Using a Regression Intercept in Cointegrated Pairs Trading
Summary
The document discusses whether to include an intercept when estimating the hedge ratio for a pairs trade. For two integrated asset price series that are cointegrated, it presents a regression with a slope and constant, where the residual is stationary. Rearranging the regression shows that the tradable spread is centered around the constant, which represents its long-run equilibrium level.
The intercept may be omitted in some workflows because spread normalization can estimate and subtract its mean when constructing a z-score. In that formulation, the residual is centered before scaling by its standard deviation to generate signals. The document also notes that omitting the constant imposes a zero-centered spread assumption, while including it allows persistent level differences. It offers general modeling guidance but no empirical comparison of strategy performance, and the choice depends on the assumed relationship and subsequent signal construction.
Key ideas
- For cointegrated integrated series, a regression intercept represents the long-run equilibrium level of the spread.
- The regression slope serves as the hedge ratio, while the residual captures deviations from equilibrium.
- A z-score process may account for the spread mean during normalization, which explains why some approaches omit the intercept.
- Omitting the intercept assumes the spread is centered at zero, so the choice should match the model and signal design.
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Full text
# Should we include constant in linear regression in pairs trading?
# Should we include constant in linear regression in pairs trading?
Should we include constant in linear regression while calculating hedge ratio for pairs trading strategy?
## Answer by Pleb (score 2)
https://quant.stackexchange.com/a/71550
### Yes
For two asset-price processes $Y_t$ and $X_t$ that are both $I(1)$ and cointegrated, the error-term, $z_t$, of the linear regression model,
$$ Y_t = \mu + \beta \cdot X_t + z_t $$
is $I(0)$ (see Verbeek, M. (2008). A guide to modern econometrics, p. 315). In the above regression, $\beta$ denotes the hedge ratio and $\mu$ denotes the long-run equilibrium value which can be seen by rearranging the regression and obtaining the spread to be used in your pairs trading strategy:
$$ \underset{Spread}{\underbrace{Y_t - \beta X_t}} = \mu + z_t $$
The reason $\mu$ is occasionally omitted from the regression is due to the fact that, it's estimated when calculating the Z-score of the spread (ie. normalizing the spread). Here, $z_t$ has zero mean and variance $\sigma^2$ and as such, the normalization of the spread is $\frac{z_t-\mu}{\sigma}$, which we use to derive the signals for the trading strategy.
## Answer by Content_Quantinsti (score 0)
https://quant.stackexchange.com/a/80637
Yes, including a constant term (intercept) in the linear regression model for calculating the hedge ratio in pairs trading is typically advisable, but the decision depends on the assumptions you make about the relationship between the two assets.
With a Constant (Intercept):
Including a constant allows for a more flexible relationship between the two assets, where their prices may not move in perfect synchronization. The hedge ratio will reflect not only the relative movement of the assets but also account for any persistent price level difference between them. This is more realistic in many cases since financial time series often have slight differences in levels due to dividends, transaction costs, or other idiosyncratic factors. Without a Constant:
If you assume a strictly linear relationship with no fixed difference (i.e., that the two assets should move perfectly in tandem after adjusting for the hedge ratio), you would omit the constant term. This can simplify the model, but it assumes that the spread between the two assets is mean-reverting around zero, which might not always hold in practice. In practice, many pairs traders include the constant when calculating the hedge ratio, as it tends to produce more accurate results in reflecting the true relationship between assets. You can always test both approaches and evaluate which one leads to better performance in your strategy.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.