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Asymmetric Regression Effects and the Limits of Derived Volatility Inputs

Article Quant Q&A · Author: Ajk
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Summary

The document considers regression when a dependent variable responds differently to increases and decreases in a predictor. It notes that a dummy variable and interaction with the predictor may not resolve the issue, then raises a separate concern from the example under discussion: using volatility measures as if they were raw observations. Such measures are statistics constructed from underlying data, and treating them as ordinary data can invalidate the regression software’s assumptions about sampling distributions and inference.

The answer also points out that many volatility measures are nonlinear. If the goal is only to estimate point responses, rather than retain valid inference or prediction, the model may need to reflect the mathematics of the raw variables that produced the measures. This is a brief warning rather than a worked regression method: it does not prescribe a particular asymmetric specification, explain how to build one, or establish that the proposed adjustment solves the original problem.

Key ideas

  • A predictor’s increase and decrease effects may require more than a simple direction dummy and interaction.
  • Volatility measures are derived statistics, not raw observations.
  • Treating summary statistics as ordinary regression data can distort inferential and predictive results.
  • Nonlinear volatility measures require careful specification when modeling responses.

Tags

StatisticsMachine learningVolatility
Full text
# Modeling independent variables that have an asymmetric impact on the dependent variable


# Modeling independent variables that have an asymmetric impact on the dependent variable












I'm trying to regress a dependent variable on an independent variable that has an asymmetric impact. E.g., the dependent variable is much more responsive to an increase in the independent variable than it is to a similar decrease. Tried putting a dummy variable to indicate increases and decreases and then have that as an interaction term with the independent variable, but that did not seem to completely solve my problem. Any help would be much appreciated.

## Answer by Dave Harris (score 0, accepted)

https://quant.stackexchange.com/a/43421

Okay, there are a couple of different problems based on your comments.

First, volatility measures are statistics and they are not data. You are using them as data. At one level this is okay in that you could have incorporated all the raw data to create them into your regression, but it is going to totally mess up all of your inferential statistics and all predictive measures.

The software that is used in the various forms of regression presumes you are inputting data only. From this, they estimate the sampling distribution of the statistics. Summary statistics used as data create completely incorrect math. The package assumes that you are calculating $f(x)$, when you are really calculating $f(x,g(y))$ or $f(g(y))$. Second, most volatility measures are non-linear. If you don't mind giving up the inferential value and predictive value and only want point response measures, you need to match the math of the underlying variables that make up those statistics to the new level.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.