Interpreting Weekday Dummy Regressions for Returns

Summary

The document compares two regressions for testing whether average returns differ by weekday. One includes an intercept and four weekday indicators, leaving one day as the reference category. The other includes an indicator for each weekday and omits the intercept, so each coefficient directly represents the fitted mean return for its day. The question asks how to read coefficient p-values and whether the full set of indicators creates multicollinearity.

The answer suggests that both parameterizations can produce equivalent fitted weekday means and significance results when interpreted consistently. In the intercept model, a weekday’s mean is represented by the intercept plus its indicator coefficient, while in the no-intercept model it is represented by that day’s coefficient. It also notes dependence among weekday indicators as a potential collinearity concern. This is a brief, tentative answer rather than a full regression derivation: it does not specify the exact hypothesis tests or diagnose the practical severity of collinearity for a dataset.

Key ideas

• With an intercept, the omitted weekday serves as the reference category for the indicator coefficients.
• Without an intercept, each weekday coefficient represents that day’s fitted mean return.
• Equivalent fitted values can result from the two parameterizations when coefficients are interpreted appropriately.
• The response flags dependence among weekday indicators but does not quantify its impact or detail specific tests.

Tags

# Testing day of the week effect


# Testing day of the week effect












I am currently reading a bit about testing day of the weeks effects. I saw two different model specifications and wonder how to interpret the results.

The first model type includes only 4 dummies for e.g., Mo till Thu, and an intercept:

$Return_t=\beta_0+\beta_1D_{1t}+\beta_2D_{2t}+\beta_3D_{3t}+\beta_4D_{4t}+\epsilon_t$

The second model type includes 5 dummies for all weekdays and no intercept.

$Return_t=\beta_1D_{1t}+\beta_2D_{2t}+\beta_3D_{3t}+\beta_4D_{4t}+\beta_5D_{5t}+\epsilon_t$

I have two questions:

Could you explain the difference in the interpretation of the p-values for the models (what does it mean when one of the dummies in model 1 is significant, what does it mean in model 2)? In my opinion the model 2 suffers from multicollinearity, because the dummies are linear dependent, is this correct?

Thanks for your help!

## Answer by RA334 (score 1)

https://quant.stackexchange.com/a/20928

1) I'm going from memory here so someone may want to confirm that I'm thinking about this correctly - but the two models will end up with the same results and significance levels - in the first model, the intercept acts as the reference day, such that the average effect of $D_d=\beta_0+\beta_d$. In the second model you should get the same effect, however, it will be equal to simply $\beta_d$.

2) I don't think one model suffers from collinearity issues and the other one doesn't - both are likely to suffer, given that the correlation between one day and another day will be negative. It will come down to how concerned you are with multicollinearity.

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