Consider a series like CPI (inflation), which I know is composed of a series of component prices (e.g. meat prices, grain prices, non-food prices, etc.), which in turn are also composed of a series of component prices (e.g. average meat prices are a combination of pork, beef, and chicken prices).
If I wanted to use a regression to find the components of CPI and their weightings, then is it better to use the final components (regress CPI against pork, beef, and chicken prices), or is it better to create fitted version of the middle components, then regress against CPI (regress average meat prices against pork, beef, and chicken prices, then fit a meat price series, then regress CPI against the fitted meat price series)?
Also, I should note that some series which were significant when fitting the interim components in the layered method - the latter method - lose their significance when the regression is flattened. So is is possible that the latter, layered method will return a better result because it includes the effect of more components?
I have tried this with actual data and the latter method gives better results.