Clearly there is a strong relationship between credit spreads and equity prices (both theoretically and empirically). But how would one go about formulating a regression which seeks to explain this relationship?
To keep it narrow, let's say we have the following time series data;
S&P500, daily % changes
Generic 5 year swap yields, daily differences
Generic 5 year high yield credit spread index, daily differences
The objective is to explain the daily movements in credit spreads by the changes in swap yields and equity prices.
I'm interested more in how one might think about structuring the regression (eg transforming the variables) to get the best fit from a linear regression, rather than the theoretical economic underpinnings or choice of data comprising this particular problem.
In particular i want to know how one might transform the inputs to be able to cover the case of when credit spreads are high (sensitivity to equity changes is also high) and when credit spreads are low (sensitivity to equity changes is also low), or if a different approach is warranted.
Among the potential other issues are that equity price changes are fat tailed, and have negative skew. Swap yields and credit spreads are both mean reverting (and lower bounded). Credit spreads depend on swap yields and equity prices, but swap yields also depend on equity prices.
Does it make sense to maybe do something like;
transform each series by dividing by it's volatility (or it's implied volatility - additional non-linear issues with that?)
regression on ranks rather than underlying data
transform each series with a sigmoid function, possibly in combination any of the above
I appreciate this might be a wide topic, but am very interested to here how people with knowledge of time series statistics and finance might approach it.