Abstract
The availability of scanner data from large-scale retailers makes the construction of a continuously updated system of price indexes over space and time for an important share of household consumption expenditures possible. However, building a coherent (transitive) system of price indexes across space and time involves issues that are irrelevant for bilateral price indexes or multilateral price indexes only over space. Some of these issues were discussed by Hill (2004), but in my opinion the most important has been ignored. Indeed, it is very likely that the same commodity is differently priced across space, but in the long run the movements of its prices will be similar (stable) in space. So it is quite natural to ask price indexes for pairs of space situations not to diverge over time if the prices of each single commodity in the basket remain approximatively pairwise proportional in the two sites. In this work, we give a definition of the test of stability preservation, starting from the stochastic properties that panels of price time series seem to obey to. Then, many different approaches to the construction of the system of indexes are analysed in order to identify those that pass the test. The selected systems are applied both to simulated and to real-world data collected in four supermarkets located in the city of Milan for a time span of 24 months.
In this work I use data that the Statistical Office of the City of Milan has obtained from supermarkets expressly for this research. I thank Flavio Necchi (Comune di Milano), supermarket managers I cannot mention for preserving data anonymity and my colleague Marco Fattore for our frequent discussions on the theme and for sharing with me and Flavio the effort of fostering our project and convincing the other partners.
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Notes
- 1.
MT stands for measurement ton a unit of volume used for measuring the cargo of a ship, truck, train, or other freight carrier, equal to exactly 40 cubic feet, or approximately 1.1326 cubic meters.
- 2.
The ADF, and Elliot, Rothenberg and Stock (1996) DF-GLS and point-optimal tests were applied. The stationarity test KPSS leads to the same conclusions.
- 3.
The Law of One Price implies that prices of tradable goods should be equal in the long run, but in order to allow some kind of intra-country Balassa-Samuelson effect we relax the identity to proportionality.
- 4.
The reader not acquainted with the concept of cointegration should refer to any recent text on time series econometrics. A thorough and mathematically rigourous treatment of cointegration in the framework of vector autoregressive processes may be found in Johansen (1995).
- 5.
In order to leave the concept open enough we can take the definition of Granger (1995) “a process may be said to have an attractor if there is some mechanism that produces an inclination to return to some value – usually its mean”.
- 6.
Recall that the CPI index for country k has generally the form
${\rm CPI}_k = \sum_n \frac{p_{n,k,t}}{p_{n,k,0}}w_{n}$for some weights \(\{w_n\}\). Thus, if log-prices are modelled as integrated processes then \(\log\,p_{n,k,t} = \Sigma_{s=1}^{t} \varepsilon_{n,k,s} + \log{p_{n,k,0}}\), and \(\log (p_{n,k,t}/p_{n,k,0})\) is an integrated process that starts form zero (at time \(t=0\)), exactly as in our simulation.
- 7.
A Dickey-Fuller test was applied imposing non deterministic regressors, since under the alternative of mean reversion the mean of the log of the price indexes should be 0, being the price pairs almost identical.
- 8.
For brevity’s sake, the quantities and the exact processes’ formulas are not reported in the paper, but the software developed for carrying out all the simulations is available from the author on request. The code is written in Ox, so the reader interested in running the software should download the Ox interpreter from http://www.doornik.com.
References
Balk, B. M. (1996). A comparison of ten methods for multilateral international price and volume comparison. Journal of Official Statistics, 12(2), 199–222.
Balk, B. M. (2008). Price and quantity index numbers. Models for measuring aggregate change and difference. New York: Cambridge University Press.
Diewert, W. E. (2002). Similarity and dissimilarity indexes: An axiomatic approach (Discussion Paper No. 02-10). Department of Economics, University of British Columbia.
ECLAC. (1978). Series Históricas del Crecimiento de Améerica Latina. Santiago, Chile: Economic Commission for Latin America and the Caribbean.
Elliot, G., Rothenberg, T. J., & Stock, J. H. (1996). Efficient tests for an autoregressive unit root. Econometrica, 64(4), 813–836.
Eltetö, O., & Koves, P. (1964). On an index number computation problem in international comparison (in Hungarian), Statisztikai Szemle, 42, 507–518.
Ermini, L., & Granger, C. W. J. (1993). Some generalizations on the algebra of I(1) processes. Journal of Econometrics, 58, 369–384.
Eurostat. (1995, December 18–20). Improving the quality of price indices. Proceedings of the International Seminar held in Florence, Commission of the European Community.
Ferrari, G., Laureti, T., & Mostacci, F. (2005). Time-space harmonization of consumer price indexes in Euro-zone countries. International Advances in Economic Research, 11, 359–378. DOI: 10.1007/s11294-005-2275-7
Geary, R. C. (1958). A note on the comparison of exchange rates and PPPs between countries. Journal of the Royal Statistical Society Series A, 121(1), 97–99.
Gerardi, D. (1982). Selected problems of inter-country comparisons on the basis of the experience of the EEC. Review of Income and Welth, 28, 253–259.
Gini, C. (1924). Quelques Considéretions au Sujet de la Construction des Nombres.
Indices des Prix et des Questions Analogues. Metron, 4, 3–162.
Granger, C. W. J. (1995). Modelling nonlinear relationships between extended-memory variables. Econometrica, 63(2), 265–279.
Hill, R. J. (1999). Comparing price levels across countries using minimum spanning trees. Review of Economics and Statistics, 81(1), 135–142.
Hill, R. J. (2004). Constructing price indexes across space and time: the case of the European union. American Economic Review, 94(5), 1379–1410.
Johansen, S. (1995). Likelihood-based inference in cointegrated vector autoregressive models. Oxford: Oxford University Press.
Khamis, S. H. (1972). A new system of index numbers for national and international purposes. Journal of the Royal Statistical Society, Series A, 135(1), 96–121.
Martini, M. (1992). I numeri indice in un apporccio assiomatico. Milan: Giuffré Editore.
Martini, M. (2001). Numeri indice per il confronto nel tempo e nello spazio. Milan: CUSL.
Martini, M., & Zavanella, B. (1995). Problems of space and time consistency between spatial and temporal indeces. Proceedings of the Internartional Seminar Improving the Quality of Price Indeces: CPI & PPP, Commission of the European Communities, 371–390.
Rao, D. S. P. (1990). A system of log-change index numbers for multilateral comparisons. In J. Salazar-Carrillo & D. S. P. Rao (Eds.), Comparisons of prices and real products in Latin America, Contribution to Economic Analysis Series. Amsterdam: North Holland.
Rao, D. S. P. (2001). Integration of CPI and PPP: Methodological issues, feasibility and recommendations, Joint World Bank-OECD Seminar on PPP (pp. 1–26). Washington, DC.
Rao, D. S. P. (2005). On the equivalence of weighted country-product-dummy (CPD) method and the Rao-system for multilateral comparison. Review of Income and Wealth, 51(4), 571–580.
Summers, R. (1973). International comparisons based upon incomplete data. Review of Income and Wealth, 19(1), 1–16.
Szulc, B. (1964). Index numbers of multilateral regional comparison (in Polish). Przeglad Statysticzny, 3, 239–254.
van Veelen, M. (2002). An impossibility theorem concerning multilateral international comparison of volumes. Econometrica, 70(1), 369–375.
Zavanella, B. (1996). International comparisons and construction of optimal graphs. In S. Camiz & S. Stefani (Eds.), Proceedings of the conferences on matrices and graphs: Theory and applications to economics, University of Brescia, Italy, 8th June 1993, 22 June 1995 (pp. 176–199). River Edge NJ: World Scientific.
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Pelagatti*, M.M. (2010). Price Indexes across Space and Time and the Stochastic Properties of Prices. In: Biggeri, L., Ferrari, G. (eds) Price Indexes in Time and Space. Contributions to Statistics. Physica-Verlag HD. https://doi.org/10.1007/978-3-7908-2140-6_5
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