For instance, if one is running a regression with deflated prices for a given year and one of the independent variables is a price index for a given sector, does it make any sense to 'deflate' this variable or somehow extract the content that might be related to the general economy (assuming I am deflating with a more agregated variable, like a consumer price index)? One might think of demand estimation or supply estimation, if that helps.
My first thought was that I shouldn't bother, but now I'm thinking of running an auxiliary regression with the agreggated price index and the sectoral price index as a dependent variable and save the residuals -- does that make sense?
Edit: Added some more intuition, as follows:
Think of an equation of a residual demand such as $q_t$$=$$\beta p_t$+$\epsilon_t$, where $q_t$ are the quantities, $p_t$ is the price and $\epsilon_t$ is an estochastic error. But since $Cov(p_t,\epsilon_t)≠0$ -- the price $p_i$ is inherently endogenous in a demand function because of the existence of another equation defining price and quantities, the supply equation --, one should estimate demand adequately through instrumental variables estimation. Eligible instruments for demand are supply shifters, such as the company cost. But one could also use some sort price index of a sector as an instrument, a sector that sells inputs to the company out of which you are estimating the residual demand. The thing is that this price index might be related to the price index that you are using for deflating your variables. In that situation, how do you extract these fluctuations related to the general economy out of your price index being used as an instrument?