Cointegration Analysis of Cryptocurrencies

Johansen cointegration test on BTC, ETH, and SOL log prices. The residual (spread) of the first cointegrating vector is plotted below.

Why Pick the Largest Eigenvalue?

In the Johansen cointegration test, the eigenvalues are sorted in descending order, and each one corresponds to a different potential cointegrating vector. The magnitude of the eigenvalue represents the strength and stability of the corresponding cointegrating relationship.

1. Strongest Relationship: The largest eigenvalue corresponds to the linear combination of the time series that is "most stationary." The resulting spread has the strongest tendency to revert to its mean.
2. Statistical Significance: The test statistics (Trace and Maximum Eigenvalue tests) are functions of these eigenvalues, helping determine how many significant cointegrating relationships exist.
3. Practical Application: For pairs trading, we want the most reliable long-run equilibrium. The vector associated with the largest eigenvalue gives the most mean-reverting portfolio.

Should You Use Log Prices?

Yes, using log prices is generally recommended for cointegration analysis in finance:

1. Percentage vs. Absolute Changes: Log prices work with relative percentage changes rather than absolute dollar amounts, which is crucial when assets trade at vastly different scales.
2. Variance Stabilization: Log transformation helps stabilize variance in heteroscedastic financial data.
3. Linearization: Real-world economic relationships between assets are often multiplicative; logarithms convert these into linear additive relationships suitable for the Johansen test.