Choosing a Regression Intercept for Return Hedging
Summary
The document explains that whether to include an intercept in a return regression depends on the hedge objective. Regressing the returns of an asset on other assets produces hedge weights, but the intercept changes what the fit is optimizing.
For minimizing expected squared replication error, the stated recommendation is to omit the intercept, so the hedge cannot rely on an arbitrary constant bias. For minimizing the variance of the hedge portfolio, the intercept should be included; omitting it need not minimize variance unless the bias happens to be zero. The answer says these relationships hold in sample. Similar out-of-sample behavior depends on historical patterns continuing and is subject to finite-sample estimation error. No numerical example or empirical comparison is supplied, so the guidance is framed around the choice of objective rather than a demonstrated performance result.
Key ideas
- The intercept choice depends on the hedge objective.
- Omitting the intercept targets minimum expected squared replication error.
- Including the intercept is appropriate when minimizing hedge portfolio variance.
- Out-of-sample results may differ because historical behavior can change and estimates are imprecise.
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# Fitting intercept in a regression hedging model # Fitting intercept in a regression hedging model If I want to hedge asset A and therefore regress its log returns on one or multiple other assets, then the resulting betas give me portfolio weights with which I can (hopefully) replicate the returns of asset A. In the regression model should I be fitting the intercept? Generally speaking I often read that it is inadvisable to not fit an intercept but in this case if I want to isolate purely the returns of one on the other then it seems to make sense to me to not fit an intercept. The returns of the assets I'm looking at aren't perfectly centered around 0 but theyre pretty close, what is the correct approach here? Thanks ## Answer by Richard Hardy (score 1, accepted) https://quant.stackexchange.com/a/81950 If your goal is to replicate the returns on asset A in terms of minimal expected squared error, then I agree that you should exclude the intercept. Otherwise, you would end up replicating the variance in A while allowing for an arbitrary bias (say, +1% or -2%, whatever happens to be the case in the data). If your goal is to minimize the variance of the returns on the hedge portfolio (which is an often considered baseline case / textbook example in hedging), you should include the intercept. By excluding it, you would replicate the returns on asset A in terms of minimal expected squared error but not minimal variance (unless you are lucky and the bias is exactly zero). These relationships hold in sample. Assuming the historical behavior of the time series will continue into the future, you should obtain similar results out of sample (with a margin of error due to estimation imprecision from your finite sample).
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.