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AJAE Appendix for “Brand-Supermarket Demand for Breakfast Cereals and Retail
Competition”
Benaissa Chidmi and Rigoberto A. Lopez
Date: July 2006
Note: The material contained herein is supplementary to the article named in the title and
published in the American Journal of Agricultural Economics (AJAE).
Appendix: Estimation Procedure
The integral in equation (5) does not have a closed formula and cannot be solved
analytically. It must be solved numerically by simulation methods. The basic idea is to
choose parameters that minimize the distance between the predicted and observed market
shares. To this end, the estimation objective is to
(A1)
Min s ( p, x, ) S ,
where s (.) denotes predicted market shares (from equation 5) and S denotes observed
market shares. However, this approach implies a costly non-linear minimization
procedure because most parameters enter (A1) in a non-linear manner. To avoid this
difficulty, Berry (1994) suggests inverting the market share function giving the mean
utility valuation
(see equation 4) that equates the predicted market shares with
observed market shares s( . ;
2
)
S. , where
2
( , , , ) is a vector of parameters
that enter the indirect utility function non-linearly.
Once the mean utility valuation
is obtained (from a standard logit model, for
example), the next step is to define the error term as the deviation from that mean. That
is,
(A1)
j
j
(S j ;
2
) ( pj
xj ).
(A2)
The error is then interacted with instruments to form the objective function to minimize
using the generalized methods of moments (GMM) estimation.
More specifically, following Nevo (2000), the estimation steps are:
Step 1: Prepare the data to use for the estimation. The data are of two categories: (1) the
market data, including market shares, prices, product characteristics, advertising and
promotion, and (2) the distributions of consumer characteristics, including demographic
characteristics D and the unobserved consumer characteristics
that enter in the random
part of the indirect utility.
Step 2: Given starting values for
distributions of D and
2
, the mean valuation utility
and the draws from the
, compute the predicted market shares. Nevo (2001) propose
using the smooth estimator that makes use of the extreme value distribution on f ( ) to
integrate ’s analytically. The predicted market shares are then approximated by
(A3)
s j ( p , x, , ;
2)
1
n
n
sij
i 1
1
n
exp(
n
j
ij
)
,
J
i 1
1
exp(
m
im
)
m 1
where n is the number of draws from the distributions of D and
per time period. In our
case, n=100 per time period or 3,500 for all 35 periods in the sample.
Step3: The computation of the predicted market shares will allow computing the mean
utility valuation
that equates the predicted market shares with observed market shares.
This is an iterative step and is solved numerically due to the non-linearity of the inversion
of the equation s( . ;
2
)
S. . BLP (1995) suggest using the contracting mapping theorem,
which yields:
t 1
(A3)
t
ln(S ) ln(s( p, x, ;
2
(A4)
),
where s (.) is the predicted market shares computed by equation A3 and T is the smallest
integer such that
T
T 1
is smaller than some tolerance level.
Step 4: The error term is computed using the equation
X 1 1 , where
is the mean
utility valuations computed in the previous step; X 1 is a matrix that regroups the variables
that enter the indirect utility function linearly, including the observed product
characteristics and the price; and
1
( , ) is the vector of parameters that enter linearly
in the indirect utility function.
The error term is then interacted with the instruments to form the objective
function to be minimized using the GMM estimation
(A3)
( )' ZA 1 Z ' ( ) ,
f
(A5)
where
( 1,
2
) is the vector of parameters to be estimated; Z is a matrix of
instruments; and A is a consistent estimate of E[ Z '
which are the parameters reported in Table 1.
' Z ] . The solution to A5 yields
*
References
Berry, S. T. 1994. “Estimating Discrete-Choice Models of Product Differentiation.” Rand
Journal of Economics 25(2):242-262.
Berry, S., J. Levinsohn and A. Pakes. 1995. “Automobile Prices in Market Equilibrium.”
Econometrica 63(4):841-890.
Nevo A. 2000. “A Practitioner’s Guide to Estimation of Random Coefficients Logit
Models of Demand,” Journal of Economics and Management Strategy, 9, No. 4,
pp.513-548.
_____. 2001. “Measuring Market Power in the Ready-To-Eat Cereal Industry.”
Econometrica 69(2): 307-342.