I'm interested in demand system estimation, however often times we only have expenditure data to work with. This is fine for estimating the linear expenditure system, however for for estimating much more interesting and useful demand systems like the Almost Ideal Demand System and Rotterdam demand system we require information on prices.

In short my question is:how do we recover information on prices from expenditure data in a time series when price and quantity aren't separated from each other (i.e we only have knowledge of spending not the separate quantity and price information)

The following method I'm proposing is a follows:

Step 1: Note that mi=p_ix_i that is the expenditure on good ii is equal to price of good ii times its quantity . most expenditure data is indexed by time so a more accurate representation of this relationship is: m_{it}=p_{it}x_{it}

Step 2:In a competitive market where prices are stable the increase in expenditure on good i comes from an increase demand for the product. With this in mind we will run a regression of our expenditure on good i on time to capture this "ideal world data:

Step 3: obtain the fitted values for the regression model \hat{m}_{it}=α0+α1*t which is a simple regression which tells us how expenditure should evolve over time ignoring information on prices because we are assuming things are stable for the time being. If prices are constant throughout time this is the same as \hat{x}^it=α+α1*t since a transformation of x_{it} by a scaler wont affect our estimates of what α1α1 is:

Step 4: Obtain price data we divide the actual expenditure data by the fitted expenditure data which was expenditure as evolving according to time. \hat{p}_{it}=mit/\hat{m}_{it} or more specifically from our relationship in step 3:

\hat{p}_{it}=m_{it}/\hat{x}_{it}

Using relationship we can generate a series of data which is a stand in for the underlying price data.

I'm not sure if this is silly or something that useful for someone who has a hard time finding data. Does this make sense?