TRIMMED MEAN
PCE INFLATION
Jim Dolmas
Research Department
Working Paper 0506

FEDERAL RESERVE BANK OF DALLAS

Trimmed mean PCE inflation
Jim Dolmas †Federal Reserve Bank of Dallas
July 25, 2005

Abstract
Research over the past decade has led to improved measures of core inflation in the Consumer Price Index, or CPI. This
paper discusses the application of some of the insights and techniques of that line of research to the Federal Reserve
Board of Governors’ preferred inflation gauge, the price index for Personal Consumption Expenditures (PCE). The result
is a new measure of core PCE inflation—the trimmed mean PCE—and a somewhat different characterization of theeconomy’s recent inflation experience.
Compared to the story told by the usual “excluding food & energy” measure, the trimmed mean PCE tells us that thelows reached in 2003 weren’t quite so low and that the highs reached in mid‐2004 were really a bit higher. On a 12‐month
basis, the new measure suggests that core PCE inflation is currently about ½ a percentage point higher than what is being
indicated by the “excluding food & energy” inflation rate.

1. Introduction
Speaking of the challenge in interpreting monthly inflation numbers during his tenure on theFederal Reserve Board, former Vice Chairman Alan Blinder has said
“The name of the game then was distinguishing the signal from the noise, which was often
difficult. The key question on my mind was typically: What part of each monthly observation
on inflation is durable and what part is fleeting?” 1
Blinder’s conception of a component of monthly inflation that is “durable” as opposed to“fleeting”, that represents “signal” rather than “noise”, corresponds to what most economists callcore inflation. Core inflation, understood in this way, represents the underlying trend in inflation,
once transitory swings have been smoothed out. Because what is transitory and what is lasting
can only be known with the benefit of hindsight, the true core inflation rate for any given monthcannot be known with certainty until well after the fact. In real time—as the data arrive andpolicy decisions need to be made—the best that economists can do is to estimate the core rate of
inflation.

Research Department, Federal Reserve Bank of Dallas, 2200 North Pearl St., Dallas, TX 75201;
jim.dolmas@dal.frb.org. I would like to thank Mark Wynne and Evan Koenig for numerous
helpful comments at various stages of this research; any remaining errors are my own. The views
expressed in this paper represent those of the author, and do not necessarily reflect the views of
the Federal Reserve Bank of Dallas or the Federal Reserve System.
1 Alan Blinder, “Commentary on ‘Measuring short‐run inflation for central bankers’”, Federal
Reserve Bank of St. Louis Review, May/June 1997.
†1

Research over the past decade—discussed in more detail below—has led to improved measuresof core inflation in the Consumer Price Index, or CPI. To the best of my knowledge, however, theinsights and techniques of that line of research have yet to be applied to the Federal Reserve
Board of Governors’ preferred inflation gauge, the price index for Personal Consumption
Expenditures (PCE). 2 The aim of this paper is to rectify that situation. The result is a newmeasure of core PCE inflation—the trimmed mean PCE—and a somewhat different
characterization of the economy’s recent inflation experience.
The remainder of the paper is organized as follows. In section 2, I try to provide some heuristic
motivation for the use of trimmed means by examining the PCE components which experienced
the largest price changes in a typical month. I argue that simply excluding all food and energy
items—the conventional method for estimating core inflation—fails to exclude some highly
volatile items, while potentially throwing out some useful information. In section 3, I make amore rigorous case for trimming, showing that the distributions of individual component pricechanges that go into the PCE inflation rate are both fat‐tailed and left‐skewed. A long tradition in
statistics recommends the use of robust measures of location, like the trimmed mean, for
distributions having fat tails, while skewness suggests that asymmetric trimming may be
optimal. In section 4 I formulate the optimal trimming problem, which entails minimizing ameasure of distance between the trimmed mean inflation rate and a proxy for “true” core
inflation. Since there is some uncertainty about the correct core proxy, I consider several. I also
examine the gains in accuracy that arise from looking at longer price‐change horizons. With the
optimally trimmed mean PCE inflation rate in hand, I proceed, in section 5, to highlight how the
new measure describes the US’s recent inflation experience and how that description differs fromthe one given by the usual “excluding food and energy” core measure. Section 6 offers someconclusions and caveats.

2. Motivation: heuristics
Measures of inflation that exclude food and energy prices are probably the most well‐known
estimators of core inflation. In fact, the “excluding food & energy” measures (XFE for short) areoften spoken of as if they were synonymous with core inflation. Properly speaking, though, they
represent just one of many potential core measures. To be sure, because of the high short‐run
volatility of some food and energy prices, there is some rationale for excluding those prices from

Since February, 2000, the Federal Reserve Board’s semi‐annual monetary policy reports toCongress have described the Board’s outlook for inflation in terms of the PCE. Prior to that, theinflation outlook was presented in terms of the CPI. In explaining their preference for the PCE,
the Board stated: “The chain‐type price index for PCE draws extensively on data from theconsumer price index but, while not entirely free of measurement problems, has several
advantages relative to the CPI. The PCE chain‐type index is constructed from a formula thatreflects the changing composition of spending and thereby avoids some of the upward bias
associated with the fixed‐weight nature of the CPI. In addition, the weights are based on a morecomprehensive measure of expenditures. Finally, historical data used in the PCE price index canbe revised to account for newly available information and for improvements in measurement
techniques, including those that affect source data from the CPI; the result is a more consistentseries over time.” (Federal Reserve Board of Governors, Monetary Policy Report to the Congress,
February 17, 2000)

2

2

a measure of core inflation, but, as research over the past decade has made clear, much better
estimates can be made by taking a more rigorous approach to the problem of which prices toinclude and which to exclude. 3
To see some evidence for the claim made in the previous sentence, consider the following datafrom March, 2005. Of the more than 200 expenditure categories that go into the PCE, Table 1
shows the 10 categories with the biggest increases in price from February 2005 to March 2005. 4
Note that the price changes shown in this table are not annualized—these are one‐month
percentage changes. By way of comparison, the change in the overall PCE price index fromFebruary to March was +0.46%.
Table 1. 10 biggest price increases in March, 2005
Component:
Gasoline & other motor fuelPurchased fuel oil
Airline service
Hotels & motels
Medical services: Labs
Farm fuelPurchased liquid petroleum gas
Miscellaneous personal services
Watch, clock & jewelry repair
Laundry & garment repair

Change from prior month:
8.0%
5.8%
4.2%
4.2%
3.2%
2.5%
2.5%
2.4%
2.4%
2.4%

Table 2 gives the corresponding list of the 10 components which experienced the largest pricedecreases in March, 2005:
Table 2. 10 biggest price decreases in March, 2005
Component:
Eggs
Fresh fruit
Women’s luggage
Men’s luggage
Intrastate toll calls
Photographic equipment
Toys, dolls & games
Household operation: Natural gas
Durable house furnishings: textiles
Lighting supplies

Change from prior month:
–4.4%
–2.6%
–1.8%
–1.8%
–1.8%
–1.8%
–1.7%
–1.7%
–1.5%
–1.5%

While it’s true that food and energy items show up a number of times on these two lists, there aremany other items as well. Moreover, not all food and energy items had price changes as large as
That more‐rigorous approach—discussed in greater detail below—was pioneered by Michael
Bryan and Stephen Cecchetti (1994). Wynne (1999) is a good survey of these methods.
4 All data used in this article are from the Bureau of Economic Analysis, via Haver Analytics. See
section 3 for more detail.
3

3

these. Some food components in particular—such as food consumed away from home—arenotoriously stable. For example, the price index for “other purchased meals”—which comprises
meals purchased at restaurants and bars—rose by just +0.15% in March. As we’ll see below, thatsort of small price volatility is typical for food purchased and eaten away from home—making its
exclusion from a measure of core inflation questionable, to say the least.
The points to take away from these tables and discussion are two‐fold. First, in any given month,excluding only food and energy items still leaves many very volatile items in the price index.
Second, excluding all food and energy items may throw out some useful information.

3. Motivation: statistics
How, then, does one approach the question of which items to exclude or include more
rigorously? In a study focusing on the CPI and the PPI, Michael Bryan, Stephen Cecchetti andRodney Wiggins make a statistical case for the use of trimmed means as a method for estimatingcore inflation. 5 That case relies on the fact that trimmed means can be more efficient estimatorsof a distribution’s location, as compared to the sample mean, when the distribution ischaracterized by heavy tails. 6 Intuitively, samples drawn from a heavy‐tailed distribution willcontain relatively large numbers of extreme values. As a result, the sample means, which aresensitive to outliers, will have a high variance. Trimming some fraction of those extreme
observations produces a less volatile estimator, a fact which Bryan, Cecchetti and Wiggins
illustrate using Monte Carlo techniques.
This section, therefore, is mainly concerned with establishing that the distributions of pricechanges which are aggregated each month to yield the PCE inflation rate are, in fact,
characterized by very heavy tails. We will also see that those distributions exhibit some degree ofleft‐skewness.
A preliminary step is to make precise how individual prices are aggregated to arrive at theoverall PCE price index. We will see that, to a first‐order approximation, the PCE inflation rate inany given month can be described as a weighted average of the rates of change of the various
component prices, with weights that change from month to month. The monthly data consistingof component price changes and associated weights—what I’ll refer to as the underlying
distributions of price changes—will constitute the basic inputs to the optimal trimming problem
described in Section 4.
Distributions of price changes
In any given month, the rate of inflation in a price index like the CPI or the PCE can be thought ofas a weighted average of the rates of change in the prices of all the goods and services that makeup the index. In contrast to the CPI, which uses expenditure weights which remain fixed for two
years, the PCE is a chain‐aggregate whose weights vary from month to month. While PCE
weights are not precisely expenditure shares, they bear a close relationship to expenditure shares.

Bryan, Cecchetti and Wiggins (1997).
There is an old literature in statistics on the use of trimmed means as robust estimators oflocation. See, for example, Crow and Siddiqui (1967).

5
6

4

In particular, to a first approximation, the weight a component receives in this month’s PCE is anaverage of (1) its expenditure share last month and (2) what its expenditure share would be if
consumers bought this month’s quantities at last month’s prices.

{

}

N

Let Pi ,t , Qi ,t i =1 denote a list of prices and real quantities for N component items at date t . The

gross rate of growth in the PCE price index, p , from t to t + 1 obeys the Fisher ideal index

formula:∑Q P ∑Q P∑Q P ∑Q P

p t +1
=
pt

i

i ,t

i ,t +1

i ,t

i ,t

i

i ,t +1 i ,t +1

i

i

.

(1)

i ,t +1 i ,tDefine the 1‐month inflation rates in the overall index and for each of the N component prices in
the usual way by

π t +1 =p t +1−1pt

and

π i ,t +1 =Pi ,t +1
− 1.
Pi ,t

Then, (1) implies the approximation

π t +1 ≅ ∑ w i ,t +1π i ,t +1
i

where

w i ,t +1 =1 Qi ,t Pi ,t
1 Qi ,t +1 Pi ,t+.
2 ∑i Qi ,t Pi ,t 2 ∑i Qi ,t +1 Pi ,t(2)

That is to say, the PCE inflation rate in any given month is a weighted average of the rates ofchange of the constituent components, with weights, given by (2), that vary over time. I willoften loosely refer to an item’s weight as its “share” in the PCE, though one should bear in mind
that this is only half true (literally). As (2) shows, a component’s weight at date t + 1 is anaverage of its expenditure share at t and what it’s share would be, hypothetically, if consumers
bought t + 1 ’s quantities at t ’s prices.
I refer to a list of component price changes and associated weights,{(w ,π ) : i = 1,2,...N}, as
i ,t

i ,t

a distribution of price changes, and, henceforward, I take those distributions to be the primitive
objects of study.

5

Before proceeding to characterize the kurtosis and skewness of the underlying distributions ofprice changes, a word on the data is in order.
The data
All the data used in this study are publicly available. The source for the detailed underlying
price and quantity data—as well as the headline and XFE inflation data used below—is the
Bureau of Economic Analysis. The data on the detailed components of the PCE index are asreported in Tables 2.4.4U and 2.4.6U in the “Underlying Detail Tables” section of the BEA’s
website. 7 The sample period I consider runs from January 1977 to December 2004. Somejustification of that choice is in order.
The degree of disaggregation of the items in the “underlying detail” data available from BEA
varies depending on the category of commodity and the sample period. Some components havethe nature of individual goods (e.g., “eggs”), while others are themselves aggregates (e.g., “fresh
fruit”). Data of any degree of disaggregation are available from 1959, and more recent data are
available at a higher degree of disaggregation then earlier data. Data at the highest degree of
disaggregation (roughly 220 components) are available only since 1990.
As a way of balancing the trade‐off between having a longer sample and having moredisaggregated data, I chose 1977–2004 as the sample period to use. The year 1977 is notable inthat it is the earliest year with prices and quantities for “computers, peripherals & software” and“video equipment and media”. The 1977–2004 sample consists of price and quantity data on 186
component items. A complete list of the components is given in Appendix A.
Kurtosis
The degree of kurtosis characterizing the distributions is most important for establishing theusefulness of the trimmed mean to measure core. This section will establish that the distributionsof monthly price changes underlying the PCE are, in fact, leptokurtotic.
Focus, for a moment, on the distribution of price changes for a single month. To ease notation,
temporarily drop time subscripts as well. Letting π =∑ w π and σ = ∑ w (π − π ) ,
N

N

i =1

i

i

i =1

2

i

i

the standard measure of the kurtosis of the distribution is the normalized fourth moment∑ w (π − π ) ,
K ≡
N

i =1

1

4

i

i

σ4

which would equal 3 for a standard normal distribution. For the monthly distributions of pricechanges associated with the PCE, the average kurtosis over the period 1977–2004 is 40.56.
1Since K is itself potentially sensitive to extreme values, I consider several “robust” kurtosis
measures as well. Generally, these measures—as well as some of the robust skewness measures
discussed below—involve the quantiles of the monthly distributions. Appendix B gives explicit
7

www.bea.doc.gov/bea/dn/nipaweb/nipa_underlying/Index.asp.

6formulae for the alternative measures. A detailed discussion of the advantages of these robust
measures, along with an economic application, is presented in Kim and White (2004).
Table 3 gives the average values of all the kurtosis measures, along with the values of thestatistics under a standard normal distribution.
Table 3. Measures of kurtosis in PCE price distributions, 1977–2004
Kurtosis measure:
Monthly average
Standard
40.56

Value under normality
3

Moors

2.02

1.23

Hogg

4.96

2.59

Crow‐Siddiqui

9.67

2.91

According to any of the four measures, the monthly distributions of price changes exhibit excess
kurtosis. To put some perspective on these numbers, consider that, for the problem of estimating
the location parameter of a single distribution, Hogg recommended trimming 37.5% from eachtail of the sample whenever his kurtosis measure (the third in Table 3) exceeded 3.2. (SeePrescott, 1978, for a discussion of Hogg’s recommendation and an alternative criterion whichcalls for a 30% trim from each tail when Hogg’s kurtosis exceeds 3.)
Skewness
The presence of skewness is inessential to the statistical case for trimming, which is based on thepresence of excess kurtosis. A finding of skewness would, however, suggest that we might notwish to constrain our trim to be symmetric. This is intuitive if one considers skewness inconjunction with an extreme form of symmetrically trimmed mean—the median. For many
distributions, skewness (as conventionally measured) is associated with a discrepancy between
the distribution’s mean and median. Suppose that the distributions of price changes underlying
the PCE are, on average, left‐skewed. It is then likely that the monthly median PCE inflation rate
will, on average, exceed the monthly mean PCE inflation rate (i.e., the headline PCE inflationrate). If true core PCE inflation is understood as a measure of the trend of the monthly headline
rates, then it is likely that the median will, on average, exceed the true core rate. In other words,
the median will be a biased estimate of core PCE inflation.
The conventional measure of skewness is the standardized third moment:∑ w (π − π )
S ≡
N

1

i =1

3

i

i

σ3

As was the case with the conventional measure of kurtosis, the conventional measure ofskewness is highly sensitive to outliers. Thus, as I did with kurtosis, I here consider several
robust measures of skewness. Formulae for the alternative measures are given in Appendix B,
and the reader is again referred to Kim and White (2004) for a discussion of their merits relative
to the conventional measure of skewness.

7

Consideration of robust measures is important in this case, as the three robust measures indicate
left‐skewness, while the conventional measure indicates a slight right‐skewness, on average, inthe monthly distributions of price changes. Table 4 presents the average monthly value, over theperiod 1977–2004, of the various skewness statistics, as well as their average absolute values andaverages since 1995. All of these measures would be zero for symmetric distributions.
Table 4. Measures of skewness in PCE price distributions, 1977–2004
Skewness measure:
Monthly average Average of absolutevaluesStandard
+0.36
3.22

Average since 1995
+0.47

Bowley

–0.09

0.24

–0.24

Groenveld‐Meeden

–0.06

0.20

–0.14

Pearson

–0.04

0.09

–0.07

The numbers given in the first line of the table mirror a finding of Bryan, Cecchetti and Wiggins(1997) in their examination of the distributions of price changes underlying the CPI: whenskewness is measured in the standard way, the monthly distributions exhibit considerable
absolute skewness, but the skewness changes sign from month to month in such a way that the
average amount is quite small. In fact, when skewness is measured in the standard way, 51% of
the monthly distributions exhibit negative skewness, while 49% exhibit positive skewness. The
average skewness of the distributions is a small positive number.
The story told by the robust measures is slightly different. In particular, average skewness isnegative according to all the robust measures. While the average skewness numbers are (inabsolute magnitude) smaller than the corresponding average absolute values, the differences areless pronounced than is the case with the standard measure. This may be due to the fact that theincidence of positive and negative skewness is less evenly divided with the robust measures ascompared to the standard measure: according to any of the three robust measures, the monthly
distributions are negatively skewed 58% of the time, versus 51% of the time for the standard
measure.
To put these numbers in perspective, note that all three alternatives are bounded between –1 and+1. The Bowley measure, for example, is the difference between the distance from a distribution’s
75% quantile to its median and the distance from the median to the 25% quantile, expressed as afraction of the distribution’s inter‐quartile range. The first entry in the second line of the tablethus has the following interpretation: on average, the 25% quantile of the monthly distributions isfurther from the median than the 75% quantile by an amount equal to 9% of the inter‐quartile
range.
Pearson’s measure is the difference between a distribution’s mean and its median, expressed as afraction of the distribution’s standard deviation, while Groenveld and Meeden’s is the samedifference, expressed as a fraction of the average absolute deviation between price changes in the
distribution and the distribution’s median.
Interestingly, the three robust measures indicate that the price change distributions have become
more negatively skewed in recent years, particularly since the mid‐1990s. The last column oftable 3 gives the average monthly skewness values under all four measures in distributions from8

1995–2004. Figure 1 plots times series of the three robust measures for the 1977–2004 period; togive some sense of the general trends, the figure presents six‐month moving averages of the threemeasures.

Figure 1: Skewness of monthly price change distributions
6‐month moving averages, 1977:07‐2004:12
BowleyGroenveld‐Meeden

Pearson

0.4
0.3
0.2
0.1
0
‐0.1
‐0.2
‐0.3
‐0.4
‐0.5
‐0.6
ʹ77

ʹ80

ʹ83

ʹ86

ʹ89

ʹ92

ʹ95

ʹ98

ʹ01

ʹ04

Source: Authorʹs calculations using Bureau of Economic Analysis data

Because the robust measures show more evidence of systematic skewness, particularly in thelatter part of the sample, I do not impose a symmetry constraint on the optimal trimming
problem. The structure of that problem, and the resulting trim proportions, are described in thenext section.

4. The optimal trim
Calculating the trimmed mean PCE inflation rate for a given month involves looking at the pricechanges for each of the individual components of personal consumption expenditures. The
individual price changes are sorted in ascending order, and a certain fraction of the most extreme
observations in both tails of the distribution are thrown out, or “trimmed”. The trimmed mean
inflation rate is then calculated as a weighted average of the remaining components. 8

As noted in section 3, in the discussion of skewness, the median price change in a distribution
can be thought of as an extreme form of trimmed mean. It corresponds to the limiting case where
nearly all the price changes in the upper and lower halves of the distribution are trimmed,
leaving only the price change of the single component exactly in the middle. Thus, the familiar
weighted median CPI, which is produced by the Federal Reserve Bank of Cleveland, is also an
extreme type of trimmed mean inflation rate.
89More precisely, let{(w ,π ) : i = 1,2,...N} denote the distribution of component price changes
i ,t

i ,t

N components are ordered such thatIˆFor α ∈ [0 ,1] , let it (α ) = min I : ∑i =1 w i ,t ≥ α , and consider afrom a given month, and assume that theπ 1,t ≤ π 2 ,t ≤ L ≤ π N ,t .

{}trimming that drops 100α % of the weight from the left tail of each month’s distribution and100β % of the weight from the right tail of each month’s distribution. This (α , β ) ‐trimmed
mean would then be defined by

π t(α ,β ) =11−α − β∑)it (1− β ))i = it (α )w i ,t π i ,tfor each date t . The optimal trimming problem is to find the best choice of α and β .
Proxying “true” core inflation
What percentages of components should be trimmed from each tail of the monthly price‐change
distributions? The amount of data trimmed from each tail of the distribution can be chosen
optimally, to minimize the average monthly discrepancy between the trimmed mean inflation
rate and a specific proxy for true core inflation. In their CPI/PPI study, Bryan et al. used acentered 36‐month moving average of monthly inflation rates to proxy for true core inflation.
That is, the true core inflation rate in any given month was assumed to be the average of thatmonth’s inflation rate together with the inflation rates of the prior 18 months and those of thesubsequent 18 months. This is a natural choice that treats true core inflation as a smooth
underlying trend in actual inflation. I will follow Bryan et al. in using this type of trend as one
proxy for true core PCE inflation, but I will also consider two alternative core proxies, bothinformed to some degree by the monetary policy‐making process.
The first alternative is a measure of trend PCE inflation that is designed to capture the inflationthat the FOMC appears to have responded to by adjusting the Federal Funds rate. This is whatmight be called an operational definition of core inflation: core inflation is the inflation thatpolicy‐makers have typically responded to. As with any trend measure, this measure isconstructed by throwing out a certain amount of high frequency movement in monthly PCE
inflation. In this particular case, I choose the cut‐off frequency between transitory movements ininflation and long‐run trend so as to maximize the correlation between the resulting trend series
and the Federal Funds rate target set by the FOMC.
To be precise, this proxy for true core inflation is obtained by applying a band‐pass filter tomonthly PCE inflation. The correlation‐maximizing cut‐off described above turns out to be 39
months. Roughly speaking, application of the band‐pass filter discards all movements in PCE
inflation of duration less than 39 months. 9
39 months is in the neighborhood of what is often taken to be the high‐frequency cut‐off
defining business cycles, which many authors, beginning with Baxter and King (1999), have taken
to be movements in aggregate variables having a frequency between 3 and 8 years. Our core
proxy can thus be thought of as identifying movements in core inflation with movements at or
below business cycle frequency. The filter I apply is the default filter suggested by Christiano
and Fitzgerald (2003).

9

10

The second alternative derives from what policy‐makers often describe as their idealized signal
for adjusting policy—movements in future inflation. This proxy measure is a moving average,
like that of Bryan et al., but its value in each month is the average of inflation in that month andthe coming 24 months.
Figure 2 shows all three alternative core proxies, together with actual monthly PCE inflation,from 1979 to the end of 2002. 10

Figure 2: PCE inflation and three proxies for true core PCE
inflation
Percent, annualized

Actual monthly PCE inflation

36‐month MA

Trend fit to Fed Funds rate

Forward‐looking

20

15

10

5

0
ʹ79

ʹ81

ʹ83

ʹ85

ʹ87

‐5

ʹ89

ʹ91

ʹ93

ʹ95

ʹ97

ʹ99

ʹ01

Source: Bureau of Economic Analysis and authorʹs calculations

The optimal trimming problem
Given a proxy for true core inflation, the optimal trimming problem chooses the trim points(α , β ) to minimize the root mean square distance between the trimmed mean and the core

proxy. Letting {π t }t =1 denote the core proxy series, we solve:Tmin (α ,β ) T −1 ∑t =1 (π t(α ,β ) − π t ) .
T

2

Christiano and Fitzgerald recommend that users of their band‐pass filter drop two years’ worthof observations from the beginning and end of the sample because of the imprecision with which
the trend is estimated near the sample end‐points. I follow that recommendation here, so that
our original 1977–2004 sample period yields proxies for true core inflation over the period 1979–
2002.

10

11

Following such a procedure, I find that the optimal amount of trimming is somewhat sensitive tothe choice of core proxy, though it is always asymmetric. For all three core proxies, the optimal
trim discards more data from the upper tails of the distributions than from the lower tails.
The first three rows of Table 5 show the optimal trims from the lower and upper tails, in percent,for the three core concepts described above. 11 The total amount trimmed—the sum of the lower
and upper trims—ranges from about 31% to 50%. By comparison, the food and energy items
excluded from the conventional XFE measure of core PCE inflation amount to only about 20% of
total expenditures.
Table 5. Optimal trimming for various core proxies
Core proxy:
Lower tail trim:
36‐month centered moving average
20.6%

Upper tail trim:
25.3%

Trend correlated with Fed Funds rate

14.4%

16.4%

Forward‐looking moving average

19.5%

30.5%

Three equally likely

19.4%

25.4%

Without strong prior beliefs about which core proxy is the right one, how does one choose among
the three suggested trims? One way to deal with the sensitivity of the recommended trimming to
the core proxy used is to calculate the trimming that is optimal in some average sense against all
three alternatives. More precisely, we can choose (α , β ) to minimize the expected root meansquare error, assuming each of the three core proxies is equally likely to be the correct one. That
problem is:∑ 3 T ∑ (π (α β ) − π )1min (α , β )h =1 , 2 , 3{ }where π t

h T
for
t =1

−1

T

t =1

,

t

h 2
t

h = 1,2 ,3 represent the three proxies for true core inflation.

The results of this calculation are given in the last line of Table 5. 12 In the remainder of the article,
unless indicated otherwise, the optimally trimmed mean will employ the trimming given in thelast line of Table 5.
Just based on the four sets of optimal trimming proportions in Table 5, one can make a reasonableguess as to how three of the four trimmed mean inflation rates will rank. For example, the left
trims for the forward‐looking moving average and “Three equally likely” cases are almost
identical, while the right trim is much larger for the forward‐looking moving average case. We

As discussed in Section 3, the sample period runs from the beginning of 1977 to the end of 2004.
The calculation of the band‐pass filtered proxy core measure entails dropping two years’ worthof observations from the beginning and end of the sample, so that the root mean square error
criterion is minimized using series running from 1979 to 2002.
12 The numbers are not that different from what one obtains simply by averaging the optimal
lower and upper trims given in the first three lines of the table. That simple averaging gives
18.2% and 24.1% as the lower and upper trims.
11

12

would thus expect the trimmed mean inflation rate for the forward‐looking moving averagetrimming to be below the trimmed mean inflation rate from the “Three equally likely” trimming.
Conversely, the right trims for the “Three equally likely” and 36‐month moving average arenearly identical, while the left trim is larger for the 36‐month moving average case. We would
thus expect the trimmed mean inflation rate for the 36‐month moving average trimming to
exceed the trimmed mean inflation rate from the “Three equally likely” trimming. Putting thesetwo parts together, we get the following ranking of inflation rates:
Forward‐looking moving average < Three equally likely < 36‐month moving average.
As will be shown in Section 5, the data bear out this intuition (see Figure 8). Based simply on thetrimming proportions, we cannot say where the “Trend fit to Fed Funds rate” case will fall
relative to the other three.
So, which goods get trimmed?
As suggested above, some food components, like food purchased and consumed away from
home, are rarely excluded when one approaches the trimming problem rigorously. This is afeature of the inflation data which Bryan and Cecchetti, op. cit., highlighted in their study of the
CPI, and it is true of the PCE as well. Figure 3 illustrates this. The figure shows the monthly
inflation rate for the PCE component “Other purchased meals” (which comprises meals at
restaurants and bars) together with the upper and lower “trim points” for the optimally trimmed
mean, from 1990 to the end of 2004.

Figure 3: Food away from home is generally not trimmed
Percent, annualized
14
12

Inflation rate for ʺother purchased mealsʺ

10Upper trim points

8
6
4
2
0
‐2
‐4

Lower trim points

‐6
‐8
ʹ90

ʹ91

ʹ92

ʹ93

ʹ94

ʹ95

ʹ96

ʹ97

ʹ98

ʹ99

ʹ00

ʹ01

ʹ02

ʹ03

ʹ04

Source: Bureau of Economic Analysis and authorʹs calculations

13

The “trim points” have the following interpretation. In each month, items whose prices rose by
more than the upper trim points in the chart are excluded from the optimally trimmed mean thatmonth, as are items whose prices fell by more (or rose by less) than the lower trim points. There
are only a handful of months, out of this period of 14 years, in which the purchased meals
component was excluded from the optimally trimmed mean.
Food items of this sort are well‐represented among the components least often excluded from theoptimally trimmed mean. Table 6 gives the top 20 “least‐often‐excluded” components, for the
sample period 1977–2004. Food items actually occupy five of the top ten spots, with “Other
purchased meals” coming in first. Out of a sample of 335 months, it’s excluded only 13 times.
The other dominant category in the least‐often‐excluded list is housing, which shows up in
various forms.
Table 6. The 20 least‐often‐excluded components, 1977–2004
Component:
Number of months excluded (out of 335):
Other purchased meals
13
Owner‐occupied stationary homes17
Casino gambling
34
Tenant‐occupied stationary homes
35
Purchased meals: elementary & secondary schools
41
Purchased meals: higher education
41
Food furnished to employees: military
41
Tenant‐occupied mobile homes
41
Food furnished to employees: civilian
42
Club and fraternity housing
51
Auto repair
52
Tenant group room and board
53
Tenant group employee lodging
54
Owner‐occupied mobile homes
58
Military clothing76
Domestic service paid in cash
80
Household operation, not elsewhere classified
86
Social welfare including child care
88
Medical care: other professional services
91
Dry‐cleaning
92
Table 7 gives a corresponding list of the top 20 “most‐often‐excluded” items. Food items figure
prominently here, too, with “Fresh vegetables” topping the list. Fuels, financial services andelectronics items are also prominent on the list.
Table 7. The 20 most‐often‐excluded components, 1977–2004
Component:
Number of months excluded (out of 335):
Fresh vegetables
314
Eggs
312
Computers and peripherals
309
Food produced and eaten on farms
302
Airline services299

14

Brokerage charges & investment counseling
SoftwarePurchased fuel oil
Fresh fruit
Poultry
Video equipment, excluding TVs
Gasoline & other motor fuelAuto insurance net premiums
Farm fuelPurchased liquid petroleum gas & other fuelDurable house furnishings: textiles
TVs
Infants’ clothing
Commercial bank imputed interest
Semi‐durable house furnishings

297
294
293
292
285
284
282
281
281
275
272
272
271
269
267

How well does the trimmed mean perform?
How much better is the trimmed mean measure of PCE inflation in comparison to the usual XFE
measure? Just as Bryan, Cecchetti and their co‐authors found regarding the CPI, one can showthat the optimally trimmed mean performs much better as an estimator of true core PCE inflation
than the usual XFE measure.
In the data running from 1979 through 2002, the gain in accuracy from using the optimallytrimmed mean rather than the XFE measure ranges from 0.72 to 0.85 percentage points, at annual
rate, depending on the choice of core concept. Assuming the three alternative core proxies areequally likely to be correct, the gain in accuracy is about 0.76 annualized percentage points. That
is to say, compared to the usual XFE measure, on average the monthly trimmed mean measure
would be expected to come closer to true monthly core inflation by roughly ¾ of a percentage
point, when the inflation rates are expressed in annual terms.
The results in the last paragraph compare the performance of 1‐month inflation rates, which arequite volatile relative to the slower‐moving core series. This is true for both the optimallytrimmed mean and the XFE measure, though less so for the former. Cecchetti (1997), looking at
the CPI, emphasized the additional noise‐reduction that can be had by examining longer‐horizon
inflation rates. Cecchetti’s point is equally valid with regard to the PCE. Looking at 3‐, 6‐, or 12‐
month inflation rates improves the accuracy of both the trimmed mean and the XFE measures asgauges of core inflation.
Table 8 shows the value of our measure of fit—the root mean square error, or RMSE—forinflation horizons of 1, 3, 6 and 12 months, for both the optimally trimmed mean and the XFE
measure. The RMSEs are calculated for the case where the correct core proxy is uncertain, and15

the three alternative proxies are equally likely. The optimal trim is calculated under the same
assumption, so it matches the last line of Table 5 (19.4% lower tail trim, 25.4% upper tail trim). 13
Table 8. Root mean square errors for various inflation horizons
1‐month3‐monthOptimally trimmed mean
0.99
0.77
Excluding food & energy
1.75
1.15

6‐month0.74
1.00

12‐month
0.79
1.06

For the trimmed mean and the XFE measure, 6‐month changes give the smallest RMSE—i.e., thehighest accuracy in gauging core inflation. While the longer horizons benefit the XFE measure
more than the trimmed mean, the latter is still the more accurate core inflation gauge. For the 3‐
month inflation horizon, the relative gain in accuracy from using the trimmed mean is almost
0.40 percentage points, and for either the 6‐month or 12‐month horizons, the gain in accuracy is
still more than ¼ of a percentage point, a not‐insignificant difference.
The optimally trimmed mean also performs better than the XFE measure in terms of its mean, oraverage, error, as can be seen in Table 9. 14 Table 9 is analogous in structure to Table 8, but eachentry in the table is now an inflation measure’s average error, in percentage points:
Table 9. Average errors for various inflation horizons
1‐month3‐monthOptimally trimmed mean
0.03
0.05
Excluding food & energy
0.21
0.20

6‐month0.08
0.23

12‐month
0.13
0.28

Figure 4 gives a visual sense of how the trimmed mean performs relative to the XFE measure.
The chart shows the annualized 6‐month inflation rates in the two measures, together with theaverage of three alternative proxies to true core inflation. The series are shown for the fullsample period used in the calculations of the optimal trim—i.e., the data run from the beginningof 1979 to the end of 2002.

The 3‐, 6‐, and 12‐month inflation rates for the trimmed mean are had by cumulating the
optimally trimmed series of 1‐month rates to obtain a price index, then taking 3‐, 6‐ and 12‐month
annualized percentage changes of that price index.
14 To see the relevance of this point, suppose that true core inflation is zero in two consecutive
months. Imagine that one measure (call it A) estimates core inflation as being +¼% in each of thetwo months, while a second (B) estimates it at +1% in the first month and –1% in the second
month. Then B would have a higher RMSE than A—on average, B is 1 percentage point awayfrom the truth, versus ¼ of a percentage point for A—but it would have a smaller mean error
than A. B’s mean, or average, error is zero (the +1 and –1 cancel out) compared to A’s mean errorof ¼ of a percentage point. If the trimmed mean and XFE measures followed this pattern—one
better in terms of RMSE, the other better in terms of average error—we might be hard pressed to
say which was the better measure. Fortunately, Tables 8 and 9 show the trimmed mean is betteron both dimensions.
1316Figure 4: Trimmed mean, Ex Food & Energy, and proxied truecore PCE inflation
Trimmed mean, 6-mo. change

Excluding food & energy, 6-mo. change

Average of 3 core proxies

Percent, annualized
12

10

8

6

4

2

0
ʹ79

ʹ80

ʹ81

ʹ82

ʹ83

ʹ84

ʹ85

ʹ86

ʹ87

ʹ88

ʹ89

ʹ90

ʹ91

ʹ92

ʹ93

ʹ94

ʹ95

ʹ96

ʹ97

ʹ98

ʹ99

ʹ00

ʹ01

ʹ02

Source: Bureau of Economic Analysis and authorʹs calculations

One may wonder about the role played here by the asymmetric nature of the optimal trim—after
all, analyses like that of Bryan, Cecchetti and Wiggins focus on trims which are constrained to besymmetric. The costs in RMSE from imposing a symmetry constraint over the sample data usedhere vary, depending on the core concept and the differencing horizon, from negligible (“Trendcorrelated with Fed Funds rate” core proxy, 3 or 6 month horizon) to 0.22 annualized percentage
points (“Forward‐looking moving average” core proxy, 12 month horizon). For the benchmark“Three equally likely” case, the costs in RMSE from imposing symmetry are small, ranging from4 to 7 percentage points, depending on the horizon. The costs in terms of average errors, though,are quite large. The average errors for the best symmetrically trimmed inflation rate in the“Three equally likely” case range from 0.31 annualized percentage points for the 1‐monthinflation horizon to 0.42 percentage points for the 12‐month inflation horizon. They are, in fact,
higher at every horizon than the corresponding numbers for the XFE measure shown in Table 9.

5. Recent U.S. inflation
Figures 5–7 show the recent behavior of the trimmed mean PCE inflation rate, together with themore common “excluding food & energy” (XFE) inflation rate. The figures show the 3‐, 6‐ and12‐month inflation rates according to both measures, from the beginning of 2002 through May
2005. Two things to note in particular in the figures are the lower volatility at each horizon of thetrimmed mean measure, compared to the XFE measure, and the fact that the trimmed mean
inflation rate is generally above the XFE rate over this period. At a 12‐month horizon (Figure 7),
for example, the trimmed mean inflation rate has run about ½ a percentage point higher than the
XFE rate for roughly the past year‐and‐a‐half.

17Some other salient points are as follows.
1.

While both the trimmed mean and XFE inflation rates decline in 2003, in all three charts,
the lows hit by the trimmed mean measure are not nearly as low as those reached by the
XFE measure. For example, the 3‐month trimmed mean inflation never rate falls below
1% in 2003, while the XFE rate is below 1% in 5 out of the 12 months. The minima of the
6‐ and 12‐month trimmed mean rates are nearer 1.5%.

2.

Both the trimmed mean and XFE inflation rates began to climb in early 2004. The highsreached in mid‐2004, however, are both higher and more sustained in the trimmed mean
measure than in the XFE measure.

3.

Inflation decelerated in the second half of 2004, according to either the trimmed mean of
the XFE inflation measure. This shows up as a decline in the 3‐ and 6‐month inflation
rates and a stabilization in the 12‐month rates. The 3‐ and 6‐month trimmed mean rates
bottom out around 1.5%, compared with around 1% for the 3‐ and 6‐month XFE troughs.
Similarly, the 12‐month trimmed mean rate stabilizes at about 2%, or ½ a percentage
point higher than the 12‐month XFE rate.

4.

While the 12‐month inflation rates in both the trimmed mean and XFE measures lookstable, the 3‐ and 6‐month rates show some pick‐up in inflation since late 2004.

Figure 5: Trimmed mean and Ex Food & Energy PCE inflation;
3‐month changes
Percent, annualized
3.0

2.5

2.0

Trimmed mean

1.5

Ex Food & Energy

1.0

0.5

0.0
Jan‐02

Jul‐02Jan‐03

Jul‐03Jan‐04

Jul‐04Jan‐05

Source: Bureau of Economic Analysis and authorʹs calculations

18

Figure 6: Trimmed mean and Ex Food & Energy PCE inflation;
6‐month changes
Percent, annualized
3.0

2.5

2.0

Trimmed mean

1.5

Ex Food & Energy

1.0

0.5

0.0
Jan‐02

Jul‐02Jan‐03

Jul‐03Jan‐04

Jul‐04Jan‐05

Source: Bureau of Economic Analysis and authorʹs calculations

Figure 7: Trimmed mean and Ex Food & Energy PCE inflation;
12‐month changes
Percent
3.0

2.5

2.0

Trimmed mean

1.5

Ex Food & Energy

1.0

0.5

0.0
Jan‐02

Jul‐02Jan‐03

Jul‐03Jan‐04

Jul‐04Jan‐05

Source: Bureau of Economic Analysis and authorʹs calculations

19

The trimmed mean in these figures is the optimal trimmed mean under the assumption that thethree core proxies described in Section 4 are all equally likely to be correct. What if one hasstrong priors about which of the three alternative proxies—the 36‐month centered moving
average, the band‐pass trend fit to the Federal Funds rate or the two‐year forward moving
average—is correct? Figure 8 plots the recent behavior of the 6‐month trimmed mean inflation
rate for each of the four trimmings given in Table 5.

Figure 8: Optimally trimmed PCE inflation under alternative
proxies for true core inflation; 6‐month changes
Percent, annualized
3.0

2.5

2.0
Three equally likely
36‐month MA

1.5

Trend fit to FFR
Forward‐looking

1.0

0.5

0.0
Jan‐02

Jul‐02Jan‐03

Jul‐03Jan‐04

Jul‐04Jan‐05

Source: Bureau of Economic Analysis and authorʹs calculations

While the patterns over time in the four series shown are virtually identical, there is difference inlevels, particularly for the case of the forward‐looking core proxy vis‐à‐vis the other three.
Compared to the benchmark “Three equally likely” case, the optimal trimmed mean inflation rate
for the forward‐looking core is lower, on average for the period shown, by about 0.21 percentage
points. The trimmed mean rate for the 36‐month moving average core concept is relatively
higher, on average, by about 0.09 percentage points, while that for the trend fit to the Fed Funds
rate is higher, on average, by about 0.05 percentage points. 15

6. Conclusions, caveats and directions for further work
This paper proposes a superior measure of core PCE inflation based on trimmed mean
techniques. Trimmed means have previously been applied to the CPI (and comparable priceindices in other countries, such as the UK’s Retail Price Index), but not, thus far, to the PCE, thepreferred inflation gauge of the Federal Reserve Board.

From January 1995 to May 2005, the average discrepancies (relative to the “Three equally
likely” case) are: forward‐looking, –0.19 percentage points; 36‐month moving average, +0.09
percentage points; trend fit to Fed Funds rate, +0.04 percentage points.
1520

As in the work by Bryan, Cecchetti and Wiggins (1997), the statistical case for the application oftrimmed means to the PCE is based on the evidence of extreme excess kurtosis in the underlying
distributions of price changes that are aggregated to give the PCE inflation rate. I also presented
some evidence of systematic skewness in the distributions, with that skewness becoming more
pronounced in more recent years. This pattern is apparent only when we move beyond thestandard measure of skewness to consider more robust (i.e., outlier‐resistant) measures. Thus—
and in contrast to Bryan et al.—I do not constrain the optimal trim to be symmetric.
The optimal trimming problem, as defined here, chooses the trimming proportions to minimize ameasure of distance (the RMSE) between the trimmed mean inflation rate and a proxy for truecore inflation. I follow Bryan et al. in using a centered 36‐month moving average of actual PCE
inflation to proxy for true core PCE inflation, but I also consider two alternative proxies. I findthat the optimal trims are somewhat sensitive to the choice of core proxy. To deal with thissensitivity, I calculate the optimal trims for the problem of minimizing the expected RMSE,
treating each of the three core proxies as equally likely. I use the resulting trimmed mean
inflation rate to characterize recent US inflation and find that the story told by the trimmed meandiffers somewhat from that told by the more‐conventional “excluding food & energy” measure ofcore inflation.
There are a few caveats to this analysis, which suggest directions for future work.
First, the trimmed mean technique is but one of several recent approaches to the estimation ofcore inflation. One alternative is to treat core inflation as a type of unobserved common trend orcommon factor driving the rates of price change of all the individual components underlying a
price index. This approach has been explored by Quah and Vahey (1995) and others. Other
alternatives—such as weighting component items according to the persistence or variability of
their price movements—have also been explored. Wynne (1999) and Mankikar and Paisley(2002) are good surveys of these alternative approaches. Comparing the performance of thesealternatives to the trimmed mean is an interesting avenue for further research.
A second caveat pertains specifically to the PCE. Unlike the CPI, the PCE is frequently revised as
more comprehensive data become available or as methodological improvements are introduced.
These revisions give rise to a discrepancy between past inflation as it looks today (given current
vintage data) and as it looked in real time (given the data available at the time). Ideally, thesample used to calculate the optimal trimming proportions would be a real‐time data set, since
our concern is with how to gauge movements in core inflation from the real‐time data we’rehanded each month. Unfortunately, the real‐time data needed for such an exercise has not been
compiled.
Finally, it should be noted that the price series for several components of the PCE are based not
on market transaction price data, but on imputations of the cost to consumers of non‐priced
services. For example, many consumers do not pay explicit fees to banks for various sorts ofbanking services. Rather, consumers pay for the services in the form of lower interest rates
earned on deposits. Imputing prices for such services is an inexact science and it is not clear towhat extent central banks should pay attention to the behavior of these imputed prices when
reckoning the overall pace of inflation. To address concerns about the proper role of imputationsin the measurement of inflation, the BEA has recently begun to publish a “market‐based PCE”

21

price index that excludes most imputations. While one of the largest imputed components,
imputed commercial banking costs, is also one of the most often excluded items from theoptimally trimmed mean (see Table 7), I’ve not considered more generally the extent to which the
optimally trimmed mean either excludes or includes non‐market‐based prices. The constructionof a trimmed mean market‐based PCE would thus seem to be a natural and fruitful direction forfurther work.

Appendix A
The following table lists the 186 underlying components whose prices and quantities are used inconstructing the monthly distributions of price changes that constitute the basic data for thetrimmed mean. To give some sense of the relative importance of the various components, thetable also gives each component’s expenditure share for the month of December 2004, the lastmonth of the sample.
Component name

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22

Durable goods
New domestic autos
New foreign autos
Net purchases of used autos
Trucks, new and net used
Recreational vehicles
Motor vehicle tires & tubes
Motor vehicle accessories & parts
Furniture, mattresses & bedsprings
Major household appliances
Small electric appliances
China, glassware, tableware & utensils
Television receivers
Video equipment & media
Audio equipment
Records, tapes, disksMusical instruments
Computers and peripherals
SoftwareFloor coverings
Clocks, lamps & furnishings
Blinds, rods and other durable furnishings
Writing equipment

Expenditure share in percent,December 2004 16
0.77
0.51
0.69
2.85
0.19
0.30
0.40
0.90
0.37
0.07
0.41
0.20
0.14
0.34
0.22
0.05
0.46
0.14
0.23
0.42
0.07
0.04

The shares given are as a percent of total spending on all items listed in the table, which isslightly less than total PCE, since price data are lacking for three items: net foreign travel, net
foreign remittances and household insurance. An entry of 0.00 means the item’s share rounded
to less than 0.01 percent. Shares may not sum to 100 due to rounding.

16

22

23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68

Tools, hardware and supplies
Outdoor equipment & supplies
Ophthalmic products & orthopedic appliances
Guns
Sporting equipment
Photographic equipment
BicyclesMotorcycles
Pleasure boats
Pleasure aircraft
Jewelry & watches
Books & maps
Non‐durable goods
Cereals
Bakery products
Beef & veal
Pork
Other meat
Poultry
Fish & seafood
Eggs
Fresh milk & cream
Processed dairy products
Fresh fruit
Fresh vegetables
Processed fruit & vegetables
Juices & non‐alocoholic beverages
Coffee, tea & beverage materials
Fats & oils
Sugar & sweets
Other foods
Pet food
Beer & ale, at home
Wine & brandy, at home
Distilled spirits, oat home
Elementary & secondary school lunch
Higher education school lunch
Other purchased meals
Alcohol in purchased meals
Food furnished to employees: civilian
Food furnished to employees: military
Food produced & consumed on farms
Shoes
Clothing for females
Clothing for infants
Sewing goods for females
Luggage for females

0.14
0.03
0.29
0.03
0.36
0.05
0.05
0.17
0.22
0.02
0.69
0.49
0.39
0.66
0.36
0.32
0.25
0.46
0.17
0.08
0.19
0.45
0.27
0.40
0.28
0.72
0.17
0.14
0.47
1.28
0.34
0.66
0.21
0.18
0.08
0.11
4.74
0.58
0.12
0.01
0.01
0.65
1.79
0.14
0.08
0.04

23

69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114

Clothing for males
Sewing goods for males
Luggage for males
Standard clothing issued to military personnel
Gasoline & other motor fuelLubricants
Fuel oilLiquefied petroleum gas & other fuelFarm fuelTobacco products
Soap
Cosmetics & perfumes
Other personal hygiene goods
Semi‐durable house furnishings
Cleaning preparations
Lighting supplies
Paper products
Prescription drugs
Non‐prescription drugs
Medical supplies
Gynecological products
Toys, dolls & games
Sport supplies, including ammunition
Film & photo supplies
Stationery & school supplies
Greeting cards
Magazines & sheet music
Newspapers
Flowers, seeds & potted plants
Services
Owner‐occupied mobile homes
Owner‐occupied stationary homes
Tenant‐occupied mobile homes
Tenant‐occupied stationary homes
Tenant landlord durables
Rental value of farm dwellings
Hotels & motels
Clubs & fraternity housing
Higher education housing
Elementary & secondary school housing
Tenant group room & board
Tenant group employee lodging
Electricity
Gas
Water & sewerage maintenance
Refuse collection
Local & cellular telephone

1.21
0.01
0.02
0.00
2.69
0.04
0.15
0.12
0.00
1.06
0.06
0.19
0.40
0.48
0.44
0.12
0.34
2.62
0.37
0.05
0.04
0.56
0.17
0.04
0.09
0.11
0.24
0.23
0.24
0.48
10.41
0.08
3.01
0.07
0.14
0.54
0.01
0.15
0.00
0.01
0.00
1.48
0.75
0.59
0.16
1.24

24

115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161

Intrastate toll calls
Interstate toll calls
Domestic service, cash
Domestic service, in‐kindMoving & storage
Rug & furniture cleaning
Electrical repair
Re‐upholstery & furniture repair
PostageHousehold operation, n.e.c.
Motor vehicle repair
Motor vehicle rental, leasing & otherBridge, tunnel, ferry & toll roads
Auto insurance
Mass transit systems
Taxicabs
Intercity railway
Intercity buses
Airlines
Other transportation services
Physicians
Dentists
Other professional medical services
Non‐profit hospitals
For‐profit hospitals
Government hospitals
Nursing homes
Health insurance
Motion picture theaters
Legitimate theater, opera, etc.
Spectator sports
Radio & television repair
Clubs & fraternal organizations
Sightseeing
Private flying
Bowling & billiards
Casino gambling
Other commercial participant amusements
Pari‐mutuel net receipts
Other recreation services
Shoe repair
Dry‐cleaning
Laundry & garment repair
Beauty shops, health clubs
Barber shops
Watch, clock & jewelry repair
Miscellaneous personal services

0.12
0.20
0.25
0.02
0.16
0.03
0.04
0.03
0.14
0.22
1.66
0.63
0.09
0.61
0.11
0.05
0.01
0.02
0.36
0.08
3.89
0.93
2.70
4.35
0.89
1.54
1.23
1.36
0.11
0.16
0.17
0.05
0.28
0.07
0.01
0.04
0.77
0.24
0.06
2.04
0.01
0.08
0.07
0.52
0.03
0.02
0.43

25

162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186

Brokerage charges & investment counseling
Bank service charges, trust services & safe deposit
box rental
Commercial bank imputed services
Other financial institutions imputed services
Expense of handling life insurance & pension
plans
Legal services
Funeral & burial expenses
Labor union expenses
Professional association expenses
Employment agency fees
Money orders
Classified ads
Tax return preparation services
Personal business services, n.e.c.
Private higher education
Public higher education
Elementary & secondary schools
Nursery schools
Commercial & vocational schools
Foundations & non‐profit research
Political organizations
Museums & libraries
Foundations to religion & welfare
Social welfare, including child care
Religion

1.07
1.11
1.22
1.24
1.13
1.03
0.21
0.14
0.07
0.03
0.03
0.01
0.11
0.06
0.67
0.81
0.36
0.14
0.46
0.19
0.06
0.12
0.15
1.70
0.69

Appendix B
This appendix provides explicit formulae for the various robust kurtosis and skewness measuresused in Section 3. These measures are discussed in more detail in a note by Kim and White(2004), who also perform Monte Carlo simulations illustrating the sensitivity to outliers of thestandard kurtosis and skewness measures and the robustness of the alternative measures.
Measures of kurtosis
To describe the three robust kurtosis measures with an economy of notation, fix a distribution ofprice changes {(wi , π i ) : i = 1,2 ,K N} and assume π 1 ≤ π 2 ≤ L ≤ π N . For α ∈ [0 ,1] let{}Iiˆ(α ) = min I : ∑i =1 w i ≥ α and q(α ) = π iˆ (α ) . In words, q(α ) is the 100α % quantile of the

distribution.
Then, Moors’ kurtosis measure is defined as26

q(7 / 8 ) − q( 5 / 8) + q( 3 / 8) − q(1 / 8).
q( 3 / 4) − q(1 / 4)

K2 =Hogg’s measure is defined as

U (.95 ) − L(.05 ),U (.50 ) − L(.50 )

K3 =
where

U (β ) =1∑ q(α )

β α ≥βandL( β ) =1∑ q(α ) .

β α ≤βLastly, there is Crow and Siddiqui’s measure:K4 =q(.975 ) − q(.025).
q(.75) − q(.25)

Measures of skewness
The skewness measure due to Bowley, described verbally in section 3, is defined formally by

S2 =q(3 / 4 ) + q(1 / 4) − 2q(1 / 2).
q( 3 / 4) − q(1 / 4)

The measure of Groenveld and Meeden is given byS3 =π − q(1 / 2)
∑i=1 wi π i − q(1 / 2 )Nwhere π denotes the mean of the distribution—i.e., π =∑ w π . Finally, Pearson’s measure
N

i =1

i

i

is given byS4 =π − q(1 / 2 )σwhere σ =∑ w (π − π ) .
N

i =1

2

i

i

27

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