1

Benson Sim

UN STATISTICS DIVISION

Demonstration on Rebasing and Linking of National AccountsWorkshop on the Methodological Review of

Benchmarking, Rebasing and Chain-Linking of Economic Indicators

24-26 August 2011, Vientiane, Lao People’s Democratic Republic

2

Outline of Demonstration• Rebasing of national accounts using hypothetical

dataset• Calculate volume measures of GDP and

Laspeyres volume indexes using a specific base year (Table 2)

• Rebase entire GDP series using new base year (Table 3)

• Rebase GDP series from new base year onwards with no linking (Table 4)

• Rebase GDP series from new base year onwards with linking (Table 5)

• Calculate annually-chained volume indexes and measures of GDP (Tables 6 to 8)

• Conclusion• Extensive reference will be made to Excel tables

3

Demonstration of RebasingTable 1

Assumptions • 2 single (homogenous) products, X and Y, in an

economy• X and Y are most detailed products available (i.e.,

no further breakdown possible)• Real growth rates of X and Y will be the same in all

situations• Statistical agency has available information on

price indexes and value at current prices of X and Y• X shows rising prices and Y shows falling prices• Example of X: service• Example of Y: information technology good

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DemonstrationTable 2

• Currently, statistical agency calculates volume measures of entire GDP series using Year 1 as base year

• For each product (X and Y), calculate volume measure by deflating nominal value by price index for each year• Example: Volume measure of X for Year 3 is

233.1100 224.1

104.0

Nominal value for Year 3 from Table 1×100

Price index for Year 3 from Table 1

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DemonstrationTable 2

• Sum up deflated values of X and Y to obtain volume measures of GDP

• Calculate real growth rates of X, Y, and GDP• Calculate Laspeyres volume index for GDP with Year

1 as reference year (i.e., volume index for Year 1 will have value of 100.0) • Example: Laspeyres volume index for GDP for

Year 3 is

429.2100 122.6

350.0

Volume measure for Year 3 from Table 2×100

Volume measure for Year 1 from Table 2

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DemonstrationTable 3

• Statistical agency realises that present base year (Year 1) has become outdated and decides to rebase GDP

• Statistical agency calculates volume measures of entire GDP series using Year 6 as new base year as it finds that Year 6 is a normal year with no dramatic changes

• For each product (X and Y), compute price indexes with Year 6 as reference period (i.e., price indexes for Year 6 will have value of 100.0) • Example: Price index for X for Year 9 is

116.0100 105.4

110.1

Price index for Year 9 from Table 1×100

Price index for Year 6 from Table 1

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DemonstrationTable 3

• For each product (X and Y), calculate volume measure by deflating nominal value by price index for each year• Example: Volume measure of X for Year 9 is

318.0100 301.8

105.4

Nominal value for Year 9 from Table 1×100

Price index for Year 9 from Table 3

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DemonstrationTable 3

• Sum up volume measures of X and Y to obtain volume measures of GDP

• Calculate real growth rates of X, Y, and GDP• Compute Laspeyres volume index for GDP with Year

6 as reference year (i.e., volume index for Year 6 will have value of 100.0) • Example: Laspeyres volume index for GDP for

Year 9 is

486.8100 125.7

387.2

Volume measure for Year 9 from Table 3×100

Volume measure for Year 6 from Table 3

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DemonstrationTable 3

• Consistent time series for volume measures of GDP• Real growth rates of X and Y are same as those in

Table 2 because X and Y are most detailed products available

• Volume measures of GDP are additive, but real GDP growth rates have been revised throughout and are much lower

• Economic history has been rewritten• As mentioned earlier, this method of doing rebasing

is not recommended

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DemonstrationTable 4

• Statistical agency realises that the rebasing method in Table 3 is not appropriate as economic history has been rewritten

• Statistical agency decides to calculate volume measures of GDP from Year 6 using Year 6 as base year and volume measures of GDP from Years 1 to 6 using Year 1 as base year

• Volume measures of X, Y, and GDP from Years 1 to 6 will be same as those in Table 2

• Volume measures of X, Y, and GDP from Years 6 to 10 will be same as those in Table 3

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DemonstrationTable 4

• Combine the two sub-series to get entire time series (see bottom of table)

• Insert break (|) between volume measures for Years 5 and 6 to indicate they are calculated using different base years

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DemonstrationTable 4

• Question• Which real GDP growth rate for Year 6 do we

use?• Real GDP growth rate calculated using Year 1 as

base year (11.6%)?• Real GDP growth rate calculated using Year 6 as

base year (7.4%)? • Answer

• Real GDP growth rate calculated using Year 1 as base year (11.6%)

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DemonstrationTable 4

• Why?• Volume measures of GDP calculated using

Year 6 as base year only start from Year 6, not Year 5

• Volume measures of GDP from Years 1 to 6 calculated using Year 1 as base year are available

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DemonstrationTable 4

• Real growth rates of X and Y are same as those in Table 2 because X and Y are most detailed products available

• Volume measures of GDP are additive• Real GDP growth rates from Years 2 to 6 are the same as

those in Table 2• Real GDP growth rates from Year 7 are lower than those

in Table 2• Periodic rebasing ensures real GDP growth rates in

recent years are calculated with weights more representative of new base year

• But, inconsistent time series for volume measures of GDP

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DemonstrationTable 5

• To solve problem with inconsistent volume measures of GDP, statistical agency decides to link the two sub-series in Table 4 using Year 6 as reference year

• Volume measures of X, Y, and GDP from Years 6 to 10 will be same as those in Table 4

• Calculate volume measures of X, Y, and GDP and volume indexes of GDP separately before Year 6 by using their real growth rates at prices of Year 1 to extrapolate backwards their levels for Year 6

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DemonstrationTable 5

• Example: Volume measure of X for Year 5 is

271.1262.6

3.21

100

Real growth rate for Year 6 from Table 51+

100

Volume measure for Year 6 from Table 5×100

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DemonstrationTable 5

• Example: Volume measure of X for Year 4 is

262.6254.0

3.41

100

Real growth rate for Year 5 from Table 51+

100

Volume measure for Year 5 from Table 5×100

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DemonstrationTable 5

• Example: Volume measure of GDP for Year 5 is

387.2346.9

11.61

100

Real growth rate for Year 6 from Table 51+

100

Volume measure for Year 6 from Table 5×100

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DemonstrationTable 5

• Example: Volume measure of GDP for Year 4 is

346.9310.5

11.71

100

Real growth rate for Year 5 from Table 51+

100

Volume measure for Year 5 from Table 5×100

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DemonstrationTable 5

• Example: Laspeyres volume index for GDP for Year 5 is

100.0100 89.6

11.61

100

Real growth rate for Year 6 from Table 51+

100

Laspeyres volume index for Year 6 from Table 5×100

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DemonstrationTable 5

• Example: Laspeyres volume index for GDP for Year 4 is

89.6100 80.2

11.71

100

Real growth rate for Year 5 from Table 51+

100

Laspeyres volume index for Year 5 from Table 5×100

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DemonstrationTable 5

• Real growth rates of X and Y are same as those in Table 2 because X and Y are most detailed products available

• Real GDP growth rates from Years 2 to 6 are the same as those in Table 2

• Real GDP growth rates from Year 7 are lower than those in Table 2

• Laspeyres volume indexes for GDP are periodically chained

• But, volume measures of GDP are non-additive from Years 1-5 (i.e., X and Y do not add up to GDP) due to the linking of components and GDP independently

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DemonstrationTable 5

• Nevertheless, it is preferable to rebase GDP and then link the volume series so as not to create breaks

• Explain in methodological notes and metadata why volume measures of GDP are non-additive after rebasing and linking

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Demonstration

• Statistical agency decides to calculate experimental annually-chained volume indexes and measures of GDP to see how real growth rates would look like

• Statistical agency decides to do this using the Fisher index number formula

• Statistical agency also decides to express the volume measures using Year 6 as reference year

• Statistical agency will need to calculate annually-chained Laspeyres and Paasche volume indexes as the annually-chained Fisher volume indexes are the geometric mean (i.e., square root) of the product of these two volume indexes

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DemonstrationTable 6

• Compute annually-chained Laspeyres volume indexes and measures as follows

• For each product (X and Y), calculate volume measure for Year t at prices of previous year (t-1) separately by using their real growth rates to extrapolate their nominal levels for Year t• Example: Volume measure of X for Year 2 at price

of Year 1 is

3.2210.0 1 216.7

100

Real growth rate for Year 2 from Table 2Nominal value for Year 1 from Table 1 1+

100

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DemonstrationTable 6

• Example: Volume measure of X for Year 3 at price of Year 2 is

3.5221.2 1 228.8

100

Real growth rate for Year 3 from Table 2Nominal value for Year 2 from Table 1 1+

100

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DemonstrationTable 6

• Sum up resultant volume measures of X and Y to obtain volume measures of GDP at prices of previous year (t-1)

• Compute Laspeyres volume index for GDP at prices of previous year (t-1) • Example: Laspeyres volume index for GDP for

Year 2 is

386.0100 110.3

350.0

Volume measure for Year 2 from Table 6×100

Nominal value for Year 1 from Table 1

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DemonstrationTable 6

• Example: Laspeyres volume index for GDP for Year 3 is

393.3100 110.2

357.0

Volume measure for Year 3 from Table 6×100

Nominal value for Year 2 from Table 1

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DemonstrationTable 6

• Compute annually-chained Laspeyres volume index for GDP using Year 6 as reference year • Example: Annually-chained Laspeyres volume

index for GDP for Year 7 is

107.9100.0 107.9

100

Index for Year 6 from Table 6Laspeyres volume index for Year 7 from Table 6

100

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DemonstrationTable 6

• Example: Annually-chained Laspeyres volume index for GDP for Year 8 is

107.3107.9 115.8

100

Index for Year 7 from Table 6Laspeyres volume index for Year 8 from Table 6

100

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DemonstrationTable 6

• Compute annually-chained Laspeyres volume measures for X, Y, and GDP using Year 6 as reference year

• Thus, annually-chained Laspeyres volume measures for X, Y and GDP for Year 6 will be same as corresponding nominal values in Table 1• Example: Annually-chained Laspeyres volume

measure for GDP for Year 7 is

107.9387.2 417.9

100

Value for Year 6 from Table 6Laspeyres volume index for Year 7 from Table 6

100

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DemonstrationTable 6

• Example: Annually-chained Laspeyres volume measure for GDP for Year 8 is

• Compute real growth rates of annually-chained Laspeyres volume measures for X, Y, and GDP

107.3417.9 448.4

100

Chained value for Year 7 from Table 6Laspeyres volume index for Year 8 from Table 6

100

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DemonstrationTable 6

• Real growth rates of X and Y are same as those in Table 2 because X and Y are most detailed products available

• Real GDP growth rates from Year 3 onwards are lower than those in Table 2

• Annually-chained Laspeyres volume measures of GDP are non-additive (i.e., X and Y do not add up to GDP) for all years except Years 6 and 7

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DemonstrationTable 7

• Compute annually-chained Paasche volume indexes and measures as follows

• For each product (X and Y), calculate volume measure for Year t at prices of following year (t+1) separately by using the reciprocal of their real growth rates to extrapolate their nominal levels for Year t+1• Example: Volume measure of X for Year 1 at the

price of Year 2 is

1221.2 214.4

3.21

100

1Nominal value for Year 2 from Table 1Real growth rate for Year 2 from Table 2

1+100

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DemonstrationTable 7

• Example: Volume measure of X for Year 2 at price of Year 3 is

1233.1 225.3

3.51

100

1Nominal value for Year 3 from Table 1Real growth rate for Year 3 from Table 2

1+100

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DemonstrationTable 7

• Sum up resultant volume measures of X and Y to obtain volume measures of GDP at prices of following year (t+1)

• Compute Paasche volume index for GDP at prices of following year (t+1) • Example: Paasche volume index for GDP for Year

2 is

357.0100 109.3

326.7

Nominal value for Year 2 from Table 1×100

Volume measure for Year 1 from Table 7

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DemonstrationTable 7

• Example: Paasche volume index for GDP for Year 3 is

364.4100 109.2

333.7

Nominal value for Year 3 from Table 1×100

Volume measure for Year 2 from Table 7

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DemonstrationTable 7

• Compute annually-chained Paasche volume index for GDP using Year 6 as reference year • Example: Annually-chained Paasche volume

index for GDP for Year 7 is

107.3100.0 107.3

100

Index for Year 6 from Table 7Paasche volume index for Year 7 from Table 7

100

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DemonstrationTable 7

• Example: Annually-chained Paasche volume index for GDP for Year 8 is

106.8107.3 114.6

100

Index for Year 7 from Table 7Paasche volume index for Year 8 from Table 7

100

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DemonstrationTable 7

• Compute annually-chained Paasche volume measures for X, Y, and GDP using Year 6 as reference year

• Thus, annually-chained Paasche volume measures for X, Y and GDP for Year 6 will be same as corresponding nominal values in Table 1• Example: Annually-chained Paasche volume

measure for GDP for Year 7 is

107.3387.2 415.4

100

Value for Year 6 from Table 7Paasche volume index for Year 7 from Table 7

100

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DemonstrationTable 6

• Example: Annually-chained Paasche volume measure for GDP for Year 8 is

• Compute real growth rates of annually-chained Paasche volume measures for X, Y, and GDP

106.8415.4 443.7

100

Chained value for Year 7 from Table 7Paasche volume index for Year 8 from Table 7

100

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DemonstrationTable 7

• Real growth rates of X and Y are same as those in Table 2 because X and Y are most detailed products available

• Real GDP growth rates are lower than those in Table 2 and Table 6

• Annually-chained Paasche volume measures of GDP are non-additive (i.e., X and Y do not add up to GDP) for all years except Years 5 and 6

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DemonstrationTable 8

• Compute annually-chained Fisher volume indexes and measures as follows

• Compute Fisher volume indexes for GDP as geometric mean (square root) of corresponding Laspeyres and Paasche volume indexes• Example: Fisher volume index for GDP for Year 2 is

12 109.8110.3 109.3

1

2Laspeyres volume index from Table 6 Paasche volume index from Table 7

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DemonstrationTable 8

• Example: Fisher volume index for GDP for Year 3 is

12 109.7110.2 109.2

1

2Laspeyres volume index from Table 6 Paasche volume index from Table 7

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DemonstrationTable 8

• Compute annually-chained Fisher volume index for GDP using Year 6 as reference year • Example: Annually-chained Fisher volume index

for GDP for Year 7 is

107.6100.0 107.6

100

Index for Year 6 from Table 7Fisher volume index for Year 7 from Table 8

100

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DemonstrationTable 8

• Example: Annually-chained Fisher volume index for GDP for Year 8 is

• Annually-chained Fisher volume index for GDP can also be computed as the geometric mean (square root) of corresponding annually-chained Laspeyres and Paasche volume indexes

107.0107.6 115.2

100

Index for Year 7 from Table 7Fisher volume index for Year 8 from Table 8

100

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DemonstrationTable 8

• Compute annually-chained Fisher volume measures for X, Y, and GDP using Year 6 as reference year

• Thus, annually-chained Fisher volume measures for X, Y and GDP for Year 6 will be same as corresponding nominal values in Table 1• Example: Annually-chained Fisher volume

measure for GDP for Year 7 is

107.6387.2 416.7

100

Value for Year 6 from Table 8Fisher volume index for Year 7 from Table 8

100

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DemonstrationTable 8

• Example: Annually-chained Fisher volume measure for GDP for Year 8 is

• Compute real growth rates of annually-chained Fisher volume measures for X, Y, and GDP

107.0416.7 446.0

100

Chained value for Year 7 from Table 8Fisher volume index for Year 8 from Table 8

100

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DemonstrationTable 8

• Real growth rates of X and Y are same as those in Table 2 because X and Y are most detailed products available

• Real GDP growth rates are lower than those in Table 2 and between those in Table 6 and Table 7

• Real GDP growth rates computed using annually-chained Fisher index number formula provide most appropriate picture of economic growth as it uses weights from two consecutive years

• Annually-chained Fisher volume measures of GDP are non-additive (i.e., X and Y do not add up to GDP) for all years except Year 6

• Calculation of annually-chained Fisher volume indexes is demanding exercise as it involves many steps

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DemonstrationConclusion

• Real growth rates of X and Y are same in all situations because X and Y are most detailed products available

• Periodic rebasing helps to ensure real GDP growth rates in recent years are calculated with weights which are more representative of new base year

• Calculating annually-chained GDP volume measures (as recommended by 2008 SNA) helps to ensure real GDP growth rates are calculated with even more representative weights

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DemonstrationConclusion

• However, computation of annually-chained GDP volume indexes and measures (especially, Fisher) is resource-intensive and computationally-demanding exercise

• Also, additivity of GDP is not preserved• If statistical agencies are currently not in a position

to calculate annually-chained GDP volume measures, they should consider periodic rebasing with linking so that consistent time-series volume measures of GDP can be produced

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DemonstrationConclusion

• Base year should not be more than 5 years old• Statistical agencies should explain clearly in

methodological notes and metadata why periodic rebasing with linking will result in volume measures of GDP becoming non-additive

• At the earliest convenience, statistical agencies should switch to computing annually-chained volume measures of GDP

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Thank You