SUT - Time series
Annual Cycle of SUT Compilation
So far our attention has focused on the SUT in current values. One may also wish to compile a SUT in values of the previous year. This is typically done within a SUT compilation cycle. Such a cycle may consist of two phases:
Benchmark compilation, say once every five years
Updating for non-benchmark years using indexes
The following table illustrates the idea of annual updating:
Updating year T on the basis of a table in year T-1 (“T-1/T compilation”) may be split up in three revisions:
To compile the final estimates for the year 2007, the base year is loaded with the (final) estimates for the year 2006
To compile semi-final estimates for the year 2008, for the base year the final estimates for 2007 will be used
To compile the provisional estimates for 2009, we use the semi-final estimates for 2008 as base year
Thus, to integrate Annual Accounts compilation with SUT compilation within the restrictions on timeliness commonly followed is to follow a methodology based on the use of a final SUT for year T (e.g. SUT 2006), for obtaining semi-final and provisional SUTs for T+1 and T+2 using whatever data is available, together with price and volume index data. The basic idea behind the compilation cycle approach is very simple. To enable the easy use of the proper deflation methods and to avoid discontinuities in compilation it is convenient to compile a SUT for a certain year using the previous year estimate as basis. So we compile SUTs in pairs. The last one is the basis for the next and so on. Further, two or three pairs can be placed in a consecutive order defining the regular annual compilation cycle going from final to provisional estimate. This guarantees that even for the provisional estimate a clear link is established with the most recent detailed final estimate. Let us assume that the final estimate for a year comes available 3 years after the calendar year. The compilation cycle approach to be completed in the year T can be illustrated as follows:
The final estimate (T-3) will have to come available early in T using the final estimate for year (T-4) as base year
The semi-final estimate for year (T-2) will be compiled using the final estimate for year (T-3) as base year
The provisional estimate for year (T-1) will be compiled using the semi-final estimate for year (T-2) as basis
The actual updating mechanism may be formally introduced by using the following symbols:
Values T-1 in prices T-1 (previous year in current prices: Prv)
Values T in prices T-1 (current year in prices of previous year: Con)
Values T in prices T (current year in current prices: Cur)
Laspeyres volume indices with weights of the previous year: QIdx
Paasche price indices with weights of the current year: PIdx
The following calculations then become possible:
Con = Cur / PIdx
Con = Prv * QIdx
PIdx = Cur / Con
QIdx = Con / Prv
VIdx = Cur / Prv (value index)
VIdx = PIdx * QIdx
Since there is redundancy (2 ways to calculate Con and VIdx ; comparison with implicit PIdx and QIdx with given index data) this constitutes a powerful way to check the SUT data and the index data. The above calculations can be performed with all transactions available in the SUT. It is most important for:
Output method: P1 (output), P2 (intermediate consumption) and VA = P1 – P2 (value added)
Expenditure method: VA = sum of final use components (domestic and by rest of the world)
For each of these major macro-economic aggregates one can calculate Con, VIdx, QIdx, PIdx on the basis of data for Prv and Cur and index data at detailed product / activity level. Note that PIdx is calculated according to the Paasche formula and QIdx according to the Laspeyres formula as is recommended. In this case the calculation VIdx = PIdx * QIdx will yield a proper value index.
Note that the T-1/T setup yields chained indexes as is recommended, which can be linked together to obtain index series.
In such an updating scheme, taxes and margins are typically determined "endogenously”, i.e. given by ratios depending on other parts of the SUT, such as output. For constant price calculations ratios of the previous year are used, for current values estimates ratios of the current year.
Before we look more closely into constant price calculations, let's look at an example.
An example in NA Builder
The following example demonstrates a simple implementation for an updating procedure from T to T+1:
Here all data except the margins are inflated first using a set of price indexes relating T+1 to T (* operation). Next the column totals are constrained against the annual national accounts totals (ANA row). The margins are calculated separately based on the margin-to-output ratios for T and applied to the T+1 output. The negative margins for the trade and transport rows are based on the proportiponal distribution for T, so that the column total remains zero. Manual adjustments can be made for T+1 if needed. Final discrpeancies are removed using the RAS procedure on P2 only, or on P2 and household consumption. Cerains cells can be exempted from the RAS if needed. The following diagrams illustrates this procedure:
Whereas in the above example no data other than price indexes and ANA totals are needed, it is more realistic and better to use whatever exogeneous data are available. Typically, these are data for imports and exports, for taxes and subsidies, for the final domstic consumption categories and for the capital formation categories, perhaps excluding changes in inventories. In this case the procedure can be modified as follows:
Whereas the above examples generate T+1 tables in current values, the following modification will yield the tables in previous years' prices (using the ANA estimates in previous years' prices):
A similar procedure can be used for backcasting SUTs from T to T-1 as well, e.g.:
Using suitable external templates and formulas it is easy to generate summary views of T -> T+1 changes, such as:
A simple implementation of these procedures can be found in the following framework:
Indexes play an important role in compiling time-series of SUTs, as well as in other NA areas, such as quarterly NA. NA Builder has several special features for working with index numbers.
Some background information on indexes can be found here
First, rather than aggregating partitions using addition, alternatve aggregation methods can be specified for sheets, such as weighted averages, using weights stored in another sheet. This way Laspeyres indexes (typically used in NA) can be correctly aggregated. An example:
Also, there are some rules for working with indexes:
COMPARE: Given ranges for current values in periods T-1 and T, and a range with values in T in prices of T-1 ("constant values"), this rule sets up implicit value, volume and price indexes referenced to T-1
INDEX: Calculates indexes from the values in the range in arg.1 either vertically (arg.3 = 1) or horizontally (arg.3 = 2); results will be written to the range in arg.2; the lag of the index is in arg.4 (default 1); the offset for the reference period (=100) is in arg.6 (default 1); the length of the reference period is in arg.5 (default 1); the results will be multiplied with the scalefactor in arg.7; if arg.8 = 1 then the results will be set up as formulas
National accounts in constant prices
How do we derive National Accounts aggregates in constant prices? First, note that properly speaking we should say: National Accounts aggregates in prices of the previous year, or in prices of a another particular year, any number of years back. But it is rather common practice to speak about National Accounts in constant prices (or values) and we will also often do so. Also note that the theory so far is for elementary goods and services, whereas for National Accounts we have aggregates of goods and services. Hence, we should further develop our theory to also encompass indexes for aggregates – we will do this later – but before going into more theory let us at least see the direction into which all of this is going.
Our goal is to express the new quantity in the old price. In the figure above, this is the sum of the areas of the rectangles A and B. The important part here is B which is equal to ∆q.p(1). So we must value the quantity change (or volume change) at prices of the previous year. Let us call this value: Con(t), with the word Con standing for “constant prices” (prices previous year) and the argument t indicating the year. Similarly, we introduce the functional term Cur(t), denoting the current value in period t i.e. the value in this years’ prices. With this terminology in place, we can express the above method (step 1 in the procedure discussed after the figure) as follows:
Con(t) = Cur(t-1). q(t,t-1) (7)
Or in words: the constant prices value in year t (i.e. the value expressed in prices of year t-1) is equal to the current value in year t-1 multiplied with the volume index relating the two years. This method of deriving constant prices values is called the extrapolation method. Given relation (6) above (v(t,t-1)=q(t,t-1).p(t,t-1)), we also have an alternative. To see this, substitute the slightly rewritten (6) into (7):
Con(t) = Cur(t-1). v(t,t-1)/p(t,t-1)
Given the definition of a value index: v(t,t-1) = Cur(t) / Cur(t-1), we can write this as:
Con(t) = Cur(t-1).(Cur(t) / Cur(t-1))/p(t,t-1)
After crossing out Cur(t-1) we have:
Con(t) = Cur(t)/p(t,t-1) (8)
So we get the same constant prices value in period t by taking the current value in t and dividing it by the price index relating the two years. This is our second method and is called the method of deflation.
Method 1: extrapolation with volume indexes
It is time for some examples. First, the extrapolation method. Here we need current values in the previous year and volume indexes. The following table gives the application of this method for two goods, labeled A and B.
The last column is obtained by multiplying columns two and three and dividing by a factor 100. Note that in order to apply this method you only need accounting data (in the Cur(t) column) for the previous year . These are extrapolated to the current year with volume indexes.
Method 2: deflation with price indexes
The deflation method is illustrated in the following table.
Note that in order to apply this method you have to compile accounting data for the current year in the last Cur(t) column, something that was not needed for the extrapolation method. Deflating these current values with price indexes will yield the same constant price values as in the previous example (in the Con(t) column).
We see that there are two methods for obtaining the same result. This hints at data redundancy. If you use the extrapolation method to derive Con(t), and then collect Cur(t) from independent National Accounts data sources there is no need to apply deflation. In fact, you can then use these two values to derive the price index, since another way of writing (8) is:
p(t,t-1) = Cur(t)/Con(t)
Such a price index is called an implicit price index, since it is not collected as part of price statistics, but calculated from value data. Alternatively, you can use the deflation method to obtain Con(t) and use an alternative form of (7) to derive the volume index:
q(t,t-1) =Con(t)/Cur(t-1)
We now have calculated an implicit volume index. Which alternative to use depends on data availability. If you only have data for the previous year, then you apply the extrapolation method and calculate the implicit price index. Once you have data for the current year as well, you can (and in fact: should) use the deflation method and calculate the implicit volume index. Comparing the implicit price and volume indexes with the real indexes coming from price and production statistics will give useful feedback about the suitability of the value data. This is a very useful tool for balancing the national accounts.
Aggregating Indexes
The question to be addressed now is: how can we derive a volume index for the total of two elementary goods A and B, given individual volume indexes for A and B separately? Using the theory developed for elementary indexes we can write the indexes for goods A and B as follows:
qA(21) = [qA(2) ⁄ qA(1)].100
qB(21) = [qB(2) ⁄ qB(1)].100
As usual in our formalism we have:
q for quantity (we are assuming elementary goods, taken as physical items for which the quantity dimension applies; hence we can decompose values into quantity and prices, which was the underlying assumption earlier; once we introduce indexes for aggregates we will use the word volume instead of quantity);
A or B denoting the good
(21) indicating the change (period 2 with respect to period 1)
The above indexes are (elementary) link indexes, as explained earlier.
What is a suitable index for the total (or aggregate) of A and B? Obviously. such an index is dependent on the relative importance of each good. We can incorporate this observation in our formalism by writing the index for the aggregate as a weighted average of both link indexes as follows:
qAB(21) (t) = qAB(2) ⁄ qAB(1) = wA(t) . [qA(2) ⁄ qA(1)] + wB(t) . [qB(2) ⁄ qB(1)] (9)
Here the “AB” in the superscript denotes the aggregate. Also the index has an argument t indicating the period for which the weights w are derived. These weights are for the individual goods A and B. The sum of both weights should of course be unity.
We have to decide on two issues: what do we use for weights and for what period do we calculate the weights (i.e. which value for t do we take, 1 or 2). The first issue is easy to decide: let us use the values of the goods (as products of prices and quantities):
vA(t) = pA(t) . qA(t)
vB(t) = pB(t) . qB(t)
We can now write the weights (for period t) for both goods as follows:
wA(t) = vA(t) ⁄ [vA(t) + vB(t)]
wB(t) = vB(t) ⁄ [vA(t) + vB(t)]
It is easy to confirm that the sum of both weights is indeed unity.
The more interesting question is what period t to use for the weights: t=1 (the “old” period) or t=2 (the “new” period). Since we are considering two periods (here labeled 1 and 2) we can use either period. Indeed it turns out that both possibilities are valid. Let us first investigate the choice t=1 in the next section.
Laspeyres Indexes
The above weights for t=1 become:
wA(1) = vA(1)) ⁄ [vA(1) + vB(1)] = [pA(1) . qA(1)] ⁄ [vA(1) + vB(1)]
wB(1) = vB(1)) ⁄ [vA(1) + vB(1)] = [pB(1) . qB(1)] ⁄ [vA(1) + vB(1)]
We can plug these expressions into equation (9) above and write for t=1:
Lq(2,1) = qAB(21)(1) = {[pA(1) . qA(1)] ⁄ [vA(1) + vB(1)]} . [qA(2) ⁄ qA(1)] + {[pB(1) . qB(1)] ⁄ [vA(1) + vB(1)]} . [qB(2) ⁄ (qB(1)]
which can be simplified to:
Lq(2,1) = [pA(1) . qA(2) + pB(1) . qB(2)] ⁄ [vA(1) + vB(1)]
Note the following:
The q1 terms for both A and B can be crossed out in the two parts that are summed in the equation
The aggregate index is labeled Lq2.1. without superscript AB (so that we can easily generalize to aggregates of more than two goods) and without argument t=1 (because this is part of the meaning of the L prefix and therefore redundant); also, for the sake of transparency, the two periods in the subscript are separated by a “,”
We have added the letter L since this index is known as a Laspeyres index. If we write v1 in the denominator explicitly, we obtain the more familiar expression for the Laspeyres index for q:
Lq(2,1) = [(pA(1) . qA(2) + pB(1) . qB(2)] ⁄ [pA(1) . qA(1) + pB(1) . qB(1)]
Note that the above expression has q(2) in the numerator (the upper part of the fraction) and q(1) in the denominator (the lower part of the fraction). All q’s are multiplied with p’s from period 1 only (the “old” period). This latter fact is the defining characteristic of a Laspeyres index.
The example in the following table may be useful to illustrate how such a Laspeyres quantity index works.
Given are value data for goods A and B in 2009 and link quantity indexes for A (103) and B (105) in 2010, denoting the change in 2010 with respect to 2009. The objective is to calculate the Laspeyres index for the total. According to equation (1) this is done by calculating the weighted average of the two link indexes, with the weights being based on the values for 2009 of both goods (100/300 for A and 200/300 for B). Hence, the Laspeyres index is given by:
Lq(2010,2009) = 100⁄300.103 + 200⁄300 . 105 = 104.3
Note that this can be rewritten as:
Lq(2010,2009) = {(100.103⁄100) + (200.105⁄100)} . (100⁄300) = 104.3
When written in this last form, the recipe for calculating the index becomes explicit:
For each individual good (i.e. for each row in the table) multiply the current value in 2009 (column “Cur”) with the link index (column “q_2010,2009”) and divide by 100; these values are stored in the last column of the table (“Con”);
Add the values in the last “Con” column for goods A and B (103 + 210 = 313);
Divide this total by the total value for 2009 (313/300);
Multiply with 100.
This method is easy to generalize to more than two goods: simply add more rows to the above table, and for each new good collect data on the value in 2009 and the link quantity index for 2010. Note that we have called the last column in the table “Con” on purpose, since this is in effect the extrapolation method for obtaining constant price estimates in 2010, with prices of 2009 for the individual goods A and B, as discussed in part 1 of this paper. The new element here is that we now also have such a constant price estimate for the total of A and B (313) based on the same extrapolation method: multiply previous period current value with a suitable volume index. The point here is that the Laspeyres index given above is such a suitable volume index. Note that we will speak of a “volume” index for the aggregate and not of a quantity index, since the aggregate may be composed of individual goods (or services) that are incommensurable. e.g. because of different physical units.
Price Indexes
Just like we had current values for goods A and B for 2009, we can collect current values for goods A and B for 2010. We can then calculate implicit price indexes for goods A and B, as demonstrated in the following table.
Recall that an implicit price index is calculated as follows:
p(2010,2009) = Cur(2010) / Con(2010)
This can be done for good A (104.9 = 108/103) and for good B (109.5 = 230/210) but also for the aggregate (108 = 338/313). The following question then arises: is this implicit price index for the aggregate also a Laspeyres index?
To answer this question we must first see what a Laspeyres price index is, as opposed to the quantity index we saw before in equation (9). Changing q into p we obtain the Laspayres price index:
Lp(2,1) = wA(t) . [pA(2) ⁄ pA(1)] + wB(t) . [pB(2) ⁄ pB(1)]
Using similar arithmetic as before we can rewrite this as:
Lp(2,1) = [(pA(2) . qA(1) + pB(2) . qB(1)] ⁄ [pA(1) . qA(1) + pB(1) . qB(1)]
The above expression now has p(2) in the numerator and p(1) in the denominator, as expected. And all p’s are multiplied with q’s from period 1 only, because it is a Laspeyres index.
But is the above implicit price index such a Laspeyres price index? Let’s find out. The easiest way to do this is to remember the following fundamental fact about value, price and volume indexes:
Value index = Volume index multiplied with price index
We know that the volume index is a Laspeyres index and the value index for the aggregate is given by:
v(2)/v(1) = [(pA(2) . qA(2) + pB(2) . qB(2)] ⁄ [pA(1) . qA(1) + pB(1) . qB(1)]
So let us insert these facts in the above fundamental relation:
[(pA(2) . qA(2) + pB(2) . qB(2)] ⁄ [pA(1) . qA(1) + pB(1) . qB(1)] = [(pA(1) . qA(2) + pB(1) . qB(2)] ⁄ [pA(1) . qA(1) + pB(1) . qB(1)] . price index
Rewriting this we get:
price index = Pp(2,1) = [(pA(2) . qA(2) + pB(2) . qB(2)] ⁄ [pA(1) . qA(2) + pB(1) . qB(2)]
For the above price index we have p(2) in the numerator and p(1) in the denominator as before. But all p’s are now multiplied with q’s from period 2 (the “new” period), instead of period 1 as was the case for the Laspayeres price index. So the conclusion is that our price index is not a Laspeyres price index. Instead, such an index is called a Paasche price index. This is denoted in the formula with the letter P in front of p(2,1). The essential difference between the two types of indexes is that a Laspeyres index has weights from period 1 and the Paasche index has weights from period 2. One would be tempted to write the Paasche price index for our example as a weighted average of the price link indexes, with weights from period 2, as follows:
wA(2) . [pA(2) ⁄ (pA(1)] + wB(2) . [pB(2) ⁄ (pB(1) =? Pp(2,1)
Inserting the expressions for the weights and for Pp2,1 as introduced earlier, we find this to be not true. Although not as a simple weighted average, there are other ways of writing the Paasche price index that make explicit the values weights from period 2, but we will not go into these here. The point now is that a Laspeyres volume index always goes together with a Paasche price index (implicit or not) in order to come to a proper value index. Similarly, we can introduce the Laspeyres price index and the Paasche volume index. These two indexes also combine to a proper value index. However, this combination is not useful for National Accounts, so we will not go into the details. It is also worth noting that other types of indexes exist, besides Laspeyres and Paasche, e.g. the Fisher index. Again, we will not go into details here.
Extending the time interval: chain linking
So far our theory for aggregate volume indexes has assumed two periods, commonly (but not necessarily) denoting years. We can easily extend the treatment to include more periods. Let us add another year (2011) to the above example, as in the following table.
For 2011 there are new data in current values for goods A and B and for the volume link indexes. It is important to note that these link indexes relate 2011 to 2010, the previous year, and not to 2009, the original year. In a similar manner as before the Laspeyres volume index for 2011 is 96.6. This index gives the change in 2011 with respect to 2010 for the aggregate (i.e. 2010 = 100). In part 1 we discussed briefly the technique of “chaining” the two indexes, as in the following formula:
q(2011,2009) = q2011,2010) . q(2010,2009) / 100 = 96.6 . 104.3 = 100.7
Alternatively, we can set up the Laspeyres procedure with weights for 2009. In this case we must chain the individual link indexes for A and B and obtain the constant price estimates in 2011 in 2009 prices. The following table gives the details of this calculation.
In this table the chained index for A is 103 . 102 = 105.1. Applying this to the current value in 2009 we obtain the constant price value (value in prices 2009) in 2011 of 105.1.
Note that this Laspeyres index, with weights from 2009, is different from the index we obtained by chain linking the Laspeyres indexes in table 3 (100.8 versus 100.7). For both indexes we have “2009 = 100”, i.e. the reference year 2009 is the same. But in the second calculation the year of the weights (which is called the base year) is 2009 for both the 2010 and the 2011 indexes, whereas in the original calculation the base year is the previous year. I.e. for the 2010 calculation the base year is 2009 and for the 2011 calculation the base year is 2010. Both options for calculating the q2011,2009 index are possible. An advantage of the “fixed base year” approach (100.8) is that the weights are fixed so that the overall volume change consists of true volume changes (here construed as quantity changes) in the underlying “basket” of goods, where the relative importance of each item in the basket is fixed. However, with the passage of time these weights run the risk of becoming outdated. The advantage of the other “previous year base year” chain linking approach (100.7) is that the weights are always kept up-to-date, since they come from the previous year.
Non-additivity
We now have a practical method to calculate volume indexes for aggregates based on volume link indexes for the individual items that make up the aggregate. For these “elementary” items (such as A and B in the above examples) we can calculate implicit price indexes if the volume indexes are known. The reverse is also possible (and in fact recommended as we will discuss later): one can collect price indexes for elementary items and – using the deflation method – deflate the current values for the second period to come to constant price estimates. Implicit volume indexes can then be calculated; these implicit volume indexes are then the basis for the usual Laspeyres calculation of the aggregate volume index. So for the elementary items of the aggregate we either use volume indexes and apply the extrapolation method or we use price indexes and use the deflation method. In both cases the volume index for the aggregate is calculated according to the Laspeyres formula (preferably the chain linked version). Finally, the price index for the aggregate is calculated as implicit index, automatically being – as we have seen – of the Paasche index type.
There is one draw-back to this procedure, which can be illustrated in the following table with the example data used earlier.
Based on the individual volume indexes for A and B we calculate as before the Laspeyres index for the aggregate for 2010 (104.3) and for 2011 (the chained version, 100.7). We then apply these volume indexes to the current value total for 2009 (300) to obtain the constant price total in 2010 (313) and 2011 (302.2). However, adding up the constant price estimates in 2011 for A (105.1) and for B (197.4) gives us 302.5, which is different. This is the problem of “non-additivity”. It is important to remember that the value 302.2 is correct, based on the volume index for the aggregate. The fact that the individual constant price values do not add up to this total should be taken as a necessary negative side effect of this method. Data should be published with the non-additivity in place, with a suitable explanation to users as to why this is so. The data should not be modified to salvage additivity. Note that for 2010 the problem will not appear, since the Laspeyres volume index 104.3 is constructed in such a way that the constant value estimate for the total (313) is exactly equal to the individual constant value estimates for A (103) and for B (210).
Summary
For individual goods and services with well defined prices, value changes can be decomposed into quantity changes and price changes; for aggregated groupings of goods and services value changes can be decomposed into volume changes and price changes;
When measuring these changes with indexes, we can multiply volume (quantity) indexes with price indexes and obtain value indexes;
National accounts data usually pertain to aggregated data (e.g. output, intermediate consumption and value added by ISIC categories; even 4-digit ISIC groups consists of many different activities, producing different goods and services); some of these aggregates are taken as “elementary”, i.e. they are not sums of still smaller items; and some are totals of elementary items (e.g. the ISIC 2-digit division totals of ISIC 4-digit group items);
There are two methods for obtaining constant price estimates of a national accounts aggregates, based on given indexes for elementary items: extrapolation method (needed: current values of the previous year and volume link indexes), deflation method (needed: current values of the current year and price link indexes);
Whatever method is used, the other index for the elementary item can be calculated as an implicit index (extrapolation method: implicit price index; deflation method: implicit volume index);
The core part of the national accounts calculation is the derivation of the Laspeyres index for the aggregate of the elementary items (e.g. GDP as the aggregate of ISIC section value added contributions and net taxes); both fixed base year and chain-linked versions are possible; chain-linking is usually preferred;
A draw-back of the chain-linked Laspeyres volume index for the aggregate is that the constant price estimates for the elementary items may not add up to the constant price estimate for the aggregate (problem of non-additivity).