# Deriving annually linked quarterly Laspeyres-type volume indexes

6.22 There are several ways of linking annually weighted quarterly Laspeyres-type volume indexes. Annex A to this chapter describes the three methods outlined in 2008 SNA, including the one-quarter overlap method which is used in the ASNA.

6.23 After linking, the quarterly chain volume estimates are benchmarked to their annual counterparts. This benchmarking serves two purposes:

- It overcomes the inconsistency arising from the different linking methods required to compile quarterly chain volume estimates versus annual chain volume estimates; and
- It ensures the quarterly chain volume estimates are consistent with the data from the annual S-U tables. The Supply-Use tables are explained in more detail in Chapter 7.

6.24 The one-quarter overlap method involves calculating a link factor using overlap values for a single quarter. To link the four quarters of year \( y-1\) at year \( y-2\) average prices with the four quarters of year \( y\) at year \( y-1\) average prices, a one-quarter overlap can be created for either the fourth quarter of year \(y-1\) or the first quarter of year \( y\). The link factor derived from an overlap for the fourth quarter of year \(y-1\):

\(\large{ = \frac{{\sum\limits_{i = 1}^n {P_i^{y - 1}q_i^{4,\left( {y - 1} \right)}} }}{{\sum\limits_{i = 1}^n {P_i^{y - 2}q_i^{4,\left( {y - 1} \right)}} }}}\)

6.25 Multiplying the quarterly values for year \( y-1\) at year \( y-2\) average prices with this link factor puts them on to a comparable valuation basis with the quarterly estimates for year \( y\) at year \(y-1\) prices.