# Annual arguments

By default, `get_chain_linked()` computes the necessary annual series by taking mean of each year like `aggregate.ts(x, FUN=mean)`. However, sometimes the annual index numbers are not equal to the mean of the subannual periods and, in that case, the chain-linked series will be wrongly computed. For these cases, the function includes the argument `x_a`, for us to introduce the correct annual indices.

Let us work through an example. With the given `gdp_current` we can compute the `gdp_constant` series

    ref_year_mean <- window(gdp_current,start = c(2020,1), end = c(2020,4)) |> mean()
    gdp_constant <- ref_year_mean * gdp_volume / 100

and then, there are two possible definitions of the annual deflator:

1. we can compute the quarterly price index and take the average value for each year

    gdp_deflator <- gdp_current / gdp_constant * 100
    gdp_deflator_mean <- aggregate.ts(gdp_deflator, FUN = mean)

2. we can obtain the annual series of current prices and constant prices by adding up the quarters, and then obtain a price index by dividing these

    gdp_deflator_sum <- aggregate.ts(gdp_current) / aggregate.ts(gdp_constant) * 100

The results of the definitions differ even from the second decimal place on.1

    dplyr::near(gdp_deflator_mean, gdp_deflator_sum, tol = 1e-2) |> all()
    #> [1] FALSE

Suppose now that we like better the second definition, because the resulting volumes are additive. Then, in order to obtain the price index for previous year prices, we must include `gdp_deflator_sum` as an argument. The result of not including it is, of course, different.

    gdp_deflator_pyp <- get_pyp(gdp_deflator, gdp_deflator_sum)
    gdp_deflator_pyp_wrong <- get_pyp(gdp_deflator)
    dplyr::near(gdp_deflator_pyp, gdp_deflator_pyp_wrong, tol = 1e-2) |> all()
    #> [1] FALSE

1. In general, they will not be the same. For current prices \(\{C^{y,q}\}\), constant prices \(\{K^{y,q}\}\), and price indices \(\{IP^{y,q} = C^{y,q}/K^{y,q}\}\), the expression \[\frac{\sum_{q=1}^4C^{y,q}}{\sum_{q=1}^4K^{y,q}} = \sum_{q=1}^4\frac{K^{y,q}}{\sum_{s=1}^4K^{y,s}}IP^{y,q}\] is different from \(\frac{1}{4}\sum_{q=1}^4IP^{y,q}\) since, in general, \(K^{y,q}/\sum_{s=1}^4K^{y,s} \neq \frac{1}{4}\).