Estimation

library(dplyr)
library(tidyr)
library(gt)
library(ggplot2)
library(OECDsppps)
library(pricelevels)
library(data.table)

Overview

The estimation of subnational PPPs (sPPPs) starts with the item-level prices that are progressively aggregated to higher levels. In addition to aggregate subnational price indices, sub-indices, for example, at the COICOP division level, can highlight more granular regional price level differences. This process aligns with current recommendations; see ICP (2021), World Bank (2013) and European Union/OECD (2024) for more information.

Estimation steps

The estimation steps are:

Estimation of basic headings using item-level prices, where price data are aggregated up to the level of basic headings, generally without the use of expenditure weights (unless such information is available, for instance, when retail scanner data provide detailed transaction-level records).
Estimation of higher level aggregates using basic heading indices to higher levels of the classification hierarchy, at which point household expenditure data are accessible and can be applied as weighting factors

Estimation methods

The choice of estimation method depends on the availability of data and the analytical objectives of the subnational PPP exercise. When the aim is to ensure cross-country comparability and to exploit micro-level price information, the Country-Product-Dummy (CPD) - Gini-Éltetö-Köves-Szulc (GEKS) approach offers a flexible framework for estimating basic heading indices (ICP 2021).

In contrast, the Eurostat OECD method (Jevons-GEKS) methodology employed at the national level imposes more stringent data requirements. Such requirements may be more difficult to meet in the context of deriving subnational PPPs based on existing microdata, particularly regarding the representativeness of individual products across all regions (European Union/OECD 2024).

The estimation procedure in this vignette follows the CPD-GEKS approach and highlights its similarities and differences with the Jevons-GEKS approach whenever instructive. For a more comprehensive discussion on price indices see World Bank (2013) and European Union/OECD (2024).

Data used

Data used in this vignette is taken from official UK microdata from the United Kingdom Office for National Statistics (ONS). Similar data was recently used in Hearne and Bailey (2025) and is publicly available:

uk_cpi() is a snipped of the UK CPI microdata containing two products: White sliced loaf branded 750 grams (COICOP 1010103) and carpenter hourly rate (COICOP 410518).
uk_hhx() is a snipped of the regional UK household expenditure data …🚧 work in progress….

1 Estimation of basic headings using item-level prices

1.1 Overview

The CPD method is a regression-based approach for estimating price parities. The underlying statistical model is

\[ p_{ij} = PPP_j \times p_i \times \epsilon_{ij} \tag{1}\]

where \(PPP_j\) is the purchasing power parity of an arbitrary region \(j\), (\(r = 1,...,j,...,R\)), \(p_i\) is the average regional price of an arbitrary commodity \(i\), (\(n = 1, ..., i, ... N\)), and \(\epsilon_{ij}\) is a independently and identically distributed random variable.¹ Taking logs of Equation 1 yields

\[ \begin{aligned} ln p_{ij} & = ln PPP_j + ln p_i + ln \epsilon_{ij} \\ & = \alpha_j + \gamma_i + ln \varepsilon_{ij} \end{aligned} \tag{2}\]

where \(\alpha_j\) is the the price level of region \(j\) relative to all other regions in the comparison. \(\alpha_j\) can also be expressed relative to a reference region, for example, the national price level. Then, \(\alpha_j\) represents the subnational purchasing power parity of region \(j\) given by \(\hat{PPP}_j = exp(\hat{\alpha}_j)\).

1.2 Estimation

The CPD model in Equation 2 may be interpreted as a fixed-effects specification, in which country effects yield estimates of subnational purchasing power parities, while commodity-specific effects generate estimates of subnational price levels. The model can be written as a regression equation in which all explanatory variables take the form of dummy indicators for each region and commodity.

\[ \begin{aligned} ln p_{ij} = & \alpha_1 D_1 + ... + \alpha_j D_j + ... +\alpha_R D_R + \\ & \eta_1 \mathcal{D}_1 + ... + \eta_i \mathcal{D}_i + ... + \eta_N \mathcal{D}_N + \varepsilon_{ij} \end{aligned} \tag{3}\]

and \(\varepsilon_{ij}\) are independently and identically (normally) distributed with a zero mean and variance \(\sigma^2\), that is, \(\varepsilon_{ij} \sim N(0, \sigma^2)\). The variables of interest, \(PPP_j\), can be estimated through the parameters \(\hat{\alpha}_j\) using ordinary least squares (OLS); see Section 1.1.3 Implementation.

1.3 Implementation

This section describes the implementation of the CPD using different approaches from simple cross-tabulations, standard OLS as well as using the pricelevels package (Weinand 2025) and finally the function estim_cpd() of this package using examples 1 and 2.

Examples 3 to 6 discuss further functionalities of estim_cpd().

Example 1: One product, two regions

1.3.1 Using cross-tabulations

# Data
df1 <- data.table(
  region = as.factor(c(1, 2, 1, 2)),
  product = as.factor(c(1, 1, 1, 1)),
  price = c(25, 28, 23, 26)
)

In the CPD regression model, the intercept corresponds to the cross-regional average.

# Calculate cross-regional price average
df1 |>
  as_tibble() |>
  summarise(mean(price))
#> # A tibble: 1 × 1
#>   `mean(price)`
#>           <dbl>
#> 1          25.5
log(25.5)
#> [1] 3.238678

And the coefficient estimate is the price ratio of the average regional prices.

# Calculate regional price averages and price relative
df1 |>
  as_tibble() |>
  group_by(region) |>
  summarise(mean(price))
#> # A tibble: 2 × 2
#>   region `mean(price)`
#>   <fct>          <dbl>
#> 1 1                 24
#> 2 2                 27
27 / 24
#> [1] 1.125

1.3.2 Using `pricelevels`

The same results can be obtained using cpd() from the pricelevels package.

# With pricelevels - estimation with respect to regional average
df1[, cpd(p = price, r = region, n = product, q = NULL, base = NULL)]
#>        1        2 
#> 0.942809 1.060660
1.060660 / 0.942809
#> [1] 1.125

# With pricelevels - estimation with respect to region 1
df1[, cpd(p = price, r = region, n = product, q = NULL, base = "1")]
#>     1     2 
#> 1.000 1.125

# With pricelevels - estimation output
df1[, cpd(
  p = price, r = region, n = product, q = NULL, base = NULL,
  simplify = FALSE
)]
#> 
#> Call:
#> stats::lm(formula = cpd_mod, data = pdata, singular.ok = FALSE)
#> 
#> Coefficients:
#> (Intercept)        lnP.1  
#>     3.23695     -0.05889
exp(-0.05889)
#> [1] 0.9428105

1.3.3 Using standard OLS

The same can be achieved with a simple OLS regression

# With OLS
pdata <- df1

# Model: add intercept to for price levels relative to base
cpd_mod <- log(price) ~ region + 1

# Transformation: equiv to "mean centring" of continuous predictor: shifting
## dummy encoding to -1/1 intercept is mean across all prices.
## Region1 is the 'main effect', i.e., the difference between levels of a given
## factor (region) across all other factors.
contrasts(x = pdata$region) <- contr.sum(levels(pdata$region))
colnames(contrasts(x = pdata$region)) <- levels(pdata$region)[-nlevels(pdata$region)]

# OLS regression
out <- lm(formula = cpd_mod, data = pdata)
out
#> 
#> Call:
#> lm(formula = cpd_mod, data = pdata)
#> 
#> Coefficients:
#> (Intercept)      region1  
#>     3.23617     -0.05898
exp(dummy.coef(out)[["region"]])
#>        1        2 
#> 0.942723 1.060757
exp(3.23617 - 0.05898)
#> [1] 23.97928
exp(3.23617 + 0.05898)
#> [1] 26.98146

Example 2: Two products, two regions

The procedure is identical for the more general case.

# Data
df2 <- data.table(
  region = as.factor(c(1, 2, 1, 2)),
  product = as.factor(c(1, 1, 2, 2)),
  price = c(25, 28, 23, 26)
)

1.3.4 Using `pricelevels`

# With pricelevels} ------
## Estimation with respect to regional average
df2[, cpd(p = price, r = region, n = product, q = NULL, base = NULL)]
#>        1        2 
#> 0.942723 1.060757
1.060757 / 0.942723
#> [1] 1.125205

## Estimation with respect to region 1
df2[, cpd(p = price, r = region, n = product, q = NULL, base = "1")]
#>        1        2 
#> 1.000000 1.125205

## Estimation output
df2[, cpd(
  p = price, r = region, n = product, q = NULL, base = NULL,
  simplify = FALSE
)]
#> 
#> Call:
#> stats::lm(formula = cpd_mod, data = pdata, singular.ok = FALSE)
#> 
#> Coefficients:
#>     pi.1      pi.2     lnP.1  
#>  3.27554   3.19680  -0.05898
exp(-0.05898)
#> [1] 0.9427256

1.3.5 Using standard OLS

# OLS ------
## Data
pdata <- df2
## Model
cpd_mod <- log(price) ~ product + region - 1

## Transformation: equiv to "mean centring"
contrasts(x = pdata$region) <- contr.sum(levels(pdata$region))
colnames(contrasts(x = pdata$region)) <- levels(pdata$region)[-nlevels(pdata$region)]

## OLS regression
out <- lm(formula = cpd_mod, data = pdata)
out
#> 
#> Call:
#> lm(formula = cpd_mod, data = pdata)
#> 
#> Coefficients:
#> product1  product2   region1  
#>  3.27554   3.19680  -0.05898
exp(dummy.coef(out)[["region"]])
#>        1        2 
#> 0.942723 1.060757

1.3.6 Integration in : `estim_cpd()`

provides the function estim_cpd() for CPD estimation. The function provides nummerically identical results as the previously discussed estimations and provides further functionalities; see the examples below as well as the documentation of estim_cpd() for more information.

Example 3: Generic - Multiple products, and regions, with and without weights

# Generate data with pricelevels -------
set.seed(123)
R <- 5 # number of regions
B <- 5 # number of product groups
N <- 5 # number of products
dt1 <- pricelevels::rdata(R = R, B = B, N = N)

CPD with no weights using cpd() in pricelevels (Weinand 2025) and estim_cpd().

# Estimating sPPPs with `pricelevels`, no weights --------
dt1[, cpd(p = price, r = region, n = product)]
#>         1         2         3         4         5 
#> 1.0163465 0.8543248 1.1667509 0.9950373 0.9920137

# Estimating sPPPs with `estim_cpd()`, no weights ---------
dt1 |>
  estim_cpd(
    region = "region",
    product = "product",
    price = "price"
  ) |>
  pull("sPPP")
#>         1         2         3         4         5 
#> 1.0163465 0.8543248 1.1667509 0.9950373 0.9920137

CPD with weights using cpd() in pricelevels (Weinand 2025) and estim_cpd().

# Estimating sPPPs with `pricelevels`, with weights --------
dt1[, cpd(p = price, r = region, n = product, w = weight)]
#>         1         2         3         4         5 
#> 1.0187925 0.8460806 1.1784210 0.9964223 0.9880038

# Estimating sPPPs with `estim_cpd()`, with weights ---------
dt1 |>
  estim_cpd(
    region = "region",
    product = "product",
    price = "price",
    weights_cpd = "weight"
  ) |>
  pull("sPPP")
#>         1         2         3         4         5 
#> 1.0187925 0.8460806 1.1784210 0.9964223 0.9880038

Example 4: Complete regression output

The function estim_cpd() also has the option to export extended regression output of the CPD model with argument output = "Full", which summarises the key information of the estimate CPD model: It provides the ‘Regression output` as well as the individual ’Residuals’ of the CPD regression.

Information in the extended regression output is used to support the validation of CPD-based subnational PPPs at the basic-heading level; see Validation vignette.

# Estimating sPPPs with `estim_cpd()` ---------
full_est <- dt1 |>
  estim_cpd(
    region = "region",
    product = "product",
    price = "price",
    output = "Full"
  )

## Regression output
full_est[["Regression output"]] |>
  gt() |>
  fmt_number(decimals = 2) |>
  sub_missing(missing_text = "")

region	sPPP	estimate	std.error	statistic	p.value	r.squared	adj.r.squared	sigma	df	df.residual	Number of products per region
1	1.02	0.02	0.01	1.41	0.16						25.00
2	0.85	−0.16	0.01	−13.68	0.00						25.00
3	1.17	0.15	0.01	13.40	0.00						25.00
4	1.00	0.00	0.01	−0.43	0.67						25.00
5	0.99
Aggregate summary statistics						1.00	1.00	0.06	29.00	96.00	126.00


## Residuals
full_est[["Residuals"]] |>
  head() |>
  gt() |>
  fmt_number(decimals = 4) |>
  sub_missing(missing_text = "")

region	.fitted	.resid	.std.resid
1	2.7628	−0.0022	−0.0384
2	2.5891	0.1242	2.2037
3	2.9008	−0.1351	−2.3972
4	2.7416	0.0063	0.1117
5	2.7386	0.0068	0.1201
1	2.9156	−0.0076	−0.1353

Example 5: Duplicate region-product price pairs defaults

By default, estim_cpd() aggregates the price quotes up to region-product pairs using unweighted means whenever there are duplicate region-product pairs found in data and no weights provided. This is identical to the bahaviour of cpd() in pricelevels

# Take UK CPI microdata containing duplicate region-product pairs ---------
red <- uk_cpi |>
  filter(Year == "2018") |>
  select(
    region = "Region",
    product = "Product code",
    price = "Reference quantity price"
  ) |>
  mutate(
    region = as.factor(region),
    product = as.factor(product)
  )

# Estimating sPPPs with `estim_cpd()` ---------
red |>
  estim_cpd() |>
  pull("sPPP")
#> Duplicate region-product pairs found in data and no weights provided: Data is aggregated to region-product pairs using unweighted means.
#>             East Midlands           East of England                    London 
#>                 0.9291930                 1.0171431                 1.3164839 
#>                     North                North West          Northern Ireland 
#>                 0.9631195                 0.9757530                 0.9888085 
#>                  Scotland                South East                South West 
#>                 1.0521466                 0.9977087                 0.9331900 
#>                     Wales             West Midlands Yorkshire and the Humberl 
#>                 0.8612814                 1.0457363                 0.9802726

# Estimating sPPPs with `pricelevels` --------
as.data.table(red)[, cpd(p = price, r = region, n = product)]
#> Warning: Duplicated observations found and aggregated
#>             East Midlands           East of England                    London 
#>                 0.9291930                 1.0171431                 1.3164839 
#>                     North                North West          Northern Ireland 
#>                 0.9631195                 0.9757530                 0.9888085 
#>                  Scotland                South East                South West 
#>                 1.0521466                 0.9977087                 0.9331900 
#>                     Wales             West Midlands Yorkshire and the Humberl 
#>                 0.8612814                 1.0457363                 0.9802726

Example 6: Duplicate region-product price pairs with aggregation weights

estim_cpd() also provides the option to add aggregation weight in case duplicate region-product pairs found in data through the weights argument; see estim_cpd() for more information.

# Estimating sPPPs with `estim_cpd()`, with aggregation weights ---------
## No weights
red |>
  mutate(w = 1) |>
  estim_cpd(weights = "w") |>
  pull("sPPP")
#> Duplicate region-product pairs found in data and no weights provided: Data is aggregated to region-product pairs using weighted means, with weights provided in `weights`.
#>             East Midlands           East of England                    London 
#>                 0.9291930                 1.0171431                 1.3164839 
#>                     North                North West          Northern Ireland 
#>                 0.9631195                 0.9757530                 0.9888085 
#>                  Scotland                South East                South West 
#>                 1.0521466                 0.9977087                 0.9331900 
#>                     Wales             West Midlands Yorkshire and the Humberl 
#>                 0.8612814                 1.0457363                 0.9802726

## Random weights
set.seed(123)
red |>
  mutate(w = runif(nrow(red))) |>
  estim_cpd(weights = "w") |>
  pull("sPPP")
#> Duplicate region-product pairs found in data and no weights provided: Data is aggregated to region-product pairs using weighted means, with weights provided in `weights`.
#>             East Midlands           East of England                    London 
#>                 0.9273710                 1.0071653                 1.3190512 
#>                     North                North West          Northern Ireland 
#>                 0.9643659                 0.9701565                 0.9970916 
#>                  Scotland                South East                South West 
#>                 1.0489943                 1.0051813                 0.9315536 
#>                     Wales             West Midlands Yorkshire and the Humberl 
#>                 0.8701807                 1.0331752                 0.9852731

Example 7: Duplicate region-product price pairs without aggregation

estim_cpd() also provides the option to run the CPD method on the raw data, that is, keeping duplicate region-product pairs found in the raw data by setting weights = 'raw' argument; see estim_cpd() for more information.

red |>
  estim_cpd(weights = "raw") |>
  pull("sPPP")
#> Duplicate region-product pairs found in data and `weights == 'raw'`: Raw data is used with no additional aggregation to region-product pairs.
#>             East Midlands           East of England                    London 
#>                 0.9462097                 1.0266122                 1.2122656 
#>                     North                North West          Northern Ireland 
#>                 0.9913797                 0.9667091                 0.9616980 
#>                  Scotland                South East                South West 
#>                 1.0780491                 1.0025071                 0.9247737 
#>                     Wales             West Midlands Yorkshire and the Humberl 
#>                 0.8891412                 1.0400253                 0.9969142

2 Estimation of higher level aggregates using basic heading indices

🚧 Additional sections remain work in progress.

References

European Union/OECD. 2024. Eurostat-OECD Methodological Manual on Purchasing Power Parities (2023 Edition). OECD Publishing, Paris. https://doi.org/10.2785/384854.

Hearne, David, and David Bailey. 2025. “Regional Prices Reconsidered.” Regional Studies, Regional Science 12 (1): 338–56. https://doi.org/10.1080/21681376.2025.2475115.

ICP. 2021. “A Guide to the Compilation of Subnational Purchasing Power Parities (PPPs).” https://thedocs.worldbank.org/en/doc/5064f2288436664bc8f9811c8a5b8c55-0050022021/original/Guide-Subnational-PPPs.pdf.

Weinand, Sebastian. 2025. Pricelevels: Spatial Price Level Comparisons. https://doi.org/10.32614/CRAN.package.pricelevels.

World Bank. 2013. Measuring the Real Size of the World Economy: The Framework, Methodology, and Results of the International Comparison Program ICP. Washington DC: World Bank. https://thedocs.worldbank.org/en/doc/927971487091799574-0050022017/original/ICPBookeBookFINAL.pdf.