Most price indexes are made with a two-step procedure, where period-over-period elemental indexes are calculated for a collection of elemental aggregates at each point in time, and then aggregated according to a price index aggregation structure. These indexes can then be chained together to form a time series that gives the evolution of prices with respect to a fixed base period. This package contains a collection of functions that revolve around this work flow, making it easy to build standard price indexes in R.
The purpose of this vignette is to give an introductory example for how to use the core functionality in this package to make a standard price index. Subsequent vignettes go into more details on advanced topics, often referencing the example in this vignette.
Matched-sample index
In this vignette we’ll be calculating a matched-sample index, where a fixed set of businesses each provide prices for a collection of products over time. The products reported by a businesses can change over time, but the set of businesses is fixed for the duration of the sample. Each businesses has a weight that is established when the sample is drawn, and represents a particular segment of the economy.
The usual approach for calculating a matched-sample index starts by
computing an elemental index for each business as an equally-weighted
geometric mean of price relatives (i.e., a Jevons index). From there,
index values for different segments of the economy are calculated as an
arithmetic mean of the elemental indexes, using the businesses-level
weights (either a Young or Lowe index, depending how the weights are
constructed; see vignette("adjust-weights")).
The ms_prices dataset has price data for five businesses
over four quarters, and the ms_weights dataset has the
weight data. Note that these data have fairly realistic patterns of
missing data and are emblematic of the kinds of survey data used to make
price indexes.
## period business product price
## 1 202001 B1 1 1.14
## 2 202001 B1 2 NA
## 3 202001 B1 3 6.09
## 4 202001 B2 4 6.23
## 5 202001 B2 5 8.61
## 6 202001 B2 6 6.40
ms_weights## business classification weight
## 1 B1 11 553
## 2 B2 11 646
## 3 B3 11 312
## 4 B4 12 622
## 5 B5 12 330The elemental_index() function makes, well, elemental
indexes, using information on price relatives, elemental aggregates
(businesses), and time periods (quarters). By default it makes a Jevons
index, but any bilateral generalized-mean index is possible (see
vignette("index-number-formulas") for more details). The
only wrinkle is that price data here are in levels, and not relatives,
but the price_relative() function can make the necessary
conversion.
elementals <- ms_prices |>
transform(
relative = price_relative(price, period = period, product = product)
) |>
elemental_index(relative ~ period + business, na.rm = TRUE)
elementals## Period-over-period price index for 4 levels over 4 time periods
## 202001 202002 202003 202004
## B1 1 0.8949097 0.3342939 NaN
## B2 1 NaN NaN 2.770456
## B3 1 2.0200036 1.6353355 0.537996
## B4 NaN NaN NaN 4.576286As with most functions in R, missing values are
contagious by default. Setting na.rm = TRUE in
elemental_index() means that missing price relatives are
ignored, which is equivalent to imputing these missing relatives with
the value of the elemental index for the respective businesses (i.e.,
parental or overall mean imputation). Other types of imputation are
covered in vignette("imputation").
The elemental_index() function returns a special index
object, and there are a number of methods for working with these
objects. For example, the resulting indexes to be extracted like a
matrix, even though it’s not a matrix.1
elementals[, "202004"]## Period-over-period price index for 4 levels over 1 time periods
## 202004
## B1 NaN
## B2 2.770456
## B3 0.537996
## B4 4.576286
elementals[c("B1", "B3"), ]## Period-over-period price index for 2 levels over 4 time periods
## 202001 202002 202003 202004
## B1 1 0.8949097 0.3342939 NaN
## B3 1 2.0200036 1.6353355 0.537996With the elemental indexes out of the way, it’s time to make a
price-index aggregation structure that maps each business to its
position in the aggregation hierarchy. The only hiccup is unpacking the
digit-wise classification for each businesses that defines the
hierarchy. That’s the job of the expand_classification()
function.
ms_weights[c("level1", "level2")] <-
expand_classification(ms_weights$classification)
pias <- ms_weights[c("level1", "level2", "business", "weight")] |>
as_aggregation_structure()It is now simple to aggregate the elemental indexes according to this
aggregation structure with the aggregate() function. As
with the elemental indexes, missing values are ignored by setting
na.rm = TRUE, which is equivalent to parentally imputing
missing values. Note that, unlike the elemental indexes, missing values
are filled in to ensure the index can be chained over time.
index <- aggregate(elementals, pias, na.rm = TRUE)
index## Period-over-period price index for 8 levels over 4 time periods
## 202001 202002 202003 202004
## 1 1 1.3007239 1.0630743 2.734761
## 11 1 1.3007239 1.0630743 1.574515
## 12 1 1.3007239 1.0630743 4.576286
## B1 1 0.8949097 0.3342939 1.574515
## B2 1 1.3007239 1.0630743 2.770456
## B3 1 2.0200036 1.6353355 0.537996
## B4 1 1.3007239 1.0630743 4.576286
## B5 1 1.3007239 1.0630743 4.576286Chaining
The elemental_index() function makes period-over-period
elemental indexes by default, which are then aggregated to make a
period-over-period index. Chaining an index is the process of taking the
cumulative product of each of these period-over-period indexes to make a
time series that compares prices to a fixed base period.
The chain() function can be used to chain the values in
an index object.
chained_index <- chain(index)
chained_index## Fixed-base price index for 8 levels over 4 time periods
## 202001 202002 202003 202004
## 1 1 1.3007239 1.3827662 3.7815355
## 11 1 1.3007239 1.3827662 2.1771866
## 12 1 1.3007239 1.3827662 6.3279338
## B1 1 0.8949097 0.2991629 0.4710366
## B2 1 1.3007239 1.3827662 3.8308934
## B3 1 2.0200036 3.3033836 1.7772072
## B4 1 1.3007239 1.3827662 6.3279338
## B5 1 1.3007239 1.3827662 6.3279338This gives almost the same result as directly manipulating the index as a matrix, except that the former returns an index object (not a matrix).
Chained indexes often need be to rebased, and this can be done with
the rebase() function. For example, rebasing the index so
that 202004 is the base period just requires dividing the chained index
by the slice for 202004.
rebase(chained_index, chained_index[, "202004"])## Fixed-base price index for 8 levels over 4 time periods
## 202001 202002 202003 202004
## 1 0.2644428 0.3439671 0.3656626 1
## 11 0.4593084 0.5974334 0.6351161 1
## 12 0.1580295 0.2055527 0.2185178 1
## B1 2.1229774 1.8998731 0.6351161 1
## B2 0.2610357 0.3395354 0.3609514 1
## B3 0.5626806 1.1366169 1.8587499 1
## B4 0.1580295 0.2055527 0.2185178 1
## B5 0.1580295 0.2055527 0.2185178 1Working with indexes
Once an index has been calculated, it usually needs to be turned into a table of index values. This can be done by either coercing an index into a matrix
as.matrix(chained_index)## 202001 202002 202003 202004
## 1 1 1.3007239 1.3827662 3.7815355
## 11 1 1.3007239 1.3827662 2.1771866
## 12 1 1.3007239 1.3827662 6.3279338
## B1 1 0.8949097 0.2991629 0.4710366
## B2 1 1.3007239 1.3827662 3.8308934
## B3 1 2.0200036 3.3033836 1.7772072
## B4 1 1.3007239 1.3827662 6.3279338
## B5 1 1.3007239 1.3827662 6.3279338or a data frame
as.data.frame(chained_index)## period level value
## 1 202001 1 1.0000000
## 2 202001 11 1.0000000
## 3 202001 12 1.0000000
## 4 202001 B1 1.0000000
## 5 202001 B2 1.0000000
## 6 202001 B3 1.0000000
## 7 202001 B4 1.0000000
## 8 202001 B5 1.0000000
## 9 202002 1 1.3007239
## 10 202002 11 1.3007239
## 11 202002 12 1.3007239
## 12 202002 B1 0.8949097
## 13 202002 B2 1.3007239
## 14 202002 B3 2.0200036
## 15 202002 B4 1.3007239
## 16 202002 B5 1.3007239
## 17 202003 1 1.3827662
## 18 202003 11 1.3827662
## 19 202003 12 1.3827662
## 20 202003 B1 0.2991629
## 21 202003 B2 1.3827662
## 22 202003 B3 3.3033836
## 23 202003 B4 1.3827662
## 24 202003 B5 1.3827662
## 25 202004 1 3.7815355
## 26 202004 11 2.1771866
## 27 202004 12 6.3279338
## 28 202004 B1 0.4710366
## 29 202004 B2 3.8308934
## 30 202004 B3 1.7772072
## 31 202004 B4 6.3279338
## 32 202004 B5 6.3279338It is also sometimes useful to get the price-updated weights used to aggregate the index; these can be calculated by first updating the aggregation structure with the aggregated index, then made into a table.
update(pias, index) |>
as.data.frame()## level1 level2 ea weight
## 1 1 11 B1 260.4832
## 2 1 11 B2 2474.7571
## 3 1 11 B3 554.4886
## 4 1 12 B4 3935.9748
## 5 1 12 B5 2088.2182