Make elemental/elementary price indexes

Compute period-over-period (chainable) or fixed-base (direct) elemental price indexes, with optional percent-change contributions for each product.

Usage

elemental_index(x, ...)

# Default S3 method
elemental_index(x, ...)

# S3 method for class 'numeric'
elemental_index(
  x,
  ...,
  period = gl(1, length(x)),
  ea = gl(1, length(x)),
  weights = NULL,
  product = NULL,
  chainable = TRUE,
  na.rm = FALSE,
  contrib = FALSE,
  r = 0
)

# S3 method for class 'data.frame'
elemental_index(x, formula, ..., weights = NULL, product = NULL)

elementary_index(x, ...)

Arguments

x

Period-over-period or fixed-base price relatives. Currently there are methods for numeric vectors (which can be made with price_relative()) and data frames.

...

Further arguments passed to or used by methods.

period

A factor, or something that can be coerced into one, giving the time period associated with each price relative in x. The ordering of time periods follows of the levels of period, to agree with cut(). The default makes an index for one time period.

ea

A factor, or something that can be coerced into one, giving the elemental aggregate associated with each price relative in x. The default makes an index for one elemental aggregate.

weights

A numeric vector of weights for the price relatives in x, or something that can be coerced into one. The default is equal weights. This is evaluated in x for the data frame method.

product

A character vector of product names, or something that can be coerced into one, for each price relative in x when making percent-change contributions. The default uses the names of x, if any; otherwise, elements of x are given sequential names within each elemental aggregate. This is evaluated in x for the data frame method.

chainable

Are the price relatives in x period-over-period relatives that are suitable for a chained calculation (the default)? This should be FALSE when x contains fixed-base relatives.

na.rm

Should missing values be removed? By default, missing values are not removed. Setting na.rm = TRUE is equivalent to overall mean imputation.

contrib

Should percent-change contributions be calculated? The default does not calculate contributions.

r

Order of the generalized mean to aggregate price relatives. 0 for a geometric index (the default for making elemental indexes), 1 for an arithmetic index (the default for aggregating elemental indexes and averaging indexes over subperiods), or -1 for a harmonic index (usually for a Paasche index). Other values are possible; see gpindex::generalized_mean() for details.

formula

A two-sided formula with price relatives on the left-hand side, and time periods and elemental aggregates (in that order) on the right-hand side.

Value

A price index that inherits from piar_index. If chainable = TRUE then this is a period-over-period index that also inherits from chainable_piar_index; otherwise, it is a fixed-based index that inherits from direct_piar_index.

Details

When supplied with a numeric vector, elemental_index() is a simple wrapper that applies gpindex::generalized_mean(r)() and gpindex::contributions(r)() (if contrib = TRUE) to x and weights grouped by ea and period. That is, for every combination of elemental aggregate and time period, elemental_index() calculates an index based on a generalized mean of order r and, optionally, percent-change contributions. Product names should be unique within each time period when making contributions, and, if not, are passed to make.unique() with a warning. The default (r = 0 and no weights) makes Jevons elemental indexes. See chapter 8 (pp. 175–190) of the CPI manual (2020) for more detail about making elemental indexes, or chapter 9 of the PPI manual (2004), and chapter 5 of Balk (2008).

The default method simply coerces x to a numeric vector prior to calling the method above. The data frame method provides a formula interface to specify columns of price relatives, time periods, and elemental aggregates and call the method above.

The interpretation of the index depends on how the price relatives in x are made. If these are period-over-period relatives, then the result is a collection of period-over-period (chainable) elemental indexes; if these are fixed-base relatives, then the result is a collection of fixed-base (direct) elemental indexes. For the latter, chainable should be set to FALSE so that no subsequent methods assume that a chained calculation should be used.

By default, missing price relatives in x will propagate throughout the index calculation. Ignoring missing values with na.rm = TRUE is the same as overall mean (parental) imputation, and needs to be explicitly set in the call to elemental_index(). Explicit imputation of missing relatives, and especially imputation of missing prices, should be done prior to calling elemental_index().

Indexes based on nested generalized means, like the Fisher index (and superlative quadratic mean indexes more generally), can be calculated by supplying the appropriate weights with gpindex::nested_transmute(); see the example below. It is important to note that there are several ways to make these weights, and this affects how percent-change contributions are calculated.

elementary_index() is an alias for elemental_index() as this is more common in the literature.

References

Balk, B. M. (2008). Price and Quantity Index Numbers. Cambridge University Press.

ILO, IMF, UNECE, OECD, and World Bank. (2004). Producer Price Index Manual: Theory and Practice. International Monetary Fund.

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

von der Lippe, P. (2007). Index Theory and Price Statistics. Peter Lang.

See also

price_relative() for making price relatives for the same products over time, and carry_forward() and shadow_price() for imputation of missing prices.

as_index() to turn pre-computed (elemental) index values into an index object.

chain() for chaining period-over-period indexes, and rebase() for rebasing an index.

aggregate() to aggregate elemental indexes according to an aggregation structure.

as.matrix() and as.data.frame() for coercing an index into a tabular form.

Examples

library(gpindex)

prices <- data.frame(
  rel = 1:8,
  period = rep(1:2, each = 4),
  ea = rep(letters[1:2], 4)
)

# Calculate Jevons elemental indexes

elemental_index(prices, rel ~ period + ea)
#> Period-over-period price index for 2 levels over 2 time periods 
#>          1        2
#> a 1.732051 5.916080
#> b 2.828427 6.928203

# Same as using lm() or tapply()

exp(coef(lm(log(rel) ~ ea:factor(period) - 1, prices)))
#> eaa:factor(period)1 eab:factor(period)1 eaa:factor(period)2 eab:factor(period)2 
#>            1.732051            2.828427            5.916080            6.928203 

with(
  prices,
  t(tapply(rel, list(period, ea), geometric_mean, na.rm = TRUE))
)
#>          1        2
#> a 1.732051 5.916080
#> b 2.828427 6.928203

# A general function to calculate weights to turn the geometric
# mean of the arithmetic and harmonic mean (i.e., Fisher mean)
# into an arithmetic mean

fw <- grouped(nested_transmute(0, c(1, -1), 1))

# Calculate a CSWD index (same as the Jevons in this example)
# as an arithmetic index by using the appropriate weights

elemental_index(
  prices,
  rel ~ period + ea,
  weights = fw(rel, group = interaction(period, ea)),
  r = 1
)
#> Period-over-period price index for 2 levels over 2 time periods 
#>          1        2
#> a 1.732051 5.916080
#> b 2.828427 6.928203