Bilateral price indices

Description

Calculation of bilateral price indices. Currently, the following ones are implemented (see below in alphabetic order).

Usage

banerjee(p, r, n, q, base=NULL, settings=list())

bmw(p, r, n, base=NULL, settings=list())

carli(p, r, n, base=NULL, settings=list())

cswd(p, r, n, base=NULL, settings=list())

davies(p, r, n, q, base=NULL, settings=list())

drobisch(p, r, n, q, w=NULL, base=NULL, settings=list())

dutot(p, r, n, base=NULL, settings=list())

fisher(p, r, n, q, w=NULL, base=NULL, settings=list())

geolaspeyres(p, r, n, q, w=NULL, base=NULL, settings=list())

geopaasche(p, r, n, q, w=NULL, base=NULL, settings=list())

geowalsh(p, r, n, q, w=NULL, base=NULL, settings=list())

geoyoung(p, r, n, q, w=NULL, base=NULL, settings=list())

harmonic(p, r, n, base=NULL, settings=list())

jevons(p, r, n, base=NULL, settings=list())

laspeyres(p, r, n, q, w=NULL, base=NULL, settings=list())

lehr(p, r, n, q, base=NULL, settings=list())

lowe(p, r, n, q, base=NULL, settings=list())

medgeworth(p, r, n, q, base=NULL, settings=list())

paasche(p, r, n, q, w=NULL, base=NULL, settings=list())

palgrave(p, r, n, q, w=NULL, base=NULL, settings=list())

svartia(p, r, n, q, w=NULL, base=NULL, settings=list())

toernqvist(p, r, n, q, w=NULL, base=NULL, settings=list())

theil(p, r, n, q, w=NULL, base=NULL, settings=list())

uvalue(p, r, n, q, base=NULL, settings=list())

walsh(p, r, n, q, w=NULL, base=NULL, settings=list())

young(p, r, n, q, base=NULL, settings=list())

Arguments

Details

Before calculations start, missing values are removed from the data. Duplicated observations for r and n are aggregated, that is, duplicated prices p and weights w are averaged and duplicated quantities q added up. If there is more than one region in the data, products with prices in only one region r are removed.

The weights w must represent expenditure shares defined as w_i^r = p_i^r q_i^r / \sum_{j=1}^{N} p_j^r q_j^r. They are internally (re-)normalized such that they add up to 1 for each region r.

Value

A named vector of price levels.

Author(s)

Sebastian Weinand

References

ILO, IMF, OECD, UNECE, Eurostat and World Bank (2020). Consumer Price Index Manual: Concepts and Methods. Washington DC: International Monetary Fund.

Examples

# sample complete price data:
set.seed(123)
dt1 <- rdata(R=3, B=1, N=5)

# compute jevons and toernqvist index:
dt1[, jevons(p=price, r=region, n=product, base="1")]
dt1[, toernqvist(p=price, r=region, n=product, q=quantity, base="1")]

# compute lowe index using quantities of region 2:
dt1[, lowe(p=price, r=region, n=product, q=quantity, base="1",
           settings=list(qbase="2"))]

# add price data:
dt2 <- rdata(R=4, B=1, N=4)
dt2[, "region":=factor(region, labels=4:7)]
dt2[, "product":=factor(product, labels=6:9)]
dt <- rbind(dt1, dt2)
dt[, is.connected(r=region, n=product)] # non-connected now

# compute jevons index with base region 1:
dt[, jevons(p=price, r=region, n=product, base="1")]

# change base region:
dt[, jevons(p=price, r=region, n=product, base="4")]

CPD and NLCPD methods

Description

The function cpd() estimates regional price levels by the Country-Product-Dummy (CPD) method, originally developed by Summers (1973). Auer and Weinand (2025) recently proposed a generalization of the CPD method. This nonlinear CPD method (NLCPD method) is implemented in the function nlcpd().

Usage

cpd(p, r, n, q=NULL, w=NULL, base=NULL, simplify=TRUE, settings=list())

nlcpd(p, r, n, q=NULL, w=NULL, base=NULL, simplify=TRUE, settings=list(), ...)

Arguments

Details

Before calculations start, missing values are removed from the data. Duplicated observations for r and n are aggregated, that is, duplicated prices p and weights w are averaged and duplicated quantities q added up. If there is more than one region in the data, products with prices in only one region r are removed.

If q is provided, expenditure shares are derived as w_i^r = p_i^r q_i^r / \sum_{j=1}^{N} p_j^r q_j^r and used as weights in the regression. If only w is provided, the weights w are (re-)normalized by default. If the weights w do not represent expenditure shares, the (re-)normalization can be turned off by settings=list(norm.weights=FALSE).

Value

For simplify=TRUE, a named vector of (unlogged) regional price levels. Otherwise, for cpd(), a lm-object containing the full regression output, and for nlcpd() the full output of nls.lm() plus element w.delta.

Method

The CPD method is a linear regression model that explains the logarithmic price of product i in region r, \ln p_i^r, by the general product price, \ln \pi_i, and the overall price level, \ln P^r:

\ln p_i^r = \ln \pi_i + \ln P^r + u_i^r

The NLCPD method inflates the CPD model by product-specific elasticities \delta_i:

\ln p_i^r = \ln \pi_i + \delta_i \ln P^r + u_i^r

Both methods require a normalization of the estimated price levels \widehat{\ln P^r} to avoid multicollinearity. If base=NULL, the normalization \sum_{r=1}^{R} \widehat{\ln P^r}=0 is used by cpd() and nlcpd(); otherwise, one price level is set to 0. The NLCPD method additionally imposes the restriction \sum_{i=1}^{N} w_i \widehat{\delta_i}=1, where the weights w_i can be defined by settings$w.delta. In nlcpd(), one \widehat{\delta_i}-parameter is derived residually from this restriction instead of being estimated.

Author(s)

Sebastian Weinand

References

Auer, L. v. and Weinand, S. (2025). The Country-Product-Dummy Method With Product-Specific Spatial Price Variation. Review of Income and Wealth, 71: e70005.

Summers, R. (1973). International Price Comparisons based upon Incomplete Data. Review of Income and Wealth, 19 (1), 1-16.

See Also

lm, dummy.coef, nls.lm

Examples

# sample complete price data:
set.seed(123)
R <- 3 # number of regions
B <- 1 # number of product groups
N <- 5 # number of products
dt1 <- rdata(R=R, B=B, N=N)

# compute expenditure share weighted cpd and nlcpd index:
dt1[, cpd(p=price, r=region, n=product, q=quantity)]
dt1[, nlcpd(p=price, r=region, n=product, q=quantity)]

# set individual start values in nlcpd():
par.init <- list("lnP"=setNames(rep(0, R), 1:R),
                 "pi"=setNames(rep(2, N), 1:N),
                 "delta"=setNames(rep(1, N), 1:N))
dt1[, nlcpd(p=price, r=region, n=product, q=quantity, par=par.init)]

# use lower and upper bounds on parameters:
dt1[, nlcpd(p=price, r=region, n=product, q=quantity,
            lower=unlist(par.init)-0.1, upper=unlist(par.init)+0.1)]

# change internal calculation of start values:
dt1[, nlcpd(p=price, r=region, n=product, q=quantity, settings=list(self.start="s2"))]

# add price data:
dt2 <- rdata(R=4, B=1, N=4)
dt2[, "region":=factor(region, labels=4:7)]
dt2[, "product":=factor(product, labels=6:9)]
dt <- rbind(dt1, dt2)
dt[, is.connected(r=region, n=product)] # non-connected now

# compute expenditure share weighted cpd and nlcpd index:
dt[, cpd(p=price, r=region, n=product, q=quantity, base="1")]
dt[, nlcpd(p=price, r=region, n=product, q=quantity, base="1")]

# compare with toernqvist index:
dt[, toernqvist(p=price, r=region, n=product, q=quantity, base="1")]


# computational speed in nlcpd() usually increases if use.jac=TRUE:
set.seed(123)
dt3 <- rdata(R=20, B=1, N=30)
system.time(m1 <- dt3[, nlcpd(p=price, r=region, n=product, q=quantity,
                              settings=list(use.jac=FALSE), simplify=FALSE,
                              control=minpack.lm::nls.lm.control("maxiter"=200))])
system.time(m2 <- dt3[, nlcpd(p=price, r=region, n=product, q=quantity,
                              settings=list(use.jac=TRUE), simplify=FALSE,
                              control=minpack.lm::nls.lm.control("maxiter"=200))])
all.equal(m1$par, m2$par, tol=1e-05)

GEKS method

Description

The function index.pairs() computes bilateral index numbers for all pairs of regions. Based on that, the function geks() derives regional price levels using the GEKS method proposed by Gini (1924, 1931), Elteto and Koves (1964), and Szulc (1964).

Usage

index.pairs(p, r, n, q=NULL, w=NULL, settings=list())

geks(p, r, n, q=NULL, w=NULL, base=NULL, simplify=TRUE, settings=list())

Arguments

Details

Before calculations start, missing values are removed from the data. Duplicated observations for r and n are aggregated, that is, duplicated prices p and weights w are averaged and duplicated quantities q added up. If there is more than one region in the data, products with prices in only one region r are removed.

The weights w must represent expenditure shares defined as w_i^r = p_i^r q_i^r / \sum_{j=1}^{N} p_j^r q_j^r. They are internally (re-)normalized such that they add up to 1 for each region r.

Value

For index.pairs(), a data.table with variables base (the base region), region (the comparison region), and the price level between the two regions (variable names defined by settings$type).

For geks(), a named vector or matrix of (unlogged) regional price levels if simplify=TRUE. Otherwise, for simplify=FALSE, a lm-object containing the full regression output.

Method

The GEKS method is a two-step approach. First, prices are aggregated into bilateral index numbers, P^{sr}, using the index formulas given in type. This is done for all pairs of regions s and r via the function index.pairs(). Second, these bilateral index numbers are transformed into transitive index numbers, P^r, by estimating the following regression model:

\ln P^{sr} = \ln \left( P^r / P^s \right) + u^{sr}

The quantities q or weights w are used within the aggregation of prices into index numbers (first stage) while the subsequent transformation of these index numbers (second stage) usually does not rely on any weights (but can if specified in settings$wmethod).

Author(s)

Sebastian Weinand

References

Gini, C. (1924). Quelques Considerations au Sujet de la Construction des Nombres Indices des Prix et des Questions Analogues. Mentron, 4 (1), 3-162.

Gini, C. (1931). On the Circular Test of Index Numbers. International Statistical Review, 9 (2), 3-25.

Elteto, O. and Koves, P. (1964). On a Problem of Index Number Computation Relating to International Comparison. Statisztikai Szemle, 42, 507-518.

Szulc, B. J. (1964). Indices for Multiregional Comparisons. Przeglad Statystyczny, 3, 239-254.

See Also

bilateral.index

Examples

# example data:
set.seed(123)
dt1 <- rdata(R=3, B=1, N=5)

### Index pairs

# matrix of bilateral index numbers:
Pje <- dt1[, index.pairs(p=price, r=region, n=product, settings=list(type="jevons"))]
# if the underlying index satisfies the country-reversal
# test (like the Jevons index), the price index numbers of
# the upper-right triangle are the same as the inverse of
# the price index numbers of the lower-left triangle.
all.equal(Pje$jevons[3], 1/Pje$jevons[7]) # true
# hence, one could set all.pairs=FALSE without loosing any
# information. however, this is no longer true for indices
# that do not satisfy this test (like the Carli index):
Pca <- dt1[, index.pairs(p=price, r=region, n=product, settings=list(type="carli"))]
all.equal(Pca$carli[3], 1/Pca$carli[7]) # false

### GEKS method

# for complete price data (no gaps), the jevons index is transitive.
# hence, no adjustment is needed by the geks approach, which is
# why the index numbers are the same:
all.equal(
  dt1[, geks(p=price, r=region, n=product, base="1", settings=list(type="jevons"))],
  dt1[, jevons(p=price, r=region, n=product, base="1")]
) # true

# this is no longer true when there are gaps in the data:
dt1.gaps <- dt1[!rgaps(region, product, amount=0.25), ]
all.equal(
  dt1.gaps[, geks(p=price, r=region, n=product, base="1", settings=list(type="jevons"))],
  dt1.gaps[, jevons(p=price, r=region, n=product, base="1")]
) # now, differences

# weighting at the second step of GEKS can be done with respect
# to the intersection of products for each pair of region:
dt1.gaps[, geks(p=price, r=region, n=product, base="1",
                settings=list(type="jevons", wmethod="obs"))]

# add price data:
dt2 <- rdata(R=4, B=1, N=4)
dt2[, "region":=factor(region, labels=4:7)]
dt2[, "product":=factor(product, labels=6:9)]
dt <- rbind(dt1, dt2)
dt[, is.connected(r=region, n=product)] # non-connected now

# compute all index pairs and geks:
require(data.table)
as.matrix(dcast(
  data=dt[, index.pairs(p=price, r=region, n=product)],
  formula=base~region,
  value.var="jevons"), rownames="base")
dt[, geks(p=price, r=region, n=product, base="1", settings=list(type="jevons"))]

Gerardi index

Description

Calculation of regional price levels using the multilateral Gerardi index (Eurostat, 1978).

Usage

gerardi(p, r, n, q, w=NULL, base=NULL, simplify=TRUE, settings=list())

Arguments

Details

Before calculations start, missing values are removed from the data. Duplicated observations for r and n are aggregated, that is, duplicated prices p and weights w are averaged and duplicated quantities q added up. If there is more than one region in the data, products with prices in only one region r are removed.

The weights w must represent expenditure shares defined as w_i^r = p_i^r q_i^r / \sum_{j=1}^{N} p_j^r q_j^r. They are internally (re-)normalized such that they add up to 1 for each region r.

Value

For simplify=TRUE, a named vector of regional price levels. Otherwise, for simplify=FALSE, a list containing the named vector of international product prices and regional price levels.

Author(s)

Sebastian Weinand

References

Balk, B. M. (1996). A comparison of ten methods for multilateral international price and volume comparisons. Journal of Official Statistics, 12 (1), 199-222.

Eurostat (1978), Comparison in real values of the aggregates of ESA 1975, Publications Office, Luxembourg.

Examples

require(data.table)

# example data:
set.seed(123)
dt1 <- rdata(R=3, B=1, N=5)

# Gerardi price index:
dt1[, gerardi(p=price, q=quantity, r=region, n=product)]

# add price data:
dt2 <- rdata(R=4, B=1, N=4)
dt2[, "region":=factor(region, labels=4:7)]
dt2[, "product":=factor(product, labels=6:9)]
dt <- rbind(dt1, dt2)
dt[, is.connected(r=region, n=product)] # non-connected now

# compute expenditure share weights:
dt[, "share" := price*quantity/sum(price*quantity), by="region"]

# Gerardi index with quantites or expenditure share weights:
dt[, gerardi(p=price, q=quantity, r=region, n=product)]
dt[, gerardi(p=price, w=share, r=region, n=product)]

Multilateral systems of equations

Description

Calculation of regional price levels using the

• Geary-Khamis method (Geary, 1958; Khamis, 1972): gkhamis()

• Iklé method (Iklé, 1972; Dikhanov, 1997; Balk, 1996): ikle()

• Rao system (Rao, 1990): rao()

• Rao-Hajargasht method (Rao and Hajargasht, 2016): rhajargasht()

Usage

gkhamis(p, r, n, q=NULL, base=NULL, simplify=TRUE, settings=list())

ikle(p, r, n, q=NULL, w=NULL, base=NULL, simplify=TRUE, settings=list())

rao(p, r, n, q=NULL, w=NULL, base=NULL, simplify=TRUE, settings=list())

rhajargasht(p, r, n, q=NULL, w=NULL, base=NULL, simplify=TRUE, settings=list())

Arguments

Details

Before calculations start, missing values are removed from the data. Duplicated observations for r and n are aggregated, that is, duplicated prices p and weights w are averaged and duplicated quantities q added up. If there is more than one region in the data, products with prices in only one region r are removed.

The weights w must represent expenditure shares defined as w_i^r = p_i^r q_i^r / \sum_{j=1}^{N} p_j^r q_j^r. They are internally (re-)normalized such that they add up to 1 for each region r.

Value

For simplify=TRUE, a named vector of regional price levels.

For simplify=FALSE, a list containing the named vector of international product prices and regional price levels, the number of iterations until convergence, and the achieved difference at convergence.

Method

The Geary-Khamis, Iklé and Rao-Hajargasht methods as well as the Rao system have in common that they set up a system of interrelated equations of international product prices and price levels, which is solved iteratively. It is only the definition of the international product prices and price levels that differ between the methods (see also package vignette).

In their original form, the four methods use quantities (or weights). However, Rao and Hajargasht (2016, p. 417) show that similar solutions exist for the unweighted definitions of international product prices and price levels. In the package, this is implemented in the functions where

• gkhamis(q=NULL) corresponds to a multilateral Dutot index;

• ikle(q=NULL, w=NULL) to a multilateral Harmonic mean index;

• rao(q=NULL, w=NULL) to a multilateral Jevons index;

• rhajargasht(q=NULL, w=NULL) to a multilateral Carli index.

Author(s)

Sebastian Weinand

References

Balk, B. M. (1996). A comparison of ten methods for multilateral international price and volume comparisons. Journal of Official Statistics, 12 (1), 199-222.

Diewert, W. E. (1999). Axiomatic and Economic Approaches to International Comparisons. In: International and Interarea Comparisons of Income, Output and Prices, edited by A. Heston and R. E Lipsey. Chicago: The University of Chicago Press.

Dikhanov, Y. (1994). Sensitivity of PPP-based income estimates to the choice of aggregation procedures. The World Bank, Washington D.C., June 10, paper presented at 23rd General Conference of the International Association for Research in Income and Wealth, St. Andrews, Canada.

Geary, R. C. (1958). A Note on the Comparison of Exchange Rates and Purchasing Power Between Countries. Journal of the Royal Statistical Society. Series A (General), 121 (1), 97–99.

Iklé, D. M. (1972). A new approach to the index number problem. The Quarterly Journal of Economics, 86 (2), 188-211.

Khamis, S. H. (1972). A New System of Index Numbers for National and International Purposes. Journal of the Royal Statistical Society. Series A (General), 135 (1), 96–121.

Rao, D. S. P. (1990). A system of log-change index numbers for multilateral comparisons. In: Comparisons of prices and real products in Latin America. Contributions to Economic Analysis Series, edited by Salazar-Carrillo and Rao. Amsterdam: North-Holland Publishing Company.

Rao, D. S. P. and G. Hajargasht (2016). Stochastic approach to computation of purchasing power parities in the International Comparison Program. Journal of Econometrics, 191 (2016), 414-425.

Examples

require(data.table)

# example data:
set.seed(123)
dt1 <- rdata(R=3, B=1, N=5)

# Gery-Khamis price index can be obtained in two ways:
dt1[, gkhamis(p=price, q=quantity, r=region, n=product, settings=list(solve="iterative"))]
dt1[, gkhamis(p=price, q=quantity, r=region, n=product, settings=list(solve="matrix"))]

# gkhamis(), ikle() and gerardi() yield same results if quantites the same:
dt1[, "quantity2" := 1000*rleidv(product)]
dt1[, gkhamis(p=price, r=region, n=product, q=quantity2)]
dt1[, gerardi(p=price, r=region, n=product, q=quantity2)]
dt1[, ikle(p=price, r=region, n=product, q=quantity2)]
dt1[, "quantity2":=NULL]

# add price data:
dt2 <- rdata(R=4, B=1, N=4)
dt2[, "region":=factor(region, labels=4:7)]
dt2[, "product":=factor(product, labels=6:9)]
dt <- rbind(dt1, dt2)
dt[, is.connected(r=region, n=product)] # non-connected now

# compute expenditure share weights:
dt[, "share" := price*quantity/sum(price*quantity), by="region"]

# Ikle index with quantites or expenditure share weights:
dt[, ikle(p=price, q=quantity, r=region, n=product)]
dt[, ikle(p=price, w=share, r=region, n=product)]

Price data characteristics

Description

Price data typically consist of prices (and purchased quantities) for multiple products and regions. Since not all products are usually available in all regions, the data exhibit gaps. In some situations, the gaps can lead to non-connected data, which prevents a price comparison between all regions.

The following functions are available to derive the characteristics of a data set:

• is.connected() checks if all regions in the data are connected either directly or indirectly by some bridging region

• neighbors() divides the regions into groups of connected regions

• connect() is a simple wrapper of neighbors(), connecting the data using the group of regions with the maximum number of observations

• gaps() computes the (percentage) number of gaps in the data

• pairs() derives the number of available bilateral index pairs

• properties() derives data characteristics for each group of connected regions

Usage

is.connected(r, n)

neighbors(r, n, simplify=FALSE)

connect(r, n)

gaps(r, n, relative=TRUE)

pairs(r, n)

properties(r, n)

Arguments

Details

Before calculations start, missing values are removed from the data. Duplicated observations for r and n are counted as one observation. Products with prices in only one region r do not provide meaningful information for interregional comparisons. Such products are therefore not considered by gaps(), pairs() and properties(). This approach follows the default treatment of all index functions in this package.

Following World Bank (2013, p. 98), a "price tableau is said to be connected if the price data are such that it is not possible to place the countries in two groups in which no item priced by any country in one group is priced by any other country in the second group".

Value

The function

• is.connected() prints a single logical indicating if the data is connected or not

• connect() returns a logical vector of the same length as r and n

• neighbors() gives a list or vector of connected regions

• pairs() returns a single numeric for the number of bilateral pairs

• gaps() returns a single numeric for the percentage of data gaps

The function properties() provides a data.table with the following variables:

Author(s)

Sebastian Weinand

References

World Bank (2013). Measuring the Real Size of the World Economy: The Framework, Methodology, and Results of the International Comparison Program. Washington, D.C.: World Bank.

Examples

### connected price data:
set.seed(123)
dt1 <- rdata(R=4, B=1, N=3)

dt1[, is.connected(r=region, n=product)] # true
dt1[, neighbors(r=region, n=product, simplify=TRUE)]
dt1[, gaps(r=region, n=product)]
dt1[, pairs(r=region, n=product)]
dt1[, properties(r=region, n=product)]

### non-connected price data:
dt2 <- data.table::data.table(
          "region"=c("a","a","h","b","a","a","c","c","d","e","e","f",NA),
          "product"=c(1,1,"bla",1,2,3,3,4,4,5,6,6,7),
          "price"=runif(13,5,6),
          stringsAsFactors=TRUE)

dt2[, is.connected(r=region, n=product)] # false
with(dt2, neighbors(r=region, n=product))
dt2[, properties(region, product)]
# note that the first two observations are treated as one
# while the observation [NA,7] is dropped. Observation [a,2]
# is still included even though it does not provide valueable
# information for interregional comparisons (the product is
# observed in only one region)

# connect the price data:
dt2[connect(r=region, n=product),]

Spatial price indices

Description

Calculation of multiple spatial price indices at once.

Usage

# list all available price indices:
list.indices()

# compute all price indices:
pricelevels(p, r, n, q=NULL, w=NULL, base=NULL, settings=list())

Arguments

Details

Before calculations start, missing values are removed from the data. Duplicated observations for r and n are aggregated, that is, duplicated prices p and weights w are averaged and duplicated quantities q added up. If there is more than one region in the data, products with prices in only one region r are removed.

The weights w must represent expenditure shares defined as w_i^r = p_i^r q_i^r / \sum_{j=1}^{N} p_j^r q_j^r. They are internally (re-)normalized such that they add up to 1 for each region r.

Value

A matrix of price levels where the rows contain the index methods and the columns the regions.

Author(s)

Sebastian Weinand

Examples

# sample complete price data:
set.seed(123)
dt1 <- rdata(R=3, B=1, N=5)

# compute specific unweighted price indices:
dt1[, pricelevels(p=price, r=region, n=product, base="1",
                  settings=list(type=c("jevons","cswd","bmw")))]

# compute all unweighted price indices:
dt1[, pricelevels(p=price, r=region, n=product, base="1")]

# compute the price indices using all methods:
dt1[, pricelevels(p=price, r=region, n=product, q=quantity, base="1")]

# add price data:
dt2 <- rdata(R=4, B=1, N=4)
dt2[, "region":=factor(region, labels=4:7)]
dt2[, "product":=factor(product, labels=6:9)]
dt <- rbind(dt1, dt2)
dt[, is.connected(r=region, n=product)] # non-connected now

# compute all unweighted indices:
dt[, pricelevels(p=price, r=region, n=product, base="1")]

# change base region:
dt[, pricelevels(p=price, r=region, n=product, base="4")]

Calculation of price ratios

Description

Calculation of regional price ratios per product with flexible setting of base prices.

Usage

ratios(p, r, n, q=NULL, w=NULL, base=NULL, settings=list())

Arguments

Details

If there are k=1,\ldots,K^{r}_{n} duplicated prices for product n in region r, these are averaged using the quantities q (or similarly as a weighted arithmetic mean using the weights w) if provided:

\bar{p}^{r}_n = \sum_{k=1}^{K^r_n} p^r_{nk} q^r_{nk} \Big/ \sum_{k=1}^{K^r_n} q^r_{nk}

Price ratios are then derived for each product n by p_n^r \Big/ \frac{1}{R} \sum_{s=1}^{R} \bar{p}_{n}^s if base=NULL and by p_n^r \big/ \bar{p}^{base}_n otherwise.

Value

A numeric vector of the same length as p. If base is not NULL and static=FALSE, the attribute attr(x, "base") is added to the output, providing the respective base region for each product.

Author(s)

Sebastian Weinand

Examples

### (1) unique price observations

set.seed(123)
dt1 <- rdata(R=3, B=1, N=4)
levels(dt1$region) <- c("a","b","c")

# calculate price ratios by product:
dt1[, ratios(p=price, r=region, n=product, base="b")]


### (2) unique price observations and missing base region


# drop two observations:
dt2 <- dt1[-c(5,10), ]

# now, region 'a' is base for product 2:
dt2[, "pr" := ratios(p=price, r=region, n=product, base="b",
                     settings=list(static=FALSE))]

# base regions are stored in attributes:
attr(dt2$pr, "base")

# with static base, NAs are produced:
dt2[, "pr_static" := ratios(p=price, r=region, n=product, base="b")]


### (3) duplicated prices

# insert duplicates and missings:
dt3 <- rbind(dt1[c(2,3),], dt1[-c(11),])
dt3[1:2, c("price","quantity") := list(price*1.1, quantity*0.95)]
anyDuplicated(dt3, by=c("region","product"))

# duplicated prices are divided by the weighted average base prices:
dt3[, ratios(p=price, r=region, n=product, q=quantity, base="b",
             settings=list(static=FALSE))]

Simulate random price and quantity data

Description

Simulate random price and quantity data for a specified number of regions (r=1,\ldots,R), product groups (b=1,\ldots,B), and individual products (n=1,\ldots,N_{b}) using the function rdata().

The generation of prices follows the NLCPD model (see nlcpd()), while expenditure share weights for product groups can be sampled using the function rweights(). Purchased quantities are assigned to individual products. Moreover, random sales and gaps (using the function rgaps()) can be introduced in the simulated data.

Usage

rgaps(r, n, amount=0, prob=NULL, pairs=FALSE, exclude=NULL)

rweights(r, b, type=~1)

rdata(R, B, N, gaps=0, weights=~b+r, sales=0, settings=list())

Arguments

Details

The function rgaps() ensures that gaps do not lead to non-connected price data (see is.connected()). Therefore, it could happen that the amount of gaps specified in rgaps() is only approximate, in particular, in cases where certain regions and/or products should additionally be excluded from exhibiting gaps by exclude.

If rgaps(pairs=FALSE), the minimum number of observations for a connected data set is R+N-1. Otherwise, for rgaps(pairs=TRUE), this number is defined by 2N+\text{max}(0, R-N-1).

Note that setting sales>0 in function rdata() distorts the initial price generating process. Consequently, parameter estimates may deviate stronger from their true values. Note also that the expenditure share weights weight represent the relevance of product groups as (often) derived from national accounts and other data sources. Therefore, they cannot be derived from the simulated prices and quantities in the data, which would represent the expenditure shares of the individual products.

Value

Function rgaps() returns a logical vector of the same length as r where TRUEs indicate gaps and FALSEs no gaps.

Function rweights() returns a numeric vector of (non-negative) expenditure share weights of the same length as r.

Function rdata() returns a data.table with the following variables:

or a list with the simulated data and its underlying parameter values, if settings=list(par.add=TRUE).

Author(s)

Sebastian Weinand

Examples

# simulate price data for ten regions and five product groups
# containing three individual products each:
set.seed(1)
dt <- rdata(R=10, B=5, N=3)
boxplot(price~paste(group, product, sep=":"), data=dt)

# simulate price data for ten regions and five product groups
# containing one to five individual products:
set.seed(1)
dt <- rdata(R=10, B=5, N=c(1,2,3,4,5))
boxplot(price~paste(group, product, sep=":"), data=dt)

# simulate price data for three product groups (with one
# product each) in four regions:
dt <- rdata(R=4, B=3, N=1)

# add expenditure share weights:
dt[, "w1" := rweights(r=region, b=group, type=~1)] # constant
dt[, "w2" := rweights(r=region, b=group, type=~b)] # product-specific
dt[, "w3" := rweights(r=region, b=group, type=~b+r)] # product-region-specific

# weights add up to 1:
dt[, list("w1"=sum(w1),"w2"=sum(w2),"w3"=sum(w3)), by="region"]

# introduce 25% random gaps:
dt.gaps <- dt[!rgaps(r=region, n=product, amount=0.25), ]

# weights no longer add up to 1 in each region:
dt.gaps[, list("w1"=sum(w1),"w2"=sum(w2),"w3"=sum(w3)), by="region"]

# approx. 25% random gaps, but keep observation for product "n2"
# in region "r1" and all observations in region "r2":
no_gaps <- data.frame(r=c("r1","r2"), n=c("n2",NA))

# apply to data:
dt[!rgaps(r=region, n=product, amount=0.25, exclude=no_gaps), ]

# or, directly, in one step:
dt <- rdata(R=4, B=3, N=1, gaps=0.25, settings=list("gaps.exclude"=no_gaps))

# introduce systematic gaps:
dt <- rdata(R=15, B=1, N=10)
dt[, "prob" := data.table::rleidv(product)] # probability for gaps increases per product
dt.gaps <- dt[!rgaps(r=region, n=product, amount=0.25, prob=prob), ]
plot(table(dt.gaps$product), type="l")