and you lose power. Information criteria — AIC, BIC — are the standard tools. BIC tends to choose a more parsimonious model, which is often preferred because over-parameterisation inflates standard errors.

Examiners notice whether you justify your lag length. A sentence about checking the correlogram of the residuals or reporting the AIC is worth marks.

5 · Stationarity, trend-stationarity and difference-stationarity

Not all trending series have unit roots. A trend-stationary process has a deterministic time trend:

y_t = α + βt + ε_t, ε_t ~ I(0)

Shocks are transitory. The series reverts to the trend line. Detrend it — regress on t and take residuals — and you have a stationary series.

A difference-stationary process has a stochastic trend:

y_t = y_t−1 + ε_t

Shocks are permanent. Detrending does not make it stationary; differencing does.

The distinction is not cosmetic. If you detrend a difference-stationary series, the residuals still have a unit root and any regression using them is spurious. If you difference a trend-stationary series, you introduce a non-invertible moving-average component and lose information. The Dickey–Fuller test with a trend term is designed to tell them apart.

Worked example — testing UK quarterly GDP

Consider the log of UK quarterly GDP from 1995Q1 to 2019Q4 — 100 observations. The series trends upward, so we run the ADF test with an intercept and a linear trend. The estimated equation is:

Δy_t = 0.41 − 0.002t − 0.18y_t−1 + 0.32Δy_t−1 + 0.19Δy_t−2 + ε̂_t

Step 1 — Identify the coefficient of interest. The coefficient on y_t−1 is γ̂ = −0.18. The ADF test statistic is the t-ratio on this coefficient: t_γ̂ = −0.18 / 0.07 = −2.57.

Step 2 — Choose the critical value. With T = 100 and a trend included, the 5% critical value is approximately −3.45. The 10% critical value is approximately −3.15.

Step 3 — Compare. −2.57 > −3.15, so we cannot reject the null at even the 10% level. The test says log GDP has a unit root.

Step 4 — Perturbation. First-difference the series and test again. Run the ADF on Δy_t with an intercept only (no trend, because the differenced series shows no obvious trend). The estimated equation is:

Δ²y_t = 0.001 − 0.84Δy_t−1 + 0.11Δ²y_t−1 + ε̂_t

Step 5 — Compute the new test statistic. γ̂ = −0.84, with standard error 0.12. t_γ̂ = −7.00.

Step 6 — Critical value for the differenced test. With an intercept only and T ≈ 99, the 1% critical value is approximately −3.51. −7.00 < −3.51, so we reject the null decisively.

Step 7 — Interpretation. Log GDP fails the unit-root test in levels but passes in first differences. The series is integrated of order one, I(1). It is difference-stationary, not trend-stationary. Any regression of log GDP on another I(1) variable — say, log consumption — must be checked for cointegration. If the variables are not cointegrated, the regression is spurious.

Figure 1 — A stationary AR(1) reverts to zero; a random walk driven by the same shocks wanders off and stays off.

Getting tangled in which critical values to use? The Dickey–Fuller distribution is where time-series marks are most often won and lost. A one-on-one time series econometrics tutor in London works through it with you until the ADF output reads like plain English. Book a trial session.

Practice

Q1. You estimate Δy_t = −0.05y_t−1 + ε̂_t with T = 200. The standard error on γ̂ is 0.02.
(a) Compute the DF test statistic.
(b) The 5% critical value for this specification is −1.95. Do you reject the unit-root null?
(c) What does your conclusion imply about the order of integration of y_t?

Q2. An ADF test with intercept and trend on 80 observations yields t_γ̂ = −3.10. The 5% critical value is −3.46. A second ADF on the first-differenced series (intercept only) yields t_γ̂ = −5.20, with a 5% critical value of −2.90. Classify the series: I(0), I(1), or I(2)?

Q3. A researcher runs an ADF test with 4 lags on a sample of 120. The p-value for the fourth lag’s t-statistic is 0.31. The AIC for the 4-lag model is −3.42; for the 3-lag model it is −3.47; for the 2-lag model it is −3.44. Which lag length would you choose, and why?

Answers: Q1 (a) t = −0.05/0.02 = −2.50; (b) −2.50 < −1.95, so reject the null — the series is stationary, I(0); (c) It is integrated of order zero. Q2: Cannot reject unit root in levels (−3.10 > −3.46), reject in first differences (−5.20 < −2.90), so the series is I(1). Q3: Choose 3 lags — it minimises the AIC (−3.47 is the lowest). The insignificant fourth lag and the AIC both point to 3. BIC would likely also favour 3 or even 2.

Key takeaways

A stationary series has constant mean, variance and autocovariance. Standard regression theory depends on this. Without it, t– and F-statistics are unreliable.
A unit root means ρ = 1 in y_t = ρy_t−1 + ε_t. Shocks are permanent and the variance grows with time.
The Dickey–Fuller test regresses Δy_t on y_t−1 and tests γ = 0. The test statistic does not follow the t-distribution — use the Dickey–Fuller critical values.
The augmented DF test adds lagged differences to absorb serial correlation. Lag length matters; use information criteria to choose it.
Trend-stationary series are I(0) around a deterministic trend. Difference-stationary series are I(1) and must be differenced. The ADF with a trend term distinguishes them.

Why London students choose our time series econometrics tutoring

LSE, King’s and UCL course alignment: our tutors know the specific metrics sequences at each institution — from LSE’s EC220/EC221 through to King’s advanced time-series options and UCL’s ECON0019. Your problem sets, Stata exercises and past papers are the starting point.
One-on-one format: every session is private, so you work at your pace through the material that is tripping you up — whether that is the Dickey–Fuller distribution, lag-length selection or the link between unit roots and cointegration.
Exam-first preparation: sessions work through past papers with marking schemes in view, because the “use the right critical values” point is where metrics marks are won and lost.

FAQ

Q: What is a unit root?
A: A unit root means the coefficient on the lagged dependent variable equals one. In y_t = ρy_t−1 + ε_t, a unit root is ρ = 1. Shocks have permanent effects and the series does not revert to a mean. The variance grows over time, and standard regression results break down.

Q: How do I interpret the Dickey–Fuller test?
A: The null hypothesis is a unit root (γ = 0). If your test statistic is more negative than the critical value, reject the null and conclude the series is stationary. If not, you cannot reject the unit root. Always use the Dickey–Fuller critical values, not the standard normal or t-tables.

Q: What is the difference between the DF and ADF tests?
A: The DF test assumes the error term is white noise. The ADF test adds lagged differences of the dependent variable to mop up any serial correlation. In practice, you almost always need the ADF version because economic time series have autocorrelated errors.

Q: How many lags should I include in an ADF test?
A: Use an information criterion — AIC or BIC — or a general-to-specific testing procedure. Start with a generous lag length and drop insignificant lags. BIC tends to choose fewer lags, which can preserve power. Always check that the residuals are approximately white noise after estimation.

Q: What happens if I use the wrong critical values?
A: You over-reject the null. The Dickey–Fuller distribution is shifted left relative to the standard normal. A t-ratio of −2.0 might look significant against ±1.96, but the 5% DF critical value is around −2.86. Using the normal table makes you think the series is stationary when it is not.

Q: Do I need to know the Dickey–Fuller distribution derivation?
A: At undergraduate level, no. You need to know that the distribution is non-standard, that critical values are more negative than the normal, and that the test is one-sided (reject only for large negative values). The Wiener process derivation is graduate material.

Book a time series econometrics tutor in London

Unit roots, Dickey–Fuller critical values and the difference between trend-stationarity and difference-stationarity — these are the concepts that separate a 2:1 from a First in econometrics. A one-on-one session turns the ADF output into something you can read and argue with. Tell us your university and module, and we will match you with the right tutor this week.