Chain-linking

For QNA there are three different methods to chain-link:

• Annual overlap method

• One quarter overlap method

• Over-the-year method

We will illustrate these three methods with a very simple example, using quarterly production data and annual price data for the years 2010 – 2012 for two goods, here called A and B. Annual data are also available for 2009. The example is based on the examples 9.4a, 9.4b and 9.4c of the IMF QNA Manual (2001).

Annual overlap method

In this method the average price data from the previous year are used as weights for each of the quarters in the current year, with the linking factors being derived from the annual data. The following table presents a simple example of the annual overlap method. For both goods A and B we have data on quantities (Q) in columns (1) and (2) and prices (P) in columns (3) and (4). There are rows for quarters Q1, Q2, Q3 and Q4 and a row for the year (Y). Note that we use annual average prices in columns (3) and (4).

Column (5) gives the total annual value (multiplying price and quantity) for both goods (e.g. for 2010: 3594 = 282 * 5.5 + 227 * 9). Column (6) gives the total quarterly value using the previous year price (e.g. for Q1, 2012: 953.8 = 85.5*4 + 53.2 * 11.5). In column (7) we calculate the “links”, by comparing the quarterly value in previous years’ prices with the “quarterized” annual value in current prices of the previous year (e.g. for Q1, 2012: 100.96 = 100 * 953.8 / (3779/4)). “Quarterized” means dividing the annual value equally over the quarters by dividing by 4. Finally, we chain-link these individual links into the series in column (8) given now in prices of the year 2009 by multiplying the links (e.g. for Q1, 2012: 109.6 = 100.96 * 103.26 * 105.14 / (100*100)). Column (9) gives the percentage change of the quarter with respect to the previous quarter. Note that the annual indexes in columns (7) and (8) are averages of the quarterly indexes.

The annual overlap technique is a “time consistent” method in the sense that the average of quarterly volume indexes is equal to the annual index, as we see in the following table.

In this table current annual values are given in row (1) (corresponding to column (5) of the earlier table) and values in prices of the previous year (PPY) are given in row (2) (corresponding to column (6) of the earlier table). The links in row (3) are calculated by dividing the PPY value by the current value of the previous year (e.g. for 2010: 105.14 = 100 * 3336 / 3173). The annual chain-linked series (expressed in changes with respect to the reference year 2009) is obtained by multiplying the links (e.g. for 2011: 108.56 = 105.14 * 103.26 / 100). Time consistency means that for each year the annual index of row (4) equals the corresponding annual average of the quarterly indexes in column (8) of the earlier table, which is indeed true in this example. It can be shown that this is always true for the annual overlap method. As we will see below, this is not true for the other methods.

In practice we usually do not work with quantity and price data as in columns (1), (2), (3) and (4) of the first table. Instead we use a variety of techniques to prepare PPY data as in column (6). The chain-linking can then be done as in columns (7) and (8). Applying the chain-linked indexes to the current value for 2009 in column (5) will convert the index series into a value series.

One quarter overlap method

For the one quarter overlap method one quarter of the year (e.g., the fourth quarter) is compiled at both the average prices of the current year and the average prices of the previous year, which than provides the linking factor for the current year. This technique gives the smoothest transition between each link, in contrast to annual overlap technique that may introduce a “step” between each link (in the first quarter) due to the change from one annual link to the next.

The following table presents the calculations for this method, using the same data as before.

For Q4 two values are calculated: in current years’ prices (e.g. Q4 2011: 948.8 = 83.1 * 4 + 53.6 * 11.5) and in previous years’ prices (e.g. Q4 2011: 939.45 = 83.1 * 5.5 + 53.6 * 9). The links for all quarters are then calculated with respect to this Q4 value (e.g. Q1 2012: 100.53 = 100 * 953.8 / 948.8). The chained index is then built up from the links via multiplication of the link with the Q4 value of the previous year (e.g. Q1 2012: 111.60 = 100.53 * 111.01 / 100).

The annual indexes are the same as before. The index for the first year 2010 is the same as the average of the quarterly indexes, but those for 2011 and 2012 are not. The quarterly chained indexes are no longer “time consistent” with the annual ones, which is a disadvantage of this method. The advantage of this method is that the “step-problem” has been eliminated, with Q4 serving as “pivot”.

Over-the-year method

For the over-the-year method all quarters (rather than only Q4) are compiled at the weighted average prices of both the current and the previous year. This technique doesn’t meet time consistency as well, even though the differences are smaller than in one the quarter overlap method and it is also affected by the “step” problem.

The following table presents the calculations for this method, again using the same data as before.

For each quarter two values are calculated: in current years’ prices (e.g. Q2 2011: 943.4 =78.3 * 4 + 54.8 * 11.5) and in previous years’ prices (e.g. Q2 2011: 923.85 = 78.3 * 5.5 + 54.8 *9). The links for all quarters are then calculated with respect to the value of the corresponding quarter a year earlier (e.g. Q2 2012: 101.64 = 100 * 958.85 / 943.4). The chained index is then built up from the links via multiplication of the link with the value of the corresponding quarter a year earlier (e.g. Q2 2012: 109.49 = 101.64 * 103.15 * 104.43 / (100 * 100)).

The annual indexes are again the same as before. As for the previous method the index for the first year 2010 is the same as the average of the quarterly indexes, but those for 2011 and 2012 are not, although they are much closer than for the one quarter overlap method.

The three chained index series have been plotted in the following graph.

The step problem affects the AO and OY methods, but not the QO method. Time consistency is met by the AO method, but not by the QO and OY methods, although for the OY method differences between the annual indexes and the averages of the quarterly indexes are typically very small. With respect to the step problem QO is the preferred method, with respect to the time consistency AO is the preferred method. AO is the method of choice for many countries.