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.
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$\begingroup$ Are you regressing the variable levels or returns ? $\endgroup$– EzyJan 9, 2019 at 0:13
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$\begingroup$ I'm regressing the variable levels. $\endgroup$– AjkJan 9, 2019 at 0:56
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$\begingroup$ If it at all helps, the independent variable(s) are different flavors of vol--close on close, intra-day, and implied vol. $\endgroup$– AjkJan 9, 2019 at 2:42
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1$\begingroup$ Have you tried regressing the changes of the dep against 2 terms being the positive changes of the indep and the negative changes of the indep ? $\endgroup$– EzyJan 9, 2019 at 15:31
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$\begingroup$ @Ezy that’s an interesting suggestion. I have a similar issue at present. Thanks for the idea $\endgroup$– udayJan 11, 2019 at 4:17
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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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$\begingroup$ Not always an easy solution. Entire slew of decision tree based ML techniques (like Gradient Boosting) that were originally designed for classification is being used by tons of people for all sorts of prediction merely because the independent variable has a nonlinear dependence. Transforming volatility to equivalent normal distributions (feature transformation) doesn’t solve it. This is a real problem. @Ezy ‘s comment is interesting, perhaps one way to avoid ML tree algos. $\endgroup$– udayJan 11, 2019 at 4:38
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$\begingroup$ I don't think this is that type of problem. Asymmetric responses usually come from three places in economics. The frustrating one, of course, is Keynes sticky prices. The second is misspecification. The third is from an omitted variable similar to a catastrophe set. $\endgroup$ Jan 11, 2019 at 18:14
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$\begingroup$ Thanks. I think the specific problem is most analogous to sticky prices. As for using volatility as a data input, could you expand upon what you see as issue and how you'd correct this if you wanted to maintain predictive value. $\endgroup$– AjkJan 11, 2019 at 19:16
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$\begingroup$ Just for some additional color, this dependent variable is from an entity that publishes it and only resets it infrequently. This entity has specifically stated that it uses volatility when determining the level of measure it publishes. $\endgroup$– AjkJan 11, 2019 at 19:22
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$\begingroup$ It is totally fine to use volatility estimates in your predictors $\endgroup$– EzyJan 12, 2019 at 4:14