A. Aggregation at the Basic Heading Level²

The 2005 ICP was based on prices collected for about 1,000 product specifications (PSs) grouped into 155 basic (expenditure) headings (BHs) in 146 countries divided into six regions. Separate product lists were used for each region to facilitate comparability, with an additional product list priced in 18 “ring” countries to enable inter-regional comparisons. The CPD method was used to estimate BH parities in four regions—Africa, Asia Pacific, West Asia, and South America³—and the EKS* method (see Diewert, 1999 and World Bank (2007), Chapter 11) was used for the OECD/Eurostat and the Commonwealth of Independent States (CIS) regions.⁴ The CPD method was used for (C=) 48 African; 23 Asia-Pacific; 11 West Asian; and 10 South America countries.

In each country the 1,000 PSs were grouped into 155 BHs; for example, (N=) 8 detailed PSs or types of poultry comprised the BH “poultry.” A PS for poultry may be a half lb. packet of non-branded, frozen, free range, de-boned, chicken breasts with skin sold in supermarkets—hereafter the term “product” and PS will be used interchangeably. For each BH for each region, say poultry for Africa, the average prices for each of the 8 products in each of the 48 African countries were used as the left-hand-side variable in a CPD regression and 48- 1=47 country and 8-1=7 product dummies on the right. The coefficients on the country dummies provided parity estimates for each of the 155 BHs for the 48 countries in Africa, and similarly for the other regions employing the CPD method. These BH parity estimates were then aggregated to PPPs using methods outlined in the World Bank’s ICP Handbook (2007) and Diewert (2008).

Of particular concern to the CPD aggregation is that for the 2005 ICP (i) average prices⁵ across outlets for each of the N products in the C countries were used, thus ignoring between outlet price variation in the regression, with an attendant loss of efficiency and potential omitted (outlet) variable bias; and (ii) the use of average prices across outlets negated the opportunity to effectively include quality adjustments into the aggregation procedure resulting in bias from either a loss of representativity, due to the omission of such non-comparable products, or due to the inclusion of non-comparable replacement item prices as if they were comparable, an issue we now turn to.

B. Checklists, Missing Observations, Non-Comparable Replacements, and Quality Adjustments

The 2005 ICP used highly detailed “tight” PSs or checklists based on detailed structured product descriptions (SPDs) to describe the price-determining characteristics of the products to be priced. ⁶This in turn enabled consistent cross-country matched price comparisons of like with like. But tight PSs make it less likely that comparable products will be found in different outlets/countries for matched price comparisons. The problem of missing price observations is accentuated by the very tightness of the PS. If a comparable match to the specification could not be found in an outlet in a country, the observation is either ignored, or the prices of non-comparable replacements used on the assumption, rightly or wrongly, that the difference in price due to the quality differentials were insignificant.⁷ Deaton and Heston (2008) note that lower quality items in poor countries may end up matched to higher quality items in rich countries, leading to an understatement of price levels in poor countries. For many goods the outlets sampled in poor countries may be closer to “dollar-stores” than to the typical outlet in the US. There is thus a trade-off between having tight specifications and poor coverage of items sampled, and loose specifications with price differences tainted by quality differences (Silver and Heravi, 2007a).

A proposal of this paper is for price collectors to select non-comparable replacements if a comparable item is not found, and note on their checklists the nature and extent of the difference from the original PS, say, a skinless chicken breast instead of one with skin. The CPD regression, with prices on the left hand side and country and product dummies on the right, can be extended to include the quality characteristics on the right to partial out the effect of the quality differences. This would serve to increase the representativity and efficiency of the resulting parameter estimates on the country dummies. The resurgence of interest in the 2005 ICP round in the CPD method and the innovation in the round of the use of detailed specifications provides a basis to deal with this problem—a hedonic CPD method. The ICP (2007) Handbook advocates the merits of a hedonic CPD especially with regard to the use of loose specifications for certain product areas:

“As already noted, the advantage of using loose specifications is that the number of price observations collected may be greatly increased. With very large sets of price observations it may be feasible to adjust for quite substantial differences in quality by employing hedonics.

One promising development is the combination of hedonics with the country product dummy method, or CPD, of estimating the parities for the basic heading, as both use the same type of multiple regression analysis.... …The use of loose specifications is therefore a real possibility that needs to be further researched for ICP purposes, but as yet there is little experience or evidence to demonstrate how well it works in practice and how robust the estimates are.”

Exclusive reliance on tight specifications may result in the basic heading PPPs being based on a quite restricted set of prices. The sample of products generated by the matched product approach based on tight specifications is far from random. There is a risk that unknown biases may be introduced.”⁸ (World Bank, Chapter 5, page 8).

The use of a hedonic country dummy framework for international price comparisons is not new—Heravi, Heston and Silver (2003)—though a hedonic country product dummy regression has, to the author’s knowledge, only been applied for inter-area PPP estimates within the United States, as outlined later. Given the importance of incorporating quality variations into PPP estimates there is, however, no formal evaluation as to how this can best be done. This paper addresses this issue.

A. A CPD Regression Using Averages Across Outlets

Does the use of product averages as in equation (2) affect the CPD estimates?

Kmenta (1986) demonstrates that OLS parameter estimates based on group means are unbiased, though the disturbances are likely to be heteroskedastic and estimates inefficient, as a result of varying within group sample sizes. Such heteroskedasticity is avoided if the number of observations (outlets sampled for each product group in each country) is the same, but this is not so for the ICP. However, a WLS estimator with weights equal to the square root of the number of outlets sampled within each group, a readily available metric for ICP, should minimize this loss of efficiency and is thus advisable.

Dickens (1990) argued that the above analysis requires an assumption that regression errors for the individual observations within groups are independent and identically distributed. To assume disturbances are independent is to make the unlikely assumption that individual outlet prices within a product group and country share no common unobserved characteristics. Dickens (1990) further argues that where grouping is by a common characteristic, weighting by group size can lead to more, rather than less, heteroskedasticity, producing inefficient parameter estimates and biased estimates of the standard errors. Of course simple tests for heteroskedasticity can be run to see if this is the case.

Kmenta (1986) also compared the variances of the parameter estimates using OLS on ungrouped data and WLS on grouped data finding that there will always be some further of loss of efficiency even if the disturbances are homoskedastic. Little efficiency will be lost if the within group variation is small relative to the between group variation in the (unobserved) Xs and there will be no efficiency loss if all Xs are the same within a group. Also there will be spurious increases in the ${\overline{R}}^2$ due to the grouping.

The effect of efficiency losses on the estimated parameters may not be trivial. Machado and Santos Silva (2001) found for a hedonic regression of rental price for digital computers on their characteristics the coefficients on the dummy variable for 1963 compared with 1960, equivalent to a country parity, were (in logarithms) -0.594 (OLS) and -0.211 (WLS)—a 45 percent compared with 19 percent fall. The difference between parameter estimates using OLS and WLS (weights based on number of observations) estimators was substantial even in this case where disturbances were homoskedastic.

There is a further issue with a CPD formulation. If there is a latent unobserved variable that say has a higher values in some countries, the parity estimate will be biased and inconsistent as a result of multicollinearity between the omitted variable and ${\gamma }_c^{*}$. Further, even if there is no multicollinearity, between the parity estimate and the omitted variable, say it is between the omitted variable and the product dummies ${\beta }_n^{*}$, say chicken breasts of a certain brand are sold in better quality stores with stricter adherence to freshness than other poultry products, then the nature of omitted variable bias is that the parity estimate will be unbiased, but ${\alpha }_1^{*}$, upon which the parity estimate is benchmarked, will be biased and inconsistent and the variance of the parity estimate, upward biased (Kmenta, 1986).

Thus grouping can result in efficiency losses due to heteroskedasticity that may be mitigated against or aggravated by the use of a WLS estimator and that may also arise even if disturbances are homoskedastic. Further, spurious ${\overline{R}}^2$ detract from the normal array of diagnostics for detecting heteroskedasticity. The reliance on only country and product dummies also serves to preclude the introduction of quality variations when an item is missing. We thus turn to consider the inclusion of quality characteristics in a hedonic CPD based on grouped data. Such characteristics will be part of the PS of a product and price collectors can simply record any deviations from these specifications so that average changes in the quality of the product can be measured for a group.

B. A Hedonic CPD Regression Using Averages Across Outlets

Can average values of the quality characteristics be entered as explanatory variables in a CPD regression using averages?

Consider (for each BH) a regression akin to equation (2) based on g=1,…,G country product groups where ${\overline{X}}_{\mathit{\text{gj}}}={\displaystyle {\sum }_{k\in g}}{X}_{\mathit{\text{kj}}}/K_g$ are means of each j explanatory variable within each country product group:

In this formulation we include quality characteristics in a CPD regression based on grouped averages. If, for example, chicken breasts in country 1 were on average better quality in some measurable way than in country 2, then the parity estimates ${\gamma }_c^{*}$ would be conditioned on the higher values of ${\overline{X}}_j$. Again tests for heteroskedasticity would need to be employed since sample sizes may vary between countries and product groups and WLS estimators employed, and again, even for homskedastic disturbances, the resulting estimators may be inefficient compared with regressions on ungrouped data. Yet further issue arises.

Machado and Santos Silva (2006) consider the case where the selection of groups is endogenous, in the sense that selection depends on unobserved characteristics affecting the dependent variable, something likely in the context of this paper; product and country grouping should affect price. Machado and Santos Silva (2006) and Dhrymes, and Lleras-Muney (2006) demonstrate that if group selection is endogenous, consistent and efficient estimates can only be obtained if covariate characteristics are fixed within groups and WLS is used, otherwise the estimates will not be consistent. Such fixing of the covariates is at odds with the needs of this paper, that is to utilize methods that condition parity estimates on quality variation. However, the dependence underlying the endogeneity is conditioned on the ${\overline{X}}_{\mathit{\text{gj}}}$. Endogeneity bias will be minimal if most price-determining variables are included, that is, if the correlation between the endogenous grouping variable and the disturbance is small (Dhrymes, and Lleras-Muney, 2006).¹² Issues of endogeneity are considered further in the context of a panel structure to the data in the next section. A salient compromise is to utilize further stratification, include factors other than product and country (say, location and outlet type) in the regression as dummy variables. The prices would be averaged over these finer grouped data.

C. A Hedonic CPD Regression Based Only on Selected Stratifying Factors, not Covariates

Can we simply include the categorical hedonic quality characteristics within X_j in a CPD regression, say for location and outlet-type, effectively a country product outlet-type location dummy (CPOtLD) method?

Assume that ${\overline{X}}_j=[{\overline{X}}_j^1,{\overline{X}}_j^2]$ for which ${\overline{X}}_j^1$ are a selection of categorical variables such as outlet type (say supermarket, open-market store, department store, specialized store, discount store, other) and location (say capital/major city, other cities, towns, rural areas) and ${\overline{X}}_j^2$ the remaining categorical variables and all covariates. The inclusion of only ${\overline{X}}_j^1$ in the regression is equivalent to stratifying by these variables. That is, calculating average prices for, these groups and entering these averages into the left-hand-side of CPD regression with appropriate dummies for outlet-type and location on the right-hand side, along with the country and product dummies.¹³ The parameters estimates on the country dummies are inefficient in not taking into account variation within these groupings, and potentially suffer from omitted covariate bias in excluding the ${\overline{X}}_j^2$.

Dhrymes, and Lleras-Muney (2006) consider whether it makes any difference, in terms of asymptotic efficiency, to use a finer or coarser grouping, say if “outlet-type’ is included is there any benefit from expanding the set of groups or loss from consolidating them? They find that finer groups always yield more efficient estimates of the structural parameters of interest.

Hill (2009) undertook some empirical work on the improving the 2005 ICP parity estimates for the Asia-Pacific region by using location (urban/rural) and store-type averages with corresponding dummies in an extended-CPD framework. The results, while preliminary, found the coefficients on such variables over the 85 BHs studied to be, more often than not, statistically significant and to impact on the parity estimates. However, the signs on these included variables were often unexpected and raised concern about the reliability of the country coding used for these variables. An alternative approach is for the price collector to identify how the non-comparable replacement differs from the specification. The desk officer then makes an explicit adjustment to its price for these quality deviations using the principles outlined below.

D. Explicit Quality Estimates

Why should we use hedonic estimates to value the quality difference: why not use outlet specific estimates from comparable products on the outlet shelf or based on option costs?

First, such an approach was used in the 2005 ICP for the case where the package size differed. If the PS was for a say 1 kg. bag of a specified quality of rice, and only a 0.5 kg. bag was on the shelf, then the country desk official would make a quality adjustment by multiplying the price by two. Thus, in this limited case, quality adjustments were incorporated into the ICP. Second, the approach can be easily developed using the checklists from which any change in the required specification is immediately apparent to the price collector. The difference in price arising from a unit change in quality may be apparent from observing the prices of other varieties of differing quality in that outlet. Since the replacement is to be compared with the missing specification for the specific outlet, there is a case for using the outlet estimate of the change in price of a unit quality characteristic, since outlet-specific factors are kept constant. However, there may not be a sufficient variety of models of the product on a store shelf. For PPP comparisons, use can be made of a wider sample of price-quality observations and the coefficients from hedonic regressions.

Product experts, as opposed to price collectors, may be used to judge the value of quality differentials. Sticking to tight PSs proved to be highly problematic for “machinery and equipment goods” in the 2005 ICP. Participating countries were asked to price as many products and product types as possible including the “preferred” make and model and also one or two “alternative” models where available. “Unspecified” models were also priced if neither preferred or alternative models were available and, in fact, these eventually made up about 40 percent of the total observations in Asia with substantial overlap. Of the submitted data, experts determined whether prices could be used in spite of minor technical variations, and also whether and how adjustments could be made to prices for more significant quality variation. For example, the price of a tractor without roll-over protection (ROP) could be adjusted by including the cost of the ROP, if the latter was part of the PS (Burdette, 2007). Also: “As stressed by the Asian core country experts, the next generation of SPDs should include more technical characteristics to support hedonic type analysis.” (Burdette, 2007, p. 8).

E. Explicit Hedonic Quality Adjustments

Why not undertake country-specific hedonic adjustments using the results from hedonic regressions estimated for individual countries?

In (3) the coefficients on the quality characteristics for each country are constrained to be the same. This is an essential characteristic of a dummy variable hedonic regression-based method. By constraining slope parameters to be the same, the difference between the country intercepts, as a measure of the price parity, is invariant to the value of the X_jc characteristics (Triplett, 2004). Yet, say a product available in country 1 with a PS of X_j1 = 10 was unavailable in some outlet(s) in country 2 and replacement(s) found with X_j2 = 12. We could simply run a hedonic regression just using country 2 data and make an explicit quality adjustment to the (log of the) price in country 2 of ${\widehat{\rho }}_{j2}^{*}$ (12 – 10) where ${\widehat{\rho }}_{j2}^{*}$ is an estimated coefficient from a hedonic regression of ${\overline{p}}_{j2}^{*}\text{on}{\overline{X}}_{j2}$ using country 2 outlet data, that is:

A quality-adjusted price for country 2 can be directly compared to the actual country 1 price using a desirable index number formula without any need for a CPD regression.

Alternatively, the explicit quality adjustment ${\widehat{\rho }}_{j1}^{*}$ (10 −12) is made to the price of country 1 where ${\widehat{\rho }}_{j1}^{*}$ is an estimated coefficient on X_j1 from a hedonic regression estimated using country 1 data. However, the two methods will produce different results unless ${\widehat{\rho }}_{j1}^{*}={\widehat{\rho }}_{j2}^{*}$. Given no a priori preference for either approach, a symmetric average of the two approaches is deemed appropriate (Feenstra, 1995) and indeed can be formulated so as to correspond to a superlative index number formula (Diewert, 2005).

The estimates of ${\widehat{\rho }}_{\mathit{\text{jc}}}^{*}$ suffer from all the defects of using grouped data outlined above and, further, may have insufficient or limited degrees of freedom. One possibility is to estimate the ${\widehat{\rho }}_{\mathit{\text{jc}}}^{*}$ from individual data and then incorporate the estimates into the hedonic CPD using a 2-stage least squares estimator. Dhrymes and Lleras-Muney (2006) demonstrate how efficiency gains can arise from the use of a mixed-2SLS (M2SLS) estimator where the dependent variable, our prices, would only be available for groups, whereas endogenous regressors are available at the individual level. Instruments would be required for the first stage estimation, but the idea of working with some variables at the outlet level and others at the grouped level begs the question as to why work at the grouped level in the first place. We now scratch this itch and advocate the use of ungrouped data and introduce some further estimation issues at the ungrouped level. Most importantly, we take formal account of the panel structure of the data.

A. A CPD Regression with Outlet Interaction Terms

Can we simply include the interaction term ${\delta }_{\mathit{\text{knc}}}^{*}$ in the CPD as in equation (5), effectively a country product outlet dummy (CPOD) method?

Omitted variable bias would arise if say higher quality products sold in higher quality outlets were sold in some countries compared with others, a multicollinearity between the country and product/outlet quality effects. The inclusion of ${\delta }_{\mathit{\text{knc}}}^{*}$ would remove such bias, but be very demanding in terms of degrees of freedom, especially since the same outlets would not necessarily be in the same countries, though more than one PS, say type of poultry, may be available in the same outlet. Further, we would have the incidental parameter problem—as KN increases for fixed C, the coefficients on the dummy variables would not be consistent since the number of these parameters increase as KN increases. (Baltagi 2005).

B. A Hedonic CPD Regression

Can we simply include the hedonic quality characteristics in a CPD regression instead of the interaction term, as in equation (6)?

Equation (6) can be seen to be a fixed effects (FE) panel estimator where the variation is over c = 1,....,C countries as opposed to more usually over time t=1,……,T, though the principles remain the same. Note that we still include in equation (6) the N=8-1=7 product dummies for the different types of poultry, but each of these products will have a detailed PS, including size, brand, free range or otherwise, skinless, frozen, sold unpackaged, and so forth, and outlet variables, such as market trader, independent butcher, supermarket, hypermarket, capital city, major city, rural area and so forth. The inclusion of X_j are as parsimonious proxies for the outlet/PS interaction terms ${\delta }_{\mathit{\text{knc}}}^{*}$. We can improve on the specification of (6) by also having quality and product interaction terms:

Thus if, for example, a certain brand of poultry may have a price premium, but the premium may be more for some products, say pre-pared ready-to-cook chicken dinners, than say frozen chicken breasts. The number of ${\tau }_{\mathit{\text{nj}}}^{*}$ estimates can be reduced by appropriate tests.

Of course, if there are no product replacements and the characteristics remain the same across countries—they are country-invariant—then there is no between country variation. The ${\lambda }_j^{*}$ cannot be identified by the FE estimator and, indeed, there is no need for them to be identified since there is no need for quality adjustments. However, where there are non-comparable product replacements, the CPD hedonic formulations in (6) and (7) can be used to provide estimates of country price parities which incorporate quality-adjusted non-comparable replacements.

Empirical work of this nature is sparse. Aten (2003 and 2006) undertook inter-area price comparisons using U.S. data on nearly 200 item strata for 38 U.S. geographical (metropolitan) areas using a hedonic CPD—her “long” method. In Aten (2006) the hedonic CPD estimates are rerun for about 50 item strata, but the CPD for the remaining strata, so there are no published direct comparisons between the hedonic CPD and CPD methods. Aten subsequently compared¹⁴ the hedonic CPD with the CPD estimates on 50 item strata and found the (unweighted geometric mean) difference in the 38 inter-area price level comparisons to range from -11 percent to 16 percent. Of the 38 inter-area comparisons, nine showed a substantial difference between the hedonic CPD and CPD estimates of more than +7 percent or less than -7 percent, though a further nine area comparisons showed smaller differences of between +2 and -2 percent.

C. A Pooled Cross-Country Hedonic Regression

Why not exclude the product fixed effects and use OLS on pooled data?

Early studies of U.S. inter-area price comparison just used the quality characteristics of the product in an outlet to control for product variation—Kokoski, Cardiff, and Moulton (1994) and Kokoski, Moulton and Zieschang (1999), that is, they pooled the data and excluded product fixed effects.

Such a model is given by:

which exclude the fixed effect dummies from equation (6) and also the interaction terms in (7) and thus has potential omitted variable bias. The extent of any bias arising from such omissions will vary between BHs and is an empirical issue.

The OLS assumption that ${\nu }_{\mathit{\text{nc}}}^{*}\sim \text{iid}(0,{\sigma }^2)$ for all n and c ignores the panel structure of the data. In (8): ${\nu }_{\mathit{\text{nc}}}^{*}={\beta }_n^{*}+{\eta }_{\mathit{\text{nc}}}^{*}\text{where}{\eta }_{\mathit{\text{nc}}}^{*}$ are uncorrelated with X_jc and ${\beta }_n^{*}$ are the product-specific effects. The ${\beta }_n^{*}$ are distinguished as a component of the error term ${\nu }_{\mathit{\text{nc}}}^{*}$ since observations on the same product are likely to be more similar than observations from different products. If ${cov}({\beta }_n^{*},{\eta }_{\mathit{\text{jc}}}^{*})=0$ then (5) can be estimated as a RE panel estimator using GLS.¹⁵

OLS estimates of (8) would still be asymptotically unbiased, however the standard errors of the estimates would be understated and the OLS estimates would not be as efficient as GLS ones. Moulton (1986) compared OLS estimates and GLS estimates for three panel applications: a hedonic housing model, a housing demand model, and an earnings function, finding substantial, and for the housing demand study, dramatic, downward bias in the OLS standard errors and thus misleading inferences. The precision of the estimates suffered particularly when average group size and intra-group error correlation were large. The latter would be particularly true when there are variables with repeated values within a group, as is likely for the quality variables X_kjc.

Evans et al. (1995) estimated the coefficient on ‘concentration’ in a concentration-price regression using panel data on 1,000 city pairs of airlines carriers over specific routes over 21 quarters. They rejected the null hypothesis of concentration (x_jc) being unrelated to the route-specific effect $({\beta }_n^{*})$ i.e. they rejected the null of ${cov}({\beta }_n^{*},X_{\mathit{\text{jc}}}^{*})=0$. As a result the FE model was preferred and the pooled OLS estimates inefficient. Such decisions as to which estimator to use can have a major impact on the results. The pooled OLS estimated coefficient was 0.166 compared with 0.230 for the fixed effects estimated coefficient, the difference increasing (0.214:OLS and 0.577:FE) when instrumental variable estimators were used (since concentration was also correlated with the error term).

Heravi, Heston and Silver (2003) provided a number of estimates of price parities for television sets including a formulation such as (6). However, they used scanner data for three countries and found limited matching of models across countries thereby justifying a pooled OLS estimator. More particularly, Silver and Heravi (2007a) use the formulation in equation (6) to investigate the trade-off between tight specifications and the poor coverage of the items compared, and loosely defined specifications to allow for greater coverage but resulting in inappropriate quality comparisons. They provide an analytical framework and empirical results, based on a simulation of loosening the specifications, identifying conditions under which the bias from poor coverage from tight specifications may be offsetting or compounding, and thus severe.

D. The Choice of Estimator for a Hedonic CPD Regression

Can we simply use a fixed effects OLS estimator for (6) and (7)?