10  Monthly Labor Review • July 2010

PPI and CPI Seasonal Adjustment

Jonathan C. Weinhagen,

Jeffrey S. Wilson,

and

Steven M. Muri

Jonathan C. Weinhagen, Jeffrey

S. Wilson, and Steven M. Muri

are economists in the Office of

Prices and Living Conditions,

Bureau of Labor Statistics. Email:

weinhagen.jonathan@bls.gov,

wilson.jeff@bls.gov, or muri.

steven@bls.gov

I

n November 2006, the first-released1

seasonally adjusted producer price 

index (PPI) for gasoline jumped 17.9 

percent, while the seasonally adjusted 

consumer price index (CPI) for gasoline 

fell 1.6 percent. The disparity between 

the two series resulted from differences in 

the indexes’ seasonal factors for gasoline. 

Seasonal factors are applied to unadjusted 

indexes to remove within-year seasonal 

patterns from time series, allowing for 

more comparable month-to-month index 

analysis.

An initial investigation indicated that 

methodological differences in the PPI and 

CPI were responsible for the differences in 

seasonal factors for gasoline. In particular, 

the PPI and CPI differed in their selection 

of data points to model as interventions 

for the series. Intervention modeling is 

used during the estimation of seasonal 

factors to remove the effects of nonseasonal events that can distort the observed 

seasonal patterns of an index.

In 2007, the Bureau of Labor Statistics 

(BLS, the Bureau) undertook an effort to 

further investigate differences in seasonal 

adjustment methods between the PPI and 

the CPI, with an eye toward developing a 

more unified approach to seasonal adjustPPI and CPI seasonal adjustment:

an update

A new update of BLS seasonal adjustment procedures ensures

that differences between seasonally adjusted PPI and CPI series 

are due to the underlying unadjusted data, and not to differences

in seasonal adjustment methods; in a further improvement, PPI

and CPI analysts are now coordinating their efforts

ment. PPI analysts and CPI analysts worked 

together and with a group of BLS time-series 

experts to develop that approach.

Several methodological changes were implemented as a result of the investigation. A 

new set of processes was introduced to build 

intervention models for time series, to assist 

with the selection of time series for intervention modeling, and to identify similar PPI

and CPI time series for joint modeling. This 

article presents the updated seasonal adjustment procedures.

The next section provides an overview of 

general PPI and CPI modeling procedures. 

Then, the third and fourth sections describe 

the processes used by the CPI and PPI, before 

and after the investigation, to select time series for intervention modeling and to develop 

intervention models. Following that, the fifth 

section presents time series selected for joint 

modeling work and the sixth section gives examples of series that were affected by the new 

seasonal adjustment procedures.

PPI and CPI seasonal adjustment 

The Bureau publishes seasonally adjusted PPI 

and CPI time-series data on a monthly basis. 

Both the PPI and the CPI utilize direct and 

indirect seasonal adjustment methods. Direct 
Monthly Labor Review • July 2010  11

seasonal adjustment is accomplished by applying seasonal 

factors to unadjusted data to remove within-year seasonal 

patterns. Indirect adjustment is a method of seasonal adjustment used for aggregate series; in this method, two or 

more directly adjusted component indexes are combined 

into higher level time series.

In the PPI, commodity-based and stage-of-processing 

indexes are eligible for seasonal adjustment. In 2009, the 

Bureau published 1,226 commodity-based PPIs and 52 

stage-of-processing PPIs. The vast majority of the PPI commodity data is directly seasonally adjusted. By contrast, 

the Bureau seasonally adjusts all of its stage-of-processing 

PPIs by means of an indirect method. The CPI’s entire set 

of 368 item-level indexes is eligible for seasonal adjustment. The Bureau uses direct adjustment for its lower level 

CPI indexes and indirect adjustment for all upper level aggregate indexes. Lower level indexes track price change 

for specific commodities over time, whereas upper level 

indexes track price change for groupings of lower level 

commodity indexes. Upper level indexes are constructed 

by using consumer expenditure weight data to combine 

lower level indexes.

Direct adjustment. The Bureau tests all PPI and CPI series 

that are eligible for direct adjustment for seasonality, and 

if seasonality is found, the series are seasonally adjusted. 

Both seasonality testing and direct seasonal adjustment 

are accomplished with the use of X-12 ARIMA, a software 

package published by the U.S. Census Bureau for seasonal 

adjustment applications. Seasonal adjustments are based 

on the X-11 variant of the Census II seasonal adjustment 

method.2 X-11 is a filter-based approach, employing moving averages to estimate trend and seasonal components 

in turn. Components are refined through several iterations 

of weighted moving averages. X-12 ARIMA uses a multiplicative time-series decomposition model by default:

 Yt = TtStIt.

In this model, Yt is the value of the observed series, Tt 

represents the trend-cycle component, St is the seasonal 

component, and It is the irregular component. The multiplicative model is appropriate when a series has increasing 

variation with time, as is often seen with PPI and CPI series. To enable the use of symmetric moving-average filters 

on a series, X-12 ARIMA uses an ARIMA (Auto-Regressive 

Integrated Moving Average) modeling facility to forecast 

and backcast observations at the endpoints of the data.

Among the many diagnostics that are available for assessing the quality and stability of seasonal adjustments 

are F-tests for the presence of stable and moving seasonality and quality control statistics from X-11.

3

 Data that 

facilitate graphical analysis also are available, including 

the unadjusted and adjusted series frequency spectra.4 Exhibit A-1 in the appendix provides a summary of diagnostic tools examined and used by BLS seasonal adjustment 

analysts, as well as a list of frequently employed graphs.

The Bureau utilizes three primary measures to determine whether a particular PPI or CPI should be seasonally adjusted: F(s), M7, and Q. F(s) is a measure of stable 

seasonality, M7 determines the amount of moving seasonality relative to the amount of stable seasonality, and Q

is a weighted average of several diagnostic statistics. (See 

exhibit A-1 in the appendix for quality control statistical 

seasonality thresholds.)

Indexes that are found to exhibit a level of seasonality 

warranting adjustment are directly adjusted by applying 

a seasonal factor to the unadjusted index according to the 

formula

 

where 

Is is the seasonal index value, 

Iu is the unadjusted index value, and 

SF is the seasonal factor.

Seasonal factors indicate the seasonal pattern of a time 

series and are derived from historical unadjusted data. The 

Bureau typically uses 8 years of unadjusted monthly data 

in developing factors and testing seasonality for both the 

PPI and the CPI. 

Intervention analysis. Nonseasonal events such as natural 

disasters or wars can distort the underlying seasonal pattern of an index. Intervention analysis entails estimating 

and removing the effects of these events from indexes prior to testing them for seasonality and developing seasonal 

factors. The goals of intervention analysis are to determine 

whether a seasonal pattern exists and to correctly estimate 

seasonal factors in spite of any distortion that might arise 

in the pattern. The Bureau applies intervention analysis to 

selected directly adjusted PPI and CPI indexes. (See later.) 

The Bureau uses X-12 ARIMA to conduct both CPI and 

PPI intervention analysis, a method in which ARIMA

models that include prespecified intervention variables 

are estimated for a time series. These variables are used to 

identify the statistical significance and relative effects of 

(100), SF

u

s

I I =
PPI and CPI Seasonal Adjustment

12  Monthly Labor Review • July 2010

nonseasonal events on time series. In cases where a nonseasonal event is found to significantly affect a time series, 

the effects of the event can be removed from the original 

time series by using the estimated coefficients from the 

ARIMA model. Three types of intervention variables are 

employed: outliers, level shifts, and ramps.

Outlier variables are specified as AO = 1 for t = t

0

 and 

AO = 0 for t ≠ t

0

, where t

0

 is the month of the nonseasonal 

event.

Level-shift variables are specified as LS = –1 for t < t

0 and 

LS = 0 for t ≥ t

0

, where t

0

 is the month of the nonseasonal 

event.

Ramp variables are specified as RP = –1 for t ≤ t

0

, RP = 

[(t – t

0

)/(t

1

 – t

0

)] – 1 for t

0

 < t < t

1

, and RP = 0 for t ≥ t

0

, 

where t

0

 is the first data point of the nonseasonal event 

and t

1

 is the last data point.

After nonseasonal effects are removed from the original 

time series, standard direct seasonal adjustment methods 

as described earlier are applied to the indexes to test for 

seasonality and to develop seasonal factors.

Indirect adjustment. High-level indexes, such as the PPI 

for Finished Goods and the All Items CPI, are indirectly 

seasonally adjusted by aggregating lower level series that 

are components of higher level indexes. Seasonally adjusted components are used when available (that is, when the 

lower level index was shown to be seasonal and a seasonal 

index was calculated); otherwise, unadjusted indexes are 

used.

The Bureau indirectly adjusts all of its PPI stage-of-processing indexes, as well as any indexes that are aggregates 

of intervention indexes. In this manner, interventions estimated for lower level indexes are indirectly included in 

aggregate indexes. The Bureau indirectly seasonally adjusts 

the All Items CPI index and 54 other aggregate series.5 

Yearly revisions and projected factors. Each year, with the 

release of the January data, the PPI and CPI seasonal factors are recalculated to reflect price movements that occurred during the just-completed calendar year. Seasonal 

factors are recalculated 5 years back, and all seasonally 

adjusted data are updated on the basis of these new factors. For example, in January 2007 factors were recalculated from 1999–2006 data and seasonal data from 

2002–06 were updated in accordance with the new set 

of factors. After the yearly revision, the PPI and the CPI

for the upcoming year are calculated with the previous 

year’s set of seasonal factors. For instance, the 2006 factors, from the January 2007 revision, are used to calculate 

indexes throughout 2007. 

Selection of intervention candidates 

Pre-2007 candidate selection procedures for the PPI. More 

than 1,200 PPI indexes are currently eligible for direct seasonal adjustment. Conducting intervention modeling on 

this entire set of indexes is not feasible because of resource 

constraints. Consequently, the Bureau performs intervention modeling on only a relatively small set of PPIs, referred to as intervention candidates.

Prior to 2007, PPIs were selected as intervention candidates on the basis of four criteria: the index must have 

been a six-digit commodity index, the index must have 

been a currently seasonally adjusted index that was going 

to fall out of seasonal adjustment due to failing quality 

control statistics for 3 consecutive years, the index must 

have had a relative importance of greater than 1 percent 

of a major stage-of-processing index, and there must have 

been an identifiable shock causing a distortion in the seasonal pattern of the index.

Indexes meeting these four criteria were added to the 

set of PPI intervention candidates. Once an index was so 

added, it typically remained a candidate unless the Bureau 

believed that the index no longer exhibited any statistically significant seasonal pattern.

CPI’s pre-2007 candidate selection procedures. The CPI 

conducted a yearly analysis of all its time series to determine which of them to include as intervention candidates. 

Among the factors analyzed were the candidate status of 

the series the previous year, information from commodity 

analysts, the results of a visual inspection of all eligible 

series, large events (such as a hurricane) that could affect 

unrelated series, and substantial changes in diagnostic 

behavior. In cases where the analysis indicated benefits 

from intervention modeling, the series was included as a 

candidate.

Updated candidate selection procedures. As mentioned earlier, 

prior to 2007 the Bureau utilized a relative-importance rule 

in selecting PPI candidates. This rule ensured that intervention work was directed toward important and visible series. 

The PPI continues to utilize the 1.0-percent relative-importance rule, and the CPI has now implemented a similar rule. 

The rule requires candidates to have a relative importance 

of 0.5 percent of the U.S. city average All Items CPI or be a 

subset of an already qualifying component series.

Previously, the Bureau would consider adding a PPI as 

a candidate only if it were a currently seasonally adjusted index that was in danger of becoming unadjusted as 

a result of exhibiting failing quality control statistics for 
Monthly Labor Review • July 2010  13

3 consecutive years, whereas all CPIs were analyzed each 

year as potential candidates. The Bureau then removed 

the requirement that a PPI needed to fail the quality control criteria for 3 consecutive years in order to become a 

candidate. Now all PPIs and CPIs that meet the relativeimportance criteria are examined each year. This change 

improves seasonal factor estimation by ensuring that all 

important indexes have the potential to become intervention candidates.

Automatic outlier detection (including level-shift detection) is now implemented for both the PPI and the CPI as 

a primary tool for assisting in candidate selection. X-12 

automatic outlier and level-shift detection is a regressionbased program that searches for and identifies statistically 

significant intervention variables. Each year, the program 

is run on all series that are eligible for intervention analysis. (That is, they meet the relative-importance criteria.) 

Indexes for which automatic outlier detection finds significant interventions are then analyzed further as potential intervention candidates.

Intervention modeling 

Pre-2007 PPI modeling procedures. Before 2007, the Bureau utilized a number of sources of information in developing PPI intervention models, including analyses of extreme values detected by X-12, graphical analyses of time 

series, analyst price notes, and quality control statistics. 

Although all of these sources of information were used to 

develop the models, analyses of extreme values detected 

by X-12 received the most consideration in the overall PPI

procedure.

Generally, the Bureau sought to include a minimal 

number of interventions in a PPI model, modeling the 

least number of interventions necessary to allow the series 

to pass the seasonality thresholds. In fact, in cases where 

an intervention candidate passed quality control thresholds without any intervention modeling, program rules 

precluded the Bureau from including any interventions in 

the seasonal model.

In selecting potential interventions, the Bureau included 

the most significant interventions in its PPI models and 

required the absolute value of all t-statistics to be greater 

than 3.0 for intervention variables. All ramps, level shifts, 

and outliers utilized in the models required economic explanations. The Bureau did not, and still does not, publish 

these explanations, but does document them for internal 

use.

Pre-2007 CPI modeling procedures. To develop intervention models, CPI seasonal adjusters analyzed a number 

of diagnostic statistics, including seasonal factor graphs, 

first-difference graphs, quality control statistics, prior adjusted series, and automatic outlier detection. These diagnostic statistics were analyzed to determine the effects of 

modeling specific data points on CPI seasonal factors, as 

well as to determine whether the modeling of these data 

points was statistically supported. The Bureau generally 

required the absolute value of t-statistics for all CPI intervention variables to be greater than 3.0, but would accept 

lower t-statistics in some cases. The Bureau also examined 

market data and commodity analyst price notes to help 

identify potential CPI intervention points.

After seasonal adjusters developed potential intervention models, CPI adjusters consulted with commodity 

analysts to try to reach a consensus on a “best” intervention model. The Bureau usually did not place the same 

importance on the CPI as it did on the PPI in attempting 

to include a minimal number of interventions in a model; 

therefore, the CPI models that were developed often were 

larger than the PPI models. Nor did the Bureau apply to 

the CPI the PPI’s constraint of not modeling interventions in cases where the candidate passed quality control 

thresholds without any intervention modeling.

The CPI required, and still requires, that all ramps and 

level shifts included in seasonal models be accompanied 

by economic explanations, which are published yearly in 

the January CPI detailed report. Outliers did not need 

specific explanations, but events surrounding an outlier 

were often noted. 

Updated modeling procedures. The Bureau has developed a 

multistep process for modeling both PPI and CPI intervention series. The first step is to use X-12 automatic outlier 

and level-shift detection to identify potential intervention points. The use of this tool provides the Bureau’s PPI 

and CPI programs with a statistically based and replicable 

means for identifying potential outliers.

X-12, however, does not search for ramps, which the 

Bureau uses as a modeling tool for both the PPI and the 

CPI. Therefore, additional analysis is implemented to select intervention points. This analysis includes graphical 

examination of the original time series, the study of price 

trend analysis developed by PPI industry analysts and CPI 

commodity analysts for internal BLS use, communication 

with commodity and industry analysts, examination of 

interventions modeled in previous years, and analysis of 

residuals from the ARIMA model. A combination of these 

tools is used by PPI and CPI seasonal adjusters to identify 

potential intervention models for a series.
PPI and CPI Seasonal Adjustment

14  Monthly Labor Review • July 2010

The Bureau requires statistical and economic justifications for all modeled PPI and CPI interventions. Intervention variables should generally have t-statistics with 

absolute values greater than 3, and all modeled interventions should have economic explanations. In addition, the 

Bureau changed its policy regarding intervention modeling for PPI candidates that have passing quality control 

statistics. Now the PPI uses the CPI method of allowing 

intervention modeling for candidates whose quality control statistics exhibit seasonality. Allowing this kind of 

modeling ensures that nonseasonal events will be modeled for each index and that the effects of these events will 

not distort seasonal factors.

Further steps are taken for PPI and CPI series that exhibit a great deal of similarity. For these series, seasonal 

adjusters from the programs meet to compare and discuss 

models. The goal is to coordinate intervention decisions 

between the two programs for similar series. Communication also allows the programs to benefit from each 

other’s expertise and helps to avoid large discrepancies in 

modeling decisions. Such discrepancies can lead to situations in which similar series exhibit consistent unadjusted 

movements but different seasonally adjusted movements. 

(The next section, on concordance series, describes the 

process for selecting these comparable series.)

Once several potential models have been developed, information criteria are used for further comparison. Information criteria help determine the appropriate number of 

estimated parameters to include in a model. The criteria 

weigh the benefits of adding variables to the model by numerically rewarding the increase in fit generated by an additional variable but numerically penalizing the model for 

the loss in degrees of freedom associated with the additional parameter. Two frequently used information criteria are the Akaike and Bayesian criteria. Akaike, however, 

has been shown to asymptotically overstate the number of 

parameters to include in a model. The Bayesian attempts 

to correct for this deficiency by being stricter than Akaike 

in terms of penalizing the loss of degrees of freedom. Both 

criteria are analyzed during modeling, but the Bayesian is 

given more weight because of Akaike’s tendency to select 

models with too many parameters. Carefully analyzing 

information criteria helps to ensure that models are not 

overfit, a condition that tends to make the models excessively complex and exhibit poor out-of-sample predictive 

performance.

After final drafts of the intervention models are completed, CPI and PPI analysts meet jointly to discuss the 

series that have been selected for seasonal adjustment coordination. Any changes resulting from this meeting are 

incorporated into the final models.

Concordance series

To assist in seasonal adjustment coordination between the 

PPI and CPI, a series concordance was developed. The concordance identifies PPI and CPI series eligible for intervention analysis that might benefit from data exchange or 

coordinated seasonal adjustment. In order to be included 

in the concordance, a series must be eligible for intervention analysis in both programs.

Using the PPI 1.0-percent relative-importance rule as a 

starting point for the concordance, analysts identified all 

PPI series having a relative importance of greater than 1.0 

percent of either the finished, intermediate, or crude goods 

indexes as potential concordance series. For 2007, 50 series were identified on the basis of the relative-importance 

criterion.6 Along with these 50 series, any current PPI 

seasonal candidates that did not pass the 1.0-percent rule 

were included, bringing the total number of series to 58. 

(The current set of PPI intervention candidates includes 

several indexes that make up less than 1.0 percent of a 

major stage of processing. Historically, these indexes have 

been intervention candidates, and the Bureau chose to 

keep them as such.)

CPI series were then matched to the 58 PPI series on 

the basis of similarity of title. After the initial matching, 

corresponding indexes were further analyzed to determine whether they exhibited similar enough movements 

to warrant PPI-CPI seasonal adjustment coordination. It 

turned out that many series with somewhat similar titles 

between programs proved imperfect matches and were 

not included in the final concordance table. The final concordance table is updated annually to reflect changes in 

relative-importance calculations and potential new series 

that arise each year. This table is presented as exhibit 1.

Examples

This section presents examples of series that were analyzed 

and, in most cases, substantially affected by the modified 

PPI and CPI seasonal adjustment procedures.

Gasoline. In the 2006 annual seasonal adjustment revision, the Bureau tested its three PPI gasoline series 

(regular, midpremium, and premium) for seasonality, and 

the quality control statistics for all three series indicated 

seasonality. Consequently, no interventions were modeled because of the program’s rule prohibiting intervention work on series whose quality control statistics met 
Monthly Labor Review • July 2010  15

 Producer Price Index (PPI) and Consumer Price Index (CPI) final concordance list

PPI finished-goods series greater than 1 percent of major stageof-processing index and adjusted by intervention analysis 

seasonal adjustment (IASA)

Closest CPI series

Item 

code Title

Relative 

importance

Seasonally 

adjusted

IASA, 

2006

Item 

code Title

Seasonally 

adjusted

IASA, 

2006

054121 Residential electric power 7.84 Y N SEHF01 Electricity Y Y

141101 Passenger cars 4.10 Y N SETA01 New vehicles Y Y

057104 Unleaded regular gasoline 4.00 Y N SS47014 Gasoline, unleaded

regular

Y Y

055121 Residential natural gas 3.00 Y Y SEHF02 Utility (piped) gas

service

Y Y

022105 Other meats, fresh, frozen,

or canned

.60 Y N SEFE Other meats Y N

057103 Unleaded premium

gasoline

.76 Y N SS47016 Gasoline, unleaded

premium

Y Y

057105 Unleaded midpremium

gasoline

.36 Y N SS47015 Gasoline, unleaded

midgrade

Y Y

057302 Home heating oil and

other distillates (fuel oil

#2)

.74 Y Y SEHE01 Fuel oil Y Y

054321 Industrial electrical power 2.76 Y N SEHF01 Electricity Y Y

054221 Commercial electrical

power

4.72 Y N SEHF01 Electricity Y Y

016101 Milk for fluid use 4.70 Y N SEFJ01 Milk Y N

Exhibit 1.

 Chart 1. Seasonal factors for gasoline, 2006 revision, 2004–06

Seasonal

factor

115

110

105

100

95

90

85

80

Seasonal

factor

Jan Mar May Jul Sep Nov Jan Mar May Jul Sep Nov Jan Mar May Jul Sep Nov

2004 2005 2006

115

110

105

100

95

90

85

80

PPI gasoline,

unleaded regular

CPI gasoline,

all ty
PPI and CPI Seasonal Adjustment

16  Monthly Labor Review • July 2010

the predetermined seasonality thresholds. The CPI, by 

contrast, included 22 intervention variables in its seasonal 

model for gasoline. 

The PPI and CPI seasonal factors for gasoline from the 

2006 revision differed substantially as a result of the different intervention models used by the two programs. 

Chart 1 presents the seasonal factors for CPI gasoline 

(all types) and PPI unleaded regular gasoline. Recall that 

seasonal factors reflect the expected seasonal pattern of a 

time series on the basis of historical data and are applied 

to the unadjusted data to create seasonal data.

The PPI and CPI seasonal factors exhibit similar seasonal 

patterns (rising in the spring and declining throughout 

fall and winter), but differ in terms of volatility and size. 

In addition, during several months the seasonal factors 

project opposite directional movements in the gasoline 

indexes. For example, from October to November, the 

PPI expects a large seasonal decrease in gasoline prices, 

whereas the CPI anticipates a small seasonal increase.

For the 2007 annual seasonal adjustment revision, the 

Bureau implemented the updated PPI and CPI modeling procedures discussed in the previous two sections. 

The Bureau was no longer precluded from modeling PPI

gasoline series data points as interventions in spite of 

quality control statistics that indicated seasonality in the 

unmodeled series. The Bureau also relied more heavily on 

automatic outlier and level-shift detection, ARIMA model 

residual analysis, information criteria, analyst input, and 

cross-program coordination to develop both its PPI and 

CPI intervention models. As a result, the PPI intervention 

model increased in size, whereas the CPI model decreased. 

Six intervention variables were in the 2007 PPI model, 

compared with none in the 2006 model. The number of 

intervention variables in the CPI model fell from 22 in 

2006 to 4 in 2007. The 2007 PPI and CPI gasoline intervention models also shared several variables; for example, 

both modeled a ramp from March 2003 to May 2003.

Chart 2 presents the seasonal factors for PPI and CPI 

gasoline resulting from the 2007 seasonal revision. Comparing the two charts shows that the 2007 revision’s seasonal factors (derived with the use of updated procedures) 

resulted in much more similar seasonal factors than those 

produced during the 2006 revision. The 2007 revision’s PPI

and CPI seasonal factors, shown in chart 2, are closer in 

terms of size and volatility than the 2006 revision’s factors, 

presented in chart 1. The discrepancy in seasonal factors 

from October to November, present in the 2006 revision’s 

gasoline factors, is corrected in the 2007 revision. The PPI

and CPI factors both project that gasoline prices will show 

a seasonal decline from October to November, based on 

the 2007 seasonal revision.

Table 1 compares the 2007 monthly percent changes in 

 Chart 2. Seasonal factors for gasoline, 2007 revision, 2005–07

Seasonal

factor

115

110

105

100

95

90

85

80

Seasonal

factor

Jan Mar May Jul Sep Nov Jan Mar May Jul Sep Nov Jan Mar May Jul Sep Nov

2005 2006 2007

PPI gasoline,

unleaded regu

CPI gasoline,

all ty

115

110

105

100

95

90

85

80
Monthly Labor Review • July 2010  17

the PPI and CPI for gasoline, calculated with seasonal factors from the 2006 and 2007 seasonal revisions. The table 

also displays the absolute values of the difference between 

the monthly CPI and PPI percent changes for both the 

2006 and 2007 revisions.

Table 1 shows that the absolute difference between 

the percent changes in the PPI and CPI for gasoline was 

smaller in 9 of the 12 months of 2007 when calculated 

on the basis of seasonal factors from the 2007 revision as 

opposed to seasonal factors from the 2006 revision. The 

average absolute difference for 2007 fell from 7.2 percent, 

on the basis of the 2006 revision, to 4.6 percent, on the 

basis of the 2007 revision. (Recall that the 2007 revision 

used the updated seasonal adjustment procedures, whereas the 2006 revision did not.)

Natural gas. In 2006, on the basis of the existing seasonal 

adjustment procedure, the PPI modeled only one intervention for natural gas, whereas the CPI model included 

eight ramps and one outlier. Chart 3 compares the CPI 

and PPI seasonal factors from the 2006 revision.

Although the PPI and CPI seasonal factors appear somewhat similar in terms of their overall pattern, several substantial differences are present between the two sets of 

factors. Most important, CPI factors project that natural 

gas prices will rise in April, May, and June, whereas PPI 

factors project a decline in those 3 months.

For the January 2007 seasonal revision, the Bureau used 

the updated modeling procedures for both PPI and CPI 

natural gas, resulting in much more similar intervention 

models than those from the 2006 revision. The CPI model 

consisted of three ramps and one outlier, the PPI model 

four level shifts and two outliers. All periods modeled as 

interventions for the CPI were also modeled by the PPI. 

The PPI model, however, included interventions in two 

additional periods.

Chart 4 presents PPI and CPI natural gas factors based 

on the 2007 seasonal revision. The two indexes’ natural 

gas seasonal factors from the 2007 revision were substantially more similar to each other than those from the 2006 

revision (shown in chart 3). Factors from both programs 

projected similar natural gas seasonal pricing patterns for 

most of the year, including the previously problematic period from April through June.

Table 2 compares the 2007 monthly percent changes for the 

PPI and CPI for natural gas, calculated with seasonal factors 

from the 2006 and 2007 seasonal revisions. The table shows 

that the absolute difference between the percent changes in 

the PPI and CPI for natural gas was smaller in 9 of the 12 

months of 2007, calculated on the basis of seasonal factors 

from the 2007 revision as opposed to seasonal factors from 

the 2006 revision. The average absolute difference for 2007 

fell from 2.6 percent, on the basis of the 2006 revision, to 1.9 

percent, on the basis of the 2007 revision. In all 3 months 

in which seasonal factors projected counterdirectional movement in the 2006 revision (April, May, and June), absolute 

Comparison of Producer Price Index (PPI) and 

Consumer Price Index (CPI) percent changes in 

gasoline, 2006 revision and 2007 revision

[1-month percent changes]

Month in 

2007

2006 revision 2007 revision

CPI PPI Absolute 

difference CPI PPI Absolute difference

January........... –3.2 –13.4 10.2 –2.5 –5.7 3.2

February......... .5 6.1 5.6 –.7 7.6 8.3

March.............. 10.6 7.9 2.7 6.7 5.5 1.2

April................. 4.8 8.2 3.4 2.4 5.2 2.8

May.................. 10.8 8.8 2.0 5.7 7.2 1.5

June................. –1.2 –1.9 .7 –.1 –1.4 1.3

July................... –1.9 3.9 5.8 .2 2.4 2.2

August............ –5.1 –14.0 8.9 –2.6 –8.0 5.4

September.... .6 8.1 7.5 2.5 1.8 .7

October.......... 1.3 –3.8 5.1 .9 1.7 .8

November..... 9.5 37.2 27.7 12.1 28.9 16.8

December...... 1.1 –5.2 6.3 2.9 –8.5 11.4

Average

absolute

difference .. ... ... 7.2 ... ... 4.6

Table 1. Comparison of Producer Price Index (PPI) and 

Consumer Price Index (CPI) percent changes in 

natural gas, 2006 revision and 2007 revision

[1-Month percent changes]

 Month in 

2007

2006 revision 2007 revision

CPI PPI Absolute 

difference CPI PPI Absolute difference

January.......... –3.0 –2.4 0.6 –3.0 –1.4 1.6

February........ 5.0 3.1 1.9 3.0 2.9 .1

March............. 3.0 5.0 2.0 2.2 3.7 1.5

April................ –1.0 .6 1.6 –1.0 .4 1.4

May................. –.9 .8 1.7 .5 .7 .2

June................ –.1 2.2 2.3 .8 1.9 1.1

July.................. –1.7 2.9 4.6 –1.8 3.6 5.4

August........... –4.2 –7.5 3.3 –2.4 –6.4 4.0

September... –1.0 1.9 2.9 –.3 –.2 .1

October......... .7 –2.0 2.7 .7 –1.7 2.4

November.... .9 –4.0 4.9 .6 –1.8 2.4

December..... 2.3 –.7 3.0 .5 –2.1 2.6

Average

absol

difference .. ... ... 2.6 ... ... 

Table 2.
PPI and CPI Seasonal Adjustment

18  Monthly Labor Review • July 2010

 Chart 3. Seasonal factors for natural gas, 2006 revision, 2004–06

Seasonal

factor

106

105

104

103

102

101

100

99

98

97

96

Seasonal

factor

Jan Mar May Jul Sep Nov Jan Mar May Jul Sep Nov Jan Mar May Jul Sep Nov

2004 2005 2006

 CPI utility

piped gas

PPI residential natural gas

106

105

104

103

102

101

100

99

98

97

96

 Chart 4. Seasonal factors for natural gas, 2007 revision, 2005–07

Seasonal

factor

Seasonal

factor

Jan Mar May Jul Sep Nov Jan Mar May Jul Sep Nov Jan Mar May Jul Sep Nov

2005 2006 2007

PPI residential natural gas

CPI utility piped gas

106

105

104

103

102

101

100

99

98

97

96

106

105

104

103

102

101

100

99

98

97

96

differences lessened subsequent to the 2007 revision.

Cars. The Bureau publishes a CPI time series for new 

cars and an analogous PPI series for passenger cars. Despite the fact that they both have the word “cars” in their 

title, the two series behave differently.7 The seasonality 

of the CPI for new cars corresponds to a changeover in 

model each September. The corresponding PPI model 
Monthly Labor Review • July 2010 19

DIFFERENCES IN SEASONAL ADJUSTMENT TREATMENTS

for similar data series can be driven by differences either in 

underlying series data or in the procedures used to arrive 

at the seasonal adjustment. The Bureau of Labor Statistics recently updated its PPI and CPI seasonal adjustment 

procedures to ensure that differences in seasonal series are 

a result of the underlying unadjusted data, as opposed to 

differences in seasonal adjustment methods. 

Procedures were updated for the selection of intervention candidates as well as intervention modeling. For 

candidate selection, both programs now implement a 

relative-importance criterion, utilize automatic outlier 

detection, and allow all series that meet the relative-importance threshold to become candidates for seasonal 

adjustment. For intervention modeling, both programs 

now use a standard set of tools, including automatic outlier detection, information criteria, graphical analysis, and 

residual analysis.

To improve PPI and CPI seasonal adjustment further, a 

specific set of series was identified in which PPI and CPI 

seasonal adjusters coordinated their efforts. This collaborative approach to the seasonal adjustment of important 

series, such as volatile energy commodities, allows the Bureau to present the most consistent treatment possible of 

seasonal adjustment. 

 Chart 5. Consumer Price Index (CPI) for new cars and Producer Price Index (PPI) indexes for passenger cars, 

1999–2007

Index1

144

140

136

132

128

124

120

Index1

PPI passenger cars

CPI new cars

1999 2000 2001 2002 2003 2004 2005 2006 2007

144

140

136

132

128

124

120

1 CPI: 1982–84 = 100; PPI: 1982 = 100.

changeover, by contrast, generally occurs in October and 

exhibits a more abrupt transition compared with the CPI 

series, and the PPI series is generally more volatile.8 The 

application of rebates to vehicle prices in each program is 

also different: PPI pricing measures consumer and dealer 

cash rebates, as well as low-interest financing offers, while 

the CPI uses an average of manufacturer and dealer rebates for each model over the previous 30 days. The latter 

approach affects both the magnitude and timing of the 

impact of rebates on CPIs relative to PPIs. Also, the CPI 

ceased directly measuring financing incentives as a part of 

its vehicle series in 1999.

Chart 5 contains the published unadjusted data for the 

preceding PPI and CPI car series. The chart reveals differences in trend, seasonality, and overall volatility between 

the series. A recurrent strong seasonal pattern is apparent 

in both series, but the timing and magnitude of seasonal 

peaks and troughs are different.

Although the Bureau publishes PPI and CPI series for 

cars, the differences between the two price series render 

explicit collaboration on seasonal adjustment intervention modeling difficult. In the end, it was decided to share 

relevant information on events affecting the automobile 

market, but not to attempt full PPI and CPI collaboration 

on seasonal adjustment intervention modeling for cars.
PPI and CPI Seasonal Adjustment

20  Monthly Labor Review • July 2010

1 The first release of a seasonally adjusted price index occurs along 

with the regular monthly release of unadjusted indexes. Indexes in 

the current year are adjusted by means of seasonal factors from the 

corresponding month of the previous year.

2

 See Julius Shiskin, Allan H. Young, and John C. Musgrave, “The 

X-11 Variant of the Census Method II of the Seasonal Adjustment 

Program,” Technical Paper no. 15 (U.S. Department of Commerce, 

Bureau of the Census, revised February 1967).

3 See John Lothian and Marietta Morry, “A Set of Quality Control 

Statistics for the X-11-ARIMA Seasonal Adjustment Method” (Ottawa, 

ON, Statistics Canada, October 1978).

4

 The spectrum, or spectral density, graph measures relative contributions 

of frequencies to overall fluctuations in the series. The x-axis measures 

time, in cycles per quarter. Seasonal effects in quarterly data can be observed at frequencies of 0.25 and 0.5 cycle per quarter. The y-axis, or 

ordinate, is 10 times the logarithm of the spectrum amplitudes for the 

first difference of the series. (For details on the spectrum diagnostics in 

X-12-ARIMA, see David F. Findley, Brian C. Monsell, William R. Bell, 

Mark C. Otto, and Bor-Chung Chen, “New Capabilities and Methods 

of the X-12-ARIMA Seasonal Adjustment Program,” Journal of Business 

and Economic Statistics, April 1998, pp. 127–77, on the Internet at www.

census.gov/ts/papers/jbes98.pdf (visited July 22, 2010).)

5

 For additional information on indirect CPI adjustment, see “Aggregation of Dependently Adjusted Seasonally Adjusted Series” (Bureau of Labor Statistics, no date), on the Internet at www.bls.gov/

cpi/cpisatn2001.pdf (visited July 22, 2010).

6

 The year 2007 was the focus of this analysis because that year was 

the first year the Bureau updated the seasonal adjustment procedures 

described in this article. 

7

 For a detailed discussion of their differences, see Maria Bustinza, 

Daniel Chow, Thaddious Foster, Tod Reese, and David Yochum, “Price 

measures of new vehicles: a comparison,” Monthly Labor Review, July 

2008, pp. 19–32. 

8 Ibid., p. 20.

Exhibit A-1. Seasonal adjustment diagnostics from X-12-ARIMA

X-11 seasonality metrics

Diagnostic Description Seasonality criterion

F(s) One-way analysis-of-variance (ANOVA) test for presence of stable

seasonality F(s) ≥ 7.0

F(m) One-way ANOVA test for presence of moving seasonality F(m) ≤ 3.0

M7 Amount of moving seasonality relative to amount of stable

seasonality M7 < 1.0

Q Weighted average of M1–M11 quality statistics Q < 1.0

Model evaluation and selection diagnostics

 Diagnostic Description Criterion

AIC Akaike information criterion, a measure of goodness of fit Minimize

BIC Bayesian information criterion, a selection method for models with

different numbers of parameters Minimize

Graphic analysis

Description

Original series and seasonally adjusted series for X-12-ARIMA output

Graphic plot of seasonal factors for X-12-ARIMA output

Original and prior adjusted seasonal series, for jobs with intervention analysis seasonal adjustment models

Spectra of the original and of the differenced seasonally adjusted series

Seasonal factors by month

APPENDIX: Diagnostic tools and frequently employed graphs

Notes