The document discusses the space-time (in)consistency between national accounts GDP statistics and purchasing power parity (PPP) exchange rates. It finds that while non-homotheticity and chain index errors could theoretically cause inconsistencies, empirical evidence suggests data errors are likely the primary cause of observed inconsistencies. The author argues that improving data quality and developing more sophisticated methods of combining national accounts and PPP data could help reduce inconsistencies. Comments praise the paper's ambition but raise questions about potential biases in the author's tests and suggest alternative approaches like state space models warrant further investigation.

1. The Space-Time (In)consistency of the
System of National Accounts:
Causes and Cures
Reproductions of this material, or any parts of it, should refer to the IMF Statistics Department as the source.
Real Sector Division
IMF Statistics Department
Nicholas Oulton, Centre for Economic Performance
London School of Economics
Discussant: Kim Zieschang, IMF
International Association for Research in Income and Wealth, Rotterdam, 24th-30th August 2014
The views expressed are the author’s and should not be attributed to the IMF staff or its Executive
Board

2. Real Sector Division
IMF Statistics Department
Summary
 Here, “space-time (S-T) consistency” means that the PPPs obtained in (say)
2005 by extrapolating 1980 PPPs using the GDP deflators from each
country’s national accounts will be the same as the actual 2005 PPPs.
There are often large S-T discrepancies in published PPP statistics and
national temporal GDP series.
 The paper notes that if the underlying aggregator functions for both
national temporal price change and bilateral PPPs are homothetic then
there will be no discrepancies in principle
 The paper argues—on the basis of applying a Laspeyres-perspective Konüs
principle (based on estimates from a PIGLOG flexible utility function)—that
neither non-homotheticity nor chain index approximation errors are the
most important causes of space-time inconsistency. Instead, data errors
are likely to be the most important cause.
2

3. Real Sector Division
IMF Statistics Department
Space-time consistency defined
3
For countries and and years and ,
consistency requires that:
B B
 
/
/
B / A B /
A t r
t r A A
  
t r
B
Pt : GDP deflator for country B in year
t
PPPB / A : PPP for country B relative to country A in year
t
t
A B r t
P P
PPP PPP
P P
 

4. Real Sector Division
IMF Statistics Department
S-T inconsistency in national accounts statistics for
53 countries in both 1980-2005 PWT
GDP per head in 2005,
1980=100 Growth of GDP per head, % p.a.
National
Accounts
Deflator
Extrapolation
PWT
8.0
Ratio,
column
2
÷
column
1
National
Accounts
Deflator
Extrapolation
PWT
8.0
Difference,
column 5
minus
column 4
(1) (2) (3) (4) (5) (6)
Mean 159.9 168.8 1.069 1.54 1.66 0.13
s.d. 75.0 87.8 0.323 1.62 1.84 1.04
Source: Penn World Table 8.0 and author (Oulton) calculations.
4

5. Real Sector Division
IMF Statistics Department
25
20
15
10
5
0
Frequency
Inconsistency index in 2005 (1980=100)
53 countries in the ICP in both 1980 and 2005.
0.500 1.000 1.500 2.000 2.500
incon
Source: Penn World Table 8.0.
Note: Incosistency index: ratio of PPP-based GDP per head to NA-based GDP per head.
5

6. Real Sector Division
IMF Statistics Department
Consequential S-T inconsistency: Argentina
Source: Penn World Table 8.0.
Note: 2005 figures use 2005 PPP. 1980 national accounts figure is
projected back from 2005 figure using national accounts growth rate of
real GDP per head. 1980 PWT figure uses 1980 PPP.
6
Real GDP per head,
US $ at PPP
Growth rate,
% p.a.
National
accounts
Penn World
Table
National
accounts
Penn World
Table
1980 8,938 3,253 --- ---
2005 9,702 9,702 0.33 4.37

7. Real Sector Division
IMF Statistics Department
Should we expect consistency?
 The GDP aggregator is unlikely to be homothetic
 The consistency formula involves both PPPs and GDP
deflators.
• PPPs are routinely adjusted to impose transitivity via all
comparison “paths” involving 2 or more countries (circularity).
• GDP deflators use domestic weights only for over-time
comparisons.
 Measurement error in the price and share data
underlying (temporal) GDP deflators and (cross-sectional)
PPPs may be significant.
7

8. Real Sector Division
IMF Statistics Department
Divisia index numbers and S-T consistency
8
Definition of growth rate of Divisia price index
for year (country) relative to year (country) , ( , ):
D
ln ( , ) ln ( )
N i
i i
 
: ( )

1
D
( ) : expenditure share of ith product in year (country)
i
t r P t r
d P t r d p t
w t
dt dt
w t t

9. Real Sector Division
IMF Statistics Department
Divisia index numbers
Divisia index numbers satisfy:
1. P × Q = V (product test)
2. Consistency in aggregation
However,
1. They are defined on continuous time (and space) and
have to be approximated by a discrete chain index, e.g.
chain Fisher or chain Törnqvist.
2. Divisia index numbers are line integrals. So a Divisia
index may violate the circularity test (transitivity of all
indirect comparisons) if it is not path-independent (i.e.,
the aggregator is not homothetic)
9

10. Real Sector Division
IMF Statistics Department
The Divisia index is a line integral

 
 
   


 
p
   


 

 p

 

w p X p
G t
t r t r
10
d p
N t D i
P t r w d
i 1
i r
d
N ( t
)
w d p
i 1 i i ( r
)
ln ( )
ln ( , ) ( )
( ) ln ( )
d
( , ) ln
G
: a particular path between endpoints ( ) and (r)
( ), ( ) : price vectors at time ( )
( , ) : vector of expen
p p
p p
w p X diture shares, dependent on prices
and possibly other factors X

11. Real Sector Division
IMF Statistics Department
Main theoretical result
Proposition 2:
If the Divisia price index is path-independent
(homotheticity holds), then, in the absence of data errors,
comparisons made using Divisia price indices are consistent
across space and time.
Corollary 2.1:
Even if the Divisia price index is not path-independent, a
Konüs index (which holds either utility constant) still yields
space-time consistency.
11

12. Real Sector Division
IMF Statistics Department
Intuition: the 2-good case
We can get from p(B,t) to p(A,t) either by path G or by path IHJ
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p(A,r)
p(B,r)
p(B,t)
p(A,t)
p2
p1
G
I
J
H
p(B,t), p(A,r): price vector in country B (A) at time t (r); Paths G, H: cross-section.
Paths I, J: time series

13. Real Sector Division
IMF Statistics Department
Main theoretical result, continued
So if there is inconsistency in practice, this must be due to
one or more of the following:
1. Path-dependence (non-homotheticity)
2. Chain index approximation error
3. Errors in price indices and/or PPPs
13

14. Real Sector Division
IMF Statistics Department
Which is the most important cause of
inconsistency? (1)
Non-homotheticity
Ratio of PPP for consumption using [Laspeyres perspective] compensated [Konüs]
shares to PPP using actual shares (141 countries in 2005)
Source: World Bank 2005 ICP. Compensated shares assume real income 41%
lower in 2005, i.e. same as in 1980. 100 products within household
consumption. See Oulton (2012c) for method.
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Index
number Mean
Std.
Dev. Min Max
Chained
Törnqvist 1.60 0.51 0.75 2.48
Chained
Fisher 1.11 0.16 0.63 1.57

15. Real Sector Division
IMF Statistics Department
Which is the most important cause of
inconsistency? (2)
Chain index approximation error
PPPs for household consumption:
ratio of chained Fisher to chained Törnqvist
Source: World Bank 2005 ICP. 100 products.
15
N Mean Std. Dev. Min Max
All
countrie
s 141 1.26 0.23 0.93 1.70
Less
than or
equal to
median 71 1.46 0.12 1.17 1.70
Above
median 70 1.06 0.09 0.93 1.25

16. Real Sector Division
IMF Statistics Department
Which is the most important cause of
inconsistency? (3)
 Non-homotheticity and chain index approximation error
are of roughly the same size as the inconsistency index.
So both might explain inconsistency.
 But the correlation (simple or multiple) between the two
and the inconsistency index is low and not significant.
 So data errors seem the most likely cause of
inconsistency.
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17. Real Sector Division
IMF Statistics Department
Comments [1]
 “In practice, the main source of inconsistency appears to
be errors in domestic and international prices (PPPs)”
• Hard to disagree with this as a significant contributing factor
 “Comparison resistant” GDP components such as General
government services and explicit and implicit (owners’) rentals on
dwellings.
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18. Real Sector Division
IMF Statistics Department
Comments [2]
 Questions on Corollary 2.1:
• If the data are not generated from a fixed reference utility, Konüs (compensated demand)
process between PPP benchmark years, won’t there be “Laspeyres perspective” bias if
setting reference utility in the earlier benchmark year or “Paasche perspective” bias if
setting reference utility in the later benchmark year?
• Should a “Fisher” concept be adopted under which the two perspectives are geometrically
averaged?
• Comment: One reason we chain (use practical Divisia) is that we think the underlying
aggregator is shifting from “delta to delta” on path G, that is, we don’t want to impose
path independence (particularly in the time domain). The implication seems to be an
automatic discrepancy between extrapolated and direct PPPs
 Question: Do we really need multi-year transitivity (circularity, path
independence) for the T (time) component of S-T price-volume comparisons?
(History is static.)
• So a little drift from extrapolated PPPs appears desirable. Transitivity, in some sense, seems
to be required for fitness for use of cross-country, point in time (PPP) comparisons (but
maybe not on all paths).
18

19. Real Sector Division
IMF Statistics Department
Comments [2]
 “Progress in removing inconsistency requires more sophisticated ways of combining
data sources, weighting each in accordance with its estimated reliability.”
• [1] For example, existing PPP transitivity adjustments could be more inconsistent than necessary
with national time series
 ICP uses Gini-Elteto-Koves-Schultz (GEKS): No time series information and no implied revision of
prior PPP benchmarks except through GDP revisions, but “strong transitivity”
 Would we do better using “path dependent,” “PPP chaining” methods such as Robert Hill’s (1999)
“spanning trees”? This would help if unadjusted PPPs align to time series better than transitivity
adjusted PPPs; effectively results in spatial path dependent “weak transitivity”
19

20. Real Sector Division
IMF Statistics Department
Comments [3]
• [2] The time series/cross section state space RRD framework of Rambaldi,
Huynh, and Prasada-Rao (this session)—with parameter settings favoring
adjustment of raw PPP benchmarks rather than country growth rates—seems to
offer a promising way forward in improving the information content of the
current PPP benchmark while easing communication of what is an otherwise
technically dense method to users. Under RRD, the current PPP benchmark
depends on all current and prior data, and implies revisable prior PPP
benchmarks (just like all other national accounts statistics). It involves
incorporating more conditioning information (regressors) to predict the PPPs
in linear econometric models , so is more data intensive than existing
procedures
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21. Real Sector Division
IMF Statistics Department
Quibble
 On the title of the paper and usage throughout, the
“SNA” is an accounting standard and not inconsistent in
the sense of this paper, but “national accounts statistics”
may be.
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22. Real Sector Division
IMF Statistics Department
Kudos
 This paper is interesting and ambitious, and makes some
headway in getting at the sources of PPP extrapolation
drift (or in some cases, extrapolation caroming) by novel
use of Divisia principle applied to space as well as time.
 The compensated share angle on estimating the impact
of nonhomotheticity and the chain approximation error
relative to Divisia is clever, but looked like a lot of work
and results in an informal consistency test (still could be
worth doing); note earlier questions concerning
Laspeyres perspective or Paasche perspective biases.
 The main useful result is the conclusion that the key PPP
extrapolation drift issue is data quality, not homotheticity
or chain approximation error.
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