In a linear regression whose components can also be broken down, is it better to do multi-layered regression, or flatten to final components?

Question

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.

Answer 1

Somewhere there's a saying "Model what you're interested in." I think you need to determine at what level you're most interested in the parameters. I'd also like to see you define what you mean by "better result." If it were me I'd probably want to obtain best estimates at the more specific level so that I could isolate the contributions of different, specific indices as much as possible. But whatever you do, watch for collinearity. I'd take a close look at a correlation matrix and/or at a printout of zero-order, partial, and part correlations, and be prepared to run several iterations of regression before drawing any conclusions about the relative contributions of different variables. I'd also want to obtain partial regression plots to see more about the nature of each X-Y relationship than a coefficient alone can give.