new price data is often easier than collecting both new price data and new quantity data, so calculating the Laspeyres index for a new period tends to require less time and effort than calculating these other indices for a new period. In practice, price indices regularly compiled and released by national statistical agencies are of the Laspeyres type, due to the above-mentioned difficulties in obtaining current-period quantity or expenditure data. Calculating indices from expenditure data Sometimes, especially for aggregate data, expenditure data are more readily available than quantity data. For these cases, the indices can be formulated in terms of relative prices and base year expenditures, rather than quantities. Here is a reformulation for the Laspeyres index: Let be the total expenditure on good c in the base period, then (by definition) we have and therefore also . We can substitute these values into our Laspeyres formula as follows: A similar transformation can be made for any index. Chained vs unchained calculations The above price indices were calculated relative to a fixed base period. An alternative is to take the base period for each time period to be the immediately preceding time period. This can be done with any of the above indices. Here is an example with the Laspeyres index, where is the period for which we wish to calculate the index and is a reference period that anchors the value of the series: Each term answers the question "by what factor have prices increased between period and period ". These are multiplied together to answer the question "by what factor have prices increased since period ". The index is then the result of these multiplications, and gives the price relative to period prices. Chaining is defined for a quantity index just as it is for a price index. Index number theory Price index formulas can be evaluated based on their relation to economic concepts (like cost of living) or on their mathematical properties. Several different tests of such properties have been proposed in index number theory literature. W.E. Diewert summarized past research in a list of nine such tests for a price index , where and are vectors giving prices for a base period and a reference period while and give quantities for these periods. Identity test: The identity test basically means that if prices remain the same and quantities remain in the same proportion to each other