Making a monthly or quarterly Lowe index with annual expenditure or revenue weights requires adjusting these weights so that the implicit annual quantity vector is used as the fixed basket when making the index with the usual two-step procedure. This can be done by price updating the annual weights to the base period of the index, and it serves as a good example of extending the functions in this package.
Let’s start by making some annual weights and quarterly indexes for a year.
set.seed(54321)
library(piar)
# Make an aggregation structure.
# 1
# |-----------|-----------|
# 11 12 13
# |---+---| |---+---| |---+---|
# 111 121 121 122 131 132
pias <- data.frame(
level1 = rep(1, 12),
level2 = rep(c(11, 12, 13), each = 4),
level3 = rep(c(111, 112, 121, 122, 131, 132), each = 2),
ea = sprintf("B%02d", 1:12),
weight = 1:12
) |>
as_aggregation_structure()
pias## Aggregation structure for 12 elemental aggregates with 3 levels above the elemental aggregates
## level1 level2 level3 ea weight
## 1 1 11 111 B01 1
## 2 1 11 111 B02 2
## 3 1 11 112 B03 3
## 4 1 11 112 B04 4
## 5 1 12 121 B05 5
## 6 1 12 121 B06 6
## 7 1 12 122 B07 7
## 8 1 12 122 B08 8
## 9 1 13 131 B09 9
## 10 1 13 131 B10 10
## 11 1 13 132 B11 11
## 12 1 13 132 B12 12# Make elemental indexes over 4 quarters.
elementals <- matrix(
runif(12 * 4, 0.4, 1.2),
nrow = 12,
dimnames = list(sprintf("B%02d", 1:12), paste0("Q", 1:4))
) |>
as_index()
elementals## Period-over-period price index for 12 levels over 4 time periods
## Q1 Q2 Q3 Q4
## B01 0.7432063 0.4362399 1.1441564 1.1277327
## B02 0.7987443 0.9221768 0.8326890 0.9997250
## B03 0.5413539 1.1952528 0.9594449 1.0966120
## B04 0.6195148 0.9421099 1.0672886 0.8581942
## B05 0.5732081 1.1348361 0.4098346 1.1227607
## B06 1.0930889 0.7699560 1.1682339 0.5616191
## B07 0.4395281 0.8571318 0.9445658 0.9494140
## B08 0.5673079 0.7615511 0.4677992 0.7279183
## B09 0.6732899 0.5341656 1.1756704 1.1541544
## B10 0.6973270 0.4546091 0.4928318 0.8646791
## B11 0.5093139 1.1175286 1.1014889 0.8980850
## B12 0.9408381 0.6190696 0.7101399 1.1024539Adjusting the weights is simple when there are no missing elemental indexes: the weight for each elemental aggregate is just divided by the average (fixed-base) index for each quarter.
## B01 B02 B03 B04 B05 B06 B07 B08
## 2.154344 2.896610 4.819251 6.777709 11.175522 6.916156 18.543541 23.728960
## B09 B10 B11 B12
## 18.520664 30.635776 19.396354 20.059400The procedure is more complicated with missing elemental indexes as reaggregating these indexes with the newly adjusted weights will generally result in different imputations for the missing elemental indexes, which in turn gives a different adjustment for the weights. In practice this procedure is done a few times until the index values converge to a fixed point. The following function shows how to do this adjustment using the tools in this package.
# Function to adjust annual weights.
adjust_weights <- function(index,
pias,
tol = .Machine$double.eps^0.5,
max_iter = 100) {
adj_pias <- pias
for (i in seq_len(max_iter)) {
# Parentally impute missing elemental indexes.
agg_index <- aggregate(index, adj_pias, na.rm = TRUE, contrib = FALSE)
elementals <- chain(agg_index[levels(pias)[[nlevels(pias)]]])
# Compute annual elemental indexes.
pb <- rowMeans(as.matrix(elementals))
# Stop if average price-update weights are within tolerance of original
# weights; adjust otherwise.
if (max(abs(pb * weights(adj_pias) - weights(pias))) < tol) {
message(gettextf("Converged after %d iterations", i - 1))
return(adj_pias)
} else {
weights(adj_pias) <- weights(pias) / pb
}
}
warning("weights adjustment did not converge")
adj_pias
}## Converged after 2 iterations## Aggregation structure for 12 elemental aggregates with 3 levels above the elemental aggregates
## level1 level2 level3 ea weight
## 1 1 11 111 B01 2.154344
## 2 1 11 111 B02 2.896610
## 3 1 11 112 B03 4.819251
## 4 1 11 112 B04 6.777709
## 5 1 12 121 B05 11.175522
## 6 1 12 121 B06 6.916156
## 7 1 12 122 B07 18.543541
## 8 1 12 122 B08 23.728960
## 9 1 13 131 B09 18.520664
## 10 1 13 131 B10 30.635776
## 11 1 13 132 B11 28.458991
## 12 1 13 132 B12 31.046172