1. Systems of Index Numbers for International Price Comparisons Based on the Stochastic Approach Gholamreza Hajargasht D.S. Prasada Rao Centre for Efficiency and Productivity Analysis School of Economics University of Queensland Brisbane, Australia

2.  Standard formulae and the new index formula  Stochastic approach to index numbers  Derivation of index numbers using stochastic approach  Empirical application – OECD data  Concluding remarks Outline

3. Notation  We have M countries and N commodities  observed price of the ith commodity in the jth country  quantity of the ith commodity in the jth country  purchasing power parity of jth country  Average price for ith commodity  shares

4.  Commonly used methods like Laspeyres, Paasche, Fisher and Tornqvist methods are for bilateral comparisons – not transitive.  Geary-Khamis method – Geary (1958), Khamis (1970)  Elteto-Koves-Szulc (EKS) method – 1968  EKS method constructs transitive indexes from bilateral Fisher indexes  Weighted EKS index – Rao (2001) Index number methods for international comparisons

5.  Variants of Geary-Khamis method – Ikle (1972), Rao (1990)  Methods based on stochastic approach:  Country-product-dummy (CPD) method – Summers (1973)  Weighted CPD method – Rao (1995)  Generating indexes using Weighted CPD method – Rao (2005), Diewert (2005)  Using CPD to compute standard errors – Rao (2004), Deaton (2005) Index number methods - continued

6. Geary-Khamis Method F Geary (1958) and Khamis (1970) F Based on twin concepts: F PPPs of currencies - PPP j ’s F International averages of prices - P i ’s F Computations based on a simultaneous equation system:

7. Rao and Ikle variants  Rao Index Geometric Mean  Ikle Index Harmonic Mean

8. The New Index Why not an arithmetic mean Comments: For all these methods it is necessary to establish the existence of solutions for the simultaneous equations. Existence of Rao and Ikle indexes was established in earlier papers. Existence of the new index is considered in this paper.

9. Relationship between the indices and stochastic approach F The stochastic approach to multilateral indexes is based on the CPD model – discussed below. F The indexes described above are all based on the Geary-Khamis framework which has no stochastic framework. F Rao (2005) has shown that the Rao (1990) variant of the Geary-Khamis method can be derived using the CPD model. F Rest of the presentation focuses on the connection between the CPD model and the indexes described above.

10. The Law of One Price  Following Summers (1973), Rao (2005) and Diewert (2005) we consider   price of i-th commodity in j-th country purchasing power parity World price of i-th commodity random disturbance The CPD model may be considered as a “hedonic regression model” where the only characteristics considered are the “country” and the “commodity”.

11. CPD Model  Rao (2005) has shown that applying a weighted least square to the following equation results in Rao’s System where are treated as parameters where are treated as parameters  The same result is obtained if we assume a log-normal distribution for u ij and use a weighted maximum likelihood estimation approach which is the same as the weighted least squares estimator.

12. Stochastic Approach to New Index  Again  This time assume

13. Maximum Likelihood  It can be shown  Taking logs we obtain

14. Weights and M-Estimation  Define a weighted likelihood as  Then we have

15. First Order Conditions The new Index independent of r The new Index independent of r The advantage of MLE is that we can calculate standard errors for PPPs

16. Stochastic Approach to Ikle  Now we consider the model  where

17. First Order Conditions  Applying a weighted maximum likelihood we obtain the following first order conditions   i.e. Ikle’s Index

18. Standard Errors  Computation of standard errors is an important motivation for considering stochastic approach.  An M-Estimator is defined as an estimator that maximizes  It has the following asymptotic distribution  where

19.  In practice, a consistent estimator can be obtained as  Where  In special cases like the standard maximum likelihood we have therefore

20.  Many software report this as their default variance estimator  This Variance is not valid in our context and the general formula should be used  For example if we apply to a weighted linear model (e.g. CPD)  But if we apply the general formula

21. Application to OECD countries  OECD data from 1996.  The price information was in the form of PPPs at the basic heading level for 158 basic headings, with US dollar used as the numeraire currency.  The estimates of PPPs based on the new index, Ikle’s and the weighted CPD for 24 OECD countries along with their standard errors are presented in the following table.

22. Country MLE Estimates New IndexCPDIkle PPPS.EPPPS.EPPPS.E. GER 1.8870.136 2.0340.1442.1870.147 FRA 6.0920.429 6.5540.4557.0350.466 ITA 1425.96109.727 1504.02115.5091584.381119.196 NLD 1.9210.150 2.0560.1552.2050.156 BEL 35.4912.577 37.8902.69840.4502.728 LUX 33.5782.488 35.8162.61838.1912.700 UK 0.6030.043 0.6420.0440.6820.045 IRE 0.6370.051 0.6690.0550.6960.060 DNK 8.5250.586 9.1310.6159.7620.631 GRC 180.47013.452 188.48213.891196.64014.005 SPA 112.4148.304 118.5468.606124.7998.738 PRT 126.04310.400 129.03710.994130.31712.002 AUT 12.7700.881 13.7300.92814.7280.948 SUI 2.0500.168 2.1830.1772.3200.180 SWE 9.4240.686 10.0750.72010.7580.742 FIN 6.1590.432 6.5980.4537.0700.462 ICE 86.8287.000 89.5416.97592.3296.810 NOR 8.8070.684 9.2380.7369.6420.764 TUR 6304.23579.128 6321.42544.9076357.003506.991 AUS 1.2640.099 1.3330.1031.4070.104 NZL 1.4640.111 1.5300.1131.5960.115 JAP 182.03113.622 187.42914.282192.39214.780 CAN 1.1680.090 1.2290.0941.2950.096 USA 1.00

23. Conclusion  A new system for international price comparison is proposed. Existence and uniqueness established  A stochastic framework for generating the Rao, Ikle and the new index has been established.  Using the framework of M-estimators, standard errors are obtained for the PPPs from each of the methods.  Empirical application of the new approach to OECD data shows the feasibility of the approach

24. Future research  A more complete existence theorem  Statistical Comparison of the three Indices – choosing between the three distribution specifications  Bayesian Estimation and Inference seems to be promising in this context  Finding a suitable model to generate the Geary- Khamis method  Using residuals from the regression framework to generate standard errors for more standard index number formulae.