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I'm trying to recover the weights used to compute a Laspeyres quantity index. The index number's formula is:

$ Q=\frac{\sum_{i=1}^{N}{p_i^0 q_i^t}}{\sum_{i=1}^{N}{p_i^0 q_i^0}} $

where in this case $i$ indicates product, $t$ indicates observation date, $p$ is price and $q$ is quantity. In the particular case I'm working on, I have a quantity index of three products. The products and the index all use the sabe period as the base (ie are normalized to 1 in that period). I have the time series for the three products and the resulting index. I want to recover the weights in order to reproduce the index from the product's quantity data.

(I'm trying to check if the a published aggregated index is true to the product quantities that are published elsewhere. There were no doubts formerly about the index's exactitude, so I'm using "old" data to recover the weights and check if the more recent and questioned observations of the index are or are't based on the underlying product quantities. The index is monthly and there are 36-48 "safe" observations and about 60 doubtful observations thereafter. There is little to no doubt about the product quantities obtained from other sources with which I intend to verify the published index.)

It should be as simple as solving the system of equations:

$ \substack{{\begin{bmatrix} resulting index \end{bmatrix}\\3\times1}} = \substack{{\begin{bmatrix} data \end{bmatrix}\\3\times3}}\times \substack{{\begin{bmatrix} weights \end{bmatrix}\\3\times1}} \\ [data]^{-1}\times[resultingindex]=[weights] $

Where $[data]$ has products in columns and observations in rows.

I tried this and the result is that the $[weights]$ vector has both positive and negative values, which doesn't make any sense. The resulting recalculated index is equal to the published one during the safe period and shows the expected anomalies in the unsafed period, which are common to other indexes; it fits nicely (actually there are some minor differences which I attribute to the use of not exactly the same sources; on average the indexes differ on about 1% during the safe period). However, if one product with a negative weight increases its quantities, then the aggregated index falls...

I've clearly done something wrong with the math (I don't doubt my Excel implementation but if anyone wishes to check it out I can provide the spreadsheet.)

What is the proper way to do this?

Thanks!

edit: this post should be tagged index-number, I think, but I can't do it because of my rep.


additional clarifications: Just to make it clear. The quantities are units of cars, trucks and bigger trucks. The aggregate index is "vehicles production". Negative weights don't make sense.

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    I understand that only nonnegative weights make sense, but what I mean is: *shxt happens*. If you data don't allow you to back out positive weights, you have to accept this fact.2012-11-29

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