Johansen Cointegration Test and Walk-Forward Spread Validation

Worked Solution

How to Think About It: Cointegration is the time-series version of "these things move together in the long run." Three $I(1)$ series individually wander like random walks, but if there is a linear combination that is stationary, that combination is mean-reverting -- and potentially tradeable. The Johansen test is the multivariate tool for detecting how many such independent linear combinations exist (the cointegration rank). The hard part is not finding cointegration in-sample (overfit is easy) but confirming it holds out of sample.

Key Insight: The Johansen test works by examining the rank of a matrix derived from a vector error correction model (VECM). Each eigenvalue corresponds to a potential cointegrating relationship, and sequential hypothesis tests determine how many eigenvalues are significantly nonzero.

The Method:

Part (i): Johansen Test for Cointegration Rank

Fit a VECM to the vector $\mathbf{p}(t) = (p_1(t), p_2(t), p_3(t))'$:

$\Delta \mathbf{p}(t) = \Pi \mathbf{p}(t-1) + \sum_{j=1}^{k-1} \Gamma_j \Delta \mathbf{p}(t-j) + \boldsymbol{\mu} + \boldsymbol{\varepsilon}(t)$

where $\Pi = \alpha \beta'$ is the long-run impact matrix. The cointegration rank $r$ equals $\text{rank}(\Pi)$.

The test uses the eigenvalues $\hat{\lambda}_1 \geq \hat{\lambda}_2 \geq \hat{\lambda}_3 \geq 0$ of $\Pi$ (obtained via reduced-rank regression / canonical correlation analysis).

Trace test (sequential):

• $H_0: r = 0$ vs. $H_1: r \geq 1$. Test statistic: $\lambda_{\text{trace}}(0) = -T \sum_{i=1}^{3} \ln(1 - \hat{\lambda}_i)$.
• If rejected: $H_0: r \leq 1$ vs. $H_1: r \geq 2$. Test statistic: $\lambda_{\text{trace}}(1) = -T \sum_{i=2}^{3} \ln(1 - \hat{\lambda}_i)$.
• If rejected: $H_0: r \leq 2$ vs. $H_1: r = 3$. Test statistic: $\lambda_{\text{trace}}(2) = -T \ln(1 - \hat{\lambda}_3)$.

Critical values come from Johansen's tabulated distributions (not standard chi-squared). They depend on the deterministic specification (constant, trend, etc.) and the number of variables.

Max-eigenvalue test (alternative):

• $H_0: r = 0$ vs. $H_1: r = 1$. Statistic: $\lambda_{\max}(0) = -T \ln(1 - \hat{\lambda}_1)$.
• $H_0: r = 1$ vs. $H_1: r = 2$. Statistic: $\lambda_{\max}(1) = -T \ln(1 - \hat{\lambda}_2)$.

For our case: we expect to reject $r = 0$ but fail to reject $r \leq 1$, confirming exactly one cointegrating relationship.

Practical notes: Choose the lag order $k$ by information criteria (AIC/BIC) on the underlying VAR in levels. Include a constant in the cointegrating equation if the spread has a nonzero mean. Test for and correct serial correlation in the residuals.

Part (ii): Estimating the Cointegrating Vector and Spread

With $r = 1$, the matrix $\Pi = \alpha \beta'$ has rank 1. The cointegrating vector $\boldsymbol{\beta} = (\beta_1, \beta_2, \beta_3)'$ is the eigenvector corresponding to the largest eigenvalue $\hat{\lambda}_1$.

The stationary spread is:

$s(t) = \beta_1 p_1(t) + \beta_2 p_2(t) + \beta_3 p_3(t)$

Normalization: typically fix $\beta_1 = 1$ (or normalize to the asset with the largest loading), so the spread represents a dollar-neutral portfolio.

The vector $\alpha = (\alpha_1, \alpha_2, \alpha_3)'$ gives the speed-of-adjustment coefficients -- how quickly each price series responds to deviations in the spread. For a tradeable spread, at least one $\alpha_i$ should be economically meaningful (the spread mean-reverts within a reasonable time horizon).

Validation checks: - The estimated spread $s(t)$ should pass an ADF test for stationarity. - The half-life of mean reversion (from the AR(1) coefficient of $s(t)$) should be in a tradeable range (days to weeks, not years). - The loadings $\beta_i$ should be economically interpretable.

Part (iii): Walk-Forward Out-of-Sample Procedure

1. Initial training window: Use the first $W$ observations (e.g., 2 years of daily data) to estimate the cointegrating vector $\hat{\boldsymbol{\beta}}$ via Johansen.

1. Out-of-sample evaluation window: Apply the estimated $\hat{\boldsymbol{\beta}}$ to the next $H$ observations (e.g., 3 months) to construct the spread $s(t) = \hat{\beta}_1 p_1(t) + \hat{\beta}_2 p_2(t) + \hat{\beta}_3 p_3(t)$.

1. Stationarity test on OOS spread: Run an ADF test on the out-of-sample spread. Record the test statistic and $p$-value.

1. Roll forward: Advance the training window by $H$ observations (expanding or rolling) and repeat.

1. Aggregate: Track the fraction of OOS windows where the spread passes the ADF test, the OOS half-life, and any structural breaks.

Critical safeguards against look-ahead bias:

• Never use future data for estimation. The cointegrating vector at time $t$ is estimated using only data up to time $t$.
• No peeking at OOS data for tuning. Lag order, deterministic specification, and other hyperparameters are chosen within the training window only.
• Re-estimate $\hat{\boldsymbol{\beta}}$ each window. Cointegrating vectors can drift -- assuming a fixed $\boldsymbol{\beta}$ over the full sample is a common source of spurious backtesting results.
• Track the cointegration rank OOS. If the Johansen test within the training window fails to find $r = 1$ for some windows, the relationship may have broken down.

Answer: Use the Johansen trace or max-eigenvalue test to sequentially test $r = 0, 1, 2$. If $r = 1$, the cointegrating vector is the eigenvector of the largest canonical correlation, and the spread $s(t) = \boldsymbol{\beta}' \mathbf{p}(t)$ is stationary. Validate OOS with a rolling-window procedure: estimate $\hat{\boldsymbol{\beta}}$ on past data, apply it forward, and test the OOS spread for stationarity via ADF. Re-estimate each window and never use future data for any estimation step.