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1
Efficient Inflation Estimclticn
by Michael F. Bryan,
Stephen G. Cecchetti, and
Rodney L. Wiggins II
Working Paper 9707
EFFICIENT JIVFLATION ESTIMATION
by Michael F. Bryan, Stephen G. Cecchetti, and Rodney L. Wiggins I1
Michael F. Bryan is assistant vice president and economist
at the Federal Reserve Bank of Cleveland. Stephen G.
Cecchetti is executive vice president and director of
research at the Federal Reserve Bank of New York and a
research associate at the National Bureau of Econonlic
Research. Rodney L. Wiggins I1 is a member of the
Research Department of the Federal Reserve Bank of
Cleveland. The authors gratefully acknowledge the
comments and assistance of Todd Clark, Ben Craig, and
Scott Roger.
Working papers of the Federal Reserve Bank of Cleveland
are preliminary materials circulated to stimulate discussion
and critical comment. The views stated herein are those of
the authors and are not necessarily those of the Federal
Reserve Bank of Cleveland or of the Board of Governors of
the Federal Reserve System.
Federal Reserve Bank of Cleveland working papers are
distributed for the purpose of promoting discussion of
research in progress. These papers may not have been
subject to the formal editorial review accorded official
Federal Reserve Bank of Cleveland publications.
Working papers are now available electronically through the
Cleveland Fed's home page on the World Wide Web:
http:Ilwww.clev.frb.org.
August 1997
Abstract
This paper investigates the use of trimmed means as high-frequency estimators
.
of inflation. The known characteristics of price change distributions, specifically the
observation that they generally exhibit high levels of kurtosis, imply that simple averages
of price data are unlikely to produce efficient estimates of inflation. Trimmed means
produce superior estimates of 'core inflation,' which we define as a long-run centered
moving average of CPI and PPI inflation. We find that trimming 9% from each tail
of the CPI price-change distribution, or 45% from the tails of the PPI price-change
distribution, yields an efficient estimator of core inflation for these two series, although
lesser trims also produce substantial efficiency gains. Historically, the optimal trimmed
estimators are found to be nearly 23% more efficient (in terms of root-mean-square error)
than the standard mean CPI, and 45% more efficient than the mean PPI. Moreover, the
efficient estimators are robust to sample period and to the definition of the presumed
underlying long-run trend in inflation.
1
Introduction
How should we interpret month-to-month changes in the measured Consumer Price
Index? Over the years, this question has led to the construction of several measures of
what has come to be called 'core' idation. Common measures of core inflation regularly
k
remove certain components from the construction of the CPI. In the U.S., 'volatile' food
and energy price movements, are often ignored, and core inflation is synonymous with the
CPI that excludes food and energy.' ~ u ist it truly the case that food and energy price
changes never contain information about trend inflation? Or, for that matter, is it only
the volatile food and energy components that distort attempts to measure the underlying
inflation trend? Surely not. This leads us to consider how we might develop a systematic,
statistical methodology for reducing the transitory noise in measured inflation indices.
This paper follows our recent work, largely beginning with Bryan and Cecchetti (1994)'
where we investigate the estimation of aggregate consumer price inflation using trimmed
means of the distribution of price changes. These are estimators that are robust to the
distributional anomalies common to price statistics. They are order statistics that are
computed by trimming a percentage from the tails of a histogram, and averaging what
is left. For example, the sample mean trims zero percent, while the median trims fifty
percent, from each tail of the distribution of price changes.
Every student in introductory statistics learns that, when data are drawn from a
normal distribution. the sample mean is the minimum variance estimator of the first
moment. But price changes are not normally distributed. In fact. as we discuss in
Bryan and Cecchetti (1996), the cross-sectional distribution of inflation has very fat tails,
with a sample kurtosis that is often substantially above ten. Underlying leptokurtotic
distributions create inferential difficulties, as they routinely produce skewed samples. In
our earlier papers, we discuss how these problems lead to transitory movements in the
sample mean, causing it to have a high small-sample variance.
Given what we know about the distribution of price changes, what is the most efficient
'Tlic 1097 Econo7nic Report of the President is a prirne example. Chart 2-6 on page 76, arid accompanying text, use thc now comrnonplace designation of core i~iflationas the 'Consumer Price Index
excludirig the volatile foot1 arid energy comporlerits.'
estimator of the first moment of the price change distribution? How can we produce a
reduced-noise estimate of aggregate inflation at high hequencies? Our answer is to trim
the price change distribution, not by removing food and energy prices every time, but by
ignoring soine percentage of the highest and lowest price changes each month.
-.
We study monthly changes in both consumer and producer prices in the U.S. Data
availability dictate that we examine 36 components of the CPI from 1967 to 1996 and 29
components of the PPI over the same period. Throughout, we take as our benchmark
the thirty-six month centered moving average of actual inflation. We evaluate the ability
of candidate estimators to track the movements in the benchmark. Our conclusions are
that the most efficient estimate of inflation a t the consumer level comes from trimming
9% from each tail, while efficient estimation of producer prices trims 45%. By trimming
a cumulative 18% of the consumer price distribution we are able to reduce the rootmean-square-error (RhfSE) of aggregate inflation by nearly one-quarter. For the PPI,
the improvement is even more. dramatic, as the RMSE declines by over 45 percent!
The remainder of the paper is composed of five sections. Section 2 reports descriptive
statistics for the distribution of CPI and P P I price changes. Section 3 discusses the
statistical problems we attempt to overcome. Section 4 follows with by a discussion of
the Monte Carlo results that guide our choice of the optimal trimmed estimator. We
provide various robustness checks in Section 5 . These include examining changes in
sample period, changes in the degree of disaggregation of CPI data, and changes in the
benchmark. Section 6 concludes.
2
Characteristics of Price Change Distributions
By how much would the monthly measure of the consumer price index have to deviate
from its recent trend for us to be relatively certain that the trend has changed? This is
the question that is in most people's minds when the Bureau of Labor Statistics releases
the price statistics each month.* Figure 1 plots the monthly changes in consumer and
producer prices, at an annual rate, together with a three-year centered moving average,
2Cecchetti (forthco~rling)suggests a prclirninary answer to exactly this question.
FIGURE 1
CPi
PPI
Monthly with
Monthly with
36
Month Centered Moving Averoge
36 Month, C e n t e r e d M o v i n g Averoge
both for the period 1967:02 to 1997:04.~
As is evident horn the figure, the monthly changes in both of these price indices
contain substantial high-frequency noise. By this we mean that deviations of the monthly
changes from the trend are quite large and often reversed. In fact, the standard deviation
3We use 36 co~nponentsof the Consumer Price Index for Urbari Co~isumers,seasonally adjusted by the
BLS. These data are all available co~itinuously,monthly, since 1967:Ol. The housing service comporlent
is based on the rerital equivalence measure of owner occupied housing, and so prior to 1982, the series is
essentially thc experimerltal CPI-XI. The producer price is bascd on the PPI for commodities, and uses a
set of between 29 and 31 componerits. All data are seasonally adjusted using the AFUMA X-11 procedures
available with SAS. A detailed Appendix containing dcscriptiona of the sources and coristructiori of the
data sets used is available fro111the authors up011 request.
of the difference between the monthly and the moving average aggregate price change
is 6.92 percentage points for the PPI and 2.50 percentage points for the CPI (both at
annual rates). A look at the actual distributions shows that a 90% confidence interval for
the CPI is from -3.92 to +3.76 percentage points, while for the PPI it is from -10.35 to
+8.97 percentage points. In other words, since 1967, monthly changes in producer prices
\
-
have been either more than 10 percentage points below or 8 percentage points above the
thirty-six month moving average one in every ten months!
The common method of excluding food and energy simply does not help much. In
fact, the standard deviation of the difference between the CPI ex food and energy and the
thirty-six month average CPI is 2.31 percentage points, and the 90% confidence interval
shrinks slightly to [-3.73,+3.76] percentage points. By contrast, for the PPI, excluding
food and energy improves things, as the standard deviation of difference between the PPI
excluding food and energy and the 36 month centered moving average of the actual PPI
drops by about 40% to 4.14, and the 90% confidence interval shrinks by about the same
amount to [-5.94,+4.76].
In an effort to better understand the nature of the transitory fluctuations in high-
frequency inflation measurement, we begin by examining the characteristics of the price
change distributions. It is useful to pause at this stage to introduce some notation. We
define the inflation in an individual component price over an horizon k as
where pit is the index level for component i at time t. From this, we define the mean
inflation in each time period, over horizon k, as
where the rit's are relative importances that are allowed to change each month to reflect
the fact that the actual index is an arithmetic a ~ e r a g e . ~
-
41t is straightforward to show that if thc price level index utilizes fixed weights, call these uji, then the
percentage cha~igcin thc aggregate i~idcxcall bc approximated by the weighted sum of the pcrcerltage
The higher-order central moments are then
Skewness and kurtosis are the scaled third and fourth moments, respectively:
and
Table 1 reports numerous descriptive statistics for the cross-sectional distribution of
monthly price changes at - overlapping horizons of one t o thirty-six .months. Among the
noteworthy characteristics is that the distributions are often skewed. The mean absolute
in monthly CP' changes is 0.20 and in PPI changes
value of the skewaess, the mean of S,',
it is 0.04, suggesting that the distributions are nearly symmetrical on average. there is
little skewness in the distributions on average. But the standard deviation of S,'is 2.35
for the CPI and 2.36 for the PPI, implying that distributions of one-month changes are
often highly skewed. This standard deviation falls off as the.horizon increases, implying
that the distribution of longer-run changes are much less likely t o exhibit skewness5
The price change distributions also have very fat tails. The average kurtosis of
monthly changes, the average value of
K:',is 11.24 for the CPI and 10.35.for the PPI. In
fact, the xeighted kurtosis of monthly price changes is in excess of 20 about ten percent
of the time. See Figure 2.
-
These facts allow us to identify a potentially important source of high frequency noise
in the measurement of inflation. In a given month, there is a high probability of observing
some subset of prices change by a substantial amount - generating the skewness and
kurtosis that we see. But, over time, these extreme changes are balanced out,, reducing
changes in the conlpoiients, where the weights change to reflect changes in relative prices. Defining the
agg~egateprice level Pi = w i p i t , then Tit = ~ l i ( p i t / p t - ~ ) .
'For exaniple the 5th and 95th percentiles of S: for the CPI are [-3.52,4.26]. But the same percentiles
for S:6 arc [-2.39,1.93].
..
Table 1: Summary Statistics for Price Change Distributions
Deviations from 36 Month Moving Average
Consumer Prices, 1967.01 to 1996.04
36e Components
1 k = 1 ( k=3 1k=121k=24)k=36
Standard Deviation
3.36
3.14
6.64
4.06
8.83
.
5
79.80 25.49
11.81
Skewness
0.20
0.16
0.21
0.29
0.26
Average
Std. Dev.
2.35
2.15
1.51
1.38
1.41
Kurtosis
4.52
4.23
11.24
9.56
5.72
Average
7.37
3.89
3.75
4.65
8.60
Median
2.39
2.20
8.36
3.49
9.80
Std. Dev.
Producer Prices, 1967.02 to 1997.04
29-32 Com~onents
1
1
1
1
1
.
Absolute Skewness
Average
0.04
0.14
0.04
0.02
0.01
S t d . D e v . 1 2.36 2 . 1 2 1 1 . 7 4 1 1 . 5 3 1 . 4 6
Kurtosis
7.26
Average
8.80
10.35
5.47
4.03
6.38
4.89
Median
6.23
3.51
2.78
8.47
6.50
Std. Dev. 11.51
6.11
3.43
All data are at annual rates.
FIGURE 2
W e i g h t e d K u r t o s i s o f Con:;umer
Prices
Montnly Chcnges, 1 9 6 7 t o 1995
0'
the observed skewness.
One economic interpretation of these distributional. characteristics is that if price
change is costly, we will not observe the distribution of desired price changes each month.
If the size and timing of price changes are based on two-sided state-dependent rules, as
in Caballero and Engel (1991), or Caplin and Leahy (1991), what we observe will depend
on the rule used by the price-setter and the history of the shocks to desired prices. As
a result, we will rarely see prices that exactly equal the price that would be set in the
absence of any price-adjustment costs. The amount of noise decreases over longer periods,
when each price has changed numerous times. But for high frequencies of one quarter or
one month, the problem can be a'serious one.6
However, one need not necessarily attach oneself to a particular model of price-setting.
behavior in order to accept our conclusions. It is well known that a mixture of random
draws from normal distributions with differing variances will produce a leptokurtic sample. As a statistical matter, then, we can show that the mean price-change statistic is
unlikely to provide.the efficient estimate of inflation,. regardless of the price setting model
that is assumed.
We can think of two possible approaches to handling the problem. One would be to
actually model price-setting explicitly using the theory as it has been worked out. But
this has substantial drawbacks, as it requires that we actually estimate the time-varying
price change rules themselves. alternative!^, we can treat the complication presented
by state-dependent price change rules as a statistical sampling problem. We view the
monthly, skewed distributions as small-sample draws from the longer-horizon (roughly)
symmetrical population distribution. The fact that the population has such high kurtosis
leads us to consider a family of estimators that are robust to the presence of fat tails, a
topic t o which we now turn.
6
~alternative
~
i
iriterpretatiorl is implied by Balke and Wyrine (1996), who show that a multi-sector,
dyriamic general equilibrium niodel with rnoriey and flexible prices can produce similar characteristics in
ari environment of asymmetric supply shocks. A distinguishing feature of this rnodel is that the '~ioise'
in the estimator riccd not significantly diminish at lower frequencies.
3
Robust Estimation
We begin by assuming that we have available a sequence of samples from a symmetric
distribution with an unknown, and possibly changing, mean. At issue is the efficient
estimation of the mean. We consider a set of estimators called trimmed means, that
..
average centered portions of the sample. The method of averaging is to order the sample,
trim the tails of the sample distribution, and average what remains.
To calculate the (weighted) a-trimmed mean, we begin by ordering the sample,
{xl,...,z,),
and the associated weights, {wl, ...,w,).
mulative weight from 1 to i; that is,
Wi G
Next, we define VVi as the cu-
wj. From this we can determine the set
of observations to be averaged for the calculation: the its such that
& < W ii(1 - &).
We call this I,. This allows &i to compute the weighted a-trimmed mean as
There are two obvious special cases. The first is the sample mean, Zo, and the second is
the sample median, .;i.50.7
The efficientestimator of the mean, in the class of trimmed sample means, will depend
on the characteristics of the datz-generating process.8 If! for example, the data are drawn
from a normal distribution, then we know that the sample mean is the most efficient
estimator. That is, the sample inean is the estimator that has the smallest small-sample
variance.
But when the data are drawn from leptokurtic distributions - distributions with
much fatter tails than the normal - the sample mean will no longer be the most efficient
estimator of the population mean, even in the class of trimmed sample means. It is
relatively easy to see why this is so. With a fat-tailed distribution, one is more likely
to obtain a draw from one of the tails of the distribution that is not balanced by an
'See Stuart and Ord (1987) pg. 50-51 arid particularly Huber (1981) for general definitions of limitedinfluerice estimators and their propcrtics.
cxamplc, Yulc aiid Keiidall(19G8) discuss the impact of changing kurtosis on the relative efficiencv of the sample mean ant1 the saniple mcdiari. But wc know of 110 general results concerrlirig the
relativc efficiency of trirnrncd-mean cstirnators.
or
equally extreme observation in the opposite tail. That is to say, as the kurtosis of the
data-generating process increases, samples have a higher probability of being ~ k e w e d . ~
The impact of kurtosis on the efficiency of trimmed-mean estimators is straightforward
to demonstrate. To do so we construct a simple experiment in which we draw a series of
.
samples from distributions with varying kurtosis and compute the efficiency of the entire
class of trimmed-mean estimators, including the mean and the median.
In all of our experiments, the data-generating process is characterized by a two parameter distribution that is a mixture of two normals, one with unit variance, and one
with changing variance. We consider a random variable z, such that
where
Pr(s = 1) = p ,
YI
Y2
-
N ( 0 , l ) , and
N(0,A) .
With probability p draws come from a standard normal and with probability (1 - p)
they come from a N(0, A). The population mean, E(z), is zero. The kurtosis of this
distribution,
%$,varies with p and A:
We examine five cases, all with p = 0.90, and A set such that
K
= (3,10,15,20,30).
In each of our experiments, we construct 10,000 replications of 250 draws each. We
then compute the 2, for a = {0,1, ..., 49,50). This yields 10,000 estimates of all of the
trimmed-mean estimators, which we label Z3, . From these we compute the root-mean'~r-yan and Cecchetti (1096) demonstrate this point in another context. We can show that the
standard deviation of the saniplc skcwrless illcreases with the kurtosis of the-data-generating process.
square error (RMSE) and the mean absolute deviation (MAD). These are
RMSE,
=
4
T(2)'
and
MAD,
=
1
-
C I(.?;)
1
.
j
Figure 3 plots the RMSE, and the MAD, for experiments based on distributions with
varying kurtosis, K(A, p) . To adjust for the fact that the variance of the distribution also
changes with A and p, we have normalized RMSEo and MADo to one for each case. The
results clearly show that the efficient trim - the trim that minimizes either the RMSE OF
the MAD - increases with the kurtosis of the data generating process. As the kurtosis
increases from 3 to 30, the efficient trim goes from 0 to 16%.
We caution that the results from these experiments are illustrative and apply only
to the specific distributions we examine. We know of no general analytic result deriving
the optimal trimmed mean estimator as a function of the moments of the underlying
distribution and the size of the sample.
Efficient Estimation of Inflation: Preliminaries
We have now established one property of price data and a related statistical fact.
First, the cross-sectional distribution of price changes, both in the CPI and the PPI, is
fat-tailed. Second, trimmed-means are the efficient estimator of the mean of a leptokurtic
distribution. We now combine these two insights and ask what is the most efficient
estimator of inflation?
We begin with a preliminary examination of the data using a simple Monte Carlo
experiment based on actual price data. In order to judge efficiency, we need to have a
measure of the population mean we are trying to estimate. Following Cecchetti (forthcoming), we choose the thirty-six month centered moving average of actual inflation.
This is an approximation of the long-term trend in inflation that is likely to be what
FIGURE 3
RMSE of Trimmed Estimators as Kurtosis Changes
Trim
MAD of Trimmed Estimators as Kurtosis Changes
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0
5
10
15
20
25
Trim
30
35 -
40
45
50
FIGURE. 4: Consumer Prices
Efficiency of ~ r i & n e dEstimators, Monte Carlo ~ e s u l t s
Zoo:
h n ~ 3 r .S O U O TE
~r r o r
Meo-
Abso1u:e
Dev;ct;o-
people have in mind when they attempt to construct measures they label core inflation.
To proceed, we take the deviation of monthly component price changes from this
thirty-six month centered moving average of inflation. For the CPI, we use 36 components
of the CPI-U over the period 1967.02 to 1997.04, with its 1985 weights. To simplify the
experiments, we set the relative importances (Tit) equal to the 1985 weights (wi).
and
leave them fixed throughout. For the PPI, we use a reduced set of 27 components
also available over the 1967.02 to 1997.04 sample and their fixed 1982 weights. After
subtracting each price change from the thirty-six month moving average change in the
appropriate index, we have two matrices of relative price changes.
In each experiment, we randomly draw a series of samples by taking one observation
for each of the component time-series - one draw from each column in the relative-pricechange matrix. This is a bootstrap procedure from which we generate 10,000 samples,
each with 36 relative price changes-for CPI data, or 29 relative price changes for P P I
data. We then compute the two measures of efficiency - the root-mean-squared error
(RMSE) and the mean absolute deviation (MAD).
The results are reported in Figures 4 and 5. The weighted means are found to be the
least efficient of all of the estimators. The efficiency of the inflation estimates greatly
FIGURE 5: Producer Prices
~ f f i c ' i e n cof~Tkiinmed Estimators, Monte Carlo Results
Roo:
Met?.
S o ~ i o - eE r r o r
Meor,
A b s o l u t e
D e v i c z i a m
improves with even very small trims from the sample. For example, in the case of the
CPI, trimming as little as 3% from each tail of cross-sectional distribution improves the
efficiency o f t h e estimator by over 15%. The most efficient estimator for monthly CPI
data was the 7% trimmed mean where the efficiency gain is approximately 20%, although
trims in the neighborhood of this estimator perform nearly as well.lo
For the PPI, however, much larger trims of the sample distribution are necessary to
achieve the efficient estimator. The optimal trim, which occurs in the range of 40%, has
an RMSE that is only one-third that of the sample mean!
'"The technique we suggest here is appropriate for cases in which the price-change distributioris are
symnietrical on average. We know of instances where this is not the case. For example, Roger's (1997)
examination of New Zealand price data reveals a persistent, positive skewness in the price chaxlge distribution that produces a bias in the trinimed estimators of the mean. Roger constructs trimmed estimators
centered on the meal1 perccntile, or the percentile of the distribution corresponding to the Inearl of the
distribution. That is, for New Zealarid price data, Roger trims the tai!s of the distribution asymmetrically, centering on the 57th percentile. In this way, the trimmed estimator is ax1 mlbiased estimate of
the CPI trend in New Zealand. Roger's insight implies a procetlure in which the trim arid centering
parameter are chosen jointly to minimize either the RiiISE or MAD criterion, subject to the esti~nator
being unbiased in the sample.
Efficient Inflation ~ s t i m aion:'
t
Historical Data
We now move to a more complete examination of the actual data. Here we will
compare the relative efficiency of trimmed estimators using the historical time series,
.
taking account of the changes in the relative importances [the rit's in equation (2)] over
time. That is to say, we will compute the weighted distributions of inflation each month,
where the weights vary based on changes in relative prices as well as the periodic rebasing
done by the Bureau of Labor Statistics roughly once per decade.
In Section 5.1, we look for the optimal trimmed mean estimator using the entire 1967
to 1997 sample currently available. Are the results of the previous section robust t o
several obvious changes in methods? We examine three cases. In the first, reported in
Section 5.2, we study more disaggregated CPI data over a shorter sample period. In
Section 5.3. we look at the implications of changing the measurement benchmark from
the thirty-six month centered moving average of actual inflation to moving averages of
from twenty-four to sixty months. Finally, in Section 5.4, we study estimator stability
by looking at optimal trims over varying sample periods.
W>conclude this section with
a summary and comparison of the trimmed means with the inflation measures that
arbitrarily exclude food and energy.
5.1
The Baseline Case
In this section we consider the time-series characteristics of the trimmed-mean estimators. We calculate the RMSE and the MAD for each trimmed estimator using monthly
historical component price data. That is, we compute the trimmed-mean estimators
of inflation month-by-month, and compare their deviations from the centered thirty-six
month moving average. The results, reproduced in Figure 6 for the CPI, and Figure 7
for the PPI, are virtually identical to those in the Monte Carlo experiments shown in
Figures 4 and 5."
It is easy to see how much inflation measures are stabilized by trimming. Figure 8
"Throughout this scction, the PPI data set uses a set of c:ornpo~~ents
that varies from 29 to 31 in
number, dependir~gOII data availability
-
FIGURE 6: Coniumer Prices
Efficiency of Trimmed Estimators, Historical Data
Roo:
M e a n
Sadare
Error
Meor,
A3solute
FIGURE 7: Producer Prices
Efficiency of Trimmed Estimators, Historical Data
3ev:c:;o-
FIGURE 8
Monthly CPI Estimators
13.00
annualized percent change
-.-----7%-trimmed
mean
-36 mo. centered.
-------
------ -
-------------------------
-------------------------
moving average
-------------------------
Monthly PPI Estimators
30.00
25-00
20.00
annualized percent change
PPI (finished goods)
L--------------------------
!----
----------
!, li
i;
-
I
-40% trim
1
-36 mo. centered
i
-
-
- ------------------
]
Table 2: Comparison of I d a t i o n Estimators
n
Mean (zo)
ex Food& Energy
Median (zsoj
Optimal Trim
Trim at Opt.
CPI
1967 to 1997
RMSE MAD
2.50
1.76
2.31
1.62
2.04
1.51
1.93
1.31
9%
9%
PPI
1967 to 1997
RMSE MAD
6.91
4.27
4.14
2.55
3.98
2.55
3.80
2.52
40%
45%
AU values are computed from monthly changes as annual rates. Deviations are from the 36month centered moving average.
The optimal trim is the trim that minimizes either RMSE, or MAD,.
plots the mean, the thirty-six month centered moving average, and the efficient trimmed
estimator for monthly CPI and PPI data for the January 1990 to December 1996 period.
Table 2 compares the properties of a number of commonly used estimators for consumer and producer price inflation. Focusing first on the CPI, we note that excluding
food and energy produces little improvement in efficiency. The CPI excluding food and
energy is only slightly more efficient thm the CPI-U itself, reducing the RhlSE from 2.50
to 2.31. But trimming clearly helps. Trimming 9% of the cross-sectional distribution of
consumer prices reduces the RMSE by just under 23 percent.'*
For producer prices, the improvements are even more dramatic. Using the long sample
period, we find that trimming 40% of the distribution from each tail improves the RMSE
by over 45 percent. Excluding food and energy from the PPI reduces the RMSE by less
than 40 percent. l3
1 2 ~ r y d e nand Carlson (1994) also note that this trim produces'the minimum time-series variance of
any trimmed-mean estimator ovcr the 1967 to 1994 period.
13A common tec:ti~liquefor rcdlicing the noise in the high frequency inflation estimates uses timeseries averages. IVe have coildlicted experirnerits that combine trimming with timc-averaging. IVc note
that averaging the conlponent price change data prior t o trimming, or pre-trim averaging, decreases
g
to produce a minimum RMSE estimator of the inflation trentl. For
the amount of t r i ~ n ~ n i nnecessary
example, using thrce-month avcragc price changes of component CPI data, the minimum RMSE of the
inflation trend is found by trimming 6% from the tails of thc pricc change distribution, compared to the
9% trims required of monthly data. Similar results were found for post-trim averages, where we average
the monthlv trimmed means. That is, if we calculate the trirnmed estimators, and conipute a $month
average of that res~ilt,thc nlininlu~nRMSE estimate of the inflation trend is foiind by trimming 6% from
5.2
More Disaggregated Data
The price statistics are collected a t a much more disaggregated level than what we
have used thus far. Does the optimal trim change with the level of aggregation? The
experiments in Section 3 suggest that the answer to this question will depend on what
happens to the kurtosis of the cross-section distribution of price changes as we vary the
level of aggregation.
To examine this issue, we assembled a data set composed of between 142 and 175
components of the CPI-U from 1978 to 1996. The number of series (and the relative
importance of each series) varies each month depending on availability. The weighted
kurtosis of these data is much higher than that for the 36 component dataset examined
in the previous section. For monthly changes, for example, Table 1 reports that inflation
in the 36 components of the CPI-U.has median kurtosis of 9.68. By contrast, the kurtosis
in the more disaggregated data set has a median cf 43.1!
As in Section 5.1, we construct, using historical data, the RMSE and MAD for each
of the trimmed estimators, from a = 0 to 50. These provide a gauge of the efficiency
gains from trimming the outlying tails of the price-change distribution. The results in
Figure 9 confirm that, in the case of consumer prices, the efficient estimation of inflation
requires more trimming when more disaggregated data are used. In this experiment, the
optimal trim is 16%, at which point the RMSE is cut nearly in half. But again, virtually
any trimming helps. For example, trimming 9% from each tail
- t-he optimal
amount
for the 36 component data set - reduces the RJIISE by about 40%.
The practical implications of this exercise are fairly important. We have found that
since the kurtosis of the price-change distributions depends on the level of disaggregation, so does the optimal trim. As a result, implementation of these techniques for the
production of a core inflation index will depend critically on the exact dataset used.
each tail of thc pricc change distribution. Even a t relativelv low frequencies, some amount of tri~n~riing
of the price change distributiori sccnis warranted. For example, using a 6-month compo~ientpricc change
and a Gmonth avcragc of thc trimmed cstirnators, the 1ni11imumRIvISE estimator of thc CPI treiid is
obtained by trimming 5% fro111cach tail of the price change distribution. These alternative srrioothing
techniques address a soniewhat diffcrcnt questio~ifrom the o11o posed in this paper: How much new
i ? lcavc thc i~ivestigationof this important area for
information does a monthly price report c o ~ i t a i ~ We
future research.
FIGURE 9:
Consumer Prices, 142 to 175 Components
Efficiency of Trimmed Estimators; Historical Data
R o o f
5.3
W e o n
Sauore
E--or
M e w - .
At>so.cte
Dcv;c:;on
Changes in the Benchmark
As we noted at the outset of the previous section, in order to assess efficiency, we must
specify a goal: What is it we would ideally like to measure? Our second robustness check
involves deviating from the thirty-siu month centered moving average as the benchmark.
Table 3 reports optimal trims as a function of the length of the moving average
specified for the benchmark, similar to those in Sections 4 and 5.1 for the optimal trim.
Included are the optimal trims using the Ivlonte Carlo nlethods, as well as those for the
historical data. The table also reports an informal confidence interval constructed as the
set of trims with RMSE or MAD within five percent of the minimum. For example, using
the historical data in the case of the 36 components CPI data and the thirty-six month
centered moving average benchmark, the minimum RMSE of 1.93 occurs at a trim'of 9%
(see Table 2). The fourth line in the first bottom panel of Table 3 reports that all of the
trims between 5% and 48% have an RMSE below 1.93*1.05=2.03.14
Several patterns emerge from these results. First, the 'point estimate' of the optimal
trim does not vary as we change the benchmark. But the approximate confidence intervals
14Note that there is 110 reason for the approximate confiderlce intervals to be either syrnrnetrical or
continuous. T h e ones reported in Table 3 all happerl to be contirinous.
Table 3: Optimal Trim for Changes in the Benchmark
Monte Carlo Results
CPI
-
PPI
MA
RMSE
MAD
RMSE
MAD
24
0.07
(0.03,0.35)
0.07
(0.03,0.44)
0.06
(0.03,0.42)
0.06
(0.03,0.41)
0.07
(0.03,O. 17)
0.07
(0.03,O. 17)
0.07
(0.03,0.17)
0.07
(0.03,0.17)
0.43
(0.3 1,0.50)
0.41
(0.31,0.50)
0.43
(0.31,0.50)
0.42
(0.30,0.50)
0.45
(0.33,0.50)
0.43
(0.33,0.50)
0.46
(0.34,0.50)
0.45
(0.33,0.50)
36
48
60
Historical Data
n
CPI
36 Components
1967 to 1997
RMSE
MAD
0.09
(0.05,0.25)
0.09
(0.05,0.48)
0.09
(0.05,0.50)
0.09
(0.05,0.50)
0.09
(0.05,0.17)
0.09
(0.05,0.19)
0.09
(0.05,0.21)
0.09
(0.05,0.23)
Numbers in parentheses are trims with
use 10,M)O replications.
PPI
29 to 31 Components
1967 to 1997
RMSE
MAD
0.40
(0.25,0.49)
0.40
(0.25,0.50)
0.43
(0.25,0.50)
0.43
(0.25,0.50)
0.45
(0.30,0.50)
0.45
(0.31,0.50)
0.45
(0.29,0.50)
0.49
(0.27,0.50)
CPI
142 to 175 Components
1978 to 1996
RMSE
MAD
0.14
(0.08,0.23).
0.16
(0.10,0.24)
0.17
(0.1 1,0.25)
0.18
(0.12,0.26)
0.16
(0.09,0.24)
0.17
(0.11,0.26)
0.17
(0.12,0.25)
0.18
(0.12,0.28)
RMSE or MAD within 5% of the value at the minimum. Monte Carlo experiments
FIGURE 10: Consumer
Prices, 36 Components
Efficiency of Trimmed Estimators, Changing Sample
4.L1SE.
-57-
o'
Mi-;-u-
MAD.
-57-
o f
rvli-:mum
have a tendency to grow as the degree of the moving average increases. Second, for the
PPI, there is little difference between the 'optimal trim' and the median. In all cases but
one, the R%ISE and MAD of the median are well within the 5% standard. Finally, for
CPI at both levels of aggregation there is a large benefit to trimming a small amount.
5.4
Variations in the Sample Period
Next, we examine the sensitivity of the results to the sample period.. This is analogous
to asking whether the underlying distributional characteristics of the data are stable. To
do this, we perform a series of Monte Carlo experiments comparable to those in Section 4,
but instead of using the full sample from which to draw, we use rolling ten year samples.
For example, in the case of the CPI we compute the optimal trim based on data from
1967 to 1976, then from 1968 to 1977, moving forward twelve months at a time.
Figures 10 and 11report the results of these experiments. Each figure has a horizontal
line at the optimal trim calculated using the full sample, together with a second line
plotting the optimal trim based on each of the ten year samples. The horizontal axis
shows the final date of the sample. To give some sense of precision, the X's in the figures
represent the approximate confidence intervals constructed as all of the trims such that
FIGURE 11: Producer Prices
Efficiency of Trirnmeh Estimators, Changing Sarriple
the criterion, RMSE or MAD, is within 5 percent of the minimum.
The RMSE and MAD of the optimal full-sample trim are nearly always within 5
percent of the minimum value for the 10 year sub-samples. In fact, for the CPI, using
the mean absolute deviation (MAD) criteria, the optimal trim is never outside of this
rough confidence bound. For the PPI, there are thirty-six 10 year sub-periods. Using the
RMSE criteria, the optimal full sample trim of 40 percent is within the confidence band
in 33 of the 36 cases.
5.5
Summary and Comparisons
Given that the "CPI excluding food and energy" is the measure of core inflation in
common use, it is useful to compare this measure of core inflation to ours. We do this is
two ways. First, we ask which components we are trimming. And second, we look at a
closer comparison of various candidate measures based on the RMSE criteria used above.
Table 4 examines which components we are trimming. For each month, we counted the
frequency at which some portion of the weight of each component was trimmed using the
optimal trim - 9% for the CPI and 40% for the PPI. We also note which components are
systematically excluded by the 'ex food and energy' measures (highlighted in bold-faced
Table 4: Frequency That a Component is Trimmed: CPI 9% trim
11
1
CPI Component
Fruits and vegetables
Motor fuel
Fuel oil and other household fuel commodities
Used cars, etc.
111fantsand toddlers apparel
Meats, poultry, fish and eggs
LVomens and girls apparel
Public transportation
Other appareI commodities
Other private transportation commodities
Gas and electricity (energy services)
Tobacco and smoking products
Dairy products
Other private transportation services
h1ens and boys apparel
Other utilities and public services
Personal and educational services
Toilet goods and personal care appliances
Medical care services
Other food at home
Footwear
Cereals and bakery products
School books and supplies
Kew vehicles
Housekeeping supplies
Housefurnishings
Entertainment services
Medical care commodities
Shelter
Housekeeping services
Entcrtairiment commodities
Persorial care services
Alcoholic beverages
Apparel services
Allto maintenance and repair
Food away from home
h1can of All Iterns
I
Mean of Food & Energy
Average
Relative
Importance
2.26
3.82
0.80
2.27
0.11
4.61
2.71
1.41
0.58
0.67
3.35
1.63
1.92
3.35
1.90
2.30
1.96
0.93
5.20
3.01
1.02
1.86
0.48
3.64
1.37
4.00
1.88
1.01
25.24
1.80
2.37
0.94
1.73
0.92
1.37
5.58
I
Percent of Sample
period that a
portion of the good
is trimmed
69.61
67.13
59.94
58.84
54.97
54.70
43.09
40.33
37.85
37.85
34.81
33.43
28.73
24.59
23.48
23.20
22.65
20.99
20.72
19.06
19.06
17.96
17.96
17.13
16.5'7
16.30
15.47
14.92
12.98
9.67
7.46
7.18
6.91
5.25
3.87
3.31
26.89
39.93
Table 5: Frequency That a Component is Trimmed: P P I 40% trim
"
PPI Component
Farm p r o d u c t s
Fats and oils
M e a t s , p o u l t r y , and fish
Prepared animal feeds
F u e l s and r e l a t e d p r o d u c t s a n d p o w e r
hlletals and metal products
Hides, skins, leather, and related products
Lumber and wood products
Sugar and confectionery
Electronic computers and computer equipme
Transportation equipment
Chemicals and allied products
P r o c e s s e d f r u i t s and v e g e t a b l e s
Dairy products
C e r e a l and b a k e r y p r o d u c t s
Miscellaneous p r o c e s s e d f o o d s
hIisccllaneous Instruments
B e v e r a g e s and b e v e r a g e m a t e r i a l s
Motor vehicles and equipment
Miscellaneous products
Electrical machinery and equipment
I Construction machinery and equipment
-4gricultural machinery and equipment
Textile products arid apparel
Rubber and plastic products
Pulp, paper, and allied products
Nonnietallic mineral products
hliscellaneous machinery
Special industry machinery and equipment
Furniture arid household durables
Gcrieral purpose machiriery and equipment
Metalworking machiriery and equipment
AIcan of All Items
Xlcari of Foocl & Energy
I
Average
Relative
Importance
7.47
0.42
3.56
1.22
12.16
11.86
0.81
2.40
1.04
0.65
8.88
6.86
0.75
1.72
1.58
1.15
0.55
1.90
7.01
3.47
4.54
0.74
0.58
5.33
2.56
6.82
2.75
1.73
1.19
2.98
2.06
1.24
Percent of Sample
period that a
portion of the good
is trimmed
98.90
97.52
96.97
96.14
96.14 .
92.84
90.08
88.98
87.88
86.75
86.78
86.23
85.67
85.40
83.20
82.64
82.09
81.54
81.54
80.72
78.73
78.51
77.41
77.13
77.13
76.03
74.10
72.45
71.35
70.80
69.42
66.39
83.05
90.08
I
type). The results show that we often trim some of the food and energy prices. Indeed,
for the CPI, food and energy components are trimmed from the efficient estimator nearly
40% of the time - nearly one and one-half times as frequently as the average component.
Still, some food and energy goods, notably food away from home, appear to provide an
.
efficient signal of core inflation as we define it here. In fact, of the 36 CPI components
considered, food away from home was the least likely to be trimmed. Moreover, many
non-food, non-energy goods appear tq provide little information about the economy's
inflation trend. Notable among these are used cars and infant and toddler apparel that
are likely to be trimmed out of the efficient estimator nearly twice as frequently as the
average good (the average component is trimmed out of the 9% trimmed mean in 27%
of the months in the sample).
The components most likely to be included in the calculation of the efficient CPI
estimator include a wide variety of services and the shelter component which, despite its
hugh average relative importance of 25.24, is likely to be on one of the trimmed tails of
the price change distribution only about 13% of the time.
Similarly for the PPI, food and ener,v goods tend to be trimmed from the efficient
estimator a disproportionately large share of the time. But some food components, such
as beverages and beverage materials and miscellaneous processed foods, are trimmed at
the same frequency as the average component. The least frequently trimmed component,
metalworking machinery and equipment, i s still trimmed about two-thirds of the time.
This is a relatively low proportion when one considers that, for any given month, 80% of
the price change distribution is trimmed to produce an efficient estimator for PPI core
inflation.
Finally, in Figure 12 we plot the ratio of the RMSE of various measures to the RMSE
of the CPI-U and PPI themselves over different sample periods. For example, for the
ten-year period ending July 1995, the RMSE for the CPI 'ex food and energy' was 57.8%
than of the CPI-U itself - about the same as that of the median. But the RMSE of the
9% trim was 42.5% of the RMSE of the CPI-U. The main result is that, for the CPI, the
9% trim is always more efficient that the CPI excluding food and energy. But for the
optimally trimmed PPI and the PPI 'ex food and energy' are very close.
FIGURE 12: Comparison,of Various Estimators
Efficiency with Changing Sample
In this paper we challenge the conventional wisdom that core inflation can be measured by simply excluding food and energy horn monthly price data. We show that
price change distributions are highly leptokurtic, or 'fat,-tailed,' and so commonly used
measures, such as the sample-mean, are inefficient estimators of the population mean of
interest. We demonstrate that trimmed-mean estimators significantly improve the efficiency of inflation estimates. Furthermore! we are able to show that as the kurtosis of
the distribution increases, efficiency dictates trimming an increasing percentage of the
sample.
We proceed to apply these insights to inflation data. For consumer prices beginning
in 1967, we find that trimming 9% from each tail of the cross-sectional price-change
distribution produces the minimum root-mean-square error and minimum mean-absolute
deviation estimate of monthly inflation. This estimator provides efficiency improvements
on the order of 23 percent relative to the mean. By contrast, the CPI excluding food
and energy provides virtually no efficiency improvement at all.
More disaggregated data amplify the difficulties, as the kurtosis of the distributions
increases. Moving from a dataset composed of 36 components of the CPI to one with
185 components beginning in 1978, we show that the optimal trim nearly doubles to
16%. Here we find an efficiency gain of nearly 50 percent (although the sample period
is substantially shorter). For producer prices beginning in 1947, where price-change
.
distributions are more leptokurtic, trimming 40% to 50% from each tail produces the
most efficient estimate of monthly aggregate price movements and improves efficiency by
over 40 percent relative to the mean.
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