The empirical demand system stems from a well-developed microeconomic model of consumer choice. Let be the quantity consumed of retail frozen seafood product i, where i = 1, …, n. Then x is a n × 1 vector with elements . Further, let be the elements of the n × 1 vector q, where is the perceived quality of good . Perceived product quality may be influenced by a myriad of non-price, non-income factors including, but not limited to, product labels, the media, food safety recalls, advertising, brand image, and an event such as the sinking of the Deepwater Horizon. Let represent a non-price, non-income information index characterizing the quality of seafood product i such that ; higher levels of bad news lead to a lower level of perceived quality. More generally, we let the vector q be a function of the vector s, or q(s).
As is the case for most applied demand studies, data is typically unavailable to construct a complete demand system [
18]. Thus, we assume the consumer’s utility function is weakly separable between retail frozen seafood and all other goods. In our problem, the individual consumer chooses
x to maximize:
subject to the linear budget constraint:
where
is the utility function,
p′ is a 1 ×
n vector of prices of retail frozen seafood, and
M is total expenditure for retail frozen seafood.
The solution to the consumer’s problem results in a vector of
n Marshallian or uncompensated demand functions
with the usual properties [
18]. Because perceived quality is a function of the information index or
q(
s), we may express the Marshallian demand functions as
so that the Marshallian demands now include a vector of shift parameters based on the information index.
Substituting Equation (4) into the utility function, we obtain the indirect utility function
. Others in the literature begin their model development with essentially this expression for the indirect utility function [
19]. Inverting the indirect utility function, we obtain the consumer’s expenditure function
By applying Shephard’s lemma to the expenditure function
we obtain the
n Hicksian demand functions and express them in expenditure share form in the
n × 1 vector
w. The presence of the informational shift variables
s in Equation (6) presents a difficult problem when estimating
w.
We represent
w using the corrected Linear Approximate Almost Ideal Demand System (LA-AIDS) model [
11,
20]. The expenditure share (
) for the
ith frozen seafood product, is given by
where the usual unobservable, nonlinear AIDS price index is replaced by the loglinear analog of the Laspeyres price index for constant base period shares
[
20]. It is given by
The use of translating and scaling techniques have long been used to incorporate shift variables such as demographics into singular expenditure systems without violating Closure Under Unit Scaling (CUUS) [
21,
22]. The notion of CUUS is maintained when the estimated parameters, such as the usual
parameters in the Almost Ideal Demand System [
11], do not depend on the data’s scaling, especially the scaling of the data related to the shift variables themselves [
6,
23].
3.1. Econometric Estimation and Autocorrelation Correction in A Singular System
Following Berndt and Savin [
24], with appropriate substitutions and addition of subscripts representing weekly time periods, the demand model of retail frozen seafood given by Equation (7) may be rewritten more compactly as
where
is a
n × 1 vector of conditional expenditure shares of frozen seafood,
is a
n ×
K matrix of unknown parameters,
is
K × 1 vector of explanatory variables, and
is a
n × 1 vector of stochastic disturbances governed by the following process
for time
t = 2, …,
T,
R is a
n ×
n matrix of unknown parameters, and
is a
n × 1 vector of residuals. Further it is assumed
is distributed
for
t = 2, …,
T.
Let t’ be a 1× n vector of ones. Because the demand model of retail frozen seafood is singular (i.e., its shares sum to one), for t = 1, …, T. The adding up conditions also imply , for t = 1, …, T and, since and are independent, t’R = k’. The final result indicates the n column sums of R equal the same constant.
The autocorrelation correction procedure for singular equation systems as developed by Berndt and Savin [
24] is quite flexible and subsumes several interesting special cases. When the
n ×
n elements of matrix
R are set to zero, this represents the case of no autocorrelation such that
and
. For the present data set this assumption is implausible and, hence, introduces an omitted variable bias in the matrix of parameter estimates Π. If the
n elements on the diagonal of matrix
R are restricted to be the same constant and the off-diagonal elements are restricted to all be zeros, this single parameter estimate for serial correlation correction will equal
k’ since
t’R =
k’. This parsimonious assumption is maintained for the present study. It is noted
R may be kept in its most general form with
n2 unique elements. For the present study, the full matrix over-parameterizes the model.
In this empirical application, consider the case of four frozen retail fish products ordered as follows: catfish, tilapia, perch, and all other fish. It is noted the data supplier combines both freshwater and saltwater perch into one seafood type. Also, in this second empirical demand application, we considered the case of four frozen retail shellfish products ordered as follows: shrimp, crawfish, mussels, and all other shellfish. For each model, this results in
n = 4 conditional expenditure share equations. Since the system is singular as the shares sum to one, the 4
th equation is dropped from the estimation. Equations (12) and (13), with the 4
th equation dropped may be rewritten as
and
for
t = 2, …,
T. Since
R4 is now a 3 × 4, Equations (14) and (15) are not estimable. Recognizing
, this is remedied by Berndt and Savin [
24] by the following transformation
so that
is now a 3 × 4. Now the
n − 1 column sums in
each equal zero. Substituting
into Equation (15) we obtain
Further substituting Equation (17) into Equation (14), we obtain the estimable, theoretically consistent, conditional Almost Ideal Demand System model of retail frozen seafood as given by
for
t = 2, …,
T. Using the PROC MODEL routine in the SAS 9.4 ETS module, we jointly estimate the parameters in
and
using nonlinear seemingly unrelated regressions (SUR) [
25]. An iterated SUR approach was not used due to lack of stability in the likelihood ratio tests for non-price, non-income informational shifters. However, it should be noted the iterated SUR and SUR led to very similar parameter estimates and levels of statistical significance with the former being only slightly more efficient. This model is highly nonlinear since
and
enter into Equation (18) as a product. It is noted
is distributed
for
t = 2, …,
T [
24,
25]. Finally, as mentioned previously,
is given in its diagonal form for first-order autocorrelation correction. The parameter estimates for
and
for both frozen fish and frozen shellfish are reported and discussed in the Empirical Results section.
3.2. Hypothesis Testing of Consumer Response to Information
Germane to this study is the cross-equation hypothesis test in which the three equations manifested in Equation (18) are estimated with Equation (9) versus the restricted model where Equation (9) is replaced with
for
i = 1, …, 3 such that
. The restricted model imposes the null hypothesis that the trend has no impact on the aggregate consumer behavior in the market for retail frozen seafood. This test is considered to be far superior to a simple inspection of the parameter by parameter asymptotic
t-statistics, especially in small samples. Using any single-equation approach, it is not possible to comprehensively test information effects on the demand system overall. The procedure used to test this cross-equation restriction is a likelihood ratio test [
25]. The likelihood ratio statistic for our model is given by
where
is the objective function of the SUR multiplied by the number of time periods net of any lags,
is
for the estimated restricted model where the covariance matrix is held constant from the estimated unrestricted model, and
is
for the unrestricted model. The test statistic is distributed asymptotically chi-square with
degrees of freedom where
is the number of estimated parameters in the unrestricted model and
is the number of estimated parameters in the restricted model. If
LR is less than the chi-square critical value for some alpha level of significance then we fail to reject the null hypothesis and conclude the restricted and unrestricted models are statistically no different. The outcome of the hypothesis test would quantify whether or not the trend affected the demand for the frozen seafood products. In addition to this test, the time trend in Equation (9) is replaced with an indicator variable set to one during the entire duration of the Deepwater Horizon event and zero otherwise. Other informational shift variables could be incorporated into the model [
7,
8]. However, those tests are beyond the scope of the present study and the subject of future research.