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Effects of Extreme Values on Price Indexes: The Case of the Air
Travel Price Index
JANICE LENT*
ABSTRACT
This paper examines the effects of extreme price values on the
Fisher and Törnqvist index formulas. Using a simple model, we first
consider the impact of outliers on the unweighted arithmetic,
harmonic, and geometric means of a collection of values. Then, under
the same model, we investigate the effect of a single extremely high
or low price on the price index formulas (weighted means). Further
investigation using Taylor series approximations leads to some
general conclusions regarding the relative robustness of the Fisher
and Törnqvist indexes. These are illustrated with empirical results
based on airfare data from the U.S. Department of Transportation's
Origin and Destination Survey.
KEYWORDS: Price index, extreme value, Taylor series. JEL Categories: C43, C13, E31.
INTRODUCTION
Many economists have come to favor the "superlative" Fisher and
Törnqvist price indexes over the more traditional Laspeyres formula
(see, e.g., Diewert 1976; Aizcorbe and Jackman 1993). The U.S.
Bureau of Labor Statistics recently began publishing a new price
index series targeting the Törnqvist formula. The choice between the
Fisher and Törnqvist formulas may be based on a variety of factors,
including other price index formulas currently in use by the
organization producing the index and the relative sensitivity of the
two formulas to extreme values. In this study, we compare the Fisher
and Törnqvist formulas with respect to the latter
criterion-sensitivity to extreme values.
Extreme-valued price ratios often occur as a result of deep
discounts or "free" promotional goods or services. Such
extreme-valued ratios can be either large or small, depending on
whether the discounted price appears in the numerator or denominator
of the price ratio. Less often, extremely high prices appear with
converse effects. The Laspeyres formula is sometimes criticized as
sensitive to extreme values, because it is based on an arithmetic
mean of the price ratios. We will see, however, that such
sensitivity depends on the direction of the outlying value (high or
low), as well as on the weights used in the selected mean.
In the next section, we consider the effect of an extreme value
on the unweighted arithmetic, harmonic, and geometric means. The
third section contains a discussion of the corresponding effects on
the Fisher and Törnqvist index formulas under differing assumptions
regarding the correlation between the expenditure-share weights and
the prices. This correlation is related to the elasticity of
substitution (i.e., the extent to which consumers shift their
purchases toward lower priced items when relative prices change).
The fourth section presents an empirical example: the case of air
travel index estimates computed using data from the Passenger Origin
and Destination Survey collected by the Bureau of Transportation
Statistics. The extreme-valued price ratios in this application
resulted from a change in data-collection procedures and are in this
sense artificial. They do, however, provide an opportunity to
compare the performances of the different index formulas under the
conditions represented by the application. We summarize our
conclusions in the final section.
EFFECTS OF EXTREME VALUES ON UNWEIGHTED MEANS
The following simple model shows the effects of an extreme value
on three types of unweighted means. Let x1, ...,
xn be a collection of nonnegative values,
where xi = µ for i = 1, ...,
n - 1, while xn = y µ for
some factor y > 0; that is, xn is
an outlier in the collection. We define the unweighted arithmetic,
harmonic, and geometric means, respectively, as follows:
,
,
and
.
For M ∈ {A, H, G}, let
.
Then
, , and
.
We first consider the rate at which the various means approach
µ as n approaches infinity. For fixed y, we
have
as n → ∞. For the harmonic mean also,
as n → ∞, and, similarly,
as n → ∞. Thus, as n becomes large, all three of
the means approach µ at approximately the same rate. Their
behavior in the presence of an outlier differs, however, under
various assumptions about the outlier itself. If we suppose that
n is fixed, we may follow derivations similar to those above
to arrive at the results, which are summarized in table
1.
The results shown in table 1 for
fG(y) may lead us to conclude that
price index formulas based on the geometric mean are, overall, the
most robust formulas available; at the very least, they represent a
sensible choice when both high and low outliers are expected to
occur. By contrast, while A is robust to low outliers, it is
sensitive to high outliers; similarly, H is robust to high
outliers but sensitive to low ones. In most applications, however,
price indexes are not computed as unweighted means. In the next
section, we examine the effect of expenditure-share weights on the
Laspeyres, Paasche, Fisher, and Törnqvist indexes, with special
emphasis on the latter two.
EFFECTS OF EXTREME VALUES ON PRICE INDEXES
Price Index Formulas
We begin by presenting several population index formulas. The
Laspeyres index measuring price change between time periods 1 and 2
is defined as
,
where pjt denotes the price of item
j at time t ∈ {1,2}, qjt
denotes the quantity of item j purchased at time
t,
,
and N denotes the number of items in the target
population. The weight wjt is the
expenditure share for item j in period t; the
price ratios
pj2/pj1
are often called price relatives. Clearly L is the
arithmetic mean of the price relatives with weights representing
first period expenditure shares. The Paasche index is a harmonic
mean of the price ratios, with second period expenditure-share
weights:
.
The Fisher index is simply defined as , while the Törnqvist is a geometric mean of the
price ratios with weights representing the averages of the period 1
and period 2 expenditure shares, shown as
,
where w j,1,2 = (w
j1 + w
j2 ) /2.
Extreme Values and the Elasticity of Substitution
To examine the effects of an outlier on the indexes described
above, suppose we have a collection of n items priced in time
periods 1 and 2. Suppose further that for j = 1, ...,
n we have pj1 =
qj1 = 1 and that for j =
1, ..., n - 1 we also have
pj2 = 1, while
pn2 = y (i.e., we assume
for simplicity that the µ above is 1.) For t ∈ {1,2} ,
let xjt =
pjtqjt, the
expenditure level for item j in period t. We wish to
allow the quantity of an item purchased to vary in response to price
change and an assumed elasticity level. When
pj2 =
pj1, we assume that
qj2 =
qj1. Otherwise, let
where ≤ τ ≤ 1, and τ is assumed constant. Then
.
We define the elasticity τ in this way, because τ
provides a convenient means of examining the effects of extreme
values under conditions of high and low elasticity, defined
relatively. Note that higher values of τ indicate less impact
of price change (represented by the price ratios) on second period
item-level expenditure levels.
For j = 1, ..., n - 1, we have
qj2 = qj1 = 1;
and
qn2. = y-τ
The resulting first and second period expenditure-share weights
are as follows:
;
;
and
.
The "average weights" used in the Törnqvist index are
,
and
.
Note that when τ is small (low or zero elasticity) and
y is large,
wn 1 < wn, 1,
2 , (1)
so the Laspeyres index gives less weight to high outliers than
does the Törnqvist index. Similarly, when τ and y are
both small,
wn 2 < wn, 1,
2 , ,(2)
indicating that the Paasche index gives less weight to low
outliers than the Törnqvist. Under conditions of low elasticity, we
therefore observe the following phenomena: although the Laspeyres
index, based on the arithmetic mean, is sensitive to high outliers,
it assigns them weights that are low relative to the Törnqvist
weights, while the Paasche index, a harmonic mean, assigns lower
weights to low outliers. The weights in the Laspeyres and Paasche
indexes can therefore be expected to compensate, at least partially,
for the sensitivity of the arithmetic and harmonic means to high and
low outliers, respectively.
Under this simple model, the values of the Laspeyres, Paasche,
Fisher, and Törnqvist indexes are as follows:
;
;
;
and
.
Both the Fisher and Törnqvist indexes are known as superlative
indexes, because economic theory suggests that they approximate a
true cost of living index under relatively weak assumptions
regarding economic conditions (Diewert 1987). (In the application
considered in the next section, these indexes should be viewed as
cost of flying indexes rather than as cost of living indexes.) We,
therefore, focus on the relative robustness of F (n,
y, τ) and T (n, y, τ)
under the assumptions τ = 1 and τ = 0. The value
τ = 1 indicates that consumers shift their purchases toward
items (or item categories) whose relative prices have decreased
between periods 1 and 2, while τ close to zero represents the
case of little or no change in buying behavior in response to price
change.
First consider the case τ = 1, where a value of τ
represents the assumption that consumers alter the quantities of the
items they purchase so as to maintain the same level of expenditure
on each item-a situation corresponding to a fairly high level of
elasticity. In this case, we have, for fixed n and large
y,
, (3)
while
. (4)
So, for reasonably large n, T is more robust than
F in the presence of high outliers. For the case of low
outliers, we have
(5)
and
(6)
for fixed n as y approaches 0. Under the simple
model, we may therefore conclude that, with regard to robustness,
conditions of high elasticity favor the Törnqvist index over the
Fisher.
With τ = 0, we have
,
and
.
Note that F (n, y, 0) = L (n,
y, 0) = P (n, y, 0). For fixed n
and large y,
, (7)
while
. (8)
As a rough rule of thumb, the above approximations suggest that
T is likely to outperform F whenever outliers are as
large as n2. The relative robustness of T
and F thus depends on the relative values of y and
n, which may, in turn, depend on the aggregation level being
considered. Equations (4) and (8) also indicate that, for large
values of n, T is much more robust to high outliers
under high elasticity than it is under low elasticity. For low
outliers, however, the elasticity assumption has less impact on
T. With n fixed and y small, we have
, (9)
and
, (10)
revealing that, under conditions of low elasticity, T is
more sensitive to low outliers than F. Equations (5), (6),
(9), and (10) suggest that T is somewhat more robust to low
outliers for τ = 0 than for τ = 1, while F is
much more robust.
The above results lead us to conclude that, under conditions of
low elasticity, the Fisher index may often be more robust to
outliers than the Törnqvist: the Fisher is more robust to low
outliers and, when n is sufficiently large relative to any
price ratios in the dataset, the Fisher is also more robust to high
outliers. Conditions of higher elasticity (τ close to 1)
render both indexes more robust to extremely high values. Under
conditions of high elasticity, the Törnqvist is preferable to the
Fisher, because it is less sensitive to both high and low
outliers.
The numerical examples shown in appendix A illustrate these
conclusions. Tables
A1 and A2
give values of the Fisher and Törnqvist indexes under the single
outlier scenario described above. (Note that these are not random
values produced by a Monte Carlo simulation but simply the values of
the functions F (n, y, τ) and T
(n, y, τ) for the given parameters.) Table A1
gives index values under the assumption that τ = 1 (high
elasticity). Under this assumption, the Törnqvist is clearly more
robust than the Fisher to both high and low outliers.
Table A2 shows values of the indexes under the assumption that
τ = 0. The bold numbers in this table highlight points at
which y becomes large enough, relative to n, to render
the Törnqvist index better than the Fisher for approximating the
mean µ = 1 in the presence of a high outlier. As expected,
the turning points occur as y approaches
n2. The examples in table A2 also illustrate that,
under low elasticity, both indexes are more sensitive to high
outliers and less sensitive to low outliers than they are under high
elasticity.
Taylor Series Results
The single outlier model employed in the previous subsections
does not, of course, account for the data complexity often
encountered in practical applications. Here, we look beyond the
single outlier model to examine Taylor series expansions that shed
further light on the relative robustness of the Fisher and Törnqvist
indexes under the general assumption of low elasticity. Following
and expanding on the development of Lent and Dorfman (2004a), we
assume that the price indexes are computed from a collection of
expenditure share weights and sub-indexes
Ig, which here take the place of the price
ratios
pj2/pj1
in the previous subsection. Each Îg is an
aggregate of the ratios
pj2/pj1
for all items j in a particular item category g. In
practice, the standard formulas are often applied in this two-stp
fashion. The categories into which we divide the items may be
defined according to item characteristics, geographic area of
purchase, or both.
We begin by defining some notation. For t ∈ {1,2}and for
each item category g, let
,
and let
, ,
and
.
Next, with wg defined as the Törnqvist
weights,
,
let
, ,
and
.
We expand each of the superlative indexes about the point at
which all of the sub-indexes Ig equal the
mean µ. The relevant partial derivatives are given in
appendix B. From the general form of the third-order approximation
given by Lent and Dorfman (2004a) for a geometric mean, we have the
following approximation of the Törnqvist index:
. (11)
The second-order approximation for the Fisher index is
(12)
Thus, to the second order, we have
. (13)
Consider the relative values of
wg1 and
wg2 in the presence of high
outliers among the Ig and high correlation
between the Ig and the
wg2 (the case of low
elasticity). Under these conditions, we are likely to have
wg2 >
wg1 for large values of
(Ig - µ)2 and thus
TI - FI > 0.
Similarly, in the presence of low outliers, we are likely to have
wg2 <
wg1 for large values of
(Ig - µ)2, resulting in
negative values of TI -
FI. Thus, the approximation shown in
equation (13) indicates that, under conditions of low elasticity,
the Fisher index may be more robust than the Törnqvist to both high
and low outliers.
AN EMPIRICAL EXAMPLE
For the air travel price index series, the apparent elasticity of
substitution is low-in some cases, even negative. The series,
therefore, exemplify only the behavior of the different indexes
under conditions of low elasticity (τ close to 0). Note that
the elasticity reflected in the data, rather than the actual
elasticity, is the quantity that affects the performance of the
indexes; Dorfman et al. (1999) showed that the elasticity reflected
in sample survey data need not always equal the true population
elasticity.
The air travel price index series shown in figures 1, 2, 3, 4, 5, and 6 are based on data from the Bureau of Transportation Statistics'
quarterly Origin and Destination (O&D) Survey. The sample for
the O&D Survey comprises about 10% of all passenger itineraries
having some U.S. component (i.e., itineraries that include at least
one flight arriving at or departing from a U.S. airport) and
includes about 6 to 7 million itineraries per quarter. Data items
collected include trip route, class of service (e.g., coach, first
class), and transaction fare including taxes. Note that the scales
differ across figures, so comparisons across figures are distorted
in some cases.
When goods and services are sampled for the purpose of estimating
a price index, the sample items generally remain in the sample over
an extended time period (e.g., two years) unless they are taken off
the market by the retailer. The stable sample allows comparison of
prices across time for identical items. Ratios of prices in
different time periods for individual items are the building blocks
of the traditional price index estimators.
In the O&D Survey, however, the sampling is performed
independently for each reference quarter. Since the itineraries
selected in a given quarter may not match those selected for a
previous or subsequent quarter, we developed and tested a two-stage
process for matching categories of itineraries across quarters and
comparing average prices within categories across time. The ratio of
average prices for different time periods is called a unit value
index. These sub-indexes are then aggregated by the Fisher,
Törnqvist, and other index formulas. The index series are based only
on data from sample itineraries flown on domestic carriers and are
chained quarterly.1
Lent and Dorfman (2004b) provide a more detailed description of the
index estimation methodology.
The figures show the Laspeyres, Paasche, Fisher, and Törnqvist
index series for various classes of service and for all classes
combined. Note that, in all of the figures, the Paasche series runs
either slightly below the Laspeyres series or even (for business
class service) above the Laspeyres, indicating low or negative
elasticity of substitution. Lent and Dorfman (2004a) describe a
method of estimating the elasticity of substitution; elasticity
estimates computed by their method, measuring elasticity of
substitution between unit value categories as described above, run
close to 0 for these data. Although air travel passengers readily
substitute one carrier for another in response to fare changes,
little substitution across trip origin/destination pairs occurs.
Since origin/destination pairs far outnumber carriers, this
substitution behavior leads to low overall estimates of elasticity
of substitution between the unit value categories.
In examining figures 1 through 6, it is important to note that
the Class of Service variable in the O&D Survey was redefined
and standardized in 1997-98. (Formerly, air carriers had used a
variety of service classifications in reporting this information, so
the variable values had to be recoded by the Bureau of
Transportation Statistics.) We therefore expect some unusual data
values to affect the index series during this period; indeed, many
of the series display a visible break between the fourth quarter of
1997 and the first quarter of 1998. These breaks may be exacerbated,
because a lower percentage of the O&D Survey observations were
"matched" across time during 1997-98 (see Lent and Dorfman 2004b for
a description of the across-time matching method), resulting in
lower than usual effective sample sizes.
Figures 1 and 2 show the series for all classes of service
combined and for restricted coach class (by far the largest class),
respectively. The series in figure
2 behave in typical fashion: the Laspeyres series runs just
above the others, displaying a slight upward drift, while the
Paasche shows a similar downward drift, and the two superlative
series run between them, closely tracking each other. This type of
behavior results from the large number of observations and because
the 1997-98 break has relatively little impact on these series. Figure
1 is similar to figure 2, except for the noticeably larger
effect of the 1997-98 change, which lifts the Törnqvist series
slightly above the others. Recall that, under conditions of low
elasticity, the Törnqvist index is often more sensitive to outliers
than the Fisher.
Index series for other classes of service (categories comprising
fewer observations) are shown in figures
3, 4, 5,
and 6.
For the unrestricted first and restricted first class indexes
(figures 3 and 4), the Laspeyres series runs very slightly above the
Paasche, indicating low but positive elasticity. For the
unrestricted first class series, the 1997-98 break sends the
Törnqvist above the other series, while the Törnqvist for restricted
first class is "bumped down" and runs well below the others for 1998
and subsequent years. In both cases, the Törnqvist continues to
roughly parallel the Fisher after the break, indicating that unusual
data values generated the level shifts. Note also that the
Törnqvist's upward shift for unrestricted first class is noticeably
less severe than its downward shift for restricted first class,
perhaps due to its greater robustness to high outliers than to low
ones.
The business class index series (figures 5 and 6) display the
relatively rare phenomenon of negative elasticity. The Paasche
series runs above the Laspeyres, indicating that consumers are
shifting their purchases toward higher priced services as
relative prices change. It is important to note that sample survey
data may not always reflect true population elasticity; in this
case, the class-of-service categories are coarsely defined, and many
different types of restrictions may apply to tickets in the
restricted categories. (Restrictions may include, for example, a
requirement of advance ticket purchase or, in the case of roundtrip
itineraries, a Friday or Saturday night stay at the destination.)
Elasticity estimates based on these data reflect substitution
between these categories but not within them (for the
same route and carrier) and may therefore suffer a downward bias.
On the other hand, since business class service is typically paid
for by a third party (i.e., the passenger's employer), very low
elasticity is expected. Some business class passengers may even
choose higher priced tickets assuming that "you get what you pay
for," and such behavior could also explain the negative elasticity
indicated. Under negative elasticity, quantities purchased are
positively correlated with price change, and this correlation may
cause expenditure shares to increase dramatically when prices
increase. The Törnqvist index, whose weights are average expenditure
shares, therefore assigns large weights to some high price ratios.
Apart from the negative elasticity, the movements of the business
class series appear similar to that of the first class series, that
is, the Törnqvist index is shifted up or down during the 1997-98
period, while the other series are less affected by the unusual
values. Table
A3 in appendix A shows unweighted percentiles of the
distributions of the unit value indexes for the first class and
business class categories over the crucial period. The outliers are
clearly sufficient in number and severity to impact the tails of the
sample distributions.
CONCLUSIONS
Practitioners may often consider robustness to outliers an
important criterion in selecting a price index formula, especially
for item categories such as airfares, in which extreme prices may
regularly result from frequent flyer awards and other price
discriminatory discounts. Although price index formulas based on
different types of means inherit the relative robustness of these
means, the weights applied in price index calculation also play a
crucial role. This paper shows that, under conditions of low
elasticity of substitution, the high correlation between the weights
and the price ratios may offset the sensitivity of the Laspeyres and
Paasche indexes, making the Fisher a more attractive option than the
Törnqvist. The choice between index formulas is therefore more
complex than the mere selection of an arithmetic, harmonic, or
geometric mean. It requires information on the elasticity of
substitution reflected in the data as well as an estimate of the
magnitude of outliers (high or low) that can be expected.
ACKNOWLEDGMENT
Comments from Alan Dorfman of the Bureau of Labor Statistics
resulted in significant improvements to this paper.
REFERENCES
Aizcorbe, A. and P. Jackman. 1993. The Commodity
Substitution Effect in CPI Data, 1982-1991: Anatomy of a Price
Change. Monthly Labor Review , December 1993:25-33.
Washington, DC: U.S. Government Printing Office.
Diewert, W.E. 1976. Exact and Superlative Index
Numbers. Journal of Econometrics 4:115-145.
_____. W.E. 1987. Index Numbers. The New Palgrave:
A Dictionary of Economics. Edited by J. Eatwell, M. Milate, and
P. Newman. London, England: MacMillan.
Diewert, W.E., R. Feenstra, and W. Alterman. 1999.
International Trade Price Indexes and Seasonal Commodities.
Washington, DC: U.S. Department of Labor, Bureau of Labor
Statistics.
Dorfman, A., S. Leaver, and J. Lent. 1999. Some
Observations on Price Index Estimators. Proceedings of the
Federal Conference on Survey Methodology, Arlington, VA,
November 1999.
Lent, J. 2003. Chain Drift in Experimental Price Index
Series for Air Travel, Proceedings of the 2003 Joint Statistical
Meetings, Section on Survey Research Methods, CD. Alexandria,
VA: American Statistical Association.
Lent, J. and A. Dorfman. 2004a. Using a Weighted
Average of the Jevons and Laspeyres Indexes to Approximate a
Superlative Index. Working paper.
_____. 2004b. A Transaction Price Index for Air
Travel. Working paper.
APPENDIX B
Partial Derivatives
To derive equations (11) and (12), the function
FI is expanded around the point I =
μ = (μ, ., μ). The general formula
for the third-order Taylor expansion is
.
For a derivation of the third-order expansion of
TI (equation (11)), see Lent and Dorfman
(2004a). The first- and second-order partial derivatives of
FI evaluated at I =
μ (used in the derivation of equation (12)) are as
follows:
END NOTES
1Price index chaining is done
by estimating long-term price changes as products of shorter term
changes (links). Quarterly chaining can lead to "chain drift," as
seen in the Laspeyres and Paasche series in the figures in this
section. For more information on chain drift in the airfare indexes,
see Lent 2003.
ADDRESS FOR CORRESPONDENCE
*J. Lent, Bureau of Transportation Statistics, Research and Innovative Technology Administration, U.S. Department of Transportation, 400 7th Street SW, Washington, DC 20590. Email: janice.lent@dot.gov
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