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Why Level Relationships Often Fail to Predict Financial Returns

Article Quant Q&A · Author: adrCoder
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Summary

The document examines why a regression may show a significant relationship between a financial variable in levels and another variable, while the relationship disappears when using log returns. It cautions that significance in levels alone is generally weak evidence of predictive power for trading.

The explanation contrasts modeling changes directly with modeling levels. A relationship between changes can accumulate into a smoother relationship between levels, while differentiating a level relationship can amplify noise in the residuals and overwhelm the apparent signal. The answer suggests that level correlations may instead reflect omitted or confounding influences. It offers a conceptual argument rather than empirical tests or a formal diagnostic procedure, so it does not establish that every level-based relationship is spurious. The practical implication is to assess whether the relationship predicts future changes, rather than relying on in-sample significance between levels.

Key ideas

  • Significance between financial variables in levels does not by itself establish predictive power for returns.
  • A relationship between changes can appear smoother when accumulated into levels.
  • Differentiating a level relationship may amplify residual noise and obscure any predictive signal.
  • Omitted variables can create historical associations between levels.
  • Evaluate whether a model predicts future changes before using it for trading.

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# Raw (level) variable is significant while log return is not significant


# Raw (level) variable is significant while log return is not significant












I know this might be an "amateur" question, but I am pretty surprised to see the following fact:

I have a dependent variable, let's call it Y.

Then I have an independent variable, let's call it X.

X is the log return of another quantity, while Y can be either used as a raw (level -- X1) or a log return ( X2 ).

So, my results are significant for the level of the independent variable (X1), but not the log return (X2).

What is going on ? It is fine to use the level (X1) for my results, right ?

Thanks

## Answer by Brian B (score 1)

https://quant.stackexchange.com/a/20979

In practice, when you encounter a relationship between historical financial variables that looks good on levels but not on returns, the model you get from it essentially always fails to be predictive.

I generally think of this as being due to the historical relationship arising from some confounding third (plus fourth and fifth...) variable effects that have not made it into the model.

Consider a relationship that works in return space

$$ \Delta y = \beta \Delta x + \epsilon $$

then in integrating the processes we get

$$ y(T)-y(0) = \beta(x(T)-x(0)) + \int_0^T \epsilon \, dt $$

and it will be a fairly clean relationship since integration is a smoothing process.

On the other hand, if we have $$ y = \beta x + \epsilon $$ then we differentiate to get increments

$$ \frac{dy}{dt} = \beta \frac{dx}{dt} + \frac{d\epsilon}{dt} $$ which on the face of it looks promising. However, taking the derivative is a "roughening" operation, so any kind of noise in $\epsilon$ gets "blown up" by the differentiation process, essentially swamping the model with noise terms.

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.