Cointegration (Phillips-Ouliaris + Johansen)

What is Cointegration (Phillips-Ouliaris + Johansen)?

Two or more non-stationary (I(1)) series are cointegrated when a linear combination is stationary — they share a long-run equilibrium and shouldn't be modeled in pure differences (that discards the level relationship). Phillips-Ouliaris tests H₀ of no cointegration from a levels regression's residuals, with proper (non-standard) critical values.

Johansen's system approach estimates the cointegrating rank r directly: the trace and max-eigenvalue statistics test successive nulls (r = 0, ≤ 1, …). It handles more than one cointegrating vector, where the single-equation residual test can't.

When should I use Cointegration (Phillips-Ouliaris + Johansen)?

• Deciding between a VAR-in-differences and a vector error-correction model.
• Testing long-run links (e.g. spot vs futures, consumption vs income).

What data does it need?

≥ 2 numeric I(1) series (time-ordered) + Johansen deterministic terms and lag length K.

What does it report?

Phillips-Ouliaris statistic vs critical values; Johansen trace and max-eigen statistics with 10/5/1% critical values per rank.

What does it assume?

• Series are individually I(1) — check with the stationarity test first.
• Johansen results are sensitive to the lag length and deterministic specification.

How do I interpret the result?

Johansen: read top-down; the first null you can't reject gives the rank r. r ≥ 1 means cointegration — model with a VECM.

See also

• Stationarity tests (ADF + KPSS)
• Vector autoregression (VAR) + Granger causality

References

• Johansen (1991). Estimation and hypothesis testing of cointegration vectors. Econometrica 59.
• Pfaff (2008). Analysis of Integrated and Cointegrated Time Series.