Time-series econometrics

Spurious regression, co-integration & forecasting

Theory and evidence from U.S. real disposable income and consumption.

DPIC96 × PCECC961960Q1–2026Q2Five research questions

A rainy view of Big Ben in London — Hendry's rainfall example motivates the difference between a convincing regression and a credible relationship.

An unexpected relationship

The model looked strong—then Hendry revealed the variable

Historical Bank of England five-pound note — British prices and British rainfall provide the historical setting.

Rain in London — British prices and British rainfall provide the historical setting.

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

A quadratic regression gave a good fit and significant coefficients.

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

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.

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 — Independent innovations generate β̂ = 0.455, R² = 0.372, and conventional p < 0.001; the residual is not stationary.

Scatter plot and OLS line for independent random walks — 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.

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

Random walk → first difference

Differencing removes the accumulated path

X_t = X_t−1 + u_tEach new shock stays in every later level.

→

ΔX_t = u_tThe first difference contains only the new shock.

If ΔX_t is I(0), then X_t 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 — The levels drift, but their estimated long-run gap stays bounded.

u_t = Y_t − α − βX_t

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 — 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

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

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.

A strong level regression is not enough; the long-run combination must be statistically stable.

Empirical design · FRED / U.S. BEA

Income and consumption share persistent long-run growth

Indexed U.S. real disposable income and real consumption with the forecast test start marked — The two quarterly series rise together, but formal tests must decide whether their gap is stable.

266QUARTERS
1960Q1–2026Q2

258TRAINING
THROUGH 2024Q2

8TEST QUARTERS
2024Q3–2026Q2

We use natural logs. Every unit-root test, lag choice, co-integration test, and model estimate uses the training sample only.

Empirical evidence · Step 1

Both variables behave like I(1) processes

Log income · level

p = .273

ADF with trend does not reject a unit root.

H₀: UNIT ROOT

Δ log income

p < .001

The first difference rejects a unit root.

RESULT: STATIONARY CHANGE

Log consumption · level

p = .800

ADF with trend does not reject a unit root.

H₀: UNIT ROOT

Δ log consumption

p < .001

The first difference rejects a unit root.

RESULT: STATIONARY CHANGE

Non-stationary levels and stationary first differences support the I(1) classification required for standard co-integration tests.

Empirical evidence · Steps 2 and 3

Two main tests identify one long-run relation

Engle–Granger · intercept

p = .00036

Reject the null of no co-integration.

H₀: NO CO-INTEGRATION

Engle–Granger · trend

p = .00473

The conclusion survives a trend specification.

H₀: NO CO-INTEGRATION

Johansen trace

rank = 1

Reject rank zero; do not reject rank at most one.

VAR LAG 4 · BIC

Maximum eigenvalue

rank = 1

The second rank test reaches the same main conclusion.

5% CRITICAL VALUES

Engle–Granger and the prespecified Johansen tests agree, although Johansen rank changes under some alternative specifications.

Empirical evidence · Error correction

Income carries the significant adjustment toward equilibrium

Estimated income-consumption equilibrium error returning around zero — The estimated equilibrium error is bounded around zero in the training sample.

Income adjustmentα = −0.1616
p < .001

Consumption adjustmentα = +0.0445
p = .081

Residual dynamicsWHITENESS p = .540
NO STRONG AUTOCORRELATION

Under this normalization, income responds significantly to the previous equilibrium error. Consumption adjustment is not significant at 5%.

Empirical evidence · Forecasting

The VECM wins this eight-quarter holdout

Actual income and consumption compared with VECM, differenced VAR, and ARIMA forecasts — Fixed-origin forecasts use the same 2024Q3–2026Q2 test period for every model.

Income · VECMRMSE 225.8 · MAE 196.0
41.6% lower RMSE than drift

Consumption · VECMRMSE 114.3 · MAE 96.5
6.7% lower RMSE than drift

Benchmark setDIFFERENCED VAR
ARIMA(1,1,0)
RANDOM WALK WITH DRIFT

Answers to the five research questions

A stable residual turns shared growth into useful information

Why spurious?

Persistent unrelated trends can create significant OLS results.

What is co-integration?

I(1) variables have a stationary long-run combination.

Why equilibrium?

The deviation stays bounded and enters the next period's adjustment.

How tested?

ADF classifies the series; Engle–Granger and Johansen test stability.

How forecast?

VECM combines short-run changes with the previous equilibrium error.

The income–consumption evidence supports one long-run relation and useful VECM forecasts, but the eight-quarter test window limits generalization.

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

Co-movement in levels does not prove a stable equilibrium.

The research route

Five questions move from illusion to forecasting