Analysis of Divergence between Chained CPI-U and Regular CPI-U
for the All_US–All_Items Indexes (2000-2002)
Owen J. Shoemaker
U.S. Bureau of Labor Statistics, 2 Mass Ave., NE, Room 3655, Washington, DC 20212
[email protected]
available for both time t as well as for time t-n.
The C-CPI-U can thus be labeled a
“superlative” index. By contrast the Regular
CPI-U uses weights that are, at a minimum, at
least two years old, and uses a combination
(Hybrid) of Geomeans and Laspeyres formulas
as its final estimator. The set of All_US–
All_Items Chained C-CPI-U index results
continue to diverge (lower) from Regular CPIU index results. We investigate the nature of
this divergence. We also analyze the two
different weight structures, possible response
biases, and the standard errors that we
calculate for these indexes.
Key Words: Tornqvist; Stratified Random
Group; Superlative Index
Any opinions expressed in this paper are those
of the author and do not constitute policy of
the Bureau
In February, 2004, the BLS calculated and
published its third annual set of C-CPI-U
indexes --- for the 12 months of 2002. The CCPI-U (Chained Consumer Price Index –
Urban) is calculated and published every year,
with a one year lag, using a Tornqvist formula,
and its set of weights are updated yearly, so
that a unique set of monthly weights are
1.08
Fig 1. Regular INDEX vs. Chained INDEX
All-US--All-Items (Jan'00 - Dec'02)
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Price Change (PC) = (INDEX – 1) * 100
% DIFF = (PCR / PCC – 1) * 100
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months. There is a clear divergence between
the two indexes, but the question is not that
there is a divergence (with the Chained CPI-U
tracking consistently lower than the Regular
CPI-U) but whether that divergence is growing
and whether it is inappropriately large.
1. Divergence of Regular vs. Chained
“Superlative” indexes
BLS’ new Chained Index, in its final and
“Superlative” form, is published once a year, in
the January-February time frame, with a oneyear time lag. For example, the 12 months of
Final Chained Indexes for the Year 2002 was
published in early 2004. The new Chained
Index was launched in January 2000 and we
now have three full years of superlative
indexes (2000-2002) to look at. In Fig 1 above
we compare the Chained Index with BLS’
official (Regular) CPI-U for these same 36
1.01
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Fig 3. Regular vs. Chained INDEX
All-US--All-Items (2000)
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Fig 2. Regular vs. Chained INDEX
All-US--All-Items (Jan'00-Dec'01)
As Fig 1 demonstrates, the divergence does not
appear to be growing.
The percentage
difference in price change stays steady or even
declines. But the second issue is salient, as the
graphs below illustrate.
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Fig 5. Regular vs. Chained INDEX
All-US--All-Items (2002)
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Fig 4. Regular vs. Chained INDEX
All-US--All-Items (2001)
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Fig 2 demonstrates the 2-year trend between
the two indexes, but is not particularly
instructive.
It is the yearly difference
between the respective 12-month price
changes that tells the tale. There is an
anticipated and even predicted lowering of
the All-US–All-Items Index if BLS’ new
Superlative (Chained) Index is used instead
of Regular CPI. The 12-month difference is
not expected to be any more than “0.4
percentage points per annum”1 This estimate
of “substitution bias” inherent in the then
current CPI-U is from the 1996 Boskin
Commission Report. At most a Superlative
Index should be exhibiting a lowered index
of no more than 0.4 percentage points.
However, in 2000, the first year of the
Superlative Index, the 1-year difference was
1
See Boskin, et. al. Final Report on the Advisory
Commission to Study the Consumer Price Index.
Committee on Finance, United States Senate: S.
Prate. 104-72, December 1996, p. 44.
0.76, or nearly twice what was estimated to
be an upper bound of bias inherent in the
official (i.e., regular) CPI. In 2001 and even
more clearly in 2002 that difference dipped
well within appropriate bounds (0.34 for
2001 and 0.30 in 2002). So the over-sized
divergence between the two indexes seems to
be attributable to something untoward
occurring in the 2000 data, either with
Regular CPI or with the Superlative Index.
spread across all eight major groups. But
returning to Fig 6, which contains the eight
Major Group graphs for the two earlier years,
we see immediately the two possible sources
of out-sized divergences: in Education &
Communications and in Recreation. The
other six major groups display little or no
divergence at all, even after a full two years.
We will investigate the two possible sources
of divergence separately.
2.
Divergences between Regular and
Superlative in the 8 Major Groups
3.
Regular vs. Superlative Weight
Differences (in Telephone Services)
The CPI, whether in its Regular or
Superlative version, is an aggregation
structure, consisting of 211 Item-Strata and
38 Index-Areas (PSUs). The aggregation
system builds up from each Item-Stratum–
Index-Area combination, up thru higher
aggregates until we get to the eight Major
Groups, which then aggregate up to the AllUS Cities–All-Items Index.
So if we
breakdown the two indexes (Regular CPI and
Superlative CPI) into smaller and smaller
sub-aggregate units we may be able to
uncover the source of the over-sized
differences observed in the earlier indexes.
Now these differences may turn out to be
evenly spread over the entire structure and so
not easily discernible to a data analysis, but
such turns out not to be the case. At each
sub-aggregate stage of the breakdown the
further comparisons reveal clearly where the
differences lie.
The largest divergence between the two
indexes seems to come from within
Education & Communication.
Without
displaying all the pertinent comparison
graphs, as we decompose down to the ItemStratum level, we will simply state the results
at each successive stage.
First, we
decompose down to Education versus
Communication at the All-US Cities level.
Here, Education reveals no divergence, with
all the divergence occurring in the
Communication aggregate. Then within this
Communication sub-aggregate there are three
Expenditure Classes (ECs): Postage and
Delivery Services, Telephone Services and
Information Services Other Than Telephone.
Of these three ECs only Telephone Services
shows any divergence between the two
indexes.
Telephone Services further
decomposes down to its three basis level
Item-Strata: Local Telephone Services, Long
Distance Services and Cellular Services. Yet
here at the lowest Item-Stratum level
divergence disappears. All three Item-Strata
show little or no divergence between the two
indexes --- even though the EC they
constitute does show clear divergence. It
would seem we are stymied in our data
analysis search. However, there is one telling
anomaly: Local Services indexes are steeply
rising over this two year period, while the
Cellular Services indexes are steeply
declining. Conceivably, if there were to be
found a clear discrepancy between the two
respective sets of weights between the
Regular and Superlative Indexes here, that
might explain the anomaly. As it turns out,
such is the case.
On the following two pages (see Fig 6 and
Fig 7), we graph the comparative subaggregate indexes for the All-US Cities by
Major Group. In Fig 7, for the year 2002, as
expected, there appear to be no discernible
divergences at the Major Group level, with
the possible exception of Recreation.
However, closer analysis of the Recreation
graph shows that the apparent difference
there is just that: apparent. The range of the
Recreation index is so narrow that the small
discrepancy between the Regular and
Superlative indexes here is actually smaller
than for most of the others. Effectively the
small (and expected) 0.30 difference between
the two 12-month indexes in 2002 is evenly
Fig 6. REGULAR vs. CHAINED INDEXES
(Jan ‘00–Dec ‘01)
All-US Cities by Major Group
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Fig 7. REGULAR vs. CHAINED INDEXES – 2002
All-US Cities by Major Group
APPAREL
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Several important characteristics distinguish
the Superlative Index from the Regular
Index. For one thing, the Superlative Index
employs a Superlative Index Formula, the
Tornqvist, which, in simplest terms, is the
geometric mean of two Geomeans formulas,
one at time t and the other at time t-1.
Regular CPI uses aggregate weights that are
anywhere from two to four years old,
whereas the Superlative Index uses timely
up-to-date weights for both time t and time
t-1.
The resultant differences are not
usually dramatic, but occasionally, like here,
they are.
The
The percentage weights in this one small EC
(Telephone
Services)
vary
quite
dramatically across the three years of data
we have been investigating. In Fig 8 we see
the minimal percentage of the weight within
this EC that was being attributed to Cellular
Services, particularly in the years 20002001, while Local Telephone Services
carries nearly half the weight within the EC.
These 2000-2001 percentages are reflective
of weights from the years 1996-1998. Even
when a new set of aggregate weights (from
1999-2000) are introduced and used in the
2002 Regular Indexes, the percentage for
Cellular Services is still quite small.
IX
Superlative
T
I , A, r [ t −1;t ]
1-month
formula
 ixit,a ,r
= ∏  t −1

i∈I , a∈A  ix i , a , r




is
 sit, a , r + sit ,−a1, r


2





,
where ix is a elementary price index, and s
is a monthly expenditure share.
The Regular Index uses a Laspeyres
formula at this its higher level of
calculation. (Note that both Regular and
Superlative use the same basic-level price
relatives in their respective computations.)
But the critical difference between the two
indexes is in their respective weights. Both
indexes draw from the same pool of
Consumer Expenditure weights, but the
2
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Fig 8. Regular (%) Weights --- Telephone Services (2000-2002)
0
10
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1 = Local Services
2 = Long Distance
3 = Cellular Services
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Fig 9. Chained (%) Weights --- Telephone Services (2000-2002)
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2 = Long Distance
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Compare these percentages to the ever-growing
percentage of Cellular Services (Fig 9) weights
from the Superlative system (which are
reflecting the most current expenditures for
these respective services).
The minimal
weight given to the steep-declining Cellular
Services in the Regular Index resulting in a
larger weight going to the steep-rising Local
Phone Services has been producing a Regular
Index in this sector of the index considerably
higher than its Superlative counterpart. While
it is difficult to claim that these weight
percentage differences account for the entirety
of the index discrepancy at the higher levels in
the Education & Communication Major Group,
they are clearly the main source of this
difference.
reveals any divergence.
Fig 10, on the
following page, shows the dramatic divergence
that occurs in the Item-Stratum, Audio
Equipment, in the April to May period in 2000.
The Regular CPI shoots up nearly 0.1
percentage points while the Superlative Index
for that same time period actually goes down a
little. This is not a large growing divergence
but a dramatic one-time surge in the Regular
CPI that is not matched in the Superlative.
What happened was that a Hedonic quality
adjustment, made on some CD Player prices in
San Diego, in that time frame, was allowed in
the PRC (Price Relative Calculation) and
produced in this one Item-Stratum in the San
Diego PSU a 3.24 price relative, a more than
three-fold increase.
4. Divergence due to Response Bias
The second Major Group which displays a
clear divergence between its Regular and
Superlative indexes is Recreation (see Fig 6
again). While this difference (0.1 percentage
points) is only one-sixth as large as the
difference obtaining in Education &
Communication, it is still over-sized.
Recreation consists of seven Expenditure
Classes (ECs). Only one of these ECs shows
any divergence: Video and Audio. Moreover,
as we continue the narrowing-down process, as
we did before in Section 3, we find that only
one of the seven Item-Strata (Audio
Equipment) within the Video and Audio EC
5. Comparative Analysis using Standard
Errors
In an earlier paper2, the Regular and Chained 1, 2, 6- and 12-month price changes for 20002001 were tested for significant differences
using simple paired t-tests. The differences
were all found to be significant at an α = .025
level (in fact at an α = .01 level). Standard
errors were also calculated for the new
Superlative 1-, 2-, 6- and 12-month price
changes for 2000 and 2001, and the
methodology explained in the paper3. Using
the same Stratified Random Groups Method,
new standard errors for 2002 have been
2
Shoemaker, Owen J. “Estimation and Comparison of
Chained CPI-U Standard Errors with Regular CPI-U
Results (2000-2001)”. ASA Proceedings, December
2003.
3
Ibid.
The question needs to be asked as to why this
one out-sized price relative in San Diego did
not produce the same dramatic bump-up using
the Superlative Index as it did in the Regular
CPI, since the same price relative went into
both calculations. The partial answer is that the
Superlative system is able, thanks to its time
lag, to smooth the off-cycle indexes (using a
geometric averaging, which transforms a 2month price relative of 3.24 to two 1-month
price relatives of 1.8).
Moreover, the
geometric nature of the Superlative formula
seems to smooth the sharp price relative spikes
within its calculation better than does the
Laspeyres formula that is used in the Regular
CPI.
calculated. The results remain in line with the
earlier results (average 2000-2001 12-month
SEs ≈ .11 and average 2002 12-month SEs ≈
.12). Since inflation rates are promulgated as
per annum results, the most pertinent results are
the 12-month standard errors. In Fig 11 below
we use these standard errors to construct 95%
Confidence Intervals around our Superlative
price change results, and then graphically
compare those to the Regular CPI results.
While the two indexes remain significantly (α =
.025) apart from each other, the gap is clearly
narrowing in the last 12 months, i.e., in 2002.
Fig 10. Regular vs. Chained INDEX (2000-2002) All-US—Audio Equipment
1.00
1.05
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Fig 11. 12-Mon Price Changes (Dec'00--Dec'02)
3.5
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•
•
•
The divergence between Regular CPIU and Chained CPI-U did spike
steeply in 2000, but settled down in
2001 and 2002 to a difference well
within the 0.4 percentage point upper
bound bias expectation.
Data analysis traced the 2000 spike to
Cellular Services (in EDUC/COMM)
and
Audio
Equipment
(in
RECREATION).
The
newer
larger
up-to-date
Superlative Index weights used in the
deflating Cellular Services item
stratum increased the difference
between the Chained and Regular
Indexes in 2000 and 2001.
The Audio Equipment item stratum
(from San Diego) that surged in April-
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+
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C
R
+
+
C
+
C
+
+
C
R
+
C
R
+
C
+
+
+
C
+
+
M
A
6. Summary
•
R
+
R
+
R
+
R
+
•
•
M
J
J
A
S
O
N
D
May 2000 induced the Regular CPI-U
to jump dramatically while inducing
the Chained CPI-U not to go up at all.
Standard errors calculated for the AllUS–All-Items Chained Index were
used to construct Confidence Intervals
for the 12-month Superlative price
changes. The Superlative Index price
changes continue to be significantly
lower than their Regular Index
counterparts, but the gap is narrowing.
Further investigation may show even
less significant differences between
the two indexes in the smaller subaggregations