Specification Issues and Confidence Intervals in Unilateral Price Effects
Analysis
Oral Capps, Jr.
Texas A&M University
Jeffrey Church
University of Calgary
Alan Love
Texas A&M University
November 2, 2001
Draft=18
Abstract
This paper contributes to the economics and econometrics literature on unilateral effects analysis. It introduces the Rotterdam demand system; considers the effect and importance of demand restrictions for minimizing the mean squared error of price simulations; demonstrates that approximate price changes are often misleading indicators of
exact price changes; and uses bootstrap techniques to determine confidence intervals and
standard errors for simulated price changes, allowing determination not only of their economic significance but also of their statistical significance.
Keywords: Unilateral Effects, Merger Simulation, AIDS and Rotterdam Demand Systems
JEL Classification: L13, L4
Please address all correspondence to
Jeffrey Church, Department of Economics, University of Calgary, Calgary, Alberta, Canada,
T2N 1N4 Fax: 1-403-282-5262. email: jrchurch@ucalgary.ca
We would like to thank two anonymous referees for their extensive comments.
1. Introduction
Section 7 of the Clayton Act prohibits mergers whose effect “may be substantially to lessen
competition.” The Department of Justice and Federal Trade Commission Merger Enforcement
Guidelines identify two possible anti-competitive effects. The first is that a merger, by removing a competitor, may facilitate the ability of the remaining firms in the market to coordinate
their behaviour in a manner that reduces competition and facilitates the exercise of (collective)
market power. The second is the potential for the merger to increase the ability of the merged
entity to exercise market power and unilaterally raise prices.1
A firm’s market power is constrained by consumers’ ability to substitute to products supplied by competing firms. The extent of demand-side substitution depends on whether consumers can, and will, switch to other products in response to a price rise (or other manifestation
of market power). The extent of supply-side substitution depends on whether consumers can
find alternative suppliers of the same product in response to a price increase (or other manifestation of market power).2 A firm can unilaterally exercise market power if these possibilities
for substitution are limited, and likely to remain so, for an extended period of time. A merger
may lead to a unilateral increase in market power if the effect of the merger is to eliminate
a significant avenue of substitution. If the products of merging firms are significant competitive constraints for each other, then by eliminating that constraint, the merger may lead to an
increase in market power and a substantial lessening of competition.
Until recently merger analysis in the United States was concerned with coordinated effects.3
The structural analysis typically applied involved, (i) defining an antitrust market, (ii) determining the increase in concentration as a result of the merger, and (iii) assessing barriers to
entry. If barriers to entry are high and the increase in concentration (as measured by the
Hirschman-Herfindahl Index) sufficiently large, the merger was presumed to lead to a substantial lessening of competition. This approach led naturally enough to making the question of
market definition virtually paramount in merger cases.
When products are differentiated, however, this approach is problematic. Product differentiation results in controversies over which differentiated products are “in” the market and
which are “out” of the market. The importance of the determination arises because if the
market is defined too narrowly, then high market shares will overstate market power, since the
market excludes substitutes that impose important competitive constraints. Too broad a market
1The
distinction between unilateral and coordinated anti-competitive effects in antitrust corresponds to the
distinction between the effect of a merger on the Nash equilibrium to the stage game (unilateral) and the subgame
perfect Nash equilibrium to the dynamic game where the stage game is repeated (coordinated).
2The Merger Enforcement Guidelines in the United States state that “A price increase can be made unprofitable
by consumers either switching to other products or switching to the same product produced by firms at other
locations.” This corresponds to our distinction between demand and supply- side substitution. The extent of
supply side substitution opportunities may depend on the ease of entry and product repositioning.
3Baker (1997b, p. 21) observes that “Unilateral theories are now by far the most common in the internal
analyses of the antitrust agencies, particularly among agency economists.”
1
definition results in low market shares that understate market power, since the market includes
products that are not close substitutes and do not exert a significant competitive constraint.
More fundamentally, the general approach of defining products to be “in” or “out” implicitly assumes that those “in” are equally effective competitors and those “out” provide no
competitive constraint. Market shares do not account for competition at the margin between
products in and out of the market and they do not provide any information regarding the nature
of the competitive constraint and extent of competition between specific products when competition is localized. Thus the use of market shares which fail to distinguish between products
that substitute in different degrees can lead to very misleading conclusions.4
Recent theoretical developments in industrial organization, the dramatic decline in the cost
of computing, and access to retail scanner data have resulted in the development of unilateral
effects analysis which forgoes market definition and attempts to predict the price effect of
mergers between producers of differentiated products through price or merger simulations.5
The choice of a demand system and its estimation is the “front-end” of a merger simulation
(Werden 1997a). Choices for demand systems depend on the richness of available data, but
the logit and the Almost Ideal Demand System (AIDS) have been used for litigation purposes
and their use is well- documented in the literature.6 The “back-end” of a merger simulation
assumes that pre-merger prices and market shares correspond to a (static) Nash equilibrium in
prices and constant marginal costs. Given an estimated demand system the post- merger prices
can be forecast by computing the Nash equilibrium in prices post merger.
This paper makes a number of contributions and extensions to the economics and econometrics of unilateral effects analysis. We introduce the Rotterdam demand system and compare
its application with the AIDS demand system which has been frequently used. To the typical
“back-end” price simulations we add a bootstrap procedure. The bootstrap procedure allows
us to calculate confidence intervals and standard errors for the predicted price changes and
assess their statistical significance. The bootstrap procedure also allows us to assess the appropriateness of imposing demand restrictions in estimation by considering the trade-off between
bias and efficiency. The analysis indicates that approximate price changes are often misleading indicators of the extent of exact price changes. Given the low cost of computing power,
this discrepancy means that continued reliance on simple formulas that approximate the price
changes of a merger is not warranted.
In the next section we introduce the two demand systems and discuss their properties. In
section 3 we explain the mechanics behind the “back-end”, or price simulations, being careful
to distinguish between exact and approximate price increases. In Section 4 unilateral effects
analysis and our extensions are illustrated by considering hypothetical mergers between brands
4See
Werden and Froeb (1996) for a detailed discussion of the limitations of structural analysis when applied
to differentiated products.
5Early contributions are Hausman, Leonard, and Zona (1994) and Werden and Froeb (1994, 1996). Recent
surveys in the antitrust literature include Hausman and Leonard (1997) and Werden (1997b).
6See Hausman and Leonard (1997) and Werden (1997b) for case examples.
2
of heat and serve spaghetti sauces. Concluding comments are made in Section 5.
2. Demand Systems and Elasticities
The choice of demand systems can potentially have a material effect on the price simulations.
Two characteristics that recommend a demand system for use in merger simulations are that
(i) it be second-order flexible, the defining characteristic of which is that for each brand the
cross-price elasticities of the demand system are estimated from the data7 and (ii) elasticities
are not constant, but vary as prices change, typically rising as price increases.8 We follow
Hausman, Leonard, and Zona and others who have used the AIDS specification. In addition
we also introduce into merger simulation the use of the Rotterdam specification.
2.1
LA/AIDS
=( )
pi qi =X , pi is the price of brand
Let wi denote the expenditure share of brand i, where wi
i, qi the quantity of brand i demanded, and X is total expenditure on the group of brands. The
AIDS specification when there are N brands is
wi
=
i
+
where P is a price index defined as
N
X
j =1
ij ln (pj ) + i ln (X=P )
(1)
ln (P ) = + Pj j ln (pj )+ Pj Pi ij ln (pi )ln (pj ).
1
2
0
The restrictions on the parameters of the AIDS demand system implied by demand theory
N
N
N
P
P
P
are, for the adding-up constraint
i
i
ij
,
, and
; for homogeneity
i=1
i=1
i=1
N
P
ij
; and for symmetry ij ji .9
j=1
=1
=0
=0
=0
=
7See
Hausman and Leonard (1997, p. 327). It is for this reason that the use of the logit model, which by
assumption constrains the pattern of substitution, is problematic. Logit demand assumes that all cross-price
elasticities with respect to the price of a given brand are identical. See, however, Werden and Froeb (1996) for
the reason why using the logit model might be acceptable when time constraints and/or data constraints preclude
using other more suitable demand systems.
8See Crooke et al (1999) for an analysis of the importance of the “curvature” of different specifications of
demand for extrapolating post-merger prices. Here “curvature” refers to how quickly elasticity increases as price
rises. A rapid increase in elasticity will quickly curtail price increases post-merger.
9Adding-up
requires that
N
P
i=1
pi qi(p; X )
=
X
and homogeneity that
qi(p; X ) is the demand function for brand i.
3
qi(p; X )
=
qi(p; X )8 i where
The highly non-linear nature of P means that in practice when an AIDS system is estiP w p :10 When P is used
P
mated Stone’s index (P) is used instead where
i
i
i
instead of P the estimated demand system is known as the “linear approximate AIDS” or
LA/AIDS model. While use of P simplifies estimation of the demand system, it is problematic
from a theoretical perspective since the expenditure share equations that comprise (1) are no
longer a reduced form. Substituting in P for P (1) becomes:
ln ( ) =
wi = i +
X
j
ln ( )
X
ij ln (pj ) + i ln (X ) ; i(
j
wj ln (pj ));
(2)
which is a system of N equations in N unknowns. The reduced form expenditure shares are
given by:
!
!
N
N
P
P
xi 1 + j ln (pj ) ; i
xj ln (pj )
j6 i
j6 i
wi =
N
1 + P ln (p )
=
=
i=1
where xi is
xi = i +
i
(3)
i
N
X
ij ln (pj ) + i ln (X ) :
(4)
j=1
The use of Stone’s index means that there are a number of possible alternative formulas
for the own- and cross-price uncompensated, or Marshallian, elasticities. One alternative is to
use those of the AIDS model, a second alternative commonly used is an approximation that
assumes expenditure shares remain constant, and a third alternative is to use the elasticities of
the LA/AIDS model.11
10See
Deaton and Muelbauer (1980a, p. 316) and Green and Alston (1990). Often the convergence problem
can be traced to 0 , the parameter associated with subsistence consumption. Indeed, the exact AIDS model could
not be estimated using the spaghetti sauce data because of convergence problems with 0.
11The AIDS elasticities are
"AIDS
ij
=
;ij
+ (
ij = wi ) ; (i = wi) j
+
N
X
k=1
!
kj ln (pk )
(5)
where ij is the Kronecker delta equal to 1 when i = j and 0 otherwise. The common approximation is method
(iii) in Green and Alston (1990) and it holds exactly when expenditure shares are constant: d ln P=d ln pi = wi.
The LA/AIDS elasticities when this assumption is made are
"ALA=AIDS
= ; ij
ij
4
+
ij ; i wj
:
wi
(6)
The appropriate formula for use in merger simulations is the one that insures internal consistency between parameter estimation and elasticity estimates. In the exact price simulations
that follow, it is the LA/AIDS specification for the demand system that is used, suggesting a
preference for the LA/AIDS elasticities and not the other two alternatives.12
From (3) and (4) the own-price and cross-price elasticities of the LA/AIDS specification
are:13
"LA=AIDS
ii
= ;1 ;
1+i
N
P
i
i ln (pi )
!
!
N
N
P
P
ii 1 + k ln (pk ) ; i
ki ln (pk )
k6 i
k
6
i
!
!
+
N
N
P
P
xi 1 + k ln (pk ) ; i
xk ln (pk )
and
"LA=AIDS
ji
= 1
=
=
k6=i
k6=i
(7)
!
!!
N
N
P
P
xj i ; xi j + ji 1 +
k ln (pk ) ; j
ki ln (pk )
k6 j !
k6 j !
N
N
P
P
xj 1 +
k ln (pk ) ; j
xk ln (pk )
=
=
;
1+i
N
P
i
= 1
k6= j
i ln (pi )
=
k 6= j
:
(8)
12The issue of the correct elasticity to use when the LA/AIDS model is estimated
has been considered in Green
and Alston (1990), Alston, Foster, and Green (1994), and Buse (1994). As Buse notes the choice depends on
whether it is the AIDS or LA/AIDS demand specification that is thought to be the correct model. If the correct
model is LA/AIDS then the LA/AIDS elasticities should be used. However, the LA/AIDS models is not an
integrable demand system and is typically used not because it represents the outcome of consumers’ choices, but
because it is not subject to the same convergence problems associated with estimating the AIDS model. It is
thus an approximation, so the choice of which elasticity formula to use depends, presumably, on which formula
using the LA/AIDS coefficient estimates yields the most accurate estimate of the AIDS elasticities. The Monte
Carlo results of Alston, Foster and Green (1994) suggest that both the Green and Alston approximation and the
LA/AIDS elasticities provide reasonable estimates of the true AIDS elasticities when used in conjunction with
LA/AIDS estimates. Buse (1994) argues that the common approximation gives the best estimates. He also argues
that the AIDS elasticity formulas are preferable to the common approximates when the LA/AIDS estimates of
the intercepts are adjusted appropriately. The adjustment requires information on subsistence consumption levels;
information not likely to be readily available in most merger cases.
13It can be demonstrated that these are equivalent to the semi-reduced forms found in Buse (1994).
5
2.2
Rotterdam
Another price system which is sufficiently flexible to capture variations in demand behaviour,
allows the imposition of restrictions derived from demand theory (symmetry and homogeneity), and allows for the derivation of price changes due to merger activity is the Rotterdam
model. Although the Rotterdam model is not derived from an underlying utility function or
expenditure function, it satisfies the integrability conditions when symmetry and homogeneity
are imposed. The Rotterdam model also offers the advantage of handling non-stationary data
due to the fact that price and quantity variables are expressed in terms of log differences. The
absolute price version of the Rotterdam model (Theil 1980) is
wit d ln (qit)
= ai + bi
N
X
j=1
wjt d ln (qjt )
+
N
X
j =1
cij d ln (pjt )
(9)
ln (qit) = ln (qit=qit; ) and d ln (pjt ) = ln (pjt=pjt; ).
N
P
The parameter restrictions implied by demand theory are, for the adding-up constraint
ai =
i
N
N
N
0, i P cij = 0, and i P bi = 1; for homogeneity j P cij = 0; and for symmetry cij = cji.
where for estimation purposes d
1
1
= 1
= 1
= 1
= 1
The own and cross price uncompensated, or Marshallian, demand elasticities in the Rotterdam model are14
cij ; wj bi
"ij
:
(10)
wi
Similar to the LA/AIDS system, (9) is not a reduced form. In what follows we require
reduced form share equations or, since the price simulations are solved numerically, a system
of equations which defines the share equations. The N shares are solved as part of the following
system of N equations. The first N equations are found by solving (9) for qit:
=
3
qit
where Zit is defined as
Zit
=
ai
+ bi
=
X
j 6= i
e
Zit
(wit (1
; bi ))
wji d ln (qjt)
qit;1
+
N
X
j=1
(11)
cij d ln (pjt ) :
(12)
The second set of N equations are simply the definition the share of expenditure for good i:
wit
=
pit qit
:
X
(13)
The final set of N equations are the equations that comprise the Rotterdam demand system—
(9). The three sets of N unknowns are wit, qit , and wit d
qit .
ln ( )
14The
expenditure elasticity of brand i is "iX
=
bi = wi:
6
2.3
Conditional vs. Unconditional Elasticities
The LA/AIDS and Rotterdam specifications discussed above model demand as a function of
total expenditure on the group of brands. The elasticity formulas are therefore conditional on
expenditure being held constant. As a result they underestimate elasticity measures since they
do not capture the effect that an increase in the price of brand i will decrease total expenditure
on all brands. Depending on data availability, consumer demand can be estimated in two
stages. At the first stage, or level, demand for total consumption of the product (say of heat
and serve spaghetti sauces) is modeled as a function of the (deflated) price index for heat and
serve spaghetti sauce, income, and other determinants of demand, e.g. demographic variables,
time trend, etc. This stage determines total expenditure on spaghetti sauce, which is allocated
in the second-stage using the demand system chosen.15
3. Bertrand-Nash Equilibrium and Price Simulations
Estimates of a demand system can be used, along with maintained assumptions regarding the
nature of competition and the behavior of marginal costs, to estimate the price effects of a
merger. It is assumed both pre- and post-merger that competition between the producers of the
differentiated products is over prices—competition is Bertrand. Observed prices and market
shares pre-merger are assumed to correspond to a Nash equilibrium in prices and marginal
costs are assumed to be constant over the relevant range of output.16
3.1
Duopoly of Single Brand Firms
=
(
)
pi qi p; X ; i qi where i is the
The profit for a firm that produces a single brand is i
(constant) marginal cost for firm i. Pre-merger the following profit-maximization condition
for firm i must hold in the Nash equilibrium:
Li
= pi ;p i = ; " 1(p) ;
i
ii
(14)
where Li is the Lerner index for firm i.
The simplest case is a pre-merger duopoly where each firm produces a single brand. Postmerger the profit-maximizing prices of the two brands of the merged entity will satisfy:
!
!
pi ; i
p ;
"ii (p)wi (p) + j j "ji (p)wj (p)
pi
pj
15See
= ;wi(p)
(15)
Hausman, Leonard, and Zona (1994) for discussion and illustration using data on beer consumption in
the United States. See also Edgerton (1997) on the estimation of elasticities in multistage demand functions.
16This section is based on Hausman, Leonard, and Zona (1994).
7
for i; j
= 1; 2 ; i 6=
j , or
pm ; Lmi = i m i
pi
= ; " 1(p) ; Lmj ""ji((pp))wwj((pp))
ii
ii
i
(16)
where Lm
i is the Lerner index for product i post-merger. Since the Lerner index for product
j , wi , wj , and, assuming that the two goods are substitutes, "ji are all positive, the effect
of the merger is to increase the market power and prices of both products.17 The merger
eliminates a source of competitive discipline on the merged firms: the constraint on a firm’s
incentive to raise its price is substitution by consumers to other products. Post-merger that
constraint is relaxed since the merged entity internalizes, or benefits from, the substitution
to the other differentiated product: raising the price of product i raises demand for product
j. The greater the margin the merged firm earns on the other product, the greater the crossprice elasticity of demand, and the relatively more important the other product—the larger
its expenditure share—the greater the exercise of market power post-merger relative to the
pre-merger equilibrium.
Using (15) to determine post-merger prices requires information about marginal cost. Marginal
costs can be backed out by substituting prevailing prices and estimated own-price elasticities
into (14) for each brand and solving for its marginal cost, i . Assuming marginal cost is constant, these values of i , as well as the equations that define the four price elasticities and the
share equations as functions of prices, can be substituted into (15) yielding a system of two
non-linear equations in the two unknowns, p1 and p2 , which can be solved (numerically) for
the two post-merger prices.18
Alternatively, the price-effect of the merger can be approximated by assuming that elasticities and shares will be unchanged post-merger. Under this assumption (16) is a system of two
equations in two unknowns (the post-merger Lerner indices) and the post-merger Lerner Index
; is
for product i
wj " ; "
jj
wi ji
m
:
Li
(17)
"jj "ii ; "ji "ij
Using (14) and (17) the approximate percentage increase in the price of brand ifrom the merger
of the two firms and coordinated pricing of the two brands i and j is
=1 2
=
pmi
; pi =
p
i
;
"ii wwji
:
"ii "ji "ij + wwji ; "jj (1 + "ii )
"ji "ij
17As we will see in the simulation results this claim assumes stability of the best- response functions.
18The elasticity and share expressions depend on the demand system used. For example, in the case
(18)
of the
LA/AIDS demand system we use (7) and (8) for the elasticities and (3) for the share equations. In the case of
the Rotterdam system we use (10) for the elasticities and for the share equations the system of 6 equations in 6
unknowns defined by (9), (11), and (13).
8
3.2
Oligopoly with Multibrand Firms
More generally we would expect that there would be more than two firms and that some, if not
all, firms would produce more than one brand. Suppose that there are M firms which produce
K1 ; K2 ; :::KM products. The pre-merger Nash equilibrium in prices requires that the following
profit- maximizing condition be satisfied for each of the k 2 Ki products firm i produces:
!
!
Ki p ; X
pk ; k
j
j
"kk +
"jk wj
pk
pj
j 6=k
=
= ; wk :
(19)
wk Lk be the product of the Lerner Index and expenditure share of brand k , E
Let sk
define a block diagonal matrix
2
3
E = 64
=
E1
0
0 7
5
:
(20)
Em
with blocks Ei Ki x Ki of own and cross price elasticities for the products of firm i, and s be
the vector of (weighted) shares. Then the pre-merger equilibrium is defined by the following
system of equations:
E 0 s ;w
(21)
The post-merger elasticity matrix EM differs from E by the deletion of the diagonal block
=
for the two merging firms (i and j) and the addition of a new diagonal block of dimension
Ki Kj x Ki Kj for the merged entity. The post-merger equilibrium is defined by
0
E M sM ;wM :
(22)
( + ) ( + )
=
To determine exact price increases (21) is solved for the marginal costs for all brands. The
elasticity and share equations as a function of prices, as well as the values for marginal cost,
are substituted into (22), which is then solved numerically for the post-merger prices.
An approximate price increase can be determined by assuming that expenditure shares and
elasticities are unchanged post merger. These assumptions mean that systems (21) and (22)
are linear. Inverting E in (21) we have:
s = ;(E 0);1 w
(23)
=
and the pre-merger Lerner index for product k is Lk sk =wk . Similarly using (22),
0
sM ; E M ;1 wM
= (
)
and the post-merger Lerner index for product k is LM
k
the price of a merging brand is
= sMk =wkM . The percentage change in
pk =pk = (1 ; Lk ) = 1 ; LMk ; 1
9
:
(24)
The price change for all non-merging brands is 0. The value of the approximation depends
on the change in the own and cross-price elasticities as the merged entity raises its prices. In
general we would expect demand to become more elastic and the cross-price elasticities to
increase as prices rise.19 In addition, we also expect that the prices of non-merging brands
will change as well. As we show below these price changes may be both economically and
statistically significant.
The simulation exercise requires identifying which observations correspond to the premerger equilibrium. The use of mean values for price and quantity, while natural in many
applications, may not be appropriate if they are not representative of recent industry experience. As Werden (1997b) observes, the theoretically correct benchmark parallels what would
prevail in the absence of the merger. Usually this will be a recent time period, but it should be
long enough to average out transitory/seasonal effects.
4. The Bootstrap: Determining the Statistical Significance of the Estimated Price Increases
Provided the products of the merging firms are substitutes (and assuming stability), and barring efficiencies that reduce marginal cost, post-merger their prices will increase. However,
while a quantitative forecast of the price effects of a merger represents a considerable advance
in merger analysis by precisely identifying the “substantial lessening of competition”, it is a
false precision. Of more interest is either a confidence interval of the price change or an assessment of the statistical significance of the price change. These assessments can be obtained
by bootstrapping the simulated price changes.
According to Efron (1982, p. 28), the “charm” of the bootstrap is that it can “be applied to complicated situations where parametric modeling and/or theoretical analysis is hopeless.” When only two products are involved in a merger, asymptotic standard errors associated
with the approximate simulated price effects can be computed using the delta method applied
to (18). However, in most merger cases involving manufacturers of differentiated products,
each firm produces multiple related brands and products. Hence, approximate price simulations must proceed using the method described in equations (19) - (24). These equations
involve highly non-linear transformations of the elasticities and applying the parametric delta
method would be, at best, highly complicated and, at worse, “hopeless.” Implementation of the
delta method to evaluate the statistical properties of exact price simulations would similarly be
highly complicated, or impossible, since computing exact price changes for even two products
involves solving pre-merger and post-merger prices for all products in the demand system.20
19The original proponents of (24) to estimate price effects of a merger are Hausman, Leonard, and Zona (1994).
20These simulations are obtained by simultaneously solving a system of first-order profit maximization equa-
tions like (14) and (15).
10
Instead, we turn to Efron’s (1979) bootstrapping procedure. The general idea of the bootstrap is to assess the variability of estimates by resampling the original data. In this way, many
“pseudo-data sets” can be obtained and the standard errors associated with any particular estimates, or transformation of estimates, can be computed through simple summary statistics.
The key to obtaining reasonable estimates of standard errors is to make sure that the pseudodata are generated from regression residuals in a manner that is consistent with the stochastic
structure of the original model. In this way, standard errors are generated using the model’s
own assumptions and the Monte Carlo distribution of observed errors are used to approximate
the distribution of the unobservable true model.
Freedman and Peters (1984) have developed a bootstrap procedure for dynamic linear systems of equations. Here, we use a simple modification of their procedure. The first step
involved is to estimate either the Rotterdam or LA/AIDS demand system using an autocorrelation correction so that residuals are independently and identically distributed.21 From these
estimates, a matrix of residuals, E are obtained where each column represents residuals from a
particular equation. Second, pseudo-data are obtained by randomly drawing rows of residuals
with replacement, assigning mass =T to each row, where T is the number of observations.
Drawing data in this way preserves contemporaneous correlation among residuals. Third,
pseudo-data are obtained by recursively solving the structural equations corrected for autocorrelation. Fourth, new parameter estimates, elasticity estimates and price simulations are
obtained from each pseudo-data set and stored.22 Fifth, steps two through four are repeated
one thousand times. Sixth, standard errors and bias are computed using Efron’s (1982) formulas.
Efron’s formula for the standard deviation is
^
1
0P
1
B b
^
^
BB b ; CC
^ = B
@ (B ; 1) CA
= 1
21In
1
=2
(25)
the illustrative example of the next section involving spaghetti sauce a correction for first-order serial
correlation was required. First-order serial correlation is likely a common problem in estimating demand systems
from scanner data.
22In some cases, pseudo- data from a particular draw results in exact simulated price effects that do not converge. In general these draws must be discarded. However, excessive problems with convergence would induce
bias since too many simulations would be discarded. Indeed, discarding results is akin to imposing curvature
restrictions since only models with appropriate curvature in the demand system are guaranteed convergence. To
reduce the number of nonconverging exact price simulations, we imposed a minimum price level of 0.25. This
minimum is substantially below the actual price levels in the data and was never the final solution value for any
pseudo-data set.
11
where B is the number of pseudo-data sets (in our application it is 1,000) and
^
=
B b
P
^
b=1
B
:
^
When applied to the simulated price changes b is the price change for pseudo data set b
calculated at mean values. Efron’s formula for the bias is
Bias
^
= ^ ; ^
(26)
where is the original parameter estimate obtained from the full data set.
A problem arises in choosing the initial values for lagged endogenous variables. Freedman
and Peters use the same initial data point for each pseudo-data set. Here, we follow Rayner
(1990) and build initial values from exogenous data and randomly drawn residuals. For example in the LA/AIDS specification the initial observation for each lagged endogenous variable
is obtained by solving:
= ^i +
wib1
N
X
^ij ln (pj 1 )
+ ^i ln (X
= P1 )
+
1
X
^l e^1 ; l :
(27)
j= 1
l=0
In practice, we use fifteen lags to approximate infinity. This approximation works well
when the estimated autocorrelation parameter, ^, is relatively small.
For the Rotterdam demand system the initial values are obtaining using:
wi2 d ln (qi2)b
= ^ai + ^bi
N
X
j=1
wj 2 d ln (qj 2)
+
1
N
X
j=1
c^ij d ln (pj 2 )
+
X
15
l=0
^l e^2 ; l
(28)
The pseudo-data set for the LA/AIDS specification is then obtained by recursively solving
to T:
the following structural equation for periods t
=2
witb
= ^ i +
N
X
j= 1
^ij ln (pjt )
+ ^i ln (Xt = Pt)
0
1
N
X
+^ @witb ; ; ^i ; ^ij ln (pjt ; ) ; ^i ln (Xt ; = Pt ; )A
1
1
j= 1
+ e^it:
1
(29)
1
The pseudo-data set for the Rotterdam specification is obtained by recursively solving the
to T:
following structural equation for t
=3
12
witd ln
(qit)b
X
X
= ^ai + ^bi wjtd ln (qjt) + c^ij d ln (pjt )
N
N
j=1
j =1
(30)
0
1
N
N
X
X
+^ @wit; d ln (qit; )b ; a^i ; ^bi wjt; d ln (qjt; ) ; c^ij d ln (pjt; )A
1
1
j=1
+^eit:
1
1
j =1
The key to obtaining reasonable standard error estimates using the bootstrap procedure is
to generate pseudo-data from regression residuals in a fashion consistent with the stochastic
structure of the original model. As a result actual data on prices, quantities, and expenditure
shares are used on the right-hand sides of (29) and (30).23
5. An Example: Hypothetical Mergers in Spaghetti Sauce
In this section we illustrate the economic and econometric issues associated with unilateral
effects analysis by considering hypothetical mergers of producers of spaghetti sauce in the
United States.
5.1
The Data and Stationarity
The data set on prices and quantity of heat and serve spaghetti sauce was collected by Information Resources Inc. (IRI) and consists of weekly sales (in dollars) and movement on
average per store for the period June 3, 1991 to May 31, 1992 (52 weeks). The information
was collected from approximately 1,700 supermarkets located in 51 different market areas in
the United States. IRI standardizes sales and movement information to account for differences
in size of containers (15 to 40 ounce containers).24 Price is therefore dollars per standardized
unit and movement, or quantity, is the number of standardized units of spaghetti sauce i sold in
23See the discussion in Section
24In many industries there are
5.2 regarding endogeniety of the regressors and identification.
far too many brands to make estimation and simulation tractable, even with
abundant scanner data. One response to this is to impose multistage budgeting by aggregating brands with similar
characteristics into segments and assuming separability across segments at the same stage. For instance in the case
of beer, the first stage is demand for beer; the second stage segments are premium, light, imported, and domestic;
the third stage segments consist of the second stage segments disaggregated into their component brands and their
relationships to each other modelled using either an AIDS or Rotterdam demand system. See Rubinfeld (2000)
for a discussion of segmentation and its effects on price elasticities. The problem with having segments at the
bottom level is that it constrains the pattern of substitution between segments to work through the allocation of
expenditure from changes in a segment’s price index at the preceding level. This assumption puts a downward
bias on the cross-price elasticities among brands in different segments. A second alternative taken here is to mask
product differentiation across the products of a firm. The brands of a firm are aggregated across its product line
and package formats (size).
13
1
Brand
Mean Price Mean Quantity Mean Expenditure Share
Classico (1)
2.38
36.11
9.75%
Hunt’s (2)
1.06
85.02
10.07%
Ragu (3)
1.74
216.60
41.77%
Prego (4)
1.84
134.54
27.92%
Private Label (5)
1.29
42.50
6.13 %
Newman’s (6)
2.18
17.70
4.35%
Table 1: Spaghetti Sauce Variable Means
Prices
Quantities
Market Share
First
Level
First
Level
First
Brand
Difference
Difference
Difference
Classico (1)
-2.47
-4.35*
-1.88
-4.42*
-1.43
-6.45*
Hunt’s (2)
-3.32**
-6.54*
-2.98** -7.67*
-4.51*
-7.34*
Ragu (3)
-2.74*** -5.77*
-2.25
-7.03*
-1.83
-5.92*
Prego (4)
-3.04**
-6.08*
-2.72*** -6.51*
-2.63*** -5.87*
Private Label (5) -4.74*
-7.32*
-4.18*
-8.05*
-7.38*
-8.21*
Newman’s (6)
-2.05
-3.67*
-1.84
-5.71*
-2.41
-5.42*
*** Significant at 10% level; ** Significant at 5% level; *Significant at 1% level.
Specification with constant, two lags, and no time trend.
Level
Table 2: ADF Test Results
week t.25 The brands of spaghetti sauces are (1) Classico, (2) Hunt’s, (3) Ragu, (4) Prego, (5)
Private Label, and (6) Newman’s Own. Mean price, quantity, and expenditure share for each
brand are shown in Table 1.
25Retail
scanner data are collected at the point of purchase, yet usually the transaction of interest is between
manufacturers of differentiated products who likely sell at wholesale. Implicitly in price simulations the link
between “front-end” estimation and “back-end” simulation involves assuming that retailers are not strategic actors. As Werden notes (1997b) two assumptions can be made regarding the behaviour of retailers/distributors: (i)
retailers have fixed absolute margins, in which case inferred marginal costs include the marginal cost of retailing;
or (ii) retailers have constant mark-ups. The former assumption is implicitly made when retail price and quantity
data are used for both the front and back end. The later assumption requires that when determining marginal costs
of the manufacturers, wholesale prices first be found by backing retailer margins out of retail prices. Of course
the appropriateness of either assumption depends on the actual nature of retailer margins.
14
5.1.1
Stationarity
Table 2 reports augmented Dickey-Fuller test results for stationarity. The results indicate that
non-stationarity is a potential problem for the levels of prices, quantities and market shares
for three of the brands—Classico, Ragu, and Newman’s—as well as the market share and
quantity for Prego. On the other hand the results indicate that all variables are stationary in
first differences. Total expenditure is also non-stationary in level (ADF test statistic -2.05), but
stationary in first differences (ADF test statistic -7.83). Indications of non-stationarity suggest
that the LA/AIDS specification, which is in levels, is problematic.
5.2
Empirical Results
The demand systems estimated are identical to (2) (for LA/AIDS) and (9) (for Rotterdam),
with the addition of quarterly dummies based on the calendar year and a random error term.
The quarterly dummies are added to capture the possibility that there are seasonal shifts in
demand.26 Both the LA/AIDS model and the Rotterdam model are estimated using an iterated
seemingly unrelated regression (ITSUR) technique, with an allowance for serial correlation
(AR(1)) in the disturbance terms of all equations (Berndt and Savin 1975).
Estimation of either the LA/AIDS or Rotterdam demand system raises concerns regarding
identification since prices are included as independent regressors. The usual approach of finding cost-based instruments may be difficult if the demand system consists of a relatively large
number of brands. A further difficulty with instruments is that they may not be available in
the frequency of the scanner data. Hausman, Leonard, and Zona (1994) have suggested two
alternatives. The first is that if scanner data are available from different geographic markets,
then price data from one geographic region can be used as instruments for other regions.27 The
second is to observe that prices may in fact not be endogenous, but instead are set by retailers
prior to consumers making their purchase decisions.28 This assumption is consistent with the
nature of scanner data: retailers set their price and at that price their supply is perfectly elastic.
This is the implicit assumption we make.
5.2.1
Demand System Estimates
Information on goodness-of-fit and serial correlation for the LA/AIDS and Rotterdam estimation, with and without the restrictions implied by demand theory imposed, are found in
26The
addition of the three dummy variables means that the intercepts of the equations in the demand systems
need to be adjusted in the simulations and the bootstrap by adding the dummies and their estimated coefficients
to (27) through (30).
27This suggestion is appropriate if sales promotions and other local demand shocks for a brand are uncorrelated
across geographic regions. The use of price data in other geographic markets is described in Hausman, Taylor
and Zona (1994, pp. 164-165) and is an implementation of Hausman and Taylor (1981).
28The need to use instruments can be tested using Hausman’s specification test (Hausman 1978).
15
Equation
Classico (1)
Hunt’s (2)
Ragu (3)
Prego (4)
Private Label (5)
Newman’s (6)
Rotterdam
R-Squared Durbin-Watson
0.5662
2.082
0.9226
2.226
0.9390
2.178
0.8546
2.056
0.9278
2.093
0.8190
2.167
LA/AIDS
R-Squared Durbin-Watson
0.8039
2.271
0.6646
1.978
0.7520
1.977
0.6516
1.974
0.7749
1.956
0.8297
1.966
Table 3: Unrestricted Rotterdam and LA/AIDS R2 and D-W Statistics
Equation
Classico (1)
Hunt’s (2)
Ragu (3)
Prego (4)
Private Label (5)
Rotterdam
R-Squared Durbin-Watson
0.5411
2.058
0.9113
2.306
0.9342
2.148
0.8454
1.985
0.9242
2.079
LA/AIDS
R-Squared Durbin-Watson
0.6883
1.519
0.6296
2.168
0.6838
1.753
0.6088
1.920
0.7521
2.098
Table 4: Restricted Rotterdam and LA/AIDS R2 and D-W Statistics
Tables 3 and 4. The values for R2 indicate that a considerable amount of variation in expenditure shares is explained by each demand system. The only notable exceptions are for the
Restricted Rotterdam model, where the R2 for the Classico equation is only 0.5411 and for
the Restricted LA/AIDS demand system which in general does not fit the data nearly as well
as the other three systems. In the Unrestricted Rotterdam case 21/48 of the demand system
parameters are statistically significant at the 5 or 10% level; for the Unrestricted LA/AIDS
specification 24/48 are significant; for the Restricted Rotterdam 18/25 are significant; and for
the Restricted LA/AIDS specification 15/25 are significant. In the Restricted Rotterdam case,
of the 7 insignificant coefficients, 5 are intercept terms.
An important issue is whether the restrictions implied by demand theory hold. Likelihood
ratio tests shown in Table 5 indicate that for the LA/AIDS demand system, symmetry and
homogeneity are individually rejected, and, as a result, both symmetry and homogeneity together do not hold. On the other hand, there is strong support for accepting symmetry for
the Rotterdam specification and limited support for accepting the null that homogeneity and
homogeneity and symmetry together hold.
The rejection of the demand restrictions for the LA/AIDS specification will introduce a
trade-off in the price simulations between bias and efficiency. Rejection of the restrictions
16
( )
( )
( )
Rotterdam
LA /AIDS
p-value
2
p-value
14.770 0.1406 32.183* 0.0004
10.769 0.0562 29.441* 0.0000
2
Symmetry 210
Homogeneity 25
Symmetry
and Homogeneity 215 24.723
* Significant at 5% level.
0.0538
50.757*
0.0000
Table 5: Testing Demand Restrictions
suggests that the unrestricted estimates should be used; however use of these estimates, as we
will show, may result in the simulated prices having very large variances. Using the restricted
estimates will introduce bias, but at a considerable gain in efficiency.
5.2.2
Elasticities
The price simulations depend upon the own and cross-price elasticities derived from the estimated demand systems. These elasticities and their standard errors are found in Tables 6
through 7.29 In all cases the own price elasticities are negative, elastic, and statistically significant at the 5% level. With regard to the cross-price elasticities 8/30 are statistically significantly
different from zero at the 5 or 10% level for the Unrestricted Rotterdam system and 23/30 are
for the Restricted Rotterdam system—with 4 of the insignificant coefficients involving Classico. For the LA/AIDS elasticities 12/30 are significant for the unrestricted LA/AIDS system
and 14/30 are for the restricted LA/AIDS system. All the significant cross-price elasticities are
positive, except for the finding of a significant negative effect on the demand for Classico from
an increase in the price of Hunt’s in the unrestricted LA/AIDS system. For the unrestricted estimates there are, however, quite a few negative cross-price elasticities that are insignificantly
different from zero. One of the effects of imposing the restrictions is that there are only two
negative cross-price elasticities for the LA/AIDS specification—both involving Classico—and
none for the Rotterdam specification. While the own-price elasticities are similar (except for
the restricted LA/AIDS Classico estimate), the cross-price elasticities can differ substantially
between the unrestricted and restricted estimates. Comparing across the two restricted demand
systems the estimates appear roughly similar except for those related to Classico.
29The
reported standard errors are based on the delta method. The mean elasticity estimate and its standard error from the bootstrap are similar to the ITSUR estimates. The reported elasticities are conditional on expenditure
being held constant.
17
Classico
Unrestricted Restricted
Classico
-2.46*
-2.56*
SE
0.64
0.41
Hunt’s
-0.20
0.14
SE
0.41
0.20
Ragu
-0.03
0.15**
SE
0.34
0.08
Prego
0.33
0.12
SE
0.39
0.11
Private Label 0.89*
0.40**
SE
0.45
0.23
Newman’s
0.42
0.43**
SE
0.28
0.23
Hunt’s
Unrestricted Restricted
-0.18
0.20
0.40
0.20
-3.16*
-2.87*
0.25
0.19
0.39**
0.13*
0.22
0.05
0.16
0.33*
0.24
0.07
0.32
0.10
0.28
0.15
0.13
0.35*
0.18
0.14
Ragu
Unrestricted Restricted
0.82**
0.88*
0.45
0.31
0.33
0.55*
0.27
0.20
-1.90*
-1.92*
0.24
0.16
0.79*
0.68*
0.27
0.17
0.60**
0.52*
0.32
0.21
0.10
0.38*
0.19
0.15
Prego
Unrestricted Restricted
Classico
0.49
0.44
SE
0.52
0.30
Hunt’s
0.58**
0.89*
SE
0.32
0.19
Ragu
0.30
0.42*
SE
0.28
0.13
Prego
-2.09*
-2.32*
SE
0.32
0.18
Private Label 0.93*
0.69*
SE
0.37
0.21
Newman’s
0.39**
0.52*
SE
0.23
0.18
Private
Unrestricted Restricted
0.10
0.28*
0.29
0.14
-0.21
0.05
0.17
0.09
0.16
0.07**
0.15
0.04
0.17
0.15*
0.17
0.04
-2.75*
-2.80*
0.20
0.14
0.07
0.18**
0.12
0.09
Newman’s
Unrestricted Restricted
-0.57
0.20*
0.80
0.10
0.72
0.13*
0.50
0.06
0.50
0.02
0.43
0.02
-0.39
0.07*
0.49
0.03
-0.16
0.12**
0.57
0.07
-3.16*
-2.56*
0.36
0.28
** Significant at 10% level; * Significant at 5% level.
Table 6: Rotterdam Elasticities
18
Classico
Unrestricted Restricted
Classico
-2.42*
-1.30*
SE
0.68
0.46
Hunt’s
0.04
-0.01
SE
0.34
0.20
Ragu
-0.19
-0.11
SE
0.30
0.09
Prego
0.30
0.17
SE
0.34
0.13
Private Label 0.81*
0.18
SE
0.35
0.22
Newman’s
0.12
0.30
SE
0.25
0.20
Prego
Unrestricted Restricted
Classico
0.26
0.63**
SE
0.52
0.37
Hunt’s
0.67*
0.88*
SE
0.34
0.20
Ragu
0.33
0.20
SE
0.27
0.12
Prego
-2.11*
-2.15*
SE
0.33
0.18
Private Label 0.94*
0.72*
SE
0.34
0.21
Newman’s
0.44**
0.60*
SE
0.24
0.18
Hunt’s
Unrestricted Restricted
-0.87*
0.08
0.40
0.21
-2.90*
-2.74*
0.26
0.18
0.48*
0.09**
0.20
0.05
0.14
0.36*
0.24
0.07
0.38
0.18
0.26
0.14
0.09
0.28*
0.19
0.14
Private
Unrestricted Restricted
-0.15
0.17
0.31
0.14
-0.03
0.10
0.18
0.09
0.23
0.04
0.15
0.03
0.13
0.18*
0.18
0.05
-2.68*
-2.72*
0.18
0.14
0.09
0.16
0.14
0.10
** Significant at 10% level; * Significant at 5% level.
Table 7: LA/AIDS Elasticities
19
Ragu
Unrestricted Restricted
0.47
0.07
0.48
0.43
0.50*
0.62*
0.23
0.19
-1.55*
-1.68*
0.21
0.20
0.50*
0.57*
0.23
0.18
0.71*
0.63*
0.23
0.20
0.53*
0.65*
0.18
0.14
Newman’s
Unrestricted Restricted
-0.89
0.17**
0.78
0.10
0.67**
0.10
0.39
0.06
0.60**
0.01
0.35
0.02
-0.62
0.10*
0.40
0.03
-0.20
0.10
0.41
0.07
-2.76*
-2.68*
0.29
0.25
Insert Table 8 Here
Table 8: Unrestricted Rotterdam Price Simulations
Insert Table 9 Here
Table 9: Restricted Rotterdam Price Simulations
5.2.3
Price Simulations
For each demand system (LA/AIDS and Rotterdam), with and without restrictions imposed,
we determine the effect on prices for each of the 15 possible two-firm mergers. We calculate
the change in prices exactly, allowing expenditure shares and elasticities to change as prices
change, and we also determine the approximate change in prices assuming elasticities and
expenditure shares remain constant. In all cases the price simulations are computed using
sample means for exogenous variables. The bootstrap provides standard deviations of the price
changes, allowing us to assess the statistical significance as well as the economic significance
of the effect of a merger on prices. Tables 8 through 11 characterize the effect of each merger
for the four model specifications.
For each specification and each merger, the relevant table shows the simulated price change,
the bootstrap standard error of the price change, and the bootstrap 90% confidence interval
based on the empirical distribution function (Davidson and MacKinnon 1993, pp. 763-766).
The 90% confidence interval is derived from the bootstrap by ordering the 1000 simulated
price changes from smallest to largest. The lower bound is observation 50, the upper bound
observation 950. Prices changes that are statistically significant at the 5 and 10% level, based
on t-tests, are also indicated.
Table 12 summarizes the findings regarding the price changes. The results indicate the
following:
The approximate price changes can be a very poor guide relative to exact price changes.
The numbers in parentheses are the common number of approximate and exact price
increases which are statistically different from zero or whose 90% confidence interval
Insert Table 10 Here
Table 10: Unrestricted LA/AIDS Price Simulations
20
Insert Table 11 Here
Table 11: Restricted LA/AIDS Price Simulations
excludes 0. The extent of overlap is not very large, indicating that the approximate
methodology is prone to being both under and over inclusive.
Of perhaps even more interest is the extent of significant price changes associated with
non-merging brands in the exact case. For the Restricted Rotterdam case, in 3 of the 15
possible mergers, there are statistically significant price increases for at least one other
brand and in all 3 cases the exact price increases for the merging brands are statistically
insignificant from 0. In the Restricted LA/AIDS case in 7 of the 15 possible mergers
there are statistically significant price increases for at least one of the non- merging
brands, though in only 1 of these cases are the exact price increases of the merging
brands statistically insignificant.
The number of positive price changes is much larger for the restricted than for the unrestricted systems, because in the unrestricted system more brands are complements. In
the restricted estimations only 3 exact price changes are negative, and none of these
is significantly different from 0. In the Restricted LA/AIDS specification the price
of Classico declines—insignificantly—in both the Classico Hunt’s and Classico-Ragu
mergers. However the cross-price elasticity between Hunt’s and Ragu with respect to
Classico is negative, indicating that they are complements.30 A bit mysterious is the
Ragu-Newman’s merger in the Restricted Rotterdam case where the price of Ragu declines, albeit insignificantly, even though the two products are substitutes. The explanation appears to be due to a violation of stability—the exact price change of all brands
except Newman’s is negative.31
The number of statistically significant price increases is considerably less than the number of predicted price increases. The number of statistically significant price increases
is much larger for the restricted demand systems. For the LA/AIDS system 50% of the
price increases are statistically significant in the restricted case compared to just over
25% in the unrestricted case.
The row labeled “Overlap SS/CI” compares the two different approaches to determining
30See
Table 7 where the relevant cross-price elasticities are shown as -.01 and -0.11 and both are statistically
insignificant.
31Approximate negative price changes are consistent with (18) since the elasticities and shares are assumed
to be constant. When there are large differences in the own-price elasticities of demand and large differences in
expenditure shares price changes can be negative even with large positive cross-price elasticities.
21
Rotterdam
Unrestricted Restricted
21
30
6
10
2
7
13
29
3 (1)
5 (3)
3 (0)
7 (3)
1
1
Approximate Positive
Statistically Significant
CI Excludes 0
Exact Positive
Statistically Significant
CI Excludes 0
Overlap SS/ CI
Negative Statistically Significant
Approximate
3
Exact
5 (1)
0
0
LA/AIDS
Unrestricted Restricted
21
28
7
16
4
11
18
28
5 (3)
14 (10)
4 (0)
14 (9)
0
9
3
7 (2)
1
0
Table 12: Summary of Price Simulations
statistical significance. It is the number of price changes for the given demand system
that are both statistically significant (at the 10% level) and whose 90% confidence interval excludes zero. The confidence intervals based on the cumulative normal distribution
in conjunction with the mean and variance obtained from the bootstrap differ from those
obtained directly using the empirical distribution function for the bootstrapped price
changes. It is evident that the simulated price changes were drawn from a distribution with fatter tails than the normal. If the simulated price effects were truly drawn
from a normal distribution, the upper and lower bound of the confidence intervals of
the simulated price changes would be narrower. Typically, one might worry about this
departure. However, in this analysis only 52 weekly observations are available. In most
cases demand systems estimated using scanner data utilize approximately three years
of weekly data. With this number of observations one would expect a closer correspondence between statistics based on large sample approximations and those that are
obtained directly from the empirical distribution function.
5.2.4
Bias vs Efficiency
The appropriateness of imposing the restrictions implied by demand theory is addressed in
the first instance by testing whether they are satisfied or not, thereby identifying the “true”
model. An alternative approach, appropriate when the objective is to make predictions, is to use
mean squared error (MSE) as the selection criteria. The MSE based on the price simulations
equals the sum of the square of bias and the variance of the price change. The bootstrap
provides not only the standard deviation of the price change—used previously to determine
22
MSE Restricted/ MSE Unrestricted
Classico-Hunt’s
Classico-Ragu
Classico- Prego
Classico- Private Label
Classico- Newman’s
Hunt’s- Ragu
Hunt’s- Prego
Hunt’s- Private Label
Hunt’s- Newman’s
Ragu- Prego
Ragu- Private Label
Ragu- Newman’s
Prego- Private Label
Prego- Newman’s
Private Label- Newman’s
Rotterdam
LA/AIDS
First Brand Second Brand First Brand Second Brand
0.0204
0.0001
4.4375
395.1357
0.1233
0.1433
73.0743
2.7047
0.0988
0.1119
33.7251
0.1524
0.0516
0.0361
0.7426
5.7351
0.7869
0.0023
132.3829
0.5268
0.6621
0.2976
421.2841
613.5124
0.4380
0.1700
0.0568
0.0005
0.0147
0.0000
0.3579
40.1529
0.0668
0.0613
0.1705
4.4463
0.0861
0.2791 1507.0555
1.2834
0.0180
0.0000
77.5508
4.6357
0.0000
0.0544
75.9180
29.3481
0.1649
0.1840
0.0052
0.1380
0.0299
0.2279
36.9865
627.6289
1.0166
0.6410
0.1389
0.7659
Table 13: MSE Comparison Rotterdam and LA/AIDS
23
their statistical significance—but also bias. Bias is the difference between mean simulated
price change from the bootstrap less the true price change. The true price change corresponds
to the price simulations at the mean of the “true” model: the unrestricted results for LA/AIDS
and the restricted results for Rotterdam. Table 13 shows the ratio of the MSE of the restricted
to the MSE of the unrestricted predicted price effect for each of the 15 possible mergers. In the
case of the LA/AIDS model the MSE for the restricted model is smaller in 13 of the 30 price
changes than the MSE for the unrestricted model. In the case of the Rotterdam demand system
the MSE of the predicted price effect of the merger based on the restricted model is smaller than
the MSE of the unrestricted model for 29 of the possible 30 price changes, which is consistent
with the Restricted Rotterdam model being “true”. In some cases the small magnitude of the
restricted MSE relative to the unrestricted MSE for the LA/AIDS demand system suggests that
the introduction of bias from imposing the constraints implied by demand theory is more than
compensated by the increased precision of the simulated price effects.32
5.3
Merger Challenges and Demand Systems
Tables 14 and 15 compare the implications for merger enforcement across the different systems. Table 14 shows the mergers which would be challenged on the basis of one statistically
significant exact price increase based on a t-test. The number of mergers subject to challenge
ranges from 3 to 11 of the possible 15 mergers. Table 15 shows the mergers which would be
challenged on the basis of the empirical confidence intervals excluding zero for at least one
exact price. The number ranges from 3 to 12 of the 15 possible mergers. In both cases the
Restricted LA/AIDS specification leads to the largest number of challenges.
Recall that the two “true” models are the restricted Rotterdam and the Unrestricted LA/AIDS
specifications. The overlap between the two of them in Table 14 is only the merger between
Ragu and Prego. The overlap using the empirical confidence intervals to assess statistical
significance is larger, the entire set of mergers challenged under the Unrestricted LA/AIDS
included in the Restricted Rotterdam set, though the number of mergers challenged in the Restricted Rotterdam set is 75% more. The discrepancy between the number of challenges and
the small size of the intersection set between the two “true” models is alarming, suggesting that
antitrust actions based on simulation models of unilateral price effects may be highly sensitive
to model specification. However, when MSE considerations are considered the differences are
markedly less.
In Tables 14 and 15 for the LA/AIDS specifications the star indicates if the significant
32To
ensure reliable exact price change simulations we imposed a convergence requirement for each price
simulation used in the bootstrap. In the unrestricted demand system simulations, convergence problems were
encountered fairly frequently. Imposition of a convergence requirement is in effect similar to imposing some
“regularity” conditions even on the unrestricted demand systems. Without imposition of convergence when solving for the exact price change in the unrestricted demand systems, relative mean squared error comparisons would
be even more favorable for the restricted systems. In results not reported here we found this to be true.
24
Unrestricted Rotterdam
Hunt’s-Ragu
Hunt’s-PL
PL -NM
Restricted Rotterdam
Classico-PL
Hunt’s-Prego
Ragu-Prego
Unrestricted LA/AIDS
Classico*-NM
Ragu*-Prego
Ragu*-NM
Prego*-NM
Hunt’s-NM*
Restricted LA/AIDS
Classico-Prego
Classico*- PL
Classico-NM
Hunt’s-Ragu
Hunt’s*-Prego*
Ragu-Prego
Ragu-PL
PL*-NM
Hunt’s-NM
Ragu-NM
Prego-NM
NW=Newman’s, PL= Private Label, *= Minimum MSE
Table 14: Merger Challenges by Demand System Based on t-test
Unrestricted Rotterdam
Hunt’s-NM
Classico-Hunt’s
Classico-NM
Restricted Rotterdam
Hunt’s-Prego
Hunt’s-PL
Hunt’s-NM
Classico-Hunt’s
Classico-Prego
Classico-NM
Prego-PL
Unrestricted LA/AIDS
Hunt’s-Prego
Hunt’s-NM
Classico-Hunt’s*
Classico-NM
Restricted LA/AIDS
Classico*- PL
Hunt’s- Ragu
Hunt’s*- Prego
Hunt’s*-PL
Hunt’s*-NM
Ragu- PL
PL *-NM
Classico-Prego*
Classico-NM*
Ragu-NM
Prego-PL*
Prego-NM
NW=Newman’s, PL= Private Label, *= Minimum MSE
Table 15: Merger Challenges by Demand System Based on CI
25
price change (in bold) has, comparing across the two LA/AIDS specifications, the smaller
MSE. From the set of 11 mergers in Table 14 that would be challenged under the Restricted
LA/AIDS specification, only in 3 cases is the MSE minimized. Adding them to the set of
mergers challenged under the Unrestricted LA/AIDS specification gives a set of 8 mergers
subject to challenge using the LA/AIDS specification. The combined set of 8 mergers based
on the lowest MSE and significant prices includes the 3 mergers identified as problematic by
the Restricted Rotterdam system. The remaining 5 mergers that would be challenged under the
LA/AIDS specification all involve Newman’s. Recall from Table 2 that Classico, Ragu, and
Newman’s own prices, quantities, and market share are non-stationary. The Rotterdam deals
with this since it is in log differences. The LA/AIDS specification fails to take account of nonstationarity.
From the set of 4 mergers in Table 15 that would be challenged under the Unrestricted
LA/AIDS specification, in only 1 case is the MSE minimized and of the 12 mergers challenged
under the Restricted LA/AIDS specification 8 involve exact price changes that are significant
and have minimum MSE. Combining the mergers from the two specifications that have exact
price changes that are significant and minimum MSE gives a set of 9 mergers that includes all
of the mergers challenged under the Restricted Rotterdam specification, except for ClassicoPrivate Label and Private Label-Newman’s. Both of these mergers involve a brand for which
its data is non-stationary.
Crooke et al (1999) demonstrate the sensitivity of price simulations to the “curvature” of
the demand curve, i.e. how quickly elasticity increases as price rises.33 Our finding that the set
of challenged mergers under the two demand systems is the same in our application—when
statistical significance, stationarity, and MSE are considered—is at odds with the hypothetical
discussion of Crooke et al (1999). Though it is true, however, that the price increases associated
with the Rotterdam systems are generally smaller than the LA/AIDS demand specifications.
6. Conclusion
In this paper we provided an introduction, discussion, and demonstration of the merger/price
simulations to determine the unilateral effect of a merger in a differentiated product market.
The analysis compared two possible demand systems, AIDS and Rotterdam. The results suggest that the Rotterdam specification warrants greater use since (i) its specification is in first
differences and price level data are likely to be non-stationary; (ii) its elasticity expressions are
unique; and (iii) the restrictions from demand theory may be more likely to hold.34
33Crooke et al point out the importance of having the same initial conditions when comparing across the
demand systems. In our application this requires the marginal costs of the different demand systems to be similar.
Except for Classico, we find that the marginal costs are very similar.
34In situations where there is no clear a priori reason to prefer one functional form over the other, final determination of the appropriate specification can be made using Pollak and Wales’ (1991) likelihood dominance
criterion or some other nonnested test procedure.
26
The results indicate as well that there can be substantial differences between exact and
approximate price increases, suggesting that the assumptions of constant market shares and
elasticities that underpin the calculation of approximate price increases are not well founded.
Our results also suggest the importance of determining the statistical significance of price
changes. Bootstrapping the standard errors of price increases allows an assessment of both
the statistical and economic significance of a price increase. Further, our results indicate that
even if the demand restrictions do not hold, it can be the case that the mean squared error
of imposing the restrictions and introducing bias is less because of a considerable gain in
efficiency.
References
[1] Alston, J., K. Foster, and R. Green. 1994. “Estimating Elasticities with the Linear Approximate Almost Ideal Demand System: Some Monte Carlo Results.” Review of Economics and Statistics 76: 351- 356.
[2] Baker, J. 1997a. “Contemporary Empirical Merger Analysis.” George Mason Law Review
5: 347-361.
[3] Baker, J. 1997b. “Unilateral Competitive Effects Theories in Merger Analysis.” Antitrust
11: 21-26.
[4] Berndt, E.K. and N.E. Savin. 1975. “Estimation and Hypothesis Testing in Singular Equation Systems with Autoregressive Disturbances.” Econometrica 43: 937-57.
[5] Blanciforti, L., and R. Green. 1983. “An Almost Ideal Demand System Incorporating
Habits: An Analysis of Expenditures on Food and Aggregate Community Groups.” Review of Economics and Statistics 65: 511-515.
[6] Buse, A. 1994. “Evaluating the Linearized Almost Ideal Demand System.” American
Journal of Agricultural Economics 76: 781-793.
[7] Crooke, P., L. Froeb, S. Tschantz, and G. Werden. 1999. “Effects of Assumed Demand
Form on Simulated Postmerger Equilibria.” Review of Industrial Organization 15: 205217.
[8] Davidson, R., and J. MacKinnon. 1993. Estimation and Inference in Econometrics. New
York: Oxford University Press.
[9] Deaton, A., and J. Muelbauer. 1980a. “An Almost Ideal Demand System.” American
Economic Review 70: 312-326.
27
[10] Deaton, A., and J. Muelbauer. 1980b. Economics and Consumer Behaviour. Cambridge:
Cambridge University Press.
[11] Edgerton, D. 1997. “Weak Separability and the Estimation of Elasticities in Multistage
Demand Systems.” American Journal of Agricultural Economics 79: 62-79.
[12] Efron, B. 1979. “Bootstrap methods: Another Look at the Jackknife.” Annals of Statistics
7: 1-26.
[13] Efron, B. 1982. The Jackknife, the Bootstrap and Other Resampling Plans. Philadelphia:
Society for Industrial and Applied Mathematics.
[14] Freedman, D. A., and S. Peters. 1984. “Bootstrapping a Regression Equation: Some
Empirical Results.” Journal of American Statistical Association 79: 97-106.
[15] Green, R., and J. Alston. 1990. “Elasticities in AIDS Models.” American Journal of Agricultural Economics 72: 442-445.
[16] Hausman, J. 1978. “Specification Tests in Econometrics.” Econometrica 46: 1251-71.
[17] Hausman, J., and G. Leonard. 1997. “Economic Analysis of Differentiated Products
Mergers Using Real World Data.” George Mason Law Review 5: 321-346.
[18] Hausman, J., G. Leonard, and J. D. Zona. 1994. “Competitive Analysis with Differenciated Products.” Annales d’ Economie et Statistique 34: 159-180.
[19] Hausman, J., and Taylor. 1981. “Panel Data and Unobservable Individual Effects.”
Econometrica 49: 1377-98.
[20] Pollak, R. A., and T. J. Wales. 1991. “The Likelihood Dominance Criterion: A New
Approach to Model Selection.” Journal of Econometrics 47: 227-242.
[21] Rayner, R. 1990. “Bootstrapping p Values and Power in the First-Order Autoregression:
A Monte Carlo Investigation.” The Journal of Business and Economic Statistics 8: 251263.
[22] Rubinfeld, D. 2000. “Market Definition with Differentiated Products: The Post/Nabisco
Cereal Merger.” Antitrust Law Journal 68: 163-182.
[23] Scheffman, D. 1992. “Statistical Measures of Market Power.” Antitrust Law Journal 60:
901-919.
[24] Theil, H. 1980. The System-Wide Approach to Microeconomics. Chicago: University of
Chicago Press.
28
[25] Werden, G. 1997a. “Simulating Unilateral Competitive Effects from Differentiated Products.” Antitrust Spring: 27-30.
[26] Werden, G. 1997b. “Simulating the Effects of Differentiated Products Mergers: A Practical Alternative to Structural Merger Policy.” George Mason Law Review 5: 363-386.
[27] Werden, G., and L. Froeb. 1994. “The Effects of Mergers in Differentiated Products Industries: Logit Demand and Merger Policy.” Journal of Law, Economics, & Organization
10: 407-426.
[28] Werden, G., and L. Froeb. 1996. “Simulation as an Alternative to Structural Merger Policy in Differentiated Products Industries.” The Economics of the Antitrust Process. ed. M.
Coates and A. Kleit. Boston: Kluwer Academic Publishers. 65-88.
29
Table 8
Approxim
mate
Exact
Approximaate
Exact
Unrestricted Rotteerdam
m
First Firm
Second Firrm
Second Firm
Merger
First Firm
Classico-Hunt's Price Change
-4.02
-12.97
-3.07
-10.98
SE
10.04
222.06
2790.73
792.12
90% CI
-16.91 10.89
-56.38 127.53 -355.95 522.77
3.87 72.04
Classico-Ragu Price Change
-5.42
-8.50
8.61
-10.50
SE
23.01
16.78
18.62
10.08
90% CI
-12.68 22.17
-8.13 14.17
-14.72 24.73
-6.36
6.41
Classico- Prego Price Change
50.72 *
-14.22
10.60
-21.46
SE
25.78
13.76
96.26
19.36
90% CI
-43.83 24.53
-10.08 29.07
17.93 195.24
-7.50 50.06
Classico- PL
Price Change
17.66 *
-9.75 *
4.64
-14.81 *
SE
5.08
3.71
3.87
6.15
90% CI
-7.62
4.90
-4.13
3.90
-5.48
4.95 -11.73
8.09
Classico- NW Price Change
1.92
-2.59
-19.96 *
-4.17
SE
4.83
4.82
7.80
228.56
90% CI
-8.30
5.51
-6.80
4.92
-6.48 16.22 30.15 96.32
Hunt's- Ragu
Price Change
68.59 *
14.12 *
5.59
-0.15
SE
2.93
6.89
24.49
9.12
90% CI
-2.62
6.12
-9.02
6.46
-38.36 30.38 -12.94 12.83
Hunt's- Prego Price Change
11.69
5.80
7.98 *
7.66
SE
21.76
4.33
3.70
33.70
90% CI
-30.08 28.11
-0.68 12.47
-4.19
4.39 -14.19 39.79
Hunt's- PL
Price Change
2.97
5.14 *
-6.25
3.22
SE
9.23
2.37
584.04
3817.80
90% CI
4.16 28.58
-4.48
2.97
-11.02 96.29
-3.75 69.79
Hunt's- NM
Price Change
1.27
-19.10 * *
33.28 *
-12.23
SE
10.17
10.41
8.89
31.67
90% CI
-9.34
7.46
1.66 24.19
-7.18 11.29 -14.34 33.28
Ragu- Prego
Price Change
59.26 *
10.96
41.12
14.19
SE
1.39
6.89
68.02
13.65
90% CI
-1.22
3.12
-5.95 10.53
-15.47 72.70 -11.18 19.81
Ragu- PL
Price Change
6.06
4.48
52.24
22.77
SE
4.13
80.20
247.86
1515.03
90% CI
-11.19
2.08
-14.72 36.20
-9.44 197.11 -17.69 41.55
Ragu- NM
Price Change
1.36
-47.51
-643.60 *
-16.41
SE
7.27
232084.65
46.41
12.45
90% CI
-6.37
9.75
-17.85 52.11
-45.51 35.09
-9.76
9.38
Prego- PL
Price Change
10.70
8.09
31.38
20.49
SE
29.37
15.31
14656.96
15.52
90% CI
-11.84 16.68
-6.59 30.23 -994.04 1008.53
-4.04 22.98
Prego- NM
Price Change
-0.34
-29.73 *
-35.68 *
-7.09 *
SE
3.84
3.60
3.21
2.33
90% CI
-5.41
6.07
-4.16
5.09
-4.13
3.24
-3.72
1.84
PL- NM
Price Change
0.82
4.84 * *
-3.72
1.62
SE
18.69
2.72
19.85
5.22
90% CI
-19.46 24.25
-3.16
2.79
-52.85 11.51 -10.25
5.97
PL= Private Labeel, NM= Newmaan's, **Siggnificant at 10% levell, *Significant at 5% level
Table 9
Approxim
mate
Exact
Restricted Rotterddam
m
First Firm
m
Merger
First Firm
Classico-Hunt's Price Change
3.63
2.97
SE
5.69
31.81
90% CI
-3.63 14.70
-3.63
Classico-Ragu Price Change
31.56 *
11.14
SE
11.49
9.07
90% CI
-6.17 29.20
-7.58
Classico- Prego Price Change
10.72
15.03
SE
22.82
10.16
90% CI
-4.65 52.12
-4.68
Classico- PL
Price Change
7.34
8.21 *
SE
8.29
4.16
90% CI
-8.59 15.79
-5.14
Classico- NW Price Change
6.03
0.82
SE
7.34
5.24
90% CI
-7.83 12.32
-6.53
Hunt's- Ragu
Price Change
19.04 *
9.01
SE
0.46
6.98
90% CI
1.22
2.67
-9.13
Hunt's- Prego Price Change
32.89
6.76 *
SE
22.91
2.93
90% CI
-22.94 49.06
2.90
Hunt's- PL
Price Change
1.18
0.17
SE
2.29
0.67
90% CI
2.72 10.46
2.50
Hunt's- NM
Price Change
3.63
0.36
SE
6.68
5.70
90% CI
-3.52 12.18
7.30
Ragu- Prego
Price Change
43.56 *
23.39 *
SE
0.73
4.17
90% CI
1.72
4.07
-5.87
Ragu- PL
Price Change
3.89
0.43
SE
3.31
10.78
90% CI
-2.35
8.28
-6.17
Ragu- NM
Price Change
1.92
-3.42
SE
3.71
12.65
90% CI
-4.89
4.97 -10.96
Prego- PL
Price Change
5.65
4.92
SE
8.41
6.35
90% CI
-8.61 16.47
-0.61
Prego- NM
Price Change
2.96
2.16
SE
7.77
5.55
90% CI
-6.97 13.97
-5.74
PL- NM
Price Change
3.06
0.05
SE
8.14
5.56
90% CI
-2.08 21.46
-6.21
14.70
17.89
22.33
6.08
7.83
12.08
12.62
4.67
25.24
5.34
26.68
27.61
16.31
8.65
10.68
Approxim
mate
Exact
Second Firm
Second Firm
4.47
3.25
10.73
6.09
-5.29 28.80
0.79 18.79
13.85 * *
2.28
8.23
6.17
-5.34 19.08
-6.40 10.59
5.56
12.54
8.68
13.09
5.98 34.35 17.97 59.30
11.85 *
10.18 *
4.01
4.89
-2.40
9.68
-1.83 13.99
13.76
3.81
11.45
11.00
-5.43 29.62 28.94 64.08
7.04
3.09
26.40
5.28
-20.58 60.40
-3.17 12.82
14.05 *
0.96
1.57
14.17
-2.21
1.58
-8.10 35.08
1.69
0.42
19.34
4.22
-11.48 46.91
-0.58 12.65
7.88
2.32
4.84
8.63
-5.34
7.09
-6.33 19.44
45.07 *
25.59 *
13.92
9.40
-12.34 30.34
-6.81 18.74
15.94 * *
6.66
8.79
7.74
1.65 30.67
-5.68 17.95
7.30
0.64
4.57
4.92
11.63 26.52
-4.49
6.75
21.30 *
11.67
2.20
7.66
2.51
9.16
0.62 24.20
14.54 *
7.61
5.06
7.11
-3.97
7.26
-5.88 14.11
4.13
0.06
25.28
4.36
-11.93 63.30
-1.19 11.92
PL= Private Labeel, NM= Newmaan's, **Siggnificant at 10% levvel, *Signnificant at 5% level
Table 10
AIDS
Unrestricted LA/A
Merger
Classico-Hunt's Price Change
SE
90% CI
Classico-Ragu Price Change
SE
90% CI
Classico- Prego Price Change
SE
90% CI
Classico- PL
Price Change
SE
90% CI
Classico- NW Price Change
SE
90% CI
Hunt's- Ragu
Price Change
SE
90% CI
Hunt's- Prego Price Change
SE
90% CI
Hunt's- PL
Price Change
SE
90% CI
Hunt's- NM
Price Change
SE
90% CI
Ragu- Prego
Price Change
SE
90% CI
Ragu- PL
Price Change
SE
90% CI
Ragu- NM
Price Change
SE
90% CI
Prego- PL
Price Change
SE
90% CI
Prego- NM
Price Change
SE
90% CI
PL- NM
Price Change
SE
90% CI
Approxim
mate
m
First Firm
0.69
5.52
-8.24
9.26
-27.72
35.59
-17.30 24.68
44.09
34.16
-51.92 33.77
13.57 *
6.88
2.50 22.09
0.24
9.58
-16.73
6.00
305.68
1038.13
2.30 115.04
12.05
14.78
-28.74
1.91
4.66
24.67
3.19 32.03
1.18
47.75
-107.76 25.17
65.77 *
1.73
-1.20
4.31
17.40 *
5.91
-8.81 10.34
22.75 *
1.20
-2.62
1.05
10.04
8.74
4.08 27.84
-1.83
6.86
-15.32
4.90
1.31
47.77
-137.71 18.61
Exact
First Firm
m
2.64
194.51
-60.69
-18.76 *
9.28
-10.30
22.47
60.08
-113.48
8.50
8.66
-20.20
6.95 *
1.21
-1.18
-134.77 *
7.09
-2.06
8.25
89.32
6.79
2.91
3.62
-8.65
8.89
26.18
2.28
75.89 *
0.94
-2.22
-48.58 *
6.07
-10.81
77.17 *
8.51
-16.01
6.48
94.38
-206.85
8.04 *
0.94
-1.20
1.05
2.96
-1.33
64.22
18.36
77.31
4.47
1.42
5.72
74.33
2.53
33.08
0.70
2.89
7.26
72.73
1.35
3.60
Approximaate
Exact
Second Firrm
Second Firm
-15.68
-10.49
535.95
20.46
-278.18 304.56
3.62 39.32
3.93
6.52
1045.58
3.99
-25.56 313.33
-4.76
3.81
5.07
4.60
10625.69
71.96
-1578.51 1693.17 -13.25 75.65
-6.27 *
-2.25
0.78
6.85
-1.63
0.91 -27.95
-5.27
-32.02
-23.18
21.19
56.68
-46.09 11.15 23.26 136.13
22.53
-65.08 *
29.44
2.98
-53.56
1.42
-4.71
1.07
9.95 *
6.37
0.75
333.36
-0.48
1.60
-9.86 14.31
-0.95
-0.29
55.43
53.84
-1.06 31.46
-6.78 43.17
45.10 *
78.90 *
8.43
35.69
-18.39
5.75 -16.38 83.92
53.48 *
36.90
5.49
23.76
-10.68
3.00 -55.70 21.57
191.79
-87.26 *
27428.44
43.71
-1416.38 1365.35 -47.44 46.02
-174.23 *
-88.52 *
33.51
16.95
-54.66 22.22 -39.36 10.66
22.81
15.03
3837.67
40.72
-567.12 412.24 -15.36 68.12
-51.07 *
-39.75 *
11.25
2.18
-28.52
5.66
-0.36
4.07
-5.86
-2.75
20.24
5.13
-61.43
1.68 -11.57
4.78
PL= Private Labeel, NM= Newmaan's, **Siggnificant at 10% levvel, *Signnificant at 5% level
Table 11
Approxim
mate
Exact
Restricted LA/AIDS
m
First Firm
m
Merger
First Firm
Classico-Hunt's Price Change
-1.73
-0.94
SE
669.57
409.74
90% CI
-49.69 72.92 -194.80 77.26
Classico-Ragu Price Change -47.79 *
-42.08
SE
1.00
75.85
90% CI
-0.34
1.98
-4.94 22.94
Classico- Prego Price Change 963.08 *
315.09 * *
SE
308.06
189.98
90% CI
-187.00 43.86 -207.98 270.69
Classico- PL
Price Change
20.59 *
15.94 *
SE
3.04
0.64
90% CI
3.76 13.37
0.25
2.25
Classico- NW Price Change
27.96 *
20.72 *
SE
6.59
2.35
90% CI
-9.43 12.98
-1.36
2.54
Hunt's- Ragu
Price Change
15.18 *
10.65 *
SE
2.45
0.66
90% CI
1.81
8.59
1.81
8.59
Hunt's- Prego Price Change
46.18 *
28.63 *
SE
3.70
6.19
90% CI
-1.27
2.86
4.88 21.71
Hunt's- PL
Price Change
2.57
1.67
SE
3.90
1.78
90% CI
3.05 14.78
2.06
7.74
Hunt's- NM
Price Change
2.96
1.92
SE
5.81
8.59
90% CI
3.93 17.67
8.89 36.04
Ragu- Prego
Price Change
46.60 *
39.37 *
SE
2.06
2.53
90% CI
-0.28
6.31
-2.84
2.87
Ragu- PL
Price Change
6.13
4.88 *
SE
4.28
0.19
90% CI
-1.67 11.42
0.01
0.62
Ragu- NM
Price Change
4.15
3.34
SE
2.65
6.57
90% CI
-2.66
2.71 -15.82
4.73
Prego- PL
Price Change
7.79 *
5.95
SE
3.16
6.78
90% CI
4.10 13.04
-2.30 19.61
Prego- NM
Price Change
4.18
3.15
SE
10.36
2.90
90% CI
-21.93 13.86
-2.71
4.27
PL- NM
Price Change
2.60
1.62 * *
SE
5.79
0.94
90% CI
-4.95 13.91
0.71
3.62
Approxim
mate
Second Firm
3.66
1317.37
-399.15 440.43
1.40
1.27
-0.64
2.38
23.15 *
9.87
1.06 32.31
15.00 *
0.10
-0.01
0.31
23.32 *
1.20
0.24
4.12
10.90 *
0.34
-0.22
0.85
18.58 *
0.15
-0.11
0.36
3.73
12.78
-18.08 16.35
5.70 *
0.71
0.33
2.62
22.97 *
6.38
-15.55
3.36
9.89
10.63
-3.81 30.54
4.46
29.62
1.34 66.16
32.23 *
8.96
-7.84 21.24
24.40 *
1.25
0.92
4.65
3.33
6.74
7.09 25.62
PL= Private Labeel, NM= Newmaan's, **Siggnificant at 10% levvel, *Signnificant at 5% level
Exact
Second Firm
2.23
406.53
-49.85 85.85
2.81
5.42
-2.02 11.80
24.53
19.81
23.44 86.00
9.53
11.43
-12.03 23.09
14.85
15.70
25.69 76.16
8.70 *
0.07
-0.04
0.18
14.17 *
0.12
-0.07
0.29
2.41
341.13
-0.21 95.21
3.64 *
0.39
0.18
1.45
20.77
21.55
-39.81 38.44
6.85 *
0.44
0.15
1.49
3.29 *
1.15
0.83
3.94
19.57
14.43
8.89 39.35
14.74 *
2.00
2.07
8.53
1.53
1.34
-0.37
4.01