Superlative aggregation
In piar: Price Index Aggregation

All of the examples so far have used a single set of weights to aggregate an index. Although this is by far the most common case, it is not suitable for aggregating a superlative index where there is more than one set of aggregation weights that change every period.

Aggregating a Paasche index

Let's start by making a chained Paasche index from 6 elemental indexes over 4 time periods to see how to deal with time-varying aggregation weights. This is similar to the example of building an index across multiple basket in vignette("multiple-baskets"), except that the weights change each period.

library(piar)

set.seed(12345)

# Make 6 elemental indexes over 4 time periods.
elementals <- matrix(c(rep(1, 6), runif(6 * 3, 0.8, 1.2)), nrow = 6) |>
  as_index() |>
  set_levels(paste0("B", 1:6))

head(elementals)

# Make aggregation weights over 4 time periods.
#            1
#      |-----+-----|
#      11          12
#  |---+---|   |---+---|
#  B1  B2  B3  B4  B5  B6

weights <- data.frame(
  level1 = 1,
  level2 = rep(11:12, each = 3),
  ea = levels(elementals),
  weights = runif(4 * 6, 100, 200),
  period = rep(1:4, each = 6)
)

head(weights)

The key tools to deal with time-varying aggregation weights are the stack() and unstack() functions. stack() appends a later index series onto an earlier one for the same levels, whereas unstack() pulls apart an index series for many periods into a collection of one-period indexes.^[split() can also be used for this.] These functions allow the aggregation to be done with a map-reduce.

The first step to making a Paasche index is to unstack the elemental indexes into a list of elemental indexes for each period. (Trying to make the elemental indexes period-by-period can be dangerous when there are missing values.)

elementals <- unstack(elementals)

This is followed by making a sequence of aggregation structures for each set of weights.

paasche_pias <- split(
  weights[c("level1", "level2", "ea", "weights")],
  weights[["period"]]
) |>
  lapply(as_aggregation_structure)

Computing the Paasche index for each period is now just a case of mapping the aggregate() function to each elemental index and aggregation structure, and then reducing the result with the stack() function.

paasche <- Map(
  aggregate,
  elementals,
  paasche_pias,
  na.rm = TRUE,
  include_ea = FALSE,
  r = -1
) |>
  Reduce(stack, x = _)

paasche

Making a Fisher index

Making a chained Fisher index requires two sets of weights: base-period weights to make the Laspeyres index and current-period weights to make the Paasche index. As with the Paasche index this can be done with a map-reduce, except now by passing two aggregation structures to the aggregate() function instead of one.

laspeyres_pias <- paasche_pias[c(1, 1, 2, 3)]

fisher <- Map(
  aggregate,
  elementals,
  pias = laspeyres_pias,
  pias2 = paasche_pias,
  na.rm = TRUE,
  include_ea = FALSE
) |>
  Reduce(stack, x = _)

fisher

This gives the same result as calculating the Laspeyres and Paasche indexes individually, and then calculating the Fisher index manually.

laspeyres <- Map(
  aggregate,
  elementals,
  pias = laspeyres_pias,
  na.rm = TRUE,
  include_ea = FALSE
) |>
  Reduce(stack, x = _)

sqrt(as.matrix(laspeyres) * as.matrix(paasche))