Index aggregation, rates of change, and contributions
The HICP is a chain-linked Laspeyres-type index (European Union 2016). The (unchained) price indices in each calendar year refer to December of the previous year, which is the price reference period. These price indices are chain-linked to the existing index using December to obtain the HICP. The HICP indices currently refer to the index reference period 2015=100. Monthly and annual change rates can be derived from the price indices. The contributions of the price changes of individual items to the annual rate of change can be computed by the “Ribe contributions”. More details can be found in European Commission (2024, chap. 8).
Index aggregation
The all-items index is a weighted average of the items’ subindices. However, because the HICP is a chain index, the subindices can not simply be aggregated. They first need to be unchained, i.e., expressed relative to December of the previous year. These unchained indices can then be aggregated as a weighted average. Since the Laspeyres-type index is consistent in aggregation, the aggregation can be done stepwise from the bottom level to the top or directly in one step.
In the following example, the euro area HICP is computed directly in
one step and also stepwise through all higher-level indices. For that
purpose, the monthly price indices and item weights are first downloaded
from Eurostat’s database. The two datasets are then merged. Second, the
price indices are unchained using the function unchain().
Based on the derived ECOICOP tree, the unchained price indices are
aggregated in one step using the Laspeyres-type index,
chain()ed, and finally rebase()d to the index
reference period 2015. A comparison to the published all-items index
values shows only small differences due to rounding (since the published
index numbers in Eurostat’s database are rounded and not available with
all decimals).
# import monthly price indices:
prc <- hicp.dataimport(id="prc_hicp_midx",
filter=list(unit="I15", geo="EA"),
date.range=c("2014-12", NA))
prc[, "time":=as.Date(paste0(time, "-01"))]
prc[, "year":=as.integer(format(time, "%Y"))]
prc[, "coicop" := gsub(pattern="^CP", replacement="", x=coicop)]
setnames(x=prc, old="values", new="index")
# unchain price indices:
prc[, "dec_ratio" := unchain(x=index, t=time), by="coicop"]
# import item weights:
inw <- item.weights[geo=="EA", list(coicop,geo,time,values)]
inw[, "time":=as.integer(time)]
setnames(x=inw, old=c("time","values"), new=c("year","weight"))
# derive coicop tree:
inw[ , "tree":=tree(id=coicop, w=weight, settings=list(w.tol=0.1)), by=c("geo","year")]
# merge price indices and item weights:
hicp.data <- merge(x=prc, y=inw, by=c("geo","coicop","year"), all.x=TRUE)
hicp.data <- hicp.data[year <= year(Sys.Date())-1,]
# compute all-items HICP in one aggregation step:
hicp.own <- hicp.data[tree==TRUE,
list("laspey"=laspeyres(x=dec_ratio, w0=weight)),
by="time"]
setorderv(x=hicp.own, cols="time")
hicp.own[, "chain_laspey" := chain(x=laspey, t=time, by=12)]
hicp.own[, "chain_laspey_15" := rebase(x=chain_laspey, t=time, t.ref="2015")]
# add published all-items HICP for comparison:
hicp.own <- merge(x=hicp.own,
y=hicp.data[coicop=="00", list(time, index)],
by="time",
all.x=TRUE)
plot(index-chain_laspey_15~time,
data=hicp.own, type="l",
xlab="Time", ylab="Difference (in index points)")
title("Difference between published index and own calculations")
abline(h=0, lty="dashed")
Similarly, the (unchained) price indices are also
aggregate()d stepwise, which produces in addition to the
all-items index all higher-level subindices. A comparison to the
all-items index that has been computed in one step shows no differences.
This highlights the consistency in aggregation of the indices.
User-defined functions can be passed to aggregate() as
well, which allows aggregation using any weighted or unweighted index
formula.
# compute all-items HICP stepwise through all higher-levels:
hicp.own.all <- hicp.data[is.coicop(coicop),
aggregate(x=dec_ratio, w0=weight, grp=coicop, index=laspeyres),
by="time"]
setorderv(x=hicp.own.all, cols="time")
hicp.own.all[, "chain_laspey" := chain(x=laspeyres, t=time, by=12), by="grp"]
hicp.own.all[, "chain_laspey_15" := rebase(x=chain_laspey, t=time, t.ref="2015"), by="grp"]
# compare all-items HICP from direct and step-wise aggregation:
agg.comp <- merge(x=hicp.own.all[grp=="00", list(time, "index_stpwse"=chain_laspey_15)],
y=hicp.own[, list(time, "index_direct"=chain_laspey_15)],
by="time")
# no differences -> consistent in aggregation:
nrow(agg.comp[abs(index_stpwse-index_direct)>1e-4,])
#> [1] 0Rates of change and contributions
The indices show the price change between a comparison period and the
index reference period. However, data users are more often interested in
monthly and annual rates of change. Monthly change rates are computed by
dividing the index in the current period by the index one month before,
while annual change rates are derived by comparing the index in the
current month to the index in the same month one year before. Both can
be easily derived using function rates(). Contributions of
the price changes of individual items to the annual rate of change can
be computed by the Ribe or Kirchner contributions as implemented in
function contrib().
# compute annual rates of change for the all-items HICP:
hicp.data[, "ar" := rates(x=index, t=time, type="annual"), by=c("geo","coicop")]
# add all-items hicp:
hicp.data <- merge(x=hicp.data,
y=hicp.data[coicop=="00", list(geo,time,index,weight)],
by=c("geo","time"), all.x=TRUE, suffixes=c("","_all"))
# ribe decomposition:
hicp.data[, "ribe" := contrib(x=index, w=weight, t=time, x.all=index_all, w.all=weight_all), by="coicop"]
# annual change rates over time:
plot(ar~time, data=hicp.data[coicop=="00",],
type="l", xlab="Time", ylab="", ylim=c(-2,12))
lines(ribe~time, data=hicp.data[coicop=="01"], col="red")
title("Contributions of food to overall inflation")
legend("topleft", col=c("black","red"), lty=1, bty="n",
legend=c("Overall inflation (in %)", "Contributions of food (in pp-points)"))