At What Price? Conceptualizing and Measuring Cost-of-Living and Price Indexes (2002) / Chapter Skim
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Appendix: Statistical Definition and Estimation of Price Indexes
Appendix: Statistical Definition and Estimation of Price Indexes
Pages 283-292
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From page 283... ...
We define the price indexes motivated by our concepts and demonstrate briefly the survey data required to estimate the indexes. To begin, consider a simple world in which there is only one good and two time periods, base and comparison and a static universe of households (HH)
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From page 284... ...
= Yt Y y o NtQtPt NoQoPo -T TT — N q P' where TN is the trend in the total HH count, Tq is again the trend in average units per HH, and Tp is again the trend in average price per unit. As above, TP may be called the price index and Tq the unit volume index.
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From page 285... ...
The trend in average units is weighted by comparison prices, and thus one might view Tpq as a Paasche index of unit volume. Since the trend in average prices is weighted by base units volume, one might thus view Top as a Laspeyres price index.
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From page 286... ...
Partition the set of goods marketed at the base period by GO = GOQ U G and partition the set of goods marketed at the comparison period by Gt = G U GtE, where GOQ denotes exiting goods, GtE denotes entering goods, and G denotes both continuing and linkable goods. Continuing goods are marketed in both time periods, while exiting goods appear in the base period but not in the comparison, and entering goods appear in the comparison but not in the base.
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From page 287... ...
Then building on the above, the trend in total dollar volume can be decomposed as Ty = TNTpqTLpT = TNTLqTppT = TNTFqTFpTR 1 Alternative price indexes based on continuing and linkable goods are given byT~p is Laspeyres price index, Top is the Paasche price index, and TFP is the Fisher price index. All price indexes discussed here extend naturally to a time series of .
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From page 288... ...
Thus, plutocratic weights are ratios of means and democratic weights are means of ratios. Similar weighting yields IPP, a democratic Paasche price index, and IFP' a democratic Fisher price index.
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From page 289... ...
At each time period, assume that BLS collects unit volume and prices for all buying occasions for all goods from each HH, i, in the sample. Comprehensive data of this kind are not currently collected by any BLS survey.
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From page 290... ...
From these basic estimated totals, one can consistently estimate the ratios A ytG R
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From page 291... ...
Let so and St denote probability samples of stores, let the subscript k index the store, and let Wko and Wkt denote survey weights corresponding to the unbiased estimator of a population total at times 0 and t, respectively. It is easy to imagine estimators A Ygt ~ WktYght A Ygo = of total dollar volume and ~ WkO YgkO kesO Qgt ~ Wkt Yght kiss QgO = ~ WkOYgkO kesO of total unit volume.
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From page 292... ...
In this factor, c(Dgio,Y+io) is anAestimator of the covariance between HH share and total HH consumption, and Y++O is an estimator of mean total consumption per HH in the base period.
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