Cointegration in Layman’s Terms…

Assets that are cointegrated often offset each other's risks as their prices move together over the long term, despite potential short-term deviations. This is due to an economic linkage or fundamental connection that ensures a long-term equilibrium in their price movements.

For instance, two companies within the same supply chain might be similarly impacted by variations in commodity prices, consumer demand, or regulatory changes. In the short term, asset prices can deviate due to temporary market sentiments or other transient factors. However, the underlying economic connections ensure a reversion to the mean over time.

In pairs trading, one asset is often bought while the other is sold, exploiting short-term price deviations. Profits are made when prices revert to their long-term equilibrium. Similarly, cointegrated assets are used for hedging. An investor can hold opposite positions in two cointegrated assets to offset risks, anticipating that a loss in one will be compensated by a gain in the other due to their long-term price correlation.

To determine if two assets are cointegrated, one common approach is the Engle-Granger two-step method.

Step 1: Estimate the Long-Run Relationship

1. Gather historical price data of the two assets in question.
2. Convert this data to log prices if needed to stabilize variance.
3. Use simple linear regression to estimate the long-term relationship between these assets.

The linear relationship can be expressed as:
Yt = α + βXt + εt

Here:
- Yt and Xt are the log prices of two assets at time t.
- α and β are the coefficients to be estimated.
- εt is the error term.

Step 2: Test for Stationarity of the Residuals

1. Extract the residuals from the regression model.
2. Use the Augmented Dickey-Fuller (ADF) test to check if these residuals are stationary. The null hypothesis in the ADF test is the presence of a unit root, indicating a non-stationary time series.
3. If the p-value is less than a predetermined significance level, the null hypothesis is rejected, indicating that the assets are cointegrated.

Notes: 
- Log returns are preferable because they are additive over time and normally distributed, simplifying analysis.
- Stationarity of the residuals indicates that the assets’ price divergences are temporary, and they revert to a mean.
- Stationarity means that statistical properties like mean, variance, and autocorrelation are consistent over time.
- A low p-value in the ADF test indicates rejection of the null hypothesis, confirming cointegration.

Visual Interpretation

The first chart below presents the residuals from the long-run relationship estimated between Asset X and Asset Y. If the residuals fluctuate around a constant mean without a visible trend and exhibit constant variance over time, this is a sign of stationarity, indicating cointegration between the assets.

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