Abstract
Most countries use either the Jevons or Carli index for the calculation of their Consumer Price Index (CPI) at the lowest (elementary) level of aggregation. The choice of the elementary formula for inflation measurement does matter and the effect of the change of the index formula was estimated by the Bureau of Labor Statistics (2001). It has been shown in the literature that the difference between the Carli index and the Jevons index is bounded from below by the variance of the price relatives. In this article, we extend this result, comparing expected values and variances of these sample indices under the assumption that prices are described by a geometric Brownian motion (GBM). We provide formulas for their biases, variances and mean-squared errors.
References
1.
Balk, B.M. 2005. “Price Indexes for Elementary Aggregates: The Sampling Approach.” Journal of Official Statistics 21(4): 675–699. Available at: https://www.scb.se/contentassets/ca21efb41fee47d293bbee5bf7be7fb3/price-indexes-for-elementary-aggregates-the-sampling-approach.pdf (accessed September 2020).
2.
Barlow, M.T. 2002. “A Diffusion Model for Electricity Prices.” Mathematical Finance 12 (4): 287–298. DOI: https://doi.org/10.1111/j.1467-9965.2002.tb00125.x.
3.
Białek, J. 2013. “Measuring Average Rate of Return of Pensions: A Discrete, Stochastic and Continuous Price Index Approaches.” International Journal of Statistics and Probability 2(4): 56–63. DOI: https://doi.org/10.5539/ijsp.v2n4p56.
4.
Białek, J. 2015. “Generalization of the Divisia price and quantity indices in a stochastic model with continuous time.” Communications in Statistics: Theory and Methods 44(2): 309–328. DOI: https://doi.org/10.1080/03610926.2014.968738.
5.
Boskin, M.S. 1996. (Chair) Advisory Commission to Study the Consumer Price Index. “Towards a More Accurate Measure of the Cost of Living.” Final report for the Senate Finance Committee. Washington D.C., Available at: https://www.ssa.gov/history/reports/boskinrpt.html.
6.
Boskin, M.S., E.R. Dulberger, R.J. Gordon, Z. Griliches, and D.W. Jorgenson. 1998. “Consumer prices in the consumer price index and the cost of living.” Journal of Economic Perspectives 12(1): 3–26. DOI: https://doi.org/10.1257/jep.12.1.3.
7.
Bureau of Labor Statistics. 2001. The experimental CPI using geometric means (CPI-UXG).
8.
Carruthers, A.G., D.J. Sellwood, and P.W. Ward. 1980. “Recent developments in the retail price index.” The Statistician 29(1): 1–32. DOI: https://doi.org/10.2307/2987492.
9.
Carli, G. 1804. Del valore e della proporzione de’metalli monetati. In: Scrittori Classici Italiani di Economia Politica 13: 297–336. Available at: https://books.google.it/books?id=v31JAAAAMAAJ.
10.
Cobb, B.R., R. Rumi, and A. Salmeron. 2012. “Approximation the Distribution of a Sum of Log-normal Random Variables”. Paper presented at the Sixth European Workshop on Probabilistic Graphical Models, Granada, Spain, 2012. Available at: http://leo.ugr.es/pgm2012/proceedings/eproceedings/cobb_approximating.pdf.
11.
Consumer Price Index Manual. Theory and practice. 2004. ILO/IMF/OECD/UNECE/Eurostat/The World Bank, International Labour Office (ILO), Geneva. Available at: https://www.ilo.org/wcmsp5/groups/public/–-dgreports/–-stat/documents/presentation/wcms_331153.pdf.
12.
Dalén, J. 1992. “Computing Elementary Aggregates in the Swedish Consumer Price Index.” Journal of Official Statistics 8(2): 129–147. Available at: https://www.scb.se/-contentassets/ca21efb41fee47d293bbee5bf7be7fb3/computing-elementary-aggregates-in-the-swedish-consumer-price-index.pdf (accessed September 2020).
13.
Dalén, J. 1994. “Sensitivity Analyses for Harmonising European Consumer Price Indices”: 147–171 in International Conference on Price Indices: Papers and Final Report, First Meeting of the International Working Group on Price Indices , November 1994, Ottawa, Statistics Canada. Available at: https://www.ottawagroup.org/Ottawa/ottawagroup.nsf/home/Meetingþ1/$file/1994þ1stþMeetingþ-þDal%C3%A9n+J%C3%B6rgenþ-þSensitivityþAnalysesþforþHarmonisingþEuropeanþConsumerþPrice+Indices.pdf.
14.
Dalén, J. 1999. “A Note on the Variance of the Sample Geometric Mean. Research Report 1.” Department of Statistics, Stockholm University, Stockholm.
15.
Diewert, W.E. 1995. “Axiomatic and economic approaches to elementary price indexes.” Discussion Paper No. 95-01. Department of Economics, University of British Columbia, Vancouver, Canada. Available at: https://economics.ubc.ca/files/2013/06/pdf_paper_erwin-diewert-95-01-axiomatic-economic-approaches.pdf.
16.
Diewert, W.E. 2012. “Consumer price statistics in the UK.” Office for National Statistics, Newport. Available at: https://www.unece.org/fileadmin/DAM/stats/documents/ece/ces/ge.22/2014/WS1/WS1_1_Diewert_on_Diewert_Consumer_Price_Statistics__in_the_UK_v.7__06.08__Final.pdf.
17.
Dorfman, A.H., S. Leaver, and J. Lent. 1999. “Some observations on price index estimators.” U.S. Bureau of Labor Statistics (BLS) Statistical Policy Working Paper 29(2). Washington D.C. Available at: https://www.bls.gov/osmr/research-papers/1999/st990080.htm.
18.
Eichhorn, W., and J. Voeller. 1976. “Theory of the Price Index.” Lecture Notes in Economics and Mathematical Systems 140. Berlin-Heidelberg-New York: Springer-Verlag. DOI: https://doi.org/10.1007/978-3-642-45492-9.
19.
Evans, B. 2012. International comparison of the formula effect between the CPI and RPI. Office for National Statistics. Newport. Available at: https://pdfs.semanticscholar.org/6667/3228f9665c9cd31011da1ef932eee394afa9.pdf?_ga=2.36133664.1832035111.1583498850-1342646647.1583498850. Published online in 2012.
20.
Fenton, L.F. 1960. “The sum of log-normal probability distributions in scatter transmission systems.” IRE Transactions on Communications Systems 8(1): 57–67. DOI: https://doi.org/10.1109/tcom.1960.1097606.
21.
Gajek, L., and M. Kałuszka. 2004. “On the average rate of return in a continuous time stochastic model.” Working paper. Technical University of Lodz, Poland. Available at: https://www.researchgate.net/publication/270892015_ON_THE_AVERAGE_RATE_OF_RETURN_IN_A_CONTINUOUS_TIME_STOCHASTIC_MODEL.
22.
Greenlees, J.S. 2001. “Random errors and superlative indexes.” Working Paper 343. Bureau of Labour Statistics, Washington D.C. Available at: https://www.bls.gov/pir/-journal/gj09.pdf.
23.
Hardy, G.H., J.E. Littlewood, and G. Polya. 1934. Inequalities . Cambridge: Cambridge University Press. DOI: https://doi.org/10.2307/3605504.
24.
Hong-Bae, K., and P. Tae-Jun. 2015. “The Behavior Comparison between Mean Reversion and Jump Diffusion of CDS Spread.” Eurasian Journal of Economics and Finance 3(4): 8–21. DOI: https://doi.org/10.15604/ejef.2015.03.04.002.
25.
Hu, Y. 2000. “Multi-dimensional geometric Brownian motions, Onsager-Machlup functions, and applications to mathematical finance.” Acta Mathematica Scientia 20(3): 341–358. DOI: https://doi.org/10.1016/s0252-9602(17)30641-0.
26.
Hull, J. 2018. Options, Futures, and other Derivatives (10 ed.). Boston: Pearson.
27.
Jakubowski, J., A. Palczewski, M. Rutkowski, and L. Stettner. 2003. Matematyka finansowa. Instrumenty pochodne. Warszawa: Wydawnictwa Naukowo-Techniczne.
28.
Jevons, W.S. 1865. “On the variation of prices and the value of the currency since 1782.” J. Statist. Soc. Lond . 28: 294–320. DOI: https://doi.org/10.2307/2338419.
29.
Kou, S.G. 2002. “A jump-diffusion model for option pricing.” Management Science 48: 1086–1101. DOI: https://doi.org/10.1287/mnsc.48.8.1086.166.
30.
Kühn, R., and P. Neu. 2008. “Intermittency in an interacting generalization of the geometric Brownian motion model.” Journal of Physics A: Mathematical and Theoretical 41 (2008): 1–12. DOI: https://doi.org/10.1088/1751-8113/41/32/324015.
31.
Levell, p. 2015. “Is the Carli index flawed?: assessing the case for the new retail price index RPIJ.” J. R. Statist. Soc. A 178(2): 303–336. DOI: https://doi.org/10.1111/rssa.12061.
32.
McClelland, R., and M. Reinsdorf. 1999. “Small Sample Bias in Geometric Mean and Seasoned CPI Component Indexes.” Bureau of Labor Statistics Working Paper No. 324, Washington D.C. Available at: https://stats.bls.gov/osmr/research-papers/1999/pdf/ec990050.pdf.
33.
Meade, N. 2010. “Oil prices – Brownian motion or mean reversion? A study using a one year ahead density forecast criterion.” Energy Economics 32 (2010): 1485–1498. DOI: https://doi.org/10.1016/j.eneco.2010.07.010.
34.
Moulton, B.R., and K.E. Smedley. 1995. “A comparison of estimators for elementary aggregates of the CPI.” Paper present at Second Meeting of the International Working Group on Price Indices. Stockholm, Sweden. Available at: https://www.ottawagroup.org/Ottawa/ottawagroup.nsf/4a256353001af3ed4b2562bb00121564/8e98d9c3-d6e9363eca25727500004401/$FILE/1995%202nd%20Meeting%20-%20A%20comparison%20of%20esti.
35.
Nwafor, C.N., and A.A. Oyedele. 2017. “Simulation and Hedging Oil Price with Geometric Brownian Motion and Single-Step Binomial Price Model.” European Journal of Business and Management 9(9): 68–81. Available at: https://researchonline.gcu.ac.uk/files/24776117/Simulation_of_Crude_Oil_Prices_EJBM_Vol.9_No.pdf.
36.
Office for National Statistics. 2013. National Statistician announces outcome of consultation on RPI. Office for National Statistics, Newport. Available at: https://webarchive.nationalarchives.gov.uk/20160111163943/http:/www.ons.gov.uk/ons/dcp29904_295002.pdf.
37.
Oksendal, B. 2003. Stochastic Differential Equations: An Introduction with Applications. Berlin: Springer. DOI: https://doi.org/10.1007/978-3-642-14394-6_5.
38.
Privault, N. 2012. “An Elementary Introduction to Stochastic Interest Rate Modeling.” Advanced Series on Statistical Science & Applied Probability: Volume 16, Word Scientific. DOI: https://doi.org/10.1142/8416.
39.
Ross, S.M. 2014. “Variations on Brownian Motion.” In Introduction to Probability Models (11th ed.), 612–14. Amsterdam: Elsevier.
40.
Schultz, B. 1995. “Choice of price index formulae at the micro-aggregation level: The Canadian Empirical evidence.” (Version updated in June 1995) Paper presented at the first meeting of the Ottawa Group on Price indices , Canada. Available at: https://www.ottawagroup.org/Ottawa/ottawagroup.nsf/home/Meeting+1/$file/1994%201st%20-Meeting%20-%20Schultz%20Bohdan%20-%20Choice%20of%20Price%20Index%20Formulae%20at%20the%20Micro-Aggregation%20Level%20The%20Canadian%20Empirical%20Evidence%202nd%20Edition.pdf.
41.
Silver, H., and S. Heravi. 2007. “Why elementary price index number formulas differ: Evidence on price dispersion.” Journal of Econometrics , 140 (2007): 874–883. DOI: https://doi.org/10.1016/j.jeconom.2006.07.017.
42.
UK Statistics Authority. 2013. Consultation on the Retail Prices Index. Statement. UK Statistics Authority, London . Available at: http://www.statisticauthority.gov.uk/news/statement-consultation-on-the-retail-prices-index-10012013.pdf.
43.
Von der Lippe, p. 2007. Index Theory and Price Statistics. Peter Lang Verlag. DOI: https://doi.org/10.3726/978-3-653-01120-3.
44.
White, A.G. 1999. “Measurement Biases in Consumer Price Indexes.” International Statistical Review , 67(3): 301–325. DOI: https://doi.org/10.1111/j.1751-5823.1999.tb00451.
45.
Yu-Sheng H., and W. Cheng-Hsun 2011. “A Generalization of Geometric Brownian Motion with Applications.” Communications in Statistics – Theory and Methods , 40(12): 2081–2103. DOI: https://doi.org/10.1080/03610921003764167.