Hendry's classic econometric warning

Can rainfall explain inflation?

A convincing regression can still describe the wrong economic relationship.

Theoretical introductionSpurious regression → co-integration

A rainy view of Big Ben in London
Fig. 1 The photograph gives context; it is not econometric evidence.

An unexpected relationship

The model looked strong—then Hendry revealed the variable

Historical Bank of England five-pound note Rain in London
Fig. 2 British prices and British rainfall provide the historical setting.
01

UK prices and a variable C showed a close nonlinear relationship.

02

A quadratic regression gave a good fit and significant coefficients.

03

With an AR(1) error model, the fit became “spectacular.”

04

C was cumulative rainfall in the United Kingdom.

This case warns about misleading econometric relationships. It is not automatic evidence of co-integration.

Why ordinary inference can fail

Persistence can turn accidental movement into false confidence

1 · REMEMBERS

Persistent levels

Earlier shocks remain inside later observations.

2 · ALIGNS

Apparent co-movement

Two unrelated paths can move together for many periods.

3 · INFLATES

Misleading tests

Conventional t-tests use the wrong reference behavior.

4 · MISSTATES

False economic story

A high R² becomes an explanation the data do not justify.

Fig. 3 The formula can be familiar while its time-series assumptions fail.

Controlled demonstration · synthetic data

Independent random walks still produce a significant OLS slope

Two independent synthetic random walks Scatter plot and OLS line for independent random walks
Fig. 4 Independent innovations generate β̂ = 0.455, R² = 0.372, and conventional p < 0.001; the residual is not stationary.

Why stationarity matters

Three paths can rise, but they require different treatments

I(0): stable level

Mean and variance stay broadly stable. Shocks fade.

Trend-stationary

A predictable trend changes the mean. Remove the trend.

I(1): stochastic trend

Each shock shifts the future path. Difference once.

Fig. 5 A visible upward path alone does not identify the type of trend.

Random walk → first difference

Differencing removes the accumulated path

Xt = Xt−1 + utEach new shock stays in every later level.
→
ΔXt = utThe first difference contains only the new shock.
Fig. 6 If ΔXt is I(0), then Xt is I(1).

Differencing prevents a naive level regression, but it may remove a genuine long-run relation.

From spurious regression to co-integration

Co-integration asks whether the gap stays stable

Synthetic co-integrated series and their stable spread
Fig. 7 The levels drift, but their estimated long-run gap stays bounded.
ut = Yt − α − βXt

X and Y may each be I(1). They are co-integrated when this linear combination is I(0).

Boundary: statistical stability does not prove causation.

The diagnostic difference

The residual reveals whether co-movement is stable

Persistent residual beside a mean-reverting residual
Fig. 8 A spurious residual wanders; a co-integrating residual returns toward a stable mean.

Long-run equilibrium and short-run correction

Yesterday's gap creates today's adjustment

Last period

Y is 10 units above equilibrium.

+10
×

Adjustment speed

λ = −0.30

−0.30
=

This period

Correction contribution

−3
Fig. 9 The error correction model combines a long-run gap with short-run changes.

How to test the relationship

The tests form a sequence, not a checklist

1 · CLASSIFY

ADF

Is each series I(0) or I(1)?

H₀: unit root

2 · TEST GAP

Engle–Granger

Is one estimated residual stationary?

H₀: no co-integration

3 · TEST RANK

Johansen

How many long-run relations exist?

H₀: rank ≤ r

4 · MODEL

ECM / VECM

How does the system correct and forecast?

Estimate after rank choice

Fig. 10 Integration order and co-integration evidence come before equilibrium correction.

Choose the method to match the design

Both tests ask whether a long-run combination is stable

Engle–Granger

  • Estimate one long-run equation.
  • Test its estimated residual.
  • Use residual-based critical values.
  • Simple for one relation.

Johansen

  • Model all variables jointly.
  • Use trace and max-eigen tests.
  • Allow several long-run relations.
  • Choose lags and deterministic terms first.
Fig. 11 A strong level regression is not enough; the long-run combination must be statistically stable.

Time-series econometrics

Spurious regression
or co-integration?

Shared movement is not enough. The residual must stay stable for the long-run relationship to be credible.

DPIC96 × PCECC96FRED data10–15 minutes

Two rising series with an unstable gap Two paths rise together, but the distance between them changes over time. gap changes Both series rise but the distance may drift time →
Fig. 1 Co-movement in levels does not prove a stable equilibrium.

The research route

Five questions move from illusion to forecasting

01

Spurious regression

Why can unrelated trends look significant?

02

Co-integration

When is a level relationship meaningful?

03

Equilibrium correction

How do short-run gaps close?

04

Statistical tests

What do ADF, Engle–Granger, and Johansen test?

05

Forecasting

Does the equilibrium improve predictions?

The logic is sequential: test persistence before trusting levels, then test equilibrium before using a VECM.

THEORY → TESTS → EVIDENCE → FORECASTS

Question 1 · Spurious regression

Persistence creates false confidence

Mechanism of spurious regression Persistent shocks create accidental co-movement, which makes ordinary p-values misleading. Persistent shockstoday changes tomorrowand many later periods allows Accidental alignmentindependent paths candrift in one direction inflates t-statisticlooks largep looks small misleads False economic storythe usual reference distributionno longer applies two unrelated random walks
Fig. 2 The regression formula looks ordinary; its sampling logic is not.
Yt = α + βXt + εt
01

Observations are strongly dependent across time.

02

OLS mistakes shared drift for repeated evidence.

03

The usual t-test no longer has its usual meaning.

A small p-value can describe a statistical artifact, not an economic relationship.