## _wp09271 - Annex 2 demonstrates this e

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### Random effects estimator
- Model specification and assumptions:
  - A RE panel model treats product and country-specific variation as random draws from respective zero mean distributions, i.e. (cov,0)*nj Xβ = and (cov,0)*cj Xγ =.
  - Assumes exogeneity of all the regressors and both product and country effects, unlike the FE model that allows for endogeneity of all the repressors.
  - Is estimated using GL or ML estimators.
  - Uses both within and between product variation across countries and therefore does not require product-specific dummy variables.
- Properties and comparisons:
  - A panel FE estimator as in (4) and (5) provides consistent estimates of *j aS δ even if the RE model is valid.
  - RE is more efficient than FE when both within and between variation are informative.
  - FE uses only within-product variation and equivalently regresses differences in knc p on differences in j X.
  - Potential disadvantage of FE: with measurement error the FE coefficients are downwards biased since within estimation may be based on small changes relative to the measurement error.
- Monte Carlo and practical guidance:
  - Baltagi (1981) Monte Carlo findings:
    - Using the mean squared error (MSE) criterion, there was always a gain from performing two-stage generalized least squares (GLS) compared with ordinary least squares (OLS).
    - All the GLS estimators considered outperformed OLS and had fairly close MSEs.
    - Recommendation: use GLS where appropriate (Baltagi, 2005).
  - Guidance:
    - Use more than one GLS procedure and, if results differ widely, test the specifications of the model.
    - A priori there is reason to use a CPD FE estimator, but a Hausman (1978) test should be used to decide whether to reject a RE specification.

### Hausman and Taylor estimator
- Focus and motivation:
  - Address potential correlation between the individual outlet-item interaction variable, *j δ in equation (3), with the explanatory variables jc X.
  - Treat the individual outlet-item interaction effect as a latent unmeasured omitted variable.
- Consequences of correlation:
  - If the latent interaction term is correlated with measured X, OLS and GLS estimates of *j δ are biased and inconsistent.
- Variable partitioning (Hausman and Taylor, 1981):
  - Divide jc X into endogenous variables 1jc X that are correlated with the latent interaction term and exogenous variables 2jc X that are uncorrelated.
  - Further divide each into country-invariant and country-varying subsets: 1ci jc X and 1cv jc X; 2ci jc X and 2cv jc X.
- Estimator mechanics and advantages:
  - FE estimator wipes out country-invariant 1ci jc X and 2ci jc X variables.
  - GLS RE estimator ignores endogeneity from the interaction term and produces biased and inconsistent estimates of parameters on 1jc X.
  - Hausman and Taylor propose an instrumental variable estimator that uses 2jc X exogenous variables as instruments.
  - Country-varying 1jc X that are uncorrelated with the latent interaction term serve two functions:
    - (i) as deviations from individual mean they produce unbiased estimates of *j δ, and
    - (ii) using the individual means they provide valid instruments for estimating parameters of country-invariant jc X uncorrelated with the interaction term.
  - Advantages:
    - HT is an improvement on FE: more efficient and produces unbiased estimates of parameters of country-invariant variables when assumptions hold.
    - HT requires at least as many exogenous country-varying variables as endogenous ones; otherwise HT is identical to FE because parameters on country-invariant variables cannot be estimated.

### Tests for choosing among estimators
- Key testing steps and recommendations:
  - Test whether product dummies in the “P” term of any CPD serve any purpose (null hypothesis **n f ββ =).
  - Baltagi (1981) shows the Chow test performs poorly due to invalid homoskedastic cross-sectional effects assumption; proposes alternatives including the Roy statistic.
  - Hausman and Taylor (1981) provide a specification test for the non-correlation assumptions required for an RE model.
  - Baltagi, Bresson, and Pirotte (2003) recommend:
    - First, undertake a standard Hausman test based on FE versus RE estimators and use the RE estimator if the Hausman test is not rejected.
    - If the Hausman test is rejected, undertake a second test based on the difference between FE and HT estimators, and use HT if some variables (but not all) are correlated with individual effects.
    - Otherwise use an FE estimator.

### Summary (V. SUMMARY)
- Context and econometric issues:
  - The 2005 ICP used the regression-based CPD approach to derive BH parity estimates and raised econometric issues because of panel structure.
  - Major concerns:
    - Use of grouped data in 2005 ICP (average prices across outlets for each of the N products) ignored between-outlet price variation, causing efficiency loss and potential omitted variable bias.
    - 2005 ICP innovation: detailed product specifications for each item price, enabling more precise matching and identification of non-comparable replacements for missing items and their quality differences.
- Recommendations and arguments:
  - Include non-comparable replacement items in a hedonic CPD for the 2011 ICP round, combining detailed product descriptions and regression-based parity estimates.
  - If grouped data continue to be used (advised against), collect and include non-comparable replacements and incorporate them into a hedonic CPD using group averages of quality characteristics.
  - Most price-determining characteristics should be included in a hedonic CPD to be consistent and mitigate omitted variable bias.
  - Further stratification of a CPD regression by outlet-type and location is a limited case of including price-determining variables; finer coding is preferable to coarser coding for efficiency.
  - Explicit adjustments for non-comparable replacements (e.g., using information from price collectors or experts) are less preferred econometrically; hedonic methods are preferred to incorporate quality characteristics directly.
  - Hedonic CPD (possibly with country-product interaction terms) is argued to be preferable to separate CPD and hedonic approaches, with consideration of panel estimator choice (FE, RE, HT).
  - Shortcomings of grouped data can be remedied by using the ungrouped (individual) data from which grouped data were derived.

### Annex 1. Characteristic Price Index Numbers
- Concept and use:
  - Characteristic price (hedonic) index numbers do not use a regression model for parity estimates, but do use a regression for the quality adjustment.
  - Useful when there is large turnover in new models and little panel structure in the data (e.g., personal computers).
- Formulation and properties:
  - Form: quantity of characteristics held constant for each product in each country, valued by country 2 hedonic parameters in the numerator and country 1 hedonic parameters in the denominator.
  - Example formulas when holding country 1 or country 2 quantities constant: (A1.1) uses sums of *j X ˆ δˆ and j X ˆ δˆ terms as in the source.
  - Superlative index number formulae can be defined as symmetric means of the two formulas or symmetric mean-value (Silver and Heravi, 2007b).
- Key conceptual distinction:
  - Regression-based (equations (5)) holds parameters constant so that difference in intercepts is invariant to any value of X.
  - Characteristic price indices (equation (8)) allow parameters to change while holding quantities constant; ˆ1 *j δ and ˆ2 *j δ differ.
- Comparative findings:
  - DHS (2008) and Silver and Heravi findings:
    - Numerical difference between the two methods depends on the exponent of the product of (expenditure-share weighted country) differences in coefficients, mean values of X, and the relative characteristics variance-covariance matrix.
    - Differences increase as the product market diversifies into distinct bundles of characteristics (e.g., high-end and bottom-of-the-market offerings).

### Annex 2. An Intuition for a Hedonic CPD
- Setup and equations:
  - Bilateral comparison c = 1, 2 with product defined by country-invariant quality characteristics X'12nn nc XXX = = nc Z1 3 n Z that change over products but not over countries, and a single country-varying characteristic that may change over products and countries (example: 3 GB PC observed in country 1 and non-comparable replacement 4 GB PC in country 2).
  - Consider for countries c = 1, 2: *ncnc *Z ω λ + 'nc X + *** ncnnc e ω β **** nc D p δ γ α + + = 221 where ***ˆˆˆ 122 α α γ − = * are the estimated parities and ˆ δ * and ˆ λ * n n * n ZX 111 ω λ + + * nn * n ZX 222 ω λ + + ***12ncnn are constrained to be the same across the two countries. (Equations labeled (A.2.1), (A.2.2), (A.2.3) in source.)
- Interpretation and intuition:
  - Equations for countries 1 and 2 given by ***n p11 δ + = and ***n p22 δ + = respectively.
  - Difference *n *n *n p p p 12 − = Δ' nc is, by construction of X and Z, given by (A.2.4): *nn ** eZ Δ Δ λ α + + 1 nc X *n ** n p α Δ − = 2.
  - Dropping country-invariant changes and product-specific effects β yields identities shown in (A.2.5).
  - Key intuition: for a given product n, the price parity measured by a hedonic CPD estimator is, after taking expectations, the cross-country difference in the average price levels of comparable products (the constant change) plus any cross-country difference in quality characteristics between the original product specification and the non-comparable replacement’s product specification each multiplied by constrained estimates of their hedonic valuation.
  - The FE panel estimator is equivalent to OLS regression of differences in (A.4) and offers computational convenience when there are many products (Davidson and MacKinnon, 1993).
  - The hedonic CPD FE intuition: parity = average comparable-product price difference + quality-difference × hedonic valuations.

*Source: _wp09271 - Annex 2 demonstrates this e (PDF chapter/section).*

### Annex 2 demonstrates this e

### _wp09271 - Annex 2 demonstrates this e

### Random effects estimator
- A RE panel model treats product and country-specific variation as random draws from respective zero mean distributions, i.e. (cov,0)*nj Xβ = and (cov,0)*cj Xγ =.
- RE model assumptions and properties:
  - Assumes exogeneity of all the regressors and both product and country effects, unlike the FE model that allows for endogeneity of all the repressors.
  - Is estimated using GL or ML estimators.
  - A panel FE estimator as in (4) and (5) provides consistent estimates of *j aS δ even if the RE model is valid.
  - The RE model uses both within and between product variation across countries and therefore does not require product-specific dummy variables; it is more efficient in such circumstances.
- Baltagi (1981) Monte Carlo findings:
  - Using the mean squared error (MSE) criterion, there was always a gain from performing two-stage generalized least squares (GLS) compared with ordinary least squares (OLS).
  - All the GLS estimators considered outperformed OLS and had fairly close MSEs.
  - Recommendation: use GLS where appropriate (Baltagi, 2005).
- Practical notes:
  - A FE model does not use dummy variables but equivalently regresses differences in knc p on differences in j X; FE uses only within-product variation.
  - Potential disadvantage of FE: with measurement error the FE coefficients are downwards biased since within estimation may be based on small changes relative to the measurement error.
- Guidance:
  - Use more than one GLS procedure and, if results differ widely, test the specifications of the model.
  - A priori there is reason to use a CPD FE estimator, but a Hausman (1978) test should be used to decide whether to reject a RE specification.

### Hausman and Taylor estimator
- Focus:
  - Address potential correlation between the individual outlet-item interaction variable, *j δ in equation (3), with the explanatory variables jc X.
  - The individual outlet-item interaction effect is treated as a latent unmeasured omitted variable.
- Consequences of correlation:
  - If the latent interaction term is correlated with measured X, OLS and GLS estimates of *j δ are biased and inconsistent.
- Variable partitioning (Hausman and Taylor, 1981):
  - Divide jc X into endogenous variables 1jc X that are correlated with the latent interaction term and exogenous variables 2jc X that are uncorrelated.
  - Further divide each into country-invariant and country-varying subsets: 1ci jc X and 1cv jc X; 2ci jc X and 2cv jc X.
- Estimator properties:
  - FE estimator wipes out country-invariant 1ci jc X and 2ci jc X variables.
  - GLS RE estimator ignores endogeneity from the interaction term and produces biased and inconsistent estimates of parameters on 1jc X.
  - Hausman and Taylor propose an instrumental variable estimator that uses 2jc X exogenous variables as instruments.
  - Country-varying 1jc X that are uncorrelated with the latent interaction term serve two functions:
    - (i) as deviations from individual mean they produce unbiased estimates of *j δ, and
    - (ii) using the individual means they provide valid instruments for estimating parameters of country-invariant jc X uncorrelated with the interaction term.
- Advantages:
  - HT is an improvement on FE: more efficient and produces unbiased estimates of parameters of country-invariant variables when assumptions hold.
  - HT requires at least as many exogenous country-varying variables as endogenous ones; otherwise HT is identical to FE because parameters on country-invariant variables cannot be estimated.

### Tests for choosing among estimators
- Key testing steps and findings:
  - Test whether product dummies in the “P” term of any CPD serve any purpose (null hypothesis **n f ββ =).
  - Baltagi (1981) shows the Chow test performs poorly due to invalid homoskedastic cross-sectional effects assumption; proposes alternatives including the Roy statistic.
  - Hausman and Taylor (1981) provide a specification test for the non-correlation assumptions required for an RE model.
  - Baltagi, Bresson, and Pirotte (2003) recommend:
    - First, undertake a standard Hausman test based on FE versus RE estimators and use the RE estimator if the Hausman test is not rejected.
    - If the Hausman test is rejected, undertake a second test based on the difference between FE and HT estimators, and use HT if some variables (but not all) are correlated with individual effects.
    - Otherwise use an FE estimator.

### Summary (V. SUMMARY)
- The 2005 ICP used the regression-based CPD approach to derive BH parity estimates and raised econometric issues because of panel structure.
- Major concerns:
  - Use of grouped data in 2005 ICP (average prices across outlets for each of the N products) ignored between-outlet price variation, causing efficiency loss and potential omitted variable bias.
  - 2005 ICP innovation: detailed product specifications for each item price, enabling more precise matching and identification of non-comparable replacements for missing items and their quality differences.
- Recommendations and arguments:
  - Include non-comparable replacement items in a hedonic CPD for the 2011 ICP round, combining detailed product descriptions and regression-based parity estimates.
  - If grouped data continue to be used (advised against), collect and include non-comparable replacements and incorporate them into a hedonic CPD using group averages of quality characteristics.
  - Most price-determining characteristics should be included in a hedonic CPD to be consistent and mitigate omitted variable bias.
  - Further stratification of a CPD regression by outlet-type and location is a limited case of including price-determining variables; finer coding is preferable to coarser coding for efficiency.
  - Explicit adjustments for non-comparable replacements (e.g., using information from price collectors or experts) are less preferred econometrically; hedonic methods are preferred to incorporate quality characteristics directly.
  - Hedonic CPD (possibly with country-product interaction terms) is argued to be preferable to separate CPD and hedonic approaches, with consideration of panel estimator choice (FE, RE, HT).
  - Shortcomings of grouped data can be remedied by using the ungrouped (individual) data from which grouped data were derived.

### Annex 1. Characteristic Price Index Numbers
- Characteristic price (hedonic) index numbers:
  - Do not use a regression model for parity estimates, but do use a regression for the quality adjustment.
  - Useful when there is large turnover in new models and little panel structure in the data (e.g., personal computers).
  - Form: quantity of characteristics held constant for each product in each country, valued by country 2 hedonic parameters in the numerator and country 1 hedonic parameters in the denominator.
  - Example formulas when holding country 1 or country 2 quantities constant: (A1.1) uses sums of *j X ˆ δˆ and j X ˆ δˆ terms as in the source.
  - Superlative index number formulae can be defined as symmetric means of the two formulas or symmetric mean-value (Silver and Heravi, 2007b).
- Key conceptual distinction:
  - Regression-based (equations (5)) holds parameters constant so that difference in intercepts is invariant to any value of X.
  - Characteristic price indices (equation (8)) allow parameters to change while holding quantities constant; ˆ1 *j δ and ˆ2 *j δ differ.
- DHS (2008) and Silver and Heravi findings:
  - Numerical difference between the two methods depends on the exponent of the product of (expenditure-share weighted country) differences in coefficients, mean values of X, and the relative characteristics variance-covariance matrix.
  - Differences increase as the product market diversifies into distinct bundles of characteristics (e.g., high-end and bottom-of-the-market offerings).

### Annex 2. An Intuition for a Hedonic CPD
- Setup (bilateral comparison c = 1, 2):
  - Product defined by country-invariant quality characteristics X'12nn nc XXX = = nc Z1 3 n Z that change over products but not over countries, and a single country-varying characteristic that may change over products and countries (example: 3 GB PC observed in country 1 and non-comparable replacement 4 GB PC in country 2).
  - Consider for countries c = 1, 2: *ncnc *Z ω λ + 'nc X + *** ncnnc e ω β **** nc D p δ γ α + + = 221 where ***ˆˆˆ 122 α α γ − = * are the estimated parities and ˆ δ * and ˆ λ * n n * n ZX 111 ω λ + + * nn * n ZX 222 ω λ + + ***12ncnn are constrained to be the same across the two countries. (Equations labeled (A.2.1), (A.2.2), (A.2.3) in source.)
- Interpretation:
  - Equations for countries 1 and 2 given by ***n p11 δ α + = and ***n p22 δ α + = respectively.
  - Difference *n *n *n p p p 12 − = Δ' nc is, by construction of X and Z, given by (A.2.4): *nn ** eZ Δ Δ λ α + + 1 nc X *n ** n p α Δ − = 2.
  - Dropping country-invariant changes and product-specific effects β yields identities shown in (A.2.5).
- Key intuition:
  - For a given product n, the price parity measured by a hedonic CPD estimator is, after taking expectations, the cross-country difference in the average price levels of comparable products (the constant change) plus any cross-country difference in quality characteristics between the original product specification and the non-comparable replacement’s product specification each multiplied by constrained estimates of their hedonic valuation.
  - The FE panel estimator is equivalent to OLS regression of differences in (A.4) and offers computational convenience when there are many products (Davidson and MacKinnon, 1993).
  - The hedonic CPD FE intuition: parity = average comparable-product price difference + quality-difference × hedonic valuations.

*Source: _wp09271 - Annex 2 demonstrates this e (PDF chapter/section).*

### References

### _wp09271 - References

### Methodology and Price/Quantity Indexes
- Aizcorbe, A., Corrado, C. and Doms M. (2000). Constructing Price and Quantity Indexes for High Technology Goods, Industrial Output Section, Division of Research and Statistics, Board of Governors of the Federal Reserve System, July.
- Balk, Bert M., (2005). “Price Indexes for Elementary Aggregates: The Sampling Approach,” Journal of Official Statistics, Vol. 21, No. 4, pp. 675–99.
- Diewert, W. Erwin (1999). Axiomatic and Economic Approaches to International Comparisons. In International and Interarea Comparisons of Income, Output and Prices, Alan Heston and Robert E. Lipsey (eds.) Studies in Income and Wealth, 61, 13–87, Chicago: University of Chicago Press.
- Diewert, W. Eerwin (2002). Hedonic Regressions: A Review of Some Unresolved Issues, Mimeo, Department of Economics, University of British Columbia.
- Diewert, W. Erwin (2004). “Elementary Indices,” in Consumer Price Index Manual: Theory and Practice, (Geneva: International Labour Office) Chapter 20, pages 355–70. Available via the Internet at www.ilo.org/public/english/bureau/stat/guides/cpi/index.htm.
- Diewert, W. Erwin (2005) Weighted Country Product Dummy Variable Regressions and Index Number Formulae, Review of Income and Wealth, 51, 4, December, 561–70.
- Diewert, W. Erwin (2008). New Methodology for Linking Regional PPPS, ICP Bulletin, 5, 2, August, 1, 10–21.
- Diewert, W. Erwin, Heravi, Saeed, and Silver, Mick (2008). Hedonic Imputation Indexes Versus Time Dummy Hedonic Indexes, NBER Working Paper 14018. Forthcoming in W. Erwin Diewert, John Greenlees, and Charles R. Hulten (eds.) Price Index Concepts and Measurement, NBER, Chicago: University of Chicago Press, 278–337, 2010.
- Triplett, Jack E., (2004). Handbook on Quality Adjustment of Price Indexes for Information and Communication Technology Products, OECD Directorate for Science, Technology and Industry, Draft, OECD, Paris.
- Burdette, Steve (2007). Comparing the Prices of Machinery and Equipment Across the Globe, ICP Bulletin, 4, 2, August, 5–8.

### Hedonic Methods, Scanner Data, and Quality Adjustment
- Berndt, E.R. and Rappaport N.J. (2001). Price and Quality of Desktop and Mobile Personal Computers: A Quarter-Century Historical Overview, American Economic Review, 91, 2, 268–273.
- Heravi, Saeed, Heston, Alan and Silver, Mick (2003). Using Scanner Data to Estimate Country Price Parities: An Exploratory Study. Review of Income and Wealth, 49,1, 1–22, March.
- Ioannides, Christos and Silver, Mick (1999). Estimating Exact Hedonic Indexes: An Application to U.K. Television Sets, Journal of Economics, Zeitschrift Für NationalÖkonomie, 69, 1.
- Kokoski, Mary, Cardiff, Patrick and Brent Moulton (1994). Interarea Price Indices for Consumer Goods and Services: An Hedonic Approach Using CPI Data, Working Paper No. 256, available from the Office of Prices and Living Conditions, July.
- Kokoski, Mary F., Moulton, Brent R. and Zieschang, KimberlyD. (1999). Interarea Price Comparisons for Heterogeneous Goods and Several Levels of Commodity Aggregation. In A Heston and R Lipsey eds., International and Interarea Comparisons of Prices, Output and Productivity, Committee on Research on Income and Wealth, NBER, Chicago: University of Chicago Press, pp. 123–66.
- Pakes A. (2003). A Reconsideration of Hedonic Price Indexes with an Application to PCs, The American Economic Review, 93, 5, December, 1576–93.
- Silver, Mick (2002). The Use of Weights in Hedonic Regressions : The Measurement of Quality-Adjusted price Changes, Mimeo, Cardiff Business School, Cardiff University.
- Silver, Mick (2007). Why Elementary Price Index Number Formulas Differ: Price Dispersion and Product Heterogeneity.” (with S. Heravi), Journal of Econometrics, 140, 2, 874–83, October.
- Silver, Mick and Heravi, Saeed (2005). A Failure in the Measurement of Inflation: Results from a Hedonic and Matched Experiment Using Scanner Data.” Journal of Business and Economic Statistics, 23, 3, 269–281, July.
- Silver, Mick and Heravi, Saeed (2007a). Purchasing Power Parity Measurement and Bias from Loose Item Specifications in Matched Samples: An Analytical Model and Empirical Study, Journal of Official Statistics, 21, 3, 463–487.
- Silver, Mick and Heravi, Saeed (2007b). Hedonic Indexes: A Study of Alternative Methods. In E.R. Berndt and C. Hulten (eds.) Hard-to-Measure Goods and Services: Essays in Honour of Zvi Griliches, pp. 235–268, NBER/CRIW, Chicago: University of Chicago Press, 2008.
- Silver, Mick and Saeed Heravi (2007c), Hedonic Imputation Indexes and Time Dummy Hedonic Indexes, Journal of Business and Economic Statistics, 25:2, 239–246.
- Machado, José A. F. and Joa[tilde on a]o M.C. Santos Silva (2001) Identification with Averaged data and Implications for Hedonic Regression Studies, Economic Research Department Working Paper 10-01, November.

### International Comparisons, Purchasing Power Parities, and Interarea Price Levels
- Aten, Bettina. H., (1996). Evidence of Spatial Autocorrelation in International Prices, Review of Income and Wealth, June, 42, 2, 149–63.
- Aten, Bettina (2003) “Report on Interarea Price Levels, 2003,” Working Paper No. 2005-11, Bureau of Economic Analysis, May.
- Aten, Bettina, H. (2006) Interarea Price Levels: An Experimental Methodology, Monthly Labor Review, September, 129, 9, 47–61.
- Cuthbert, J. and M. Cuthbert. (1988). On Aggregation Methods of Purchasing Power Parities, Working Paper No. 56, November, Paris: OECD.
- Deaton, Angus and Alan Heston (2008). Understanding PPPs and PPP-based national accounts, paper presented at CRIW-NBER Summer Workshop, July 14–15, 2008.
- Rao, D.S. Prasada (2002). On the Equivalence of Weighted Country-Product-Dummy (CPD) Method and the Rao System for Multilateral Price Comparisons, School of Economics, University of New England, Armidale, Australia, March.
- Rao, D.S. Prasada, (2004). The Country-Product-Dummy Method: A Stochastic Approach to the Computation of Purchasing Power Parities in the ICP. Paper presented at the SSHRC International Conference on Index Number Theory and the Measurement of Prices and Productivity, Vancouver, Canada, June 30–July 3.
- Diewert, W. Erwin (2008). New Methodology for Linking Regional PPPS, ICP Bulletin, 5, 2, August, 1, 10–21.
- World Bank, (2007). International Comparisons Program Handbook, 2003–2006, Washington D.C.: World Bank. Available by clicking on: www.worldbank.org/data/icp

### Econometrics, Panel Data, and Statistical Methods
- Baltagi, Badi H. (1981). A Two-Way Error Component Model, Journal of Econometrics, 17, 21–49.
- Baltagi, Badi H. (2005). Econometric Analysis of Panel Data, Chichester: John Wiley & Sons.
- Baltagi Badi H., Bresson, Georges and Alaine Pirotte (2003). Fixed Effects, Random Effects or Hausman-Taylor? A Pretest Estimator, Economic Letters, 79, 361–369.
- Baltagi, Badi H. and Dan Levin (1986) Estimating Dynamic Demand for Cigarettes Using Panel Data: The Effects of Bootlegging, Taxation and Advertising Reconsidered, The Review of Economics and Statistics, 68, 1, February, 148–155.
- Davidson, R. and J.G. MacKinnon (1993). Estimation and Inference in Econometrics, Oxford: University Press.
- Dhrymes, Phoebus J. and Adriana Lleras-Muney (2006) Estimation of Models with grouped and Ungrouped Data by Means of “2SLS”, Journal of Econometrics, 133, 1–29.
- Dickens, William T. (1990). Error Components in Grouped Data: Is It Ever Worth Weighting? The Review of Economics and Statistics, 72, 2, May, 328–333.
- Druska, V. and Horrace, W.C. (2004). Generalised Moments Estimation for Spatial Panel Data: Indonesia Rice Farming, American Journal of Agricultural Economics, 86 ,1 February, 185–198.
- Evans, W. N., Froeb, L. M. and Werden, G.J. (1993). Endogeneity in the concentration-price relationship: causes consequences and cures, Journal of Industrial Economics, XLI, 4, December, 431–438.
- Hausman, J.A. (1978). Specification Tests in Econometrics, Econometrica, 46, 1251 – 1271.
- Hausman, J. A. and Taylor, W.E. (1981). Panel Data and Unobservable Individual Effects, Econometrica, 49, 1377–1398.
- Johnston, J. and Dinardo, J. (1997). Econometric Methods, Fourth Edition, New York: McGraw-Hill.
- Kmenta, Jan (1986). Elements of Econometrics, 2nd Edition. New York: Maxwell Macmillan International Edition.
- Moulton, Brent (1986). Random Group Effects and the Precision of Regression Estimates, Journal of Econometrics, 32, 385–397.
- Sickles, R.C. (2005). Panel Estimators and the Identification of Firm-Specific Efficiency Levels in Parametric and Non-Parametric Settings, Journal of Econometrics, June, 126, 2, 305–34.
- Zellner, A. (1962). An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests of Aggregation Bias, Journal of the American Statistical Association, 57, 348–368.
- Teekens, R. and Koerts, J. (1972) Some Statistical Implications of the Log Transformations of Multipicative Models, Econometrica, 40, 5, 793–819.

### Additional empirical and theoretical contributions
- Aten, Bettina (2003) “Report on Interarea Price Levels, 2003,” Working Paper No. 2005-11, Bureau of Economic Analysis, May.
- Summers, R. (1973). International Comparisons with Incomplete Data, The Review of Income and Wealth, March.
- Triplett, Jack. E. and McDonald, R.J. (1977) Assessing the Quality Error in Output Measures: The Case of Refrigerators, The Review of Income and Wealth, 23, 2, 137–156.
- Rao, D.S. Prasada (2002). On the Equivalence of Weighted Country-Product-Dummy (CPD) Method and the Rao System for Multilateral Price Comparisons, School of Economics, University of New England, Armidale, Australia, March.
- Machado, José A. F. and Joa[tilde on a]o M.C. Santos Silva (2001) Identification with Averaged data and Implications for Hedonic Regression Studies, Economic Research Department Working Paper 10-01, November.

*Source: _wp09271 - References*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2009/_wp09271.pdf_